Utility-based hybrid fuzzy axiomatic design and its application in supply chain finance decision making with credit risk assessments

Computers in Industry 114 (2020) 103144

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Utility-based hybrid fuzzy axiomatic design and its application in supply chain finance decision making with credit risk assessments Xingli Wu, Huchang Liao ∗ Business School, Sichuan University, Chengdu 610064, China

a r t i c l e

i n f o

Article history: Received 13 July 2019 Received in revised form 6 October 2019 Accepted 12 October 2019 Keywords: Multiple Criteria decision-making Supply chain risk management Fuzzy axiomatic design Ordinal aggregation Credit risk assessment

a b s t r a c t The fuzzy axiomatic design (FAD) is an attractive multiple criteria analysis approach due to its novelty in considering the functional requirement of each criterion. However, there are some challenges of the FAD approach in solving multiple experts multiple criteria decision-making problems with hybrid information. This paper dedicates to filling this research gap by introducing a utility-based hybrid FAD approach with an ordinal aggregation method. To do so, methods to calculate the possibility of achieving the functional requirements are firstly proposed to deal with five widely used evaluation expressions. In addition, a target-based weighted average operator is integrated with the FAD method so as to determine the whole ranking set of alternatives. Finally, given different evaluation standards used by different individuals, to avoid losing original opinions in information aggregation process, an ordinal aggregation method is developed to integrate the subordinate rank sets derived by individual experts into a collective one. An application to a credit risk assessment problem in supply chain finance is provided and the comparative analyses are conducted to highlight the developed method. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Multiple criteria decision-making (MCDM) aims to select the optimal solution or derive a ranking set from predetermined alternatives according to their performances on a set of criteria. This issue exists in various fields, such as engineering, manufacturing, management and economic. To deal with the MCDM problems, the performance of each alternative with respect to each criterion, which may be expressed as ordinal information or cardinal information, should be measured and aggregated (Kahraman & Cebi, 2009). The MCDM problems with ordinal information are constructed in special situations in which detailed evaluations are hard to obtain. Anyway, cardinal information is more accurate than ordinal information. There are mainly two categories of techniques to deal with cardinal evaluations (Liao et al., 2018b). The first one focuses on measuring the performance of each alternative by comparing with the ideal solution. There are three representativeness methods, including the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method (Lai et al., 1994; Celik et al., 2009b), the VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method (Opricovic & Tzeng, 2004; Abdel-Baset et al., 2019) and the MULTIMOORA (MULTIplicative Multi-Objective Optimization by Ratio Analysis) method (Brauers & Zavadskas, 2010). This type of approaches is easy to understand; however, the ideal solution, as a reference point, is invalid to describe alternatives’ merits if all alternatives perform good or bad under a criterion. The second category is the outranking method which focuss on portraying the relations between pairwise alternatives as preference, indifference and incomparable relations. The PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) (Behzadiana et al., 2010) and the ELECTRE (ELimination Et Choix Traduisant la REalité-ELimination and Choice Expressing the Reality) (Roy, 1968) are two representative outranking methods. Briefly speaking, the above-mentioned methods dedicate to comparing alternatives with either the ideal solution or other alternatives to score their performances. However, the functional requirement, as a minimal constraint of each criterion, is ignored by these methods. Functional requirement is fundamental in practical MCDM problems. For example, the candidates for recruiting are required to perform at least medium under each index such as experience, professional competence and communicative competence, or he will be screened directly.

∗ Corresponding author. E-mail addresses: [email protected] (W. Xingli), [email protected] (H. Liao). https://doi.org/10.1016/j.compind.2019.103144 0166-3615/© 2019 Elsevier B.V. All rights reserved.

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Fortunately, the functional requirement has been effectively addressed by the axiom design (AD) approach proposed by Suh (1990). Unlike the other MCDM methods, the AD depicts the performances of alternatives by comparing with the functional requirement of each criterion. The significant advantage of the AD approach is that the derived optimal solution meets the minimal constraints of all criteria (Büyüközkan & Göcer, 2017), and then, there is a low risk of failure in case we use the selected alternative. Considering the uncertain information in most MCDM problems, Kulak & Kahraman (2005a) enhanced the AD method by combining it with fuzzy set theory and named the new method as fuzzy AD (FAD). Since then, the FAD approach has attracted growing attention (Kulak et al., 2010; Büyüközkan & Göcer, 2017). However, because of the short time, there are still some research gaps on the FAD approach: (1) The problem on solving hybrid MCDM problems The MCDM problems usually contain not only quantitative criteria such as “price”, but also qualitative criteria such as “quality” (Özceylan et al., 2016). The values of quantitative criteria can be scored as crisp numbers or interval numbers while the values of qualitative criteria are hard to score in numerical numbers but can be represented in linguistic values directly (Ko, 2015). Therefore, the decision matrix is usually composed by hybrid information. The FAD approach has been extended by incorporating different linguistic representation models separately to solve qualitative MCDM problems (Akay et al., 2011; Büyüközkan & Göcer, 2017; Kahraman et al., 2018). However, they only can deal with the problems in which the evaluations are expressed in single linguistic form. As far as we know, only Cheng et al. (2017) solved the hybrid MCDM problems by combining the FAD with the TOPSIS method. But their integrated method loses the advantage of the FAD approach and consequently the selected alternative may not be able to achieve the functional requirements. The first goal of this study is to improve the FAD approach to tackle the hybrid MCDM problems. There are five commonly-used representation forms, including crisp numbers, interval numbers, single linguistic terms, adjacent linguistic terms and continuous interval-valued linguistic terms. The first two models are used to express the values of quantitative criteria. Linguistic evaluations are usually expressed as single linguistic terms for simplicity. In some special cases where there is hesitancy when making evaluations, we can use adjacent linguistic terms to express the uncertain opinion (Rodríguez et al., 2012). On the contrary, experts may tend to make precise linguistic evaluations which can be expressed as continuous interval-valued linguistic terms (Liao et al., 2018b). The five aforementioned models have different expression forms and semantics. Therefore, the method to determine the probability of achieving functional requirements are different when evaluations are expressed by different models. We can determine the probabilities for different situations with the help of graphic analysis for each model, which makes the method easy to use. In addition, the probabilities derived from different models can be uniformed to [0,1]. Then, the hybrid information can be addressed simultaneously in an aggregation process. (2) The problem on deriving the ranking of alternatives The FAD approach is effective to select the optimal alternative that has the biggest probability to achieve the functional requirements of all criteria, but it is limited in ranking all alternatives. In practical MCDM problems, we may obtain many alternatives which have the same ability of achieving the functional requirements by the FAD approach, but there may be significant differences among these alternatives in terms of comprehensive performances. In this sense, the real ranking of alternatives cannot be determined only by the FAD approach. Therefore, the second goal of our study is to further analyze the performance of alternatives and derive the full ranking set. The main idea is to rank alternatives according to their probabilities of achieving the functional requirements at first; then when two alternatives have the same probability, they should be ranked based on their comprehensive values. Multiple criteria utility theory is effective to derive the comprehensive performance by aggregating criterion values. There are mainly three aggregation types, including complete compensation, incomplete compensation and non-compensation (Liao & Wu, 2019). Since the alternatives can be classified by the FAD method according to their negative or positive performance under each criterion, we introduce a target-based weighted average operator which is fully compensatory to measure the comprehensive performances of alternatives. The benefit, cost and division criteria are considered, and both numerical numbers and linguistic terms can be handled by this operator. (3) The problem on solving MCDM problems with multiple experts Given the limitations of individuals’ experience and knowledge, a group of experts are usually invited to make evaluations for MCDM problems to enhance the reliability of final results. When solving multiple expert MCDM (MEMCDM) problems by the FAD approach, the experts’ evaluations were aggregated by weighted average operators (Celik et al., 2009b; Cebi & Kahraman, 2010; Kannan et al., 2015). In reality, numbers or words have different meanings to different experts since the experts have different standards in evaluations (Li et al., 2017). If we aggregate the experts’ opinions into collective ones directly, the actual meanings of the evaluations given by each individual expert may be misled. Hence, the last goal of this research is to improve the FAD approach to cope with the MEMCDM problems and make sure that not only the original opinions of each expert are retained in computation process but also the final results are satisfactory to most experts. To avoid the distortion of individualized evaluations in the aggregation process, we argue that the opinion of group experts can be trade-off by integrating the subordinate ranking sets derived from each expert. The difficulty is to integrate the ordinal information. The ORESTE (organísation, rangement et Synthèse de données relarionnelles, in French) method originated by Roubens (1982) is excellent to address the decision matrix composed by ordinal information. Its greatest feature is that it does not need accurate criterion weights (Liao et al., 2018a; Wu & Liao, 2018). In most practical situations, it is cumbersome to assign weights to experts, especially in the case of many experts, but is easy to order these experts. Given this, motivated by the aggregation operator in ORESTE, we develop an ordinal aggregation method to make trade-off between experts. In brief, this study dedicates to developing an integrated method to solve MEMCDM problems with hybrid information. Our idea is motivated from the FAD approach, the utility theory and the ORESTE method, but not limited to them. The proposed approach is named as

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Fig. 1. The publications involving the AD approach for solving MCDM problems.

the utility-based hybrid FAD approach with ORESTE, and is abbreviated as the Ulti-HFAD-ORESTE method. This paper can be highlighted by the following contributions: • We analyze how to determine the probabilities of achieving the functional requirements for different expression models. The results are normalized so as to address hybrid information. • We present a ranking method considering both the probability of achieving functional requirements and comprehensive performances of alternatives. In this way, the selected alternative not only dominants others overall but meets the requirements of decision-makers on each criterion. • We propose an ordinal aggregation technique to integrate the subordinate ranks obtained from each expert into the collective ranks. The process of aggregating experts’ evaluations is avoided. • Based on a developed credit risk criteria system, a case about selecting small and medium-sized enterprises in supply chain finance (SCF) on behalf of a commercial bank in China for investment is solved by the proposed method. Comparative analyses between the proposed method and the TOPSIS, VIKOR, MULTIMOORA, PROMETHEE and ELECTRE methods are provided. The remainder of this paper is formed as follows. Section 2 reviews the FAD method, the credit risk assessment in SCF, and the art of the FAD approach. The Ulti-HFAD-ORESTE method is developed in Section 3. Section 4 illustrates a case study of credit risk evaluation in SCF and highlights the proposed method through comparative analyses. The paper ends with some conclusions in Section 5. 2. Literature review The FAD approach is reviewed at first since it is the theoretical basis of this paper. Then, the credit risk assessment in SCF as the application base is surveyed. The art of the FAD approach is recalled for further presentation. 2.1. Survey on the developments and applications of the FAD approach The AD technique proposed by Suh (1990) aims to guide the process of product design or system design. It consists of two axioms. The Independence Axiom refers to the independence of functional requirements. That is to say, each functional requirement in a design solution can be satisfied without affecting the other functional requirements. The Information Axiom represents the degree to which the solution meets functional requirements. The greater the probability of successfully achieving functional requirements, the better the solution is. These two axioms can be applied in isolation from each other. The Independence Axiom is mainly employed in design domain, while the Information Axiom is a practical tool to solve general MCDM problems. Kulak et al. (2010) reviewed the literature on both axioms of the AD approach from 1990 to 2009 and concluded that the research on Information Axiom for MCDM problems would increase. Büyüközkan and Göcer (2017) summarized the developments and applications of the AD approach on MCDM from 2010 to 2014. From Web of Science, we explored 366 publications which are indexed by Science Citation Index Expanded or Social Sciences Citation Index when we use “axiomatic design” as the searching strategy, and explored 97 publications when we use “axiomatic design” and “decision making” as the searching strategy. After investigating these publications, we found that only 69 papers used the AD approach to solve MCDM problems and 5 papers were related to both Independence Axiom and Information Axiom. Despite that the researches on Information Axiom are far less than those on Independence Axiom, the Information Axiom has aroused growing concerns in terms of solving MCDM problems in recent years. This trend is shown clearly in Fig. 1. In this paper, we only focus on the Information Axiom of the AD approach. To incorporate imprecision in practical decision making, Kulak & Kahraman (2005a, 2005b) extended the AD method by combining it with fuzzy set theory (Zadeh, 1965). From then on, the qualitative MCDM problems can be solved by the FAD method, and the number of studies using AD technique in MCDM have been observably increased as shown in Fig. 1. To further enrich the applications, the FAD approach has been combined with fuzzy set extensions and new approaches have been proposed, such as the interval type-2 FAD approach (Akay et al., 2011), intuitionistic FAD approach (Büyüközkan & Göcer, 2017) and the trapezoidal intuitionistic FAD approach (Kahraman et al., 2018). Considering the weights of criteria, Kulak & Kahraman (2005b) modified the FAD approach and proposed the weighted FAD approach. However, the function to calculate the weighted information in this approach does not satisfy the monotonicity. To overcome this defect, some scholars used the weighted average formula to add the weights of criteria (Chen et al., 2016; Zheng, et al., 2017; Büyüközkan & Göcer, 2017). To solve the MCDM problems with hierarchical index system, Kahraman & Cebi (2009) introduced the hierarchical FAD approach by considering the weights of criteria at each level of the hierarchy. Given that different MCDM methods have different advantages, many

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Table 1 The enhanced FAD approaches from theoretical level. Ref.

Integrated method

How to improve the FAD approach

Seiti et al. (2018) Kahraman et al. (2018)

– Trapezoidal intuitionistic fuzzy set

Deng and Jiang (2018) Zheng et al. (2017) Büyüközkan & Göcer (2017) Cheng et al. (2017)

Evidence theory Rough set Intuitionistic fuzzy set TOPSIS

C¸akır (2016) Chen et al. (2015)

SMART –

Büyüközkan et al. (2012) Celik et al. (2009a) Kahraman & Cebi (2009)

Fuzzy AHP Fuzzy QFD and fuzzy AHP –

Considering both optimistic and pessimistic risks in evaluations Extend the FAD approach into trapezoidal intuitionistic fuzzy environment by a defuzzification process Develop an evidential FAD approach Combine rough set with the FAD approach Extend the FAD approach into intuitionistic fuzzy environment Develop a heterogeneous FAD approach to handle benefit, cost and deviation criteria Use the SMART method to determine the weights of criteria in the FAD approach Extend the FAD approach into hybrid environment by considering both the fuzzy and random properties of criteria Propose a two-phase process by the FAD approach Introduce the FAD approach into the framework of fuzzy QFD Develop a hierarchical FAD approach by considering more than one level of criteria

Fig. 2. Distribution of the application fields of the FAD approach.

researchers integrated the FAD method with other MCDM techniques to improve its effectiveness. Most of these papers were concentrated on combining the FAD method with the analytic hierarchy process (AHP) in which the AHP was used to determine the weights of criteria (Cebi & Kahraman, 2010; Büyüközkan et al., 2012; Büyüközkan & Göcer, 2017; Büyüközkan et al., 2017). C¸akır (2016) combined the FAD approach with the fuzzy SMART (Simple Multi Attribute Rating Technique) method in which the SMART method was used to calculate the criteria weights while the FAD method was used to rank alternatives. Celik et al. (2009a) introduced the FAD method into the framework of quality function deployment (QFD) to select the best alternatives. The papers that aim to improve the FAD approach from theoretical level are summarized in Table 1. The literature of the FAD approach shows various application areas on MCDM problems, which can be visualized in Fig. 2. The papers whose impact factors of the journals are lower than 1 are omitted here. From the above literature review, we can draw the following conclusions: (1) The FAD approach improved the theory and applications of the traditional AD approach. Since the FAD approach was proposed, the literature on applying this method to MCDM problems increased significantly. (2) The existing literature mainly focused on the applications of the FAD approach. Only few papers enhanced this method from the theoretical level. (3) The FAD approach is effective for solving complex MCDM problems with large numbers of criteria and alternatives. It solved many practical problems successfully, such as matching demanders and suppliers (Chen et al., 2016), constructing approval mechanism for ship design project (Cebi et al., 2010), selecting material handling equipment (Kulak, 2005) and model selection in ship management (Celik & Er, 2009). 2.2. Survey on the credit risk assessment of small and medium-sized enterprises in SCF Small and medium-sized enterprises are small in scale, poor in market competitiveness and risk resistance. They have narrow channels ¨ ¨ difficulty(Lekkakos & Serrano, 2016). The SCF is a new financing for raising funds. Their developments are severely restricted by financing mode which can help the small and medium-sized enterprises get out of this problem. The supply chain is regarded as an organic whole chain for developing financing services by combining financial institutions with small and medium-sized enterprises and the core enterprises (Zhang, 2016). The credit business of the small and medium-sized enterprises on the chain is carried out based on the operations of the supply chain, while traditional credit concerns the scale, credit level and pledge assets of individual enterprises (Liu et al., 2015). There

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Table 2 The advantages of the SCF compared with the traditional finance. Property

SCF

Traditional finance

Financing object Financing mode Relationship between bank and small and medium-sized enterprises Service efficiency Service effect

Small and medium-sized enterprises on supply chain Open credit for the whole supply chain Long-term partnership

Individual enterprise Single enterprise credit Debtor-creditor relationship

High Solve the capital demand in supply and marketing process, and enhance the effectiveness of supply chain

Low Alleviate the financial difficulties of individual enterprise

¨ ¨ the small and medium-sized enterprises compared with the traditional are some advantages of the SCF for solving financing difficultyof finance (Wuttke et al., 2016; Zhu et al., 2017), which can be illustrated in Table 2. Although the cash flow of real transaction volume as a guarantee of repayment is conducive to the prevention and control of credit risk, the SCF itself still has risks, such as moral risk, supervision and realizable risk of collateral, and supply chain products’ legal and operational risk. With the development of SCF, a large number of breach and fraud cases occurred, which brought huge losses to commercial banks (Zhang et al., 2015). The identification and evaluation of credit risks in SCF have become key tasks for risk management and control before financing. It is critical for commercial banks to determine whether to finance a small and medium-sized enterprise in a supply chain or not and which company should give priority to finance according to the credit risks in SCF. Zhang et al. (2015) applied the support vector machine method and backpropagation (BP) neural network, respectively, to assess the credit risks in SCF. Considering both quantitative and qualitative criteria, Zhao et al. (2018) presented a risk evaluation system and utilized the AHP method to determine the comprehensive risk to finance to small and medium-sized enterprises in online SCF. Given the fuzziness of evaluations, Chen (2012) applied the fuzzy ordinal regression support vector machine to assess the credit risks of small and medium-sized enterprises in SCF. Zhang (2016) solved an MCDM problem concerning selecting an appropriate small and medium-sized enterprise in a supply chain to finance considering credit risks with the evaluations being expressed in intuitionistic fuzzy numbers. In summary, the existed literature mainly focused on assessing credit risk intensity of a single small and medium-sized enterprise in SCF. However, without comparative analysis among the small and medium-sized enterprises, the results are limited in terms of objectivity and reliability. The MCDM method is effective to cope with the risk evaluation problems since the small and medium-sized enterprises can be compared with others to highlight their performances under several risk factors. Despite that Zhang (2016) proposed an MCDM method to address this problem, the evaluation expression model used by Zhang (2016) did not conform to people’s cognition and expression habits. 2.3. The art of the FAD approach There are two fundamental concepts in the FAD approach (Kulak & Kahraman, 2005a, 2005b): “design range” and “system range”. The “design range” refers to the acceptable range of a design (criterion), and it is determined by experts according to their perceptions. The “system range” refers to the real feature of a possible design system, and it corresponds to the performance of an alternative. The acceptable solution exists at the intersection of “design range” and “system range”. This overlap is called “common range”. Fig. 5 shows the relationships among these concepts under the fuzzy linguistic environment. The probability of achieving the functional requirements can be described by the ratio of “common range” and “system range”, which is shown as: pi =

common range system range

(1)

The information content, Ii , is defined based on the probability of success. The FAD approach states that the design with the minimum information content should be the best design. The entropy function can measure the information in terms of uncertainty. Given that Suh’s entropy (Suh, 1990) does not require the summation of probabilities as 1, it is a suitable technique to measure the information content Ii based on the probability pi in the FAD approach as: Ii = log2

1

(2)

pi

Considering the different importance of criteria, Kulak & Kahraman (2005b) modified the information content as:

⎧ 1/wj ⎪ , if 0 ≤ log2 (1/pij ) < 1 ⎪ ⎨ log2 (1/pij )  w ˜Iij = log2 (1/pij ) j , if log2 (1/pij ) > 1 ⎪ ⎪ ⎩ wj ,

(3)

if log2 (1/pij ) = 1

where wj is the weight of criterion cj , pij is the probability of achieving the functional requirement of criterion cj for alternative ai . There is a problem that the function of Eq. (3) is not monotonic. That is to say, if 0 ≤ log2 (1/pij ) < 1, log2 (1/ptj ) > 1 and log2 (1/pkj ) = 1, the equation ˜Itj > ˜Ikj > ˜Iij is not established. For example, let log2 (1/pij ) = 0.9, log2 (1/ptj ) = 1.1, log2 (1/pkj ) = 1 and wj = 0.5, by Eq. (3), we obtain ˜Iij = 0.81, ˜Itj = 1.05 and ˜Ikj = 0.5. In this case, the original information is misled by Eq. (3). To avoid this drawback, the weighted

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mean operator was applied to calculate the weighted information content by some researchers (Chen et al., 2016, Zheng, et al., 2017, Büyüközkan & Göcer, 2017). This is a simple but effective operator, shown as:



Iij = wj log2 1/pij



(4)

The collective information content of alternative ai with respect to n criteria is obtained as: ICi =

n

Iij

(5)

j=1

The alternatives are ranked in ascending order of the collective information contents. If alternative ai cannot achieve the “design range” on criterion cj at all, that is to say, the common range on criterion cj equals to 0, then, Iij = ∞ and ICi = ∞. In this case, we cannot select ai as an excellent alternative even though ai can achieve the “design range” of other criteria completely. In other words, the excellent alternative must meet the minimum functional requirements with respect to all criteria. This is in line with the actual demands for MCDM problems, and is a unique feature of the FAD approach compared with other MCDM methods. 3. The framework of the utility value-based hybrid FAD approach with ORESTE method In this section, a detail description about the MEMCDM problems with hybrid information is presented first. Then, the presentation of the Ulti-HFAD-ORESTE method is given. Finally, we provide the procedure of the proposed method. 3.1. Description of the MEMCDM problems with hybrid information A general MEMCDM problem consists of a finite set of alternatives, A = {a1 , · · ·ai , · · ·, am }, and a set of criteria, C = {c1 , · · ·cj , · · ·, cn },

T

with a weight vector W = w1 , · · ·wj , · · ·, wn . A group of experts, E = {e1 , · · ·eq , · · ·, eQ }, are invited to make assessments toward these alternatives with respect to the criteria. The values of deterministic quantitative criteria are measured as crisp numbers or interval numbers and thus can be gathered without any subjective evaluations. The values of random quantitative criteria are judged by experts and usually expressed as numerical numbers. Experts prefer to evaluate the qualitative criteria in linguistic terms which conform to their cognitive perceptions and expressing habits (Wu and Liao, 2019). (q) (q) (q) Each expert, eq , needs to determine the functional requirement (design range) of each criterion, denoted as DR = {dr1 , · · ·drj , · · ·, drn }, (q)

and the performance of each alternative (system range), srij , with respect to the random quantitative criteria or qualitative criteria. The “design range” has the same expression type as the “system range” corresponding to each individual. That is to say, if one expert evaluates the “design range” of a criterion as numerical number (or linguistic term), then, the “system ranges” of alternatives on this (q) criterion are expressed as numerical numbers (or linguistic terms). The individual decision matrix, IDm×n , can be established by all (q)

srij (i = 1, 2, · · ·, m and j = 1, 2, · · ·, n) as:

⎡

(q)

sr11

(q)

sr12

···

(q)

sr1n

⎤

⎢ (q) ⎥ (q) (q) ⎢ sr21 sr22 · · · sr2n ⎥ ⎢ ⎥ (q) IDm×n = ⎢ ⎥ .. . ⎢ .. .. .. ⎥ ⎣ . ⎦ . . (q)

srm1

(q)

srm2

···

(6)

(q)

srmn

3.2. The Ulti-HFAD-ORESTE method This section proposes the Ulti-HFAD-ORESTE method to make up the limitations of the FAD approach for solving MEMCDM problems with hybrid information. First, methods are introduced to calculate the probability of achieving functional requirements under different evaluation context. In addition, a target-based weighted average operator is proposed and added to the FAD approach to calculate the comprehensive performances of alternatives. Finally, an ordinal aggregation method is introduced to integrate subordinate ranks considering different importance of experts. 3.2.1. The hybrid FAD approach with distinct information representation forms The FAD approach only can solve the evaluations expressed as the trigonometric fuzzy numbers (Kulak & Kahraman, 2005a, 2005b) or single linguistic terms (Kahraman & Cebi, 2009) whose membership functions were predefined. In addition, little research addressed how to determine the “common range” and how to calculate the probability of achieving functional requirements. This part aims to fill this gap (q) by analyzing five widely-used evaluation tools, respectively. Methods to determine pij are introduced and the hybrid FAD approach is presented. Case 1.

: The “system ranges” are crisp numbers, i.e.,

(q) pij



=

(q)

(q)

∈ drj

1,

if srij

0,

if srij ∈ / drj

(q)

(q)

(7)

Fig. 3 illustrates the relationships among “design range”, “system range” and “common range” when the “system ranges” are represented (q) (q) (q) (q) (q) by crisp numbers. In Fig. 3, drj = [I − , I + ], x1 ∈ [I − , I + ] and x2 ∈ / [I − , I + ]. If srij = x1 , then, pij = 1; if srij = x2 , then, pij = 0.

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Fig. 3. The area of “common range” when “system ranges” are crisp numbers.

Fig. 4. The area of “common range” when “system range” is an interval number.

Fig. 5. The area of “common range” when “system range” is single linguistic term.

Case 2.

: The “system ranges” are interval numbers, i.e.,

(q)

pij =

⎧ 0, if x+ ≤ I − or x− ≥ I + ⎪ ⎪ ⎪ ⎪ + − ⎪ ⎨ x − I , if x+ > I − and x− < I − + − x −x

⎪ ⎪ I + − x− ⎪ , if x− < I + and x+ > I + ⎪ ⎪ ⎩ x+ − x− − − + + 1,

where

(q) drj

Case 3.

(8)

if x ≥ I and x ≤ I (q)

= [I − , I + ] and srij = [x− , x+ ]. Fig. 4 shows the case when “system range” is an interval number.

: The “system ranges” are singleton linguistic terms

It is flexible for experts to express their evaluations on qualitative criteria in linguistic terms rather than numerical numbers (Wu et al., 2018). To make linguistic evaluations, we should predefine a linguistic term set (LTS) which is a set of possible linguistic terms of a linguistic variable. The uniform and symmetrical LTS, denoted as S = {s− , · · ·, s0 , · · ·, s }, is useful and widely-used where  is a positive integer. Here we let  = 3 and S = {s−3 = vey bad, s−2 = bad,s−1 = a little bad, s0 = medium, s1 = a little good, s2 = good, s3 = vey good} be an LTS. Suppose that the “design range” is “at least a little good”, and it is illustrated as the area encircled by blue lines while the “system

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Fig. 6. The area of “common range” when “system ranges” are two adjacent linguistic terms.

Fig. 7. The area of “common range” when “system range” is a continuous interval-valued linguistic term.

range” is illustrated as the area encircled by bold lines in Figs. 5–7 . The overlap between the “system range” and the “design range” is the “common range”. If the evaluation of “system ranges” are expressed as singleton linguistic terms (Liao et al., 2018b), there are three situations about the (q) relationships among “design range”, “system range” and “common range”, which can be illustrated by Fig. 5. The value of pij in different situations can be represented as:

(q)

pij =

⎧ ⎪ ⎨ 1,

(q)

(q)

if srij ⊆ drj

(q) (q) (q) (q) / ∅ and srij ⊂ / drj 1/4, if srij ∪ drj = ⎪ ⎩ (q) (q)

0,

if srij ∪ drj

(9)

=∅

(q)

(q)

If the “system range” is “very bad”, “bad” or “a little bad”, then the “common range” is zero and there is srij ∪ drj

(q)

= ∅. In this case, pij = 0

as illustrated in Fig. 5(a). If the “system range” is “medium”, then there is (the area of common range)/(the area of system range) = 1/4, (q) (q) (q) (q) (q) and this is the only case that srij ∪ drj = / ∅ and srij ⊂ / drj . In this case, pij = 1/4 as stated in Fig. 5(b). If the “system range” is “a little (q)

(q)

good”, “good” or “very good”, then the “common range” is the same as the “system range”. There is srij ⊆ drj This case is illustrated by Fig. 5(c). Case 4.

: The “system ranges” are two adjacent linguistic terms

(q)

and we can obtain pij = 1.

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

9

Due to the complex of the objective things, experts may hesitate among more than one adjacent linguistic term when making assessments (Rodríguez et al., 2012). For example, one may deem the reliability of an equipment is “between a little high and high”. To describe this kind of complex linguistic evaluations, Rodríguez et al. (2012) presented a text free grammar to generate the possible linguistic expressions and introduced the semantics of these expressions. Fig. 6 illustrates the possible situations of the “common range” when the (q) evaluations on “system range” are composed by two adjacent linguistic terms. Fig. 6(a) shows that srij = between bad and a little bad; (q)

(q)

in this case, srij ∪ drj (q) srij (q) srij

(q)

= between a little bad and medium and (q) drj

(q)

(q)

= ∅ then pij = 0. Fig. 6(b) and Fig. 6(c) show the situation that srij ∪ drj (q) pij

= 1/2 with

(q) srij

(q) srij

(q)

(q)

(q)

= / ∅ and srij ⊂ / drj . pij = 1/8 with

= between medium and a little good. Fig. 6(d) shows the situation that (q)

⊆ and = between a little good and good. In this case, pij = 1. Despite that not all possible values of “system range” and “design range” are listed in Fig. 6 for a linguistic variable, all possi(q) (q) bility values of pij are analyzed when the “system range” is expressed as two adjacent linguistic terms. We observe that if srij = (q)

(q)

between very bad and bad, then it belongs to the first scenario and we have pij = 0; if srij = between good and very good, then it belongs (q)

to the last scenario and we have pij = 1. In the cases that the “system range” is expressed by more than two adjacent linguistic terms, the (q)

number of possibility values pij increases. We can analyze these situations in a similar way as the situations that the “system ranges” are expressed as singleton linguistic terms or two adjacent linguistic terms. Case 5.

: The “system ranges” are continuous interval-valued linguistic terms

In some cases, experts may have deep understanding for alternatives with rich knowledge and experience. Then, they may give precise linguistic evaluations that the possible values of a linguistic variable are subordinate intervals of the interval LTS S = [s− , s ] (Liao et al., 2018b). For example, one may evaluate the quality of a product as “30 % proportion higher than medium and 20 % proportion lower than a little good”. This evaluation can be expressed as the continuous interval-valued linguistic element [s0.3 , s0.8 ]. Fig. 7 illustrates the “common range” of the continuous interval-valued linguistic evaluation [s0.3 , s0.8 ]. Unlike the situations illustrated in Fig. 5 and Fig. 6, it is not easy to determine the area of the “common range” directly when “system range” is expressed as the continuous interval-valued linguistic term. We need to calculate the area of the overlapping part based on the membership functions of the linguistic terms. In Fig. 7, the transverse coordinates which represent the semantics of the corresponding linguistic terms are between 0 and 1. Generally, let S = {s− , · · ·, s0 , · · ·, s } be an LTS. We can calculate the semantic of a linguistic term by (Liao et al., 2018b): x(s˛ ) = ( + ˛)/2

(10)

where ˛ is the subscript of the linguistic term s˛ that s˛ ∈ S. Example 1. Let  = 3, by Eq. (10), we obtain x(s0.3 ) = 3.3/6 and x(s0.8 ) = 3.8/6. Then, the function of the line L1 in Fig. 7 is u1 (x) = 6(x − 3/6) of the two lines is (0.65, 0.9). Then, the area and the function of the line L2 is u2 (x) = -6(x − 3.8/6)+1. The coordinate of the intersection

4.8 3 1 3.8 3.3 2.3 of the “common range” is 12 × ( 4.8 − ) × 0.9 = 0.135. The area of the “system range” is × − + − × 1 = 0.25. Thus, 6 6 2 6 6 6 6 (q)

pij = 0.135/0.25 = 0.54. The work of this part can be summarized as follows: (q)

(1) This part presents the techniques to determine the value of pij when the “system range” is expressed as crisp number, interval number, singleton linguistic term, adjacent linguistic terms and continuous interval-valued linguistic term, respectively. These are five widely-used information representation forms. Therefore, experts can be flexible to express their opinions when using the hybrid FAD approach. (q) (2) We can find that the number of possible values of pij is small, especially when the “system range” is represented by crisp number, (q)

singleton linguistic term or two adjacent linguistic terms. By Eqs. (4) and (5), we obtain that if pij = 0 (j ∈ {1, 2, · · ·, n}), then ICi = ∞; if (q)

pij = 1 (j = 1, 2, · · ·, n), then ICi = 0. We may obtain many alternatives whose collective information contents are ∞ or 0 for an MCDM problem by the hybrid FAD approach. Therefore, we introduce a method in the next part to analyze the comprehensive performance of alternatives and derive the ranking of them. 3.2.2. Determining subordinate ranks with the target-based weighted average operator This part aims to solve the problems that how to determine the comprehensive performance of alternatives and how to combine the collective information content determined by FAD method with the collective information content to derive the ranking set. There are mainly three kinds of aggregation operators to derive the comprehensive performance of alternatives (Wu et al., 2018). The first one is the weighted maximum value-based operator which captures the worst performance of an alternative under all criteria. The second one is the weighted geometric operator which has an incomplete compensability that the bad performance under some criteria cannot be fully compensated by the good performance under other criteria. This kind of aggregation operator pays attention to the poor side of alternatives. The weighted average operator is a completely compensatory operator that the bad performances of an alternative under some criteria can be completely compensated by the good performances under other criteria. It is observed that the bad performances of alternatives are captured by the hybrid FAD approach comprehensively. The alternative which can successfully achieve the functional requirements of all criteria has a high rank, while the alternative which cannot meet the functional requirement of one criterion achieves a low rank. That is, with the help of the FAD method, the compensatory efferent among criteria have not a negative influence on aggregation results. Therefore, it is appropriate to depict the comprehensive performances of

10

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

alternatives by the completely compensatory operator. Motivated by the target-based aggregation operator (Hafezalkotob & Hafezalkotob, 2015), we introduce the target-based weighted average operator to calculate the utility values, shown as:

⎛

⎞⎞    (q) (q)  srij − taj  ⎟  ⎟ ⎠⎠ i = 1, 2, · · ·, m  (q) (q)  max srij − taj 

⎛

⎜ ⎜ ⎝wj ⎝1 − n

(q)

Ui

=

j=1

(q)

where Ui

(q)

to eq . taj (q)

1) taj 2)

(q) taj

= =

(11)

i=1,2,···,m

(q)

is the utility value of alternative ai corresponding to eq , srij is the system range of alternative ai on criterion cj corresponding

is a target value of criterion cj and can be determined by (q)

max srj

i=1,2,···,m

(q) min srj i=1,2,···,m

if cj is a benefit criterion; if cj is a cost criterion;

(q)

3) taj is a medium value which can be determined by experts if cj is a deviation criterion whose best value is neither the maximum value nor the minimum value. For example, the ideal “temperature” is not high or low but between 36◦ and 37◦ .

It is observed that

(q) taj

may not equal to

(q) drj .

1−

 (q) (q)  srij −taj    represents the normalized value of srij(q) under criterion cj . The  max srij(q) −ta(q) j

i=1,2,···,m (q)

alternatives can be ranked in descending order of Ui . (q)

(q)

To calculate srij − taj , we express the values of “system range” and “design range” in interval forms. The numerical value can be

expressed as the interval [x− , x+ ], especially x− = x+ if the “system range” is a crisp number. The linguistic evaluation can be expressed as the interval [s˛− , s˛+ ] where ˛+ , ˛− ∈ [−, ]. Especially, if the “system range” is expressed as several adjacent linguistic terms such − + − + , s˛ ] that s˛ = s˛ and s˛ = s˛+v . as {s˛ , s˛+1 , · · ·, s˛+v } where s˛ , s˛+1 · · ·, s˛+v ∈ {s− , · · ·, s }, then it can be represented as the interval [s˛ [s˛ , s˛+v ] is the envelope of {s˛ , s˛+1 , · · ·, s˛+v } (Rodríguez et al., 2012). We denote the target “system range” as [x∗− , x∗+ ] for numerical ∗− ∗+ , s˛ ] for linguistic evaluations. Therefore, based on the distance measure of interval numbers (Liao et al., 2018b), the numbers and [s˛ (q) (q) operation of srij − taj can be unified as: (q)

(q)

srij − taj

⎧ ⎪ ⎪ ⎨

= 1 ∗(q)+

2 max (xj

(q)−

− xij

)

     ∗(q)− (q)−   ∗(q)+ (q)+  (q) (q)− (q)+ (q) ∗(q)− ∗(q)+ − xij  + xj − xij  , if srij = [xij , xij ] and taj = [xj , xj ] xj

(12)

    ⎪  ∗(q)+ ⎪ (q)−  (q)+  (q) ij(q)− ij(q)+ (q) ∗j(q)− ∗j(q)+ ⎩ 1 ˛∗(q)− − ˛ + ˛ − ˛    , if srij = [s˛ , s˛ ] and taj = [s˛ , s˛ ] j ij j ij 2 i=1,2,···,n

∗(q)−

(q)−

∗(q)+

(q)+

∗j(q)−

ij(q)−

∗j(q)+

ij(q)+

, ˛ij , ˛j and ˛ij are the subscripts of the linguistic terms s˛ , s˛ , s˛ and s˛ , respectively, and they belong to where ˛j [−, ]. (q) (q) Based on the collective information content ICi and the utility value Ui , the ranking principle to derive the subordinate rank set, R(q) = {r (q) (a1 ), r (q) (a2 ), · · ·, r (q) (am )}, of all alternatives associated to each expert can be presented as: (q) (q) 1) if ICi > ICk , then, r (q) (ai ) > r (q) (ak ) ; (q)

2) if ICi (q)

(q)

= ICk , then, (q)

-if Ui

> Uk , then, r (q) (ai ) < r (q) (ak );

-if

< Uk , then, r (q) (ai ) > r (q) (ak );

-if

(q) Ui (q) Ui

(q)

=

(q) Uk ,

then, r (q) (ai ) = r (q) (ak ).

3.2.3. Derive the group ranking result by the ordinal aggregation method This section introduces an ordinal aggregation method to integrate the subordinate rank sets of all experts into a collective one. The ORESTE method only requires the initial rankings of alternatives under each criterion and the initial rankings of criteria to rank alternatives. The core of the ORESTE is to integrate the initial ranks of alternatives determined by their performance under each criterion and the initial ranks of criteria determined by their importance into global initial ranks by a weighted Euclidean distance formula. Comparing with three well-known MCDM methods including TOPSIS, VIKOR and ELECTRE, Liao et al. (2018a) highlighted the superiority of the ORESTE method concerning the visualization, the simple calculation process and the wide application scope. Due to the difference of knowledge and experience, experts play different importance on an MEMCDM problem and the subordinate rank sets of alternatives derived by the experts may be different. However, it is difficult to assign a precise weight to an expert according to his knowledge and experience, which is subjective and hard to measure but it is easy to rank them. Therefore, we need to aggregate Q subordinate ranking sets R(q) = {r (q) (a1 ), r (q) (a2 ), · · ·, r (q) (am )} (q = 1, 2, · · ·, Q ) into a collective one considering the importance rank set of experts IR = {r(e1 ), r(e2 ), · · ·, r(eQ )}. This is an ordinal-based MCDM problem that we can deem each expert as a criterion. It coincides with the conditions of the ORESTE method. Below, we present an ordinal aggregation method to integrate Q subordinate rank sets into a collective one based on the idea of the ORESTE method:

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

11

Fig. 8. Visualization of the Ulti-HFAD-ORESTE method.

Calculate the global preference score of each alternative with respect to each expert by a weighted Euclidean distance formula as Eq. (13). gp(q) (ai ) =



2

ς r (q) (ai )



2

+ (1 − ) r(eq )

(13)

where ς is a parameter, implying the relative importance between r (q) (ai ) and r(eq ) for calculating the global preference score. Without loss of generality, let ς = 0.5. The global ranking, gr (q) (ai ), of each alternative with respect to each expert can be determined as: (a) gr (q) (ai ) > gr (q) (ak ), if gp(q) (ai ) > gp(q) (ak ); (b) gr (q) (ai ) = gr (q) (ak ), if gp(q) (ai ) = gp(q) (ak ). Note. The rule of Besson’s mean rank (Roubens, 1982) is utilized here. If ai ranks at the u th position, then the rank of ai is r(ai ) = u ; if both ai and ak rank at the u th position, then their ranks are r(ai ) = r(ak ) = (u + (u + 1))/2. Aggregate the global ranks of each alternative associated to all experts by GR(ai ) =

Q

gr (q) (ai )

(14)

q=1

The collective rank set R = {r(a1 ), r(a2 ), · · ·, r(am )} can be obtained in ascending order of GR(ai ) (i = 1, 2, · · ·, m). The subordinate rank set can reflect each expert’s original opinions. The importance of experts is considered by Eq. (13) and all experts’ subordinate rank sets are integrated by Eq. (14). Therefore, the collective rank set derived by the ordinal aggregation method can be satisfied by most experts. 3.3. Procedure of the Ulti-HFAD-ORESTE method There are three parts of the proposed method. We first use the hybrid FAD approach to distinguish the alternatives according to their ability of achieving the functional requirement of each criterion. In this way, the risk is low to make decision since the best alternative derived by the hybrid FAD method performs not bad under each criterion. Then, a target-based weighted average operator is utilized to calculate the utility values of alternatives. The comprehensive performance of an alternative can be derived by this technique. Considering both the collective information contents and the utility values, we can obtain the subordinate rank sets of alternatives, which are then integrated into a collective one by the ordinal aggregation technique method considering the preference order of experts. For the convenience of understanding and application, we present the procedure of the proposed Ulti-HFAD-ORESTE method below and further illustrate it in Fig. 8 intuitively. Step 1. (Generate the MEMCDM problem) Determine the candidate alternatives, ai (i = 1, 2, · · ·, m). Define the evaluation criteria, cj (j = 1, 2, · · ·, n), and determine their quality of deterministic quantitative, random quantitative or qualitative, and their form of benefit, cost or deviation. Then, invite experts, eq (q = 1, 2, · · ·, Q ), to evaluate the performance of each alternative with respect to each quantitative (q)−

criterion, expressed as [xij

(q)+

, xij

ij(q)−

], and each qualitative criterion, expressed as [s˛

ij(q)+

, s˛

]. The experts should determine the “design

12

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

Table 3 The criteria system for credit risk evaluation on small and medium-sized enterprises in SCF. First level criterion

Second level criterion

Property

Form

Reference

Operation conditions of small and medium-sized enterprises Credit status of small and medium-sized enterprises

Management ability c1 Profit ability c2 Debt paying ability c3 Credit rating c4 Industry growth rate c5 Degree of information sharing c6 Cooperation level c7 Debt paying ability c8 Credit rating c9

Qualitative Qualitative Qualitative Random quantitative Deterministic quantitative Qualitative Qualitative Qualitative Random quantitative

Benefit Benefit Benefit Benefit Benefit Benefit Benefit Benefit Benefit

Zhang et al. (2015) Chen (2012) Zhang et al. (2015) Zhang et al. (2015) Chen (2012) Zhao et al. (2016) Zhao et al. (2016) Zhang et al. (2015) Zhang et al. (2015)

Quality of the supply chain Credit status of leading enterprise

∗(q)−

range” of each quantitative criterion in the form of [xj

∗(q)+

, xj

∗i(q)−

] and each qualitative criterion in the form of [s˛

∗i(q)+

, s˛

] . Finally, the

(q)

individual decision matrix, IDm×n , is constructed corresponding to each expert based on their evaluations. (q)

Step 2. (Apply the hybrid FAD approach) Compute the probability pij of achieving the functional requirement of each criterion by Eq. (q)

(q)

(1) according to different expression forms of evaluations. Calculate the weighted information content Iij by Eq. (4). Then, integrate Iij (q)

(j = 1, 2, · · ·, n) to collective information content ICi

for each alternative by Eq. (5). (q)

Step 3. (Use the target-based weighted average operator) Determine the target value of each criterion, taj . Calculate the distance (q)

(q)

between “system range” srij and the target value of each criterion taj

(q)

by Eq. (12); compute the utility value Ui (q) ICi

of each alternative by

(q) Ui

Eq. (11). Determine the subordinate rank set R(q) = {r (q) (a1 ), r (q) (a2 ), · · ·, r (q) (am )} considering both and according to the ranking principle proposed in Sect. 3.2.2. Step 4. (Use the ordinal aggregation method) Calculate the global preference score of each alternative with respect to each expert by Eq. (13). Then compute the global ranks by Eq. (14) and determine the collective rank set R = {r(a1 ), r(a2 ), · · ·, r(am )}. Output the rank set and end the procedure. Managerial implications: We should point out that the proposed MEMCDM method has a wide range of applications. This method is not limited to dealing with complex decision-making problems which consist with hybrid information, functional requirements of criteria and group experts. It can also solve simple problems like the classical MCDM methods such as the TOPSIS, VIKOR and PROMETHEE. If the decision matrix is composed by the single form of information, then in Step 2, one of the corresponding methods mentioned in Eqs. (7)–(10) can be used to calculate the probabilities of achieving functional requirements; while if the decision matrix consists of more than one kind of information, we can select the corresponding techniques to calculate the probabilities. If there is not necessary to consider the functional requirements of criteria, then Step 2 can be removed. If all alternatives can be ranked in Step 2, then Step 3 can be left out. If there is only one expert to make evaluation, then we do not need to run Step 4. In brief, the proposed method is flexible. In addition, the proposed method can be applied to different walks of life. For example, when choosing a suitable job, one needs to consider salary, working environment, development opportunities, location and company prospects comprehensively and requires, and he/she may require that the salary is at least $20,000 a year and the location is near to the home. The proposed method can help him/ her to select a job which meets these additional requirements and performs well under all criteria. For another example, when the government is planning water resource development, they should measure cost, probability of water shortage, energy (reuse factor), flood control, land and forest utilization, and water quality. According to the government’s budget, obviously, the cost needs to be controlled within a certain range, which is a functional requirement. Furthermore, we can find that the salary, cost and probability of water shortage are quantitative but the working environment, development opportunities, land and forest utilization are qualitative. These practical decision-making problems show the reasonability of the hypotheses in this paper. 4. Case study: SCF decision making based on the credit risk assessments of small and medium-sized enterprises 4.1. Case description In China, small and medium-sized enterprises account for 99 % of the total number of enterprises. They make a great contribution to the development of national economy with 60 % GDP, 50 % tax and 80 % urban employment (Zhu et al., 2017). The SCF has great potential in Chinese market given that it can expand customer base and provide new channels for stabilizing high-end customers for commercial banks. The profit of commercial banks in developing SCF business is much higher than that of traditional financial businesses. CD bank is a medium-sized and nationwide commercial bank in China. Its Chengdu branch, located in Chengdu city, Sichuan province, has a relatively short time and a small scale in the marketing of large customers. Its target customers are mainly small and medium-sized enterprises. Most small and medium-sized enterprises in China have small scale of operation and little collateral assets, which makes them difficult to meet the internal credit requirements of the bank. To speed up business development, the Chengdu branch of CD bank decides to develop SCF business. There are six small and medium-sized enterprises, {a1 , a2 , a3 , a4 , a5 , a6 }, from different supply chains applying for a loan from this commercial bank. Due to financial constraints, the bank decides to invest in two of them who have the lowest comprehensive credit risk. There are many credit risk criteria in SCF that have been used in existing papers (Zhang et al., 2015; Chen, 2012; Zhao et al., 2016). Considering the strong correlation among some risk criteria and low importance for some of them, we streamline the criteria and establish the criteria system in Table 3 for credit risk evaluation of small and medium-sized enterprises in SCF. We analyze the credit risk from four aspects which make up the first level of the criteria system. The operation conditions and credit status of small and medium-sized enterprises are critical for financing, and they are also considered in traditional finance. If the industry growth is good and the cooperation

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

13

between enterprises is closed, not only is the funds and goods’ turnover high in supply chain but also the profitability and debt paying ability of small and medium-sized enterprises is good. Therefore, the quality of the supply chain is important in SCF. Considering that the credit strength of leading enterprise in supply chain plays an importance role in SCF, we should consider it when making evaluation. The evaluation criteria {c1 , c2 , c3 , c4 , c5 , c6 , c7 , c8 , c9 } are listed in Table 3. The value of deterministic quantitative criterion c5 for each enterprise is precisely known. Four experts {e1 , e2 , e3 , e4 } are invited to evaluate the performance of these six small and medium-sized enterprises with respect to the random quantitative criteria {c4 , c9 } and the qualitative criteria {c1 , c2 , c3 , c6 , c7 , c8 }. Each expert also assigns a “design range” for each criterion. The weight vector of the criteria is determined by the experts as W = (0.12, 0.18, 0.12, 0.18, 0.05, 0.05, 0.12, 0.08, 0.10)T . The LTS for evaluating the six qualitative criteria is unified as {s−3 = very low, s−2 = low, s−1 = a little low, s0 = medium, s1 = a little high, s2 = high, s3 = veryhigh}. The evaluations given by four experts are listed in Tables A1–A4 in Appendix, respectively. Translating them into interval numbers or interval-valued linguistic terms, we can establish four individual decision matrices as ID(1) , ID(2) , ID(3) and ID(4) . (1)

drj

a1 a2 ID

(1)

=

a3 a4 a5

⎡

(2)

a1 a2 ID

(2)

=

a3 a4 a5

⎡

(3)

a1 a2 ID(3) =

a3 a4 a5

⎡

(4)

a1 a2 ID(4) =

a3 a4 a5 a6

(6, 10]

[20%, +∞]

[s1 , s3 ]

[s1 , s2 ]

[6, 7]

[30%, 30%]

[s1 , s1 ]

[s0 , s2 ]

[5, 10]

[52%, 52%]

[s0 , s0 ]

[s2 , s2 ]

[7, 9]

[73%, 73%]

[s1 , s1 ]

[s1 , s1 ]

[7, 7]

[10%, 10%]

[s2 , s2 ]

[s2 , s3 ]

[5, 7]

[25%, 25%]

[4, 6]

[41%, 41%]

[s1 , s3 ]

(7, 10]

[s1 , s1 ]

[s1 , s3 ]

[7, 8]

[s0 , s0 ]

⎥ ⎥ ⎥ [s1 , s1 ] [s0 , s1 ] [7, 8] ⎥ ⎥ ⎥ (s2 , s3 ] [s3 , s3 ] [8, 9] ⎥ ⎥ ⎥ [s−3 , s0 ) [s−2 , s−2 ] [5, 8] ⎥ ⎥ ⎥ [s1 , s1 ] [s2 , s2 ] [8, 10] ⎦

[s2 , s2 ]

[s3 , s3 ]

[s0 , s0 ]

[8, 9]

[s−1 , s0 ]

[s0 , s3 ]

[s0 , s3 ]

[s1 , s3 ]

(6, 10]

[20%, +∞]

[s0 , s3 ]

[s0 , s3 ]

[s1 , s3 ]

(6, 10]

(s1 , s3 ]

[6, 8]

[30%, 30%]

[s−1 , s−0.5 ]

[s0 , s0 ]

[s1 , s3 ]

[5, 7]

[s1 , s1 ]

[5, 7]

[52%, 52%]

[s−1 , s−1 ]

[s1 , s1 ]

[s1 , s1 ]

[5, 7]

[s2 , s2 ]

[8, 8]

[73%, 73%]

(s2 , s2 ]

[s2 , s3 ]

[s3 , s3 ]

[8, 9]

[s0 , s1 ]

[4, 7]

[10%, 10%]

[s3 , s3 )

[s−3 , s0 ]

[s−3 , s−3 ]

[6, 8]

[s3 , s3 ]

[5, 7]

[25%, 25%]

[s0 , s0 ]

[s−2 , s−1 ]

[s−1 , s0 ]

[6, 8]

[s0 , s0 ]

[5, 5]

[41%, 41%]

[s3 , s3 ]

[s3 , s3 ]

[s1 , s1 ]

[7, 8]

[s−1 , s−1 ]

[s1 , s1 ]

[s0 , s3 ]

[s1 , s3 ]

[s0 , s0 ]

⎡

⎤

[s1 , s3 ]

[s1 , s1 ]

[s1 , s3 ]

⎢ ⎢ [s0 , s0 ] [s2.4 , s2.8 ] [s1 , s1 ] ⎢ ⎢ [s , s ] [s , s ] [s , s ] 0 0 0 0 ⎢ −1 −1 ⎢ ⎢ [s2 , s2 ] [s , s ] [s 1 1 3 , s3 ] ⎢ ⎢ ⎢ [s−2 , s−2 ] [s−1 , s−1 ] [s1 , s1 ] ⎢ ⎢ [s2 , s3 ] [s2 , s2 ] ⎣ [s3 , s3 ]

a6

drj

[s1 , s3 ]

[s−1 , s−1 ]

⎢ ⎢ [s0.2 , s0.7 ] [s−1 , s1 ] ⎢ ⎢ [s , s ] [s , s ] −1 0 ⎢ 0 0 ⎢ ⎢ [s3 , s3 ] [s 2 , s2 ] ⎢ ⎢ ⎢ [s−3 , s−3 ] [s0 , s0 ] ⎢ ⎢ ⎣ [s2.2 , s2.9 ] [s2 , s2 ]

a6

drj

[s1 , s3 ]

⎢ [s1 , s2 ] ⎢ [s1 , s1 ] ⎢ ⎢ [s , s ] [s1 , s1 ] ⎢ 0 0 ⎢ ⎢ [s3 , s3 ] (s2 , s3 ] ⎢ ⎢ ⎢ [s−2 , s−2 ] [s−1 , s1 ] ⎢ ⎢ [s2 , s3 ] ⎣ [s2 , s2 ]

a6

drj

[s0 , s3 ]

[s0 , s3 ]

[s0 , s0 ]

[s0 , s3 ]

[s1 , s1 ]

[s0 , s3 ]

⎢ [s1 , s2 ] [s1 , s1 ] ⎢ [s2 , s2 ] ⎢ ⎢ [s , s ] [s , s ] [s , s ] 0 0 −1 1 ⎢ −1 −1 ⎢ ⎢ [s1 , s1 ] [s , s ] [s 2 2 2 , s3 ] ⎢ ⎢ ⎢ [s−3 , s−3 ] [s−1 , s0 ] [s0 , s1 ] ⎢ ⎢ [s3 , s3 ] [s2 , s3 ] ⎣ [s2 , s2 ] [s−1 , s−1 ]

[s2 , s2 ]

[s−2 , s0 ]

[6, 10]

[20%, +∞]

[s0 , s3 ]

[s1 , s3 ]

[s2 , s3 ]

[7, 10]

[6, 7]

[30%, 30%]

[s0 , s0 ]

[s1 , s1 ]

[s2 , s2 ]

[7, 7]

[5, 8]

[52%, 52%]

[s−1 , s−1 ]

[s0 , s2 ]

[s0.5 , s0.9 ]

[6, 8]

[8, 9]

[73%, 73%]

[s2 , s2 ]

[s3 , s3 ]

[s3 , s3 ]

[8, 9]

[7, 9]

[10%, 10%]

[s3 , s3 ]

[s−2 , s−1 ]

[s−2 , s−2 ]

[5, 8]

[8, 8]

[25%, 25%]

[s1 , s1 ]

[s2 , s2 ]

[s2 , s2 ]

[8, 8]

[5, 5]

[41%, 41%]

[s3 , s3 ]

[s2 , s2 ]

[s−1 , s−1 ]

[8, 8]

[6, 10]

[20%, +∞]

[s0 , s3 ]

[s1 , s3 ]

[s1 , s3 ]

[7, 10]

[6, 7]

[30%, 30%]

[s0 , s0 ]

[s−1 , s−1 ]

[s2 , s2 ]

[7, 7]

[5, 7]

[52%, 52%]

[s1 , s1 ]

[s0 , s0 ]

[s1 , s1 ]

[6, 8]

[7, 8]

[73%, 73%]

[s0 , s2 ]

[s2 , s3 ]

[s0 , s1 ]

[6, 8]

[7, 7]

[10%, 10%]

[s3 , s3 ]

[s−1 , s−1 ]

[s−3 , s−3 ]

[5, 8]

[6, 7]

[25%, 25%]

[s−1 , s0 ]

[s0 , s0 ]

[s1 , s1 ]

[7, 8]

[6, 6]

[41%, 41%]

[s2 , s2 ]

[s3 , s3 ]

[s−1 , s−1 ]

[8, 8]

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Given that there are both quantitative and qualitative criteria for credit risk evaluation and the experts have an acceptable risk level range for each criterion, the proposed Ulti-HFAD-ORESTE method may be powerful to evaluate the credit risks of these small and medium-sized enterprises in SCF. Its calculation results can provide some references for the Chengdu branch of CD commercial bank.

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W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

Table 4 The calculation results derived by the Ulti-HFAD-ORESTE method. e1

e2 (1)

a1 a2 a3 a4 a5 a6

(1)

e3 (2)

(2)

e4 (3)

(3)

Collective (4)

(4)

ICi

Ui

r (1) (ai )

ICi

Ui

r (2) (ai )

ICi

Ui

r (3) (ai )

ICi

Ui

r (4) (ai )

r(ai )

0 0.39 0 ∞ 0.28 ∞

0.56 0.45 0.92 0.23 0.70 0.40

2 4 1 6 3 5

0.48 0.56 0 ∞ 0.78 ∞

0.46 0.35 0.83 0.24 0.63 0.46

2 3 1 6 4 5

0 ∞ 0 ∞ 0 ∞

0.53 0.27 0.85 0.18 0.78 0.35

3 5 1 6 2 4

∞ 0.83 0.18 ∞ 0.29 ∞

0.46 0.23 0.75 0.15 0.65 0.46

4.5 3 1 6 2 4.5

3 4 1 6 2 5

Table 5 The results calculated by the hybrid FAD approach without the target-based weighted average operator. e1

e2 (1)

a1 a2 a3 a4 a5 a6

e3 (2)

e4 (3)

Collective (4)

ICi

r (1) (ai )

ICi

r (2) (ai )

ICi

r (3) (ai )

ICi

r (4) (ai )

r(ai )

0 0.39 0 ∞ 0.28 ∞

1.5 4 1.5 5.5 3 5.5

0.48 0.56 0 ∞ 0.78 ∞

2 3 1 5.5 4 5.5

0 ∞ 0 ∞ 0 ∞

2 5 2 5 2 5

∞ 0.83 0.18 ∞ 0.29 ∞

5 3 1 5 2 5

3 4 1 5 2 6

Table 6 Calculation results derived by five well-known MCDM methods. TOPSIS

a1 a2 a3 a4 a5 a6

VIKOR

MULTIMOORA

PROMETHEE

ELECTRE

RC i

r(ai )

GU i

IRi

r(ai )

RS i

RP i

FS i

r(ai )

NDi

r(ai )

CI i

r(ai )

0.56 0.51 0.92 0.23 0.70 0.40

3 4 1 6 2 5

0.44 0.55 0.08 0.77 0.30 0.60

0.09 0.11 0.03 0.18 0.12 0.18

2.5 4 1 6 2.5 5

0.40 0.38 0.51 0.27 0.43 0.35

0.09 0.11 0.03 0.77 0.12 0.18

0.52 0.52 0.61 0.38 0.52 0.48

2 4 1 6 3 5

0.13 −0.73 3.89 −2.99 1.23 −1.53

3 4 1 6 2 5

2.59 2.19 4.39 0.97 3.07 1.74

2.5 4 1 6 2.5 5

4.2. Solving the case by the Ulti-HFAD-ORESTE method This part solves the case by the proposed method. As the individual decision matrices have been given above, we start the computation procedure from Step 2. Step 2. The information content of each alternative with respect to each criterion is calculated by Eq. (5). The collective information (q) contents, ICi (i = 1, 2, · · ·, 6), of the alternatives derived by each expert are shown in Table 4. (q)

Step 3. Since all criteria are benefit forms, we can obtain the target value of each criterion by taj

=

(q)

max srj . The utility values of

i=1,2,···,m

the alternatives are computed by Eqs. (11) and (12) based on the four individual decision matrices and are listed in Table 4. According to (q) (q) the ranking principle in terms of ICi and Ui , we obtain the subordinate ranking sets of the alternatives, which are also shown in Table 4. Step 4. Based on experts’ knowledge and experience, their importance relations are e1 = e3 e4 e2 , then IR = {1.5, 4, 1.5, 3}. By Eqs. (13) and (14), we obtain the collective ranking as a3 a5 a1 a2 a6 a4 . Thus, a3 and a5 are two most appropriate enterprises for the Chengdu branch of CD bank to invest. They not only can achieve the functional requirements of all criteria with low risk, but also perform well overall according to the experts’ collective opinions. 4.3. Comparative analyses Comparative analyses are conducted in this section to highlight the advantages of the developed method. 4.3.1. Solving the case by the hybrid FAD approach without target-based weighted average operator This part illustrates the effectiveness of the target-based weighted average operator in aiding the hybrid FAD approach to derive the subordinate rank set of alternatives. The results derived by the hybrid FAD approach without the target-based weighted average operator are shown in Table 5. Comparative analysis: Despite that there is little difference between the collective rank sets of alternatives in the cases that using or not using the target-based weighted average operator, the subordinate rank sets are quite different. Taking the results obtained from expert e3 as an example, alternatives a1 , a3 and a5 are ranked at the top while alternatives a2 , a4 and a6 are ranked at the bottom when we only use the hybrid FAD approach. However, there is a large gap among a1 , a3 and a5 , or among a2 , a4 and a6 in terms of the comprehensive performances. For instance, a3 performs better than a1 under the most criteria such that a3 dominates a1 intuitively. This importance information cannot be reflected by the hybrid FAD approach in the final ranking while the target-based weighted average operator makes up for this drawback of the hybrid FAD approach. From Table 6, we can find that the alternatives which have the same order when using the hybrid FAD approach can be further ranked by the target-based weighted average operator according to their collective performances.

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

15

4.3.2. Solving the case by the hybrid FAD approach with the aggregated evaluations of experts To highlight the reliability of the ordinal aggregation technique to integrate the subordinate rank sets derived from the individual experts, this part solves the case by the hybrid FAD approach with the aggregated evaluations of experts, and makes comparative analysis with the proposed method. According to the importance relations among the experts, we assign their weights as ω(e1 ) = 0.35, ω(e2 ) = 0.1, ω(e3 ) = 0.35 and ω(e4 ) = 0.2. We aggregate the experts’ evaluations by Eq. (15) if cj is quantitative or by Eq. (16) if cj is qualitative (Liao et al., 2018b).



[xij− , xij+ ]

=

Q

(q)− ω(eq )xij ,

q=1

 ij− ij+ [s˛ , s˛ ]

=

Q

Q



(q)+ ω(eq )xij

q=1

ij(q)− ω(eq )s˛ ,

q=1

Q

(15)

 ij(q)+ ω(eq )s˛

(16)

q=1

The group decision matrix is obtained as drj a1 a2 ID = a3 a4 a5 a6

⎡

[s0 , s3 ] ⎢ [s0.8 , s0.8 ] ⎢ [s , s ] ⎢ −0.6 −0.6 ⎢ [s2.3 , s2.3 ] ⎢ ⎢ [s−2.3 , s−2.3 ] ⎣ [s , s ] 2.4 2.4 [s−0.7 , s−0.7 ]

[s0.7 , s3 ] [s1.3 , s2.2 ] [s0.3 , s0.4 ] (s1.7 , s2 ] [s−0.9 , s0 ] [s2 , s2.9 ] [s0.9 , s0.9 ]

[s0.8 , s3 ] [s1 , s1.6 ] [s−0.1 , s1 ] [s2.4 , s2.6 ] [s0.7 , s1 ] [s2.3 , s2.7 ] [s−0.4 , s0.4 ]

[6, 10] [6, 7.1] [5, 8.4] [7.5, 8.7] [6.7, 7.7] [6.3, 7.4] [4.9, 5.6]

[20%, +∞] [30%, 30%] [52%, 52%] [73%, 73%] [10%, 10%] [25%, 25%] [41%, 41%]

[s0.4 , s3 ] [s0.3 , s0.3 ] [s−0.3 , s−0.3 ] [s1.3 , s1.7 ] [s2.7 , s2.7 ) [s0.2 , s0.4 ] [s2.5 , s2.5 ]

[s0.9 , s3 ] [s0.5 , s0.5 ] [s0.5 , s1.2 ] [s2.4 , s3 ] [s−2.3 , s−0.6 ] [s0.9 , s1 ] [s2.7 , s2.7 ]

[s1.4 , s3 ] [s1.6 , s2.5 ] [s0.5 , s1 ] [s2.4 , s2.6 ] [s−2.3 , s−2.3 ] [s1.5 , s1.6 ] [s−0.5 , s−0.5 ]

⎤

[6.9, 10] [6.8, 7.4] ⎥ [6.3, 7.9] ⎥ ⎥ [7.6, 8.8] ⎥ ⎥ [5.1, 8] ⎥ ⎦ [7.6, 8.7] [7.9, 8.4]

Following Steps 2 and 3 of the proposed method, the results derived by the hybrid FAD approach with the aggregated evaluations of experts are: IC1 = 0.07, IC2 = 0.63, IC3 = 0, IC4 = ∞, IC5 = 0.01 and IC6 = ∞; U1 = 0.55, U2 = 0.39, U3 = 0.93, U4 = 0.22, U5 = 0.77 and U6 = 0.41. Based on the ranking principle, we obtain a3 a5 a1 a2 a6 a4 . Comparative analysis: The final rank set of the alternatives derived by the hybrid FAD approach with the aggregated evaluations of experts is the same as the final rank set derived by the proposed method, which implies the reliability of our method. From the calculation process, however, we find that there are some limitations of aggregating experts’ evaluations directly compared with integrating the subordinate rank sets obtained from the individual experts: 1) Different evaluation standards of experts are ignored; 2) The aggregation process likes a “black box” without clear interpretation and the aggregated evaluations are not visible, especially for large-scale group where there are great differences among the experts’ opinions, but it is intuitive to reflect each expert’s opinions on alternatives according to the subordinate rank sets; 3) The aggregated evaluation is sensitive to crisp weights of experts, but the result is stable when using the ordinal aggregation technique; 4) It is difficult to obtain experts’ crisp weights but is easy to obtain their ranks. 4.3.3. Solving the case by other MCDM methods To highlight the advantages of the proposed method, in this section, we solve the case by five well-known MCDM methods mentioned in the introduction, i.e., TOPSIS, VIKOR, MULTIMOORA, PROMETHEE and ELECTRE, and then make comparative analyses. For the convenience of comparison, here we only take the individual decision matrix given by e1 as an illustration. The results are shown in Table 6. Comparative analysis: The rankings of alternatives derived by these MCDM methods are the same except the ranks between a1 and a5 . From the individual decision matrix, ID(1) , we can find that a5 performs better with respect to most criteria but performs worse under criteria c4 and c6 , while a1 performs slightly better under all criteria. The dominance relations between a1 and a5 are different by different methods. Both the TOPSIS and the PROMETHEE deduce a1 ≺ a5 ; the VIKOR works out that a1 is indifferent to a5 ; the ELECTRE derives that a1 is incomparable to a5 , but both the MULTIMOORA and the proposed method infer a1 a5 . Differences in principles of these MCDM methods result in different calculation results. The TOPSIS method determines the utility values of alternatives by comparing with the positive and negative ideal solutions. The optimal alternative with the biggest relative closeness, RC i , is closest to the positive ideal solution and farthest from the negative ideal solution. The PROMETHEE method focuses on conducting pairwise comparisons to derive the positive dominant flows and the negative dominant flows. The optimal alternative with the largest net dominant flow (NDi ) dominates others overall. The weighted averaging operator is applied by both TOPSIS and PROMETHEE methods to aggregate the performances of alternatives under all criteria. By this operator, the bad performances of a5 under c4 and c6 are completely compensated by the good performances under other criteria so that its’ comprehensive utility value is higher than that of a1 . In practical decision-making problems, however, there is a big risk to take an alternative as the optimal one which performs good overall but very bad under some criteria. The utility values of alternatives in the VIKOR method are integrated by the “group utility value (GU i )” and the “individual regret value (IRi )”. However, it is difficult to determine the relative importance between GU i and IRi . For the case presented above, if we let the weight of GU i satisfy w(GU i ) < 0.5, then a1 a5 ; if we let w(GU i ) = 0.5, then a1 = a5 ; if we let w(GU i ) > 0.5, then a1 ≺ a5 . The result is sensitive to this weight. Similarity, based on pairwise comparisons, the outranking relations (preference, indifference and incomparability) are derived by comparing concordance index (CI i ) and discordance index without aggregation in the ELECTRE method. Despite that the concordance index of a5 is better than a1 , the discordance index of a5 is worse than a1 . Therefore, a5 is incomparable with a1 in ELECTRE. To enhance the robustness of the result, the weighted geometric operator is utilized in the MULTIMOORA method. Considering the results (RS i , RP i and FS i ) derived by three kinds of aggregation operators, the MULTIMOORA method derives a1 a5 since there is bad performance of a5 under some criteria. In conclusion, the existing methods concentrate on depicting the comprehensive performances of alternatives by comparing with the positive ideal solution, negative ideal solution or other alternatives with respect to each criterion. But the functional requirement of each criterion is ignored in these methods. The positive ideal solution for a criterion may be not able to meet the functional requirement of each criterion while the negative ideal solution may achieve the functional requirement of some criteria. In the aforementioned case, a5 cannot meet the functional requirement of c6 completely while a1 can meet the functional requirements of all criteria. In practice, experts tend

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W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

Table 7 Characteristics of different MCDM methods. Method

Reference point

Mathematical calculation

Information type

Computing process

Computing time

Stability

Optimal solution

TOPSIS

WAO

Short

Stable

Visualized

Short

Instable

MULTIMOORA

Positive ideal solution

Visualized

Short

Stable

Close to positive ideal solution Close to positive ideal solution Perform best overall

PROMETHEE

Each alternative

Not visualized

Long

Stable

Dominate others

ELECTRE

Each alternative

Not visualized

Long

Instable

Dominate others

The proposed method

Functional requirements

Qualitative or quantitative Qualitative or quantitative Qualitative or quantitative Qualitative or quantitative Qualitative or quantitative Qualitative and quantitative

Visualized

VIKOR

Positive and negative ideal solutions Positive ideal solution

Visualized

Short

Stable

Meet functional requirement of each criterion

WAO and WMO WAO, WMO and WGO WAO WAO and WMO WAO

Note. WAO, weighted average operator; WMO, weighted maximum operator; WGO, weighted geometric operator.

to select a1 rather than a5 despite of the better comprehensive performance of a5 since they are reluctant to take risks. In the proposed Ulti-HFAD-ORESTE method, both the satisfaction on functional requirement of each criterion and the comprehensive performances of alternatives are considered. The characteristics of the above mentioned MCDM methods are concluded in Table 7. In summary, the advantages of the proposed Ulti-HFAD-ORESTE method can be highlighted as follows: (1) Hybrid information can be solved. The proposed method is flexible to deal with hybrid evaluations with numerical numbers and linguistic terms. But most MCDM methods are only able to solve either quantitative evaluations or qualitative evaluations by combined with the separate evaluation representation models. (2) Three forms of criteria are considered. The proposed method is efficient to cope with benefit, cost and deviation criteria. But the deviation criteria, which exist objectively, are ignored by most MCDM methods. (3) The risk of making decision is low. The optimal solution derived by the proposed Ulti-HFAD-ORESTE method can meet the functional requirement of each criterion so that the risk of the failure is very low when applying the optimal solution in practice. (4) The computing process is visualized and the result is stable. The relations between “system range” and “design range” are clear. The stability of the rank set is enhanced by depicting the comprehensive performances of alternatives. 5. Conclusions The functional requirement is usually predefined for each criterion in practical MEMCDM problems. The alternative which is below the minimal requirement of a criterion cannot be selected in decision-making process, otherwise it will suffer a big risk of failure in operation. The functional requirement is considered in the FAD approach so as to reduce decision risk but ignored by most MCDM methods. However, the FAD method is limited in 1) selecting the optimal alternative, 2) dealing with single type of evaluation representation, 3) single person decision making. Motivated by the idea of the utility theory and ORESTE method, this paper developed a hybrid MEMCDM approach, named the Ulti-HFAD-ORESTE method, which can retain the advantages of the FAD and make up for its shortcomings. Firstly, we argued that a practical MCDM problem consists of both quantitative and qualitative criteria such that the decision matrix is generally composed by hybrid evaluations, such as crisp numbers, interval numbers, and linguistic terms. Methods to determine the probabilities of alternatives in achieving the functional requirements of criteria were developed for different representation models and the results were normalized so as to unify hybrid information. In addition, we proposed an alternative ranking method based on both the probabilities to achieve functional requirements and the comprehensive values. The measuring on the probabilities to achieve functional requirements can distinguish the merits and demerits of an alternative under each criterion. If one attribute value cannot meet the functional requirement, then the overall value of this alternative is low. In this way, we can get low-risk decision-making results, but we cannot further distinguish the alternatives when they are good under all criteria or have bad values under at least one criterion. Therefore, we introduced an attribute value aggregation method which can derive the comprehensive value of alternatives to supplement the FAD method for ranking all alternatives. The comparison analysis based on the case study illustrated that the aggregation method can improve the FAD method to obtain reliable results. Furthermore, we presented an ordinal aggregation technique to deal with the problem which involves a group of experts with different evaluating standards. Comparing with the cardinal aggregation technique, the ordinal form can avoid the distortion of individualized evaluations. Finally, the advantages of the Ulti-HFAD-ORESTE method were highlighted in terms of visualization, stability, information type and reliability by comparing with five well-known MCDM methods including TOPSIS, VIKOR, MULTIMOORA, PROMETHEE and ELECTRE. However, for solving hybrid decision matrix, this paper is limited to five kinds of evaluation representation models. Some complex representation models have been proposed and aroused wide applications, such as the probabilistic linguistic term set (Liao et al., 2019a), which should be addressed to improve the generality of our method. According to psychophysics, the semantic of linguistic terms may be unevenly distributed when evaluating the values of some criteria. For example, the gap between “a little high” and “high” may be smaller than the gap between “high” and “very high”. Therefore, how to calculate the probability of achieving the functional requirements under uneven distributed linguistic evaluation context is an important and interesting research topic that needs to be further investigated in the future. Combining the linguistic scale function (Liao et al., 2019b) with the membership function of linguistic terms may be a workable

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

17

way to meet our psychological cognition. Furthermore, an experimental set-up using several data sets should be established to illustrate the conclusions about the strengths and weaknesses of the proposed method.

Declaration of Competing Interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgements The work was supported by the National Natural Science Foundation of China (No. 71771156), and the 2019 Sichuan Planning Project of Social Science (No. SC18A007).

Appendix A

Table A1 The evaluations given by expert e1 . c1

c2

c3

c4

c5

c6

c7

c8

c9

30%

At least a little high A little high

52%

Medium

A little high

a3

Very high

More than high

73%

A little high

More than high

At least a little high At least a little high Between medium and a little high Very high

More than 7

Between 6 and 7 Higher than 5

At least a little high A little high

Medium

At least a little high Between a little high and high Between medium and high High

At least20%

a2

At least a little high Between a little high and high A little high

More than 6

a1

At least medium A little high

Between 8and 9

a4

Low

10%

High

Lower than medium

Low

Between 5 and 8

a5

High

Between a little low and a little high At least high

25%

Medium

A little high

High

At least 8

a6

A little low

A little high

Between 5 and 7 Between 4 and 6

41%

High

Very high

Medium

Between 8 and 9

(1)

drj

A little high

At least high Between a little low and medium

Between 7 and 9 7

Between 7 and 8 Between 7 and 8

Table A2 The evaluations given by expert e2 . c1

c2

c3

c4

c5

c6

c7

c8

c9

At least medium Between a little low and a little high

At least a little high More than a little high

More than 6

At least 20% 37%

At least a little high At least a little high

More than 6

Between 6 and 8

A little high

Between 5 and 7

52%

A little high

A little high

Between 5 and 7

a3 a4

Very high Very low

Between a little low and medium High Medium

At least medium Between a little low and medium and close to a little low A little low

At least medium Medium

a2

At least medium 20% higher than medium and 30% lower than a little high Medium

8 Between 4 and 7

83% 10%

High Very high

At least high Lower than medium

20% higher than high and 10% lower than very high A little low

High

Between 5 and 7

25%

Medium

Between low and a little low

Very high Between a little low and medium A little high

Between 8 and 9 Between 6 and 8

a5

High Between medium and a little high Very high

A little high

Medium

5

41%

Very high

Very high

A little high

Between 7 and 8

(2) drj

a1

a6

Between 5 and 7

Between 6 and 8

18

W. Xingli and H. Liao / Computers in Industry 114 (2020) 103144

Table A3 The evaluations given by expert e3 . c1

c2

c3

c4

c5

c6

c7

c8

c9

At least a little high A little high

At least 20%

At least 7

37%

At least a little high A little high

At least high

Between 6 and 7

At least medium Medium

High

7

a2

A little low

At least a little high 40% higher than high and 20% lower than very high Medium

At least 6

a1

At least medium Medium

Medium

Between 5 and 8

52%

A little low

Between medium and high

Between 6 and 8

a3

High

A little high

Very high

83%

High

Very high

Between 8 and 9

a4

Low

A little low

A little high

10%

Very high

Between 5 and 8

Very high

High

25%

A little high

High

8

a6

Medium

Between high and very high Medium

Between low and a little low High

Low

a5

Between 8 and 9 Between 7 and 9 8

50% higher than medium and 10% lower than a little high Very high

A little high

5

41%

Very high

High

A little low

8

(3)

drj

Table A4 The evaluations given by expert e4 . c1

c2

c3

c4

c5

c6

c7

c8

c9

At least medium A little high

At least 20% 30%

At least a little high At least low

At least a little high High

At least 7 7

A little low

Between 6 and 7 Between 5 and 7

At least medium Medium

a2

At least medium Between a little high and high Medium

At least 6

a1

At least medium High

52%

A little high

Medium

A little high

Between 6 and 8

a3

A little high

High

Between 7 and 8

73%

Between high and very high

Very Low

10%

A little low

Between 5 and 8

High

Between medium and a little high Very high

7

a5

Between a little low and medium At least high

Between medium and a little high Very low

Between 6 and 8

a4

Between medium and high Very high

Between 6 and 7

25%

medium

A little high

Between 7 and 8

a6

A little low

High

Between low and Medium

6

41%

Between a little low and medium High

Very high

A little low

8

(1)

drj

Between a little low and a little high Between high and very high

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