Multi-item single source ordering problem with transportation cost: A Lagrangian decomposition approach

As a part of supply chain management literature and practice, it has been recognized that there can be significant gains in integrating inventory and transportation decisions. The problem we tackle here is a common one both in retail and production sectors where several items have to be ordered from a single supplier. We assume that there is a finite planning horizon to make the ordering decisions for the items, and in this finite horizon the retailer or the producer knows the demand of each item in each period. In addition to the inventory holding cost, an item-base fixed cost associated with each item included in the order, and a piecewise linear transportation cost are incurred. We suggest a Lagrangean decomposition based solution procedure for the problem and carry out numerical experiments to analyze the value of integrating inventory and transportation decisions under different scenarios.     

基于多种运输渠道的经济批量问题的多项式时间算法

为了集中管理, 降低成本, 增强竞争优势, 供应商通常只负责生产,而把产品的配送业务外包给某个配送中心, 由配送中心按照零售商的需求决定发货运输的方式和时间.这样供应商, 配送中心和零售商就构成了二级供应链系统.本文研究的是运输方式为不同参数的全单位数量折扣结构时, 二级供应链系统的经济批量问题.分析了最优解的性质,并对此问题的一种特殊情形给出一个多项式时间算法.     

Minimizing the Total Cost in an Integrated Vendor—Managed Inventory System

In this paper we consider a complex production-distribution system, where a facility produces (or orders from an external supplier) several items which are distributed to a set of retailers by a fleet of vehicles. We consider Vendor-Managed Inventory (VMI) policies, in which the facility knows the inventory levels of the retailers and takes care of their replenishment policies. The production (or ordering) policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The cost includes the fixed and variable production costs at the facility, the inventory costs at the facility and at the retailers and the transportation costs, that is the fixed costs of the vehicles and the traveling costs. We study two different types of VMI policies: The order-up-to level policy, in which the order-up-to level quantity is shipped to each retailer whenever served (i.e. the quantity delivered to each retailer is such that the maximum level of the inventory at the retailer is reached) and the fill-fill-dump policy, in which the order-up-to level quantity is shipped to all but the last retailer on each delivery route, while the quantity delivered to the last retailer is the minimum between the order-up-to level quantity and the residual transportation capacity of the vehicle. We propose two different decompositions of the problem and optimal or heuristic procedures for the solution of the subproblems. We show that, for reasonable initial values of the variables, the order in which the subproblems are solved does not influence the final solution. We will first solve the distribution subproblem and then the production subproblem. The computational results show that the fill-fill-dump policy reduces the average cost with respect to the order-up-to level policy and that one of the decompositions is more effective. Moreover, we compare the VMI policies with the more traditional Retailer-Managed Inventory (RMI) policy and show that the VMI policies significantly reduce the average cost with respect to the RMI policy.     

A two-level supply chain with partial backordering and approximated Poisson demand

We consider a two-level supply chain with a number of identical, independent ‘retailers’ at the lower echelon and a single supplier at the upper echelon controlled by continuous review inventory policy (R, Q). Each retailer experiences Poisson demand with constant transportation times. We assume constant lead time for replenishing supplier orders from an external warehouse to the supplier and unsatisfied retailer orders are backordered in the supplier. We assume that the unsatisfied demand is partially backordered in the identical retailers. The partially backordering policy is implemented in the identical retailers using an explicit control parameter ‘b’ which limits the maximum number of backorders allowed to be accumulated during the lead time. We develop an approximate cost function to find optimal reorder points for given batch sizes in all installations, the optimal value of b in the identical retailers and the related accuracy is assessed through simulation.     

Optimal retailer's ordering policies in the EOQ model under trade credit financing

The main purpose of this note is to modify the assumption of the trade credit policy in previously published results to reflect the real-life situations. All previously published models implicitly assumed that the supplier would offer the retailer a delay period, but the retailer would not offer the trade credit period to his/her customer. In most business transactions, this assumption is debatable. In this note, we assume that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer's replenishment model. Furthermore, we assume that the retailer's trade credit period offered by supplier M is not shorter than the customer's trade credit period offered by retailer N(M?N). Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal ordering policies. Then a theorem is developed to determine efficiently the optimal ordering policies for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this note.     

A two-echelon inventory model with lost sales

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R_i) control policy. The warehouse operates an (SQ, (S − 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L_i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.     

Analysis and design of returns policies from a supplier's perspective

This paper considers the problem of designing a returns policy in a supply chain from a supplier's perspective. The supply chain considered here is assumed to have one supplier and one retailer who serves a random demand of a product with a short life cycle. The retailer can return all the unsold products to the supplier with a partial refund. We found that if the retailer behaviour is rational, that is, ordering the optimal quantity to maximize its expected profit, then both retailer and supplier could benefit from the returns policy. Furthermore, we established that the optimal buyback price is independent of the mean of the random demand, but the variance of the demand has a significant impact on setting the optimal buyback price. The higher the variance the higher the optimal buyback price and the larger the profit gain of both parties. Numerical studies are employed to help understand the benefits of returns policies for the supplier, the retailer, and the whole supply chain.     

A two-echelon base-stock inventory model with Poisson demand and the sequential processing of orders at the upper echelon

We consider a two-echelon inventory system with a number of non-identical, independent ‘retailers’ at the lower echelon and a single ‘supplier’ at the upper echelon. Each retailer experiences Poisson demand and operates a base stock policy with backorders. The supplier manufactures to order and holds no stock. Orders are produced, in first-come first-served sequence, with a fixed production time. The supplier therefore functions as an M/D/1 queue. We are interested in the performance characteristics (average inventory, average backorder level) at each retailer. By finding the distribution of order lead time and hence the distribution of demand during order lead time, we find the steady state inventory and backorder levels based on the assumption that order lead times are independent of demand during order lead time at a retailer. We also propose two alternative approximation procedures based on assumed forms for the order lead time distribution. Finally we provide a derivation of the steady state inventory and backorder levels which will be exact as long as there is no transportation time on orders between the supplier and retailers. A numerical comparison is made between the exact and approximate measures. We conclude by recommending an approach which is intuitive and computationally straightforward.     

Direct shipping and the dynamic single-depot/multi-retailer inventory system

In this paper we study a single-depot/multi-retailer system with independent stochastic stationary demands, linear inventory costs, and backlogging at the retailers over an infinite horizon. In addition, we also consider the transportation cost between the depot and the retailers. Orders are placed each period by the depot. The orders arrive at the depot and are allocated and delivered to the retailers. No inventory is held at the depot. We consider a specific policy of direct shipments. That is, a lower bound on the long run average cost per period for the system over all order/delivery strategies is developed. The simulated long term average cost per period of the delivery strategy of direct shipping with fully loaded trucks is examined via comparison to the derived lower bound. Simulation studies demonstrate that very good results can be achieved by a direct shipping policy.     

Consignment contracting: Who should control inventory in the supply chain?

Consignment is a popular form of business arrangement where supplier retains ownership of the inventory and gets paid from the retailer based on actual units sold. The popularity of such an arrangement has come with some continued debates on who should control the supply chain inventory, the supplier or retailer. This paper aims at shedding light on these debated issues. We consider a single period supply chain model where a supplier contracts with a retailer. Market demand for the product is price-sensitive and uncertain. The supplier decides his consignment price charged to the retailer for each unit sold, and the retailer then chooses her retail price for selling the product. We study and compare two different consignment arrangements: The first allows the retailer to choose the supply chain inventory, together with her retail price, and is labeled as a Retailer Managed Consignment Inventory (RMCI) program; and the second calls for the supplier to decide the inventory, together with his consignment price, and is labeled as a Vendor Managed Consignment Inventory (VMCI) program. We show that with an RMCI program, the supply chain loses at least 26.4% of its first-best (expected) profit, while with VMCI, it loses just or no more than 26.4% of the first-best profit. Second, we demonstrate that both programs lead to an equal split of the corresponding channel profit between the supplier and the retailer. These results indicate that it is beneficial both to the supplier and to the retailer when delegating the inventory decision to the supplier rather than to the retailer in the channel.     

Multi-period price promotions in a single-supplier, multi-retailer supply chain under asymmetric demand information

This paper considers a two-stage supply chain in which a supplier serves a set of stores in a retail chain. We consider a two-stage Stackelberg game in which the supplier must set price discounts for each period of a finite planning horizon under uncertainty in retail-store demand. As a mechanism to stimulate sales, the supplier offers periodic off-invoice price discounts to the retail chain. Based on the price discounts offered by the supplier, and after store demand uncertainty is resolved, the retail chain determines individual store order quantities in each period. Because the supplier offers store-specific prices, the retailer may ship inventory between stores, a practice known as diverting. We demonstrate that, despite the resulting bullwhip effect and associated costs, a carefully designed price promotion scheme can improve the supplier’s profit when compared to the case of everyday low pricing (EDLP). We model this problem as a stochastic bilevel optimization problem with a bilinear objective at each level and with linear constraints. We provide an exact solution method based on a Reformulation-Linearization Technique (RLT). In addition, we compare our solution approach with a widely used heuristic and another exact solution method developed by Al-Khayyal (Eur. J. Oper. Res. 60(3):306–314, 1992) in order to benchmark its quality.     

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.     

Myopic control of stochastic inventories with intermittent updates: continuous <Emphasis Type="Italic">versus</Emphasis> periodic replenishment

A manufacturer who is responsible for supplying a retailer with a single product is considered. The retailer sells the product in response to stochastic demand and provides the manufacturer with periodic updates about his inventories. Replenishing the retailer's inventory under two myopic base-stock policies is addressed. These policies, referred to as vendor managed inventory, represent a relatively new approach to allocating responsibility in the replenishment process. Specifically, the manufacturer, who is responsible for the retailer's inventories, can replenish them either continuously at any point in time or periodically, at one point in time for each period. The myopic replenishment policies that are considered are of a base-stock type. It is shown that the selected policies become optimal as the number of review periods tends to infinity. Furthermore, the two replenishment alternatives are compared in terms of both base-stock levels and expected costs, including those for inventory holding/shortage and transportation costs. Although continuous rather than periodic replenishment is evidently more expensive in terms of transportation costs, it is shown that even when the transportation cost constitutes more than 55% of the total average cost, it may still be preferable to replenish continuously rather than periodically.     

A solution approach to the inventory routing problem in a three-level distribution system

We consider the infinite horizon inventory routing problem in a three-level distribution system with a vendor, a warehouse and multiple geographically dispersed retailers. In this problem, each retailer faces a demand at a deterministic, retailer-specific rate for a single product. The demand of each retailer is replenished either from the vendor through the warehouse or directly from the vendor. Inventories are kept at both the retailers and the warehouse. The objective is to determine a combined transportation (routing) and inventory strategy minimizing a long-run average system-wide cost while meeting the demand of each retailer without shortage. We present a decomposition solution approach based on a fixed partition policy where the retailers are partitioned into disjoint and collectively exhaustive sets and each set of retailers is served on a separate route. Given a fixed partition, the original problem is decomposed into three sub-problems. Efficient algorithms are developed for the sub-problems by exploring important properties of their optimal solutions. A genetic algorithm is proposed to find a near-optimal fixed partition for the problem. Computational results show the performance of the solution approach.     

A note on: Optimal ordering policy for stock-dependent demand under progressive payment scheme

In a recent paper, Soni and Shah [Soni, H., Shah, N. H. (2008). Optimal ordering policy for stock-dependent demand under progressive payment scheme. European Journal of Operational Research 184(1), 91–100] developed a model to find the optimal ordering policy for a retailer with stock-dependent demand and a supplier offering a progressive payment scheme to the retailer. This note corrects some errors in the formulation of the model of Soni and Shah. It also extends their work by assuming that the credit interest rate of the retailer may exceed the interest rate charged by the supplier. Numerical examples illustrate the benefits of these modifications.     

Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. We assume that the supplier would offer the retailer a delay period and the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model under the replenishment rate is finite. Furthermore, we assume that the retailer’s trade credit period offered by supplier M is not shorter than the customer’s trade credit period offered by retailer N (M ? N). Since the retailer cannot earn any interest in this situation, M < N.     

Real-time order tracking for supply systems with multiple transportation stages

This paper presents a stochastic model that evaluates the value of real-time shipment tracking information for supply systems that consist of a retailer, a manufacturer, and multiple stages of transportation. The retailer aggregates demand for a single product from end customers and places orders on the manufacturer. Orders received by the manufacturer may take several time periods before they are fulfilled. Shipments dispatched by the manufacturer move through multiple stages before they reach the retailer, where each stage represents a physical location or a step in the replenishment process. The lead time for a new order depends on the number of unshipped orders at the manufacturer’s site and the number and location of all shipments in transportation. The analytic model uses real-time information on the number of orders unfulfilled at the manufacturer’s site, as well as the location of shipments to the retailer, to determine the ordering policy that minimizes the long-run average cost for the retailer. It is shown that the long-run average cost is lower with real-time tracking information, and that the cost savings are substantial for a number of situations. The model also provides some guidelines for operating this supply system under various scenarios. Numerical examples demonstrate that when there is a lack of information it is better for the retailer to order every time period, but with full information on the status in the supply system it is not always necessary for the retailer to order every time period to lower the long-run average cost.     

Demand seasonality in retail inventory management

We investigate the value of accounting for demand seasonality in inventory control. Our problem is motivated by discussions with retailers who admitted to not taking perceived seasonality patterns into account in their replenishment systems. We consider a single-location, single-item periodic review lost sales inventory problem with seasonal demand in a retail environment. Customer demand has seasonality with a known season length, the lead time is shorter than the review period and orders are placed as multiples of a fixed batch size. The cost structure comprises of a fixed cost per order, a cost per batch, and a unit variable cost to model retail handling costs. We consider four different settings which differ in the degree of demand seasonality that is incorporated in the model: with or without within-review period variations and with or without across-review periods variations. In each case, we calculate the policy which minimizes the long-run average cost and compute the optimality gaps of the policies which ignore part or all demand seasonality. We find that not accounting for demand seasonality can lead to substantial optimality gaps, yet incorporating only some form of demand seasonality does not always lead to cost savings. We apply the problem to a real life setting, using Point-of-Sales data from a European retailer. We show that a simple distinction between weekday and weekend sales can lead to major cost reductions without greatly increasing the complexity of the retailer’s automatic store ordering system. Our analysis provides valuable insights on the tradeoff between the complexity of the automatic store ordering system and the benefits of incorporating demand seasonality.     

An efficient heuristic for inventory control when the customer is using a (s,S) policy

We consider a capacitated supply chain in which the supplier has the information of the (s,S) policy used by the retailer as well as the end-customer demand distribution. For the resulting inventory control problem at the supplier, optimal policies and structural properties were presented by Gavirneni et al. (Management Sci. 45(1) (1999) 16). They detailed an efficient solution procedure for the uncapacitated problem and resorted to computationally expensive infinitesimal perturbation analysis (IPA) for the capacitated situation. In this paper, we study a heuristic, based on the uncapacitated solution, for the capacitated situation. A detailed computational study showed that this heuristic is very efficient in that the costs increased by only 3.3% on the average. The heuristic was especially effective at higher capacities, lower holding costs, and extreme values of demand variance.     

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.