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An IEW-MOORA approach for solving MCDM problems.

1. Introduction

Multiple Criteria Decision Making (MCDM) problems are characterized by the existence of two or more criteria, which are to be optimized simultaneously. However due to the conflicting nature of the criteria, it may not be possible to reach the ideal solution, which optimizes all the criteria, among the set of possible alternatives. The merit of MCDM techniques is that they consider both qualitative parameters as well as the quantitative ones. MCDM includes many solution techniques such as Simple Additive Weighting (SAW), Weighting Product (WP) (Hwang, 1981), and Analytic Hierarchy Process (AHP) (Saaty, 1980). The problem of allocating the weights of criteria when there is no preference is an open research area. Many scholars tried to tackle this problem by various techniques like Information Entropy Weight method (El-Santawy, 2012), the weighted average operator (OWA), and several other methods (El-Santawy and Ahmed, 2013).

In Information Theory, Entropy is a measure of the uncertainty in a random variable. In our context, it suits the problem of allocating weights in MCDM by quantification of the expected value of the information contained in a criterion.

In this paper we try to address this problem by employing Information Entropy Weight (IEW) method to allocate weights, and then combining the proposed method to a well-known technique called Multi-Objective Optimization on the basis of Ratio Analysis (MOORA). The new approach so-called IEW-MOORA can be used when no preference among the criteria is considered. Also, it is validated and illustrated by ranking the alternatives of a given numerical example.

The rest of this paper is organized as follows: Section 2 is made for the MOORA approach, the proposed Information Entropy Weight method is illustrated in section 3, in section 4 a numerical example is given for validation, and finally section 5 is made for conclusion.

2. MOORA

A MCDM problem can be concisely expressed in a matrix format, in which columns indicate criteria (attributes) considered in a given problem; and in which rows list the competing alternatives. Specifically, a MCDM problem with m alternatives ([A.sub.1], [A.sub.2], ..., [A.sub.m]) that are evaluated by n criteria ([C.sub.1], [C.sub.2], ..., [C.sub.n]) can be viewed as a geometric system with m points in n-dimensional space. An element [x.sub.ij] of the matrix indicates the performance rating of the ith alternative [A.sub.i], with respect to the jth criterion [C.sub.j], as shown in Eq. (1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Brauers first introduced the MOORA method in order to solve various complex and conflicting decision making problems (Brauers, 2004). The MOORA method starts with a decision matrix as shown in Eq. (1). The procedure of MOORA for ranking alternatives can be described as following: Step 1: Compute the normalized decision matrix by vector method as shown in Eq. (2)

[x.sup.*.sub.ij] = [x.sub.ij]/[square root of [m.summation over (i=1)][x.sup.2.sub.ij]] i = 1, ..., m; j = 1, ..., n. (2)

Step 2: Calculate the composite score as illustrated in Eq. (3)

[z.sub.i] = [b.summation over (j=1)] [x.sup.*.sub.ij] - [n.summation over (j=b+1)] [x.sup.*.sub.ij] i = 1, ..., m. (3)

where [b.summation over (j=1)] [x.sup.*.sub.ij] and [b.summation over (j=b+1)] [x.sup.*.sub.ij] are for the benefit and non-benefit (cost) criteria, respectively. If there are some attributes more important than the others, the composite score becomes

[z.sub.i] = [b.summation over (j=1)] [w.sub.j] [x.sup.*.sub.ij] - [n.summation over (j=b+1)] [w.sub.j] [x.sup.*.sub.ij] i = 1, ..., m. (4)

where [W.sub.j] is the weight of jth criterion.

Step 3: Rank the alternative in descending order.

Recently, MOORA has been widely applied for dealing with MCDM problems of various fields, such as economy control (Brauers and Zavadskas, 2006), contractor selection (Brauers, 2008), and consulting firms ranking (ElSantawy and Zean El-Dean, 2012).

3. Information Entropy Weight

In MCDM, the weight of a criterion reflects its importance. The Information Entropy Weight (IEW) based on the information entropy of raw data (Zhang, 2011) is combined to MOORA method in this paper. The rest of this section is made for illustrating the new method of allocating weights. Range standardization was done to transform different scales and units among various criteria into common measurable units in order to compare their weights.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

D'=[(x').sub.mxn] is the matrix after range standardization; max [x.sub.ij], min [x.sub.ij] are the maximum and the minimum values of the criterion (j) respectively, all values in D' are (0 [less than or equal to] [x'.sub.ij] [less than or equal to] 1). So, according to the normalized matrix D'= [(x').sub.mxn] the information entropy is calculated as shown in the following steps. First in order to avoid the insignificance of ln [f.sub.ij] in Eq. (7) [f.sub.ij] is stipulated as shown in Eq. (6):

[f.sub.ij] = 1 + [x'.sub.ij]/[[SIGMA].sup.m.sub.i=1](1 + [x'.sub.ij]) (6)

[H.sub.j] = - ([m.summation over (i=1)] [f.sub.ij]ln[f.sub.ij]) I = 1,2, ..., m; j =1,2, ..., n. (7)

After calculating the variation degree ([H.sub.j]), the deviation degree of the criterion (j) noted by ([G.sub.j]) is computed as in Eq. (8)

[G.sub.j] = 1 [H.sub.j] j = 1,2, ..., n (8)

It is obvious that ([G.sub.j]) is greater if the value of ([H.sub.j]) is smaller. Consequently if the ([G.sub.j]) is higher, the information entropy ([H.sub.j]) is lower, which indicates that the more the information criterion (j) provided, the greater weight given to the criterion (j). The weight ([W.sub.j]) of the criterion (j) is defined as:

[w.sub.j] = [G.sub.j]/[[SIGMA].sup.n.sub.j=1] [G.sub.j] = 1 - [H.sub.j]/n - [[SIGMA].sup.n.sub.j=1][H.sub.j] (9)

where j = 1,2, ..., n

4. Numerical Example

In the provided example, no preference exists among criteria; the absence of weights allocated to criteria is tackled by applying the Information Entropy Weight (IEW) method to assign weights to criteria. In this section, an example of six alternatives to be ranked through comparing five criteria is presented to explain the method proposed. As shown in Table 1, the six alternatives, their performance ratings with respect to all criteria, and the utility types of all criteria are presented.

In the above example, there are no weights specified for the criteria by the decision maker, so the proposed method will be applied. Table 2 illustrates the range standardization done to decision matrix as in Eq. (5).

Table 3 shows the values of the variation degree ([H.sub.j]), the deviation degree ([G.sub.j]), and the weight assigned to each criterion ([W.sub.j]) based on information entropy as in Eqs. (7, 8, and 9).

By applying the procedure of MOORA as illustrated in section 2; the benefit, cost, and composite scores are obtained for all alternatives as shown in Table 4. The second alternative should be selected because it has the maximum composite score.

5. Conclusion

In this paper, the MOORA method is combined to the Information Entropy method to constitute a new approach called IEW-MOORA in order to rank the alternatives when no preference is found (i.e. no weights are provided for the criteria). The proposed method assign weights to criteria relative to the quantity of information gained. This process is suitable for MCDM and can be extended to other applications and methods.

References

Brauers, W. K. M. (2008), "Multi-objective contractor's ranking by applying the MOORA method", Journal of Business Economics and Management, 2: 245-255.

Brauers, W. K. M. and Zavadskas, E. K. (2006), "The MOORA method and its application to privatization in a transition economy", Control and Cybernetics, 35(2): 445-469.

Brauers, W. K. M. (2004), Optimization methods for a stakeholder society, A revolution in economic thinking by multiobjective optimization, Kluwer Academic Publishers, Boston.

El-Santawy, M. F. and Ahmed, A. N. (2013), "A Standard Deviation Approach for Allocating Weights in Multi-Criteria Decision Making", Wulfenia Journal, 20(12): 336-341.

El-Santawy, M. F. and Zean El-Dean, R. A. (2012), "Selection of a Consulting Firm by Using SDV-MOORA", Life Science Journal 9(2s):126-128.

El-Santawy, M. F. (2012) "A VIKOR Method for Solving Personnel Training Selection Problem", INTERNATIONAL JOURNAL OF COMPUTING SCIENCE, ResearchPub, 1(2): 9-12.

Hwang, C.L. and Yoon, K. (1981), Multiple Attributes Decision Making Methods and Applications, Heidelberg: Springer, Berlin.

Saaty, T.L. (1980), The Analytic Hierarchy Process, McGraw-Hill, NewYork.

Zhang, H., Gu, C.L., Gu, L.W. and Zhang, Y. (2011), "The evaluation of tourism destination competitiveness by TOPSIS & information entropy--A case in the Yangtze River Delta of China", Tourism Management, 32 : 443-451.

Mohamed F. El-Santawy

Department of Operations Research

Institute of Statistical Studies and Research (ISSR)

Cairo University, Egypt

E-mail: lost_zola@yahoo.com
Table 1. Decision matrix

                  [C.sub.1]   [C.sub.2]   [C.sub.3]

Utility type         Max         Min         Max

Alternative 1        48          14           5
Alternative 2        56          20          20
Alternative 3        23          24          10
Alternative 4        58          12          12
Alternative 5        69          17          10
Alternative 6        23          10           8

                  [C.sub.4]   [C.sub.5]

Utility type         Min         Max

Alternative 1        14          56
Alternative 2        18          44
Alternative 3        22          50
Alternative 4        24          36
Alternative 5        16          32
Alternative 6        17          45

Table 2. Range standardized decision matrix

                [C.sub.1]   [C.sub.2]   [C.sub.3]

Alternative 1   0.543       0.286       0
Alternative 2   0.717       0.714       1
Alternative 3   0           1           0.333
Alternative 4   0.761       0.143       0.467
Alternative 5   1           0.5         0.333
Alternative 6   0           0           0.2

                [C.sub.4]   [C.sub.5]

Alternative 1   0           1
Alternative 2   0.4         0.5
Alternative 3   0.8         0.75
Alternative 4   1           0.167
Alternative 5   0.2         0
Alternative 6   0.3         0.542

Table 3. Weights assigned to criteria

             [H.sub.j]    [G.sub.j]   [W.sub.j]

[C.sub.1]    1.58583977   -0.585840    0.20090
[C.sub.2]    1.58966902   -0.589669    0.20221
[C.sub.3]    1.58414753   -0.584148    0.20032
[C.sub.4]    1.57794522   -0.577945    0.19819
[C.sub.5]    1.57851685   -0.578517    0.19839

Table 4. Ranking lists and scores

                 Benefit      Cost     Composite
                 criteria   criteria     score      Rank

Alternative1      0.2162     0.1288      0.0875      3
Alternative2      0.3118     0.1753      0.1365      1
Alternative3      0.1985     0.2121     -0.0136      6
Alternative4      0.2451     0.1619      0.0831      4
Alternative5      0.2422     0.1520      0.0902      2
Alternative6      0.1755     0.1221      0.0535      5
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Title Annotation:multiple criteria decision making; Information Entropy Weight-Multi-Objective Optimization on the basis of Ratio Analysis
Author:El-Santawy, Mohamed F.
Publication:Computing and Information Systems
Geographic Code:1USA
Date:Feb 1, 2014
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