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An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its application to financial decision making
Accepted Manuscript Title: An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its application to financial decision making Author: Ting-Yu Chen PII: DOI: Reference:
S1568-4946(18)30370-3 https://doi.org/10.1016/j.asoc.2018.06.036 ASOC 4952
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
7-9-2017 13-5-2018 21-6-2018
Please cite this article as: Chen T-Yu, An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its application to financial decision making, Applied Soft Computing Journal (2018), https://doi.org/10.1016/j.asoc.2018.06.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its
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application to financial decision making
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Ting-Yu Chen
Professor, Graduate Institute of Business and Management, College of Management, Chang Gung University
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Adjunct Professor, Department of Industrial and Business Management,
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College of Management, Chang Gung University
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Adjunct Research Fellow, Department of Nursing, Linkou Chang Gung
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Memorial Hospital
33302, Taiwan
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Address: No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City
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Tel: +886-3-2118800 ext. 5678 Fax: +886-3-2118500
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E-mail: [email protected]
May 13, 2018
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Highlights
Representation of highly uncertain information using PF/IVPF sets. Development of risk attitude-based score functions for PF/IVPF values. Construction of risk attitudinal assignment models in the PF/IVPF context.
Enrichment of the outranking methodology under complex uncertainty. Application to financial decision-making problems. 1
Comparative analysis with other outranking approaches.
ABSTRACT
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The aim of this paper is to develop a novel outranking method using a risk attitudinal assignment model under complex uncertainty based on Pythagorean fuzzy (PF) sets and interval-valued Pythagorean fuzzy (IVPF) sets. Additionally, this paper applies the proposed methodology to address financial decision-making problems to investigate its applicability and effectiveness in the real world. These developed methods and techniques can provide a new viewpoint for capturing the amount of information conveyed by PF and IVPF values and can fully take into account the decision maker’s attitude towards risk-taking in the face of high uncertainty. Considering the diverse attitudes toward risk neutrality, risk seeking, and risk aversion, this paper introduces novel measures of risk attitude-based score functions within the PF/IVPF environments and investigates their desirable and useful properties. These properties provide a means of establishing linear orders and admissible orders in the PF/IVPF contexts. This means that the proposed measures can overcome some drawbacks and ambiguities of the previous techniques through score and accuracy functions and can address incomparable PF/IVPF information more effectively. As an application of the risk attitude-based score functions, this paper develops novel risk attitudinal assignment models to establish a useful outranking approach for solving multiple criteria decision-making problems. Two algorithms for PF and IVPF settings are developed based on the concepts of a precedence frequency matrix and a precedence contribution matrix for conducting multiple criteria evaluation and the ranking of alternatives. A financing decision on aggressive/conservative policies of working capital management is presented to demonstrate the applicability of the proposed outranking approach in real situations. Moreover, a comparison to the technique of risk attitudinal ranking methods is investigated to validate the advantages of the proposed methodology. Furthermore, a comparative analysis with a newly developed outranking method, the IVPFELECTRE (for the elimination and choice translating reality), is conducted via an application to the investment problem of software development projects. Compared with these benchmark approaches, the proposed methods can produce a more reasonable and persuasive result for ranking the order relationships of alternatives in the highly uncertain context. The practical studies and comparative discussions demonstrate an excellent performance of the developed methodology that is effective and flexible enough to accommodate more-complex decision-making environments.
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Keywords: outranking method; risk attitudinal assignment model; (interval-valued) Pythagorean fuzzy set; financial decision making; risk attitude-based score function; working capital management.
1. Introduction Multiple criteria decision analysis (MCDA) problems often involve vague or imprecise decision information due to the subjective nature and uncertainty of human judgments. In particular, MCDA within changeable, complicated, and/or unpredictable environments is a challenging task for decision makers. The theory of 2
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Pythagorean fuzziness is useful to represent the uncertain information contained in MCDA problems. Based on several valuable risk attitude-related concepts, this study attempts to develop a new outranking methodology to help decision makers conduct multiple criteria evaluation and decision analysis under complex uncertainty. Decision makers often deal with problems that involve vague or imprecise information. In response, fuzzy sets have extended deterministic MCDA methods to address uncertainty and imprecision in real situations [1]. Many generalizations have been proposed to the ordinary fuzzy sets to model different kinds of imprecise and subjective information. One such generalization is Atanassov’s intuitionistic fuzzy (A-IF) sets [3], which are becoming increasingly popular and widely used in the MCDA field [1, 32], and A-IF applications are growing in popularity by the day. Adding to this trend, Yager [45-48] introduced a nonstandard fuzzy set named Pythagorean fuzzy (PF) sets, which are an extension of A-IF sets. PF sets, which are characterized by degrees of membership and nonmembership, must meet the condition that the square sum of the membership and nonmembership degrees is less than or equal to one, while the sum of the two degrees can be more than one [14, 20, 23, 39, 50, 51]. Because of such a relaxed constraint condition, the space of PF values is evidently larger than that of A-IF values [30, 38, 50, 51]. From this perspective, not only can PF sets depict imprecise and ambiguous information, which A-IF sets can capture, but they can also model more-complex uncertainty in practical situations, which the latter cannot describe. Interval-valued Pythagorean fuzzy (IVPF) sets, which are parallel to Atanassov’s interval-valued intuitionistic fuzzy (A-IVIF) sets [4], are an extension of PF sets [25, 35, 50, 51]. IVPF sets permit the degrees of membership and non-membership of a given set to have an interval value within [0, 1]. The only requirement is that the square sum of the respective upper bounds of the interval-valued membership and non-membership degrees is less than or equal to one. Analogously, IVPF sets are capable of managing more-complex uncertainty than A-IVIF sets. PF sets and IVPF sets have wider application potentials because of their great ability to address strong fuzziness, ambiguity, and inexactness during the decision-making process. To date, the application of the PF/IVPF theory to MCDA has been studied from a number of different perspectives, such as the following: PF aggregation operators using confidence levels [18] or using Einstein t‐ norms and t‐ conorms [19], a projection model based on PF geometric Bonferroni mean operators [24], Pythagorean uncertain linguistic partitioned Bonferroni mean operators [26], symmetric PF weighted geometric/averaging operators [27], a PF stochastic method based on the prospect theory and regret theory [33], generalized PF point weighted averaging operators [37], a PF set-based group decision-making method [30], a PF group decision-making method with probabilistic information and an ordered weighted averaging (OWA) approach [49], IVPF linguistic weighted operators based on an IVPF linguistic variable set [14], an improved accuracy function for ranking IVPF information [16, 17], an extended technique for order preference by similarity to ideal solution (TOPSIS) based on hesitant PF sets [23], and an IVPF VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method [11]. In these applications of PF sets and IVPF sets, how to tell the difference between the magnitudes of PF/IVPF values is a critical issue for ranking or evaluating the order relationship of PF/IVPF information. As is well known, managing uncertainty to improve decision quality is an essential issue of MCDA processes, especially within a complex and volatile business 3
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environment. However, nowadays, many phenomena in practical circumstances, such as the presence of new technologies and increasingly dynamic organizational factors, are intricate and unpredictable [13]. It is especially the process of globalization and technological change that has led to the emergence of a complex network of relationships in the business environment. Moreover, a free market economy means increased complexity and uncertainty of factors affecting the financial standing of entities [22], and it is necessary to predict various influences in the highly competitive and globalized marketplace [29]. Consider the issue of consumer credit scoring in financial management. Due to great uncertainties in changeable markets, predicting both business and consumer bankruptcy is a highly imprecise and ambiguous task [22]. This implies that relevant judgments and evaluations may be difficult to measure or quantify along the decision-making cycle. Meanwhile, decision makers often face uncontrollable or unforeseeable risks resulting from a rapidly changing environment. For these reasons, the uncertainty, fuzziness, and vagueness inherent in the information structure of decision-making processes make rigorous mathematical models inappropriate for solving real-world MCDA problems, especially in an intricate and unpredictable business context. According to the discussions above, there are two critical reasons for decision makers to support the employment of PF/IVPF sets in MCDA. First, decision makers are faced with hesitation in numerous down-to-earth situations [32]. As a result, MCDA problem are often characterized by a high degree of uncertainty (e.g., lack of information) and ambiguity (e.g., lack of clarity or consistency of information). Compared with other nonstandard fuzzy sets, such as the widely used A-IF and AIVIF sets, the application of PF and IVPF sets can accommodate much higher degrees of uncertainty and thus have a more-powerful capability in addressing complicated and ambiguous MCDA problems. Second, high degrees of uncertainty in complicated environments often bring about high risks, resulting in many unfavorable decision consequences. In particular, a rapidly changing business environment contributes to greater complexity and difficulty in decision making. Because the judgment in evaluating alternative actions entails strong uncertainty and fuzziness, the methodology used for MCDA must provide appropriate support for these problem characteristics. Employing PF/IVPF sets can facilitate modeling the effects of highorder uncertainties associated with a complex decision environment. Subjective judgments and evaluations can be conveniently realized with the use of PF/IVPF values. Following the aforementioned motivations, this paper attempts to develop a novel outranking decision-making method for addressing MCDA problems under complex uncertainty involving PF/IVPF information. The purpose of this paper is to develop a novel outranking approach that utilizes risk attitudinal assignment models based on an extended concept of risk attitude-based score functions to manage MCDA problems within the PF/IVPF environments. First, motivated by the risk attitudinal technique, this paper introduces new measures of risk attitude-based score functions based on the amount of information conveyed by PF/IVPF values. Some useful and valuable properties of the proposed measures are also investigated to establish a linear order and an admissible order in the PF/IVPF contexts. These new measures have an excellent capability to characterize PF/IVPF information as comparable values to address the essential issue of ranking difficulty in highly uncertain circumstances. Next, this paper employs the risk attitude-based score functions to determine the precedence ranks among alternatives with respect to each criterion. Based on the obtained criterion-wise precedence relationships, this paper 4
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identifies the precedence frequency matrix and the precedence contribution matrix. A new outranking methodology using effective risk attitudinal assignment models is further developed to solve MCDA problems involving PF/IVPF information. Moreover, this paper utilizes practical financial decision-making problems to demonstrate the applicability and usefulness of the proposed outranking methods. Finally, this paper implements a comparative analysis with the other relevant outranking methods to examine the validity and superiority of the developed methods. Compared with the related literature, the main innovation and contributions of this paper in theory and practice contain development of risk attitude-based score functions for comparing PF/IVPF information, construction of risk attitudinal assignment models in the PF/IVPF contexts, and model application to financial decision making. First, considering the attitudinal character towards risk-taking, this paper presents novel measures of risk attitude-based score functions for the sake of effectively characterizing PF/IVPF information. In particular, the developed measures not only possess desirable and useful properties but also provide a means of establishing a linear order and an admissible order to overcome difficulties in ranking incomparable PF/IVPF values. These features are of great value in distinguishing among the magnitudes of PF/IVPF values and yielding the corresponding precedence relationships. Second, by determining criterion-wise precedence rankings among alternatives based on the risk attitude-based score functions, this paper presents useful concepts of a precedence frequency matrix and a precedence contribution matrix. Moreover, two effective risk attitudinal assignment models are established for solving MCDA problems within the PF and IVPF environments. The proposed risk attitudinal assignment models give the decision maker full support in representing imprecise knowledge and incorporating diverse risk attitudes into the existing assignment-based methodology, which leads to addressing many practical MCDA problems in a more appropriate and flexible way. Third, this paper extends the range of applications of the proposed methodology and makes it accommodate complex financial decisionmaking problems. Financing decisions on aggressive/conservative policies is very crucial for the efficient management of working capital. This paper illustrates how the proposed approach can be applied to the real-world problem of selecting a financing policy in the medical and health care fields. The practical usefulness and contributions of the developed concepts and models can be supported by this financial decisionmaking application. Furthermore, a comparative analysis with the other outranking method for solving an investment problem of software development projects is carried out to demonstrate the advantages and practicality of the proposed approach. The remainder of this paper is organized as follows. Section 2 briefly reviews some basic definitions and concepts of PF sets and IVPF sets. Section 3 proposes a useful concept of risk attitude-based score functions that take the decision maker’s risk attitude and the amount of information associated with PF/IVPF values into account. This section also investigates some valuable properties of the developed concept in detail. Section 4 constructs an outranking approach for addressing MCDA problems involving PF/IVPF information. Based on the precedence frequency and precedence contribution matrices, two effective algorithms with novel risk attitudinal assignment models are established to determine the outranking relationships among alternatives. Section 5 applies the proposed methodology to a practical problem concerning a financing decision on aggressive/conservative working capital policies of a medical institution to demonstrate its feasibility and applicability. Some discussions of the sensitivity analysis and a comparison to the technique of risk 5
attitudinal ranking methods are provided as well. To examine the advantages of the proposed methodology, further comparative analyses with a newly developed outranking method within the IVPF environment are conducted to address an investment problem. Finally, Section 6 presents the conclusions.
2. Basic concepts This section introduces some basic concepts related to PF sets and IVPF sets that are used throughout this paper.
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Definition 1. [24, 45-48, 49] A PF set P is defined as a set of ordered pairs of membership and non-membership in a finite universe of discourse X and is given as follows:
P x, P ( x), P ( x) x X ,
(1)
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which is characterized by the degree of membership P : X[0, 1] and the degree of
non-membership P : X[0, 1] of the element x X to the set P with the condition: 0 P ( x) P ( x) 1 . 2
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Let p (P ( x), P ( x)) denote a PF value. The degree of indeterminacy relative to P for each x X is defined as follows:
P ( x) 1 P ( x) P ( x) . 2
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Definition 2. [34, 36, 38, 50-52] Let p1 ( (P1 ( x), P1 ( x)) ) and p2 ( (P2 ( x), P2 ( x)) ) be two PF values in X. The distance between p1 and p2 is defined as follows: 2 2 2 2 2 2 1 d ( p1 , p2 ) P1 ( x) P2 ( x) P1 ( x) P2 ( x) + P1 ( x) P2 ( x) . (4) 2 Definition 3. [18, 19, 33, 38, 52] Let p be a PF value in X. The score function s( p) and the accuracy function a( p) of p are separately defined as follows: s( p) P ( x) P ( x) ,
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a( p) P ( x) P ( x) .
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Definition 4. Let p, p1, and p2 be three PF values in X. The lattice ( LPF , LPF ) of nonempty PF values in X is defined as follows: LPF
( x), P
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2
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with the partial order LPF , which is defined as, for any p1 LPF and p2 LPF ,
p1 LPF p2 if and only if P1 ( x) P2 ( x) and P1 ( x) P2 ( x) . The top and bottom
elements of ( LPF , LPF ) are (1,0) and (0,1), respectively. Theorem 1. [52] For any three PF values p, p1, and p2 in X, the score function s( p) and the accuracy function a( p) satisfy the following properties: (T1.1) 1 s( p) 1 (boundedness of s function); 6
(T1.2) 0 a( p) 1 (boundedness of a function); (T1.3) if p1 LPF p2 , then s( p1 ) s( p2 ) . Definition 5. [18, 19, 38, 52] Let p1 and p2 be two PF values in X. A prioritized comparison procedure within the PF environment is defined as follows: (D5.1) if s( p1 ) s( p2 ) , then p1 p2 ; (D5.2) if s( p1 ) s( p2 ) , then p1
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p2 ; (D5.3) if s( p1 ) s( p2 ) and a( p1 ) a( p2 ) , then p1 p2 ; (D5.4) if s( p1 ) s( p2 ) and a( p1 ) a( p2 ) , then p1 p2 ; (D5.5) if s( p1 ) s( p2 ) and a( p1 ) a( p2 ) , then p1 ~ p2 .
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Definition 6. [14, 16, 17, 25, 35, 51] Let Int([0, 1]) denote the set of all closed subintervals of the unit interval [0, 1]. An IVPF set P is defined as a set of ordered pairs of membership and non-membership in a finite universe of discourse X and is given as follows: (8) P x, P ( x), P ( x) x X ,
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which is characterized by the interval of the membership degree: P : XInt([0, 1]), x X P ( x) ( [P ( x ), P (x )]) [0,1] and the interval of the non-membership degree: P : XInt([0, 1]), x X P ( x) ( [ P ( x ), P (x )]) [0,1]
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Let p (P ( x), P ( x)) ([P ( x), P ( x)],[ P ( x), P ( x)]) denote an IVPF value. The
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Definition 7. [14, 35, 51] Let p1 ( ([ P1 ( x), P1 ( x)],[ P1 ( x), P1 ( x)]) ) and p2 ( ([ P2 ( x), P2 ( x)],[ P2 ( x), P2 ( x)]) ) be two IVPF values in X. The distance between
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Definition 8. [16, 17, 35] Let p be an IVPF value in X. The score function S ( p) and the accuracy function A( p) of p are separately defined as follows: 2 2 2 2 1 S ( p) P ( x) P ( x) P ( x) P ( x) , (14) 2
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Definition 9. Let p , p1 , and p2 be three IVPF values in X. The lattice
( LIVPF , LIVPF ) of nonempty IVPF values in X is defined as follows:
LIVPF P ( x), P ( x) P ( x), P ( x) [0,1], P ( x) P ( x) 1 2
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p1 LIVPF p2 if and only if P ( x) P ( x) , P ( x) P ( x) , P ( x) P ( x) , and
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Theorem 2. [35] For any three IVPF values p , p1 , and p2 in X, the score function S ( p) and the accuracy function A( p) satisfy the following properties: (T2.1) 1 S ( p) 1 (boundedness of S function); (T2.2) 0 A( p) 1 (boundedness of A function);
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(T2.3) if p1 LIVPF p2 , then S ( p1 ) S ( p2 ) .
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Definition 10. [16, 17] Let p1 and p2 be two IVPF values in X. A prioritized comparison procedure within the IVPF environment is defined as follows: (D10.1) if S ( p1 ) S ( p2 ) , then p1 p2 ; (D10.2) if S ( p1 ) S ( p2 ) , then p1 p2 ; (D10.3) if S ( p1 ) S ( p2 ) and A( p1 ) A( p2 ) , then p1 p2 ; (D10.4) if S ( p1 ) S ( p2 ) and A( p1 ) A( p2 ) , then p1 p2 ; (D10.5) if S ( p1 ) S ( p2 ) and A( p1 ) A( p2 ) , then p1 ~ p2 .
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3. A novel concept of risk attitude-based score functions
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Different decision makers have diverse attitudes towards risk-taking, such as risk seeking and risk aversion, in real-world MCDA problems. Incorporating the decision maker’s risk attitude into the developed outranking technique not only facilitates practical applications of the proposed methodology but also characterizes morereasonable subjective judgments in the decision-making processes. From the perspective of risk-taking, this paper develops novel risk attitude-based score functions within the PF/IVPF environments and investigates their desirable and useful properties in this section. The proposed methodology is built upon a solid base on theory and concepts, as depicted in Figure 1. This figure summarizes relevant theoretical bases of risk attitude-based score functions, linear orders, and admissible orders. To conduct a comprehensive comparative analysis, this paper utilizes certain numerical examples that were adopted from Garg [16, 17, 19], Liang et al. [25], Peng and Dai [33], and Zhang [50, 51] to examine the applicability and effectiveness of the proposed risk attitude-based score functions for ranking PF/IVPF information. Detailed comparative results and discussions are presented in Appendix A. These new 8
concepts form the core structure of the proposed risk attitudinal assignment models in the subsequent study. Relevant theoretical bases of risk attitude-based score functions
Relevant theoretical bases of linear orders and admissible orders
Definitions 11 and 14 introduce novel measures of risk attitude-based score functions within the PF/IVPF environments.
Definitions 12 and 15 construct binary relations by means of the risk attitude-based score functions.
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Remarks 1 and 3 present the meanings of the parameters within the risk attitudebased score functions.
Theorems 4 and 7 examine the properties of reflexivity, antisymmetry, transitivity, and completeness possessed by linear orders via the risk attitude-based score functions.
Definitions 13 and 16 establish admissible orders within the PF/IVPF environments.
Theorems 3 and 6 investigate some useful properties possessed by the risk attitudebased score functions.
Theorems 5 and 8 show that the binary relations based on the risk-neutral score functions are admissible orders.
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Remarks 2 and 4 explain the fundamental inner psychological characteristics of optimism and pessimism embedded in the risk attitude-based score functions.
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Figure 1. Theoretical bases of the proposed methodology.
3.1 New risk attitude-based score function of PF values
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The concept of PF sets is an effective tool for handling uncertainty in real-world applications, especially in the field of MCDA. However, the complexity of manipulating PF information makes it troublesome to directly employ the relevant concept in formulating the developed outranking methodology. Motivated by the techniques of risk attitudinal ranking methods [21, 42], this paper introduces a novel measure to characterize PF values as comparable real numbers to address the ranking issue within the PF environment. Guo [21] proposed a useful measure to compare AIF values based on the amount of information using a geometrical representation of A-IF sets to overcome some drawbacks and ambiguities in the existing approach via score functions and accuracy functions. Moreover, Guo [21] introduced the idea of an attitudinal-based extension of his developed technique. This paper presents an extended concept of the risk attitude-based score function for appropriately characterizing PF information and facilitating an effective ranking procedure in the proposed risk attitudinal assignment model. Yager [44] extended the applicability of the OWA operator by considering a situation in which the argument is a continuous interval [ , ] , i.e., the so-called continuous ordered weighted averaging (C-OWA) operator, which is expressed as follows: 1 dQ( y ) FQ ([ , ]) (17) y dy . 0 dy 9
In the formula above, FQ ([ , ]) is characterized by the basic unit-interval monotonic (BUM) function Q: [0, 1][0, 1] with the condition: (1) Q(0) 0 ; (2) Q(1) 1 ; and (3) Q( y1 ) Q( y2 ) if y1 y2 . Additionally, Yager [44] proved that FQ ([ , ]) could be derived in the following way: FQ ([ , ]) Q( y)dy 1 , 1
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where Q( y)dy and [0,1] . In particular, is an attitudinal character of the 0
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BUM function Q that can be extracted from the decision maker’s attitude. Motivated by Yager’s approach, Guo [21] designated Q( y) y t ( t 0 ) to capture the decision maker’s risk attitude via the BUM function Q. Accordingly, the following results were obtained: 1 1 , (19) y t dy 0 t 1 1 1 t . (20) FQ ([ , ]) 1 t 1 t 1 t 1 Note that lim FQ ([ , ]) and lim FQ ([ , ]) . Moreover, FQ ([ , ]) t 0
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( ) 2 when t 1 . Because the attitudinal character (determined by t in Guo’s proposal) is capable of describing the type and level of the decision maker’s attitude, Guo [21] conceived it as a ratio of the hesitation margin assigned to the membership function in accordance with the decision maker’s characteristics. Furthermore, Guo [21] developed a novel attitudinal-based measure that can fully take into account both the amount of information associated with A-IF values and the attitude of the decision maker involved. Considering the usefulness of the attitudinal character , this paper attempts to extend the applicability of the attitudinal-based measure in the A-IF context to a more general PF environment. This allows us to introduce a new concept of the risk attitude-based score function that is established in a manner analogous to Guo’s risk attitudinal technique.
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Definition 11. Let p be a PF value in X. Let a parameter t denote the risk attitude of a decision maker and t 0 . The risk attitude-based score function rt ( p) of p is defined as follows:
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t 1 2 2 2 rt ( p) 1 P ( x) P ( x) . P ( x) t 1 t 1
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Remark 1. The parameter t represents a decision maker’s attitude towards risktaking, where t (1 ) / . More specifically, the decision maker’s risk attitude can be described as risk-seeking if 0 t 1 (i.e., 0.5 1 ), risk-neutral if t 1 (i.e., 0.5 ), and risk-averse if t 1 (i.e., 0 0.5 ). The risk attitude-based score function can be equivalently rewritten via the attitudinal character as follows:
2 2 2 r ( p) 1 1 P ( x) P ( x) P ( x) .
(22)
Remark 2. In addition to the risk attitude, the parameter t can reflect fundamental inner psychological characteristics of optimism and pessimism in some way. 10
Optimism and pessimism are fundamental constructs that express how people respond to their perceived environment and how they construe and affect subjective judgments [7, 9]. According to Definition 11, some typical cases are discussed as follows: (1) When 0 t 1 , more than half of the degree of indeterminacy has been assigned to the degree of membership. This implies that there is relatively less uncertainty left in the indeterminacy part. Because optimistic decision makers interpret their decision situations positively and expect favorable outcomes [7, 9], the case of 0 t 1 represents a progressive behavior and can reflect the decision maker’s dispositional optimism. (2) When t 1 , the risk attitude-based score function becomes:
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1 2 2 2 1 (23) r1 ( p) 1 P ( x) P ( x) P ( x) , 2 2 i.e., the so-called risk-neutral score function. This case can reflect the decision maker’s neutral attitude towards risk-taking. (3) When t 1 , less than half of the degree of indeterminacy has been assigned to the degree of membership, i.e., relatively more uncertainty is left in the indeterminacy part. Pessimistic decision makers expound their decision situations negatively and anticipate unfavorable outcomes [7, 9]. Thus, this case represents a conservative behavior and can reflect the decision maker’s dispositional pessimism. (4) The cases of t 0 and t can reflect complete (or extreme) optimism and pessimism, respectively.
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Theorem 3. For a PF value p in X, the risk attitude-based score function rt ( p) satisfies the following properties: (T3.1) 0 rt ( p) 1 (boundedness of rt function); (T3.2) if p=(0, 1), then rt ( p) 0 for all t (zero property of rt function);
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(T3.3) if p=(1, 0), then rt ( p) 1 for all t (one property of rt function);
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(T3.4) s( p) rt ( p) for all t. Proof. The proofs of (T3.2) and (T3.3) are trivial. (T3.1) According to Definition 1, it is easy to see that 0 ( P ( x))2 1 and
0 (P ( x))2 ( P ( x))2 1 . Moreover, because t 0 , it is known that
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0 t (t 1) 1 and 0 1 (t 1) 1 . One can infer that 0 (t (t 1) )( P ( x))2 1 and 0 ( P ( x))2 (1 (t 1) )( P ( x))2 1 . It can be obtained that 0 rt ( p) 1 . Hence, (T3.1) is proved. (T3.4) For all t 0 , one can easily prove the following results:
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( x)
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P ( x) P ( x) P ( x) s( p). 2
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2
Thus, (T3.4) is valid. This completes the proof. In general, the ranking of PF values is nontrivial because no natural linear orders 11
exist among them. The following definition and theorem provide us with a means of employing the risk attitude-based score function rt ( p) to establish a linear order within the PF environment. Definition 12. Let p1 and p2 be two PF values in X. Let a binary relation LPF r ( t ) be defined as “being smaller than or indifferent from” based on the risk attitude-based score function rt . That is, p1 LPF r ( t ) p2 if and only if rt ( p1 ) rt ( p2 ) .
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Theorem 4. Let p, p1, and p2 be three PF values in X. Let ( LPF r (t ) , LPF r ( t ) ) denote a
lattice of nonempty PF values in X, in which LPF r (t ) {(P ( x), P ( x))|P ( x), P ( x)
[0,1],(P ( x))2 ( P ( x))2 1} with the binary relation LPF r ( t ) . Moreover, the top and
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bottom elements of ( LPF r (t ) , LPF r ( t ) ) are (1,0) and (0,1), respectively. Then, the binary relation LPF r ( t ) is a linear order.
Proof. For the reflexivity, because rt ( p) rt ( p) , p LPF r ( t ) p for each p in X. Thus, the binary relation LPF r ( t ) is reflexive. For the antisymmetry, if both rt ( p1 ) rt ( p2 )
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and rt ( p2 ) rt ( p1 ) hold, it is clearly known that rt ( p1 ) rt ( p2 ) . This implies that if
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p1 LPF r ( t ) p2 and p2 LPF r ( t ) p1 , then p1 LPF r ( t ) p2 ; thus, LPF r ( t ) is antisymmetric.
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For the transitivity, the relations p LPF r ( t ) p1 and p1 LPF r ( t ) p2 indicate
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rt ( p) rt ( p1 ) and rt ( p1 ) rt ( p2 ) , respectively. Because rt ( p) , rt ( p1 ) , and rt ( p2 ) are comparable real numbers, one obtains rt ( p) rt ( p1 ) rt ( p2 ) . Therefore, both p LPF r ( t ) p1 and p1 LPF r ( t ) p2 imply that p LPF r ( t ) p2 for all p, p1, and p2 in X; that
ED
is, LPF r ( t ) is transitive. Accordingly, the binary relation LPF r ( t ) is a partial order because it is reflexive, antisymmetric, and transitive. Moreover, the lattice ( LPF r (t ) , LPF r ( t ) ) is a partially ordered set. On the other hand, it is obvious that any
PT
two p1 and p2 in X are comparable via their corresponding risk attitude-based score functions rt ( p1 ) and rt ( p2 ) ; i.e., either rt ( p1 ) rt ( p2 ) or rt ( p2 ) rt ( p1 ) holds. It
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follows that the partial order LPF r ( t ) is complete. Consequently, the binary relation
LPF r ( t ) is a linear order. This completes the proof. It is worth mentioning that the binary relation LPF r ( t ) using the proposed risk
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attitude-based score function rt is not only a linear order but also an admissible order in the case of t 1 (i.e., a risk-neutral attitude). Bustince et al. [6] extended the usual partial order between intervals to present the concept of an admissible order as a total order. Guo [21] also proposed the concept of admissible orders within A-IF and A-IVIF environments. In a manner similar to that of Bustince et al. [6] and Guo [21], this paper defines the concept of an admissible order in the PF context. To make things concrete, an order on LPF is admissible if it is linear and refines the order 12
LPF . Definition 13. Let ( LPF , LPF ) be a partially ordered set. The order
is called an
admissible order within the PF environment if (D13.1) is a linear order on LPF ; (D13.2) for any p1 , p2 LPF , p1 p2 whenever p1 LPF p2 . Theorem 5. Let p1 and p2 be two PF values in X, where p1 and p2 belong to LPF with
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the partial order LPF . If p1 LPF p2 , then r1 ( p1 ) r1 ( p2 ) . Moreover, the binary relation LPF r (1) is an admissible order on LPF .
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Proof. Applying Definition 12 and Theorem 4, it is known that the risk-neutral score function r1 : LPF [0,1] induces on LPF a relation LPF r (1) given by
p1 LPF r (1) p2 if and only if r1 ( p1 ) r1 ( p2 ) . Furthermore, this relation is a linear order
on LPF . According to (3) and (23), r1 ( p1 ) can be calculated in the following way:
2
P1
2
( x)
2 1 1 P1 ( x) P1 ( x) 2
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2 1 r1 ( p1 ) 1 1 P1 ( x) P1 ( x) 2
2
2 2 1 2 2 1 1 1 1 1 P1 ( x) P1 ( x) P1 ( x) P1 ( x) 2 2 2 2 2 2
2
P1
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1 1 P1 ( x) 4
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2 ( x) 1 P1 ( x)
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2
P1
2 ( x)
2 2 4 1 1 P1 ( x) P1 ( x) . 2 Similarly, one has r1 ( p2 ) (1 2) (1 ( P ( x))2 )2 ( P ( x))4 . It is easy to show that
ED
2
2
r1 ( p1 ) r1 ( p2 ) because P1 ( x) P2 ( x) and P1 ( x) P2 ( x) according to the
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premise condition of p1 LPF p2 . Moreover, p1 LPF r (1) p2 holds because
r1 ( p1 ) r1 ( p2 ) . Therefore, the binary relation LPF r (1) based on the risk-neutral score
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function r1 refines the partial order LPF ; that is, p1 LPF p2 implies p1 LPF r (1) p2 . According to Definition 13, the binary relation LPF r (1) is admissible in the PF context. In addition, for the lattice ( LPF , LPF r (1) ) , LPF is a chain because the relation
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LPF r (1) is a linear order on LPF . This completes the proof.
3.2 New risk attitude-based score function of IVPF values This subsection extends the proposed new concepts of a risk attitude-based score function and an admissible order from a PF environment to a more general IVPF environment. Definition 14. Let p be an IVPF value in X. Let a parameter t denote the risk 13
attitude of a decision maker and t 0 . The risk attitude-based score function Rt ( p) of p is defined as follows:
1 2 2 2 2 t Rt ( p) 1 P ( x) P ( x) P ( x) P ( x) 2 2 t 1
2 2 1 P ( x) P ( x) t 1
(24)
1 2
.
1 2
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Remark 3. Because t (1 ) / , the risk attitude-based score function can be equivalently rewritten using the attitudinal character as follows: 2 2 2 2 1 1 R ( p) 1 P ( x) P ( x) P ( x) P ( x) 2 2 P ( x) P ( x) . 2
2
(25)
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Remark 4. Some cases of the t value in Definition 14 are examined as follows: (1) When 0 t 1 (or equivalently, 0.5 1 ), more than half of the interval of the indeterminacy degree has been assigned to the interval of the membership degree. Because relatively less uncertainty is left in the indeterminacy part, the case of 0 t 1 can reflect the decision maker’s optimistic attitude towards risktaking in the IVPF context. (2) When t 1 (or equivalently, 0.5 , i.e., a neutral attitude), the risk attitudebased score function R1 ( p) becomes the risk-neutral score function, as follows:
ED
2 2 2 2 1 1 R1 ( p) 1 P ( x) P ( x) P ( x) P ( x) 2 4
(26) 1 2 2 2 1 P ( x) P ( x) . 2 (3) When t 1 (or equivalently, 0 0.5 ), less than half of the interval of the indeterminacy degree has been assigned to the interval of the membership degree. Relatively more uncertainty is left in the indeterminacy part. This case can reflect the decision maker’s pessimistic attitude towards risk-taking in the IVPF context. (4) The cases of t 0 and t can respectively reflect the decision maker’s completely (or extremely) optimistic and pessimistic attitude towards risk-taking in the IVPF context.
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Theorem 6. For an IVPF value p in X, the risk attitude-based score function Rt ( p) satisfies the following properties: (T6.1) 0 Rt ( p) 1 (boundedness of Rt function); (T6.2) if p ([0,0], [1,1]), then Rt ( p) 0 for all t (zero property of Rt function); (T6.3) if p ([1,1], [0,0]), then Rt ( p) 1 for all t (one property of Rt function); (T6.4) S ( p) Rt ( p) for all t. Proof. The proofs of (T6.2) and (T6.3) are trivial. 14
(T6.1) With the use of Definition 6, one has 0 ( P ( x))2 1, 0 ( P ( x ))2 1 , that 0 t (t 1) 1 and 0 1 (t 1) 1 because t 0 . Observe that: 2 2 t 0 P ( x) P ( x) 1 , 2 t 1 2 2 1 0 P ( x) P ( x) 1 , t 1 2 2 1 0 P ( x) P ( x) 1 . t 1 It can be obtained that 0 Rt ( p) 1 ; hence, (T6.1) is proved. (T6.4) For all t 0 , it is easily verified that: 2 2 2 2 t 1 1 P ( x) P ( x) 1 P ( x) P ( x) 2 t 1 2
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0 (P ( x))2 +( P ( x))2 1 , and 0 (P ( x))2 +( P ( x)) 2 1 . Moreover, it is known
2 2 2 2 1 1 P ( x) P ( x) 1 P ( x) P ( x) 2 2 2 2 2 2 2 1 1 P ( x) P ( x) P ( x) P ( x) P ( x) P ( x) , 2 2 2 2 2 2 2 2 P ( x) P ( x) t 1 1 P ( x) P ( x) P ( x) P ( x) . Thus, one can conclude that: Rt ( p)
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1
2 2 2 2 2 2 1 t 1 P ( x) P ( x) P ( x) P ( x) P ( x) P ( x) 1 2 2 t 1 t 1
2 2 2 2 2 2 1 1 1 ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) P P P P P P 2 2 2
ED
2 2 2 2 1 P ( x) P ( x) P ( x) P ( x) =S ( p). 2
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Therefore, (T6.4) is valid. This completes the proof. Because no natural linear orders exist among IVPF values, this paper provides the following definition and theorem to establish a linear order within the IVPF environment by means of the risk attitude-based score function Rt ( p) .
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Definition 15. Let p1 and p2 be two IVPF values in X. Let a binary relation
LIVPF R ( t ) be defined as “being smaller than or indifferent from” based on the risk
attitude-based score function Rt . That is, p1 LIVPF R ( t ) p2 if and only if
Rt ( p1 ) Rt ( p2 ) . Theorem 7. Let p , p1 , and p2 be three IVPF values in X. Let ( LIVPF R (t ) , LIVPF R ( t ) ) 15
denote a lattice of nonempty IVPF values in X, in which LIVPF R (t ) {(P ( x),
P ( x))|P ( x), P ( x) [0,1],(P ( x))2 ( P ( x))2 1} with the binary relation LIVPF R ( t ) . Moreover, the top and bottom elements of ( LIVPF R (t ) , LIVPF R ( t ) ) are ([1,1], [0,0]) and ([0,0], [1,1]), respectively. Then, the binary relation LIVPF R ( t ) is a linear order. Proof. For the reflexivity, because Rt ( p) Rt ( p) , p LIVPF R ( t ) p for each p in X. The binary relation LIVPF R ( t ) is reflexive. For the antisymmetry, if both
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Rt ( p1 ) Rt ( p2 ) and Rt ( p2 ) Rt ( p1 ) hold, it is easy to see that Rt ( p1 ) Rt ( p2 ) . This indicates that if p1 LIVPF R ( t ) p2 and p2 LIVPF R ( t ) p1 , then p1 LIVPF R ( t ) p2 . One can obtain that LIVPF R ( t ) is antisymmetric. For the transitivity, the relations
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p LIVPF R ( t ) p1 and p1 LIVPF R ( t ) p2 indicate Rt ( p) Rt ( p1 ) and Rt ( p1 ) Rt ( p2 ) ,
respectively. Because Rt ( p) , Rt ( p1 ) , and Rt ( p2 ) are comparable real numbers, an immediate consequence of Rt ( p) Rt ( p1 ) Rt ( p2 ) can be obtained. As a result, if p LIVPF R ( t ) p1 and p1 LIVPF R ( t ) p2 , then p LIVPF R ( t ) p2 for all p , p1 , and p2 in X.
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That is, LIVPF R ( t ) is transitive. Therefore, the binary relation LIVPF R ( t ) is a partial
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order because it is reflexive, antisymmetric, and transitive. Additionally, the lattice ( LIVPF R (t ) , LIVPF R ( t ) ) is a partially ordered set. In particular, any two p1 and p2 in X
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are comparable via their corresponding risk attitude-based score functions Rt ( p1 ) and Rt ( p2 ) . Apparently, either Rt ( p1 ) Rt ( p2 ) or Rt ( p2 ) Rt ( p1 ) holds. One can
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conclude that the partial order LIVPF R ( t ) is complete. Accordingly, the binary relation
ED
LIVPF R ( t ) is a linear order. This completes the proof.
PT
The risk-neutral score function R1 (i.e., t 1 in Rt ) is not only a linear order but also an admissible order. The following definition and theorem provide us with a means of establishing an admissible order within the IVPF environment. To be precise, an order on LIVPF is admissible if it is linear and refines the order LIVPF .
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Definition 16. Let ( LIVPF , LIVPF ) be a partially ordered set. The order admissible order within the IVPF environment if (D16.1) is a linear order on LIVPF ;
is called an
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(D16.2) for any p1 , p2 LIVPF , p1 p2 whenever p1 LIVPF p2 . Theorem 8. Let p1 and p2 be two IVPF values in X, where p1 and p2 belong to LIVPF with the partial order LIVPF . If p1 LIVPF p2 , then R1 ( p1 ) R1 ( p2 ) . Moreover, the binary relation LIVPF R (1) is an admissible order on LIVPF . Proof. Recalling Definition 15 and Theorem 7, it is easy to see that the risk-neutral score function R1 : LIVPF [0,1] induces on LIVPF a relation LIVPF R (1) given by 16
p1 LIVPF R (1) p2 if and only if R1 ( p1 ) R1 ( p2 ) . This relation is also a linear order on
LIVPF . By means of (12) and (26), R1 ( p1 ) can be computed as follows: 2 2 2 2 1 1 R1 ( p1 ) 1 1 P1 ( x) P1 ( x) 1 P1 ( x) P1 ( x) P1 ( x) 2 4
2
1
2 2 2 2 2 1 ( x) 1 P1 ( x) P1 ( x) 1 P1 ( x) P1 ( x) 2
P1
2
2 2 2 2 1 1 1 1 P1 ( x) P1 ( x) P1 ( x) P1 ( x) 1 P1 ( x) 2 4 2
1
2 ( x) ( x) ( x) 2
P1
P1
2
P1
2
2 2 2 2 2 1 1 1 P1 ( x) P1 ( x) P1 ( x) P1 ( x) 4 4
1
2 2 2 2 1 P1 ( x) P1 ( x) . 4 Analogously, one obtains:
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2
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2 2 2 2 2 1 1 R1 ( p2 ) 1 P2 ( x) P2 ( x) P2 ( x) P2 ( x) 4 4
1
A
ED
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2 2 2 2 1 P2 ( x) P2 ( x) . 4 It is verified that R1 ( p1 ) R1 ( p2 ) because P1 ( x) P2 ( x) , P1 ( x) P2 ( x) ,
P ( x) P ( x) , and P ( x) P ( x) , according to the premise condition of 1
2
1
2
PT
p1 LIVPF p2 . Moreover, p1 LIVPF R (1) p2 holds because R1 ( p1 ) R1 ( p2 ) . Thus, the binary relation LIVPF R (1) based on the risk-neutral score function R1 refines the
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partial order LIVPF ; that is, p1 LIVPF p2 implies p1 LIVPF R (1) p2 . According to Definition 16, the binary relation LIVPF R (1) is admissible within the IVPF environment. Furthermore, for the lattice ( LIVPF , LIVPF R (1) ) , LIVPF is a chain because
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the relation LIVPF R (1) is a linear order on LIVPF . This completes the proof.
4. Outranking methods with risk attitudinal assignment models Based on the risk attitude-based score functions, this section develops an outranking approach using novel risk attitudinal assignment models for managing MCDA problems involving PF/IVPF information. Some useful concepts, such as the precedence frequency and precedence contribution matrices, are introduced to facilitate the modeling processes. This section provides two effective algorithms for 17
solving the problems of multiple criteria evaluation and selection in the PF/IVPF contexts. The issue of choosing an appropriate tool to determine linear orders is very important for many MCDA applications. As proven in Theorems 4 and 7, the binary relations LPF r ( t ) and LIVPF R ( t ) based on the risk attitude-based score functions rt and Rt , respectively, are linear orders. In particular, according to Theorems 5 and 8, the binary relations LPF r (1) and LIVPF R (1) are admissible orders on LPF and LIVPF ,
4.1 Proposed methodology in the PF context Consider an MCDA problem in which Z z1 , z2 ,
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respectively. Bustince et al. [6] have demonstrated that the concept of an admissible order is a useful tool for providing a total order. Therefore, the risk-neutral score functions r1 and R1 can provide total orders in the PF and IVPF contexts, respectively. Accordingly, this paper attempts to employ the risk attitude-based score functions to identify criterion-wise precedence ranks among alternatives and develop novel risk attitudinal assignment models for addressing MCDA problems within the PF/IVPF environments.
, zm is a set of m ( m 2 )
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feasible alternatives and C c1 , c2 , , cn is a finite set of n ( n 2 ) criteria. In reallife decision problems, the decision maker’s evaluations related to the performance of alternatives with respect to each criterion are often expressed by linguistic terms comprising vagueness and uncertainty. These uncertain, vague and hesitant judgments can be described more comprehensively by using the PF/IVPF sets. In the PF context, let a PF value pij ( ij , ij ) denote the evaluative rating of an alternative zi Z with respect to a criterion c j C , such that ij [0,1] , ij [0,1] , and
ED
0 (ij )2 ( ij )2 1 . In particular, ij and ij represent the degree to which zi
performs well or poorly, respectively, in terms of cj. The indeterminacy degree that 2 2 corresponds to each pij is determined as ij 1 ( ij ) ( ij ) . The MCDA problem
PT
within the PF environment can be concisely expressed in the following PF decision matrix: c1 c2 cn
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z1 ( 11 , 11 ) ( 12 , 12 ) ( 1n , 1n ) (27) z ( , ) ( 22 , 22 ) ( 2 n , 2 n ) p pij 2 21 21 mn zm ( m1 , m1 ) ( m 2 , m 2 ) ( mn , mn ) and the weight vector (w1 , w2 , , wn )T of the criteria, where the importance weight
wj of cj satisfies the normalization conditions: w j [0,1] and
n j 1
wj 1 .
MCDA problems can be deemed assignment problems because the MCDA task is generally related to the ordering of alternatives based on a priority ranking system; that is, MCDA can be considered the assigning of alternatives to a rank of order. It is often to face a practical problem that independence among criteria does not exactly hold. Criteria are generally dependent on each other in a complex, dynamic, and 18
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highly subjectively way [43]. In particular, when value dependency, preferential dependency, or anchor dependency exists among the criteria in question, it is hard to expect the criteria take the separable additive form [12, 43]. The assignment-based model provides an effective and convenient way to overcome this difficulty, because it does not directly combine the evaluative ratings across various criteria. Specifically, the assignment-based model combines all criterion-wise rankings into an overall ranking which achieves a best compromise among component rankings. Originated from the core structure of the assignment-based model [10, 12, 43], this paper employs the proposed risk attitude-based score function rt to construct a novel risk attitudinal assignment model and to develop an outranking method for addressing the MCDA problem. The proposed outranking methodology involving PF information is summarized in the following algorithmic procedure. Algorithm I. (Proposed methods in a PF setting) Step I-1: Determine risk attitude-based score functions for each evaluative rating. Set the t value as a parameter for representing the decision maker’s risk attitude, where 0 t 1 , t 1 , and t 1 for risk-seeking, risk-neutral, and riskaverse attitudes, respectively. Applying Definition 11, the risk attitude-based score function rt ( pij ) of the evaluative rating pij for each zi Z and c j C is calculated as follows:
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2 2 2 t 1 rt ( pij ) 1 ij ij ij . t 1 t 1 Step I-2: Identify a precedence frequency matrix based on criterion-wise precedence ranks. According to the properties proven in Theorems 4 and 5, it is known that the binary relation LPF r ( t ) is a linear order for all t; moreover, the binary
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relation LPF r (1) is an admissible order. Thus, it is appropriate to utilize the concept
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PT
ED
of risk attitude-based score functions to determine the criterion-wise precedence ranks among alternatives. More specifically, the m alternatives can be ranked in terms of each c j based on the descending order of the rt ( pij ) values. Next, one can establish a precedence frequency matrix whose entries measure the frequency at which each alternative is assigned a particular rank among all criterion-wise precedence relationships of the alternatives. In the PF context, let an m m square non-negative matrix f denote the precedence frequency matrix. The entry f i k ( i, k 1, 2, , m ) in f represents the frequency that zi is ranked k-th among all of the criterion-wise precedence rankings by means of the rt ( pij ) values. The matrix f is expressed as follows: 1st 2nd m-th
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f11 1 f f 2 zm f m1 z1 z2
f12 f 22 f m2
f1m f 2m , f mm
(28)
where the entry f i k satisfies the properties of
m i 1
fi k n for k 1, 2,
,m. 19
m k 1
fi k n for i 1, 2,
, m and
Step I-3: Construct a precedence contribution matrix via combining the importance weights. Based on the precedence frequency matrix f, a more comprehensive concept of the precedence contribution matrix is established to adequately utilize preference information about the criteria. Let an m m square non-negative matrix denote the precedence contribution matrix in the PF context, which is expressed as follows: 1st 2nd m-th
1m 2m . mm
(29)
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11 12 1 2 22 zm m1 m2 z1 z2
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Note that the entry ik ( i, k 1, 2, , m ) in represents a measure of the concordance among all of the criteria in assigning the alternative zi a rank k. Let c j1 , c j2 , , and c j k (with the importance weights w j1 , w j2 , , and w j k , fi
fi
respectively) represent the corresponding criteria for which zi is ranked k-th. To be precise, the entry ik is determined by aggregating the products of multiplying the fi
, and rt ( pij k ) , respectively, corresponding to f i k . That
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rt ( pij1 ) , rt ( pij2 ) rt ( pij2 ),
, and w j k by the risk attitude-based score functions fi
is,
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fi k
w j rt ( pij ) 1
(30)
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k i
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importance weights w j1 , w j2 ,
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PT
ED
for i, k 1, 2, , m , where 0 ik 1 . Step I-4: Establish a risk attitudinal assignment model using a permutation matrix. For the precedence contribution matrix , the larger the contribution revealed by entry ik is, the greater the concordance will be from assigning zi to the k-th overall precedence rank. Accordingly, this paper constructs a simple and k effective risk attitudinal assignment model to solve for an aggregate ranking. Let i denote a binary variable that is restricted to be either 0 or 1. In the PF context, let an m m square matrix denote a permutation matrix whose element ik =1 if zi is k assigned to the overall rank k; otherwise, i =0. The matrix is expressed as follows: 1st 2nd m-th
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11 12 1m 1 (31) 2 22 2m . 1 zm m m2 mm Note that zi should be assigned to only one rank in the overall precedence ranking; m thus, k 1 ik 1 . Similarly, a given rank k should only have one alternative assigned z1 z2
to it; hence,
m i 1
ik 1 . To find the overall precedence ranking that yields the 20
m
largest value of
i 1
ik , this paper utilizes a zero-one linear programming
m k k 1 i
format to establish the following risk attitudinal assignment model: m m max ik ik i 1 k 1 m
subject to
k 1
k i
1, i 1, 2,
, m,
k i
1, k 1, 2,
, m,
m
i 1
(32)
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ik 0 or 1 for all i and k . Step I-5: Acquire an outranking solution by solving for the optimal permutation matrix. The optimal permutation matrix ˆ can be obtained by solving the risk attitudinal assignment model in (32). To determine an outranking solution, the overall precedence orders of the alternatives can be easily determined by multiplying Z by ˆ , as follows: ˆ11 ˆ12 ˆ1m 1 ˆ2 ˆ22 ˆ2m ˆ Z z1 , z2 , , zm . (33) 1 2 ˆmm ˆm ˆm This overall ranking is an ultimate priority ranking that produces the best compromise among all of the criterion-wise precedence rankings of the alternatives.
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4.2 Proposed methodology in the IVPF context
ED
The outranking method using a risk attitudinal assignment model involving PF information can be extended to the more general IVPF environment. For an MCDA problem in the IVPF context, the evaluative rating of an alternative zi Z with
respect to a criterion c j C is expressed as an IVPF value pij [ ij , ij ],[ ij , ij ] ,
PT
where [ ij , ij ] Int([0, 1]), [ ij , ij ] Int([0, 1]), and 0 (ij )2 ( ij )2 1 . The intervals [ ij , ij ] and [ ij , ij ] represent the flexible degrees where zi performs
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well or poorly, respectively, in terms of cj. The interval of the indeterminacy degree that corresponds to each pij is computed as ij [ ij , ij ] 1 (ij )2 ( ij ) 2 , 1 (ij )2 ( ij ) 2 . The MCDA problem involving IVPF information can be concisely expressed in the following IVPF decision matrix:
21
mn
c1
, , , z1 11 11 11 11 z , , , 2 21 21 21 21 zm , , , m1 m1 m1 m1 and the weight vector (w1 , w2 ,
c2
cn
(34)
, 1n , 1n , 1n 22 , 22 , 22 , 22 2n , 2n , 2n , 2n m 2 , m 2 , m 2 , m 2 mn , mn , mn , mn n T , wn ) , such that w j [0,1] and j 1 w j 1 . 12
, 12 , 12 , 12
1n
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p pij
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Based on a risk attitudinal assignment model via the risk attitude-based score function Rt , the algorithmic procedure of the proposed outranking methodology involving IVPF information is described, as follows. Algorithm II. (Proposed methods in an IVPF setting) Step II-1: Determine risk attitude-based score functions for each evaluative rating. Set the t value according to the decision maker’s risk attitude. Applying Definition 14, the risk attitude-based score function Rt ( pij ) of the evaluative rating
t 1 1 . 2 ij
2 ij
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2 2 1 t ij ij 1 2 2 t 1
2 ij
2 ij
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Rt ( pij )
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pij for each zi Z and c j C is calculated as follows:
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Step II-2: Identify a precedence frequency matrix based on criterion-wise precedence ranks. By the properties demonstrated in Theorems 7 and 8, one concludes that the binary relation LIVPF R ( t ) is a linear order for all t; moreover, the
ED
binary relation LIVPF R (1) is an admissible order. It follows that the m alternatives can
PT
be ranked with respect to each c j based on the descending order of the Rt ( pij ) values. Based on the obtained criterion-wise precedence ranks among alternatives, one can establish the precedence frequency matrix F in a manner analogous to the matrix f in (28). Let an m m square non-negative matrix F denote the precedence
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frequency matrix in the IVPF context. The entry Fi ( i, k 1, 2, , m ) in F represents the frequency with which zi is ranked k-th among all of the criterion-wise precedence rankings according to the Rt ( pij ) values. The matrix F is expressed as follows: 1st 2nd m-th
F11 1 F F 2 zm Fm1
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z1 z2
F12 F22 Fm2
k
F1m F2m , Fmm
(35)
k where the entry Fi satisfies the properties of
m i 1
Fi k n for k 1, 2,
m k 1
Fi k n for i 1, 2,
, m and
,m.
Step II-3: Construct a precedence contribution matrix via combining the 22
importance weights. Let an m m square non-negative matrix denote the precedence contribution matrix in the IVPF context, which is expressed as follows: 1st 2nd m-th
11 12 1 22 2 zm m1 m2 z1 z2
1m 2m . mm ( i, k 1, 2, , m ) in represents a measure of the
(36)
k
Fik
importance weights w j1 , w j2 ,
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Here, the entry i concordance among all of the criteria in assigning the alternative zi a rank k according to the precedence frequency matrix F. Let c j1 , c j2 , , and c j (with the
, and w j k , respectively) represent the corresponding
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k criteria for which zi is ranked k-th. The entry i is determined in the following way: Fi k
w j Rt ( pij ) k i
1
(37)
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for i, k 1, 2, , m , where 0 ik 1 . Step II-4: Establish a risk attitudinal assignment model using a permutation matrix. For the precedence contribution matrix , the larger the contribution revealed by the entry i is, the greater concordance will be from assigning zi to the k-th overall precedence rank. Thus, a useful risk attitudinal assignment model is
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constructed to solve for an aggregate ranking. Let i denote a binary variable that is restricted to be either 0 or 1. In the IVPF context, let an m m square matrix k
denote a permutation matrix whose element i =1 if zi is assigned to the overall rank
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k; otherwise, i =0. The matrix is expressed as follows: 1st 2nd m-th k
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11 12 1m 1 2 22 2m . zm m1 m2 mm It can be easily checked that z1 z2
(38)
m k 1
ik 1 because zi should be assigned to only one
rank in the overall precedence ranking. Moreover, one has
m i 1
ik 1 because a
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m m max ik ik i 1 k 1 m
subject to
k 1
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1, i 1, 2,
, m,
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1, k 1, 2,
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i 1
(39)
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ik 0 or 1 for all i and k . Step II-5: Acquire an outranking solution by solving for the optimal permutation ˆ can be determined by solving the risk matrix. The optimal permutation matrix attitudinal assignment model in (39). To acquire an outranking solution, the overall ˆ , as precedence orders of the alternatives can be derived by multiplying Z by follows: ˆ11 ˆ12 ˆ1m 1 2 ˆ 2m ˆ 2 ˆ 2 ˆ Z z1 , z2 , , zm (40) . m ˆ ˆ m1 ˆ m2 m
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5. Model application to financial decision making
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The proposed outranking methodology is a powerful tool for dealing with information that is characterized by hesitancy and vagueness. This section presents a financial decision-making application related to working capital management for a medical institution, along with a sensitivity analysis and a comparison to the technique of risk attitudinal ranking methods. Moreover, the developed approach is applied to an investment problem. A comparative study is also conducted with a newly developed outranking method that is an extension of the elimination and choice translating reality (ELECTRE) method to examine the advantages of the proposed methodology.
5.1 Financing policy of working capital management
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Efficient management of working capital is essential in the overall corporate strategy to improve firms’ profitability and liquidity, to create shareholder value, and to gain sustainable competitiveness [15, 28, 40]. Managing working capital, whose main components are accounts payable, accounts receivable, and inventory, means managing the day-to-day short-term operations of a firm [2, 28]. Working capital management plays a pivotal role for corporate finance due to its significant influence on firms’ liquidity and profitability [5, 31, 40, 41]. More precisely, higher working capital levels give firms a chance to increase sales and obtain discounts for early payments, which can enhance firms’ value [5]. However, higher working capital levels often require external financing (e.g., financing through debt). Thus, firms often incur additional financing expenses, which might lead to credit risks or the threat of bankruptcy [5, 40]. Combining these positive and negative perspectives regarding working capital effects, working capital management involves the risk and return trade-off, which is related to aggressive or conservative working capital policies. Financing policy refers to the financing models implemented by firms to satisfy 24
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working capital requirements. The financing policies of working capital management can generally be determined as aggressive or conservative [41]. In an aggressive working capital policy, the firm invests a small amount of capital in current assets in order to achieve higher gains on fixed assets [40, 41]. It utilizes higher levels of normally lower-cost short-term debt and less long-term capital, which may lead to the risk of short-term liquidity [5, 40]. Accordingly, it is considered the most risky and, potentially, the most profitable working capital management policy [28, 40]. Contrary to the aggressive policy, a conservative working capital policy relies on a greater volume of long-term debt in relation to the financing from short-term sources [41]. It causes the firm to define the liquidity requirements, which leads to a low liquidity risk position and profitability. Thus, the conservative policy reduces the risk of bankruptcy through investing a large amount of capital in current assets [28, 40, 41]. Simply stated, an aggressive working capital policy has a positive impact on profitability but a negative impact on liquidity, whereas a conservative working capital policy has a negative impact on profitability but a positive impact on liquidity [28]. In comparison with regular firms, those in the medical and health care industries possess several distinct characteristics that render financing decision making during working capital management more difficult and complex for managers. Ideally, medical institutions should be able to receive sufficient capital from the provision of treatment and care services to cover associated costs. To achieve this objective, managers must skillfully handle the four stages of the working capital cycle: (i) receive cash; (ii) use this cash to purchase resources (for instance, sanitary consumables and hiring) and pay accounts; (iii) use these resources to provide medical and care services; and (iv) bill patients and collect payments to continue the working capital cycle. However, under Taiwan’s current national health insurance (NHI) system, associated payments are usually received two or more months after patients or third-party payers have been billed. Most hospitals rely on third-party payers to pay patient bills; accounts receivable represent approximately 75% of the current assets of hospitals. Although patient deductibles show a gradually increasing trend, with the increasing amounts patients are burdened with, the risk of uncollectible accounts increases as well. In addition, because most accounts receivable are from third-party payers, owning accounts receivable with large sums implies the potential loss of other investment opportunities for earning returns. Hospital managers must be able to conduct evaluations based on the institution’s financial situation and environment, select suitable financial policies to ensure sufficient working capital, and maximize mobility and profitability. For managers, these are highly difficult and complex tasks. In short, for medical institutions, working capital significantly affects mobility, profitability, and competitiveness. Ensuring the possession of sufficient working capital to meet daily demands is essential to the operation of medical institutions. The key to doing so successfully lies in the selection of financial policies. In the light of above, the developed outranking methodology is applied to select a proper financing policy in the medical and health care fields. It is anticipated that the application results will both facilitate efficient management of working capital and support the practicality and effectiveness of the proposed approach.
5.2 Problem statement Financing decisions on aggressive/conservative working capital policies is a challenging problem for firms in an intricate and unpredictable business environment. Because of the particularity of the health care system and, specifically, the complexity 25
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of the NHI system in Taiwan, the selection of short-term financial policies becomes a highly complicated and ambiguous MCDA problem. This study focuses on the problem of selecting an appropriate financing policy of working capital management for a medical institution with the largest scale in Taiwan. To validate the practical usefulness and effectiveness of the developed approach, this subsection illustrates how the proposed concepts and models can be applied to address the financing decision in the medical and health care fields. The medical institution investigated in this case study is the Chang Gung Memorial Hospital (CGMH). CGMH offers the largest and most comprehensive health care services in Taiwan, being composed of a network of seven hospital branches. The main branch of CGMH is located in Linkou. Linkou CGMH is a multispecialty hospital and is the largest medical center in Taiwan. It contains approximately 4,000 beds and a total of 29 specialty centers. It is also the largest academic center ever accredited by the Joint Commission International (JCI) in 20142017. Working capital management is important to Linkou CGMH because it can grant this medical center financial flexibility and reduce its dependence on external sources of finance, especially under the government regulations of Taiwan’s NHI system. More generally, this importance is even greater for a medical institution if most of its institutional assets are current and it is highly reliant on current liabilities. Determining the optimal financing level of working capital is of great importance to Linkou CGMH. Sources of funds for financing are divided into longand short-term sources. Long-term sources of funds include long-term debt, shareholders’ equity, and spontaneous current liabilities, whereas short-term sources of funds include short-term loans, notes payable, discounted notes receivable, and other loans backed by current assets. Based on the differences between sources of funds and application methods, financing arrangements can be broadly divided into three types of basic policy: aggressive, conservative, and moderate. Crucial characteristics of these three types of basic policies are shown in Figure 2.
Figure 2. The characteristics of three types of basic financing policies. In an aggressive financing policy, short-term funds are used to finance temporary 26
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current assets and a portion of permanent current assets (i.e., short-term funds used to finance long-term funds), whereas the remaining permanent current assets and fixed assets are financed using long-term funds. Because this method utilizes less long-term and more short-term funds for financing, the higher interest rates required of longterm fund financing can usually be reduced, thus decreasing the financing cost. However, fluctuations in short-term interest rates are greater than those of long-term interest rates, and there is the risk of short-term interest rates increasing. Additionally, the stress of short-term debt repayment is high. Therefore, when this approach is adopted, it is necessary to evaluate whether interest rates demonstrate a rapidly increasing trend and to ensure that channels for short-term fund financing are continuous in order to consider both profitability and security. A conservative financing policy maintains a more cautious approach toward funding and primarily uses long-term funds to finance permanent current assets, fixed assets, and a portion of temporary current assets (i.e., long-term funds used to finance short-term funds). Only when the fund demand of temporary current assets exceeds the long-term funds used for financing are short-term funds used to finance the remaining portion. Although adopting this approach results in higher security and more-stable sources of funds, the financing costs are increased and the profit levels are reduced. In contrast, a moderate financing policy uses long-term funds to finance permanent current assets and fixed assets (i.e., long-term funds used to finance longterm funds), whereas temporary current assets are financed by short-term funds (i.e., short-term funds used to finance short-term funds). When adopting this approach, it is necessary to maintain the highest possible consistency between the expiration dates of short-term funds and long-term liabilities. However, because there are too many uncertain factors in practice, adopting this approach may result in gaps in funding. This case study designed five types of alternative plans suitable for use in working capital management based on the three aforementioned types of basic policy. It is important to note that the risk tolerance of medical and health care providers influences these working capital strategies. If providers are in a medical market with a lower number of competitors, the working capital demand can be more precisely predicted because the regional medical service demand is more stable. Therefore, medical institutions tend to adopt aggressive financing policies. By contrast, if providers are in an environment with more competitors, the regional medical service demand is less definite and stable; consequently, providers tend to adopt conservative financing policies. The three general principles in deciding whether to use short-term or long-term loans to meet working capital requirements are as follows: (i) using short-term loans to meet short-term working capital requirements; (ii) using long-term loans to meet long-term working capital requirements; and (iii) using a mixed strategy for capital requirements that are volatile, that is, using long-term loans to meet a certain amount of demand, then using short-term loans when short-term requirements emerge. This is because a mixed-type moderate financing policy is more suitable to an environment that is highly volatile. However, because the range of a moderate financing policy is wider and more difficult to define, in order to be more appropriate and accurate, this case study further divides the moderate financing policy into “aggressive-leaning,” “equally aggressive and conservative,” and “conservativeleaning” based on distributions of aggressive and conservative orientations. This case study comprises five plans related to working capital financing policy: aggressive dominant, aggressive-leaning, balanced aggressive and conservative, conservative27
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leaning, and conservative dominant. These are shown in Figure 3. To carefully evaluate these five types of alternative plans, managers consider the following six evaluation criteria: (1) Cash reserves A conservative financing policy involves an excess of cash reserves and very few short-term loans. This type of policy can reduce the likelihood that medical institutions will experience a financial crisis, and it eliminates concerns about short-term liquidation difficulties. However, using excess funds for unprofitable projects such as cash or securities will lower profit levels and incur holding costs. Managers must focus on the financial situations of medical institutions to carefully evaluate the amount of cash reserve needed to achieve maximized profitability and minimized risk and cost.
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Figure 3. Five alternatives for a financing decision of working capital management.
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(2) Maturity hedging Many managers will attempt to coordinate the expiration dates of current assets and liabilities and use short-term bank loans to finance sanitary consumables. Then, they will use long-term financing to purchase fixed assets, avoiding to the greatest extent possible the use of short-term loans to finance longterm assets. However, this method of maturity hedging creates a situation where frequent refinancing is needed, and managers must bear the risk of interest rate fluctuations. (3) Interest rate fluctuation When managers are evaluating which type of financing strategy to use, it is necessary to consider relative long- and short-term interest rate levels. Short-term interest rates are usually lower than long-term ones. However, the volatility of short-term interest rates is high; when short-term loans expire, if another loan is desired, there may be a risk of increased interest rates. Although long-term interest rates ensure more stable sources of funds, financing with long-term interest rates results in higher costs, on average. (4) Financial leverage Financial leverage and working capital management are closely correlated in medical institutions; this correlation is usually measured using the ratio of total current liabilities to total assets. An increase in the ratio of current liabilities to 28
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total assets results in a higher leverage ratio, greater short-term liquidation stress, and an increase in the risk that medical institutions face. (5) Return on Assets The degree of aggression in financing policy is often negatively correlated with the return on assets (ROA) of medical institutions; even an increase in shortterm liabilities for financing may negatively affect ROA. Because a conservative financing policy uses more long-term loans, medical institutions have a longer time to fulfill financial obligations, which positively affects ROA. However, when short-term interest rates exhibit declining trends, because financing costs decrease, an aggressive financing policy positively affects ROA. Therefore, when managers evaluate the ROA performance of each plan, it is necessary to first predict trends in short-term interest rates. (6) Financing costs Financing costs are the associated costs incurred by medical institutions in order to increase funds or the cost of fund use and fund raising. Financing costs differ according to the sources of funding. For instance, the interest paid differs when bonds are issued and when different bank loans are applied for. In addition, opportunity cost is a type of financing cost; it refers to using a particular type of resource for a particular purpose and the maximum gain of other purposes that is given up as a result. Although medical institutions that use their own funds do not need to pay costs such as interest, they will lose potential profits gained from investing these funds. Notably, interdependence exists among the aforementioned six criteria, and the interactive relationship between each renders it difficult for managers to make a definite evaluation of the performance of the plans in each criterion (because criteria evaluation values are influenced by the associated effects of other related criteria). For instance, interest rate fluctuations will directly influence financing costs. If future interest rates increase, aggressive-leaning policies will be influenced by increases in short-term interest rates and will lead to increased financing costs. In addition, maturity hedging and interest rate fluctuation are interdependent. Generally, shorter periods result in greater interest rate fluctuations; by contrast, longer debt expiration dates result in lesser interest rate fluctuations. Moreover, managers’ degrees of preference for risk influence how criteria values are defined. Considering the imprecise and uncertain information associated with a complex medical circumstance, this paper attempts to employ the developed Algorithm II based on the IVPF theory to address this issue.
5.3 Application of the proposed approach
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The proposed methodology with Algorithm II was employed to help Linkou CGMH select the most appropriate financing policy of working capital management. This MCDA problem is defined by five financing policies and six criteria for evaluating the alternatives. Specifically, the set of alternatives is denoted by Z={z1 (aggressive dominant policy), z2 (aggressive-leaning policy), z3 (balanced aggressive and conservative policy), z4 (conservative-leaning policy), z5 (conservative dominant policy)}. The set of criteria is denoted by C={c1 (cash reserves), c2 (maturity hedging), c3 (interest rate fluctuation), c4 (financial leverage), c5 (return on assets), c6 (financing costs)}. The weight vector of the criteria was given by the authority of Linkou CGMH in advance as (0.10, 0.05, 0.20, 0.10, 0.25, 0.30)T. This study conducted a survey to collect three managers’ opinions and judgments about the 29
performance evaluation of the five alternatives with respect to each criterion. Based on a statistical inference approach, Chen [8] developed an interval estimation method with finite population correction to construct the membership functions of A-IVIF sets. By extending Chen’s statistical inference approach to the IVPF environment, this study estimated respective intervals of membership and non-membership degrees according to the survey results, which can reflect managers’ knowledge and expertise. The IVPF decision matrix p ( [ pij ]56 ) was constructed as follows: c1
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z1 ([0.15, 0.26],[0.74, 0.92]) z2 ([0.27, 0.44],[0.66, 0.82]) p z3 ([0.57, 0.69],[0.46, 0.57]) z4 ([0.78, 0.86],[0.13, 0.28]) z5 ([0.92, 0.96],[0.11, 0.17])
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([0.29, 0.37],[0.75, 0.81]) ([0.42, 0.56],[0.48, 0.59])
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([0.49, 0.61],[0.38, 0.58]) ([0.53, 0.68],[0.42, 0.47]) ([0.71, 0.82],[0.19, 0.30]) ([0.84, 0.91],[0.15, 0.27]) ([0.78, 0.88],[0.11, 0.19])
([0.73, 0.82],[0.21, 0.33])
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([0.89, 0.91],[0.05, 0.15]) ([0.56, 0.66],[0.38, 0.45]) c5
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z1 ([0.22, 0.36],[0.64, 0.76]) ([0.85, 0.96],[0.07, 0.19]) ([0.41, 0.56],[0.34, 0.45]) z2 ([0.34, 0.48],[0.59, 0.67]) ([0.74, 0.86],[0.12, 0.21]) ([0.67, 0.79],[0.23, 0.38]) z3 ([0.55, 0.67],[0.41, 0.53]) ([0.68, 0.74],[0.26, 0.35]) ([0.88, 0.95],[0.09, 0.13]) . z4 ([0.68, 0.79],[0.27, 0.34]) ([0.28, 0.35],[0.72, 0.86]) ([0.81, 0.87],[0.11, 0.25]) z5 ([0.78, 0.86],[0.23, 0.29]) ([0.07, 0.14],[0.82, 0.93]) ([0.43, 0.59],[0.52, 0.65])
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1 1 1 2 2 2 2 2 2 1 . ij ij ij ij ij ij 2 4 2 The calculated results are presented in the top part of Table 1. For example, for p45 (=([0.28, 0.35], [0.72, 0.86])), the indeterminacy interval was given by 45
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[ 1 0.352 0.862 , 1 0.282 0.722 ] [0.37, 0.63]. It follows that R1 ( p45 )
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Table 1. Some results for the financing decision on working capital policies. zi/rank c1 c2 c3 c4 c5 c6 Results of the risk-neutral score function R1 ( pij ) z1 0.3893 0.4642 0.6055 0.4854 0.9110 0.6153 z2 0.4943 0.6419 0.6786 0.5516 0.8217 0.7667 z3 0.6873 0.7935 0.8831 0.6786 0.7510 0.9192 z4 0.8367 0.8456 0.8004 0.7702 0.4515 0.8530 z5 0.9419 0.9050 0.6818 0.8363 0.3293 0.6091 Criterion-wise precedence ranks by sorting the R1 ( pij ) values 1st 2nd 3rd
z5 z4 z3
z5 z4 z3
z3 z4 z5
z5 z4 z3 30
z1 z2 z3
z3 z4 z2
4th 5th
z2 z1
z2 z1
z2 z1
z2 z1
z4 z5
z1 z5
In Step II-2, for each c j C , the five alternatives were ranked by sorting each
R1 ( pij ) value in descending order. The results of the criterion-wise precedence ranks are revealed in the bottom part of Table 1. Taking c6 as an example, based on the descending order of the R1 ( pi 6 ) values, it can be seen that p56 LIVPF R (1) p16 LIVPF R (1)
p26 LIVPF R (1) p46 LIVPF R (1) p36 , and thus, the precedence ranking is
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z3 z4 z2 z1 z5 with respect to c6. Accordingly, the precedence ranks of z1 , z2 , , z5 are ranked fourth, third, first, second, and fifth, respectively. Using the
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5 k 1
Fi k 6 for i 1, 2,
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Fi k 6 for
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z1 F11 F12 F13 F14 F15 1 0 0 1 4 z2 F21 F22 F23 F24 F25 0 1 1 4 0 F z3 F31 F32 F33 F34 F35 2 0 4 0 0 . z4 F41 F42 F43 F44 F45 0 5 0 1 0 z5 F51 F52 F53 F54 F55 3 0 1 0 2 4 For example, F2 =4 because the alternative z2 is ranked fourth with respect to c1, c2,
In Step II-3, the entries i for each i, k {1, 2, ,5} were calculated to establish the following precedence contribution matrix : 1st 2nd 3rd 4th 5th
13 23 33 43 53
14 24 34 44 54
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12 22 32 42 52
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15 0.2278 25 0.0000 35 0.4524 45 0.0000 55 0.2231
0.0000 0.0000 0.1846 0.2318 0.2054 0.2300 0.2724 0.0000 0.0000 0.3640 0.0000 0.0000 . 0.6190 0.0000 0.1129 0.0000 0.0000 0.1364 0.0000 0.2651
1 Consider 51 as an example. It is known that F5 3 ; moreover, the corresponding
criteria for which z5 is ranked first consist of c1, c2, and c4. Thus, let c j1 c1 ,
c j2 c2 , and c j (i.e., c j ) c4 , in which w j1 w1 0.10, w j2 w2 0.05, and
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w j3 w4 0.10. The entry 51 was computed as follows:
51 1 w j R1 ( pij ) 0.1 0.9419 0.05 0.9050 0.10 0.8363 0.2231 . 3
In Step II-4, a permutation matrix whose entry ik {0,1} for each i, k {1, 2, ,5} was defined as follows:
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z1 11 12 13 14 15 z2 21 22 23 24 25 z3 31 32 33 34 35 . z4 41 42 43 44 45 z5 51 52 53 54 55 Using the zero-one linear programming format in (39), this study constructed the following risk attitudinal assignment model: 0.2278 11 0.0000 12 +0.0000 13 +0.1846 14 +0.2318 15 1 2 3 4 5 0.0000 2 0.2054 2 +0.2300 2 +0.2724 2 +0.0000 2 max 0.4524 31 0.0000 32 +0.3640 33 +0.0000 34 +0.0000 35 1 2 3 4 5 +0.0000 4 0.6190 4 +0.0000 4 +0.1129 4 +0.0000 4 +0.2231 1 0.0000 2 +0.1364 3 +0.0000 4 +0.2651 5 5 5 5 5 5 ,5,
(41)
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ik 0 or 1 for all i and k . In Step II-5, the optimal objective value 1.7511 was acquired by solving the above model using a branch-and-bound algorithm. Moreover, ˆ14 =ˆ 23 ˆ 31 ˆ 42 ˆ 55 1 , and the other ˆ ik 0 . Employing (40), the overall precedence orders of the five financing policies was determined by multiplying Z by ˆ , i.e.: 0 0 0 1 0 0 0 1 0 0 Z ˆ z1 , z2 , z3 , z4 , z5 1 0 0 0 0 z3 , z4 , z2 , z1 , z5 . 0 1 0 0 0 0 0 0 0 1 Therefore, the ultimate priority ranking of the five financing policies is z3 z4 z2 z1 z5 in the risk-neutral case. That is, the balanced aggressive and conservative policy (z3) is the best choice that produces the overall compromise among all of the criterion-wise precedence rankings of the alternatives. Furthermore, this paper conducted a sensitivity analysis to observe the influence of the risk attitudinal parameter t on the solution results. Detailed analysis and discussions are presented in Appendix B. In addition to the illustrative case of t=1, this paper considered diverse attitudes of risk seeking (t=0.000001, 0.25, 0.5, and 0.75) and risk aversion (t=5, 10, 100, and 1000000). The obtained results of the criterion-wise precedence rankings and the overall precedence rankings are revealed in Table B.1 of Appendix B. According to the sensitivity analysis based on Table B.1, this paper concludes that the proposed outranking methodology can not only fully take into account the decision maker’s attitude towards risk-taking but also yield reasonable and persuasive results under complex uncertainty based on Pythagorean fuzziness. 32
5.4 Comparison to the risk attitudinal ranking method Because the proposed risk attitude-based score function was originated from Guo’s developed technique [21], this paper further conducts a comparative analysis with Guo’s approach to validate the effectiveness and advantages of the proposed methodology in this subsection. It is noted that the technique of Guo’s risk attitudinal ranking methods aims to tackle A-IF and A-IVIF information. To facilitate a consistent comparison, this paper generates the evaluative rating data using the AIVIF format. Let pij0 ( [ij0 , ij0 ],[ ij0 , ij0 ] ) denote an A-IVIF evaluative rating based on
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ij ij ij ij p , , , , ij ij ij ij ij ij ij ij ij ij ij ij
(42)
ij ij , . ij ij ij ij ij ij
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financing decision on working capital policies, the A-IVIF decision matrix p 0 ( [ pij0 ]56 ) was constructed as follows:
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z1 ([0.0970, 0.1765],[0.4788, 0.6245]) z2 ([0.1655, 0.2706],[0.4046, 0.5043]) p0 z3 ([0.3332, 0.4044],[0.2689, 0.3341]) z4 ([0.5124, 0.5490],[0.0854, 0.1787]) z5 ([0.6543, 0.7098],[0.0782, 0.1257])
([0.1774, 0.2263],[0.4589, 0.4954]) ([0.2962, 0.3526],[0.2297, 0.3353]) ([0.4499, 0.5101],[0.1204, 0.1866]) ([0.5179, 0.5846],[0.0730, 0.1262]) ([0.6388, 0.6291],[0.0359, 0.1037])
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z1 ([0.2515, 0.3234],[0.2874, 0.3407]) ([0.1378, 0.2167],[0.4010, 0.4575]) z2 ([0.3142, 0.3970],[0.2490, 0.2744]) ([0.2045, 0.2797],[0.3549, 0.3904]) z3 ([0.5558, 0.6088],[0.0992, 0.1806]) ([0.3259, 0.3896],[0.2429, 0.3082])
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z1 ([0.5894, 0.7081],[0.0485, 0.1402]) ([0.2568, 0.3283],[0.2130, 0.2638]) z2 ([0.4863, 0.5602],[0.0789, 0.1368]) ([0.4172, 0.4785],[0.1432, 0.2301]) z3 ([0.4183, 0.4446],[0.1599, 0.2103]) ([0.6127, 0.6965],[0.0627, 0.0953]) . z4 ([0.1713, 0.2213],[0.4404, 0.5438]) ([0.5414, 0.5631],[0.0735, 0.1618]) z5 ([0.0480, 0.0993],[0.5624, 0.6596]) ([0.2547, 0.3432],[0.3080, 0.3781]) Based on Guo’s proposed attitudinal-based measure [21] within the A-IVIF environment, the risk attitude-based score function Rt( pij0 ) of pij0 is defined as
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The risk attitude-based score function Rt ( pij ) was replaced by Rt( pij0 ) , and then the proposed Algorithm II was employed to address the MCDA problem of aggressive/conservative financing policies for Linkou CGMH. The obtained results based on Guo’s developed technique [21] and the proposed attitudinal-based measure are contrasted in Figures 4-6. Specifically, Figure 4 presents the comparisons of overall precedence ranks concerning the five financing policies with respect to different values of the risk attitudinal parameter t. Figure 5 illustrates the changes of overall precedence rankings for each alternative under different t values. Based on the comparison results in Figures 4 and 5, one can easily observe that the ranking orders obtained by Guo’s technique and the proposed approach are the same when the parameter t takes the values of 1, 5, 10, 100, and 1000000. To be precise, the ultimate priority rankings z3 z4 z2 z1 z5 and z3 z4 z2 z5 z1 were determined in the risk-neutral and risk-averse settings, respectively, regardless of the employment of the risk attitude-based score functions Rt( pij0 ) and Rt ( pij ) . Nevertheless, there are obvious differences between the overall precedence rankings through using Rt( pij0 ) and Rt ( pij ) in a risk-seeking situation.
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Figure 4. Contrast of overall precedence ranks under different t values. As illustrated in Figure 4 (a) and Figure 5 (a), the same ultimate priority ranking (i.e., z3 z4 z2 z1 z5 ) was acquired in the risk-seeking and risk-neutral settings based on Guo’s attitudinal-based measure. In contrast, as depicted in Figure 4 (b) and Figure 5 (b), two different ultimate ranking results z1 z4 z3 z2 z5 and
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z3 z4 z2 z1 z5 in the settings of risk-seeking and risk-neutral attitudes, respectively, were determined based on the proposed measure. It is worth mentioning that the cases with a risk-seeking attitude (i.e., t=0.000001, 0.25, 0.5, and 0.75) more or less reflect the decision maker’s dispositional optimism. This implies that the aggressive dominant policy (z1) is possibly more likely to be the best ranked in a riskseeking situation. However, the risk attitude-based score function Rt( pij0 ) based on Guo’s measure cannot produce an acceptable and credible result because the identical ultimate ranking was generated regardless of risk-seeking and risk-neutral situations. These findings indicate that Guo’s developed technique may lack the applicability to tackle MCDA problems under complex uncertainty. Unlike Guo’s approach, the obtained results through using the proposed risk attitude-based score function Rt ( pij ) are intuitively reasonable and convincing in theory and in practice.
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Furthermore, Figure 6 contrasts the overall precedence ranks of the five financing policies under three risk attitudes. As presented in Figure 6 (a), the overall precedence ranks of the alternatives z2 (with the third rank), z3 (with the first rank), and z4 (with the second rank) are unchanging despite the risk attitudes. The only difference is that the alternatives z1 is the fourth ranked in risk-seeking and riskneutral situations, while it is the fifth ranked in a risk-averse situation. The outcomes of the alternatives z5 is just the opposite. Namely, z5 is the last ranked in risk-seeking and risk-neutral situations and the fourth ranked in a risk-averse situation. In light of the pattern in Figure 6 (a), the application results based on Guo’s developed technique are less sensitive for the decision maker’s distinct attitudes towards risk-taking. More concretely, despite diverse attitudes toward risk neutrality, risk seeking, and risk aversion, the outcomes of the overall precedence ranks for each alternative are very similar through the use of Guo’s attitudinal-based measure. This implies that the risk attitude-based score function Rt( pij0 ) cannot reflect the influences of different risktaking attitudes on overall precedence ranks of the alternatives. As opposed to Guo’s approach, the proposed methodology is sensitive for distinct attitudes toward risk neutrality, risk seeking, and risk aversion. As revealed in Figure 6 (b), except z4, the overall precedence ranks of the alternatives are more or less different in diverse risk36
taking settings. More precisely, the alternative z1 is the first, fourth, and fifth ranked in risk-seeking, risk-neutral, and risk-averse situations, respectively. Moreover, z2 is the fourth ranked in a risk-seeking situation and the third ranked in risk-neutral and riskaverse situations; z3 is the third ranked in a risk-seeking situation and the best ranked in risk-neutral and risk-averse situations. Finally, z5 is the fifth ranked in risk-seeking and risk-neutral situations and the fourth ranked in a risk-averse situation. Thus, contrary to the use of Rt( pij0 ) , the proposed risk attitude-based score function
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Rt ( pij ) can yield more-reasonable and more-persuasive results. Based on the comparative discussions with respect to Figures 4-6, it can be concluded that the proposed outranking methodology can provide a useful tool for reflecting the influence of the decision maker’s attitude towards risk-taking in the face of complex uncertainty.
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(a) Results based on the risk attitude-based score function Rt( pij0 )
(b) Results based on the risk attitude-based score function Rt ( pij ) Figure 6. Overall precedence ranks of alternatives under three risk attitudes.
5.5 Comparison to IVPF-ELECTRE using an investment problem This subsection investigates an investment problem and conducts a comparative analysis with the most well-known and widely used outranking methodology, 37
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ELECTRE, to demonstrate the advantages of the proposed approach. It is worth noting that the MCDA models can be generally divided into scoring, compromising, and outranking models. Both the proposed approach and the ELECTRE methodology belong to the outranking model. This was the main reason for selecting ELECTRE as a comparative approach. Another reason was its popularity in practice, because numerous methods in the ELECTRE family have been successfully applied in the MCDA field. To date, in the literature on the ELECTRE methodology, little research has focused on the development of MCDA methods based on PS sets or IVPS sets. Based on the concepts of score functions and accuracy functions, Peng and Yang [35] employed the core structure of ELECTRE to propose a new IVPF-ELECTRE method for handling multiple criteria group decision-making problems involving IVPF information. Peng and Yang’s work is a seminal study of extending the ELECTRE methodology to the IVPF environment. In this regard, this paper deems it a benchmark method for a comparative study. More precisely, this paper chose the IVPF-ELECTRE method as a comparative approach to validate the strength and practicality of the proposed outranking methodology based on a risk attitudinal assignment model in the IVPF context. In general, the better alternative has the higher score function or the higher accuracy function in the case where the alternatives have the same score. In line with the concepts of score and accuracy functions defined for IVPF values, Peng and Yang [35] constructed the IVPF concordance/discordance sets to define the concordance/discordance indices. In the IVPF-ELECTRE procedure, these sets are divided into the IVPF strong, medium, and weak concordance/discordance sets according to various scenarios defined by score and accuracy functions. To reflect the relative dominance of a certain alternative over a competing alternative, Peng and Yang determined the concordance index by means of the sum of the weights associated with those criteria and relations that are contained in the IVPF strong, medium, and weak concordance sets. Moreover, to reflect the relative difference of an alternative with respect to a competing alternative, Peng and Yang identified the discordance index in terms of discordance criteria that are contained in the IVPF strong, medium, and weak discordance sets. The concordance/discordance Boolean matrices and the outranking matrix are then established to indicate the order of relative superiority of alternatives. Peng and Yang [35] provided an illustrative application concerning an investment problem to verify the effectiveness of their developed approach. This application involves three experts (i.e., project manager, CEO, and chairman) in an investment company evaluating five software development projects according to four criteria. The set of alternatives is denoted by Z={z1 (a mail development project), z2 (a game development project), z3 (a browser development project), z4 (a music player development project), z5 (a microblog development project)}. The set of criteria is denoted by C={c1 (economic feasibility), c2 (technological feasibility), c3 (staff feasibility), c4 (period feasibility)}. Peng and Yang [35] employed their developed IVPF weighted average operator to aggregate individual IVPF decision matrices provided by each expert to determine the aggregated decision matrix. Detailed data of the evaluative rating pij in the aggregated decision matrix are provided in the top part of Table 2. Moreover, Peng and Yang [35] applied the maximizing deviation method to determine the objective weights of the four criteria. They combined the objective weights and the weights of the three experts to acquire the importance 38
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weights of the criteria. The computed results are expressed as a weight vector (0.29, 0.23, 0.26, 0.22)T. To compare the application results on a consistent basis, this paper employed the proposed approach with Algorithm II to analyze the same data in Table 2. Assume that, without loss of generality, t 1 (i.e., risk neutrality) due to a lack of relevant information about the decision maker’s risk attitude in this case. The results of the risk-neutral score function R1 ( pij ) and the corresponding precedence ranks are listed in the middle part of Table 2. However, it can be observed that the alternatives z2 and z5 are tied with respect to the criterion c4. That is, the precedence ranking of z1 z2 ~ z5 z3 z4 was acquired in terms of c4 because R1 ( p14 ) R1 ( p24 ) R1 ( p54 ) R1 ( p34 ) R1 ( p44 ) . To avoid confusing computations of the precedence frequency in the matrix F, this paper reproduces c4 by splitting it into two extended criteria: c4(1) and c4(2). Moreover, the original ranking z1 z2 ~ z5 z3 z4 is equalized as z1 z2 z5 z3 z4 and z1 z5 z2 z3 z4 for c4(1) and c4(2), respectively. An adjusted version of the matrix F was obtained as follows: 1st 2nd 3rd 4th 5th 1 0 0 0 1 1 1 2 1 1 2 0 . 1 1 1 2 1 2 1 1 5 5 Meanwhile, k 1 Fi k i 1 Fi k n 1 5 for all i and k. Note that each of the
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Table 2. Data and application results in the investment problem. Data of the evaluative rating pij c1 c2 ([0.69, 0.79], [0.21, 0.31]) ([0.70, 0.80], [0.20, 0.30]) ([0.55, 0.65], [0.35, 0.45]) ([0.35, 0.45], [0.55, 0.65]) ([0.57, 0.67], [0.33, 0.43]) ([0.56, 0.66], [0.34, 0.44]) ([0.62, 0.72], [0.28, 0.38]) ([0.43, 0.53], [0.47, 0.57]) ([0.53, 0.63], [0.37, 0.47]) ([0.54, 0.64], [0.36, 0.46]) c3 c4 z1 ([0.65, 0.75], [0.25, 0.35]) ([0.63, 0.73], [0.27, 0.37]) z2 ([0.32, 0.42], [0.58, 0.68]) ([0.56, 0.66], [0.34, 0.44]) z3 ([0.69, 0.79], [0.21, 0.31]) ([0.54, 0.64], [0.36, 0.46]) z4 ([0.56, 0.66], [0.34, 0.44]) ([0.45, 0.55], [0.45, 0.55]) z5 ([0.54, 0.64], [0.36, 0.46]) ([0.56, 0.66], [0.34, 0.44]) Results of the risk-neutral score function R1 ( pij ) and the precedence ranks
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ik 0 or 1 for all i and k . By solving the above model, the overall precedence ranking of the five software development projects is z1 z4 z5 z3 z2 ; thus, the best choice is the mail development project (i.e., z1). Compared with the obtained decision graph depicted by Peng and Yang [35], the IVPF-ELECTRE method rendered the outranking relationships z1 z2 , z1 z4 , z1 z5 , z3 z2 , z3 z5 , and z4 z2 . Moreover, Peng and Yang concluded that z1 is superior to z2, z4, and z5. Therefore, the benchmark method and the proposed method provided very similar ranking results. However, the IVPF-ELECTRE method cannot differentiate the outranking relationships between z1 and z3, z2 and z5, z3 and z4, and z4 and z5. This implies that the overall precedence orders of z1 and z3 cannot be confirmed via the IVPF-ELECTRE method. Thus, the decision maker may be doubtful regarding whether z1 is the best choice, which may result in decision difficulty. In contrast, the proposed outranking method with Algorithm II yielded the ranking result z1 z4 z5 z3 z2 , which can clarify the ultimate priority orders of all alternatives such that one can conclude that z1 is indeed superior to the other four alternatives. It is clear that the proposed method has the capability of producing acceptable and reasonable ranking results. In contrast, it is troublesome to make a final decision 40
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for decision makers when using the IVPF-ELECTRE method. The reason for this finding could be confusion among the outranking relationships of pairs z1 and z3, z2 and z5, z3 and z4, and z4 and z5 according to the solution results by IVPF-ELECTRE. Unlike the benchmark method, the developed outranking method is perfectly capable of yielding a total order of the alternatives, which is essential for MCDA applications and decision-making in reality. Because the proposed method utilizes a risk attitudinal assignment model, tied situations among alternatives hardly occur in the final solution results. More specifically, the optimal permutation matrix ˆ can be determined by solving the risk attitudinal assignment model that is expressed using a simple zero-one linear programming model. Because of the constraint conditions in the risk attitudinal assignment model, the obtained matrix ˆ can explicitly differentiate the ultimate priority orders of all alternatives and contribute the certain outranking relationships of alternatives that have similar overall evaluation values. Consider the application results regarding the investment problem of software development projects as an example. As opposed to the benchmark method of IVPF-ELECTRE, the proposed approach can indeed distinguish the ultimate priority orders of similar or indefinite alternatives (i.e., z1 z3 , z5 z2 , z4 z3 , and z4 z5 ) and provide a definite overall precedence ranking of all alternatives (i.e., z1 z4 z5 z3 z2 ) for substantive decision support. Therefore, differentiation of the ultimate priority orders among various alternatives, especially among similar or indefinite alternatives, is a distinct advantage of the proposed method. It is noteworthy that the decision approach of the proposed methodology is quite different from the IVPF-ELECTRE method. Essentially, the IVPF-ELECTRE method renders a partial rank of alternatives by considering concordance and discordance. In contrast, the proposed methodology does not utilize the measures of concordance and discordance (i.e., the concordance/discordance indices) to determine the outranking relationships between alternatives. More specifically, the IVPF-ELECTRE simultaneously employs the score and accuracy functions to construct the IVPF strong, medium, and weak concordance/discordance sets and then determine the IVPF concordance/discordance indices. As opposed to such complex computations in the IVPF-ELECTRE, the proposed methodology takes a straightforward and effective manner based on the risk attitude-based score function instead of concordance and discordance measures. Recall that the comparison results (see Tables A.1 and A.2) in Appendix A have demonstrated the validity and reasonability of the proposed risk attitude-based score functions rt and Rt . Compared with the existing score and
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information and facilitating subjective judgments and evaluations. In brief, unlike the IVPF-ELECTRE based on score and accuracy functions, the proposed methodology incorporates the decision maker’s attitude towards risk-taking into the analytical model by means of the developed concepts of risk attitude-based score functions and risk attitudinal assignment models. Furthermore, the developed techniques possess certain flexibility and adaptability via different settings of the parameter t according to individual risk perceptions in the face of high uncertainty.
6. Conclusions
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The theory of PF/IVPF sets provides an intuitive and feasible way of modeling imprecise information and quantifying the ambiguous nature of subjective judgments. The employment of the PF/IVPF theory to address MCDA problems is one of the most promising directions in real-world applications. Many useful models and methods have been developed to utilize PF/IVPF sets to accommodate more-complex decision-making environments. In contrast to the existing techniques, this paper has established a novel outranking approach for managing the sophisticated data expressed by PF/IVPF sets in a simple and effective manner. The proposed risk attitudinal assignment models based on the developed risk attitude-based score functions have a great capability to deal with complex ambiguity and imprecision expressed as PF/IVPF values, which are flexible in reflecting the uncertainty and hesitation associated with the decision makers’ subjective assessments and different attitudes toward risk. Several interesting theoretical properties of the developed concepts have been explored to provide more-robust theoretical contributions. As an application of the proposed methodology, a real-world problem concerning a financing decision for working capital management was investigated to verify its feasibility and applicability. The practicality and advantages of the developed model and techniques have been supported through the application results in realistic situations, the sensitivity analysis of the risk attitudinal parameter, and the comparative discussions with relevant approaches. In addition to useful theoretical insights, the proposed approach has practical effectiveness, and it contributes to the financial decision-making field. Furthermore, this paper selected the IVPF-ELECTRE as a benchmark method for conducting a comparative study because it is the most closely related outranking-based methodology in cases with a high degree of uncertainty. Concerning an application to the investment problem of software development projects, the advantages and practicality of the proposed approach were verified through a comparison analysis of the obtained results yielded by the IVPFELECTRE method and the proposed Algorithm II. Unlike the concordance and discordance analysis in the IVPF-ELECTRE procedure, Algorithm II utilizes an effective and straightforward approach based on the risk attitudinal technique (e.g., risk attitude-based score functions and risk attitudinal assignment models). The developed approach can not only accurately distinguish between the PF/IVPF information but also incorporate the decision maker’s diverse risk-taking attitudes. Compared with the IVPF-ELECTRE method, Algorithm II can produce a more reasonable and persuasive result for ranking the order relationships of alternatives in the IVPF context. In summary, the proposed approach is very suitable for managing MCDA problems with uncertain information using PF/IVPF sets. In particular, the risk attitude-based score functions corresponding to various settings of the attitude 42
parameter can confer upon the risk attitudinal assignment models a certain flexibility and adaptability in addressing real-life MCDA problems.
Acknowledgments
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The author acknowledges the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The author is grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 105-2410-H182-007-MY3) and Chang Gung Memorial Hospital (BMRP 574 and CMRPD2F0202) during the study completion.
Conflict of interest statement
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The author declares that there is no conflict of interest regarding the publication of this paper.
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Appendix A: Comparative discussion of score functions
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A comparative analysis and discussions were conducted to validate the applicability and effectiveness of the risk attitude-based score functions for ranking PF/IVPF values. The numerical test examples were adopted from Garg [16, 17, 19], Liang et al. [25], Peng and Dai [33], and Zhang [50, 51]. The comparison results of the score functions (i.e., s( p) vs. rt ( p) and S ( p) vs. Rt ( p) ) within the PF/IVPF environments are separately summarized in the following two tables. In these comparative analyses, different t values consisting of t 1 (risk neutrality), t 0.5 (risk seeking), and t 2 (risk aversion) were designated during the computation process of rt ( p) and Rt ( p) .
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The comparison results of s( p) , r1 ( p) , r0.5 ( p) , and r2 ( p) in the PF context are presented in Table A.1. As sketched here, the existing score function s( p) cannot tell the difference between the magnitudes of p1 and p2, because s( p1 ) s( p2 ) in all numerical cases. According to the comparison rule provided in Definition 5, it is necessary to determine the accuracy functions when the score functions fail to distinguish between two PF values. Consider the case of p1 (0.7, 0.4) and
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Similarly, it can be obtained that r0.5 ( p1 ) r0.5 ( p2 ) and r2 ( p1 ) r2 ( p2 ) by means of the risk-seeking score function ( t 0.5 ) and the risk-averse score function ( t 2 ), respectively. These rankings are the same as the result yielded by the accuracy function a( p) . Note that a particular result of r1 ( p1 ) r0.5 ( p1 ) r2 ( p1 ) 0.7454 was
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5 3 (i.e., 0.7454) for all t. Nevertheless, the risk attitude-based score function rt can still effectively distinguish between the PF values even in this special case. Considering all of the numerical examples in Table A.1, the comparison results indicate that the proposed rt ( p) can distinguish the difference in PF values when the existing score function s( p) does not work. Moreover, as opposed to the two-stage comparisons (i.e., simultaneously taking the score and accuracy functions), the proposed risk attitude-based score function is straightforward and effective. Table A.2 summarizes the comparison results of S ( p) , R1 ( p) , R0.5 ( p) , and
R2 ( p) in the IVPF context. As observed in the outcomes, one can clearly see that the existing score function S ( p) cannot well distinguish IVPF values in most numerical cases. An exception can be found in the case of p1 ([1 10,(1 4) 1 5],[ 3 10, 44
(1 4) 3 5]) and p2 ([0,(1 10) 3 2],[(1 10) 7 2,1 5]) (adopted from Garg [16]).
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p1 p2 according to the comparison rule described in Definition 10. Meanwhile, the same ranking results were obtained using the proposed risk attitude-based score functions (i.e., R1 ( p1 )>R1 ( p2 ) , R0.5 ( p1 )>R0.5 ( p2 ) , and R2 ( p1 )>R2 ( p2 ) ). However, consider a similar example adapted from this case. If the PF values were modified as follows: p1 ([1 40,(1 4) 1 5],[ 3 10,(1 4) 3 5]) and p2 ([3 40,(1 10) 3 2], [(1 10) 7 2,1 5]) , then S ( p1 ) S ( p2 ) 0.0272 . It is unable to give the correct
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justification to p1 and p2 , because S ( p1 ) S ( p2 ) . Accordingly, the decision maker must continue to use the accuracy function for further comparisons. The results A( p1 ) 0.0403 and A( p2 ) 0.0478 were obtained; thus, p1 p2 . In contrast, with
p2 because R1 ( p1 ) 0.5030 and R1 ( p2 ) 0.5048 in a risk-neutral situation, R0.5 ( p1 ) 0.6630 and R0.5 ( p2 ) 0.6636 in a risk-seeking situation, and R2 ( p1 ) 0.3429 and R2 ( p2 ) 0.3460 in a risk-averse situation. This means that the ranking procedure using the proposed Rt ( p) is simpler and more effective than the current score function-based approach. Next, consider the two cases adopted from Garg [17]. From Table A.2, the same result of p1 ~ p2 was obtained regardless of whether the score function-based approach or the proposed risk attitude-based score function was used. Additionally, the accuracy functions of p1 and p2 are equal. To observe other comparison results in more-diverse instances, this paper extended Peng and Dai’s PF cases [33] to the IVPF environment and modified partial degrees within p1 and p2 , as shown in Table A.2. More specifically, the case of p1 ([0.5,0.6],[0.5,0.6]) and
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the use of the proposed Rt ( p) , it is easy to conclude that p1
p2 ([0.6,0.7],[0.6, 0.7]) was partially adapted from the case of p1 (0.5,0.5) and p2 (0.6,0.6) . Moreover, the case of p1 ([0.6,0.7],[ 0.4, 0.4]) and
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p2 ([0.3465,0.5],[0.4, 0.4]) as well as the case of p1 ([0.7,0.7],[ 0.3, 0.4]) and p2 ([0.5,0.5], [0.245,0.4]) was partially adapted from the case of
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p1 (0.7, 0.4) and p2 (0.5,0.4) . Due to the obtained results of S ( p1 ) S ( p2 ) in the three adapted cases, it can be concluded that the existing score function is unable to distinguish between p1 and p2 . In contrast, the developed Rt ( p) performs well in such cases because it can tell the difference between the magnitudes of p1 and p2 . Therefore, the comparative analysis demonstrates that the proposed risk attitude-based score function is capable of providing a relatively correct justification to the ranking of IVPF values during the decision-making process.
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Table A.1. Comparison results of the score functions within the PF environment. Results of r1 ( p) Results of s( p) Data source PF values r1 ( p1 ) 0.5408
r0.5 ( p1 ) 0.6591
r2 ( p1 ) 0.4216
s( p2 ) 0.1000
r1 ( p2 ) 0.6185
r0.5 ( p2 ) 0.6708
r2 ( p2 ) 0.5657
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
s( p1 ) 0.0000
r1 ( p1 ) 0.6124
r0.5 ( p1 ) 0.6972
r2 ( p1 ) 0.5270
s( p2 ) 0.0000
r1 ( p2 ) 0.6557
r0.5 ( p2 ) 0.7040
r2 ( p2 ) 0.6072
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
s( p1 ) 0.0900
r1 ( p1 ) 0.7177
r0.5 ( p1 ) 0.7367
r2 ( p1 ) 0.6986
s( p2 ) 0.0900
r1 ( p2 ) 0.6199
r0.5 ( p2 ) 0.7189
r2 ( p2 ) 0.5206
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
s( p1 ) 0.1100
r1 ( p1 ) 0.7204
r0.5 ( p1 ) 0.7428
r2 ( p1 ) 0.6979
s( p2 ) 0.1100
r1 ( p2 ) 0.6684
r0.5 ( p2 ) 0.7344
r2 ( p2 ) 0.6022
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
p1 ( 3 3, 2 3)
s( p1 ) 0.1111
r1 ( p1 ) 0.6573
r0.5 ( p1 ) 0.7324
r2 ( p1 ) 0.5821
p2 (2 3, 3 3)
s( p2 ) 0.1111
r1 ( p2 ) 0.7027
r0.5 ( p2 ) 0.7407
r2 ( p2 ) 0.6646
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
p1 ( 5 3, 2 3)
s( p1 ) 0.1111
r1 ( p1 ) 0.7454
r0.5 ( p1 ) 0.7454
r2 ( p1 ) 0.7454
p2 (2 3, 3 3)
s( p2 ) 0.1111
r1 ( p2 ) 0.7027
r0.5 ( p2 ) 0.7407
r2 ( p2 ) 0.6646
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
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p2 (0.6, 0.6)
p1 (0.7, 0.4)
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p2 (0.5, 0.4)
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p1 (0.7, 0.38)
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p2 (0.5477, 0.6324)
Peng and Dai [33]
Results of r2 ( p)
s( p1 ) 0.1000
p1 (0.3162, 0.4472)
Garg [19]
Results of r0.5 ( p)
p2 (0.6, 0.5)
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p2 ([ 9 4, 10 4],[ 2 4, 5 4])
p1 ([0.2,0.4],[0.2,0.4]) p2 ([0.5,0.6],[0.5,0.6])
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p1 ([ 7 4, 11 4],[1 4, 5 4])
p1 ([1 10,(1 4) 1 5],[ 3 10,(1 4) 3 5])
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p1 ([0,0.5],[0.1,0.7]) p2 ([0.3,0.4],[0.5,0.5])
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p2 ([0,(1 10) 3 2],[(1 10) 7 2,1 5]) Garg [17]
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p1 ([0.1,0.2],[0.4,0.5])
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p2 ([0.1,0.2],[1
Case adapted from Garg [16]
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p1 ([1 40,(1 4) 1 5],[ 3 10,(1 4) 3 5])
p2 ([3 40,(1 10) 3 2],[(1 10) 7 2,1 5]) p1 ([0.5,0.6],[0.5,0.6])
p2 ([0.6,0.7],[0.6,0.7])
p1 ([0.6,0.7],[ 0.4, 0.4]) p2 ([0.3465,0.5],[0.4,0.4]) p1 ([0.7,0.7],[ 0.3, 0.4]) p2 ([0.5,0.5],[0.245,0.4])
Results of R1 ( p)
Results of R0.5 ( p)
Results of R2 ( p)
S ( p1 ) 0.3750
R1 ( p1 ) 0.7756
R0.5 ( p1 ) 0.8176
R2 ( p1 ) 0.7336
S ( p2 ) 0.3750 S ( p1 ) S ( p2 ) S ( p1 ) 0.0000 S ( p2 ) 0.0000 S ( p1 ) S ( p2 ) S ( p1 ) 0.0225 S ( p2 ) 0.0300
R1 ( p2 ) 0.7893 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5477 R1 ( p2 ) 0.6344 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5053 R1 ( p2 ) 0.5034
R0.5 ( p2 ) 0.8209 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6815 R0.5 ( p2 ) 0.7011 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6646 R0.5 ( p2 ) 0.6627
R2 ( p2 ) 0.7578 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.4137 R2 ( p2 ) 0.5674 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.3460 R2 ( p2 ) 0.3441
S ( p1 ) S ( p2 )
R1 ( p1 )>R1 ( p2 )
R0.5 ( p1 )>R0.5 ( p2 )
R2 ( p1 )>R2 ( p2 )
S ( p1 ) 0.1250 S ( p2 ) 0.1250 S ( p1 ) S ( p2 ) S ( p1 ) 0.1800 S ( p2 ) 0.1800
R1 ( p1 ) 0.5484 R1 ( p2 ) 0.5484 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5021 R1 ( p2 ) 0.5021
R0.5 ( p1 ) 0.6548 R0.5 ( p2 ) 0.6548 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6326 R0.5 ( p2 ) 0.6326
R2 ( p1 ) 0.4410 R2 ( p2 ) 0.4410 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.3702 R2 ( p2 ) 0.3702
S ( p1 ) S ( p2 ) S ( p1 ) 0.0272 S ( p2 ) 0.0272 S ( p1 ) S ( p2 ) S ( p1 ) 0.0000 S ( p2 ) 0.0000 S ( p1 ) S ( p2 )
R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5030 R1 ( p2 ) 0.5048 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.6344 R1 ( p2 ) 0.6801 R1 ( p1 ) R1 ( p2 )
R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6630 R0.5 ( p2 ) 0.6636 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.7011 R0.5 ( p2 ) 0.7062 R0.5 ( p1 ) R0.5 ( p2 )
R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.3429 R2 ( p2 ) 0.3460 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.5674 R2 ( p2 ) 0.6538 R2 ( p1 ) R2 ( p2 )
S ( p1 ) 0.0250
R1 ( p1 ) 0.6839
R0.5 ( p1 ) 0.7142
R2 ( p1 ) 0.6534
S ( p2 ) 0.0250 S ( p1 ) S ( p2 ) S ( p1 ) 0.1400 S ( p2 ) 0.1400 S ( p1 ) S ( p2 )
R1 ( p2 ) 0.5871 R1 ( p1 )>R1 ( p2 ) R1 ( p1 ) 0.7242 R1 ( p2 ) 0.6226 R1 ( p1 )>R1 ( p2 )
R0.5 ( p2 ) 0.6971 R0.5 ( p1 )>R0.5 ( p2 ) R0.5 ( p1 ) 0.7516 R0.5 ( p2 ) 0.7296 R0.5 ( p1 )>R0.5 ( p2 )
R2 ( p2 ) 0.4767 R2 ( p1 )>R2 ( p2 ) R2 ( p1 ) 0.6967 R2 ( p2 ) 0.5154 R2 ( p1 )>R2 ( p2 )
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Table A.2. Comparison results of the score functions within the IVPF environment. Data source Results of S ( p) IVPF values
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Appendix B: Sensitivity analysis in the financing problem
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This appendix conducts a sensitivity analysis with respect to the risk attitudinal parameter t for the financing decision on working capital policies. Regarding diverse attitudes of risk seeking (in which the t vales were set as 0.000001, 0.25, 0.5, and 0.75), risk neutrality (i.e., t=1), and risk aversion (in which the t vales were set as 5, 10, 100, and 1000000), the results of the sensitivity analysis are presented in Table B.1, consisting of the criterion-wise precedence rankings and the overall precedence rankings in different settings of the parameter t.
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Table B.1. Computation results with different values of the parameter t. t value Criterion-wise precedence ranking Overall precedence ranking The setting of a risk-seeking attitude 0.000001 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 0.25 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 0.5 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 0.75 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 The setting of a risk-neutral attitude 1 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z1 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 The setting of a risk-averse attitude 5 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1 10 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1 100 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1 1000000 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1
Concerning the setting of a risk-seeking attitude, the same ultimate priority ranking, i.e., z1 z4 z3 z2 z5 , was acquired among the four cases. This result is evidently different from the overall ranking that was obtained based on the risk48
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neutral setting. Additionally, the ultimate priority ranking z3 z4 z2 z5 z1 was derived among the four risk-averse cases. It can be observed that this ranking is very similar to the risk-neutral result. In particular, the balanced aggressive and conservative policy (z3) is the best ranked in risk-neutral and risk-averse situations. In contrast, the best-ranked alternative is the aggressive dominant policy (z1) in a riskseeking situation. It is worthwhile to mention that these comparison results are intuitively reasonable and acceptable because the parameter t can more or less reflect the inner psychological characteristics of optimism and pessimism. As described in Remark 2, the settings of 0 t 1 , t 1 , and t 1 represent progressive, neutral, and conservative behaviors, respectively. Specifically, the cases of t=0.000001, 0.25, 0.5, and 0.75 can reflect the decision maker’s dispositional optimism in some way. Thus, it can be anticipated that the aggressive dominant policy is ranked the best in the four cases. Analogously, the result that the best choice is the balanced aggressive and conservative policy can be expected if the decision maker possesses a neutral attitude towards risk-taking. It is interesting to find that the balanced aggressive and conservative policy is also ranked the best in the cases of t=5, 10, 100, and 1000000. Nevertheless, the precedence relationships between z1 and z5 are different in the settings of risk neutrality and risk aversion, i.e., z1 z5 for the neutral case and z5 z1 for the four risk-averse cases. Such results are reasonable because the setting of t 1 reflects the decision maker’s dispositional pessimism. Accordingly, the conservative dominant policy generally performs better than the aggressive dominant policy for the decision maker with a risk-averse attitude. Based on the comparative discussions with respect to Table B.1, it can be concluded that the proposed outranking methodology can fully take into account the decision maker’s attitude towards risk-taking, and it can produce intuitively appealing and persuasive results in the face of high uncertainty.
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S1568-4946(18)30370-3 https://doi.org/10.1016/j.asoc.2018.06.036 ASOC 4952
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
7-9-2017 13-5-2018 21-6-2018
Please cite this article as: Chen T-Yu, An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its application to financial decision making, Applied Soft Computing Journal (2018), https://doi.org/10.1016/j.asoc.2018.06.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
An outranking approach using a risk attitudinal assignment model involving Pythagorean fuzzy information and its
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application to financial decision making
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Ting-Yu Chen
Professor, Graduate Institute of Business and Management, College of Management, Chang Gung University
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Adjunct Professor, Department of Industrial and Business Management,
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College of Management, Chang Gung University
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Adjunct Research Fellow, Department of Nursing, Linkou Chang Gung
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Memorial Hospital
33302, Taiwan
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Address: No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City
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Tel: +886-3-2118800 ext. 5678 Fax: +886-3-2118500
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E-mail: [email protected]
May 13, 2018
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Highlights
Representation of highly uncertain information using PF/IVPF sets. Development of risk attitude-based score functions for PF/IVPF values. Construction of risk attitudinal assignment models in the PF/IVPF context.
Enrichment of the outranking methodology under complex uncertainty. Application to financial decision-making problems. 1
Comparative analysis with other outranking approaches.
ABSTRACT
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The aim of this paper is to develop a novel outranking method using a risk attitudinal assignment model under complex uncertainty based on Pythagorean fuzzy (PF) sets and interval-valued Pythagorean fuzzy (IVPF) sets. Additionally, this paper applies the proposed methodology to address financial decision-making problems to investigate its applicability and effectiveness in the real world. These developed methods and techniques can provide a new viewpoint for capturing the amount of information conveyed by PF and IVPF values and can fully take into account the decision maker’s attitude towards risk-taking in the face of high uncertainty. Considering the diverse attitudes toward risk neutrality, risk seeking, and risk aversion, this paper introduces novel measures of risk attitude-based score functions within the PF/IVPF environments and investigates their desirable and useful properties. These properties provide a means of establishing linear orders and admissible orders in the PF/IVPF contexts. This means that the proposed measures can overcome some drawbacks and ambiguities of the previous techniques through score and accuracy functions and can address incomparable PF/IVPF information more effectively. As an application of the risk attitude-based score functions, this paper develops novel risk attitudinal assignment models to establish a useful outranking approach for solving multiple criteria decision-making problems. Two algorithms for PF and IVPF settings are developed based on the concepts of a precedence frequency matrix and a precedence contribution matrix for conducting multiple criteria evaluation and the ranking of alternatives. A financing decision on aggressive/conservative policies of working capital management is presented to demonstrate the applicability of the proposed outranking approach in real situations. Moreover, a comparison to the technique of risk attitudinal ranking methods is investigated to validate the advantages of the proposed methodology. Furthermore, a comparative analysis with a newly developed outranking method, the IVPFELECTRE (for the elimination and choice translating reality), is conducted via an application to the investment problem of software development projects. Compared with these benchmark approaches, the proposed methods can produce a more reasonable and persuasive result for ranking the order relationships of alternatives in the highly uncertain context. The practical studies and comparative discussions demonstrate an excellent performance of the developed methodology that is effective and flexible enough to accommodate more-complex decision-making environments.
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Keywords: outranking method; risk attitudinal assignment model; (interval-valued) Pythagorean fuzzy set; financial decision making; risk attitude-based score function; working capital management.
1. Introduction Multiple criteria decision analysis (MCDA) problems often involve vague or imprecise decision information due to the subjective nature and uncertainty of human judgments. In particular, MCDA within changeable, complicated, and/or unpredictable environments is a challenging task for decision makers. The theory of 2
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Pythagorean fuzziness is useful to represent the uncertain information contained in MCDA problems. Based on several valuable risk attitude-related concepts, this study attempts to develop a new outranking methodology to help decision makers conduct multiple criteria evaluation and decision analysis under complex uncertainty. Decision makers often deal with problems that involve vague or imprecise information. In response, fuzzy sets have extended deterministic MCDA methods to address uncertainty and imprecision in real situations [1]. Many generalizations have been proposed to the ordinary fuzzy sets to model different kinds of imprecise and subjective information. One such generalization is Atanassov’s intuitionistic fuzzy (A-IF) sets [3], which are becoming increasingly popular and widely used in the MCDA field [1, 32], and A-IF applications are growing in popularity by the day. Adding to this trend, Yager [45-48] introduced a nonstandard fuzzy set named Pythagorean fuzzy (PF) sets, which are an extension of A-IF sets. PF sets, which are characterized by degrees of membership and nonmembership, must meet the condition that the square sum of the membership and nonmembership degrees is less than or equal to one, while the sum of the two degrees can be more than one [14, 20, 23, 39, 50, 51]. Because of such a relaxed constraint condition, the space of PF values is evidently larger than that of A-IF values [30, 38, 50, 51]. From this perspective, not only can PF sets depict imprecise and ambiguous information, which A-IF sets can capture, but they can also model more-complex uncertainty in practical situations, which the latter cannot describe. Interval-valued Pythagorean fuzzy (IVPF) sets, which are parallel to Atanassov’s interval-valued intuitionistic fuzzy (A-IVIF) sets [4], are an extension of PF sets [25, 35, 50, 51]. IVPF sets permit the degrees of membership and non-membership of a given set to have an interval value within [0, 1]. The only requirement is that the square sum of the respective upper bounds of the interval-valued membership and non-membership degrees is less than or equal to one. Analogously, IVPF sets are capable of managing more-complex uncertainty than A-IVIF sets. PF sets and IVPF sets have wider application potentials because of their great ability to address strong fuzziness, ambiguity, and inexactness during the decision-making process. To date, the application of the PF/IVPF theory to MCDA has been studied from a number of different perspectives, such as the following: PF aggregation operators using confidence levels [18] or using Einstein t‐ norms and t‐ conorms [19], a projection model based on PF geometric Bonferroni mean operators [24], Pythagorean uncertain linguistic partitioned Bonferroni mean operators [26], symmetric PF weighted geometric/averaging operators [27], a PF stochastic method based on the prospect theory and regret theory [33], generalized PF point weighted averaging operators [37], a PF set-based group decision-making method [30], a PF group decision-making method with probabilistic information and an ordered weighted averaging (OWA) approach [49], IVPF linguistic weighted operators based on an IVPF linguistic variable set [14], an improved accuracy function for ranking IVPF information [16, 17], an extended technique for order preference by similarity to ideal solution (TOPSIS) based on hesitant PF sets [23], and an IVPF VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method [11]. In these applications of PF sets and IVPF sets, how to tell the difference between the magnitudes of PF/IVPF values is a critical issue for ranking or evaluating the order relationship of PF/IVPF information. As is well known, managing uncertainty to improve decision quality is an essential issue of MCDA processes, especially within a complex and volatile business 3
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environment. However, nowadays, many phenomena in practical circumstances, such as the presence of new technologies and increasingly dynamic organizational factors, are intricate and unpredictable [13]. It is especially the process of globalization and technological change that has led to the emergence of a complex network of relationships in the business environment. Moreover, a free market economy means increased complexity and uncertainty of factors affecting the financial standing of entities [22], and it is necessary to predict various influences in the highly competitive and globalized marketplace [29]. Consider the issue of consumer credit scoring in financial management. Due to great uncertainties in changeable markets, predicting both business and consumer bankruptcy is a highly imprecise and ambiguous task [22]. This implies that relevant judgments and evaluations may be difficult to measure or quantify along the decision-making cycle. Meanwhile, decision makers often face uncontrollable or unforeseeable risks resulting from a rapidly changing environment. For these reasons, the uncertainty, fuzziness, and vagueness inherent in the information structure of decision-making processes make rigorous mathematical models inappropriate for solving real-world MCDA problems, especially in an intricate and unpredictable business context. According to the discussions above, there are two critical reasons for decision makers to support the employment of PF/IVPF sets in MCDA. First, decision makers are faced with hesitation in numerous down-to-earth situations [32]. As a result, MCDA problem are often characterized by a high degree of uncertainty (e.g., lack of information) and ambiguity (e.g., lack of clarity or consistency of information). Compared with other nonstandard fuzzy sets, such as the widely used A-IF and AIVIF sets, the application of PF and IVPF sets can accommodate much higher degrees of uncertainty and thus have a more-powerful capability in addressing complicated and ambiguous MCDA problems. Second, high degrees of uncertainty in complicated environments often bring about high risks, resulting in many unfavorable decision consequences. In particular, a rapidly changing business environment contributes to greater complexity and difficulty in decision making. Because the judgment in evaluating alternative actions entails strong uncertainty and fuzziness, the methodology used for MCDA must provide appropriate support for these problem characteristics. Employing PF/IVPF sets can facilitate modeling the effects of highorder uncertainties associated with a complex decision environment. Subjective judgments and evaluations can be conveniently realized with the use of PF/IVPF values. Following the aforementioned motivations, this paper attempts to develop a novel outranking decision-making method for addressing MCDA problems under complex uncertainty involving PF/IVPF information. The purpose of this paper is to develop a novel outranking approach that utilizes risk attitudinal assignment models based on an extended concept of risk attitude-based score functions to manage MCDA problems within the PF/IVPF environments. First, motivated by the risk attitudinal technique, this paper introduces new measures of risk attitude-based score functions based on the amount of information conveyed by PF/IVPF values. Some useful and valuable properties of the proposed measures are also investigated to establish a linear order and an admissible order in the PF/IVPF contexts. These new measures have an excellent capability to characterize PF/IVPF information as comparable values to address the essential issue of ranking difficulty in highly uncertain circumstances. Next, this paper employs the risk attitude-based score functions to determine the precedence ranks among alternatives with respect to each criterion. Based on the obtained criterion-wise precedence relationships, this paper 4
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identifies the precedence frequency matrix and the precedence contribution matrix. A new outranking methodology using effective risk attitudinal assignment models is further developed to solve MCDA problems involving PF/IVPF information. Moreover, this paper utilizes practical financial decision-making problems to demonstrate the applicability and usefulness of the proposed outranking methods. Finally, this paper implements a comparative analysis with the other relevant outranking methods to examine the validity and superiority of the developed methods. Compared with the related literature, the main innovation and contributions of this paper in theory and practice contain development of risk attitude-based score functions for comparing PF/IVPF information, construction of risk attitudinal assignment models in the PF/IVPF contexts, and model application to financial decision making. First, considering the attitudinal character towards risk-taking, this paper presents novel measures of risk attitude-based score functions for the sake of effectively characterizing PF/IVPF information. In particular, the developed measures not only possess desirable and useful properties but also provide a means of establishing a linear order and an admissible order to overcome difficulties in ranking incomparable PF/IVPF values. These features are of great value in distinguishing among the magnitudes of PF/IVPF values and yielding the corresponding precedence relationships. Second, by determining criterion-wise precedence rankings among alternatives based on the risk attitude-based score functions, this paper presents useful concepts of a precedence frequency matrix and a precedence contribution matrix. Moreover, two effective risk attitudinal assignment models are established for solving MCDA problems within the PF and IVPF environments. The proposed risk attitudinal assignment models give the decision maker full support in representing imprecise knowledge and incorporating diverse risk attitudes into the existing assignment-based methodology, which leads to addressing many practical MCDA problems in a more appropriate and flexible way. Third, this paper extends the range of applications of the proposed methodology and makes it accommodate complex financial decisionmaking problems. Financing decisions on aggressive/conservative policies is very crucial for the efficient management of working capital. This paper illustrates how the proposed approach can be applied to the real-world problem of selecting a financing policy in the medical and health care fields. The practical usefulness and contributions of the developed concepts and models can be supported by this financial decisionmaking application. Furthermore, a comparative analysis with the other outranking method for solving an investment problem of software development projects is carried out to demonstrate the advantages and practicality of the proposed approach. The remainder of this paper is organized as follows. Section 2 briefly reviews some basic definitions and concepts of PF sets and IVPF sets. Section 3 proposes a useful concept of risk attitude-based score functions that take the decision maker’s risk attitude and the amount of information associated with PF/IVPF values into account. This section also investigates some valuable properties of the developed concept in detail. Section 4 constructs an outranking approach for addressing MCDA problems involving PF/IVPF information. Based on the precedence frequency and precedence contribution matrices, two effective algorithms with novel risk attitudinal assignment models are established to determine the outranking relationships among alternatives. Section 5 applies the proposed methodology to a practical problem concerning a financing decision on aggressive/conservative working capital policies of a medical institution to demonstrate its feasibility and applicability. Some discussions of the sensitivity analysis and a comparison to the technique of risk 5
attitudinal ranking methods are provided as well. To examine the advantages of the proposed methodology, further comparative analyses with a newly developed outranking method within the IVPF environment are conducted to address an investment problem. Finally, Section 6 presents the conclusions.
2. Basic concepts This section introduces some basic concepts related to PF sets and IVPF sets that are used throughout this paper.
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Definition 1. [24, 45-48, 49] A PF set P is defined as a set of ordered pairs of membership and non-membership in a finite universe of discourse X and is given as follows:
P x, P ( x), P ( x) x X ,
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which is characterized by the degree of membership P : X[0, 1] and the degree of
non-membership P : X[0, 1] of the element x X to the set P with the condition: 0 P ( x) P ( x) 1 . 2
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Let p (P ( x), P ( x)) denote a PF value. The degree of indeterminacy relative to P for each x X is defined as follows:
P ( x) 1 P ( x) P ( x) . 2
(3)
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Definition 2. [34, 36, 38, 50-52] Let p1 ( (P1 ( x), P1 ( x)) ) and p2 ( (P2 ( x), P2 ( x)) ) be two PF values in X. The distance between p1 and p2 is defined as follows: 2 2 2 2 2 2 1 d ( p1 , p2 ) P1 ( x) P2 ( x) P1 ( x) P2 ( x) + P1 ( x) P2 ( x) . (4) 2 Definition 3. [18, 19, 33, 38, 52] Let p be a PF value in X. The score function s( p) and the accuracy function a( p) of p are separately defined as follows: s( p) P ( x) P ( x) ,
(5)
a( p) P ( x) P ( x) .
(6)
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Definition 4. Let p, p1, and p2 be three PF values in X. The lattice ( LPF , LPF ) of nonempty PF values in X is defined as follows: LPF
( x), P
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( x) P ( x), P ( x) [0,1], P ( x) P ( x) 1 2
2
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with the partial order LPF , which is defined as, for any p1 LPF and p2 LPF ,
p1 LPF p2 if and only if P1 ( x) P2 ( x) and P1 ( x) P2 ( x) . The top and bottom
elements of ( LPF , LPF ) are (1,0) and (0,1), respectively. Theorem 1. [52] For any three PF values p, p1, and p2 in X, the score function s( p) and the accuracy function a( p) satisfy the following properties: (T1.1) 1 s( p) 1 (boundedness of s function); 6
(T1.2) 0 a( p) 1 (boundedness of a function); (T1.3) if p1 LPF p2 , then s( p1 ) s( p2 ) . Definition 5. [18, 19, 38, 52] Let p1 and p2 be two PF values in X. A prioritized comparison procedure within the PF environment is defined as follows: (D5.1) if s( p1 ) s( p2 ) , then p1 p2 ; (D5.2) if s( p1 ) s( p2 ) , then p1
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p2 ; (D5.3) if s( p1 ) s( p2 ) and a( p1 ) a( p2 ) , then p1 p2 ; (D5.4) if s( p1 ) s( p2 ) and a( p1 ) a( p2 ) , then p1 p2 ; (D5.5) if s( p1 ) s( p2 ) and a( p1 ) a( p2 ) , then p1 ~ p2 .
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Definition 6. [14, 16, 17, 25, 35, 51] Let Int([0, 1]) denote the set of all closed subintervals of the unit interval [0, 1]. An IVPF set P is defined as a set of ordered pairs of membership and non-membership in a finite universe of discourse X and is given as follows: (8) P x, P ( x), P ( x) x X ,
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which is characterized by the interval of the membership degree: P : XInt([0, 1]), x X P ( x) ( [P ( x ), P (x )]) [0,1] and the interval of the non-membership degree: P : XInt([0, 1]), x X P ( x) ( [ P ( x ), P (x )]) [0,1]
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with the following condition: 2 2 0 P ( x) P ( x) 1 .
Let p (P ( x), P ( x)) ([P ( x), P ( x)],[ P ( x), P ( x)]) denote an IVPF value. The
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interval of the indeterminacy degree relative to P is defined as follows: 2 2 2 2 P ( x) [ P ( x), P ( x)]= 1 P ( x) P ( x) , 1 P ( x) P ( x) .
(12)
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Definition 7. [14, 35, 51] Let p1 ( ([ P1 ( x), P1 ( x)],[ P1 ( x), P1 ( x)]) ) and p2 ( ([ P2 ( x), P2 ( x)],[ P2 ( x), P2 ( x)]) ) be two IVPF values in X. The distance between
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p1 and p2 is defined as follows: 2 2 2 1 D( p1 , p2 ) P ( x) P ( x) P ( x) P ( x) 4
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Definition 8. [16, 17, 35] Let p be an IVPF value in X. The score function S ( p) and the accuracy function A( p) of p are separately defined as follows: 2 2 2 2 1 S ( p) P ( x) P ( x) P ( x) P ( x) , (14) 2
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2 2 2 2 1 P ( x) P ( x) P ( x) P ( x) . 2
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Definition 9. Let p , p1 , and p2 be three IVPF values in X. The lattice
( LIVPF , LIVPF ) of nonempty IVPF values in X is defined as follows:
LIVPF P ( x), P ( x) P ( x), P ( x) [0,1], P ( x) P ( x) 1 2
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with the partial order LIVPF , which is defined as, for any p1 LIVPF and p2 LIVPF , 1
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p1 LIVPF p2 if and only if P ( x) P ( x) , P ( x) P ( x) , P ( x) P ( x) , and
( x) ( x) . The top and bottom elements of ( LIVPF , LIVPF ) are ([1,1], [0,0]) and P1
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Theorem 2. [35] For any three IVPF values p , p1 , and p2 in X, the score function S ( p) and the accuracy function A( p) satisfy the following properties: (T2.1) 1 S ( p) 1 (boundedness of S function); (T2.2) 0 A( p) 1 (boundedness of A function);
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(T2.3) if p1 LIVPF p2 , then S ( p1 ) S ( p2 ) .
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Definition 10. [16, 17] Let p1 and p2 be two IVPF values in X. A prioritized comparison procedure within the IVPF environment is defined as follows: (D10.1) if S ( p1 ) S ( p2 ) , then p1 p2 ; (D10.2) if S ( p1 ) S ( p2 ) , then p1 p2 ; (D10.3) if S ( p1 ) S ( p2 ) and A( p1 ) A( p2 ) , then p1 p2 ; (D10.4) if S ( p1 ) S ( p2 ) and A( p1 ) A( p2 ) , then p1 p2 ; (D10.5) if S ( p1 ) S ( p2 ) and A( p1 ) A( p2 ) , then p1 ~ p2 .
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3. A novel concept of risk attitude-based score functions
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Different decision makers have diverse attitudes towards risk-taking, such as risk seeking and risk aversion, in real-world MCDA problems. Incorporating the decision maker’s risk attitude into the developed outranking technique not only facilitates practical applications of the proposed methodology but also characterizes morereasonable subjective judgments in the decision-making processes. From the perspective of risk-taking, this paper develops novel risk attitude-based score functions within the PF/IVPF environments and investigates their desirable and useful properties in this section. The proposed methodology is built upon a solid base on theory and concepts, as depicted in Figure 1. This figure summarizes relevant theoretical bases of risk attitude-based score functions, linear orders, and admissible orders. To conduct a comprehensive comparative analysis, this paper utilizes certain numerical examples that were adopted from Garg [16, 17, 19], Liang et al. [25], Peng and Dai [33], and Zhang [50, 51] to examine the applicability and effectiveness of the proposed risk attitude-based score functions for ranking PF/IVPF information. Detailed comparative results and discussions are presented in Appendix A. These new 8
concepts form the core structure of the proposed risk attitudinal assignment models in the subsequent study. Relevant theoretical bases of risk attitude-based score functions
Relevant theoretical bases of linear orders and admissible orders
Definitions 11 and 14 introduce novel measures of risk attitude-based score functions within the PF/IVPF environments.
Definitions 12 and 15 construct binary relations by means of the risk attitude-based score functions.
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Remarks 1 and 3 present the meanings of the parameters within the risk attitudebased score functions.
Theorems 4 and 7 examine the properties of reflexivity, antisymmetry, transitivity, and completeness possessed by linear orders via the risk attitude-based score functions.
Definitions 13 and 16 establish admissible orders within the PF/IVPF environments.
Theorems 3 and 6 investigate some useful properties possessed by the risk attitudebased score functions.
Theorems 5 and 8 show that the binary relations based on the risk-neutral score functions are admissible orders.
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Remarks 2 and 4 explain the fundamental inner psychological characteristics of optimism and pessimism embedded in the risk attitude-based score functions.
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Figure 1. Theoretical bases of the proposed methodology.
3.1 New risk attitude-based score function of PF values
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The concept of PF sets is an effective tool for handling uncertainty in real-world applications, especially in the field of MCDA. However, the complexity of manipulating PF information makes it troublesome to directly employ the relevant concept in formulating the developed outranking methodology. Motivated by the techniques of risk attitudinal ranking methods [21, 42], this paper introduces a novel measure to characterize PF values as comparable real numbers to address the ranking issue within the PF environment. Guo [21] proposed a useful measure to compare AIF values based on the amount of information using a geometrical representation of A-IF sets to overcome some drawbacks and ambiguities in the existing approach via score functions and accuracy functions. Moreover, Guo [21] introduced the idea of an attitudinal-based extension of his developed technique. This paper presents an extended concept of the risk attitude-based score function for appropriately characterizing PF information and facilitating an effective ranking procedure in the proposed risk attitudinal assignment model. Yager [44] extended the applicability of the OWA operator by considering a situation in which the argument is a continuous interval [ , ] , i.e., the so-called continuous ordered weighted averaging (C-OWA) operator, which is expressed as follows: 1 dQ( y ) FQ ([ , ]) (17) y dy . 0 dy 9
In the formula above, FQ ([ , ]) is characterized by the basic unit-interval monotonic (BUM) function Q: [0, 1][0, 1] with the condition: (1) Q(0) 0 ; (2) Q(1) 1 ; and (3) Q( y1 ) Q( y2 ) if y1 y2 . Additionally, Yager [44] proved that FQ ([ , ]) could be derived in the following way: FQ ([ , ]) Q( y)dy 1 , 1
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0
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where Q( y)dy and [0,1] . In particular, is an attitudinal character of the 0
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BUM function Q that can be extracted from the decision maker’s attitude. Motivated by Yager’s approach, Guo [21] designated Q( y) y t ( t 0 ) to capture the decision maker’s risk attitude via the BUM function Q. Accordingly, the following results were obtained: 1 1 , (19) y t dy 0 t 1 1 1 t . (20) FQ ([ , ]) 1 t 1 t 1 t 1 Note that lim FQ ([ , ]) and lim FQ ([ , ]) . Moreover, FQ ([ , ]) t 0
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( ) 2 when t 1 . Because the attitudinal character (determined by t in Guo’s proposal) is capable of describing the type and level of the decision maker’s attitude, Guo [21] conceived it as a ratio of the hesitation margin assigned to the membership function in accordance with the decision maker’s characteristics. Furthermore, Guo [21] developed a novel attitudinal-based measure that can fully take into account both the amount of information associated with A-IF values and the attitude of the decision maker involved. Considering the usefulness of the attitudinal character , this paper attempts to extend the applicability of the attitudinal-based measure in the A-IF context to a more general PF environment. This allows us to introduce a new concept of the risk attitude-based score function that is established in a manner analogous to Guo’s risk attitudinal technique.
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Definition 11. Let p be a PF value in X. Let a parameter t denote the risk attitude of a decision maker and t 0 . The risk attitude-based score function rt ( p) of p is defined as follows:
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t 1 2 2 2 rt ( p) 1 P ( x) P ( x) . P ( x) t 1 t 1
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Remark 1. The parameter t represents a decision maker’s attitude towards risktaking, where t (1 ) / . More specifically, the decision maker’s risk attitude can be described as risk-seeking if 0 t 1 (i.e., 0.5 1 ), risk-neutral if t 1 (i.e., 0.5 ), and risk-averse if t 1 (i.e., 0 0.5 ). The risk attitude-based score function can be equivalently rewritten via the attitudinal character as follows:
2 2 2 r ( p) 1 1 P ( x) P ( x) P ( x) .
(22)
Remark 2. In addition to the risk attitude, the parameter t can reflect fundamental inner psychological characteristics of optimism and pessimism in some way. 10
Optimism and pessimism are fundamental constructs that express how people respond to their perceived environment and how they construe and affect subjective judgments [7, 9]. According to Definition 11, some typical cases are discussed as follows: (1) When 0 t 1 , more than half of the degree of indeterminacy has been assigned to the degree of membership. This implies that there is relatively less uncertainty left in the indeterminacy part. Because optimistic decision makers interpret their decision situations positively and expect favorable outcomes [7, 9], the case of 0 t 1 represents a progressive behavior and can reflect the decision maker’s dispositional optimism. (2) When t 1 , the risk attitude-based score function becomes:
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1 2 2 2 1 (23) r1 ( p) 1 P ( x) P ( x) P ( x) , 2 2 i.e., the so-called risk-neutral score function. This case can reflect the decision maker’s neutral attitude towards risk-taking. (3) When t 1 , less than half of the degree of indeterminacy has been assigned to the degree of membership, i.e., relatively more uncertainty is left in the indeterminacy part. Pessimistic decision makers expound their decision situations negatively and anticipate unfavorable outcomes [7, 9]. Thus, this case represents a conservative behavior and can reflect the decision maker’s dispositional pessimism. (4) The cases of t 0 and t can reflect complete (or extreme) optimism and pessimism, respectively.
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Theorem 3. For a PF value p in X, the risk attitude-based score function rt ( p) satisfies the following properties: (T3.1) 0 rt ( p) 1 (boundedness of rt function); (T3.2) if p=(0, 1), then rt ( p) 0 for all t (zero property of rt function);
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(T3.3) if p=(1, 0), then rt ( p) 1 for all t (one property of rt function);
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(T3.4) s( p) rt ( p) for all t. Proof. The proofs of (T3.2) and (T3.3) are trivial. (T3.1) According to Definition 1, it is easy to see that 0 ( P ( x))2 1 and
0 (P ( x))2 ( P ( x))2 1 . Moreover, because t 0 , it is known that
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0 t (t 1) 1 and 0 1 (t 1) 1 . One can infer that 0 (t (t 1) )( P ( x))2 1 and 0 ( P ( x))2 (1 (t 1) )( P ( x))2 1 . It can be obtained that 0 rt ( p) 1 . Hence, (T3.1) is proved. (T3.4) For all t 0 , one can easily prove the following results:
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t 1 2 2 2 rt ( p) 1 P ( x) P ( x) P ( x) t 1 t 1
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( x) P
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( x) 2
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P
( x)
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( x)
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P ( x) P ( x) P ( x) s( p). 2
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Thus, (T3.4) is valid. This completes the proof. In general, the ranking of PF values is nontrivial because no natural linear orders 11
exist among them. The following definition and theorem provide us with a means of employing the risk attitude-based score function rt ( p) to establish a linear order within the PF environment. Definition 12. Let p1 and p2 be two PF values in X. Let a binary relation LPF r ( t ) be defined as “being smaller than or indifferent from” based on the risk attitude-based score function rt . That is, p1 LPF r ( t ) p2 if and only if rt ( p1 ) rt ( p2 ) .
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Theorem 4. Let p, p1, and p2 be three PF values in X. Let ( LPF r (t ) , LPF r ( t ) ) denote a
lattice of nonempty PF values in X, in which LPF r (t ) {(P ( x), P ( x))|P ( x), P ( x)
[0,1],(P ( x))2 ( P ( x))2 1} with the binary relation LPF r ( t ) . Moreover, the top and
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bottom elements of ( LPF r (t ) , LPF r ( t ) ) are (1,0) and (0,1), respectively. Then, the binary relation LPF r ( t ) is a linear order.
Proof. For the reflexivity, because rt ( p) rt ( p) , p LPF r ( t ) p for each p in X. Thus, the binary relation LPF r ( t ) is reflexive. For the antisymmetry, if both rt ( p1 ) rt ( p2 )
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and rt ( p2 ) rt ( p1 ) hold, it is clearly known that rt ( p1 ) rt ( p2 ) . This implies that if
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p1 LPF r ( t ) p2 and p2 LPF r ( t ) p1 , then p1 LPF r ( t ) p2 ; thus, LPF r ( t ) is antisymmetric.
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For the transitivity, the relations p LPF r ( t ) p1 and p1 LPF r ( t ) p2 indicate
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rt ( p) rt ( p1 ) and rt ( p1 ) rt ( p2 ) , respectively. Because rt ( p) , rt ( p1 ) , and rt ( p2 ) are comparable real numbers, one obtains rt ( p) rt ( p1 ) rt ( p2 ) . Therefore, both p LPF r ( t ) p1 and p1 LPF r ( t ) p2 imply that p LPF r ( t ) p2 for all p, p1, and p2 in X; that
ED
is, LPF r ( t ) is transitive. Accordingly, the binary relation LPF r ( t ) is a partial order because it is reflexive, antisymmetric, and transitive. Moreover, the lattice ( LPF r (t ) , LPF r ( t ) ) is a partially ordered set. On the other hand, it is obvious that any
PT
two p1 and p2 in X are comparable via their corresponding risk attitude-based score functions rt ( p1 ) and rt ( p2 ) ; i.e., either rt ( p1 ) rt ( p2 ) or rt ( p2 ) rt ( p1 ) holds. It
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follows that the partial order LPF r ( t ) is complete. Consequently, the binary relation
LPF r ( t ) is a linear order. This completes the proof. It is worth mentioning that the binary relation LPF r ( t ) using the proposed risk
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attitude-based score function rt is not only a linear order but also an admissible order in the case of t 1 (i.e., a risk-neutral attitude). Bustince et al. [6] extended the usual partial order between intervals to present the concept of an admissible order as a total order. Guo [21] also proposed the concept of admissible orders within A-IF and A-IVIF environments. In a manner similar to that of Bustince et al. [6] and Guo [21], this paper defines the concept of an admissible order in the PF context. To make things concrete, an order on LPF is admissible if it is linear and refines the order 12
LPF . Definition 13. Let ( LPF , LPF ) be a partially ordered set. The order
is called an
admissible order within the PF environment if (D13.1) is a linear order on LPF ; (D13.2) for any p1 , p2 LPF , p1 p2 whenever p1 LPF p2 . Theorem 5. Let p1 and p2 be two PF values in X, where p1 and p2 belong to LPF with
IP T
the partial order LPF . If p1 LPF p2 , then r1 ( p1 ) r1 ( p2 ) . Moreover, the binary relation LPF r (1) is an admissible order on LPF .
SC R
Proof. Applying Definition 12 and Theorem 4, it is known that the risk-neutral score function r1 : LPF [0,1] induces on LPF a relation LPF r (1) given by
p1 LPF r (1) p2 if and only if r1 ( p1 ) r1 ( p2 ) . Furthermore, this relation is a linear order
on LPF . According to (3) and (23), r1 ( p1 ) can be calculated in the following way:
2
P1
2
( x)
2 1 1 P1 ( x) P1 ( x) 2
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2 1 r1 ( p1 ) 1 1 P1 ( x) P1 ( x) 2
2
2 2 1 2 2 1 1 1 1 1 P1 ( x) P1 ( x) P1 ( x) P1 ( x) 2 2 2 2 2 2
2
P1
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1 1 P1 ( x) 4
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2 ( x) 1 P1 ( x)
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2
P1
2 ( x)
2 2 4 1 1 P1 ( x) P1 ( x) . 2 Similarly, one has r1 ( p2 ) (1 2) (1 ( P ( x))2 )2 ( P ( x))4 . It is easy to show that
ED
2
2
r1 ( p1 ) r1 ( p2 ) because P1 ( x) P2 ( x) and P1 ( x) P2 ( x) according to the
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premise condition of p1 LPF p2 . Moreover, p1 LPF r (1) p2 holds because
r1 ( p1 ) r1 ( p2 ) . Therefore, the binary relation LPF r (1) based on the risk-neutral score
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function r1 refines the partial order LPF ; that is, p1 LPF p2 implies p1 LPF r (1) p2 . According to Definition 13, the binary relation LPF r (1) is admissible in the PF context. In addition, for the lattice ( LPF , LPF r (1) ) , LPF is a chain because the relation
A
LPF r (1) is a linear order on LPF . This completes the proof.
3.2 New risk attitude-based score function of IVPF values This subsection extends the proposed new concepts of a risk attitude-based score function and an admissible order from a PF environment to a more general IVPF environment. Definition 14. Let p be an IVPF value in X. Let a parameter t denote the risk 13
attitude of a decision maker and t 0 . The risk attitude-based score function Rt ( p) of p is defined as follows:
1 2 2 2 2 t Rt ( p) 1 P ( x) P ( x) P ( x) P ( x) 2 2 t 1
2 2 1 P ( x) P ( x) t 1
(24)
1 2
.
1 2
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Remark 3. Because t (1 ) / , the risk attitude-based score function can be equivalently rewritten using the attitudinal character as follows: 2 2 2 2 1 1 R ( p) 1 P ( x) P ( x) P ( x) P ( x) 2 2 P ( x) P ( x) . 2
2
(25)
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Remark 4. Some cases of the t value in Definition 14 are examined as follows: (1) When 0 t 1 (or equivalently, 0.5 1 ), more than half of the interval of the indeterminacy degree has been assigned to the interval of the membership degree. Because relatively less uncertainty is left in the indeterminacy part, the case of 0 t 1 can reflect the decision maker’s optimistic attitude towards risktaking in the IVPF context. (2) When t 1 (or equivalently, 0.5 , i.e., a neutral attitude), the risk attitudebased score function R1 ( p) becomes the risk-neutral score function, as follows:
ED
2 2 2 2 1 1 R1 ( p) 1 P ( x) P ( x) P ( x) P ( x) 2 4
(26) 1 2 2 2 1 P ( x) P ( x) . 2 (3) When t 1 (or equivalently, 0 0.5 ), less than half of the interval of the indeterminacy degree has been assigned to the interval of the membership degree. Relatively more uncertainty is left in the indeterminacy part. This case can reflect the decision maker’s pessimistic attitude towards risk-taking in the IVPF context. (4) The cases of t 0 and t can respectively reflect the decision maker’s completely (or extremely) optimistic and pessimistic attitude towards risk-taking in the IVPF context.
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Theorem 6. For an IVPF value p in X, the risk attitude-based score function Rt ( p) satisfies the following properties: (T6.1) 0 Rt ( p) 1 (boundedness of Rt function); (T6.2) if p ([0,0], [1,1]), then Rt ( p) 0 for all t (zero property of Rt function); (T6.3) if p ([1,1], [0,0]), then Rt ( p) 1 for all t (one property of Rt function); (T6.4) S ( p) Rt ( p) for all t. Proof. The proofs of (T6.2) and (T6.3) are trivial. 14
(T6.1) With the use of Definition 6, one has 0 ( P ( x))2 1, 0 ( P ( x ))2 1 , that 0 t (t 1) 1 and 0 1 (t 1) 1 because t 0 . Observe that: 2 2 t 0 P ( x) P ( x) 1 , 2 t 1 2 2 1 0 P ( x) P ( x) 1 , t 1 2 2 1 0 P ( x) P ( x) 1 . t 1 It can be obtained that 0 Rt ( p) 1 ; hence, (T6.1) is proved. (T6.4) For all t 0 , it is easily verified that: 2 2 2 2 t 1 1 P ( x) P ( x) 1 P ( x) P ( x) 2 t 1 2
SC R
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0 (P ( x))2 +( P ( x))2 1 , and 0 (P ( x))2 +( P ( x)) 2 1 . Moreover, it is known
2 2 2 2 1 1 P ( x) P ( x) 1 P ( x) P ( x) 2 2 2 2 2 2 2 1 1 P ( x) P ( x) P ( x) P ( x) P ( x) P ( x) , 2 2 2 2 2 2 2 2 P ( x) P ( x) t 1 1 P ( x) P ( x) P ( x) P ( x) . Thus, one can conclude that: Rt ( p)
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1
2 2 2 2 2 2 1 t 1 P ( x) P ( x) P ( x) P ( x) P ( x) P ( x) 1 2 2 t 1 t 1
2 2 2 2 2 2 1 1 1 ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) P P P P P P 2 2 2
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2 2 2 2 1 P ( x) P ( x) P ( x) P ( x) =S ( p). 2
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Therefore, (T6.4) is valid. This completes the proof. Because no natural linear orders exist among IVPF values, this paper provides the following definition and theorem to establish a linear order within the IVPF environment by means of the risk attitude-based score function Rt ( p) .
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Definition 15. Let p1 and p2 be two IVPF values in X. Let a binary relation
LIVPF R ( t ) be defined as “being smaller than or indifferent from” based on the risk
attitude-based score function Rt . That is, p1 LIVPF R ( t ) p2 if and only if
Rt ( p1 ) Rt ( p2 ) . Theorem 7. Let p , p1 , and p2 be three IVPF values in X. Let ( LIVPF R (t ) , LIVPF R ( t ) ) 15
denote a lattice of nonempty IVPF values in X, in which LIVPF R (t ) {(P ( x),
P ( x))|P ( x), P ( x) [0,1],(P ( x))2 ( P ( x))2 1} with the binary relation LIVPF R ( t ) . Moreover, the top and bottom elements of ( LIVPF R (t ) , LIVPF R ( t ) ) are ([1,1], [0,0]) and ([0,0], [1,1]), respectively. Then, the binary relation LIVPF R ( t ) is a linear order. Proof. For the reflexivity, because Rt ( p) Rt ( p) , p LIVPF R ( t ) p for each p in X. The binary relation LIVPF R ( t ) is reflexive. For the antisymmetry, if both
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Rt ( p1 ) Rt ( p2 ) and Rt ( p2 ) Rt ( p1 ) hold, it is easy to see that Rt ( p1 ) Rt ( p2 ) . This indicates that if p1 LIVPF R ( t ) p2 and p2 LIVPF R ( t ) p1 , then p1 LIVPF R ( t ) p2 . One can obtain that LIVPF R ( t ) is antisymmetric. For the transitivity, the relations
SC R
p LIVPF R ( t ) p1 and p1 LIVPF R ( t ) p2 indicate Rt ( p) Rt ( p1 ) and Rt ( p1 ) Rt ( p2 ) ,
respectively. Because Rt ( p) , Rt ( p1 ) , and Rt ( p2 ) are comparable real numbers, an immediate consequence of Rt ( p) Rt ( p1 ) Rt ( p2 ) can be obtained. As a result, if p LIVPF R ( t ) p1 and p1 LIVPF R ( t ) p2 , then p LIVPF R ( t ) p2 for all p , p1 , and p2 in X.
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That is, LIVPF R ( t ) is transitive. Therefore, the binary relation LIVPF R ( t ) is a partial
N
order because it is reflexive, antisymmetric, and transitive. Additionally, the lattice ( LIVPF R (t ) , LIVPF R ( t ) ) is a partially ordered set. In particular, any two p1 and p2 in X
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are comparable via their corresponding risk attitude-based score functions Rt ( p1 ) and Rt ( p2 ) . Apparently, either Rt ( p1 ) Rt ( p2 ) or Rt ( p2 ) Rt ( p1 ) holds. One can
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conclude that the partial order LIVPF R ( t ) is complete. Accordingly, the binary relation
ED
LIVPF R ( t ) is a linear order. This completes the proof.
PT
The risk-neutral score function R1 (i.e., t 1 in Rt ) is not only a linear order but also an admissible order. The following definition and theorem provide us with a means of establishing an admissible order within the IVPF environment. To be precise, an order on LIVPF is admissible if it is linear and refines the order LIVPF .
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Definition 16. Let ( LIVPF , LIVPF ) be a partially ordered set. The order admissible order within the IVPF environment if (D16.1) is a linear order on LIVPF ;
is called an
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(D16.2) for any p1 , p2 LIVPF , p1 p2 whenever p1 LIVPF p2 . Theorem 8. Let p1 and p2 be two IVPF values in X, where p1 and p2 belong to LIVPF with the partial order LIVPF . If p1 LIVPF p2 , then R1 ( p1 ) R1 ( p2 ) . Moreover, the binary relation LIVPF R (1) is an admissible order on LIVPF . Proof. Recalling Definition 15 and Theorem 7, it is easy to see that the risk-neutral score function R1 : LIVPF [0,1] induces on LIVPF a relation LIVPF R (1) given by 16
p1 LIVPF R (1) p2 if and only if R1 ( p1 ) R1 ( p2 ) . This relation is also a linear order on
LIVPF . By means of (12) and (26), R1 ( p1 ) can be computed as follows: 2 2 2 2 1 1 R1 ( p1 ) 1 1 P1 ( x) P1 ( x) 1 P1 ( x) P1 ( x) P1 ( x) 2 4
2
1
2 2 2 2 2 1 ( x) 1 P1 ( x) P1 ( x) 1 P1 ( x) P1 ( x) 2
P1
2
2 2 2 2 1 1 1 1 P1 ( x) P1 ( x) P1 ( x) P1 ( x) 1 P1 ( x) 2 4 2
1
2 ( x) ( x) ( x) 2
P1
P1
2
P1
2
2 2 2 2 2 1 1 1 P1 ( x) P1 ( x) P1 ( x) P1 ( x) 4 4
1
2 2 2 2 1 P1 ( x) P1 ( x) . 4 Analogously, one obtains:
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2
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2 2 2 2 2 1 1 R1 ( p2 ) 1 P2 ( x) P2 ( x) P2 ( x) P2 ( x) 4 4
1
A
ED
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2 2 2 2 1 P2 ( x) P2 ( x) . 4 It is verified that R1 ( p1 ) R1 ( p2 ) because P1 ( x) P2 ( x) , P1 ( x) P2 ( x) ,
P ( x) P ( x) , and P ( x) P ( x) , according to the premise condition of 1
2
1
2
PT
p1 LIVPF p2 . Moreover, p1 LIVPF R (1) p2 holds because R1 ( p1 ) R1 ( p2 ) . Thus, the binary relation LIVPF R (1) based on the risk-neutral score function R1 refines the
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partial order LIVPF ; that is, p1 LIVPF p2 implies p1 LIVPF R (1) p2 . According to Definition 16, the binary relation LIVPF R (1) is admissible within the IVPF environment. Furthermore, for the lattice ( LIVPF , LIVPF R (1) ) , LIVPF is a chain because
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the relation LIVPF R (1) is a linear order on LIVPF . This completes the proof.
4. Outranking methods with risk attitudinal assignment models Based on the risk attitude-based score functions, this section develops an outranking approach using novel risk attitudinal assignment models for managing MCDA problems involving PF/IVPF information. Some useful concepts, such as the precedence frequency and precedence contribution matrices, are introduced to facilitate the modeling processes. This section provides two effective algorithms for 17
solving the problems of multiple criteria evaluation and selection in the PF/IVPF contexts. The issue of choosing an appropriate tool to determine linear orders is very important for many MCDA applications. As proven in Theorems 4 and 7, the binary relations LPF r ( t ) and LIVPF R ( t ) based on the risk attitude-based score functions rt and Rt , respectively, are linear orders. In particular, according to Theorems 5 and 8, the binary relations LPF r (1) and LIVPF R (1) are admissible orders on LPF and LIVPF ,
4.1 Proposed methodology in the PF context Consider an MCDA problem in which Z z1 , z2 ,
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respectively. Bustince et al. [6] have demonstrated that the concept of an admissible order is a useful tool for providing a total order. Therefore, the risk-neutral score functions r1 and R1 can provide total orders in the PF and IVPF contexts, respectively. Accordingly, this paper attempts to employ the risk attitude-based score functions to identify criterion-wise precedence ranks among alternatives and develop novel risk attitudinal assignment models for addressing MCDA problems within the PF/IVPF environments.
, zm is a set of m ( m 2 )
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feasible alternatives and C c1 , c2 , , cn is a finite set of n ( n 2 ) criteria. In reallife decision problems, the decision maker’s evaluations related to the performance of alternatives with respect to each criterion are often expressed by linguistic terms comprising vagueness and uncertainty. These uncertain, vague and hesitant judgments can be described more comprehensively by using the PF/IVPF sets. In the PF context, let a PF value pij ( ij , ij ) denote the evaluative rating of an alternative zi Z with respect to a criterion c j C , such that ij [0,1] , ij [0,1] , and
ED
0 (ij )2 ( ij )2 1 . In particular, ij and ij represent the degree to which zi
performs well or poorly, respectively, in terms of cj. The indeterminacy degree that 2 2 corresponds to each pij is determined as ij 1 ( ij ) ( ij ) . The MCDA problem
PT
within the PF environment can be concisely expressed in the following PF decision matrix: c1 c2 cn
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z1 ( 11 , 11 ) ( 12 , 12 ) ( 1n , 1n ) (27) z ( , ) ( 22 , 22 ) ( 2 n , 2 n ) p pij 2 21 21 mn zm ( m1 , m1 ) ( m 2 , m 2 ) ( mn , mn ) and the weight vector (w1 , w2 , , wn )T of the criteria, where the importance weight
wj of cj satisfies the normalization conditions: w j [0,1] and
n j 1
wj 1 .
MCDA problems can be deemed assignment problems because the MCDA task is generally related to the ordering of alternatives based on a priority ranking system; that is, MCDA can be considered the assigning of alternatives to a rank of order. It is often to face a practical problem that independence among criteria does not exactly hold. Criteria are generally dependent on each other in a complex, dynamic, and 18
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IP T
highly subjectively way [43]. In particular, when value dependency, preferential dependency, or anchor dependency exists among the criteria in question, it is hard to expect the criteria take the separable additive form [12, 43]. The assignment-based model provides an effective and convenient way to overcome this difficulty, because it does not directly combine the evaluative ratings across various criteria. Specifically, the assignment-based model combines all criterion-wise rankings into an overall ranking which achieves a best compromise among component rankings. Originated from the core structure of the assignment-based model [10, 12, 43], this paper employs the proposed risk attitude-based score function rt to construct a novel risk attitudinal assignment model and to develop an outranking method for addressing the MCDA problem. The proposed outranking methodology involving PF information is summarized in the following algorithmic procedure. Algorithm I. (Proposed methods in a PF setting) Step I-1: Determine risk attitude-based score functions for each evaluative rating. Set the t value as a parameter for representing the decision maker’s risk attitude, where 0 t 1 , t 1 , and t 1 for risk-seeking, risk-neutral, and riskaverse attitudes, respectively. Applying Definition 11, the risk attitude-based score function rt ( pij ) of the evaluative rating pij for each zi Z and c j C is calculated as follows:
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2 2 2 t 1 rt ( pij ) 1 ij ij ij . t 1 t 1 Step I-2: Identify a precedence frequency matrix based on criterion-wise precedence ranks. According to the properties proven in Theorems 4 and 5, it is known that the binary relation LPF r ( t ) is a linear order for all t; moreover, the binary
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relation LPF r (1) is an admissible order. Thus, it is appropriate to utilize the concept
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PT
ED
of risk attitude-based score functions to determine the criterion-wise precedence ranks among alternatives. More specifically, the m alternatives can be ranked in terms of each c j based on the descending order of the rt ( pij ) values. Next, one can establish a precedence frequency matrix whose entries measure the frequency at which each alternative is assigned a particular rank among all criterion-wise precedence relationships of the alternatives. In the PF context, let an m m square non-negative matrix f denote the precedence frequency matrix. The entry f i k ( i, k 1, 2, , m ) in f represents the frequency that zi is ranked k-th among all of the criterion-wise precedence rankings by means of the rt ( pij ) values. The matrix f is expressed as follows: 1st 2nd m-th
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f11 1 f f 2 zm f m1 z1 z2
f12 f 22 f m2
f1m f 2m , f mm
(28)
where the entry f i k satisfies the properties of
m i 1
fi k n for k 1, 2,
,m. 19
m k 1
fi k n for i 1, 2,
, m and
Step I-3: Construct a precedence contribution matrix via combining the importance weights. Based on the precedence frequency matrix f, a more comprehensive concept of the precedence contribution matrix is established to adequately utilize preference information about the criteria. Let an m m square non-negative matrix denote the precedence contribution matrix in the PF context, which is expressed as follows: 1st 2nd m-th
1m 2m . mm
(29)
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11 12 1 2 22 zm m1 m2 z1 z2
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Note that the entry ik ( i, k 1, 2, , m ) in represents a measure of the concordance among all of the criteria in assigning the alternative zi a rank k. Let c j1 , c j2 , , and c j k (with the importance weights w j1 , w j2 , , and w j k , fi
fi
respectively) represent the corresponding criteria for which zi is ranked k-th. To be precise, the entry ik is determined by aggregating the products of multiplying the fi
, and rt ( pij k ) , respectively, corresponding to f i k . That
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rt ( pij1 ) , rt ( pij2 ) rt ( pij2 ),
, and w j k by the risk attitude-based score functions fi
is,
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fi k
w j rt ( pij ) 1
(30)
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k i
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importance weights w j1 , w j2 ,
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PT
ED
for i, k 1, 2, , m , where 0 ik 1 . Step I-4: Establish a risk attitudinal assignment model using a permutation matrix. For the precedence contribution matrix , the larger the contribution revealed by entry ik is, the greater the concordance will be from assigning zi to the k-th overall precedence rank. Accordingly, this paper constructs a simple and k effective risk attitudinal assignment model to solve for an aggregate ranking. Let i denote a binary variable that is restricted to be either 0 or 1. In the PF context, let an m m square matrix denote a permutation matrix whose element ik =1 if zi is k assigned to the overall rank k; otherwise, i =0. The matrix is expressed as follows: 1st 2nd m-th
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11 12 1m 1 (31) 2 22 2m . 1 zm m m2 mm Note that zi should be assigned to only one rank in the overall precedence ranking; m thus, k 1 ik 1 . Similarly, a given rank k should only have one alternative assigned z1 z2
to it; hence,
m i 1
ik 1 . To find the overall precedence ranking that yields the 20
m
largest value of
i 1
ik , this paper utilizes a zero-one linear programming
m k k 1 i
format to establish the following risk attitudinal assignment model: m m max ik ik i 1 k 1 m
subject to
k 1
k i
1, i 1, 2,
, m,
k i
1, k 1, 2,
, m,
m
i 1
(32)
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ik 0 or 1 for all i and k . Step I-5: Acquire an outranking solution by solving for the optimal permutation matrix. The optimal permutation matrix ˆ can be obtained by solving the risk attitudinal assignment model in (32). To determine an outranking solution, the overall precedence orders of the alternatives can be easily determined by multiplying Z by ˆ , as follows: ˆ11 ˆ12 ˆ1m 1 ˆ2 ˆ22 ˆ2m ˆ Z z1 , z2 , , zm . (33) 1 2 ˆmm ˆm ˆm This overall ranking is an ultimate priority ranking that produces the best compromise among all of the criterion-wise precedence rankings of the alternatives.
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4.2 Proposed methodology in the IVPF context
ED
The outranking method using a risk attitudinal assignment model involving PF information can be extended to the more general IVPF environment. For an MCDA problem in the IVPF context, the evaluative rating of an alternative zi Z with
respect to a criterion c j C is expressed as an IVPF value pij [ ij , ij ],[ ij , ij ] ,
PT
where [ ij , ij ] Int([0, 1]), [ ij , ij ] Int([0, 1]), and 0 (ij )2 ( ij )2 1 . The intervals [ ij , ij ] and [ ij , ij ] represent the flexible degrees where zi performs
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well or poorly, respectively, in terms of cj. The interval of the indeterminacy degree that corresponds to each pij is computed as ij [ ij , ij ] 1 (ij )2 ( ij ) 2 , 1 (ij )2 ( ij ) 2 . The MCDA problem involving IVPF information can be concisely expressed in the following IVPF decision matrix:
21
mn
c1
, , , z1 11 11 11 11 z , , , 2 21 21 21 21 zm , , , m1 m1 m1 m1 and the weight vector (w1 , w2 ,
c2
cn
(34)
, 1n , 1n , 1n 22 , 22 , 22 , 22 2n , 2n , 2n , 2n m 2 , m 2 , m 2 , m 2 mn , mn , mn , mn n T , wn ) , such that w j [0,1] and j 1 w j 1 . 12
, 12 , 12 , 12
1n
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p pij
SC R
Based on a risk attitudinal assignment model via the risk attitude-based score function Rt , the algorithmic procedure of the proposed outranking methodology involving IVPF information is described, as follows. Algorithm II. (Proposed methods in an IVPF setting) Step II-1: Determine risk attitude-based score functions for each evaluative rating. Set the t value according to the decision maker’s risk attitude. Applying Definition 14, the risk attitude-based score function Rt ( pij ) of the evaluative rating
t 1 1 . 2 ij
2 ij
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2 2 1 t ij ij 1 2 2 t 1
2 ij
2 ij
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Rt ( pij )
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pij for each zi Z and c j C is calculated as follows:
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Step II-2: Identify a precedence frequency matrix based on criterion-wise precedence ranks. By the properties demonstrated in Theorems 7 and 8, one concludes that the binary relation LIVPF R ( t ) is a linear order for all t; moreover, the
ED
binary relation LIVPF R (1) is an admissible order. It follows that the m alternatives can
PT
be ranked with respect to each c j based on the descending order of the Rt ( pij ) values. Based on the obtained criterion-wise precedence ranks among alternatives, one can establish the precedence frequency matrix F in a manner analogous to the matrix f in (28). Let an m m square non-negative matrix F denote the precedence
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frequency matrix in the IVPF context. The entry Fi ( i, k 1, 2, , m ) in F represents the frequency with which zi is ranked k-th among all of the criterion-wise precedence rankings according to the Rt ( pij ) values. The matrix F is expressed as follows: 1st 2nd m-th
F11 1 F F 2 zm Fm1
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z1 z2
F12 F22 Fm2
k
F1m F2m , Fmm
(35)
k where the entry Fi satisfies the properties of
m i 1
Fi k n for k 1, 2,
m k 1
Fi k n for i 1, 2,
, m and
,m.
Step II-3: Construct a precedence contribution matrix via combining the 22
importance weights. Let an m m square non-negative matrix denote the precedence contribution matrix in the IVPF context, which is expressed as follows: 1st 2nd m-th
11 12 1 22 2 zm m1 m2 z1 z2
1m 2m . mm ( i, k 1, 2, , m ) in represents a measure of the
(36)
k
Fik
importance weights w j1 , w j2 ,
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Here, the entry i concordance among all of the criteria in assigning the alternative zi a rank k according to the precedence frequency matrix F. Let c j1 , c j2 , , and c j (with the
, and w j k , respectively) represent the corresponding
SC R
Fi
k criteria for which zi is ranked k-th. The entry i is determined in the following way: Fi k
w j Rt ( pij ) k i
1
(37)
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for i, k 1, 2, , m , where 0 ik 1 . Step II-4: Establish a risk attitudinal assignment model using a permutation matrix. For the precedence contribution matrix , the larger the contribution revealed by the entry i is, the greater concordance will be from assigning zi to the k-th overall precedence rank. Thus, a useful risk attitudinal assignment model is
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k
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constructed to solve for an aggregate ranking. Let i denote a binary variable that is restricted to be either 0 or 1. In the IVPF context, let an m m square matrix k
denote a permutation matrix whose element i =1 if zi is assigned to the overall rank
ED
k
k; otherwise, i =0. The matrix is expressed as follows: 1st 2nd m-th k
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11 12 1m 1 2 22 2m . zm m1 m2 mm It can be easily checked that z1 z2
(38)
m k 1
ik 1 because zi should be assigned to only one
rank in the overall precedence ranking. Moreover, one has
m i 1
ik 1 because a
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given rank k should only have one alternative assigned to it. The following risk attitudinal assignment model is formulated to determine the overall precedence m m ranking that yields the largest value of i 1 k 1ik ik :
23
m m max ik ik i 1 k 1 m
subject to
k 1
k i
1, i 1, 2,
, m,
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(39)
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ik 0 or 1 for all i and k . Step II-5: Acquire an outranking solution by solving for the optimal permutation ˆ can be determined by solving the risk matrix. The optimal permutation matrix attitudinal assignment model in (39). To acquire an outranking solution, the overall ˆ , as precedence orders of the alternatives can be derived by multiplying Z by follows: ˆ11 ˆ12 ˆ1m 1 2 ˆ 2m ˆ 2 ˆ 2 ˆ Z z1 , z2 , , zm (40) . m ˆ ˆ m1 ˆ m2 m
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5. Model application to financial decision making
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The proposed outranking methodology is a powerful tool for dealing with information that is characterized by hesitancy and vagueness. This section presents a financial decision-making application related to working capital management for a medical institution, along with a sensitivity analysis and a comparison to the technique of risk attitudinal ranking methods. Moreover, the developed approach is applied to an investment problem. A comparative study is also conducted with a newly developed outranking method that is an extension of the elimination and choice translating reality (ELECTRE) method to examine the advantages of the proposed methodology.
5.1 Financing policy of working capital management
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Efficient management of working capital is essential in the overall corporate strategy to improve firms’ profitability and liquidity, to create shareholder value, and to gain sustainable competitiveness [15, 28, 40]. Managing working capital, whose main components are accounts payable, accounts receivable, and inventory, means managing the day-to-day short-term operations of a firm [2, 28]. Working capital management plays a pivotal role for corporate finance due to its significant influence on firms’ liquidity and profitability [5, 31, 40, 41]. More precisely, higher working capital levels give firms a chance to increase sales and obtain discounts for early payments, which can enhance firms’ value [5]. However, higher working capital levels often require external financing (e.g., financing through debt). Thus, firms often incur additional financing expenses, which might lead to credit risks or the threat of bankruptcy [5, 40]. Combining these positive and negative perspectives regarding working capital effects, working capital management involves the risk and return trade-off, which is related to aggressive or conservative working capital policies. Financing policy refers to the financing models implemented by firms to satisfy 24
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working capital requirements. The financing policies of working capital management can generally be determined as aggressive or conservative [41]. In an aggressive working capital policy, the firm invests a small amount of capital in current assets in order to achieve higher gains on fixed assets [40, 41]. It utilizes higher levels of normally lower-cost short-term debt and less long-term capital, which may lead to the risk of short-term liquidity [5, 40]. Accordingly, it is considered the most risky and, potentially, the most profitable working capital management policy [28, 40]. Contrary to the aggressive policy, a conservative working capital policy relies on a greater volume of long-term debt in relation to the financing from short-term sources [41]. It causes the firm to define the liquidity requirements, which leads to a low liquidity risk position and profitability. Thus, the conservative policy reduces the risk of bankruptcy through investing a large amount of capital in current assets [28, 40, 41]. Simply stated, an aggressive working capital policy has a positive impact on profitability but a negative impact on liquidity, whereas a conservative working capital policy has a negative impact on profitability but a positive impact on liquidity [28]. In comparison with regular firms, those in the medical and health care industries possess several distinct characteristics that render financing decision making during working capital management more difficult and complex for managers. Ideally, medical institutions should be able to receive sufficient capital from the provision of treatment and care services to cover associated costs. To achieve this objective, managers must skillfully handle the four stages of the working capital cycle: (i) receive cash; (ii) use this cash to purchase resources (for instance, sanitary consumables and hiring) and pay accounts; (iii) use these resources to provide medical and care services; and (iv) bill patients and collect payments to continue the working capital cycle. However, under Taiwan’s current national health insurance (NHI) system, associated payments are usually received two or more months after patients or third-party payers have been billed. Most hospitals rely on third-party payers to pay patient bills; accounts receivable represent approximately 75% of the current assets of hospitals. Although patient deductibles show a gradually increasing trend, with the increasing amounts patients are burdened with, the risk of uncollectible accounts increases as well. In addition, because most accounts receivable are from third-party payers, owning accounts receivable with large sums implies the potential loss of other investment opportunities for earning returns. Hospital managers must be able to conduct evaluations based on the institution’s financial situation and environment, select suitable financial policies to ensure sufficient working capital, and maximize mobility and profitability. For managers, these are highly difficult and complex tasks. In short, for medical institutions, working capital significantly affects mobility, profitability, and competitiveness. Ensuring the possession of sufficient working capital to meet daily demands is essential to the operation of medical institutions. The key to doing so successfully lies in the selection of financial policies. In the light of above, the developed outranking methodology is applied to select a proper financing policy in the medical and health care fields. It is anticipated that the application results will both facilitate efficient management of working capital and support the practicality and effectiveness of the proposed approach.
5.2 Problem statement Financing decisions on aggressive/conservative working capital policies is a challenging problem for firms in an intricate and unpredictable business environment. Because of the particularity of the health care system and, specifically, the complexity 25
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of the NHI system in Taiwan, the selection of short-term financial policies becomes a highly complicated and ambiguous MCDA problem. This study focuses on the problem of selecting an appropriate financing policy of working capital management for a medical institution with the largest scale in Taiwan. To validate the practical usefulness and effectiveness of the developed approach, this subsection illustrates how the proposed concepts and models can be applied to address the financing decision in the medical and health care fields. The medical institution investigated in this case study is the Chang Gung Memorial Hospital (CGMH). CGMH offers the largest and most comprehensive health care services in Taiwan, being composed of a network of seven hospital branches. The main branch of CGMH is located in Linkou. Linkou CGMH is a multispecialty hospital and is the largest medical center in Taiwan. It contains approximately 4,000 beds and a total of 29 specialty centers. It is also the largest academic center ever accredited by the Joint Commission International (JCI) in 20142017. Working capital management is important to Linkou CGMH because it can grant this medical center financial flexibility and reduce its dependence on external sources of finance, especially under the government regulations of Taiwan’s NHI system. More generally, this importance is even greater for a medical institution if most of its institutional assets are current and it is highly reliant on current liabilities. Determining the optimal financing level of working capital is of great importance to Linkou CGMH. Sources of funds for financing are divided into longand short-term sources. Long-term sources of funds include long-term debt, shareholders’ equity, and spontaneous current liabilities, whereas short-term sources of funds include short-term loans, notes payable, discounted notes receivable, and other loans backed by current assets. Based on the differences between sources of funds and application methods, financing arrangements can be broadly divided into three types of basic policy: aggressive, conservative, and moderate. Crucial characteristics of these three types of basic policies are shown in Figure 2.
Figure 2. The characteristics of three types of basic financing policies. In an aggressive financing policy, short-term funds are used to finance temporary 26
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current assets and a portion of permanent current assets (i.e., short-term funds used to finance long-term funds), whereas the remaining permanent current assets and fixed assets are financed using long-term funds. Because this method utilizes less long-term and more short-term funds for financing, the higher interest rates required of longterm fund financing can usually be reduced, thus decreasing the financing cost. However, fluctuations in short-term interest rates are greater than those of long-term interest rates, and there is the risk of short-term interest rates increasing. Additionally, the stress of short-term debt repayment is high. Therefore, when this approach is adopted, it is necessary to evaluate whether interest rates demonstrate a rapidly increasing trend and to ensure that channels for short-term fund financing are continuous in order to consider both profitability and security. A conservative financing policy maintains a more cautious approach toward funding and primarily uses long-term funds to finance permanent current assets, fixed assets, and a portion of temporary current assets (i.e., long-term funds used to finance short-term funds). Only when the fund demand of temporary current assets exceeds the long-term funds used for financing are short-term funds used to finance the remaining portion. Although adopting this approach results in higher security and more-stable sources of funds, the financing costs are increased and the profit levels are reduced. In contrast, a moderate financing policy uses long-term funds to finance permanent current assets and fixed assets (i.e., long-term funds used to finance longterm funds), whereas temporary current assets are financed by short-term funds (i.e., short-term funds used to finance short-term funds). When adopting this approach, it is necessary to maintain the highest possible consistency between the expiration dates of short-term funds and long-term liabilities. However, because there are too many uncertain factors in practice, adopting this approach may result in gaps in funding. This case study designed five types of alternative plans suitable for use in working capital management based on the three aforementioned types of basic policy. It is important to note that the risk tolerance of medical and health care providers influences these working capital strategies. If providers are in a medical market with a lower number of competitors, the working capital demand can be more precisely predicted because the regional medical service demand is more stable. Therefore, medical institutions tend to adopt aggressive financing policies. By contrast, if providers are in an environment with more competitors, the regional medical service demand is less definite and stable; consequently, providers tend to adopt conservative financing policies. The three general principles in deciding whether to use short-term or long-term loans to meet working capital requirements are as follows: (i) using short-term loans to meet short-term working capital requirements; (ii) using long-term loans to meet long-term working capital requirements; and (iii) using a mixed strategy for capital requirements that are volatile, that is, using long-term loans to meet a certain amount of demand, then using short-term loans when short-term requirements emerge. This is because a mixed-type moderate financing policy is more suitable to an environment that is highly volatile. However, because the range of a moderate financing policy is wider and more difficult to define, in order to be more appropriate and accurate, this case study further divides the moderate financing policy into “aggressive-leaning,” “equally aggressive and conservative,” and “conservativeleaning” based on distributions of aggressive and conservative orientations. This case study comprises five plans related to working capital financing policy: aggressive dominant, aggressive-leaning, balanced aggressive and conservative, conservative27
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leaning, and conservative dominant. These are shown in Figure 3. To carefully evaluate these five types of alternative plans, managers consider the following six evaluation criteria: (1) Cash reserves A conservative financing policy involves an excess of cash reserves and very few short-term loans. This type of policy can reduce the likelihood that medical institutions will experience a financial crisis, and it eliminates concerns about short-term liquidation difficulties. However, using excess funds for unprofitable projects such as cash or securities will lower profit levels and incur holding costs. Managers must focus on the financial situations of medical institutions to carefully evaluate the amount of cash reserve needed to achieve maximized profitability and minimized risk and cost.
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Figure 3. Five alternatives for a financing decision of working capital management.
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(2) Maturity hedging Many managers will attempt to coordinate the expiration dates of current assets and liabilities and use short-term bank loans to finance sanitary consumables. Then, they will use long-term financing to purchase fixed assets, avoiding to the greatest extent possible the use of short-term loans to finance longterm assets. However, this method of maturity hedging creates a situation where frequent refinancing is needed, and managers must bear the risk of interest rate fluctuations. (3) Interest rate fluctuation When managers are evaluating which type of financing strategy to use, it is necessary to consider relative long- and short-term interest rate levels. Short-term interest rates are usually lower than long-term ones. However, the volatility of short-term interest rates is high; when short-term loans expire, if another loan is desired, there may be a risk of increased interest rates. Although long-term interest rates ensure more stable sources of funds, financing with long-term interest rates results in higher costs, on average. (4) Financial leverage Financial leverage and working capital management are closely correlated in medical institutions; this correlation is usually measured using the ratio of total current liabilities to total assets. An increase in the ratio of current liabilities to 28
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total assets results in a higher leverage ratio, greater short-term liquidation stress, and an increase in the risk that medical institutions face. (5) Return on Assets The degree of aggression in financing policy is often negatively correlated with the return on assets (ROA) of medical institutions; even an increase in shortterm liabilities for financing may negatively affect ROA. Because a conservative financing policy uses more long-term loans, medical institutions have a longer time to fulfill financial obligations, which positively affects ROA. However, when short-term interest rates exhibit declining trends, because financing costs decrease, an aggressive financing policy positively affects ROA. Therefore, when managers evaluate the ROA performance of each plan, it is necessary to first predict trends in short-term interest rates. (6) Financing costs Financing costs are the associated costs incurred by medical institutions in order to increase funds or the cost of fund use and fund raising. Financing costs differ according to the sources of funding. For instance, the interest paid differs when bonds are issued and when different bank loans are applied for. In addition, opportunity cost is a type of financing cost; it refers to using a particular type of resource for a particular purpose and the maximum gain of other purposes that is given up as a result. Although medical institutions that use their own funds do not need to pay costs such as interest, they will lose potential profits gained from investing these funds. Notably, interdependence exists among the aforementioned six criteria, and the interactive relationship between each renders it difficult for managers to make a definite evaluation of the performance of the plans in each criterion (because criteria evaluation values are influenced by the associated effects of other related criteria). For instance, interest rate fluctuations will directly influence financing costs. If future interest rates increase, aggressive-leaning policies will be influenced by increases in short-term interest rates and will lead to increased financing costs. In addition, maturity hedging and interest rate fluctuation are interdependent. Generally, shorter periods result in greater interest rate fluctuations; by contrast, longer debt expiration dates result in lesser interest rate fluctuations. Moreover, managers’ degrees of preference for risk influence how criteria values are defined. Considering the imprecise and uncertain information associated with a complex medical circumstance, this paper attempts to employ the developed Algorithm II based on the IVPF theory to address this issue.
5.3 Application of the proposed approach
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The proposed methodology with Algorithm II was employed to help Linkou CGMH select the most appropriate financing policy of working capital management. This MCDA problem is defined by five financing policies and six criteria for evaluating the alternatives. Specifically, the set of alternatives is denoted by Z={z1 (aggressive dominant policy), z2 (aggressive-leaning policy), z3 (balanced aggressive and conservative policy), z4 (conservative-leaning policy), z5 (conservative dominant policy)}. The set of criteria is denoted by C={c1 (cash reserves), c2 (maturity hedging), c3 (interest rate fluctuation), c4 (financial leverage), c5 (return on assets), c6 (financing costs)}. The weight vector of the criteria was given by the authority of Linkou CGMH in advance as (0.10, 0.05, 0.20, 0.10, 0.25, 0.30)T. This study conducted a survey to collect three managers’ opinions and judgments about the 29
performance evaluation of the five alternatives with respect to each criterion. Based on a statistical inference approach, Chen [8] developed an interval estimation method with finite population correction to construct the membership functions of A-IVIF sets. By extending Chen’s statistical inference approach to the IVPF environment, this study estimated respective intervals of membership and non-membership degrees according to the survey results, which can reflect managers’ knowledge and expertise. The IVPF decision matrix p ( [ pij ]56 ) was constructed as follows: c1
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z1 ([0.15, 0.26],[0.74, 0.92]) z2 ([0.27, 0.44],[0.66, 0.82]) p z3 ([0.57, 0.69],[0.46, 0.57]) z4 ([0.78, 0.86],[0.13, 0.28]) z5 ([0.92, 0.96],[0.11, 0.17])
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([0.29, 0.37],[0.75, 0.81]) ([0.42, 0.56],[0.48, 0.59])
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([0.49, 0.61],[0.38, 0.58]) ([0.53, 0.68],[0.42, 0.47]) ([0.71, 0.82],[0.19, 0.30]) ([0.84, 0.91],[0.15, 0.27]) ([0.78, 0.88],[0.11, 0.19])
([0.73, 0.82],[0.21, 0.33])
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([0.89, 0.91],[0.05, 0.15]) ([0.56, 0.66],[0.38, 0.45]) c5
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z1 ([0.22, 0.36],[0.64, 0.76]) ([0.85, 0.96],[0.07, 0.19]) ([0.41, 0.56],[0.34, 0.45]) z2 ([0.34, 0.48],[0.59, 0.67]) ([0.74, 0.86],[0.12, 0.21]) ([0.67, 0.79],[0.23, 0.38]) z3 ([0.55, 0.67],[0.41, 0.53]) ([0.68, 0.74],[0.26, 0.35]) ([0.88, 0.95],[0.09, 0.13]) . z4 ([0.68, 0.79],[0.27, 0.34]) ([0.28, 0.35],[0.72, 0.86]) ([0.81, 0.87],[0.11, 0.25]) z5 ([0.78, 0.86],[0.23, 0.29]) ([0.07, 0.14],[0.82, 0.93]) ([0.43, 0.59],[0.52, 0.65])
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Consider the risk-neutral case (i.e., t 1 ) as an illustrative example. In Step II-1, the risk-neutral score function of the evaluative rating pij was determined in the following manner:
1 1 1 2 2 2 2 2 2 1 . ij ij ij ij ij ij 2 4 2 The calculated results are presented in the top part of Table 1. For example, for p45 (=([0.28, 0.35], [0.72, 0.86])), the indeterminacy interval was given by 45
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[ 1 0.352 0.862 , 1 0.282 0.722 ] [0.37, 0.63]. It follows that R1 ( p45 )
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0.5 [1 0.25 (0.372 0.632 )][0.282 0.352 0.5 (0.372 0.632 )] 0.4515.
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Table 1. Some results for the financing decision on working capital policies. zi/rank c1 c2 c3 c4 c5 c6 Results of the risk-neutral score function R1 ( pij ) z1 0.3893 0.4642 0.6055 0.4854 0.9110 0.6153 z2 0.4943 0.6419 0.6786 0.5516 0.8217 0.7667 z3 0.6873 0.7935 0.8831 0.6786 0.7510 0.9192 z4 0.8367 0.8456 0.8004 0.7702 0.4515 0.8530 z5 0.9419 0.9050 0.6818 0.8363 0.3293 0.6091 Criterion-wise precedence ranks by sorting the R1 ( pij ) values 1st 2nd 3rd
z5 z4 z3
z5 z4 z3
z3 z4 z5
z5 z4 z3 30
z1 z2 z3
z3 z4 z2
4th 5th
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z2 z1
z2 z1
z2 z1
z4 z5
z1 z5
In Step II-2, for each c j C , the five alternatives were ranked by sorting each
R1 ( pij ) value in descending order. The results of the criterion-wise precedence ranks are revealed in the bottom part of Table 1. Taking c6 as an example, based on the descending order of the R1 ( pi 6 ) values, it can be seen that p56 LIVPF R (1) p16 LIVPF R (1)
p26 LIVPF R (1) p46 LIVPF R (1) p36 , and thus, the precedence ranking is
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z3 z4 z2 z1 z5 with respect to c6. Accordingly, the precedence ranks of z1 , z2 , , z5 are ranked fourth, third, first, second, and fifth, respectively. Using the
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5 k 1
Fi k 6 for i 1, 2,
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Fi k 6 for
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z1 F11 F12 F13 F14 F15 1 0 0 1 4 z2 F21 F22 F23 F24 F25 0 1 1 4 0 F z3 F31 F32 F33 F34 F35 2 0 4 0 0 . z4 F41 F42 F43 F44 F45 0 5 0 1 0 z5 F51 F52 F53 F54 F55 3 0 1 0 2 4 For example, F2 =4 because the alternative z2 is ranked fourth with respect to c1, c2,
In Step II-3, the entries i for each i, k {1, 2, ,5} were calculated to establish the following precedence contribution matrix : 1st 2nd 3rd 4th 5th
13 23 33 43 53
14 24 34 44 54
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12 22 32 42 52
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z1 11 z2 21 z3 31 z4 41 z5 51
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15 0.2278 25 0.0000 35 0.4524 45 0.0000 55 0.2231
0.0000 0.0000 0.1846 0.2318 0.2054 0.2300 0.2724 0.0000 0.0000 0.3640 0.0000 0.0000 . 0.6190 0.0000 0.1129 0.0000 0.0000 0.1364 0.0000 0.2651
1 Consider 51 as an example. It is known that F5 3 ; moreover, the corresponding
criteria for which z5 is ranked first consist of c1, c2, and c4. Thus, let c j1 c1 ,
c j2 c2 , and c j (i.e., c j ) c4 , in which w j1 w1 0.10, w j2 w2 0.05, and
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w j3 w4 0.10. The entry 51 was computed as follows:
51 1 w j R1 ( pij ) 0.1 0.9419 0.05 0.9050 0.10 0.8363 0.2231 . 3
In Step II-4, a permutation matrix whose entry ik {0,1} for each i, k {1, 2, ,5} was defined as follows:
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z1 11 12 13 14 15 z2 21 22 23 24 25 z3 31 32 33 34 35 . z4 41 42 43 44 45 z5 51 52 53 54 55 Using the zero-one linear programming format in (39), this study constructed the following risk attitudinal assignment model: 0.2278 11 0.0000 12 +0.0000 13 +0.1846 14 +0.2318 15 1 2 3 4 5 0.0000 2 0.2054 2 +0.2300 2 +0.2724 2 +0.0000 2 max 0.4524 31 0.0000 32 +0.3640 33 +0.0000 34 +0.0000 35 1 2 3 4 5 +0.0000 4 0.6190 4 +0.0000 4 +0.1129 4 +0.0000 4 +0.2231 1 0.0000 2 +0.1364 3 +0.0000 4 +0.2651 5 5 5 5 5 5 ,5,
(41)
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ik 0 or 1 for all i and k . In Step II-5, the optimal objective value 1.7511 was acquired by solving the above model using a branch-and-bound algorithm. Moreover, ˆ14 =ˆ 23 ˆ 31 ˆ 42 ˆ 55 1 , and the other ˆ ik 0 . Employing (40), the overall precedence orders of the five financing policies was determined by multiplying Z by ˆ , i.e.: 0 0 0 1 0 0 0 1 0 0 Z ˆ z1 , z2 , z3 , z4 , z5 1 0 0 0 0 z3 , z4 , z2 , z1 , z5 . 0 1 0 0 0 0 0 0 0 1 Therefore, the ultimate priority ranking of the five financing policies is z3 z4 z2 z1 z5 in the risk-neutral case. That is, the balanced aggressive and conservative policy (z3) is the best choice that produces the overall compromise among all of the criterion-wise precedence rankings of the alternatives. Furthermore, this paper conducted a sensitivity analysis to observe the influence of the risk attitudinal parameter t on the solution results. Detailed analysis and discussions are presented in Appendix B. In addition to the illustrative case of t=1, this paper considered diverse attitudes of risk seeking (t=0.000001, 0.25, 0.5, and 0.75) and risk aversion (t=5, 10, 100, and 1000000). The obtained results of the criterion-wise precedence rankings and the overall precedence rankings are revealed in Table B.1 of Appendix B. According to the sensitivity analysis based on Table B.1, this paper concludes that the proposed outranking methodology can not only fully take into account the decision maker’s attitude towards risk-taking but also yield reasonable and persuasive results under complex uncertainty based on Pythagorean fuzziness. 32
5.4 Comparison to the risk attitudinal ranking method Because the proposed risk attitude-based score function was originated from Guo’s developed technique [21], this paper further conducts a comparative analysis with Guo’s approach to validate the effectiveness and advantages of the proposed methodology in this subsection. It is noted that the technique of Guo’s risk attitudinal ranking methods aims to tackle A-IF and A-IVIF information. To facilitate a consistent comparison, this paper generates the evaluative rating data using the AIVIF format. Let pij0 ( [ij0 , ij0 ],[ ij0 , ij0 ] ) denote an A-IVIF evaluative rating based on
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hesitation degree corresponding to pij0 , where [ ij0 , ij0 ] Int([0, 1]). According to the transformation procedure introduced by Chen [11], the A-IVIF evaluative rating pij0 and its hesitation degree ij0 are determined as follows:
ij ij ij ij p , , , , ij ij ij ij ij ij ij ij ij ij ij ij
(42)
ij ij , . ij ij ij ij ij ij
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financing decision on working capital policies, the A-IVIF decision matrix p 0 ( [ pij0 ]56 ) was constructed as follows:
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z1 ([0.0970, 0.1765],[0.4788, 0.6245]) z2 ([0.1655, 0.2706],[0.4046, 0.5043]) p0 z3 ([0.3332, 0.4044],[0.2689, 0.3341]) z4 ([0.5124, 0.5490],[0.0854, 0.1787]) z5 ([0.6543, 0.7098],[0.0782, 0.1257])
([0.1774, 0.2263],[0.4589, 0.4954]) ([0.2962, 0.3526],[0.2297, 0.3353]) ([0.4499, 0.5101],[0.1204, 0.1866]) ([0.5179, 0.5846],[0.0730, 0.1262]) ([0.6388, 0.6291],[0.0359, 0.1037])
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z1 ([0.2515, 0.3234],[0.2874, 0.3407]) ([0.1378, 0.2167],[0.4010, 0.4575]) z2 ([0.3142, 0.3970],[0.2490, 0.2744]) ([0.2045, 0.2797],[0.3549, 0.3904]) z3 ([0.5558, 0.6088],[0.0992, 0.1806]) ([0.3259, 0.3896],[0.2429, 0.3082])
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z1 ([0.5894, 0.7081],[0.0485, 0.1402]) ([0.2568, 0.3283],[0.2130, 0.2638]) z2 ([0.4863, 0.5602],[0.0789, 0.1368]) ([0.4172, 0.4785],[0.1432, 0.2301]) z3 ([0.4183, 0.4446],[0.1599, 0.2103]) ([0.6127, 0.6965],[0.0627, 0.0953]) . z4 ([0.1713, 0.2213],[0.4404, 0.5438]) ([0.5414, 0.5631],[0.0735, 0.1618]) z5 ([0.0480, 0.0993],[0.5624, 0.6596]) ([0.2547, 0.3432],[0.3080, 0.3781]) Based on Guo’s proposed attitudinal-based measure [21] within the A-IVIF environment, the risk attitude-based score function Rt( pij0 ) of pij0 is defined as
follows:
1 t 1 ij0 ij0 ij0 ij0 ij0 ij0 . 1 2 2 t 1 t 1
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(44)
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The risk attitude-based score function Rt ( pij ) was replaced by Rt( pij0 ) , and then the proposed Algorithm II was employed to address the MCDA problem of aggressive/conservative financing policies for Linkou CGMH. The obtained results based on Guo’s developed technique [21] and the proposed attitudinal-based measure are contrasted in Figures 4-6. Specifically, Figure 4 presents the comparisons of overall precedence ranks concerning the five financing policies with respect to different values of the risk attitudinal parameter t. Figure 5 illustrates the changes of overall precedence rankings for each alternative under different t values. Based on the comparison results in Figures 4 and 5, one can easily observe that the ranking orders obtained by Guo’s technique and the proposed approach are the same when the parameter t takes the values of 1, 5, 10, 100, and 1000000. To be precise, the ultimate priority rankings z3 z4 z2 z1 z5 and z3 z4 z2 z5 z1 were determined in the risk-neutral and risk-averse settings, respectively, regardless of the employment of the risk attitude-based score functions Rt( pij0 ) and Rt ( pij ) . Nevertheless, there are obvious differences between the overall precedence rankings through using Rt( pij0 ) and Rt ( pij ) in a risk-seeking situation.
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Figure 4. Contrast of overall precedence ranks under different t values. As illustrated in Figure 4 (a) and Figure 5 (a), the same ultimate priority ranking (i.e., z3 z4 z2 z1 z5 ) was acquired in the risk-seeking and risk-neutral settings based on Guo’s attitudinal-based measure. In contrast, as depicted in Figure 4 (b) and Figure 5 (b), two different ultimate ranking results z1 z4 z3 z2 z5 and
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z3 z4 z2 z1 z5 in the settings of risk-seeking and risk-neutral attitudes, respectively, were determined based on the proposed measure. It is worth mentioning that the cases with a risk-seeking attitude (i.e., t=0.000001, 0.25, 0.5, and 0.75) more or less reflect the decision maker’s dispositional optimism. This implies that the aggressive dominant policy (z1) is possibly more likely to be the best ranked in a riskseeking situation. However, the risk attitude-based score function Rt( pij0 ) based on Guo’s measure cannot produce an acceptable and credible result because the identical ultimate ranking was generated regardless of risk-seeking and risk-neutral situations. These findings indicate that Guo’s developed technique may lack the applicability to tackle MCDA problems under complex uncertainty. Unlike Guo’s approach, the obtained results through using the proposed risk attitude-based score function Rt ( pij ) are intuitively reasonable and convincing in theory and in practice.
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Figure 5. Change of overall precedence rankings for each alternative.
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Furthermore, Figure 6 contrasts the overall precedence ranks of the five financing policies under three risk attitudes. As presented in Figure 6 (a), the overall precedence ranks of the alternatives z2 (with the third rank), z3 (with the first rank), and z4 (with the second rank) are unchanging despite the risk attitudes. The only difference is that the alternatives z1 is the fourth ranked in risk-seeking and riskneutral situations, while it is the fifth ranked in a risk-averse situation. The outcomes of the alternatives z5 is just the opposite. Namely, z5 is the last ranked in risk-seeking and risk-neutral situations and the fourth ranked in a risk-averse situation. In light of the pattern in Figure 6 (a), the application results based on Guo’s developed technique are less sensitive for the decision maker’s distinct attitudes towards risk-taking. More concretely, despite diverse attitudes toward risk neutrality, risk seeking, and risk aversion, the outcomes of the overall precedence ranks for each alternative are very similar through the use of Guo’s attitudinal-based measure. This implies that the risk attitude-based score function Rt( pij0 ) cannot reflect the influences of different risktaking attitudes on overall precedence ranks of the alternatives. As opposed to Guo’s approach, the proposed methodology is sensitive for distinct attitudes toward risk neutrality, risk seeking, and risk aversion. As revealed in Figure 6 (b), except z4, the overall precedence ranks of the alternatives are more or less different in diverse risk36
taking settings. More precisely, the alternative z1 is the first, fourth, and fifth ranked in risk-seeking, risk-neutral, and risk-averse situations, respectively. Moreover, z2 is the fourth ranked in a risk-seeking situation and the third ranked in risk-neutral and riskaverse situations; z3 is the third ranked in a risk-seeking situation and the best ranked in risk-neutral and risk-averse situations. Finally, z5 is the fifth ranked in risk-seeking and risk-neutral situations and the fourth ranked in a risk-averse situation. Thus, contrary to the use of Rt( pij0 ) , the proposed risk attitude-based score function
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Rt ( pij ) can yield more-reasonable and more-persuasive results. Based on the comparative discussions with respect to Figures 4-6, it can be concluded that the proposed outranking methodology can provide a useful tool for reflecting the influence of the decision maker’s attitude towards risk-taking in the face of complex uncertainty.
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(a) Results based on the risk attitude-based score function Rt( pij0 )
(b) Results based on the risk attitude-based score function Rt ( pij ) Figure 6. Overall precedence ranks of alternatives under three risk attitudes.
5.5 Comparison to IVPF-ELECTRE using an investment problem This subsection investigates an investment problem and conducts a comparative analysis with the most well-known and widely used outranking methodology, 37
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ELECTRE, to demonstrate the advantages of the proposed approach. It is worth noting that the MCDA models can be generally divided into scoring, compromising, and outranking models. Both the proposed approach and the ELECTRE methodology belong to the outranking model. This was the main reason for selecting ELECTRE as a comparative approach. Another reason was its popularity in practice, because numerous methods in the ELECTRE family have been successfully applied in the MCDA field. To date, in the literature on the ELECTRE methodology, little research has focused on the development of MCDA methods based on PS sets or IVPS sets. Based on the concepts of score functions and accuracy functions, Peng and Yang [35] employed the core structure of ELECTRE to propose a new IVPF-ELECTRE method for handling multiple criteria group decision-making problems involving IVPF information. Peng and Yang’s work is a seminal study of extending the ELECTRE methodology to the IVPF environment. In this regard, this paper deems it a benchmark method for a comparative study. More precisely, this paper chose the IVPF-ELECTRE method as a comparative approach to validate the strength and practicality of the proposed outranking methodology based on a risk attitudinal assignment model in the IVPF context. In general, the better alternative has the higher score function or the higher accuracy function in the case where the alternatives have the same score. In line with the concepts of score and accuracy functions defined for IVPF values, Peng and Yang [35] constructed the IVPF concordance/discordance sets to define the concordance/discordance indices. In the IVPF-ELECTRE procedure, these sets are divided into the IVPF strong, medium, and weak concordance/discordance sets according to various scenarios defined by score and accuracy functions. To reflect the relative dominance of a certain alternative over a competing alternative, Peng and Yang determined the concordance index by means of the sum of the weights associated with those criteria and relations that are contained in the IVPF strong, medium, and weak concordance sets. Moreover, to reflect the relative difference of an alternative with respect to a competing alternative, Peng and Yang identified the discordance index in terms of discordance criteria that are contained in the IVPF strong, medium, and weak discordance sets. The concordance/discordance Boolean matrices and the outranking matrix are then established to indicate the order of relative superiority of alternatives. Peng and Yang [35] provided an illustrative application concerning an investment problem to verify the effectiveness of their developed approach. This application involves three experts (i.e., project manager, CEO, and chairman) in an investment company evaluating five software development projects according to four criteria. The set of alternatives is denoted by Z={z1 (a mail development project), z2 (a game development project), z3 (a browser development project), z4 (a music player development project), z5 (a microblog development project)}. The set of criteria is denoted by C={c1 (economic feasibility), c2 (technological feasibility), c3 (staff feasibility), c4 (period feasibility)}. Peng and Yang [35] employed their developed IVPF weighted average operator to aggregate individual IVPF decision matrices provided by each expert to determine the aggregated decision matrix. Detailed data of the evaluative rating pij in the aggregated decision matrix are provided in the top part of Table 2. Moreover, Peng and Yang [35] applied the maximizing deviation method to determine the objective weights of the four criteria. They combined the objective weights and the weights of the three experts to acquire the importance 38
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weights of the criteria. The computed results are expressed as a weight vector (0.29, 0.23, 0.26, 0.22)T. To compare the application results on a consistent basis, this paper employed the proposed approach with Algorithm II to analyze the same data in Table 2. Assume that, without loss of generality, t 1 (i.e., risk neutrality) due to a lack of relevant information about the decision maker’s risk attitude in this case. The results of the risk-neutral score function R1 ( pij ) and the corresponding precedence ranks are listed in the middle part of Table 2. However, it can be observed that the alternatives z2 and z5 are tied with respect to the criterion c4. That is, the precedence ranking of z1 z2 ~ z5 z3 z4 was acquired in terms of c4 because R1 ( p14 ) R1 ( p24 ) R1 ( p54 ) R1 ( p34 ) R1 ( p44 ) . To avoid confusing computations of the precedence frequency in the matrix F, this paper reproduces c4 by splitting it into two extended criteria: c4(1) and c4(2). Moreover, the original ranking z1 z2 ~ z5 z3 z4 is equalized as z1 z2 z5 z3 z4 and z1 z5 z2 z3 z4 for c4(1) and c4(2), respectively. An adjusted version of the matrix F was obtained as follows: 1st 2nd 3rd 4th 5th 1 0 0 0 1 1 1 2 1 1 2 0 . 1 1 1 2 1 2 1 1 5 5 Meanwhile, k 1 Fi k i 1 Fi k n 1 5 for all i and k. Note that each of the
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Table 2. Data and application results in the investment problem. Data of the evaluative rating pij c1 c2 ([0.69, 0.79], [0.21, 0.31]) ([0.70, 0.80], [0.20, 0.30]) ([0.55, 0.65], [0.35, 0.45]) ([0.35, 0.45], [0.55, 0.65]) ([0.57, 0.67], [0.33, 0.43]) ([0.56, 0.66], [0.34, 0.44]) ([0.62, 0.72], [0.28, 0.38]) ([0.43, 0.53], [0.47, 0.57]) ([0.53, 0.63], [0.37, 0.47]) ([0.54, 0.64], [0.36, 0.46]) c3 c4 z1 ([0.65, 0.75], [0.25, 0.35]) ([0.63, 0.73], [0.27, 0.37]) z2 ([0.32, 0.42], [0.58, 0.68]) ([0.56, 0.66], [0.34, 0.44]) z3 ([0.69, 0.79], [0.21, 0.31]) ([0.54, 0.64], [0.36, 0.46]) z4 ([0.56, 0.66], [0.34, 0.44]) ([0.45, 0.55], [0.45, 0.55]) z5 ([0.54, 0.64], [0.36, 0.46]) ([0.56, 0.66], [0.34, 0.44]) Results of the risk-neutral score function R1 ( pij ) and the precedence ranks
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Consider 53 as an example. The alternative z5 is ranked third among all adjusted criterion-wise precedence rankings with respect to the criterion c2 (with the weight w2) and the extended criterion c4(1) (with the weight 0.5 w4 ). Moreover,
F53 2 from the obtained matrix F. It can be shown with the use of (37) that
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ik 0 or 1 for all i and k . By solving the above model, the overall precedence ranking of the five software development projects is z1 z4 z5 z3 z2 ; thus, the best choice is the mail development project (i.e., z1). Compared with the obtained decision graph depicted by Peng and Yang [35], the IVPF-ELECTRE method rendered the outranking relationships z1 z2 , z1 z4 , z1 z5 , z3 z2 , z3 z5 , and z4 z2 . Moreover, Peng and Yang concluded that z1 is superior to z2, z4, and z5. Therefore, the benchmark method and the proposed method provided very similar ranking results. However, the IVPF-ELECTRE method cannot differentiate the outranking relationships between z1 and z3, z2 and z5, z3 and z4, and z4 and z5. This implies that the overall precedence orders of z1 and z3 cannot be confirmed via the IVPF-ELECTRE method. Thus, the decision maker may be doubtful regarding whether z1 is the best choice, which may result in decision difficulty. In contrast, the proposed outranking method with Algorithm II yielded the ranking result z1 z4 z5 z3 z2 , which can clarify the ultimate priority orders of all alternatives such that one can conclude that z1 is indeed superior to the other four alternatives. It is clear that the proposed method has the capability of producing acceptable and reasonable ranking results. In contrast, it is troublesome to make a final decision 40
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for decision makers when using the IVPF-ELECTRE method. The reason for this finding could be confusion among the outranking relationships of pairs z1 and z3, z2 and z5, z3 and z4, and z4 and z5 according to the solution results by IVPF-ELECTRE. Unlike the benchmark method, the developed outranking method is perfectly capable of yielding a total order of the alternatives, which is essential for MCDA applications and decision-making in reality. Because the proposed method utilizes a risk attitudinal assignment model, tied situations among alternatives hardly occur in the final solution results. More specifically, the optimal permutation matrix ˆ can be determined by solving the risk attitudinal assignment model that is expressed using a simple zero-one linear programming model. Because of the constraint conditions in the risk attitudinal assignment model, the obtained matrix ˆ can explicitly differentiate the ultimate priority orders of all alternatives and contribute the certain outranking relationships of alternatives that have similar overall evaluation values. Consider the application results regarding the investment problem of software development projects as an example. As opposed to the benchmark method of IVPF-ELECTRE, the proposed approach can indeed distinguish the ultimate priority orders of similar or indefinite alternatives (i.e., z1 z3 , z5 z2 , z4 z3 , and z4 z5 ) and provide a definite overall precedence ranking of all alternatives (i.e., z1 z4 z5 z3 z2 ) for substantive decision support. Therefore, differentiation of the ultimate priority orders among various alternatives, especially among similar or indefinite alternatives, is a distinct advantage of the proposed method. It is noteworthy that the decision approach of the proposed methodology is quite different from the IVPF-ELECTRE method. Essentially, the IVPF-ELECTRE method renders a partial rank of alternatives by considering concordance and discordance. In contrast, the proposed methodology does not utilize the measures of concordance and discordance (i.e., the concordance/discordance indices) to determine the outranking relationships between alternatives. More specifically, the IVPF-ELECTRE simultaneously employs the score and accuracy functions to construct the IVPF strong, medium, and weak concordance/discordance sets and then determine the IVPF concordance/discordance indices. As opposed to such complex computations in the IVPF-ELECTRE, the proposed methodology takes a straightforward and effective manner based on the risk attitude-based score function instead of concordance and discordance measures. Recall that the comparison results (see Tables A.1 and A.2) in Appendix A have demonstrated the validity and reasonability of the proposed risk attitude-based score functions rt and Rt . Compared with the existing score and
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information and facilitating subjective judgments and evaluations. In brief, unlike the IVPF-ELECTRE based on score and accuracy functions, the proposed methodology incorporates the decision maker’s attitude towards risk-taking into the analytical model by means of the developed concepts of risk attitude-based score functions and risk attitudinal assignment models. Furthermore, the developed techniques possess certain flexibility and adaptability via different settings of the parameter t according to individual risk perceptions in the face of high uncertainty.
6. Conclusions
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The theory of PF/IVPF sets provides an intuitive and feasible way of modeling imprecise information and quantifying the ambiguous nature of subjective judgments. The employment of the PF/IVPF theory to address MCDA problems is one of the most promising directions in real-world applications. Many useful models and methods have been developed to utilize PF/IVPF sets to accommodate more-complex decision-making environments. In contrast to the existing techniques, this paper has established a novel outranking approach for managing the sophisticated data expressed by PF/IVPF sets in a simple and effective manner. The proposed risk attitudinal assignment models based on the developed risk attitude-based score functions have a great capability to deal with complex ambiguity and imprecision expressed as PF/IVPF values, which are flexible in reflecting the uncertainty and hesitation associated with the decision makers’ subjective assessments and different attitudes toward risk. Several interesting theoretical properties of the developed concepts have been explored to provide more-robust theoretical contributions. As an application of the proposed methodology, a real-world problem concerning a financing decision for working capital management was investigated to verify its feasibility and applicability. The practicality and advantages of the developed model and techniques have been supported through the application results in realistic situations, the sensitivity analysis of the risk attitudinal parameter, and the comparative discussions with relevant approaches. In addition to useful theoretical insights, the proposed approach has practical effectiveness, and it contributes to the financial decision-making field. Furthermore, this paper selected the IVPF-ELECTRE as a benchmark method for conducting a comparative study because it is the most closely related outranking-based methodology in cases with a high degree of uncertainty. Concerning an application to the investment problem of software development projects, the advantages and practicality of the proposed approach were verified through a comparison analysis of the obtained results yielded by the IVPFELECTRE method and the proposed Algorithm II. Unlike the concordance and discordance analysis in the IVPF-ELECTRE procedure, Algorithm II utilizes an effective and straightforward approach based on the risk attitudinal technique (e.g., risk attitude-based score functions and risk attitudinal assignment models). The developed approach can not only accurately distinguish between the PF/IVPF information but also incorporate the decision maker’s diverse risk-taking attitudes. Compared with the IVPF-ELECTRE method, Algorithm II can produce a more reasonable and persuasive result for ranking the order relationships of alternatives in the IVPF context. In summary, the proposed approach is very suitable for managing MCDA problems with uncertain information using PF/IVPF sets. In particular, the risk attitude-based score functions corresponding to various settings of the attitude 42
parameter can confer upon the risk attitudinal assignment models a certain flexibility and adaptability in addressing real-life MCDA problems.
Acknowledgments
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The author acknowledges the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The author is grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 105-2410-H182-007-MY3) and Chang Gung Memorial Hospital (BMRP 574 and CMRPD2F0202) during the study completion.
Conflict of interest statement
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The author declares that there is no conflict of interest regarding the publication of this paper.
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Appendix A: Comparative discussion of score functions
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A comparative analysis and discussions were conducted to validate the applicability and effectiveness of the risk attitude-based score functions for ranking PF/IVPF values. The numerical test examples were adopted from Garg [16, 17, 19], Liang et al. [25], Peng and Dai [33], and Zhang [50, 51]. The comparison results of the score functions (i.e., s( p) vs. rt ( p) and S ( p) vs. Rt ( p) ) within the PF/IVPF environments are separately summarized in the following two tables. In these comparative analyses, different t values consisting of t 1 (risk neutrality), t 0.5 (risk seeking), and t 2 (risk aversion) were designated during the computation process of rt ( p) and Rt ( p) .
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The comparison results of s( p) , r1 ( p) , r0.5 ( p) , and r2 ( p) in the PF context are presented in Table A.1. As sketched here, the existing score function s( p) cannot tell the difference between the magnitudes of p1 and p2, because s( p1 ) s( p2 ) in all numerical cases. According to the comparison rule provided in Definition 5, it is necessary to determine the accuracy functions when the score functions fail to distinguish between two PF values. Consider the case of p1 (0.7, 0.4) and
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Similarly, it can be obtained that r0.5 ( p1 ) r0.5 ( p2 ) and r2 ( p1 ) r2 ( p2 ) by means of the risk-seeking score function ( t 0.5 ) and the risk-averse score function ( t 2 ), respectively. These rankings are the same as the result yielded by the accuracy function a( p) . Note that a particular result of r1 ( p1 ) r0.5 ( p1 ) r2 ( p1 ) 0.7454 was
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obtained in the case adopted from Zhang [51]. More specifically, all of the rt ( p1 ) values are equal to 0.7454 for all t 0 in this case. The reason for the special result is the null degree of indeterminacy relative to p1 ( 5 3, 2 3) ; namely, P1 ( x) 0 . This will bring about rt ( p) (1 t (t 1) 02 )(( P ( x)) 2 1 (t 1) 02 ) P ( x)
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5 3 (i.e., 0.7454) for all t. Nevertheless, the risk attitude-based score function rt can still effectively distinguish between the PF values even in this special case. Considering all of the numerical examples in Table A.1, the comparison results indicate that the proposed rt ( p) can distinguish the difference in PF values when the existing score function s( p) does not work. Moreover, as opposed to the two-stage comparisons (i.e., simultaneously taking the score and accuracy functions), the proposed risk attitude-based score function is straightforward and effective. Table A.2 summarizes the comparison results of S ( p) , R1 ( p) , R0.5 ( p) , and
R2 ( p) in the IVPF context. As observed in the outcomes, one can clearly see that the existing score function S ( p) cannot well distinguish IVPF values in most numerical cases. An exception can be found in the case of p1 ([1 10,(1 4) 1 5],[ 3 10, 44
(1 4) 3 5]) and p2 ([0,(1 10) 3 2],[(1 10) 7 2,1 5]) (adopted from Garg [16]).
In this case, the results S ( p1 ) 0.0225 and S ( p2 ) 0.0300 were acquired; thus,
p1 p2 according to the comparison rule described in Definition 10. Meanwhile, the same ranking results were obtained using the proposed risk attitude-based score functions (i.e., R1 ( p1 )>R1 ( p2 ) , R0.5 ( p1 )>R0.5 ( p2 ) , and R2 ( p1 )>R2 ( p2 ) ). However, consider a similar example adapted from this case. If the PF values were modified as follows: p1 ([1 40,(1 4) 1 5],[ 3 10,(1 4) 3 5]) and p2 ([3 40,(1 10) 3 2], [(1 10) 7 2,1 5]) , then S ( p1 ) S ( p2 ) 0.0272 . It is unable to give the correct
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justification to p1 and p2 , because S ( p1 ) S ( p2 ) . Accordingly, the decision maker must continue to use the accuracy function for further comparisons. The results A( p1 ) 0.0403 and A( p2 ) 0.0478 were obtained; thus, p1 p2 . In contrast, with
p2 because R1 ( p1 ) 0.5030 and R1 ( p2 ) 0.5048 in a risk-neutral situation, R0.5 ( p1 ) 0.6630 and R0.5 ( p2 ) 0.6636 in a risk-seeking situation, and R2 ( p1 ) 0.3429 and R2 ( p2 ) 0.3460 in a risk-averse situation. This means that the ranking procedure using the proposed Rt ( p) is simpler and more effective than the current score function-based approach. Next, consider the two cases adopted from Garg [17]. From Table A.2, the same result of p1 ~ p2 was obtained regardless of whether the score function-based approach or the proposed risk attitude-based score function was used. Additionally, the accuracy functions of p1 and p2 are equal. To observe other comparison results in more-diverse instances, this paper extended Peng and Dai’s PF cases [33] to the IVPF environment and modified partial degrees within p1 and p2 , as shown in Table A.2. More specifically, the case of p1 ([0.5,0.6],[0.5,0.6]) and
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p2 ([0.3465,0.5],[0.4, 0.4]) as well as the case of p1 ([0.7,0.7],[ 0.3, 0.4]) and p2 ([0.5,0.5], [0.245,0.4]) was partially adapted from the case of
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p1 (0.7, 0.4) and p2 (0.5,0.4) . Due to the obtained results of S ( p1 ) S ( p2 ) in the three adapted cases, it can be concluded that the existing score function is unable to distinguish between p1 and p2 . In contrast, the developed Rt ( p) performs well in such cases because it can tell the difference between the magnitudes of p1 and p2 . Therefore, the comparative analysis demonstrates that the proposed risk attitude-based score function is capable of providing a relatively correct justification to the ranking of IVPF values during the decision-making process.
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Table A.1. Comparison results of the score functions within the PF environment. Results of r1 ( p) Results of s( p) Data source PF values r1 ( p1 ) 0.5408
r0.5 ( p1 ) 0.6591
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r0.5 ( p2 ) 0.6708
r2 ( p2 ) 0.5657
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r1 ( p1 ) 0.6124
r0.5 ( p1 ) 0.6972
r2 ( p1 ) 0.5270
s( p2 ) 0.0000
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r0.5 ( p2 ) 0.7040
r2 ( p2 ) 0.6072
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s( p1 ) 0.0900
r1 ( p1 ) 0.7177
r0.5 ( p1 ) 0.7367
r2 ( p1 ) 0.6986
s( p2 ) 0.0900
r1 ( p2 ) 0.6199
r0.5 ( p2 ) 0.7189
r2 ( p2 ) 0.5206
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
s( p1 ) 0.1100
r1 ( p1 ) 0.7204
r0.5 ( p1 ) 0.7428
r2 ( p1 ) 0.6979
s( p2 ) 0.1100
r1 ( p2 ) 0.6684
r0.5 ( p2 ) 0.7344
r2 ( p2 ) 0.6022
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
p1 ( 3 3, 2 3)
s( p1 ) 0.1111
r1 ( p1 ) 0.6573
r0.5 ( p1 ) 0.7324
r2 ( p1 ) 0.5821
p2 (2 3, 3 3)
s( p2 ) 0.1111
r1 ( p2 ) 0.7027
r0.5 ( p2 ) 0.7407
r2 ( p2 ) 0.6646
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
p1 ( 5 3, 2 3)
s( p1 ) 0.1111
r1 ( p1 ) 0.7454
r0.5 ( p1 ) 0.7454
r2 ( p1 ) 0.7454
p2 (2 3, 3 3)
s( p2 ) 0.1111
r1 ( p2 ) 0.7027
r0.5 ( p2 ) 0.7407
r2 ( p2 ) 0.6646
s( p1 ) s( p2 )
r1 ( p1 ) r1 ( p2 )
r0.5 ( p1 ) r0.5 ( p2 )
r2 ( p1 ) r2 ( p2 )
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p2 (0.5477, 0.6324)
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Results of r2 ( p)
s( p1 ) 0.1000
p1 (0.3162, 0.4472)
Garg [19]
Results of r0.5 ( p)
p2 (0.6, 0.5)
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p2 ([ 9 4, 10 4],[ 2 4, 5 4])
p1 ([0.2,0.4],[0.2,0.4]) p2 ([0.5,0.6],[0.5,0.6])
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Garg [16]
p1 ([ 7 4, 11 4],[1 4, 5 4])
p1 ([1 10,(1 4) 1 5],[ 3 10,(1 4) 3 5])
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p1 ([0,0.5],[0.1,0.7]) p2 ([0.3,0.4],[0.5,0.5])
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p2 ([0,(1 10) 3 2],[(1 10) 7 2,1 5]) Garg [17]
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p1 ([0.1,0.2],[0.4,0.5])
20 ,0.6])
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p2 ([0.1,0.2],[1
Case adapted from Garg [16]
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p1 ([1 40,(1 4) 1 5],[ 3 10,(1 4) 3 5])
p2 ([3 40,(1 10) 3 2],[(1 10) 7 2,1 5]) p1 ([0.5,0.6],[0.5,0.6])
p2 ([0.6,0.7],[0.6,0.7])
p1 ([0.6,0.7],[ 0.4, 0.4]) p2 ([0.3465,0.5],[0.4,0.4]) p1 ([0.7,0.7],[ 0.3, 0.4]) p2 ([0.5,0.5],[0.245,0.4])
Results of R1 ( p)
Results of R0.5 ( p)
Results of R2 ( p)
S ( p1 ) 0.3750
R1 ( p1 ) 0.7756
R0.5 ( p1 ) 0.8176
R2 ( p1 ) 0.7336
S ( p2 ) 0.3750 S ( p1 ) S ( p2 ) S ( p1 ) 0.0000 S ( p2 ) 0.0000 S ( p1 ) S ( p2 ) S ( p1 ) 0.0225 S ( p2 ) 0.0300
R1 ( p2 ) 0.7893 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5477 R1 ( p2 ) 0.6344 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5053 R1 ( p2 ) 0.5034
R0.5 ( p2 ) 0.8209 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6815 R0.5 ( p2 ) 0.7011 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6646 R0.5 ( p2 ) 0.6627
R2 ( p2 ) 0.7578 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.4137 R2 ( p2 ) 0.5674 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.3460 R2 ( p2 ) 0.3441
S ( p1 ) S ( p2 )
R1 ( p1 )>R1 ( p2 )
R0.5 ( p1 )>R0.5 ( p2 )
R2 ( p1 )>R2 ( p2 )
S ( p1 ) 0.1250 S ( p2 ) 0.1250 S ( p1 ) S ( p2 ) S ( p1 ) 0.1800 S ( p2 ) 0.1800
R1 ( p1 ) 0.5484 R1 ( p2 ) 0.5484 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5021 R1 ( p2 ) 0.5021
R0.5 ( p1 ) 0.6548 R0.5 ( p2 ) 0.6548 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6326 R0.5 ( p2 ) 0.6326
R2 ( p1 ) 0.4410 R2 ( p2 ) 0.4410 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.3702 R2 ( p2 ) 0.3702
S ( p1 ) S ( p2 ) S ( p1 ) 0.0272 S ( p2 ) 0.0272 S ( p1 ) S ( p2 ) S ( p1 ) 0.0000 S ( p2 ) 0.0000 S ( p1 ) S ( p2 )
R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.5030 R1 ( p2 ) 0.5048 R1 ( p1 ) R1 ( p2 ) R1 ( p1 ) 0.6344 R1 ( p2 ) 0.6801 R1 ( p1 ) R1 ( p2 )
R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.6630 R0.5 ( p2 ) 0.6636 R0.5 ( p1 ) R0.5 ( p2 ) R0.5 ( p1 ) 0.7011 R0.5 ( p2 ) 0.7062 R0.5 ( p1 ) R0.5 ( p2 )
R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.3429 R2 ( p2 ) 0.3460 R2 ( p1 ) R2 ( p2 ) R2 ( p1 ) 0.5674 R2 ( p2 ) 0.6538 R2 ( p1 ) R2 ( p2 )
S ( p1 ) 0.0250
R1 ( p1 ) 0.6839
R0.5 ( p1 ) 0.7142
R2 ( p1 ) 0.6534
S ( p2 ) 0.0250 S ( p1 ) S ( p2 ) S ( p1 ) 0.1400 S ( p2 ) 0.1400 S ( p1 ) S ( p2 )
R1 ( p2 ) 0.5871 R1 ( p1 )>R1 ( p2 ) R1 ( p1 ) 0.7242 R1 ( p2 ) 0.6226 R1 ( p1 )>R1 ( p2 )
R0.5 ( p2 ) 0.6971 R0.5 ( p1 )>R0.5 ( p2 ) R0.5 ( p1 ) 0.7516 R0.5 ( p2 ) 0.7296 R0.5 ( p1 )>R0.5 ( p2 )
R2 ( p2 ) 0.4767 R2 ( p1 )>R2 ( p2 ) R2 ( p1 ) 0.6967 R2 ( p2 ) 0.5154 R2 ( p1 )>R2 ( p2 )
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Table A.2. Comparison results of the score functions within the IVPF environment. Data source Results of S ( p) IVPF values
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Appendix B: Sensitivity analysis in the financing problem
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This appendix conducts a sensitivity analysis with respect to the risk attitudinal parameter t for the financing decision on working capital policies. Regarding diverse attitudes of risk seeking (in which the t vales were set as 0.000001, 0.25, 0.5, and 0.75), risk neutrality (i.e., t=1), and risk aversion (in which the t vales were set as 5, 10, 100, and 1000000), the results of the sensitivity analysis are presented in Table B.1, consisting of the criterion-wise precedence rankings and the overall precedence rankings in different settings of the parameter t.
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Table B.1. Computation results with different values of the parameter t. t value Criterion-wise precedence ranking Overall precedence ranking The setting of a risk-seeking attitude 0.000001 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 0.25 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 0.5 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 0.75 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z1 z4 z3 z2 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 The setting of a risk-neutral attitude 1 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z1 z5 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z5 z2 z1 c6: z3 z4 z2 z1 z5 The setting of a risk-averse attitude 5 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1 10 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1 100 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1 1000000 c1: z5 z4 z3 z2 z1 c4: z5 z4 z3 z2 z1 z3 z4 z2 z5 z1 c2: z5 z4 z3 z2 z1 c5: z1 z2 z3 z4 z5 c3: z3 z4 z2 z5 z1 c6: z3 z4 z2 z5 z1
Concerning the setting of a risk-seeking attitude, the same ultimate priority ranking, i.e., z1 z4 z3 z2 z5 , was acquired among the four cases. This result is evidently different from the overall ranking that was obtained based on the risk48
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neutral setting. Additionally, the ultimate priority ranking z3 z4 z2 z5 z1 was derived among the four risk-averse cases. It can be observed that this ranking is very similar to the risk-neutral result. In particular, the balanced aggressive and conservative policy (z3) is the best ranked in risk-neutral and risk-averse situations. In contrast, the best-ranked alternative is the aggressive dominant policy (z1) in a riskseeking situation. It is worthwhile to mention that these comparison results are intuitively reasonable and acceptable because the parameter t can more or less reflect the inner psychological characteristics of optimism and pessimism. As described in Remark 2, the settings of 0 t 1 , t 1 , and t 1 represent progressive, neutral, and conservative behaviors, respectively. Specifically, the cases of t=0.000001, 0.25, 0.5, and 0.75 can reflect the decision maker’s dispositional optimism in some way. Thus, it can be anticipated that the aggressive dominant policy is ranked the best in the four cases. Analogously, the result that the best choice is the balanced aggressive and conservative policy can be expected if the decision maker possesses a neutral attitude towards risk-taking. It is interesting to find that the balanced aggressive and conservative policy is also ranked the best in the cases of t=5, 10, 100, and 1000000. Nevertheless, the precedence relationships between z1 and z5 are different in the settings of risk neutrality and risk aversion, i.e., z1 z5 for the neutral case and z5 z1 for the four risk-averse cases. Such results are reasonable because the setting of t 1 reflects the decision maker’s dispositional pessimism. Accordingly, the conservative dominant policy generally performs better than the aggressive dominant policy for the decision maker with a risk-averse attitude. Based on the comparative discussions with respect to Table B.1, it can be concluded that the proposed outranking methodology can fully take into account the decision maker’s attitude towards risk-taking, and it can produce intuitively appealing and persuasive results in the face of high uncertainty.
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