Dynamic Lot-sizing Problem Research Articles

Fixed-quantity dynamic lot sizing using simulated annealing

In this paper, simulated annealing (SA) is applied to the deterministic dynamic lot-sizing problem with batch ordering and backorders. Batch ordering requires orders that are integer multiples of a fixed quantity that is larger than 1. The performance of the developed SA heuristic is compared to that of a genetic algorithm (GA) and a modified silver-meal (MSM) heuristic developed in the literature, based on the frequency of obtaining the optimum solution and the percentage average deviation from the optimum solution. In addition, the effects of three factors on the performance of the SA, GA, and the MSM are investigated in a 23 factorial experiment. The investigated factors are the demand pattern, the batch size, and the length of the planning horizon. Results indicate that the SA heuristic has the best performance, followed by GA, in terms of the frequency of obtaining the optimum solution and the average deviation from the optimum solution. SA is also the most robust of the investigated heuristics as its performance is only affected by the length of the planning horizon.

TECHNICAL NOTE—Solving Linear Cost Dynamic Lot-Sizing Problems in O(n log n) Time

In this paper, we study capacitated dynamic lot-sizing problems with or without backorders, under the assumption that production costs are linear, that is, there are no setup costs. These two dynamic lot-sizing problems (with or without backorders) are minimum-cost flow problems on an underlying network that possess a special structure. We show how the well-known successive shortest-path algorithm for the minimum-cost flow problem can be used to solve these problems in O(n2) time, where n is the number of periods in the dynamic lot-sizing problems, and how, with the use of dynamic trees, we can solve these problems in O(n log n) time. Our algorithm also extends to the dynamic lot-sizing problem with integer variables and convex production costs with running time O(n log n log U), where U is the largest demand value.

The assortment problem with dynamic deterministic demands and stationary stocking-substitution policies

We consider multi-period one-dimension assortment problems in which demands for all sizes meet the conditions for a dynamic demand economic lot size problem, the same set of sizes are stocked and the same substitution rules are followed in all periods (stationary stocking-substitution policies), and substitution costs have a 'scrap' component and, possibly, either unit-dependent or unit-independent fixed costs. We develop the form of an optimal solution for certain assumptions about the nature of the cost functions and for the relationships among demand patterns for the different sizes. Solution algorithms are developed for both segmented and quasi-segmented stocking-substitution policies and computation-reduction results are developed.

A BACKWARD APPROACH LPC-BASED CONSTRUCTIVE HEURISTIC

ABSTRACT In this paper, we propose a heuristic for the well-known single-level un-capacitated dynamic lot-sizing problem. The proposed heuristic is based on the average period cost criterion introduced by the heuristic from Silver and Meal [18], also known as Least Period Cost (LPC). Two desirable features of the proposed heuristic are simplicity and the heuristic's ability to complement LPC effectively. We examine the effectiveness of the proposed heuristic by using a set of 20 benchmarking test problems attributed to Baker [2] and Berry [3]. Furthermore, we perform a large-scale simulation involving 80,000 randomly generated problems to evaluate the proposed heuristic against several popular heuristics in the literature. Our computational results show that the proposed heuristic is able to achieve highly competitive results.

Single-Item Dynamic Lot-Sizing Models With Bounded Inventory and Outsourcing

This paper addresses a real-life lot-sizing problem which can be considered a single-item dynamic lot-sizing problem with bounded inventory. The particularity is that the demand of a period can be entirely or partially outsourced with an outsourcing cost. The goal is to minimize the total cost of production, setup, inventory holding, and outsourcing. The cost functions are linear but time-varying. We assume that the unit production cost is constant or nonincreasing over time. The problem is shown to be solvable in a strongly polynomial time with a dynamic-programming approach. The proposed algorithm can solve problems of sizes of up to 400 periods in less than 2 ms on a 1.4-GHz Pentium IV processor.

Forecast Horizons for a Class of Dynamic Lot-Size Problems Under Discrete Future Demand

We present structural and computational investigations of a new class of weak forecast horizons—minimal forecast horizons under the assumption that future demands are integer multiples of a given positive real number—for a specific class of dynamic lot-size (DLS) problems. Apart from being appropriate in most practical instances, the discreteness assumption offers a significant reduction in the length of a minimal forecast horizon over the one using the classical notion of continuous future demands. We provide several conditions under which a discrete-demand forecast horizon is also a continuous-demand forecast horizon. We also show that the increase in the cost resulting from using a discrete minimal forecast horizon instead of the classical minimal forecast horizon is modest. The discreteness assumption allows us to characterize forecast horizons as feasibility/optimality questions in 0-1 mixed-integer programs. On an extensive test bed, we demonstrate the computational tractability of the integer programming approach. Owing to its prevalence in practice, our computational experiments emphasize the special case of integer future demands.

On the stochastic uncapacitated dynamic single-item lotsizing problem with service level constraints

In this paper, we consider the uncapacitated single-item dynamic lotsizing problem with stochastic period demands and backordering. We present a model formulation that minimizes the setup and holding costs with respect to a constraint on the probability that the inventory at the end of any period does not become negative ( α service level) and, alternatively, to a fill rate constraint ( β service level). In contrast to earlier model formulations which consider the cycle α service level ( α c ) and which approximate the on hand inventory by the net inventory, we include the exact on hand inventory into the model formulation. Therefore, the models are also applicable in situations with very low service levels.

Dynamic lot-sizing problem with demand time windows and container-based transportation cost

In this paper, we study the dynamic lot-sizing problem with demand time windows and container-based transportation cost. For each particular demand, there are corresponding earliest and latest times, and the duration between such earliest and latest times is the demand time window. If a demand is satisfied by a delivery within demand time window, then there is no holding or backlogging cost incurred. Our purpose is to satisfy demand at a minimum total cost, including setup cost, procurement cost, container cost, and inventory holding cost.

Solving Large-scale Profit Maximization Capacitated Lot-size Problems by Heuristic Methods

This paper introduces a simple heuristic for a quadratic programming sub-problem within a Lagrangean relaxation heuristic for a dynamic pricing and lot-size problem. This simple heuristic is demonstrated to work well on both ‘standard problem instances’ from the CLSP-literature, as well as on very large-scale cases. Additionally, we introduce price constraints within the framework of dynamic pricing, discuss their relevance in a real world market modelling, and demonstrate their applicability within this algorithmic framework.

Two modifications of the least cost per period heuristic for dynamic lot-sizing

This paper proposes two constructive heuristics for the well-known single-level uncapacitated dynamic lot-sizing problem. The proposed heuristics, called net least period cost (nLPC) and nLPC(i), are developed by modifying the average period cost concept from Silver and Meal's heuristic, commonly known as least period cost (LPC). An improved tie-breaking stopping rule and a locally optimal decision rule are proposed in the second heuristic to enhance performance. We test the effectiveness of the proposed heuristics by using 20 benchmarking test problems frequently used in the literature. Furthermore, we perform a large-scale simulation study involving three factors, 50 experimental conditions, and 100 000 randomly generated problems to evaluate the proposed heuristics against LPC and six other well-known constructive heuristics in the literature. The simulation results show that both nLPC and nLPC(i) produce average holding and setup costs lower than or equal to those of LPC in every one of the 50 experimental conditions. The proposed heuristics also outperform each of the six other heuristics evaluated in all experimental conditions, without an increase in computational requirements. Lastly, considering that both nLPC and nLPC(i) are fairly simple for practitioners to understand and that lot-sizing heuristics have been commonly used in practice, there should be a very good chance for practical applications of the proposed heuristics.

Computing Minimal Forecast Horizons:An Integer Programming Approach

In this paper, we use integer programming (IP) to compute minimal forecast horizons for the classical dynamic lot-sizing problem (DLS). As a solution approach for computing forecast horizons, integer programming has been largely ignored by the research community. It is our belief that the modelling and structural advantages of the IP approach coupled with the recent significant developments in computational integer programming make for a strong case for its use in practice. We formulate some well-known sufficient conditions, and necessary and sufficient conditions (characterizations) for forecast horizons as feasibility/optimality questions in 0–1 mixed integer programs. An extensive computational study establishes the effectiveness of the proposed approach.

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.

Modeling industrial lot sizing problems: a review

In this paper we give an overview of recent developments in the field of modeling deterministic single-level dynamic lot sizing problems. The focus of this paper is on the modeling of various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production–distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research.

A polynomial algorithm for the production/ordering planning problem with limited storage

This paper concerns the dynamic lot size problem where the storage capacity is limited and shortages are allowed. The planning horizon is divided into T periods and, for each period, concave functions to define the holding/stockout and production costs are considered. It is proved that the results derived in a previous work for the dynamic lot size problem assuming time-varying storage capacities remain valid for the case with backlogging.

Genetic algorithm and Hopfield neural network for a dynamic lot sizing problem

The dynamic lot sizing plays an important role when the demand of an item varies with time. This paper addresses a dynamic lot sizing problem (DLSP) with capacity constraint and discount price structure. Although the well-known dynamic programming (DP) of Wagner-Within is capable of providing an optimal solution for single stage lot sizing problems, it suffers from its high computational complexity. This limits the use of DP in practical problems that are generally larger in size. Two meta-heuristics, genetic algorithm (GA) and Hopfield neural network (HNN) are designed for DLSP to get best trade-off between solution quality and computational time. The DP algorithm is modeled to derive the optimal solution to the experimental problem. The optimality of GA and HNN are tested by comparing the percentage deviation of GA and HNN results against the optimal solution derived using DP.

A heuristic algorithm for a multi-product dynamic lot-sizing and shipping problem

This paper analyzes a dynamic lot-sizing problem, in which the order size of multiple products and a single container type are simultaneously considered. In the problem, each order (product) placed in a period is immediately shipped by some containers in the period and the total freight cost is proportional to the number of containers used. It is also assumed that backlogging is not allowed. The objective of this study is to simultaneously determine the lot-sizes and the transportation policy that minimizes the total costs, which consist of production cost, inventory holding cost, and freight cost. Because this problem is NP-hard, a heuristic algorithm with an adjustment mechanism is proposed based on the optimal solution properties. The computational results from a set of simulation experiment are also presented.

Dynamic lot size problems with one-way product substitution

We consider two multi-product dynamic lot size models with one-way substitution, where the products can be indexed such that a lower-index product may be used to substitute for the demand of a higher-index product. In the first model, the product used to meet the demand of another product must be physically transformed into the latter and incur a conversion cost. In the second model, a product can be directly used to satisfy the demand for another product without requiring any physical conversion. Both problems are generally computationally intractable. We develop dynamic programming algorithms that solve the problems in polynomial time when the number of products is fixed. A heuristic is also developed, and computational experiments are conducted to test the effectiveness of the heuristic and the efficiency of the optimal algorithm.

An O(T log T) Algorithm for the Dynamic Lot Size Problem with Limited Storage and Linear Costs

In this paper the dynamic lot size problem with time varying storage capacities and linear costs is addressed. Like in the uncapacitated version, this problem can be formulated as a network flow problem. Considering the properties of the underlying network, we devise an O(T log T) greedy algorithm to obtain optimal policies and we report computational results for randomly generated problems.

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.

Solving the Dynamic Lot-Sizing Problem with Safety Stocks and Limited Inventories based on New Properties

Solving the Dynamic Lot-Sizing Problem with Safety Stocks and Limited Inventories based on New Properties