Article
Single-Manufacturer Multi-Retailer Supply Chain Models with
Discrete Stochastic Demand
Yaqing Xu
1
, Jiang Zhang
2
, Zihao Chen
1
and Yihua Wei
3,
*
Citation: Xu, Y.; Zhang, J.; Chen, Z.;
Wei, Y. Single-Manufacturer
Multi-Retailer Supply Chain Models
with Discrete Stochastic Demand.
Sustainability 2021, 13, 8271.
https://doi.org/10.3390/su13158271
Academic Editor: João Carlos de
Oliveira Matias
Received: 16 June 2021
Accepted: 20 July 2021
Published: 23 July 2021
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4.0/).
1
School of Economics and Management, Xidian University, Xi’an 710071, China; yaqingxu2002@sina.com (Y.X.)
2
Robert B. Willumstad School of Business, Adelphi University, Garden City, NY 11530, USA;
zhang@adelphi.edu
3
School of Management, Shanghai University, Shanghai 200444, China
* Correspondence: weiyh2001@aliyun.com
Abstract:
Although there are highly discrete stochastic demands in practical supply chain prob-
lems, they are seldom considered in the research on supply chain systems, especially the single-
manufacturer multi-retailer supply chain systems. There are no significant differences between
continuous and discrete demand supply chain models, but the solutions for discrete random demand
models are more challenging and difficult. This paper studies a supply chain system of a single
manufacturer and multiple retailers with discrete stochastic demands. Each retailer faces a random
discrete demand, and the manufacturer utilizes different wholesale prices to influence each retailer’s
ordering decision. Both Make-To-Order and Make-To-Stock scenarios are considered. For each
scenario, the corresponding Stackelberg game model is constructed respectively. By proving a series
of theorems, we transfer the solution of the game model into non-linear integer programming model,
which can be easily solved by a dynamic programming method. However, with the increase in the
number of retailers and the production capacity of manufacturers, the computational complexity of
dynamic programming drastically increases due to the Dimension Barrier. Therefore, the Fast Fourier
Transform (FFT) approach is introduced, which significantly reduces the computational complexity
of solving the supply chain model.
Keywords:
decentralized supply chain; multiple retailers; discrete demand; Fast Fourier Transform
1. Introduction
Many studies in supply chain management focus on one manufacturer/supplier and
one retailer setting; those studies have produced abundant results in the supply chain
integration, supply chain coordination, and supply chain inventory management, etc. The
studies in one manufacturer/supplier and multiple retailers in the decentralized setting are
more complicated and challenging. Though there are attempts at supply chain integration,
many decentralized supply chain decisions remain vitally important in the practices.
Most supply chain models consider the uncertain demand as a continuous random
variable, but demands for certain products such as luxury automobiles, large appliances,
expensive jewelry, watches, etc., are discrete random variables. Though continuous and
discrete demand–supply chain models have no significant differences, the discrete ran-
dom demand models are considered as stochastic, dynamic games between supplier and
retailers. The solutions for such models are more challenging and difficult.
This paper studies a single-manufacturer multiple-retailers supply chain system. Each
retailer faces a different random discrete demand, and the manufacturer offers differential
wholesale prices to each retailer to influence each retailer’s ordering quantity, thereby
maximizing their expected profit. Both Make-To-Order and Make-To-Stock scenarios are
considered. For each scenario, the original stochastic game problems are transferred into
discrete optimization problems. Discrete optimization problems can be generally solved
by dynamic programming methods. However, as the problem dimension increases, the
Sustainability 2021, 13, 8271. https://doi.org/10.3390/su13158271 https://www.mdpi.com/journal/sustainability
