Multi-item Capacitated Lot-sizing Problem Research Articles

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.

Motivations and analysis of the capacitated lot-sizing problem with setup times and minimum and maximum ending inventories

This paper first analyzes the negative impact of the end-of-horizon effect when solving the capacitated multi-item lot-sizing problem with setup costs and times on a rolling horizon. Maximum ending inventories for items and a global minimum ending inventory are considered to define a new optimization problem whose optimal solutions are much less impacted by the end-of-horizon effect. Then, a generation scheme is proposed to create new instances with initial inventories and ending inventories. This scheme relies on the analysis of the cyclical production planning problem to derive relevant parameters. Computational experiments are carried out to compare the solutions obtained for original instances of the literature and for the new instances, and to analyze the relevance of the new instances on a rolling horizon.

Design of efficient multiobjective binary PSO algorithms for solving multi‐item capacitated lot‐sizing problem

In this paper, a multi-item capacitated lot-sizing problem with setup times and backlogging (MICLSP_SB) is addressed with respect to two conflicting objective functions. This nondeterministic polynomial time (NP)-hard problem consists of finding optimal production plans while minimizing, simultaneously, the total cost and the total inventory level. To effectively solve the considered problem, two new versions of multiobjective binary particle swarm optimization are designed. In the first proposed version “standard multiobjective particle swarm optimization” (S-MOBPSO) algorithm, two main contributions are introduced. First, a new particle encoding and decoding method is developed to treat the MICLSP_SB decision variables. Second, the overall violation minimization method and the constraint handling method are combined to effectively handle the problem constraints. To further enhance the overall performance, an improvement procedure that minimizes the capacity constraints violations is developed and embedded into the S-MOBPSO algorithm. This second version is named improved MOBPSO (I-MOBPSO). An illustrative example is presented to explain the application of the proposed algorithms. Five performance metrics are considered to measure and evaluate the effectiveness of the proposed algorithms. Experimental results demonstrate the efficiency of the S-MOBPSO and I-MOBPSO algorithms compared to each other as well as the basic MOBPSO and nondominated sorting genetic algorithm II.

A fix-and-optimize heuristic for the capacitated multi-item stochastic lot-sizing problem

This study addresses the stochastic multi-item capacitated lot-sizing problem. Here, it is assumed that all items are produced on a single production resource and unmet demands are backlogged. The literature shows that the deterministic version of this problem is NP-Hard. We consider the case where period demands are time-varying random variables. The objective is to determine the minimum expected cost production plan so as to meet stochastic period demands over the planning horizon. We extend the mixed integer programming formulation introduced in the literature to capture the problem under consideration. Further, we propose a fix-and-optimize heuristic building on an item-period oriented decomposition scheme. We then conduct a numerical study to evaluate the performance of the proposed heuristic as compared to the heuristic introduced by Tempelmeier and Hilger [16]. The results clearly show that the proposed fix-and-optimize heuristic arises as both cost-efficient and time-efficient solution approach as compared to the benchmark heuristic.

Constant Approximation Algorithm for Nonuniform Capacitated Multi-Item Lot Sizing via Strong Covering Inequalities

We study the nonuniform capacitated multi-item lot-sizing problem. In this problem, there is a set of demands over a planning horizon of T discrete time periods, and all demands must be satisfied on time. We can place an order at the beginning of each period s, incurring an ordering cost Ks. In this order, we can order up to Cs units of products. On the other hand, carrying inventory from time to time incurs an inventory holding cost. The goal of the problem is to find a feasible solution that minimizes the sum of ordering and holding costs. Levi et al. [Levi R, Lodi A, Sviridenko M (2008) Approximation algorithms for the capacitated multi-item lot-sizing problem via flow-cover inequalities. Math. Oper. Res. 33(2):461–474.] gave a two-approximation for the problem when the capacities Cs are the same. Extending the result to the case of nonuniform capacities requires new techniques as pointed out in the discussion section of their paper. In this paper, we solve the problem by giving a 10-approximation algorithm for the capacitated multi-item lot-sizing problem with general capacities. The constant approximation is achieved by adding an exponential number of new covering inequalities to the natural facility location–type linear programming (LP) relaxation for the problem. Along the way of our algorithm, we reduce the lot-sizing problem to two generalizations of the classic knapsack-covering problem. We give LP-based constant approximation algorithms for both generalizations via the iterative rounding technique.

New integration of preventive maintenance and production planning with cell formation and group scheduling for dynamic cellular manufacturing systems

This paper proposes a new integration method for cell formation, group scheduling, production, and preventive maintenance (PM) planning problems in a dynamic cellular manufacturing system (CMS). The cell formation sub-problem aims to form part families and machine groups, which minimizes the inter-cell material handling, under-utilization, and relocation costs. The production planning aspect is a multi-item capacitated lot-sizing problem accompanied by sub-contracting decisions, while the group scheduling problem deals with the decisions on the sequential order of the parts and their corresponding completion times. The purpose of the maintenance sub-problem is to determine the availability of the system and the time when the noncyclical perfect PM must be implemented to reduce the number of corrective actions. Numerical examples are generated and solved by Bender’s decomposition pack in GAMS to evaluate the interactions of the proposed model. Statistical analysis, based on a nonparametric method, is also used to study the behavior of the model’s cost components in two different situations. It is shown that by adding the PM planning decisions to the tactical decisions of the dynamic CMS, the optimal configuration and production plans of the system are heavily affected. The results indicate that omitting the PM actions increases the number of sudden failures, which leads to a higher total cost. Finally, it is concluded that the boost in the total availability of the dynamic CMS is one of the main advantages of the proposed integrated method.

Optimisation of multi-plant capacitated lot-sizing problems in an integrated supply chain network using calibrated metaheuristic algorithms

In this paper, a mathematical model for a multi-item multi-period capacitated lot-sizing problem in an integrated supply chain network composed of multiple suppliers, plants and distribution centres is developed. The combinations of several functions such as purchasing, production, storage, backordering and transportation are considered. The objective is to simultaneously determine the optimal raw material order quantity, production and inventory levels, and the transportation amount, so that the demand can be satisfied with the lowest possible cost. Transfer decisions between plants are made when demand at a plant can be fulfilled by other production sites to cope with the under-capacity and stock-out problems of that plant. Since the proposed model is NP-hard, a genetic algorithm is used to solve the model. To validate the results, particle swarm optimisation and imperialist competitive algorithm are applied to solve the model as well. The results show that genetic algorithm offers better solution compared to other algorithms.

Integrated planning of production and maintenance for imperfect system with subcontracting strategies

The purpose of this article is to deal with subcontracting strategies in the context of production, maintenance and quality integration. We study the multi-item capacitated lot-sizing problem for a production system composed of a single machine. The production system is considered imperfect, producing both conforming and non-conforming items. However, the deterioration of the system is a function of the time and production rate, which affects the quality of the manufactured items. Consequently, a quality control strategy is established, the aim is to inspect, adjust and control the manufactured items. To solve our problem, an evolutive optimization approach is proposed, namely the genetic algorithm (GA). Then, in order to adjust the parameters of GA, we use the Taguchi method. This article is one of the few documents dealing with integrated production management, maintenance and quality under subcontracting constraints that takes into account the complex aspect of the multi-item manufacturing industry. Then, a sensitivity analysis is also carried out to illustrate the robustness of the proposed control policy. Finally, we compare our results with the literature to validate our approach and highlight the advantage of subcontracting in minimizing costs.

An effective multi-objective particle swarm optimization for the multi-item capacitated lot-sizing problem with set-up times and backlogging

ABSTRACTIn this article, a multi-item capacitated lot-sizing problem (MICLSP) with consideration of set-up times and backlogging, which is one of the key issues in production planning, is addressed. The inventory is formulated as a separate objective function representing its level instead of integrating it into a total cost function for effective control of this objective. The proposed model attempts to minimize simultaneously the total cost and the total inventory level. The developed model shows high performance compared to its mono-objective version. An efficient multi-objective particle swarm optimization (MOPSO) is developed to generate a representative set of Pareto-optimal solutions. A numerical example is given to illustrate the application of the proposed algorithm. The algorithm’s effectiveness is proven based on two experiments conducted on a set of 28 instances. The experiments compare the proposed algorithm’s results with those of the augmented epsilon-constraint method and the basic version of MOPSO.

Effective matheuristics for the multi-item capacitated lot-sizing problem with remanufacturing

Effective Mixed Integer Programming (MIP) based matheuristics for the multi-item capacitated lot-sizing problem with remanufacturing (CLSP-RM) are proposed in this paper. This NP-hard problem consists of determining an optimal plan for the capacitated production and remanufacture of multiple items in order to satisfy their deterministic dynamic demands over a discrete time horizon. Two constructive approaches are proposed: a column generation rounding heuristic followed by a Linear Programming (LP)/MIP repairing mechanism, and a relax-and-fix method. A fix-and-optimize local search procedure is also applied to improve the quality of the solutions obtained by the constructive techniques. Computational results show that the new approach combining a column generation rounding heuristic with a fix-and-optimize local search procedure is very competitive with the state-of-the-art heuristic. Specifically, the obtained solutions are at least as good as the best known in the literature for 86.85% of the instances, with new best solutions encountered for 69.72% of them. Moreover, all obtained solutions are within 5.54% of optimality, showing the robustness of the newly proposed approach.

Partial objective inequalities for the multi-item capacitated lot-sizing problem

In this paper, we study a mixed-integer programming model of the single-level multi-item capacitated lot-sizing problem (MCLSP), which incorporates shared capacity on the production of items for each period throughout a planning horizon. We derive valid bounds on the partial objective function of the MCLSP formulation by solving the first t periods of the problem over a subset of all items, using dynamic programming and integer programming techniques. We also develop algorithms for strengthening these valid inequalities by back-lifting techniques. These inequalities can be utilized within a cutting-plane algorithm, in which we perturb the partial objective function coefficients to identify violated inequalities to the MCLSP polytope. Our computational results show that the envelope inequalities are very effective for the MCLSP instances with different capacity and cost characteristics, when compared to the (l, S) inequalities.

A hybrid genetic algorithm and variable neighborhood search for multi-family capacitated lot-sizing problem

The paper presents a genetic algorithm (GA) hybridized with variable neighborhood search (VNS) to solve multi-item capacitated lot-sizing multi-family problem with setup times. The problem has a practical application in production planning e.g., in foundry industry, so test cases for computational experiments were based on the data from the real production process in a foundry. The VNS algorithm is used after a certain number of GA generations for all individuals in the population to improve solutions. The presented method applied for large instances of the problem outperforms both a dedicated genetic algorithm and a CLPEX Solver-based rolling horizon methods known from the literature.

Variable neighborhood formulation search approach for the multi-item capacitated lot-sizing problem with time windows and setup times

In this paper we suggest a new variant of Variable neighborhood search designed for solving Mixed integer programming problems. We call it Variable neighborhood formulation search (VNFS), since both neighborhoods and formulations are changed during the search. VNS deals with integer variables, while an available (commercial) solver is responsible for continues variables and the objective function value. We address the multi-item capacitated lotsizing problem with production time windows and setup times, under the non-customer specific case. This problem is known to be NP-hard and can be formulated as a mixed 0-1 program. Neighborhoods are induced from the Hamming distance in 0-1 variables, while the objective function values in the corresponding neighborhoods are evaluated using different mathematical programming formulations of the problem. The computational experiments show that our approach is more effective and efficient when compared with the existing methods from the literature.

MIP Formulations and Metaheuristics for Multi-Item Capacitated Lot-Sizing Problem with Non-Customer Specific Production Time Windows and Setup Times

Our research focuses on the development of two cooperative approaches for resolution of the multi-item capacitated lot-sizing problems with time windows and setup times (MICLSP-TW-ST). In this paper we combine variable neighborhood search and accurate mixed integer programming (VNS-MIP) to solve MICLSP-TW-ST. It concerns so a particularly important and difficult problem in production planning. This problem is NP-hard in the strong sense. Moreover, it is very difficult to solve with an exact method; it is for that reason we have made use of the approximate methods. We improved the variable neighborhood search (VNS) algorithm, which is efficient for solving hard combinatorial optimization problems. This problem can be viewed as an optimization problem with mixed variables (binary variables and real variables). The new VNS algorithm was tested against 540 benchmark problems. The performance of most of our approaches was satisfactory and performed better than the algorithms already proposed in the literature.

Fix-and-Optimize and Variable Neighborhood Search Approaches for Stochastic Multi-Item Capacitated Lot-Sizing Problems

We discuss stochastic multi-item capacitated lot-sizing problems with and without setup carryovers (also known as link lot size), S-MICLSP and S-MICLSP-L. The two models are motivated from a real-world steel enterprise. To overcome the nonlinearity of the models, a piecewise linear approximation method is proposed. We develop a new fix-and-optimize (FO) approach to solve the approximated models. Compared with the existing FO approach(es), our FO is based on the concept of “k-degree-connection” for decomposing the problems. Furthermore, we also propose an integrative approach combining our FO and variable neighborhood search (FO-VNS), which can improve the solution quality of our FO approach by diversifying the search space. Numerical experiments are performed on the instances following the nature of realistic steel products. Our approximation method is shown to be efficient. The results also show that the proposed FO and FO-VNS approaches significantly outperform the recent FO approaches, and the FO-VNS approaches can be more outstanding on the solution quality with moderate computational effort.

Multi-item capacitated lot-sizing problem in a flow-shop system with energy consideration

This paper discusses a multi-item capacitated lot-sizing and scheduling problem in a flow-shop system with energy consideration. A mixed integer linear programming is formulated. The planning horizon is defined by a set of periods where each one is characterized by a length, an electricity price, a maximal allowed power and an external demand. Since the capacitated lot-sizing problems are NP-hard, a fix-and-relax heuristic is developed. To evaluate the effectiveness of the proposed heuristic, computational experiments are presented and numerical results are discussed and analyzed. The obtained results are promising.

A bi-objective multi-item capacitated lot-sizing model: Two Pareto-based meta-heuristic algorithms

Lot-sizing problems (LSP) form a class of production planning problems in which available quantities are always considered as decision variables in the production plan. The goal of this paper is to present a multi-item capacitated lot-sizing problem (MICLSP) with setup times, safety stock deficit costs, demand shortage costs – both backorder and lost sale states – and different manners of production. Although a considerable amount of research has concentrated on model development and solution procedures in the terms of single-objective problems in the past decade, to make the model more realistic this paper develops a bi-objective mathematical programming model with two conflicting objectives including: (1) minimizing the total cost considered by the production plans including production costs with different manners of production, inventory costs, safety stock deficit costs, shortage costs and setup costs; (2) minimizing the required storage space. Considering that the proposed model is NP-hard, we propose two novel Pareto-based multi-objective meta-heuristic algorithms called multi-objective vibration damping optimization (MOVDO) and the non-dominated ranking genetic algorithm (NRGA) for the literature on LSP. In order to validate the performance of the proposed MOVDO and NRGA, a non-dominated sorting genetic algorithm (NSGA-II), one of the most common multi-objective meta-heuristic algorithms, is applied. The optimal solutions are also reported to justify the results. Finally, we calibrate both algorithms by robust response surface methodology (RSM); then, the results are analysed on some test problems, both graphically and statistically.

Novel Pareto-based meta-heuristics for solving multi-objective multi-item capacitated lot-sizing problems

Capacitated production lot-sizing problems (LSPs) are challenging problems to solve due to their combinatorial nature. We consider a multi-item capacitated LSP with setup times, safety stocks, and demand shortages plus lost sales and backorder considerations for various production methods (i.e., job shop, batch flow, or continuous flow among others). We use multi-objective mathematical programming to solve this problem with three conflicting objectives including: (i) minimizing the total production costs; (ii) leveling the production volume in different production periods; and (iii) producing a solution which is as close as possible to the just-in-time level. We also consider lost sales, backorders, safety stocks, storage space limitation, and capacity constraints. We propose two novel Pareto-based multi-objective meta-heuristic algorithms: multi-objective vibration damping optimization (MOVDO) and a multi-objective harmony search algorithm (MOHSA). We compare MOVDO and MOHSA with two well-known evolutionary algorithms called the non-dominated sorting genetic algorithm (NSGA-II) and multi-objective simulated annealing (MOSA) to demonstrate the efficiency and effectiveness of the proposed methods.

A hybrid heuristic approach for production planning in supply chain networks

Planning distributed manufacturing facilities is one of the most challenging tasks in the supply chain management. This paper proposes a production planning algorithm for the multi-level, multi-item capacitated lot-sizing problem (MLCLSP) in a supply chain network that takes back order into account. MLCLSP is a mixed integer linear programming (MIP) problem and is NP-hard. This paper presents an efficient, hybrid, heuristic algorithm named greedy rolling horizon search (GRHS) that combines a rolling horizon local search heuristic with an exact linear program (LP) solver. Computational experiments show that GRHS performs well in terms of total costs and computational time and is superior to existing meta-heuristics, such as tabu search, simulated annealing, and genetic algorithms.

A biased random key genetic algorithm approach for inventory-based multi-item lot-sizing problem

In this article, we have explored multi-item capacitated lot-sizing problem by addressing the backlogging and associated high penalty costs incurred. At the same time, penalty cost for exceeding the resource capacity has also been taken into account. Penalty cost related to both backlogging and overutilizing capacity has been included in main objective function. The main objective is to achieve such a solution that minimizes the total cost. The ingredients of total cost are the setup cost, production cost, inventory holding cost and aforementioned both the penalty costs. To solve this computationally complex problem, a less explored algorithm “biased random key genetic algorithm” has been applied. To the best of our knowledge, this research presents the first application of biased random key genetic algorithm to a lot-sizing problem. To test the effectiveness of proposed algorithm, extensive computational tests are conducted. The encouraging results show that the proposed algorithm is an efficient tool to tackle such complex problems. A comparative study with other existing heuristics shows the supremacy of proposed algorithm in terms of quality of the solution, number of generation and computational time.