Multi-level Lot Sizing Research Articles

A hybrid binary particle swarm optimization for large capacitated multi item multi level lot sizing (CMIMLLS) problem

The lot sizing problem deals with finding optimal order quantities which minimizes the ordering and holding cost of product mix. when multiple items at multiple levels with all capacity restrictions are considered, the lot sizing problem become NP hard. Many heuristics were developed in the past have inevitably failed due to size, computational complexity and time. However the authors were successful in the development of PSO based technique namely iterative improvement binary particles swarm technique to address very large capacitated multi-item multi level lot sizing (CMIMLLS) problem. First binary particle Swarm Optimization algorithm is used to find a solution in a reasonable time and iterative improvement local search mechanism is employed to improvise the solution obtained by BPSO algorithm. This hybrid mechanism of using local search on the global solution is found to improve the quality of solutions with respect to time thus IIBPSO method is found best and show excellent results.

A rolling horizon-based heuristic to solve a multi-level general lot sizing and scheduling problem with multiple machines (MLGLSP_MM) in job shop manufacturing system

Article history: Received December 10, 2013 Received in revised format 12 March 2014 Accepted April 24 2014 Available online April 25 2014 This article addresses multi-level lot sizing and scheduling problem in capacitated, dynamic and deterministic cases of a job shop manufacturing system with sequence-dependent setup times and costs assumptions. A new mixed-integer programing (MIP) model with big bucket time approach is provided to the problem formulation. It is well known that the capacitated lot sizing and scheduling problem (CLSP) is NP-hard. The problem of this paper that it is an extent of the CLSP is even more complicated; consequently, it necessitates the use of approximated methods to solve this problem. Hence, two new mixed integer programming-based approaches with rolling horizon framework have been used to solve this model. To evaluate the performance of the proposed model and algorithms, some numerical experiments are conducted. The comparative results indicate the superiority of the second heuristic. © 2014 Growing Science Ltd. All rights reserved.

A variable neighborhood search with an effective local search for uncapacitated multilevel lot-sizing problems

In this study, we improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. The improvement is twofold. First, we developed an effective local search method known as the Ancestors Depth-first Traversal Search (ADTS), which can be embedded in the VNS to significantly improve the solution quality. Second, we proposed a common and efficient approach for the rapid calculation of the cost change for the VNS and other generate-and-test algorithms. The new VNS algorithm was tested against 176 benchmark problems of different scales (small, medium, and large). The experimental results show that the new VNS algorithm outperforms all of the existing algorithms in the literature for solving uncapacitated MLLS problems because it was able to find all optimal solutions (100%) for 96 small-sized problems and new best-known solutions for 5 of 40 medium-sized problems and for 30 of 40 large-sized problems.

Multi-level lot sizing and job shop scheduling with compressible process times: A cutting plane approach

This paper proposes an integer linear programming formulation for a simultaneous lot sizing and scheduling problem in a job shop environment. Among others, one of our realistic assumptions is dealing with flexible machines which enable the production manager to change their working speeds. Then, a number of valid inequalities are developed based on problem structures. As the valid inequalities can help in reducing the non-optimal parts of the solution space, they are dealt with as some cutting planes. The proposed cutting planes are used to solve the problem in (i) cut-and-branch, and (ii) branch-and-cut approaches. The performance of each cutting plane is investigated with CPLEX 12.2 on a set of randomly-generated test data. Then, some performance criteria are identified and the proposed cutting planes are ranked by TOPSIS method.

A Lagrangian relaxation-based method and models evaluation for multi-level lot sizing problems with backorders

The capacitated multi-level lot sizing problem with backorders has received a great deal of attention in extant literature on operations and optimization. The facility location model and the classical inventory and lot sizing model with (ℓ, S) cuts have been proposed to formulate this problem. However, their comparative effectiveness has not yet been explored and is not known. In this paper, we demonstrate that on linear programming relaxation, the facility location formulation yields tighter lower bounds than classical inventory and lot sizing model. It further shows that the facility location formulation is computationally advantageous for deriving both lower and upper bounds. The results are expected to provide guidelines for choosing an effective formulation during the development of solution procedures. We also propose a Lagrangian relaxation-based heuristic along with computational results that indicate its competitiveness with other heuristics and a prominent commercial solver, Cplex 11.2.

Two parameter-tuned metaheuristic algorithms for the multi-level lot sizing and scheduling problem

Two parameter-tuned metaheuristic algorithms for the multi-level lot sizing and scheduling problem

A Note on “A MAX-MIN Ant System for Unconstrained Multi-Level Lot-Sizing Problems”

Pitakaso et al.[1] presented an ant systems based on random cumulative Wagner-Whitin (RCWW) (RCWW-STVS) for uncapacitated multilevel lot-sizing (MLLS) problems and gave out a computational result for Dellaert's instance [2] with a random variable r = 0.43. The result is not quite right. This paper highlighted the error and presented a revision to the result.

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 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.

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.

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).

Um modelo de otimização para o problema integrado de dimensionamento de lotes e programação da produção em fábricas de refrigerantes

O presente artigo apresenta, modela matematicamente e soluciona um problema multi-nível integrado de dimensionamento de lotes e programação da produção em um ambiente industrial com máquinas paralelas que apresentam restrições de capacidade, custos e tempos de preparo dependentes da seqüência. O problema é motivado pela realidade encontrada em alguns setores industriais, em particular o de fabricação e engarrafamento de bebidas. Nesse tipo de indústria a produção envolve dois níveis interdependentes com decisões relativas à armazenagem das matérias-primas e ao engarrafamento das bebidas. As diversas matérias-primas são armazenadas em tanques de onde escoam para as linhas de engarrafamento. O desafio é determinar simultaneamente o dimensionamento e a programação de custo mínimo das matérias-primas nos tanques e o envasamento de bebidas nas linhas, onde tempos e custos de trocas dependem do tipo de item previamente armazenado e envasado. É proposto um modelo matemático inteiro-misto que introduz diversas restrições combinadas que até então costumavam ser tratadas separadamente na literatura. A não existência de testes com modelos similares nos obrigou a criar um conjunto de instâncias para avaliar o modelo e as técnicas de solução propostas. As instâncias foram solucionadas otimamente por meio do pacote computacional GAMS/Cplex. A solução exata se mostrou viável apenas em instâncias de pequena dimensão devido à complexidade do problema em estudo. Os resultados computacionais obtidos pelo GAMS/Cplex são apresentados e analisados.

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.

Simultaneous lot sizing and scheduling for multi-product multi-level production

The paper deals with the multi-level lot sizing and scheduling problem for job shop production in capacitated, dynamic and deterministic cases. Demand is to be fulfilled without backlogging. The multi-level lot sizing and scheduling models of the literature are small bucket problems such as the discrete lot sizing and scheduling problem (DLSP) and the proportional lot sizing and scheduling problem (PLSP). In contrast we develop a big bucket model for multi-product multi-level production. The multi-level general lot sizing and scheduling problem with multiple machines (MLGLSP_MM) is based on the general lot sizing and scheduling problem (GLSP) for single-level production. Our focus is on minimising the sum of sequence dependent setup costs, inventory costs, production costs and the costs of maintaining the setup conditions of the machines. Inventory costs will be calculated exactly. In addition sequence dependent setup times are considered. Finally, our approach minimises the lead-time of semi-finished goods and the throughput time of finished goods.

Comparative approaches to equipment scheduling in high volume factories

Mail processing and distribution centers (P&DCs) are large factories that accept, sort, and sequence mail in preparation for delivery. A central problem in these facilities is how to schedule the equipment over the day to ensure batch production under tight equipment and workforce constraints. The problem falls into the general category of multi-level lot sizing and is notoriously difficult to solve. No exact algorithms exist that are efficient and can consistently provide high-quality solutions. In the paper, we investigate two specialized approaches for solving this problem. The first is a piece-by-piece LP-based heuristic and the second is a Benders decomposition. The heuristic uses the LP fractional solution as a target and attempts to find an integer solution that is as close to it as possible by minimizing the L 1 -norm. The procedure consists of solving the LP relaxation of the original model and two reduced mixed-integer programs. The process is similar to what has been called ‘piece-by-piece decomposition’ in nonlinear programming. While the goal there is to find better initial starting points for the nonlinear code, our goal is to improve solution quality. Computational testing using data provided by the Dallas P&DC confirmed the superiority of the piece-by-piece LP-based heuristic with respect to branch and bound, which proved to be computational expensive, and the Benders algorithm which took several hours to find feasible solutions. In general, the LP-based heuristic was an order of magnitude faster than branch and bound and provided solutions that were able to process more than 99.75% of the daily volume.

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.

Multilevel Lot Sizing with a Genetic Algorithm Under Fixed and Rolling Horizons

This paper proposes a genetic algorithm (GA) as a heuristic for multilevel lot sizing, combined with the Bit Mod heuristic developed in this context, and invoking adaptive probabilities for crossover and mutation. The influence of various parameters under fixed and rolling horizons is detailed. Design of experiments methodology is used in this connection. In the rolling horizon, the behaviour of the GA and other rules are compared with and without freezing the plan. The performance of the GA is compared with the cost-modified Wagner Whitin algorithm and cost-modified silver meal methods. The superiority of the proposed method is discussed, and case studies are given.