Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

An (s,S) random lifetime inventory model with a positive lead time

Positive lead times substantially complicate the modeling and analysis of inventory systems with finite shelf lifetimes and they have not been sufficiently addressed in the existing literature. In this paper, we analyze an (s, S) continuous review model with a positive lead time. We assume an exponential lifetime and an exponential lead time. Matrix-geometric solutions can be obtained for the steady state probability distribution of the inventory level. We then derive the total expected cost function. We carry out numerical studies and gain insights to the selection of system parameters. The findings on the impact of a positive lead time on the optimal reorder point and reorder batch size will be useful in developing strategies in handling inventory problems with finite lifetimes and positive lead times.     

Production-inventory systems in stochastic environment and stochastic lead times

We consider a production-inventory system where the production and demand rates are modulated by a finite state Continuous Time Markov Chain (CTMC). When the inventory position (inventory on hand – backorders+inventory on order) falls to a reorder point r, we place an order of size q from an external supplier. We consider the case of stochastic leadtimes, where the leadtimes are i.i.d. exponential(μ) random variables, and orders may or may not be allowed to cross. We derive the distribution of the inventory level, and analyze the long run holding, backlogging, and ordering cost rate per unit time. We use simulation to study the sensitivity of the system to the distribution of the lead times.     

Managing Dynamic Inventory Systems with Product Returns: A Markov Decision Process

This paper presents a Markov decision process for managing inventory systems with Markovian customer demand and Markovian product returns. Employing functional analysis, we prove the existence of the optimal replenishment policies for the discounted-cost and average-cost problems when demand, returns, and cost functions are of polynomial growth. Our model generalizes literature results by integrating Markovian demand, Markovian returns, and positive replenishment lead times. In particular, the optimality of the reorder point, order-up-to policies is proved when the order cost consists of fixed setup and proportional cost components and the inventory surplus cost is convex. We then make model extensions to include different cost components and to differentiate returned products from new ones. Finally, we derive managerial insights for running integrated closed-loop supply chains. At the aggregate level, returns reduce effective demand while many structural characteristics of inventory models are intact. A simple heuristic for managing systems with returns is to still utilize literature results without returns, but effective demand is lower than customer demand.     

Sourcing from suppliers with random yield for price-dependent demand

We study a multi-period inventory planning problem. In each period, the firm under consideration can source from two possibly unreliable suppliers for a price-dependent demand. Our analysis suggests that the optimal procurement policy is neither a simple reorder-point policy nor a complex one without any structure, as previous studies suggest. Instead, we prove the existence of a reorder point for each supplier. No order is placed to that supplier for any inventory level above the reorder point and a positive order is issued to that supplier for almost every inventory level below the reorder point. We characterize conditions under which the optimal policy reveals monotone response to changes in the inventory level. Furthermore, two special cases of our model are examined in detail to demonstrate how our analysis generalizes a number of well-known results in the literature.     

Inventory control for point-of-use locations in hospitals

Most inventory management systems at hospital departments are characterised by lost sales, periodic reviews with short lead times, and limited storage capacity. We develop two types of exact models that deal with all these characteristics. In a capacity model, the service level is maximised subject to a capacity restriction, and in a service model the required capacity is minimised subject to a service level restriction. We also formulate approximation models applicable for any lost-sales inventory system (cost objective, no lead time restrictions etc). For the capacity model, we develop a simple inventory rule to set the reorder levels and order quantities. Numerical results for this inventory rule show an average deviation of 1% from the optimal service levels. We also embed the single-item models in a multi-item system. Furthermore, we compare the performance of fixed order size replenishment policies and (R,?s,?S) policies.     

(s, S) Inventory systems with lost sales and Markov renewal demands

In the study of stochastic inventory systems it is generally assumed that the demand epochs are renewable in nature. The present paper deals with a single-item (s, S) inventory model with a finite number of different types of demands, in which the demand epochs form a Markov renewal process. The lead times are exponentially distributed and the demands that occur during stockout periods are not backordered. For this model the transient and limiting inventory level distributions are computed. Also the theory of point processes is used to obtain the mean reorder and shortage rates and their limiting values. For the special case of renewal demands, the problem of minimizing the long-run expected cost rate is analysed.     

Lead time and ordering cost reductions in continuous review inventory systems with partial backorders

This paper investigates the impact of ordering cost reduction on the modified continuous review inventory systems involving variable lead time with a mixture of backorders and lost sales. The objective is to simultaneously optimise the order quantity, ordering cost, reorder point and lead time. We first assume the lead time demand follows a normal distribution, then relax this assumption to consider the distribution free case where only the mean and variance of lead time demands are known. An algorithm procedure of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.     

A Nomogram for Stock Control

A well-known replenishment policy for an inventory subject to probabilistic demand gives rise to the problem of determining an order quantity and a reorder level which minimize the average cost of keeping inventory. If the problem can be expressed in terms of a simple approximative model which includes a restriction on back-orders and represents demand in the replenishment lead time as normally distributed, it is possible to construct a "set-square index" nomogram which accomplishes a major part of the necessary computations. The procedure for using the nomogram and the principles of its construction are described.     

Monotonicity properties of a class of stochastic inventory systems

We consider inventory systems which are governed by an (r,q) or (r,nq) policy. We derive general conditions for monotonicity of the three optimal policy parameters, i.e., the optimal reorder level, order quantity and order-up-to level, as well as the optimal cost value, as a function of the various model primitives, be it cost parameters or complete cost rate functions or characteristics of the demand and leadtime processes. These results are obtained as corollaries from a few general theorems, with separate treatment given to the case where the policy parameters are continuous variables and that where they need to be restricted to integer values. The results are applied both to standard inventory models and to those with general shelf age and delay dependent inventory costs.     

An optimal (Q,r) policy for a multipart assembly system under stochastic part procurement lead times

We consider an EOQ-type model for a simple production system, where a number of parts are acquired to produce a single product and the part procurement lead times are random. Assembly is instantaneous and takes place intermittently in batches but cannot start until all the parts are available. The problem is to simultaneously determine when to order each part and what lot size to produce, namely to determine the reorder point for each part and the assembly lot size, so that the average total cost per unit time, composed of the setup cost, inventory holding costs for the parts and the assembled product, and the shortage cost for the assembled product, is minimized. We then develop a tailormade solution method for this problem to obtain a global optimal solution by taking advantage of the structure of the problem formulation, where the nonlinear programming problem is decomposed into a family of subproblems parametrized by the average assembly delay time. Numerical experiments are then conducted for the case of twopart problems, and some interesting observations are made.     

Inventory models with variable lead time and present value

The figures for inventory make up a huge proportion of a company's working capital. Because of this, we formulated the optimal replenishment policy considering the time value of money to represent opportunity cost. In this article, we provide a mixed inventory model, in which the distribution of lead time demand is normal, to consider the time value. First, the study tries to find the optimal reorder point and order quantity at all lengths of lead time with components crashed to their minimum duration. Secondly, we develop a method to insure the uniqueness of the reorder point to locate the optimal solution. Finally, some numerical examples are given to illustrate our findings.     

A Model for Determining Tactical Parameters for Materials Requirements Planning Systems

This paper develops an approximate model for simultaneously determining reorder intervals and planned lead times for MRP systems. Unlike previous research, this model explicitly considers commonality, non-instantaneous production, multiple work centres (with multiple machines and with limited capacity), and queue inventory carrying cost. Four heuristic solution procedures for the non-linear integer optimization problem are proposed and compared with an exact branch-and-bound algorithm on a set of forty test problems. The results suggest that two of the heuristics are both effective and efficient.     

Simple approximations for the reorder point in periodic and continuous review (s, S) inventory systems with service level constraints

In this paper we consider a general class of (s, S) inventory systems including periodic review and continuous review systems. We allow for stochastic lead times for replenishment orders provided that the probability of orders crossing in time is negligible for the relevant (s, S) control rules. In accordance with common practice we emphasize on service level constraints rather than assuming given stockout costs. In particular we consider the service measure requiring that a specified fraction of the demand is met directly from stock on hand. The purpose of this paper is to present practically useful approximations for the recorder point s such that the required service level is achieved. By a simple and direct approach, a unifying treatment of the general class of (s, S) inventory systems considered is given. We obtain for the first time tractable approximations for the continuous review (s, S) inventory system with undershoots of the reorder point. Also, we discuss 2-moments approximations obtained by fitting normal respectively gamma distributions to the empirical demand distributions. Extensive numerical experience with the approximations is reported, including results about the sensitivity of the reorder point to the higher moments of the demand distributions.     

Stock Control with Two Suppliers and Normal Lead Times

This paper helps practitioners to compute the mean and variance of the lead time demand distribution when two suppliers are used simultaneously to replenish stock of a single item. The lead times of each supplier are assumed to be normally distributed and two replenishment orders are placed, one with each supplier, at the same time. The results indicate that the reorder level required to give a specific probability of no stock-out during replenishment is lower when using two suppliers simultaneously. Tables have also been prepared to help practitioners determine the minimum sizes of replenishment orders when two suppliers are used and replenishment orders are placed at the same time.     

Back Ordering in Inventory Control

An approach to determining the empirical relationship between time out of stock and back ordering is outlined. Using this relationship the expected lost sales in a reorder level system of inventory control are derived for continuous and block demand depletion, fixed and variable lead times. Their applicability to a generalized inventory model is discussed.     

Reducing lost-sales rate on the stochastic inventory model with defective goods for the mixtures of distributions

In this paper, we assume that the demands of different customers are not identical in the lead time. Thus, we investigate a continuous review inventory model involving controllable lead time and a random number of defective goods in buyer’s arriving order lot with partial lost sales for the mixtures of distributions of the lead time demand to accommodate more practical features of the real inventory systems. Moreover, we analyze the effects of increasing investment to reduce the lost sales rate when the order quantity, reorder point, lost sales rate and lead time are treated as decision variables. In our studies, we first assume that the lead time demand follows the mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. By analyzing the total expected cost function, we develop an algorithm to obtain the optimal ordering policy and the optimal investment strategy for each case. Finally, we provide numerical examples to illustrate the results.     

可替代产品库存模型的研究

市场上，很多产品之间可相互替代，某种产品缺货时，可用另一种产品替代，也可以重新进货以满足顾客的需求。我们的目的是：从销售商的角度，讨论这两个因素对库存策略的影响。我们建立了这类问题有两个产品的单调期的利润最大化模型。证明了问题的解的存在性，给出了目标函数是凹函数和子模函数的充分条件，讨论了求解的方法和各参数对库存的影响。通过对几种特殊情况的讨论和比较，证明了替代和再订货可以提高利润并且可减少库存总量。     

Competitive strategies and market segmentation for suppliers with substitutable products

This research studies the competition between two coexisting suppliers in a two-echelon supply chain. The suppliers have different inventory cost structures (holding cost and setup cost). Each supplier offers one type of the two substitutable products to multiple buyers. Buyers’ preferences between the substitutable products differ. Each buyer has a particular order profile (order frequency and quantity). A buyer chooses between the suppliers based on the prices offered by both suppliers and his/her own preference. A Hotelling-type model is used to describe buyers’ preferences for the products. We are able to describe the conditions for buyers to switch between the suppliers, and therefore spot the buyer groups that may or may not switch when the suppliers compete. Pricing strategies for different buyer groups are suggested to the competitive suppliers accordingly. Furthermore, equilibrium prices, market segments, and overall profits for the suppliers are revealed based on Game Theory. An algorithm is also proposed to forecast buyers’ reactions to suppliers’ pricing strategies given the buyers’ order profiles and preferences between the substitutable products.