A note on the deterministic EPQ with partial backordering

In a recent paper published in Omega [Pentico DW, Drake MJ, Toews C. The deterministic EPQ with partial backordering: a new approach. Omega 2009; 37(3):624–36], the authors proposed an EPQ model with partial backordering. In the EPQ model, a critical value of backordering rate was developed, below which the optimal production policy is either to lose all sales or meet all demand, above which the optimal policy is to allow stockouts with partial backordering and meet fractional demand. However, there exists a flaw in the critical value of backordering rate, which will be amended in this note.     

On the optimality of the optimal policies for the deterministic EPQ with partial backordering

In a note published in Omega [Zhang RQ. A note on the deterministic EPQ with partial backordering. Omega 2009;37(5):1036–8], the amended decision procedure for the Pentico et al.'s EPQ with partial backordering (EPQ-PBO) is proposed, by developing another critical value of the backordering rate. However, there is a case when a decision made with this amended procedure is not optimal, which will be shown in this paper. A new decision procedure will be proposed based on the derived necessary and sufficient conditions for considering the policy of losing all sales or the policy to meet all demand as possible optimal decisions. The proposed decision procedure is adapted for one of the extensions of the EPQ-PBO.     

A note on “The EPQ with partial backordering and phase-dependent backordering rate”

Pentico et al. [The EPQ with partial backordering and phase-dependent backordering rate. Omega 2011;39(5):574-7] recently proposed an EPQ model with partial backordering. They tried to determine the minimum average total cost per year from the three cases: to lose all sales, to meet all demand, and to allow stockouts with partial backordering. In this paper, we solve the same model without differential calculus and present a different decision procedure. Without the comparison of those costs, a numerical example is illustrated to determine the optimal production policy by our criteria.     

The deterministic EPQ with partial backordering: A new approach

Several authors have developed models for the EOQ when only a percentage of stockouts will be backordered. Most of these models are complicated, with equations unlike those for the EOQ with full backordering. In this paper we extend work by Pentico and Drake [The deterministic EOQ with partial backordering: a new approach. European Journal of Operational Research 2008; in press] that developed equations for the EOQ with partial backordering that are more like those for the EOQ with full backordering to develop a comparable model for the EPQ with partial backordering.     

The EPQ with partial backordering and phase-dependent backordering rate

In previous work we established a closed-form optimal stocking strategy for an EPQ model with partial backordering at a constant rate β. Here we extend this work to allow for the possibility that the percentage of demand backordered will increase when production starts again. We show how our previous model can be adapted to find the optimal decision variable values for this new assumption and develop the condition that the initial value of β must meet for partial backordering to be optimal.     

An arborescent inventory model in a supply chain system

This study develops an arborescent (tree-like) inventory model with a constant demand rate. By considering the integration of the producer, distributors and retailers, a mathematical model and an economic ordering policy are developed. It can be shown that the integrated approach results in an impressive cost-reduction compared with an independent decision by the partial party. A numerical example of an arborescent inventory system is attached to demonstrate the economic ordering policy with the help of mathematical software - Maple V.     

Myopic Analysis for Multi‐Echelon Inventory Systems with Batch Ordering and Nonstationary/Time‐Correlated Demands

We provide an exact myopic analysis for an N‐stage serial inventory system with batch ordering, linear ordering costs, and nonstationary demands under a finite planning horizon. We characterize the optimality conditions of the myopic nested batching newsvendor (NBN) policy and the myopic independent batching newsvendor (IBN) policy, which is a single‐stage approximation. We show that echelon reorder levels under the NBN policy are upper bounds of the counterparts under both the optimal policy and the IBN policy. In particular, we find that the IBN policy has bounded deviations from the optimal policy. We further extend our results to systems with martingale model of forecast evolution (MMFE) and advance demand information. Moreover, we provide a recursive computing procedure and optimality conditions for both heuristics which dramatically reduces computational complexity. We also find that the NBN problem under the MMFE faced by one stage has one more dimension for the forecast demand than the one faced by its downstream stage and that the NBN policy is optimal for systems with advance demand information and stationary problem data. Numerical studies demonstrate that the IBN policy outperforms on average the NBN policy over all tested instances when their optimality conditions are violated.     

Constrained production lot-size model with trade-credit policy: 'A comparison geometric programming approach via Lagrange'

In this paper our main objective is to investigate a deterministic inventory production lot-size model with a permissible delay in payment under a restriction. We analyse our deterministic inventory model under a restriction which will be assumed as the average inventory level. In fact we use in our analysis two approaches: the geometric programming approach; and the Lagrange method. Then a comparison between these two approaches is performed, which is our aim. Finally we deduce some previously published works of other researchers as special cases.     

Sourcing from Multiple Suppliers for Price‐Dependent Demands

We analyze a model that integrates demand shaping via dynamic pricing and risk mitigation via supply diversification. The firm under consideration replenishes a certain product from a set of capacitated suppliers for a price‐dependent demand in each period. Under deterministic capacities, we derive a multilevel base stock list price policy and establish the optimality of cost‐based supplier selection, that is, ordering from a cheaper source before more expensive ones. With general random capacities, however, neither result holds. While it is optimal to price low for a high inventory level, the optimal order quantities are not monotone with respect to the inventory level. In general, a near reorder‐point policy should be followed. Specifically, there is a reorder point for each supplier such that no order is issued to him when the inventory level is above this point and a positive order is placed almost everywhere when the inventory level is below this point. Under this policy, it may be profitable to order exclusively from the most expensive source. We characterize conditions under which a strict reorder‐point policy and a cost‐based supplier‐selection criterion become optimal. Moreover, we quantify the benefit from dynamic pricing, as opposed to static pricing, and the benefit from multiple sourcing, as opposed to single sourcing. We show that these two strategies exhibit a substitutable relationship. Dynamic pricing is less effective under multiple sourcing than under single sourcing, and supplier diversification is less valuable with price adjustments than without. Under limited supply, dynamic pricing yields a robust, long‐term profit improvement. The value of supply diversification, in contrast, mainly comes from added capacities and is most significant in the short run.     

A New Two‐Bin Policy for Inventory Systems with Differentiated Demand Classes

We consider an inventory system under continuous review with two demand classes that are different in terms of service level required (or penalty cost incurred for backordering of demand). Prior literature has proposed the critical level rationing (CLR) policy under which the demand from the lower priority class is backordered once inventory falls below the critical level. While this reduces the penalty cost for the higher demand class, the fill rate achieved for the lower priority demand class gets compromised. In this study, we propose a new class of two‐bin (2B) policy for the problem. The proposed 2B policy assigns separate bins of inventory for the two demand classes. The demand for each class is fulfilled from its assigned bin. However, when the bin intended for the higher demand class is empty, the demand from the higher class can still be fulfilled with the inventory from the other bin. The advantage of the 2B policy is that better fill rates are achieved, especially for the lower demand class. Computational results show that the proposed policy is able to provide a much higher service level for the lower priority class demand without increasing the total cost too much and without affecting the service level for the higher priority class. When a service level constrained optimization problem is considered, the 2B policy dominates the CLR policy when the service level difference for the two classes is not too high or the service levels required for both the classes are relatively lower.     

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.     

无外部市场条件下中间产品转移价格的博弈分析与决策

本文首先用反例证明了文^[1]中在各分厂成本函数和最终产品需求函数已知的条件下,给出的中间产品最优转移价格的两个计算公式并不总是成立的,给出了这两个计算公式成立的充要条件,并分析了公司关于中间产品转移价格决策的动态一致性问题。接着,利用完全信息动态博弈对中间产品转移价格的确定问题做了进一步的分析研究。最后,针对文^[1]中给出的中间产品最优转移价格的两个计算公式不成立的情况,讨论了中间产品转移价格的确定问题。     

Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost

In this paper, an economic order quantity inventory model is analyzed, considering that the unit cumulative holding cost has two significant components: a fixed cost which represents the cost of accommodating the item in the warehouse and a variable cost given by a potential function of the length of time over which the item is held in stock. Shortages are allowed and, during the stockout period, only a fraction of demand is partially backordered. The backordering cost includes a fixed cost and a cost linearly dependent on the length of time for which backorder exists. A solution procedure is developed for determining the optimal inventory policy. Moreover, to illustrate the effects of some parameters on the optimal policy and the minimum total inventory cost, a numerical study is developed.     

An EOQ inventory model for items with Weibull distribution deterioration,ramp type demand rate and partial backlogging

In this paper, an EOQ inventory model is presented depleted not only by demand but also by Weibull distribution deterioration, in which the demand rate is assumed that with a ramp type function of time. In the model, shortages are allowed partial backlogging and the backlogging rate is variable and is dependent on waiting time for the next replenishment. The method is illustrated by three numerical examples, and sensitivity analysis of the optimal solution with respect to parameters of the system is carried out.     

A mixed inventory model with variable lead time and random supplier capacity

In this paper, we have studied analytically the implication of a controllable lead-time and a random supplier capacity on the continuous review inventory policy, in which the order quantity, reorder point and lead-time are decision variables. Two models are considered: the normal lead-time demand and lead-time demand is distributed free. For both cases, after formulating the general model, some properties of the optimal ordering policy have been developed. Particularly, we have shown that the expected annual total cost is a unimodal function and quasi-convex in the order quantity. When the variable capacity distribution is exponential, we develop effective procedures for finding the optimal solutions. Furthermore, the effects of parameters are also performed.     

Point‐of‐Use Hybrid Inventory Policy for Hospitals

Modern point‐of‐use technology at hospitals has enabled new replenishment policies for medical supplies. One of these new policies, which we call the hybrid policy, is currently in use at a large U.S. Midwest hospital. The hybrid policy combines a low‐cost periodic replenishment epoch with a high‐cost continuous replenishment option to avoid costly stockouts. We study this new hybrid policy under deterministic and stochastic demand. We develop a parameter search engine using simulation to optimize the long‐run average cost per unit time and, via a computational study, we provide insights on the benefits (reduction in cost, inventory, and number of replenishments) that hospitals may obtain by using the hybrid policy instead of the commonly used periodic policies. We also use the optimal hybrid policy parameters from the deterministic analysis to propose approximate expressions for the stochastic hybrid policy parameters that can be easily used by hospital management.      

Impact of Demand Uncertainty on Stability of Supplier Alliances in Assembly Models

In their recent paper, Nagarajan and Sošić study an assembly supply chain in which n suppliers sell complementary components to a downstream assembler, who faces a price‐sensitive deterministic demand. Suppliers may form alliances, and each alliance then sells a kit of components to the assembler and determines the price for that kit. The assembler buys the components (kits) from the alliances and sets the selling price of the product. Nagarajan and Sošić consider three modes of competition—supplier Stackelberg, vertical Nash (VN), and assembler Stackelberg models—which correspond to different power structures in the market, and study stable supplier alliances when the assembler faces linear and isoelastic demand. In this paper, we study the impact that demand uncertainty has on stability results obtained in Nagarajan and Sošić. We first analyze models in which all decisions are made before the uncertainty is resolved, and show that the alliance of all suppliers remains stable when demand is isoelastic, or under Stackelberg models when demand is linear. However, demand uncertainty may change stability results when both parties make decisions simultaneously (VN model) and demand is linear. We then extend our results by considering scenarios in which some decisions may be postponed and made after the actual demand is known. When the ordering quantity can be determined after observing the true demand, we show that stable outcomes correspond to those obtained in the deterministic case and uncertainty has no impact on coalition stability; if only the assembler's pricing decision is postponed, we need additional conditions for stability results to carry over in the additive demand model.     

The impact of distributions of uncertain lumpy demand on inventories

Most studies on inventory management under demand uncertainty analyse the detrimental effects of demand variability (i.e. its coefficient of variation). In contrast, the consequences of the shape of demand distribution (e.g. its multi-modality and asymmetry) have received only minor attention. The aim of the present paper is to assess the impact of the shape of demand distribution on inventories. In particular, it considers contexts in which demand distributions have more than one 'mode' and are asymmetric, thus making requirements lumpy. The analysis is based on a simulation model which, in order to ensure comparable conditions, keeps all the other factors constant (i.e. ceteris paribus), and compares different demand distributions with the same mean and coefficient of variation. Results show that that demand shape is a key driver of inventory investment. Specifically: (i) the magnitude of the impact of multi-modality and asymmetry is comparable to that of variability; (ii) multi-modality and asymmetry can have either a negative or positive effect depending on the target service level sought. The managerial implications of these findings are addressed in the conclusions.     

Comparing reactive Kanban and reactive CONWIP

This paper compares a reactive Kanban system to a reactive CONWIP system under conditions of unstable changes in demand using simulation experiments. After an introduction, a model of the JIT ordering systems, the Kanban and the CONWIP systems, is constructed. In order to obtain the fundamental information for developing a control rule of buffer size, the performance of the two types of the JIT ordering systems is analysed under various stable-demand conditions by simulation experiments. Based on the results, the reactive JIT ordering systems are proposed, and the performance of the proposed systems is investigated. The results showed that both of the proposed systems can react to unstable changes in demand and maintain the mean waiting time of demand at less than the required level. In the reactive Kanban system, the total of the mean work-in-process inventories becomes much less than that in the traditional Kanban system without controlling buffer size. However, in the reactive CONWIP system, the total of the mean work-in-process inventories becomes much more than or nearly equal to that in the traditional CONWIP system without controlling buffer size under the strongly correlated or the weakly correlated processing times, respectively. Based on the results, it can be claimed that, in the proposed systems, the reactive Kanban system is more effective to react to unstable changes in demand than the reactive CONWIP system.     

EOQ with independent endogenous supply disruptions

We consider an inventory installation, controlled by the periodic review base stock (S, T) policy and facing a fixed-rate deterministic demand which, if unsatisfied, is backordered. The supply process is unreliable, so supply deliveries may fail according to an independent Bernoulli process; we refer to such failures reflecting the supply service quality and being internal to the supply chain, as endogenous disruptions. We seek to jointly determine the two policy variables, so to minimize long-run average cost. While an approximate model for this problem was recently analyzed, we present an exact analysis, valid for two common accounting schemes for inventory cost evaluation: continuous and end-of-cycle costing. After developing a unified (and exact) average cost model for both costing schemes, the cost for each scheme is analyzed. In both cases, the optimal policy variables and cost prevail in closed-form, having an identical structure to those of EOQ (with backorders). In fact, under continuous costing, the optimal solution reduces to EOQ for perfect supply. Analytical properties, demonstrating the impact of deteriorating supply quality on the optimal policy, are established. Moreover, computations reveal the cost impact of deploying heuristics that either ignore supply disruptions or rely on inaccurate costing information.