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APPLICATION OF FINITE SAMPLING POINTS IN PROBABILITY BASED MULTI - OBJECTIVE OPTIMIZATION BY MEANS OF THE UNIFORM EXPERIMENTAL DESIGN
Maosheng Zhenga, Haipeng Tengb, Yi Wangc, Jie Yud
a Northwest University, School of Chemical Engineering, Xi'an, People's Republic of China, e-mail: [email protected], corresponding author, ORCID iD: https://orcid.org/0000-0003-3361-4060
b Northwest University, School of Chemical Engineering, Xi'an, People's Republic of China, e-mail: [email protected], ORCID iD: https://orcid.org/0000-0003-2987-7415
c Northwest University, School of Chemical Engineering, Xi'an, People's Republic of China, e-mail: [email protected], ORCID iD: https://orcid.org/0000-0001-6711-0026
d Northwest University, School of Life Science, Xi'an, People's Republic of China, e-mail: [email protected],
ORCID iD: https://orcid.org/0000-0001-6606-5462
DOI: 10.5937/vojtehg70-37087; https://doi.org/10.5937/vojtehg70-37087
FIELD: Mathematics, Materials ARTICLE TYPE: Original scientific paper
Abstract:
Introduction/purpose: An approximation for assessing a definite integral is continuously an attractive topic owing to its practical needs in scientific and engineering areas. An efficient approach for preliminarily calculating a definite integral with a small number of sampling points was newly developed to get an approximate value for a numerical integral with a complicated integrand. In the present paper, an efficient approach with a small number of sampling points is combined to the novel probability-based multi-objective optimization (PMOO) by means of uniform experimental design so as to simplify the complicated definite integral in the PMOO preliminarily.
Methods: The distribution of sampling points within its single peak domain is deterministic and uniform, which follows the rules of the uniform design method and good lattice points; the total preferable probability is the unique and deterministic index in the PMOO.
Results: The applications of the efficient approach with finite sampling points in solving typical problems of PMOO indicate its rationality and convenience in the operation.
Conclusion: The efficient approach with finite sampling points for assessing a definite integral is successfully combined with PMOO by means of the uniform design method and good lattice points.
Key words: preferable probability, multi-objective optimization, finite sampling points, simplifying evaluation, uniform design method.
Introduction
Recently, an efficient approach for assessing a definite integral with a small number of sampling points has been proposed based on the uniform experimental design method and the good lattice point from the viewpoint of practical application (Yu et al, 2022) preliminarily. It indicated that the efficient evaluation of a definite integral for a periodical function in its single peak domain can be obtained by using 11 sampling points in one dimension, 17 sampling points in two dimensions, and 19 sampling points in three dimensions with a small relative error preliminarily. The fundamental of the finite sampling points (FSPs) for assessing a definite integral was the rules of uniform and deterministic distribution of the FSPs according to the good lattice point (Hua & Wang, 1981; Fang, 1980; Fang, et al, 1994, 2018; Ripley, 1981; Wang & Fang, 2010), or the so-called "quasi - Monte Carlo method" (QMC).
The so-called "curse of dimensionality" problem was broken in the publication of the calculating results of Paskov & Traub (1995) by using Halton sequences and Sobol sequences for accounting a ten - tranche CMO (Collateralized Mortgage Obligation) in high dimensions, reaching even 360 dimensions. Their findings were that QMC methods performed very well as compared to simple MC methods, as well as to antithetic MC methods (Tezuka, 1998, 2002; Paskov & Traub, 1995; Paskov, 1996; Sloan & Wozniakowski, 1998). Afterwards, a lot of similar phenomena were found in different evaluations for pricing problems by using different types of low-discrepancy sequences (Tezuka, 1998). All these consequences provide a powerful support to using the QMC with finite sampling points to conduct a definite integral numerically.
In the present paper, the newly developed efficient approach for assessing a definite integral with a small number of sampling points is combined to the novel probability - based multi - objective optimization (PMOO) so as to simplify the complicated definite integral in PMOO. The novel PMOO aims to overcome the shortcomings of personal and subjective factors in the previous multi - object optimizations, so a novel
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concept of preferable probability and the corresponding assessment are developed (Zheng, 2022; Zheng et al, 2021, 2022). The preferable probability is used to reflect the preferablity degree of the candidate in the optimization, all performance utility indicators of candidates are divided into beneficial or unbeneficial types according to their features in the selection, and each performance utility indicator contributes to one partial preferable probability quantitatively. The total preferable probability is the product of all partial preferable probabilities in the viewpoint of probability theory, which is the overall consideration of various response variables simultaneously so as to reach a compromised optimization. The total preferable probability is the unique deterministic index in the optimal process comparatively. Appropriate achievements have been obtained.
Essence of the uniform experimental design method
The uniform experimental design method (UED) was proposed by Fang & Wang (1994, 2018) and the essence of the UED contains:
A) Uniformity. The sampling points for an experiment are evenly distributed in the input variable (parameter) space, so the term "space filling design" is widely used in the literature. The UED arranges the test design (test point, sampling points in space) through a uniform design table, which is deterministic without any randomness.
B) Overall Mean Model. The UED is to hope that the test point can give the minimum deviation of the total mean value of the output (response) variable from the actual total mean value.
C) Robust. The UED design can be applied to a variety of situations and is robust to model changes.
D) Following basic procedures are involved in the UED:
1) Total Mean Model
It assumes that there exists a deterministic relationship between the input independent variables X1, X2, x3, ..., Xs and the response y by
Furthermore, it supposes that the experiment domain is the unit cube Cr = [0, 1]r, the total mean value the response y on Cr is,
If m sampling points pi, p2, p3, ..., Pm are taken on Cr, then the mean value of y on these m sampling points is
y = f (xvx2'x3,---xr), X = {Xi, X2, X3, ..., Xr} eC.
(2)
_ i m
y(Dm) = - £ f (Pj). (3)
In Eq.(3), Dm = {pi, p2, p3, ..., pm} represents a design of these m sampling points.
Fang & Wang (1994, 2018) proved that if the sampling points pi, p2, p3, ..., pm are uniformly distributed on the domain Cr, the deviation
E(y) - y(Dm) of the sampling point set on Cr and Dm is the smallest
approximately.
2) Uniform Design Table
Fang & Wang (1994, 2018) and Wang & Fang (2010) developed a Uniform Design Table for the proper utilization of the UED which can be employed by anyone to arrange their sampling points. However, the preliminarily necessary number of sampling points was not clarified by Fang in their UED. Here in this paper, the number of sampling points suggested in the article of Yu et al (2022) is adopted for our utilization.
3) Regression
Regression is the next procedure to complete the optimum.
For our purpose, the total preferable probability and the approximate expression for the response y' = f'(x1, x2, x3, ..., xr) can be obtained through data fitting, which is close to the true model (Fang & Wang, 1994, 2018).
The application of uniform design is becoming more and more extensive these years, including a successful application of the uniform experimental design in the Chinese Missile Design and Ford Motor Company of the USA, and the number of successful cases is increasing.
Combination of finite sampling points with the probability-based multi-objective optimization by means of the uniform experimental design
The above statements indicate the remarkable features of the UED, i.e., the uniform distribution of experiment / sampling points within the test domain and the small number of tests, fully representative of each point, and an easy to perform regression analysis. So here the Finite Sampling Points method is combined with the novel probability-based multi-objective optimization by means of the uniform experimental design and the good lattice point (GLP) to simplify the complicated data processing preliminarily in the following section.
In order to demonstrate the combination of finite sampling points with the probability-based multi-objective optimization, some typical examples are given in the following sections in detail.
1) Multi-objective optimization of tower crane boom tie rods
Qu et al (2004) conducted the multi - objective optimization of tower crane boom tie rods by the fuzzy optimization model.
Through a careful analysis, they set the minimum mass W(X) of the boom tie rod and the minimum angular displacement Q(X) of the boom as the multiple objectives, and obtained the following model,
W(X) = 208.323xi + 433.868x2, (4)
= 2.0288 xlO. (5) 9.8621 ^ + 5.3471 x2
The constraint conditions are,
0.003379< xi < 0.005805, (6)
0.003379 < X2 < 0.005468. (7)
According to the optimal requirements of W(X) and Q(X), both W(X) and d(X) are unbeneficial indexes (Qu et al, 2004) which have "the smaller the better" features in the optimization.
Thus, according to the probability-based multi-objective optimization (Zheng, 2022; Zheng et al, 2021, 2022), the partial preferable probabilities of W(X) and d(X) are expressed as
Pw = WWmax + Wmin - W(X)], (8)
Pb= p '[9max +Qmin - 9(X)], (9)
In Eqs. (8) and (9), @w, Wmin, and Wmax express the normalization factor, the minimum and maximum values of the index W(X), respectively; fa, dmm, and 9max indicate the normalization factor, the minimum and maximum values of the index d(X), individually.
Simultaneously,
Pw -1--(10)
J [Wmax + Wmn - W(X)]dXi ' dX2
Jx1l. x2u
Pd »x (11)
J PL, +0m,n -0(X)] ' dXi ' dx2
Jx1l. x2u
In Eqs. (8) and (9), xil, xiu, X2L and хги express the lower limit and the upper limit of Xi and X2 in their domain, respectively.
According to the common procedure, the subsequent thing is to substitute Eqs. (4) and (5) into Eqs. (8) through (11) with the constraints of Eqs. (6) and (7) to conduct the evaluations. It can be seen that the assessments are tediously long and complicated due to the sophisticated integration. However, if we use the finite sampling points algorithm proposed by Yu et al (2022), the approximate assessments of the definite integral in Eqs. (10) and (11) can be simplified with the finite numbers of discrete sampling points.
According to Yu et al (2022), 17 discrete sampling points are suggested for the two independent variables Xi and X2 preliminarily. So the Uniform Design Table of UV(175) is taken to conduct the approximate assessment. The designed results for the 17 discrete sampling points are shown in Table 1 together with the calculated consequences of W(X) and d(X), in which X10 and X20 indicate the original positions from the Uniform Design Table UV(175) for the [1, 17] * [1, 17] domain.
Table 2 shows the evaluation results of this problem.
Table 1 - Designed results U*i7(175) together with the calculated consequences of
W(X) and 6(X)
Таблица 1 - Полученные результаты U*i7(175) вместе с рассчитанными последствиями W(X) и 9(X) Табела 1 - Про]ектовани резултати U*i7(175) за^едно са израчунатим последицама W(X) и 9(X)
No. X10 X20 X1 / m2 X2 / m2 W / T Q/°
1 1 7 0.003450 0.004178 2.5314 0.0036
2 2 14 0.003593 0.005038 2.9343 0.0033
3 3 3 0.003736 0.003686 2.3776 0.0036
4 4 10 0.003879 0.004546 2.7805 0.0032
5 5 17 0.004021 0.005407 3.1834 0.0030
6 6 6 0.004164 0.004055 2.6267 0.0032
7 7 13 0.004307 0.004915 3.0296 0.0030
8 8 2 0.004449 0.003563 2.4729 0.0032
9 9 9 0.004592 0.004424 2.8758 0.0029
10 10 16 0.004735 0.005284 3.2788 0.0027
11 11 5 0.004877 0.003932 2.7220 0.0029
No. X10 X20 X1 / m2 X2 / m2 W / T Q/°
12 12 12 0.005020 0.004792 3.1250 0.0027
13 13 1 0.005163 0.003440 2.5682 0.0029
14 14 8 0.005306 0.004301 2.9712 0.0027
15 15 15 0.005448 0.005161 3.3741 0.0025
16 16 4 0.005591 0.003809 2.8174 0.0027
17 17 11 0.005734 0.004669 3.2203 0.0025
Table 2 - Evaluation results of this problem Таблица 2 - Результаты оценки данной проблемы Табела 2 - Резултати процене овог проблема
Partial preferable probability Total
No. Pw(X) Рв(Х) Ptx103 Rank
1 0.0659 0.0471 3.1006 16
2 0.0576 0.0536 3.0905 17
3 0.0690 0.0473 3.2642 13
4 0.0608 0.0538 3.2703 12
5 0.0525 0.0592 3.1091 15
6 0.0639 0.0540 3.4512 8
7 0.0557 0.0593 3.3037 11
8 0.0671 0.0542 3.6333 5
9 0.0588 0.0595 3.4993 7
10 0.0506 0.0639 3.2346 14
11 0.0620 0.0596 3.6957 3
12 0.0537 0.0641 3.4426 9
13 0.0651 0.0598 3.8930 1
14 0.0569 0.0642 3.6514 4
15 0.0486 0.0680 3.3052 10
16 0.0600 0.0643 3.8609 2
17 0.0518 0.0681 3.5246 6
Table 2 shows that the preliminarily assessed result of the total preferable probability of sampling point No. 13 exhibits the maximum in the first glance, so the optimal configuration could be around sampling point No. 13.
As to sampling point No. 13, the optimal mass Woptim. of the boom tie rod and the optimal angular displacement 0optim. of the boom are 2.5682
tons and 0.0029° at xi = 0.0052 m2 and X2 = 0.0034 m2, which are better than those of Qu's (2004) results of 2.8580 tons, and 0.0026° at xi = 0.0058 m2 and X2 = 0.0038 m2, comprehensively.
Moreover, regression can be applied for further optimization. The regressed result of the total probability Pt with respect to xi and X2 is
PtxICP = 8.2971 - 249.4110xi - 304.5570x2 - 0.0978x10-1xi1 -0.0083x10-1X21, (12)
R2 = 0.9362. (13)
The regressed result of the W with respect to xi and x2 is
W= 2.89x10-15 + 208.3230x1 + 433.8680x2, (14)
R2 = 1. (15)
The regressed result of the total probability d with respect to x1 and x2
is
d = 0.0035 - 0.1459x1 - 0.2412x2 - 5.7700x1a6x{1 -1.4000x10-7x2-1,
(16)
R2 = 0.9941. (17)
The optimal result of the regressed formula of Eq. (12) being maximum is Pt'x103 = 3.8890 at x1 = 0.0058 m2 and x2 = 0.0034 m2; the corresponding values for optimal Wand Q are, W* = 2.6754 tons, Q* = 0.0028°, which are much better than those of Qu's results as well.
2) Multi-objective optimization with a single input variable
It is certain that multi-objective optimization with a single input variable is a very simple problem and direct assessment can be conducted.
The simple example is that the optimal solution of the min f1(x) = x2 together with min f2(x) = (x - 2)2 simultaneously within the range of x e [5, 7], which was discussed by Huang & Chen (2009) with tediously long and complex evolutionary computations of Pareto optimization.
Here, by using the probability-based multi-objective optimization, the problem can be reanalyzed and the partial preferable probability for f1(x) and /2(x) can be expressed as,
Pn = (49 - x2)/432, P/2 = [49 - (x - 2)2]/432. (14)
Thus, the total preferable probability Pt = P/vP/2 takes its maximum value at x = 1 distinctly; therefore, the simultaneous minimum values of /1(x) and /2(x) are compromisingly equaled to 1. Obviously, the assessing process is much simpler than that of complex evolutionary computations
of Pareto optimization (Huang & Chen, 2009).
Furthermore, if the sampling point method is used, 11 sampling points can be employed for the assessment preliminarily (Yu, et al, 2022). The uniformly distributed sampling points are shown in Table 3 in their domain x e [-5, 7] together with the value of Ptand their ranking.
Table 3 - The positions of the distribution of the sampling points in the integral domain [-5, 7] together with the value of Pt and their ranking Таблица 3 - Положения распределения точек выборки в интегральной области [-5, 7] вместе со значением Pt и их ранжированием Табела 3 - Позици^е дистрибуцц'е тачака узорковаъа у домену интеграла [-5, 7] за^едно са вредношПу Pt и ъихово рангираъе
No Location of point Ptx 102 Rank
1 -4.45455 0.114658 6
2 -3.36364 0.408543 5
3 -2.27273 0.722118 4
4 -1.18182 0.991634 3
5 -0.09091 1.171558 2
6 1.00000 1.234568 1
7 2.09091 1.171558 2
8 3.18182 0.991634 3
9 4.27273 0.722118 4
10 5.36364 0.408543 5
11 6.45455 0.114658 6
Again, the maximum value for Pt is located at x = 1 exactly.
Discussion
1) On the number of the discrete sampling points in the evaluation
In the literature of Yu et al (2022), it is suggested roughly but not proven mathematically that 17 and 19 sampling points are proper preliminarily for evaluating a complicated integral.
Here, we would stress the following. In accordance wih Hua and Wang (1081) and Fang and Wang (1994), as to the GLP, the discrepancy of the low-discrepancy point set is O(p"1(logp)s"1) for the s -dimension with the prime number p, so if we take 11 GLPs for a 1 -dimensional problem, the value of O(1/11) « 0.0909, i.e., less than 10%; analogically, for a 2 - dimensional problem, if we adopt to use 17 GLPs,
the value of O(p-1(logp)s-1) is approximately O(17"1(log17)1) « 0.0724, which is near to the situation of 1 - dimensional problem; while for a 3 -dimensional problem, if we take 19 GLPs, the approximate result of O(p-1(logp)s) is O(19-1(log19)2) « 0.0861, which is close to the situation of a 1 - dimensional problem as well. However, if we accept 23, 29, 31 or even 41 GLPs for 3-d, the consequences for O(p-1(logp)s-1) are 0.0806, 0.0737, 0.0717, or 0.0634, respectively, which are nearly the same as that of 19 GLPs basically.
The successful results of assessing complicated definite integrations realize the applicability of the approximation from the point of view of engineering practice. Perhaps the abstruse physical detail is related to the spatial correlation of spatial sampling points, which was pointed by Ripley (1981) and worth to be further explored by mathematicians.
2) On the combination of the finite sampling points in probability-based multi-objective optimization by means of the Uniform Experimental Design
The newly developed efficient approach for preliminarily assessing a definite integral with a small number of sampling points can be combined with the novel probability-based multi-objective optimization (PMOO), provided the discrete specimen points are uniformly and deterministically distributed within the domain according to the rules of the GLP and the UED. The optimal results in the present paper for typical examples indicate the advantages of this treatment. However, further applications and mathematical intensions of the appropriate algorithm for assessing numerical integration developed newly are needed to be deeply explored in future.
Besides, in order to improve the precision of approximate maximum by using discrete sampling point method, sequential algorithm for optimization can be combined with the probability - based multi -objective optimization in its discreterization (Zheng et al, 2022).
Conclusion
From the above discussion, the efficient approach for preliminarily calculating a definite integral with a small number of sampling points is successfully combined with the novel probability-based multi-objective optimization (PMOO) so as to simplify the complicated calculation of a definite integral in PMOO. The Uniform Experimental Design method and the good lattice point are involved in the combination, thus significantly simplifying complicated data processing by approximation.
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ПРИМЕНЕНИЕ КОНЕЧНЫХ ТОЧЕК ВЫБОРКИ В МНОГОЦЕЛЕВОЙ ОПТИМИЗАЦИИ, ОСНОВАННОЙ НА ВЕРОЯТНОСТИ С ПОМОЩЬЮ ЕДИНОЙ ЭКСПЕРИМЕНТАЛЬНОЙ РАЗРАБОТКИ
Maoшенг Чжэна, корреспондент, Хайпэн TeHa, Йи Вона, Джи Йюб
Северо-западный политехнический университет, г. Сиань, Народная Республика Китай
а факультет химической инженерии
б факультет естественных наук
РУБРИКА ГРНТИ: 27.47.00 Математическая кибернетика, 27.47.19 Исследование операций, 81.09.00 Материаловедение, 45.09.00 Электротехнические материалы ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: Аппроксимация для оценки определенного интеграла не перестает привлекать внимание ученых, ввиду своего практического применения в различных областях инженерных наук. Недавно был разработан эффективный подход к вычислению определенного интеграла с небольшим числом точек выборки для получения приблизительного значения численного интеграла со сложным подынтегральным выражением. В данной работе в целях упрощения сложного определенного интеграла в МООВ был применен эффективный подход с небольшим числом точек выборки, объединенный с новой многоцелевой оптимизацией, основанной на вероятности (MOOB) с помощью единой экспериментальной разработки.
Методы: Распределение точек выборки в пределах области с одним пиком является детерминированным и равномерным, что
соответствует правилам метода единой разработки и точек идеальной решетки; общая предпочтительная вероятность является уникальным и детерминированным индексом в МООВ.
Результаты: Применение эффективного подхода с конечными точками выборки при решении типовых проблем в МООВ указывает на его рациональность и удобство в эксплуатации.
Выводы: Эффективный подход с конечными точками выборки для оценки определенного интеграла успешно комбинируется с МООВ с помощью метода единой разработки и точек идеальной решетки.
Ключевые слова: предпочтительная вероятность, многоцелевая оптимизация, конечные точки выборки, упрощение оценки, единый метод разработки.
ПРИМЕНА КОНАЧНИХ ТАЧАКА УЗОРКОВАНА У ВИШЕКРИТЕРШУМС^ ОПТИМИЗАЦИИ ЗАСНОВАНОJ НА ВЕРОВАТНОЪИ ПОМОЪУ УНИФОРМНОГ ЕКСПЕРИМЕНТАЛНОГ
ДИЗАJНА
Maoшенг Ценга, аутор за преписку, Хаипенг Тенг3, Jи Ванга, Ъе иуб Универзитет Северозапад, Си]ан, Народна Република Кина а Факултет хеми]ског инженерства
б Факултет природних наука
ОБЛАСТ: математика, матери]али ВРСТА ЧЛАНКА: оригинални научни рад
Сажетак:
Увод/цил>: Апроксимаци}а процене коначног интеграла не преста}е да буде привлачна тема захваъу}уЪи свор] практично] примени у научним и инженерским областима. Недовно }е развц'ен ефикасан приступ израчунавану одре^еног интеграла с малим бро]ем тачака узоркована како би се добила приближна вредност за нумерички интеграл са компликованим интеграндом. У овом раду ефикасан приступ с малим бро]ем тачака узоркована комбинован }е са новом вишекритери]умском оптимизациям заснованом на вероватноЬи (ПМОО) помогу униформног експерименталног дизана с циъем да се по}едностави компликовани одре^ени интеграл у ПМОО.
Методе: Дистрибуцща тачака узоркована унутар подруч]а издво}еног врха детерминистичка }е и униформна, што следи из правила метода униформног дизана и тачака добре решетке.
Укупна пожеъна вероватноЯа je jeduHcmeeHu и детерминистички индекс у ПМОО.
Резултати: Примене ефикасног приступа с коначним тачкама узорковаъа за решаваъе типичних проблема у ПМОО указуjу на иегову рационалност и погодност при операц^ама. Закъучак: Ефикасан приступ с коначним тачкама узорковаъа за оцену одре^еног интеграла успешно се комбину/е са ПМОО помогу метода униформног дизаjна и тачака добре решетке.
Къучне речи: пожеъна вероватноЬа, вишекритер^умска оптимизац^а, коначне тачке узорковаъа, поjeдностав^ива^e евалуац^е, метод униформног дизаjна.
Paper received on / Дата получения работы / Датум приема чланка: 22.03.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 22. 06. 2022.
Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 24. 06. 2022.
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