Optimizing multi-stage production with constant lot size and varying numbers of batches

The model presented in this paper assumes that a uniform lot size is produced through a series of manufacturing stages, with a single set-up and without interruption at each stage. Transportation of partial lots, called batches, is allowed between stages after the whole batch is completed. The batch sizes must be equal at any particular stage, but the optimal number of equal-sized batches may differ across stages. Of course, the set-up costs, the inventory-holding costs and the transportation costs influence both the optimal batch-sizes at the various stages and the uniform lot size. An optimization method for this deterministic model is developed and is illustrated by an example.     

Optimizing multi-stage production with constant lot size and varying number of unequal sized batches

This paper describes a model for a multi-stage production/inventory system in which a uniform lot size is produced through all stages with a single setup and without interruption at each stage. Partial lots, called batches, may be transported to the next stage upon completion. The number of the unequal sized batches may differ across stages. Considering setup costs, inventory holding costs, and transportation costs, an optimization method is developed to determine the economic lot size and the optimal batch sizes for each stage. The method is illustrated by a computational example and further numerical simulations.     

Lot Splitting in Stochastic Flow Shop and Job Shop Environments*

In recent years many firms have been implementing small lot size production. Lot splitting breaks large orders into smaller transfer lots and offers the ability to move parts more quickly through the production process. This paper extends the deterministic studies by investigating various lot splitting policies in both stochastic job shop and stochastic flow shop settings using performance measures of mean flow time and the standard deviation of flow time. Using a computer simulation experiment, we found that in stochastic dynamic job shops, the number of lot splits is more important than the exact form of splitting. However, when optimal job sizes are determined for each scenario, we found a few circumstances where the implementation of a small initial split, called a “flag,” can provide measurable improvement in flow time performance. Interestingly, the vast majority of previous research indicates that methods other than equal lot splitting typically improves makespan performance. The earlier research, however, has been set in the static, deterministic flow shop environment. Thus, our results are of practical interest since they show that the specific method of lot splitting is important in only a small set of realistic environments while the choice of an appropriate number of splits is typically more important.     

AN ECONOMIC PRODUCTION QUANTITY MODEL WITH LEARNING AND FORGETTING CONSIDERATIONS

The focus of this work is on the effects of learning on economic production quantity in batch production systems. We assumed that both unit variable manufacturing time and setup time follow a learning curve. We modified the classical Economic Production Quantity model to incorporate these two types of learning phenomena. We also incorporated the forgetting effect in our model so that a fraction of the learning is lost between consecutive lots. We developed a dynamic program to obtain the optimal solution to the problem. We investigated the nonincreasing lot size property and used it to improve the efficiency of our dynamic program. We consider a special case of the model in which all lot sizes are assumed equal. After theoretical treatment, we carried out a computational study of the effect of assuming equal lot sizes on the optimal solutions. The results of our examples strongly indicate that the assumption of equal lot sizes not only simplifies the determination of the optimal solutions, but also provides close approximations to the optimal solutions.     

Optimal Inspection of Imports to Prevent Invasive Pest Introduction

The United States imports more than 1 billion live plants annually—an important and growing pathway for introduction of damaging nonnative invertebrates and pathogens. Inspection of imports is one safeguard for reducing pest introductions, but capacity constraints limit inspection effort. We develop an optimal sampling strategy to minimize the costs of pest introductions from trade by posing inspection as an acceptance sampling problem that incorporates key features of the decision context, including (i) simultaneous inspection of many heterogeneous lots, (ii) a lot‐specific sampling effort, (iii) a budget constraint that limits total inspection effort, (iv) inspection error, and (v) an objective of minimizing cost from accepted defective units. We derive a formula for expected number of accepted infested units (expected slippage) given lot size, sample size, infestation rate, and detection rate, and we formulate and analyze the inspector's optimization problem of allocating a sampling budget among incoming lots to minimize the cost of slippage. We conduct an empirical analysis of live plant inspection, including estimation of plant infestation rates from historical data, and find that inspections optimally target the largest lots with the highest plant infestation rates, leaving some lots unsampled. We also consider that USDA‐APHIS, which administers inspections, may want to continue inspecting all lots at a baseline level; we find that allocating any additional capacity, beyond a comprehensive baseline inspection, to the largest lots with the highest infestation rates allows inspectors to meet the dual goals of minimizing the costs of slippage and maintaining baseline sampling without substantial compromise.     

Lot‐sizing Decisions Under Limited‐time Price Reduction*

We consider the optimal lot‐sizing policy for an inventoried item when the vendor offers a limited‐time price reduction. We use the discounted cash flow (DCF) approach in our analysis, thereby eliminating the sources of approximation found in most of the earlier studies that use an average annual cost approach. We first characterize the optimal lot‐sizing policies and their properties, then develop an algorithm for determining the optimal lot sizes. We analytically demonstrate that the lot sizes derived using an average annual cost approach for the different variants of the problem are, in general, larger than the DCF optimum. While DCF analysis is more rigorous and yields precise lot sizes, we recognize that the associated mathematical models and the solution procedure are rather complex. Since simple and easy‐to‐understand policies have a strong practical appeal to decision makers, we propose a DCF version of a simple and easy‐to‐implement heuristic called the “Early Purchase” (EP) strategy and discuss its performance. We supplement our analytical developments with a detailed computational analysis and discuss the implications of our findings for decision making.     

Optimal batch size and investment in multistage production systems with scrap

In this paper, we study the problem of selecting the optimum production batch size in multistage manufacturing facilities with scrap and determining the optimal amount of investment. We analyse the effect of investment for quality improvement on the reduction of the proportion of defectives, and the effect of this reduction on processing cost, setup cost, holding cost, and profit loss. The quality characteristic of the product manufactured is assumed to be normally distributed with a mean equal to the target value. The purpose of the investment is to reduce the variance of the quality characteristic and hence the proportion of defectives. The model assumes known demand, which must be satisfied completely, scrappage at each stage and profit loss due to scrap. Using this model, the optimal values of the production quantity and the proportion of defectives for minimizing the total cost are obtained. The optimal investment is then obtained using the relationship between the investment and the proportion of defectives.     

Production lot sizing with process deterioration and machine breakdown under inspection schedule

The paper develops integrated production, inventory and maintenance models for a deteriorating production system in which the production facility may not only shift from an ‘in-control’ state to an ‘out-of-control’ state but also may break down at any random point in time during a production run. In case of machine breakdown, production of the interrupted lot is aborted and a new production lot is started when the on-hand inventory is depleted after corrective repair. The process is inspected during each production run to examine the state of the production process. If it is found in the ‘in-control’ state then either (a) no action is taken except at the time of last inspection where preventive maintenance is done (inspection policy-I) or (b) preventive maintenance is performed (inspection policy-II). If, however, the process is found to be in the ‘out-of-control’ state at any inspection then restoration is done. The proposed models are formulated under general shift, breakdown and repair time distributions. As it is, in general, difficult to find the optimal production policy under inspection policy-I, a suboptimal production policy is derived. Numerical examples are taken to determine numerically the optimal/suboptimal production policies of the proposed models, to examine the sensitivity of important model parameters and to compare the performance of inspection and no inspection policies.     

WIP inventories in the simultaneous determination of optimal production and purchase lot sizes

This paper studies an integrated production and purchasing lot sizing model with work-in-process WIP inventory. In this model, the single product is made in a multiprocess manufacturing system. The raw materials are procured from outside sources and are converted gradually into the product. A solution procedure is developed to simultaneously find the optimal lot sizes for the product and its raw materials and the corresponding total relevant cost. It is shown that if the cost of WIP inventory is considered in the production lot size computation, the optimal lot sizes of the product as well as those of the raw materials could be altered significantly.     

Optimal Food Safety Sampling Under a Budget Constraint

Much of the literature regarding food safety sampling plans implicitly assumes that all lots entering commerce are tested. In practice, however, only a fraction of lots may be tested due to a budget constraint. In such a case, there is a tradeoff between the number of lots tested and the number of samples per lot. To illustrate this tradeoff, a simple model is presented in which the optimal number of samples per lot depends on the prevalence of sample units that do not conform to microbiological specifications and the relative costs of sampling a lot and of drawing and testing a sample unit from a lot. The assumed objective is to maximize the number of nonconforming lots that are rejected subject to a food safety sampling budget constraint. If the ratio of the cost per lot to the cost per sample unit is substantial, the optimal number of samples per lot increases as prevalence decreases. However, if the ratio of the cost per lot to the cost per sample unit is sufficiently small, the optimal number of samples per lot reduces to one (i.e., simple random sampling), regardless of prevalence. In practice, the cost per sample unit may be large relative to the cost per lot due to the expense of laboratory testing and other factors. Designing effective compliance assurance measures depends on economic, legal, and other factors in addition to microbiology and statistics.     

Analysis of a batch machine fed by an unreliable discrete machine

This paper analyses the behaviour of a manufacturing line consisting of two machines in series where the first machine processes lots one at a time, and is subject to failure and takes a random amount of time to repair when it fails, and the second machine is a perfectly reliable batch machine. A control limit policy is adopted to determine lot sizes for the batch machine. When the batch machine completes processing, if the number of lots in the buffer is greater than or equal to the critical number (Q?), all the lots in the buffer are loaded immediately, otherwise the batch machine waits until Q lots are accumulated. An embedded discrete time Markov-chain approach is proposed, and recursive approaches are developed to derive necessary performance measures. A numerical example explains how to obtain the optimal value of a critical number minimizing the cost functions.     

Economic manufacturing quantities under conditions of learning and forgetting (EMQ/LaF)

Abstract

Abstract. It has been empirically observed that productivity improves as production continues due to system 'learning’, but that it deteriorates once the activity is stopped due to system 'forgetting’. Both learning and forgetting follow an exponential form with a 'doubling factor’ ranging between 0.75 and 0.98. We review and critique two previously proposed models, correct some minor errors in them, and expand one of them to accommodate a finite horizon. We also propose a new model that is more in harmony with the established learning function, for the determination of the optimal number and size of the lots in the finite and infinite horizon. The methodology used throughout is dynamic programming. We investigate the impact of all three models on the optimal lot sires and their costs, and establish the functional relations between the total cost and the various factors affecting them.     

PROCEDURES FOR DETERMINING RELATIVE FREQUENCIES OF PRODUCTION/ORDER IN MULTISTAGE ASSEMBLY SYSTEMS*

We show that the problems of determination lot sizes in a multistage assembly system for the case of instantaneous production and constant demand for the end item can be reduced to the problems of determining relative frequenceis of production/order for items at each production stage. We further show that such frequencies are independent of the demand levels. Optimal and near-optimal solution procedures for this reduced problem are provided. The near-optimal procedure successively treats each stage of production as a final production stage while simulatenously incorporating decisions made at lower stages into decisions made at higher stages. Experimental results show that the near-optimal procedure results in optimal solutions 75 percent of the time and performs considerably better than representative heuristics available in the literature. Further, its performance is relatively less susceptible to product/structural characteristics of the system.     

A decision support system for a third-party coordinator managing supply chain with demand uncertainty

Global supply chains reduce cost but increase lead times, complexities and uncertainties. Retailers in consumer products industry are getting shorter lead time to respond to market demand. To meet this challenge, many rely on third party supply chain managers (SCMs) for economically supplying required quantities of finished products quickly. However, due to shorter ‘time to market’, the SCM has to procure raw materials and start production process based on expected demand. Since SCM absorbs financial penalties associated with under- and over-estimation of demand from retailer, finding an optimal production lot size and product customisation strategy are essential to an SCM's operation. We develop a profit maximisation model and provide a close-form solution that allows an SCM to calculate optimal production lot size. The model is used to examine profitability of postponing product customisation. Finally, the effect of demand variation on SCM's profitability is explored.     

“双随机、一公开”产品质量监管模型研究

在市场监管领域全面推行“双随机、一公开”监管，是党中央、国务院作出的重大决策部署，是市场监管理念和方式的重大创新。本文首次构建“双随机、一公开”产品质量监管的理论模型，理清“双随机”与“一公开”及各参与主体之间的作用关系，分析市场主体名录库和执法检查人员名录库中抽查企业与检查人员的群体行为演化机理，得出针对“双随机、一公开”监管的激励机制。研究发现：通过“双随机”抽取检查人员和抽查企业并“公开”产品质量信息，需配以恰当的激励机制才能有效制约抽检企业的假冒伪劣行为和检查人员的无效抽检行为；当公开声誉较高时，对检查人员实施质检激励或对抽查企业实施质量激励均可以使得系统良性演化，但如果公开声誉较低，需同时对检查人员和抽查企业实施质检与质量双重激励；质检激励机制设计时需对标准质量和假冒伪劣对应的检查人员分别给予质检激励，质量激励机制设计时需对有效抽检和无效抽检对应的抽查企业分别给予质量激励；质检激励和质量激励下系统稳定点的演化轨迹存在很大差异，质检激励先使得检查人员演化到有效抽检，然后通过公开声誉带动抽查企业演化到标准质量，而质量激励先使抽查企业演化到标准质量，然后通过公开声誉带动检查人员演化到有效抽检，如果公开声誉较低，这种传动机制会被阻断导致系统不会演化到理想状态。最后本文为完善和创新“双随机、一公开”监管提出了针对性的政策建议。     

A flexibly structured lot sizing heuristic for a static remanufacturing system

An effective planning of lot sizes is a key strategy to efficiently manage a combined manufacturing/remanufacturing system in the presence of substantial setup costs. Due to its complex interdependencies, optimal policies and solutions have not been identified so far, but several heuristic approaches have been analyzed in recent contributions. The main heuristic shortcuts are forcing equally sized lot sizes over the planning horizon as well as imposing a specific cycle structure, i.e., a sequence of manufacturing batches is followed by a sequence of remanufacturing batches. We are instead proposing a flexibly structured heuristic that allows for differently sized remanufacturing batches. We show in a comprehensive numerical study that our approach outperforms other existing approaches in more than half of all instances by up to 17%.     

Lot scheduling problems of multiple items in the shop with both receiving and delivery just in time

Abstract. This paper deals with lot scheduling problems of multiple items processed in the shop with either a single machine or heterogeneous machines, under the condition that parts to be processed must arrive at the production line at the right times in the right quantities, and that completed parts must be delivered at their due date. Each of the problems is divided into subproblems of lot sizing and scheduling the resulting lots. Solution procedures solving the subproblems are separately optimal. However, since one of the subproblems is not independent of the other, the proposed algorithms adopting these procedures are to be considered heuristic. An iterative procedure is provided to improve solutions. The performance measure used is that of minimizing the total actual flow time considering both receiving just in time and delivery just in time. Numerical examples are presented to show the implementation of the algorithms.     

Towards optimised robotic assembly

Assembly by robot is destined to have a major international impact over the next few years, yet the optimal design, analysis and economic evaluation of a potential robotic assembly application is currently extremely difficult. On looking at the current state-of-the-art in each stage of the hierarchy which a prospective robotic assembly project must pass through, it becomes clear that the whole procedure is closely analogous to attempting to paint an ‘ideal’ picture—both from the point of view of tasks which must be performed and also the level of complexity involved. Unfortunately, unlike with painting, there is not yet any wealth of experience on which roboticists can draw in their quest for ‘ideal’ assembly. Nevertheless, work at Imperial College aims at systematically uncovering some of the underlying principles of robotic assembly which if left to discovery through trial and error might take years—years which industry simply does not have.     

Optimal and Heuristic Models for Lot Splitting in a Flow Shop

This paper investigates optimal lot-splitting policies in a multiprocess flow shop environment with the objective of minimizing either mean flow time or makespan. Using a quadratic programming approach to the mean flow time problem, we determine the optimal way of splitting a job into smaller sublots under various setup times to run time ratios, number of machines in the flow shop, and number of allowed sublots. Our results come from a deterministic flow shop environment, but also provide insights into the repetitive lots scheme using equal lot splits for job shop scheduling in a stochastic environment. We indicate those conditions in which managers should implement the repetitive lots scheme and where other lot-splitting schemes should work better.