Stochastic optimization using grey wolf optimization with optimal computing budget allocation

doi:10.1016/j.asoc.2021.107154

Applied Soft Computing

Volume 103, May 2021, 107154

https://doi.org/10.1016/j.asoc.2021.107154 Get rights and content

Highlights

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A new optimal computing budget allocation model is built.
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The search efficiency of the grey wolf algorithm is improved.
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The novel approach solves stochastic optimization problem more efficiently.
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Numerical testing confirms the improvement of the search efficiency.

Abstract

Stochastic optimization problems exist widely in many manufacturing and service systems. Due to the stochastic nature, these problems usually have no analytical solutions and are difficult to solve. This research proposes a hybrid approach that integrates the grey wolf optimization algorithm and the simulation optimization framework. In this hybrid approach, the grey wolf optimization algorithm is used to search for candidate solutions from the solution space, while the simulation helps the algorithm to identify the desired solutions such that the search is guided to more promising regions. To enhance the efficiency of simulation, this work designs a computing budget allocation rule that helps the grey wolf optimization algorithm to select the elite candidate solutions in each iteration. The proposed computing budget allocation rule is then integrated with the grey wolf optimization algorithm to solve stochastic optimization problems. Numerical experiments confirm that the proposed computing budget allocation rule performs better than extant allocation rules, and can find the better solution for stochastic optimization problems using fewer iterations by integration with the grey wolf optimization algorithm.

Introduction

Most of the real-world engineering optimization problems involve a great deal of uncertainty in the form of randomness [1], [2], [3], [4], [5], [6]. For example, in remanufacturing systems, it is difficult to obtain the accurate information of an end-of-life product when we disassemble it because we do not know its exact depreciation and abrasion it received during its use [7]. Hence, disassembly sequence planning and disassembly line balancing problems are usually formulated as stochastic optimization problems [8], [9]. Stochastic optimization problems are usually intractable due to their randomness and the large solution space [10], [11], [12]. Thus, it is difficult to find their optimal solutions. To overcome the difficulty, one can find an algorithm that can search promising solutions from a large solution space, and design a stochastic simulation approach to evaluate the candidate solutions accurately.

Population-based meta-heuristics are useful optimization algorithms in solving complex optimization problems because they do not rely on the problem structure [13], [14]. They start with a set of solutions, which is known as the initial population. Elite solutions are selected from this population to generate solutions for the next population. By repeating the above process, the approximately optimal solution can be found for the optimization problem. In recent years, a novel population-based meta-heuristic, called grey wolf optimization (GWO), was introduced to solve continuous optimization problems in deterministic environments [15], [16], [17], [18]. GWO mimics a predation process of grey wolves where an optimal solution of the optimization problem is assumed to be a prey [16], [17], [18], [19], [20], [21], [22], [23]. Due to its easy implementation and superior search ability, GWO has been successfully employed to solve various deterministic optimization problems [24], [25], [26].

In the case of deterministic optimization problems, the performance of a candidate solution can be evaluated with certainty. However, due to the stochastic nature of stochastic optimization problems, the performance of a candidate solution must be evaluated multiple times [27], [28], [29]. Simulation is one of the most frequently used approach to evaluate the performance of a solution in stochastic optimization problems that are usually highly complex [30]. Although the true performance of a solution can be obtained if we conduct a very large number of simulations, the computation resources are always limited. Thus, the number of simulations runs for each solution is the key factor that affects the accuracy of evaluation.

In this research, we aim to apply GWO to solve stochastic optimization problems. Accurate evaluation of each candidate solution can guide GWO to search for more promising regions so as to increase its search efficiency. Therefore, we have to use the limited computation resources to evaluate the objective function values of solutions as accurately as possible. By analysing the procedure of GWO, we found that GWO needs to choose the top 3 best solutions and the best solution among all the candidate solutions at each iteration. In fact, determining the desired solutions from the population can be viewed as a ranking and selection problem that aims to select the top $m$ solutions and the best solution from a finite number of alternatives [31]. Prior studies on ranking and selection problem can be categorized into three major approaches, i.e., the indifference-zone (IZ), the value of information (VIP) and the optimal computing budget allocation (OCBA). The IZ approach assumes that the performance difference of the best solution and other solutions is equal or greater than a given threshold, and its goal is to find a feasible approach such that a pre-specified probability of correct selection can be achieved [32], [33], [34]. The VIP describes the evidence of correct selection employing the Bayesian posterior distribution. The simulation budget allocation is determined via maximizing the value of information based on decision theory [35]. The OCBA approach intelligently allocates the simulation budget according to both mean and variance of candidate solutions in order to maximize the probability of correct selection [29], [36]. A comprehensive comparison of the performance among the three procedures was conducted in [37], and it concluded that no approach can dominate the others in all scenarios.

OCBA, which is developed from ordinal optimization [38], [39], is an efficient method for determining the computing resource allocation such that the best solution can be selected with the highest probability of correct selection under a fixed computing budget [40]. It sequentially determines the computation resources among candidate solutions based on the newly updated sample mean and variance. At each iteration, the ratios of computation resources among solutions are calculated according to sample means and variances. Hence, OCBA can choose the desired solutions from multiple alternative solutions more effectively and correctly. In recent years, the OCBA framework has been extended for different types of ranking and selection problems, including subset selection problems [41], [42], [43], [44], ranking related problems [45], [46], [47], and multi-objective selection problems [48].

Existing studies did not consider choosing the top $m$ best solutions and the best one simultaneously from multiple alternative solutions. This new type of ranking and selection procedure is needed when we use GWO to solve stochastic optimization problems. To improve the search efficiency of GWO to do such work, we need to derive a new computing budget allocation rule. The contribution of this research is threefold. Firstly, motivated by using GWO in solving stochastic optimization problems, a new computing budget allocation rule of selecting the top $m$ best solutions and the best one from a finite number of alternatives is derived. Secondly, a hybrid approach that integrates GWO and the suggested computing budget allocation rule is proposed to solve the stochastic optimization problems using simulation. Lastly, the efficiency of the GWO has been improved by integrating with the suggested computing budget allocation rule. The improvement has been confirmed by the numerical experiments.

The rest of this paper is organized as follows. The details of GWO are introduced in Section 2. In Section 3, we formulate the new ranking and selection problem, and derive the approximated asymptotically optimal allocation rule. The hybrid approach that integrates GWO and the suggested computing budget allocation rule is proposed. Numerical experiments are given in Section 4. Section 5 concludes this research and suggests some possible future research.

Section snippets

Grey Wolf optimization algorithm

GWO is a novel population-based meta-heuristic that is initially introduced by Mirjalili et al. [15] to solve continuous optimization problems in deterministic environments. It is inspired by the social life of wolves with hierarchy structures. In a grey wolf pack, a leader located in the first level is named as alpha ( $α$ ) wolf. The leader makes decisions about the whole wolf pack such as hunting and sleeping. Other members within the pack must obey its rules. Therefore, alpha wolf has the

Computing budget allocation rule

According to GWO algorithm, ${\vec{π}}_{α}$ represents the best individual in the population, ${\vec{π}}_{β}$ and ${\vec{π}}_{δ}$ are the top 2 best individuals except ${\vec{π}}_{α}$ . At each iteration, the top 3 best individuals are chosen. The best one is used to update ${\vec{π}}_{α}$ , and the rest are employed to update ${\vec{π}}_{β}$ and ${\vec{π}}_{δ}$ without ranking requirements. In stochastic optimization problems, the desired individuals cannot be selected with certainty due to the inherent noises of using simulation to evaluate the objective function. In order to

Experimental study

In this section, numerical experiments are carried out to verify the performance of OCBA-mb and the effectiveness of GWO with OCBA-mb. Equal allocation (EA) and proportional to variance (PTV) rules are used as benchmark for comparison. They have been widely used for comparison in the existing studies [49], [50], [51], [52]. Equal allocation (EA) rule is the simplest allocation rule that can be implemented easily. It is usually used as a benchmark method for comparison for all kinds of ranking

Conclusion and future work

This research proposes an integrating approach of grey wolf optimization (GWO) and simulation to solve stochastic optimization problems. By analysing GWO, we suggested a computing budget allocation rule that is named as OCBA-mb to help GWO to find the desired individuals as accurately as possible with a fixed simulation budget in each iteration. By integrating OCBA-mb with GWO, numerical experiments have shown that significant improvement in the search efficiency of GWO in solving stochastic

CRediT authorship contribution statement

Yaping Fu: Writing - original draft, Software, Formal analysis. Hui Xiao: Methodology, Conceptualization, Writing - review & editing. Loo Hay Lee: Supervision, Investigation, Validation. Min Huang: Writing - review & editing, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported in part by National Natural Science Foundation of China under grant numbers 61703220, 71971176 and 71620107003, the Applied Basic Research Program of Sichuan Province, China under grant number 2020YJ0027, the China Postdoctoral Science Foundation Funded Project under grant number 2019T120569, the Shandong Province Outstanding Youth Innovation Team Project of Colleges and Universities of China under grant number 2020RWG011, and the Fundamental Research Funds for the

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Stochastic optimization using grey wolf optimization with optimal computing budget allocation

Highlights

Abstract

Introduction

Section snippets

Grey Wolf optimization algorithm

Computing budget allocation rule

Experimental study

Conclusion and future work

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Appl. Soft Comput.

Inform. Sci.

European J. Oper. Res.

Comput. Ind. Eng.

Transp. Res. E

Adv. Eng. Softw.

Expert Syst. Appl.

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Expert Syst. Appl.

Knowl.-Based Syst.

Appl. Soft Comput.

Transport. Res. D

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J. Clean. Prod.

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