Optimal Mixed Batch Shipment Policy with Variable Safety Factor for the Single-Vendor Single-Buyer Production-Inventory System

This study investigates the production and inventory problem involving stochastic demand in a single-vendor single-buyer integrated system. The stochastic model is constructed and is controlled by both the reorder and shipping points with mixed geometric and equal batch shipment policy and variable safety factor. The minimum cost model is transformed into the maximum order quantity model. Consequently, the structure of mixed geometric and equal shipments can easily be obtained. The structure of the mixed geometric and equal shipments comprises number of batches, number of geometric shipments, and ratio of equal sized shipments to the size of the first shipment. The problem is solved using the proposed algorithm that determines the economic lot size, the structure of the mixed geometric and equal shipments, and the safety factor. An illustrative example demonstrates the effectiveness of the algorithmic procedures.     

Routing,ship size,and sailing frequency decision-making for a maritime hub-and-spoke container network

This study formulates a two-objective model to determine the optimal liner routing, ship size, and sailing frequency for container carriers by minimizing shipping costs and inventory costs. First, shipping and inventory cost functions are formulated using an analytical method. Then, based on a trade-off between shipping costs and inventory costs, Pareto optimal solutions of the two-objective model are determined. Not only can the optimal ship size and sailing frequency be determined for any route, but also the routing decision on whether to route containers through a hub or directly to their destination can be made in objective value space. Finally, the theoretical findings are applied to a case study, with highly reasonable results. The results show that the optimal routing, ship size, and sailing frequency with respect to each level of inventory costs and shipping costs can be determined using the proposed model. The optimal routing decision tends to be shipping the cargo through a hub as the hub charge is decreased or its efficiency improved. In addition, the proposed model not only provides a tool to analyze the trade-off between shipping costs and inventory costs, but it also provides flexibility on the decision-making for container carriers.     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

An improved solution to the replenishment policy for the EMQ model with rework and multiple shipments

Recently, Chiu et al. (2012) [1] present an alternative optimization procedure to derive the optimal replenishment lot size for an economic manufacturing quantity (EMQ) model with rework and multiple shipments. This inventory model was proposed by Chiu et al. (2011) [2]. Both papers do not consider the determining of the number of shipments. This paper determines both the optimal replenishment lot size and the optimal number of shipments jointly. The solution of this paper is better than the solutions of Chiu et al.  and .     

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.     

Optimal transportation policies for production/inventory systems with an unreliable and a reliable carrier

In this paper, we consider a periodic-review make-to-order production/inventory system with two outbound transportation carriers: One carrier is reliable, the other carrier is less reliable but more economical. The objective is to find the optimal shipping policy that minimizes the total discounted transportation, inventory, and customer waiting costs. Under several scenarios, we characterize the optimal policy and present the structural properties for the optimal control parameters and the key performance measures. Our results provide managerial insights on how a manufacturer can effectively manage its transportation carriers and product shipment. We also discuss several possible extensions of the model.     

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.     

A comprehensive extension of an integrated inventory model with ordering cost reduction and permissible delay in payments

Huang (2010) [1] proposed an integrated inventory model with trade credit financing in which the vendor decides its production lot size while the buyer determines its expenditure to minimize the annual integrated total cost for both the vendor and the buyer. In this paper, we extend his integrated supply chain model to reflect the following four facts: (1) generated sales revenue is deposited in an interest-bearing account for the buyer, (2) the buyer’s interest earned is not always less than or equal to its interest charged, (3) the total number of shipments in one lot size is the vendor’s decision variable to minimize the cost, and (4) it is vital to have a discrimination term which can determine whether the buyer’s replenishment cycle time is less than the permissible delay period or not. We then derive the necessary and sufficient conditions to obtain the optimal solution, and establish some theoretical results to characterize the optimal solution. Finally, numerical examples are presented to illustrate the proposed model and its optimal solution.     

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.     

Economic selection of process mean for single-vendor single-buyer supply chain

Process mean selection for a container-filling process is an important decision in a single-vendor single-buyer supply chain. Since the process mean determines the vendor’s conforming and yield rates, it influences the vendor–buyer decisions regarding the production lot size and number of shipments delivered from the vendor to buyer. It follows, therefore, that these decisions should be determined simultaneously in order to control the supply chain total cost. In this paper, we develop a model that integrates the single-vendor single-buyer problem with the process mean selection problem. This integrated model allows the vendor to deliver the produced lot to buyer in number of unequal-sized shipments. Moreover, every outgoing item is inspected, and each item failing to meet a lower specification limit is reprocessed. Further, in order to study the benefits of using this integrated model, two baseline cases are developed. The first of which considers a hierarchical model where the vendor determines the process mean and schedules of production and shipment separately. This hierarchical model is used to show the impact of integrating the process mean selection with production/inventory decisions. The other baseline case is studied in the sensitivity analysis where the optimal solution for a given process is compared to the optimal solution when the variation in the process output is negligible. The integrated model is expected to lead to reduction in reprocessing cost, minimal loss to customer due to the deviation from the optimum target value, and consequently, providing better products at reduced cost for customers. Also, a solution procedure is devised to find the optimal solution for the proposed model and sensitivity analysis is conducted to investigate the effect of the model key parameters on the optimal solution.     

Improved rounding procedures for the discrete version of the capacitated EOQ problem

We consider a transportation problem where different products have to be shipped from an origin to a destination by means of vehicles with given capacity. The production rate at the origin and the demand rate at the destination are constant over time and identical for each product. The problem consists in deciding when to make the shipments and how to fill the vehicles, with the objective of minimizing the sum of the average transportation and inventory costs at the origin and at the destination over an infinite horizon. This problem is the well known capacitated EOQ (economic order quantity) problem and has an optimal solution in closed form. In this paper we study a discrete version of this problem in which shipments are performed only at multiples of a given minimum time. It is known that rounding-off the optimal solution of the capacitated EOQ problem to the closest lower or upper integer value gives a tight worst-case ratio of 2, while the best among the possible single frequency policies has a performance ratio of 5/3. We show that the 5/3 bound can be obtained by a single frequency policy based on a rounding procedure which considers classes of instances and, for each class, identifies a shipping frequency by rounding-off in a different way the optimal solution of the capacitated EOQ problem. Moreover, we show that the bound can be reduced to 3/2 by using two shipping frequencies, obtained by a rounding procedure, in one class of instances only.     

Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity – the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.     

An integrated inventory model with quality improvement and two-part credit policy

Trade credit financing plays a vital role in current business operations. Vendors extend payment dates to encourage sales, and buyers are not required to pay immediately after receiving products. This is equivalent to receiving a price reduction. Besides, buyers reduce their need for capital from bank loans. In addition, a number of defective products are produced during the production process. The number of defective items influences the on-hand inventory levels of buyers, service levels, and frequency of orders. To ensure that the analysis incorporates a realistic production environment, we developed an integrated inventory model with a two-part trade credit and considered an imperfect production process that can be improved by capital investment. The objective was to determine the optimal ordering, shipping, and quality improvement policies to maximize joint total profit. An iterative algorithm was established to determine the optimal strategy. Furthermore, a sensitivity analysis was conducted to examine the effects of changing parameter values on the optimal solution.     

单一生产商单一零售商供应链中最优生产与运输协调策略研究

在供应链背景下,易腐品的生产运输协调决策具有非常重要的现实意义,也是近几年研究的热点问题。本文在前人研究的基础上,通过引入安全库存并将其作为决策变量,拓展了易腐品生产运输协调决策模型;并证明了最佳运输批量序列一定是单调不减的及第一次补货时一定要清空库存;进而证明了最佳运输批量序列的具体形式并提出了一种运输策略。本文的目标是在新的运输策略下最大化单一生产商单一零售商供应链系统的利润,找到最佳运输批量、最佳生产时间以及运输次数。最后,本文进行了算例分析,用来验证本文提出的模型。     

Operations planning for a multi-stage kanban system

A multi-stage production line which operates under a just-in-time production philosophy with linear demand is considered here. The first workstation processes the raw materials after receiving them from suppliers, a kanban mechanism between the workstations transports the work-in-process to the succeeding workstation, and after processing them, delivers the finished products to a buyer or a warehouse. The problem is to find optimally the number of raw material orders, kanbans circulated between workstations, finished goods shipments to the buyers, and the batch size for each shipment (lot). A cost function is developed based on the costs incurred due to the raw materials, the work-in-process between workstations, and the finished goods. Optimal number of raw material orders that minimizes the total cost is obtained, which is then used to find the optimal number of kanbans, finished goods shipments, and the batch sizes for shipments. Numerical examples are used to demonstrate the computations of optimal parameters, and to configure the kanban mechanism on a timescale. Several avenues for future research are also indicated.     

A (Q, R) inventory replenishment model with two delivery modes

Although splitting shipments across multiple delivery modes typically increases total shipping costs as a result of diseconomies of scale, it may offer certain benefits that can more than offset these costs. These benefits include a reduction in the probability of stockout and in the average inventory costs. We consider a single-stage inventory replenishment model that includes two delivery modes: a cheaper, less reliable mode, and another, more expensive but perfectly reliable mode. The high-reliability mode is only utilized in replenishment intervals in which the lead time of the less-reliable mode exceeds a certain value. This permits substituting the high-reliability mode for safety stock, to some degree. We characterize optimal replenishment decisions with these two modes, as well as the potential benefits of simultaneously using two delivery modes.     

需求依赖于展示区库存水平的易腐品订购和转运策略

本文主要研究易腐品零售商的订货和转运策略。零售商的库存分为两部分,即展示区/货架库存和仓库库存。零售商定期向供应商订货,零售商收到订购的商品首先将其中一部分商品存放在展示区中,余下的部分储存在仓库。展示区的空间是有限的,并且需求依赖于展示区商品的库存量。本文首先建立了以平均利润最大化为目标的库存优化模型并对模型最优解的存在性进行了分析,然后得到了求解最优订购量、转运量、转运时间间隔以及再订购点的算法,最后给出了不同参数条件下的算例。     

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.     

Inventory allocation and shipping when demand temporarily exceeds production capacity

We address the concept of an integrated inventory allocation and shipping model for a manufacturer with limited production capacity and multiple types of retailers with different backorder/waiting and delivery costs. The problem is to decide how to allocate and deliver produced items when the total retailer demand exceeds the production capacity, so that total retailer backorder and delivery costs are minimized. Our analytical model provides optimal allocation and shipping policies from the manufacturer’s viewpoint. We also investigate the allocation strategy of a manufacturer competing with other retailers to directly sell to end consumers.