Multi-level Lot-sizing Problem Research Articles

Neighborhood search techniques for solving uncapacitated multilevel lot-sizing problems

In this paper, several neighborhood search techniques for solving uncapacitated multilevel lot-sizing problems are investigated. We introduce three indexes: distance, changing range, and changing level that have great influence on the searching efficacy of neighborhood search techniques. These insights can help develop more efficient heuristic algorithms. As a result, we have developed an iterated neighborhood search (INS) algorithm that is very simple but that demonstrates good performance when tested against 176 benchmark instances under different scales (small, medium, and large), with 25 instances having been updated with new best known solutions in the computing experiments.

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.

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions.

A reduced variable neighborhood search algorithm for uncapacitated multilevel lot-sizing problems

Multilevel lot-sizing (MLLS) problems, which involve complicated product structures with interdependence among the items, play an important role in the material requirement planning (MRP) system of modern manufacturing/assembling lines. In this paper, we present a reduced variable neighborhood search (RVNS) algorithm and several implemental techniques for solving uncapacitated MLLS problems. Computational experiments are carried out on three classes of benchmark instances under different scales (small, medium, and large). Compared with the existing literature, RVNS shows good performance and robustness on a total of 176 tested instances. For the 96 small-sized instances, the RVNS algorithm can find 100% of the optimal solutions in less computational time; for the 40 medium-sized and the 40 large-sized instances, the RVNS algorithm is competitive against other methods, enjoying good effectiveness as well as high computational efficiency. In the calculations, RVNS updated 7 (17.5%) best known solutions for the medium-sized instances and 16 (40%) best known solutions for the large-sized instances.

Modeling of Multi-Level Capacitated Lot-Size Scheduling Problem

Problem statement: Lot-size is the clustering of items for transportation or manufacturing processes occurring at the same time. The issue in lot-size problem is to design production processes so that the feasible production quantities are equal to customer demand quantities and the timing of production is such that inventory positions are almost zero. Approach: In this study, we explore the multi-level lot-size and scheduling problem. It is on a multi-level capacitated lot-size problem or known as the multi-level lot-size problem with bottlenecks. Two models were introduced to solve the multi-level lot-size problem namely the Billington model and Alf Kimms model. Using these models, a simple heuristic method was designed to solve a multi level capacitated lot sizing scheduling problem. Results: In this study, we showed that Alf Kimms model is more efficient than Billington model. The result given by Alf Kimms model is always feasible without further modification unlike the Billington model. This is due to the way the constraint is devised to ensure the inventory balance. The constraint used in Alf Kimms model ensures that inventory in hand is always sufficient to fulfill the demand occurred in each period. However, the use of this constraint in Billington model is to ensure that the total production for each item is always greater than or equal to the total external demand in the time horizon. Therefore, without some form of modification, the result given by Billington model will be infeasible production plan. Conclusion: A comparative study between these models shows that both models were successfully devised to solve capacitated multi-level lot sizing problem with the objective function to minimize the total holding costs and setup- cost. This study also shows that the production schedule will always start at the last period because this will give the lowest costs and it also shows that Alf Kimms model gives a set of feasible sub-optimal schedule compare to Billington model.

Discrete Particle Swarm Optimization for the Multi-Level Lot-Sizing Problem

This paper presents a Discrete Particle Swarm Optimization (DPSO) approach for the Multi-Level Lot-Sizing Problem (MLLP), which is an uncapacitated lot sizing problem dedicated to materials requirements planning (MRP) systems. The proposed DPSO approach is based on cost modification and uses PSO in its original form with continuous velocity equations. Each particle of the swarm is represented by a matrix of logistic costs. A sequential approach heuristic, using Wagner-Whitin algorithm, is used to determine the associated production planning. The authors demonstrate that any solution of the MLLP can be reached by particles. The sequential heuristic is a subjective function from the particles space to the set of the production plans, which meet the customer’s demand. The authors test the DPSO Scheme on benchmarks found in literature, more specifically the unique DPSO that has been developed to solve the MLLP.

Solving large unconstrainted multi level lot-sizing problem by a binary particle swarm optimization

In this paper, we present a binary particle swarm optimization (PSO) algorithm for unconstrained multi-level lot-sizing problem (MLLS), which is a production planning problem in materials requirements planning (MRP) system. The problem aims to find production planning which take the minimization of total setup costs and inventory holding costs. We develop a binary PSO to solve the problems. In our algorithm we firstly randomly find a set of initial solution. Then, we use the particles to find solution according to standard mechanism of PSO. After that, we apply hybrid selection mechanism to restart the algorithm. Hybrid selection is a kind of general restart mechanism in PSO but beside simply restart by randomly generate new solution we introduce good solutions which are generated from standard single level lot sizing problem such as Wagner and Whitin algorithm, Silver and Meal heuristic to combine with the current best solution and resulting of the hybrid selection we will get new population which will be used as in the standard binary PSO. Finally we apply local search with the global best solution to improve the solution quality. The hybrid PSO shows good experiment results and outperforms all other approaches in term of average quality of solution.

A variable neighborhood search based approach for uncapacitated multilevel lot-sizing problems

In this paper, an effective approach based on the variable neighborhood search (VNS) algorithm is presented to solve the uncapacitated multilevel lot-sizing (MLLS) problems with component commonality and multiple end-items. A neighborhood structure for the MLLS problem is defined, and two kinds of solution move policies, i.e., move at first improvement (MAFI) and move at best improvement (MABI), are used in the algorithm. A new rule called Setup shifting is developed to conduct a more efficient neighborhood search for the MLLS problems. Computational studies are carried out on two sets of benchmark problems. The experimental results show that the VNS algorithm is capable of solving MLLS problems with good optimality and high computational efficiency as well, outperforming most of the existing algorithms in comparison.

Incorporating a database approach into the large-scale multi-level lot sizing problem

Traditionally, optimization for large-scale multi-level lot sizing (MLLS) problems always encountered heavy computational burden. Scholars also indicated that “whatever the optimal method chosen to solve the MLLS problem, standard optimization packages were still faced with computer memory constraints and computational limits that prevented them from solving realistic size cases”. Therefore, the main purpose of this paper is to propose an optimal method to reduce the computer memory while solving the large-scale MLLS problems. The optimal method is designed to implement on a database entirely because the demand for computer memory can be reduced significantly by means of the utilization of database storage. An example is given to illustrate the proposed method and computation capability is tested for the MLLS problems with up to 1000 levels and 12 periods.

A Comparison of Particle Swarm Optimizations for Uncapacitated Multilevel Lot-sizing Problems(Theory and Methodology)

A Comparison of Particle Swarm Optimizations for Uncapacitated Multilevel Lot-sizing Problems(Theory and Methodology)

A Simple Heuristic Algorithm Based on Segmentation to Solve Multilevel Lot-sizing Problems

We have developed an effective heuristic algorithm based on segmentation to solve the multilevel lot-sizing problems in a series assembly product structure (Kaku and Xu 2006, Kaku, Li and Xu 2008). The heuristic algorithm segments the solution space and guides the search direction in which a better solution is able to exist, by using the solution structure information of the multilevel lot-sizing problem, so that better performance can be achieved. In this paper, we extend the heuristic algorithm to general case of multilevel lot sizing problem with time-invariant cost structures and no restrictive assumption on the product structure. Comparing with the results obtained by the genetic algorithm (developed by Dellaert and Jeunet 2000, Dellaert, Jeunet and Jonard 2000), the effectiveness of the heuristic algorithm is shown by simulation experiments.

Solving uncapacitated multilevel lot-sizing problems using a particle swarm optimization with flexible inertial weight

The multilevel lot-sizing (MLLS) problem is a key production planning problem in materials requirements planning (MRP) system. The MLLS problem deals with determining the production lot-sizes of various items appearing in the product structure over a given finite planning horizon to minimize the production cost, the inventory carrying cost, the back ordering cost and etc. This paper proposed a particle swarm optimization (PSO) algorithm for solving the uncapacitated MLLS problem with assembly structure. All the mathematical operators in our algorithm are redefined and the inertial weight parameter can be either a negative real number or a positive one. The feasibility and effectiveness of our algorithm are investigated by comparing the experimental results with those of a genetic algorithm (GA).

A MAX-MIN ant system for unconstrained multi-level lot-sizing problems

In this paper, we present an ant-based algorithm for solving unconstrained multi-level lot-sizing problems called ant system for multi-level lot-sizing algorithm (ASMLLS). We apply a hybrid approach where we use ant colony optimization in order to find a good lot-sizing sequence, i.e. a sequence of the different items in the product structure in which we apply a modified Wagner–Whitin algorithm for each item separately. Based on the setup costs each ant generates a sequence of items. Afterwards a simple single-stage lot-sizing rule is applied with modified setup costs. This modification of the setup costs depends on the position of the item in the lot-sizing sequence, on the items which have been lot-sized before, and on two further parameters, which are tried to be improved by a systematic search. For small-sized problems ASMLLS is among the best algorithms, but for most medium- and large-sized problems it outperforms all other approaches regarding solution quality as well as computational time.

Genetic algorithm-based integrated production planning considering manufacturing partners

This paper deals with an integrated production planning problem in the presence of manufacturing partners. In this environment, although most of the products and parts are produced in local plants, they can also be purchased from any desirable manufacturing partners at reasonable costs. The objective of this study is to provide efficient integrated production plans for both local plants with finite production capacity and manufacturing partners, while minimizing the total production costs. For solving this problem, we formulated an integrated production planning problem by modifying a multi-level lot-sizing (MLLS) problem, and proposed an efficient genetic-algorithm-based heuristic for minimizing the costs of production, inventory, and subcontracting with manufacturing partners. The proposed heuristic consists of a unique chromosome structure, a chromosome-generation method, and genetic operators. The experimental analysis shows that the proposed heuristic generates quite good solutions at low computational costs, in comparison with a commercial optimization package.

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.

An alternative to safety stock policies for multi-level rolling schedule MRP problems

We study the impact of positive lead times on the multi-level lot-sizing problem in a rolling schedule environment. We show how stockout situations may arise even in a context of deterministic demand. We therefore develop a procedure to avoid such stockouts and we compare its performance through a simulation study to a safety stock strategy. Simulation results show the superiority of the proposed procedure.

METAHEURISTIC APPROACHES FOR THE MULTILEVEL RESOURCE-CONSTRAINED LOT-SIZING PROBLEM WITH SETUP AND LEAD TIMES

We propose the use of metaheuristics for the resource-capacitated multilevel lot-sizing problem with general product structures, setup costs, setup times, and lead times. Initially, we develop a heuristic which moves production in time in order to obtain feasible solutions with good quality. Strategies for the short-term memory and long-term memory of tabu search are then included to guide the search of the subordinate heuristic for new, feasible, and better solutions. Simulated annealing components are embedded into tabu search in order to improve its performance. For small problems, the solutions provided by tabu search and the hybrid metaheuristic are compared to optimal solutions and for larger problems, the quality of the solutions is evaluated against a lower bound generated by Lagrangean relaxation.

Single-point stochastic search algorithms for the multi-level lot-sizing problem

Among the most common decisions in manufacturing and distribution companies are probably those regarding Material Requirements Planning. However, that firms are daily confronted with these decisions does not mean they are easy to handle. The multi-level lot-sizing (MLLS) problem is a combinatorial optimization problem which can only be solved optimally within reasonable delays when small instances are considered. This has motivated the search for heuristic techniques achieving a satisfactory balance between computational demands and cost effectiveness. In particular, the MLLS problem has characteristic features that have permitted the development of specific methods: interdependencies exist among stages in the product structure. In this paper, we examine the performance of single point stochastic techniques and compare them to several problem specific algorithms that exist in the literature. A large set of 280 variants of stochastic search algorithms is designed and applied to a variety of problems of small and large size. We find that these techniques, despite their simplicity and the widespread belief that they are generally efficient, only seldom outperform problem-specific algorithms, and when they do so it is usually associated with a much longer execution time. We also exhibit an efficient combination of search and annealing which is found able to produce significant and consistent improvements over problem-specific algorithms.

Randomized multi-level lot-sizing heuristics for general product structures

We consider the multi-level lot-sizing (MLLS) problem as it occurs in material requirements planning systems, with no capacity constraints and a time-invariant cost structure. Many heuristics have been developed for this problem, as well as optimal solution methods which are applicable only to small instances. Few heuristic approaches however have been specifically built to address the MLLS problem with general product structures of large size. In this paper we develop randomized versions of the popular Wagner–Whitin algorithm [Management Science 5 (1958) 89] and the Silver–Meal technique [Production and Inventory Management 14 (1973) 64] which can easily handle product structures with numerous common parts. We also provide randomized variants of more sophisticated MLLS heuristics such as Graves’ multi-pass method [TIMS Studies in the Management Sciences 16 (1981) 95], a technique due to Bookbinder and Koch [Journal of Operations Management 9 (1990) 7] and that of Heinrich and Schneeweiss [Multi-Stage Production Planning and Control, Lecture Notes in Economics and Mathematical Systems, Springer, 1986, p. 150]. The resultant heuristics are based on original randomized set-up cost modifications designed to account for interdependencies among stages. The effectiveness of the proposed algorithms is tested through a series of simulation experiments reproducing common industrial settings (product structures of large size with various degrees of complexity over long horizons). It is concluded that the randomized version of the Graves algorithm outperforms existing heuristics in most situations. The randomization of the Wagner–Whitin algorithm proved to be the best single-pass method while only requiring a low computational effort.

Multilevel Lot Sizing with Setup Times and Multiple Constrained Resources: Internally Rolling Schedules with Lot-Sizing Windows

In this paper a new time-oriented decomposition heuristic is proposed to solve the dynamic multi-item multilevel lot-sizing problem in general product structures with single and multiple constrained resources as well as setup times. While lot-sizing decisions are made sequentially within an internally rolling planning interval (or lot-sizing window), capacities are always considered over the entire planning horizon. For each submodel a model formulation based on the “Simple Plant Location” representation is developed. These mixed-integer linear submodels are solved by standard mathematical programming software even for relatively large test instances. Extensive computational tests show that the heuristic proposed provides a better solution quality than a well-known special purpose heuristic.