Economic production quantity with the presence of imperfect quality and random machine breakdown and repair based on the artificial bee colony heuristic

This article considers a production-inventory system consisting of a single imperfect unreliable machine. The items manufactured by the system are either perfect items or imperfect items, which require a rework to be restored to perfect quality. The rework rate is permitted to be different from the production rate if the rework process is different from the main manufacturing process. The fraction of the number of imperfect items is random following a general distribution function. The time to failure of the machine is random, following a general distribution function. If the machine fails before the lot is completed, the production is interrupted and the machine repair is started immediately. A random machine repair time is assumed, with a general distribution function. Unlike a common assumption in the literature, after the repair of the machine is completed, the production resumes. During the machine repair, a shortage can occur. A single-variable expected average cost function is derived to find the optimal lot size. Because of the complexity in the model, the ABC heuristic is proposed and implemented to find a near optimal value for the lot size. The article also provides a sensitivity analysis of the model's key parameters. It has been observed that the lot interruption-resumption policy leads to smaller lot sizes.     

Bayesian analysis of hybrid life tests with exponential failure times

Summary A hybrid life test procedure is discussed from the Bayesian viewpoint. A total ofn items is placed on test, failed items are either not replaced or are replaced, and the test is terminated either when a pre-chosen number,K, of items have failed, or when a pre-determined time on test has been reached. Posterior and predictive distributions are obtained under the assumption of an exponential failure distribution, and point and interval estimates are given for the mean life and the life of an untested item. The results are applied to a numerical example.     

A Lot-sizing Model for Just-in-time Manufacturing

We consider the problem of determining lot sizes of multiple items that are manufactured by a single capacitated facility. The manufacturing facility may represent a bottleneck processing activity on the shop floor or a storeroom that provides components to the shop floor. Items flow from the facility to a downstream facility, where they are assembled according to a specified mix. Just-in-time (JIT) manufacturing requires a balanced flow of items, in the proper mix, between successive facilities. Our model determines lot sizes of the various items based on available capacity and four attributes of each item: demand rate, holding cost, set-up time and processing time. Holding costs for each item accrue until the appropriate mix of items is available for shipment downstream. We develop a lot-sizing heuristic that minimizes total holding cost per time unit over all items, subject to capacity availability and the required mix of items.     

An EMQ model with inspections and random machine failures

In the classical Economic Manufacturing Quantity (EMQ) model, it is assumed that all items produced are of perfect quality and the production facility never breaks down. However, in real production, the product quality is usually a function of the state of the production process which may deteriorate over time and the production facility may fail randomly. In this paper, we study the effect of machine failures on the optimal lot size and on the optimal number of inspections in a production cycle. The formula for the long-run expected average cost per unit time is obtained for a generally distributed time to failure. An optimal production/inspection policy is found by minimising the expected average cost.     

Constrained Lot-Sizes under Conditions of Inventory Growth

When average aggregate inventory levels are constrained to equal a constant level over time, optimal lot sizes can be identified which strike a balance between holding costs and set-up costs among items which form the aggregate. However, when it is desirable to change aggregate inventory levels over time, assumptions implicit in the traditional formulation are violated. The procedure proposed generates lot sizes which are consistent not only with the current average aggregate inventory level but also with its projected growth over the planning horizon. Comparison is made to lot sizes generated by the misapplication of traditional lot sizing methods to the inventory growth situation.     

The EOQ with defective items and partially permissible delay in payments linked to order quantity derived algebraically

Most researchers established their inventory lot-size models under trade credit financing by assuming that the supplier offers the retailer fully permissible delay in payments and the products received are all non-defective. However, in the real business environment, it often can be observed that the supplier offers the retailer a fully permissible delay in payments only when the order quantity is greater than or equal to the predetermined quantity Q _d . In addition, an arriving order lot usually contains some defective items due to imperfect production processes or other factors. To capture this reality, the paper extends Huang (2007) economic order quantity (EOQ) model with partially permissible delay in payments to consider defective items. We formulate the proposed problem as a profit maximization EOQ model in which the replenishment cycle time is the decision variable. Then we use the arithmetic-geometric mean inequality approach to determine the optimal solution under various situations. An algorithm to obtain the optimal solution is also provided. Finally, the numerical examples and sensitivity analysis are given to illustrate the results.     

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.     

Batch sizing and job sequencing on a single machine

We study a single-machine scheduling problem in which the items to be processed have to be batched as well as sequenced. Since processed items become available in batches, flow times are defined to be the same for all items in the same batch. A constant set-up delay is incurred between consecutive batches. For any fixed, but arbitrary item sequence, we present an algorithm that finds a sequence of batches such that the total flow time of the items is minimized; we prove that for a set ofn items, the algorithm runs inO(n) time. We show that, among all sequences, the one leading to the minimum flow time has the items in non-decreasing order of running times. Thus, the optimal algorithm for the combined problem, called thebatch-sizing problem, runs inO(n logn) time. We also prove that this algorithm yields an improved solution to a scheduling problem recently studied by Baker [1].     

An instantaneous replenishment model under the effect of a sampling policy for defective items

We propose to study a EOQ-type inventory model with unreliable supply, with each order containing a random proportion of defective items. Every time an order is received, an acceptance sampling plan is applied to the lot, according to which only a sample is inspected instead of the whole lot. If the sample conforms to the standards, i.e. if the number of imperfect items is below an “acceptance number”, no further screening is performed. Otherwise, the lot is subject to 100% screening. We formulate an integer non-linear mathematical program that integrates inventory and quality decisions into a unified profit model, to jointly determine the optimal lot size and optimal sampling plan, characterized by a sample size, and an acceptance number. The optimal decisions are determined in a way to achieve a certain average outgoing quality limit (AOQL), which is the highest proportion of defective items in the outgoing material sold to customers. We provide a counter-example demonstrating that the expected profit function, objective of the mathematical program, is not jointly concave in the lot and sample size. However, we show that for a given sampling plan, the expected profit function is concave in the lot size. A solution procedure is presented to compute the optimal solution. Numerical analysis is provided to gain managerial insights by analyzing the impact of changing various model parameters on the optimal solution. We also show numerically that the optimal profit determined using this model is significantly higher when compared to the optimal profit obtained using Salameh and Jaber (2000)’s [1] model, indicating much higher profits when acceptance sampling is used.     

Reliability Characteristics of Alternative Checkup Schedules for Detecting Hidden Failures

We consider a group of N identical objects (machines, etc.) working in similar conditions. In a certain important part of these objects, a specific failure may appear. It has a hidden nature and can be revealed only during special examination (checkup). A schedule for carrying out these checks is fixed in advance. When the failure is revealed for the first time in some object, special reinforcement measures are taken for all functioning objects in order to eliminate the appearance of this failure in the future. Each checkup schedule can be characterized by several reliability indices such as probability of discovering the failure before a certain time, expected time of failure detection, expected time of operation with a hidden failure, etc. We present formulae for computing all of these indices, for a general type schedule, and compare numerically two types of schedules: one when all items undergo the checkup at the same time and the second in which the checkup times are different for different items (by preserving the same checkup frequency). We demonstrate that under certain circumstances, such as N being large, the second type schedules have better reliability indices.     

Determining the optimum process mean based on quadratic quality loss function and rectifying inspection plan

In 1996, Pulak and Al-Sultan presented a rectifying inspection plan for determining the optimum process mean. However, they did not consider the quality cost for the product within the specification limits and did not point out whether the non-conforming items in the sample of accepted lot is replaced or eliminated from the lot. In this paper, we propose a modified Pulak and Al-Sultan’s model with quadratic quality loss function of product within the specification limits. Assume that the non-conforming items in the sample of accepted lot are replaced by conforming ones. Finally, the numerical results and sensitivity analysis of parameters of modified model and those of Pulak and Al-Sultan are provided for illustration.     

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.     

The last departure time from an M t /G/∞ queue with a terminating arrival process

This paper studies the last departure time from a queue with a terminating arrival process. This problem is motivated by a model of two-stage inspection in which finitely many items come to a first stage for screening. Items failing first-stage inspection go to a second stage to be examined further. Assuming that arrivals at the second stage can be regarded as an independent thinning of the departures from the first stage, the arrival process at the second stage is approximately a terminating Poisson process. If the failure probabilities are not constant, then this Poisson process will be nonhomogeneous. The last departure time from an M _t /G/∞ queue with a terminating arrival process serves as a remarkably tractable approximation, which is appropriate when there are ample inspection resources at the second stage. For this model, the last departure time is a Poisson random maximum, so that it is possible to give exact expressions and develop useful approximations based on extreme-value theory.      

Inventory of differential items selling at two shops under single management

Inventory of differential (both good and defective) items purchased in a lot for two shops under a single management is considered here. In the primary shop, the lot is received and separated, only good units are sold and shortages are allowed which are backlogged from the good units specially purchased at the beginning of the next cycle. The separated defective units are continuously trasnsferred and sold at a reduced price from the secondary shop. This inventory model is formulated and illustrated numerically for three scenarios depending upon the time periods of the shops.     

Economic lot scheduling problem with imperfect production processes

In this paper, we model the effects of imperfect production processes on the economic lot scheduling problem (ELSP). It is assumed that the production facility starts in the in-control state producing items of high or perfect quality. However the facility may deteriorate with time and shifts at a random time to an out of control state and begins to produce nonconforming items. A mathematical model is developed for ELSP taking into account the effect of imperfect quality and process restoration. Numerical examples are presented to illustrate important issues related to the developed model.     

A production lot size inventory model for deteriorating items and arbitrary production and demand rates

In this paper, we present a method for finding the optimal replenishment schedule for the production lot size model with deteriorating items, where demand and production are allowed to vary with time in an arbitrary way and in which shortages are allowed. The method is illustrated by a numerical example.     

Lot streaming for quality control in two-stage batch production

We consider a two-stage batch manufacturing process in which the first stage shifts out-of-control at iid exponential times after starting in control. To improve quality, a production batch at Stage 1 is subjected to lot streaming: it is divided into sublots that are processed at Stage 1 and then passed one-by-one to Stage 2 for simultaneous inspection and processing. In any sublot, Stage 1 produces good items before the shift and bad items after. The state of Stage 1 is known as soon as a bad item is encountered in Stage 2, at which time Stage 1 is re-set to the in-control state. We examine both cases of continuous first-stage and continuous second-stage production. For each case we examine both LIFO and FIFO inspection and processing policies at Stage 2. We use nonlinear programming to develop lot streaming policies which minimize the expected number of defective items for LIFO and FIFO policies. We also develop simple approximately optimal policies and compare the output performance of optimal, approximately optimal and equal-lot policies (when applicable) in a numerical example.     

The clustered flow-shop problem

This paper considers a generalization of the classical flow shop problem wheren the items, grouped intok fixed sequences (clusters) are processed onm machines. A permutation is being sought that minimizes the completion time of processing all items. This paper develops conditions when the clustered problem can be reduced to a classical case. It also provides sufficient optimality conditions for an approximate solution generated by a method for the two machine clustered problem.     

考虑外包的单产品再制造批量决策

在已有动态批量决策问题的基础上,考虑了再制造与外包这两个因素建立了单产品的再制造批量决策模型。对考虑外包的单产品再制造批量决策问题最优解的性质进行了讨论,在此基础上利用动态规划的思想给出了时间复杂性为O(T4)的算法,并用C++实现了本算法,实验表明本算法是有效的。     

On a finite horizon production lot size inventory model for deteriorating items: An optimal solution

A generalized production lot size inventory model for deteriorating items over a finite planning horizon is considered. The demand, production, and deteriorating rates are assumed to be known and continuous functions of time. Shortages are allowed and completely backlogged. The conditions under which the system total cost attains its (unique) global minimum are derived. An example which illustrate the applicability of theoretical results is also introduced.