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.     

Partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation

In this paper, the traditional inventory lot-size model is extended to allow not only for general partial backlogging rate but also for inflation. The assumptions of equal cycle length and constant shortage length imposed in the model developed by Moon et al. [Moon, I., Giri, B.C., Ko, B., 2005. Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting, European Journal of Operational Research 162(3), 773–785] are also relaxed. For any given number of replenishment cycles the existence of a unique optimal replenishment schedule is proved and further the convexity of the total cost function of the inventory system in the number of replenishments is established. The theoretical results here amend those in Yang et al. [Yang, H.L., Teng, J.T., Chern, M.S., 2001. Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand, Naval Research Logistics 48(2), 144–158] and provide the solution to those two counterexamples by Skouri and Papachristos [Skouri, K., Papachristos, S., 2002. Note on “deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand” by Yang et al. Naval Research Logistics 49(5), 527–529.]. Finally we propose an algorithm to find the solution, and obtain some managerial results by using sensitivity analyses.     

Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging

In this paper, a deterministic inventory model for deteriorating items with price-dependent demand is developed. The demand and deterioration rates are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. Under these assumptions, for any given selling price, we first develop the criterion for the optimal solution for the replenishment schedule, and prove that the optimal replenishment policy not only exists but also is unique. If the criterion is not satisfied, the inventory system should not be operated. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use numerical examples to illustrate the algorithm.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

需求随价格变化的具有折扣的易变质物品的库存模型

本文研究了在需求随价格变化及物品易变质的条件下，当供应商给予数量折扣时的库存问题。证明了当供应商给予数量折扣时，零售商的需求量是增大的，并给出了供应商给予数量折扣时零售商的订货量和订货周期的计算方法。对物品变质率和需求价格敏感系数对零售商的订货量、订货周期、出售价格和单位时间利润的影响进行了数值分析，并给出了数值算例。     

一个带有增加量价格折扣的两货栈库存模型

已有的确定性两货栈(其中一个是自己货栈(OW);另一个是租用货栈(RW))库存模型通常不考虑增加量价格折扣,然而在实际生活中,增加量价格折扣却是促使库存管理者加大订货量的一个重要原因.本文通过考虑增加量价格折扣而将两货栈系统作了进一步扩展,在采用间隔式运输模式运送RW的物品到OW的情形下,建立了一个带有增加量价格折扣并允许短缺的两货栈库存模型,提供了一种寻求最优库存策略的简单方法.     

Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate

In this paper, a deterministic inventory model for deteriorating items with two warehouses is developed. A rented warehouse is used when the ordering quantity exceeds the limited capacity of the owned warehouse, and it is assumed that deterioration rates of items in the two warehouses may be different. In addition, we allow for shortages in the owned warehouse and assume that the backlogging demand rate is dependent on the duration of the stockout. We obtain the condition when to rent the warehouse and provide simple solution procedures for finding the maximum total profit per unit time. Further, we use a numerical example to illustrate the model and conclude the paper with suggestions for possible future research.     

考虑随机模糊缺陷率且允许缺货的EOQ模型

针对实际库存管理中的产品缺陷问题,研究了含随机模糊缺陷率且允许缺货的经济订购批量(EOQ)模型,并运用随机模糊理论将其转化为确定模型,设计了随机模糊模拟仿真算法进而确定了其最优订购策略.数值算例分析了缺陷率对最优订货量和最优利润的影响.     

Optimal order size to take advantage of a one-time discount offer with allowed backorders

In this paper, we develop an inventory model for determining the optimal ordering policies for a buyer who operates an inventory policy based on an EOQ-type model with planned backorders when the supplier offers a temporary fixed-percentage discount and has specified a minimum quantity of additional units to purchase. A distinguishing feature of the model is that both fixed and linear backorder costs are included, whereas previous works include only the linear backordering cost. A numerical study is performed to provide insight into the behavior of the model.     

An optimal replenishment policy for deteriorating items with stock-dependent demand and relaxed terminal conditions under limited storage space

This paper deals with the problem of determining the optimal replenishment policy for deteriorating items with stock-dependent demand in which the terminal condition of zero-ending inventory is relaxed. In the model, shortages are allowed and partial backlogging/lost sales. That is, the zero/non-zero ending inventory models are considered simultaneously. The items in stock are displayed to the customers in shelves with limited storage capacity. In theoretical analysis, the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions under various cases are shown. We then also provide a simple algorithm to find the optimal solutions for various situations. Further, a couple of numerical examples are presented to demonstrate the developed model and solution procedure, and several management insights are obtained from the numerical examples. Finally, sensitivity analysis of the optimal solution with respect to major parameters is also carried out.     

The optimal pricing and ordering policy for an integrated inventory model when trade credit linked to order quantity

In traditional inventory models, it is implicitly assumed that the buyer must pay for the purchased items as soon as they have been received. However, in many practical situations, the vendor is willing to provide the buyer with a permissible delay period when the buyer’s order quantity exceeds a given threshold. Therefore, to incorporate the concept of vendor–buyer integration and order-size-dependent trade credit, we present a stylized model to determine the optimal strategy for an integrated vendor–buyer inventory system under the condition of trade credit linked to the order quantity, where the demand rate is considered to be a decreasing function of the retail price. By analyzing the total channel profit function, we developed some useful results to characterize the optimal solution and provide an iterative algorithm to find the retail price, buyer’s order quantity, and the numbers of shipment per production run from the vendor to the buyer. Numerical examples and sensitivity analysis are given to illustrate the theoretical results, and some managerial insights are also obtained.     

Joint optimization of price and order quantity with shortages for a two-warehouse system

In developing the best strategy for real-world applications, the vendor must have some knowledge of the buyers’ behavior such as response to shortages and price increases. Using this knowledge, he can develop a policy that will ensure the largest net profit. Considering the fact, a two-warehousing inventory model has been developed where the demand is price-sensitive under the bulk release rule. Stockouts are allowed and are fully backlogged. Moreover, the transportation cost is taken to be dependent on the transported units. The model jointly optimizes the selling price and the order quantity by maximizing the system profit. Results have been validated with the help of a numerical example.      

Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments

This paper deals with an economic production quantity (EPQ) inventory model with reworkable defective items when a given multi-shipment policy is used. In this work, it is assumed that in each cycle, the rework process of all defective items starts when the regular production process finishes. After the rework process, a portion of reworked items fails. This portion becomes scrap and only the perfect finished items can be delivered to customers at the end of rework process. A profit function is derived to model the inventory problem and it is shown that the profit function is concave. Due to the complexity of the optimization problem, an algorithm is developed to determine the optimal values of manufacturing lot size and price such that the long-run average profit function is maximized. Furthermore, two special cases are identified and explained. Finally, a numerical example is given to illustrate the applicability of the proposed inventory model.     

Reacting in real-time to production contingencies in a capacitated flexible cell

We present a PC-based procedure for the real-time revision of production and inventory schedules in a capacitated flexible manufacturing cell, so as to counter the impact of production contingencies (defined as unanticipated disturbances that force a revision in the schedule). In capacity constrained production systems, a reallocation of capacity among the various manufactured items coupled with a backordering of the end-item must occur to counter such contingencies. However, by taking advantage of existing and planned inventories in the current schedule generated using Materials Requirement Planning (MRP), it is possible to delay the onset of these end-item backorders, and provide the scheduler with additional time to explore other corrective measures such as using overtime and sub-contracting, before resorting to backordering. The objectives in our problem are to delay the end-item backorders as far as possible, minimize the magnitude of these backorders, and maximize the savings in inventory costs that may result from these backorders. The algorithm is capable of handling problems of large size and is computationally efficient. The implementation of the algorithm is extremely simple and requires only three items of data input in an open format as prompted by the system.     

An inventory model where backordered demand ratio is exponentially decreasing with the waiting time

We analyze an inventory system with a mixture of backorders and lost sales, where the backordered demand rate is an exponential function of time the customers wait before receiving the item. Stockout costs (backorder cost and lost sales cost) include a fixed cost and a cost proportional to the length of the shortage period. A procedure for determining the optimal policy and the maximum inventory profit is presented. This work extends several inventory models of the existing literature.     

需求依赖于服务水平的易变质品库存策略研究

本文研究需求依赖于上一周期服务水平、缺货时订单部分损失的两周期易变质品库存问题。分别考虑一次订货和多次订货两种情况,以平均利润最大化为目标构建库存模型,证明了模型解的存在性和唯一性,得到了最优库存服务水平和最优补货策略。最后,通过算例给出两个模型的应用,对重要参数进行了灵敏度分析,并且将两种模型的结果进行了对比分析。结果表明:订单损失率的增加会提高服务水平,但会使得利润降低;顾客期望服务水平的提高会降低第一阶段的服务水平,同时使利润减少;单位库存持有成本或变质率的增加会降低服务水平和平均利润。通常情况,企业通过多次订货能获得更大的利润,而只有当库存持有成本极小时,一次订购才能够获得更大的利润。同时,结果也表明:服务水平对库存策略有较大的影响,因此在进行库存决策时考虑服务水平具有重要的作用。     

An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting

This paper derives an inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting over a finite planning horizon. We show that the total cost function is convex. With the convexity, a simple solution algorithm is presented to determine the optimal order quantity and the optimal interval of the total cost function. The results are discussed with a numerical example and particular cases of the model are discussed in brief. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.     

Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns

This paper deals with an economic production quantity inventory model for non-instantaneous deteriorating items under inflationary conditions considering customer returns. We adopt a price- and time-dependent demand function. Also, the customer returns are considered as a function of both price and demand. The effects of time value of money are studied using the Discounted Cash Flow approach. The main objective is to determine the optimal selling price, the optimal replenishment cycles, and the optimal production quantity simultaneously such that the present value of total profit is maximized. An efficient algorithm is presented to find the optimal solution. Finally, numerical examples are provided to solve the presented inventory model using our proposed algorithm, which is further clarified through a sensitivity analysis. The results of analysing customer returns provide important suggestions to financial managers who use price as a control to match the quantity sold to inventory while maximizing revenues. The paper ends with a conclusion and an outlook to future studies.     

Optimal inventory policy for specialty chemical products in multiple packages

Many products are inventoried and sold in multiple outer packages, which causes the manufacturer or distributor to maintain a segmented inventory. Oftentimes, this scenario leads to shortages in some packages and over-stock in other package types. The shortages are usually linked to specific shortage costs, and the over-stock is always associated with some sort of unnecessary holding cost that either erodes profit margins or results in increased costs for the consumer. In this paper, the inventory policy for a specialty chemical manufacturer's cleaner, which is inventoried in four unique packages, is studied. A mathematical model is developed to account for the costs associated with initial procurement, holding and repackaging (shortage), which are the primary costs associated with the inventory system. Based upon this model, an optimal inventory policy is generated that reduces the need for repackaging, without requiring a prohibitive amount of safety stock. A specific historical case is summarized, and the results of this example are compared with the results that the new inventory policy would have produced, given the same circumstances.     

Optimizing multi-item multi-period inventory control system with discounted cash flow and inflation: Two calibrated meta-heuristic algorithms

A mixed binary integer mathematical programming model is developed in this paper for ordering items in multi-item multi-period inventory control systems, in which unit and incremental quantity discounts as well as interest and inflation factors are considered. Although the demand rates are assumed deterministic, they may vary in different periods. The situation considered for the problem at hand is similar to a seasonal inventory control model in which orders and sales happen in a given season. To make the model more realistic, three types of constraints including storage space, budget, and order quantity are simultaneously considered. The goal is to find optimal order quantities of the products so that the net present value of total system cost over a finite planning horizon is minimized. Since the model is NP-hard, a genetic algorithm (GA) is presented to solve the proposed mathematical problem. Further, since no benchmarks can be found in the literature to assess the performance of the proposed algorithm, a branch and bound and a simulated annealing (SA) algorithm are employed to solve the problem as well. In addition, to make the algorithms more effective, the Taguchi method is utilized to tune different parameters of GA and SA algorithms. At the end, some numerical examples are generated to analyze and to statistically and graphically compare the performances of the proposed solving algorithms.