Optimal production cycles, procurement schedules, and joint investment in an imperfect production system

In this paper, we consider the simultaneous determination of production cycles for the end product, procurement schedules for its input materials, and joint investment in setup reduction and process quality improvement for a production system with imperfect production processes. In the analysis, we assume that setup reduction and process quality are functions of capital expenditure and that the input materials, which are purchased from outside suppliers, are gradually converted into the product during manufacture. We derive a solution procedure to find the optimal production cycle time, procurement schedules, joint investment, and the corresponding total relevant cost. We present numerical examples to illustrate the procedure and to delineate the relationships among production cycle times for the end product, the procurement schedules for its input materials, and setup reduction and quality improvement.     

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.     

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 economic production lot size model in an imperfect production system

The paper investigates an EPL (Economic Production Lotsize) model in an imperfect production system in which the production facility may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time. The basic assumption of the classical EPL model is that 100% of produced items are perfect quality. This assumption may not be valid for most of the production environments. More specifically, the paper extends the article of Khouja and Mehrez [Khouja, M., Mehrez, A., 1994. An economic production lot size model with imperfect quality and variable production rate. Journal of the Operational Research Society 45, 1405–1417]. Generally, the manufacturing process is ‘in-control’ state at the starting of the production and produced items are of conforming quality. In long-run process, the process shifts from the ‘in-control’ state to the ‘out-of-control’ state after certain time due to higher production rate and production-run-time.The proposed model is formulated assuming that a certain percent of total product is defective (imperfect), in ‘out-of-control’ state. This percentage also varies with production rate and production-run time. The defective items are restored in original quality by reworked at some costs to maintain the quality of products in a competitive market. The production cost per unit item is convex function of production rate. The total costs in this investment model include manufacturing cost, setup cost, holding cost and reworking cost of imperfect quality products. The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.     

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.     

EPQ with Process Capability and Quality Assurance Considerations

The classical economic production quantity (EPQ) model assumes that items produced are of perfect quality and that the unit cost of production is fixed. However, in realistic situations, product quality is never perfect but is directly affected by the production processes and the quality assurance programme. In addition, the unit production cost is not fixed but increases with quality assurance expenses. We present an EPQ model with imperfect production processes and quality-dependent unit production cost. After discussion of the procedure for determining the optimal solution, a sensitivity analysis of the impacts of the cost parameters on the optimal solution is provided. Finally, the problems associated with cost estimation are addressed.     

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)     

The impact of inspection errors,imperfect maintenance and minimal repairs on an imperfect production system

This paper develops an integrated model of production lot-sizing, maintenance and quality for considering the possibilities of inspection errors, preventive maintenance (PM) errors and minimal repairs for an imperfect production system with increasing hazard rates. In this study, a PM activity is imperfect in that a production system cannot be recovered as good as new and might cause the production system to shift to the out-of-control state with a certain probability. Numerical analyses are used to simulate the effect of changes in various parameters on the optimal solution for which the time that the process remains in the in-control state is assumed to follow a Weibull distribution. In addition, we investigate the effects of inspection errors and PM errors on the minimum total cost of the optimal inspection interval, inspection frequency and production quantity.     

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.     

An inventory model with reliability in an imperfect production process

The paper analyzes an economic manufacturing quantity (EMQ) model with price and advertising demand pattern in an imperfect production process under the effect of inflation. If the machine goes through a long-run process, it may shift from in-control state to out-of-control state. As a result, the system produces imperfect items. The imperfect items are reworked at a cost to make it as new. The production of imperfect quality items increases with time. To reduce the production of the imperfect items, the systems have to more reliable and the produced items depend on the reliability of the machinery system. In this direction, the author considers that the development cost, production cost, material cost are dependent on reliability parameter. Considering reliability as a decision variable, the author constructs an integrated profit function which is maximized by control theory. A numerical example along with graphical representation and sensitivity analysis are provided to illustrate the model.     

An economic order quantity with imperfect quality and quantity discounts

Previous studies in the issue of inventory models with imperfect quality assumed the defectives could be sold in a batch by the end of the inspection process and the manufacturing systems were push systems. However, the above assumptions may not be true in the pull system in which buyer is powerful. Therefore, in this paper, we develop a new inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier. Based on the concept of powerful buyer, there are three considerations included in this new model: (1) the order quantity is manufactured at one setup and is shipped over multiple deliveries, (2) the defectives are screened out by a 100% inspection for each shipment but sold in a batch by the end of inspection at the last shipment of each cycle, and (3) the supplier offers quantity discounts to response the request of the powerful buyer. Further, an algorithm is developed to help the powerful buyer to determine the optimal order policy accurately and quickly. Two numerical examples are available in this paper to illustrate the proposed model and algorithm. Besides, based on the numerical examples, a sensitivity analysis is made to investigate the effects of four important parameters (the inspection rate, the defective rate, the receiving cost, and the ordering cost) on the optimal solution.     

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.     

Random fuzzy EOQ model with imperfect quality items

This paper investigates an economic order quantity (EOQ) problem with imperfect quality items, where the percentage of imperfect quality items in each lot is characterized as a random fuzzy variable while the setup cost per lot, the holding cost of each unit item per day, and the inspection cost of each unit item are characterized as fuzzy variables, respectively. In order to maximize the expected long-run average profit, a random fuzzy EOQ model is constructed. Since it is almost impossible to find an analytic method to solve the proposed model, a particle swarm optimization (PSO) algorithm based on the random fuzzy simulation is designed. Finally, the effectiveness of the designed algorithm is illustrated by a numerical example.     

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.     

An integrated inventory model with quality improvement and two-part credit policy

Trade credit financing plays a vital role in current business operations. Vendors extend payment dates to encourage sales, and buyers are not required to pay immediately after receiving products. This is equivalent to receiving a price reduction. Besides, buyers reduce their need for capital from bank loans. In addition, a number of defective products are produced during the production process. The number of defective items influences the on-hand inventory levels of buyers, service levels, and frequency of orders. To ensure that the analysis incorporates a realistic production environment, we developed an integrated inventory model with a two-part trade credit and considered an imperfect production process that can be improved by capital investment. The objective was to determine the optimal ordering, shipping, and quality improvement policies to maximize joint total profit. An iterative algorithm was established to determine the optimal strategy. Furthermore, a sensitivity analysis was conducted to examine the effects of changing parameter values on the optimal solution.     

Process mean determination under constant raw material supply

Setting the mean (target value) for a production process is an important decision for a producer when material cost is a significant portion of production cost. Because the process mean determines the process conforming rate, it affects other production decisions, including, in particular, production setup and raw material procurement policies. In this paper, we consider the situation in which the product of interest is assumed to have a lower specification limit, and the items that do not conform to the specification limit are scrapped with no salvage value. The production cost of an item is a linear function of the amount of the raw material used in producing the item, and the supply rate of the raw material is finite and constant. Furthermore, it is assumed that quantity discounts are available in the raw material cost and that the discounts are determined by the supply rate. Two types of discounts are considered in this paper: incremental quantity discounts and all-unit quantity discounts. A two-echelon model is formulated for a single-product production process to incorporate the issues associated with production setup and raw material procurement into the classical process mean problem. Efficient solution algorithms are developed for finding the optimal solutions of the model.     

考虑中间库存缓冲区的多目标设备不完美预防维修策略研究

针对考虑库存缓冲区的多目标设备维修问题,以设备维修能力为约束条件,获得随机故障设备的不完美预防维修策略。首先,利用准更新过程,表示出设备的随机故障次数。其次,结合设备故障次数表达式,以最大设备可用度和最小生产总成本为多目标构建不完美预防维修模型,使用粒子群算法求解,优化设备可用度与生产总成本,获得更新周期内的库存量和预防维修周期两个决策变量的最优值。最后,通过算例分析,验证了多目标不完美预防维修模型的可用性。     

Inventory and investment in setup and quality operations under Return On Investment maximization

In this paper, we construct and analyze a Return On Investment (ROI) maximization model for inventory and capital investment in setup and quality operations under an investment budget constraint. Specifically, we show how such an ROI maximization model can be formulated and derive analytical results such as the conditions under which the inventory is reduced and for the determination of the unique global optimal solution. In addition, by applying the Reformulation-Linearization Technique (RLT), we show via numerical examples how this nonconvex optimization model can be solved effectively and how RLT may produce superior results to those from the conventional Cut Across the Board Rule (CABR). Various managerial insights are provided throughout the paper. For example, as the investment budget increases (or decreases), a fundamental shift of investment strategies (setup cost reduction vs. quality improvement) may be necessary so as to maximize ROI.     

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.     

多阶段生产系统最优生产批量和原材料订购决策模型

Sarker和Parija(1996)建立了生产系统最优生产批量和原材料订购决策模型。然而他们的模型仅局限于单阶段生产系统,本文将他们的模型扩展到多阶段生产系统,我们首先建立了使整个多阶段生产系统总成本最小的各阶段最优生产批量、原材料订购批量及阶段之间的运输批量模型,然后分析了原材料订购费、半成品运费及设备安装费的敏感性。最后,我们结合实例综合分析了原材料订购费、半成品运输费和设备安装费的变化及最小值点取整后对原材料订购决策、最优生产批量和总成本的影响。