A mathematical model for order splitting in a multiple supplier single-item inventory system

In this paper we develop a mathematical model which considers multiple-supplier single-item inventory systems. The lead times of the suppliers and demand arrival rate are random variables. All shortages are backordered. Continuous review (s, Q) policy has been assumed. When the inventory level hits the reorder level, the total order is split among n suppliers. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost and shortage cost is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared to the relevant models in the literature. In addition, some managerial insights are observed.     

A new order splitting model with stochastic lead times for deterioration items

In unreliable supply environments, the strategy of pooling lead time risks by splitting replenishment orders among multiple suppliers simultaneously is an attractive sourcing policy that has captured the attention of academic researchers and corporate managers alike. While various assumptions are considered in the models developed, researchers tend to overlook an important inventory category in order splitting models: deteriorating items. In this paper, we study an order splitting policy for a retailer that sells a deteriorating product. The inventory system is modelled as a continuous review system (s, Q) under stochastic lead time. Demand rate per unit time is assumed to be constant over an infinite planning horizon and shortages are backordered completely. We develop two inventory models. In the first model, it is assumed that all the requirements are supplied by only one source, whereas in the second, two suppliers are available. We use sensitivity analysis to determine the situations in which each sourcing policy is the most economic. We then study a real case from the European pharmaceutical industry to demonstrate the applicability and effectiveness of the proposed models. Finally, more promising directions are suggested for future research.     

Simultaneous coordination of order quantity and reorder point in a two-stage supply chain

In this paper, we investigate the simultaneous coordination of order quantity and reorder point in a two-stage supply chain (SC). While coordination of order quantity has received much attention in the supply chain management literature, coordination of the reorder point has been less-studied. The retailer's reorder point has a direct impact on product availability and customer service level (CSL) and therefore has a great impact on SC profitability. Our proposed model adopts a two-stage SC with stochastic demand and lead times over multiple periods. The proposed coordination model assures global optimization of order quantity–reorder point decisions. Using a pricing scheme with a discount factor, we extract conditions in which both downstream and upstream members have sufficient motivation to participate in the coordination scheme. Numerical experiments demonstrate that the proposed model can achieve channel coordination. Results of the modeling and analyses show that coordination of both reorder point and order quantity can lead to increased SC profitability as well as CSL improvement.     

Headquarter-centered common sourcing management through order coordination and consolidation

It is common for subsidiaries of a group company to use the same types of components for producing similar products. Different subsidiary companies may well procure such components from the same suppliers. This paper studies two sourcing management models. One is the Subsidiary-Autonomous Sourcing Management (SD-ASM) where subsidiaries manage their inventories and place purchasing orders independent of each other. The other is the Headquarter-centered Common Sourcing Management (HQ-CSM) where purchasing orders of subsidiaries are processed centrally through some kind of headquarter coordination. In the SD-ASM model, each subsidiary places replenishment orders at a time interval corresponding to their economic order quantity (EOQ). In the HQ-CSM model, two purchasing order management policies are examined. One is the Order Coordination policy in which common replenishment epochs or time periods are proposed by the headquarter and the subsidiaries are encouraged to coordinate the timing of their orders based on the common replenishment epochs. The other is the Order Consolidation policy in which the subsidiaries combine the quantity of their orders and the headquarter places a combined order with the supplier. In the Order Coordination policy, classic RAND heuristic is used to find the best common replenishment epoch and the best replenishment timing of each subsidiary. In the Order Consolidation policy, the optimal order quantity of the combined order is obtained from a mathematical model. The combined order is then allocated to the subsidiaries according to a proportional allocation rule. A series of numerical studies is conducted to compare the costs of the SD-ASM and HQ-CSM policies. The results show that HQ-CSM outperforms SD-ASM in terms of cost and robustness against demand uncertainties. This achievement is largely due to the economies of process (synergistic ordering process), the economies of scale (large order quantity with price discount) and risk pooling effect (transshipments). The results also reveal that the Order Consolidation policy with a combined order always performs better than the Order Coordination policy with common replenishment epochs especially in face of high demand uncertainties and high service level in the global market.     

A fuzzy solution approach for multi objective supplier selection

In today’s severe competitive environment the selection of appropriate suppliers is a significantly important decision for effective supply chain management. Appropriate suppliers reduce purchasing costs, decrease production lead time, increase customer satisfaction and strengthen corporate competitiveness. In this study a multiple sourcing supplier selection problem is considered as a multi objective linear programming problem. Three objective functions are minimization of costs, maximization of quality and maximization of on-time delivery respectively. In order to solve the problem, a fuzzy mathematical model and a novel solution approach are proposed to satisfy the decision maker’s aspirations for fuzzy goals. The proposed approach can be efficiently used to obtain non-dominated solutions. A numerical example is given to illustrate how the approach is utilized.     

Some remarks on an optimal order quantity and reorder point when supply and demand are uncertain

In the reports in the literature on inventory control, the effects of the random capacity on an order quantity and reorder point inventory control model have been integrated with lead time demand following general distribution. An iterative solution procedure has been proposed for obtaining the optimal solution. However, the resulting solution may not exist or it may not guarantee to give a minimum to the objective cost function, the expected cost per unit time. The aim of this study was to introduce a complete solution of the order quantity/reorder point problem, optimality, properties and bounds on the optimal order quantity and reorder point. The two most appealing distributions of lead time demand, normal and uniform distributions, in conjunction with an exponentially distributed capacity, are used to illustrate our findings in determining the optimal order quantity and reorder point.     

Optimal order limit policy under cost effectiveness criterion

The reorder point/reorder quantity policies referred as (r, Q) policies are widely used in industry and extensively studied in the literature. It should be noted, however, that the shortage often occurs in practical situations even if the optimal reorder point and reorder quantity maximizing or minimizing any optimality criteria are rationally determined. In this paper, we present a mathematical model to adjust the shortage period by the emergency order. The problem is to choice the timing for the expedited order. Applying the concept on repair limit policy in the context of reliability theory, the optimal order limit policy maximizing an optimality criterion called cost effectiveness is analytically derived in the framework of (r, Q) inventory system. Finally, some examples for stochastic lead times are given to explain the numerical procedure to obtain the optimal policy.     

Newsvendor problems with demand forecast updating and supply constraints

This study investigates an extension of the newsvendor model with demand forecast updating under supply constraints. A retailer can postpone order placement to obtain a better demand forecast with a shorter supply lead time. However, the manufacturer would charge the retailer a higher cost for a shorter lead time and set restrictions on the ordering times and quantities. This prevents retailers from taking full advantage of demand forecast updating to improve profits. In studying the manufacturer-related effects, two supply modes are investigated: supply mode A, which has a limited ordering time scale, and supply mode B, which has a decreasing maximum ordering quantity. For supply mode A, it is proven under justifiable assumptions that a retailer should order either as early or as late as possible. For supply mode B, an algorithm is proposed to simplify the ordering policy by appropriately relaxing the ordering quantity restrictions. Numerical analysis is conducted to investigate the influence of product and demand parameters on the value of demand forecast updating in the two supply modes. A comparison of the different supply scenarios demonstrates the negative effects of increased purchasing cost and ordering time and quantity restrictions when demand forecast updating is implemented.     

Joint determination of order quantity and reorder point of continuous review model under quantity and freight rate discounts

The increased emphasis on transportation costs has enhanced the need to develop models with transportation consideration explicitly. However, in stochastic inventory models, the transportation cost is considered implicitly as part of fixed ordering cost and thus is assumed to be independent of the size of the shipment. As such, the effect of the transportation and purchasing costs are not adequately reflected in final planning decisions. In this paper, transportation and purchasing considerations are integrated with continuous review inventory model. The objective is to view the system as an integrated whole and determine the lot size and reorder point which minimize the expected total cost per unit time. In addition, procedures are developed to solve the proposed models. Numerical experiments are also performed to explore the effect of key parameters on lot size, reorder point and expected total cost. The new models have a significant impact on lot size, reorder point and expected total cost. Savings up to 17.15% of the expected total cost are realized when using the proposed models.     

Analyzing an inventory system with multiple reorder points and periodic replenishment

This paper investigates multiple reorder point, periodic replenishment systems similar to those utilized on board some U.S. naval vessels for Selected Item Management (SIM). This inventory system is of interest since it involves both regular and priority replenishment, three reorder points, and cyclical restocking of regular orders. A simulation model is developed to study the impact of demand distribution, cycle time, priority order leadtime, and the quantity of units ordered on the performance of the system. It is found that the frequency of service and days without shortages performance measures can be maximized by setting the reorder points at equal intervals between the zero level and the high level. Also, a power approximation model is presented that estimates the high level that will lead to a desired percent days without shortage percentage.     

Sourcing with random yields and stochastic demand: A newsvendor approach

We studied a supplier selection problem, where a buyer, while facing random demand, is to decide ordering quantities from a set of suppliers with different yields and prices. We provided the mathematical formulation for the buyer's profit maximization problem and proposed a solution method based on a combination of the active set method and the Newton search procedure. Our computational study shows that the proposed method can solve the problem efficiently, and is able to generate interesting and insightful results that lead us to various managerial implications.

Scope and purpose

In today's globally competitive environment, decision makers in supply chains face numerous challenges particularly regarding the selection of suppliers or outsourcing partners. To assist in this endeavor, we examined a double-layered supply chain where a buyer facing the end users has the option of selecting among a cohort of suppliers. The available suppliers may have different yield rates and unit costs. The buyer has to decide, given the stochastic nature of the problem's governing parameters, whether or not to order from each supplier, and if so how much. We developed a ‘newsvendor-style’ model for the problem, and proposed a solution algorithm for it. Numerical studies were performed to provide some insights for supplier selection and order quantity decisions.     

Optimization of cost and service level in the presence of supply chain disruption risks: Single vs. multiple sourcing

The coordinated supplier selection and customer order scheduling in the presence of supply chain disruption risks is studied for single and multiple sourcing strategies. Given a set of customer orders for products, the decision maker needs to select a single supplier or a subset of suppliers for purchasing parts required to complete the customer orders, and schedule the orders over the planning horizon, to mitigate the impact of disruption risks. The suppliers are located in different geographic regions and the supplies are subject to different types of disruptions: to random local disruptions of each supplier individually, to random regional disruptions of all suppliers in the same region simultaneously and to random global disruptions of all suppliers simultaneously. For any combination of suppliers hit by different types of disruptions, a formula for calculating the corresponding disruption probability is developed. The obtained combinatorial stochastic optimization problem is formulated as a mixed integer program with conditional value-at-risk as a risk measure. The problem objective is either to minimize expected worst-case cost or to maximize expected worst-case customer service level, i.e., the expected worst-case fraction of customer orders filled on or before their due dates. The risk-averse solutions that optimize worst-case performance of a supply chain under disruptions risks are compared for the two sourcing strategies and the two objective functions. Numerical examples and computational results are presented and some managerial insights on the choice between the two sourcing strategies are reported.     

(Q,r,L) Inventory model with defective items

This paper assumes that an arrival order lot may contain some defective items, and the number of defective items is a random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity, the reorder point and the lead time are decision variables. In our studies, we first assume that the lead time demand follows a normal distribution, and then relax the assumption about the form of the distribution function of the lead time demand and apply the minimax distribution-free procedure to solve the problem. We develop an algorithm procedure to obtain the optimal ordering strategy. Furthermore, the effects of parameters are also included.     

期货市场存在下原材料混合采购策略研究

在原材料价格与需求不确定下,研究制造商结合运营方式与期货合约的混合采购策略。利用期望效用理论建立了一个风险规避制造商的决策模型。研究结果表明,制造商的最优合约采购量随着原材料需求的增大而增加,但与其波动无关。最优采购量与风险规避度或现货价格波动性之间的关系取决于现货期望价格与期货价格（或预定价格）两者大小关系。采用混合策略后制造商的期望效用水平得到提高。     

结合期权合约与现货市场的原材料采购风险管理

通过建立一个两时点的采购模型,研究在原材料价格波动下一个风险规避制造商的最优组合采购策略.制造商可利用期权合约购买原材料,或者从现货市场采购.研究结果表明,最优期权购买数量受到风险规避程度、期权费以及执行价格的影响.最优购买数量与决策者风险规避程度之间的关系受到期权定价影响;最优购买数量随着期权购买费或执行价格的增加而减少.最后,通过比较说明,采用组合采购策略可以提高制造商的均值-方差效用水平.     

Evaluation of sourcing contracts in wood supply procurement using simulation

Procurement operations in forest companies are exposed to various risks, which may increase procurement costs. Examples of risks are the contract's unreliability and contract breach. Deterministic planning models cannot perfectly reflect the complexities of real-world applications in the presence of sourcing risks when the future is uncertain. In practice, forest industries use contracts to guarantee the wood supply. Monthly forecasts are prepared for the delivery volume and are primarily based on the experience of procurement staff and the total volume of contracts. Missed deliveries in the contracts lead to a mismatch between the supply and demand for wood fiber and increase the costs of procurement as a result of high inventory costs or expensive purchases from the open market. Previous studies on simulating the impact of sourcing risks on procurement operations have been conducted; however, none have addressed the selection of procurement contracts in the presence of sourcing risks. In forest industry, numerous suppliers are available with sourcing contracts. Each contract possesses its own characteristics such as flexibility, volume, schedule, and price of delivery. A Monte Carlo simulation approach is implemented to analyze the behavior of a deterministic planning approach. Random events are generated by formulating different types of sourcing risks, having either short- or long-term impact. The simulation is embedded with a deterministic planning model in each period. Results showed that management of sourcing risks is easier with flexible contracts than with fixed contracts, despite their higher purchasing cost.     

Coordinating supplier׳s reorder point: A coordination mechanism for supply chains with long supplier lead time

In this paper, the issue of the upstream stochastic lead time in supply chain (SC) is investigated. A coordination mechanism is developed for reducing the harmful effect of upstream lead time. The supplier stochastic lead time can substantially harm the whole supply chain service level, especially when it is accumulated with downstream stochastic lead times. In this study, aggregation of both the supplier and the retailer stochastic lead time is analyzed in a two-stage supply chain (SC). To dampen harmful effects of a long aggregate lead time, a ‘per order extra payment’ model is developed for convincing the supplier to increase its reorder point. Numerical experiments show that coordinating the supplier׳s reorder point creates a significant profit for the whole supply chain. In addition, the proposed model is capable of optimizing the supplier׳s reorder point and fairly sharing the extra benefits. Some conditions are also extracted, under which the proposed model shows good performance.     

Supply chain coordination based on return contracts with a threshold ordering quantity

Return contracts are commonly used by companies selling products with short life cycles and highly uncertain demand. Current research on return contracts assumes suppliers are responsible for all surplus products. In practice, retailers tend to order more than necessary and leave suppliers with large after‐season returns. To mitigate the problem, a new type of return contract with a threshold ordering quantity has been developed by some enterprises. Under these contracts, suppliers specify a threshold for retailers’ ordering quantity. They buy back only the portion in excess of the threshold. In this paper, we show that this new type of contract can achieve two objectives: (a) the supply chain is coordinated, and (b) both the supplier and the retailer can gain more profit than they can gain under a wholesale‐price‐only contract. The new contract does not require any manipulation of wholesale prices. This makes it more acceptable in practice by supply chain members. We also illustrate our findings in a numerical example.     

An optimal batch size for a production system operating under periodic delivery policy

A manufacturing system that procures raw materials from suppliers in a lot and processes them into a finished product is considered in this research. An ordering policy is proposed for raw materials to meet the requirements of a production facility which, in turn, must deliver finished products demanded by outside buyers at fixed points in time. First, a general cost model is formulated considering both raw materials and finished products. Then, using this model, a simple procedure is developed to determine an optimal ordering policy for procurement of raw materials, and the manufacturing batch size, to minimize the total cost of meeting customer demands in time. The dependent relationships between production batch size and rawmaterial purchasing quantity, and various delivery patterns considered in recent literature are critically reviewed. The quality of the solution is evaluated. Numerical examples are provided.     

A mixed inventory model with variable lead time and random back-order rate

In this paper, we study the determination of the optimal lead time, reorder point and order quantity considering that the back-order probability of a demand made during a stock-out period depends on the interval from the moment in which the order is placed until the next replenishment. Two models are analysed for the specification of the back-order probability: exponential functions and piecewise constant functions. The distribution of the lead time demand is assumed to be Poisson. An algorithm for the determination of the optimal order quantity, reorder point and lead time is given. A numerical example is presented to illustrate the results.