Journal of Industrial Engineering Research in Production Systems

Journal of Industrial Engineering Research in Production Systems

Modeling Supply Chain Management Problem as a Bi-Level Stackelberg Game and Solving it Using Karush-Kahn-Tucker Optimality Conditions

Document Type : Research Paper

Authors
Narges Mirazizi 1
Hassan Hassanpour 1
Javad Tayyebi 2
1 Department of Mathematics, Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran
2 Department of Industrial Engineering, Faculty of Computer and Industrial Engineering, Birjand University of Technology, Birjand, Iran
10.22084/ier.2026.30601.2198
Abstract
In real-world supply chain problems, we sometimes encounter problems with two suppliers at two different levels, one of which plays the role of leader and the other of follower due to their position in the supply of goods, so the problem is formulated as a bi-level Stackelberg game model. For example, some goods are produced domestically, but the entire market demand is not met and the shortage of goods must be met through imports, and some goods are not produced domestically (or have a small production) and the entire market demand is met through imports, and smugglers may also import goods, which leads to negative economic and social consequences. In the paper two categories of the supply problem between two suppliers at two different levels. In the first category, the producer is at the first level and the importer is at the second level, and the suppliers' goal is to maximize their profits. In the second group, the importer is at the first level and the smuggler is at the second level. The importer pursues two goals: maximizing his profit and preventing smuggling by minimizing the smuggler's profit, while the smuggler seeks to maximize his profit, so we are faced with an obstacle problem. These two types of problems are modeled as bi-level Stackelberg game problems and then converted into one-level models using the Karush-Kahn-Tucker optimality conditions and solved. Finally, examples of both models are presented to explain the method and a sensitivity analysis is also performed.
Keywords
Stackelberg Game Problem
Bi-Level Game
Karush-Kahn-Tucker's Conditions
Supply Chain Management
Subjects
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Journal of Industrial Engineering Research in Production Systems
Volume 13, Issue 27
September 2026
Pages 5-19