Robustness of inventory replenishment and customer selection policies for the dynamic and stochastic inventory-routing problem

When inventory management, distribution and routing decisions are determined simultaneously, implementing a vendor-managed inventory strategy, a difficult combinatorial optimization problem must be solved to determine which customers to visit, how much to replenish, and how to route the vehicles around them. This is known as the inventory-routing problem. We analyze a distribution system with one depot, one vehicle and many customers under the most commonly used inventory policy, namely the (s,S), for different values of s. In this paper we propose three different customer selection methods: big orders first, lowest storage first, and equal quantity discount. Each of these policies will select a different subset of customers to be replenished in each period. The selected customers must then be visited by a vehicle in order to deliver a commodity to satisfy the customers' demands. The system was analyzed using public benchmark instances of different sizes regarding the number of customers involved. We compare the quality and the robustness of our algorithms and detailed computational experiments show that our methods can significantly improve upon existing solutions from the literature.     

A study on inventory replenishment policies in a two-echelon supply chain system

Inventory control plays an important role in supply chain management. Properly controlled inventory can satisfy customers’ demands, smooth the production plans, and reduce the operation costs; yet failing to budget the inventory expenses may lead to serious consequences. The bullwhip effect, observed in many supply chain management cases, causes excessive inventory due to information distortion, i.e. the order amount is exaggerated while a minor demand variation occurs, and the information amplified dramatically as the supply chain moves to the upstream. In this paper, one of the main causes of bullwhip effect, order batching, is considered. A simplified two-echelon supply chain system, with one supplier and one retailer that can choose different replenishment policies, is used as a demonstration. Two types of inventory replenishment methods are considered: the traditional methods (the event-triggered and the time-triggered ordering policies), and the statistical process control (SPC) based replenishment method. The results show that the latter outperforms the traditional method in the categories of inventory variation, and in the number of backlog when the fill-rate of the prior model is set to be 99%. This research provides a different approach to inventory cost-down other than the common methods like: information sharing, order batch cutting, and lead time reduction. By choosing a suitable replenishment policy, the number of backorder and the cost of inventory can be reduced.     

Integrated inventory control and inspection policies with deterministic demand

In this paper, we develop integrated inventory inspection models with and without replacement of nonconforming items. Inspection policies include no inspection, sampling inspection, and 100% inspection. We consider a buyer who places an order from a supplier. When a lot is received, the buyer uses some type of inspection policy. The fraction nonconforming is assumed to be a random variable following a beta distribution. Both the order quantity and the inspection policy are decision variables. In the inspection policy involving determining sampling plan parameters, constraints on the buyer and manufacturer risks are set in order to obtain a fair plan for both parties. A solution procedure for determining the operating policies for inventory and inspection consisting of order quantity, sample size, and acceptance number is proposed. Numerical examples are presented to conduct a sensitivity analysis for important model parameters and to illustrate important issues about the developed models.     

Simulation-based optimization of process control policies for inventory management in supply chains

A simulation-based optimization framework involving simultaneous perturbation stochastic approximation (SPSA) is presented as a means for optimally specifying parameters of internal model control (IMC) and model predictive control (MPC)-based decision policies for inventory management in supply chains under conditions involving supply and demand uncertainty. The effective use of the SPSA technique serves to enhance the performance and functionality of this class of decision algorithms and is illustrated with case studies involving the simultaneous optimization of controller tuning parameters and safety stock levels for supply chain networks inspired from semiconductor manufacturing. The results of the case studies demonstrate that safety stock levels can be significantly reduced and financial benefits achieved while maintaining satisfactory operating performance in the supply chain.     

Multi-criteria inventory classification with reference items

The ABC method is a well-known approach to classify inventory items into ordered categories, such as A, B and C. As emphasized in the literature, it is reasonable to evaluate the inventory classification problem in the multi-criteria context. From this point of view, it corresponds to a sorting problem where categories are ordered. Here, one important issue is that the weights of the criteria and categorization preferences can change from industry to industry. This requires the analysis of the problem in a specific framework where the decision maker (expert)’s preferences are considered. In this study, the preferences of the decision maker are incorporated into the decision making process in terms of reference items into each class. We apply two utility functions based sorting methods to the problem. We perform an experiment and compare results with other algorithms from the literature.     

带有两次订购机会且两阶段需求相关的Newsboy模型

首先建立需求形式为一般随机变量情形下零售商的两次订购决策模型,并从理论上证明了在两个阶段需求相互独立情形下,零售商期望利润函数的下凹性及最优订购策略的存在唯一性,从而弥补了已有模型在寻求最优订购策略时采用数值方法所存在的缺憾;然后,建立两阶段需求相关情形下带有两次订购的Newsboy模型,给出了模型的分析解.数值实例验证了模型的求解过程,并得到了相关管理启示.     

The value of modeling with reference effects in stochastic inventory and pricing problems

We analyze a periodic review inventory system in which the random demand is contingent on the current price and the reference price. The reference price captures the price history and acts as a benchmark against which the current price is compared. The randomness is due to additive and multiplicative random terms. The objective is to maximize the discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We study three key issues using numerical computation and simulation. First, we study the effects of reference price mechanism on the total expected profit. It is shown that high dependence on a good history increases the profit. Second, we investigate the value of dynamic programming and show that the firm that ignores the dynamic structure suffers from the revenue. Third, we analyze the value of estimating the correct demand model with reference effects. We observe that this value is significant when the inventory related costs are low.     

Retailer's optimal ordering policies with trade credit financing

In this article, we extended Goyal's model to develop an Economic Order Quantity (EOQ) model in which the supplier offers the retailer the permissible delay period M, and the retailer in turn provides the trade credit period N (with N?≤?M) to his/her customers. In addition, we assume that (1) the retailer's selling price per unit is necessarily higher than its unit cost, and (2) the interest rate charged by a supplier or a bank is not necessarily higher than the retailer's investment return rate. We then establish an appropriate EOQ model with trade credit financing, and provide an easy-to-use closed-form solution to the problem. Furthermore, we find it is possible that a well-established buyer may order a lower quantity and take the benefit of the permissible delay more frequently, which contradicts to the result by the previous researchers. Finally, we perform some sensitivity analyses to illustrate the theoretical results and obtain some managerial results.     

Generalized spare ordering policies with allowable inventory time

In this paper, a generalized ordering policy of a one-unit system with age-dependent minimal repair and random lead time is considered. We treat the general order-replacement model with two decision variables: ordering time and allowable inventory time. The allowable inventory period is measured from the time instant that the ordered spare is delivered. By introducing costs due to ordering, repairs, shortage, and holding, we derive the expected cost per unit time in the long run as a criterion of optimality, and seek the optimum policy by minimizing that cost. And we can prove under some mild conditions that there exists a finite and unique optimum policy. Various special cases are discussed.     

Solving fuzzy problems in operations research: inventory control

 In this paper we are interested in generating good approximate solutions to single item, N-period, fuzzy inventory control problems. It is a fuzzy inventory control problem since some of the parameters (ordering cost, holding cost, penalty cost) can be fuzzy numbers. We consider three cases: (1) demand is known each period; (2) demand is unknown, and fuzzy each period; and (3) demand is fuzzy and backordering is allowed. We employ an evolutionary algorithm to search out good approximate solutions.     

Two-echelon stochastic inventory system with returns and partial backlogging

This paper investigates a two-echelon stochastic inventory system with returns and partial backlogging; our proposed model, which is realistic for practical situations, might be applicable in many manufacturing planning situations. The objective is to find a continuous-review inventory control policy that minimises the total expected annual cost of the system. A computer code using the software Mathematica 5.2 is developed to derive the optimal solution. Additionally, we discuss the sensitivity of the optimal solution together with the changes of the values of the parameters associated with the model for decision-making. The results are illustrated with the help of four numerical examples.     

一种利用切比雪夫逼近求解库存最优控制的方法

针对供应链中库存随着需求的变化可能导致的积压和对生产(或采购)产生的不利影响,为更好地协调生产(或采购)并减少产品库存,研究了一类基于库存约束和动态时变需求下的多品种、多周期、多循环的生产与库存的最优控制模型.结合最优控制理论,给出一种采用切比雪夫多项式逼近和高斯-切比雪夫数值积分对库存最优控制问题进行数值求解的方法.实例分析表明该方法是可行的.     

Random network models and sensitivity algorithms for the analysis of ordering time and inventory state in multi-stage supply chains

Supply chains in reality face a highly dynamic and uncertain environment, especially the uncertain end-customer demands and orders. Since the condition of product market changes frequently, the tasks of order management, product planning, and inventory management are complex and difficult. It is imperative for companies to develop new ways to manage the randomness and uncertainty in market demands. Based on the graphical evaluation and review technique, this paper provides a simple but integrated stochastic network mathematical model for supply chain ordering time distribution analysis. Then the ordering time analysis model is extended so that the analysis of inventory level distribution characteristics of supply chain members is allowed. Further, to investigate the effects of different end-customer demands on upstream orders and relative inventory levels, model-based sensitivity analysis algorithms for ordering fluctuations and inventory fluctuations are developed. A detailed numerical example is presented to illustrate the application of the proposed models to a multi-stage supply chain system, and the results of which shows the effectiveness and flexibility of the proposed stochastic network models and algorithms in order and inventory management.     

Coordinating two-period ordering and advertising policies in a dynamic market with stochastic demand

In this paper, we study the optimal two-stage advertising and ordering policies and the channel coordination issues in a supply chain composed of one manufacturer and one retailer. The manufacturer sells a short-life-cycle product through the retailer facing stochastic demand in dynamic markets characterised by price declines and product obsolescence. Following a two-period newsvendor framework, we develop two members’ optimal ordering and advertising models under both the centralised and decentralised settings, and present the closed-form solutions to the developed models as well. Moreover, we design a two-period revenue-sharing contract, and develop sufficient conditions such that the channel coordination can be achieved and a win–win outcome can be guaranteed. Our analysis suggests that the centralised decision creates an incentive for the retailer to increase the advertising investments in two periods and put the purchase forward, but the decentralised decision mechanism forces the retailer to decrease the advertising investments in two periods and postpone/reduce its purchase in the first period. This phenomenon becomes more evident when demand variability is high.     

On the performance of the base-stock inventory system under a compound Erlang demand distribution

In this paper, we propose a new method for determining the optimal base-stock level in a single echelon inventory system where the demand is a compound Erlang process and the lead-time is constant. The demand inter-arrival follows an Erlang distribution and the demand size follows a Gamma distribution. The stock is controlled according to a continuous review base-stock policy where unfilled demands are backordered. The optimal base-stock level is derived based on a minimization of the total expected inventory cost. A numerical investigation is conducted to analyze the performance of the inventory system with respect to the different system parameters and also to show the outperformance of the approach that is based on the compound Erlang demand assumption as compared to the classical Newsboy approach. This work allows insights to be gained on stock control related issues for both slow and fast moving stock keeping units.     

An application of fuzzy set theory to inventory control models

A method for solving an inventory control problem, of which input data are described by triangular fuzzy numbers will be presented here. The continuous review model of the inventory control problem with fuzzy input data will be focused in, and a new solution method will be presented. For the reason that the result should be a fuzzy number because of fuzzy input data, and the certain number about order quantity is prefered in the real-world, it is necessary to transform the fuzzy result to crisp one. The interval mean value concept is used here to help to solve this problem. Under the condition of total cost minimum, the interval order quantity maximum can be obtained.     

Considering lost sale in inventory routing problems for perishable goods

This paper presents a mathematical model for an inventory routing problem (IRP). The model is especially designed for allocating the stock of perishable goods. It is assumed that the age of the perishable inventory has a negative impact on the demand of end customers and a percentage of the demand is considered as lost sale. The proposed model balances the transportation cost, the cost of inventory holding and lost sale. In addition to the usual inventory routing constraints, we consider the cost of lost sale as a linear or an exponential function of the inventory age. The proposed model is solved to optimality for small instances and is used to obtain lower bounds for larger instances. We have also devised an efficient meta-heuristic algorithm to find good solutions for this class of problems based on Simulated Annealing (SA) and Tabu Search (TS). Computational results indicate that, for small problems, the average optimality gaps are less than 10.9% and 13.4% using linear and exponential lost sale functions, respectively. Furthermore, we show that the optimality gaps found by CPLEX grow exponentially with the problem size while those obtained by the proposed meta-heuristic algorithm increase linearly.     

一种多周期随机需求生产/库存控制方法

为了合理地对库存进行管理,使得产品的存贮、生产和缺货等费用的总和最小,建立了一种多周期随机需求生产／库存模型．该模型采用（s,Q）策略对生产和库存进行控制,即当成品库存降至S时准备生产,生产量为Q．通过对模型费用函数特性的分析,设计了一种最优生产控制算法,根据该算法可以得出系统的最优生产准备点和最优生产量．理论分析和计算结果表明,该方法可以有效地减小系统生产和库存的平均费用．     

Optimal joint replenishment,delivery and inventory management policies for perishable products

In this paper we analyze the optimal joint decisions of when, how and how much to replenish customers with products of varying ages. We discuss the main features of the problem arising in the joint replenishment and delivery of perishable products, and we model them under general assumptions. We then solve the problem by means of an exact branch-and-cut algorithm, and we test its performance on a set of randomly generated instances. Our algorithm is capable of computing optimal solutions for instances with up to 30 customers, three periods, and a maximum age of two periods for the perishable product. For the unsolved instances the optimality gap is always small, less than 1.5% on average for instances with up to 50 customers. We also implement and compare two suboptimal selling priority policies with an optimized policy: always sell the oldest available items first to avoid spoilage, and always sell the fresher items first to increase revenue.