Solving Fuzzy MCDM by Subtracting Benefit Criteria from Cost Criteria

Abstract This paper suggests a fuzzy MCDM (multiple criteria decision-making) approach, where ratings of alternatives versus criteria and weights of criteria are assessed in fuzzy numbers or linguistic values represented by fuzzy numbers. Criteria are classified into cost and benefit ones. In the proposed method, the ratings assigned by decision makers to each alternative versus each criterion and the weights assigned by decision makers to each criterion are averaged. The averaged cost and benefit ratings are further normalized into comparable scales respectively. The membership function of subtracting the summation of weighted normalized benefit ratings from that of weighted normalized cost ratings for each alternative can be developed by interval arithmetic of fuzzy numbers. The fuzzy number ranking method of centroid is then applied to determine the ordering of the alternatives. A numerical example of robot selection demonstrates feasibility of the proposed method.


Introduction
Fuzzy multiple criteria decision-making (MCDM) is a powerful tool for evaluation and selection of alternatives versus different criteria, where ratings of alternatives under different criteria and the importance weights of criteria are usually assessed in fuzzy numbers or linguistic values (Zadeh, 1975) represented by fuzzy numbers. Numerous fuzzy MCDM methods have been investigated. A review of many of these methods can be found in Carlsson and Fuller (1996), Ribeiro (1996), Chu and Varma (2012), and Moghimi and Anvari (2014). Some other recent methods can be found in (Afkham, et al., 2012;Akdag, et al., 2014;Büyüközkan, et al., 2011;Chung et al., 2015;Ghorbani, et al., 2013;Govindan, et al., 2013). However, clear development for the membership function of the aggregation of the fuzzy weighted ratings of each alternative cannot be found in the above works. This limitation deters their applicability to real world problems.
To resolve the above problem, a new fuzzy MCDM approach for alternative selection is suggested. In this work, criteria are categorized into cost and benefit ones. Cost criteria have the property: the smaller, the better. Conversely, benefit criteria have the property: the larger, the better. Via the proposed method, the ratings given by decision-makers to each alternative under each criterion and the weights given by decision-makers to each of the criteria are averaged. Since criteria may have incommensurable units (Chen and Hwang, 1992), all averaged cost and benefit ratings are further normalized into comparable scales respectively before weighted. Obviously, if the summation of the weighted normalized cost ratings (SWNCR) is smaller and/or the summation of the weighted normalized benefit ratings (SWNBR) is larger, the higher priority the alternative will have. The larger value the SWNBR offsets the SWNCR, the higher priority the alternative will have. Thus, the concept of subtracting SWNBR from SWNCR is used to evaluate the alternative performance (or suitability). If final fuzzy evaluation value is smaller, priority of the alternative is higher. Using interval arithmetic of fuzzy numbers can develop the membership function of subtracting SWNBR from SWNCR for each alternative. Because the final fuzzy evaluation values are still fuzzy numbers, then a ranking method is needed.
Many fuzzy number ranking methods have been proposed since fuzzy set theory was introduced by Zadeh in 1965. A comparison of many ranking methods can be seen in Wang and Kerre (2001) and Wang and Lee (2008). Some recent works can be found in (Asady, 2010;Ezzati, et al., 2012; Hari Ganesh and Jayakumar, 2014; Jafarian and Rezvani, 2013; Rao and Shankar, 2013;Sharma, 2015;Thorani, et al., 2012). Despite the merits, some methods are computational complex and others are difficult to present connection by formula between the ranking procedure and the final fuzzy evaluation values of alternatives under fuzzy MCDM model. To resolve the above limitations, this work applies the centroid method (Yager, 1981)  ) where ( ) The fuzzy number A is a triangular fuzzy numbers if its membership function A f is given by (Laarhoven and Pedrycz, 1983): where a, b and c are real numbers.

Definition 2.
The α-cut of fuzzy number A can be defined as (Kaufmann and Gupta, 1991) Given fuzzy numbers A and B, By the interval arithmetic, some main operations of A and B can be expressed as follows (Kaufmann and Gupta, 1991):

A New Fuzzy MCDM Approach
Assume that a committee of k decision-makers (i.e., D 1 ,D 2 ,...,D k ) is responsible for evaluating m alternatives (i.e., A 1 ,A 2 ,...,A m ) under n selection criteria (i.e., C 1 ,C 2 ,...,C n ), where the ratings of alternatives versus each of the criteria as well as the weights of all criteria are assessed in fuzzy numbers or linguistic values (Zadeh, 1975) represented by triangular fuzzy numbers. Both cost criteria (C), j=1~h, and benefit criteria (B), j=h+1~n, are considered.

Average ratings and Perform Normalization
Let ijt The averaged rating, C assessed by the committee of k decision-makers can be evaluated as: respectively. This method preserves the property in which the ranges of normalized triangular fuzzy numbers belong to [0,1]. The normalization of the averaged ratings, i.e. ij x 's, is as follows: (10) where ij

Average Weights
, of criterion j C assessed by the committee of k decision-makers can be evaluated as:

Develop Membership Function
The membership function of subtracting the SWNBR from the SWNCR for each alternative, i.e. We now have two simplified equations to solve, namely: Only roots in [0,1] will be retained in (14) and (15). The left membership function, i.e.

Determine the Ordering of Alternatives
Herein, an intuitive ranking method, the centroid index (Yager, 1981) According to the proposed concept, the smaller the ( ) i A C , the higher priority the alternative i A will have.
Thus, for any two final fuzzy numbers

Numerical Example
Assume that a manufacturing company needs a robot to perform a material-handling task (Liang and Wang, 1993). After preliminary screening, three robots R 1 , R 2 and R 3 are chosen for further evaluation. A committee of three decision-makers D 1 , D 2 and D 3 is formed to conduct the evaluation and selection of the three robots. Two cost criteria, purchase cost (C 1 ) and positioning accuracy (C 2 ), and two benefit criteria, programming flexibility (C 3 ) and man-machine interface (C 4 ), are considered.
Further assume that the decision-makers use the linguistic rating set S={VP, P, F, G, VG}, where VP=Very Poor=(0,0,3), P=Poor=(0,3,5), F=Fair= (2,5,8), G=Good= (5,7,10), and VG=Very Good= (7,10,10)  assessed by decision-makers for the three robots are presented in Table 1 and the suitability ratings of three robots versus benefit criteria are also shown in Table 1. The importance weights of the criteria are displayed in Table 2. By Eqs. (9) and (10), the averaged rating and normalized averaged rating of each robot i R under each criterion from the decision-making committee can be obtained as also shown in Table 1. By Eq. (11), the averaged weights of the criteria from the decision-making committee can be obtained as also shown in Table 2.

Conclusions
This work suggests a new fuzzy MCDM algorithm, which is established based on subtracting the summation of weighted normalized benefit ratings from that of weighted normalized cost ratings for each alternative. Using interval arithmetic of fuzzy numbers can then clearly develop the membership function of the final aggregation of fuzzy numbers. The centroid method of fuzzy number ranking has been applied to help determine ordering of all alternatives.
The suggested method provides an extension to the fuzzy MCDM techniques available. A numerical example has demonstrated feasibility and computational process of the proposed method.