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.     

An arithmetic-geometric mean inequality approach for determining the optimal production lot size with backlogging and imperfect rework process

Some classical studies on economic production quantity (EPQ) models with imperfect production processes have focused on determining the optimal production lot size. However, these models neglect the fact that the total production-inventory costs can be reduced by reworking imperfect items for a relatively small repair and holding cost. To account for the above phenomenon, we take the out of stock and rework into account and establish an EPQ model with imperfect production processes, failure in repair and complete backlogging. Furthermore, we assume that the holding cost of imperfect items is distinguished from that of perfect ones, as well as, the costs of repair, disposal, and shortage are all included in the proposed model. In addition, without taking complex differential calculus to determine the optimal production lot size and backorder level, we employ an arithmetic-geometric mean inequality method to determine the optimal solutions. Finally, numerical examples and sensitivity analysis are analyzed to illustrate the validity of the proposed model. Some managerial insights are obtained from the numerical examples.     

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.     

A generalized geometric-programming solution to “An economic production quantity model with flexibility and reliability considerations”

The classical economic production quantity (EPQ) model assumes that items are produced by a perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be perfected cost-free, the set-up cost can be reduced by investment in flexibility improvement. In this paper, we propose an EPQ model with a flexible and imperfect production process. We formulate this inventory decision problem using geometric programming (GP), establish more general results using the arithmetic-geometric mean inequality, and solve the problem to obtain a closed-form optimal solution. Following the theoretical treatment, we provide a numerical example to demonstrate that GP has potential as a valuable analytical tool for studying a certain class of inventory control problems. Finally we discuss some aspects of sensitivity analysis of the optimal solution based on the GP approach.     

An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α-cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α-cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new 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.     

Economic Production Lot Size Model with Variable Production Rate and Imperfect Quality

The classical economic production lot size (EPL) model assumes a constant production rate that is predetermined and inflexible, and perfect quality. Recent models have removed the assumption of perfect quality while maintaining the inflexible production rate assumption. Production rates in many cases, such as orders filled by a machine, can be changed. Moreover, unit production cost and process quality depend on the production rate. In this paper, we extend the EPL model to cases where the production rate is a decision variable. Unit production cost becomes a function of the production rate. Also, the quality of the production process deteriorates with increased production rate. We solve the proposed model for special cost and quality functions and illustrate the results with a numerical example. The results show that, for cases where increases in the production rate lead to a significant deterioration in quality, the optimal production rate may be smaller than the rate that minimizes unit production cost. For cases where quality is largely independent of the production rate, the optimal production rate may be larger than the rate that minimizes unit production cost.     

Determining the optimal production policy for a system with imperfect production and shortage

The classic EPQ model assumes that items are produced of perfect quality and no shortage is permitted. In the real world situation, however, due to process deterioration or other factors, the occurrence of imperfect quality items is inevitable. This paper develops an extended economic production quantity (EPQ) model with imperfect production, shortage, and imperfect rework. We assume that the quality scan is conducted during the production. The scanned imperfect items are classified as the repairable and scrap. We consider that not all of the repairable items can be restored to meet the specified quality standard. Only some portion of defective items can be restored as normal items, the other results in defective, due to repair failure, can be sold at a discounted price to a secondary market. The renewal reward theorem is utilized to deal with the variable cycle length. The production quantity and the shortage level are determined in an optimal manner so as to minimize the average system cost. A numerical example is used to demonstrate its practical usage. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Production–shipment policy for EPQ model with quality assurance and an improved delivery schedule

This article is concerned with determining the production–shipment policy for an economic production quantity model with quality assurance and an improved delivery schedule. We extend a recent work by Chiu et al. [Y.-S.P. Chiu, C.-A.K. Lin, H.-H. Chang, and V. Chiu, Mathematical modeling for determining economic batch size and optimal number of deliveries for EPQ model with quality assurance, Math. Comput. Model. Dyn. Sys. 16 (4) (2010), pp. 373–388] by incorporating an alternative delivery plan that aims at lowering the inventory holding cost for both supplier and buyer in such an integrated inventory system. Mathematical modelling along with Hessian matrix equations is used, and as a result the optimal production batch size and optimal number of deliveries are derived. A numerical example is provided to demonstrate the practical use of the results and the significant savings in stock holding costs for both vendor and buyer.     

An EOQ model with process reliability considerations

The classical economic order quantity (EOQ) model assumes that items produced are of perfect quality and that the unit cost of production is independent of demand. However, in realistic situations, product quality is never perfect, but is directly affected by the reliability of the production process. In this paper, we consider an EOQ model with imperfect production process and the unit production cost is directly related to process reliability and inversely related to the demand rate. In addition, a numerical example is given to illustrate the developed model. Sensitivity analysis is also performed and discussed.     

Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance

This study integrates maintenance and production programs with the economic production quantity (EPQ) model for an imperfect process involving a deteriorating production system with increasing hazard rate: imperfect repair and rework upon failure (out of control state). The imperfect repair performs some restorations and restores the system to an operating state (in-control state), but leaves its failure until perfect preventive maintenance (PM) is performed. There are two types of PM, namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas are obtained for deriving the expected total cost. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM, setup, breakdown and holding costs.     

An EMQ model with inspections and random machine failures

In the classical Economic Manufacturing Quantity (EMQ) model, it is assumed that all items produced are of perfect quality and the production facility never breaks down. However, in real production, the product quality is usually a function of the state of the production process which may deteriorate over time and the production facility may fail randomly. In this paper, we study the effect of machine failures on the optimal lot size and on the optimal number of inspections in a production cycle. The formula for the long-run expected average cost per unit time is obtained for a generally distributed time to failure. An optimal production/inspection policy is found by minimising the expected average cost.     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

Optimal production run time for two-stage production system with imperfect processes and allowable shortages

This paper considers a two-stage production system with imperfect processes. Shortages are allowed, and the unsatisfied demand is completely backlogged. In addition, the capital investment in process quality improvement is adopted. Under these assumptions, we first formulate the proposed problem as a cost minimization model where the production run time and process quality are decision variables. Then we develop the criterion for judging whether the optimal solution not only exists but also is unique. If the criterion is not satisfied, the production system should not be opened. An algorithm for the computations of the optimal solutions is also provided. Finally, a numerical example and sensitivity analysis are carried out to illustrate the model.     

报童模型的最优解及其解空间研究

文章从经典报童模型出发,找到了使收益最大化的报童模型最优解及其存在的解空间。在分析最优解存在条件的基础上,研究了单位生产成本和单位缺货惩罚成本对最优解的解空间的影响。在此基础上,进一步分析了如何通过控制单位生产成本和单位缺货惩罚成本,影响最优解存在条件的方法。最后,在该领域其他学者的实际算例的基础上,提出了分别通过调节单位生产成本和单位缺货惩罚成本,以及同时调节单位生产成本和单位缺货惩罚成本,从而影响企业生产决策的三种方法。文章结果可以指导相关学者选择适当的报童模型算例,且实际算例表明该方法在企业管理方面也有较好的效果和应用前景。     

On optimal batch sizing in a multi-stage production system with rework consideration

Recently, Sarker et al. [Sarker, B.R., Jamal, A.M.M., Mondal, S., 2008. Optimal batch sizing in a multi-stage production system with rework consideration. European Journal of Operational Research 184(3) 915–929] presented an EPQ inventory model for a multi-stage manufacturing system with rework process; basically they proposed two operational inventory policies. In the paper, there are some mathematical expressions which are to be corrected. At first, this paper presents the mathematical expressions corrected and the appropriate solution to the numerical example. We also established the closed forms for the optimal total inventory cost, the conditions for which there is an optimal solution, and the mathematical expressions for determining the total additional cost for working with a non optimal solution for both policies that were not given by Sarker et al. (2008).     

An Optimal Batch Size for a Production System Operating Under a Fixed-Quantity,Periodic Delivery Policy

A manufacturing system which procures raw materials from suppliers and processes them to convert to finished products is considered here. This paper develops an ordering policy for raw materials to meet the requirements of a production facility which, in turn, must deliver finished products demanded by outside buyers at fixed interval points in time. First, a general cost model is developed considering both supplier (of raw material) and buyer (of finished products) sides. This model is used to determine an optimal ordering policy for procurement of raw materials, and the manufacturing batch size to minimize the total cost for meeting equal shipments of the finished products, at fixed intervals, to the buyers. The total cost is found to be a piece-wise convex cost function. An interval that contains the optimal solution is first determined followed by an optimization technique to identify the exact solution from this interval.     

Optimal preventive maintenance of a production system with an intermediate buffer

In this paper we consider a model consisting of a deteriorating installation that transfers a raw material to a production unit and a buffer which has been built between the installation and the production unit. The deterioration process of the installation is considered to be nonstationary, i.e. the transition probabilities may depend not only on the working conditions of the installation but on its age as well. The problem of the optimal preventive maintenance of the installation is considered. Under a suitable cost structure it is shown that, for fixed age of the installation and fixed buffer level, the optimal policy is of control-limit type. When the deterioration process is stationary, an efficient Markov decision algorithm operating on the class of control-limit policies is developed. There is strong numerical evidence that the algorithm converges to the optimal policy. Two generalizations of this model are also discussed.     

On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time

This paper considers the impact of random machine breakdowns on the classical Economic Production Quantity (EPQ) model for a product subject to exponential decay and under a no-resumption (NR) inventory control policy. A product is manufactured in batches on a machine that is subject to random breakdowns in order to meet a constant demand over an infinite planning horizon. The product is assumed to have a significant rate of deterioration and time to deterioration is described by an exponential distribution. Also, the time-to-breakdown is a random variable following an exponential distribution. Under the NR policy, when a breakdown occurs during a production run, the run is immediately aborted. A new run will not be started until all available inventories are depleted. Corrective maintenance of the production system is carried out immediately after a breakdown and it takes a fixed period of time to complete such an activity. The objective is to determine the optimal production uptime that minimizes the expected total cost per unit time consisting of setup, corrective maintenance, inventory carrying, deterioration, and lost sales costs. A near optimal production uptime is derived under conditions of continuous review, deterministic demand, and no shortages.     

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.