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.     

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.     

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.     

Economic production quantity model with a shifting production rate

Typical models for determining the economic production quantity (EPQ) assume perfect product quality and perfect production processes. Deteriorating processes may affect production systems in several ways. They may decrease the quality of the items produced, cause production stoppage and breakdowns and/or reduce the production rate due to production process inefficiency. The purpose of this paper is to present an EPQ model that incorporates the effect of shifts in production rate on lot sizing decisions due to speed losses. The cycle starts with a certain production rate and after a random time, the production rate shifts to a lower value. A mathematical model to determine the optimal production policy under these conditions is developed and analyzed. Numerical examples are presented for illustrative purposes.     

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.     

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.     

An optimal production and inspection strategy with preventive maintenance error and rework

This study considers the integrated problem of production, preventive maintenance (PM), inspection, and inventory for an imperfect production process where rework and PM error exist. PM is performed when the process is in a controlled state. The correct implementation of PM results in a lower system failure rate, whereas a PM error results in the system shifting to the out-of-control state with a certain probability. The age of the system after PM is correlated with the level of PM performed. When the process in an out-of-control state produces a certain percentage of non-conforming items, we assume that a certain proportion of the non-conforming items can be reworked into conforming items. In a deteriorating production system, we determine the optimal inspection interval, inspection frequency, and production quantity that will yield the maximal unit expected profit. Numerical analyses are used to investigate the effectiveness of imperfect PM and to explore the effect of rework and PM error on profit.     

Economic lot sizing in a production system with random demand

An extended economic production quantity model that copes with random demand is developed in this paper. A unique feature of the proposed study is the consideration of transient shortage during the production stage, which has not been explicitly analysed in existing literature. The considered costs include set-up cost for the batch production, inventory carrying cost during the production and depletion stages in one replenishment cycle, and shortage cost when demand cannot be satisfied from the shop floor immediately. Based on renewal reward process, a per-unit-time expected cost model is developed and analysed. Under some mild condition, it can be shown that the approximate cost function is convex. Computational experiments have demonstrated that the average reduction in total cost is significant when the proposed lot sizing policy is compared with those with deterministic demand.     

Supply chain coordination with defective items and quantity discount

This study develops an integrated inventory system involving defective items and quantity discount for optimal pricing and ordering strategies. The model analysed in this study is one in which the buyer orders a quantity, the vendor produces more than buyer's order quantity in order to reduce set-up cost, and then he/she offers an all-units quantity discount to the buyer. Our objective is to determine the optimal order quantity, retail price, mark-up rate, and the number of shipments per production run from the vendor to the buyer, so that the entire supply chain joint total profit incurred has a maximum value. Furthermore, an algorithm of finding the optimal solution is developed. Numerical examples are provided to illustrate the theoretical results.     

Multi-product production quantity model with repair failure and partial backordering

In this paper, a production quantity model with random defective items, service level constraints and repair failure is studied. The existence of only one machine results in limited production capacity and partial backordering. The aim of this research is to determine the optimal cycle length, optimal production quantity and optimal backordered quantity of each product such that the expected total cost (holding, shortage, production, setup, defective items and repair costs) is minimized. Two numerical examples and sensitivity analysis are provided to illustrate the practical usage of the proposed method.     

A three-echelon supply chain coordination with quantity discounts for multiple items

This study derives an integrated model for three-echelon supply chain (i.e. single-supplier, single-vendor and multi-retailer) to minimise the joint costs of all participants associated with holding costs, ordering costs, and purchasing costs. We particularly emphasise that quantity discounts and joint replenishment policies can be utilised as coordination mechanisms among all parties. In addition, a linearisation approach and mixed integer method are developed not only to obtain an executable integer solution for a finite planning horizon problem but also to simplify the conventional multiple-step procedure used in the quantity discount. The result indicates that the coordination and consolidation of orders can lead to the total cost reduction. Finally, a numerical example is given to illustrate the benefits of orders coordination and consolidation.     

Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP,SFS, SA,and WCA

A multi-product economic production quantity model with several real-world technical and physical constraints is developed in this paper. The cost function includes ordering, holding, backordering, lost sale, and the cost caused by unused space in the warehouse. The goal is to minimize the total inventory cost, where shortages are allowed and partially backordered with fixed and linear costs. The aim is to determine the length of the inventory cycle, the length of positive inventory period, and the backordering rates of the products during the shortage period in order to minimize the total inventory costs while satisfying all constraints. Due to complexity and non-linearity of the proposed model, sequential quadratic programming (SQP), stochastic fractal search (SFS), simulated annealing (SA), and water cycle algorithm (WCA) are utilized for solution. Ninety numerical examples in small, medium, and large sizes are solved to evaluate the efficiency of the solution methods. The performances of the solution methods are compared statistically. Besides, sensitivity analysis is performed to determine the effect of change in the main parameters of the problem on the objective function value and decision variables.     

An economic order quantity model of imperfect quality items with partial backlogging

This article deals with an economic order quantity (EOQ) model in which a certain percentage of a lotsize is of imperfect quality products. This percentage follows a uniform distribution function. During the inspection of the total lot-size, a stock-out situation may occur. In a stock-out situation, a partial fraction of the demand is adjusted by partial back ordering and the rest of the demand is considered as a case of lost sales. The associated integrated expected profit function is generalised for the general distribution function of imperfect quality products. Then the objective function is maximised. Also, three special cases of the general model are studied. A suitable numerical example is provided to illustrate the model and the solution procedure. Comparison between the general and special cases are also shown with the help of numerical examples. Sensitivity analysis of the optimal solutions with respect to all individual parameters of the general model is carried out.     

Economic optimization for an off-line inspection, disposition and rework model

An economic off-line inspection, disposition, and rework (IDR) model for a batch produced from an unreliable production system was recently proposed by Wang et al. [Economic optimization of off-line inspection with rework consideration, European Journal of Operational Research, 194 (2009) 807–813], where the process is assumed to possess a discrete general shift distribution. Unfortunately, there are some important flaws in the proposed IDR model; in particular, while obtaining optimal IDR policy, they incorrectly assumed that the process shift distribution had the memoryless property. As a result, the purpose of this paper is to reformulate the IDR model, and to develop a solution procedure to find the optimal IDR policy.     

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.     

Optimum process mean and manufacturing quantity settings for serial production system under the quality loss and rectifying inspection plan

In 1994, Al-Sultan presented a single sampling plan applied in determining the optimum process mean for two machines in a serial production system. However, Al-Sultan did not consider the quality cost for the product within the specification limits, pointed out that the non-conforming items in the sample of accepted lot is replaced or eliminated from the lot, and proposed an integrated model with production and quality. In this study, the author considers the problem of quality loss for the modified Al-Sultan’s model with k machines in a serial production system based on a single sampling rectifying inspection plan. Taguchi’s symmetric quadratic quality loss function is applied in evaluating the product quality. Then, the author further proposes a modified and integrated economic manufacturing quantity (EMQ) model based on the application of the modified Al-Sultan’s model for obtaining the maximum expected total profit of product per unit of time. The numerical results show that the price of an accepted products sold has the most important effect on both the process means and the expected total profit per unit of time.     

Optimal order policy in response to announced price increase for deteriorating items with limited special order quantity

When a supplier announces an impending price increase due to take effect at a certain time in the future, it is important for each retailer to decide whether to purchase additional stock to take advantage of the present lower price. This study explores the possible effects of price increases on a retailer's replenishment policy when the special order quantity is limited and the rate of deterioration of the goods is assumed to be constant. The two situations discussed in this study are as follows: (1) when the special order time coincides with the retailer's replenishment time and (2) when the special order time occurs during the retailer's sales period. By analysing the total cost savings between special and regular orders during the depletion time of the special order quantity, the optimal order policy for each situation can be determined. We provide several numerical examples to illustrate the theories in practice. Additionally, we conduct a sensitivity analysis on the optimal solution with respect to the main parameters.     

Optimizing a bi-objective multi-product EPQ model with defective items,rework and limited orders: NSGA-II and MOPSO algorithms

In this paper, a bi-objective multi-products economic production quantity (EPQ) model is developed, in which the number of orders is limited and imperfect items that are re-workable are produced. The objectives of the problem are minimization of the total inventory costs as well as minimizing the required warehouse space. The model is shown to be of a bi-objective nonlinear programming type, and in order to solve it two meta-heuristic algorithms namely, the non-dominated sorting genetic algorithm (NSGA-II) and multi-objective particle swarm optimization (MOPSO) algorithm, are proposed. To verify the solution obtained and to evaluate the performance of proposed algorithms, two-sample t-tests are employed to compare the means of the first objective value, the means of the second objective values, and the mean required CPU time of solving the problem using two algorithms. The results show while both algorithms are efficient to solve the model and the solution qualities of the two algorithms do not differ significantly, the computational CPU time of MOPSO is considerably lower than that of NSGA-II.     

Optimization of the finite production rate model with scrap,rework and stochastic machine breakdown

This study employs mathematical modeling along with a recursive searching algorithm to determine the optimal run time for an imperfect finite production rate model with scrap, rework, and stochastic machine breakdown. In real-life manufacturing systems, generation of defective items and machine breakdown are inevitable. The objective of this paper is to address these issues and to be able to derive the optimal production run time. It is assumed that the proposed manufacturing system produces defective items randomly, a portion of them is considered to be scrap, and the other portion can be repaired through rework. Further, the proposed system is subject to random breakdown and when it occurs, the abort/resume (AR) policy is adopted. Under such an inventory control policy, the production of the interrupted lot will be resumed immediately when machine is fixed and restored. Mathematical modeling along with a recursive searching algorithm is used for deriving the replenishment policy for such a realistic production system.