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Incorporate seagull optimization into ordinal optimization for solving the constrained binary simulation optimization problems

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Abstract

Constrained binary simulation optimization problems (CBSOP) are optimization problems with binary variables and stochastic objective function subject to given constraints. Solving the CBSOP by conventional optimization algorithms becomes highly time-consuming when the problem size is increased. Although the ordinal optimization (OO) theory provides a reliable framework to solve CBSOP, the constraints still limit the efficiency and competitiveness of the OO theory. In this work, an approach incorporating binary seagull optimization into ordinal optimization, abbreviated as BSOO, is developed for solving the CBSOP in a reasonable time. The BSOO comprises three essential components: emulator, exploration, and exploitation. First of all, the regularized minimal-energy tensor product B-splines are regarded as an emulator to estimate the performance of a solution. Next, the binary seagull optimization algorithm is utilized to determine N exceptional solutions from the decision space. Finally, the reformed optimal computing budget allocation is employed to find an illustrious solution from the N exceptional solutions. To verify the proposed method, the BSOO is applied for finding the optimal layout of shortcuts for maximizing the capacity of the sorting conveyor system in a reasonable time. Experimental results of the BSOO are compared to five heuristic methods. The BSOO outperforms the five heuristic methods even after the latter took more than 30 times the CPU time that was consumed by BSOO upon completion. Test results reveal that the BSOO can be adopted in a real-time application of the sortation system.

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Abbreviations

\(x = [x_{1} , \ldots ,x_{K} ]^{T}\) :

A binary solution

\(f(x)\) :

Performance metric of the system

\({\text{E}}\left[ {f({\mathbf{x}})} \right]\) :

Expectation of system’s performance

\(h_{i} ({\mathbf{x}})\) :

The ith constrained function

\(d_{i}\) :

The ith service level

\(\overline{f}({\mathbf{x}})\) :

Sample mean

L :

The number of replications

\(f_{\ell } ({\mathbf{x}})\) :

The estimate of the \(\ell\) th replication

\(\eta\) :

Penalty weight

\(F({\mathbf{x}})\) :

Penalized objective function

\(pe_{i} ({\mathbf{x}})\) :

Quadratic penalty function

\(L_{a}\) :

The replications of the accurate estimate

\( \, F_{a} ({\mathbf{x}})\) :

Penalized objective function using an accurate estimate

P :

The number of basis functions

\(w_{i}\) :

Coefficient of the B-spline

\(\Phi_{i} ({\mathbf{x}})\) :

B-splines basis functions

\({\text{Q}}\) :

n Coefficients vector

\(B_{d} ({\mathbf{x}})\) :

Vector of n B-spline piecewise polynomial functions

\({{\varvec{\Phi}}}({\mathbf{x}}_{i} )\) :

Mapping vector of the spline coefficients

\({\mathbf{w}}\) :

The spline coefficients vector

H :

The discretized matrix

\(\xi\) :

The regularization term on gradient training data

\(\delta\) :

The regularization term penalizing the norm of the spline coefficients

A :

Factor for controlling the movement behavior of the seagulls

B :

Factor for balancing between exploration and exploitation

\(\Psi\) :

Total number of seagulls

\(t_{\max }\) :

The required iterations

\({\mathbf{x}}_{i}^{t}\) :

The position of the ith seagull at iteration t

\({\mathbf{z}}_{i}^{t}\) :

The search position of the ith seagull that does not collide with other seagulls at iteration t

\({\mathbf{y}}_{i}^{t}\) :

The search direction of the ith seagull toward the elite seagull at iteration t

\({\mathbf{D}}_{i}^{t}\) :

The distance between the ith seagull and the elite seagull at iteration t

\({\mathbf{V}}_{i}^{t}\) :

The speed of the ith seagull at iteration t

\({\mathbf{x}}^{*}\) :

The position of the best-so-far elite seagull

\(A_{\min }\) :

The lower bound of A

\(A_{\max }\) :

The upper bound of A

\(B_{\min }\) :

The lower bound of B

\(B_{\max }\) :

The upper bound of B

\(\alpha_{k}\) :

The profit of knapsack example

\(\beta_{k}\) :

The weight of knapsack example

\(C_{b}\) :

The allowable computing effort

\(N\) :

Number of exceptional solutions

\(L_{0}\) :

The basic replications

\(L_{i}\) :

The replications allocated to the \(i\) th exceptional solution

Δ:

An additional computing budget

\(s\) :

A time-reducing parameter

\({\mathbf{X}} = [x_{i,j} ]_{I \times J}\) :

A solution matrix

\(x_{i,j}\) :

A 0–1 variable from start point i to endpoint j

\(\lambda\) :

The arrival interval rate of the parcel

\({\text{E}}\left[ {f({\mathbf{X}},\lambda )} \right]\) :

Throughput of the system

D :

The required number of the shortcut

\(\overline{f}({\mathbf{X}},\lambda )\) :

The sample mean of throughput

\(F({\mathbf{X}},\lambda )\) :

A penalized objective function

\(\Pi\) :

Number of randomly chosen samples

\(\Omega\) :

A representative subset

\(r\) :

The rank of an illustrious solution in \(\Omega\)

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Acknowledgements

Shih-Cheng Horng is now a professor of the Department of Computer Science and Information Engineering at Chaoyang University of Technology, Taiwan, R.O.C. Shieh-Shing Lin is currently a professor of the Department of Electrical Engineering at St. John's University, Taiwan, R.O.C.

Funding

This research work is supported in part by the Ministry of Science and Technology in Taiwan, R.O.C., under Grant MOST111-2221-E-324–021.

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, Chaoyang University of Technology, 168 Jifong E. Rd, Wufeng, Taichung, 41349, Taiwan

    Shih-Cheng Horng & Shieh-Shing Lin

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  1. Shih-Cheng Horng
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Contributions

All authors contributed to the study’s conception and design. In addition, SC and SS performed material preparation, model building, and analysis. SC wrote the first draft of the manuscript. Finally, all authors read and approved the final manuscript.

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Correspondence to Shih-Cheng Horng.

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Horng, SC., Lin, SS. Incorporate seagull optimization into ordinal optimization for solving the constrained binary simulation optimization problems. J Supercomput 79, 5730–5758 (2023). https://doi.org/10.1007/s11227-022-04880-y

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