A Smoothing Algorithm for a New Two-Stage Stochastic Model of Supply Chain Based on Sample Average Approximation

Liu Yang,Xiao-Jiao Tong,Yao Xiong

doi:10.1155/2017/5681502

Abstract

We construct a new two-stage stochastic model of supply chain with multiple factories and distributors for perishable product. By introducing a second-order stochastic dominance (SSD) constraint, we can describe the preference consistency of the risk taker while minimizing the expected cost of company. To solve this problem, we convert it into a one-stage stochastic model equivalently; then we use sample average approximation (SAA) method to approximate the expected values of the underlying random functions. A smoothing approach is proposed with which we can get the global solution and avoid introducing new variables and constraints. Meanwhile, we investigate the convergence of an optimal value from solving the transformed model and show that, with probability approaching one at exponential rate, the optimal value converges to its counterpart as the sample size increases. Numerical results show the effectiveness of the proposed algorithm and analysis.

Highlights

A supply chain planning (SCP) is a network of suppliers, manufacturing plants, warehouses, and distribution channels organized to acquire raw materials, convert these raw materials into specified final products, and distribute these products to customers
Most real SCP problems are characterized by numerous sources of technical and commercial uncertainty; critical parameters such as customer demands, suppliers, and resource capacities are quite uncertain
There are four advantages in our models and methods: (i) we introduce a stochastic dominance (SSD) constraint to describe the preference consistency of the risk taker while minimizing the expected cost of company; (ii) we transfer the two-stage problem into an equivalent one-stage problem which can be solved more ; (iii) the smoothing sample average approximation (SAA) method can avoid the infinite constraints in the transformed models and the size of the smoothing algorithm model will not increase as the sample grows; (iv) the smoothing algorithm can get the global optimal solution because the smoothing function maintains the original convexity

Summary

Introduction

A supply chain planning (SCP) is a network of suppliers, manufacturing plants, warehouses, and distribution channels organized to acquire raw materials, convert these raw materials into specified final products, and distribute these products to customers. Beginning with the seminal work of Geoffrion and Graves [1] on multicommodity distribution system design, a large number of optimization-based approaches have been proposed for the problem of SCP; see, for example, [2,3,4]. Most real SCP problems are characterized by numerous sources of technical and commercial uncertainty; critical parameters such as customer demands, suppliers, and resource capacities are quite uncertain. There are a few research works addressing comprehensive SCP with twostage stochastic models. As far as we know, Tsiakis et al considered a two-stage stochastic programming model for SCP under demand uncertainty, where the authors developed a large-scale mixed-integer linear programming model; see [5]

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Results

Conclusion