Original scientific papers
This work is licensed under Creative Commons Attribution 4.0 International.
Received: 22 March 2022
Accepted: 24 June 2022
DOI: https://doi.org/10.5937/vojtehg70-37087
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.
Keywords: preferable probability, multi–objective optimization, finite sampling points, simplifying evaluation, uniform design method..
Pезюме:
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 (MOOВ) 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.
Abstract:
Увод/циљ: Апроксимација процене коначног интеграла не престаје да буде привлачна тема захваљујући својој практичној примени у научним и инжењерским областима. Недовно је развијен ефикасан
приступ израчунавању одређеног интеграла с малим бројем тачака узорковања како би се добила приближна вредност за нумерички интеграл са компликованим интеграндом. У овом раду ефикасан приступ с малим бројем тачака узорковања комбинован је са новом вишекритеријумском оптимизацијом заснованом на вероватноћи (ПМОО) помоћу униформног експерименталног дизајна с циљем да се поједностави компликовани одређени интеграл у ПМОО.
Методе: Дистрибуција тачака узорковања унутар подручја издвојеног врха детерминистичка је и униформна, што следи из правила метода униформног дизајна и тачака добре решетке. Укупна пожељна вероватноћа је јединствени и детерминистички индекс у ПМОО.
Резултати: Примене ефикасног приступа с коначним тачкама узорковања за решавање типичних проблема у ПМОО указују на његову рационалност и погодност при операцијама.
Закључак: Ефикасан приступ с коначним тачкама узорковања за оцену одређеног интеграла успешно се комбинује са ПМОО помоћу метода униформног дизајна и тачака добре решетке.
Keywords: пожељна вероватноћа, вишекритеријумска оптимизација, коначне тачке узорковања, поједностављивање евалуације, метод униформног дизајна.
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 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:
-
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.
-
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.
-
Robust. The UED design can be applied to a variety of situations and is robust to model changes.
-
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 
and the response y by
(1)Furthermore, it supposes that the experiment domain is the unit cube 
, the total mean value the response y on 
is,
(2)If 
sampling points 
are taken on 
then the mean value of y on these 
sampling points is
(3)In Eq.(3), 
represents a design of these m sampling points.
Fang & Wang (1994, 2018) proved that if the sampling points 
, pm are uniformly distributed on the domain 
the deviation 
of the sampling point set on 
and 
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 
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 
of the boom tie rod and the minimum angular displacement 
of the boom as the multiple objectives, and obtained the following model,
(4)
(5)The constraint conditions are,
(6)
(7)According to the optimal requirements of 
and 
, both 
and 
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 
and 
are expressed as
(8)
(9)In Eqs. (8) and (9), 
, 
, and 
express the normalization factor, the minimum and maximum values of the index 
, respectively; 
, 
, and 
indicate the normalization factor, the minimum and maximum values of the index 
, individually.
Simultaneously,
(10)
(11)In Eqs. (8) and (9), 
and 
express the lower limit and the upper limit of 
and 
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
and 
preliminarily. So the Uniform Design Table of 
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 
and 
, in which 
and 
indicate the original positions from the Uniform Design Table 
for the 
domain.
Table 2 shows the evaluation results of this problem.
Designed results U*17(175) together with the calculated consequences of W(X) and θ(X)
Таблица 1 – Полученные результаты U*17(175) вместе с рассчитанными последствиями W(X) и θ(X)
Табела 1 – Пројектовани резултати U*17(175) заједно са израчунатим последицама W(X) и θ(X)
Evaluation results of this problem
Таблица 2 – Результаты оценки данной проблемы
Табела 2 – Резултати процене овог проблема
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 
of the boom tie rod and the optimal angular displacement 
of the boom are 2.5682 tons and 0.0029° at
and 
, which are better than those of Qu’s (2004) results of 2.8580 tons, and 
and 
, comprehensively.
Moreover, regression can be applied for further optimization. The regressed result of the total probability 
with respect to 
and 
is
(12)
(13)The regressed result of the 
with respect to 
and 
is
(14)
(15)The regressed result of the total probability 
with respect to 
and 
is
(16)
(17)The optimal result of the regressed formula of Eq. (12) being maximum is 
; the corresponding values for optimal 
and 
are, 
tons,
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
together with
simultaneously within the range of 
, 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 
and 
can be expressed as,
(14)Thus, the total preferable probability 
takes its maximum value at 
distinctly; therefore, the simultaneous minimum values of 
and 
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 
together with the value of 
and their ranking.
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 – Позиције дистрибуције тачака узорковања у домену интеграла [-5, 7] заједно са вредношћу Pt и њихово рангирање
Again, the maximum value for 
is located at 
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 
for the s – dimension with the prime number p, so if we take 11 GLPs for a 1 – dimensional problem, the value of 
, i.e., less than 10%; analogically, for a 2 – dimensional problem, if we adopt to use 17 GLPs, the value of 
is approximately 
, 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 
is 
, 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 
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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ARTICLE TYPE: Original scientific paper
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