Determining the optimal run time for EPQ model with scrap,rework, and stochastic breakdowns

This paper is concerned with determination of optimal run time for an economic production quantity (EPQ) model with scrap, rework, and stochastic machine breakdowns. In real life manufacturing systems, generation of defective items and random breakdown of production equipment are inevitable. In this study, a portion of the defective items is considered to be scrap, while the other is assumed to be repairable. Total production-inventory cost functions are derived respectively for both EPQ models with breakdown (no-resumption policy is adopted) and without breakdown taking place. These cost functions are integrated and the renewal reward theorem is used to cope with the variable cycle length. Theorems on conditional convexity of the integrated overall costs and bounds of the production run time are proposed and proved. We conclude that the optimal run time falls within the range of bounds and it can be pinpointed by the use of the bisection method based on the intermediate value theorem. Numerical example is provided to demonstrate its practical usages.     

Production lot sizing with process deterioration and machine breakdown

The paper presents a generalized economic manufacturing quantity model for an unreliable production system in which the production facility may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time (when it starts producing defective items) and may ultimately break down afterwards. If a machine breakdown occurs during a production run, then corrective repair is done; otherwise, preventive repair is performed at the end of the production run to enhance the system reliability. The proposed model is formulated assuming that the time to machine breakdown, corrective and preventive repair times follow arbitrary probability distributions. However, the criteria for the existence and uniqueness of the optimal production time are derived under general breakdown and uniform repair time (corrective and preventive) distributions. The optimal production run time is determined numerically and the joint effect of process deterioration, machine breakdowns and repairs (corrective and preventive) on the optimal decisions is investigated for a numerical example.     

Mathematical modelling for determining economic batch size and optimal number of deliveries for EPQ model with quality assurance

The classic economic production quantity (EPQ) model assumes a continuous inventory-issuing policy for satisfying product demand and a perfect production for all items produced. However, in a real-life vendor–buyer integrated system, a multi-delivery policy is often used in lieu of continuous issuing policy and it is inevitable to generate defective items during a production run. This study addresses these issues by incorporating multiple deliveries of the finished batch, customer's inventory-holding cost and manufacturer's quality assurance cost into an EPQ model with the imperfect reworking of random defective items. Mathematical modelling and analyses are employed. Convexity of the long-run expected cost function is proved by the use of Hessian matrix equations, and the closed-form solutions in terms of the optimal lot size and optimal number of deliveries are obtained. The research results are demonstrated with a numerical example with a discussion on its practical usage.     

Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging

This paper derives the optimal replenishment policy for imperfect quality economic manufacturing quantity (EMQ) model with rework and backlogging. The classic EMQ model assumes that all items produced are of perfect quality. However, in real‐life manufacturing settings, generation of imperfect quality items is almost inevitable. In this study, a random defective rate is assumed. All items produced are inspected and the defective items are classified as scrap and repairable. A rework process is involved in each production run when regular manufacturing process ends, and a rate of failure in repair is also assumed. Unit disposal cost and unit repairing and holding costs are included in our mathematical modelling and analysis. The renewal reward theorem is employed in this study to cope with the variable cycle length. The optimal replenishment policy in terms of lot‐size and backlogging level that minimizes expected overall costs for the proposed imperfect quality EMQ model is derived. Special cases of the model are identified and discussed. Numerical example is provided to demonstrate its practical usage. Copyright © 2006 John Wiley & Sons, Ltd.     

Joint determination of production run length and number of standbys in a deteriorating production process

Two economic manufacturing quantity models with unrepairable and repairable standby key modules are proposed in this study that determine the economic production run length and the economic number of standbys in a deteriorating production process, where the key module of the production unit deteriorates over time and incurs some portion of defective items. For the model with unrepairable standbys, the active key module, once deteriorating, is replaced by a standby and the module itself is disposed. For the model with repairable standbys, the deteriorating key module is replaced by a standby and the module is then sent to the service center for maintenance. When completing the maintenance, it then joins the standbys for later production use. By minimizing the annual cost, which takes into account setup cost, holding cost, costs due to standbys and defective items, the economic production run length and the economic number of standbys are obtained for each of the proposed models.     

生产线随机崩坍情形下的变质品EPQ策略

变质品生产过程,可能率先出现"次品"的不稳定生产情形,随后机器崩坍;生产状态稳定性迁移时机、机器崩坍时间、维修时间皆乃随机变量;同时,企业无法观测当期需求,只能根据前期需求而随机地安排启动生产时刻.理论模型及数值算例皆表明,此种情况下,企业可以非等周期生产,存在组织生产次数(N)与生产率(P)的优解.敏感度分析看出,当需求拖后率增加、变质率+次品率降低时,企业成本显著降低,但首期生产启动时刻、生产率几乎没有变化.     

An optimal burn‐in policy based on a degradation model

Burn‐in tests help manufacturers detect defective items and remove them before being sold to customers. In a competitive marketplace, cost is a major consideration and not employing a burn‐in test may result in higher and needless expenses. With this in mind, we consider degradation‐based burn‐in tests in which the degradation path follows a Wiener process and weak items are identified when the process crosses a piecewise linear function. We also study linear functions as a special case of such a piecewise linear barrier. Within this setup, we apply a cost model to determine the optimal burn‐in test. Finally, we discuss an illustrative example using GaAs laser degradation data and present an optimal burn‐in test for it.     

Optimal production time and number of maintenance actions for an imperfect production system under equal-interval maintenance policy

This paper deals with the optimal production/maintenance (PM) policy for a deteriorating production system which may shift from the in-control state to the out-of-control state while producing items. The process is assumed to have a general shift distribution. Under the commonly used maintenance policy, equal-interval maintenance, the joint optimizations of the PM policy are derived such that the expected total cost per unit time is minimized. Different conditions for optimality, lower and upper bounds and uniqueness properties on the optimal PM policy are provided. The implications of another commonly used policy, to perform a maintenance action only at the end of the production run, are also discussed. Structural properties for the optimal policy are established so that an efficient solution procedure is obtained. In the exponential case, some extensions of the results obtained previously in the literature are presented. A numerical example is provided to illustrate the solution procedure for the optimal production and maintenance policy.     

Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

An integrated inventory model for a deteriorating item with allowable shortages and credit linked wholesale price

This study proposes a single manufacturer, single retailer integrated inventory model that includes deterioration and shortages in the retailer’s inventory. The manufacturer’s production process is assumed to be imperfect as it produces a certain percentage of defective items. The retailer performs a 100  % screening process immediately on receiving a lot from the manufacturer and returns the detected defective items to the manufacturer in the next delivery. The manufacturer disposes the defective items and incurs a disposal cost. To increase sales, (s)he offers a trade credit to the retailer. The retailer’s wholesale price varies linearly with the credit period. The objective is to determine the optimal replenishment cycle time, the time of running out of stock, the length of the credit period and the number of lots from the manufacturer to the retailer so as to maximize the total profit of the integrated system. A solution algorithm is designed and illustrated through numerical examples. Furthermore, a sensitivity analysis is carried out to study the influence of the model-parameters on the optimal solution.     

Optimal production run length for products sold with warranty

This article studies the optimal production run length for a deteriorating production system in which the products are sold with free minimal repair warranty. The deterioration process of the system is characterized by a two-state continuous-time Markov chain. For products sold with free minimal repair warranty, we show that there exists a unique optimal production run length such that the expected total cost per item is minimized. Since there is no closed form expression for the optimal production run length, an approximate solution is derived. In addition, three special cases which provide bounds for searching the optimal production run length are investigated and some sensitivity analysis is carried out to study the effects of the model parameters on the optimal production run length. Finally, a numerical example is given to evaluate the performance of the optimal production run length.     

On the convexity for the expected total cost per unit time of the EPQ model with scrap,rework and stochastic machine breakdown

This study explores the economic production quantity model with scrap, rework and stochastic machine breakdown. The main purpose of this paper is twofold:(P1) This paper will adopt the rigorous methods of mathematics to demonstrate that the expected total cost per unit time is convex on all positive numbers to improve the conditional convexity in Theorem 1 of Chiu et al. (2010) [7].(P2) This paper gives the concrete proof to provide bounds for the optimal production run time to remove the logical shortcomings of mathematics presented in proof of Theorem 2 of Chiu et al. (2010) [7].     

A note on “Production lot sizing with quality screening and rework”

Recently, Moussawi-Haidar et al. (2016) considered a production process with random supply and an inspection process performed during and at the end of production. Two economic production quantity models with defective items were developed, in which Model 1 assumes that defective items are sold at a discounted price at the end of inspection process, and Model 2 assumes that defective items are reworked at a cost at the end of inspection process. In the paper, there are some mathematical expressions which are to be corrected. We first present the mathematical expressions corrected and establish the necessary conditions for which there is an optimal solution. We next provide the correct solutions to the numerical example.     

Supply chain model for a deteriorating product with time-varying demand and production rate

The paper develops a two-echelon supply chain model with a single-buyer and a single-vendor. The buyer sells a seasonal product over a short selling period and its inventory is subject to deterioration at a constant rate over time. The vendor's production rate is dependent on the buyer's demand rate, which is a linear function of time. Also, the vendor's production process is not perfectly reliable; it may shift from an in-control state to an out-of-control state at any time during a production run and produce some defective (non-conforming) items. Assuming that the vendor follows a lot-for-lot policy for replenishment made to the buyer, the average total cost of the supply chain is derived and an algorithm for finding the optimal solution is developed. The numerical study shows that the supply chain coordination policy is more beneficial than those policies obtained separately from the buyer's and the vendor's perspectives.     

Optimal deteriorating items production inventory models with random machine breakdown and stochastic repair time

This study develops deteriorating items production inventory models with random machine breakdown and stochastic repair time. The model assumes the machine repair time is independent of the machine breakdown rate. The classical optimization technique is used to derive an optimal solution. A numerical example and sensitivity analysis are shown to illustrate the models. The stochastic repair models with uniformly distributed repair time tends to have a larger optimal total cost than the fixed repair time model, however the production up time is less than the fixed repair time model. Production and demand rate are the most sensitive parameters for the optimal production up time, and demand rate is the most sensitive parameter to the optimal total cost for the stochastic model with exponential distribution repair time.     

Inventory models for defective items incorporating marketing decisions with variable production cost

This paper investigates the finite replenishment inventory models of a single product with imperfect production process. In this process, a certain fraction or a random number of produced items are defective. These non-conforming items are rejected or reworked or if they reached to the customer, refunded. Here, a generalised unit cost function is formulated incorporating the several factors like raw material, labour, replenishment rate and others factors of the manufacturing system. The rate of replenishment is considered to be a variable. The selling price of an unit is determined by a mark-up over the production cost. Optimum production of the product is suggested to have maximum profit using a gradient based mathematical programming technique for optimization. Finally, numerical examples are given to illustrate the results and the significant features of the production system. As a particular case, the result of the perfect system (without defective items) are obtained. Also, the effect of changes in the selling rate, defectiveness, production cost and other parameters on the optimal average profit are graphically presented. Some interesting decisions regarding production policy are established.     

Joint determination of optimal safety stocks and production policy for an imperfect production system

In this article, we develop an imperfect economic manufacturing quantity (EMQ) model for an unreliable production system subject to process deterioration, machine breakdown and repair and buffer stock. The basic model is developed under general process shift, machine breakdown and repair time distributions. We suggest a computational algorithm for determination of the optimal safety stock and production run time which minimize the expected cost per unit time in the steady state. For a numerical example, we illustrate the outcome of the proposed model and perform a sensitivity analysis with respect to the model-parameters which have direct influence on the optimal decisions.     

A production–inventory model in an imperfect production process

The paper develops a model to determine the optimal product reliability and production rate that achieves the biggest total integrated profit for an imperfect manufacturing process. The basic assumption of the classical Economic Manufacturing Quantity (EMQ) model is that all manufacturing items are of perfect quality. The assumption is not true in practice. Most of the production system produces perfect and imperfect quality items. In some cases the imperfect quality (non conforming) items are reworked at a cost to restore its quality to the original one. Rework cost may be reduced by improvements in product reliability (i.e., decreasing in product reliability parameter). Lower value of product reliability parameter results in increase development cost of production and also smaller quantity of nonconforming products. The unit production cost is a function of product reliability parameter and production rate. As a result, higher development cost increases unit production cost. The problem of optimal planning work and rework processes belongs to the broad field of production–inventory model which deals with all kinds of reuse processes in supply chains. These processes aim to recover defective product items in such a way that they meet the quality level of ‘good item’. The benefits from imperfect quality items are: regaining the material and value added on defective items and improving the environment protection. In this point of view, a model is introduced here to guide a firm/industry in addressing variable product reliability factor, variable unit production cost and dynamic production rate for time-varying demand. The paper provides an optimal control formulation of the problem and develops necessary and sufficient conditions for optimality of the dynamic variables. In this purpose, the Euler–Lagrange method is used to obtain optimal solutions for product reliability parameter and dynamic production rate. Finally, numerical examples are given to illustrate the proposed model.     

An EPQ model with setup cost and process quality as functions of capital expenditure

This paper considers an economic production quantity (EPQ) model with imperfect production processes, in which the setup cost and process quality are functions of capital expenditure. The mathematical model is derived to investigate the effects of an imperfect production process on the optimal production cycle time when capital investment strategies in setup reduction and process quality improvement are adopted. An efficient procedure is developed to find the optimal production run length, setup cost and process quality. Finally, a numerical example is provided to illustrate the theoretical results. Some managerial implications are also included.     

Batching deteriorating items with applications in computer communication and reverse logistics

We study two deterministic scheduling problems that combine batching and deterioration features. In both problems, there is a certain demand for identical good quality items to be produced in batches. In the first problem, each batch is assigned an individual machine that requires a cost and a time to be activated. All the machines are identical, work in parallel, and always produce good quality items. All the items are available at time zero and they deteriorate while waiting for production. Deterioration results in a linear increase of time and cost of production. In the second problem, there is a single machine that produces good quality as well as defective items in batches. Each batch is preceded by a setup time and requires a setup cost. Defective items have to be reworked on the same machine. They deteriorate while waiting for rework. At a time to be decided, the machine switches from production to rework defective items of the current batch. After rework, every defective item has the required good quality. In both problems, the objective is to find batch partitioning such that a linear combination of the production cost and production completion time is minimized. The two problems are observed at computer service providers and also reverse logistics. In computer service providers, machines and items correspond to communication service channels and information transfer tasks, respectively. We reduce both problems to minimizing a function of one variable representing the number of batches. In an optimal solution of either problem, there are at most two different batch sizes. Linear time algorithms are proposed for both problems.