New approaches for the uncapacitated three-level lot-sizing and replenishment problem with a distribution structure

We consider the uncapacitated three-level lot-sizing and replenishment problem with a distribution structure. In this NP-hard problem, a single production plant sends the produced items to replenish warehouses from where they are dispatched to the retailers in order to satisfy their demands over a finite planning horizon. The goal of the problem is to determine an integrated production and distribution plan minimizing the total costs, which comprehends fixed production and transportation setup as well as variable inventory holding costs. We describe new valid inequalities both in the space of a standard mixed integer programming (MIP) formulation and in that of a new alternative extended MIP formulation. We show that using such extended formulation, valid inequalities having similar structures to those in the standard one allow achieving tighter linear relaxation bounds. Furthermore, we propose a preprocessing approach to reduce the size of a multi-commodity MIP formulation and a multi-start randomized bottom-up dynamic programming based heuristic. Computational experiments indicate that the use of the valid inequalities in a branch-and-cut approach significantly increase the ability of a MIP solver to solve instances to optimality. Additionally, the valid inequalities for the extended formulation outperform those for the standard one in terms of number of solved instances, running time and number of enumerated nodes. Moreover, the proposed heuristic is able to generate solutions with considerably low optimality gaps within very short computational times even for large instances. Combining the preprocessing approach with the heuristic, one can achieve an increase in the number of solutions solved to optimality within the time limit together with significant reductions on the average times for solving them.

[1]  Y. B. Park,et al.  An integrated approach for production and distribution planning in supply chain management , 2005 .

[2]  Kerem Akartunali,et al.  A computational analysis of lower bounds for big bucket production planning problems , 2012, Computational Optimization and Applications.

[3]  Laurence A. Wolsey,et al.  Production Planning by Mixed Integer Programming , 2010 .

[4]  Nova Indah Saragih,et al.  A heuristic method for location-inventory-routing problem in a three-echelon supply chain system , 2019, Comput. Ind. Eng..

[5]  Laurence A. Wolsey,et al.  Uncapacitated lot-sizing: The convex hull of solutions , 1984 .

[6]  Hande Yaman,et al.  A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands , 2012, Oper. Res..

[7]  Laurence A. Wolsey,et al.  Approximate extended formulations , 2006, Math. Program..

[8]  Laurence A. Wolsey,et al.  MIP formulations and heuristics for two-level production-transportation problems , 2012, Comput. Oper. Res..

[9]  Laurence A. Wolsey,et al.  Multi-item lot-sizing problems using strong cutting planes , 1991 .

[10]  Jean-François Cordeau,et al.  A comparison of formulations for a three-level lot sizing and replenishment problem with a distribution structure , 2017, Comput. Oper. Res..

[11]  Jully Jeunet,et al.  Randomized multi-level lot-sizing heuristics for general product structures , 2003, Eur. J. Oper. Res..

[12]  Stefan Helber,et al.  A Fix-and-Optimize Approach for the Multi-Level Capacitated Lot Sizing Problems , 2010 .

[13]  Mauricio G. C. Resende,et al.  Greedy Randomized Adaptive Search Procedures , 1995, J. Glob. Optim..

[14]  Albert P. M. Wagelmans,et al.  Economic Lot Sizing: An O(n log n) Algorithm That Runs in Linear Time in the Wagner-Whitin Case , 1992, Oper. Res..

[15]  Feng Chu,et al.  A solution approach to the inventory routing problem in a three-level distribution system , 2011, Eur. J. Oper. Res..

[16]  Harvey M. Wagner,et al.  Dynamic Version of the Economic Lot Size Model , 2004, Manag. Sci..

[17]  Jesus O. Cunha,et al.  A computational comparison of formulations for the economic lot-sizing with remanufacturing , 2016, Comput. Ind. Eng..

[18]  Abdulrahim Shamayleh,et al.  Three stage dynamic heuristic for multiple plants capacitated lot sizing with sequence-dependent transient costs , 2019, Comput. Ind. Eng..

[19]  Haldun Süral,et al.  The one-warehouse multi-retailer problem: reformulation, classification, and computational results , 2012, Ann. Oper. Res..

[20]  Jean-François Cordeau,et al.  The production routing problem: A review of formulations and solution algorithms , 2015, Comput. Oper. Res..

[21]  Laurence A. Wolsey,et al.  Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems , 1993 .

[22]  K. Rameshkumar,et al.  Application of particle swarm intelligence algorithms in supply chain network architecture optimization , 2012, Expert Syst. Appl..

[23]  Jesus O. Cunha,et al.  On reformulations for the one-warehouse multi-retailer problem , 2016, Ann. Oper. Res..

[24]  Simin Zhang,et al.  Production and Distribution Planning in Danone Waters China Division , 2018, Interfaces.

[25]  Laurence A. Wolsey,et al.  Uncapacitated two-level lot-sizing , 2010, Oper. Res. Lett..

[26]  L. Cárdenas-Barrón,et al.  An optimal solution to a three echelon supply chain network with multi-product and multi-period , 2014 .