Approach of Solving Multi-objective Programming Problem by Means of Probability Theory and Uniform Experimental Design

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INTRODUCTION
Multi-objective programming (MOP) is a branch of mathematical programming, which studies the optimization of more than one objective function [1]. The idea of multiobjective programming sprouts in 1776 in the study of utility theory in economics. In 1896, economist Pareto first proposed the multi-objective programming problem in the study of economic balance, and gave a simple idea, which was later called Pareto optimal solution. In 1947, von Neumann and Morgenstern mentioned the multi-objective programming problem in their work on game theory, which attracted much more attentions to this problem. In 1951, Koopmans proposed the multi-objective optimization problem in the activity analysis of production and distribution, and proposed the concept of Pareto optimal solution for the first time. In the same year, Kuhn and Tucker gave the concept of Pareto optimal solution of vector extremum problem from the perspective of mathematical programming. The sufficient and necessary conditions for the existence of this solution are also studied. Debreu's discussion on evaluation equilibrium in 1954 and Harwicz's research on multi-objective optimization problems in topological vector spaces in 1958 laid the foundation for the establishment of this discipline. In 1968, Johnsen published the first monograph on multi-objective decision-making models. Until the 1970s and 1980s, the basic theory of multiobjective programming was finally established after the efforts of many scholars, which made it a new branch of applied mathematics [2].
There are generally the following methods for solving multi-objective programming: one is the method of transforming multiple objectives into a single objective that is easier to solve, such as main objective method, linear weighting method, ideal point method, etc. The other is called hierarchical sequence method, that is, the target is given a sequence according to its importance, and each time the next target optimal solution is found in the previous target optimal solution set, until the common optimal solution is obtained.
The main target method takes a certain f1(x) as the main target, and the other p-1 are non-main targets. At this time, it is hoped that the main target will reach the maximum value, and the remaining targets should meet certain conditions; the linear weighting method will assign the same weight to the objective functions f 1 (x), f 2 (x), ..., f p (x) respectively. The coefficient ω j , perform a linear weighted sum to obtain a new evaluation function, , then the multiobjective problem becomes a single-objective problem, but normalization is required when the dimensions are different; for a linear programming problem with multiple objectives, the decision maker hopes to achieve to, these goals in turn under these constraints by means of minimizing the total deviation from the target values, which is the problem to be solved by goal planning [1]. In practical engineering systems, such as many nonlinear, multi-variable, multi-constraint and multiobjective optimization problems in power systems, the existing mathematical methods have limited ability to optimize these problems, and the obtained solutions are not satisfactory [2].
Above discussions indicate that the normalization and the introductions of subjective factors are indispensable treatment in the above "additive" algorithms to transfer diverse criteria into a "unique criterion", and the final result depends on the normalization process significantly [3]. Different normalization methods could result in complete differences in the consequence. Besides, beneficial performance index and unbeneficial performance index are treated in non-equivalent or inconsistent manners in some algorithms. In addition, the "additive" algorithm in the multiobjective optimization is corresponding to the form of "union" from the viewpoint of set theory. So, above algorithms could be seen as a semi-quantitative approach in some sense.
Recently, a probability-based method for multi-objective optimization (PMOO) was proposed to solve the intrinsic problems of subjective factors in previous multiobjective optimizations [3][4][5]. A brand new idea of favorable probability was proposed to reflect the favorable degree of a performance index in the optimization in the PMOO. The PMOO aims to treat the simultaneous optimization of multiple objectives in the viewpoint of probability theory. In the novel methodology of PMOO, all performance utility indicators of alternatives are preliminarily divided into two types, i.e., beneficial or unbeneficial types according to their functions and preference in the optimization; each performance utility indicator of an alternative contributes to a partial favorable probability quantitatively. Moreover, the product of all partial favorable probabilities produces the total favorable probability of an alternative, which thus transfers the multi-objective optimization problem into a single-objective optimization one rationally. In this paper, it regulates a rational approach of multiobjective programming by means of probability theory, discrete uniform experimental design, and sequential algorithm for optimization. Furthermore, examples for illumination of this approach are given.

NEW APPROACH OF SOLVING MULTI-OBJECTIVE PROGRAMMING PROBLEM
The rational approach of multi-objective programming is conducted by the combination of probability theory, discrete uniform experimental design, and sequential algorithm for optimization integrally.
The probability-based method for multi-objective optimization is used to conduct conversion of the multiobjective optimization problem into a single-objective optimization one in the viewpoint of probability theory.
The discrete uniform experimental design is used to supply an efficient sampling to simplify the conversion, which is especially important for the goal functions in multiobjective programming problem being continuous functions. Sequential algorithm for optimization is employed to carry out further optimization.

Probability Theory Based Treatment
In the viewpoint of probability theory, the entire event of appearance of "simultaneous optimization of multiobjective" is corresponding to the product of the each individual objective (event). Therefore, the usual term "the higher the better" for the utility index of performance indicator needs to be expressed quantitatively in term of probability theory, which stimulates us to seek a proper expression for the term "the higher the better" in probability theory quantitatively. A brand new idea of "favorable probability" was proposed in [3][4][5] to interpret the preference degree of the candidate in the selection, i.e., it uses the term "favorable probability" to characterize the preference degree of the utility index of a performance indicator quantitatively in the optimization.
As to the multi-objective programming problem, each goal is indeed an objective of the PMOO. All performance utility indicators of alternatives are preliminarily divided into two types, i.e., beneficial or unbeneficial types according to their functions and preference in the optimization; thus, the subsequent process of PMOO can be employed rationally.
In Eqs. (1) and (2), X ij is the value of utility index of performance indicator; n expresses the number of the performance indicator; m expresses the number of the alternative in the evaluation. j X represents the arithmetic average of the value of utility index of performance indicator X ij over index i for specific j; X jmax and X jmin indicate the maximum and minimum values of X ij over index i for specific j, respectively.
Moreover, the total / overall favorable probability of an alternative is written as The total / overall favorable probability of an alternative, i.e. Eq. (3), thus transfers the multi-objective optimization problem into a single-objective optimization one in viewpoint of probability theory for the simultaneous optimization of multiple objectives rationally.

Discrete Uniform Experimental Design and Sequential Algorithm for Optimization
Since the goal functions in multi-objective programming problem are usually continuous ones, discretization can be used to conduct the simplified treatment for the simplicity.
As was stated in [6], the methodologies of good lattice point (GLP) and uniform experimental design (UED) make the discretization possible and practical. The methodologies of GLP and UED are based on number theory, which could supply effective assessment for a definite integral with finite sampling points [6,7]. The finite sampling points are uniformly distributed within the integral domain with lowdiscrepancy [8,9]. The characteristic of the uniformly distributed point set makes the convergence much faster than Monte Carlo sampling [8,9], which thus has been promising a very good algorithm in approximate calculations with a surname -"quasi -Monte Carlo Method". Fang specially developed uniform design and uniform design table for the proper using of UED [10]. Sequential uniform design or sequential algorithm for optimization (SNTO) can be used to conduct further optimization for the multi-objective programming problem due to its similarity to problem of multi-objective optimization [6,8].
Finally, the multi-objective programming problem is conducted by means of the probability -based multiobjective optimization and discrete uniform experimental design straightforward.

APPLICATIONS
In this section, two examples are given to illuminate the applications of the regulated approach in solving multiobjective programming problem by means of probability theory and discrete uniform experimental design.

Production with Maximum Profits and Least Pollutions
A factory produces two kinds of products α and β during the planning period. Each product consumes three different resources, A, B, and C [1]. The unit consumption of resources for each product, the limit of various resources, the unit price, unit profit and unit pollution caused by each product are shown in Tab. 1 [1]. Assume that all products can be sold. Now, the problem is how to arrange production which can maximize profit and output value, and cause the least pollution. Solution: Assume the output of products α and β are x 1 and x 2 , respectively, the mathematical model of the problem is as following with s. t. (restraint) conditions, Max f 1 (x) = 70x 1 + 120x 2 , Max f 2 (x) = 400x 1 + 600x 2 , Min f 3 (x) = 3x 1 + 2x 2 , s. t. 9x 1 + 4x 2 ≤ 240, 4x 1 + 5x 2 ≤ 200, 3x 1 + 10x 2 ≤ 300, Since this problem is with two input variables, says, x 1 and x 2 , according to literatures [6] and [10], at least 17 uniformly distributed sampling points are needed to conduct the discretization with uniform experimental design within the working domain. Here we try to employ the uniform table U*24(24 9 ) to perform the discretization, the consequences are shown in Tab. 2.
From Tab. 2, it can be seen that 5 sampling points are excluded due to the restraint of the s. t. conditions, and 19 sampling points are within the working domain of the s. t. conditions, which meets the requirement of at least 17 uniformly distributed sampling points within the domain of the s. t. conditions. In this problem, both the goal functions f1(x) and f 2 (x) are beneficial indexes, while the goal functions f 3 (x) is an unbeneficial index.  Tab. 3 shows the results of the assessments with PMOO, P f1 , P f2 and P f3 represent the partial favorable probabilities of functions f 1 , f 2 and f 3 at the corresponding discretized sampling points, respectively; P t expresses the total / overall favorable probability of each alternative. From Tab. 3, it can be seen that the sampling point No. 2 exhibits the maximum value of total favorable probability. Therefore, further optimization by using sequential uniform design is conducted around the sampling pint No. 2 of the Tab. 2.
Tab. 4 shows the results of the assessments by using sequential uniform design for further optimization, in which c (t) = (Max P t (i-1) -Max P t (i))/Max P t (i-1) expresses the relative error of the maximum total favorable probability at i th sequential step. If we assume a pre-assigned value δ = 2% for c (t) , then the final optimal consequences for this multiobjective optimization problem are f 1Opt. = 3591.927, f 2Opt. = 17962.24 and f 3Opt. = 59.9609 at the 5 th step with "coordinates" 1 x * = 0.0521 and 2 x * = 29.9023. Obviously, 1 x * and 2 x * approach to 0 and 30 at ultimate limit, respectively, which corresponds to optimum values of f 1Opt. = 3600, f 2Opt. = 18000 and f 3Opt. = 60, individually. Table 4 Results of the assessments by using sequential uniform design with U*24(24 9 ) Step Domain Optimum "coordinates" Value of goal Max. total favorable probability P t ×10 5

Production with Maximum Profits and One Output
A factory produces two products: A and B. The profit of producing each piece of A is 4 ¥RMB, and the profit of producing each piece of B is 3 ¥RMB. The processing time of each piece of A is twice as long as that of each piece of B. If the whole time is used to process B, 500 pieces of B can be produced for per day. The factory's daily supply of raw materials is only enough to produce a total of 400 pieces of A and B. Besides, the product A is a tight-fitting product that sells very well. Now, the problem is how to arrange the daily outputs of A and B so that the factory can obtain the maximum profit under the existing conditions. Solution: Let's first set x1 = daily output of product A, x 2 = daily output of product B [11]. Then, it gets following mathematical model, Since this problem is with two input variables, says, x 1 and x 2 , again at least 17 uniformly distributed sampling points could be used to conduct the discretization with uniform experimental design within the working domain [6,10]. Here we try to use the uniform table U*31(31 10 ) to perform the discretization, the consequences are shown in Tab. 5. From Tab. 5, it can be seen that 14 sampling points are excluded due to the restraint of the s. t. conditions, and 17 sampling points luckily are within the domain of the s. t. conditions, which satisfies the requirement of at least 17 uniformly distributed sampling points within the domain of the s. t. conditions. In this problem, both the goal functions f1(x) and f 2 (x) are beneficial indexes. Tab. 6 shows the results of the assessments with PMOO, P f1 and P f2 indicate the partial favorable probabilities of functions f 1 and f 2 at the corresponding discretized sampling points, individually; P t represents the total / overall favorable probability of each alternative. From Tab. 6, it can be seen that the sampling point No. 25 exhibits the maximum value of total favorable probability. Therefore, further optimization by using sequential uniform design is conducted around the sampling point No. 25 of the Tab. 5.  Tab. 7 shows the results of the assessments by using sequential uniform design for further optimization. Again, set a pre-assigned value δ = 2% for c (t) , then the final optimal consequences for this multi-objective optimization problem are f 1Opt. = 1000.56 and f 2Opt. = 249.597 at the 6 th step with "coordinates" 1 x * = 249.597 and 2 x * = 0.7258. Analogically, the tendencies of 1 x * and 2 x * are 250 and 0 at ultimate limit, respectively, which leads to optimum values of f 1Opt. = 1000 and f 2Opt. = 250, separately. Table 7 Results of the assessments by using sequential uniform design with U*31(31 10 ) Step Domain Optimum coordinates Value of goal Max. total favorable

DISCUSSION
In the past, the multi-objective programming problem was solved usually by using "linear weighting method" [1, 2], i.e., "additive algorithm" in the previous approaches to transfer the multiple objectives into a single objective one, which is the intrinsic problem in principle in the viewpoint of probability theory with "union set" in essence [3]. Or some approaches even took certain objectives as restraint conditions to solve the multi-objective programming problem [1, 2], which obviously deviates from the original intention of multi-objective programming problem in the spirit of the "simultaneous optimization of multiple objectives" essentially.
While, the probability-based method for multiobjective optimization attempts to treat the simultaneous optimization of multiple objectives in the viewpoint of probability theory, which is the proper methodology for multi-objective optimization [3][4][5]. Therefore, the consequences of the previous approaches are incomparable to the results of the probability-based method for multiobjective optimization due to their intrinsic problem.

CONCLUSION
By using probability-based multi-objective optimization for the simultaneous optimization of multiple objectives, discrete uniform experimental design for performing simplification, and the sequential algorithm for conducting further optimization, the multi-objective programming problem can be conducted rationally. The approach properly takes the simultaneous optimization of each objective of multi-objective programming problem into account, which reflects the essence of the multi-objective programming naturally, and creates a new way.

Conflict Statement
There is no conflict of interest.