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 multiobject
plans. In multiattribute 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 multiattribute 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 = {P_{1}, P_{2},…, P_{n}} and m indexes (or named objectives), I = {I_{1}, I_{2},…, I_{m}}, the evaluation value of plan P_{i} related with the index I_{j} is defined as a_{ij} = (i = 1,2,…n, j = 1,2,…m). The assessment matrix is A = [a_{ij}]_{nxm}. For its subjective and objective weights assignment, the index assessment vector is W^{1},…,W^{l}. The kth weights vector is W^{k} = (w^{k}_{1}, w^{k}_{1},…, w^{k}_{m}). Before the calculation, the assessment matrix should be normalized and the matrix is defined as R = [r_{ij}]_{nxm}.
Assume Ob_{1}, Ob_{2}, Ob_{3} are sets of subscript of cost, benefit and fix type indexes, respectively, then, the relationship between them is Ob_{1}∪Ob_{2}∪Ob_{3} = {1,2,…, m}, and Ob_{s}∩Ob_{t} = ø, (s≠t, s,t = 1,2,3).
For the cost type index I_{j}, the element in normalization matrix R is:
For the benefit type index I_{j}, the element in normalization matrix R is:
For the fix type index I_{j}, the element in normalization matrix R is:
where, p_{j}^{min} = min {p_{ij}i = 1,2,…,n},
p_{j}^{max} = max {p_{ij}i = 1,2,…,n}, denote
minimum value and maximum value of evaluation index I_{j}. In Eq.
3, q_{ij} = p_{ij}α_{j} and α_{j}
is the idea value. q_{j}^{min} = min {q_{ij}i = 1,2,…,n},
q_{j}^{max} = min {q_{ij}i = 1,2,…,n} denote minimum
value and maximum value of evaluation index I_{j} 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 Ob_{j}’s entropy is:
where, H = {h_{ij}}_{nxm} and:
Compute the objective’s weight vector according to method of the subjective and objective assignment. The vector is w = (w_{1}, w_{2},…,w_{m}) and the element of the vector is:
Normalize the weights vector and then the combination weights of the jth index is:
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:
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 P_{i} and idea objective is:
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,
For the benefit type index, the element in normalization matrix is:
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).
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 (I_{1}), travel distance (I_{2}), cost (I_{3}), risk (I_{4}), reliability (I_{5}), flow (I_{6}). 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 (I_{5}) and flow (I_{6}) belong to benefit type indexes while others belong to cost type indexes. Consequently, the matrix R is
Expert’s subjective weight vector is assumed to be
Apply the coefficient into equation to get the combination assessment value of four plans.
where V_{i} is the combination assessment value of plan P_{i}.
The value of V_{i} 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 multiattribute decision application.