Rank Similarity based MADM Method Selection

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1 Rank Similarity based MADM Method Selection Subrata Chakraborty School of Electrical Engineering and Computer Science CRC for Infrastructure and Engineering Asset Management Queensland University of Technology Queensland 4000, Australia Chung-Hsing Yeh Clayton School of Information technology Faculty of Information Technology Monash University Victoria 3800, Australia Abstract Selecting the most suitable Multiple Attribute Decision Making (MADM) method for a given MADM problem is a challenge for the decision maker. When there are several suitable MADM methods available for the problem, the challenge is even greater. We present a novel MADM method selection approach based on the Spearman s rank correlation. The approach will help the decision maker in selecting the most preferred MADM method from a set of suitable and acceptable methods. The most preferred MADM method is the one that produces the most preferred outcome. The most preferred outcome is the one which is closest to all other outcomes. The closeness between the ranking outcomes are measured in terms of the similarity between them. I. INTRODUCTION Multiattribute decision making (MADM) problems are diverse greatly in terms of the decision information, the decision context and the applications. With the availability of multiple suitable MADM methods M k (k = 1,2,...,K) for a given MADM problem θ, selecting the most suitable one is an extremely challenging task [1] [2]. Several comparative and simulation based studies suggest the suitability of certain methods under given decision settings [3] [4] [5] [6]. Under certain decision contexts, the decision maker may use the results of these studies for method evaluation and selection. In a decision context where a suitable and acceptable set of MADM methods M k is available for a given problem θ, the decision maker needs to select the most preferred one among them. This decision context considered can be specified as: (a) The MADM problem involves one single decision maker only. (b) The decision maker can use a set of suitable MADM methods for a given problem. All these methods produce acceptable outcomes, but the decision maker must choose one method as the most preferred one among them. Current challenges for this decision context include: (a) Previous studies consider that only one method is suitable for a given problem and all other methods are not acceptable [3] [7] [8] [9] [2] [10]. A new method selection approach is required which can compare a set of suitable methods to find the most preferred one. (b) Objective performance measures need to be developed to find the most preferred method from a set of suitable methods. In this study a novel method selection approach is developed to select the most preferred method from a set of suitable and acceptable MADM methods M k for a given problem θ. The approach considers the similarities between the ranking outcomes produced by the given set of suitable MADM methods M k. In the following sections we describe the general MADM problem and introduce the new approach, followed by a numerical example to illustrate the approach. II. METHODOLOGY DEVELOPMENT A. General MADM problem The general MADM problem θ involves the following: (a) A set of Q decision makers D q ;q = 1,2,...,Q. (b) A set of I alternatives A i ;i = 1,2,...,I. (c) A set of J attributes C j ;j = 1,2,...,J. (d) A set of J attribute weights W j ;j = 1,2,...,J. (e) Performance ratings x ij ;i = 1,2,...,I;j = 1,2,...,J. The MADM problem θ may have only one decision maker D q (Q = 1) or a group of decision makers D q (Q 1). The set of decision alternatives A i includes various decision options the decision maker is considering for the given decision problem which are to be evaluated and ranked. For example, a buyer may have several options while buying a car. The set of attributes C j are the selection criteria the decision maker considers while evaluating the decision alternatives A i. For example, the car buyer may evaluate the car options based on price, comfort, mileage and performance. The set of attribute weights W j represents the relative importance of the attributes C j to the decision maker. For example, the car buyer may consider that price is more important than comfort and performance, hence will have a higher attribute weight. The attribute weights are presented as a vector W as shown in Equation 1. The performance rating x ij represents the assessment scores provided by the decision maker D q for each alternative A i with respect to each attribute C j. All the performance ratings x ij for all the alternatives A i in relation to all the attributes C j can be represented as a decision matrix X as shown in Equation 2, where rows and columns represent alternatives and attributes respectively [11] [12] [2]. W = {W j };j = 1,2,...,J (1)

2 X = x 11 x x 1J x 21 x x 2J x I1 x I2... x IJ (2) where i = 1,2,...,I and j = 1,2,...,J. With the set of alternatives A i and the set of attributes C j defined, the general MADM problem θ can be represented as a combination of the decision matrix X and the weight vector W by using Equations 1 and 2 as θ = {X,W} (3) To solve the MADM problem θ, a number of suitable MADM methods M k (k = 1,2,...,K) are available. The MADM methods M k generally require a normalisation procedure and an aggregation technique. The normalisation procedures are used to transform the performance rating x ij to a comparable unit as they may have diverse measurement units. The aggregation technique is applied to combine normalised performance ratings with the attribute weights W j to obtain an overall value V i (i = 1,2,...,I) for each alternative A i. The overall value V i is used to evaluate and rank the decision alternatives A i. B. Rank Similarity and Method Evaluation The new method selection approach is developed for dealing with the decision context where each of the outcomes produced by a set of suitable MADM methods M k are considered valid and acceptable to the decision maker. The most preferred method is to be chosen from the set of suitable MADM methods M k depending on the most preferred ranking outcome. The solution space is considered to be limited, as it consists of the ranking outcomes produced by a specific set of suitable MADM methods M k for the given problem θ. Hence, the most preferred outcome must be among the ranking outcomes produced. For a given MADM problem θ, the solution space consists of different ranking outcomes O k (k = 1,2,...,K) produced by each suitable Method M k, which are all valid and acceptable to the decision maker. The most preferred MADM method is the one that produces the most preferred outcome. The most preferred outcome is the one which is closest to all other outcomes. The closeness between the ranking outcomes can be measured in terms of the similarity between them. In this new approach, the similarity between two ranking outcomes is measured by using the rank correlation coefficient [13]. The method which produces the outcome most similar to all other outcomes is the most preferred method for the given problem. C. The Rank Correlation Coefficient The rank correlation coefficient is widely used as a measurement of association between different ranks [14] [15]. It has been successfully applied in various studies to test the sensitivity and significance of certain information in different MADM problem settings [4] [7] [16]. The rank correlation coefficient between two ranks can be defined as 6 I ρ = 1 d 2 i i=1 I 3 1 ;i = 1,2,...,I (4) where d i is the difference between the ranks for the decision alternatives A i (i = 1,2,...,I). D. Rank Similarity Index The rank similarity index (RSI) is developed as a measure of decision outcome similarity for an MADM method with all the other suitable MADM methods in the set of acceptable methods. This measure indicates the relative closeness of a method with other methods in terms of ranking outcome similarity. The RSI is the average of the rank correlation coefficients between a ranking outcome and all the other ranking outcomes. The method with the largest RSI indicates that the ranking outcome it produces is most similar or closest to all other outcomes, hence the most preferred one. The rank similarity index can be obtained through four stages as shown in Figure 1. Fig. 1. Rank similarity index methodology

3 Stage I: Generate the rank matrix (R k ): This step involves solving the decision problem with each MADM method in the acceptable set and obtaining the ranking outcomes. The outcomes are presented as a matrix called the rank matrix (R k ), formed by combining the ranking outcomes O k produced for the alternatives A i by Methods M k as shown in Equation 5. R k = r 11 r r 1K r 21 r r 2K r I1 r I2... r IK (5) where r ik (1 r ik I) represents the rank of alternative A i (i = 1,2,...,I) by using Method M k (k = 1,2,...,K). Stage II: Calculate rank correlation (RC) between ranking outcomes: The rank correlations for a Method M k in relation to each of the other remaining Methods M h (h = 1,2,...,K;k h) are calculated by applying Equations 4 and 5 as RC kh = ρ(o k,o h ) (6) where k = 1,2,...,K;h = 1,2,...,K;k h. Stage III: Calculate the rank similarity index (RSI) for each method: The rank similarity index for Method M k can be calculated by taking the average of correlations calculated by Equation 6 as K RC kh h=1 RSI k = K where k = 1,2,...,K;h = 1,2,...,K;k h. Stage IV: Find the largest rank similarity index (RSI + ): The largest RSI from the set of RSIs calculated by Equation 7 can be obtained as (7) RSI + = maxrsi k (8) where k = 1,2,...,K. The method with the largest rank similarity index (RSI + ) is the most preferred one for the given MADM problem. III. NUMERICAL EXAMPLE A. Methods Used in the Example In this example, variants of the three widely used MADM methods are used, including: (a) the simple additive weighting (SAW) [17] [18] [19], (b) the technique for order preference by similarity to ideal solution (TOPSIS) [11], and (c) the weighted product (WP) [20] [21] [22]. 1) SAW method: The SAW method, also known as the weighted sum method, is probably the best known and most widely used MADM method [11]. The basic logic of the SAW method is to obtain a weighted sum of the performance ratings of each decision alternative over all the attributes. The overall weighted preference value is used as the basis for comparison between the alternatives. This method involves the following two steps: Step 1 (Obtain the normalised decision matrix:) The performance ratings x i j in the decision matrix X shown in Equation 2 are normalised by applying linear scale transformation (Max) normalisation technique. This procedure divides the performance ratings x i j for alternatives A i with respect to each attribute C j by the maximum performance rating for that attribute. For benefit and cost attributes, the normalised performance rating y ij (i = 1,2,...,I;j = 1,2,...,J) is obtained by Equations 9 and 10 respectively. y ij = x ij x max j y ij = 1 x ij x max j (9) (10) where x max j is the maximum performance rating among alternatives for attribute C j. The normalised performance ratings y ij can be arranged as a matrix shown in Equation 11. Y = y 11 y y 1J y 21 y y 2J y I1 y I2... y IJ (11) Step 2 (Obtain the overall preference value): The overall preference value for alternative A i can be obtained by combining the attribute weights W j from Equation 1 with the normalised performance rating from Equation 11 as V i = J W j y ij (12) Where V i (i = 1,2,...,I) is the overall preference value of decision alternative A i [11] [23]. An alternative with a greater overall value V i will receive a higher ranking. 2) TOPSIS method: The TOPSIS method has been used extensively to solve various practical MADM problems, due to its simplicity, computational efficiency and the ability to measure the performances of the decision alternatives in simple mathematical form [24]. In TOPSIS, an index known as similarity to positive-ideal solution is defined by combining the closeness to the positive-ideal solution and remoteness to the negative-ideal solution. This index is used to rank the alternatives [11] [23]. We will refer to the index as the overall preference value in order to maintain uniformity with other methods used. The TOPSIS method involves the following steps. Step 1 (Calculate the normalised performance ratings:) The performance ratings x ij in the decision matrix X shown in Equation 2 are normalised by applying vector normalisation procedure. In this procedure, each performance rating x ij in

4 the decision matrix X is divided by its norm. The normalised value y ij (i = 1,2,...,I;j = 1,2,...,J) is obtained by y ij = x ij I x 2 ij i=1 (13) This procedure has the advantage of converting all attributes into dimensionless measurement unit, thus making interattribute comparison easier. But it has the drawback of having non-equal scale length leading to difficulties in straightforward comparison [25] [5]. The normalised performance ratings y ij can be given as a matrix similar to Equation 11. Step 2 (Calculate weighted normalised performance rating:) The weight W j from Equation 1 is combined with the normalised decision matrix Y from Equation 11 to get the weighted normalised performance rating v ij (i = 1,2,...,I;j = 1,2,...,J) shown in Equation 14. The weighted normalised decision matrix is shown in Equation 15. v ij = W j y ij (14) V = v 11 v v 1J v 21 v v 2J v I1 v I2... v IJ (15) Step 3 (Identify the positive-ideal and negative-ideal solutions): The set of positive-ideal solution A and negativeideal solution A are identified from Equation 15 in terms of weighted normalised performance ratings as where A = [v 1,v 2,...,v J] (16) A = [v 1,v 2,...,v J ] (17) { vj maxvij if j is a benefit attribute = minv ij if j is a cost attribute { v j = minvij maxv ij if j is a benefit attribute if j is a cost attribute Step 4 (Calculate separation measure): The separation measures for each decision alternative A i is calculated using n-dimensional Euclidean distance. The separation (distance) of each alternative from the positive-ideal solution A and negative-ideal solution A can be obtained by Equation 18 and 19 respectively. Si = J (v ij vj )2 (18) S i = J (v ij v j )2 (19) Step 5 (Obtain the overall preference value): The overall preference value V i for each alternative A i can be calculated as S i V i = S i +Si (20) A higher value of V i indicates a higher ranking for alternative A i. 3) WP method: With given performance ratings x ij in the decision matrix X as shown in Equation 2 and Weights W j for attribute C j (j = 1,2,...,J) as shown in Equation 1, the overall preference value V i for alternatives A i is calculated as V i = J x wj ij (21) The alternatives A i are ranked according to the value of V i. A higher V i value indicates a higher ranking for the alternative A i. Table I shows the nine suitable MADM methods evaluated in this example. These methods include four variants of SAW, four variants of TOPSIS and the WP method. The SAW and TOPSIS variants were developed using 4 different normalisation procedures [26]. TABLE I SET OF MADM METHODS USED IN THE EXAMPLE MADM method Normalisation procedure Aggregation technique M 1 N 1 : Vector normalisation SAW M 2 N 2 : Linear scale (max-min) SAW M a 3 N 3 : Linear scale (max) SAW M 4 N 4 : Linear scale (sum) SAW M b 5 N/A WP M c 6 N 1 : Vector normalisation TOPSIS M 7 N 2 : Linear scale (max-min) TOPSIS M 8 N 3 : Linear scale (max) TOPSIS M 9 N 4 : Linear scale (sum) TOPSIS a This is conventional SAW method b This is conventional WP method c This is conventional TOPSIS method B. The Example To illustrate the rank similarity based method selection approach, the decision matrix from the graduate fellowship applicants ranking case is used [25]. Table II shows the decision matrix. The attributes weights for the decision problem are given as W = (0.3,0.1,0.3,0.15,0.15).

5 TABLE II DECISION MATRIX USED Attributes Alternatives C 1 C 2 C 3 C 4 C 5 A A A A A A The methods shown in Table I produce different ranking outcomes which are shown as a rank matrix in Table III by applying Equation 5. The rank correlation coefficients for each method with respect to other methods are calculated by applying Equation 6 on Table III, and the results are shown in Table IV. The rank similarity index is calculated by applying Equation 7 on Table IV and the results are shown in Table V. TABLE III RESULTANT RANK MATRIX Methods AlternativesM 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 A A A A A A TABLE IV RANK CORRELATION COEFFICIENTS BETWEEN RANKS M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 M 1 1 (.086) M 2 (.086) (.543) (.257) M M (.086).143 (.086).086 M (.543) (.2) (.429) (.257).2 M (.086) (.2) M (.429) (.086) M (.086) (.257) M (.257) (.086) TABLE V RANK SIMILARITY INDEX FOR MADM METHODS M 1 M 2 M 3 M 4 M 5 M 6 M 7 M 8 M 9 RSI From Table V, we can select the largest RSI using Equation 8 as RSI + = RSI(M 3 ) = This suggests that Method M 3 produces the ranking outcome most similar to the ranking given by all other methods. Hence, this method is the most preferred one for the given MADM problem under the decision context considered. These results can be used in conjunction with other decision contexts where the decision maker is considering multiple contexts and can select a method which most satisfies all the contexts. In this particular example, it is observed that the conventional SAW method (i.e. Method M 3 ) is the best performer, and conventional TOPSIS method (i.e. Method M 6 ) and WP method (i.e. method M 5 ) do not perform well. This highlights the need for a change in the way the existing method comparison and selection studies are conducted. These results reinforce the argument that MADM methods considered for selection should not just include the ones originally developed or commonly applied (such as M 3, M 5 and M 6 in Table I). Instead, the comparisons must be done at more detail levels including normalization procedures and aggregation techniques, wherever possible. IV. CONCLUSION The rank similarity based MADM method selection approach developed provides an efficient, yet simple context dependent approach for a given MADM problem. Although the illustrated example has used the variants of SAW, TOPSIS and WP methods only, the approach is applicable for selecting from any set of MADM methods capable of producing a complete ranking outcome. The importance of applying a problem specific method selection approach for any given MADM problem rather than the generalised selection approach has also been highlighted. REFERENCES [1] S. Chakraborty and C.-H. Yeh, Consistency comparison of normalization procedures in multiattribute decision making, WSEAS Transactions on Systems and Control, vol. 2, no. 2, pp , [2] C.-H. Yeh, The selection of multiattribute decision making methods for scholarship student selection, International Journal of Selection and Assessment, vol. 11, no. 4, pp , [3] L. Simpson, Do decision makers know what they prefer?: Mavt and electre ii, Journal of the Operational Research Society, vol. 47, no. 7, pp , [4] S. H. Zanakis, A. Solomon, N. Wishart, and D. S, Multi-attribute decision making: A simulation comparison of select methods, European Journal of Operational Research, vol. 107, no. 3, pp , [5] D. L. Olson, Comparison of three multicriteria methods to predict known outcomes, European Journal of Operational Research, vol. 130, no. 3, pp , [6] S. Chakraborty and C.-H. Yeh, A simulation comparison of normalization procedures for topsis, in Proceedings of the International Conference on Computers and Industrial Engineering (CIE39). IEEE, 2009, pp [7] E. Triantaphyllou and A. Sanchez, A sensitivity analysis approach for some deterministic multi-criteria decision making methods, Decision Sciences, vol. 28, no. 1, pp , [8] A. Guitouni and J.-M. Martel, Tentative guidelines to help choosing an appropriate mcda method, European Journal of Operational Research, vol. 109, no. 2, pp , [9] C.-H. Yeh, A problem-based selection of multi-attribute decisionmaking methods, International Transaction in Operational Research, vol. 9, no. 2, pp , [10] K. Cho, Multicriteria decision methods: An attempt to evaluate and unify, Mathematical and Computer Modelling, vol. 37, no. 9-10, pp , [11] C. L. Hwang and K. Yoon, Multiple Attribute Decision Making: Methods and Applications: A State of the Art Survey. Berlin, Heidelberg, New York: Springer-Verlag, [12] V. Belton and T. J. Stewart, Multiple Criteria Decision Analysis: An Integrated Approach. Boston/ Dordrecht/ London: Kluwer Academic Publishers, [13] C. Spearman, The proof and measurement of association between two things, The American Journal of Psychology, vol. 15, no. 1, pp , [14] M. Kendall, Rank Correlation Methods, 3rd ed. [15] K. S. Raju and C. R. S. Pillai, Multicriterion decision making in river basin planning and development, European Journal of Operational Research, vol. 112, no. 3, pp , 1999.

6 [16] M. Yurdakul and T. I. C. Yusuf, Application of correlation test to criteria selection for multi criteria decision making (mcdm) models, International Journal of Advanced Manufacturing Technology, vol. 40, no. 3-4, pp , [17] C. W. Churchman and R. L. Ackoff, An approximate measure of value, Journal of the Operations Research Society of America, vol. 2, no. 2, pp , [18] K. R. MacCrimmon, Decision making among multiple-attribute alternatives: A survey and consolidated approach, ARPA, Tech. Rep. RAND Memorandum RM-4823-ARPA, [19] A. J. Klee, The role of decision models in the evaluation of competing environmental health alternatives, Management Science, vol. 18, no. 2, pp. B52 B67, [20] P. W. Bridgman, Dimensional Analysis. New Haven, CT: Yale University Press, [21] M. K. Starr, Production Management. Englewood Cliffs, NJ: Prentice- Hall, [22] K. P. Yoon, The propagation of errors in multiple-attribute decision analysis: A practical approach, Journal of the Operational Research Society, vol. 40, no. 7, pp , [23] M. Zeleny, Multiple Criteria Decision Making. New York: Mcgraw- Hill, [24] C.-H. Yeh and Y.-H. Chang, Modelling subjective evaluation for fuzzy group multicriteria decision making, European Journal of Operational Research, vol. 194, no. 2, pp , [25] K. P. Yoon and C.-L. Hwang, Multiple Attribute Decision Making: An Introduction. London, New Delhi: Thousand Oaks, Sage Publications, [26] S. Chakraborty and C.-H. Yeh, Comparing normalization procedures in multiattribute decision making under various problem settings, in Fifth International Conference on Information Technology in Asia (CITA07), 2007, pp

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