Multi-level Capacitated Lot-sizing Problem Research Articles

A matheuristic approach for the multi-level capacitated lot-sizing problem with substitution and backorder

The lot-sizing problem aims at determining the products to be produced and their quantities for each time period, which is a difficult problem in production planning. This problem becomes even more complicated when practical aspects such as limited production capacity, bill of materials, and item substitution are considered. In this paper, we study a new variant of the lot-sizing problem, called the multi-level capacitated lot-sizing problem with substitution and backorder. Unlike previous studies, this variant considers substitutions at both the product and component levels, which is based on the real needs of manufacturers to increase planning flexibility. Backorders are allowed, but should be delivered within a certain time limitation. We formulate this problem using a mathematical programming model. A matheuristic approach is proposed to solve the problem. This first generates an initial feasible solution using a relax-and-fix algorithm, and then improves it using a hybrid fix-and-optimise algorithm. The proposed algorithm is calibrated with a full factorial design of experiments, and its efficiency is well validated. Finally, through extensive numerical experiments, we analyse the properties of this new lot-sizing problem, such as the effect of substitution options, and the influence of backorder time limitation, and provide several useful managerial insights for manufacturing companies to save costs in production planning.

New construction heuristic for capacitated lot sizing problems

We consider the classical single-level multi-item capacitated lot-sizing problem (CLSP) which is the core model for production planning. It is NP-hard and if setup operations consume capacity the feasibility problem itself is NP-complete. Several construction heuristics have been proposed in the research literature, but none of them achieves a sufficient solution quality and generality at the same time - meaning that they can be applied to different variations of the problem easily. We propose a general two-step construction heuristic (2-SCH) which sorts the customer orders in the first step and iteratively adds them to a preliminary production plan in the second step. Hence, various problem classes can be solved easily and fast. Computational experiments on the CLSP without setup times show that the 2-SCH outperforms the well-known Dixon–Silver (Dixon & Silver, 1981) and the ABC heuristics Maes and Van Wassenhove (1986) and provides better results than the genetic programming approach proposed recently by Hein et al. (2018). We also apply it to the CLSP with setup times where it outperforms the construction heuristic proposed by Trigeiro (1989). Finally, we show the flexibility of the 2-SCH by applying it to the CLSP with backorders and the multi-level CLSP as well as the single-level CLSP in an online environment.

Adaptive Genetic Algorithm Based on Fuzzy Reasoning for the Multilevel Capacitated Lot-Sizing Problem with Energy Consumption in Synchronizer Production

The multilevel capacitated lot-sizing problem (MLCLSP) is a vital theoretical problem of production planning in discrete manufacturing. An improved algorithm based on the genetic algorithm (GA) is proposed to solve the MLCLSP. Based on the solution results, the distribution of energy consumption in a synchronous production case is analyzed. In the related literature, the GA has become a much-discussed topic in solving these kinds of problems. Although the standard GA can make up for the defects of the traditional algorithm, it will lead to the problems of unstable solution results and easy local convergence. For these reasons, this research presents an adaptive genetic algorithm based on fuzzy theory (fuzzy-GA) to solve the MLCLSP. Firstly, the solving process of the MLCLSP with the fuzzy-GA is described in detail, where algorithms for key technologies such as the capacity constraint algorithm and the algorithm of solving fitness value are developed. Secondly, the auto-encoding of decision variables for MLCLSPs is studied; within this, the decision variables of whether to produce or not are encoded into a hierarchical structure based on the bill of material; combined with external demand, the decision variables of lot-sizing are constructed. Thirdly, the adaptive optimization process of parameters of the GA for the MLCLSP based on fuzzy theory is expounded, in which membership function, fuzzy rule, and defuzzification of the MLCLSP is mainly presented. Experimental studies using the processed dataset collected from a synchronizer manufacturer have demonstrated the merits of the proposed approach, in which the energy consumption distribution of the optimized production plan is given. The optimal lot-sizing is closer to the average value of the optimal value compared with the standard GA, which indicates that the proposed fuzzy-GA approach has better convergence and stability.

Distance and matching-induced search algorithm for the multi-level lot-sizing problem with substitutable bill of materials

This study examines the multi-level capacitated lot-sizing problems with a special focus on item replaceability. Two linear programming models are established using different variable definitions and the McCormick envelope. These are further strengthened using some valid inequalities. In large-scale instances, the problem becomes computationally difficult to solve because of its non-deterministic polynomial-time hardness. Therefore, a matching-induced search algorithm to solve the problem is proposed. The algorithm leverages the matching between the rounded relaxation and the incumbent solution values to fix a few binary variables. It also applies the relax-and-fix heuristic to solve the reduced-size problems and progressively improve solution quality. The algorithm is further enhanced by using both the matching information of historical feasible solution values and the distance between the relaxation and incumbent solution values. Extensive computational experiments are conducted to test the efficiency of the models and algorithms. Experimental results show that increasing the number of substitutable items on the bill of materials (BOM) or increasing the overlapping ratios among the BOMs, or both, can effectively reduce the total cost. The proposed enhanced algorithm can perform better than the relax-and-fix and some other heuristics in the literature.

A heuristic for the multi-level capacitated lot sizing problem with inventory constraints

ABSTRACTIn this paper, we present a heuristic for the multi-level capacitated lot-sizing problem (MLCLSP) with inventory constraints. This problem reflects one of the most difficult lot-sizing problems and has a high practical relevance. After addressing a mathematical definition a heuristic from the literature is enhanced to provide a solution approach. The heuristic starts calculating a solution for the uncapacitated problem and then performs backward and forward moves of the lot sizes. The approach can be used to control the work in process represented by inventories. Computational results based on different test instances are described and analysed. An example illustrates the approach of the heuristic.

A Markovian approach for multi-level multi-product multi-period capacitated lot-sizing problem with uncertainty in levels

This paper considers a multi-level, multi-item, multi-period capacitated lot-sizing problem with sequence-dependent family set-up times, set-up carry over and uncertainty in levels due to uncertainty in inspection, rework and scrap. In this study, we, first, determined total processing time for each product of each family. Then, expected number of times associated with visiting each level of each product as well as amount of raw materials are calculated. We developed a mixed integer linear programming model with a numerical example and sensitivity analysis.

A Hyperheuristic Approach for Dynamic Multilevel Capacitated Lot Sizing with Linked Lot Sizes for APS implementations

This study is concerned with solving real-life sized APS problems practically. Specifically, the problem of Multilevel Capacitated Lot Sizing Problem with linked lot sizes (MLCLSP-L) is considered. The problem is a classical, practical and notoriously hard problem. We propose a new modeling technique for MLCLSP-L based on a GA-driven hyperheuristic, which enables modeling of some issues previously not modeled. Proposed model uses an indirect representation by allowing GA search through a space of low level heuristics. Each one of the low level heuristics is simple and determines the detailed production plan of a machine in a period. The solution is constructed through combination of these low level heuristics. New model is demonstrated by solving moderate size test problem along with software developed.

A novel fuzzy stochastic multi-objective linear programming for multi-level capacitated lot-sizing problem: a real case study of a furniture company

This paper develops a fuzzy stochastic multi-objective linear programming model for a multi-level, capacitated lot-sizing problem (ML-CLSP) in a mixed assembly shop. The proposed model aims to minimize the total cost consisting of total variable production cost, inventory cost, backorder cost, and setup cost while maximizing the resource utilization rate simultaneously. To cope with inherent mixed fuzzy stochastic uncertainty associated with input data, e.g., the demand and process-related parameters, they are treated as fuzzy stochastic parameters. We conducted a numerical example from literature to illustrate the efficiency of the proposed method against other ones. To validate the expediency of the proposed ML-CLSP and solution method, a real case study was executed in a furniture company. The results demonstrate the usefulness of the proposed model and its solution approach.

A relax-and-fix with fix-and-optimize heuristic applied to multi-level lot-sizing problems

In this paper, we propose a simple but efficient heuristic that combines construction and improvement heuristic ideas to solve multi-level lot-sizing problems. A relax-and-fix heuristic is firstly used to build an initial solution, and this is further improved by applying a fix-and-optimize heuristic. We also introduce a novel way to define the mixed-integer subproblems solved by both heuristics. The efficiency of the approach is evaluated solving two different classes of multi-level lot-sizing problems: the multi-level capacitated lot-sizing problem with backlogging and the two-stage glass container production scheduling problem (TGCPSP). We present extensive computational results including four test sets of the Multi-item Lot-Sizing with Backlogging library, and real-world test problems defined for the TGCPSP, where we benchmark against state-of-the-art methods from the recent literature. The computational results show that our combined heuristic approach is very efficient and competitive, outperforming benchmark methods for most of the test problems.

Fix-and-optimize and variable neighborhood search approaches for multi-level capacitated lot sizing problems

In this paper, a new fix-and-optimize (FO) approach is proposed for two dynamic multi-level capacitated lot sizing problems (MLCLSP), the MLCLSP without setup carryover and the MLCLSP with setup carryover. Given an MIP model of a lot sizing problem, the approach iteratively solves a series of sub-problems of the model until no better solution can be found. Each sub-problem re-optimizes a subset of binary decision variables determined based on the interrelatedness of binary variables in the constraints of the model, while fixing the values of the other binary variables. Based on the FO, a variable neighbourhood search (VNS) approach for the MLCLSP without setup carryover is also developed, which can further improve the solution obtained by the FO by diversifying the search space. Numerical experiments on benchmark instances show that both our FO and VNS approaches can obtain a better solution for most instances compared with that found by the fix-and-optimize approach proposed by Helber and Sahling (International Journal of Production Economics 2010;123:247–256).

BFO: a hybrid bees algorithm for the multi-level capacitated lot-sizing problem

This paper presents a hybrid heuristic based on the bees algorithm combined with the fix-and-optimize heuristic to solve the multi-level capacitated lot-sizing problem. The bees algorithm can be used as a new method to determine the sequence in which to apply the partition in the fix-and-optimize approach. This new manner of choosing the partition adds diversity to the solution pool and yields different local optima solutions after some iterations. The bees-and-fix-and-optimize (BFO) algorithm attempts to avoid these local optima by performing random search in accordance with the concept of bees algorithm. The BFO has yielded good results for instances from the literature and, in most cases, the results are superior to the best results provided by approaches presented in recent literature. They show that this construction concept is advantageous and illustrate the efficiency of hybrid methods composed of matheuristics and metaheuristics. Furthermore, the BFO approach is a general-purpose heuristic that can be applied to solve other types of production planning problems.

Lead time considerations for the multi-level capacitated lot-sizing problem

The classical multi-level capacitated lot-sizing problem formulation is often not suitable to correctly capture resource requirements and precedence relations. Depending on lead time assumptions, either the model provides infeasible production plans or plans with costly needless inventory. We tackle this issue by explicitly modeling these two aspects and the synchronization of batches of products in the multi-level lot-sizing and scheduling formulation. Two models are presented; one considering batch production and the other one allowing lot-streaming. Comparisons with traditional models demonstrate the capability of the new approach in delivering more realistic results. The generated production plans are always feasible and cost savings of 30–40 percent compared to classical models are observed.

A variable neighborhood decomposition search algorithm for multilevel capacitated lot-sizing problems

Abstract The multi-level capacitated lot-sizing problem (MLCLSP) has been an important part of material requirement planning (MRP) system. In this paper we combine variable neighborhood decomposition search and accurate mixed integer programming (VNDS-MIP) to solve MLCLSP. This method is based on the variable neighborhood search, and with the use of exact LP/MIP solvers ILOG CPLEX, it's proved to be very efficient in solving MLCLSP problem. Computational experiments are carried out on three classes of benchmark instances and VNDS-MIP shows good performance in solving the lot-sizing problem. For the 300 small-sized instances, the VNDS-MIP algorithm can find 97.67% of the optimal solutions in several seconds; for the 150 medium-sized instances, the VNDS-MIP algorithm is better than the other methods.

Quantitative Modeling of the Capacitated Multi-Level Production-Inventory Problem in Petroleum Industry

This research deals with the multi-level capacitated lot-sizing problem originating from the production system of a large typical oil refinery. A deterministic mixed integer programming model based on the minimization of setup, production, transportation and inventory costs is formulated to determine the production, inventory and transportation quantities in each time period under complicated material-balance and capacity constraints. A numerical example is subsequently provided to illustrate the broad applicability of the proposed model.

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

Abstract 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.

A fix-and-optimize approach for the multi-level capacitated lot sizing problem

This paper presents an optimization-based solution approach for the dynamic multi-level capacitated lot sizing problem (MLCLSP) with positive lead times. The key idea is to solve a series of mixed-integer programs in an iterative fix-and-optimize algorithm. Each of these programs is optimized over all real-valued variables, but only a small subset of binary setup variables. The remaining binary setup variables are tentatively fixed to values determined in previous iterations. The resulting algorithm is transparent, flexible, accurate and relatively fast. Its solution quality outperforms those of the approaches by Tempelmeier/Derstroff and by Stadtler.

Rolling-horizon and fix-and-relax heuristics for the multi-product multi-level capacitated lotsizing problem with sequence-dependent setups

This paper discusses the multi-product multi-level capacitated lotsizing and scheduling problem with sequence-dependent setups. An exact formulation of the problem is provided as a mixed-integer program which is impractical to solve in reasonable computing time for non-small instances. To solve non-small instances of the problem, MIP-based heuristics are provided. To test the accuracy of heuristics, two lower bounds are developed and compared against the optimal solution. The trade-offs between schedule quality and computational time of heuristics are also provided.

Dynamic capacitated lot-sizing problems: a classification and review of solution approaches

This paper presents a review of four decades of research on dynamic lot-sizing with capacity constraints. We discuss both different modeling approaches to the optimization problems and different algorithmic solution approaches. The focus is on research that separates the lot-sizing problem from the detailed sequencing and scheduling problem. Our conceptional point of reference is the multi-level capacitated lot-sizing problem (MLCLSP). We show how different streams of research emerged over time. One result is that many practically important problems are still far from being solved in the sense that they could routinely be solved close to optimality in industrial practice. Our review also shows that currently mathematical programing and the use of metaheuristics are particularly popular among researchers in a vivid and flourishing field of research.

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.