Piecewise linear approximations for the static–dynamic uncertainty strategy in stochastic lot-sizing

In this paper, we develop a unified mixed integer linear modelling approach to compute near-optimal policy parameters for the non-stationary stochastic lot sizing problem under static–dynamic uncertainty strategy. The proposed approach applies to settings in which unmet demand is backordered or lost; and it can accommodate variants of the problem for which the quality of service is captured by means of backorder penalty costs, non-stockout probabilities, or fill rate constraints. This approach has a number of advantages with respect to existing methods in the literature: it enables seamless modelling of different variants of the stochastic lot sizing problem, some of which have been previously tackled via ad hoc solution methods and some others that have not yet been addressed in the literature; and it produces an accurate estimation of the expected total cost, expressed in terms of upper and lower bounds based on piecewise linearisation of the first order loss function. We illustrate the effectiveness and flexibility of the proposed approach by means of a computational study.     

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.     

Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates

This paper considers an inventory system with non-instantaneous deteriorating item in which demand rate is a function of advertisement of an item and selling price. This paper aids the retailer in maximizing the total profit by determining optimal inventory and marketing parameters. In contrast to previous inventory models, an arbitrary holding cost rate and arbitrary deterioration rate have been incorporated to provide general framework to the model. First, a mathematical model is formulated and then some useful theoretical results have been framed to characterize the optimal solutions. The necessary and sufficient conditions for the existence and uniqueness of the optimal solutions are also derived. An algorithm is designed to find the optimum solutions of the proposed model. Numerical examples are included to illustrate the algorithmic procedure and the effects of key parameters are studied to analyze the behavior of the model.     

EFFECTS OF INTEGRATING ORDER/BACKORDER QUANTITY AND PRICING DECISIONS

We consider a monopolistic situation where the retailer aims to find the profit-maximizing selling price, and the order and backorder quantity of an item. The price and demand are assumed to be inversely related. We develop a simple optimal approach, the proportion-balancing algorithm (PBA), which utilizes the proportions of cost components to sales revenue. We demonstrate that the PBA can handle three commonly used demand functions. However, applicability of the PBA depends on how the price-demand relationship is specified. A sensitivity analysis shows that the effects of changes in cost parameters on the order and backorder quantities can be very different from those of the classical economic order quantity model with backorders.     

An Inventory Model with Planned Shortages and Price Dependent Demand

Previous research has yielded a procedure for a retailer to determine the optimal lot size and selling price when a supplier offers all-unit quantity discounts and demand is a decreasing function of price. In this paper, we extend that research by allowing for shortages. An algorithm is presented that determines the optimal lot size, order level, and selling price for a class of demand functions, including the constant price-elasticity and linear demand functions.     

Joint inventory and technology selection decisions

In some manufacturing systems, the type of production technology selected dictates the lot size, or sets an upper limit on it. In this paper, we formulate a model in which the lot sizing decision has to be made at the technology selection stage. Furthermore, the cost of the technology depends on the lot size it can produce. The problem is addressed under two different assumptions about demand. First demand is assumed to be constant. A closed-form expression for the optimal lot size is derived for this case. Second, demand is assumed to be linearly decreasing. Using an accurate approximation to the total cost function, a closed-form expression for the optimal lot size is derived. We illustrate the models with numerical examples.     

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.     

提前期、构建成本和短缺量拖后率均可控的EOQ模型

传统库存模型通常将提前期和构建成本视为不可控制。事实上可以通过追加投资缩短提前期和降低构建成本。缺货期间,为减少订单丢失量和补偿顾客的损失,供应商会给予一定的价格折扣。现实库存系统中,容易得到需求的期望值和标准差,但较难得到其分布规律。基于此,考虑短缺量拖后率与价格折扣和缺货期间库存水平相关,提出了一种需求为任意分布且提前期和构建成本均可控的EOQ模型,证明了模型存在唯一最优解,给出了一种寻优算法。数值仿真分析表明,一般情况下,压缩提前期和降低构建成本能降低订购批量和安全库存,降低库存总成本;短缺量拖后系数和缺货概率对库存总成本影响较大,企业应尽量降低缺货概率,尤其在短缺量拖后系数较小时。     

Determining Optimal Selling Price and Lot Size When the Supplier Offers All-Unit Quantity Discounts*

This paper is concerned with finding the optimal price and lot size for a retailer purchasing a product for which the supplier offers all-unit quantity discounts. Demand for the product is assumed to be a decreasing function of price, and a procedure is developed for finding the optimal price and lot size for a class of demand functions. The procedure then is applied to two common demand functions: (1) the constant price-elasticity function, and (2) the linear demand function.     

Note on inventory models with Weibull distribution deterioration

This article explores the inventory model with a general demand rate function in which both the Weibull distributed deterioration and partial backlogging are considered. The inventory model discussed here is based on the important finding by Wu [Wu, K.S., 2001. An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging. Production Planning and Control, 12, 787–793]. There are four parts in our research. First, we derive the analytical framework of the inventory model for a general demand rate function1 1.?Based on detailed suggestions of Professor Wafik H. Iskander, Email: whiskander@mail.wvu.edu . Second, for the ramp type demand, we improve Wu's model to find the criterion to guarantee the existence and uniqueness of the optimal solution. Third, we develop a new model to compensate for the missing case in Wu's article. Fourth, we combine our results to show that our findings are applicable to the ramp type demand inventory model, so that the optimal solution is independent of the demand function. Finally, some numerical examples and graphs are provided to illustrate our discovery and demonstrate the application of our analytical framework.     

Inventory Rationing for Multiple Class Demand under Continuous Review

We consider how a firm should ration inventory to multiple classes in a stochastic demand environment with partial, class‐dependent backlogging where the firm incurs a fixed setup cost when ordering from its supplier. We present an infinite‐horizon, average cost criterion Markov decision problem formulation for the case with zero lead times. We provide an algorithm that determines the optimal rationing policy, and show how to find the optimal base‐stock reorder policy. Numerical studies indicate that the optimal policy is similar to that given by the equivalent deterministic problem and relies on tracking both the current inventory and the rate that backorder costs are accumulating. Our study of the case of non‐zero lead time shows that a heuristic combining the optimal, zero lead time policy with an allocation policy based on a single‐period profit management problem is effective.     

Inventory Control when the Lead‐time Changes

A single‐echelon inventory system with continuous review and Poisson demand is considered. There are standard linear holding and backorder costs but no ordering or set‐up costs. We study a change in the lead‐time, which is rather typical in connection with application of a Just‐In‐Time philosophy. Our main focus is a lead‐time decrease but we also consider a lead‐time increase. Due to the lead‐time change, the optimal steady state solution will also, in general, change. We consider the transient problem of minimizing the costs when bringing the system from its original steady state to the new steady state.     

A note on optimal price,warranty length and production rate for free replacement policy in static demand markets

This note provides some extensions on optimal policies on price, warranty length and production rate proposed by Wu et al. (2009) [1] (Wu, C.C.; Chou, C.Y.; Huang C. Optimal price, warranty length and production rate for free replacement policy in the static demand market. Omega 2009; 37: 29–39.) More specifically, the assumptions of the original paper is extended in this note from the production rate being positive and the second derivative of the demand function with respect to price and warranty-length being negative to no restrictions on both.     

基于CRRA效用准则的资产负债管理

本文在连续时间不完备市场框架下,考虑了投资者终端时刻资产负债比率的期望效用最大化问题。假设金融市场由1个无风险资产与多个风险资产构成,其中风险资产的价格过程由几何布朗运动刻画;投资者在整个投资时间水平内面临一个由几何布朗运动刻画的外生负债。利用随机动态规划方法,给出了相应的HJB方程与验证定理,并得到了最优投资策略与最优值函数的解析表达式。进一步,通过敏感性分析与数值算例发现：（1）外生负债的预期增长率与当前时刻的资产负债比率对最优投资策略没有影响;（2）在不考虑外生负债时,在最优策略下,投资到风险资产上的资金比例随着风险资产波动率或相对风险厌恶系数的增大而减小,而在考虑外生负债时,并非如此,只有满足一定条件时最优投资策略才是风险资产波动率或相对风险厌恶系数的减函数;（3）不考虑外生负债时,最优值函数是投资时间水平与风险资产预期收益率的增函数,风险资产波动率的减函数,但在考虑外生负债时该结论只在各参数满足一定关系时才成立,否则结论相反。     

A joint optimal pricing and order quantity model under parameter uncertainty and its practical implementation

We consider a robust optimization model of determining a joint optimal bundle of price and order quantity for a retailer in a two-stage supply chain under uncertainty of parameters in demand and purchase cost functions. Demand is modeled as a decreasing power function of product price, and unit purchase cost is modeled as a decreasing power function of order quantity and demand. While the general form of the power functions are given, it is assumed that parameters defining the two power functions involve a certain degree of uncertainty and their possible values can be characterized by ellipsoids. We show that the robust optimization problem can be transformed into an equivalent convex optimization which can be solved efficiently and effectively using interior-point methods. In addition, we propose a practical implementation of the model, where the stochastic characteristics of parameters are obtained from regression analysis on past sales and production data, and ellipsoidal representations of the parameter uncertainties are obtained based on a combined use of genetic algorithm and Monte Carlo simulation. An illustrative example is provided to demonstrate the model and its implementation.     

考虑缺货的模糊库存模型及其优化求解

研究了考虑缺货时多模糊参数的库存模型优化求解问题。采用梯形模糊数对库存问题进行了描述。当订货量为常数时,基于函数原理,采用梯级平均综合表示法进行了解模糊,对允许适度缺货的库存模型进行了建模和优化求解。当订货量为模糊数时,将扩展拉格朗日法与反证法相结合,解决了不等式约束库存求解问题,结果表明运用模糊数学来解决库存问题是合理可行的。     

A joint optimal pricing and order quantity model under parameter uncertainty and its practical implementation

We consider a robust optimization model of determining a joint optimal bundle of price and order quantity for a retailer in a two-stage supply chain under uncertainty of parameters in demand and purchase cost functions. Demand is modeled as a decreasing power function of product price, and unit purchase cost is modeled as a decreasing power function of order quantity and demand. While the general form of the power functions are given, it is assumed that parameters defining the two power functions involve a certain degree of uncertainty and their possible values can be characterized by ellipsoids. We show that the robust optimization problem can be transformed into an equivalent convex optimization which can be solved efficiently and effectively using interior-point methods. In addition, we propose a practical implementation of the model, where the stochastic characteristics of parameters are obtained from regression analysis on past sales and production data, and ellipsoidal representations of the parameter uncertainties are obtained based on a combined use of genetic algorithm and Monte Carlo simulation. An illustrative example is provided to demonstrate the model and its implementation.     

爱尔朗型按订单装配系统最优生产——库存控制策略研究

本文研究具有复杂装配结构的爱尔朗型按订单装配(ATO)系统的组件生产与库存优化控制问题。系统涉及多种组件,一个最终产品和多类客户需求。在此系统中,各种组件的生产时间服从爱尔朗分布,各类客户的需求为泊松到达过程。针对不同客户需求类型:产品需求与独立组件需求且同为销售损失型,建立基于马尔可夫决策过程(MDP)的平均总成本模型,应用动态规划方法求解最优策略。仿真模拟方法实现最优策略,并通过数值实验分析多生产阶段和系统参数对最优策略的影响。研究结果表明,爱尔朗型生产时间ATO系统的最优策略为状态依赖型策略,即组件的生产与库存分配由动态基础库存水平值和动态库存配给水平值控制。对于任一组件,其基础库存水平值和库存配给水平值均随着生产阶段的增加而降低,且生产阶段对基础库存水平值和平均总成本的影响较显著。     

Optimal price,warranty length and production rate for free replacement policy in the static demand market

In the present paper, a decision model is developed for producers in the static demand market to determine the optimal price, warranty length and production rate of a product to maximize profit based on the pre-determined life cycle. The free renewal warranty policy is considered under which failed products are renewed before the end of warranty length at no cost to consumers. The expected number of renewals based on warranty length is derived for Weibull life distributed products. The objective function includes both demand and cost functions, where production cost, warranty cost and inventory cost are involved. A solution approach using the maximum principle is described, and is applied to two specific cases of markets. The first case of market considers positive discount rate, and the second case of market considers zero discount rate. The economic sensitivity analysis is conducted to evaluate the effect of model parameters on the optimal solution. Some conclusions are drawn based on the sensitivity analysis.     

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.