Capacitated Lot-sizing Problem Research Articles

Planning approaches for stochastic capacitated lot-sizing with service level constraints

We investigate a stochastic capacitated lot-sizing problem whose optimal solution requires the integration of dynamic safety stock planning into lot-sizing. Then, we introduce an integrated mixed-integer linear program with service-level constraints. The integrated model endogenously sets dynamic safety stocks over replenishment cycles of different lengths determined by the model. Since there is limited capacity, soft service-level constraints are introduced to guarantee a feasible solution. In the experimental study, we compare the performance of the integrated model to the stochastic dynamic program and the widely-used sequential approach. If available capacity increases, the integrated model closes the gap to the lower bound approximated by using a stochastic dynamic program. If capacity is limited, the integrated model outperforms the sequential approach because it yields identical service levels with lower inventories. However, in the case of sufficient flexibility (capacity), we identify a major shortcoming of the integrated models: They can generate excessive safety stock if the re-planning opportunities under rolling horizon planning are ignored. To overcome this problem, we extend the integrated model to account for those re-planning opportunities.

A note on: “A hybrid Benders approach for coordinated capacitated lot-sizing of multiple product families with set-up times,” by T. Bayley, H. Süral, and J. Bookbinder

In a recent paper by Bayley et al. [2018. “A Hybrid Benders Approach for Coordinated Capacitated Lot-Sizing of Multiple Product Families with Set-Up Times.” International Journal of Production Research 56 (3): 1326–1344] on a capacitated lot-sizing problem, a Benders-based approach is described. Strengthening the master problem is based on inequalities that should be valid. We provide insight indicating that this is not the case for those inequalities proposed by Bayley et al. (2018).

A linearisation approach to the stochastic dynamic capacitated lotsizing problem with sequence-dependent changeovers

We present a mixed-integer linear programming formulation that simultaneously optimises lot sizes and production sequences on a capacity constrained machine with sequence-dependent changeovers subject to stochastic dynamic demand while at the same time satisfying a fill rate constraint. To tackle the non-linearity of the exact formulation, we introduce a piecewise linearisation technique both for the expected inventory on hand and for the backorder functions that uses the target service level and the parameters of the demand distribution to assign breakpoints to the most promising intervals of the linearisation domain. We show that our strategy leads to lower cost and to more conservative production plans, in comparison to techniques recommended by earlier research. In addition, we discuss why any breakpoint selection strategy that does not exclude the concave region for , is prone to be outperformed by the approach we present. Finally, we propose a Relax-and-Fix with Fix-and-Optimize heuristic, and show based on the broad set of instances from Haase, Knut, and Alf Kimms [2000. “Lot sizing and scheduling with sequence-dependent setup costs and times and efficient rescheduling opportunities.” International Journal of Production Economics 66 (2): 159–169], that it is more effective than a state-of-the-art solver in terms of run time and solution quality.

A population-based metaheuristic for the capacitated lot-sizing problem with unrelated parallel machines

This paper deals with the capacitated lot-sizing problem with multiple items, setup time and unrelated parallel machines. The aim of the article is to propose an efficient population-based metaheuristic to tackle this problem. In the proposed approach, a dynamic constructive heuristic is proposed for generating a set of initial solutions. A path-relinking strategy between solutions in the current population is integrated in the diversification procedure, allowing us to explore the solution space further. The population is then intensified using a local search procedure which uses an operator based on machine capacity relaxation coupled with a repair procedure. The proposed approach is evaluated from the instances introduced in Toledo, F. M. B., and V. A. Armentano (2006.“A Lagrangian-Based Heuristic for the Capacitated Lot-Sizing Problem in Parallel Machines.” European Journal of Operational Research 175 (2): 1070–1083). The results of the experiment show that the proposed approach yields competitive results in terms of solution quality for most of the instances.

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.

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

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

Joint optimization of dynamic pricing and lot-sizing decisions with nonlinear demands: Theoretical and computational analysis

This paper addresses a capacitated lot-sizing problem with pricing decisions. The considered problem consists of planning the production of different products during several time periods with setup costs. Unlike the classical version of the capacitated lot sizing problem, the demand for the products is not fixed but price-sensitive in this problem. The demand function is assumed to be nonlinear. The decisions consist of establishing the best strategy for production and inventory, and the best price policy. We propose an improved mathematical formulation by extending some previous works using new lower and upper bounds to reduce the solution space. We also introduce new heuristic methods to provide near-optimal solutions. These methods are tested on several instances from previous studies. The obtained results illustrate the efficiency of these methods.

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.

MINIMALISASI BIAYA SIMPAN DAN BIAYA SETUP PADA MULTIPLE PRODUK: SIMULASI DENGAN CAPACITATED LOT SIZING PROBLEM

Lot sizing problem in production planning aims to optimize production costs (processing, setup and holding cost) by fulfilling demand and resources capacity costraint. The Capacitated Lot sizing Problem (CLSP) model aims to balance the setup costs and inventory costs to obtain optimal total costs. The object of this study was a plastic component manufacturing company. This study use CLSP model, considering process costs, holding costs and setup costs, by calculating product cycle and setup time. The constraint of this model is the production time capacity and the storage capacity of the finished product. CLSP can reduce the total production cost by 4.05% and can reduce setup time by 46.75%. 
 Keyword: Lot size, CLSP, Total production cost.

A branch-and-price approach for a biomass feedstock logistics supply chain design problem

This article addresses a biomass feedstock logistics supply chain design problem, which comprises a multi-period facility location problem, a special case of a single-item parallel-facilities capacitated lot-sizing problem, and a network flow problem. We formulate this problem as a mixed-integer program and propose a branch-and-price-based method for its solution that relies on effective implementation strategies. Our computational investigation reveals efficacy of the proposed method in obtaining near-optimal solutions for large-sized problem instances over direct solution of our model formulation of the problem by CPLEX. Several useful managerial insights resulting from our analysis are also presented.

A practical assessment of risk-averse approaches in production lot-sizing problems

ABSTRACTThis paper presents an empirical assessment of four state-of-the-art risk-averse approaches to deal with the capacitated lot-sizing problem under stochastic demand. We analyse two mean-risk models based on the semideviation and on the conditional value-at-risk risk measures, and alternate first and second-order stochastic dominance approaches. The extensive computational experiments based on different instances characteristics and on a case-study suggest that CVaR exhibits a good trade-off between risk and performance, followed by the semideviation and first-order stochastic dominance approach. For all approaches, enforcing risk-aversion helps to reduce the cost-standard deviation substantially, which is usually accomplished via increasing production rates. Overall, we can say that very risk-averse decision-makers would be willing to pay an increased price to have a much less risky solution given by CVaR. In less risk-averse settings, though, semideviation and first-order stochastic dominance can be appealing alternatives to provide significantly more stable production planning costs with a marginal increase of the expected costs.

Modeling the Capacitated Multi-Level Lot-Sizing Problem under Time-Varying Environments and a Fix-and-Optimize Solution Approach.

In this study, we investigated the time-varying capacitated lot-sizing problem under a fast-changing production environment, where production factors such as the setup costs, inventory-holding costs, production capacities, or even material prices may be subject to continuous changes during the entire planning horizon. Traditional lot-sizing theorems and algorithms, which often assume a constant production environment, are no longer fit for this situation. We analyzed the time-varying environment of today’s agile enterprises and modeled the time-varying setup costs and the time-varying production capacities. Based on these, we presented two mixed-integer linear programming models for the time-varying capacitated single-level lot-sizing problem and the time-varying capacitated multi-level lot-sizing problem, respectively, with considerations on the impact of time-varying environments and dynamic capacity constraints. New properties of these models were analyzed on the solution’s feasibility and optimality. The solution quality was evaluated in terms of the entropy which indicated that the optimized production system had a lower value than that of the unoptimized one. A number of computational experiments were conducted on well-known benchmark problem instances using the AMPL/CPLEX to verify the proposed models and to test the computational effectiveness and efficiency, which showed that the new models are applicable to the time-varying environment. Two of the benchmark problems were updated with new best-known solutions in the experiments.

Model formulations for the capacitated lot-sizing problem with service-level constraints

We present deterministic model formulations for the capacitated lot-sizing problem, including service-level constraints that address both the periodic and cyclic $$\alpha $$ service levels, as well as the $$\beta $$, $$\gamma $$, and $$\eta $$ service levels. We assume that service levels for individual products are given and controlled over a given reporting period (i.e., one year). These deterministic model formulations may be used when all data are deterministic and decision makers intend to exploit given service levels to minimize setup and holding costs. A further application is to provide lower bounds for capacitated lot-sizing problems with stochastic demand and given service-level constraints. In contrast to well-known stochastic or robust optimization approaches, there are new proposals in the literature that do not require scenario modeling and thus have much greater potential for use in solving (large-scale) real-world production planning problems. However, an evaluation of the quality of solutions resulting from these new approaches is difficult. For this purpose, lower bounds showing the best possible (ideal) solution should be of great help. In a computational study, we provide insight into the computational efforts associated with deterministic model formulations with service-level constraints. Finally, lower bounds generated by the deterministic model with $$\beta $$ service-level constraints are compared with the results of a rolling schedule strategy addressing a stochastic lot-sizing problem with given $$\beta $$ service-level constraints. The resultant difference in the objective function values (costs) defines the uncertainty gap. We demonstrate its increase with forecast inaccuracy as well as machine utilization.

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.

The item dependent stockingcost constraint

In a previous work we introduced a global StockingCost constraint to compute the total number of periods between the production periods and the due dates in a multi-order capacitated lot-sizing problem. Here we consider a more general case in which each order can have a different per period stocking cost and the goal is to minimise the total stocking cost. In addition the production capacity, limiting the number of orders produced in a given period, is allowed to vary over time. We propose an efficient filtering algorithm in O(n log n) where n is the number of orders to produce. On a variant of the capacitated lot-sizing problem, we demonstrate experimentally that our new filtering algorithm scales well and is competitive wrt the StockingCost constraint when the stocking cost is the same for all orders.

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 robust capacitated lot sizing problem with setup times and overtime decisions with backordering allowed under demand uncertainty

Abstract This research work considers the multi-product multi-period capacitated lot-sizing problem with setup times and overtime with backordering allowed (CLSPSTOB) under demand uncertainty by means of a budget-uncertainty set robust optimization. We first present a mixed integer linear programming (MILP) model that describes the deterministic CLSPSTOB with production costs, setup costs, inventory holding costs, backordering costs, and overtime costs. Then, a robust optimization approach is applied to deal with the demand uncertainty with unknown probability distribution, and a robust counterpart model based on duality theory is obtained. Some numerical experiments are presented for the robust capacitated lot-sizing problem with setup times and overtime with backordering allowed (RCLSPSTOB).

An Optimal Policy for the Capacitated Lot Sizing Problem with financing of the Working Capital Requirement

Abstract We study a capacitated version of the classical lot-sizing problem that incorporates financial aspect to the operational decisions. Namely, we consider a case in which the manufacturer does not have enough financing capacity to cover the expenses of the operations and must use bank loans to finance its operations before receiving payment from its customers. We propose a mathematical model for this problem and exhibit a non speculative assumption that extends the Wagner-Whitin cost structure to this class of problem. We show that when this assumption is satisfied, the problem reduces to finding a shortest path in an acyclic graph, which implies that it can be solved using a dynamic programming approach. A short numerical study also show that adding financing considerations to a capacitated lot-sizing problem may modify the production sequence of the optimal policy.

Joint pricing and lot-sizing problem with variable capacity

Abstract This paper presents an extension to the classical capacitated lot-sizing problem. Demand for the products is price-dependent and modelized as an isoelastic function. The capacity is not considered as a parameter anymore, it becomes a decision variable for the company as the production, inventory and pricing decisions. This variable capacity leads to two variants of the problem: one with a time-dependent capacity and the other one a uniform capacity over the periods. We propose heuristic methods, which are based on Lagrangian multipliers, to solve variations of the problem. The methods provide high-quality results.