A Network Flow Algorithm for Solving Generalized Assignment Problem

Yongwen Hu,Qunpo Liu,M Javaid

doi:10.1155/2021/5803092

Abstract

The generalized assignment problem (GAP) is an open problem in which an integer k is given and one wants to assign k ′ agents to k k ′ ≤ k jobs such that the sum of the corresponding cost is minimal. Unlike the traditional K -cardinality assignment problem, a job can be assigned to many, but different, agents and an agent may undertake several, but different, jobs in our problem. A network model with a special structure of GAP is given and an algorithm for GAP is proposed. Meanwhile, some important properties of the GAP are given. Numerical experiments are implemented, and the results indicate that the proposed algorithm can globally and efficiently optimize the GAP with a large range cost.

Highlights

Assignment problem (AP) is one of the fundamental combinatorial optimization problems with various applications in real life. e classical AP is proved to be an NP-hard problem, and it deals with a situation of assigning n jobs to n agents such that each job must be processed by exactly one agent and vice versa
It is well known that AP is thought of as a special min-cost flow problem, which suggests that classical AP or generalized assignment problem (GAP) can be solved by some methods with network flow theory
The performance of the proposed algorithm is demonstrated by numerical experiments and we compare its performance with a PP-Lapjv algorithm [31] and bees algorithm for GAP. e bees algorithm is proposed by Ozbakir et al [23] for the complex integer optimization problem. e PP-Lapjv algorithm is an improved version of Lapjv algorithm [32, 33]. e GAP is transformed into a balanced AP when we use the PP-Lapjv algorithm for solving GAP

Summary

Introduction

Assignment problem (AP) is one of the fundamental combinatorial optimization problems with various applications in real life. e classical AP is proved to be an NP-hard problem, and it deals with a situation of assigning n jobs to n agents such that each job must be processed by exactly one agent and vice versa. Dell’Amico and Martello [4] considered a generalization of AP (called K-cardinality linear assignment problem) where one wants to assign k (out of m ) agents to k (out of n ) jobs (k ≤ min(m, n)) so that the sum of the corresponding cost is minimal. We consider a more generalized assignment problem (GAP) in which a cost matrix Am×n and positive integer k(0 < k ≤ min{m, n}) are given and one wants to assign k′ agents to k(k′ ≤ k) jobs so that the sum of the corresponding cost is minimal. Dell’Amico and Martello developed some algorithms for solving K-cardinality linear assignment problem by min-. It is well known that AP is thought of as a special min-cost flow problem, which suggests that classical AP or GAP can be solved by some methods with network flow theory.

Mathematical Formulation and Network Flow Model

Proposed Algorithm for Solving GAP

Numerical Experiments

Conclusion and Further