Deterministic inventory lot-size models with shortages for fluctuating demand and unit purchase cost

The classical economic order quantity (EOQ) model assumes not only a constant demand rate but also a fixed unit purchasing cost. In today's time‐based competition, the unit cost of a high‐tech product declines significantly over its short product life cycle while its demand increases. Therefore, using the classical EOQ formulation for a high‐tech product will cause varying magnitudes of error. In addition, the cost of purchases as a percentage of sales is often substantial. Consequently, adding the purchasing strategy into the EOQ model is vital. In this paper, we assume that not only the demand function but also the unit purchase cost is fluctuating with time. We then provide an easy‐to‐use algorithm to find the optimal replenishment number and schedule. In a numerical example, we show that the total cost obtained by our proposed model is 32.4% less expensive than that obtained by the classical EOQ model.     

A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate

The purpose of this paper is to extend [Ouyang, L. Y., Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13] periodic review inventory model with variable lead time by considering the fuzziness of expected demand shortage and backorder rate. We fuzzify the expected shortage quantity at the end of cycle and the backorder (or lost sales) rate, and then obtain the fuzzy total expected annual cost. Using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. For the proposed model, we provide a solution procedure to find the optimal review period and optimal lead time in the fuzzy sense so that the total expected annual cost in the fuzzy sense has a minimum value. Furthermore, a numerical example is provided and the results of fuzzy and crisp models are compared.     

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.     

Discounting decision for enterprises with high fixed cost and low variable cost

Higher sales are the key to increased profit, especially for low variable cost industries. The use of quantity discount to promote business is an important way to increase sales. This study evaluates the use of discount, prepaid, multiple‐purchase credit to increase sales. The purpose is to develop a strategy to maximize the unit revenue by simultaneously determining the discount rate and utility. A numerical example and sensitivity analysis are given to illustrate the theory.     

Inventory model with a mixture of back orders and lost sales

Very few researchers have considered inventory models with partial backlogging. The models developed earlier considered a fraction of demand to be backordered while the remaining fraction is lost during the stock out period. In this paper we have developed an inventory model with partial backlogging, redefining the demand rate at a particular instant as a function of the amount of orders already backlogged at that instant of time. Infinite replenishment rate and zero lead time are assumed. Expressions for optimum order quantity and optimuim value of maximum inventory are obtained by minimizing the total system costs. The model is illustrated with a numerical example, including sensitivity analysis with respect to the backlogging parameter.     

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.     

The (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">l</Emphasis>) model for stochastic demand under Intuitionistic fuzzy aggregation with Bonferroni mean

This paper investigates a hill type economic production-inventory quantity (EPIQ) model with variable lead-time, order size and reorder point for uncertain demand. The average expected cost function is formulated by trading off costs of lead-time, inventory, lost sale and partial backordering. Due to the nature of the demand function, the frequent peak (maximum) and valley (minimum) of the expected cost function occur within a specific range of lead time. The aim of this paper is to search the lowest valley of all the valley points (minimum objective values) under fuzzy stochastic demand rate. We consider Intuitionistic fuzzy sets for the parameters and used Intuitionistic Fuzzy Aggregation Bonferroni mean for the defuzzification of the hill type EPIQ model. Finally, numerical examples and graphical illustrations are made to justify the model.     

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.     

A mixed inventory model with variable lead time and random back-order rate

In this paper, we study the determination of the optimal lead time, reorder point and order quantity considering that the back-order probability of a demand made during a stock-out period depends on the interval from the moment in which the order is placed until the next replenishment. Two models are analysed for the specification of the back-order probability: exponential functions and piecewise constant functions. The distribution of the lead time demand is assumed to be Poisson. An algorithm for the determination of the optimal order quantity, reorder point and lead time is given. A numerical example is presented to illustrate the results.     

Mixture inventory model with back orders and lost sales for variable lead time demand with the mixtures of normal distribution

In recent papers by Ben-Daya and Raouf and by Ouyang et al. a continuous review inventory model is presented in which they considered both the lead time and the order quantity as decision variables. When the demands of the different customers do not have identical lead times, then we cannot use only a distribution (such as Ouyang et al. who used a normal distribution) to describe the demand of the lead time. Hence, we have extended the model of Ouyang et al. by considering the mixtures of normal distribution (see the book by Everitt and Hand). In addition, we also still assume that shortages are allowed. Moreover, the total amount of stock-out is considered as a mixture of back orders and lost sales during the stock-out period. Moreover, we also develop an algorithmic procedure to find the optimal order quantity and optimal lead time; the effects of parameters are also studied.     

Coordinating orders in a two echelon supply chain with controllable lead time and ordering cost using the credit period

In this research, a coordination mechanism based on a credit period in a two echelon supply chain with one buyer and one supplier, is designed. The buyer is faced with uncertain demand by coping with normal distribution. Both lead time and ordering cost for receiving his order can be reduced at an added cost; in other words, they are controllable. The optimization models with and without integration are proposed. Then a way to coordinate orders in supply chain based on the credit period so that the total cost of supply chain would be minimized is designed. By using this mechanism we also discuss how the credit period is to be determined in order to achieve channel coordination and a win-win outcome. Finally, numerical examples are solved to illustrate the theoretical results and obtain the managerial insights.     

A distribution-free newsvendor problem with nonlinear holding cost

In this paper, we analyse a single-period newsvendor model to determine the optimal order quantity where the customers’ balking occurs.This scenario occurs when the customers are opposed to buy a product for various reasons, such as decreasing quality of product, product is not as good as fresh when it reaches under a threshold level, etc. The model is investigated by assuming that the holding cost function depends on order quantity and the inventory level at which customer balking occurs depends on holding cost. The model allows partial backlogging and permits part of the backlogged shortages to turn into lost sales. We develop the model without taking any specific distributional form of demand, only assuming the mean and the variance of the distribution of demand. Finally, we illustrate the model by numerical examples and compare our distribution-free model with the specific distributional form of demand.     

Capacity-constrained supplier selection model with lost sales under stochastic demand behaviour

In today’s market conditions, volume of demand is quite uncertain and thus it is hard to estimate. In many cases, buyer is prone to use supply chain flexibility rather than inventory holding strategy to withstand demand uncertainty. We assume that the buyer releases a replenishment order to the supplier for each cycle (or period) under the contract which is mainly composed of four parameters: (1) supply cost per unit, (2) minimum order quantity, (3) order quantity reduction penalty and (4) maximum capacity of the supplier. Based on these parameters, there are two flexibility options that buyer should evaluate in the order of cycle (1) issue an order smaller than the minimum order quantity and pay the related penalty and (2) place no order and lose the sales. Hence, Q _lost emerges as a critical buyer decision, the order quantity, below which no order is placed. Total expected supply cost plus lost sales, as a function of Q _lost is presented. We derive the optimal Q _lost that minimises the total cost function. Since capacity of each supplier is finite, we then develop a supplier selection model with total cost minimisation over the suppliers subject to capacity constraint that has a stochastic nature stemming from demand behaviour. Linearisation on the model is performed using chance-constrained programming approach. From a given set of supply bids from the potential supply chain partners, the buyer is able to make a quantifiable choice.     

A hybrid of monopoly and perfect competition model for hi-tech products

For Hi-tech products, the demand rate, the component cost as well as the selling price usually decline significantly with time. In the case of perfect competition, shortages usually result in lost sales; while in a monopoly, shortages will be completely backordered. However, neither perfect competition nor monopoly exists. Therefore, there is a need to develop a replenishment model considering a hybrid of perfect competition and monopoly when the cost, price and demand are decreasing simultaneously. A numerical example and sensitivity analysis are carried out to illustrate this model. The results show that a higher decline-rate in the component cost leads to a smaller service level and a larger replenishment interval. When the component cost decline rate increases and the selling price decline rate decreases simultaneously, the replenishment interval decreases. In perfect competition it is better to have a high service level, while for the case with monopoly, keeping a low service level is better due to complete backordering.     

An inventory replenishment policy for deteriorating items with shortages and partial backlogging

It has long been assumed that the shortages in inventory systems are either completely backlogged or totally lost. However, it is more reasonable to characterize that the longer the waiting time for the next replenishment, the smaller the backlogging rate would be. Moreover, the opportunity cost due to lost sales should be considered since some customers would not like to wait for backlogging during the shortage periods. Without considering these two realistic conditions, study on the inventory modeling for deteriorating items with shortages and partial backlogging cannot be complete and general. In the present article we define an appropriate time-dependent partial backlogging rate and introduce the opportunity cost due to lost sales. Numerical examples are also presented to illustrate the effects of changes in backlogging parameter and unit opportunity cost on total cost and the optimal number of replenishments.Scope and purposeIn a recent article published in this Journal, Giri et al. (Comput. Oper. Res. 27 (2000) 495–505) implemented an existing procedure to the inventory problem of Hariga and Al-Alyan (Comput. Oper. Res. 24 (1997) 1075–83) which concerns with lot-sizing heuristic for deteriorating items with shortages allowed in all cycles except the last one. Giri et al. deviated from the traditional practice and suggested a new policy allowing shortages in all cycles over a finite planning horizon. Their numerical results indicated the proposed policy is cheaper to operate with a cost reduction up to 15%. However, they did not consider the opportunity cost due to lost sales that happen because customers would not like to wait for backlogging. Moreover, for many products with growing sales, the length of the waiting time for the next replenishment is the main factor for determining whether the backlogging will be accepted or not, and the backlogging rate is expected to be time-dependent. Thus the assumption made in Giri et al. that the backlogging rate is a fixed fraction of the total amount of shortages is not reasonable.The purpose of this paper is to present a more realistic discussion of the inventory problem for deteriorating items with time-varying demands and shortages over a finite planning horizon. In contrast to the model by Giri et al., we define an appropriate partial backlogging rate and introduce the opportunity cost due to lost sales. We attempt to complement their model as a practical and general solution for inventory replenishment problems. With these extensions, the scope of applications of the present results is expanded.     

Optimal production lot size and reorder point of a two-stage supply chain while random demand is sensitive with sales teams' initiatives

The paper develops a production-inventory model of a two-stage supply chain consisting of one manufacturer and one retailer to study production lot size/order quantity, reorder point sales teams’ initiatives where demand of the end customers is dependent on random variable and sales teams’ initiatives simultaneously. The manufacturer produces the order quantity of the retailer at one lot in which the procurement cost per unit quantity follows a realistic convex function of production lot size. In the chain, the cost of sales team's initiatives/promotion efforts and wholesale price of the manufacturer are negotiated at the points such that their optimum profits reached nearer to their target profits. This study suggests to the management of firms to determine the optimal order quantity/production quantity, reorder point and sales teams’ initiatives/promotional effort in order to achieve their maximum profits. An analytical method is applied to determine the optimal values of the decision variables. Finally, numerical examples with its graphical presentation and sensitivity analysis of the key parameters are presented to illustrate more insights of the model.     

Disruption management for a dominant retailer with constant demand-stimulating service cost

In this paper, we consider coordination model of a one-manufacturer and multi-retailer supply chain with a dominant retailer’s sales promotion opportunity and possible demand disruption. An appropriate contractual scheme can be used to fully coordinate the supply chain even if the demand disruption occurs. In our study, we also analyze how the demand disruption affects the coordination mechanism. When the demand is disrupted, the manufacturer only needs to adjust the maximum variable wholesale price and the subsidy rate under the linear quantity discount scheme. For each case of the demand disruption, we find that the higher the market share of the dominant retailer, the lower its average wholesale price will be. Meanwhile, the higher service cost leads to the higher subsidy rate provided by the manufacturer. The optimal wholesale/retail price, order quantity and subsidy rate can be greatly influenced by the demand disruption. If the disrupted amount of demand is sufficiently small, however, the manufacturer needs to take some special measures to prevent the retailers from deviating the order quantity of the original plan. To demonstrate these findings, we illustrate our propositions by numerical examples.     

Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.     

(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.