The multi-item capacitated lot-sizing problem with setup times and shortage costs

We address a multi-item capacitated lot-sizing problem with setup times and shortage costs that arises in real-world production planning problems. Demand cannot be backlogged, but can be totally or partially lost. The problem is NP-hard. A mixed integer mathematical formulation is presented. Our approach in this paper is to propose some classes of valid inequalities based on a generalization of Miller et al. [A.J. Miller, G.L. Nemhauser, M.W.P. Savelsbergh, On the polyhedral structure of a multi-item production planning model with setup times, Mathematical Programming 94 (2003) 375–405] and Marchand and Wolsey [H. Marchand, L.A. Wolsey, The 0–1 knapsack problem with a single continuous variable, Mathematical Programming 85 (1999) 15–33] results. We also describe fast combinatorial separation algorithms for these new inequalities. We use them in a branch-and-cut framework to solve the problem. Some experimental results showing the effectiveness of the approach are reported.     

Dynamic programming approximation algorithms for the capacitated lot-sizing problem

This paper provides a new idea for approximating the inventory cost function to be used in a truncated dynamic program for solving the capacitated lot-sizing problem. The proposed method combines dynamic programming with regression, data fitting, and approximation techniques to estimate the inventory cost function at each stage of the dynamic program. The effectiveness of the proposed method is analyzed on various types of the capacitated lot-sizing problem instances with different cost and capacity characteristics. Computational results show that approximation approaches could significantly decrease the computational time required by the dynamic program and the integer program for solving different types of the capacitated lot-sizing problem instances. Furthermore, in most cases, the proposed approximate dynamic programming approaches can accurately capture the optimal solution of the problem with consistent computational performance over different instances.     

Solving the capacitated lot-sizing problem with backorder consideration

In this research, we formulate and solve a type of the capacitated lot-sizing problem. We present a general model for the lot-sizing problem with backorder options, that can take into consideration various types of production capacities such as regular time, overtime and subcontracting. The objective is to determine lot sizes that will minimize the sum of setup costs, holding cost, backorder cost, regular time production costs, and overtime production costs, subject to resource constraints. Most existing formulations for the problem consider the special case of the problem where a single source of production capacity is considered. However, allowing for the use of alternate capacities such as overtime is quite common in many manufacturing settings. Hence, we provide a formulation that includes consideration of multiple sources of production capacity. We develop a heuristic based on the special structure of fixed charge transportation problem. The performance of our algorithm is evaluated by comparing the heuristic solution value to lower bound value. Extensive computational results are presented.     

A Lagrangian-based heuristic for the capacitated lot-sizing problem in parallel machines

This paper addresses the capacitated lot-sizing problem involving the production of multiple items on unrelated parallel machines. A production plan should be determined in order to meet the forecast demand for the items, without exceeding the capacity of the machines and minimize the sum of production, setup and inventory costs. A heuristic based on the Lagrangian relaxation of the capacity constraints and subgradient optimization is proposed. Initially, the heuristic is tested on instances of the single machine problem and results are compared with heuristics from the literature. For parallel machines and small problems the heuristic performance is tested against optimal solutions, and for larger problems it is compared with the lower bound provided by the Lagrangian relaxation.     

Relaxations for two-level multi-item lot-sizing problems

We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.     

A comparison of heuristics and relaxations for the capacitated plant location problem

Approaches proposed in the literature for the Capacitated Plant Location Problem are compared. The comparison is based on new theoretical and computational results. The main emphasis is on relaxations. In particular, dominance relations among the various relaxations found in the literature are identified. In the computational study, the relaxations are compared as a function of various characteristics of the test problems. Several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics, both in computing time and in the quality of the solutions found.     

An MIP-based interval heuristic for the capacitated multi-level lot-sizing problem with setup times

This paper considers the capacitated multi-level lot-sizing problem with setup times, a class of difficult problems often faced in practical production planning settings. In the literature, relax-and-fix is a technique commonly applied to solve this problem due to the fact that setup decisions in later periods of the planning horizon are sensitive to setup decisions in the early periods but not vice versa. However, the weakness of this method is that setup decisions are optimized only on a small subset of periods in each iteration, and setup decisions fixed in early iterations might adversely affect setup decisions in later periods. In order to avoid these weaknesses, this paper proposes an extended relax-and-fix based heuristic that systematically uses domain knowledge derived from several strategies of relax-and-fix and a linear programming relaxation technique. Computational results show that the proposed heuristic is superior to other well-known approaches on solution qualities, in particular on hard test instances.     

Models for capacitated lot-sizing problem with backlogging,setup carryover and crossover

Setup operations are significant in some production environments. It is mandatory that their production plans consider some features, as setup state conservation across periods through setup carryover and crossover. The modelling of setup crossover allows more flexible decisions and is essential for problems with long setup times. This paper proposes two models for the capacitated lot-sizing problem with backlogging and setup carryover and crossover. The first is in line with other models from the literature, whereas the second considers a disaggregated setup variable, which tracks the starting and completion times of the setup operation. This innovative approach permits a more compact formulation. Computational results show that the proposed models have outperformed other state-of-the-art formulation.     

A multi-item newsvendor problem with preseason production and capacitated reactive production

Given items with short life cycles or seasonal demands, one can potentially improve profits by producing during the selling season, especially when its production capacity is substantial. We develop a two-stage, multi-item model incorporating reactive production that employs a firm’s internal capacity. Production occurs in an uncapacitated preseason stage and a capacitated reactive stage. Demands occur in the reactive stage. Reactive capacities are pre-allocated to each item in the preseason stage and cannot be changed during the reactive stage. Reactive production occurs during the selling season with full knowledge of demands. The objective is expected profit maximization. Unsatisfied demand is lost. The revenue, salvage value, and production and lost sales costs are proportional. Assuming no fixed costs, we present a simple algorithm for computing optimal policies. For a model with fixed costs for allocating preseason stage production and reactive stage capacity to product families, we characterize optimal policies and develop optimal and heuristic algorithms.     

An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure

We present a fully polynomial time approximation scheme (FPTAS) for a capacitated economic lot-sizing problem with a monotone cost structure. An FPTAS delivers a solution with a given relative error ɛ in time polynomial in the problem size and in 1/ɛ. Such a scheme was developed by van Hoesel and Wagelmans [8] for a capacitated economic lot-sizing problem with monotone concave (convex) production and backlogging cost functions. We omit concavity and convexity restrictions. Furthermore, we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem.     

Models and methods for capacitated lot-sizing problems

This is a summary of the author’s PhD thesis supervised by Philippe Chrétienne and Safia Kedad-Sidhoum and defended in December 2005 at the Université Pierre et Marie Curie (Paris VI). The thesis is written in French and is available from . This work mainly deals with multi-item capacitated lot-sizing problems with setup times, shortages on demand and safety stock deficit costs. We propose a new mathematical model that includes these new constraints. Three solution approaches are considered: branch-and-cut, Lagrangean relaxation and MIP-based heuristics approaches. Experimental results showing the effectiveness and the limit of each approach are presented. This work was financed by the ANRT and DynaSys S.A.     

Dynamic knapsack sets and capacitated lot-sizing

 A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small instances provide a significant proportion of all the facet-defining inequalities. We then consider single-item capacitated lot-sizing problems, and propose the joint study of three related sets. The first models the discrete lot-sizing problem, the second the continuous lot-sizing problem with Wagner-Whitin costs, and the third the continuous lot-sizing problem with arbitrary costs. The first set that arises is precisely a dynamic knapsack set, the second an intersection of dynamic knapsack sets, and the unrestricted problem can be viewed as both a relaxation and a restriction of the second. It follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets. Some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot-sizing problems. Received: March 28, 2001 / Accepted: November 9, 2001 Published online: September 27, 2002 RID="★" ID="★" Research carried out with financial support of the project TMR-DONET nr. ERB FMRX–CT98–0202 of the European Union. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Key words. knapsack sets – valid inequalities – simultaneous lifting – lot-sizing – Wagner-Whitin costs     

A hybrid optimization approach for multi-level capacitated lot-sizing problems

Solving multi-level capacitated lot-sizing problems is still a challenging task, in spite of increasing computational power and faster algorithms. In this paper a new approach combining an ant-based algorithm with an exact solver for (mixed-integer) linear programs is presented. A MAX–MIN ant system is developed to determine the principal production decisions, a LP/MIP solver is used to calculate the corresponding production quantities and inventory levels. Two different local search methods and an improvement strategy based on reduced mixed-integer problems are developed and integrated into the ant algorithm. This hybrid approach provides superior results for small and medium-sized problems in comparison to the existing approaches in the literature. For large-scale problems the performance of this method is among the best.     

Fix-and-optimize heuristics for capacitated lot-sizing with sequence-dependent setups and substitutions

In this paper, we consider a capacitated single-level dynamic lot-sizing problem with sequence-dependent setup costs and times that includes product substitution options. The model is motivated from a real-world production planning problem of a manufacturer of plastic sheets used as an interlayer in car windshields. We develop a mixed-integer programming (MIP) formulation of the problem and devise MIP-based Relax&Fix and Fix&Optimize heuristics. Unlike existing literature, we combine Fix&Optimize with a time decomposition. Also, we develop a specialized substitute decomposition and devise a computation budget allocation scheme for ensuring a uniform, efficient usage of computation time by decompositions and their subproblems. Computational experiments were performed on generated instances whose structure follows that of the considered practical application and which have rather tight production capacities. We found that a Fix&Optimize algorithm with an overlapping time decomposition yielded the best solutions. It outperformed the state-of-the-art approach Relax&Fix and all other tested algorithm variants on the considered class of instances, and returned feasible solutions with neither overtime nor backlogging for all instances. It returned solutions that were on average only 5% worse than those returned by a standard MIP solver after 4 hours and 19% better than those of Relax&Fix.     

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.     

On multi-item economic lot-sizing with remanufacturing and uncapacitated production

In this study, we consider the multi-item economic lot-sizing problem with remanufacturing and uncapacitated production. By observing that the problem comprises several independent single-item problems, we show how very high quality feasible solutions and bounds can be obtained by solving each item separately using an effective recently proposed approach. Computational experiments demonstrate that our approach improves the best known feasible solutions and lower bounds for all the available instances. In addition, we show that 86 instances can be solved to optimality and the remaining open gap is below 0.5% for almost all the unsolved instances.     

Hybrid simulated annealing and MIP-based heuristics for stochastic lot-sizing and scheduling problem in capacitated multi-stage production system

This paper addresses lot sizing and scheduling problem of a flow shop system with capacity constraints, sequence-dependent setups, uncertain processing times and uncertain multi-product and multi-period demand. The evolution of the uncertain parameters is modeled by means of probability distributions and chance-constrained programming (CCP) theory. A new mixed-integer programming (MIP) model with big bucket time approach is proposed to formulate the problem. Due to the complexity of problem, two MIP-based heuristics with rolling horizon framework named non-permutation heuristic (NPH) and permutation heuristic (PH) have been performed to solve this model. Also, a hybrid meta-heuristic based on a combination of simulated annealing, firefly algorithm and proposed heuristic for scheduling is developed to solve the problem. Additionally, Taguchi method is conducted to calibrate the parameters of the meta-heuristic and select the optimal levels of the algorithm’s performance influential factors. Computational results on a set of randomly generated instances show the efficiency of the hybrid meta-heuristic against exact solution algorithm and heuristics.