Capacitated Dynamic Lot-sizing Problem Research Articles

An effective heuristic with evolutionary algorithm for the coordinated capacitated dynamic lot-size and delivery problem

This study considers an extensive coordinated capacitated dynamic lot-size problem that contains a delivery scheduling problem. The coordinated capacitated dynamic lot-size and delivery problem (CCLSDP) assumes that a family of products is subject to capacity constraints, and the demand without backorders is dynamic or time-varying but deterministic. The objective is to determine a replenishment and delivery schedule that minimizes the total system costs, including the family and minor setup costs, delivery costs, and inventory holding costs. We develop a six-phase heuristic (SPH) to solve the CCLSDP. The SPH contains six phases that can efficiently reschedule replenishment and delivery planning. Each phase of the SPH evaluates the cost and feasibility of the shifting operations, and determines whether to reschedule the system planning through a series of sequential shifting operations. To further enhance the overall performance, we embed SPH as a lifting procedure into evolutionary algorithms contributing to explore the search space and escapes the local optimum. Experimental results of randomly generated instances that were over a wide range of parameters indicate that the proposed heuristics could efficiently and effectively solve the complicated CCLSDP.

A simulated annealing algorithm with neighbourhood list for capacitated dynamic lot-sizing problem with returns and hybrid products

ABSTRACTThis research addresses the capacitated dynamic lot-sizing problem with returns and hybrid products (). The problem is to identify how many of each product type to produce during each period for a hybrid system with manufacturing capacity constraints. The objective of is to maximise total profit of the production system that consists of new, remanufactured and hybrid products. is a multi-period CLSP, which is modelled as a mixed-integer nonlinear programming problem. The traditional CLSP is NP-hard, and the nonlinearity of makes the problem even harder to solve. Therefore, a Simulated Annealing (SA) algorithm with a neighbourhood list (SA_NL) is proposed. By using a list of several neighbourhoods, the SA algorithm is improved. SA_NL is compared to SA, three variants of Genetic Algorithm (GA) and a Variable Neighbourhood Search (VNS) algorithm. The variants of GA are GA with one-point crossover (), GA with two-point crossover () and GA with one-point period-based crossover (). Over all instances, the results show that the proposed SA_NL outperforms SA, VNS, , and by 0.54%, 0.34%, 1.92%, 1.78% and 2.92%, respectively.

Application of fuzzy set to lot-sizing production planning with remanufacturing and heterogeneous demands

Remanufacturing, which processes end-of-life products with disassembly, testing, reprocessing, and reassembly operations in such a way that their quality and performance is restored, has attracted substantial interest in recent years. In this article, we address a capacitated dynamic lot-sizing problem with remanufacturing, in which there are heterogeneous demand streams for newly manufactured products assembled wholly by brand-new components and remanufactured ones reassembled by reprocessed components. The demand for remanufactured products could also be satisfied by new ones, as remanufactured products do not meet self-demand, but not vice versa. For this planning problem, a fuzzy mixed integer linear programming model, which considers the uncertainties of market demands, the quantity of end-of-life products available, the remanufacturable rate, selling prices, unit cost values, and capacity constraints, is developed by triangular fuzzy numbers. After that, the fuzzy mixed integer linear programming model is transformed into a crisp equivalent by clarifying fuzzy constraints and objective. The solution approach is designed by genetic algorithm, in which self-adaptive formula is adopted to resolve difficulty in obtaining optimum value of crossover probability and mutation probability. Finally, a numerical example is suggested to demonstrate the applicability and effectiveness of the proposed model and solution approach.

An effective approach to multi-item capacitated dynamic lot-sizing problems

In this study we solve the multi-item capacitated dynamic lot-sizing problem, where each item faces a series of dynamic demands, and in each period multiple items share limited production resources. The objective is to find the optimal production plan so as to minimise the total cost, including production cost, inventory holding cost, and fixed setup cost. We consider both single-level and multi-level cases. In the multi-level case, some items are consumed in order to produce some other items and therefore items face internally generated demand in addition to external demands. We propose a simple three-stage approach that is applicable to both classes of problems. In the first stage we perform preprocessing, which is designed to deal with the difficulty due to the joint setup cost (a fixed cost incurred whenever production occurs in a period). In the second stage we adopt a period-by-period heuristic to construct a feasible solution, and in the final stage we further improve the solution by solving a series of subproblems. Extensive experiments show that the approach exhibits very good performance. We then analyse how the superior performance is achieved. In addition to its performance, one appealing feature of our method is its simplicity and general applicability.

A Heuristic Solution of Multi-Item Single Level Capacitated Dynamic Lot-Sizing Problem with Setup Time

The multi-item single level capacitated dynamic lot-sizing problem consists of scheduling N items over a horizon of T periods. The objective is to minimize the sum of setup and inventory holding costs over the horizon subject to a constraint on total capacity in each period. No backlogging is allowed. Only one machine is available with a fixed capacity in each period. In case of a single item production, an optimal solution algorithm exists. But for multi-item problems, optimal solution algorithms are not available. It has been proved that even the two-item problem with constant capacity is NP-hard, that is, it is in a class of problems that are extremely difficult to solve in a reasonable amount of time. This has called for searching good heuristic solutions. For a multi-item problem, it would be more realistic to consider the setup time, since switching the machine from one item to another would require a setup time. This setup time would be independent of item sequences and this could be a very important parameter from practical point of view. The current research work has been directed toward the development of a model for multiitem problem considering this parameter. Based on the model a program has been executed and feasible solutions with some real life data have been obtained.

Heuristic genetic algorithm for capacitated production planning problems with batch processing and remanufacturing

In this paper, we analyze a version of the capacitated dynamic lot-sizing problem with substitutions and return products. Both batch manufacturing and batch remanufacturing are considered within the framework of deterministic time-varying demands in a finite time horizon, where the option of emergency procurement/outsourcing subject to a subcontract is also allowed. Setup costs are taken into account when batch manufacturing or batch remanufacturing takes place. We first apply a genetic algorithm to determine all periods requiring setups for batch manufacturing and batch remanufacturing, then develop a dynamic programming approach to provide the optimal solution to determine how many new products are manufactured or return products are remanufactured in each of these periods. The objective is to minimize the total cost, including batch manufacturing, batch remanufacturing, emergency procurement, holding and setup costs. Numerical examples illustrate the effectiveness of the approach.

Parallelism of continuous- and discrete-time production planning problems

We consider a capacitated production-inventory problem in both discrete- and continuous-time, stationary settings. In the discrete-time setting we analyze the infinite horizon capacitated dynamic lot-sizing problem, find the optimal solution and characterize its properties. For the continuous-time setting we formulate a new problem, which we claim to be an appropriate counterpart of the above discrete-time problem. No other counterpart model was found in the literature, including the vast literature on optimal control, which presumably deals with similar problems but in a continuous-time framework. The new problem formulation is the basis for a new class of models, which forms an alternative way of analyzing certain dynamic lot-sizing problems. This new alternative could sometimes be simpler than the analysis in the discrete case.

The capacitated dynamic lot size problem with variable technology

Traditionally, the technological coefficients in production models were assumed to be fixed. In recent years however, researchers have used the learning curve model to represent nonlinear technological coefficients, and the dynamic lot-sizing problem with learning in setups has received attention. This article extends the research to consider capacity restrictions in the single-level, multi-item case. The research has two goals, first, to analyze the effects of setup learning on a production schedule, and second, to investigate efficient ways of solving the resulting nonlinear integer model. Previously derived algorithms do not address the issue of capacity; thus a heuristic is developed and its solution is compared with the optimal solution, where possible, or to a lower bound solution.

Analysis Of Lagrangian Decomposition For The Multi-Item Capacitated Lot-Sizing Problem

AbstractThe multi-item capacitated dynamic lot-sizing problem consists of determining the quantity and the timing of production for several products in a finite number of periods so as to satisfy a known demand in each period and minimize the sum of the set-up, production and inventory costs without incurring backlogs. A production capacity is imposed in each period.Lagrangian decomposition is applied to two formulations of the problem. Using standard subsets of constraints, it is shown that Lagrangian decomposition yields only five genuinely different relaxations. Their values and computational complexities are compared. As the number of items grows, the five relaxations merge into two families that contrast sharply by their values, their computing times and the quality of the feasible solutions that they generate.

The capacitated dynamic lot-sizing problem with startup and reservation costs: A forward algorithm solution

Abstract Often in production environments, there are several products available to be processed on a machine. The decisions that must be made are: 1) which product should be processed next, and 2) how much time should elapse before a changeover occurs and a different product is chosen for processing. This problem is known as the multiple product cycling problem. The single product variation of this problem is addressed in this paper. It is assumed that the product to be processed next has already been chosen (decision #1). The second decision concerning production run length must now be determined. In the single product problem, there is a fixed startup cost associated with turning the machine on in a period when the machine was off in the previous period and a fixed reservation cost incurred in each period in which the machine is on. Ending inventory in a period is assessed a holding cost. The relationship of the single product problem to the multiple period problem is as follows. The startup cost is identical to the changeover cost in the multiple product problem. By processing a particular product, other products must wait. This effect is captured by the reservation cost which is the opportunity cost one would incur by having the machine dedicated to processing a particular product. A forward dynamic programming algorithm is presented which uses heuristic rules to detect empirical decision horizons. The entire problem is partitioned into smaller Subproblems which are then easily solved. The subproblems are created by the occurrence of startup regeneration points. These are periods where there is no ending inventory and the following period contains a machine startup. Computational experience is reported which shows that reductions in computational effort (measured in CPU time and the number of nodes in the decision tree evaluated) can be as high as 95 to 99% when using the decision horizon techniques and bounding procedures described in this paper. These results are based on the solution of 28 test problems that were each 20 periods in length. Because a forward algorithm was used, not only were the solutions to the 20-period problems obtained, but the solutions to the one-period problems through 19-period problems were obtained as well. Even though the procedure described in this paper cannot guarantee that it will always find the optimal solution, it did find the optimal solution in every problem tested. By being able to solve the single product problem presented in this paper quickly, the ability to rapidly generate optimal or near optimal solutions to the multiple product problem (which uses the procedure described in this paper as a subroutine) appears closer to realization.

Primal-dual approach to the single level capacitated lot-sizing problem

The Lagrangean relaxation of the single level capacitated dynamic lot-sizing problem can be solved using the primal-dual method. The algorithm has monotone and finite convergence properties. It works as a steepest ascent method. A variant of this approach is also studied. A heuristic routine used to obtain a feasible solution in each iteration is presented. Computational experiences show that this method usually yields better solutions than the subgradient method although it requires greater CPU times.

Set partitioning and column generation heuristics for capacitated dynamic lotsizing

This paper discusses set partitioning and column generation heuristics for the multi item single level capacitated dynamic lotsizing problem. Most previous research efforts reported in literature use the single item uncapacitated problem solutions as candidate plans. In this paper a different approach is proposed in which candidate plans are also generated by some well-known multi item capacitated dynamic lotsizing heuristics. The LP relaxation of a set partitioning model is solved and rounded to obtain an integer solution. A number of heuristics are subsequently applied to improve upon this initial solution. Heuristics are also used to convert the possibly fractional solution from the column generation step to a feasible integer one. Our methods are compared with other common sense heuristics. Computational experience shows that our methods perform best for problems with fewer periods than items, but are quite good on average.

An O(T2) Algorithm for the NI/G/NI/ND Capacitated Lot Size Problem

In this paper, we study a class of the capacitated dynamic lot size problem, where, over time, the setup costs are nonincreasing, the unit holding costs have arbitrary pattern, the unit production costs are nonincreasing and the capacities are nondecreasing. We investigate the properties of the optimal solution for the problem and develop the concept of candidate subplan. It is proven that only the candidate subplans need to be examined in searching for an optimal solution. A dynamic programming algorithm, incorporating the concept of candidate subplan, is then devised which has run time complexity of O(T2).

An OT2 Algorithm for the NI/G/NI/ND Capacitated Lot Size Problem

In this paper, we study a class of the capacitated dynamic lot size problem, where, over time, the setup costs are nonincreasing, the unit holding costs have arbitrary pattern, the unit production ...

The Capacitated Multi-Item Dynamic Lot-Sizing Problem

Abstract A previous article by Dogramaci, Panayiotopoulos, and Adam [1] proposed a unidirectional forward pass algorithm for the multiple-item limited capacity lot-sizing problem. The algorithm guarantees a feasible solution to any problem for which a feasible solution exists. A not unusual problem is presented here that cannot be solved to a feasible schedule by the algorithm. The difficulty is due to lumpy demand and to the algorithm's choice of shifting entire lots before split lots. A modification to the original algorithm prevents the infeasibility from occurring.

Multi Item Single Level Capacitated Dynamic Lotsizing Heuristics: A Computational Comparison (Part I: Static Case)

Abstract The multi item single level capacitated dynamic lotsizing problem consists of scheduling N items over a horizon of T periods. Demands are given and should be satisfied without backlogging. The objective is to minimize the sum of setup costs and inventory holding costs over the horizon subject to a constraint on total capacity in each period. Three simple heuristics from literature (Lambrecht and Vanderveken, Dixon and Silver, Dogramaci, Panayiotopoulos and Adam) are compared on a large set of test problems and their difference in performance is analyzed for various problem parameters.

A heuristic solution of multi-item single level capacitated dynamic lot-sizing problem

The multi-item single level capacitated dynamic lot-sizing problem consists of scheduling N items over a horizon of T periods. The objective is to minimize the sum of setup and inventory holding costs over the horizon subject to a constraint on total capacity in each period. No backlogging is allowed. Only one machine is available with a fixed capacity in each period. In case of a single item production, an optimal solution algorithm exists. But for multi-item problems, optimal solution algorithms are not available. It has been proved that even the two-item problem with constant capacity is NP (nondeterministic polynomial)-hard. That is, it is in a class of problems that are extremely difficult to solve in a reasonable amount of time. This has called for searching good heuristic solutions. For a multi-item problem, it would be more realistic to consider an upper limit on the lot-size per setup for each item and this could be a very important parameter from practical point of view. The current research work has been directed toward the development of a model for multi-item problem considering this parameter. Based on the model a program has been executed and feasible solutions have been obtained. Keywords: Heuristics, inventory, lot-sizing, multi-item, scheduling.DOI: 10.3329/jme.v38i0.893 Journal of Mechanical Engineering Vol.38 Dec. 2007 pp.1-7