Single-vendor multi-buyer integrated production-inventory model with controllable lead time and service level constraints

This paper presents an integrated production-inventory model where a vendor produces an item in a batch production environment and supplies it to a set of buyers. The buyer level demand is assumed to be independent normally distributed and lead time of every buyer can be reduced at an added crash cost. The buyers review their inventory using continuous review policy, and the unsatisfied demand at the buyers is completely backordered. A model is formulated to minimize the joint total expected cost of the vendor–buyers system to determine the optimal production-inventory policy. Since it is often difficult to estimate the stock-out cost in inventory systems, and so instead of having stock-out cost component in the objective function, a service level constraint (SLC) corresponding to each buyer is included in the model. A Lagrangian multiplier technique based algorithmic approach is proposed, which evaluates a very limited number of combinations of lead time of the buyers to find simultaneously the optimal lead time, order quantity and safety factor of the buyers and the number of shipments between the vendor and the buyers in a production cycle. Finally, a numerical example and effects of the key parameters are included to illustrate the results of the proposed model.     

Supply chain model for a deteriorating product with time-varying demand and production rate

The paper develops a two-echelon supply chain model with a single-buyer and a single-vendor. The buyer sells a seasonal product over a short selling period and its inventory is subject to deterioration at a constant rate over time. The vendor's production rate is dependent on the buyer's demand rate, which is a linear function of time. Also, the vendor's production process is not perfectly reliable; it may shift from an in-control state to an out-of-control state at any time during a production run and produce some defective (non-conforming) items. Assuming that the vendor follows a lot-for-lot policy for replenishment made to the buyer, the average total cost of the supply chain is derived and an algorithm for finding the optimal solution is developed. The numerical study shows that the supply chain coordination policy is more beneficial than those policies obtained separately from the buyer's and the vendor's perspectives.     

A coordination system for seasonal demand problems in the supply chain

This article considers a single product coordination system using a periodic review policy, participants of the system including a supplier and one or more heterogeneous buyers over a discrete time planning horizon in a manufacturing supply chain. In the coordination system, the demand of buyer in each period is deterministic, the supplier replenishes all the buyers, and all participants agree to plan replenishment to minimize total system costs. To achieve the objective of the coordination system, we make use of small lot sizing and frequent delivery policies (JIT philosophy) to transport inventory between supplier and buyers. Moreover, demand variations of buyers are allowed in the coordination system to suit real-world situations, especially for hi-tech industries. Furthermore, according to the mechanisms of minimizing the total relevant costs, the proposed method can obtain the optimal number of deliveries, shipping points and shipping quantities in each order for all participants in the coordination system.     

An Optimal Batch Size for a Production System Operating Under a Fixed-Quantity,Periodic Delivery Policy

A manufacturing system which procures raw materials from suppliers and processes them to convert to finished products is considered here. This paper develops an ordering policy for raw materials to meet the requirements of a production facility which, in turn, must deliver finished products demanded by outside buyers at fixed interval points in time. First, a general cost model is developed considering both supplier (of raw material) and buyer (of finished products) sides. This model is used to determine an optimal ordering policy for procurement of raw materials, and the manufacturing batch size to minimize the total cost for meeting equal shipments of the finished products, at fixed intervals, to the buyers. The total cost is found to be a piece-wise convex cost function. An interval that contains the optimal solution is first determined followed by an optimization technique to identify the exact solution from this interval.     

Uncertain Switching Costs and Purchase Decisions in Electronic Markets

As advances in information technologies (IT) significantly reduce the time and cost of acquiring and processing product information, some buyers that traditionally work with one or few suppliers have switched to the market environment. The rapid growth of e-commerce has led researchers to believe that a uniform shift to the electronic markets is inevitable. This article examines the issue from the perspective of uncertain supplier’s performance, switching costs, and the value of information. We show that the presence of uncertainty and switching costs favors contractual relationships between buyers and suppliers. As IT makes the market more competitive, the marginal value of information diminishes. Meanwhile, the overall effect of IT on uncertainty and switching costs is fairly limited. As a result, buyers facing high uncertain supplier’s performance and switching costs may find working with a small number of suppliers a better choice.     

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.     

Optimizing delivery lead time/inventory placement in a two-stage production/distribution system

In this paper we study a system composed of a supplier and buyer(s). We assume that the buyer faces random demand with a known distribution function. The supplier faces a known production lead time. The main objective of this study is to determine the optimal delivery lead time and the resulting location of the system inventory. In a system with a single-supplier and a single-buyer it is shown that system inventory should not be split between a buyer and supplier. Based on system parameters of shortage and holding costs, production lead times, and standard deviations of demand distributions, conditions indicating when the supplier or buyer(s) should keep the system inventory are derived. The impact of changes to these parameters on the location of system inventory is examined. For the case with multiple buyers, it is found that the supplier holds inventory for the buyers with the smallest standard deviations, while the buyers with the largest standard deviations hold their own inventory.     

Optimal policy for a dynamic,non-stationary,stochastic inventory problem with capacity commitment

This paper studies a single-product, dynamic, non-stationary, stochastic inventory problem with capacity commitment, in which a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity over the planning horizon is constrained with the capacity. The objective of the buyer is to choose the capacity at the beginning of the planning horizon and the order quantity in each period to minimize the expected total cost over the planning horizon. We characterize the structure of the minimum sum of the expected ordering, storage and shortage costs in a period and thereafter and the optimal ordering policy for a given capacity. Based on the structure, we identify conditions under which a myopic ordering policy is optimal and derive an equation for the optimal capacity commitment. We then use the optimal capacity and the myopic ordering policy to evaluate the effect of the various parameters on the minimum expected total cost over the planning horizon.     

Production–shipment policy for EPQ model with quality assurance and an improved delivery schedule

This article is concerned with determining the production–shipment policy for an economic production quantity model with quality assurance and an improved delivery schedule. We extend a recent work by Chiu et al. [Y.-S.P. Chiu, C.-A.K. Lin, H.-H. Chang, and V. Chiu, Mathematical modeling for determining economic batch size and optimal number of deliveries for EPQ model with quality assurance, Math. Comput. Model. Dyn. Sys. 16 (4) (2010), pp. 373–388] by incorporating an alternative delivery plan that aims at lowering the inventory holding cost for both supplier and buyer in such an integrated inventory system. Mathematical modelling along with Hessian matrix equations is used, and as a result the optimal production batch size and optimal number of deliveries are derived. A numerical example is provided to demonstrate the practical use of the results and the significant savings in stock holding costs for both vendor and buyer.     

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。     

Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

Solving the vendor–buyer integrated inventory system with arithmetic–geometric inequality

In the past, economic order quantity (EOQ) and economic production quantity (EPQ) were treated independently from the viewpoints of the buyer or the vendor. In most cases, the optimal solution for one player was non-optimal to the other player. In today’s competitive markets, close cooperation between the vendor and the buyer is necessary to reduce the joint inventory cost and the response time of the vendor–buyer system. The successful experiences of National Semiconductor, Wal-Mart, and Procter and Gamble have demonstrated that integrating the supply chain has significantly influenced the company’s performance and market share (Simchi-Levi et al. (2000) [1]). Recently, Yang et al. (2007) [2] presented an inventory model to determine the economic lot size for both the vendor and buyer, and the number of deliveries in an integrated two stage supply chain. In this paper, we present an alternative approach to determine the global optimal inventory policy for the vendor–buyer integrated system using arithmetic–geometric inequality.     

An integrated producer–buyer supply chain with delivery cost under stochastic transportation time

An integrated producer–buyer supply chain is used to simultaneously determine the optimum levels of the safety stock, delivery quantity, and number of shipments in this paper. The scenario is created by scheduling a single-setup at the producer with multiple deliveries to the buyer, and all shipments to the buyer are equal-sized batches. This study attempts to study the effects of delivery cost and transportation time, assumes that there is a stochastic transportation time between both producer and buyer, and that shortages are allowed. The transportation time is assumed to be Weibull distributed. The objective functions of the integrated model include the setup cost, inventory carrying cost, and delivery cost. We analyze the scenario where the delivery cost is explicitly considered in the model rather than considered as part of the fixed ordering cost or insignificant. A numerical example is also presented to demonstrate the proposed model using actual shipping rate data. In particular, the results show that when the producer's and buyer's carrying costs are low, and/or the mean time of transportation and delivery costs are high, then this can benefit both parties with regard to sharing total profit.     

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.     

Trade credit for supply chain coordination

Trade-credit is a seller’s short-term loan to the buyer, allowing the buyer to delay payment of an invoice. It has been the largest source of working capital for a majority of business-to-business firms in the United States. Numerous theories have been proposed to explain trade-credit, mainly from finance perspectives. It has also been an important issue in supply chain management. Surprisingly, most literature in supply chain management has examined the retailer’s stocking policies given a supplier’s trade-credit. This paper attempts to shed light on trade-credit from a supplier’s perspective, and presents it as a tool for supply chain coordination. Specifically, we explicitly assume firms’ financial needs for inventory. Following a Newsvendor framework, we assume that the supplier grants trade-credit and markdown allowance. Given the supplier’s offer, the retailer determines order quantity and the financing option for the inventory, either trade-credit or direct financing from a financial institution. Our result shows that the supplier’s markdown allowance alone cannot fully coordinate the supply chain if the retailer employs direct financing. Positive financing costs call for trade-credit in order to subsidize the retailer’s costs of inventory financing. Using trade-credit in addition to markdown allowance, the supplier fully coordinates the retailer’s decisions for the largest joint profit, and extracts a greater portion of the maximized joint profit.     

A Production Lot-Size Model for Perishable Items Under Two Level Trade Credit Policy for a Retailer with a Powerful Position in a Supply Chain System

This paper investigates a production lot-size inventory model for perishable items under two levels of trade credit for a retailer to reflect the supply chain management situation. We assume that the retailer maintains a powerful position and can obtain full trade credit offered by supplier yet retailer just offers the partial trade credit to customers. Under these conditions, retailer can obtain the most benefits. Then, we investigate the retailer’s inventory policy as a cost minimization problem to determine the retailer’s inventory policy. A rigorous mathematical analysis is used to prove that the annual total variable cost for the retailer is convex, that is, unique and global-optimal solution exists. Mathematical theorems are developed to efficiently determine the optimal ordering policies for the retailer. The results in this paper generalize some already published results. Finally, numerical examples are given to illustrate the theorems and obtain a lot of managerial phenomena.     

A two-echelon inventory system with transportation capacity constraint

Consider a one-warehouse multi-retailer system under constant and deterministic demand, which is subjected to transportation capacity for every delivery period. To search for the best stationary zero inventory ordering (ZIO) policy, or the best power-of-two policy, or the best nested policy, the problem is formulated as a 0–1 integer linear program in which the objective function comprises of a fixed transportation cost whenever a delivery is made and the inventory costs for both the warehouse and retailers. To overcome the transportation capacity limitation, we extend the policies to allow for staggering deliveries. It is shown that with transportation capacity constraint the non-staggering policy can have its effectiveness close to 0% from the best staggering policy and the power-of-two policy with staggering allowed can have its effectiveness close to 0% from the optimal policy. Nevertheless in general, the power-of-two policy fairs well on a number of randomly generated problems. To solve the large distribution network problem, an efficient heuristic based on the power-of-two policy with staggering of deliveries is suggested.     

A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts

This paper explores the coordination between a supplier and a buyer within a decentralized supply chain, through the use of quantity discounts in a game theoretic model. Within this model, the players face inventory and pricing decisions. We propose both cooperative and non-cooperative approaches considering that the product traded experiences a price sensitive demand. In the first case, we study the dynamics of the game from the supplier's side as the leader in the negotiation obtaining a Stackelberg equilibrium, and then show how the payoff of this player could still improve from this point. In the second case, a cooperative model is formulated, where decisions are taken simultaneously, emulating a centralized firm, showing the benefits of the cooperation between the players. We further formulate a pricing game, where the buyer is allowed to set different prices to the final customer as a reaction to the supplier's discount decisions. For the latter we investigate the difference between feasibility of implementing a retail discount given a current coordination mechanism and without it. Finally the implications of transportation costs are analyzed in the quantity discount schedule. Our findings are illustrated with a numerical example showing the difference in the players’ payoff in each case and the optimal strategies, comparing in each case our results with existing work.     

Retailer��s optimal sourcing strategy by using one major supplier and one emergent supplier

Fast-changing market demand, short product life cycle, and unpredictable events have been enforcing companies in a supply chain to respond to customers’ needs as quickly as possible. While many firms are still seeking a better inventory management within the general business environment, this paper develops an ordering decision support model within a business environment with more unpredictable events. To reduce the uncertainty, a retailer adopts a double sourcing policy with one major supplier and one emergent supplier. Through a two-period dynamic programming model formulation and the simulations based on design of experiments, we analyzed the effects of factors such as penalty cost, backup ratio, purchasing cost ratio, and demand correlation coefficient on retailer’s ordering policy and expected profit. Detailed sensitivity analysis is further conducted based on combinations of penalty cost and backup ratio. Results show that the use of an emergent supplier help increase the retailer’s expected profits. It not only increases the retailer’s upstream sourcing flexibility, but also increases the retailer’s service level with a lower inventory level. As penalty cost or backup ratio increases, the contribution of emergent supplier increases.     

An economic order quantity with imperfect quality and quantity discounts

Previous studies in the issue of inventory models with imperfect quality assumed the defectives could be sold in a batch by the end of the inspection process and the manufacturing systems were push systems. However, the above assumptions may not be true in the pull system in which buyer is powerful. Therefore, in this paper, we develop a new inventory model for items with imperfect quality and quantity discounts where buyer has exerted power over its supplier. Based on the concept of powerful buyer, there are three considerations included in this new model: (1) the order quantity is manufactured at one setup and is shipped over multiple deliveries, (2) the defectives are screened out by a 100% inspection for each shipment but sold in a batch by the end of inspection at the last shipment of each cycle, and (3) the supplier offers quantity discounts to response the request of the powerful buyer. Further, an algorithm is developed to help the powerful buyer to determine the optimal order policy accurately and quickly. Two numerical examples are available in this paper to illustrate the proposed model and algorithm. Besides, based on the numerical examples, a sensitivity analysis is made to investigate the effects of four important parameters (the inspection rate, the defective rate, the receiving cost, and the ordering cost) on the optimal solution.