Lot-sizing for a single-stage single-product production system with rework of perishable production defectives

We consider a single-stage single-product production system. Produced units may be non-defective, reworkable defective, or non-reworkable defective. The system switches between production and rework. After producing a fixed number (N) of units, all reworkable defective units are reworked. Reworkable defectives are perishable or can become technologically obsolete. We assume that the rework time and the rework cost increase linearly with the time that a unit is held in stock. Therefore, N should not be too large. On the other hand, N should not be too small either, since there are set-up times and costs associated with switching between production and rework. For a given N, we derive an explicit expression for the average profit (sales revenue minus costs). Using this expression, the optimal value for N can be determined numerically. Moreover, it is easy to perform a sensitivity analysis, as we illustrate. RID="*" ID="*"The research of Dr. Ruud H. Teunter has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. The research presented in this paper is part of the research on re-use in the context of the EU sponsored TMR project REVersed LOGistics (ERB 4061 PL 97-5650) in which take part the Otto-von-Guericke Universitaet Magdeburg (D), the Erasmus University Rotterdam (NL), the Eindhoven University of Technology (NL), INSEAD (F), the Aristoteles University of Thessaloniki (GR), and the University of Piraeus (GR). We thank the anonymous referees for their many helpful comments. Correspondence to: R. H. Teunter     

Economic Production Cycles with Imperfect Production Processes

In this paper, we study the effects of an imperfect production process on the optimal production cycle time. The system is assumed to deteriorate during the production process and produce some proportion of defective items. The optimal production cycle is derived, and is shown to be shorter than that of the classical Economic Manufacturing Quantity model. The analysis is extended to the case where the defective rate is a function of the set-up cost, for which the set-up cost level and the production cycle time are jointly optimized. Finally, we also consider the case where the deterioration process is dynamic in its nature, i.e., the proportion of defective items is not constant. Both linear, exponential, and multi-state deteriorating processes are studied. Numerical examples are provided to illustrate the derivation of the optimal production cycle time in these situations.     

Dynamic lot-sizing with rework of defective items and minimum lot-size constraints

In most production processes, defective items may result from an imperfect production system and the need of reworking them is inevitable in many production environments. Despite the great importance of rework in real-world manufacturing, the body of literature is very limited. This paper deals with the effects of defective items and rework on the Capacitated Lot-Sizing Problem (CLSP). We present a mixed-integer programming formulation of the CLSP with rework of defective items and minimum lot-size constraints on production lots. The formulation describes an imperfect production process that leads to a fraction of defective items that have to be reworked before they can be sold to customers. Detailed numerical experiments show that while the occurrence of defective items significantly increases the computational times, reasonably sized minimum lot-size constraints, besides their practical importance, can be a good strategy to accelerate the solution process.     

Optimisation of critical chain sequencing based on activities’ information flow interactions

One critique for the classic critical chain sequencing methods is that only resource constraints and logical relationships between activities are considered, while interactions of information flows are ignored. However, information flow interactions exist between almost all project activities and bring about big rework risk and rework costs in project management, especially for the long-term projects tied up with big amount of money. In this paper, we propose a new approach to sequence a project’s critical chain by considering activities’ information flow. Specifically, we use the design structure matrix to measure directions and intensities of information flows between activities. The criterion for critical chain sequencing is then to minimise the total coordination cost while considering the influence of feedbacks. The simulation results show that, as opposed to the traditional critical chain sequencing methods, our approach shortens average rework time and reduces the cost.     

Surveillance Programs for Lots in Storage

The military, and many civilian organizations, require that a sufficient supply of materials be in storage at any one time to meet certain emergency needs that may arise. However, the deterioration of these materials in storage may render them unusable. It is therefore essential that a systematic program for the surveillance of such stored items be maintained in order to insure that the supply of usable items is always adequate.

This paper investigates the appropriate time for corrective action to compensate for the deteriorating quality of a lot in storage, given the cost of appropriate corrective action and the cost of an imperfect lot when the emergency occurs as functions of the number of defective items in the lot. The critical equation is derived for general cost functions and then for specific cost functions of interest. The assumption made about the lot in storage is that the storage life of each item has a common known exponential distribution; also included is a discussion of how the results can also be applied in the initial absence of information regarding the distribution of storage life.     

Sequence dependent set-up times and job sequencing

On many machine tools the set-up times are sequence dependent; that is, the time to set-up for the new part is dependent upon the part for -which the machine is currently set-up. In scheduling lots on the machine, the set-up time is often a significant component of the total time required to complete each lot. Set-up times are frequently not available to the production scheduler, and when they are, they frequently are only averages which conceal the sequence dependency of the set-up times. In this paper, we first develop a procedure that classifies set-up operations and predicts the set-up times and we then develop a method for sequencing the jobs so as to minimize the total set-up time. An example using real world data is used to illustrate these procedures     

Minimisation of inspection and rework cost in a BLU factory considering imperfect inspection

We address an optimisation problem for minimising the cost of both the inspection and the rework incurred throughout an interconnected inspection–rework system. In order to formulate the relevant cost objective function, assuming imperfect inspection, we make a time-based flow analysis between nodes (or shops), and derive the limiting sizes of flows between nodes and the limiting fraction defective by solving a set of nonlinear balance equations. We provide an enumeration method for determining the optimal frequency of inspection cycles at the end of assembly lines which minimises the relevant cost of inspection and rework.     

Economic lot and supply scheduling problem: a time-varying lot sizes approach

We study the economic lot and supply scheduling problem (ELSSP) that arises in the distribution and manufacturing industries. The ELSSP involves the simultaneous scheduling of end-item production and inbound transportation of input materials over an infinite time horizon to minimise the average costs of inventory, production set-up and transportation. We present a new methodology based on a time-varying lot sizes approach for the ELSSP. We also provide computational experiments showing that the developed algorithm outperforms the existing heuristic for improved integrated scheduling.     

An integrated producer–buyer relationship in the environment of EMQ and JIT production systems

The philosophy of this paper is to build up a strong bond between a producer and a buyer. In this respect, an integrated producer–buyer inventory model with constant demand and small lot size is framed in two different production environments: an EMQ (economic manufacturing quantity)-based production environment and a JIT (just-in-time)-based production environment. Imperfect production with rework is considered for the producer while shortage and backlogging are introduced for the buyer. The joint total cost of the producer and buyer is minimised to determine the optimum rate of production and optimum shortage period for both cases. A comparative study is performed between these two cases using the numerical results in order to make a decision concerning the choice of a more convenient production system for small lot sizes. Sensitivity analyses are carried out together for both cases.     

Strategic Allocation of Inspection Effort In A Serial, Multi-Product Production System

For a multi-stage production system, optimal location of inspection activities is an important consideration in minimizing the inspection-related and salvage costs. Set-up and inventory carrying costs also become important factors when several products are produced on the same line due to the product changeovers involved.

In this paper we discuss the effect of these additional costs (set-up and inventory carrying) on the inspection strategy, i.e., “all or none” versus partial inspection. We suggest a shortest path heuristic to determine the strategic location of inspection activities and the production lot sizes.     

A Markovian approach for multi-level multi-product multi-period capacitated lot-sizing problem with uncertainty in levels

This paper considers a multi-level, multi-item, multi-period capacitated lot-sizing problem with sequence-dependent family set-up times, set-up carry over and uncertainty in levels due to uncertainty in inspection, rework and scrap. In this study, we, first, determined total processing time for each product of each family. Then, expected number of times associated with visiting each level of each product as well as amount of raw materials are calculated. We developed a mixed integer linear programming model with a numerical example and sensitivity analysis.     

Capacity-constrained scheduling for a logic IC final test facility

A capacity-constrained scheduling using the concept of the theory of constraints for a semiconductor Logic IC final test operation is presented. The scheduling of the IC final test considers unrelated parallel machines with multiple constraint problems. A broad product mix, variable lot sizes and yields, long and variable set-up times, as well as limited test equipment capacity characterize the operations in this test facility. Discrete event simulation models based on e-M-Plant? are developed to implement the capacity-constrained scheduling algorithm. A comparison is also made with other rules, which are combinations of the rules such as first come first serve and earliest due date for the order scheduling, and the rules such as minimum set-up time, shortest processing time and shortest set-up time plus processing time for the dispatching test equipment. The simulation results show that the proposed capacity-constrained scheduling outperforms other rules for the committed volume performance in many different operational conditions. Directions for future research are also presented.     

Lead-time reduction in a stochastic inventory system with learning consideration

We develop a continuous review inventory model where lead-time is considered as a controllable variable. Lead-time is decomposed into all its components: set-up time, processing time and non-productive time. These components reflect the set-up cost reduction, lot size lead-time interaction and lead-time crashing, respectively. The learning effect in the production process is also included in the processing time component of the lead-time. The finite investment approach for lead-time and set-up cost reduction and their joint optimization, in addition to the lot size lead-time interaction, introduce a realistic direction in lead-time management and control. A numerical example and a sensitivity analysis are presented using the design of experiments to investigate the effect of the model parameters and, in particular, those related to the different lead-time components on the the expected total cost.     

考虑设备劣化及返工的质量批量模型研究

为了得到最大质量批量，研究了返工成本边际效应下的期望利润值计算方法，将生产过程分为两个阶段：无需返工和需要返工的连续加工。给出了质量控制策略下批量计算公式以及考虑边际成本下的最大利润所对应的质量批量。采用蒙特卡洛仿真和Design Expert 8.0.6软件拟合，通过数字实验，对比了返工成本与未考虑返工成本情况下的期望利润值。研究显示，考虑返工成本边际效应下的质量批量大于一般的经济批量，所得到期望利润也最大。     

Optimal investment and lot-sizing policies for improved productivity and quality

Productivity and quality are an integrated component of the operational strategy of any firm. An increase in productivity implicitly assumes an improvement in quality. The concept of dynamic process quality control and smaller lot-size production have been employed to eliminate defective items, to reduce the cycle time of a product and to improve quality and productivity. We present a mathematical model to establish the relationship between various parameters of productivity and quality. In addition, the proposed model is used to determine the optimal levels of productivity and quality parameters such as batch sizes, and investment in set-up and process control operations. The basic criterion considered for optimizing the level of such parameters is the minimization of total system cost. The proposed model relates productivity and quality to set-up reduction, queueing of batches, batch sizes, and drift rate reduction. We conclude with an example problem to illustrate the behaviour and application of the model.     

Effect of rework,rejects and inspection on lot size with work-in-process inventory

The economic order quantity and economic production quantity models are the most commonly used inventory models in production environments for the calculation of optimum lot size. However, these models are based on the unrealistic assumption that every process produces good quality products every time. Moreover, the impact of inspection is neglected in all extended inventory control models involving work in process inventory. By taking both imperfect production and lot size inspection into consideration, this paper presents a more realistic approach for the modelling of optimum lot size and total cost with a focus on the work in process inventory. A mathematical model is derived for optimum lot size based on the minimisation of the average cost. Our approach incorporates the effect of rework, rejects and inspection on work in process inventory. The significant effect of imperfect production and inspection on optimum lot size is evaluated via numerical examples. In comparison to existing models, the proposed model is a more generalised and flexible form of inventory model for independent demands.     

The G-group heuristic to solve the multi-product,sequencing, lot sizing and scheduling problem in flow shops

This paper presents a new heuristic to solve the problem of making sequencing, lot sizing and scheduling decisions for a number of products manufactured in a flow shop environment, so as to minimize the sum of setup and inventory holding costs while a given demand is fulfilled without backlogging. The proposed solution method first determines sequencing decisions then lot sizing and scheduling decisions are simultaneously determined. This heuristic, called the G-group method, divides the set of products into G groups and requires that products belonging to the same group have the same cycle time. Also, the cycle time of each group is restricted to be an integer multiple of a basic period. For each basic period of the global cycle, the products to be produced during this period and the production sequence to be used are chosen. Then, a non-linear program is solved to determine lot sizes and to construct a feasible production schedule. To evaluate its performance, the G-group method was compared to four other methods. Numerical results show that the proposed heuristic outperforms all these methods.     

Finite difference analysis of reliability optimization through sampling techniques

This paper describes qualitative and quantitative techniques for reliability sampling plans based on failure density functions (MTTF, MTBF and MTTR) as expected values and as functions of time duration (t). Examples are provided of how the techniques are applied in determining plans for a specified lot size. These plans are to test service life, the effectiveness of reliability and MTTF, MTBF and MTTR as expected values and as functions of t and the percentage of reliable units expected in any lot size based on consumer's and producer's risks. Some graphs are constructed from which we can investigate the required aims. Several decision plans for several specified confidence limits, and variable percentages of deterioration of components, are shown for several sample sizes and specified lot sizes. The techniques used in this research, which are based on the calculus of finite differences, on statistical simulation and on the complex stochastic process for determining the relations required, are applicable to all statistical distributions, provided that the components are statistically independent.     

Lot-splitting approach of a hybrid manufacturing system under CONWIP production control: a mathematical model

This paper discusses the advantages of lot splitting in hybrid manufacturing environments where cellular and functional layouts are combined under Constant Work in Process (CONWIP) production control. The proposed model fills a research gap in the related literature by applying lot splitting and pull production simultaneously. A linear CONWIP control mathematical model that minimises the average flow time is developed in case of lot splitting. The developed model has sequence-dependent set-up times. The demand level, coefficient of variation (CV) impact and set-up time reduction effect on CONWIP production control are also investigated. The model is solved using GAMS21.6 optimisation software; the optimal backlog list, the number and size of sublots are reported. The proposed model is compared with lot production under push control in different settings as well as with two different heuristics from the literature. Experimental results indicate that in all settings, the lot splitting is more advantageous than lot production in terms of average flow time. CV has a greater effect than set-up time reduction on average flow time.     

The impact of quality considerations on material flow in two-stage inventory systems

Supply chain management literature calls for coordination between the different members of the chain. Inventory models achieve this coordination along a supply chain by making the lot size at an upstream entity an integer multiplier of the lot size at the adjacent downstream entity. Such models assume that all components produced are of acceptable quality and may cause suppliers to produce larger quantities than what is optimal. In this paper, we formulate and solve two-stage supply chain inventory models in which the proportion of defective products increases with increased production lot sizes. We show that quality considerations can lead to significant reduction in production lot sizes. In addition, the models show that most benefits to the supply chain are attained from the suppliers producing on a just-in-time basis rather than delivering to their customers just-in-time. We derive closed-form expressions for the optimal lot sizes for a two-stage supply chain under deterministic and then stochastic demand and illustrate the models with numerical examples.