# Bi-level programming problem in the supply chain and its solution algorithm

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## Abstract

Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where the upper level of a hierarchy may have his objective function and decision space partly determined by other levels. In addition, each planner’s control instruments may allow him to influence the policies at other levels and thereby to improve his own objective function. As a tool, bi-level programming is applied for modeling decentralized decisions in which two decision makers make decisions successively. In this paper, we specifically address bi-level decision-making problems with budget constraint as an attractive feature in the context of enterprise-wide supply chain. We first describe the typical bi-level linear programming problem (BLLPP) and its optimal solution to the penalty function problem, and then, a cooperative decision-making problem in supply chain is modeled as BLLPP. A particle swarm optimization-based computational algorithm is designed to solve the problem, and the numerical example is presented to illustrate the proposed framework.

## Keywords

Decentralized supply chain Bi-level linear programming Budget constraint Particle swarm optimization algorithm## Notes

### Funding

This study was funded by National Natural Science Foundation of China (Grant Numbers 71671079, 71361018) and Humanities and Social Science Foundation of Ministry of Education of China (Grant Number 15YJCZH107).

### Compliance with ethical standards

### Conflict of interest

All authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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