Optimal inventory decisions in a multiperiod newsvendor problem with partially observed Markovian supply capacities

This paper considers a multi-period news-vendor problem with partially observed supply-capacity information which evolves as a Markovian Process. The supply capacity is fully observed by the buyer when the capacity is smaller than the buyer’s ordering quantity. Otherwise, the buyer knows that the current-period supply capacity is greater than its ordering quantity. Based on these two observations, the buyer updates the future supply-capacity forecasting accordingly. With a dynamic programming formulation, we prove the existence of an optimal ordering policy. We also prove that the optimal order quantity is greater than the myopic order quantity.     

Optimal policy for a two-facility inventory problem with storage constraints and two freight modes

This paper considers a two-facility supply chain for a single product in which facility 1 orders the product from facility 2 and facility 2 orders the product from a supplier in each period. The orders placed by each facility are delivered in two possible nonnegative integer numbers of periods. The difference between them is one period. Random demands in each period arise only at facility 1. There are physical storage constraints at both facilities in each period. The objective of the supply chain is to find an ordering policy that minimizes the expected cost over a finite horizon and the discounted stationary expected cost over an infinite horizon. We characterize the structure of the minimum expected cost and the optimal ordering policy for both the finite and the discounted stationary infinite horizon problems.     

An inventory control problem for deteriorating items with back-ordering and financial considerations

This paper investigates the effects of time value of money and inflation on the optimal ordering policy in an inventory control system. We proposed an economic order quantity model to manage a perishable item over the finite horizon planning under which back-ordering and delayed payment are assumed. The demand and deterioration rates are constant. The present value of total cost during the planning horizon in this inventory system is modeled first, then a three phases solution procedure is proposed to derive the optimal order and shortage quantities, and the number of replenishment during the planning horizon. Finally, the proposed model is illustrated through numerical examples and the sensitivity analysis is reported to find some managerial insights.     

Coherent risk measures in inventory problems

We analyze an extension of the classical multi-period, single-item, linear cost inventory problem where the objective function is a coherent risk measure. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min–max type formulations. For the single period newsvendor problem, we show that the structure of the optimal solution of the risk averse model is similar to that of the classical expected value problem. For a finite horizon dynamic inventory model, we show that, again, the optimal policy has a similar structure as that of the expected value problem. This result carries over even to the case when there is a fixed ordering cost. We also analyze monotonicity properties of the optimal order quantity with respect to the degree of risk aversion for certain risk measures.     

A deterministic,multi-item inventory model with supplier selection and imperfect quality

This paper considers the scenario of supply chain with multiple products and multiple suppliers, all of which have limited capacity. We assume that received items from suppliers are not of perfect quality. Items of imperfect quality, not necessarily defective, could be used in another inventory situation. Imperfect items are sold as a single batch, prior to receiving the next shipment, at a discounted price. The demand over a finite planning horizon is known, and an optimal procurement strategy for this multi-period horizon is to be determined. Each of products can be sourced from a set of approved suppliers, a supplier-dependent transaction cost applies for each period in which an order is placed on a supplier. A product-dependent holding cost per period applies for each product in the inventory that is carried across a period in the planning horizon. Also a maximum storage space for the buyer in each period is considered. The decision maker, the buyer, needs to decide what products to order, in what quantities, with which suppliers, and in which periods. Finally, a genetic algorithm (GA) is used to solve the model.     

Optimal policy for inventory systems with capacity commitment and fixed ordering costs

This paper investigates a single-product, periodic-review, non-stationary inventory system with total maximum capacity commitment and fixed ordering costs over a finite planning horizon. We characterize the optimal ordering policy as a state-dependent ($s,S$) policy whose parameters only depend on the sum of the net inventory and the remaining capacity. We show that such policy can degenerate into two simple policies in two special cases respectively. We also derive bounds on parameters of the optimal ordering policy.     

Optimal production plans and shipment schedules in a supply-chain system with multiple suppliers and multiple buyers

This research addresses an optimal policy for production and procurement in a supply-chain system with multiple non-competing suppliers, a manufacturer and multiple non-identical buyers. The manufacturer procures raw materials from suppliers, converts them to finished products and ships the products to each buyer at a fixed-interval of time over a finite planning horizon. The demand of finished product is given by buyers and the shipment size to each buyer is fixed. The problem is to determine the production start time, the initial and ending inventory, the cycle beginning and ending time, the number of orders of raw materials in each cycle, and the number of cycles for a finite planning horizon so as to minimize the system cost. A surrogate network representation of the problem developed to obtain an efficient, optimal solution to determine the production cycle and cycle costs with predetermined shipment schedules in the planning horizon. This research prescribes optimal policies for a multi-stage production and procurements for all shipments scheduled over the planning horizon. Numerical examples are also provided to illustrate the system.     

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.     

Analysis of supply contracts with minimum total order quantity commitments and non-stationary demands

In this paper we address a buyer–supplier arrangement of particular importance: total order quantity commitment (TOQC). Under the TOQC contract that has been negotiated between the buyer and the supplier, the former agrees to an obligation to procure a certain quantity of an item from the latter over the predetermined period of time (additional quantity can be obtained maybe at a different price). The optimal inventory replenishment policy is shown to be dual order-up-to levels under a given TOQC, and the optimal TOQC is also demonstrated to be mathematically straightforward to obtain.     

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。     

线性需求合并短缺的变质性物品的生产——库存模型

本文发展了线性需求合并短缺的变质性物品的生产——库存模型，以系统平均总费用最小为目标，提供了有限计划期内的生产调整策略以便适应市场需求的变化．同时还提供了无短缺情形的相应模型，最后出示了一些数字例子     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

时变需求等量进货情形下集成供应链生产订货策略研究

构建了一个包含原料采购、生产和销售过程的集成供应链模型,研究了由原料、生产商和销售商产品构成的三层库存系统的生产订货问题。在有限的规划期内,销售商每次进货量相同,生产商按照EOQ模型采购原材料。以最小化供应链系统的总运营成本为目标,构建一个混合整数非线性规划模型,寻找销售商最优订货方案和生产商最佳生产策略。首先利用网络优化方法求解生产商的最优生产计划,其次利用定界穷举法寻求销售商最优的订货周期,给出了具体的计算方法和Matlab程序。通过算例分析验证了算法的有效性,并研究了各参数对最小费用及最优解的影响。     

Flexible supply policy with options and capacity constraints

This paper considers a class of multi-period flexible supply policies with options and capacity constraints. The main results are to characterize the optimal ordering and purchasing options policy and the minimum expected cost in a period and thereafter under the assumptions about the options and ordering quantities.     

Analysis of supply contracts with quantity flexibility

This paper explores a class of supply contracts under which a buyer receives discounts for committing to purchases in advance. The further in advance the commitment is made, the larger the discount. As time rolls forward, the buyer can increase the order quantities for future periods of the rolling horizon based on updated demand forecast information and inventory status. However, the buyer pays a higher per-unit cost for the incremental units. Such contracts are used by automobile and contract manufacturers, and are quite common in fuel oil and natural gas delivery markets. We develop a finite-horizon dynamic programming model to characterize the structure of the optimal replenishment strategy for the buyer. We present heuristic approaches to calculate the order volume in each period of the rolling horizon. Finally, we numerically evaluate the heuristic approaches and draw some managerial insights based on the findings.     

Optimal policies for inventory systems with discretionary sales, random yield and lost sales

We determine replenishment and sales decisions jointly for an inventory system with random demand, lost sales and random yield. Demands in consecutive periods are independent random variables and their distributions are known. We incorporate discretionary sales, when inventory may be set aside to satisfy future demand even if some present demand may be lost. Our objective is to minimize the total discounted cost over the problem horizon by choosing an optimal replenishment and discretionary sales policy. We obtain the structure of the optimal replenishment and discretionary sales policy and show that the optimal policy for finite horizon problem converges to that of the infinite horizon problem. Moreover, we compare the optimal policy under random yield with that under certain yield, and show that the optimal order quantity （sales quantity） under random yield is more （less） than that under certain yield.     

两种可替代物品的经济订货批量模型

研究了两种可替代物品的经济订货批量问题.在计划期内,若某一种物品发生缺货,则可以被另一种物品以一定的替代率代替补充,以库存系统的总费用最小为目标函数,证明了最优策略存在的唯一性,并给出了求解最优订购策略的算法,最后通过两个算例验证了算法的最优性.     

Integrated inventory model with quantity discount and price-sensitive demand

Quantity discount has been a subject of study for a long time; however, little is known about its effect on integrated inventory models when price-sensitive demand is placed. The objective of this study is to find the optimal pricing and ordering strategies for an integrated inventory system when a quantity discount policy is applied. The pricing strategy discussed here is one in which the vendor offers a quantity discount to the buyer. Then, the buyer will adjust his retail price based on the purchasing cost, which will influence the customer demand as a result. Consequently, an integrated inventory model is established to find the optimal solutions for order quantity, retail price, and the number of shipments from vendor to buyer in one production run, so that the joint total profit incurred has the maximum value. Also, numerical examples and a sensitivity analysis are given to illustrate the results of the model.     

Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics

The motivation for our study comes from some production and inventory systems in which ordering/producing quantities that exceed certain thresholds in a given period might eliminate some setup activities in the next period. Many examples of such systems have been discussed in prior research but the analysis has been limited to production settings under deterministic demand. In this paper, we consider a periodic-review production-inventory model under stochastic demand and incorporate the following fixed-cost structure into our analysis. When the order quantity in a given period exceeds a specified threshold value, the system is assumed to be in a “warm” state and no fixed cost is incurred in the next period regardless of the order quantity; otherwise the system state is considered “cold” and a positive fixed cost is required to place an order. Assuming that the unsatisfied demand is lost, we develop a dynamic programming formulation of the problem and utilize the concepts of quasi-K-convexity and non-K-decreasing to show some structural results on the optimal cost-to-go functions. This analysis enables us to derive a partial characterization of the optimal policy under the assumption that the demands follow a Pólya or uniform distribution. The optimal policy is defined over multiple decision regions for each system state. We develop heuristic policies that are aimed to address the partially characterized decisions, simplify the ordering policy, and save computational efforts in implementation. The numerical experiments conducted on a large set of test instances including uniform, normal and Poisson demand distributions show that a heuristic policy that is inspired by the optimal policy is able to find the optimal solution in almost all instances, and that a so-called generalized base-stock policy provides quite satisfactory results under reasonable computational efforts. We use our numerical examples to generate insights on the impact of problem parameters. Finally, we extend our analysis into the infinite horizon setting and show that the structure of the optimal policy remains similar.     

An inventory-transportation system with stochastic demand

We study a logistic system in which a supplier has to deliver a set of products to a set of retailers to face a stochastic demand over a given time horizon. The transportation from the supplier to each retailer can be performed either directly, by expensive and fast vehicles, or through an intermediate depot, by less expensive but slower vehicles. At most one time period is required in the former case, while two time periods are needed in the latter case. A variable transportation cost is charged in the former case, while a fixed transportation cost per journey is charged in the latter case. An inventory cost is charged at the intermediate depot. The problem is to determine, for each time period and for each product, the quantity to send from the supplier to the depot, from the depot to each retailer and from the supplier to each retailer, in order to minimize the total expected cost. We first show that the classical benchmark policy, in which the demand of each product at each retailer is set equal to the average demand, can give a solution which is infinitely worse with respect to the optimal solution. Then, we propose two classes of policies to solve this problem. The first class, referred to as Horizon Policies, is composed of policies which require the solution of the overall problem over the time horizon. The second class, referred to as Reoptimization Policies, is composed of a myopic policy and several rolling-horizon policies in which the problem is reoptimized at each time period, once the demand of the time period is revealed. We evaluate the performance of each policy dynamically, by using Monte Carlo Simulation.