Improving service rate for a tree-type three-echelon supply chain system with backorders at retailer's level

This research deals with a distributive or tree-type three-echelon production-distribution supply chain system with allowable backorder. Allowing backorder could reduce the total of a production-distribution system by reducing holding cost due to the lower average inventory, even though backorders carry some cost and lower a company's goodwill. The main purpose of this research is to develop replenishment policies for a tree-type three-echelon supply chain system with allowable backorder. The supply chain network is composed of a producer, multiple distributors, and multiple retailers. This research attempts to improve service rate by reducing the backorder at the retailer level. The distributors are allowed to ship product to retailer quicker in order to reduce backorder. The total cost function of the proposed model is developed. Since the total cost function contains some integer variables, differentiating the function with respect to the variables could not be used as a basis to solve for the optimal solutions. A branching search process was utilized to find the integer solutions. A numerical example is used to demonstrate improvement in service rate and total cost using the model.     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

Optimal myopic policy for a stochastic inventory problem with fixed and proportional backorder costs

In this paper, we consider a single product, periodic review, stochastic demand inventory problem where backorders are allowed and penalized via fixed and proportional backorder costs simultaneously. Fixed backorder cost associates a one-shot penalty with stockout situations whereas proportional backorder cost corresponds to a penalty for each demanded but yet waiting to be satisfied item. We discuss the optimality of a myopic base-stock policy for the infinite horizon case. Critical number of the infinite horizon myopic policy, i.e., the base-stock level, is denoted by S. If the initial inventory is below S then the optimal policy is myopic in general, i.e., regardless of the values of model parameters and demand density. Otherwise, the sufficient condition for a myopic optimum requires some restrictions on demand density or parameter values. However, this sufficient condition is not very restrictive, in the sense that it holds immediately for Erlang demand density family. We also show that the value of S can be computed easily for the case of Erlang demand. This special case is important since most real-life demand densities with coefficient of variation not exceeding unity can well be represented by an Erlang density. Thus, the myopic policy may be considered as an approximate solution, if the exact policy is intractable. Finally, we comment on a generalization of this study for the case of phase-type demands, and identify some related research problems which utilize the results presented here.     

考虑时值及通货膨胀率的多阶段变质性物品最优库存模型

本文考虑了时值及通货膨胀率下，部分短缺量拖后的变质性物品最优订购问题。在假定变质率为常数和短缺期间损失率与实际缺货量成正比的前提下，给出了寻找最优订购策略的算法，并且证明了在该策略下费用函数取得最小值。最后给出数字实例以说明本模型及求解过程。     

一类最优EOQ模型的进一步扩展

对一类经济批量订购模型作如下进一步扩展:第一,允许短缺,短缺量部分拖后供给,且短缺期间损失率与实际缺货量成正比;第二,订购费用是可变的,且线性依赖于订购量.在此假定下,研究了有限计划时间水平及常数变质率下,部分短缺量拖后的变质性物品多阶段库存问题,给出了寻找最优订购策略的算法,证明了所给最优策略的存在唯一性及在该策略下费用函数取得最小值.最后给出应用实例.     

Comparative statics of changes in risk on monotonically and partially responsive kinked payoffs

This paper derives two sets of necessary and sufficient conditions on the comparative statics of changes in risk under kinked payoff functions that are monotonically responsive and partially responsive to the realization of risk, respectively. The former case includes the newsboy problem with backorder; the latter case includes the newsboy problem without backorder and under some restrictions, also includes the optimal deductible insurance problem. Some relatively non-restrictive conditions derived from the necessary and sufficient conditions reveal that the three problems can be quite different, even though they are often viewed in the literature as being congruous. These conditions lead to simple predictions of the direction of change in any risk-averse agent’s optimal choice upon a change in risk, without assuming specific functional forms for the utility function.     

Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint

In this paper, a multiproduct single-machine production system under economic production quantity (EPQ) model is studied in which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the production defective rates are random variables, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the company. The aim of this research is to determine the optimal production quantity, the allowable shortage level, and the period length of each product such that the expected total cost, including holding, shortage, production, setup and defective items costs, is minimized. The mathematical model of the problem is derived for which the objective function is proved to be convex. Then, a derivative approach is utilized to obtain the optimal solution. Finally, two numerical examples in each of which a sensitivity analysis is performed on the model parameters, are provided to illustrate the practical usage of the proposed methodology.     

An economic order quantity model with partial backlogging under general backorder cost function

In this work, we study a single-item inventory model where shortages are allowed. A known constant fraction of the demand during the stockout period is backlogged, and the rest are lost sales. Usually, in the literature on inventory control, the unit backorder cost is considered to be a linear function of the waiting time until the customer gets the item. However, in some real-world situations, the unit cost of a backorder may not be linear. To model this situation, we develop a new approach by considering that the backlogging unit cost is a nondecreasing, continuous, and positive function of the amount of time the customers wait before receiving the item. Our objective is to maximize the average profit per unit time. An effective solution procedure to determine the optimal policy and the maximum average profit is developed. Numerical examples, which help us to understand the theoretical results, are also presented.     

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.     

Q,r Inventory Models with Time-Weighted Backorders

This paper attempts to show that the iterative methods that are used to solve a class of reorder point inventory models can be replaced by some non-iterative method. The method is described in general terms and two sufficient conditions are derived to check if the method applies to a model. As illustrations, two time-weighted backorder models are considered and they are solved for Normal and Gamma leadtime demand by the proposed method. It is observed that some of the computational advantages of the method also carry over to a class of multi-item models.     

Advanced supply chain planning with mixtures of backorders,lost sales,and lost contract

This paper investigates a supply chain system in which a supplier prepares for the selling season by building stock levels prior to the beginning of the season and shortages realized at the beginning of the season are represented as mixtures of backorders and lost sales. Backlogged items are replenished as soon as possible through an emergency procurement as opposed to waiting until the next scheduled delivery as in many continuous review scenarios, and the backorder rate is modeled as a piecewise linear function of the magnitude of the shortage. The often intangible cost associated with lost sales and customer goodwill is also quantified. In particular, the buyer and supplier are engaged in a contractual agreement and the loss of customer goodwill from the supplier’s perspective is represented as the expected cost associated with violating the conditions of the contract. The likelihood of contract cancellation is also represented as a function of the magnitude of shortage. The optimal solution is derived in closed form for the case of exponential demand distribution, and an example problem is illustrated with numerical data in order to demonstrate calculation of the optimal solution and corresponding sensitivity analysis for demand distributions in which the solution cannot be expressed in closed form.     

Modeling customer impatience in a newsboy problem with time-sensitive shortages

Customers across all stages of the supply chain often respond negatively to inventory shortages. One approach to modeling customer responses to shortages in the inventory control literature is time-dependent partial backlogging. Partial backlogging refers to the case in which a customer will backorder shortages with some probability, or will otherwise solicit the supplier’s competitors to fulfill outstanding shortages. If the backorder rate (i.e., the probability that a customer elects to backorder shortages) is assumed to be dependent on the supplier’s backorder replenishment lead-time, then shortages are said to be represented as time-dependent partial backlogging. This paper explores various backorder rate functions in a single period stochastic inventory problem in an effort to characterize a diversity of customer responses to shortages. We use concepts from utility theory to formally classify customers in terms of their willingness to wait for the supplier to replenish shortages. Under mild assumptions, we verify the existence of a unique optimal solution that corresponds to each customer type. Sensitivity analysis experiments are conducted in order to compare the optimal actions associated with each customer type under a variety of conditions. Additionally, we introduce the notion of expected value of customer patience information (EVCPI), and then conduct additional sensitivity analyses to determine the most and least opportune conditions for distinguishing between customer behaviors.     

Nonlinear Programming Analysis to Estimate Implicit Inventory Backorder Costs

In this paper, we use nonlinear programming to provide an alternative treatment of the economic order quantity problem with planned backorders. Many businesses, such as capital-goods firms that deal with expensive products and some service industries that cannot store their services, operate with substantial backlogs. In practical problems, it is usually very difficult to estimate accurately the values of the two types of backorder costs, i.e., the time-dependent unit backorder cost and the unit backorder cost. We redefine the original problem without including these backorder costs and construct a nonlinear programming problem with two service measure constraints which may be easier to specify than the backorder costs. We find that, with this different formulation of our new problem, we obtain results which give implicit estimates of the backorder costs. The alternative formulation provides an easier-to-use model and managerially meaningful results. Next, we show that, for a wide range of parameter values, it usually suffices to consider only one type of backorder cost, or equivalently, only one type of service measure constraint. Finally, we develop expressions which bracket the optimal values of the decision variables in a narrow range and provide a simple method for computing the optimal solution. In the most complicated case, this method requires finding the unique root of a polynomial.     

Optimal ordering policies for periodic-review systems with replenishment cycles

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.     

Optimal policy for an inventory system with backlogging and all-units discounts: Application to the composite lot size model

This paper examines an inventory model with full backlogging and all-units quantity discounts. The practical scenario of a salesperson offering compensation to a client so as not to lose the sale is considered. The cost of a backorder thus includes both a fixed cost and a further cost which is proportional to the length of time the said backorder exists. A first algorithm is developed to determine the optimal policy while some extensions to this algorithm are obtained that include additional conditions on the model. In particular, the well known composite lot size model, developed by Tersine, is solved, incorporating a new stockout cost and a new all-units discount. Numerical examples are provided to illustrate the application of the algorithms.     

A multi-objective wholesaler–retailers inventory-distribution model with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number

This paper develops a single wholesaler and multi retailers mixture inventory distribution model for a single item involving controllable lead-time with backorder and lost sales. The retailers purchase their items from the wholesaler in lots at some intervals throughout the year to meet the customers’ demand. Not to loose the demands, the retailers offer a price discount to the customers on the stock-out items. Here, it is assumed that the lead-time demands of retailers are uncertain in both stochastic and fuzzy sense, i.e., these are simultaneously random and imprecise. To implement this behavior of the lead-time demands, at first, these demands are assumed to be random, say following a normal distribution. With these random demands, the expected total cost for each retailer is obtained. Now, the mean lead-time demands (which are crisp ones) of the retailers are fuzzified. This fuzzy nature of the lead-time demands implies that the annual average demands of the retailers must be fuzzy numbers, suppose these are triangular fuzzy numbers. Using signed distance technique for defuzzification, the estimate of total costs for each retailer is derived. Therefore, the problem is reduced to optimize the crisp annual costs of wholesaler and retailers separately. The multi-objective model is solved using Global Criteria method. Numerical illustrations have been made with the help of an example taking two retailers into consideration. Mathematical analyses have been made for global pareto-optimal solutions of the multi-objective optimization problem. Sensitivity analyses have been made on backorder ratio and pareto-optimal solutions for wholesaler and different retailers are compared graphically.     

基于可控提前期和可变押后订单比例的库存问题研究

针对可控提前期领域现有研究中存在的不足,提出了同时考虑可控提前期和缺货对押后订单比例影响的连续盘点库存管理问题。在市场需求服从正态分布条件下建立年总成本表达式,并同时决策订货量、订货点和提前期使年总成本最小。实例计算表明我们的库存管理方法可以减少总成本,提高顾客服务水平和使库存管理更加平稳。     

在允许缺货情况下的易变质产品供应商管理库存

本首先通过解析推导给出了一个考虑一个供应商和多个不同的订货商，并且允许订货商缺货的易腐商品供应商管理库存VMI(Vendor Managed Inventory)模型；其次通过mathematica4．2软件计算。给出了算例及其最优解，并通过灵敏度分析讨论了单位缺货成本和变质率对库存决策的影响；最后给出了总结，说明了研究的意义。     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

A Multi-Objective Production Inventory Model with Backorder for Fuzzy Random Demand Under Flexibility and Reliability

In this paper, an Economic Production Quantity (EPQ) model is developed with flexibility and reliability consideration of production process in an imprecise and uncertain mixed environment. The model has incorporated fuzzy random demand, an imprecise production preparation time and shortage. Here, the setup cost and the reliability of the production process along with the backorder replenishment time and production run period are the decision variables. Due to fuzzy-randomness of the demand, expected average demand is a fuzzy quantity and also imprecise preparation time is represented by fuzzy number. Therefore, both are first transformed to a corresponding interval number and then using the interval arithmetic, the single objective function for expected profit over the time cycle is changed to respective multi-objective functions. Due to highly nonlinearity of the expected profit functions it is optimized using a multi-objective genetic algorithm (MOGA). The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.