Optimal policy for set-up time reduction in a multistage production inventory system

A batch production-inventory system consisting of multiple stages with an optimal policy of set-up time reduction and a fixed increment cost are discussed. The ratio of set-up time reduction as a decision variable under various cases of demand in the batch production-inventory model is considered. The ratio of set-up time reduction and lot size are solved simultaneously to obtain an optimal value of the total annual cost. A numerical example is presented to demonstrate the accuracy of the proposed method.     

Lot size reorder point inventory model with controllable lead time and set-up cost

This paper deals with the lead time and set-up cost reductions problem on the modified lot size reorder point inventory model in which the production process is imperfect. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of lead time to be reduced and the ordering lot size. The option of investing in reducing set-up cost is also included. Two commonly used investment cost functional forms, logarithmic and power, are employed for set-up cost reduction. We assume that the stochastic demand during lead time follows a Normal distribution. The objective is simultaneously to optimize the lot size, reorder point, set-up cost and lead time. An algorithm of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.     

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.     

Lot-sizing in a class of multistage production inventory systems

In this paper, a mathematical model for a multistage multifacility and multiproduct production inventory system is presented. The basic objective of the model is to determine the economic production quantity (EPQ) for each product-stage-facility combination by minimizing total system cost. The total system cost consists of the set-up cost and the in-process inventory carrying cost. A numerical example is solved to explain the model; a search method has been used for the optimization. The results obtained are compared with the simulation results to validate the mathematical model.     

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.     

Multiproduct production/inventory control under random demands

Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the “norm-based” approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time     

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.     

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.     

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.     

An economic production quantity model for deteriorating items with preventive maintenance policy and random machine breakdown

In recent years, many researches on economic production quantity (EPQ) models with machine breakdown and preventive maintenance have been developed, but few of them have developed integrated models for deteriorating items. In this study, we develop EPQ models for deteriorating items with preventive maintenance, random machine breakdown and immediate corrective action. Corrective and preventive maintenance times are assumed to be stochastic and the unfulfilled demands are lost sales. Two EPQ models of uniform distribution and exponential distribution of corrective and maintenance times are developed. An example and sensitivity analysis is given to illustrate the models. For the exponential distribution model, it is shown that the corrective time parameter is one of the most sensitive parameters to the optimal total cost.     

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.     

Maximising profits for an EPQ model with unreliable machine and rework of random defective items

This article deals with an economic production quantity (EPQ) model in an imperfect production system. The production system may undergo in ‘out-of-control’ state from ‘in-control’ state, after a certain time that follows a probability density function. The density function varies with reliability of the machinery system that may be controlled by new technologies, investing more costs. The defective items produced in ‘out-of-control’ state are reworked at a cost just after the regular production time. Occurrence of the ‘out-of-control’ state during or after regular production-run time is analysed and also graphically illustrated separately. Finally, an expected profit function regarding the inventory cost, unit production cost and selling price is maximised analytically. Sensitivity analysis of the model with respect to key parameters of the system is carried out. Two numerical examples are considered to test the model and one of them is illustrated graphically.     

含模糊修复速率的生产库存模型

研究了缺陷产品可修复且修复速率为模糊变量情况下的生产库存模型;给出了基于模糊结构元理论的求解模糊库存总成本最小化的方法步骤;得到了含有模糊结构元的最佳生产批量的表达式;应用Hessian矩阵的正定性验证了最小成本的存在性;分析了其与经典EPQ公式之间的联系;最后通过具体算例,证明了模型求解方法的可行性。     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

Economic production quantity (EPQ) models under an imperfect production process with shortages backordered

In this paper, we develop economic production quantity (EPQ) models to determine the optimal production lot size and backorder quantity for a manufacturer under an imperfect production process. The imperfect production process is characterised by the fraction of defective items at the time of production γ. The paper considers different cases of the EPQ model depending on (1) whether γ is known with certainty or is a random variable, and (2) whether imperfect items are drawn from inventory (a) as they are detected, (b) at the end of each production period or (c) at the end of each production cycle. Straightforward convexity results are shown and closed-form solutions are provided for the optimal order and backorder quantities for each of the cases we considered. We provide two numerical examples: one in which the defective probability follows a uniform distribution and the second which we assume follows a beta distribution, to illustrate the effects of yield variability and timing of the withdrawal of defectives on the optimal solutions. We obtain similar results for both numerical examples, which show that both the yield variability and the withdrawal timing are not critical factors.     

Alternative approach for solving replenishment lot size problem with discontinuous issuing policy and rework

Conventional approach for solving the replenishment lot size problem is by using differential calculus on the long-run average production cost function with the need to prove optimality first. Recent studies proposed an algebraic approach to the solution of classic economic order quantity (EOQ) and the economic production quantity (EPQ) models without reference to the use of derivatives. This paper extends it to the solution of a specific EPQ model as was examined by Chiu et al. [Chiu, S. W, Chen, K. -K, Lin, H. -D. Numerical method for determination of the optimal lot size for amanufacturing system with discontinuous issuing policy and rework. International Journal forNumerical Methods in Biomedical Engineering. doi: 10.1002/cnm.1369. (in Press; Published online March-10-2010).]. As a result, optimal replenishment lot size and a simplified optimal production-inventory cost formula for such a particular EPQ model can be derived without derivatives. This alternative approach may enable practitioners who with little knowledge of calculus to understand the realistic production systems with ease.     

A study for optional inspection policies in a flexible manufacturing cell

The objective of the paper is to determine the location of inspection stages in order to minimize the expected total manufacturing cost while satisfying the high-quality product demand of customers in a flexible manufacturing cell (FMC). This allows a designer to examine alternative designs in order to select those which are optimal, or near optimal, in manufacturing cost terms. To accomplish this objective, we have formulated a total manufacturing decision model consisting of machine set-up and processing costs. Inspection set-up and inspection costs, and cost due to defective. A case study and sensitivity analysis are presented to demonstrate the application of the decision model developed.     

A production inventory model with fuzzy random demand and with flexibility and reliability considerations

The classical inventory control models assume that items are produced by perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be increased without a price, its set-up cost can be reduced with investment in flexibility improvement. In this paper, a production inventory model with flexibility and reliability (of production process) consideration is developed in an imprecise and uncertain mixed environment. The aim of this paper is to introduce demand as a fuzzy random variable in an imperfect production process. Here, the set-up cost and the reliability of the production process along with the production period are the decision variables. Due to fuzzy-randomness of the demand, expected average profit of the model is a fuzzy quantity and its graded mean integration value (GMIV) is optimized using unconstraint signomial geometric programming to determine optimal decision for the decision maker (DM). A numerical example has been considered to illustrate the model.     

A heuristic for determination of optimal capacity of central storage

A system of one machine used to produce m products in batches and a central storage used for storing raw materials and finished products is considered. The system maintains enough finished products to assure no stockout will happen. The machine has a finite production rate greater than or equal to the demand rate for each product, and thus operates with periodic start-ups and shut-downs. For special case when set-up for each product requires time and every product is produced in every cycle, the best policy given limited storage space is determined, and total production cost as a function of set-up costs, carrying cost, storage cost, and material handling cost is presented. A heuristic is developed that determines “optimal feasible cycle time” such that production scheduling remains feasible, the capacity of the central storage is not exceeded, and the overall costs are minimized.     

Combining genetic approach and integer programming to solve multi-facility economic lot-scheduling problem

Economic lot-scheduling problem (ELSP) has been studied since the 1950??s. ELSP deals with the scheduling of the production of several items on a single facility in a cyclical pattern. The facility can only produce one single item at a time, and there is a set-up cost and set-up time associated with each item. Because of the rapid development of many emerging markets nowadays, many common items are produced in different places in order to satisfy the demands in different markets. This becomes the multi-facilities ELSP problems. In ELSP problems, it is known that if more items types to be produced by the facility, the production frequency of each item type will increase because of the balancing of the production rate and the demand rate. Consequently, the number of set-up time and set-up cost increases accordingly. Thus, reallocating the common items, which can be produced in any facilities, to be produced only on certain facility can certainly reduce the number of production frequency, and lead to lower related costs. The objective of this paper is to propose an optimization methodology combining Integer Programming and Genetic Algorithm to solve multi-facility ELSP problems. This paper proposes to divide the main problem into a master problem and sub-problems, which are solved by Integer Programming and Genetic Algorithm respectively. To demonstrate the significance of reallocating the common items and aggregating them to produce in certain facility, several models have been designed and tested. The comparison of the models demonstrates the reduction of the costs benefited by result of common items reallocation.