Economic production quantity model for items with continuous quality characteristic,rework and reject

The economic production quantity (EPQ) model is a well-known and commonly used inventory control technique. However, the model is built on an unrealistic assumption that all the produced items need to be of perfect quality. Having relaxed this assumption, some researchers have studied the effects of the imperfect products on the inventory control techniques. This article, thus, attempts to develop an EPQ model with continuous quality characteristic and rework. To this end, this study assumes that a produced item follows a general distribution pattern, with its quality being perfect, imperfect or defective. The analysis of the model developed indicates that there is an optimal lot size, which generates minimum total cost. Moreover, the results show that the optimal lot size of the model equals that of the classical EPQ model in case imperfect quality percentage is zero or even close to zero.     

Joint production and maintenance strategy for economic production quantity model with imperfect production processes

In this study, an economic production quantity (EPQ) model is generalized by considering maintenance and production programs for an imperfect process involving a deteriorating production system with increasing hazard rate. There are two types of preventive maintenance (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. Following a failure, the delayed repair performs some restorations and reduces production rate to restore the system into an operating state (in-control state), but leaves its lower production rate until perfect PM is performed. That is, the production run period not always starts in normal production rate. This study considers backorders, as well as loss of inventory due to the lower production rate. 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 ability, repair cost and production decreasing rate on total costs and production period.     

Optimal economic production quantity policy for randomly failing process with minimal repair,backorder and preventive maintenance

This study examines the feasibility of using an economic production quantity (EPQ) model incorporating maintenance and production programs to model an imperfect process involving a deteriorating production system. In response to failure, defective parts were produced and minimal repairs performed to create an in-control state. The conditions are studied in the case of the EPQ model undergoing a backorder owing to rejection of defective parts after a failure. Following production run period, two types of periodic preventive maintenance (PM) exist: imperfect and perfect. The probability of perfect PM being performed depends on the number of imperfect PM performed since the last renewal cycle. For the EPQ model, the optimal run time for minimising the total cost is discussed. Various special cases are considered, including the PM learning effect. Finally, this investigation presents a numerical example to illustrate the effects of PM ability, repair cost and defect number on total costs and production period. This study finds that enhancing maintenance ability reduces production related costs. The product system can be produced more efficiently using a PM program.     

Economic production quantity models for deteriorating/imperfect products and service with rework

In this paper, two economic production quantity (EPQ) models are proposed for deteriorating/imperfect items with rework process. The production process is imperfect, and imperfect quality items are reworked to become serviceable. At the same time, the remaining good quality items may deteriorate. The inspection of deteriorated items is also imperfect, so that deteriorated items may be sold to customers, which will create negative impact on corporate image. In the first model, a single production-rework plant system is considered. The optimal production times and the economic production quantities are obtained analytically. In the second model, a system that consists of n production plants and one rework plant is considered. A solution procedure is developed in order to obtain the optimal operating cost. Numerical examples are provided to compare the performance of the two systems.     

Optimal production run length with imperfect production processes and backorder in fuzzy random environment

This article considers the economic production run time problem with imperfect production processes and allowable shortages. The elapsed time until the production process shifts is assumed to be a fuzzy random variable, and fuzzy random total cost per unit time model is constructed. The expectation theory and signed distance are employed to transform the fuzzy random model into crisp model. An effective approximate algorithm is developed to search for the optimal production run length. Furthermore, numerical examples are provided to illustrate the results of proposed model.     

An optimal production lot-sizing problem for an imperfect process with imperfect maintenance and inspection time length

This article develops an integrated model in considering the situations of an imperfect process with imperfect maintenance and inspection time for the joint determination of both economic production quantity (EPQ) and preventive maintenance (PM). This imperfect process has a general deterioration distribution with increasing hazard rate. Even with periodic PM, such a production system cannot be recovered as good as new. This means that the system condition depends on how long it runs. Also, the PM level can be distinct due to the maintenance cost. For convenience, it is assumed the age of system is reduced in proportional to the PM level. Further, during a production cycle, we need an inspection to see if the process is in control. This inspection might demand a considerable amount of time. In this article, we take PM level and inspection time into consideration to optimise EPQ with two types of out-of-control states. To see how the method works, we use a Weibull shock model to show the optimal solutions for the least costs.     

Economic production quantity with rework process at a single-stage manufacturing system with planned backorders

In traditional inventory models such as the economic order quantity (EOQ) and the economic production quantity (EPQ) the sole objective is to minimize the total inventory-related costs, typically holding cost and ordering cost. These models do not consider the presence of defective products in the lot or rework of them. Recently, Jamal, Sarker, and Mondal (Jamal, A. A. M., Sarker, B. R., & Mondal, S., (2004). Optimal manufacturing batch size with rework process at single-stage production system. Computers and Industrial Engineering, 47(1), 77–89) proposed a model, which dealt with the optimum batch quantity in a single-stage system in which rework is done by addressing two different operational policies to minimize the total system cost, but their models do not consider planned backorders. In this direction, this paper develops an EPQ type inventory model with planned backorders for determining the economic production quantity for a single product, which is manufactured in a single-stage manufacturing system that generates imperfect quality products, and all these defective products are reworked in the same cycle. We also establish the range of real values of the proportion of defective products for which there is an optimal solution, and the close form for the total cost of inventory system. The use of the inventory model is illustrated with numerical examples. The classical EOQ, EPQ inventory models with or without planned backorders and Jamal, Sarker and Mondal’s model (Jamal, A. A. M., Sarker, B. R., & Mondal, S., (2004). Optimal manufacturing batch size with rework process at single-stage production system. Computers and Industrial Engineering, 47(1), 77–89) are shown to be special cases of the EPQ inventory model presented in this paper.     

The machine breakdown paradox: How random shifts in the production rate may increase company profits

The causes and effects of machine breakdowns have frequently been investigated in the past. One popular stream of research studies technical errors in production and analyzes their impact on the inventory policy of the company. In this paper, we show that random shifts in the production rate of a machine, which may occur, for example, due to technical defects, may lead to a reduction in total cost and therewith to an increase in profit. This obvious paradox may lead to situations where it is economically rational for the company to sustain a technically inefficient situation, or even to take measures to intentionally induce a shift in the production rate, for example by damaging the machine on purpose. In this paper, we illustrate this paradox by referring to an existing inventory model, and trace it back to common assumptions made in the literature.     

A mathematical model on EPQ for stochastic demand in an imperfect production system

In the paper, we develop an EPQ (economic production quantity) inventory model to determine the optimal buffer inventory for stochastic demand in the market during preventive maintenance or repair of a manufacturing facility with an EPQ (economic production quantity) model in an imperfect production system. Preventive maintenance, an essential element of the just-in-time structure, may cause shortage which is reduced by buffer inventory. The products are sold with the free minimal repair warranty (FRW) policy. The production system may undergo “out-of-control” state from “in-control” state, after a certain time that follows a probability density function. The defective (non-conforming) items in “in-control” or “out-of-control” state are reworked at a cost just after the regular production time. Finally, an expected cost function regarding the inventory cost, unit production cost, preventive maintenance cost and shortage cost is minimized analytically. We develop another case where the buffer inventory as well as the production rate are decision variables and the expected unit cost considering the above cost functions is optimized also. The numerical examples are provided to illustrate the behaviour and application of the model. Sensitivity analysis of the model with respect to key parameters of the system is carried out.     

Economic production quantity for profit maximization

In certain circumstances a relationship between unit contribution and producing output may exist in the classical economic production quantity (EPQ) production. We discuss an EPQ model with unit contribution dependent on production output and formulate the problem as signomial geometric problem (SPG) We show that a closed form optimal solution is readily obtainable for surch an SGP problem. We also illustrate that it is simple and easy to perform certain types of post-optimality sensitivity analysis on the optimal solution. An example is worked out in detail to demonstrate the viability of using geometric programming as an efficient solution method for dealing with a certain class of inventory optimization problems.     

Combining an alternative multi-delivery policy into economic production lot size problem with partial rework

This paper combines an alternative multi-delivery policy into an imperfect economic production quantity (EPQ) model with partial rework, with the purpose of reducing supplier’s stock holding cost. We extend the problem examined by Chiu et al. [Chiu, Y.-S. P, Chiu, S. W., Li, C.-Y., & Ting, C.-K. (2009). Incorporating multi-delivery policy and quality assurance into economic production lot size problem. Journal of Scientific and Industrial Research, 68(6), 505-512] by considering an n + 1 delivery policy in lieu of n multi-delivery plan for the specific EPQ model with partial rework. Under such a policy, an initial delivery of perfect (finished) items is distributed to customer for satisfying product demand during manufacturer’s regular production time and rework time. At the end of rework, fixed quantity n installments of the finished products are delivered to customer at a fixed interval of time. As a result, a closed-form optimal replenishment batch size solution to the problem is obtained. A numerical example with analysis and comparison is provided to show practical usage of the proposed model and demonstrate its significant savings in stock holding cost.     

考虑缺货的模糊随机缺陷生产系统

针对不可靠的生产过程,研究了生产故障时间为模糊随机变量且允许缺货的缺陷生产系统.建立含缺货费和模糊随机重修费的经济生产批量模型.基于可信性理论,建立其期望费用模型,揭示了费用函数的性质,并证明了使费用最小的最优生产时间的存在性和唯一性,从而确定了最优生产时间的上下界.基于此,设计了最优生产时间的二分法求解过程.最后通过算例验证了所提出模型的有效性,并分析了缺货费用、重修费用和缺陷产品比例对最优生产策略的影响.     

An economic production quantity model with random defective rate,rework process and backorders for a single stage production system

This paper revisits the economic production quantity (EPQ) model with rework process at a single-stage manufacturing system with planned backorders. It is well known that any imperfect production system of real life has random defective rates. In this direction, this paper extends an inventory model to allow random defective rates. Basically, three different inventory models are developed for three different distribution density functions such as uniform, triangular, and beta. The analytical derivation provides closed-form solution for each inventory model. We have made comparison tables of optimal results among the distribution functions. Some numerical examples and sensitivity analysis are given to illustrate the inventory models.     

An EMQ model in an imperfect production process

The intention of this article is to develop a framework of production policy (resumption and non-resumption) in order to find out optimal safety stock, optimal production rate and production lot size. It encompasses specific versions of the concept of quality and inventory model, stochastic machine breakdown and its correcting and regular repair paths with safety stocks. This framework hopefully serves to simplify answers to the important questions: How much safety stocks, production rate and production lot size are required to minimise the total expected system cost. The optimal production rate, production lot size, production run time and safety stocks are determined numerically and the joint effect of process deterioration, machine breakdown and its repair (correcting and preventive) on the optimal decision is investigated for a numerical example. Such an investigation should also yield logistics directions for the design of products and their manufacturing processes.     

A continuous review production–inventory system in fuzzy random environment: Minmax distribution free procedure

In many manufacturing systems, the production process may take some time to start the initial phase due to various reasons such as delay in installation of machines, short supply of raw materials, unavailability of workers, etc. Thus, the organization should plan accordingly so that the manufacturing process can start at the desired time. In an economic production quantity (EPQ) model, lead-time plays a significant role in ensuring that the manufacturing process starts on time. As we know, when both lead-time and demand rate are deterministic and constant, then demand during the lead-time is constant, and is referred to as zero lead-time. Moreover, when either or both of them are random variables, then lead-time demand (LTD) is a random variable. In such a case, a crucial question is: “when should the order be placed?” On the other hand, the distributional information on demand may not always be available or there may be many distribution functions in the practice, which have same mean and variance, but their frequencies are different. In this study, we develop an EPQ model in stochastic framework, wherein the distribution function of demand is unknown, but the mean and variance are known. The inventory level is continuously reviewed, and an order is placed when it reaches the reorder level. The real-life business situations are so sophisticated and floating in nature that the consideration of ‘impreciseness’ along with ‘statistical variability’ in demand parameter is more preferable. To be a part of this contingency, we further extend the model in the fuzzy random environment by considering demand rate as a fuzzy random variable (FRV). Furthermore, we mathematically analyze the cost function and propose a heuristic procedure to find the global optimum. Numerical examples with sensitivity analysis are also provided for illustration purpose.     

Optimal replenishment decision in an EPQ model with defective items under supply chain trade credit policy

The traditional inventory of the economic order quantity model assumes perfect items in an ordered lot and an infinite replenishment rate. However, such conditions are rare in actual production environments. Additionally, most studies of this problem have only considered suppliers offering the wholesaler a grace period. In practice, wholesalers often extend a fixed credit period to downstream customers as well. This study therefore proposes a production model for a lot-size inventory system with finite production rate and defective quality under the condition of two-level trade credit policy and the condition that defective items involve both imperfect quality and scrap items. Thus, optimal wholesaler replenishment decisions can be determined for defective items under two-level trade credit policy in the EPQ framework. Four theorems for determining the optimal cycle time and the results in this study can be deduced as a special case of earlier models. Finally, illustrative examples are provided to verify the theoretical results.     

Economic production quantity models for deteriorating items with price- and stock-dependent demand

Large piles of consumer goods displayed in a supermarket are often associated with on sale items to induce more sales and profits. In this paper, we first establish an economic production quantity (or EPQ) model for deteriorating items when the demand rate depends not only the on-display stock level but also the selling price per unit. In addition, we impose a ceiling on the number of on-display stocks because too much stock leaves a negative impression on the buyer and the amount of shelf/display space is limited. We then provide the necessary conditions to determine an optimal solution that maximizes profits for the EPQ model. Finally, sensitivity analysis is applied on the parameter effects of the optimal price and production run time.     

A study on negative binomial inspection for imperfect production systems

Production systems continuously deteriorate with age and usage due to corrosion, fatigue, and cumulative wear in production processes, resulting in an increasing possibility of producing defective products. To prevent selling defective products, inspection is usually carried out to ensure that the performance of a sold product satisfies the customer requirements. Nevertheless, some defective products may still be sold in practice. In such a case, warranties are essential in marketing products and can improve the unfavorable image by applying higher product quality and better customer service. The purpose of this paper is to provide manufacturers with an effective inspection strategy in which the task of quality management is performed under the considerations of related costs for production, sampling, inventory, and warranty. A Weibull power law process is used to describe the imperfection of the production system, and a negative binomial sampling is adopted to learn the operational states of the production process. A free replacement warranty policy is assumed in this paper, and the reworking of defective products before shipment is also discussed. A numerical application is employed to demonstrate the usefulness of the proposed approach, and sensitivity analyses are performed to study the various effects of some influential factors.     

A joint design of production run length,maintenance policy and control chart with multiple assignable causes

Although economic production quantity, statistical process monitoring and maintenance are three major concepts in process optimization of industrial environments, they have been often investigated separately in literature. Furthermore, in studies that consider these three concepts simultaneously, it is assumed that there is only one assignable cause in the production process. This simplified assumption is unlikely to occur in real production processes due to the usual complexity of manufacturing systems, which may lead to a poor performance in both economic and statistical criteria if the assignable cause originating the shift is different from the one anticipated at the design of the chart. To overcome these drawbacks, this paper develops an integrated model ofeconomic production quantity, statistical process monitoring and maintenance in the presence ofmultiple assignable causes. The particle swarm optimization algorithm is used to minimize the expected total cost per production cycle, subject to statistical quality constraints. Also, a comparative study is performed to illustrate the effect of considering multiple assignable causes on model’s costs. Finally, a sensitivity analysis is conducted on the expected total cost per production cycle and process variable values to extend insights into the matter.     

An economic production quantity model for a deteriorating system integrated with predictive maintenance strategy

While there has been considerable work over the years on the basic deterministic economic production quantity (EPQ) and its derivative models, there have been few extensions of these models that recognize the potential effects of machine degradation. As maintenance activities can keep machines in good operation, it should be integrated into EPQ models to meet real situations. Due to machine degradation, this paper integrates predictive maintenance into EPQ model in which autoregressive integrated moving average model is adopted to predict system’s healthy indicator. Moreover, two kinds of system out-of-control states are considered in this proposed EPQ model: in State I, the system produces non-conforming items; and in State II, the system fails. Aiming at minimizing the expected average total cost and optimizing the EPQ, suitable maintenance intervals and frequency are determined prior to any predicted failure. Finally, a case study is presented and the computational results are discussed to show the efficiency of this integrated EPQ model.