Performance Evaluation of Stochastic Multi-Echelon Inventory Systems: A Survey

Globalization, product proliferation, and fast product innovation have significantly increased the complexities of supply chains in many industries. One of the most important advancements of supply chain management in recent years is the development of models and methodologies for controlling inventory in general supply networks under uncertainty and their widefspread applications to industry. These developments are based on three generic methods: the queueing-inventory method, the lead-time demand method and the flow-unit method. In this paper, we compare and contrast these methods by discussing their strengths and weaknesses, their differences and connections, and showing how to apply them systematically to characterize and evaluate various supply networks with different supply processes, inventory policies, and demand processes. Our objective is to forge links among research strands on different methods and various network topologies so as to develop unified methodologies.Masdar Institute of Science and TechnologyNational Science Foundation (U.S.) (NSF Contract CMMI-0758069)National Science Foundation (U.S.) (Career Award CMMI-0747779)Bayer Business ServicesSAP A

Multi-echelon Inventory Control with Integrated Shipment Decisions

Rising fuel prices and increasing environmental awareness emphasizes the importance of the transportation aspect in logistics. This calls for new improved inventory control methods that consider the effects of shipment strategies in a more realistic manner. This thesis, consisting of an introduction and three scientific papers, studies how shipment decisions can be included in the inventory control of distribution systems. The systems studied in the papers consist of a central warehouse that supplies goods to a number of retailers that face stochastic customer demand. The first two papers consider a system where shipments from the central warehouse are consolidated to groups of retailers periodically. This means that replenishment orders of one or several items from different retailers are consolidated and dispatched at certain time intervals. By doing so, transportation cost savings can be realized and emissions can be reduced. This is achieved by filling the vehicles or load carriers to a higher extent and by using cheaper and more environmentally friendly, transportation modes. The first paper explicitly focuses on how to include more realistic transportation costs and emissions. This is done by obtaining the distribution of the size of an arbitrary shipment leaving the central warehouse (directly affected by the shipment frequency). It is thereby easy to evaluate any system where the transportation costs and emissions are dependent on the size of the shipment. The paper also provides a detailed analysis of a system where there is an opportunity to reserve shipment capacity on an intermodal truck-train-truck solution to at least one of the retailer groups. For this system it is shown how to jointly optimize the shipment intervals, the reserved capacities on the intermodal transportation modes and the reorder points in the system. The presented optimization procedure is applicable in three scenarios; (i) the emissions are not considered, (ii) there is a fixed cost per unit of emission, and (iii) there is a constraint on the maximum emissions per time unit. The second paper extends the analysis of a similar time-based shipment consolidation system to handle compound Poisson demand (instead of pure Poisson demand). This system has a simpler transportation cost structure, but the more general demand structure makes the model applicable for a broader array of products. The paper also extends the model to handle fill rate constraints, which further improves the practical applicability. The cost analysis is performed with a new methodology, based on the nominal inventory position. This variable is a helpful tool for analyzing the dynamics of distribution systems. Another system where this tool can be used is studied in the third paper. In this paper all stock points use installation stock (R,Q) ordering policies (batch ordering). This implies that situations can occur when only part of a requested retailer order is available at the central warehouse. The existing literature predominantly assumes that the available units are shipped immediately and the remaining units are shipped as soon as they arrive to the central warehouse, referred to as partial delivery. An alternative is to wait until the entire order is available before dispatching, referred to as complete delivery. The paper introduces a cost for splitting the order and evaluates three delivery policies; the PD policy (only partial deliveries are used), the CD policy (only complete deliveries are used), and the state-dependent MSD policy (an optimization between a partial and a complete delivery is performed for each delivery). The MSD policy is proven to perform better than both the PD and the CD policy. In a numerical study it is shown that significant savings can be made by using the MSD policy

Exact Methods for Multi-echelon Inventory Control : Incorporating Shipment Decisions and Detailed Demand Information

Recent advances in information technologies and an increased environmental awareness have altered the prerequisites for successful logistics. For companies operating on a global market, inventory control of distribution systems is often an essential part of their logistics planning. In this context, the research objective of this thesis is: To develop exact methods for stochastic inventory control of multi-echelon distribution systems incorporating shipment decisions and/or detailed demand information.The thesis consists of five scientific papers (Paper I, II, III, IV and V) preceded by a summarizing introduction. All papers study systems with a central warehouse supplying a number of non-identical local warehouses (retailers) facing stochastic demand. For given replenishment policies, the papers provide exact expressions for evaluating the expected long-run system behavior (e.g., distributions of backorders, inventory levels, shipment sizes and expected costs) and present optimization procedures for the control variables. Paper I and II consider systems where shipments from the central warehouse are consolidated to groups of retailers and dispatched periodically. By doing so, economies of scale for the transports can be reached, reducing both transportation costs and emissions. Paper I assumes Poisson customer demand and considers volume-dependent transportation costs and emissions. The model involves the possibility to reserve intermodal (train) capacity in combination with truck transports available on demand. For this system, the expected inventory costs, the expected transportation costs and the expected transport emissions are determined. Joint optimization procedures for the shipment intervals, the capacity reservation quantities, the reorder points and order-up-to levels in the system are provided, with or without emission considerations. Paper II analyses the expected costs of the same system for compound Poisson demand (where customer demand sizes may vary), but with only one transportation mode and fixed transportation costs per shipment. It also shows how to handle fill rate constraints. Paper III studies a system where all stock points use installation stock (R,Q) ordering policies (batch ordering). This implies that situations can occur when only part of a requested retailer order is available at the central warehouse. In these situations, the models in existing literature predominantly assume that available units are shipped immediately (partial delivery). An alternative is to wait until the entire order is available before dispatching (complete delivery). The paper introduces a cost for splitting the order and evaluates a system where optimal choices between partial and complete deliveries are made for all orders. In a numerical study it is shown that significant savings can be made by using this policy compared to systems which exclusively use either partial or complete deliveries. Paper IV shows how companies can benefit from detailed information about their customer demand. In a continuous review base stock system, the customer demand is modeled with independent compound renewal processes at the retailers. This means that the customer inter-arrival times may follow any continuous distribution and the demand sizes may follow any discrete distribution. A numerical study shows that this model can achieve substantial savings compared to models using the common assumption of exponential customer inter-arrival times. Paper V is a short technical note that extends the scope of analysis for several existing stochastic multi-echelon inventory models. These models analyze the expected costs without first determining the inventory level distribution. By showing how these distributions can be obtained from the expected cost functions, this note facilitates the analysis of several service measures, including the ready rate and the fill rate

An enhanced approximation mathematical model inventorying items in a multi-echelon system under a continuous review policy with probabilistic demand and lead-time

An inventory system attempts to balance between overstock and understock to reduce the total cost and achieve customer demand in a timely manner. The inventory system is like a hidden entity in a supply chain, where a large complete network synchronizes a series of interrelated processes for a manufacturer, in order to transform raw materials into final products and distribute them to customers. The optimality of inventory and allocation policies in a supply chain for a cement industry is still unknown for many types of multi-echelon inventory systems. In multi-echelon networks, complexity exists when the inventory issues appear in multiple tiers and whose performances are significantly affected by the demand and lead-time. Hence, the objective of this research is to develop an enhanced approximation mathematical model in a multi-echelon inventory system under a continuous review policy subject to probabilistic demand and lead-time. The probability distribution function of demand during lead-time is established by developing a new Simulation Model of Demand During Lead-Time (SMDDL) using simulation procedures. The model is able to forecast future demand and demand during lead-time. The obtained demand during lead-time is used to develop a Serial Multi-echelon Inventory (SMEI) model by deriving the inventory cost function to compute performance measures of the cement inventory system. Based on the performance measures, a modified distribution multi-echelon inventory (DMEI) model with the First Come First Serve (FCFS) rule (DMEI-FCFS) is derived to determine the best expected waiting time and expected number of retailers in the system based on a mean arrival rate and a mean service rate. This research established five new distribution functions for the demand during lead-time. The distribution functions improve the performance measures, which contribute in reducing the expected waiting time in the system. Overall, the approximation model provides accurate time span to overcome shortage of cement inventory, which in turn fulfil customer satisfaction

Real-time Allocation Decisions in Multi-echelon Inventory Control

Inventory control is a crucial activity for many companies. Given the recent advances in information technology, there have never been greater opportunities for coordinated inventory control across supply chain facilities. But how do we design efficient control methods and policies that take advantage of the detailed information that is now becoming available? This doctoral thesis investigates these issues within the field of inventory control theory. The objective of the research is: To develop mathematical models and policies for efficient control and increased understanding of stochastic multi-echelon inventory systems, with a focus on allocation decisions and the use of real-time information. This thesis is based on five scientific papers which are preceded by a summarizing introduction. The papers address different types of inventory distribution systems, all consisting of a central stocking facility that supplies an arbitrary number of local stocking facilities (referred to as retailers). The retailers face stochastic end customer demand. The systems are characterized by the presence of real-time inventory information, including continuously updated information on the current inventory levels at different facilities and on the locations of outstanding orders. In Paper I and Paper II we derive and evaluate different decision rules for stock allocation (known as allocation policies) for a central warehouse which applies a time based shipment consolidation strategy. The allocation policy determines how the central warehouse should distribute its stock among different retailers in case of shortages. New allocation policies that utilize real-time information are compared to the commonly used First Come - First Served policy which requires less information. In Paper III we shift focus to the delivery policy at a central warehouse which supplies multiple retailers that order in batches. When the central warehouse cannot satisfy an entire retailer order immediately, the delivery policy determines if the order should be shipped in several parts or in its entirety when all items are available. We investigate the value of using a new delivery policy that uses real-time information on when replenishments will arrive at the central warehouse. The information is used to determine the best course of action for each order placed by the retailers. We also study how to allocate safety stocks to all facilities in the system given this new policy. In Paper IV and Paper V we consider a system where retailers may receive emergency shipments from a support warehouse in combination with regular replenishments from a central warehouse/outside supplier. We investigate how safety stocks should be allocated between the retailers and the support warehouse. Furthermore, we evaluate the benefits of tracking orders in real time and using this information in the decision whether or not to request an emergency shipment

A Metaheuristic-Based Simulation Optimization Framework For Supply Chain Inventory Management Under Uncertainty

The need for inventory control models for practical real-world applications is growing with the global expansion of supply chains. The widely used traditional optimization procedures usually require an explicit mathematical model formulated based on some assumptions. The validity of such models and approaches for real world applications depend greatly upon whether the assumptions made match closely with the reality. The use of meta-heuristics, as opposed to a traditional method, does not require such assumptions and has allowed more realistic modeling of the inventory control system and its solution. In this dissertation, a metaheuristic-based simulation optimization framework is developed for supply chain inventory management under uncertainty. In the proposed framework, any effective metaheuristic can be employed to serve as the optimizer to intelligently search the solution space, using an appropriate simulation inventory model as the evaluation module. To be realistic and practical, the proposed framework supports inventory decision-making under supply-side and demand-side uncertainty in a supply chain. The supply-side uncertainty specifically considered includes quality imperfection. As far as demand-side uncertainty is concerned, the new framework does not make any assumption on demand distribution and can process any demand time series. This salient feature enables users to have the flexibility to evaluate data of practical relevance. In addition, other realistic factors, such as capacity constraints, limited shelf life of products and type-compatible substitutions are also considered and studied by the new framework. The proposed framework has been applied to single-vendor multi-buyer supply chains with the single vendor facing the direct impact of quality deviation and capacity constraint from its supplier and the buyers facing demand uncertainty. In addition, it has been extended to the supply chain inventory management of highly perishable products. Blood products with limited shelf life and ABO compatibility have been examined in detail. It is expected that the proposed framework can be easily adapted to different supply chain systems, including healthcare organizations. Computational results have shown that the proposed framework can effectively assess the impacts of different realistic factors on the performance of a supply chain from different angles, and to determine the optimal inventory policies accordingly

Multi-Echelon Inventory Optimization and Demand-Side Management: Models and Algorithms

Inventory management is a fudamental problem in supply chain management. It is widely used in practice, but it is also intrinsically hard to optimize, even for relatively simple inventory system structures. This challenge has also been heightened under the threat of supply disruptions. Whenever a supply source is disrupted, the inventory system is paralyzed, and tremenduous costs can occur as a consequence. Designing a reliable and robust inventory system that can withstand supply disruptions is vital for an inventory system\u27s performance.First we consider a basic type of inventory network, an assembly system, which produces a single end product from one or several components. A property called long-run balance allows an assembly system to be reduced to a serial system when disruptions are not present. We show that a modified version is still true under disruption risk. Based on this property, we propose a method for reducing the system into a serial system with extra inventory at certain stages that face supply disruptions. We also propose a heuristic for solving the reduced system. A numerical study shows that this heuristic performs very well, yielding significant cost savings when compared with the best-known algorithm.Next we study another basic inventory network structure, a distribution system. We study continuous-review, multi-echelon distribution systems subject to supply disruptions, with Poisson customer demands under a first-come, first-served allocation policy. We develop a recursive optimization heuristic, which applies a bottom-up approach that sequentially approximates the base-stock levels of all the locations. Our numerical study shows that it performs very well.Finally we consider a problem related to smart grids, an area where supply and demand are still decisive factors. Instead of matching supply with demand, as in the first two parts of the dissertation, now we concentrate on the interaction between supply and demand. We consider an electricity service provider that wishes to set prices for a large customer (user or aggregator) with flexible loads so that the resulting load profile matches a predetermined profile as closely as possible. We model the deterministic demand case as a bilevel problem in which the service provider sets price coefficients and the customer responds by shifting loads forward in time. We derive optimality conditions for the lower-level problem to obtain a single-level problem that can be solved efficiently. For the stochastic-demand case, we approximate the consumer\u27s best response function and use this approximation to calculate the service provider\u27s optimal strategy. Our numerical study shows the tractability of the new models for both the deterministic and stochastic cases, and that our pricing scheme is very effective for the service provider to shape consumer demand

Effective Multi-echelon Inventory Systems for Supplier Selection and Order Allocation

Successful supply chain management requires an effective sourcing strategy to counteract uncertainties in both the suppliers and demands. Therefore, determining a better sourcing policy is critical in most of industries. Supplier selection is an essential task within the sourcing strategy. A well-selected set of suppliers makes a strategic difference to an organization\u27s ability to reduce costs and improve the quality of its end products. To discover the cost structure of selecting a supplier, it is more interesting to further determine appropriate levels of inventory in each echelon for different suppliers. This dissertation focuses on the study of the integrated supplier selection, order allocation and inventory control problems in a multi-echelon supply chain. First, we investigate a non-order-splitting inventory system in supply chain management. In particular, a buyer firm that consists of one warehouse and N identical retailers procures a type of product from a group of potential suppliers, which may have different prices, ordering costs, lead times and have restriction on minimum and maximum total order size, to satisfy stochastic demand. A continuous review system that implements the order quantity, reorder point (Q, R) inventory policy is considered in the proposed model. The model is solved by decomposing the mixed integer nonlinear programming model into two sub-models. Numerical experiments are conducted to evaluate the model and some managerial insights are obtained with sensitivity analysis. In the next place, we extend the study to consider the multi-echelon system with the order-splitting policy. In particular, the warehouse acquisition takes place when the inventory level depletes to a reorder point R, and the order Q is simultaneously split among m selected suppliers. This consideration is important since it could pool lead time risks by splitting replenishment orders among multiple suppliers simultaneously. We develop an exact analysis for the order-splitting model in the multi-echelon system, and formulate the problem in a Mixed Integer Nonlinear Programming (MINLP) model. To demonstrate the solvability and the effectiveness of the model, we conduct several numerical analyses, and further conduct simulation models to verify the correctness of the proposed mathematical model