An integrated supplier selection and inventory problem with multi-sourcing and lateral transshipments

In this research, we consider the supplier selection problem of a firm offering a single product via multiple warehouses. The warehouses face stationary, stochastic demand and replenish their inventory via multiple suppliers, to be determined from a set of candidates, with varying price, capacity, quality, and disruption characteristics. Additionally, the warehouses may simultaneously replenish their inventory from other warehouses proactively. With these characteristics, the problem is a multi-sourcing, supplier selection, and inventory problem with lateral transshipments. Even though the benefits of multi-sourcing and lateral transshipments have been presented in the literature individually to mitigate risks associated with uncertain demand and disrupted supply, the intertwined sourcing and inventory decisions under these settings have not been investigated from a quantitative perspective. We develop a decomposition based heuristic algorithm, powered with simulation. While the decomposition based heuristic determines a solution with supplier selection and inventory decisions, the simulation model evaluates the objective function value corresponding to each generated solution. Experimental results show, contrary to the existing literature, inferior decisions may result when considering the selection of suppliers solely on unit and/or contractual costs. We also evaluate the impact of multi-sourcing with rare but long disruptions compared to frequent but short ones.     

Optimal Inventory Control with Dual‐Sourcing,Heterogeneous Ordering Costs and Order Size Constraints

We consider a dual‐sourcing inventory system, where procuring from one supplier involves a high variable cost but negligible fixed cost whereas procuring from the other supplier involves a low variable cost but high fixed cost, as well as an order size constraint. We show that the problem can be reduced to an equivalent single‐sourcing problem. However, the corresponding ordering cost is neither concave nor convex. Using the notion of quasi‐convexity, we partially characterize the structure of the optimal policy and show that it can be specified by multiple thresholds which determine when to order from each supplier and how much. In contrast to previous research, which does not consider order size constraints, we show that it is optimal to simultaneously source from both suppliers when the beginning inventory level is sufficiently low. We also show that the decision to source from the low‐cost supplier is not monotonic in the inventory level. Our results require that the variable costs satisfy a certain condition which guarantees quasi‐convexity. However, extensive numerical results suggest that our policy is almost always optimal when the condition is not satisfied. We also show how the results can be extended to systems with multiple capacitated suppliers.     

Coordinated Replenishments from a Common Supplier

Coordinated replenishment strategies may be implemented by jointly ordering multiple items from a common supplier. A major benefit of coordinated replenishment is that it increases the size of shipments, permitting the buyer to enjoy transportation economies without a major increase in average inventory levels. The coordinated replenishment problem is complex because side constraints govern the attainment of transportation rate breaks. The problem is further complicated by the presence of purchase quantity discount opportunities. Thus, the buyer must decide which items to order independently, which items to include in a group order, and the order quantities of each item, governed by the frequency of independent or group orders. We present a mathematical model and a heuristic solution procedure that provide analytical support to the buyer seeking to minimize total costs of replenishing multiple items from a common supplier. The relevant costs are purchase prices, ordering costs, holding costs, and transportation costs. Coordinated replenishment provides nearly a 30 percent reduction in controllable costs relative to independent control. Experimentation with the heuristic has yielded optimal solutions over 88 percent of the time. When optimality was not obtained, the mean penalty was much less than one percent. The average heuristic search was more than two orders of magnitude faster than branch and bound, even for small problems, and possessed a much tighter distribution around the mean search time.     

A Multi‐Supplier Sourcing Problem with a Preference Ordering of Suppliers

We study a sourcing problem faced by a firm that seeks to procure a product or a component from a pool of alternative suppliers. The firm has a preference ordering of the suppliers based on factors such as their past performance, quality, service, geographical location, and financial strength, which are commonly included in a supplier scorecard system. Thus, the firm first uses available inventory from supplier 1, if any, then supplier 2, if any, and so on. The suppliers differ in costs and prices. The buyer firm seeks to determine which suppliers to purchase from and in what quantities to maximize its total expected profit subject to the preference ordering constraint. We present the optimal solution to this problem, and show that it has a portfolio structure. It consists of a sub‐set of suppliers that are ordered by their underage and overage costs. This portfolio achieves a substantial profit gain compared to sourcing from a unique supplier. We present an efficient algorithm to compute the optimal solution. Our model applies to component sourcing problems in manufacturing, merchandizing problems in retailing, and capacity reservation problems in services.     

Inventory Rationing for Multiple Class Demand under Continuous Review

We consider how a firm should ration inventory to multiple classes in a stochastic demand environment with partial, class‐dependent backlogging where the firm incurs a fixed setup cost when ordering from its supplier. We present an infinite‐horizon, average cost criterion Markov decision problem formulation for the case with zero lead times. We provide an algorithm that determines the optimal rationing policy, and show how to find the optimal base‐stock reorder policy. Numerical studies indicate that the optimal policy is similar to that given by the equivalent deterministic problem and relies on tracking both the current inventory and the rate that backorder costs are accumulating. Our study of the case of non‐zero lead time shows that a heuristic combining the optimal, zero lead time policy with an allocation policy based on a single‐period profit management problem is effective.     

Dynamic Pricing and Inventory Management with Dual Suppliers of Different Lead Times and Disruption Risks

It is common for a firm to make use of multiple suppliers of different delivery lead times, reliabilities, and costs. In this study, we are concerned with the joint pricing and inventory control problem for such a firm that has a quick‐response supplier and a regular supplier that both suffer random disruptions, and faces price‐sensitive random demands. We aim at characterizing the optimal ordering and pricing policies in each period over a planning horizon, and analyzing the impacts of supply source diversification. We show that, when both suppliers are unreliable, the optimal inventory policy in each period is a reorder point policy and the optimal price is decreasing in the starting inventory level in that period. In addition, we show that having supply source diversification or higher supplier reliability increases the firm's optimal profit and lowers the optimal selling price. We also demonstrate that, with the selling price as a decision, a supplier may receive even more orders from the firm after an additional supplier is introduced. For the special case where the quick‐response supplier is perfectly reliable, we further show that the optimal inventory policy is of a base‐stock type and the optimal pricing policy is a list‐price policy with markdowns.     

The Newsboy Problem with Multiple Discounts Offered by Suppliers and Retailers

Recent extensions of the newsboy problem have focused on multiple discounts used by retailers to sell excess inventory. Earlier extensions have focused on quantity discounts offered by suppliers. An important practical extension would address a combination of the two previous extensions. In this paper, a newsboy problem with supplier quantity discounts and retailer multiple discounts is formulated and solved. The case of all-units supplier quantity discounts is addressed, and the results with numerical examples are illustrated.     

STOCK LEVELS AND DELIVERY RATES IN VENDORMANAGED INVENTORY PROGRAMS

Using the latest information technology, powerful retailers like Wal‐Mart have taken the lead in forging shorter replenishment‐cycles, automated supply systems with suppliers. With the objective to reduce cost, these retailers are directing suppliers to take full responsibility for managing stocks and deliveries. Suppliers' performance is measured according to how often inventory is shipped to the retailer, and how often customers are unable to purchase the product because it is out of stock. This emerging trend also implies that suppliers are absorbing a large part of the inventory and delivery costs and, therefore, must plan delivery programs including delivery frequency to ensure that the inherent costs are minimized. With the idea to incorporate this shift in focus, this paper looks at the problem facing the supplier who wants quicker replenishment at lower cost. In particular, we present a model that seeks the best trade‐off among inventory investment, delivery rates, and permitting shortages to occur, given some random demand pattern for the product. The process generating demand consists of two components: one is deterministic and the other is random. The random part is assumed to follow a compound Poisson process. Furthermore, we assume that the supplier may fail to meet uniform shipping schedules, and, therefore, uncertainty is present in delivery times. The solution to this transportationinventory problem requires determining jointly delivery rates and stock levels that will minimize transportation, inventory, and shortage costs. Several numerical results are presented to give a feel of the optimal policy's general behavior.     

冷轧生产批量计划与调度问题模型及算法

对冷轧生产系统特点进行分析,将冷轧生产批量计划与调度问题抽象为多阶段、多品种带有中间库的批量计划与调度问题。针对该问题建立了数学模型,通过对库存成本和调整成本惩罚系数的控制可以协调库存水平和调整次数的关系。对所建立的模型,提出了基于二进制粒子群优化与局部搜索的混合求解算法。最后,通过对企业实际生产数据的计算和分析,验证了模型和算法的可行性和有效性。     

Inventory Commitment and Prioritized Backlogging Clearance with Alternative Delivery Lead Times

We propose a model where customers are classified into two groups: short lead‐time customers who require the product immediately and long lead‐time customers to whom the supplier may deliver either immediately or in the next cycle. Unmet orders are backlogged with associated costs. Specifically, the supplier faces two problems: how the on‐hand inventories should be allocated between the two classes of customers and how the backlogged orders should be cleared when replenishments arrive. We treat the former as an inventory commitment problem and handle the latter with priority rules. We characterize and compare the inventory commitment policies with three priority rules in clearing backlogs. We also explore the optimal inventory replenishment decision and evaluate the performance of each priority rule.     

基于Stackelberg博弈的变质物品分销网络设计模型

研究了一类短销售期的变质物品的分销网络设计问题.假定零售商的缺货成本依赖于分配给为其提供服务的分销中心的库存成本,供应商在销售期末给零售商提供第二次订货机会,供应商根据零售商的订货决策确定分销中心的最优选址和确定每个分销中心为哪些零售商提供服务,从而最小化总的运作成本(选址成本,运输成本,库存成本和变质成本),其中分销中心的运输成本和库存成本依赖于零售商确定的订货数量;而零售商则根据供应商的决策确定自身的最优订货决策,利用Stackelberg博弈分析的方法,建立了一类变质物品的分销网络设计模型,并使用拉格朗日松弛算法求解,最后通过数值算例分析了模型最优解对于参数的敏感性.     

A sustainable supply chain network considering lot sizing with quantity discounts under disruption risks: centralized and decentralized models

This study proposes a framework for the main parties of a sustainable supply chain network considering lot-sizing impact with quantity discounts under disruption risk among the first studies. The proposed problem differs from most studies considering supplier selection and order allocation in this area. First, regarding the concept of the triple bottom line, total cost, environmental emissions, and job opportunities are considered to cover the criteria of sustainability. Second, the application of this supply chain network is transformer production. Third, applying an economic order quantity model lets our model have a smart inventory plan to control the uncertainties. Most significantly, we present both centralized and decentralized optimization models to cope with the considered problem. The proposed centralized model focuses on pricing and inventory decisions of a supply chain network with a focus on supplier selection and order allocation parts. This model is formulated by a scenario-based stochastic mixed-integer non-linear programming approach. Our second model focuses on the competition of suppliers based on the price of products with regard to sustainability. In this regard, a Stackelberg game model is developed. Based on this comparison, we can see that the sum of the costs for both levels is lower than the cost without the bi-level approach. However, the computational time for the bi-level approach is more than for the centralized model. This means that the proposed optimization model can better solve our problem to achieve a better solution than the centralized optimization model. However, obtaining this better answer also requires more processing time. To address both optimization models, a hybrid bio-inspired metaheuristic as the hybrid of imperialist competitive algorithm (ICA) and particle swarm optimization (PSO) is utilized. The proposed algorithm is compared with its individuals. All employed optimizers have been tuned by the Taguchi method and validated by an exact solver in small sizes. Numerical results show that striking similarities are observed between the results of the algorithms, but the standard deviations of PSO and ICA–PSO show better behavior. Furthermore, while PSO consumes less time among the metaheuristics, the proposed hybrid metaheuristic named ICA–PSO shows more time computations in all small instances. Finally, the provided results confirm the efficiency and the performance of the proposed framework and the proposed hybrid metaheuristic algorithm.

     

Pricing decisions in a multi-criteria setting for product recovery facilities

Unpredictability in the arrival time and quantity of discarded products at product recovery facilities (PRFs) and varying demand for recovered components contribute to the volatility in their inventory levels. Achieving profit under such capricious inventory levels and stringent environmental legislations remains a challenge to many PRFs. This paper presents a multi-criteria decision model to determine a pricing policy that can simultaneously address two issues: stabilize inventory fluctuations and boost profits. The model considers that PRFs passively accepts discarded products as well as acquires them proactively if necessary. Under a multi-criteria setting, the current work determines prices of reusable and recyclable components to maximize revenue and minimize product recovery costs. A genetic algorithm is employed to solve the multi-criteria decision making problem. Sensitivity analysis is performed to investigate the effect of sorting yield, disassembly yield, and reusable component yield on the profits, prices, inventory levels, and disposal quantities.     

分布式供应链中基于准时制的原油采购计划方法

本文研究了分布式供应链多供应商/多炼油厂的原油采购计划问题,建立了在有限资源约束下的准时制多目标采购优化数学模型,以实现总采购费用和供应链循环时间最小。然后,本文将非线性规划模型转化为线性规划模型,并利用层次分析法(Analytic Hierarchy Process,AHP)与多目标规划相结合的方法求解模型得到采购计划方案。数值仿真结果表明本文所提出的原油采购方法的有效性和实用性。     

A Continuous‐Review Inventory Model with Disruptions at Both Supplier and Retailer

We consider a continuous‐review inventory problem for a retailer who faces random disruptions both internally and externally (from its supplier). We formulate the expected inventory cost at this retailer and analyze the properties of the cost function. In particular, we show that the cost function is quasi‐convex and therefore can be efficiently optimized to numerically find the optimal order size from the retailer to the supplier. Computational experiments provide additional insight into the problem. In addition, we introduce an effective approximation of the cost function. Our approximation can be solved in closed form, which is useful when the model is embedded into more complicated supply chain design or management models.     

Multi-Item Grouping Algorithm Yielding Near-Optimal Logistics Cost

This article describes an algorithm used to formulate an inbound consolidation strategy when multi-items are replenished in groups and when total logistics cost is to be minimized. The importance of this algorithm is threefold (1) no optimal procedure exists for grouping multi-items when minimizing total logistics cost, (2) a complete enumeration of all possible groups (from which the optimal grouping set can be identified) is impractical due to the combinatorial nature of the grouping problem, and (3) no other heuristics have been developed that adequately reflect shipping cost in the analysis like this one does. We report the experience of using this algorithm to group and reorder 75 selected ensembles containing a total of 517 inventory items of a retail merchandising firm.     

Managing stochastic demand in an Inventory Routing Problem with transportation procurement

We study an Inventory Routing Problem in which the supplier has a limited production capacity and the stochastic demand of the retailers is satisfied with procurement of transportation services. The aim is to minimize the total expected cost over a planning horizon, given by the sum of the inventory cost at the supplier, the inventory cost at the retailers, the penalty cost for stock-out at the retailers and the transportation cost. First, we show that a policy based just on the average demand can have a total expected cost infinitely worse than the one obtained by taking into account the overall probability distribution of the demand in the decision process. Therefore, we introduce a stochastic dynamic programming formulation of the problem that allows us to find an optimal policy in small size instances. Finally, we design and implement a matheuristic approach, integrating a rollout algorithm and an optimal solution of mixed-integer linear programming models, which is able to solve realistic size problem instances. Computational results allow us to provide managerial insights concerning the management of stochastic demand.     

Dual Sourcing Under Random Supply Capacities: The Role of the Slow Supplier

Sourcing from multiple suppliers with different characteristics is common in practice for various reasons. This paper studies a dynamic procurement planning problem in which the firm can replenish inventory from a fast and a slow supplier, both with uncertain capacities. The optimal policy is characterized by two reorder points, one for each supplier. Whenever the pre‐order inventory level is below the reorder point, a replenishment order is issued to the corresponding supplier. Interestingly, the reorder point for the slow supplier can be higher than that of the fast even if the former has a higher cost, lower reliability, and smaller capacity than the latter, suggesting the possibility of ordering exclusively from an inferior slow supplier in the short term. Moreover, the firm may allocate a larger portion of the long‐term total order quantity to the slow supplier than to the fast, even if the former does not possess any cost or reliability advantage over the latter. Such phenomena, different from the observations made in previous studies, happen when the demand is uncertain and the supply is limited or unreliable. Our observations highlight the importance of incorporating both demand uncertainty and supplier characteristics (i.e., cost, lead time, capacity and uncertainty) in a unified framework when formulating supplier selection and order allocation strategies.     

PRODUCTION SCHEDULING OF CONTINUOUS FLOW LINES: MULTIPLE PRODUCTS WITH SETUP TIMES AND COSTS

The problem of production planning and setup scheduling of multiple products on a single facility is studied in this paper. The facility can only produce one product at a time. A setup is required when the production switches from one type of product to another. Both setup times and setup costs are considered. The objective is to determine the setup schedule and production rate for each product that minimize the average total costs, which include the inventory, backlog, and setup costs. Under the assumption of a constant production rate, we obtain the optimal cyclic rotation schedule for the multiple products system. Besides the decision variables studied in the classical economic lot scheduling problem (ELSP), the production rate is also a decision variable in our model. We prove that our solutions improve the results of the classical ELSP.