Supply chain models for an assembly system with preprocessing of raw materials

In this paper, we investigate the material procurement and delivery policy in a production system where raw materials enter into the assembly line from two different flow channels. The system encompasses batch production process in which the finished product demand is approximately constant for an infinite planning horizon. Two distinct types of raw materials are passed through the assembly line before to convert them into the finished product. Of the two types of raw materials, one type requires preprocessing inside the facility before the assembly operation and other group is fed straightway in the assembly line. The conversion factors are assigned to raw materials to quantify the raw material batch size required. To analyze such a system, we formulate a nonlinear cost function to aggregate all the costs of the inventories, ordering, shipping and deliveries. An algorithm using the branch and bound concept is provided to find the best integer values of the optimal solutions. The result shows that the optimal procurement and delivery policy minimizes the expected total cost of the model. Using a test problem, the inventory requirements at each stage of production and their corresponding costs are calculated. From the analysis, it is shown that the rate and direction change of total cost is turned to positive when delivery rates per batch reaches close to the optimal value and the minimum cost is achieved at the optimal delivery rate. Also, it is shown that total incremental cost is monotonically increasing, if the finished product batch size is increased, and if, inventory cost rates are increased. We examine a set of numerical examples that reveal the insights into the procurement-delivery policy and the performance of such an assembly type inventory model.     

The scheduling of deliveries in a production-distribution system with multiple buyers

In a real production and distribution business environment with one supplier and multiple heterogeneous buyers, the differences in buyers’ ordering cycles have influence on production arrangements. Consequently, the average inventory level (AIL) at the supplier’s end is affected by both the production policy and the ordering policy, typically by the scheduling of deliveries. Consequently, the average inventory holding cost is most deeply affected. In this paper, it is proposed that the scheduling of deliveries be formulated as a decision problem to determine the time point at which deliveries are made to buyers in order to minimize the supplier’s average inventory. A formulation of the average inventory level (AIL) in a production cycle at the supplier’s end using a lot-for-lot policy is developed. Under the lot-for-lot policy, the scheduling of deliveries (SP) is formulated as a nonlinear programming model used to determine the first delivery point for each buyer with an objective to minimize the sum of the product of the individual demand quantity and the first delivery time for each buyer. Thus, the SP model determines not only the sequence of the first deliveries to individual buyers, but also the time when the deliveries are made. An iterative heuristic procedure (IHP) is developed to solve the SP model assuming a given sequence of buyers. Six sequence rules are considered and evaluated via simulation.     

Analysis of the Best Double Frequency Policy in the Single Link Problem with Discrete Shipping Times

We study an inventory–transportation problem where one product has to be shipped from an origin to a destination by vehicles of given capacity over an infinite time horizon. The product is made available at the origin and consumed at the destination at the same constant rate. The intershipment time must be not lower than a given minimum value. The problem is to decide when to make the shipments and how to load the vehicles to minimize the sum of the transportation and the inventory costs at the origin and at the destination per time unit. We study the case in which the intershipment time is a multiple of the minimum value, i.e., the problem with discrete shipping times. We show that, in this case, the best double frequency policy has a tight performance bound of about 1.1603 with respect to the optimal periodic policy and of about 1.1538 with respect to the best frequency-based policy. Moreover, we show that, from the worst-case point of view, the best double frequency policy is the optimal frequency-based policy.     

A two-level hedging point policy for controlling a manufacturing system with time-delay,demand uncertainty and extra capacity

This paper focuses on the production control of a manufacturing system with time-delay, demand uncertainty and extra capacity. Time-delay is a typical feature of networked manufacturing systems (NMS), because an NMS is composed of many manufacturing systems with transportation channels among them and the transportation of materials needs time. Besides this, for a manufacturing system in an NMS, the uncertainty of the demand from its downstream manufacturing system is considered; and it is assumed that there exist two-levels of demand rates, i.e., the normal one and the higher one, and that the time between the switching of demand rates are exponentially distributed. To avoid the backlog of demands, it is also assumed that extra production capacity can be used when the work-in-process (WIP) cannot buffer the high-level demands rate. For such a manufacturing system with time-delay, demand uncertainty and extra capacity, the mathematical model for its production control problem is established, with the objective of minimizing the mean costs for WIP inventory and occupation of extra production capacity. To solve the problem, a two-level hedging point policy is proposed. By analyzing the probability distribution of system states, optimal values of the two hedging levels are obtained. Finally, numerical experiments are done to verify the effectiveness of the control policy and the optimality of the hedging levels.     

Impacts of inventory shortage policies on transportation requirements in two-stage distribution systems

This paper investigates the impacts inventory shortage policies have on transportation costs in base-stock distribution systems under uncertain demand. The model proposed demonstrates how backlogging arrangements can serve to decrease the variability of transportation capacity requirements, and hence the magnitude of transportation costs, when compared with policies that expedite demand shortages. The model shows how inventory policy decisions directly impact expected transportation costs and provides a new method for setting stock levels that jointly minimizes inventory and transportation costs. The model and solution method provide insights into the relationship between inventory decisions and transportation costs and can serve to support delivery policy negotiations between a supplier and customer that must choose between expediting and backlogging demand shortages.     

随机需求条件下的延迟发运策略模型及性质

本针对随机需求条件下物流配送中心的库存和运输联合决策问题，在基本库存和自身运输能力不足的情况下，提出对剩余客户订货需求采取部分延迟到下一期与部分利用第三方物流立即发运两相结合的策略，并在具有一般惩罚(损失)费延迟发运量限制的条件下，建立运输和库存相关总成本数学期望最小的优化模型，论证了该模型的主要性质，在此基础上很容易构造求解该类问题的优化方法。     

Single Supplier�CBuyer Integrated Inventory Model Under Multiple JIT Delivery and Stock-Dependent Demand

In this paper, an integrated inventory model for a supply chain comprising of single buyer and single supplier is studied when demand is stock-dependent and units in inventory deteriorate at a constant rate. The total cost of the integrated system consists of the transportation cost, inspection cost and the cost of less flexibility under the assumption of JIT deliveries. The total integrated cost of single-supplier and single-buyer is minimized with respect to number of inspections and deliveries, the cycle time of deliveries and the delivery size for the replenishment time. A numerical example is given to validate the model. The sensitivity analysis carried out suggests that the unit inspection cost, deterioration rate of units in inventory and stock-dependent parameter are the critical factors.     

A serial inventory system with supplier selection and order quantity allocation

Given the prevalence of both supplier selection and inventory control problems in supply chain management, this article addresses these problems simultaneously by developing a mathematical model for a serial system. This model determines an optimal inventory policy that coordinates the transfer of items between consecutive stages of the system while properly allocating orders to selected suppliers in stage 1. In addition, a lower bound on the minimum total cost per time unit is obtained and a 98% effective power-of-two (POT) inventory policy is derived for the system under consideration. This POT algorithm is advantageous since it is simple to compute and yields near optimal solutions.     

Multi-location transshipment problem with capacitated transportation

We consider coordination among stocking locations through replenishment strategies that take explicitly into consideration transshipments, transfer of a product among locations at the same echelon level. We incorporate transportation capacity such that transshipment quantities between stocking locations are bounded due to transportation media or the location’s transshipment policy. We model different cases of transshipment capacity as a capacitated network flow problem embedded in a stochastic optimization problem. Under the assumption of instantaneous transshipments, we develop a solution procedure based on infinitesimal perturbation analysis to solve the stochastic optimization problem, where the objective is to find the policy that minimizes the expected total cost of inventory, shortage, and transshipments. Such a numerical approach provides the flexibility to solve complex problems. Investigating two problem settings, we show the impact of transshipment capacity between stocking locations on system behavior. We observe that transportation capacity constraints not only increase total cost, they also modify the inventory distribution throughout the network.     

Capacitated location model with online demand pooling in a multi-channel supply chain

This paper presents a location model that assigns online demands to the capacitated regional warehouses currently serving in-store demands in a multi-channel supply chain. The model explicitly considers the trade-off between the risk pooling effect and the transportation cost in a two-echelon inventory/logistics system. Keeping the delivery network of the in-store demands unchanged, the model aims to minimize the transportation cost, inventory cost, and fixed handling cost in the system when assigning the online demands. We formulate the assignment problem as a non-linear integer programming model. Lagrangian relaxation based procedures are proposed to solve the model, both the general case and an important special case. Numerical experiments show the efficiency of our algorithms. Furthermore, we find that because of the pooling effect the variance of in-store demands currently served by a warehouse is an important parameter of the warehouse when it is considered as a candidate for supplying online demands. Highly uncertain in-store demands, as well as low transportation cost per unit, can make a warehouse appealing. We illustrate with numerical examples the trade-off between the pooling effect and the transportation cost in the assignment problem. We also evaluate the cost savings between the policy derived from the model, which integrates the transportation cost with the pooling effect, and the commonly used policy, which is based only on the transportation cost. Results show that the derived policy can reduce 1.5–7.5% cost in average and in many instances the percentage of cost savings is more than 10%.     

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.     

Single-vendor multi-buyer inventory coordination under private information

This paper considers the problem of coordinating a single-vendor multi-buyer inventory system when there are privacy restrictions in the information required to solve the problem. The objective function and cost parameters of each facility are regarded as private information that no other facilities in the system have access to. Moreover, each facility is responsible to specify its own replenishment policy. The objective is to minimize the total average setup/ordering and inventory-related cost. Solution methodologies under private and global information are developed to find two types of nested power-of-two stationary policies. The first policy assumes all the buyers must replenish simultaneously. The second policy is a more general case where the common replenishment assumption is relaxed. A simple form of information exchange is uncovered that allows the solution methodologies for private and global information yield the same results. The experimental results suggest that the performance of the proposed heuristics is comparable or better than an existing method.     

An Algorithm for Limited Capacity Inventory Problem with Staggering

This paper describes an algorithm for solving the integer programming problem, which arises in the staggering approach to limited capacity inventory systems. Numerical examples are given.     

A Column Generation Approach to Delivery Planning over Time with Inhomogeneous Service Providers and Service Interval Constraints

We consider a problem of delivery planning over multiple time periods. Deliveries must be made to customers having nominated demand in each time period. Demand must be met in each time period by use of some combination of inhomogeneous service providers. Each service provider has a different delivery capacity, different cost of delivery to each customer, a different utilisation requirement, and different rules governing the spread of deliveries in time. The problem is to plan deliveries so as to minimise overall costs, subject to demand being met and service rules obeyed. A natural integer programming model was found to be intractable, except on problems with loose demand constraints, with gaps between best lower bound and best feasible solution of up to 35.1%, with an average of 15.4% over the test data set. In all but the problem with loosest demand constraints, Cplex 6.5 applied to this formulation failed to find the optimal solution before running out of memory. However a column generation approach improved the lower bound by between 0.6% and 21.9%, with an average of 9.9%, and in all cases found the optimal solution at the root node, without requiring branching.     

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.     

An integrated inventory–transportation system with modified periodic policy for multiple products

Efficient management of a distribution system requires an integrated approach towards various logistical functions. In particular, the fundamental areas of inventory control and transportation planning need to be closely coordinated. Our model deals with an inbound material-collection problem. An integrated inventory–transportation system is developed with a modified periodic-review inventory policy and a travelling-salesman component. This is a multi-item joint replenishment problem, in a stochastic setting, with simultaneous decisions made on inventory and transportation policies. We propose a heuristic decomposition method to solve the problem, minimizing the long-run total average costs (major- and minor-ordering, holding, backlogging, stopover and travel). The decomposition algorithm works by using separate calculations for inventory and routing decisions, and then coordinating them appropriately. A lower bound is constructed and computational experience is reported.     

Characterization of facility assignment costs for a location-inventory model under truckload distribution

We consider a two-stage distribution system, where the first stage consists of potential distribution centres (DCs) and the second stage consists of geographically dispersed existing retailers. Our goal is to determine the set of open DCs and assignment of open DCs to retailers simultaneously with inventory decisions of retailers. In addition to the DC-specific fixed facility location costs, we explicitly model the inventory replenishment and holding costs at the retailers and truckload transportation costs between the DCs and the retailers. The transportation costs are subject to truck/cargo capacity, leading to an integrated location-inventory problem with explicit cargo costs. We develop a mixed-integer nonlinear model and analyse its structural properties leading to exact expressions for the so-called implied facility assignment costs and imputed per-unit per-mile transportation costs. These expressions analytically demonstrate the interplay between strategic location and tactical inventory/transportation decisions in terms of resulting operational costs. Although both the theory and practice of integrated logistics have recognized the fact that strategic and tactical decisions are interrelated, to the best of our knowledge, our paper is the first to offer closed-form results demonstrating the relationship explicitly. We propose an efficient solution approach utilizing the implied facility assignment costs and we demonstrate that significant savings are realizable when the inventory decisions and cargo costs are modelled explicitly for facility location purposes.     

Optimal EOQ Models for Deteriorating Items with Time-Varying Demand

In this paper, optimal inventory lot-sizing models are developed for deteriorating items with general continuous time-varying demand over a finite planning horizon and under three replenishment policies. The deterioration rate is assumed to be a constant fraction of the on-hand inventory. Shortages are permitted and are completely backordered. The proposed solution procedures are shown to generate global minimum replenishment schedules for both general increasing and decreasing demand patterns. An extensive empirical comparison using randomly generated linear and exponential demands revealed that the replenishment policy which starts with shortages in every cycle is the least cost policy and the replenishment policy which prohibits shortages in the last cycle exhibited the best service level effectiveness. An optimal procedure for the same problem with trended inventory subject to a single constraint on the minimum service level (maximum fraction of time the inventory system is out of stock during the planning horizon) is also proposed in this paper.     

A partition approach to the inventory/routing problem

In this study we focus on the integration of inventory control and vehicle routing schedules for a distribution system in which the warehouse is responsible for the replenishment of a single item to the retailers with demands occurring at a specific constant (but retailer-dependent) rate, combining deliveries into efficient routes. This research proposes a fixed partition policy for this type of problem, in which the replenishment interval of each of the retailers’ partition region as well as the warehouse is accorded the power of two (POT) principle. A lower bound of the long-run average cost of any feasible strategy for the considered distribution system is drawn. And a tabu search algorithm is designed to find the retailers’ optimal partition regions under the fixed partition policy proposed. Computational results reveal the effectiveness of the policy as well as of the algorithm.     

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.