Discrete Lotsizing Research Articles

Modeling lotsizing and scheduling problems with sequence dependent setups

Several production environments require simultaneous planing of sizing and scheduling of sequences of production lots. Integration of sequencing decisions in lotsizing and scheduling problems has received an increased attention from the research community due to its inherent applicability to real world problems. A two-dimensional classification framework is proposed to survey and classify the main modeling approaches to integrate sequencing decisions in discrete time lotsizing and scheduling models. The Asymmetric Traveling Salesman Problem can be an important source of ideas to develop more efficient models and methods to this problem. Following this research line, we also present a new formulation for the problem using commodity flow based subtour elimination constraints. Computational experiments are conducted to assess the performance of the various models, in terms of running times and upper bounds, when solving real-word size instances.

Discrete lot sizing and scheduling problem under batch processing constraints in the semiconductor manufacturing

Highly advanced technology products are manufactured on the series of equipment having various characteristics and requirements. The deposition process of organic light emitting diode, for example, is performed in chambers for a long manufacturing time, while various product devices belonging to the same family can be processed in sequence before the preventive maintenance schedule or the refill of chemical gas required. The deposition process plays as a bottleneck, and its productivity of the schedule is critical to the upstream low-temperature poly silicon process and the downstream encapsulation and module processes as well. The batch family scheduling problem is formulated using the mixed-integer programming (MIP) in consideration of the family setups, lot sizing, supplementary mask tools requirements, and material exhaustion requirements. It is shown that the planning and scheduling decisions can be made simultaneously in an integrated model, and that it can be implemented in the actual manufacturing line. Through the optimized model analysis, the capacity can be enhanced by 20∼30 % without losing the throughput and demand satisfaction as well.

Stochastic Programming Model for Discrete Lotsizing and Scheduling Problem on Parallel Machines

In recent years, it has been difficult for manufactures and suppliers to forecast demand from a market for a given product precisely. Therefore, it has become important for them to cope with fluctuations in demand. From this viewpoint, the problem of planning or scheduling in production systems can be regarded as a mathematical problem with stochastic elements. However, in many previous studies, such problems are formulated without stochastic factors, treating stochastic elements as deterministic variables or parameters. Stochastic programming incorporates such factors into the mathematical formulation. In the present paper, we consider a multi-product, discrete, lotsizing and scheduling problem on parallel machines with stochastic demands. Under certain assumptions, this problem can be formulated as a stochastic integer programming problem. We attempt to solve this problem by a scenario aggregation method proposed by Rockafellar and Wets. The results from computational experiments suggest that our approach is able to solve large-scale problems, and that, under the condition of uncertainty, incorporating stochastic elements into the model gives better results than formulating the problem as a deterministic model.

Determining the adaptive decision zone of discrete lot sizing model with changes of total cost

The Economic Order Quantity (EOQ) zone is beneficial for giving some latitude in picking the lot sizes in a continuous time inventory problem, but it is not suitable for a discrete time inventory problem, the discrete lot sizing (DLS) problem. In this paper, a novel enumeration method is proposed and coded as a user-friendly computerized scheduling system to “visualize” the complex DLS problems by projecting the entire feasible solutions on a two dimension space, where setup frequency and total cost are placed on the horizontal and the vertical axis respectively. First, the zone around the optimal solution in the DLS problem is demonstrated always smooth and this phenomenon is defined as DLS zone in which giving a small penalty cost from the optimal solution brings several alternative solutions for picking the lot sizes. Second, even if the penalty costs are changed, the computerized scheduling system is able to determine the adaptive decision zone and find the included alternative solutions. The flexibility in picking the lot sizes for discrete time inventory problems is significantly enhanced since decision makers are enabled to choose a preferable solution from the DLS zone.

Multiple machine continuous setup lotsizing with sequence-dependent setups

We address the short-term production planning and scheduling problem coming from the glass container industry. A furnace melts the glass that is distributed to a set of parallel molding machines. Both furnace and machine idleness are not allowed. The resulting multi-machine multi-item continuous setup lotsizing problem with a common resource has sequence-dependent setup times and costs. Production losses are penalized in the objective function since we deal with a capital intensive industry. We present two mixed integer programming formulations for this problem, which are reduced to a network flow type problem. The two formulations are improved by adding valid inequalities that lead to good lower bounds. We rely on a Lagrangian decomposition based heuristic for generating good feasible solutions. We report computational experiments for randomly generated instances and for real-life data on the aforementioned problem, as well as on a discrete lotsizing and scheduling version.

Discrete lot sizing and scheduling using product decomposition into attributes

A production planning problem, known as the discrete lot sizing and scheduling problem with sequence-dependent changeover costs, is considered. We propose a new way of modelling the production system based on the use of a multi-attribute product structure encountered in many industrial situations. The basic idea is to describe the products as combinations of physical attributes and to exploit this description to reduce the size of the mixed-integer program to be solved. The results of our computational experiments show the practical usefulness of the proposed formulation which leads to significantly improved efficiency in the solution process.

Heuristic lot size scheduling on unrelated parallel machines with applications in the textile industry

In this paper, we present an industrial problem found in a company that produces acrylic fibres to be used by the textile industry. The problem is a particular case of the discrete lot sizing and scheduling problem (DLSP). In this problem, lots of similar products must be generated and sequenced in ten unrelated parallel machines, in order to minimize tool changeovers and the quantity of fibre delivered after the required due date. The company problem is original because a changeover can occur between two lots of the same product due to tool wear. We analyse the problem in detail and present an adaptation of a heuristic found in the literature to solve it. Results obtained with the proposed heuristic are compared with results that used to be obtained by the production planner, using historical data.

Simultaneous lot sizing and scheduling for multi-product multi-level production

The paper deals with the multi-level lot sizing and scheduling problem for job shop production in capacitated, dynamic and deterministic cases. Demand is to be fulfilled without backlogging. The multi-level lot sizing and scheduling models of the literature are small bucket problems such as the discrete lot sizing and scheduling problem (DLSP) and the proportional lot sizing and scheduling problem (PLSP). In contrast we develop a big bucket model for multi-product multi-level production. The multi-level general lot sizing and scheduling problem with multiple machines (MLGLSP_MM) is based on the general lot sizing and scheduling problem (GLSP) for single-level production. Our focus is on minimising the sum of sequence dependent setup costs, inventory costs, production costs and the costs of maintaining the setup conditions of the machines. Inventory costs will be calculated exactly. In addition sequence dependent setup times are considered. Finally, our approach minimises the lead-time of semi-finished goods and the throughput time of finished goods.

A Column Generation Approach for Discrete Lotsizing and Scheduling Problem on Identical Parallel Machines

A Column Generation Approach for Discrete Lotsizing and Scheduling Problem on Identical Parallel Machines

Tight Mip Formulation for Multi-Item Discrete Lot-Sizing Problems

This paper discusses mixed-integer programming formulations of variants of the discrete lot-sizing problem. Our approach is to identify simple mixed-integer sets within these models and to apply tight formulations for these sets. This allows us to define integral linear programming formulations for the discrete lot-sizing problem in which backlogging and/or safety stocks are present, and to give extended formulations for other cases. The results help significantly to solve test cases arising from an industrial application motivating this research.

Discrete Lotsizing and Scheduling by Batch Sequencing

The discrete lotsizing and scheduling problem for one machine with sequence-dependent setup times and setup costs is solved as a single machine scheduling problem, which we term the batch sequencing problem. The relationship between the lotsizing problem and the batch sequencing problem is analyzed. The batch sequencing problem is solved with a branch & bound algorithm which is accelerated by bounding and dominance rules. The algorithm is compared with recently published procedures for solving variants of the DLSP and is found to be more efficient if the number of items is not large.

A two-phase genetic algorithm to solve variants of the batch sequencing problem

We introduce the batch sequencing problem (BSP) with item and batch availability for the single-machine and two-machine flow-shop case. We propose a genetic algorithm which solves the BSP through a decomposition into a phase I-batching and a phase II-scheduling decision. The batch sequencing problem is closely related to the discrete lotsizing and scheduling problem (DLSP). Computational experience shows that the genetic algorithm for solving the BSP favourably compares with procedures for solving the DLSP.

Strong turnpike policies in the single-item capacitated lot-sizing problem with periodical dynamic parameter

This article considers optimization problems in a discrete capacitated lot sizing model for a single product with limited backlogging. The demand as well as the holding and backlogging costs are assumed to be periodical in time. Nothing is assumed about types of the cost functions. It is shown that there exists an optimal infinite inverse policy and a strong turnpike policy. A forward algorithm for computing optimal policies relative to the class of batch ordering type policies is derived. Some backward procedure is adopted to determine a strong turnpike policy. The algorithm is simple, and it terminates after the a number of steps equal to the turnpike horizon. Some remarks on the existence of rolling horizontal plans and forecast horizons are also given. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 775–790, 1997

Solving the discrete lotsizing and scheduling problem with sequence dependent set-up costs and set-up times using the Travelling Salesman Problem with time windows

In this paper we consider the Discrete Lotsizing and Scheduling Problem with sequence dependent set-up costs and set-up times (DLSPSD). DLSPSD contains elements from lotsizing and from job scheduling, and is known to be NP-Hard. An exact solution procedure for DLSPSD is developed, based on a transformation of DLSPSD into a Travelling Salesman Problem with Time Windows (TSPTW). TSPTW is solved by a novel dynamic programming approach due to Dumas et al. (1993). The results of a computational study show that the algorithm is the first one capable of solving DLSPSD problems of moderate size to optimality with a reasonable computational effort.

Remarks on: “Some Extensions of the Discrete Lotsizing and Scheduling Problem”

Saloman et al. (Salomon, M., L. G. Kroon, R. Kuik, L. N. Van Wassenhove. 1991. Some extensions of the discrete lotsizing and scheduling problem. Management Sci. 37 801–812.) claim the NP-completeness of different variants of the discrete lot-sizing and scheduling problem (DLSP) which are differentiated by using a six-field notation. However, some of the given proofs are incorrect because the transformations employed map certain data values of the preimage problem to the number of data in the resulting instance and are hence not polynomial. As an example, we concentrate here on the 1/*/SI/G/A-optimization problem (Salomon et al. [Salomon, M., L. G. Kroon, R. Kuik, L. N. Van Wassenhove. 1991. Some extensions of the discrete lotsizing and scheduling problem. Management Sci. 37 801–812.], Theorem 3). Similar arguments pertain to the 1/*/A/G/SI-DLSP, too.

Sequential-Analysis Based Randomized-Regret-Methods for Lot-Sizing and Scheduling

Lotsizing and scheduling comprises activities which have to be done repeatedly within MRP-systems. We consider the proportional (multi-item, capacitated, dynamic) lotsizing and scheduling problem which is more general than the discrete lotsizing and scheduling problem as well as the continuous setup lotsizing problem. A simple stochastic backward algorithm with biased sampling via randomized regrets is presented. We partition the parameter space of the stochastic algorithm and choose subspaces via sequential analysis based on hypothesis testing. The new ideas provided in this paper, i.e. the randomized-regrets-based backward algorithm as well as the controlled search via sequential analysis, have three important properties: They are simple, effective, and rather general. Computational results are presented as well.

Specially Structured Uncapacitated Facility Location Problems

This paper considers a specially structured uncapacitated facility location problem. We show that several problems, including certain tool selection problems, substitutable inventory problems, supplier sourcing problems, discrete lot sizing problems, and capacity expansion problems, can be formulated as instances of the problem. We also show that the problem with m facilities and n customers can be solved in O(mn), as a shortest path problem on a directed graph.

Integrative versus separate cycle scheduling heuristics for capacitated discrete lot sizing and sequencing problems

This study investigates the effect of integrating lot sizing and lot sequencing decisions for a capacitated production system with a discrete shipping schedule. We compare integrative cycle scheduling heuristics, which explicitly consider both decisions, with traditional separate approaches. Experiments are conducted for both zero and positive setup cost situations with different daily demand variations, weekly demand patterns and number of items. The results of the total inventory cost and CPU time performance indicate the superiority of the integrative heuristics, the importance of an intelligent sequencing scheme, and the influential role of demand variations and the number of items. Insights from this study can be applied to a number of manufacturing settings, from corrugated paper making to component subassembly operations.

Proportional lotsizing and scheduling

Two multi-item capacitated dynamic lotsizing and scheduling models with a finite horizon have been established recently: the discrete lotsizing and scheduling problem as well as the continuous setup lotsizing problem. An analysis of the underlying fundamental assumptions provides the basis for introducing a new model, the proportional lotsizing and scheduling problem. We present a new backward-oriented regret-based biased random sampling method which solves the new model efficiently. The model is well suited for the incorporation of some of the extensions relevant for practice: setup times, sequence-dependent setup costs (times), multiple machines as well as multiple stages.

The single-item discrete lotsizing and scheduling problem: optimization by linear and dynamic programming

This paper considers the single-item discrete lotsizing and scheduling problem (DLSP). DLSP is the problem of determining a minimal cost production schedule, that satisfies demand without backlogging and does not violate capacity constraints. We formulate DLSP as an integer programming problem and present two solution procedures. The first procedure is based on a reformulation of DLSP as a linear programming assignment problem, with additional restrictions to reflect the specific (setup) cost structure. For this linear programming (LP) formulation it is shown that, under certain conditions on the objective, the solution is all integer. The second procedure is based on dynamic programming (DP). Under certain conditions on the objective function, the DP algorithm can be made to run very fast by using special properties of optimal solutions.