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.     

Order splitting in single sourcing with scheduled-release orders

Traditional inventory models assume that a buyer places one order with a supplier in each order cycle. A large number of researchers have studied the benefits of dual sourcing such that an order quantity is split and placed simultaneously with two suppliers. We show that many of the benefits of dual sourcing are due to order splitting rather than using two suppliers. We investigate order splitting with one supplier such that the first part of the order is sent out immediately but the second part of the order is released later (scheduled-release). Through extensive computational results, we show that in many situations where dual sourcing or the use of a cheaper supplier would be cost effective, single sourcing with order splitting using scheduled-release orders is better. The paper provides a quantitative rationale to continue with one supplier. We also summarize the qualitative reasons to prefer single sourcing or multiple sourcing.     

Analysis of a Continuous Review Stock-control Model with Multiple Suppliers

This paper analyses the use of multiple suppliers to replenish one stock item. The replenishment order is split into several portions, one for each supplier, and orders for these portions are placed simultaneously with the respective suppliers. Approximate expressions are derived for the mean and variance of the effective lead time, which is the minimum of the individual lead times of each supplier. These results give effective lead-time-demand mean and variance that are smaller than those of the individual suppliers. The results can be used to lower the operating cost of the stocking policy.     

需求随价格变化的具有折扣的易变质物品的库存模型

本文研究了在需求随价格变化及物品易变质的条件下，当供应商给予数量折扣时的库存问题。证明了当供应商给予数量折扣时，零售商的需求量是增大的，并给出了供应商给予数量折扣时零售商的订货量和订货周期的计算方法。对物品变质率和需求价格敏感系数对零售商的订货量、订货周期、出售价格和单位时间利润的影响进行了数值分析，并给出了数值算例。     

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.     

Derivation of the cost equation of a dual replenishment inventory model with two order levels

This paper investigates an inventory control system where a single product can be ordered in two separate and independent processes within an inventory cycle. The cost equation is developed by deriving the steady-state probabilities for each level of inventory and using those probabilities to define (a) the average level of inventory, (b) time between replenishments and (c) the units acquired using each replenishment. The terms of the equation are represented as a function of two order levels: the least-order level, used to determine the need for two replenishment processes, and the maximum-order level, which represents the highest level of inventory allowed.     

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.     

Determining optimal order splitting and reorder level for N-supplier inventory systems

This paper considers multiple-supplier single-item inventory systems, where the item acquisition lead times of suppliers and demand arrival are random, and backorder is allowed. The acquisition takes place when the inventory level depletes to a reorder level, and the order is split among multiple suppliers. The acquisition lead times may have different distributions, the unit purchasing prices from suppliers may be different, and thus the order quantities for different suppliers may be different. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per unit time, consisting of the fixed ordering cost, procurement cost, inventory holding cost and shortage cost, is minimized. We develop a mathematical model describing the system in detail. We also conduct extensive numerical experiments to analyze the advantages and distinct characteristics of multiple-supplier systems.     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

Are more suppliers better?: generalising the Guo and Ganeshan procedure

In multiple supplier inventory models, where several suppliers are used to replenish the stock of one item, computation of mean and variance of supplier lead times requires the knowledge of the moments of order statistics from the parent lead time distribution. This article presents a general procedure of finding the moments of supplier lead times in multiple supplier inventory models. The procedure is based on using the Generalised Lambda Distribution (GLD) to approximate the lead time distribution. Numerical examples are provided to validate our procedure. The proposed procedure has the advantage that the computations involved are very simple and can be used for any continuous unimodal lead time distribution.     

Production control and replenishment strategy with multiple suppliers

This paper considers the joint supplier selection, replenishment and manufacturing control problem in a dynamic stochastic context. This problem is characterized by conflicting interests between suppliers, the manufacturer, and clients, which raise the need for coordination and information sharing. This paper contributes to the discourse mainly by developing and resolving an integrated mathematical model leading to information sharing strategies for supplier selection, replenishments and production activities. This is an optimal control problem with state constraints and hybrid dynamics. A dynamic stochastic model is thus proposed, and the optimality conditions obtained are then solved numerically. It is shown that the problem considered leads to a modified state-dependent multi-level (s, S) policy for the supplier selection and replenishment strategy and a base-stock policy for the production activities. The fact that these control policies are coupled confirms the necessity of considering the interactions present in the system in an integrated model. The obtained results show clearly that it is always profitable to consider multiple suppliers to make replenishment and production decisions. Moreover, it is shown that the availability rates of the supply chain actors and the replenishment lead time are important parameters to consider when choosing the best supplier.     

Optimal pricing and inventory policies with reliable and random-yield suppliers: characterization and comparison

In this paper, we study the joint pricing and inventory replenishment problem for a periodic-review inventory system with random demand and dual suppliers, one of the suppliers is reliable but more expensive, the other supplier is less expensive but is unreliable with random yield. We characterize the firm’s optimal policies that simultaneously determine the optimal ordering and pricing decisions in each period over a finite planning horizon, and investigate the impacts of supply source diversification and supplier reliability on the firm and on its customers. We show that having source diversification or higher reliability of suppliers not only increases the firm’s expected profit, but also results in a lower optimal selling price, thus they benefit both the firm and its customers.     

A continuous-review inventory model with random disruptions at the primary supplier

We consider a continuous-review inventory problem for a retailer facing constant customer demand for a single product. This retailer is assumed to follow the well known and widely used order-up-to policy in making replenishment decisions, and can order from two suppliers who differ in reliability and costs. Supplier 1, the primary supplier, is cheaper, but is subject to random disruptions. Supplier 2, the backup supplier or the contingent source, is more expensive, but is perfectly reliable. If Supplier 1 is available when the inventory level at the retailer reaches the reorder point, the retailer orders from Supplier 1. Otherwise, it will wait for a while to see if Supplier 1 can recover from the disruption quickly. If so, it will still get replenishment from Supplier 1 to take advantage of its lower charge. However, the retailer will reroute to the backup supplier if Supplier 1 still does not recover from the disruption when the cap of waiting (the maximal waiting time of the retailer if Supplier 1 is disrupted) is reached. We analytically study the optimal sourcing and replenishment decisions at the retailer, and the impacts of various problem parameters on the optimal decisions. We also conduct extensive numerical experiments to compare different sourcing and replenishment decisions the retailer can make and get further managerial insights into the problem.     

Stock Rationing in a Continuous Review Two-Echelon Inventory Model

In this paper we consider a 1-warehouse, N-retailer inventory system where demand occurs at all locations. We introduce an inventory model which allows us to set different service levels for retailers and direct customer demand at the warehouse. For each retailer a critical level is defined, such that a retailer replenishment order is delivered from warehouse stock if and only if the stock level exceeds this critical level. It is assumed that retailer replenishment orders, which are not satisfied from warehouse stock, are delivered directly from the outside supplier, instead of being backlogged. We present an analytical upper bound on the total cost of the system, and develop a heuristic method to optimize the policy parameters. Numerical experiments indicate that our technique provides a very close approximation of the exact cost. Also, we show that differentiating among the retailers and direct customer demand can yield significant cost reductions.     

Order splitting in continuous review (Q,r) inventory models

A number of recent articles in the literature have argued the case, when lead time is variable, for splitting a replenishment order for Q between n suppliers by comparing this with the alternative of placing a single order for Q on one supplier. The split order compares favourably on the grounds that the arrival of the first component of a split order cannot be later than the arrival of an order from any one specified supplier. This note argues that an alternative comparison could be made with a policy of ordering Q/n from a single supplier (n times as often). It makes this comparison in the context of a continuous review (Q, r) inventory model but does so not by comparing aggregate costs but by fixing Q and the customer stock service level and comparing the average stock — an approach which is more appropriate to how many companies manage inventory in practice. We consider Poisson and deterministic demand processes, a general lead time distribution and both lost sales and backorder models.     

Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures

This paper considers a multi-supplier economic lot-sizing problem in which the retailer replenishes his inventory from several suppliers. Each supplier is characterized by one of three types of order cost structures: incremental quantity discount cost structure, multiple set-ups cost structure and all-unit quantity discount cost structure. The problem is challenging due to the mix of different cost structures. For all cases of the problem where each supplier is characterized by one of the first two cost structures, some optimality properties are proposed and optimal algorithms based on dynamic programming are designed. For the case where all suppliers are characterized by all-unit quantity discount cost structures, it is hard to design a polynomial time algorithm by the analyzed optimal properties. However, it is proved that one of its special cases can be solved in polynomial time.     

Inventory control in the presence of an electronic marketplace

This paper studies a periodic review inventory model in the presence of an electronic marketplace (EM). Emergency orders can be placed in the EM for additional cost, and excess inventory can be sold to the EM. When the order leadtime from the supplier is one period, the optimal inventory control policy is developed from a dynamic programming model of the problem. The policy is characterized by three critical inventory levels. When the order leadtime from the supplier is longer than one period, an EM policy is developed to determine the quantities of inventory to purchase from and sell to the EM in each period. Based on this EM policy, three ordering policies are proposed to determine the order quantity from the supplier. Numerical results show that significant cost reductions can be obtained by using the EM to adjust the inventory level in each period. The amount of cost reduction is greatly affected by system parameters, especially the order leadtime from the supplier and the costs for transactions in the EM.     

Approximate analysis and optimization of batch ordering policies in capacitated supply chains

Devising manufacturing/distribution strategies for supply chains and determining their parameter values have been challenging problems. Linking production management to stock keeping processes improves the planning of the supply chain activities, including material management, culminating in improved customer service levels. In this study, we investigate a multi-echelon supply chain consisting of a supplier, a plant, a distribution center and a retailer. Material flow between stages is driven by reorder point/order quantity inventory control policies. We develop a model to analyze supply chain behavior using some key performance metrics such as the time averages of inventory and backorder levels, as well as customer service levels at each echelon. The model is validated against simulation, yielding good agreement of robust performance metrics. The metrics are then used within an optimization framework to design the supply chain so as to minimize expected total system costs. The outcome of the optimization framework specifies how to move inventory throughout the supply chain and how to set inventory control parameters, i.e., reorder levels and replenishment batch sizes.     

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.