供应中断下双源采购的供应链契约设计

考虑由一个制造商和两个供应商构成的两级供应链契约设计问题,将可靠性服从随机分布的供应商作为主供应商,将完全中断供应商作为次供应商,建立以利润最大化为目标的签约和执行两阶段博弈的动态规划模型,得到制造商的最优契约设计。制造商根据产品零售价格的变化灵活做出只与一个供应商签约、与两个供应商同时签约或都不签约的决定,供应商根据契约设计内容决定自己的最优生产规模。研究表明：供应商最优生产规模与产品的单位生产成本呈负相关,与单位变动支付和单位惩罚成本呈正相关;次供应商的订货量与其本身的可靠性呈正相关,与主供应商的可靠性呈负相关;供应商的固定支付与其相应的可靠性呈负相关。     

Optimal ordering policies in response to a discount offer

Sometimes supplier offers a temporary price discount to increase cash flow or decrease the inventory level of certain items. Thus, the manufacturer may be able to improve the effectiveness of his inventory system by ordering a special quantity at this sale period. In this paper, economic order quantity (EOQ) models with a discounted price are developed to obtain the optimal ordering policy during the sale period for five different cases: (a) coincidence of sale period with replenishment time, (b) non-coincidence of sale period with replenishment time, (c) sale period is longer than a cycle, (d) discounted price as a function of the special ordering quantity, and (e) incremental discount. Each case has its own characteristics of the sale period and the discounted price. The objective is to take the maximum possible advantage from the discounted price by ordering a special quantity during the sale period. The optimal ordering policy is obtained by maximizing the difference between the two costs: Regular EOQ cost and special quantity cost during the sale period. Moreover, a comparison of different discount scenarios is developed to sense the effect of different parameters on the ordering policies. The annual gain obtained is linearly related to the discount and the on-hand remnant inventory. Numerical analyses are provided to illustrate and testify the values of the optimal special quantity. The analysis showed an impressive improvement in the effectiveness of the inventory system when a special order is placed during the sale period. The optimal special quantity is driven for each case to visualize real-life problems. Sensitivity analysis is also initiated to study the change in the total savings with respect to the variation of the special optimal quantity.     

Myopic optimal policy for a multi-period, two-delivery-lead-times, stochastic inventory problem with minimum cumulative commitment and capacity

This paper studies a single-product, multi-period, stochastic inventory problem that imposes the lower and upper bounds on the cumulative order quantity during a planning horizon and allows two delivery lead times. This model includes three features. The first one is that a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity during the planning horizon is constrained with the capacity. The second one is that the buyer agrees to purchase the product at least a certain percentage of the purchased capacity during the planning horizon. The third one is that the supplier allows the buyer to order the product with two-delivery-lead-times. We identify conditions under which a myopic ordering policy is optimal. We also develop an algorithm to calculate the optimal capacity when the minimum cumulative order quantity commitment is a certain percentage of the capacity. We then use the algorithm to evaluate the effect of the various parameters on the buyer’s minimum expected total cost during the planning horizon. Our computation shows that the buyer would benefit from the commitments and two-delivery-lead-times.     

Replenish-up-to inventory control policy with random replenishment intervals

The multi period inventory problems have been studied under two main assumptions. Continuous review assumption where an order can be made at any time depending on inventory position and periodic review assumption where an order can be initiated only at discrete time epochs. In this study, we analyze a multi period inventory problem that falls under neither of these two assumptions. In the case we consider, there are periodic replenishments but the replenishment intervals are taken to be i.i.d. random variables. This setting represents the real life cases where a supplier visits a retailer with random inter arrival times and the retailer replenishes his inventories based on a replenish-up-to-level inventory control policy. We also assume that only a certain fraction of unmet demand is backordered and the rest of it is lost.In this setting under general distribution between replenishment epochs, we show the concavity of the expected profit function and give the condition that must hold for the optimal replenish-up-to-level. We also present the specific solutions and analysis under two different distributions, namely, uniform and exponential distributions, together with some numerical examples.     

Production lot sizing with a secondary outsourcing facility

An extended economic production quantity (EPQ) model under stochastic demand is investigated in this paper, where a fixed lot sizing policy is implemented to reduce the complexity of production planning and inventory control, and outsourcing with a secondary facility is used to supplement the lot sizing policy and to cope with the random demand. The considered cost includes: setup cost for the batch production, inventory carrying cost, backorder cost when the demand cannot be met immediately during the production period, and outsourcing cost when the total demand is greater than the lot size in one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex. Extensive computational tests have illustrated that the average cost reduction of the proposed model is significant when compared with that of the classical lot sizing policy. Significant cost savings can be achieved by deploying the production lot sizing policy with an outsourcing strategy when the mean demand rate is high.     

Managing inventory with two suppliers under yield uncertainty and risk aversion

In this paper, we consider a single-product single-period inventory model in which the retailer can source from two suppliers. The primary supplier is cheaper but unreliable in the sense that it generates supply yield uncertainty, whereas the secondary supplier is perfectly reliable but more expensive. The reliable supplier's capacity is fixed and the retailer cannot order more than the quantity reserved in advance. We study the problem in the context of a risk-averse retailer who has to determine the optimal order quantity from the primary supplier and the optimal reserved quantity from the secondary supplier. We develop the model in the perspective of a low risk averse retailer and quantify the risk via an exponential utility function. We show by numerical experiments how the resulting dual sourcing strategies differ from those obtained in the risk-neutral analysis. We also examine the sensitivity of some model-parameters on the optimal decisions.     

Economic lot-sizing and dynamic quantity competition

We study a problem of dynamic quantity competition in continuous time with two competing retailers facing different replenishment cost structures. Retailer 1 faces fixed ordering costs and variable procurement costs and all inventory kept in stock is subject to holding costs. Retailer 2 only faces variable procurement costs. Both retailers are allowed to change their sales quantities dynamically over time. Following the structure of the economic order quantity (EOQ) model, retailer 1 places replenishment orders in batches and retailer 2 follows a just-in-time (JIT) policy. The objective of both retailers is to maximize their individual average profit anticipating the competitor's replenishment and output decisions. The problem is solved by a two-stage hierarchical optimization approach using backwards induction. The second-stage model is a differential game in output quantities between the two retailers for a given cycle length. At the first stage, the replenishment policy is determined. We prove the existence of a unique optimal solution and derive an open-loop Nash equilibrium. We show that both retailers follow contrary output strategies over the order cycle. The EOQ retailer, driven by inventory holding costs, decreases his market share whereas the output of the JIT retailer increases. Moreover, depending on the cost structure, the EOQ retailer might partially be a monopolist. At the first stage, the EOQ retailer determines the cycle length, anticipating the optimal output trajectories at the second stage.     

Inventory management in supply chains: A bargaining problem

This paper is focused on supply chain management from the perspective of inventory management. The coordination of order and production policies between buyers and suppliers in supply chains is of particular interest. When a buyer of an item decides independently, he will place orders based on his economic order quantity (EOQ). However, the buyer's EOQ may not lead to a favorable policy for the supplier. A cooperative order and production policy can reduce total cost significantly. Should the buyer have the dominant position to impose his EOQ on the supplier, then consequently no incentive exists for him to deviate from his EOQ in order to choose a cooperative policy. To induce the buyer to order in quantities more favorable to the supplier, the supplier could offer a cooperative policy associated by a side payment to the buyer. The research presented in this paper provides several bargaining models depending on alternative production policies of the supplier. With these bargaining models the offered cooperative policy and the offered side payment can be derived.     

A stochastic inventory model of dual sourced supply chain with lead-time reduction

Recently, the amount of literature on analyzing the effects of investment strategies to control lead times has been increasing. Issues on investment to reduce lead times are important because variability in lead time between successive stages often has a great effect on the coordination of supply chain.This paper considers dual-sourcing models with stochastic lead times and constant unit demand in which lead times are reduced at a cost that can be viewed as an investment. In order to obtain an analytically tractable model, the distributions of lead times for two suppliers are assumed to be exponential. In our two-supplier model, we will concentrate on lead times as random variables, which are made endogenous in the stochastic lead-time model through “expediting factors”, the constants of proportionality between the expedited lead times and ordinary lead times, as was done by Bookbinder and Çakanyildirim (Eur. J. Oper. Res. 115 (1999) 300).Firstly, we determine the order quantity (Q), reorder point level (r), and order splitting proportion (k₁) in the case of no lead-time reduction. Using the (Q,r) found, we decide the expediting factors and new k₁ in the case of lead-time reduction. We compare the expected total cost per unit time for the two models and investigate savings. Additionally, sensitivity analyses are conducted with respect to the various cost parameter ranges, and remarks are made for further research.     

A JIT lot-splitting model for supply chain management: Enhancing buyer–supplier linkage

This study develops a buyer–supplier coordination model to facilitate frequent deliveries in small lot sizes in a manufacturing supply chain. The proposed model, based on the integrated total relevant costs of both buyer and supplier, determines optimal order quantity, the number of deliveries/setups, and shipping quantity over a finite planning horizon in a relatively simple JIT single buyer single supplier scenario. Under deterministic conditions for a single product, we show that the optimal delivery policy adopted by both buyer and supplier in a cooperative manner can be economically beneficial to both parties. It is shown that the optimal delivery size can be unique, regardless of the order quantity and the number of deliveries. Numerical results are also presented.     

The economic production quantity (EPQ) with shortage derived algebraically

Previously, in several papers and textbooks, the classical economic order quantity (EOQ) and the economic production quantity (EPQ) formulas for the shortage case, have been derived using differential calculus and solving two simultaneous equations (derived from setting the two first partial derivatives to zero) with the need to prove optimality conditions with second-order derivatives. In a previous original piece of work, a new approach to find the EOQ with backlogging using some slight algebraic developments appeared. This paper extends the mentioned algebraic approach to the EPQ formula taking shortages into consideration within the case of only one backlog cost per unit and time unit. The final expressions provide the same formulas that are available in the classic textbooks on inventory theory.     

An improved model for supplier selection under capacity constraint and multiple criteria

Ghodsypour and O'Brien [2001. The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint. International Journal of Production Economics 73(1), 15–27] study multiple sourcing problem with multiple criteria and capacitated suppliers. Considering the buyer's quality requirements as a constraint, they develop a mixed integer non-linear programming model to find the least-cost cyclic ordering policy for the buyer. They propose to solve the model by enumerating over all possible supplier combinations. Although the problem is interesting and their analysis is correct, we point out two issues with their assumptions, namely, capacitated suppliers and cyclic ordering policy. We discuss two different capacitated supplier settings: (i) long-run average annual capacity, and (ii) exact annual capacity. First, under long-run average annual capacity assumption we propose a model which provides the same or a better solution and is much easier to solve than their model. Then, we discuss how to modify our and their models to handle exact annual capacity assumption.     

An integrated production-distribution model for a deteriorating inventory item

We develop an integrated production-distribution model for a deteriorating item in a two-echelon supply chain. The supplier’s production batch size is restricted to an integer multiple of the discrete delivery lot quantity to the buyer. Exact cost functions for the supplier, the buyer and the entire supply chain are developed. These lead to the determination of individual optimal policies, as well as the optimal policy for the overall, integrated supply chain. We outline a procedure for determining the optimal supply chain decisions with the objective of minimizing the total system cost. Our approach is illustrated through a numerical example.     

A serial supply chain of newsvendor problem with safety stocks under complete and partial information sharing

We consider a supply chain where multiple members are serially connected. The decision is to determine the ordering quantity of a member to the next upstream member in the supply chain. The basic cost model is similar to the newsvendor problem with additional consideration to safety stock. This paper presents optimal approaches for coordination of the supply chain under both complete and partial information sharing in order to maximize the total expected benefit. For complete information sharing we develop an optimal coordination algorithm. For partial information sharing, we propose an optimal coordination algorithm based on the Alternating Direction Method and the Diagonal Quadratic Approximation Method. A numerical example is discussed to show the optimal convergence of ordering quantities and discuss the properties of the proposed algorithms.     

A queueing-inventory system with two classes of customers

We consider a queueing-inventory system with two classes of customers. Customers arrive at a service facility according to Poisson processes. Service times follow exponential distributions. Each service uses one item in the attached inventory supplied by an outside supplier with exponentially distributed lead time. We find a priority service rule to minimize the long-run expected waiting cost by dynamic programming method and obtain the necessary and sufficient condition for the priority queueing-inventory system being stable. Formulating the model as a level-dependent quasi-birth-and-death (QBD) process, we can compute the steady state probability distribution by Bright–Taylor algorithm. Useful analytical properties for the cost function are identified and extensive computations are conducted to examine the impact of different parameters to the system performance measures.     

Channel coordination with the newsvendor model using asymmetric information

In this paper, after surveying short-term two-echelon supply channel coordination methods, we present an extended version of the newsvendor model in which the supplier has to fulfil all demand of the customer, even if this calls for an additional setup of production. Given uncertain demand forecast, the solution is an optimal production quantity that minimises the expected total cost including setup, inventory holding and obsolete inventory costs. Then, the model is studied in a decentralised setting where the customer has private information about the demand forecast, while the supplier knows the various cost factors. We suggest such a coordination protocol and payment scheme that provides both partners the right incentive for minimising the total cost: the customer is interested in sharing her unbiased demand forecast and uncertainty, while the supplier's rational decision concurs with the overall optimum. Hence, local decisions based on asymmetric information coordinate the channel in the global sense. The results are also demonstrated by taking some real-life test cases from an industrial study that motivated our work.     

TECHNICAL NOTE A MODIFIED EOQ MODEL WITH TEMPORARY SALE PRICE DERIVED WITHOUT DERIVATIVES

Most previous works applied differential calculus to derive optimal solutions for economic ordering policy. The formulations are suggested in this short note to show that optimal solutions can be derived algebraically without using differential calculus to treat the EOQ model with a temporary sale price.     

Concurrent pricing and lot sizing for make-to-order contract production

This paper develops a model for the simultaneous (i.e. concurrent) determination of the inventory and pricing policies of a supplier, which produces and supplies a product to a buyer, on the basis of a contractual agreement, calling for the delivery of a specific quantity periodically. Assuming that the customer is rational, i.e. it follows its own optimal purchasing policy, the objective of the supplier is to determine the product's selling price, in conjunction with an appropriate production/inventory policy, so that a predetermined gross profit level is achieved. It is further assumed that the supplier's production batch size is an integer multiple of the buyer's order quantity. In formulating a mathematical model of this situation, the interactions between the product's price, the buyer's economic order quantity and the supplier's selling price, costs and profit are taken into account. For solving this model, a simple iterative algorithm is proposed, which is illustrated through a numerical example. Sensitivity analysis performed on the model demonstrates that it is relatively robust and quite insensitive to errors in estimating the buyer's ordering to carrying cost ratio on the part of the producer.     

Optimal pricing and order quantity for the newsvendor problem with free shipping

To attract and keep customers, companies, especially those in e-business, are increasingly offering free shipping to buyers whose order sizes exceed the free shipping quantity. In this paper, given the supplier offers free shipping and the retailer faces stochastic demand, we determine the retailer's (i.e., the newsvendor's) optimal order quantity and the optimal selling price simultaneously. We consider two different ways in which price affects the demand distribution, namely price only affects the location or scale of the demand distribution. We explicitly incorporate the supplier's quantity discount and transportation cost into the models. The transportation cost function is very general, which includes those most commonly used in the literature. We numerically examine the impacts of free shipping, quantity discount, transportation cost, and demand variance on the retailer's optimal order quantity and pricing decisions. We find that even though the retailer faces uncertain demand, free shipping can effectively encourage the retailer to order more of the good and can benefit the supplier, the retailer, and the end customers. An increase in transportation cost or a decrease in purchase price will induce the retailer to order more of the good and decrease the retail price. With increasing demand variance, the retailer should order more of the good. We also find that the newsvendor can cope with demand variance by taking advantage of free shipping.     

基于收益共享的供应链中断协调研究

本文假定市场需求不确定和供给存在中断危机，同时考虑订货过剩损失和缺货成本，研究了由零售商、主供应商和备份供应商组成的供应链网络，构建了收益共享合同协调模型，通过相关参数的设定达到供应链协调。供应链协调时的供货量及整体利润均大于分散模型；备份供应商的供货量及利润随着中断危机的增大而增加，零售商利润也随之增加。