Integrated inventory control and inspection policies with deterministic demand

In this paper, we develop integrated inventory inspection models with and without replacement of nonconforming items. Inspection policies include no inspection, sampling inspection, and 100% inspection. We consider a buyer who places an order from a supplier. When a lot is received, the buyer uses some type of inspection policy. The fraction nonconforming is assumed to be a random variable following a beta distribution. Both the order quantity and the inspection policy are decision variables. In the inspection policy involving determining sampling plan parameters, constraints on the buyer and manufacturer risks are set in order to obtain a fair plan for both parties. A solution procedure for determining the operating policies for inventory and inspection consisting of order quantity, sample size, and acceptance number is proposed. Numerical examples are presented to conduct a sensitivity analysis for important model parameters and to illustrate important issues about the developed models.     

(Q,r,L) Inventory model with defective items

This paper assumes that an arrival order lot may contain some defective items, and the number of defective items is a random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity, the reorder point and the lead time are decision variables. In our studies, we first assume that the lead time demand follows a normal distribution, and then relax the assumption about the form of the distribution function of the lead time demand and apply the minimax distribution-free procedure to solve the problem. We develop an algorithm procedure to obtain the optimal ordering strategy. Furthermore, the effects of parameters are also included.     

Computational procedure of optimal inventory model involving controllable backorder rate and variable lead time with defective units

This article considers that the number of defective units in an arrival order is a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate be a control variable. In addition, we assume that the lead time demand follows a mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, three numerical examples are also given to illustrate the results.     

On the distribution free continuous-review inventory model with a service level constraint

We consider a continuous-review (Q, r) inventory model with a fill rate service constraint and relax the assumption that the distribution of lead time demand is known. We adopt a distribution free approach: We assume that only the first two moments of the lead time demand distribution are known, and then, optimize the policy parameters against the worst possible distribution. We are able to derive closed-form expressions for the optimal order quantity and reorder point.     

A note on defective units in an inventory model with sub-lot sampling inspection for variable lead-time demand with the mixture of free distributions

In a recent paper Wu and Ouyang (2000) assumed that an arriving order lot may contain some defective items and considered that the number of defective items in the sub‐lot sampled to be a random variable. They derived a modified mixture inventory model with backorders and lost sales, in which the order quantity, re‐order point, and the lead‐time were decision variables. In their studies they assumed that the lead‐time demand followed a normal distribution for the first model and relaxed the assumption about the form of the distribution function of the lead‐time demand for the second model. When the demand of the different customers is not identical with regard to the lead‐time, then one cannot use only a single distribution (such as Wu and Ouyang (2000) ) to describe the demand of the lead‐time. Hence, we extend and correct the model of Wu and Ouyang (2000) by considering the lead‐time demand with the mixed normal distributions (see Everitt and Hand (1981) , and Wu and Tsai (2001) ) for the first model and the lead‐time demand with the mixed distributions for the second model. And we also apply the minimax mixed distributions free approach to the second model. Moreover, we also develop an algorithm procedure to obtain the optimal ordering strategy for each case.     

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.     

Simultaneous determination of the optimal production–inventory and product inspection policies for a deteriorating production system

In this paper, we attempt to find a method for the optimization of production–inventory and product inspection policies for deteriorating production systems. Taking advantage of the nature of a deteriorating production system, a strategy would be not to inspect the first s items of the batch. Therefore, an inspection policy which disregards the first s (DTF-s) items of the batch is proposed. Under the DTF-s policy, we do not inspect the first s produced items but inspect only those items from the (s+1)th till the end of the production run. The objective of this study was the joint determination of the production lot size and the inspection policy s, resulting in a minimization of the expected average cost per unit time. Based on this model, the underlying conditions necessary for the existence of an optimal policy are given. Two commonly used inspection strategies, no inspection and full inspection are discussed. Under both inspection strategies, an optimal production–inventory lot is bounded by the traditional economic quantity. The case of full inspection is shown to be an extension of previously reported results. The option of investing in the process of quality improvement is also discussed. Finally, numerical examples are given to illustrate the method and its advantages in the conclusion.Scope and purposeThis paper considers the relationship between production, inventory and inspection in a deteriorating production system which may transit from the “in-control” state to the “out-of-control” state after a period of operation. Once the transition to the “out-of-control” state has occurred, it is assumed that some percentage of the items produced are defective or of substandard quality. However, in many cases, defects in a defective item can only be identified by an inspection process which carries an inspection cost. Those inspected items which are found to be defective are reworked at some cost before being shipped. On the other hand, defective items which are not inspected will be passed to the customer, incurring a much larger warranty cost. In order to operate such a system economically, tradeoffs among production setup, inventory, inspection and defective cost must be analyzed. Deterioration of the production system is an inherent process in all manufacturing industries. An understanding of the relationship among production, inventory and inspection for such systems will help managers to maintain efficient and economic control of operations.     

Inventory control model for a supply chain system with multiple types of items and minimum order size requirements

This paper considers a replenishment problem for a single buyer who orders multiple types of items from two or more heterogeneous suppliers in order to sell to end customers. The buyer periodically orders each type of item from the suppliers according to a select inventory control policy. Processing the order, each supplier enforces the policy that an order from the buyer must meet a predetermined minimum order quantity (MOQ). Therefore, the buyer must decide how much to order from each supplier considering the current inventory level, demand forecast, and MOQ requirement. The buyer's problem is formulated as an integer programming model and an efficient implementation strategy is suggested to apply the model to real problems. Numerical experiments are performed to test the validity of the proposed model as well as the efficiency of the implementation strategy. The experimental results show that this model combined with the implementation method yields a considerable cost reduction compared to the most efficient policy currently available.     

The distribution free continuous review inventory system with a service level constraint

The stochastic inventory models require the information on the lead time demand. However, the distributional information of the lead time demand is often limited in practice. We relax the assumption that the cumulative distribution function, say F, of the lead time demand is completely known and merely assume that the first two moments of F are known and finite. The distribution free approach for the inventory model consists of finding the most unfavorable distribution for each decision variable and then minimizing over the decision variable. We apply the distribution free approach to the continuous review inventory system with a service level constraint. We develop an iterative procedure to find the optimal order quantity and reorder level.     

An integrated vendor–buyer inventory model with backorder price discount and effective investment to reduce ordering cost

The single-vendor single-buyer integrated production inventory system has been an object of study for a long time, but little is known about the effect of investing in reducing ordering cost on the integrated inventory models with backorder price discount and variable lead time. The purpose of this article is to investigate in the continuous review model with backorder price discount and variable lead time to effectively increase investment and to reduce the joint expected annual total cost. The integrated strategy discussed here is one in which the buyer orders a quantity, then the vendor produces n times order quantity in each production cycle, in order to reduce setup cost. In addition, the buyer offers backorder price discounts to the customers that may motivate the customers’ desire for backorders, and buyer ordering cost can be reduced through effective investment. An integrated inventory model is established to find the optimal solutions of order quantity, ordering cost, backorder price discount, lead time, and the number of shipments from the vendor to the buyer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, numerical examples are used to demonstrate the benefits of the model.     

Two-echelon supply chain inventory model with controllable lead time and service level constraint

This paper considers a two-echelon supply chain inventory problem consisting of a single-vendor and a single-buyer. In the system under study, a vendor produces a product in a batch production environment and supplies it to a buyer facing a stochastic demand, which is assumed to be normally distributed. Also, buyer’s lead time is controllable which can be shortened at an added cost and all shortages are backordered. A model has been formulated for an integrated vendor–buyer problem to jointly determine the optimal order quantity, lead time and the number of shipments from the vendor to the buyer during a production cycle while minimizing the total expected cost of the vendor–buyer integrated system. It is often difficult to estimate the shortage cost in inventory systems. Therefore, instead of having a shortage cost term in the objective function, a service level constraint (SLC) is included in the model that requires a certain proportion of demands to be met in each cycle. An efficient procedure has been suggested to find the bounds on number of shipments and then, an algorithm is developed to obtain the optimal solution of the proposed model. A numerical example is included to illustrate the algorithmic procedure and the effects of key parameters are studied to analyze the behavior of the model. Finally, the savings of buyer and vendor are investigated from implementation of joint optimization model over the model in which they minimize their own cost independently.     

An integrated production–distribution inventory system involving probabilistic defective and errors in quality inspection under variable setup cost

This paper presents a probabilistic defective vendor–buyer integrated inventory model with the consideration of quality inspection errors at the buyer's end and setup cost as function of capital investment. An integrated inventory model is established to find the optimal solutions of lot size, setup cost, and the total number of shipments from the vendor to the buyer in one production run, so that the joint expected total cost incurred has the minimum value. We consider three types of continuous probabilistic defective function to find the associated cost of the system. The expected total cost function is derived for each of these three distributions, its convexity is proved via differential calculus. An efficient iterative algorithm is designed to obtain the optimal solution of the model. The computational effort and time are small for the proposed algorithm and it is simple to implement. Numerical examples and sensitivity analysis are used to demonstrate the application and the performance of the proposed methodology. The computational results indicate that if we make decisions with the capital investment in reducing setup cost, it will help to lower the system cost, and we obtain a significant amount of savings to increase the competitive edge in business.     

Economic ordering policy for an item with imperfect quality subject to the in-house inspection

This article considers a production/inventory system where each lot of items received or produced contains a random proportion of defective units, items of imperfect quality. The purchaser contacts a 100% inspection in order to identify the perfect (acceptable) quality items. The model examines the following two options for the imperfect quality items: sell them to a secondary market, as a single batch and at a price lower to that of new ones, or rework them at some cost and then use them as new ones to satisfy demand. After inspection, the good quality items are sent to the working inventory warehouse in batches of equal size. For both of these cases, the optimal ordering lot size and the optimal number of batches are obtained. A numerical example illustrates the solution procedure and sensitivity analysis results are reported.     

Coordinated ordering decisions for products with short lifecycle and variable selling price

The supplier–buyer coordination is an important policy in the supply chain management. The buyer in the two-echelon inventory system with regular selling season has to face the uncertainty of customer demand, supplier’s delivery time and variable price change. At the same time, the supplier has to consider the inventory holding and delay cost. The objective of this study is to develop an integrated supply chain strategy for products with short lifecycle and variable selling price to entice cooperation. The strategy must provide a win–win situation for both the supplier and the buyer. A numerical case example, sensitivity analysis and compensation mechanism are given to illustrate the model.     

Joint replenishment policy with backordering and special sale

In this paper, an inventory control model with a joint replenishment policy and a temporary discount is developed. We assume that shortage is allowed and buyer uses an economic order quantity inventory control model. Different cases based on ordering policies for the first joint replenishment if the special order is not taken, and coincidence of a special period length with a positive or negative inventory level of the last regular period length, are investigated. Furthermore, several theorems are proved through which closed-form solutions are obtained. At the end, two numerical examples illustrate the different situations that the buyer may face and sensitivity analyses for both examples are reported.     

An integrated production-inventory deteriorating model for pricing policy considering imperfect production,inspection planning and warranty-period- and stock-level-dependant demand

In marketing, enterprises try all motivated selling strategies to stimulate customers to buy a product. One of these selling strategies is a warranty policy that provides a return promise of free replacement. The buyer may place more orders because of the display of the product. An increasing demand resulting from these motivated factors influences the replenishment planning. In operational process, quality level resulting in relevant activities may cause changes of operational planning. The purpose of this study is to investigate an integrated production inventory deteriorating model considering the pricing policy, the imperfect production, the inspection planning, the warranty-period and the stock-level-dependant demand with the Weibull deterioration, partial backorder and inflation. We incorporate a single-retailer single-manufacturer cooperation from the perspectives of both the manufacturer and the retailer. The classical optimisation technique and the heuristic method are used to derive the optimum solutions. A numerical example and sensitivity analysis are presented.     

A manufacturer–buyer integrated inventory model with stochastic lead times for delivering equal- and/or unequal-sized batches of a lot

Although the subject of manufacturer–buyer integrated inventory management with deterministic lead times has received a lot of attention from researchers, the corresponding problem with stochastic lead times has been given comparatively little consideration. Recently, it has been treated in the case of an exponential distribution of lead times with the lot transferred in equal-sized batches (sub-lots). In this treatment the buyer orders the next batch when his/her stock level falls to a certain reorder point, allowing for shortages and complete backordering. The total cost benefit of solving the problem using an integrated inventory system instead of independent ones had been demonstrated. However, rather than an exponential distribution, a normal distribution of lead times seems to provide a better fit to the problem. Moreover, synchronization of the integrated production flow by generalizing the method of transferring batches of the lot might lead to a lower total cost. Based on these notions, we develop here a manufacturer–buyer integrated inventory model with a normal distribution of lead times for delivering equal- and/or unequal-sized batches of a lot. Then a solution technique to the model and hence a solution algorithm are presented. The potential benefit of the present method is illustrated with solutions of some numerical problems. The sensitivities of the solutions to variations in the parameter values are also studied.     

An inventory model for imperfect items under inflationary conditions with considering inspection errors

This paper represents a discounted cash-flow approach for an inventory model for imperfect items under inflationary conditions with considering inspection errors. The previous imperfect quality inventory studies, however, have mostly had the emphasis on developing cost-minimizing models that do not consider imperfect inspection processes and related defect sales return issues despite their practical significance. In this paper, we assume that some produced items might not be perfect and the first stage inspector of product quality control might make some inspection errors during the separation of defective and perfect items. Thus, this study proposes a profit maximizing inventory model with incorporating both imperfect production quality and two-way imperfect inspection, i.e., Type-one inspection error of falsely screening out a proportion of no defects and disposing of them like defects and Type-two inspection error of falsely not screening out a proportion of defects, thereby passing them on to customers, resulting in defect sales returns. In addition, this model includes one more stage of inspection that is after the rework process and there is no inspection error in this stage. The purpose of this model is to determine the important factors of an inventory system to optimize the present value of the total profit in the finite time horizon. Finally, a numerical example is provided to solve the presented inventory model using our proposed innovative approach, which is further clarified through a sensitivity analysis.     

A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.     

A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.