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A Novel Multi-attribute Allocation Method Based on Entropy Principle



Siliang Wang
 
ABSTRACT

This research studies the key problem of multi-attribute allocation in multi-attribute decision. Proper attribute allocation is very important in the decision process. In order to solve uncertainty from distribution of attributes, a novel multi-attribute allocation method is presented based on optimization theory and maximum entropy principle. A linear combination weights mathematical model is also proposed through mathematical derivation. Numerical results are provided using typical test data and prove the efficiency of the novel method.

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  How to cite this article:

Siliang Wang , 2012. A Novel Multi-attribute Allocation Method Based on Entropy Principle. Journal of Software Engineering, 6: 16-20.

DOI: 10.3923/jse.2012.16.20

URL: https://scialert.net/abstract/?doi=jse.2012.16.20
 
Received: November 14, 2011; Accepted: December 19, 2011; Published: March 20, 2012

INTRODUCTION

Multi index evaluation is widely used in many fields, such as social economy, engineering technology and military activities (Zaras, 2001; Hwang and Yoon, 1981; Kim and Ahn, 1999). The decision maker makes the decision based on weighing up the multi-object plans. In multi-attribute decision model, the weights of the attributes play an important role. Proper attribute allocation is the most important problem in the multi index evaluation system.

Recently, the combination assignment methods are proposed in combination of subjective and objective weights. There are two kinds of combination assignment. One is multiplication combination assignment and the other is addition combination assignment. The multiplication combination assignment is applied to the problem that many indexes exist and the distribution of the index is uniform. However, the multiplication operate would cause “multiplication effect” (Guo and Guo, 2005; Liang et al., 2005). That means the bigger index would turn bigger while the smaller turns smaller. Thus, the method of multiplication combination assignment is limited for its application (Ma et al., 1999). By contrast, the method of addition combination assignment is widely used in multi-attribute decision model. The addition combination assignment is defined linear combination weights and the most used method is subjective and objective combination assignment based on optimization theory. The studies about the linear combination weights are popular, such as combination weights assignment based on matrix theory, expert evaluation and so on (Chu et al., 1979).

From the view of mathematical statistic, the real weights of each index are considered as random variables. The coefficients of weights computed by different assignment method are sample values. All the studies cited above omit the uncertainty caused from the random variables. In order to solve the uncertainty problem, this study presents a novel linear combination assignment method which is based on optimization theory and Jaynes maximum entropy principle. A novel mathematical model is also build through the theory analysis. The final numerical example proves its feasibility.

PROBLEM DESCRIPTION

Assume n plans that are yet to be assessed, P = {P1, P2,…, Pn} and m indexes (or named objectives), I = {I1, I2,…, Im}, the evaluation value of plan Pi related with the index Ij is defined as aij = (i = 1,2,…n, j = 1,2,…m). The assessment matrix is A = [aij]nxm. For its subjective and objective weights assignment, the index assessment vector is W1,…,Wl. The kth weights vector is Wk = (wk1, wk1,…, wkm). Before the calculation, the assessment matrix should be normalized and the matrix is defined as R = [rij]nxm.

Assume Ob1, Ob2, Ob3 are sets of subscript of cost, benefit and fix type indexes, respectively, then, the relationship between them is Ob1∪Ob2∪Ob3 = {1,2,…, m}, and Obs∩Obt = ø, (s≠t, s,t = 1,2,3).

For the cost type index Ij, the element in normalization matrix R is:

(1)

For the benefit type index Ij, the element in normalization matrix R is:

(2)

For the fix type index Ij, the element in normalization matrix R is:

(3)

where, pjmin = min {pij|i = 1,2,…,n}, pjmax = max {pij|i = 1,2,…,n}, denote minimum value and maximum value of evaluation index Ij. In Eq. 3, qij = |pijj| and αj is the idea value. qjmin = min {qij|i = 1,2,…,n}, qjmax = min {qij|i = 1,2,…,n} denote minimum value and maximum value of evaluation index Ij at the type of fix. The idea plan is the best plan and its corresponding matrix is composed with all elements“1”(Xu, 2004).

According to the definition of the entropy principle (Jaynes, 1957), the plan Obj’s entropy is:

(4)

where, H = {hij}nxm and:

(5)

Compute the objective’s weight vector according to method of the subjective and objective assignment. The vector is w = (w1, w2,…,wm) and the element of the vector is:

(6)

Normalize the weights vector and then the combination weights of the jth index is:

(7)

where, W′ = w′1, w′1,…,w′Am is the subjective weight vector. As a result, the generalized distance between the objective i and idea point is:

(8)

From the Eq. 8, the decision maker can select the optimal weights assignment. However, the subjective or objective weight is only considered solely. The combination weight is often used to evaluate the system, which includes subjective and objective weights. The recent methods dealt with the combination weights are often using linear combination weights. The methods omit the uncertainty of weights. The following part of the paper presents a novel method of linear combination weights based on entropy principle and optimization theory to solve the uncertainty.

THE METHOD OF COMBINATION WEIGHTS BASED ON ENTROPY PRINCIPLE AND OPTIMIZATION THEORY

The combination weight is composed of subjective weight and objective weight, namely, assumes is the subjective weight and is the objective weight. The linear combination weight is . However, the coefficient a is defined in advance. The way of the linear combination weight still belongs to subjective weight assignment. Thus, how to assign the distribution of subjective weight and objective weight is the main aim of the study.

The generalized distance between objective Pi and idea objective is:

(9)

The aim of solution of linear combination weight vector is to ensure the value of x that can make the minimum distance between all objectives and idea objectives. That is,

(10)

For the benefit type index, the element in normalization matrix is:

(11)

On the other side, in order to solve the uncertainty, the distribution of coefficient of the linear combination weight should be defined according to the Jaynes maximum entropy principle (Li, 1987).

(12)

(13)

Through the entropy principle, the assignment of the distribution of subjective weight and objective weight is distributed in the paper.

NUMERICAL EXAMPLES AND ANALYSIS

In stochastic time varying transportation networks, route choice often relies on many factors, such as travel time (I1), travel distance (I2), cost (I3), risk (I4), reliability (I5), flow (I6). The following table gives the data for four plans (or tests) with six indexes.

Firstly, compute the normalization assessment matrix R according to Eq. 1 and 2. Among those indexes in Table 1, the reliability (I5) and flow (I6) belong to benefit type indexes while others belong to cost type indexes. Consequently, the matrix R is

(14)

Expert’s subjective weight vector is assumed to be

(15)

Apply the coefficient into equation to get the combination assessment value of four plans.

(16)

where Vi is the combination assessment value of plan Pi. The value of Vi is in the last row in Table 2.

Table 1: Expert assessment value in route choice

Table 2: Assessment result by using different weight vectors

CONCLUSION

This paper presents a novel computation method of index weight using Jaynes maximum entropy and optimization theory. We built a novel linear combination weight model based on generalized distance and entropy principle. The model not only considers the distance, but also the uncertainty of the weight coefficient. The final numerical example proves its feasibility. The future work is to practice in real multi-attribute decision application.

REFERENCES
Chu, A.T.W., R.E. Kalaba and K. Spingarn, 1979. A comparison of two methods for determining the weights of belonging to fuzzy sets. J. Optim. Theor. Applied, 27: 531-538.

Guo, C. and H. Guo, 2005. Approach of multiple attribute group decision making with different forms of preference information. Syst. Eng. Electron., 27: 63-65.

Hwang, C.L. and K. Yoon, 1981. Multiple Attribute Decision Making-Method and Applications, A State-of-the-Art Survey. Springer Verlag, New York.

Jaynes, E.T., 1957. Information theory and statistical mechanics. Phys. Rev., 106: 620-630.
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Kim, S.H. and B.S. Ahn, 1999. Interactive group decision making procedure under incomplete information. Eur. J. Oper. Res., 116: 498-507.
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Li, X.S., 1987. Entropy and Optimization. The University of Liverpool Press, UK.

Liang, L., L. Xiong and G. Wang, 2005. New method of determining the objective weight of decision makers in group decision making. Syst. Eng. Electron., 27: 652-655.

Ma, J., Z.P. Fan and L.H. Huang, 1999. A subjective and objective integrated approach to determine attributes weights. Eur. J. Oper. Res., 112: 397-404.
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Xu, X., 2004. A note on the subjective and objective integrated approach to determine attribute weights. Eur. J. Oper. Res., 156: 530-532.
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Zaras, K., 2001. Rough approximation of a preference relation by a multi-criterion stochastic dominance for determinist and stochastic evaluation problems. Eur. J. Operat. Res., 130: 305-314.
CrossRef  |  Direct Link  |  

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