Surrogate Based Derivative Free Optimization Methodology for Supply Chain Management

Surrogate based Derivative Free Optimization Methodology for Supply Chain Management

Nihar Sahay and Marianthi Ierapetritou

Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ

Simulation models are one of the most effective tools to study supply chains. Compared to analytical and mathematical programming techniques, they offer the capability to include a greater deal of fidelity. Different kinds of simulation models have been widely used to study various aspects of supply chain management. These models provide a very convenient approach to generate various “what-if” scenarios and find the optimal values of the discrete variables. However stand-alone simulation models cannot be used to optimize the continuous variables. It is necessary to couple the simulation model with an optimization approach in order to find the optimal values of the continuous variables. During the recent years, supply chains have evolved into global, highly complex networks and the overall supply chain operations are a result of numerous autonomous, adaptive and intelligent entities. A high fidelity simulation model is not only difficult to develop but also difficult to use in an optimization framework due to computational complexity involved in each function evaluation. In order to optimize the variables in these simulation models, deterministic optimization solvers cannot be used as the derivatives are unavailable. Also, since the simulations take long times to run, it is not possible to perform a large number of simulation runs in order to approximate the derivatives. Taking these factors into consideration, we propose a surrogate based derivative free optimization methodology to solve a supply chain planning problem.

Simulation based optimization approaches have been extensively studied in the literature. These approaches have been used to solve diverse problems. Jung et al.¹ propose a simulation based optimization framework to determine the safety level of each product at each production site in order to meet the required customer satisfaction. The objective is the minimization of the expected value of the cost of the supply chain. The optimization framework consists of an outer and an inner loop. The outer loop consists of a stochastic simulation coupled with a stochastic gradient based search module for finding the optimal safety level. The inner loop consists of the discrete event simulation model coupled with a deterministic planning and scheduling optimization model.  Wan et al.² propose a simulation based optimization framework that includes domain reduction, use of least square support vector machine to construct the surrogate model, and incremental sampling by maximizing Bayesian information and expected improvement. The framework is used to optimize the inventory levels in a three stage supply chain. Mele et al.³ propose a simulation based optimization framework for parameter optimization of supply chain networks. The logical rules defined for the agents in the simulation model are parametrized and then a genetic algorithm is used to optimize the parameter values. The framework is used to determine the optimal values of the parameters associated with the inventory control policy. Derivative free optimization methods have been comprehensively reviewed by Rios and Sahinidis.⁴ Development of surrogate models for solving derivative free optimization problems has also received a great deal of attention in the literature.^5-8

In this work, a surrogate based derivative-free optimization approach is proposed to optimize a supply chain planning problem. An initial global sampling is performed to construct an initial surrogate model using the expensive simulation model, which is improved by adaptive sampling. A kriging model⁹ is used to develop the surrogate approximation whereas, a trust region based framework is used to optimize the surrogate model. The proposed optimization framework addresses issues related to the number of function calls required and also the limitations related to high dimensionality. A high dimensional supply chain planning problem is used to demonstrate the effectiveness of the proposed framework. The detailed agent based simulation model that is used as the black box function is able to provide a realistic representation of the supply chain operations.  

 

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2.            Wan X, Pekny JF, Reklaitis GV. Simulation-based optimization with surrogate models—Application to supply chain management. Computers & Chemical Engineering. 2005;29(6):1317-1328.

3.            Mele FD, Guillén G, Espuña A, Puigjaner L. A Simulation-Based Optimization Framework for Parameter Optimization of Supply-Chain Networks. Industrial & Engineering Chemistry Research. 2006/04/01 2006;45(9):3133-3148.

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