Exact algorithms for multi-module capacitated lot-sizing problem, and its generalizations with two-echelons and piecewise concave production costs

• Published in: IIE transactions Vol. ahead-of-print; no. ahead-of-print; pp. 1 - 16
• Main Authors: Kulkarni, Kartik, Bansal, Manish
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.12.2023; Taylor & Francis Ltd
• Subjects: Algorithms; Computational efficiency; Cost function; fixed-parameter tractable algorithms; Linear functions; Lot sizing; Management science; Modules; multi-mode lot-sizing; Multi-module capacitated lot-sizing; n-mixing set; piecewise concave production cost; Polynomials; Production costs; Questions; Run time (computers); Subcontracting; Transportation; two-echelon capacitated lot-sizing
• ISSN: 2472-5854; 2472-5862
• DOI: 10.1080/24725854.2022.2153948

We study new generalizations of the classic capacitated lot-sizing problem with concave production (or transportation), holding, and subcontracting cost functions in which the total production (or transportation) capacity in each time period is the summation of capacities of a subset of n available modules (machines or vehicles) of different capacities. We refer to this problem as M ulti-module C apacitated L ot- S izing Problem without or with S ubcontracting, and denote it by MCLS or MCLS-S, respectively. These are NP-hard problems if n is a part of the input and polynomially solvable for n = 1. In this article we address an open question: Does there exist a polynomial time exact algorithm for solving the MCLS or MCLS-S with fixed ? We present exact fixed-parameter tractable (polynomial) algorithms that solve MCLS and MCLS-S in time for a given It generalizes algorithm of Atamtürk and Hochbaum [Management Science 47(8):1081-1100, 2001] for MCLS-S with n = 1. We also present exact algorithms for two-generalizations of the MCLS and MCLS-S: (a) a lot-sizing problem with piecewise concave production cost functions (denoted by LS-PC-S) that takes time, where m is the number of breakpoints in these functions, and (b) two-echelon MCLS that takes time. The former reduces run time of algorithm of Koca et al. [INFORMS J. on Computing 26(4):767-779, 2014] for LS-PC-S by 93.6%, and the latter generalizes algorithm of van Hoesel et al. [Management Science 51(11):1706-1719, 2005] for two-echelon MCLS with n = 1. We perform computational experiments to evaluate the efficiency of our algorithms for MCLS and LS-PC-S and their parallel computing implementation, in comparison to Gurobi 9.1. The results of these experiments show that our algorithms are computationally efficient and stable. Our algorithm for MCLS-S addresses another open question related to the existence of a polynomial time algorithm for optimizing a linear function over n-mixing set (a generalization of the well-known 1-mixing set).