A computationally efficient simulation-based optimization method with region-wise surrogate modeling for stochastic inventory management of supply chains with general network structures
Introduction
A supply chain is a complicated system containing networks of information flows and material flows (Chopra, 2010, Garcia and You, 2015, Simchi-Levi, 2008). A typical supply chain usually contains different but interdependent facility nodes for different purposes, ranging from the procurement of raw materials, product processing, delivery of finished products, etc. The efficiency of a supply chain can be evaluated from different aspects, including the economics, rapidness of service, environmental sustainability, etc. Therefore, the supply chain management is focused on improving the above metrics and it has become more and more important for manufacturing companies to achieve growth in profits (Grossmann, 2005, Relvas et al., 2006, Varma et al., 2007, Wassick et al., 2012). Among the supply chain system, the inventory plays a critical role for connecting different functional units into a highly integrated system (Zipkin, 2000). The inventory management concerns demand forecasting, physical inventory carrying and quality control. It is dedicated to coordinating the logistics with production planning, and it can also help the supply chain system to respond to various kinds of uncertainties (Michalski, 2009).
The objective of inventory management optimization under demand uncertainty is to maximize the performance of the inventory system in a stochastic environment by determining an optimal set of inventory control parameters with regard to a certain inventory control policy. This is a challenging problem since there are two key performance measures: maintaining a low operation cost and not violating the service level constraints. Combined with its stochastic nature, the problems are usually intractable (Cheng et al., 2003, Jung et al., 2004). Also, the idealized assumptions necessary for mathematical models make the solutions inapplicable for real world cases (Kochel and Nielander, 2005). In order to reproduce details such as the multi-stage nature of the network, fluctuation of demands, uncertain lead times, and the integration of versatile “intelligent” control strategies, the simulation method has become mainstream for the modeling of a real-world supply chain and has been adopted by an increasing number of companies to evaluate their performance (Shah, 2005).
However, there are still a few research challenges in simulation-based optimization approaches for stochastic inventory management. The simulation is usually computationally expensive since multiple replications are required to overcome the noise in the returned result. Also, in contrast with mathematical models, simulation provides a “what-if” response to the system inputs in which there is no accessible gradient information. Though there are general approaches for solving simulation-based optimization problems such as genetic algorithm (GA) and simulated annealing (SA), they are all metaheuristic approaches which cannot guarantee the solution's quality (Mansouri, 2006, Mele et al., 2006). An important branch of simulation-based optimization resorts to the use of surrogate models, where the black-box functions are sampled and predicted by analytical approximations ranging from linear regression to adaptive nonlinear models (Cozad et al., 2014, Wang and Shan, 2007). However, inventory control optimization has unique and unfavorable features. First, a supply chain network is usually composed of a number of facility nodes and the problem's dimension can be high if there are many control parameters to be determined. Second, both the objective and the constraints for the inventory control problem are black-boxes; hence, it is a significant burden to construct the surrogate function for each equation. In addition, many existing surrogate-based methods lack generality and are based on some premises such as normally distributed demand, divergent networks, convexity assumptions, etc. Thus, an efficient algorithm for solving more general inventory control optimization problems under uncertainty is the next step for research (Chu et al., 2015b, Jung et al., 2008).
Instead of delving into supply chains with idealized presumptions, we focus our solutions to inventory system simulators featured with more realistic considerations. The definition of “general” allows the inventory system to adopt more flexible inventory control policies and principles including multi-sourcing and asynchronous decision making. The change of inventory control parameters will not only affect the performance of each node in the system but also the expected material flow rates within the entire network. A concomitant challenge is that the simulation becomes a more unpredictable black-box response, losing benign characteristics such as convexity and the possibility to be simplified as multiple single-stage models. In this work, we propose a novel region-wise surrogate modeling method for solving the inventory management optimization problem for a general supply chain network under uncertainty. By integrating the design of a simulation experiment, simulation by objective-oriented programing, and kriging metamodeling, a trust-region framework with iterative model recalibration algorithm is introduced to tackle the general inventory control problem with multi-sourcing options and asynchronous reordering. We reduce the entire supply chain network into region-wise approximations and further extend the surrogate-based optimization framework's capability to solve a large inventory system under both demand uncertainty and lead time uncertainty.
The novelties of this work are summarized as follows:
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A novel region-wise surrogate modeling of an inventory system with complex network structure
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Trust-region based algorithm with iterative model recalibration for surrogate-based optimization
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Application of simulation-based optimization to general supply chain networks with multi-sourcing capability
The remainder of the paper is organized as follows. In Section 2, a literature review for recent studies on simulation-based optimization for inventory systems is presented. Our assumptions for the general network structure and the base-stock control policy are introduced in Section 3. Section 4 presents the problem statement of inventory management optimization in general network under uncertainty. In Section 5, the simulation method and modeling procedures are explained. The efficient region-wise surrogate modeling optimization framework is stated in Section 6, where we also briefly introduce the surrogate-based optimization method to the black-box system. During Section 7, the proposed algorithm is demonstrated by two representative case studies and compared with using GA. Section 8 is the conclusion.
Section snippets
Literature review
The theory of stochastic inventory management with different modeling methods can be referred to in a great variety of literatures (Graves et al., 2000, Porteus, 2002, Zipkin, 2000). Based on these foundational works, extensive efforts have been directed towards the management of safety stocks under demand or lead time uncertainty if the inventory system follows the base-stock control policy (Ettl et al., 2000, Inderfurth and Minner, 1998; H. L. Lee and Billington, 1993, You and Grossmann, 2008
Background
In this paper, we propose a computational framework for solving the inventory optimization problems for general supply chain networks with given structure. The inventory system is a demand-driven network, and each inventory is ruled by the base-stock policy. The customers are grouped into several sales regions and external orders can be directed to any facility in the network.
Problem statement
The supply chain is dedicated to serve all demands from the sales regions, which occur during each planning period and follow some probability distributions. Each inventory manager monitors the inventory position and places orders to its supplier nodes to maintain a certain level of base-stock once per given cycle of reorder period. Orders are processed by the suppliers with their on-hand inventory and can be either fulfilled immediately or backlogged. If the order is to be fulfilled by the
Simulation details and model formulation
To solve the simulation-based optimization, we need to design a computer experiment to simulate the operation of an inventory system. Section 5.1 illustrates the process of inventory simulation with a general network structure. In Section 5.2, we formulate the corresponding optimization problem with the simulation model.
Surrogate-based optimization framework
In this section, we propose our computationally efficient algorithm for solving the simulation-based optimization problem for the inventory system. In Section 6.1, we introduce the kriging metamodeling method as the background for our surrogate-based approach to black-box systems. In Section 6.2, the region-wise surrogate modeling method is presented, and the simulation-based problem is rephrased by its reduced order formulation. The overall iterative procedures are detailed in Section 6.3. The
Case study
We apply the presented region-wise surrogate modeling and optimization framework to two case study problems for inventory management optimization. The Monte Carlo simulation for the inventory system is programed in Java using Netbeans. The network structures, operation parameters and initial values for the decision variables are loaded from an Excel document. The Java package can be called by MATLAB and we use the DACE toolbox for estimating the nonlinear surrogate functions for input–output
Conclusion
A simulation-based optimization for optimizing inventory control of general inventory systems was proposed. We introduced the multi-sourcing capability into the inventory network where each inventory can have up to 2 supplier nodes; thus, the network can have more desirable features such as order coordination and multiple manufacturing sites. The inventory simulation was programmed by the objective oriented programing method, and we used the Monte Carlo method to estimate both the expectations
References (53)
An overview of approximation methods for large-scale dynamical systems
Annu Rev Control
(2005)- et al.
Simulation-optimization approach to clinical trial supply chain management with demand scenario forecast
Comp Chem Eng
(2012) - et al.
Design and planning under uncertainty: issues on problem formulation and solution
Comp Chem Eng
(2003) - et al.
Integrated planning and scheduling under production uncertainties: bi-level model formulation and hybrid solution method
Comp Chem Eng
(2015) - et al.
Simulation-based optimization framework for multi-echelon inventory systems under uncertainty
Comp Chem Eng
(2015) - et al.
Supply chain design and optimization: challenges and opportunities
Comp Chem Eng
(2015) - et al.
Embedding structural information in simulation-based optimization
Comp Chem Eng
(2013) - et al.
Safety stocks in multi-stage inventory systems under different service measures
Eur J Oper Res
(1998) - et al.
Integrated safety stock management for multi-stage supply chains under production capacity constraints
Comp Chem Eng
(2008) - et al.
A simulation based optimization approach to supply chain management under demand uncertainty
Comp Chem Eng
(2004)
Kriging metamodeling in simulation: a review
Eur J Oper Res
Simulation-based optimisation of multi-echelon inventory systems
Int J Prod Econ
A simulated annealing approach to a bi-criteria sequencing problem in a two-stage supply chain
Comp Chem Eng
Process industry supply chains: advances and challenges
Comp Chem Eng
A simulation-optimization framework for addressing combinatorial and stochastic aspects of an R&D pipeline management problem
Comp Chem Eng
Enterprise-wide modeling & optimization: an overview of emerging research challenges and opportunities
Comp Chem Eng
Simulation-based optimization with surrogate models – application to supply chain management
Comp Chem Eng
Addressing the operational challenges in the development, manufacture, and supply of advanced materials and performance products
Comp Chem Eng
Design of responsive supply chains under demand uncertainty
Comp Chem Eng
A trust-region framework for constrained optimization using reduced order modeling
Optim Eng
A trust-region framework for managing the use of approximation models in optimization
Struct Optim
First-order approximation and model management in optimization
Large-Scale PDE-Constrain Optim
Stochastic kriging for simulation metamodeling
Oper Res
Performance measurement in a supply chain
Int J Adv Manuf Technol
Supply chain management: strategy, planning, and operation
Introduction to derivative-free optimization
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