Multi-level Capacitated Lot-sizing Problem Research Articles

A review of tactical planning models

The aim of this paper is to give an overview on models and methods used to solve tactical planning problems. The modeling and the elaboration of the well-know tactical planning problems (master planning & scheduling, material requirement planning and multi-site planning) are discussed. These problems are modeled from two “lot sizing” models called the Capacitated Lot Sizing Problem (CLSP) and Multi Level Capacitated Lot Sizing Problem (MLCLSP). From both models, a lot of extensions has been proposed in the literature. The purpose of this paper is twofold: first, classifications of the CLSP and MLCLSP as well as their extensions are given. For each model, the major scientific contributions are mentioned. These classifications made from seventy papers give an overview of “lot sizing” models dedicated to the MPS, MRP and Multi-site and show the diversity of models. Second, from a classification, an analysis of methods used for each model is given. The instance size, best gap and reference for gap computation are given for each contribution. This work can be used to elaborate an optimization tool for tactical planning problematic such as Advanced Planning System.

A heuristic for the dynamic multi-level capacitated lotsizing problem with linked lotsizes for general product structures

In this paper, a new model formulation for the dynamic multi-level capacitated lotsizing problem with linked lotsizes is introduced. Linked lotsizes means that the model formulation correctly accounts for setup carryovers between adjacent periods if production of a product is continued in the next period. This model formulation is a good compromise between the big-bucket and small-bucket model formulation in that it inherits the stability of a big-bucket model and at least partially includes the precise description of setup operations provided by a small-bucket model. A Lagrangean heuristic is developed and tested in a numerical experiment with a set of invented data and a data set taken from industry. The solutions found show a good quality.

Combining population-based and exact methods for multi-level capacitated lot-sizing problems§

We present an ant-based algorithm to solve multi-level capacitated lot-sizing problems. We apply a hybrid approach where we use the ant system to optimize the decomposition of the problem into smaller subproblems. These subproblems, containing only a few items and periods, are solved using CPLEX. Then the overall solution is derived by consolidating the partial solutions. This hybrid approach provides superior results with respect to solution quality in comparison with the existing approaches in the literature. §This research emerged from the PhD thesis of Rapeepan Pitakaso during his stay at the University of Vienna.

Multilevel capacitated lot-sizing and resource-constrained project scheduling: an integrating perspective

Some well-known decision problems can be regarded as special cases of the resource-constrained project-scheduling problem (RCPSP) with respect to the structure of their mathematical models. These decision models include assembly line balancing, job shop scheduling and some packing problems. Here, the differences and similarities of mathematical models of the multilevel capacitated lot-sizing problem (MLCLSP) and the RCPSP are shown, and it is concluded that the RCPSP is a special case of the MLCLSP. This theoretical result is complemented by an integrated, general model formulation that allows one to coordinate customer-specific orders and make-to-stock (lot size) production at the master (production) planning level.

A lagrangian relaxation approach for supply chain planning with order/setup costs and capacity constraints

A heuristic approach is developed for supply chain planning modeled as multi-item multi-level capacitated lot sizing problems. The heuristic combines Lagrangian relaxation (LR) with local search. Different from existing LR approaches that relax capacity constraints and/or inventory balance constraints, our approach only relaxes the technical constraints that each 0-1 setup variable must take value 1 if its corresponding continuous variable is positive. The relaxed problem is approximately solved by using the simplex algorithm for linear programming, while Lagrange multipliers are updated by using a surrogate subgradient method that ensures the convergence of the dual problem in case of the approximate resolution of the relaxed problem. At each iteration, a feasible solution of the original problem is constructed from the solution of the relaxed problem. The feasible solution is further improved by a local search that changes the values of two setup variables at each time. By taking the advantages of a special structure of the lot-sizing problem, the local search can be implemented by using a modified simplex algorithm, which significantly reduces its computation time. Numerical experiments show that our approach can find very good solutions for problems of realistic sizes in a short computation time and is more effective than an existing commercial optimization code.

A multi-class multi-level capacitated lot sizing model

When demand loading is higher than available capacity, it takes a great deal of effort for a traditional MRP system to obtain a capacity-feasible production plan. Also, the separation of lot sizing decisions and capacity requirement planning makes the setup decisions more difficult. In a practical application, a production planning system should prioritize demands when allocating manufacturing resources. This study proposes a planning model that integrates all MRP computation modules. The model not only includes multi-level capacitated lot sizing problems but also considers multiple demand classes. Each demand class corresponds to a mixed integer programming (MIP) problem. By sequentially solving the MIP problems according to their demand class priorities, this proposed approach allocates finite manufacturing resources and generates feasible production plans. In this paper we experiment with three heuristic search algorithms: (1) tabu search; (2) simulated annealing, and (3) genetic algorithm, to solve the MIP problems. Experimental designs and statistical methods are used to evaluate and analyse the performance of these three algorithms. The results show that tabu search and simulated annealing perform best in the confirmed order demand class and forecast demand class, respectively.

Analysis of solution space-dependent performance of simulated annealing: the case of the multi-level capacitated lot sizing problem

This study describes an analysis of different neighbourhood transition schemes and their effects on the performance of a general purpose simulated annealing (SA) procedure in solving the dynamic multi-level capacitated lot sizing problem (MLCLSP) with general product structures. The proposed neighbourhood transition schemes are based on relaxing different types of constraints, each of which defines a different solution space. The experiments assess the influence of expanding the search space which includes infeasibilities arising from the elimination of different model restrictions. The results indicate that the performance of SA is highly dependent on the definition and the tightness of the search space. Furthermore, the increase in the number of search moves carried out by SA is shown to significantly improve the results with linearly increasing computational times. Scope and purpose Simulated annealing (SA) is a popular solution technique used in the field of combinatorial optimiziation. Due to its global search capability and the simplicity of its implementation, SA is used by many researchers and practitioners in solving difficult problems. Although the performance of SA depends on the generation of neighbours and neighbourhood transition schemes, this feature has not been studied adequately in literature for production planning problems. Here, various neighbourhood transition schemes are analysed within the context of a SA procedure designed for the multi-item multi-level capacitated lot sizing problem (MLCLSP) encountered frequently in industry and especially in material requirements planning (MRP) applications. In this problem, capacity availability and inventory feasibility should be considered simultaneously in order to avoid sub-optimal solutions. Considering these two major constraints, different neighbourhood transition schemes which define different search spaces evolve in developing the general purpose SA procedure, and their effects are observed on the quality of solutions resulting from SA. This study provides valuable guidance for selecting an appropriate solution space in the design of SA or other global/local search algorithms developed for production planning problems.

Reformulations of the shortest route model for dynamic multi-item multi-level capacitated lotsizing

The shortest route representation of the dynamic multi-item multi-level capacitated lotsizing problem is appealing due to the tight bound provided by its Linear Programming (LP) relaxation. However, it suffers from two main drawbacks: Firstly, even solving the LP relaxation of problem instances with up to 16 time periods and 40 items with standard mathematical programming software might require a prohibitive amount of computer time. Secondly, the quadratic growth of the number of variables with the number of periods restricts the solution of problem instances with many periods. Both drawbacks will be addressed in this paper by proposing reformulations of the original shortest route model. Especially we will introduce a model formulation which allows theuser to find a tradeoff between model size and tightness of the lower bound obtained by the LP relaxation by specifying the number oflook ahead periods τ. Furthermore, we will provide an iterative procedure for determining those look ahead periods which result in the tightest LP relaxation. Theoretical insights as well as computational results are provided, too.

Mixed integer programming model formulations for dynamic multi-item multi-level capacitated lotsizing

Recently there have been two new proposals to model the dynamic multi-item multi-level capacitated lotsizing problem by variable redefinitions based on either a shortest route or a simple plant location representation of the underlying decision problem. Here we will introduce an extended version of the well-known inventory and lotsize model followed by a new model formulation based on modeling the changes of end-of-period inventory levels explicitly. Secondly, we will compare the suitability of different model formulations when solving problem instances with up to 40 items and 16 periods utilizing state-of-the-art standard MIP software on a personal computer.