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Since the publication of the Goyal model in 1985, research on the modeling of inventory lot-size under trade credits has resulted in a body of literature. In this paper, we present a review of the advances in inventory literature under conditions of permissible delay in payments since 1985. We classify all related previous articles into five categories based on: (a) without deterioration, (b) with deterioration, (c) with allowable shortage, (d) linked to order quantity, and (e) with inflation. The motivations, extensions and weaknesses of various previous models have been discussed in brief to bring out pertinent information regarding model developments in the past two decades.

This paper focuses on imperfect manufacturing system with remanufacturing, manufacturing and product returned cycle for a single item which deteriorates with respect to time. The product has maximum fixed lifetime. To decrease deterioration of the product, manufacturer spends capitals on preservation technology to preserve the item. In this paper, the effect of inflation is also considered. Here, time-dependent quadratic demand is debated which is suitable for the products whose demand increases initially and afterward it starts to decrease. The objective is to minimize the total cost of manufacturer with respect to cycle time and investment for preservation technology. The model is supported with numerical example. Sensitivity analysis is done to derive insights for decision makers. Graphical result, in three dimensions, is exhibited with supervisory decision.

In this paper, we discuss why it is appropriate maximize the profits, instead of minimizing the costs, in an inventory system with an inventory-level-dependent demand rate. In addition, we restate Urban's viewpoint that the restriction of zero ending-inventory is not necessary in an inventory-level-dependent demand model. Consequently, we amend Giri and Chaudhuri's inventory model for deteriorating items by changing the objective to maximize the profits and relaxing the restriction of zero ending-inventory. Finally, we provide a couple of examples to show that both the order quantity and the profit obtained from our proposed model are significantly larger than those in Giri and Chaudhuri's model, in which the objective is to minimize the costs.

This paper studies the joint stock and production capacity rationing polices for a make-to-stock system with two partially substitutable products manufactured at two factories, respectively. Customers are classified into three types with each type further segmented into multiple classes. Type I and Type II customers are selective and accept their favorite products only. Type III accepts either of the two products and hence its demand is regarded as the flexible demand. The management has to decide whether to run or stop production and whether the various classes of demand for each type of customers have to be satisfied from available inventory or not — in which case demand is lost — in order to minimize the firm’s inventory and production costs. We formulate the problem as a make-to-stock queue system and characterize the optimal joint stock and production rationing policies as monotone switching curves. Numerical examples are given to illustrate the optimal policies and demonstrate the value of flexible demand. Moreover, we develop a simple heuristic policy and compare the costs associated with the optimal and heuristic policies for different parameters.

An inventory model is discussed with generalized ramp-type demand where the time to deterioration follows Weibull distribution. Shortages of inventories are allowed and completely backlogged. Total cost is derived by trading off setup cost, holding cost, deterioration cost, and shortage cost. The optimal replenishment policy for a single period is derived by minimizing the total cost per unit time over infinite time horizon. A numerical example is presented and sensitivity analysis is also carried out. The rationale for generalized ramp-type demand is discussed.

In the classical economic order quantity (EOQ) inventory model, it was assumed that the retailer must pay for the received items immediately. However, in practice, the supplier not only allows retailer to settle the account after a certain fixed period but also may offer a cash discount to encourage the retailer to pay for his purchases as soon as possible. On the other hand, it is common practice in most inventory systems to hold excess stocks in a rented warehouse whenever the storage capacity of the owned warehouse is insufficient. Therefore, the purpose of this paper is to establish an EOQ model with limited storage capacity, in which the supplier provides cash discount and permissible delay in payments for the retailer. In the model, we develop some useful theorems to characterize the optimal solution and provide a simple method to find the optimal replenishment cycle time and payment time. Finally, several numerical examples are given to illustrate the theoretical results and some managerial insights are also obtained.

The proposed EOQ model is a genuine attempt to manage retailer’s inventory when retailer’s stock is comprised of defective as well as constantly deteriorating items. Defective items in inventory are managed by selling them at discounted rate after a quality check process. Deterioration of items is controlled by investing suitable amount in preservation technology. The study assumes price sensitive quadratic demand incorporating the effect of inflation leading to a realistic situation. The objective of this paper is to maximize the retailer’s total profit with respect to cycle time, selling price, and preservation technology investment. Numerical examples are given to validate the model, and sensitivity analysis of inventory parameters is done to understand their effect. The outcome of this paper is applicable to goods like utility vehicles, stationary items, Fashion accessories, Cloths, Footwears, etc.

Inventory models are quite important when it comes to analyzing scenarios that occur in a variety of areas, including food, warehouses, vegetable markets, and other similar places. Within the scope of this study, we investigate both a linear and a nonlinear time-dependent inventory control model for things that are deteriorating. When it comes to conducting business in the modern day, the relationship that exists between retailers and customers is absolutely necessary for successful implementation. Consumers engage in commercial transactions with retailers in order to buy goods with the intention of increasing their earnings. The goal of this study is to calculate the number of times demand orders are placed by customers throughout the rotation time in order to achieve the highest possible overall profit for both the client and the merchant, both with and without the use of collaboration. In addition, it intends to investigate the disparity between the total profit obtained by the customer and the merchant in two different scenarios. Using analytical methods, the proposed model is evaluated in order to obtain the best possible solution to the problem. In the following section, a numerical example and a comparison study are discussed. Ultimately, a sensitivity analysis of the parameters is to be done to investigate the variations in the results of various parameters inside the model based on the optimal strategy.

Inventory models with deteriorating items have received considerable attention in recent years. In considering the deteriorating inventory with permissible delay in payments, most researchers pay attention to a single warehouse. Under conditions of permissible delay in payments, this paper develops a model to determine the optimal cycle time for a single deteriorating item that is stored in two different warehouses. A rented warehouse (RW) is used to store the excess units over the fixed capacity W of the owned warehouse (OW). The rented warehouse is assumed to charge higher unit holding cost than the OW. In this paper, we propose a two-warehouse inventory model for deteriorating items under permissible delay in payments. It is assumed that the deterioration rate in RW is the same as in OW, and the holding cost in RW is greater than that in OW. The stocks of RW are transported to OW in continuous release pattern and the transportation cost is ignored. Three theorems are developed to determine the optimal cycle time and numerical examples are given to illustrate these theorems.

This work presents an extension of the classical EOQ model for items with a fixed shelf life and a declining demand rate due to a reduction in the quality of the item in the course of its shelf-life. The demand rate, reflecting consumer preference for fresh items, is a polynomial function of the remaining time until the expiry date of the item. A mathematical model is developed to maximize the average profit per unit time, subject to shelf-life and marginal profit limitations. We prove that based on these limitations the model has a unique optimal solution and suggest a procedure for finding the optimal cycle length. A numeric example, including a sensitivity analysis, illustrates the model.

The traditional single period inventory problem assumes that the market demand is a random variable. However, as an empirical or subjective estimation, market demand is better to be regarded as an uncertain variable. This paper is concerning with single period inventory problem under two main assumptions that (i) the market demand is an uncertain variable and (ii) a setup cost and an initial stock exist. Under the framework of uncertainty theory, the optimal inventory policy for uncertain single period inventory problem with an initial stock and a setup cost is derived, which is of (s,S) type. Also, some expansions are obtained.

This paper presents a sophisticated modification of the economic production quantity (EPQ) model. It incorporates Weibull-distributed progressive deterioration, dynamic holding costs, variable demand, and strong strategies for effectively handling shortages and backlogs. Targeting products that have a short-term expiration but can still be stored for a long time without significant deterioration, the model aims to increase production quantities while lowering overall variable expenses. The extended EPQ model utilizes Maclaurin series approximations to solve ordinary differential equations, effectively capturing the dynamics of inventory under three different demand patterns. The focus of this handle’s situations when the demand for a product changes in a predictable manner over time and considers the likelihood of defective goods during the manufacturing process. This approach provides useful insights for optimizing inventory management and production planning in the manufacturing of goods that are used up or consumed. The usefulness of the model is confirmed in the study through the use of two illustrative instances. Additionally, sensitivity analysis is utilized to suggest potential areas for additional investigation.

This paper presents a stylized model to determine the optimal strategy for the integrated supplier-retailer inventory model under the condition that both the supplier and retailer have adopted a trade credit strategy. By analyzing the total channel profit function, we develop an algorithm to simultaneously determine the retailer's optimal order quantity and the number of shipment per production run from the supplier to the retailer. Our results demonstrate that the trade credit strategy is effective to supply chain system performance when customers are sensitive to the credit period length offered by the retailer. Moreover, when customers are sensitive to the credit period, if the retailer conveys partial advantage gained from the trade credit offered by the supplier to customers by suitably adjusting the customer's credit period then the entire system and every channel partner can benefit.

In this paper, an inventory model for deteriorating items with ramp-type time and price dependent consumption rate over a finite planning horizon is considered. In contrast to the traditional deterministic inventory model with static price over the entire planning horizon or fixed number of price changes over the finite time horizon, an alternative model is derived in which prices and the number of price change are to be decision variables. We show that the total profit function is concave. With the concavity, a solution procedure is presented to determine optimal prices, optimal number of pricing cycles and optimal lot size and optimal profit. We illustrate the model with numerical examples. Sensitivity analysis of the model is also carried out.

This study develops a mathematical inventory model for deteriorating item taking into account a vertical integration of a three-echelon supply chain (one supplier, one distributor, and one retailer) through strategic alliances. The objective is to minimize the joint total relevant cost for the integrated inventory model. A simple but efficient heuristic technique is used to derive the optimal solution. A numerical example and sensitivity analysis on the optimal results are presented to validate the results of the proposed integrated model. The proposed mathematical model has demonstrated how an integrated approach to decision making can achieve a global optimum and outperform three typically individual models (i.e., independent model, dominant supplier's model and dominant retailer's model).

We consider a retailer, facing uncertain supply and price-sensitive stochastic demand, who has to make stocking and pricing decisions for a given selling period. We also consider the case when the demand is price-sensitive deterministic and provide a unified framework for the model with additive errors. For both scenarios, we look at the case when the price is set before receiving the supply, called simultaneous pricing and the case when the price is set after receiving it, which is called postponed pricing. We develop a procedure for finding the optimal policy for the retailer with general distributions for the supply and the demand. To study the effect of supply uncertainty on expected profit, we conduct sensitivity analysis and develop results for both pricing scenarios and give insights. The results have important implications for a retailer in the supply chain, where a portion of the inventory may be lost due to variety of factors including mishandling and failure to meet quality standards. The findings shed light on the nature and role of prices and their relationship to supply and demand.

In this paper, we formulate a stochastic periodic-review inventory model with lead-time that considers both duration- and quantity-dependent inventory costs, and show that the optimality equation has the same format as the standard newsvendor problem with a modified demand distribution. We suggest (i) an approximation of the optimal order-up-to level, based on discretization of the inventory cost accounting, and (ii) two heuristic formulas, based on the two-moment normal approximation of the modified demand distribution, and on an approximated deterministic model. We evaluate the performances of these formulas using a Brownian motion demand process under different scenarios, and discuss their advantages and disadvantages.

Drop-shipping is a commonly adopted online-order fulfillment strategy in the Internet age. In this practice, online retailers leverage the fulfillment capabilities of suppliers to fulfill orders. On the other hand, purchase dependence is a frequent phenomenon and is characterized by the purchase of certain items together due to their unknown interior associations. Although this concept has been significantly examined in the marketing field (e.g., market basket analysis), it has largely remained unaddressed in operations management. This paper develops an $(R,T)$ model to address an environment in which unmet demand orders are partially lost and partially backordered when purchase dependence exists. The partial backorders are fulfilled by a drop-shipping option. Through computational analyses, this paper demonstrates the effect of both drop-shipping on a partial backordering and purchase dependence. The results show that more profit can be realized by utilizing a drop-shipping option under purchase dependence.