Allowable Shortage Research Articles

An EOQ model with allowable shortage under trade credit in different scenario

In present business culture, usually supplier offers a permissible delay in payments to retailer in order to stimulate his demand. However, while developing the inventory model with shortages under permissible delay in payments, it has been observed, in the literature, that researcher have not considered the fact that retailer can earn interest on the revenue generated after fulfilling the outstanding demand as soon as his new consignment arrives at the beginning of the cycle. Thus, the revenue along with the interest earn can be utilized to pay off the amount at the of credit period. At this point of time there may be two scenarios, either he has enough amounts to settle the account with the supplier or delay incurring interest charges on the unpaid/overdue balance and the determination of a retailer’s pay off time, after the expiring of credit period, is largely affected by his interest income and interest payable. Owing to these facts, a retailer cost minimization inventory model has been developed for each scenario which jointly optimizes the cycle length and stock-in period. The model has been validated with the help of numerical example. Sensitivity analysis along with its economic interpretation has been also presented.

Fuzzy rough supply chain model under inflation and credit period with stock dependent consumption rate and partial backlogging shortages via genetic algorithm

In many inventory situations, purchasers are allowed a period to pay back for the goods bought without paying any interest. Depending on the length of that payment period, the purchaser can earn interest on the sales of the inventory. This paper develops a model to determine an optimal ordering policy for deteriorating items under inflation, stock dependent demand, permissible delay of payment, allowable shortage and unfinished demand is partially backlogged in fuzzy rough environments. The present value of total cost incurred in this inventory system is developed first, then an optimal order quantity and maximum allowable shortage are obtained by using a search procedure. The effect of inflation and time value of money was investigated under given sets of inflation and discount rates. This study shows that the optimal order quantity and maximum allowable shortage vary with the difference between inflation and time discount. Computational results provide some interesting policy implications. The genetic algorithm GA is used to make decision for above inventory model. The model is illustrated with some numerical data.

Optimal production run length for an imperfect production process with allowable shortages in fuzzy random environments

In this article, we develop a model to determine the economic production run length for a deteriorating production system and allowable shortages in fuzzy random environments. Based on the credibility measure of fuzzy event, the economic production quantity model with fuzzy random elapsed time can be transformed into a crisp model when deriving the expected total cost per unit time. The optimal production run length is proved to exist and be unique. Bounds for the optimal production run length are provided. Furthermore, the bisection method searching for the optimal production run length is designed. The relevant applications of the model are outlined. Finally, a numerical example is given to confirm the proposed model.

Supply chain model with stock dependent demand, quadratic rate of deterioration with allowable shortage

The goal of this devotion of time and attention to acquiring knowledge is to develop assist calculations and prediction for supply chain (three echelons), in which an optimal ordering decision for the manufacturer–supplier–retailer chain is developed when the supplier's lead–time is considered. This paper examines deterministic inventory model with variable deterioration and stock dependent demand, where the unsatisfied demand is partially backlogged which depends on waiting time. A solution procedure is presented to determine a stock dependent demand, variable deterioration rate, shortage period and the demand during the shortage is partially backlogged which is dependent on the waiting time such that the difference between an initial outlay and the subsequent amount earned for the supply chain is maximised. Sensitivity analyses are given to illustrate the model with the help of profit maximum principle.

An Economic Order Quantity Model with Completely Backordering and Nondecreasing Demand under Two-Level Trade Credit

In the traditional inventory system, it was implicitly assumed that the buyer pays to the seller as soon as he receives the items. In today’s competitive industry, however, the seller usually offers the buyer a delay period to settle the account of the goods. Not only the seller but also the buyer may apply trade credit as a strategic tool to stimulate his customers’ demands. This paper investigates the effects of the latter policy, two-level trade credit, on a retailer’s optimal ordering decisions within the economic order quantity framework and allowable shortages. Unlike most of the previous studies, the demand function of the customers is considered to increase with time. The objective of the retailer’s inventory model is to maximize the profit. The replenishment decisions optimally are obtained using genetic algorithm. Two special cases of the proposed model are discussed and the impacts of parameters on the decision variables are finally investigated. Numerical examples demonstrate the profitability of the developed two-level supply chain with backorder.

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.

A New Mathematical Inventory Model with Stochastic and Fuzzy Deterioration Rate under Inflation

This paper develops an inventory model for items with uncertain deterioration rate, time-dependent demand rate with nonincreasing function, and allowable shortage under fuzzy inflationary situation. The goods are not deteriorating upon reception, but the deteriorating starts after elapsing a specified time. The lead time and inflation rate are both uncertain in the model. The resultant effect of inflation and time value of money is assumed to be fuzzy in nature and also we consider lead time as a fuzzy function of order quantity. Furthermore the following different deterioration rates have been considered: for the first case we consider fuzzy deterioration rate and for the second case we assume that the deterioration rate is time dependent and follows Weibull distribution with three known parameters. Since the inflation rate, deterioration rate, and the lead time are fuzzy numbers, the objective function becomes fuzzy. Therefore the estimate of total costs for each case is derived using signed distance technique for defuzzification. The optimal replenishment policy for the model is to minimize the total present value of inventory system costs, derived for both the above mentioned policies. Numerical examples are then presented to illustrate how the proposed model is applied.

An operational policy for a three-stage distributive supply chain system with retailers’ backorders

This research deals with a production-inventory problem consisting of three-stage tree-type single-producer multi-distributor and multi-retailer supply chain network where a model has been developed for achieving the lowest total cost for operating a supply chain system with allowable shortage. In this research, the shortage is assumed to occur in a form of backorder. The refrigerator manufacturing can suitably be used to illustrate the scenario which makes the situation more rational and realistic. By assumptions, the customers will not go somewhere else, the cost of lost sale is not included in this case. The total cost of this tree-type distributive supply chain system is comprised of setup costs and holding costs at all echelons, and the backorder cost at retailer level only. A replenishment policy is sought to operate the system at a minimum cost. The problem is formulated as an unconstrained mixed-integer programming problem and a newly developed doubly hybrid meta-heuristic algorithm is adopted as the solution procedure. The solutions which determine the desired replenishment policy for the producer, distributors, and retailers are obtained to minimize the total system cost. A sensitivity analysis is performed to evaluate the outcome of the proposed solution procedure presented in this research. Numerical results are generated to demonstrate the solution procedure, and future research is indicated for further study.

A Multi Item Inventory Model for Deteriorating Items with Expiration Date and Allowable Shortages

Inventory decisions in supply chain are crucial for its success. These decisions become more important for the products with expiration date. Making these decisions in inventory systems with multiple products is a challenging task for managers. Most approaches in the literature for optimizing decisions in such an environment consider only a single item inventory. This paper presents a multi item inventory model to optimize the unit time profit of inventory management for the products having an expiration date after which the product can not be sold. As on one side the shortage costs are significant, on the other side, to maintain appropriate inventory levels for such type of products and avoid shortages is a very problematic job. For validation, the model is simulated and the results are compared. This article offers an approach for optimization and thus has business significance. Keywords: Deterioration, Expiration Date, Lead Time, Multi Items, Partial Backlogging, Shortages

Multi Item Inventory Model for Deteriorating Items with Expiration Date and Allowable Shortages

Inventory decisions in supply chain are crucial for its success. These decisions become more important for the products with expiration date. Making these decisions in inventory systems with multiple products is a challenging task for managers. Most approaches in the literature for optimizing decisions in such an environment consider only a single item inventory. This paper presents a multi item inventory model to optimize the unit time profit of inventory management for the products having an expiration date after which the product can not be sold. As on one side the shortage costs are significant, on the other side, to maintain appropriate inventory levels for such type of products and avoid shortages is a very problematic job. For validation, the model is simulated and the results are compared. This article offers an approach for optimization and thus has business significance.

Optimization of blood supply chain with shortened shelf lives and ABO compatibility

In clinical practice, red blood cells (RBCs) have a maximal shelf life (MSL) of 42 days. Recent studies suggest managing RBC inventory with a more restrictive MSL. To this end, a new simulation optimization (SO) framework is proposed for blood supply chain inventory management with ABO blood group compatibility. The inventory objective is to minimize the expected system outdate rate under a predetermined maximally allowable shortage level. The proposed SO framework is incorporated with a new metaheuristic optimization algorithm, TA–TS, to identify near-optimal inventory policies in reasonably acceptable computational time. The new SO framework is shown to offer a better tradeoff between solution accuracy and computational expense. The efficiency of the proposed framework is evaluated in detail for a single-hospital single-blood center supply chain system, in which the MSL of RBC units are shortened to 7, 14 and 21 days. A recently-developed replenishment policy based on old inventory ratio (the OIR policy developed by Duan and Liao (2013b)) is used to control the freshness of the entire inventory. Three different scenarios are investigated: (1) no ABO compatible substitution; (2) ABO compatible substitution at hospital only; (3) ABO compatible substitution at both hospital and blood center. Using the proposed SO framework, we are able to identify a near-optimal solution for each of these scenarios and quantify the potential savings offered by ABO compatible substitution. In the ABO compatible substitution scenarios, there is a clear trend of the increased use of group O blood. As the MSL decreases, group O blood are needed more frequently to substitute for other ABO/Rh(D) compatible blood types. Allowing ABO/Rh(D)-compatible blood substitution helps reduce the system-wide outdate at least by 16% even under the most restrictive MSL. For more relax MSL of 14 days and 21 days, the proposed framework is capable of keeping the highest system-wide outdate rate at as little as 2%.

A vendor-buyer integrated inventory model with defective items, production cost and equal-sized delivery

We propose an integrated inventory model for vendor-buyer system considering defective items and allowable shortage backordering. The shipment of lot from vendor to buyer is made in equal size. The buyer conducts inspection to ensure the quality of the received items. Here, we incorporate production cost into the model which gives the vendor the opportunity to adjust the production rate. We develop an iterative procedure to obtain the optimal solution of the delivery quantity, frequency of delivery, number of backorder and production rate. From numerical example, it can be proved that our integrated model results in lower total cost compared to the independent model. Allowing the vendor to control the production cost by adjusting the production rate will also give total cost savings to the system. The increase in the rate of the defective items produced by vendor will lead to the increase in delivery quantity and the decrease in number of backordered items and production rate.

An EPQ Inventory Model with Allowable Shortages for Deteriorating Items under Trade Credit Policy

This paper attempts to obtain the replenishment policy of a manufacturer under EPQ inventory model with backorder. It is assumed here that the manufacturer delays paying for the received goods from the supplier and the items start deteriorating as soon as they are being produced. Based on these assumptions, the manufacturer’s inventory model is formulated, and cuckoo search algorithm is applied then to find the replenishment time, order quantity, and selling price with the objective of maximizing the manufacturer’s total net profit. Besides, the traditional inventory system is shown as a special case of the proposed model in this paper, and numerical examples are given to demonstrate better performance of trade credit. These examples are also used to compare the results of cuckoo search algorithm with genetic algorithm and investigate the effects of the model parameters on its variables and net profit.

A new age-based replenishment policy for supply chain inventory optimization of highly perishable products

A simulation optimization framework is proposed for supply chain inventory management of highly perishable products. A new replenishment policy based on old inventory ratio is developed, hence called OIR policy. It is an age-based policy using only partial age information to measure the freshness of the entire inventory. The efficiency of the new policy is evaluated in detail for a single-vendor-multi-buyer platelet (with a limited shelf life of 5 days) supply chain. The inventory objective is to minimize the expected system outdate rate under a predetermined maximal allowable shortage level. The new OIR policy is compared with two existing order-up-to policies: one is the order-up-to policy without age consideration; the other one is the “EWA” policy developed by Broekmeulen and van Donselaar (2009). The three policies are compared under both decentralized and centralized controls for different levels of the fillrate constraint. The computational results show that adopting centralized control over the whole platelet supply chain greatly helps in reducing the system expected outdate rate from 19.6% down to 1.04% on average while keeping sufficiently high fill rate at each entity. The two policies with age consideration are generally better than the policy without age consideration under both control strategies. This is particularly true for decentralized control. The new OIR policy is recommended because it is the best among all three, consistently yielding good results in all cases studied.

Retailer's Optimal Ordering Policy under Conditions of Allowable Shortage and Two-Level Trade Credit Derived without Derivatives

An inventory model was developed to determine an ordering policy for the retailer under conditions of allowable shortage and two levels of delay permitted. We present a simple algebraic method to replace the use of differential calculus for determining the retailer's optimal ordering policy. A theorem is presented to obtain the optimal order quantity, and numerical examples are given to illustrate the results obtained in this paper.

Decision making in a Fuzzy Environment with allowable shortages & chance constraints for Multi - Items

Decision making in a Fuzzy Environment with allowable shortages & chance constraints for Multi - Items

Credit financing in economic ordering policies for defective items with allowable shortages

In the classical inventory models, the common unrealistic assumption is that all the items produced are of good quality in nature. However, in realistic environment, it can be observed that there may be some defective items in an ordered lot. These items are usually picked up during the screening process and are sold as a single lot at the end of screening process. Further, it is tacitly assumed that the supplier must be paid for the items as soon as the items are received. Whereas, in today business transaction, it is common to see that the retailer is allowed some grace period before they settle the account with the supplier. Under this scenario, a new inventory model for imperfect quality items has been developed under permissible delay in payments. Shortages are allowed and fully backlogged, which are eliminated during screening process as it has been assumed that screening rate is greater than the demand rate. This model jointly optimizes the order quantity and shortages by maximizing the expected total profit. Results have been validated with the help of numerical example using Matlab 7.0.1. Comprehensive sensitivity analysis has also been presented.

Preventive maintenance and optimal buffer inventory for products sold with warranty in an imperfect production system

This paper deals with an imperfect production system with allowable shortages due to regular preventive maintenance for products sold with free minimal repair warranty. Preventive maintenance is an essential factor of the just-in-time structure that results in a shutdown of the production process for a certain period of time. During such an interruption, a buffer stock is needed to adjust the market demand. The study includes the possibility of imperfect production and determines the optimum buffer level and production run time by trading off the holding cost, shortage cost, rework cost, repair cost for warranty, labour/energy costs, material cost and cost for maintenance so that the cost per unit product is minimised.

An Integrated Production-Supply System with Uncertain Demand, Nonlinear Lead Time and Allowable Shortages

A supply chain consisting of a single-supplier and a single-buyer is modeled and compared in two different modes: non-coordinated and coordinated. The model is established based on the fact that the demand is uncertain and shortages are considered as lost sales. The buyer’s order lead time is a nonlinear function of the buyer’s order size and the number of shipments from the supplier. Quantity discount offers are used as a tool to achieve the coordination between both parties. In non-coordinated mode the total annual profits of both parties are maximized using partial derivative and a lower and an upper bound are obtained for the supplier’s wholesale price. For the coordinated mode total annual profit of the whole supply chain system is maximized using partial derivatives and coordination may increase the total annual profit of the whole system is mathematically proved. In order to encourage the both parties to coordinate, a fair profit-sharing method is proposed based on the total costs that each party incurs. The supplier’s wholesale price is evaluated such that coordination seems appealing and profitable for both parties.

Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit

Abstract This study proposes a single-supplier, single-retailer integrated inventory model that accounts for defective items that arrive in a retailer's order under a full-lot inspection policy. All defective items are returned to the supplier in next delivery. After receipt of the returned items, the supplier will classify them into two types: items that still have some worth and waste items. For those items that still have some worth, the supplier will offer customers a discount in order to minimize losses arising from these defective items. The supplier needs to pay a disposal fee for those items classified as waste items. Shortages are allowed and are fully backlogged. To encourage sales the supplier offers trade credit to the retailer. A two-echelon inventory model is established, and the decision variables include: replenishment cycle time, the time taken to run out of stock and the number of lots delivered from the supplier to the retailer. An algorithm is developed to determine the optimal supply chain strategy and numerical examples are provided to show the solution procedure. Also, a sensitivity analysis is conducted on the main parameters of the model.