Setups and effective capacity: the impact of lot sizing techniques in an MRP environment

This study investigates how lot sizing techniques influence the profit performance, inventory level, and order lardiness of an assembly job shop controlled by MRP. Four single-level lot sizing techniques are compared by simulation analysis under two levels of master schedule instability and two levels of end item demand. A second analysis investigates the influence of a multilevel lot sizing technique, the generalized constrained-K (GCK) cost modification, on the four single-level techniques at low demand and low nervousness. The analyses reveal a previously unreported phenomenon. Given the same inventory costs, the single-level lot sizing techniques generate substantially different average batch sizes. The lot sizing techniques maintain the following order of increasing average batch size (and decreasing total setup time):

economic order quantity (EOQ)

period order quantity (POQ)

least total cost (LTC)

Silver-Meal heuristic (SML)

The causes for different average batch sizes among the lot sizing techniques are analysed and explained. Demand lumpiness, inherent in multilevel manufacturing systems controlled by MRP, is found to be a major factor. The number of setups each lot sizing technique generates is the primary determinant of profit performance, inventory level, and order tardiness. EOQ, a fixed order quantity technique, is less sensitive to nervousness than the discrete lot sizing techniques. EOQ_, however, generates the smallest average batch size, and, therefore, the most setups. Since setups consume capacity, EOQ, is more sensitive to higher demand. SML generates the largest average batch sizes, and is, therefore, less sensitive to increased demand. At low demand, the other lot sizing techniques perform better on all criteria. They generate smaller batches and, therefore, shorter actual lead times. The GCK cost modification increases the average batch size generated by each lot sizing technique. GCK improves the profit and customer service level of EOQ the lot sizing technique with the smallest batches. GCK causes the other lot sizing techniques to generate excessively large batches and, therefore, excessively long actual lead times.     

THE PERFORMANCE OF A SIMPLE INCREMENTAL LOT-SIZING RULE IN A MULTILEVEL INVENTORY ENVIRONMENT

A simple incremental cost approach to lot sizing was tested in a multilevel inventory environment. The incremental approach has not previously been tested in a large-scale study involving multiple product-structure levels. Using the Wagner-Whitin (WW) algorithm as a benchmark, the simple incremental rule (IPPA) was compared to three heuristic procedures (LFL, EOQ, and POQ) frequently used in material requirements planning (MRP) lot-sizing research. The incremental rule consistently generated lower total order/setup and carrying costs than the three heuristics across the 3,200 multilevel test situations examined. In many of the test situations, the incremental rule also outperformed the WW benchmark.     

冷轧生产批量计划与调度问题模型及算法

对冷轧生产系统特点进行分析,将冷轧生产批量计划与调度问题抽象为多阶段、多品种带有中间库的批量计划与调度问题。针对该问题建立了数学模型,通过对库存成本和调整成本惩罚系数的控制可以协调库存水平和调整次数的关系。对所建立的模型,提出了基于二进制粒子群优化与局部搜索的混合求解算法。最后,通过对企业实际生产数据的计算和分析,验证了模型和算法的可行性和有效性。     

A Simple Heuristic for Lot Sizing with Setup Times

This paper presents an easily understood and computationally efficient heuristic algorithm for the capacitated lot sizing problem (CLSP), the single machine lot-sizing problem, with nonstationary costs, demands, and setup times. The algorithm solves problems with setup time or setup cost. A variation of the algorithm can solve problems when limited amounts of costly overtime are allowed. Results of experimentation indicate that the most significant effects on solution quality are due to the level of setup costs relative to holding costs and the size of the problems as determined by the number of items. Also affecting solution quality are tightness of the capacity constraint and variability of demand in a problem. When the capacity constraint is extremely tightly binding, it sometimes has difficulty finding solutions that do not require overtime.     

PRODUCTION SCHEDULING OF CONTINUOUS FLOW LINES: MULTIPLE PRODUCTS WITH SETUP TIMES AND COSTS

The problem of production planning and setup scheduling of multiple products on a single facility is studied in this paper. The facility can only produce one product at a time. A setup is required when the production switches from one type of product to another. Both setup times and setup costs are considered. The objective is to determine the setup schedule and production rate for each product that minimize the average total costs, which include the inventory, backlog, and setup costs. Under the assumption of a constant production rate, we obtain the optimal cyclic rotation schedule for the multiple products system. Besides the decision variables studied in the classical economic lot scheduling problem (ELSP), the production rate is also a decision variable in our model. We prove that our solutions improve the results of the classical ELSP.     

Integrative Cycle Scheduling Approach for a Capacitated Flexible Assembly System*

This study revisits the traditional single stage, multi-item, capacitated lot-sizing problem (CLSP) with a new integrative focus on problem structuring. Unlike past research, we develop integrative cycle scheduling approaches which simultaneously address lot-sizing, capacity, and sequencing issues. Our purposes are to (1) explore the effect of sequencing on inventory levels, (2) examine the problem of infeasibility in the economic lot scheduling problem (ELSP), and (3) provide a simple methodology of generating low-cost cycle schedules in an environment with discrete shipping, dynamic demands, limited capacity, zero setup cost, and sequence-independent setup times. Our procedures are compared to benchmark cycle scheduling approaches in terms of both inventory cost and computation time under different demand scenarios, using the operating data from a flexible assembly system (FAS) at the Ford Motor Company's Sandusky, Ohio plant.     

Minimizing tardiness on a single processor with sequence-dependent setup times: a simulated annealing approach

In today's fast-paced Just-In-Time and mass customization manufacturing in a sequence-dependent setup environment, the challenge of making production schedules to meet due-date requirements is becoming a more complex problem. Unfortunately, much of the research on operations scheduling problems has either ignored setup times or assumed that setup times on each machine are independent of the job sequence. This paper considers the problem of minimizing tardiness, a common measure of due-date performance, in a sequence-dependent setup environment. Simulated annealing was used to solve the sequencing problem, and its performance was compared with random search. Our experimental results show that the algorithm can find a good solution fairly quickly, and thus can rework schedules frequently to react to variations in the schedule. The algorithm is invaluable for ‘on-line’ production scheduling and ‘last-minute’ changes to production schedule. The results of this research also suggest ways in which more complex and realistic job shop environments, such as multiple machines with a higher number of jobs in the sequence, and other scheduling objectives can be modeled. This research also investigates computational aspects of simulated annealing in solving complex scheduling problems.     

Concepts,Theory, and Techniques A JOINT LOT-SIZING RULE FOR FIXED LABOR-COST SITUATIONS

Many American firms are implementing just-in-time production in order to minimize inventories, reduce flow time, and maximize resource utilization. These firms recognize that, in the short run, setup costs really are fixed expenses and it is available capacity which is the critical factor in determining production-run quantities. We propose using available capacity to increase the number of setups and reduce lot-size inventories. This results in improved relevant cost performance. Sugimori, Kusunoki, Cho, and Uchikawa [16] in their paper on the Toyota kanban system developed a relationship for lead time but failed to use it for lot sizing. We use this relationship to develop the joint lot-sizing rule. The efficacy of our proposed rule is demonstrated by applying it to lot-size scheduling problems at the John Deere Engine Works [14]. Extensions of the proposed rule to undercapacity situations with material-wastage costs in the setup processes and to multistate production inventory systems also are discussed.     

A heuristic model for the sequence-dependent lot scheduling problem

This paper presents a heuristic algorithm for finding a good solution for the sequence-dependent lot scheduling problem. Unlike available methods, the algorithm eliminates the need for creating new artificial problems and implementing feasibility tests. It also eliminates the tedious task of translating setup relationships into a mathematical programming formulation. The result is a conceptually simple solution technique that is practically motivated and easily implemented for use on the shop floor. Comparison of algorithm performance with published results demonstrates the efficacy of the approach.     

Lot sizing and scheduling: a survey of practices in Brazilian companies

The lot sizing and scheduling (LSS) problem has motivated much research in production planning and control. However, while there are many papers on modelling this problem, there is scant research on how it is addressed in the industry. We present a survey of Brazilian industry practices in LSS, intending to understand the problem faced by operations managers, drawing parallels with academic findings to identify gaps between theory and practice and opportunities for future research. We confirm that the main objective of practitioners in companies operating under MTS strategy is to maximise service level, followed by inventory reduction. Availability of raw material, demand uncertainty and safety stock levels, lack of decision support software and the impact of sequence-dependent set-up times are other major concerns highlighted by the practitioners. Lastly, we confirmed that despite the relevant body of research on this problem, there is still a large gap between theory and practice.     

Economic Production Lot Sizing with Periodic Costs and Overtime*

Traditional approaches for modeling economic production lot‐sizing problems assume that a single, fixed equipment setup cost is incurred each time a product is run, regardless of the quantity manufactured. This permits multiple days of production from one production setup. In this paper, we extend the model to consider additional fixed charges, such as cleanup or inspection costs, that are associated with each time period's production. This manufacturing cost structure is common in the food, chemical, and pharmaceutical industries, where process equipment must be sanitized between item changeovers and at the end of each day's production. We propose two mathematical problem formulations and optimization algorithms. The models' unique features include regular time production constraints, a fixed charge for each time period's production, and the availability of overtime production capacity. Experimental results indicate the conditions under which our algorithms' performance is superior to traditional approaches. We also test the procedures on a set of lot‐sizing problems facing a national food processor and document their potential economic benefit.     

Coordinated Capacitated Lot‐Sizing Problem with Dynamic Demand: A Lagrangian Heuristic

Coordinated replenishment problems are common in manufacturing and distribution when a family of items shares a common production line, supplier, or a mode of transportation. In these situations the coordination of shared, and often limited, resources across items is economically attractive. This paper describes a mixed‐integer programming formulation and Lagrangian relaxation solution procedure for the single‐family coordinated capacitated lot‐sizing problem with dynamic demand. The problem extends both the multi‐item capacitated dynamic demand lot‐sizing problem and the uncapacitated coordinated dynamic demand lot‐sizing problem. We provide the results of computational experiments investigating the mathematical properties of the formulation and the performance of the Lagrangian procedures. The results indicate the superiority of the dual‐based heuristic over linear programming‐based approaches to the problem. The quality of the Lagrangian heuristic solution improved in most instances with increases in problem size. Heuristic solutions averaged 2.52% above optimal. The procedures were applied to an industry test problem yielding a 22.5% reduction in total costs.     

An Experimental Analysis of Capacity-Sensitive Setup Parameters for MRP Lot Sizing

Most material requirements planning (MRP) systems apply standard costing (absorption costing) approaches to define setup costs that are used as fixed (time invariant) setup parameters in single-level lot-sizing methods. This paper presents a computationally simple approach for estimating more appropriate setup parameters based on estimates of work-center shadow prices. These setup parameters then are used in traditional single-level MRP lot-sizing procedures. The shadow price of capacity at each work center is calculated as the increase in the overall inventory carrying cost for each additional hour of capacity lost to setups. The opportunity cost of a setup for an order subsequently is determined based on the routing information for each order and is used by traditional MRP lot-sizing procedures to calculate lot sizes. A simulation experiment compares the performance period order quantity lot sizing with capacity-sensitive setup parameters with the fixed accounting-based setup parameters. The simulation replicates the planning and control functions of a typical MRP system. The results of the experiment show that capacity-sensitive setup parameters can make significant reductions in both carrying cost and lateness and can achieve many of the benefits of optimized production technology in the context of an MRP system.     

EXISTENCE AND DERIVATION OF FORECAST HORIZONS IN A DYNAMIC LOT SIZE MODEL WITH NONDECREASING HOLDING COSTS

We are concerned with a discrete-time undiscounted dynamic lot size model in which demand and the production setup cost are constant for an initial few periods and the holding cost of inventory is an arbitrary nondecreasing function assumed to be stationary (i.e., explicitly independent of time) in the same initial few periods. We show that there exists a finite forecast horizon in our model and obtain an explicit formula for it. In addition, we obtain fairly general conditions under which the existence of a solution horizon in the model implies the existence of a forecast horizon. We also derive an explicit formula for the minimal solution horizon. These results extend the earlier ones obtained for the dynamic lot size model with linearly increasing holding costs.     

A simulation of single-level MRP lot sizing heuristics: an analysis of performance by rule

This paper presents the results of a large-scale computer simulation of 12 of the standard single-level, discrete demand lot sizing heuristics. The authors present the results in 3-D illustrations which depict the performance of these heuristics on 15 individual demand patterns. This information is prefaced by a brief review of the method used to perform the simulation. The performance of each of the 12 heuristics was evaluated for 51 sets of cost parameters for each of 15 different demand patterns. This has resulted in the analysis of 9180 combinations of heuristic, demand pattern, and cost parameters. The authors believe that this, by far, represents the largest digital simulation of single-level lot sizing rules completed to date. During the past two decades, a significant amount of research investigating the economics of lot sizing single-level discrete demand patterns has been conducted. However, many of the conclusions reached by individual research efforts on this subject have differed. At various times a lot sizing heuristic has performed best in one study, only to have findings refuted in a later analysis. Overall, this has led to a certain degree of confusion and mistrust of the heuristics themselves. The authors believe that there are three major reasons why previous research efforts have reached various conclusions. First, previous studies have included only subset of the possible heuristics. Second, previous studies have used different methods of calculating holding costs. Finally, previous studies have used demand patterns so short that a large percentage of transient noise is contained in the performance data.     

Piecewise linear approximations for the static–dynamic uncertainty strategy in stochastic lot-sizing

In this paper, we develop a unified mixed integer linear modelling approach to compute near-optimal policy parameters for the non-stationary stochastic lot sizing problem under static–dynamic uncertainty strategy. The proposed approach applies to settings in which unmet demand is backordered or lost; and it can accommodate variants of the problem for which the quality of service is captured by means of backorder penalty costs, non-stockout probabilities, or fill rate constraints. This approach has a number of advantages with respect to existing methods in the literature: it enables seamless modelling of different variants of the stochastic lot sizing problem, some of which have been previously tackled via ad hoc solution methods and some others that have not yet been addressed in the literature; and it produces an accurate estimation of the expected total cost, expressed in terms of upper and lower bounds based on piecewise linearisation of the first order loss function. We illustrate the effectiveness and flexibility of the proposed approach by means of a computational study.     

Stochastic lot sizing problem with controllable processing times

In this study, we consider the stochastic capacitated lot sizing problem with controllable processing times where processing times can be reduced in return for extra compression cost. We assume that the compression cost function is a convex function as it may reflect increasing marginal costs of larger reductions and may be more appropriate when the resource life, energy consumption or carbon emission are taken into consideration. We consider this problem under static uncertainty strategy and α service level constraints. We first introduce a nonlinear mixed integer programming formulation of the problem, and use the recent advances in second order cone programming to strengthen it and then solve by a commercial solver. Our computational experiments show that taking the processing times as constant may lead to more costly production plans, and the value of controllable processing times becomes more evident for a stochastic environment with a limited capacity. Moreover, we observe that controllable processing times increase the solution flexibility and provide a better solution in most of the problem instances, although the largest improvements are obtained when setup costs are high and the system has medium sized capacities.     

Simultaneous determination of production lot size and process parameters under process deterioration and process breakdown

The purpose of this study is to combine production-inventory management with process-quality design for determining production lot size and process parameters under the possibility of process deterioration and breakdown. The total cost of such an integrated model includes: the combined setup cost (production setup and process resetting), the costs of quality loss, tolerance and mean costs for processes established, a penalty cost for process breakdown and carrying costs for cumulated inventory. The quadratic quality loss function is introduced to assess quality loss within the system. Decision variables include the initial setting (process mean) and process tolerance for process parameters determination, and production lot size for production-inventory management. The cycle time for production-inventory management is assumed to be the same as the resetting cycle for the new process-quality system. The contribution of this study lies in its development of an integrated model that enables process parameters, production lot size, and cycle time to be determined concurrently for quality and economic considerations, and at an earlier time in the process design and production management stage. An example is presented to demonstrate the proposed model.     

Duality Approaches to Economic Lot‐Sizing Games

Sharing common production, resources, and services to reduce cost are important for not for profit operations due to limited and mission‐oriented budget and effective cost allocation mechanisms are essential for encouraging effective collaborations. In this study, we illustrate how rigorous methodologies can be developed to derive effective cost allocations to facilitate sustainable collaborations in not for profit operations by modeling the cost allocation problem arising from an economic lot‐sizing (ELS) setting as a cooperative game. Specifically, we consider the economic lot‐sizing (ELS) game with general concave ordering cost. In this cooperative game, multiple retailers form a coalition by placing joint orders to a single supplier in order to reduce ordering cost. When both the inventory holding cost and backlogging cost are linear functions, it can be shown that the core of this game is non‐empty. The main contribution of this study is to show that a core allocation can be computed in polynomial time under the assumption that all retailers have the same cost parameters. Our approach is based on linear programming (LP) duality. More specifically, we study an integer programming formulation for the ELS problem and show that its LP relaxation admits zero integrality gap, which makes it possible to analyze the ELS game by using LP duality. We show that there exists an optimal dual solution that defines an allocation in the core. An interesting feature of our approach is that it is not necessarily true that every optimal dual solution defines a core allocation. This is in contrast to the duality approach for other known cooperative games in the literature.     

Multi-objective scheduling of two-job families on a single machine

This paper presents a preliminary analysis of the typical scheduling environment in semiconductor manufacturing involving multiple job families, and where more than one objective such as cycle time, machine utilization and the due-date accuracy needs to be simultaneously considered. In this study, the NP-hard problem of scheduling N independent jobs on a single testing machine with due dates and sequence-dependent setup times is addressed, where the multiple objectives are to minimize average cycle time, to minimize average tardiness, and to maximize machine utilization. A Pareto optimal solution, which is not inferior to any other feasible solutions in terms of all objectives, is generated combining the analytically optimal and conjunctive simulated scheduling approach. First, the machine-scheduling problem is modeled using the discrete event simulation approach and the problem is divided into simulation clock based lot selection sub-problems. Then, a Pareto optimal lot is selected using the compromise programming technique for multiobjective optimization at each decision instant in simulated time. With the help of a broad experimental design, this developed solution is then compared with common heuristic-dispatching rules such as SPT and EDD, which show better results for all the objectives over a wide range of problems. The developed scheduling method shows approximately 16.7% reduction in average cycle time, 25.6% reduction in average tardiness, and 21.6% improvement in machine utilization over the common dispatching rules, SPT and EDD.