Generalizing and Integrating TOPSIS and Cook
Generalizing and Integrating TOPSIS and Cook-Seiford method for multi-criteria group decision-making with both cardinal and ordinal data 1 1 2 Wu Li (李 武 ), Guanqi Guo (郭观七), and Xiaoqiang Zhou (周小强) 1 School of Information and Communication Eng ineering, Hunan Institute of Science and Technology, Yueyang 414006, China 2 School of Computer, Hunan Institute of Science and Technolog y, Yueyang 414006, China Correspondence should be addressed Wu Li; [email protected] Abstract: The T OPSIS and Cook-Seiford social choice function are generalized and integrated for multi-criteria grou p decision-making (MCGDM) with both cardinal evaluations and ordinal preferen ces o f the alternatives. Unlike traditional T OPSIS, at first, the group’s positive ideal solution and negative ideal solution under cardinal and ordinal preferen ces are defined respectively. T hus the group rankings of the alternatives with respect to each criteri a are derived from the individual preferences by the modified group T OPSIS considering the weights of decision makers under each criteria. Then the weighted distance function representing the total inconsistency between the comprehensive rankings of all alternatives and the ones under all criteria is presented after the criteria w eights are taken into account. Form the respective of minimizing the criteria-weighte d distance of the rankings, a nonlinear integer programming is developed and transform ed into an assignment problem to obtain the final rankings o f all alternatives. An illustrative c ase is presented and some comparisons on the results show the develop ed approach is practical and effective. T his study extends T OPSIS to group decision-making with ordinal preferences and generalizes Cook-Seiford social choice fun ction to multi-criteria decision-making considering the criteri a weights, and can be a novel benchmark for MCGDM with both cardinal and ordinal data. Keywords: Multi-criteria group decision-making (MCGDM); TOPSIS; Cook-Seiford social choice fun ction; Group’s ideal solution; Minimum distance method 1. Introduction Multi-criteria group decision-making (MCGDM) has wide applications in engineering, economics, management, military fields and so on. Its essence is assembling decision-making information, sorting and selecting the outcome through definite alternatives by a group o f decision makers according to multiple criteria [1-4]. As a typical MCGDM problem, for example, supplier sel ection in supply chain management need take into account multiple criteria including quality, delivery, perform ance history, warranties, price, technical capability and financial position [5] and is usually made by multiple experts in quality control, financial management or supply chain management, and related administrative o fficials. MCGDM often requires the experts to provide their qualitative/quantitative assessments for determining the perform ance of each alternative with respect to each criterion, and thus leads to the concurrent appearance o f both cardinal and ordinal preferences in the same MCGDM. For example, the experts usually are willing or easy to give their cardinal/ordinal preferences of alternatives for those qualitative/quantitative criteria. Also, it is quite natural to think that different experts can provide their evaluations by means of such different preference structures as preference orderings, utility functions, multiplicative preference relations, fuzzy preference relations, and so on [6, 7]. For group decision-making (GDM), fortunat ely, Chiclana and Delgado initiated a notable model with multiple preferences represented by means o f preference orderings, utility fun ctions and fuzzy preferen ce relations early in the late-1990s [8, 9]. Following them, Herrera [10], Mata [11], Dong [12, 6], Fan [13], Wang [14] and Xu [15] et al. have made great progress in the GDM with different preference structures. T heir methodologies can be divided into two categori es: trans formation function [8-12 ] and direct aggregation [6, 13-15] method. T he former transfo rms all preferen ces in different structures into a uniform fo rm and then aggregates the preferences unified to obtain the rankings o f all alternatives. Unlike the trans formation method, the latter obtain directly the rankings o f all alternatives by developing and solving some optimization models [13-15] or aggregating the individual preferences determinated firstly and separately. In spite of their success in dealing with the complexity resulted from multiple preferen ce form, the transformation method may lose decision info rmation and even cause preferen ce distortion, and in the meanwhile, the direct framework is a bit convoluted [6]. Moreover, almost all these work have not involved multiple criteria, i.e., little attention has been paid to MCGDM, especially with both cardinal and o rdinal preferen ces. In fact, the technique for o rder p erfo rmance by similarity to ideal solution (TOPSIS) [16] and Cook-Seiford social choice fun ction [17] can be ext ended and combinat ed fo r this problem, and this is just what we shall investigate in this paper. As a useful technique in dealing with multi-criteria decision-making (MCDM), T OPSIS argues that the ranking of alternatives will be based on the shortest distance from the positive ideal solution (PIS) and the farthest from the negative ideal solution (NIS). It simultaneously considers the distances to both PIS and NIS, and a preferen ce ord er is ranked according to their relative closeness, and a combination of these two distance measur es [16]. According to Shih et al. [18], T OPSIS has advantages such as: (i) a sound logic that represents the rationale of human choice; (ii) a scalar value that accounts fo r both the best and worst alternatives simultaneously; (iii) a simple computation process that can be easily programmed into a spreadshe et; and (iv) the perfo rmance measures o f all alternatives on attributes can be visualized on a polyhedron, at least for any two dimensions. Moreover, Zanakis et al. used the simulation comparison to show that T OPSIS has the fewest rank reversals among the eight methods in the category [19]. T hese advantages mentioned above make T OPSIS a major MCDM technique as compared with other related te chniques such as AHP and ELECT RE. In recent years, T OPSIS has been extended to GDM problems with interval data, fuzzy data or linguistic data [20-25]. T hese works generalizing TOPSIS for GDM can b e classi fied into two categories: ext ernal aggregation and intern al aggreg ation [18]. Outside the traditional TOPSIS procedure, the form er utilizes some operations to manipulate the alternative ratings and weight ratings [20, 21], or uses a social welfare function to obtain a final ranking from individual decision makers of the group [22]. T he latter aggreg ates the pr eference o f individuals within the T OPSIS procedu re [23 ]. Besides, in the external aggregation class, we can further distinguish the methods as pre-operation (i.e., mathematical operators fo r cardinal info rmation, refer to [20, 21]) and post-operation (i.e., Borda’s count or function fo r ordinal information, see [2]), which depend on whether the aggregation is done before or afte r the T OPSIS procedu re. It seems that extern al aggregation aims to provide more information to support a complex decision, and the internal aggregation focus es on an integrat ed decision making procedure. In spite of the success of T OPSIS in GDM, there are several problems we need to discuss fu rther. Firstly, these above works based on T OPSIS only dealt with the decision-making problem with crisp numbers, fuzzy data, interval data or linguistic information of alternatives. How to extend T OPSIS to decision-making with ordinal preferences has remain ed unknown. Secondly, these works either used mathematical operators to manipulate the alternative ratings and weight ratings given by individual decision makers before the T OPSIS procedure or c alculat ed the separation measures o f PIS and NIS for the group afte r obtaining the separation measures from individual positive ideal solution (IPIS) and individual negative ideal solution (INIS) within the T OPSIS procedure. Like traditional T OPSIS, that is to say, these works were bas ed on IPIS and INIS rather than group positive ideal solution (GPIS) and group negative ideal solution (GNIS). T hirdly, these works only considered the whole weights o f the decision makers without taking into account the weights o f them under each criteria. For the same criteria, in fact, the authoritativeness and validity of evalu ations of alternatives given by different decision maker may be different since the knowledge, ability, position, and familiarity with the decision-making problem o f di fferent decision maker are different. Thus, the weights of d ecision makers with respe ct to each criteri a have to be considered respe ctively. Lastly, it should be noted that traditional T OPSIS doubles the effects o f attribute weighting on the separation measures. Thus, the priorities of alternatives are overly controlled by attribute weights. For these reasons, we shall modify and generalize T OPSIS for MCGDM with both cardinal and ordinal preferen ces. As a good social choice method, Cook and Seiford [17 ] first defined a distance fun ction on the set of all preference orders given by multiple decision makers and which was b e proved to satisfy certain desirable properties. Then they developed a nonlinear integ er prog ramming minimizing the distance between the final rankings o f the alternatives and the ones given by all decision makers, and trans formed it to an assignment problem to obtain the rankings of the alternatives fo r the group. However, the method was used fo r GDM in single criteria setting. In fact, the Cook-Seifo rd method, although proposed initially for aggregating individual prefe rences in GDM, can be extended for MCDM to integrating the rankings of all alternatives under each criteri a into the comprehensive rankings. More importantly, it regarded the sum of the difference o f every alternative’s ranking in two preference orders as the distance o f the two preference orders and did not consider the decision makers’ weights. So the method proposed by Cook and Seiford may yi eld multiple results for the sam e decision-making problem. In fact, its distance measurement approach can be modifi ed. For these reasons above, in this paper we propose a novel methodology for MCGDM with both cardin al and ordinal data. Firstly, the GPIS and GNIS with respect to each criteria under cardinal and ordinal preferences are defined respe ctively, and then the alternatives’ rankings for the group under each criteria are derived by ou r modifi ed group T OPSIS. Secondly, considering all the criteria weights, we present a distance fun ction to denote the inconsistency between the compreh ensive rankings o f all alternatives and the ones unde r all criteri a. According to the idea that the ideal comprehensive ranking is the one which minimizes the criteria-weighted distance between itself and the ones under all criteria, a nonlinear integer prog ramming is developed. We transform it into an assignment problem to obtain the final rankings of all alternatives. Lastly, an illustrative case is presented and some comparisons on the results show the developed appro ach is practical and effective. T he remainder of this paper is organized as follows. T he proposed approach is introduced in the following section. Section 3 presents an illustrative cas e and some discussions on the results. Finally, several conclusions will be drawn in Section 4. 2.2. The proposed algorithm 2. Proposed framework To obtain the final rankings of all alternatives fo r the group, there are two processes to implement, namely (1) the aggregation process for aggregating the individual preferen ces under each criteri a into group rankings of the alternatives by modified group T OPSIS, and (2) the integration process fo r integrating the group rankings of th e alternatives under each crite ria into the final comp rehensiv e rankings of them by generalized Cook-Seifo rd method. (1) The aggregation process based on modifi ed group T OPSIS We present firstly the concepts o f GPIS and GNIS as follows. Definition 1. If the perform ance ratings o f a altern ative with respect to certain criteria Cn given by all decision makers are all the highest one, then the alternative is the GPIS with respect to criteria Cn , named GPISn . Definition 2. If the perform ance ratings o f a altern ative with respect to certain criteria Cn given by all decision makers are all the lowest one, then the alternative is the GNIS with respect to criteria Cn , named GNISn . Now we detail the proposed modi fied group TOPSIS. T he algorithm involves the following steps. Step 1. Determine GPI S n and GN ISn under criteri a In this section we shall develop a novel hybrid MCGDM model based on the combination o f the extended T OPSIS and Cook-Seiford social choice function in the presence o f both cardinal evaluations and ordinal preferences. Before presenting the proposed method we define and formulat e the MCGDM first. 2.1. The MCGDM problem formulation Without loss of generality and for the sake of simplicity, we put forward two reason able assumptions as follows. Firstly, we assume the individual cardinal preferen ces o f the alternatives are express ed by crisp numbers since our proposed approach can also work well fo r other cardinal preference fo rms (for example, interval data, fuzzy data, linguistic information and so on) as long as the following group T OPSIS is modified acco rdingly. Secondly, the criteria evaluated by cardinal preferences are set be benefit type and the cardinal evaluations have been standardized. Otherwise, they can be transfo rmed into benefit indexes by some appropriate normalization method. So we consider a MCGDM problem with L decision makers evalu ating M alternatives according to N Dl (l 1,2, , L) criteria. is decision maker, Am ( m 1, 2, , M ) is alternative, and Cn ( n 1,2, , N ) is criteria. n ( n 1, 2, , N ) is the weight of criteria C n N which satisfy n 1 and n 0 . n1 ln (l 1,2, , L; n 1, 2, , N ) is the weight of decision L maker Dl under criteria Cn which satisfy ln 1 l1 l is the performance rating of alternative and ln 0. rnm Am with respect to criteria C n given by decision maker l is a crisp number and fo r Dl . For cardinal evaluations rnm ordinal preferences it is the rank position of Am and r 1,2, , M . For certain n ( n 1, 2, , N ) , all r l nm l nm l constitute the group decision matrix Rn rnm as M L follows. rn11 rn21 rnL1 1 2 L r r rn 2 Rn n 2 n 2 1 2 L rnM rnM rnM Below our concern is to obtain the final rankings o f all alternatives in the opinion of the group of the decision makers. Cn ( n 1, 2, , N ) . GPISn rn1 , rn2 , , rnL , GNISn rn1 , rn2 , , rnL where l max rnm , for cardinal evaluations; rnl m , l 1,2, , L . 1, for ordinal preferences. l min rnm , for cardinal evaluations; rnl m , l 1, 2,, L . M , for ordinal preferences. Step 2. Calculate dnm and dnm respectively, i.e., the separation measure o f Am ( m 1, 2, , M ) from GPI S n and GNISn group. dnm under criteria Cn ( n 1, 2, , N ) for the L L l 1 l 1 nl ( rnml rnl ) 2 , dnm nl ( rnml rnl ) 2 Note: T he traditional T OPSIS doubles the effects o f attribute weighting on the separation measures. Thus, the priorities of alternatives are overly controlled by attribute weights. So we adopt the weighted Minkowski distance fun ction above to overcome this problem and also to enhance the reliability of decision. Step 3. Calculate d nm , the relative closeness of Am ( m 1,2, , M ) to criteria C n for the group. dnm GPISn ( n 1, 2, , N ) dnm d nm dnm under G Step 4. Obtain rnm , the group ranking position of M (P2 ) Am ( m 1, 2,, M ) under criteria Cn ( n 1, 2, , N ) . We can rank the alternatives using dnm in descending order. In j 1 k 1 M N n (rmG rnmG ) 2 , where m 1 rmG is the final group ranking n1 position of Am ( m 1, 2, , M ) . Now the integration process is composed of the following two steps. Step 1. Develop the optimization model to determinat e rmG , the final ranking position of Am ( m 1, 2, , M ) . As mentioned above, the comprehensive rankings o f all alternatives should minimize the weighted distance in definition 3. So the final rankings of them in the view of the group can be obtained by solving a nonlinear integer programming as follows. M (P1 ) N n (rmG rnmG ) 2 min m 1 s.t . n1 G m r 1,2, , M ( m 1, 2, , M ) G G rj rk ( j k; j , k 1,2, , M ) Step 2. Solve the optimization model above and obtain the rmG , the final ranking position of Am ( m 1, 2,, M ) . Since the nonlinear integer programming above is hard to solve by traditional approach to nonlinear programming or integer programming, we have to turn to other method. In fact, each alternative must be rank ed at a location among 1 to M , and each alternative can be located only one location in the above rang e, so the final rankings o f all alternatives can be obtained by solving the following assignment problem. N n ( k rnjG )2 (1) n 1 M s.t . x jk 1 ( j 1,2, , M ) other words, the larger d nm is, the more superior Am is with respect to criteria C n for the group. (2) T he integration process based on generalized Cook-Seiford method. Cook-Seiford social choice function calculates firstly the sum of the difference o f each alternative’ ranking position between the group and a certain individual ordering vector without considering decision maker weight, and regards the sum as the inconsistency of the group preference and the individual rankings. T hen it adds up all these sums to obtain the total inconsistency of the group and all individual ordinal preferences. Inspired by the Cook-Seiford method and unlike it, we first consider the criteria weights and calculat e the weighted devi ation between th e comprehensiv e ranking position and the one under all criteria o f single alternative (not ordering vector), then add up the weighted deviations of all alternatives to the total inconsistency between the compreh ensive rankings o f all alternatives and the ones under all criteria as follows. Definition 3. T he weighted distance representing the total inconsistency between the compreh ensive rankings o f all alternatives and the ones under all criteria is expr essed by M min x jk (2) k 1 M x jk 1 ( k 1,2, , M ) (3) j 1 x jk 0,1 ( j , k 1, 2, , M ) (4) N n (k rnjG ) 2 in formula (1) denotes the weighted n1 deviation between the compreh ensive ranking position and the ones under all criteria o f S j when its final ranking position is set at k . Formula (2) and (4) indicate that each alternative can be loc ated only one location, and similarly fo rmula (3) and (4) say that each location can also be located by only one alternative. x jk 1( j, k 1, 2, , M ) in the solution of (P2 ) implies that r Gj k , that is to say, the final group ranking position of S j is k . 3. Illustrative case and discussions on the results In this section we utilize a supplier selection example in hydroelectric project supply chain to illustrate the application of the developed app roach and then present some comparisons and analyses on the results. 3.1. An illustrative case Consider a MCGDM problem with 5 decision makers evaluating 4 altern atives according 4 criteri a. T he weights of the criteria and the decision makers with respect to each criteria are listed in table 1. Table1 T he weights of all criteria and decision makers under each criteria n n 1n 2n 3n 4n 5n 1 0.3 0.3 0.15 0.15 0.15 0.25 2 0.3 0.15 0.3 0.25 0.15 0.15 3 0.2 0.2 0.2 0.2 0.2 0.2 4 0.2 0.2 0.2 0.2 0.2 0.2 Suppose the decision makers g ave their cardinal evaluation fo r the fi rst and second criteria, and ordinal preferen ces for the latter two crite ria. The perform ance rating o f alternatives in respect to each criteria giv en by the decision makers constitute the group decision make matrix Rn ( n 1,2,3,4) as follows. 0.3 0.6 R1 0.2 0.7 0.5 0.4 0.3 0.2 0.2 0.3 0.3 0.7 0.5 0.4 , R2 0.6 0.7 0.4 0.4 0.8 0.5 0.6 0.5 0.6 0.6 0.3 0.6 0.6 0.4 0.7 0.9 0.4 0.8 0.2 0.6 0.5 0.6 0.8 0.3 0.8 0.5 4 3 3 4 4 3 3 2 3 4 3 2 1 2 2 2 1 4 1 1 R3 , R4 2 4 4 3 3 4 4 3 4 3 1 1 2 1 1 1 2 1 2 2 Now we use the developed approach to rank the alternatives and the procedu re is as follows. Step 1. Determine GPI S n and GNISn where n 1, 2, 3, 4 GPIS1 (0.7, 0.6,0.7, 0.7, 0.6), GNIS1 (0.2,0.3,0.3,0.3, 0.2); GPIS2 (0.8, 0.8, 0.9,0.8, 0.8), GNIS2 (0.2,0.2, 0.3, 0.4,0.4); GPIS3 (1,1,1,1,1), GNIS 3 (4,4, 4, 4,4); GPIS4 (1,1,1,1,1), GNIS4 (4, 4,4, 4, 4). Step 2. Calculate dnm , the relative closeness o f Am ( m 1, 2, 3, 4) to GPIS n (n 1, 2, 3, 4) . T he results are listed in table 2. Table 2 T he relative closeness of all alternatives to the corresponding GPIS m d1m d2m d3m d4m 1 2 3 4 0.223330 0.597171 0.413591 0.798026 0.294086 0.661637 0.428023 0.545316 0.192927 0.639355 0.320377 0.863473 0.360645 0.637771 0.192927 0.759747 G Step 3. Obtain rnm , the group ranking position of Am ( m 1, 2, 3, 4) under criteria Cn ( n 1, 2, 3, 4) . The result can be derived from the data in table 2 and listed in table 3. Table 3 T he rankings of all suppliers with respect to each criteria for the group m r1m G r2 m G r3m G r4 m 1 2 3 4 4 2 3 1 4 1 3 2 4 2 3 1 3 2 4 1 T hat is to say, the final group rankings of 4 alternatives are: A4 A2 A3 A1 So the best alternative is A4 . 3.2. Discussions on the results To show the effectiven ess of the proposed method, we shall solve the illustrative MCGDM problem again by other existing methods. For the cardinal evaluations, at first, we can use weighted sum method to rank the alternatives for the group. According to the weighted sum method, we can set L l eGnm lnm rnm (n 1, 2 ; m 1, 2,3, 4) , and then rank the l 1 alternatives using the index in descending order. In other words, the larger eGnm is, the more superior Am is for the group. All eGnm under the first and second criteria are as follows: e11G 0.32, e12G 0.5, e13G 0.415, e14G 0.6; eG21 0.42, eG22 0.675, e23G 0.495, e24G 0.6. So the rankings of the alternatives under these two criteria fo r the group are as follows: r11G 4 , r12G 2 , r13G 3, r14G 1; r21G 4 , r22G 1, r23G 3, r24G 2. Obviously the results are identical with the ones obtained by the proposed method above. Secondly, we utilize Borda’s count method [2] to obtain the rankings of the alternatives under the criteria evaluat ed by ordinal preferences. T he Borda’s count is defined as L G l bnm (M rnm ) ( n 3,4; m 1, 2, 3, 4) and similarly the G l 1 larger bGnm is, the more superior Am is for the group. T hus we have G b31G 2 , b32G 10 , b33G 4 , b34 14; Step 4. Determine rmG , the final group ranking position of Am ( m 1, 2, 3, 4) . N Let e jk n (k rnjG )2 ( j, k 1, 2,3,4) and then n1 constitute E e jk where ejk is located in the jth 4 4 row and kth column as follows. 2.82843 1.84391 0.89443 0.44721 0.83666 0.54772 1.37840 2.34521 E 2.23607 1.26491 0.44721 0.89443 0.54772 0.83666 1.51658 2.73861 Substituting E into (P2 ) , we obtain the solution: x14 x 22 x33 x41 1; x jk 0, for other j , k . G G b41G 5 , b42 11, b43 2 , b44G 12. So the rankings of the suppliers with respect to the criteria of supply capacity and aft er servic e fo r the group are as follows and they are also identical with the ones obtained by the proposed method above. r31G 4 , r32G 2 , r33G 3, r34G 1; r41G 3, r42G 2 , r43G 4 , r44G 1. It must be noted that the Borda’s count above do not take into account the weights of the decision makers since the weights of them in the illustrative case are just the same. Otherwise the weighted Borda’s count can be expressed as L G l bnm ln ( M rnm ) (n 3,4; m 1, 2, 3, 4) which shall l 1 brings about the same results. Now we calculate the final rankings o f all alternatives fo r the group by Bernardo method [2] which involves the following steps: Step 1. Defin e the consistency matrix Fn f jkn M M under criteria Cn ( n 1,2,, N ) where 1, when rnjG k ; f jkn ( j , k 1, 2, , M ) 0, otherwise. Step 2. Calculate the weighted consistency matrix N F f jk M M n Fn . defined, and thus the Cook-Seirford social choice function is extended for MCGDM. (3) T he proposed frame work can be a benchmark solution for MCGDM with both cardinal and o rdinal preferen ces since it can work w ell for other cardinal preferen ce forms as long as the group T OPSIS is modified accordingly. However, we do not involve the situation where there are multiple different preference structures even under one criteria. T hese problems will be left fo r our future study. n 1 Step 3. Obtain rmG , the comprehensive ranking position of Am ( m 1, 2, , M ) for the group by solving the following 0-1 programming problem: M (P3 ) M max f jk x jk Conflict of Interests T he authors declare that there is no con flict of interests regarding the publication of this paper. Acknowledgements j 1 k 1 M s.t . x jk 1, j 1, 2,..., M jk 1, k 1, 2,..., M T his work was supported by the National Natural Science Foundation of China (No. 61473118), Natural Science Foundation of Hunan Provin ce (No. 2015JJ2074) and Scientific Research Fund o f Hunan Provincial Edu cation Department (No. 13K102). k 1 M x j 1 x jk 0,1 , j , k 1, 2,..., M x jk 1( j, k 1, 2, , M ) in the solution of (P3 ) G j indicates r k , that is to say, the final ranking position of Aj is k . According to the procedur e above, we have 0 0.2 0.8 0 0.3 0.7 0 0 F 0 0 0.8 0.2 0 0.7 0.3 0 Substituting F into (P3 ) , we obtain the solution: x14 x22 x33 x41 1; x jk 0 , for other j , k . T he solution indicates the fin al rankings o f 4 alte rnatives in the view of the group are: A4 A2 A3 A1 The final result is also identical with the one obtained by the proposed method in this paper and this just indicates it is effective and rational. 4. Conclusions In this paper we investigate the MCGDM problem with both cardinal and ordinal preferences and develop a approach based on combination o f our extended g roup T OPSIS and generalized Cook-Seifo rd social choic e method. T he main contribution of this study may be summarized as follows. (1) Group ideal solution concepts fo r cardinal and ordinal preferences are presented, and thus a modified g roup T OPSIS without doubling the effects of attribute weighting on the separation measures is developed for MCGDM. (2) A weighted distance fun ction of ranking vectors is References [1] Z. Jiang, and Y. 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