A new solution for solving a multi-objective integer programming problem with probabilistic multi-objective optimization

Новое решение для многоцелевых задач целочисленного программирования с помощью вероятностной многоцелевой оптимизации

Ново решење проблема вишекритеријумског целобројног програмирања помоћу вишекритеријумске оптимизације засноване на вероватноћи

Maosheng Zhenga mszhengok@aliyun.com

Northwest University, China

Jie Yub yujie@nwu.edu.cn

Northwest Universit, China

This work is licensed under Creative Commons Attribution 4.0 International.

Multi-objective programming (GP) is an important branch of optimization theory. It is a mathematical method developed to solve multi-objective decision-making problems based on linear and nonlinear programmings. Since 1960s, it has been gradually developed and matured. It is widely used in economic management and planning, human resource management, government management, optimization of large - scale projects and other important areas.

The idea of multi-objective programming originated from the study of the utility theory in economics in 1776. In 1896, economist Pareto first put forward the multi-objective programming problem in the study of economic balance, and gave a simple idea which was later called the Pareto optimal solution. In 1947, von Neumann and Morgenster mentioned the multi-objective programming problem in their game theory work, which attracted more attention to this problem. In 1951, Koopmans put forward the multi-objective optimization problem in the analysis of production and sales activities, and first formed the concept of the Pareto optimal solution. In the same year, Kuhn and Tucker gave the concept of the Pareto optimal solution of the vector extremum problem from the angle of mathematical programming. The necessary and sufficient conditions for the existence of this solution are also studied. Debreu's discussion on evaluation balance in 1954 and Harwicz's research on multi-objective optimization in topological vector space in 1958 laid the foundation for the establishment of this discipline. In 1968, Johnsen published the first monograph on the multi-objective decision-making model. Until 1970s-1980s, the basic theory of multi-objective programming was finally established through the efforts of many scholars, making it a new branch of applied mathematics (Huang et al, 2017; Liu, 2014; Ying, 1988).

Up to now, there are the following general methods to solve multiobjective programming: one is to transfer multiple objectives into a single objective that is easier to solve, such as the main objective method, the linear weighting method, the ideal point method, etc.; the other method is called the hierarchical sequence method, i.e. a sequence is given according to the importance of the target, and the next target optimal solution is searched in the previous target optimal solution set every time until a common optimal solution is obtained; the third one is the main target method, which takes one f₁(x) as the main target, and the other P-1 as the non-main target. At this time, it is hoped that the main target will reach the maximum value, and other targets will meet certain conditions; the fourth one is the linear weighting method, which sets a series of weight coefficients $\omega$ _j for objective functions f_j(x), and thus a new evaluation function is obtained by linear weighted summation, which makes the multi-objective problem become a single-objective problem. However, under the condition that the dimensions of the target are different, normalization is needed. For a multi-objective linear programming problem, decision makers hope to achieve these goals in turn under these constraints by minimizing the total deviation from the target value, which is the problem to be solved by goal programming (Huang et al, 2017; Liu, 2014; Ying, 1988).

In practical engineering systems, such as many nonlinear, multi-variable, multi-constraint and multi-objective optimization problems in power systems, the existing mathematical methods have limited ability to optimize these problems, and the solutions obtained are not satisfactory.

The above discussion shows that normalization and the introduction of subjective factors in the previous methods are indispensable processes in their "additive" algorithm, and the final result depends to a great extent on the normalization method adopted after the targets with different attributes are converted into "single" targets (Zheng et al, 2021). Different normalization methods may lead to completely different results. In addition, in some algorithms, the beneficial performance index and the unbeneficial performance index are treated unequally.

From the point of view of the set theory, the "additive" algorithm in the previous methods for multi-objective optimization corresponds to the form of "union". Therefore, the above algorithm can only be regarded as a semi-quantitative method in a sense.

Recently, a probabilistic multi - objective optimization (PMOO) method has been proposed to solve the inherent problems of subjective factors of the previous methods of multi - objective optimization (Zheng et al, 2021; Zheng et al, 2022a; Zheng et al, 2022b). A brand - new concept of preferable probability is put forward to reflect the preference degree of performance indicators in project management optimization. PMOO aims to deal with multi-objective simultaneous optimization from the perspective of the probability theory. In the new methodology of PMOO, the performance utility indicators of all candidates are preliminarily divided into the beneficial category and the unbeneficial category according to their roles and preferences in optimization; each performance utility index of the candidate quantitatively contributes to a partial preferable probability; the product of all partial preferable probabilities deduces the total preferable probability of a candidate; the total preferable probability thus transfers a multi-objective problem into a single-objective one. In the evaluation, the total preferable probability of a candidate is the unique and decisive index of the candidate.

In this article, by using probabilistic multi - objective optimization and the good lattice point to conduct discrete sampling and sequential optimization for successive deep optimization, a reasonable method of multi - objective programming is formulated, and the application details of this method are illustrated with two examples.

In this section, probabilistic multi - objective optimization, good lattice point (GLP) discretization and sequential optimization are organically combined, which establishes a rational method for solving a multi-objective programming problem. The probabilistic multi - objective optimization method is used to transfer a multi - objective optimization problem into a single - objective optimization one from the perspective of the probability theory; the discretization of GLP provides an effective discrete sampling to simplify mathematical processing, which is especially important for dealing with multi - objective programming problems with continuous objective functions; and sequential optimization is used for successive deep optimization.

The systematic implementation is demonstrated in the subsections A) and B).

A) A method based on the perspective of probabilistic multi - objective optimization

From the perspective of probabilistic multi - objective optimization, the whole event with multi - objective simultaneous optimization corresponds to the product of all single objectives (events). For multi - objective programming problems, each objective can be analogically seen as a single event (Zheng et al, 2021; Zheng et al, 2022a; Zheng et al, 2022b). All performance utility indexes of the candidate are preliminarily divided into two categories: beneficial and unbeneficial, according to their role and preference of a candidate in optimization, respectively. Specifically, the assessment of the preferable probability P_ijof both beneficial indicators and unbeneficial indicators can be carried out according to the evaluation procedure in Figure 1 (Zheng et al, 2021; Zheng et al, 2022a; Zheng et al, 2022b).

B) Discrete sampling by means of the good lattice point and successive sequential optimization

In multi - objective programming problems, the objective function is usually continuous. In order to simplify mathematical processing, the discrete sampling by means of the good lattice point (GLP) can be used. As described in literature (Hua & Wang, 1981; Fang & Wang, 1994; Fang et al, 2018), the methods of good lattice point and uniform experimental design (UED) make discrete sampling possible and practical. The GLP method and UED are based on the number theory, and it can obtain an effective approximate value for a definite integral or an extreme value problem with a limited number of sampling points (Hua & Wang, 1981; Fang & Wang, 1994; Fang et al, 2018). Such a limited number of sampling points is uniformly distributed in the super space with low discrepancy. The characteristic of the uniform point set makes its convergence speed much faster than that of the Monte Carlo sampling method (Hua & Wang, 1981; Fang & Wang, 1994; Fang et al, 2018), so it is considered as an efficient approximation named quasi-Monte Carlo method. In order to use this uniformly distributed point set appropriately, Professor Fang specially developed uniform design and uniform design tables (Fang, 1994; Fang et al, 2018).

As to the successive sequential optimization of multi - objective optimization problems, a sequential optimization algorithm (SNTO) can be employed for deep optimization (Zheng et al, 2022c; Zheng et al, 2023).

Moreover, by combining probabilistic multi - objective optimization, discrete sampling, and sequential optimization, the multi - objective programming problem can be solved rationally.

At first, the requirements of integers could be given up so as to simply conduct the above procedures. Finally, the optimum solutions of the input variables must be rounded to the nearest integers, which must be withstanding the constraint conditions as well.

Figure 1
Evaluation procedure of the PMOO method

Рис. 1 – Процедура оценки метода PMOO

Слика 1 – Поступак евалуације метода РМОО

In this section, two examples are employed to illustrate the use of the above methods in solving multi - objective integer programming problems.

I) An integer programming problem that maximizes profits and minimizes pollution

A factory plans to produce two products, PV cell 1 and PV cell 2. During production, it causes certain polluting gas release into the air (Huang et al, 2017). So, in the production plan, the goals are to get maximum profit with minimum pollution at the same time. Profit, unit pollution of each product, mechanical ability, manpower resource and resource limits are shown in Table 1 (Huang et al, 2017). Therefore, the problem is how to organise production which maximizes profits and causes the least pollution.

Table 1
Resource consumption, profit and pollution of each product

Таблица 1 – Потребление ресурсов, прибыль и загрязнение каждого продукта

Табела 1 – Потрошња ресурса, профит и загађивање сваког производа

Solution

Assuming that the output of the products Cell 1 and Cell 2 is x₁ and x₂, respectively, the mathematical model and the constraint conditions of this problem are as follows,

Because there are two input variables x₁ and x₂ in this problem, at least 17 evenly distributed sampling points are needed for the discretization in the working domain according to literature (Zheng et al, 2022c). Here, we try to use the uniform design table U*₂₄(24⁹) to implement the discretization (Fang, 1994; Fang et al, 2018), and the results are shown in Table 2. As it can be seen from Table 2, five sampling points were excluded due to the limitation of the constraint conditions, and the remaining 19 sampling points were within the working area, which meets the basic requirement of at least 17 uniformly distributed sampling points within the working zone.

Additionally, in this problem, the objective function f₁(x) is a beneficial indicator while the objective function f₂(x) is an unbeneficial indicator. Table 3 shows the evaluation results of the partial preferable probabilities P_f1 and P_f2 of the objective functions f₁(x) and f₂(x) at the corresponding discrete sampling point, respectively; P_t represents the total/overall preferable probabilities of each sampling point.

As it can be seen from Table 3, sampling point No 2 shows the maximum value of the total preferable probability. Therefore, around sampling point No 2 of Table 2, sequential uniform design is adopted for successive deep optimization.

Table 2
Evaluation results of discrete sampling with U*₂₄(24⁹)

Таблица 2 – Результаты оценки дискретной выборки с U*₂4(24⁹)

Табела 2 – Резултати евалуације дискретног узорковања са U*₂₄(24⁹)

Table 3
Evaluation results of discrete PMOO using U*₂₄(24⁹)

Таблица 3 – Результаты оценки дискретного PMOO с помощью U*₂₄(24⁹)

Табела 3 – Резултати евалуације дискретне РМОО помоћу U*₂₄(24⁹)

Table 4 shows the evaluation results of the successive deep optimization using sequential uniform optimization, in which c(t) = (Max P_t⁽ⁱ¹⁾ – Max P_t⁽ⁱ⁾)/Max P_t^(i-1) represents the relative error of the maximum total preferable probability of the i-th sequential step. If we assume that the preassignment of c^(t) = 2%, the successive deep optimization can be terminated in step 3. At this time, the final optimal results of this multiobjective programming optimization problem are f_1Opt. = 42.7625 ¥RMB, f_2Opt. = 14.4 unit, while the instant input variables of the successive deep optimization in step 3 are x₁ = 0.125 and x₂ = 14.2125, respectively. Since this is an integer programming problem, the solution for x₁ and x₂ must be rounded to the nearest integers, so the values of x₁ and x₂ are 0 and 14, respectively, and the optimal values of objective functions are thus f_1Opt. = 42 ¥RMB yuan and f_2Opt. = 14 unit, individually. This result is much better than that given by Huang with a linear weighting algorithm (Huang et al, 2017).

Table 4
Evaluation results of sequential optimization with U*₂₄(24⁹) discrete sampling

Таблица 4 – Результаты оценки последовательной оптимизации с дискретной выборкой U*₂₄(24⁹)

Табела 4 – Резултати евалуације секвенцијалне оптимизације помоћу дискретног узорковања U*₂₄(24⁹)

II) Purchasing raw material for production

A factory needs to purchase certain raw material for production. There are two kinds of raw materials in the market, A and B, with unit prices of 2 ¥RMB yuan / kg and 1.5 ¥RMB yuan /kg, respectively. It is required that the total cost now should not exceed 300 ¥RMB yuan, and the raw material A should not be less than 60 kg. How to determine the best purchasing plan, spend the least money and purchase the largest amount of raw materials? The smallest weight unit is 1 kg.

Assuming that the two raw materials, A and B, are purchased in x₁ and x₂ kg, respectively, then the total cost is:

f₁(x) = 2x₁ + 1.5x₂;

The total amount of the purchased raw materials is:

Then the goal of our solution is to spend the least money to buy the most raw materials, i.e. to minimize f₁(x) while maximizing f₂(x).

At the same time, it is necessary to meet the requirements that the total cost should not exceed 300 ¥RMB yuan, and the raw materials A should not be less than 60 kg, so the constraint conditions are as follows:

Solution

Based on the above analysis, the following optimal mathematical model is given:

Because this problem has two input variables x₁ and x₂, similarly, at least 17 evenly distributed sampling points in the working domain are needed (Zheng et al, 2022c; Zheng et al, 2023). Here, we try to use the uniform test table U₃₇(37¹²) conduct the discrete sampling (Fang, 1994; Fang et al, 2018), and the results are shown in Table 5. It can be seen from Table 5 that, due to the limitation of the constraint conditions, 18 sampling points are excluded, and, luckily, 19 sampling points are within the scope of the constraint conditions, which meets the requirement of at least 17 uniformly distributed sampling points within the scope of s. t. condition. In this problem, the objective function f₁(x) is the unbeneficial indicator, and f₂(x) is the beneficial indicator.

Table 6 shows the evaluation results of the partial preferable probabilities of the functions f₁ and f₂ at the discrete sampling points, P_f1 and P_f2, respectively; P_t represents the total/overall preferable probability of each sampling point. As it can be seen from Table 6, sampling point No 2 shows the maximum value of the total preferable probability. Therefore, around the 2nd sampling point in Table 6, sequential optimization is adopted for successive deep optimization. Table 7 shows the evaluation results of the sequential optimization using the uniform design table U₃₇(37¹²). Similarly, if a pre-specified value of 0.7% is set for c^(t), then the deep optimization can be terminated in step 3. At this point, the final optimal results of this multi-objective programming optimization problem are f_1Opt. = 298.7635 ¥RMB yuan and f_2Opt. = 179.0270 kg, while the instant input variables of the successive deep optimization at step 3 are x₁ = 60.4460 kg and x₂ = 118.5810 kg. Similarly, since this is an integer programming problem, the solution for x1 and x2 must be rounded to the nearest integers, so the values of x₁ and x₂ are 60 kg and 119 kg, respectively, and the optimal values of objective functions are thus f_1Opt. = 298.5 ¥RMB and f_2Opt. = 179 kg, individually.

Table 5
Results of discretization with U₃₇(37¹²)

Таблица 5 – Результаты дискретизации с U₃₇(37¹²)

Табела 5 – Резултати дискретизације са U₃₇(37¹²)

Table 6
Evaluation results of PMOO discrete sampling with U₃₇(37¹²)

Таблица 6 – Результаты оценки дискретной выборки PMOО с U₃₇(37¹²)

Табела 6 – Резултати евалуације дискретног узорковања PMOO са U₃₇(37¹²)

Table 7
Evaluation results of sequential optimization with U₃₇(37¹²) discrete sampling

Таблица 7 – Результаты оценки последовательной оптимизации с дискретной выборкой U₃₇(37¹²)

Табела 7 – Резултати евалуације секвенцијалне оптимизације помоћу дискретног узорковања U₃₇(37¹²)

When solving multi - objective programming problems, the approaches employed in the previous work of other methods include the linear weighting method (Zheng et al, 2022c), i.e. the "additive" algorithm, which transfers multi – objective problems into single objective ones. However, from the perspective of the probability theory, this essentially means a "union", and some methods even take several objectives as constraint conditions to solve multi-objective programming problems (Zheng et al, 2022c), which is not realized to the intrinsic meaning of multi - objective programming problems, while probabilistic multi-objective optimization tries to deal with simultaneous optimization of multiple objectives from the perspective of the probability theory, which is a rational method of multi - objective optimization (Zheng et al, 2022c). Therefore, the results obtained by other methods cannot be compared with the results of probabilistic multi - objective optimization.

By using the combination of probabilistic multi - objective optimization, discrete sampling by means of the good lattice points, and successive sequential optimization to solve the multi - objective integer programming problem, we establish a reasonable scheme for solving the multi - objective programming problem. This method properly considers simultaneous optimization of many objectives in the problem, which rationally reflects the essence of simultaneous optimization of multiple objectives, and opens a new approach to the relevant problem.

FIELD: mathematics, computer science

ARTICLE TYPE: original scientific paper

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