A new product development concept selection approach based on cumulative prospect theory and hybrid-information MADM

Accepted Manuscript A New Product Development Concept Selection Approach Based on Cumulative Prospect Theory and Hybrid-Information MADM Cheng-shuo Ying, Yan-Lai Li, Kwai-Sang Chin, Hong-Tai Yang, Jie Xu PII: DOI: Reference:

S0360-8352(18)30230-4 https://doi.org/10.1016/j.cie.2018.05.023 CAIE 5229

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

18 September 2017 17 April 2018 15 May 2018

Please cite this article as: Ying, C-s., Li, Y-L., Chin, K-S., Yang, H-T., Xu, J., A New Product Development Concept Selection Approach Based on Cumulative Prospect Theory and Hybrid-Information MADM, Computers & Industrial Engineering (2018), doi: https://doi.org/10.1016/j.cie.2018.05.023

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A New Product Development Concept Selection Approach Based on Cumulative Prospect Theory and Hybrid-Information MADM

Cheng-shuo Yinga Yan-Lai Lib,c,*

Kwai-Sang China Hong-Tai Yangb,c Jie Xud,*

a Department of System Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, P.R. China b School of Traffic and Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R. China c Tangshan School of Graduate, Southwest Jiaotong University, Tangshan, Hebei 063000, P.R. China d School of Software Engineering, Northeastern University, Shenyang, Liaoning, 110004, P.R. China

Abstract: New product development (NPD) concept selection is one of the foremost phases in the early stage of NPD to identify the best alternative concept for an enterprise. In practice, not all attributes of NPD concept can be estimated precisely considering inevitable uncertainty associated with the NPD processes. Thus, the NPD concept selection is a hybrid-information multiple attribute decision making (HI-MADM) problem, in which attribute values are represented in various formats (e.g., crisp numbers, interval numbers, and linguistic terms). In the concept selection, psychological behaviors of the NPD team (NPDT) have a non-negligible influence on the accuracy of the final decision. Nevertheless, the decision behaviors are rarely considered in existing studies on the concept selection. In this paper, a risk HI-MADM method based on cumulative prospect theory (CPT) is proposed to select the NPD alternative concepts. Initially, decision information in various formats is normalized and the expectations of the NPDT are set as the corresponding reference points with considering the psychology of the NPDT. Subsequently, the gains and losses matrix relative to the reference points is constructed. Furthermore, the prospect values of concept attributes are calculated based on the value function of CPT. Then, by aggregating prospect values and attribute weights by a simple additive weighting (SAW) method, the comprehensive prospect values of alternative concepts are obtained, and then the ranking order of all concepts can be determined. Finally, a NPD case study of a new automatic dishwasher is used to illustrate the feasibility and validity of the proposed approach and meanwhile a sensitivity analysis and a comparison analysis are conducted. Key words: New product development; Cumulative prospect theory; Hybrid-information multiple attribute decision making; Concept alternatives *

Corresponding author: E-mail address: [email protected] (Yan-Lai Li), [email protected] (Jie Xu)

A New Product Development Concept Selection Approach Based on Cumulative Prospect Theory and Hybrid-Information MADM

Abstract: New product development (NPD) concept selection is one of the foremost phases in the early stage of NPD to identify the best alternative concept for an enterprise. In practice, not all attributes of NPD concept can be estimated precisely considering inevitable uncertainty associated with the NPD processes. Thus, the NPD concept selection is a hybrid-information multiple attribute decision making (HI-MADM) problem, in which attribute values are represented in various formats (e.g., crisp numbers, interval numbers, and linguistic terms). In the concept selection, psychological behaviors of the NPD team (NPDT) have a non-negligible influence on the accuracy of the final decision. Nevertheless, the decision behaviors are rarely considered in existing studies on the concept selection. In this paper, a risk HI-MADM method based on cumulative prospect theory (CPT) is proposed to select the NPD alternative concepts. Initially, decision information in various formats is normalized and the expectations of the NPDT are set as the corresponding reference points with considering the psychology of the NPDT. Subsequently, the gain and loss matrix relative to the reference points is constructed. Furthermore, the prospect values of concept attributes are calculated based on the value function of CPT. Then, by aggregating prospect values and attribute weights by the simple additive weighting (SAW) method, the comprehensive prospect values of alternative concepts are obtained, and then the ranking order of all concepts can be determined. Finally, a NPD case study of a new automatic dishwasher is used to illustrate the feasibility and validity of the proposed approach and meanwhile a sensitivity analysis and a comparison analysis are conducted. Key words: New product development; Cumulative prospect theory; Hybrid-information multiple attribute decision making; Concept alternatives

1 Introduction

Under a challenge of shrinking product life cycle and heterogeneous customer expectations for high-quality products, the continuous development of new products is crucial for entrepreneurs to keep up with customer requirements (CRs) and market dynamics. New product sales usually account for a large proportion of corporate profits, and however, the failure ratio of new product is substantial, and the cost of failure is relatively high. Thereby new product development (NPD) is widely recognized as a core product strategy with high risk and one of the most challenging components of enterprise competitiveness (Chiang & Che, 2010; Tai, 2017). A general NPD process usually starts from the formation of a NPD team (NPDT) comprised of product R&D managers, engineers, and experts from various business sectors and then goes through several sequential stages, e.g., the CRs identification, product conceptual design, industrial design and planning, prototyping, manufacturing design, and successive launch (Carlile, 2002; Kornish & Ulrich, 2011). Fig. 1 shows a common procedure of the NPD process.

Insert Figure 1 about here

Once initial conceptual designs are obtained based on the CRs identification which is conducted with the utilization of customer research methods such as conjoint analysis and Delphi method, the following processes are to weight diverse alternatives and figure out the best solution in the comprehensive goals set, i.e., product concept selection. There are over 37.9% of companies in the industry adopting a scoring methodology for the selection of product concepts for the objectives of risk reduction and a promotion of profitability (Cooper et al., 1999). Product concept selection is critical at an early stage of NPD because of its contribution to the rational deployment of scarce R&D resources (e.g., human resources, capital, equipment, time), as well as its identification of the optimal concept which best fulfill maximum CRs and design constraints imposed by the NPDT on product attributes (Aral & Walker, 2011; Ko et al., 2016; Yan et al., 2006). However, the concept selection is an intricate inherently multi-attribute decision process, where discrete alternatives are evaluated against multiple attributes such as the new product performance, lead time, manufacturing cost, quality, and so on. The NPDT can utilize multiple-attribute decision making (MADM) systems, also referred to as multi-criteria decision-making tools, to optimize the concept space

comprised of optional concepts and multiple concept attributes. Fig. 2 shows an illustrative procedure of the MADM analysis.

Insert Figure 2 about here

In the real world, the selection of NPD concepts contains significant amounts of uncertainty causing factors, which confuse the NPDT to reach the optimum choice of the alternative concepts. Basically, there are two aspects of uncertainties and ambiguities confronted. On the one hand, uncertain preferences during attributes evaluation exist on account of individual heterogeneity (Zeki Ayağ et al., 2013). On the other hand, human judgments on attributes are subjective and imprecise (Relich & Pawlewski, 2017). In particular, the decision information of some concept attributes are difficult or impossible to be deterministic in concept selection considering the complexity and ambiguity of available information (Büyüközkan & Güleryüz, 2016; Chiao, 2016), such as performance, quality, and stability. The decision data of these attribute are often represented in a form of interval numbers or fuzzy numbers rather than crisp numbers based on the nature of diverse attributes. For example, lead time and return on investment can be expressed with an interval number, while a fuzzy number or a linguistic term can be used to characterize some qualitative information, such as quality, stability, and reliability. Thereby, the decision data are generally represented in diversiform attribute values for maximum retention of the original decision information, thus guaranteeing the authenticity and reliability of the final decision solutions (Li & Wan, 2014). Therefore, the concept selection problem is typical MADM problem with decision data represented in hybrid information forms, i.e., hybrid-information MADM (HI-MADM) problem. Basically, this paper focuses on the concept selection problem with attribute values in format of crisp numbers, interval numbers, and linguistic terms. Besides, several recent psychological studies have confirmed the existence of some behavioral characteristics of DM under high risks and uncertainty, e.g., loss aversion, judgmental distortion, and reference dependence, which have a non-negligible influence on the accuracy of final solution. Nevertheless, most existing concept selection methods are on the basis of expected utility theory (EUT) without proper consideration of DM’s psychological behaviors, bringing about the selection of non-optimal concepts. Since concept selection is often conducted along with fuzziness and information incompleteness, there is an indispensable implication to consider human behaviors in the NPD concept selection for sake of effective decision support to the NPDT.

In summary, the NPD alternative concepts selection is a HI-MADM problem which encompasses DM’s psychological factors and high risks. Traditional decision-making methods cannot directly be applied to such a problem due to the multi-form attributes and the difference of the hypothesis of psychological behavior. Therefore, there are practical implications to develop an effective and efficient NPD concept selection method to address hybrid formats of decision information as well as to reflect psychological preferences of the NPDT into the final decision.

2 Literature Review

In general, several product concepts are designed with the application of systematic product development tools such as quality function deployment. Companies carrying out NPD require thorough consideration when assessing or ranking the NPD concepts because of the importance and non-negligible high risks of NPD concept selection. In recent years, several methods are developed for solving such challenging risk decision making in NPD. Some previous methods score the priority of each attribute of NPD concept as a crisp number or an interval number (Cooper et al., 1999; Hall & Nauda, 1990). Subsequently, the decision information of a quantitative NPD attribute is addressed based on a numeric scale (Chan & Wu, 2002, 2005) due to its superiority in identifying decision data under uncertainty and fuzziness, e.g., 1−5−9, where 1, 5, and 9 represent the linguistic terms “poor”, “moderate”, and “good”, respectively. Thereby, many decision analysis tools are applied to deal with the NPD concept selection, such as simple additive weighting (SAW) (Zionts & Wallenius, 1983), analytic hierarchy process (AHP) (Pun et al., 2010), analytic network process (ANP) (Lee et al., 2010), technique for order preference by similarity to an ideal solution (TOPSIS) (Shidpour et al., 2013), artificial neural network (Bouchereau & Rowlands, 2000), VlseKriterijumska Optimizacija I Kompromisno (VIKOR) algorithm (Aghajani Bazzazi et al., 2011; Tiwari et al., 2016), and so on. Moreover, product concept selection is often implemented under a fuzzy context containing high uncertainty and incomplete information (Z. Ayağ

demir, 200

y k kan

ey ogl , 2004).

Hence, some studies extensively integrate fuzzy set theory (Zadeh, 1965) into the NPD process because the utilization of fuzzy linguistic terms have merits in addressing the fuzzy nature of human assessments. Zeki Ayağ (2005) developed a fuzzy AHP-based simulation approach to evaluate product design alternatives in a NPD environment. Fan et al. (2009) constructed a fuzzy linguistic approach based on 2-tuple linguistic representation model to evaluate collaboration satisfaction of the NPDT. Augustine et al. (2010) proposed a

fuzzy framework for selecting and evolving improved concepts through a rigorous concept evaluation and convergence process with the utilization of a fuzzy inference process. Chiang and Che (2010) applied the AHP and fuzzy-DEA to construct an evaluation and ranking approach in a NPD project selection. Since it was practical to describe the attribute values with hybrid formats of information rather than with a homogeneous mathematical form, the MADM approaches under the assumption of multiform decision information had thereby become a new study hotspot (Dağdeviren, 2010 Wan

Li, 2013 Zaras, 2004).

Rao et al. (2007) presented a decision making method based on a grey relational degree to solve a problem of fuzzy hybrid multiple attribute decision making with precision numbers, intervals, and fuzzy numbers. Fan et al. (2013a) proposed an extended TODIM method to address the HI-MADM problem with the attribute values in the format of crisp numbers, intervals and fuzzy numbers. Li and Wan (2014) proposed a fuzzy linear programming method to solve heterogeneous MADM problems with generous decision information expressed by real numbers, interval numbers, and trapezoidal fuzzy numbers. Cheng et al. (2017) developed a heterogeneous axiomatic design method to solve HI-MADM problem by introducing a distance measure into axiomatic design in an anti-vibration design selection problem. The abovementioned MADM methods have made considerable contributions to the decision support for NPD concept selection. Nevertheless, the existing decision analysis methods rarely take the psychological behaviors of the NPDT into account, and most of them are derived from the EUT under a strict assumption of absolute rationality of the NPDT. However, a few psychological studies have revealed that DMs are bounded rational rather than completely rational in decision process (Charness & Rabin, 2005; Dong et al., 2015; Kahneman & Tversky, 1979; Quiggin, 1991). To characterize decision behaviors, some behavioral decision theories have been developed gradually, e.g., the prospect theory (Kahneman & Tversky, 1979), disappointment theory (Gul, 1991), cumulative prospect theory (CPT) (Gurevich et al., 2009; Tversky & Kahneman, 1992), third-generation prospect theory (Schmidt et al., 2008), regret theory (Zhang et al., 2016), and so on. In fact, CPT has been the most widely applied in various decision issues among these theories because CPT gives constructive formulas on values and weights with features of simple computing and clear logic structure (Liu et al., 2014), and that the outcomes obtained by CPT have the best consistency with practical data evidence (Bleichrodt et al., 2007). According to Kahneman and Tversky (1979) and Tversky and Kahneman (1992), there are several behavioral characteristics of NPDT under risk and uncertainty. For instance, in the selection of new product concepts, the NPDT often has a psychological prospect for each attribute such as expected return, lead time, manufacturability, and risk, i.e., reference

point (also referred as reference dependence). If an attribute value appears to be inferior to the reference point, the NPDT will be dissatisfied and consider the gap as losses. Rather, an attribute value prior to the reference point will satisfy the NPDT and the surplus will be encoded as gains. In addition, the NPDT has different attitudes pertaining to cognitive gains and losses, i.e., risk seeking in losses and risk aversion in gains (Abdellaoui et al., 2016; Abdellaoui et al., 2007). In particular, losses are more painful than commensurate gains, i.e., high loss aversion. For example, the prospective loss of $10 is much larger in magnitude than the prospective gain of $10. In addition, the outcomes with high probabilities tend to be underestimated while those with small probabilities tend to be overestimated (Han & Pinto, 2000; Kothiyal et al., 2014). In practice, these psychological behavior factors of the NPDT lead to the errors in the selection of concepts. Therefore, it deserves more attention to develop a method of incorporating CPT into decision support for the NPD concept selection. Therefore, in this paper, the potential benefit and loss scenarios of concept selection are investigated under the consideration of the NPDT’s decision behaviors and that an effective CPT-based MADM method is proposed for solving the risk decision problem in the concept selection. Notably, this paper mainly considers a HI-MADM problem where the decision information is mainly given in three formats, i.e., crisp numbers, interval numbers, and linguistic terms. The contributions of our work are summarized as follow: 

This work proposes a new CPT-based decision-making method for a concept selection considering the psychological behaviors, such as reference point, loss aversion, diminishing sensitivity, and so on, which is not considered in existing concept selection method. The method provides more utility information (e.g., gains and losses matrix and prospect values) to reflect the NPDT’s psychology.



Our work gives the normalization method, comparison method, and distance measure method for different formats of attributes, which contributes to the integration of hybrid decision information.



We illustrate the advantages of the CPT-based MADM method by a sensitivity analysis and comparison analysis with an existing EUT-based method.

The rest of this paper is organized as follows. Section 3 elaborates the formulation and solution procedure of the risk HI-MADM problem. Section 4 presents a CPT-based HI-MADM method for the concept selection. In Section 5, the proposed method is illustrated by a real concept selection in the NPD of a new automatic dishwasher and meanwhile the sensitivity analysis and comparison analysis are conducted. Lastly, Section 6 summarizes the paper with some concluding remarks.

3 Formulation and solution procedure of concept selection problem in NPD 3.1 Formulation of the HI-MADM problem in NPD concept selection In view of the functional nature of the concept attributes, the attributes considered in the HI-MADM of concept selection can be grouped into two categories, i.e., cost and profit type, both of which can be described as crisp numbers, intervals, and linguistic terms as illustrated in Fig. 3. In detail, a profit-type attribute is the attribute for maximization whose value is always the larger the better. Oppositely, a cost attribute is the attribute for minimization whose value is the smaller the better.

Insert Figure 3 about here

Supposing that there are m alternative product concepts to be evaluated against n attributes based on the identification of competing CRs in the concept selection stage of NPD. Let M  1,2,..., m and

N  1, 2,..., n . Let S  s1 , s2 ,..., sm  be a finite concept set, where si denotes the i th alternative product concept; and let A  a1 , a2 ,..., an  be a finite attribute set, where a j denotes the j th attribute. Let the subscript sets of the profit and cost attributes be denoted by N p and N c , respectively, which must satisfy N p

Nc =N as well as N p

Nc = . Let w   w1 , w2 ,...wn 

T

be the vector of attribute weights,

where w j denotes the importance degree of the attribute a j , which satisfies 1  w j  0( j  N ) and



n j 1

w j  1 . Generally, w j is provided by the NPDT directly. Let Q   q1 , q2 ,...qn 

T

be the expectation

vector of attributes, where q j is the expectation or goal of the NPDT concerning the jth attribute, which can be selected as the corresponding reference point concerning a j (see Subsection 4.2 for a more detailed discussion). Let d ij denote the evaluation value of the NPDT for product concept si with regard to the attribute a j , i  M , j  N . Then the decision matrix for the HI-MADM of alternative product concepts can be defined as

 d11 d D   dij  =  21 m n    d m1

d12 d 22 dm2

d1n  d 2 n  .   d mn 

In this paper, the decision data involved in the concept selection are represented in three various formats, i.e., crisp numbers, interval numbers, and linguistic terms for reflecting the nature of the attributes. Notably, the attribute value d ij and expectation q j with respect to the same attribute a j , j  N should be described in the unified mathematical form for sake of information consistency and comparability between them. Therefore, the attribute set A can be further divided into three subsets as Acn   A1 , A2 ,..., Ak1  ,



Ain  Ak1 1 , Ak1  2 ,..., Ak2

Acn

Ain



,

and



Alt  Ak2 1 , Ak2  2 ,..., An



,

where

1  k1  k2  n ,

such

that

Alt  A . The attributes in Acn , Ain , and Alt are respectively described as crisp numbers,

numerical intervals, and linguistic terms. Correspondingly, let Ncn = 1,2,..., k1 , Nin = k1  1, k1  2,..., k2  , and Nlt = k2  1, k2  2,..., n denote the subscript set of Acn , Ain and Alt respectively, such that Ncn

Nin

Nlt  N obviously. Moreover, the descriptions of different formats of attribute values are

further detailed as follows: (1) Crisp number. If d ij and q j are crisp numbers, indicating that precise prospective expectation and evaluation with regard to attribute a j are provided based on sufficient production information, then

q j  qj , dij  dij , j  Ncn , i  M . Without loss of generality, we assume qj  0 , dij  0 . (2) Interval number. If d ij and q j are interval numbers, which means the expectation and evaluation of the NPDT concerning the attribute a j are difficult to address exactly but probably fall in a certain low up low up numerical interval, then q j  q j and dij  dij , where q j  q j , q j  , dij  dij , dij  ,

j  Nin ,

and 0  dijlow  dijup . Besides, a crisp number can appear  qup i  M . In general, we suppose that 0  qlow j j as a special case of an interval number, e.g., qj =q j  qj , qj  . (3) Linguistic term. If d ij and q j are represented by linguistic terms, suggesting that the attribute value and expectation with regard to the attribute a j are given qualitatively rather than quantitatively. Then, q j  q j , dij  dij , j  Nlt , i  M , where q j and d ij are linguistic terms which must satisfy





q j , dij  L . Here, L  l p | p  0,1,..., E

is a pre-defined ordered linguistic term set consisting of an odd

number of linguistic terms, where E denotes the number of set elements which is usually an even number and l p denotes the p+1th linguistic term in set L. For example, when E = 6, the set can be defined as L=

{extremely poor, very poor, poor, moderate, good, very good, extremely good}. Besides, some operational laws for any two linguistic terms la , lb  L can be expressed as follows (Herrera et al., 2005): (a) If a  b , then the set is ordered: la

lb (i.e., la is not inferior to lb ). Thus, a larger subscript p

indicates a better attribute performance of the linguistic terms. (b) If a  E  b , then a negation operator is observed: neg  la   lb . (c) If la

lb , then there are max  la , lb   la and min  la , lb   lb .

3.2 Solution procedure of the concept selection problem in NPD The problem considered in this paper is, in consideration of the psychological behaviors of the NPDT, how to figure out the best concept from the available alternative concepts based on the given decision information, e.g., the finite concept set S, the decision matrix D, the expectation vector Q, and the attribute weight vector w. In this paper, an HI-MADM method based on CPT is developed to solve the concept selection problem mentioned above, in which the attributes are described in crisp numbers, interval numbers, and linguistic terms. Fig. 4 shows the solution procedure of the NPD concept selection. Firstly, the decision information in three different formats are normalized using the normalization methods for the sake of commensurability among data in diverse dimensional units. Secondly, the expectations of the NPDT are set as the reference points, based on which the type of gain or loss for each attribute is determined by the order relation comparison method. Then the distance measure between each attribute and its relative reference point is calculated by the Euclidean distance functions. Further, the gain and loss matrix is constructed by aggregating the gains and losses relative to the reference points. Afterwards, in view of the risk preferences of the NPDT toward gains and losses, the prospect values of concept attributes are calculated based on the value function of CPT. Consequently, the comprehensive prospect values of all concepts are obtained by aggregating prospect values and attribute weights based on the SAW method. Lastly, a ranking of all product concepts is determined based on the obtained comprehensive prospect values.

Insert Figure 4 about here

4 The proposed NPD concept selection approach based on CPT and HI-MADM

In this section, a detailed description of the proposed CPT-based HI-MADM approach is provided for the concept selection problem in consideration of the decision-making behaviors. The proposed method can be divided into three stages, namely transformation of three formats of decision-making information, construction of gain and loss matrix, and ranking of concept alternatives.

4.1 Transformation of three formats of decision-making information In the practical concept selection, heterogeneous formats of expectation and attribute values should be normalized to eliminate the impact of different physical dimensions on the accuracy of decision solutions. Let B  (b1 , b2 ,..., bn )T denote the normalized vector of the expectations of the NPDT, where b j denotes the normalized value of expectation pertaining to jth attribute. Correspondingly, let F   fij  denote the mn normalized decision matrix, where f ij denotes the normalized attribute value of the attribute a j . Notably, diversiform values of expectation and attribute are considered to be transformed into normalized profit-type data within the closed interval [0,1]. The corresponding normalization formulas for each format of attribute are given respectively as follows: (1) Crisp number. For the Aj  Acn , j  Ncn , the q j and d ij are described as crisp numbers, then the b j and f ij are respectively defined by  qj  oj  /  oj  oj  , j  N p  bj   ,  j  /  o j  o j  , j  N c o  q   j 

(1)

 fij  oj  /  o j  o j  , j  N p , i  M   fij   ,        o j  fij  /  o j  o j  , j  N c , i  M

(2)

Where

 

 

o  max max  d   , q , j  N ij j cn  j iM . (3)   o j  min min  dij  , qj , j  N cn iM  (2) Interval number. If the d ij and q j are described as interval numbers for Aj  Ain , i  M , j  Nin , then the b j and f ij are respectively given by

low up low up low up low   q low   j  o j  /  o j  o j  ,  q j  o j  /  o j  o j  , j  N p bij  b , b    , (4) up up low up low up low  oup  , j  Nc  q / o  o , o  q / o  o         j j j j j j j j   low low up low up low up low   dij  o j  /  o j  o j  ,  dij  o j  /  o j  o j   , i  M , j  N p   fij   fijlow , fijup    , up up up low up low up low  o j  dij  /  o j  o j  ,  o j  dij  /  o j  o j  , i  M , j  N c    low j

up j

(5)

Where

 

(3) Linguistic term. The d ij

 

 oup  max max  d up  , q up , j  N ij j in  j iM (6) .  low low olow  min min d , q , j  N  ij  j in iM  j and q j are described qualitatively as linguistic assessment terms for the

Aj  Alt , i  M , j  Nlt . Then the b j and f ij are respectively given by  j  Np  qj , bj   , neg q ,  j  j  Nc  

(7)

  dij , i  M , j  N p (8) fij   . neg dij , i  M , j  N c   Specially, the decision information in the form of linguistic terms can be transformed into triangular

 

fuzzy numbers for the convenience of calculations. Let b j   b1j , b2j , b3j  and fij   fij1 , fij2 , fij3  , i  M , j  Nlt denote the transformed expectation and decision matrix in the form of triangular fuzzy numbers,

respectively. Further, according to the fuzzy set theory, the corresponding transformation formula is given as





l  max  p  1 / E,0 , p / E, min  p  1 / E,1 .

(9)

T Consequently, the normalized expectation vector B  (b1 , b2 ,..., bn ) and decision matrix F   fij  mn

are obtained by the aforementioned normalization methods by using Eqs. (1)−(9).

4.2 Gain and loss matrix construction In consideration of decision behaviors of the NPDT, the gain and loss matrix is an effective tool to describe the NPDT’s cognitive gain and loss with respect to each attribute. The most critical stage in the construction of gains and loss matrix is the setting of the corresponding reference point. Commonly, based on the full investigation of CRs and constraints of corporate R&D resources, the NPDT has an expected goal concerning each concept attribute (Fan et al., 2013b), which is consistent with the objectivity of the concept selection, i.e., product concept with desirable attribute performance and efficient utilization of corporate R&D resource. Notably, according to the empirical

research conducted by Heath et al. (1999), the goal namely expectation serves as reference point and alters results in a manner consistent with the value function of CPT, thus setting the expectation of DM as the corresponding reference point inherits several properties of the CPT’s value function—not only loss aversion but diminishing sensitivity. Hence, this paper considers select the expectations of the NPDT as the reference point for the better integration of the value function of CPT. Then, the type of gain or loss of a concept attribute can be determined by the order relation comparison between the attribute value and the value of the relative reference point. In view of the incommensurability of diversiform attributes, the comparison processes for three attribute types are detailed respectively in the next paragraph. (1) For attribute Aj  Acn , the order relation between normalized reference point bj and normalized attribute value fij can be determined by the mathematical comparison within real number field. low up (2) For attribute Aj  Ain , let bic and fijc denote the center of the interval number b j  b j , b j  low up and fij   fij , fij  respectively. Correspondingly, let biw and fijw denote the width of b j and f ij

respectively.

Supposing

that

bic 

1 up low bj  bj  , 2

low , biw  bup j  bj

fijc 

1 up fij  fijlow   2

and

fijw  fijup  fijlow , then the order relation between b j and f ij is determined by the following criteria

(Ishibuchi & Tanaka, 1990): a. When bic  fijc , if bic  fijc , then b j  fij ; if bic  fijc , then b j  fij . b. When bic  fijc , if biw  fijw , then b j  fij ; if biw  fijw , then b j  fij ; if biw  fijw , then b j  fij

.

(3) For attribute Aj  Ala , let lijf  f  0,1, 2,..., E  1, E  and l bj  b  0,1, 2,..., E  1, E  denote the linguistic terms concerning the normalized attribute value f ij , and those concerning the reference point b j . The method of order relation comparison between f ij and b j can be described as

 fij  b j   fij  b j   fij  b j

lijf

l bj

lijf  l bj . lijf

l bj

In addition, the measurement of perceptive benefit or loss of each attribute should be determined for the construction of gain and loss matrix. The amounts of benefit and loss can be regarded as the deviation degree between the attribute values and corresponding reference point, which can be calculated by the

Euclidean distance functions for advantages of convenient calculation and accurate expression of the true distance between two points. Let R   rij  denote the distance matrix, where rij is the distance mn measurement between the attribute value f ij and its relative reference point b j . The calculation formulas (Sun et al., 2015) are given by:  fij  bj ,   2 2  1  up low  rij   fij  bup  blow  j    f ij j      2   1 2 2 2   fij1  b1j    fij2  b 2j    fij3  b3j      3 After that, the prospective gains and losses can be calculated based

i  M , j  N cn i  M , j  N in .

(10)

i  M , j  N la

on the determined type of gain and

 

loss and distance measurement between each attribute and its reference point. Let C fij

denote the gain

or loss of f ij relative to reference point b j , which can be given by

 

 rij , C  fij    rij ,

fij  b j , i  M , j  N . fij  b j , i  M , j  N

(11)

 

Here, if C fij  0 , it indicates the cognitive gain of the f ij with respect to b j ; if C fij  0 , then it means cognitive loss of the f ij with respect to b j . Then, the gain and loss matrix C ( fij )  can be mn constructed as

 C ( f11 ) C ( f12 ) C( f ) C( f ) 21 22 C  C ( fij )  =  m n   C ( f m1 ) C ( f m 2 )

C ( f1n )  C ( f 2 n )  .   C ( f mn ) 

4.3 The ranking of the alternatives According to the CPT theory proposed by Tversky and Kahneman (1992), the prospect value is an effective tool to reflect the intuitively perceived utility of an individual. Based on the gain or loss of the attribute A j ( C ( f1 j ) , C ( f 2 j ) ,…, C ( f mj ) ), the prospective utility, i.e., the prospect value of attribute A j can be

 

calculated to describe the cognitive utility of concept attributes to the NPDT. Let V fij

denote the

prospect value with regard to the attribute a j  j  N  of the concept alternative si  i  M  , then it is given based on the value function of CPT as





 C f , C  fij   0  ij   (12) V  fij    .   C  fij  , C  fij   0  where α and β denote the exponent parameters and λ denotes the loss aversion parameter, satisfying





 ,   0,1 ,   1 (Neilson & Stowe, 2002). The function for 0    1 indicates risk aversion over gains; the function for 0    1 exhibits risk seeking over losses. Therefore, a larger α indicates higher risk aversion in the loss domain, and a larger β indicates higher risk seeking in the gain domain. Notably, β is usually larger than α. Moreover, the loss aversion coefficient λ is widely recognized to be larger than 1 because of the psychological nature of the NPDT namely high level of loss aversion in the decision making (Kahneman & Tversky, 1979). In this paper, the values of α, β, and λ are determined based on previous studies (Booij & Kuilen, 2009; Tversky & Kahneman, 1992) as α=0.89, β=0.92, and λ=2.25. Based on the gains and losses of attributes, the prospect decision matrix can be built as V  V ( fij )  mn .

 

Finally, according to the prospect values V fij

and corresponding weight of attributes

 w , w ,...w  , 1

2

j

the comprehensive prospect value of the concept si , i  M can be calculated. To this end, the SAW

 

method is used to aggregate V fij

and w into the overall prospect value of concept si . Let the

comprehensive prospect value be denoted by U  si  , then it can be calculated by

U  si    (V  fij   w j ), i  M . n

(13)

j 1

Obviously, a larger comprehensive prospect value U  si  indicates that the product concept si will have a better performance. Therefore, the ranking of the finite alternative NPD concepts can be determined based on the comprehensive prospect values; moreover, the most desirable product concept for the fulfillment of CRs as well as the competitiveness improvement will be selected. In summary, an algorithm for the proposed CPT-based HI-MADM method for product concepts selection in NPD is presented in the next paragraph. Step 1. Form an NPDT, and then determine the finite product concept set S  s1 , s2 ,..., sm  , the finite attribute set A  a1 , a2 ,..., an  , the weight vector w   w1 , w2 ,...wn  , the expectation vector T

Q   q1 , q2 ,...qn  , and the evaluation matrix D  dij  . mn T

Step 2. Construct the normalized decision matrix F   fij  mn and normalized expectation vector

B  (b1 , b2 ,..., bn )T using Eqs. (1) ~ (9). Step 3. Select the expectation of the NPDT as the reference point, and construct distance matrix R   rij 

mn

using Eq. (10).

Step 4. Construct the gain and loss matrix C  C ( fij )  using Eq. (11). mn Step 5. Build the prospect decision matrix V  V ( fij )  mn based on the calculation of the prospect

 

value V fij

using Eq. (12).

Step 6. Calculate the comprehensive prospect value U  si  of the alternative concept si  i  M  using Eq. (13). Step 7. Determine the ranking of the m alternative NPD concepts according to the obtained comprehensive prospect values U  si  .

5 Case study

In this section, to illustrate the proposed method, a real case study under the R&D background of a late-model household automatic dishwasher is given. Meanwhile, the comparison analyses of the obtained outcomes are implemented.

5.1 An automatic dishwasher concept selection problem and the analysis process Increasing application of the electric appliances in the lives of residents has immensely improved the living standards of the Chinese people and meanwhile brought opportunities and challenges to the competitive household appliance market. In response to the challenges posed by the changing market, an electrical appliance maker in Sichuan province carries out the NPD of a new-type household automatic dishwasher as a new product strategy. Through the identification of CRs and subsequent product conceptual design process with the application of consumer research methods and systematic development tools, the enterprise obtains five product concepts of the automatic dishwasher for further evaluation. For selecting the best concept considering NPDT’s psychological behaviors, the proposed NPD concept selection method is used to further evaluate the concept alternatives and the procedure is summarized as follows. Step 1. A NPDT is formed with a product manager, relevant engineers, customer representatives, and

persons from different business sectors. According to CRs and characteristics of available concepts s1 , s2 , s3 , s4 , s5 , the NPDT brainstorms to generate creative ideas for attribute candidates, which are further

evaluated for the elimination of the infeasible ones. Subsequently, five attributes are determined to evaluate the five concept alternatives, as depicted in Table 1.

Insert Table 1 about here

Among the attributes exhibited in Table 1, the attributes a1 and a2 are of the cost type while the a3 , a4 , and a5 are of the profit type. The five attributes are evaluated in three formats due to the uncertainty

and information incompleteness in the NPD environment. In particular, the attribute values of a1 and a3 are described in the format of crisp numbers; the attribute values of a2 are described in the format of interval numbers; and the attribute values of a4 and a5 are represented in the format of linguistic assessment terms, with the linguistic term set determined as L= l0 , l1 , l2 , l3 , l4 , l5 , l6  , as shown in Table 2.

Insert Table 2 about here

The vector of attribute weights is provided by the NPDT as w   0.24,0.16,0.15,0.19,0.27 

T

market

survey

and

Delphi

method.

The

expectation

vector

of

the

NPDT

is

based on set

as

Q  [41, 48],[67,73],10, G,VG  . The NPDT provides the performances of a j  j  1, 2,...,5 with respect T

to the five concept alternatives, and then the decision matrix D   dij  is obtained based on the relative 55 market survey and sufficient discussion of the NPDT, as shown in Table 3.

Insert Table 3 about here

Step 2. Three formats of attribute values are normalized into the profit-type values ranged from zero to one by using Eqs. (1) ~ (8) for the elimination of the influence of different dimensional units. Besides, for the convenience of calculation and comparison of the linguistic terms, the terms can be transformed into triangular fuzzy numbers based on the membership functions using Eq. (9), as shown in Table 4.

Insert Table 4 about here

The expectation vector of NPDT is normalized through the transformation of Q   q1 , q2 ,...qn 

T

as B=

([0.429,0.679],[0.400,0.640],0.667,(0.500,0.667,0.833),(0.667,0.833,1.000)). The normalized decision matrix is constructed as

F   fij  55

1.000 0.536   0.607  0.000 0.250

[0.840,1.000] [0.120,0.400] [0.120,0.760] [0.000,0.560] [0.280,0.800]

0.333 (0.167,0.333,0.500) (0.167,0.333,0.500)  1.000 (0.833,1.000， 1.000) (0.333， 0.500,0.667)  0.000 (0.500,0.667,0.833) (0.500,0.667,0.833)  .  0.667 (0.667， 0.833,1.000) (0.833,1.000,1.000)  0.667 (0.500,0.667， 0.833) (0.500,0.667,0.833) 

Step 3. The expectations of the NPDT are set as the corresponding reference points. Then the type of gain or loss for each attribute is determined by the order relation comparison method and the distance matrix is built using Eq. (10) as 0.4636  0.1263  R   rij  = 0.1360 55   0.5675  0.3283

0.4020 0.2608 0.2154 0.2884 0.1414

0.3333 0.3333 0.6667 0.0000 0.0000

0.3333 0.2885 0.0000 0.1667 0.0000

0.5000  0.3333  0.1667  .  0.1359  0.1667 

Step 4. Subsequently, the gain and loss matrix are constructed based on the type of gain or loss of each attribute and R using Eq. (11) as  0.4636 0.4020 0.3333 0.3333  0.1263 0.2608 0.3333 0.2885  C  C ( fij )  =  0.1360 0.2154 0.6667 0.0000 55   0.5675 0.2884 0.0000 0.1667  0.3283 0.1414 0.0000 0.0000

0.5000  0.3333 0.1667  .  0.1359  0.1667 

Step 5. The prospect value of each attribute can be computed using Eq. (12) and the prospect decision matrix is constructed considering the risk preferences of the NPDT based on CPT as

V  V  fij   m n

 0.5045 0.4444 0.8189 0.8189 1.1891  0.3353 0.6533 0.3762 0.3308 0.8189      0.1694 0.5480 1.5495 0.0000 0.4328    0.1693  1.3361 0.7149 0.0000 0.2030  0.8075 0.1754 0.0000 0.0000 0.4328 

Step 6. The comprehensive prospect value of each alternative can be calculated according to Eq. (13) as

U  s1   0.4073 , U  s2   0.2868 , U  s3   0.3963 , U  s4   0.3511 , and U  s5   0.2826 .

Step 7. The ranking order of the five product concepts is obtained according to the comprehensive prospect values with respect to five concept alternatives as s5

s2

s4

s3

s1 . Apparently, the concept

alternative s5 is the best ranked with the highest comprehensive prospect value equal to -0.2826. Notably, none of the comprehensive prospect values is larger than zero, indicating that under the limitation of corporate R&D resources, the expectation of the NPDT goes far beyond the technology level and NPD capacity of enterprise. With the existing R&D capacity, the best selection within the finite concept set for the NPDT under high level of expectation is the product concept s5. Hence, the concept s5 is adopted as the new product strategy, which leads to the achievement of the expected enterprise sales and an increase in enterprise market share. This empirical case study illustrates the feasibility and effectiveness of the proposed CPT-based HI-MADM method.

5.2 Sensitivity analysis of the loss aversion coefficient In the above case of dishwasher concept selection, the sensitivity analysis of the loss aversion coefficient λ is conducted to determine the influence of the psychological loss aversion on the concept selection. The ranking orders of the dishwasher concepts in different conditions of loss aversion levels are shown in Table 5. It should be noted that the scenario where λ=1 is the ideal case where the NPDT is completely rational. As can be observed from Table 5, the variation of the loss aversion coefficient λ results in the change of concept rankings. In more detail, the ranking orders are sensitive to the change of λ and the concept s2 is regarded as the best solution in conditions of low level of loss aversion. Nevertheless, the rankings are robust as s5

s2

s4

s3

s1 and that the s5 is recognized as the optimal alternative while the λ is set

above a certain level (about 2.1502 in this case). This is mainly because the prospect values of some loss type concept attributes are underestimated under the condition of low loss aversion. However, such losses are amplified as the loss aversion coefficients increase, resulting in an increase in the rankings of the concepts with smaller loss. As stated earlier, high level of loss aversion exists widely in practical fuzzy decision making, hence the ranking results under the assumption of relatively high loss aversion have more implications.

Insert Table 5 about here

5.3 Comparison analysis with existing EUT-based VIKOR method In this subsection, we compare the selection results derived by the existing the EUT-based VIKOR method and the proposed CPT-based method in this paper. VIKOR was an popular MCDM tool initially developed by Opricovic (1998) to generate compromise solution for MCDM problem by determining ranking of alternatives rationally under the hypothesis of rational man. The application of VIKOR was then widely extended for HI-MCDM problem under fuzzy decision-making environment due to its effectiveness and efficiency (Aghajani Bazzazi et al., 2011; Prakash & Barua, 2016). Nevertheless, the existing extended selection methods based on VIKOR ranked alternatives rationally based on the particular measure of ‘closeness’ to the ‘ideal’ solution without consideration of DM’s psychological behavior. The VIKOR technique extended for fuzzy HI-MCDM by Prakash and Barua (2016) is applied to the case example of NPD concept selection. It should be noted that the identical attribute weights in the case study are utilized in the comparison analysis as w   0.24,0.16,0.15,0.19,0.27  . Moreover, the normalization T

method, distance measurement method, comparison methods for crisp numbers, interval numbers and fuzzy numbers suggested in this paper are applied in this comparison analysis for comparability and accuracy of the outcome decision based on VIKOR. Hence, the original decision information for the NPD concept alternatives are normalized as

F   fij  55

1.000 0.536   0.607  0.000 0.250

[0.840,1.000] [0.120,0.400] [0.120,0.760] [0.000,0.560] [0.280,0.800]

0.333 (0.167,0.333,0.500) (0.167,0.333,0.500)  1.000 (0.833,1.000， 1.000) (0.333， 0.500,0.667)  0.000 (0.500,0.667,0.833) (0.500,0.667,0.833)  .  0.667 (0.667， 0.833,1.000) (0.833,1.000,1.000)  0.667 (0.500,0.667， 0.833) (0.500,0.667,0.833) 

In the implementation of extended EUT-based VIKOR method, the first and foremost step is to determine the positive and negative ideal solution among the alternative set by drawing out positive and negative ideal value of all the attributes. The attribute values are compared by comparison method



    suggested in this paper and the positive ideal solution (PIS) f j  f1 j , f 2 j ,..., f nj



 solution (NIS) f j  f1 j , f 2 j ,..., f nj





and negative ideal

based on f j =max j fij and f j- =min j fij as demonstrated in Table 6.

In general, there are three closeness indexes i.e., Si , Ri , and Qi , encompassed in VIKOR. Si denotes the distance rate of ith alternative to the PIS, Ri denotes the distance rate of ith alternative to the NIS, and Qi is the comprehensive value of the ith alternative by VIKOR. These indexes can be calculated based on

Eqs. (14) ~ (16) and the computation results are obtained in Table 7.

n

Si   w j k 1

R  f j , fij 

R  f j , f j 

(14)

,

 R  f j , fij    Ri  max  w j j  R  f j , f j  

S Q v S

i



i



  Where R f j , f j



 S   S

(15)

R  R  .  1  v   R  R  

i





(16)

denotes the Euclidian distance measure between the PIS f j and NIS f j , w j

indicates the provided weight of each attribute. Besides, the S   max i Si ; S -  min i Si ; R  max i Ri ;

R-  min i Ri ; v indicates the weight of the strategy of the ‘the majority criteria’, here s ppose v = 0.5. Lastly, the ranking of the NPD concepts is determined on the basis of obtained closeness index values in descending order as provided in Table 8.

Insert Table 6 about here

Insert Table 7 about here

Insert Table 8 about here

As can be observed from Table 7 and Table 8, the concept s2 is the best ranked by both Q and S, meanwhile satisfying Q  s2   Q  s5  =0.25  1/  5  1 ,i.e., two necessary conditions for a valid solution by VIKOR are both satisfied (Opricovic, 1998). Therefore, s2 is considered as the optimal by the extended VIKOR model followed by s5. It is noticed that the ranking result obtained by the extended VIKOR is quite different from that obtained by the HI-MADM method proposed in this paper. The main difference of the results of the proposed CPT-based method and VIKOR is that the ranking order between s2 and s5 that are converse, i.e., s2 ≻ s5 based on the EUT-based VIKOR while s5 ≻ s2 obtained according to the method developed in this paper. Moreover, a further comparison between the VIKOR outcome and the results obtained in sensitivity analysis manifests that the ranking result obtained by the VIKOR has more in similarity with the result obtained with the proposed method i.e., s2 is recognized as the optimal choice, under assumption of a low level of loss aversion which means a rational DM. In summary, compared with the existing EUT-based VIKOR extended for fuzzy MCDM, the proposed CPT-based HI-MADM method has following differences and advantages. a)

The main difference of these two methods lies in the different assumptions of rationality of DM in

the decision making. The implementation process of VIKOR exposes that VIKOR method rationally provides compromise priority of alternatives without giving any consideration of psychological factors of NPDT while the proposed method considers perceptive value of NPDT in NPD concept selection by aggregating reference point and CPT which is an effective and efficient tool to reflect the NPDT’s decision behavior. Hence, different compromise solutions are given for the same case study. b)

The sensitivity analysis conducted in subsection 5.2 highlights the influential importance of the psychological behavior of NPDT on the final outcomes. There are in consequence practical and valuable implications to take into consideration psychological factor in the NPD concept selection. A deeper insight shows that the proposed method under the situation where the NPDT is assumed to be rational offers the similar solution as the EUT-based method, which not only stresses again the impacts of psychology on decision-making as well as the feasibility of the proposed method, but also demonstrates that the developed method’s ability in avoiding the decision risk of selecting a non-optimal NPD design.

c)

The developed HI-MADM method incorporates CPT for revelation of the psychological factors of NPDT in the concept selection, i.e., reference point, loss aversion, and risk behavior. In particular, the proposed method calculates the prospect value of each attribute based on relative reference point and value function of CPT, providing an opportunity that we can gain a further insight into the gain or loss, namely contribution, of each attribute of each concept to the final ranking outcome. Lastly, the CPT-based method helps discover more hidden utility information in term of perspective value rather than EUT-based methods and thus gives a more desirable and expectant outcome with a practical meaning for NPDT.

6 Conclusions

In this paper, an effective CPT-based concept selection method is developed for solving HI-MADM problem in the risk decision-making environment. The most notable characteristic of the proposed method is that it can sufficiently consider the NPDT’s psychological decision behaviors, e.g., reference point, loss aversion, and diminishing sensitivity, under the context of uncertainty and incomplete information. In this method, crisp numbers, intervals, and linguistic terms are used to describe multiple types of evaluation

information and expectations of NPDT for retaining objective and subjective decision information. Firstly, decision information is normalized into profit-type numbers within the closed interval [0,1] and the expectation of the NPDT is chosen as the reference point. Then, the gain and loss matrix is constructed on the basis of pairwise order relation comparison and distance measure between attributes and reference points. Furthermore, the value function of CPT is applied to calculate the prospect values which can reflect the perspective utility of each attribute. Consequently, the comprehensive prospect values of all concepts are derived by aggregating prospect values and attributes weights. The ranking order of concepts is generated according to the obtained comprehensive prospect values. The proposed method is illustrated by a real dishwasher concept selection problem. Then, the sensitivity analysis of loss aversion coefficient demonstrates that the final decision is sensitive to the loss aversion, but the ranking results are robust in the condition of high levels of loss aversion. The comparison analysis shows that the psychological decision behaviors play an important role in concepts selection and further illustrates the advantages of a CPT-based method over a EUT-based method. The proposed method is very flexible with a clear logic and a simple calculation procedure. Further, it provides a systematic decision method which can be applicable to system evaluation in many fields, especially in situations where diversiform decision information is provided and the NPDT’s psychology sho ld be considered, s ch as supplier selection, risk investment, equipment selection and so on. In terms of future research, two research directions are identified. Firstly, since the expectations of the NPDT are deterministic in the proposed method, the method can be extended for the situation where the expectations (i.e., reference point) are uncertain in concept selection. Second, the proposed method can be further improved for the HI-MADM problem involving deviation attributes.

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Fig. 1. Common procedure of NPD process

Fig. 2. An illustrative procedure for MADM

Fig. 3 Hybrid attribute information in the concept selection of an automatic dishwasher

Fig. 4 The solution procedure of the product design assessment method

Table 1. Determined attributes to evaluate NPD concept alternatives Attribute

Content

Unit

a1

Development cost

Million RMB

a2

Cleaning time

Minute

a3

Dishware capacity

Tableware set

a4

Stability

---

a5

Cleaning efficacy

---

Table 2 linguistic terms in L

Content

l0

l1

l2

l3

l4

l5

l6

EP

VP

P

M

G

VG

EG

(extremely

(very

(poor)

(moderate)

(good)

(very

(extremely

poor)

poor)

good)

good)

Table 3. The original decision matrix in dishwasher concept selection problem a1

a2

a3

a4

a5

s1

32

[58,62]

8

P

P

s2

45

[73,80]

12

EG

M

s3

43

[64,80]

6

G

G

s4

60

[69,83]

10

VG

EG

s5

53

[63,76]

10

G

G

Table 4. Relationships between linguistic terms and triangular fuzzy numbers Linguistic term

Triangular fuzzy number

EG

(0.833,1.000,1.000)

VG

(0.667,0.833,1.000)

G

(0.500,0.667,0.833)

M

(0.333,0.500,0.667)

P

(0.167,0.333,0.500)

VP

(0.000,0.167,0.333)

EP

(0.000,0.000,0.167)

Table 5. Results of sensitivity analysis of the loss aversion coefficient Different levels of λ

Ranking

λ=1.00 (Ideal)

s2

s1

s4

s5

s3

λ=1.50

s2

s5

s4

s1

s3

λ=2.00

s2

s5

s4

s1

s3

λ=2.25 (Reference)

s5

s2

s4

s3

s1

λ=2.5

s5

s2

s4

s3

s1

λ=3.0

s5

s2

s4

s3

s1

λ=3.5

s5

s2

s4

s3

s1

λ=4.0

s5

s2

s4

s3

s1

Table 6. Positive and negative ideal solutions for concept selection Attribute

a1

a2

a3

a4

a5

Positive ideal solution

0.000

[0.120,0.400]

0.000

(0.167,0.333,0.500)

(0.167,0.333,0.500)

Negative ideal solution

1.000

[0.840,1.000]

1.000

(0.833,1.000,1.000)

(0.833,1.000,1.000)

Table 7. Values of R, S and Q index for all dishwasher concepts

Scheme

s1

s2

s3

s4

s5

S

0.560

0.469

0.589

0.494

0.547

S-=0.469

S+= 0.589

R

0.270

0.198

0.150

0.240

0.180

R-=0.150

R+= 0.270

Q

0.878

0.199

0.500

0.477

0.449

Table 8. The ranking of the concepts by each closeness index S

Rank by S

R

Rank by R

Q

Rank by Q

s1

0.560

4

0.270

5

0.878

5

s2

0.469

1

0.198

3

0.199

1

s3

0.589

5

0.150

1

0.500

4

s4

0.494

2

0.240

4

0.477

3

s5

0.547

3

0.180

2

0.449

2

Highlights

1 An expectation of an attribute is set as the corresponding reference point. 2 The gains and losses matrix relative to the reference points is constructed. 3 The prospect values of concept attributes are calculated based on CPT.