Multi-criteria decision making with fuzzy linguistic preference relations

Multi-criteria decision making with fuzzy linguistic preference relations

Ying-Hsiu Chen

, Tien-Chin Wang

, Chao-Yen Wu

a_{Department of Information Engineering, I-Shou University, Kaohsiung 840, Taiwan} b

Department of International Business, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan

c

Department of Information Management, I-Shou University, Kaohsiung 840, Taiwan

a r t i c l e

i n f o

Article history: Received 31 July 2009

Received in revised form 19 August 2010 Accepted 1 September 2010

Available online 24 September 2010

Keywords: Fuzzy LinPreRa

Fuzzy linguistic preference relations AHP

Extent analysis method Consistency

Decision analysis

a b s t r a c t

Although the analytic hierarchy process (AHP) and the extent analysis method (EAM) of fuzzy AHP are extensively adopted in diverse fields, inconsistency increases as hierarchies of criteria or alternatives increase because AHP and EAM require rather complicated pair-wise comparisons amongst elements (attributes or alternatives). Additionally, decision makers normally find that assigning linguistic variables to judgments is simpler and more intuitive than to fixed value judgments. Hence, Wang and Chen proposed fuzzy linguistic preference relations (Fuzzy LinPreRa) to address the above problem. This study adopts Fuzzy LinPreRa to re-examine three numerical examples. The re-examination is intended to compare our results with those obtained in earlier works and to demonstrate the advan-tages of Fuzzy LinPreRa. This study demonstrates that, in addition to reducing the number of pairwise comparisons, Fuzzy LinPreRa also increases decision making efficiency and accuracy.

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1. Introduction

Multi-criteria decision making (MCDM) methods have been proposed in recent decades for assisting decision making with multiple objectives. MCDM involves determining the most suitable optimal alternative, in the sense that, when the problem involves multiple conflicting criteria, there are several Pareto-optimal solutions, but only one solution (the pre-ferred one) should be selected. Pairwise comparison is often used in the decision making process. By using pairwise compar-isons, judges are not required to explicitly define a measurement scale for each attribute[1]. Since pairwise comparison values are the judgments obtained from an appropriate semantic scale, in practice the decision-maker(s) usually give some or all pair-to-pair comparison values with an uncertainty degree rather than precise ratings. Hence, pairwise comparisons provide a flexible and realistic way to accommodate real-life data. One common MCDM method is analytic hierarchy process (AHP), which was proposed by Saaty[2]. The main advantage of AHP is its systematic organization of tangible and intangible factors, which provides a structured, yet relatively simple solution to decision-making problems[3]. The AHP has been ap-plied in many different domains, including project management[4], enterprise resource planning (ERP) system selection[5], risk assessment[6]and knowledge management tools evaluation[7].

However, due to the complexity and uncertainty of real-world decision making problems and the inherent subjectivity of human judgment, exact judgments are often unrealistic or infeasible. Decision makers often find it more natural or easier to assign linguistic variables to judgments rather than to make fixed value judgments. It is more appropriate to present data using fuzzy numbers instead of crisp numbers. Therefore, alternative methods have been proposed to improve AHP. These

0307-904X/$ - see front matter Crown Copyright  2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.09.009

⇑ Corresponding author. Address: Department of International Business, National Kaohsiung University of Applied Sciences, No. 415, Jiangong Rd., Sanmin Dist., Kaohsiung City 807, Taiwan. Tel.: +886 7 3814526x6126.

E-mail addresses:ivychen31@gmail.com(Y.-H. Chen),tcwang@cc.kuas.edu.tw,dr.deor@gmail.com(T.-C. Wang),cywu@isu.edu.tw(C.-Y. Wu).

Applied Mathematical Modelling 35 (2011) 1322–1330

Contents lists available atScienceDirect

Applied Mathematical Modelling

methods are systematic approaches to solving the alternative selection and justification problem by applying fuzzy set theory and hierarchical structure analysis.

Early work in fuzzy AHP by van Laarhoven and Pedrycz[8]compared fuzzy ratios described by triangular membership functions. Buckley[9]investigated the use of fuzzy weights and fuzzy utility to extend AHP by the geometric mean method to derive the fuzzy weight. Chang[10]introduced a new approach using triangular fuzzy numbers for a pairwise comparison scale in fuzzy AHP. Extent analysis method (EAM) has also been used for synthetic extent values of pairwise comparisons. Cheng[11]proposed a new algorithm for evaluating naval tactical missile systems using fuzzy AHP based on grade value of membership function. Cheng et al.[12]proposed a new method for evaluating weapons systems by AHP based on linguis-tic variable weight. Zhu et al.[13]discussed EAM and applications of fuzzy AHP. Leung and Cao[14]proposed a fuzzy con-sistency definition with tolerance deviation.

Among the above approaches, EAM has been employed in many applications due to its computational simplicity. For example, Bozdag et al.[15] used the EAM to evaluate computer integrated manufacturing alternatives. Kahraman et al.

[16,17]also used this approach for selecting facility locations and evaluating catering firms in Turkey. Bozbura and Beskese

[18]also applied the Chang EAM to improve the quality of prioritization of organizational capital measurement indicators under uncertain conditions. In addition to those mentioned above, numerous other studies [19–26] have applied this method.

Although AHP and EAM are employed in diverse fields, inconsistency increases as hierarchies of criteria or alternatives increase. Decision makers often have difficulty ensuring a consistent pairwise comparison between voluminous decisions because consistency ratios (CRs) are produced after the evaluation process and global acceptance criteria are limited. To ad-dress this dilemma, Herrera-Viedma et al.[27]developed consistent fuzzy preference relations to avoid the inconsistent solutions in the decision-making processes. Moreover, using AHP requires n(n  1)/2 pairwise comparisons, whereas consis-tent fuzzy preference relations require only n  1 comparisons. Wang and Chen[28–30], Chen[31]developed a method that adopts fuzzy linguistic assessment variables rather than crisp values to construct fuzzy linguistic preference relations (Fuzzy LinPreRa) matrices based on consistent fuzzy preference relations[27]. Their method assures consistency and only requires n  1 judgments from a set of n elements. In this study, Fuzzy LinPreRa is applied to re-examine three numerical examples investigated by Kahraman et al.[17], Erensal et al.[21]and Bozbura and Beskese[18]in order to compare our results with those of earlier studies and demonstrate the advantage of Fuzzy LinPreRa.

The rest of this paper is organized as follows. Section2briefly reviews the EAM on fuzzy AHP, consistent fuzzy preference relations and Fuzzy LinPreRa. Section3presents three numerical studies to demonstrate applications of Fuzzy LinPreRa. Fi-nally, discussion and concluding remarks are presented in Sections4 and 5, respectively.

2. Methodology

This section presents three different MCDM methods. The first is EAM on fuzzy AHP proposed by Chang in 1996. The sec-ond is consistent fuzzy preference relations developed by Herrera-Viedma et al.[27]. Finally, a fuzzy linguistic preference relation is introduced[28–31].

2.1. The extent analysis method (EAM) on fuzzy AHP

Let X = {x1, . . . , xn, n P 2} be an object set, and U = {g1, g2, . . . , gm} be a goal set. According to the Chang[10]extent analysis, each object is considered separately, and for each object, the analysis is carried out for each of the possible goals, gi. There-fore, m extent analysis values for each object can be obtained as follows:

e M1

gi; eM2gi; . . . ; eMmgi; i ¼ 1; 2; . . . ; n; ð1Þ

where eMj_gi ðj ¼ 1; 2; . . . ; mÞ are all triangular fuzzy numbers (TFNs), and parameters l, m and u are the least possible value, the most possible value, and the largest possible value respectively. A TFN is represented as (l, m, u).

The EAM step can be summarized as follows:

Step 1: The value of fuzzy synthetic extent with respect to the ith object is defined as

Si Xm j¼1 e Mj gi Xn i¼1 Xm j¼1 e Mj gi " #1 ; ð2Þ

where  denotes the extended multiplication of two fuzzy numbers.

Step 2: The degree of possibility of eM2¼ ðl2;m2;u2Þ P eM1¼ ðl1;m1;u1Þ is defined as

Vð eM2P eM1Þ ¼ sup yPx minð

l

l

From the perspective of pairwise comparison times in the various approaches, for ten indicators in Example 3, the EAM needsnðn1Þ

2 ¼1092 ¼ 45 pairwise comparisons for each decision matrix, which may cause an inconsistency problem. How-ever, the Fuzzy LinPreRa only requires n  1 = 10  1 = 9 comparisons, and consistency is ensured.

Additionally, Wang et al.[38]proposed that EAM is effective for showing to what degree the priority of one decision cri-terion or alternative is higher than those of all the others in a fuzzy comparison matrix. The method cannot derive priorities from a fuzzy comparison matrix. In three examples, the weights of some decision criteria were zero, which was irrational, and the criteria were excluded from the decision analysis. However, using Fuzzy LinPreRa can avoid such unreasonable conditions.

5. Conclusion

Perfect consistency is difficult to obtain in practice, particularly when measuring preferences on a set with many natives. The concept of EAM is applied to solve the fuzzy reciprocal matrix for determining the criteria importance and alter-native performance. However, as in traditional AHP, inconsistency increases as hierarchies of criteria or alteralter-natives increase. This study adopts Fuzzy LinPreRa, a new approach for handling fuzzy AHP. This method can avoid inconsistent conditions as the number of criteria (alternative) increase.

Compared to EAM, the Fuzzy LinPreRa introduced here provides greater flexibility for solving MCDM problems with preference information about alternatives and/or attributes. The EAM requires n(n  1)/2 pairwise comparisons, but the Fuzzy LinPreRa only requires n  1 comparisons and ensures their consistency. The numerical illustrations clearly demon-strate the accuracy and efficiency of this method. The consistency of the fuzzy preference relations provided by decision makers is improved, such that inconsistent solutions in decision making processes are avoided. In conclusion, in addition to reducing the number of pairwise comparison, Fuzzy LinPreRa also enhances decision making efficiency and accuracy. Acknowledgement

The authors gratefully acknowledge the Editor and anonymous reviewers for their valuable comments and constructive suggestions. This research was supported in part by the National Science Council of the Republic of China under the grant NSC 97-2410-H-151-017 and NSC 98-2410-H-151-006.

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