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A new solution for solving a multi-objective integer programming problem with probabilistic multi - objective optimization
Maosheng Zhenga, Jie Yub
a Northwest University, School of Chemical Engineering, Xi'an, People's Republic of China, e-mail: mszhengok@aliyun.com, corresponding author, ORCID iD: https://orcid.org/0000-0003-3361-4060
b Northwest University, School of Life Science, Xi'an, People's Republic of China, e-mail: yujie@nwu.edu.cn,
ORCID iD: https://orcid.org/0000-0001-6606-5462
DOI: 10.5937/vojtehg71 -41556; https://doi.org/10.5937/vojtehg71-41556
FIELD: mathematics, computer science ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: In this paper, a new solution for solving a multi-objective integer programming problem with probabilistic multi - objective optimization is formulated. Furthermore, discretization by means of the good lattice point and sequential optimization are employed for a successive simplifying treatment and deep optimization. Methods: In probabilistic multi - objective optimization, a new concept of preferable probability has been introduced to describe the preference degree of each performance utility of a candidate; each performance utility of a candidate contributes a partial preferable probability and 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. Discretization by means of the good lattice point is employed to conduct discrete sampling for a continuous objective function and sequential optimization is used to perform deep optimization. At first, the requirements of integers in the treatment could be given up so as to simply conduct above procedures. Finally, the optimal solutions of the input variables must be rounded to the nearest integers. Results: This new scheme is used to deal with two production problems, i.e., maximizing profit while minimizing pollution and determining a purchasing plan for spending as little money as possible while getting as large amount of raw materials as possible. Promising results are obtained for the above two problems from the viewpoint of the probability theory for simultaneous optimization of multiple objectives.
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Conclusion: This method properly considers simultaneous optimization of multiple objectives in multi-objective integer programming, which naturally reflects the essence of multi-objective programming, and opens a new way of solving multi-objective problems.
Key words: multi-objective optimization, integer programming problem, preferable probability, discrete sampling, sequential optimization.
Introduction
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 multi-objective 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 coj for objective functions fj(x), and thus a new evaluation function U(x) = • fj(x) 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, multivariable, 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 semiquantitative 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
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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.
Solution for solving an integer programming problem by means of probabilistic multi - objective optimization
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 Pj of 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 евалуаци^е метода РМОО
Applications
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 - Потрошъа ресурса, профит и зага^иваъе сваког производа
Content Product Limit unit
Cell i Cell 2
Resource exhaust unit per product 1 5 72
Mechanical ability exhaust per product G.5 G.25 В
Manpower resource exhaust per product G.2 G.2 4
Profit per product (¥RMB) 1 3
Pollution unit per product 1.5 1
Solution
Assuming that the output of the products Cell 1 and Cell 2 is xi and X2, respectively, the mathematical model and the constraint conditions of this problem are as follows, Max fi(x) = xi + 3x2, Min f2(x) = 1.5xi + x2,
s. t. 0.5xi + 0.25x2 < 8 (Mechanical ability), 0.2xi + 0.2x2 < 4 (Manpower resource), xi + 5x2 <72 (Resource limit), and xi, x2 > 0.
Because there are two input variables xi 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 fi(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 Pfi and Pf2 of the objective functions fi(x) and f2(x) at the corresponding discrete sampling point, respectively; Pt 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 евалуаци^е дискретное узорковаъа са U*24(249)
No Input variable Objective Note
x1 x2 f1 f2
1 0.3333 6.3 19.2333 6.8
2 1 12.9 39.7 14.4
3 1.6667 4.5 15.1667 7
4 2.3333 11.1 35.6333 14.6
5 3 2.7 11.1 7.2
6 3.6667 9.3 31.5667 14.8
7 4.3333 0.9 7.0333 7.4
8 5 7.5 27.5 15
9 5.6667 14.1 Excl.
10 6.3333 5.7 23.4333 15.2
11 7 12.3 43.9 22.8
12 7.6667 3.9 19.3667 15.4
13 8.3333 10.5 39.8333 23
14 9 2.1 15.3 15.6
15 9.6667 8.7 35.7667 23.2
16 10.3333 0.3 11.2333 15.8
17 11 6.9 31.7 23.4
18 11.6667 13.5 Excl.
19 12.3333 5.1 27.6333 23.6
20 13 11.7 Excl.
21 13.6667 3.3 23.5667 23.8
22 14.3333 9.9 Excl.
23 15 1.5 19.5 24
24 15.6667 8.1 Excl.
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 евалуацще дискретне РМОО помоПу U*24(249)
No Preferable probability
Partial Total
Pf1 Pf2 Ptx 103
1 0.0402 0.0877 3.5268
2 0.0830 0.0601 4.9936
3 0.0317 0.0870 2.7581
4 0.0745 0.0594 4.4280
5 0.0232 0.0862 2.0018
6 0.0660 0.0587 3.8749
7 0.0147 0.0855 1.2577
8 0.0575 0.0580 3.3340
10 0.0490 0.0572 2.8055
11 0.0918 0.0297 2.7277
12 0.0405 0.0565 2.2892
13 0.0833 0.0290 2.4146
14 0.0320 0.0558 1.7854
15 0.0748 0.0283 2.1139
16 0.0235 0.0551 1.2938
17 0.0663 0.0275 1.8255
19 0.0578 0.0268 1.5494
21 0.0493 0.0261 1.2857
23 0.0408 0.0254 1.0343
Table 4 shows the evaluation results of the successive deep optimization using sequential uniform optimization, in which c(t) = (Max Pt(l' i) - Max Pt(l))/Max Pt(hi) represents the relative error of the maximum total preferable probability of the l-th sequential step. If we assume that the pre-assignment of c(t) = 2%, the successive deep optimization can be terminated in step 3. At this time, the final optimal results of this multi-objective programming optimization problem are fiopt. = 42.7625 ¥RMB, f2oPt. = 14.4 unit, while the instant input variables of the successive deep optimization in step 3 are xi = 0.125 and x2 = 14.2125, respectively. Since this is an integer programming problem, the solution for xi and x2 must be rounded to the nearest integers, so the values of xi and x2 are 0 and 14, respectively, and the optimal values of objective functions are thus fiopt. = 42 ¥RMB yuan and f2Opt. = 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 евалуаци^е секвенци^алне оптимизаци^е помоПу дискретное узорковаъа U*24(249)
Step Range Instant input variable Optimal objective Max Ptx 103 о«
X1* X2* fiopt. ftopt.
0 [0, 16] x [0, 14.4] 1 12.9 39.7 14.4 4.9936
1 [0,8] x [7.2, 14.4] 0.5 13.65 41.45 14.4 4.4152
2 [0, 4] x [10.8, 14.4] 0.25 14.025 42.325 14.4 4.2855 0.0294
3 [0, 2] x [12.6, 14.4] 0.125 14.2125 42.7625 14.4 4.2034 0.0191
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 xi and X2 kg, respectively, then the total cost is: fi(x) = 2xi + 1.5x2;
The total amount of the purchased raw materials is: f2(x) = xi + x2.
Then the goal of our solution is to spend the least money to buy the most raw materials, i.e. to minimize fi(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: 2xi + 1.5x2 <300; xi > 60, x2 >0.
Solution
Based on the above analysis, the following optimal mathematical model is given:
Min fi(x) = 2xi + 1.5x2;
Max f2(x) = xi + X2;
s. t.
2xi + 1.5x2 <300;
xi > 60, x2 >0.
Because this problem has two input variables xi 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 fi(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 fi and f2 at the discrete sampling points, Pfi and Pf2, respectively; Pt 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 fiopt. = 298.7635 ¥RMB yuan and f2oPt. = 179.0270 kg, while the instant input variables of the successive deep optimization at step 3 are xi = 60.4460 kg and x2 = 118.5810 kg. Similarly, since this is an integer programming problem, the solution for xi and x2 must be rounded to the nearest integers, so the values of xi and x2 are 60 kg and 119 kg, respectively, and the optimal values of objective functions are thus fiopt. = 298.5 ¥RMB and f2Opt. = 179 kg, individually.
Table 5 - Results of discretization with Ue7(3712) Таблица 5 - Результаты дискретизации с U37(3712) Табела 5 - Резултати дискретизацще са U37(3712)
No Input variable Objective Note
X1 X2 f1 f2
1 61.2162 53.5135 202.7027 114.7297
2 63.6487 108.6490 290.2703 172.2970
3 66.0811 43.7838 197.8378 109.8649
4 68.5135 98.9189 285.4054 167.4324
5 70.9459 34.0541 192.9730 105
6 73.3784 89.1892 280.5405 162.5676
7 75.8108 24.3243 188.1081 100.1351
8 78.2432 79.4595 275.6757 157.7027
9 80.6757 14.5946 183.2432 95.2703
10 83.1081 69.7297 270.8108 152.8378
11 85.5405 4.8649 178.3784 90.4054
12 87.9730 60 265.9459 147.9730
13 90.4054 115.1351 Excl.
14 92.8378 50.2703 261.0811 143.1081
15 95.2703 105.4054 Excl.
16 97.7027 40.5405 256.2162 138.2432
17 100.1351 95.6757 Excl.
18 102.5676 30.8108 251.3514 133.3784
19 105 85.9460 Excl.
20 107.4324 21.0811 246.4865 128.5135
21 109.8649 76.2162 Excl.
22 112.2973 11.3514 241.6216 123.6486
23 114.7297 66.4865 Excl.
24 117.1622 1.6216 236.7568 118.7838
25 119.5946 56.7568 Excl.
26 122.0270 111.8919 Excl.
27 124.4595 47.0270 Excl.
28 126.8919 102.1622 Excl.
29 129.3243 37.2973 Excl.
30 131.7568 92.4324 Excl.
31 134.1892 27.5676 Excl.
32 136.6216 82.7027 Excl.
33 139.0541 17.8378 Excl.
34 141.4865 72.9730 Excl.
35 143.9189 8.1081 300 152.027
36 146.3514 63.2432 Excl.
37 148.7838 118.3784 Excl.
Table 6 - Evaluation results of PMOO discrete sampling with U3ï(3712) Таблица 6 - Результаты оценки дискретной выборки PMOO с U37(3712) Табела 6 - Резултати евалуацще дискретное узоркова^а PMOO са U37(3712)
No Preferable probability
Partial Total
Pf1 Pf2 Ptx 103
1 0.0615 0.0456 2.8059
2 0.0420 0.0685 2.8754
3 0.0626 0.0437 2.7344
4 0.0430 0.0666 2.8664
5 0.0637 0.0418 2.6586
6 0.0441 0.0647 2.8533
7 0.0647 0.0398 2.5787
8 0.0452 0.0627 2.8360
9 0.0658 0.0379 2.4945
10 0.0463 0.0608 2.8145
11 0.0669 0.0360 2.4061
12 0.0474 0.0589 2.7887
14 0.0485 0.0569 2.7588
16 0.0495 0.0550 2.7247
18 0.0506 0.0531 2.6864
20 0.0517 0.0511 2.6439
22 0.0528 0.0492 2.5971
24 0.0539 0.0473 2.5462
35 0.0398 0.0605 2.4058
Table 7 - Evaluation results of sequential optimization with Ue7(3712) discrete sampling Таблица 7 - Результаты оценки последовательной оптимизации с дискретной
выборкой U37(3712) Табела 7 - Резултати евалуаци^е секвенци^алне оптимизаци^е помоПу дискретное узоркова^а U3ï(3712)
Step Range Instant input variable Objective Max Ptx 103 c«
X1* X2* /1Opt. f2Opt.
0 [60, 150] X [0, 120] 63.6487 108.6490 290.2703 172.2970 2.8754
1 [60, 105] X [60, 120] 61.8243 114.3240 295.1351 176.1490 2.8481 0.0095
2 [60, 82] X [90, 120] 60.8919 117.1620 297.5270 178.0540 2.8102 0.0133
3 [60, 71] X [105, 120] 60.4460 118.5810 298.7635 179.0270 2.7909 0.0069
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Discussion
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.
Conclusion
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.
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Новое решение для многоцелевых задач целочисленного программирования с помощью вероятностной многоцелевой оптимизации
Маошенг Чжэна, Джи Йюб
Северо-западный политехнический университет, г. Сиань, Народная Республика Китай а факультет химической инженерии, корресподент
б факультет естественных наук
РУБРИКА ГРНТИ: 27.47.00 Математическая кибернетика,
27.47.19 Исследование операций ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: В данной статье представлено новое решение для многоцелевых задач целочисленного программирования с помощью вероятностной многоцелевой оптимизации. Кроме того, в целях сукцессивного упрощения обработки и глубокой оптимизации
используются дискретизация с помощью соответствующих узлов решетки и последовательная оптимизация. Методы: В вероятностную многоцелевую оптимизацию введена новая концепция предпочтительной вероятности для описания степени предпочтения полезности каждого кандидата. Каждая полезность характеристик кандидата вносит частичную предпочтительную вероятность, а произведение всех частичных предпочтительных вероятностей составляет общую предпочтительную вероятность кандидата. Таким образом, общая предпочтительная вероятность переводит многоцелевую проблему в одноцелевую. Дискретизация по методу узлов идеальной решетки применяется для дискретной выборки, а последовательная оптимизация — для глубокой оптимизации. Также в целях упрощения данной процедуры можно отказаться от целочисленных требований. В конце процедуры оптимальные решения введенных переменных необходимо округлить до ближайшего целого числа.
Результаты: Данный подход используется для решения двух производственных задач, а именно: максимизации прибыли при минимизации загрязнения и составления плана закупок как можно большего количества сырья при наименьших затратах. С помощью теории вероятностей вышеуказанные задачи показали многообещающие результаты одновременной оптимизации нескольких целей.
Выводы: Данное решение учитывает одновременную оптимизацию нескольких целей при многокритериальном целочисленном программировании, что, естественно, отражает суть многокритериального программирования и тем самым открывает новые возможности к решению многокритериальных задач.
Ключевые слова: многоцелевая оптимизация, задача целочисленного программирования, предпочтительная вероятность, дискретная выборка, последовательная оптимизация.
Ново решете проблема вишекритери]умског целобро]ног програмира^а помойу вишекритери]умске оптимизаци]е засноване на вероватнойи
Маошенг Ценга, Ъаи иуб
Универзитет Северозапад, Си]ан, Народна Република Кина а Факултет хеми]ског инженерства, аутор за преписку
б Факултет природних наука
ОБЛАСТ: математика, рачунарске науке КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад
Сажетак:
Увод/цил>: У раду се формулише ново решете проблема вишекритери]умског целобро]ног програмираъа помогу пробабилистичке вишекритери]умске оптимизаци}е. Тако^е, користи се дискретизацща помогу добрих тачака решетке, као и секвенци]ална оптимизаци}а ради сукцесивног по}едноставп>ивак>а и дубинске оптимизаци}е.
Методе: У пробабилистичку вишекритери}умску оптимизацщу уведен }е нови концепт пожеъне вероватноЬе како би се описао степен пожел>ности сваке по}единачне корисности перформансе неког кандидата. Свака по}единачна корисност перформансе кандидата доприноси парци]алноj пожеъноj вероватноЬи, а производ свих тих вероватноЬа чини укупну пожеъну вероватноПу кандидата. На таj начин укупна пожеъна вероватноЬа преводи вишекритери]умски проблем у ]еднокритери]умски. Дискретизациям помогу метода добрих тачака решетке врши се дискретно узорковаъе за континуалну функцщу циъа, а секвенцщалном оптимизациям дубинска оптимизаци}а. Тако^е, може се одустати од захтева целих бро}ева ради по}едноставп>ивак>а наведеног поступка. На кра}у се оптимална решена унетих вари}абли мора]у заокружити на на]ближи цели бро].
Резултати: Оваj приступ се користи за решаваъе два проблема у производи: за максимизаци}у прихода уз на]ма^е могуче зага^еъе и за креираъе плана за набавку на}веЬе количине репроматери]ала по на]мак>о] цени. ОбеЬава}уЬи резултати су добц'ени за два наведена проблема помогу теорбе вероватноЬе за истовремену оптимизаци}у више циъева.
Закъучак: Ово решете узима у обзир истовремену оптимизацщу више циъева при вишекритери]умском целобро}ном програмираъу, што природно одсликава суштину вишекритери]умског програмираъа и тиме отвара нове путеве ка решаваъу вишекритери]умских проблема.
Къучне речи: вишекритери}умска оптимизаци}а, проблем целобро]ног програмираъа, пожеъна вероватноЬа, дискретно узорковаъе, секвенци}ална оптимизаци]а.
Paper received on / Дата получения работы / Датум приема чланка: 04.12.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 23.03.2023.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 25.03.2023.
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© 2023 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
© 2023 Авторы. Опубликовано в «Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией «Creative Commons» (http://creativecommons.org/licenses/by/3.0/rs/).
© 2023 Аутори. Об]авио Во]нотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). Ово ]е чланак отвореног приступа и дистрибуира се у складу са Creative Commons licencom (http://creativecommons.org/licenses/by/3.0/rs/).
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