On alternative mixed integer programming formulations and LP-based heuristics for lot-sizing with setup times

We address the multi-item, capacitated lot-sizing problem (CLSP) encountered in environments where demand is dynamic and to be met on time. Items compete for a limited capacity resource, which requires a setup for each lot of items to be produced causing unproductive time but no direct costs. The problem belongs to a class of problems that are difcult to solve. Even the feasibility problem becomes combinatorial when setup times are considered. This difculty in reaching optimality and the practical relevance of CLSP make it important to design and analyse heuristics to nd good solutions that can be implemented in practice. We consider certain mixed integer programming formulations of the problem and develop heuristics including a curtailed branch and bound, for rounding the setup variables in the LP solution of the tighter formulations. We report our computational results for a class of instances taken from literature

The development and testing of new, single and multiple echelon, dynamic, capacitated, lot sizing heuristics

Three new heuristics and two random problem generators are introduced and tested. These heuristics and problem generators are associated with single and multiple echelon, dynamic, capacitated lot sizing problems, with or without setup times. These types of problems often occur in an MRP environment;First, the Dixon and Silver (DS) lot sizing heuristic was extended. The new, single echelon heuristic offers improved cost performance at a small computational expense. It incorporates newly developed perturbation factors that allow for multiple iterations of the DS heuristic. At the conclusion of all iterations, the lowest cost production plan is recalled. The second, single echelon heuristic developed and tested, the MG heuristic, is capable of solving large-scale, dynamic, capacitated, lot sizing problems, with or without setup times. This heuristic has three main sections: (A) Wagner-Whitin algorithm and a feasibility seeking subroutine; (B) a DS heuristic modified to allow for setup times; and (C) newly developed improvement algorithms that seek lower costs while maintaining feasibility;Comparison testing of the MG heuristic against other leading heuristics used a random problem generator to produce realistic, large-scale, single echelon problems. For even the largest group of problems tested, 4000 items and 25 periods, its average CPU time (on a DECstation 5000) was 1.0 minute. And, for all 216 problems tested, the MG heuristic\u27s average solution costs were just 0.86% higher than the best heuristic against which it was tested, at 0.026 the computation time;The new, multiple echelon heuristic utilizes multiple iterations of a sequential top-down approach that combines single echelon approaches with a feasibility feedback mechanism to higher echelons. Additionally, the heuristic incorporates two cost modification procedures: (A) Blackburn and Millen\u27s KCC procedure; and (B) newly developed holding cost adjustment factors, one for each echelon. The holding cost adjustment factors are available for application to each item on a particular echelon and assist the heuristic in finding a feasible solution to capacitated problems. Then, the best combination of factors is explored with a simulated annealing procedure. In comparison to other heuristics, encouraging test results were obtained for assembly problems produced using a new random problem generator

Meta-Heuristics for Dynamic Lot Sizing: a review and comparison of solution approaches

Proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve. Because these problems are so difficult, various solution techniques have been proposed to solve them. In the past decade, meta-heuristics such as tabu search, genetic algorithms and simulated annealing, have become popular and efficient tools for solving hard combinational optimization problems. We review the various meta-heuristics that have been specifically developed to solve lot sizing problems, discussing their main components such as representation, evaluation neighborhood definition and genetic operators. Further, we briefly review other solution approaches, such as dynamic programming, cutting planes, Dantzig-Wolfe decomposition, Lagrange relaxation and dedicated heuristics. This allows us to compare these techniques. Understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms. Finally, we discuss general guidelines for computational experiments and illustrate these with several examples

Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms

This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time

A priori reformulations for joint rolling-horizon scheduling of materials processing and lot-sizing problem

In many production processes, a key material is prepared and then transformed into different final products. The lot sizing decisions concern not only the production of final products, but also that of material preparation in order to take account of their sequence-dependent setup costs and times. The amount of research in recent years indicates the relevance of this problem in various industrial settings. In this paper, facility location reformulation and strengthening constraints are newly applied to a previous lot-sizing model in order to improve solution quality and computing time. Three alternative metaheuristics are used to fix the setup variables, resulting in much improved performance over previous research, especially regarding the use of the metaheuristics for larger instances

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

A hybrid heuristic for the multi-plant capacitated lot sizing problem with setup carry-over

This paper addresses the capacitated lot sizing problem (CLSP) with a single stage composed of multiple plants, items and periods with setup carry-over among the periods. The CLSP is well studied and many heuristics have been proposed to solve it. Nevertheless, few researches explored the multi-plant capacitated lot sizing problem (MPCLSP), which means that few solution methods were proposed to solve it. Furthermore, to our knowledge, no study of the MPCLSP with setup carry-over was found in the literature. This paper presents a mathematical model and a GRASP (Greedy Randomized Adaptive Search Procedure) with path relinking to the MPCLSP with setup carry-over. This solution method is an extension and adaptation of a previously adopted methodology without the setup carry-over. Computational tests showed that the improvement of the setup carry-over is significant in terms of the solution value with a low increase in computational time.FAPES

Lagrangean based lower bounds for a multi-plant lot-sizing problem with capacity constraints

2pInternational audienceThe paper addresses a multi-item, multi-plant lot-sizing problem with capacity restrictions. A set of facilities (plants) is available for producing some items. For each period of a discrete planning horizon, a demand is de ned for each pair of item and plant. The problem consists in producing all the demands such that the total production, inventory, setup and transfer costs is minimized. Setup production times are considered as well as capacity constraints on the production. Moreover, transfers between plants are allowed, however, the total transferred quantity between each pair of plants is upper bounded as well as the total inventory at each plant for a given period. The problem considered is NP-hard. We quote the work of Sambivasan and Yahya that describes some Lagrangean-based heuristics to solve a relaxed version of the problem where no transfer and storage capacities are considered. In the present work, we propose a Lagrangean lower bound on the optimal cost value of the problem based on the decomposition of the problem into Facility Location and Multi-Commodity Flow problems