Percentage Of Defective Items Research Articles

A Stochastic Integrated Inventory Model Single Supplier-Single Retailer in Periodic Review with Losing Flexibility Cost

Efficiency in the supply chain can be established by integrating the supplier-retailer inventory policy. This article proposes the integrated inventory model between supplier-retailer under stochastic demand. This model aims to determine the optimal review period and calculate the total inventory cost, including some defective items, backorder price discounts, and losing flexibility costs. We assume that the retailer can order 'n' times for every 'm' shipment from supplier to retailer in each production cycle under a periodic review. Stochastic conditions can cause sudden changes in orders by the retailer in large quantities, eventually forcing the supplier to reduce their setup policies. This condition makes the retailer be charged a losing flexibility cost as compensation for the reduction setup pushed by the supplier in a long-term partnership contract. Based on numerical examples and sensitivity analysis, the percentage of defective items in each shipment from supplier to retailer significantly affects integrated total inventory cost.

EFFECTS OF LEARNING ON THE ECONOMIC ORDERING POLICIESFOR DEFECTIVE ITEMS UNDER FUZZY ENVIRONMENT WITH PERMISSIBLE DELAY IN PAYMENTS

The phenomenon of learning canand has been implemented in several distinct disciplines such as environmental protection techniques, manufacturing processes as well as business transactions. As a matter of fact, learning effect is a phenomenon which occurs almost every where and enables the employees to perform new tasks with better performance after fluent repititons over a course of time. The learning effect acts as a considerable function for cost reduction. It consists of perfect and imperfect quality item’s severance from the lot and instantly selling them at different prices. This paper is an adjunct of the commendable work by Jaggi et al.(2013) which considers learning in holding costs and percentage of defective items, follows the learning curve effects underthe fuzzy environment and in addition focuses on proving the concreteness of the fact, that the percentage of defective items together with the holding costs follow the learning effect. In obedience with the economic order quantity (EOQ) model, the ordered lot has 100% perfect items, but this is only suppositional. On contemplation and practical analysis, we have arrived to learn that some lots possess defective items because of process retrogression and other such factors.This research paper refines the economic order quantity model in the context that there are defective items present in each ordered lot confirmed after initial inspection and shortages are allowed under trade credit financingin accordance with the learning effect. The defective items are separated from the considered lot through vigilant inspection and are sold again at a fixed decisive cost price. Conclusively, we de-fuzzify the total profit functions using the triangular method, and for verification of the same, numerical examples and sensitive analysis have been presented in this paper for a clear understanding. KEYWORDS: Inventory, Learning effect,Imperfect items, Shortages, Trade credit financing, Fuzzy system MSC: 90B05

A retailer inventory model when the reliability of inspection system affects the percentage of defective items which are delivered to final customers

ABSTRACTThis paper studies an inventory model system in which the inspection system reliability hinges on the reliability of its components and affects recognition of defective items over the planning horizon. The inventory model is discussed during the useful life of the inspection system and its components will be sold at the end of the useful life at the price of salvage value. The goal is choosing the internal components of the inspection system, determining the order and fixed shortage quantity, number of ordering cycles to maximize the revenue that is gained by the retailer. After that the model is formulated and discussed in details, particle swarm optimization (PSO) and genetic algorithm (GA) are used to solve the proposed model. To demonstrate the application of the proposed methodology and assessing the performances of the solution algorithms, different numerical examples are solved and compared.

Stochastic supply chain model with imperfect production and controllable defective rate

This paper investigates the impact of ordering cost reduction and process quality improvement on an integrated vendor–buyer inventory system with variable lead time in just-in-time (JIT) manufacturing environment. We assume that the vendor invests money in order to improve the production process quality and reduce the number of defectives, and buyer invests money to reduce the ordering cost. Each lot received by the buyer contains a certain percentage of defective items and it is assumed that there is an inspection activity on the part of the buyer with a fixed screening rate greater than the demand rate. Shortages are partially backlogged with a certain limit of backorder price discount. The objective of this study is to obtain the optimal decisions and the best investment policy to minimise the joint total cost incurred by the buyer and the vendor. An efficient algorithm is developed to determine the optimal policy. The model is illustrated through a numerical example. From the numerical results, it is observed that significant savings can be achieved through the reductions of lead time, ordering cost and defective rate. Some managerial insights also obtained from the numerical study.

A two-echelon supply chain inventory model with shortage backlogging, inspection errors and uniform demand under imperfect quality items

Actually, some production and inspection processes do not work perfectly, thereby producing a certain number of defective items and inspection errors. Previous imperfect-quality integrated inventory studies have mostly focused on developing models that do not consider two-way imperfect inspection under stochastic demand. Thus, this paper proposes a cost-minimising integrated vendor-buyer production-inventory model that incorporates imperfect production quality, stochastic demand, and imperfect inspection processes. The imperfect inspection processes are Type I inspection error of classifying a non-defective item as defective and Type II inspection error of classifying a defective item as non-defective. Meanwhile, the demand is assumed uniformly distributed. The objective of this research is to determine the optimal order quantity and reorder point such that the expected integrated total annual cost is minimised. In order to solve the integrated model, an optimal solution procedure is developed. A numerical example is provided to illustrate the application of the model. Besides, a sensitivity analysis is also given to investigate the effects of six important parameters (the transportation cost, the buyer's holding cost, the vendor's holding cost, the percentage of defective items, Type I error, and Type II error) on the optimal solution.

A joint replenishment problem considering multiple trucks with shipment and resource constraints

A joint replenishment problem (JRP) is presented to determine the optimal reordering policy for multi-items with a percentage of defective items. This JRP also has several constraints, such as shipment constraint, budget constraint, and transportation capacity constraint. At the meantime, multiple trucks, each with a fixed transportation cost, are considered and also order quantities of restricted items are not shared among the trucks during the shipment. The objective is to minimize the total expected cost per unit time. A two-dimensional genetic algorithm (GA) is provided to determine an optimal family cycle length and the reorder frequencies. A numerical example is presented and the results are discussed. Extensive computational experiments are performed to test the performance of the GA. The JRP is also solved by using an evolutionary algorithm (EA) and the results obtained from GA and EA are compared.

Efficient inventory control for imperfect quality items

In this paper, we present a general EOQ model for items that are subject to inspection for imperfect quality. Each lot that is delivered to the sorting facility undertakes a 100 per cent screening and the percentage of defective items per lot reduces according to a learning curve. The generality of the model is viewed as important both from an academic and practitioner perspective. The mathematical formulation considers arbitrary functions of time that allow the decision maker to assess the consequences of a diverse range of strategies by employing a single inventory model. A rigorous methodology is utilised to show that the solution is a unique and global optimal and a general step-by-step solution procedure is presented for continuous intra-cycle periodic review applications. The value of the temperature history and flow time through the supply chain is also used to determine an efficient policy. Furthermore, coordination mechanisms that may affect the supplier and the retailer are explored to improve inventory control at both echelons. The paper provides illustrative examples that demonstrate the application of the theoretical model in different settings and lead to the generation of interesting managerial insights.

A single-vendor multi-buyer integrated model with controllable lead time and quality improvement through reduction in defective items

This paper presents a single-vendor multi-buyer integrated inventory model with an imperfect production system of the vendor. The expected demand of each buyer is assumed to be dependent on selling price while lead-time demand of each buyer is assumed to be normally distributed. Further, each buyer's lead time can be reduced at an added crash cost. We extend the model by assuming that the vendor invests money in order to improve the quality of the product and reduce the percentage of defective items. The objective is to determine the optimal values of the decision variables involved so that the average total profit of the system is maximised. We relax the assumption of normality to treat the model with distribution-free approach, where only the mean and standard deviation of lead-time demand are known. Optimal results are obtained for a numerical example and effects of the key parameters are included to illustrate the results of the proposed model.

An integrated inventory model for a deteriorating item with allowable shortages and credit linked wholesale price

This study proposes a single manufacturer, single retailer integrated inventory model that includes deterioration and shortages in the retailer’s inventory. The manufacturer’s production process is assumed to be imperfect as it produces a certain percentage of defective items. The retailer performs a 100 % screening process immediately on receiving a lot from the manufacturer and returns the detected defective items to the manufacturer in the next delivery. The manufacturer disposes the defective items and incurs a disposal cost. To increase sales, (s)he offers a trade credit to the retailer. The retailer’s wholesale price varies linearly with the credit period. The objective is to determine the optimal replenishment cycle time, the time of running out of stock, the length of the credit period and the number of lots from the manufacturer to the retailer so as to maximize the total profit of the integrated system. A solution algorithm is designed and illustrated through numerical examples. Furthermore, a sensitivity analysis is carried out to study the influence of the model-parameters on the optimal solution.

Lot sizing in case of defective items with investments to increase the speed of quality control

In many cases the quality of each item in a lot is checked. Speeding up the quality checking process increases the responsiveness of the system and saves cost. The percentage of defective items is a random variable and two models are proposed. In one of the models the system remains always at the same state, while in the other one after each order cycle, the state of the system may change, thus the percentage of defective items may be different in consecutive periods. In both cases the speed of the quality checking is a variable, and procedures are provided to find the optimal lot sizes and screening speed for general and specific investment cost functions. The characteristics of the two model settings will largely be different when the percentage of defective items is high. Among the important managerial insights gained is that a high unit backlogging cost, especially spurs the system to invest more intensively into improving the quality checking process.

Solving a tri-objective supply chain problem with modified NSGA-II algorithm

In this paper, we propose the modification of an existing Multi-Objective Evolutionary Algorithm (MOEA) known as Non-dominated Sorting Genetic Algorithm-II (NSGA-II). The proposed algorithm has been applied on a tri-objective problem for a two echelon serial supply chain. The objectives considered are: (1) minimization of the total cost of a two-echelon serial supply chain and (2) minimization of the variance of order quantity and (3) minimization of the total inventory. The variance of order quantity is an important factor to consider since the variance of order quantity is used to measure the bullwhip effect which is one of the performance measures of a supply chain. The supply chain under consideration is assumed to consist of buyers and supplier. The production process at the supplier is an imperfect production process and thus produces defective items. A percentage of defective items are sold at a secondary market and the remaining defective items are repaired. We have introduced a mutation algorithm which has been embedded in the proposed algorithm. Since the proposed mutation algorithm is performed over the entire population, thus the mutation algorithm has caused the modification of the parts of the original NSGA-II. The results of the modified algorithm have been compared with those of the original NSGA-II and SPEA2 (Strength Pareto Evolutionary Algorithm 2) evolutionary algorithms for varying values of probability of crossover. The experimental results show that the proposed algorithm performs significantly better than the original NSGA-II and SPEA2.

Economic order and production quantity models without constraint on the percentage of defective items

The paper deals with a version of the economic order and production quantity models when the fraction of defective items is probability variable that either may vary from cycle to cycle, or remains the same as it was in the first period. We drop the well-known assumption intended to avoid shortages, and two lot sizing rules are derived. Relationships between the new and known lot sizing rules are analyzed and cases are identified when lot sizing rules give significantly different results.

Planning of Truncated Sequential Binomial Test via Relative Efficiency

A planning methodology is proposed for the sequential probability ratio test (SPRT) for the purpose of practical application. The SPRT is the most common acceptance test in the field of reliability and quality control. In it, the hypothesis is checked that the percentage of defective items does not exceed a specified value. Truncation is resorted to compensate for the absence of a limit on the test duration, but it complicates the planning process. Moreover, the discreteness and multidimensionality of the characteristics of such tests prevent their direct comparison and optimization. To remedy these drawbacks, quality features of the test are proposed, one of which—the relative efficiency—represents the ratio of the test's weighted average sample number till its stopping and its counterpart for the nontruncated SPRT. It facilitates solution of the problems in automatic planning of the test. Another important advantage of this relative efficiency is that it yields accurate and simple formulas for the stopping boundary. Besides, these formulas permit sound choice of the truncation level already at early stages of the planning process. A planner's algorithm and an industrial example are also included. The proposed methodology can also be applied to exponential SPRT. The advantages of tests based on the proposed methodology over those in IEC‐61123 (the binomial case) and IEC‐61124 (the exponential case) are demonstrated, and revision of the standards is recommended. Copyright © 2012 John Wiley & Sons, Ltd.

An optimal shipment strategy for imperfect items in a stock-out situation

The objective of this paper is to minimize the expected total cost by consolidating a number of batches of imperfect quality products of different cycles for a single shipment and maximize the expected average profit function. Shortages are allowed at the end of each ordering cycle and are partially backlogged. The stock-out period is assumed to be exponentially distributed. The ordered lot size and cycle length of all the ordering cycles of a shipping cycle are considered to be equal. The model is analyzed for both infinite and finite planning horizons. The percentage of defective items in a lot is assumed to follow a uniform distribution. All the fractions of imperfect quality items in the ordering cycles of each shipping phase are assumed to be independent and identically distributed. After the completion of a screening process, one portion of the perfect quality products is used to serve partial backlogging at a cost per unit and the rest of the stock is used to adjust the demand. Calculus method is used to obtain the optimal number of order cycles for shipment. Optimal shortage period, optimal lot size and expected average profit for the model are developed on both of the finite and infinite time horizons. A comparison between the our models and Maddah and Jaber’s (2008) [1] model is done through numerical studies which are also used to illustrate the models graphically.

A two-warehouse inventory model for items with imperfect quality and penalty costs under screening errors

This paper corrects an improper expression in the economic order quantity (EOQ) model with two-warehouses and imperfect quality developed by Chung et al. [2009. A twowarehouse inventory model with imperfect quality production processes. Computers & Industrial Engineering 56, 193–197]. The modified model yields a simple and corrected expression for the optimal order quantity and expected profit per unit time. This paper then extends the above idea to a case involving a 100% inspection process with screening errors that may occur under imperfect quality and two warehouses. A model, in which it includes two scenarios, with imperfect quality and penalty costs under screening errors (Type I and Type II error) is developed. The effects of percentage of defective items and screening errors on the optimal solution are investigated. A numerical example is provided for the developed model and its result reveals that the order lot size and expected total profit considering in this paper is less than Chung et al. [2009]. Managerial insights are also draw.

Optimal batch sizing, quality improvement and rework for an imperfect production system with inspection and restoration

This paper incorporates the effect of process inspection and restoration (Ben-Daya and Rahim, 2003) in an imperfect production process while considering three different scenarios (Ojha et al., 2007): 1) single lot of raw material?multiple lot of finished product?multiple delivery of the product, 2) single lot of raw material?multiple lot of finished product?single delivery of the product, 3) single lot of raw material?single lot of finished product?single delivery of the product. A numerical algorithm is proposed, and an illustrative example is provided. The optimal number of inspections, ordering quantities and expected total cost of the illustrative example are evaluated for each of the three models along with a sensitivity analysis to show the impact of the model parameters on the solution. The analysis shows that incorporating the inspection and restoration in the imperfect production process results in larger batch sizes and lower total costs. The implementation of inspection and restoration in imperfect production processes is good in cases of higher percentage of defective items produced (α), as it results in higher savings on total cost. [Submitted 03 October 2007; Revised 22 May 2008; Revised 10 November 2008; Accepted 11 November 2008]

Batch scheduling of deteriorating reworkables

The problem of scheduling the production of new and recoverable defective items of the same product manufactured on the same facility is studied. Items are processed in batches. Each batch comprises two sub-batches processed consecutively. In the first sub-batch, all the items are newly manufactured. Some of them are of the required good quality and some are defective. The defective items are remanufactured in the second sub-batch. They deteriorate while waiting for rework. This results in increased time and cost for their remanufacturing. All the items in the same sub-batch complete at the same time, which is the completion time of the last item in the sub-batch. Each remanufactured defective item is of the required good quality. It is assumed that the percentage of defective items in each batch is the same. A setup time is required to start batch processing and to switch from manufacturing to remanufacturing. The demands for good quality items over time are given. The objective is to find batch sizes such that the total setup and inventory holding cost is minimized and all the demands are satisfied. Dynamic programming algorithms are presented for the general problem and some important special cases.

An optimal production run length with imperfect production process and nondecreasing percentage defective proportion

This paper develops a model to determine an optimal run time under an imperfect production process. The shortages are allowed. It is assumed that the elapsed time until the production process shift is arbitrarily distributed and the percentage of defective items is a nondecreasing function once the system is in the out-of-control state. This paper generalizes the results presented by Chung and Hou (2003).

Cost minimizing scheduling of work and rework processes on a single facility under deterioration of reworkables

The problem of scheduling the production of new and recovering defective items of the same product manufactured on the same facility is studied. The items are produced in batches. The processing of a batch includes two stages. In the first stage, all items of a batch are manufactured and good-quality items go to the inventory to satisfy given demands. In the second stage, defective items of the same batch are reworked. After rework each item has the required good-quality. During waiting for rework defective items are assumed to deteriorate. This results in an increase in time and cost for performing rework processes. It is assumed that the percentage of defective items is the same in each batch, and that they are uniformly distributed in each batch. A setup time as well as a setup cost is required to start batch processing and to switch from production to rework. The objective is to find batch sizes such that all demands are satisfied and total setup, rework and inventory holding cost is minimized. Polynomial time algorithms are presented to solve two realistic special cases of this problem. A generalization to a multiple product case is discussed.

Batching work and rework processes with limited deterioration of reworkables

We study a deterministic problem of planning the production of new and recovering defective items of the same product manufactured on the same facility. Items of the product are produced in batches. The processing of a batch includes two stages. In the first work stage, all items of a batch are manufactured and good quality items go to the inventory to satisfy given demands. In the second rework stage, some of the defective items of the same batch are reworked. Each reworked item has the required good quality. While waiting for rework, defective items deteriorate. There is a given deterioration time limit. A defective item, that is decided not to be reworked or cannot be reworked because its waiting time will exceed the deterioration time limit, is disposed of immediately after its work operation completes. Deterioration results in an increase in time and cost for performing rework processes. It is assumed that the percentage of defective items is the same in each batch, and that they are evenly distributed in each batch. A setup time as well as a setup cost is required to start batch processing and to switch from production to rework. The objective is to find batch sizes and positions of items to be reworked such that a given number of good-quality items is produced and total setup, rework, inventory holding, shortage and disposal cost is minimized. A polynomial dynamic programming algorithm is presented to solve this problem.