Multi-item Capacitated Lot-sizing Problem Research Articles

MIP-based heuristics for multi-item capacitated lot-sizing problem with setup times and shortage costs

We address a multi-item capacitated lot-sizing problem with setup times that arises in real-world production planning contexts. Demand cannot be backlogged, but can be totally or partially lost. Safety stock is an objective to reach rather than an industrial constraint to respect. The problem is NP-hard. We propose mixed integer programming heuristics based on a planning horizon decomposition strategy to find a feasible solution. The planning horizon is partitioned into several sub-horizons over which a freezing or a relaxation strategy is applied. Some experimental results showing the effectiveness of the approach on real-world instances are presented. A sensitivity analysis on the parameters of the heuristics is reported.

Multi-item capacitated lot-sizing with demand uncertainty

We consider a stochastic version of the classical multi-item Capacitated Lot-Sizing Problem (CLSP). Demand uncertainty is explicitly modeled through a scenario tree, resulting in a multi-stage mixed-integer stochastic programming model with recourse. We propose a plant-location-based model formulation and a heuristic solution approach based on a fix-and-relax strategy. We report computational experiments to assess not only the viability of the heuristic, but also the advantage (if any) of the stochastic programming model with respect to the considerably simpler deterministic model based on expected value of demand. To this aim we use a simulation architecture, whereby the production plan obtained from the optimization models is applied in a realistic rolling horizon framework, allowing for out-of-sample scenarios and errors in the model of demand uncertainty. We also experiment with different approaches to generate the scenario tree. The results suggest that there is an interplay between different managerial levers to hedge demand uncertainty, i.e. reactive capacity buffers and safety stocks. When there is enough reactive capacity, the ability of the stochastic model to build safety stocks is of little value. When capacity is tightly constrained and the impact of setup times is large, remarkable advantages are obtained by modeling uncertainty explicitly.

CAPACITATED LOT-SIZING PROBLEM WITH SETUP TIMES, STOCK AND DEMAND SHORTAGES

We address a multi-item capacitated lot-sizing problem with setup times, stock and demand shortages. Demand cannot be backlogged, but can be totally or partially lost. Safety stock is an objective to reach rather than an industrial constraint to respect. In this paper we consider an approach based on a lagrangean relaxation of the capacity constraints to generate a lower bound of the optimal value. This relaxation induces N single-item uncapacitated lot-sizing problems, where N denotes the number of items. A polynomial algorithm is developed to solve the sub-problems optimally. Some experimental results showing the effectiveness of the approach are reported.

Learning complementarity and setup time reduction

This article studies the effects of transmission of learning in setups for the multi-item capacitated lot-sizing problem. This learning complementarity affects the scheduling of the products and the resulting model considers simultaneous decisions about lot-sizing and sequencing in a nonlinear formulation. Learning complementarity favors frequent setups and lower inventories and reaches the zero-inventory schedules even at low levels of learning. Through the use of an example the following three scenarios are examined: (i) no transmission of learning, (ii) partial transmission of learning, and (iii) complete transmission of learning. The schedules of the three scenarios are compared and the computational complexity of the nonlinear integer model considering lot-sizing and sequencing issues is discussed.

Lagrangian heuristics for the capacitated multi-item lot-sizing problem with backordering

This paper presents two algorithms for solving a network-based formulation of the capacitated multi-item lot-sizing problem with backordering. We employ Lagrangian decomposition and Lagrangian relaxation techniques which exploit the underlying network structure of the problem. In both approaches, we exploit a transportation subproblem which guarantees a primal feasible solution at every iteration of the procedure. Further, we use a primal partitioning scheme to produce additional primal feasible solutions. Valid lower bounds are obtained at each iteration of the algorithms, thereby providing a readily available ex post measure of the quality of the primal solutions. Computational analysis shows that both algorithms are quite effective, particularly when item setup and unit backorder costs are high. We also provide a means of evaluating the potential impact of permitting backorders under various problem characteristics.

An application of Lagrangean decomposition to the capacitated multi-item lot sizing proble

In this paper we present a Lagrangean decomposition technique for solving the capacitated multi-item lot sizing problem (CMLSP). The approach decomposes the problem into a transportation problem and N independent single-item uncapacitated lot sizing problems. The algorithm applies subgradient optimization to update the Lagrangean multipliers. Each iteration of the Lagrangean procedure generates two primal feasible solutions: one from the transportation subproblem, and the other from the network flow problem resulting from a primal partition produced by the solution to the N single-item problems. The Lagrangean algorithm proves to be quite effective and stable in producing good primal and dual solutions to the CMLSP.

Analysis Of Lagrangian Decomposition For The Multi-Item Capacitated Lot-Sizing Problem

AbstractThe multi-item capacitated dynamic lot-sizing problem consists of determining the quantity and the timing of production for several products in a finite number of periods so as to satisfy a known demand in each period and minimize the sum of the set-up, production and inventory costs without incurring backlogs. A production capacity is imposed in each period.Lagrangian decomposition is applied to two formulations of the problem. Using standard subsets of constraints, it is shown that Lagrangian decomposition yields only five genuinely different relaxations. Their values and computational complexities are compared. As the number of items grows, the five relaxations merge into two families that contrast sharply by their values, their computing times and the quality of the feasible solutions that they generate.

Analysis of relaxations for the multi-item capacitated lot-sizing problem

The multi-item capacitated lot-sizing problem consists of determining the magnitude and the timing of some operations of durable results for several items in a finite number of processing periods so as to satisfy a known demand in each period. We show that the problem is strongly NP-hard. To explain why one of the most popular among exact and approximate solution methods uses a Lagrangian relaxation of the capacity constraints, we compare this approach with every alternate relaxation of the classical formulation of the problem, to show that it is the most precise in a rigorous sense. The linear relaxation of a shortest path formulation of the same problem has the same value, and one of its Lagrangian relaxations is even more accurate. It is comforting to note that well-known relaxation algorithms based on the traditional formulation can be directly used to solve the shortest path formulation efficiently, and can be further enhanced by new algorithms for the uncapacitated lot-sizing problem. An extensive computational comparison between linear programming, column generation and subgradient optimization exhibits this efficiency, with a surprisingly good performance of column generation. We pinpoint the importance of the data characteristics for an empirical classification of problem difficulty and show that most real-world problems are easier to solve than their randomly generated counterparts because of the presence of initial inventories and their large number of items.

Lot-size models with backlogging: Strong reformulations and cutting planes

We examine mixed integer programming reformulations of the uncapacitated lot-sizing problem with backlogging. First we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables, leading to a known plant location reformulation and a shortest path reformulation. Each of these reformulations is strong in the sense that its linear programming relaxation solves the lot-sizing problem. Secondly we attempt to treat the problem in the space of the original variables. We give an implicit description of the convex hull of solutions, and show how the problem of finding a violated cutting plane can be solved as a linear program. We also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm. The efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi-item capacitated lot-sizing problems with backlogging. Near optimal solutions have been found to problems with 8 periods and up to 100 times.

Lagrangean Relaxation for the Multi-Item Capacitated Lot-Sizing Problem: A Heuristic Implementation

Abstract The multi-item capacitated lot-sizing problem consists of determining the magnitude and the timing of some operations of durable results for several items in a finite number of processing periods so as to satisfy a known demand in each period. The subgradient algorithm implemented to minimize the processing costs is based on a Lagrangean relaxation of the capacity constraints imposed on the resources. The method incorporates a primal partitioning scheme—with a network flow subproblem—to obtain good feasible solutions.

Strong Formulations for Multi-Item Capacitated Lot Sizing

Multi-item capacitated lot-sizing problems are reformulated using a class of valid inequalities, which are facets for the single-item uncapacitated problem. Computational results using this reformulation are reported, and problems with up to 20 items and 13 periods have been solved to optimality using a commercial mixed integer code. We also show how the valid inequalities can easily be generated as part of a cutting plane algorithm, and suggest a further class of inequalities that is useful for single-item capacitated problems.

Soil erosion

This paper deals with the dynamic multi-item capacitated lot-sizing problem under random period demands (SCLSP). Unfilled demands are backordered and a fill rate constraint is in effect. It is assumed that, according to the static-uncertainty strategy of Bookbinder and Tan [1], all decisions concerning the time and the production quantities are made in advance for the entire planning horizon regardless of the realization of the demands. The problem is approximated with the set partitioning model and a heuristic solution procedure that combines column generation and the recently developed ABCβ heuristic is proposed.