An integrated production and inventory model for a system comprising an assembly supply chain and a distribution network

This study investigates the production and inventory problem for a system comprising an assembly supply chain and a distribution network. A uniform lot size is produced uninterruptedly with a single setup at each production stage. Equal-sized batch shipment policy is applied to the whole system and the number of batches can be varied. All retailers have agreed on a joint replenishment policy with a common replenishment cycle. The objective is to determine the optimal common replenishment cycle, the number of batches of each production stage and retailer, all of which minimises the integrated total cost. Moreover, a new concept is introduced; namely, critical replenishment cycle. The replenishment cycle division (RCD) and recursive tightening (RT) methods are then developed to obtain the optimal solutions to the subject problem. Two theorems are verified to ensure the solutions obtained by the RCD and RT methods reaching the global optimum. An example is presented to illustrate the procedures involved in the RCD and RT methods.     

Joint decisions of shipment consolidation and dynamic pricing of food supply chains

This paper focuses on a perishable product supply chain with a vendor and multiple retailers. These retailers, densely spread in a distribution zone, are sensitive to price, delivery time and product quality. With the aim of optimizing the vendor’s expected long-run average profit during a shipment consolidation cycle, an analytical model is proposed for this problem. According to the upper bound expressions of the expected long-run average profit, the approximate optimal time policy and freshness-keeping cost are given based on a certain range of time parameter. Our theoretical findings are verified through a numerical case. Some useful managerial insights are obtained by analyzing the sensitivity of this model from six perspectives, which are market scenarios, types of perishable products, quality requirements of all retailers, cost parameters, line-haul time and vehicle capacity.     

Analysis of the maximum level policy in a production-distribution system

We consider a production-distribution system, where a facility produces one commodity which is distributed to a set of retailers by a fleet of vehicles. Each retailer defines a maximum level of the inventory. The production policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The overall cost is composed by fixed and variable production costs at the facility, inventory costs at both facility and retailers and routing costs. We study two different types of replenishment policies. The well-known order-up to level (OU) policy, where the quantity shipped to each retailer is such that the level of its inventory reaches the maximum level, and the maximum level (ML) policy, where the quantity shipped to each retailer is such that the inventory is not greater than the maximum level. We first show that when the transportation is outsourced, the problem with OU policy is NP-hard, whereas there exists a class of instances where the problem with ML policy can be solved in polynomial time. We also show the worst-case performance of the OU policy with respect to the more flexible ML policy. Then, we focus on the ML policy and the design of a hybrid heuristic. We also present an exact algorithm for the solution of the problem with one vehicle. Results of computational experiments carried out on small size instances show that the heuristic can produce high quality solutions in a very short amount of time. Results obtained on a large set of randomly generated problem instances are also shown, aimed at comparing the two policies.     

A decision rule for coordination of inventory and transportation in a two-stage supply chain with alternative supply sources

This paper deals with a two-stage supply chain that consists of two distribution centers and two retailers. Each member of the supply chain uses a (Q,R) inventory policy, and incurs standard inventory holding and backlog costs, as well as ordering and transportation costs. The distribution centers replenish their inventory from an outside supplier, and the retailers replenish inventory from one of the two distribution centers. When a retailer is ready to replenish its inventory that retailer must decide whether it should replenish from the first or second distribution center. We develop a decision rule that minimizes the total expected cost associated with all outstanding orders at the time of order placement; the retailers then repeatedly use this decision rule as a heuristic. A simulation study which compares the proposed policy to three traditional ordering policies illustrates how the proposed policy performs under different conditions. The numerical analysis shows that, over a large set of scenarios, the proposed policy outperforms the other three policies on average.     

Incorporating lateral transfers of vehicles and inventory into an integrated replenishment and routing plan for a three-echelon supply chain

This paper focuses on developing an integrated replenishment and routing plan that takes into account lateral transfers of both vehicles and inventory for a three-echelon supply chain system including a single plant, multiple distribution centers and multiple retailers. A mixed integer programming model to the overall system is formulated first, and then an optimization-based heuristic consisting of three major components is proposed. The purpose of the first component is to assign retailers to distribution centers, and determine routing schedules for each distribution center. And the remaining two components are corresponding to two smaller optimization models – a variant of the classical transportation problem modeled for determining vehicle transfer between distribution centers, and a variant of the conventional minimum cost network flow problem modeled for determining inventory replenishment and transfer. Experimental results reveal that the proposed algorithm is rather computational effectiveness, and the pooling strategy that considers both vehicles and inventory transfers is a worthy option in designing supply chain operations.     

随机需求下的选址-库存配送系统集成规划模型及算法

研究了随机需求条件下由单供应商、候选分拨中心和分销点构成的选址－库存问题, 分销点、分拨中心分别基于周期检查(R, s,Q)和连续检查(s, S)库存控制策略. 综合考虑库存成本、运输成本和设施成本之间的均衡关系, 建立了二级库存与无能力约束选址集成规划模型. 给出了适合求解实际规模问题的拉格朗日松弛算法, 提出了求解子问题的有效启发式方法, 改进了次梯度优化方法. 通过仿真试验验证了模型的正确性和算法的有效性. 最后讨论了相对于传统规划方法, 需求方差、服务水平、持有成本、提前期等关键库存控制参数对系统运营成本节约的影响规律.     

Design of Stochastic Distribution Networks Using Lagrangian Relaxation

This paper addresses the design of single commodity stochastic distribution networks. The distribution network under consideration consists of a single supplier serving a set of retailers through a set of distribution centers (DCs). The number and location of DCs are decision variables and they are chosen from the set of retailer locations. To manage inventory at DCs, the economic order quantity (EOQ) policy is used by each DC, and a safety stock level is kept to ensure a given retailer service level. Each retailer faces a random demand of a single commodity and the supply lead time from the supplier to each DC is random. The goal is to minimize the total location, shipment, and inventory costs, while ensuring a given retailer service level. The introduction of inventory costs and safety stock costs leads to a nonlinear NP-hard optimization problem. A Lagrangian relaxation approach is proposed. Computational results are presented and analyzed showing the effectiveness of the proposed approach.     

随机需求下的选址一库存配送系统集成规划模型及算法

研究了随机需求条件下由单供应商、候选分拨中心和分销点构成的选址-库存问题,分销点、分拨中心分别基于周期检查(R,s,Q)和连续检查(s,S)库存控制策略．综合考虑库存成本、运输成本和设施成本之间的均衡关系,建立了二级库存与无能力约束选址集成规划模型．给出了适合求解实际规模问题的拉格朗日松弛算法,提出了求解子问题的有效启发式方法,改进了次梯度优化方法．通过仿真试验验证了模型的正确性和算法的有效性．最后讨论了相对于传统规划方法,需求方差、服务水平、持有成本、提前期等关键库存控制参数对系统运营成本节约的影响规律．     

An integrated approach for logistic and vendor managed inventory in supply chain

In this paper, a quaternary policy system towards integrated logistics and inventory aspect of the supply chain has been proposed. A system of multi retailers and distributors, with each distributor following a unique policy, will be analysed. The first policy is continuous time replenishment policy where the retailers’ inventory is replenished in every time interval. In the next three policies, inventory of the retailers will be replenished by some definite policy factors. The vendor managed inventory (VMI) system is used for updating the inventory of the retailers. An order-up-to policy (q, Q) is used for updating the inventory of distributors. Total erstwhile demands to the retailer will be used to determine the amount of inventory acclivity. Furthermore, the distributors will be sending the delivery vehicles to few fellow retailers who are shortlisted according to the policy, followed by the retailers and associated distributors. On the basis of random demand that the retailers are facing from end customers and the total demand that has incurred in the supply chain, products are unloaded to the selected retailers from the delivery vehicle. The path of the delivery vehicle is retrieved by dynamic ant colony optimization. In addition, a framework has been developed to measure the end-customer satisfaction level and total supply chain cost incorporating the inventory holding cost, ordering cost and the transportation cost. The framework has been numerically moulded with different settings to compare the performance of the quadruplet policies.     

Rationing mechanisms and inventory control-policy parameters for a divergent supply chain operating with lost sales and costs of review

We consider a static divergent two-stage supply chain with one distributor and many retailers. The unsatisfied demands at the retailers’ end are treated as lost sales, whereas the unsatisfied demand is assumed to be backlogged at the distributor. The distributor uses an inventory rationing mechanism to distribute the available on-hand inventory among the retailers, when the sum of demands from the retailers is greater than the on-hand inventory at the distributor. The present study aims at determining the best installation inventory control-policy or order-policy parameters such as the base-stock levels and review periods, and inventory rationing quantities, with the objective of minimizing the total supply chain costs (TSCC) consisting of holding costs, shortage costs and review costs in the supply chain over a finite planning horizon. An exact solution procedure involving a mathematical programming model is developed to determine the optimum TSCC, base-stock levels, review periods and inventory rationing quantities (in the class of periodic review, order-up-to S policy) for the supply chain model under study. On account of the computational complexity involved in optimally solving problems over a large finite time horizon, a genetic algorithm (GA) based heuristic methodology is presented.     

Coordination of split deliveries in one-warehouse multi-retailer distribution systems

In this paper, we consider a distribution system where a warehouse is responsible for replenishing the inventories at multiple retailers by a fleet of vehicles of limited capacity. If a distribution policy of the system involves split deliveries, that is, the inventory of at least one retailer is replenished by using multiple vehicle routes, the coordination of the deliveries can further reduce the inventory cost of the retailer. We consider the coordination where two split deliveries are realized by direct shipping and multiple-stop shipping, respectively. To the best of our knowledge, this kind of coordination was never studied in the literature but can find its application in inventory routing problems. This paper proposes and analyses a class of coordination policies for the split deliveries which can reduce the inventory costs of the retailers without increasing transportation costs. A non-linear programming model is established for formulating the class of polices. Because the optimal coordination policy corresponding to an optimal solution of the model may be hard to find and/or implement, two simple but effective coordination policies are proposed. The inventory cost savings realized by the two policies are evaluated analytically and algorithmically. Our theoretical analysis and computational experiments show that both policies are effective. Under certain conditions, they can save 50% of the inventory costs at the retailers without increasing transportation costs.     

Optimal ordering decision for deteriorating items with expiration date and uncertain lead time

The ordering policy for the retailers and the suppliers is a function of deterioration, product expiration date, the supplier’s uncertain lead time, available capital constraint and the retailer’s seasonal pattern demand. We develop a deteriorating inventory replenishment model of the system and present an algorithm to derive the retailer’s optimal replenishment cycle, shortage period, order quantity and the supplier’s managing cost. Numerical example and sensitivity analysis are given to illustrate the model.     

Analysing the effectiveness of vendor-managed inventory in a single-warehouse,multiple-retailer system

This paper considers a two-stage supply chain, consisting of a single warehouse and multiple retailers facing deterministic demands, under a vendor-managed inventory (VMI) policy. It presents a two-phase optimisation approach for coordinating the shipments in this VMI system. The first phase uses direct shipping from the supplier to all retailers to minimise the overall inventory costs. Then, in the second phase, the retailers are clustered using a construction heuristic in order to optimise the transportation costs while satisfying some additional restrictions. The improvement of the system's performance through coordinated VMI replenishments against the system with direct shipping only is shown and discussed in the comparative analysis section.     

考虑多类零售商的库存与运输VMI 集成策略

研究由单批发商多零售商构成的供应商管理库存(VMI)供应链,批发商向外部供应商订货,并为价格和缺货成本不同的零售商补货.首先构造解析模型以分析如何为多类零售商确定不同的库存分配策略;然后将这一最优的库存分配策略与最优的先到先服务(FCFS)策略进行比较,同时比较相应简化的库存分配策略和简化的FCFS策略.解析和算例结果表明,简化的库存分配策略总是优于简化的FCFS策略,而最优的库存分配策略大部分情况下优于最优的FCFS策略.     

A model and methodologies for the location problem with logistical components

This paper significantly extends traditional facility location models by introducing several logistical cost components such as holding, ordering, and transportation costs in a multi-commodity, multi-location framework. Since location and logistical costs are highly inter-related, the paper provides an integrated model, and seeks to minimize total physical distribution costs by simultaneously determining optimal locations, flows, shipment compositions, and shipment cycle times. Two sophisticated heuristic methodologies, based on Lagrangian relaxation and simulated annealing, respectively, are provided and compared in an extensive computational experiment.Scope and purposeLogistics has recently acquired great significance in industry, in part due to the rapidly growing interest in Supply Chain Management. One of the important open issues in logistics is the effective integration of logistical cost components such as transportation cost with facility location models, since the two are highly inter-related in practice. In particular, locations, flows, shipment compositions, and shipment cycle times are highly inter-dependent. The determination of optimal values of these variables is crucial for minimizing physical distribution costs. This paper proposes an integrated location–consolidation model and provides two sophisticated methodologies to solve the problem. The relative performance of the two methodologies is investigated in an extensive computational experiment.     

Stochastic models for the coordinated production and shipment problem in a supply chain

In this study, we consider the coordination of transportation and production policies between a single supplier and a single retailer in a stochastic environment. The supplier controls the production, holds inventory and ships the products to the retailer to satisfy the external demand. We model the system as a Markov decision process, and show that the optimal production and transportation decisions are complex and non-monotonic. Therefore, we analyze two widely-used shipment policies in the industry as well, namely time-based and quantity-based shipment policies in addition to a hybrid time-and-quantity based shipment policy. We numerically compare the performances of these policies with respect to the optimal policy and analyze the effects of the parameters in the system.     

A computational study for common network design in multi-commodity supply chains

In this paper, we study a supply chain network design problem which consists of one external supplier, a set of potential distribution centers, and a set of retailers, each of which is faced with uncertain demands for multiple commodities. The demand of each retailer is fulfilled by a single distribution center for all commodities. The goal is to minimize the system-wide cost including location, transportation, and inventory costs. We propose a general nonlinear integer programming model for the problem and present a cutting plane approach based on polymatroid inequalities to solve the model. Randomly generated instances for two special cases of our model, i.e., the single-sourcing UPL&TAP and the single-sourcing multi-commodity location-inventory model, are provided to test our algorithm. Computational results show that the proposed algorithm can solve moderate-sized problem instances efficiently.     

A new heuristic to solve the one-warehouse N-retailer problem

We deal with a multi-echelon inventory system in which one warehouse supplies an item to multiple retailers. Customer demand arrives at each retailer at a constant rate. The retailers replenish their inventories from the warehouse that in turn orders from an outside supplier. It is assumed that shortages are not allowed and lead times are negligible. The goal is to determine replenishment policies that minimize the overall cost in the system. We develop a heuristic to compute efficient policies, which also can easily be used in a spreadsheet application. The main idea consists of finding a balance between the replenishment and the inventory holding costs at each installation. This new heuristic we compare with two other approaches proposed in the literature; the computational studies show that in most of the instances generated the new method provides lower costs.     

A convex optimization approach for solving the single-vehicle cyclic inventory routing problem

This paper investigates the mathematical structure of the Single-Vehicle Cyclic Inventory Routing Problem (SV-CIRP). The SV-CIRP is an optimization problem consisting of finding a recurring distribution plan, from a single depot to a selected subset of retailers, that maximizes the collected rewards from the visited retailers while minimizing transportation and inventory costs. It appears as fundamental building block for all variants of the cyclic inventory routing problem (CIRP). One of the main complications in developing solution methods for the SV-CIRP using the current formulations is the non-convexity of the objective function. We demonstrate how the problem can be reformulated so that its continuous relaxation is a convex optimization problem. We further examine its mathematical properties and compare our findings with statements previously done in literature. Based of these findings we propose an algorithm that solves the SV-CIRP more effectively. We present experimental results on well-known benchmark instances, for which we are able to find optimal solutions for 22 out of 50 instances and obtained new best known solutions to 23 other instances.     

Integer programming solution approach for inventory-production–distribution problems with direct shipments

We construct an integrated multi‐period inventory–production–distribution replenishment plan for three‐stage supply chains. The supply chain maintains close relationships with a small group of suppliers, and the nature of the products (bulk, chemical, etc.) makes it more economical to rely upon a direct shipment, full‐truck load distribution policy between supply chain nodes. In this paper, we formulate the problem as an integer linear program that proves challenging to solve due to the general integer variables associated with the distribution requirements. We propose new families of valid cover inequalities, and we derive a practical closed‐form expression for generating them, upon the determination of a single parameter. We study their performances through benchmarking several branch‐and‐bound and branch‐and‐cut approaches. Computational testing is performed using a large‐scale planning problem faced by a North American company.