On multi-stage production/inventory systems under stochastic demand

This paper was presented at the 1992 Conference of the International Society of Inventory Research in Budapest, as a tribute to professor Andrew C. Clark for his inspiring work on multi-echelon inventory models both in theory and practice. It reviews and extends the work of the authors on periodic review serial and convergent multi-echelon systems under stochastic stationary demand. In particular, we highlight the structure of echelon cost functions which play a central role in the derivation of the decomposition results and the optimality of base stock policies. The resulting optimal base stock policy is then compared with an MRP system in terms of cost effectiveness, given a predefined target customer service level. Another extension concerns an at first glance rather different problem; it is shown that the problem of setting safety leadtimes in a multi-stage production-to-order system with stochastic lead times leads to similar decomposition structures as those derived for multi-stage inventory systems. Finally, a discussion on possible extensions to capacitated models, models with uncertainty in both demand and production lead time as well as models with an aborescent structure concludes the paper

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

Rolling schedule approaches for supply chain operations planning

Supply Chain Operations Planning (SCOP) involves the determination of an extensive production plan for a network of manufacturing and distribution entities within and across organizations. The production plan consist of order release decisions that allocate materials and resources in order to transform these materials into (intermediate) products. We use the word item for both materials, intermediate products, and end-products. Furthermore, we consider arbitrary supply chains, i.e. the products produced by the supply chain as a whole and sold to customers consist of multiple items, where each item may in turn consists of multiple items and where each item may be used in multiple items as well. The aim of SCOP is not only to obtain a feasible production plan, but the plan must be determined such that pre-specified customer service levels are met while minimizing cost. To obtain optimal production plans we use a linear programming (LP) model. The reason we use an LP model is twofold. First, LP models can easily be incorporated in existing Advanced Planning Systems (APS). Second, while the multi-echelon inventory concept can only be used for uncapacitated supply chains and some special cases of capacitated supply chains, capacity constraints but also other restrictions can easily added to LP models. In former mathematical programming (MP) models, the needed capacity was allocated at a fixed time offset. This time offset was indicated by fixed or minimum lead times. By the introduction of planned lead times with multi-period capacity allocation, an additional degree of freedom is created, namely the timing of capacity allocation during the planned lead time. When using the LP model in a rolling schedule context, timing the capacity allocation properly can reduce the inventory cost. Although the number of studies on MP models for solving the SCOP or related problems are carried out by various researchers is enormous, only a few of these studies use a rolling schedule. Production plans are only calculated for a fixed time horizon based on the forecast of customers demand. However, since customer demand is uncertain, we emphasize the use of a rolling schedule. This implies that a production plan, based on sales forecasts, is calculated for a time interval (0; T], but only executed for the first period. At time 1, the actual demand of the first period is known, and the inventory status of the consumer products are adjusted according the actual demand. For time interval (1; T + 1], a new production plan is calculated. In this thesis, we studied the proposed LP strategy with planned lead times in a rolling schedule setting whereby we focused on the following topics: ² timing of production within the planned lead time, ² factors influencing the optimal planned lead time, ² early availability of produced items, i.e. availability of items before the end of their planned lead time, and ² balanced material allocation. In the first three studies we explore the possibilities of using planned lead times. In the first study, timing of production, we compare the situation whereby released items are produced as soon as there is available capacity with the situation whereby released items are produced as late as possible within the planned lead time. If items are produced as soon as possible, there is more capacity left for future production. Since we work with uncertain customer demand whereby demand may be larger than expected, this capacity might be very useful. A drawback of production as soon as possible are the additional work-in-process cost. The results of simulation studies show that if the utilization rates of resources and/or the variation in demand are high, producing early is better. However this is only the case if the added value between the concerned item and the end item is high. The second study deals with factors influencing the optimal planned lead time. From queuing theory it is already known that the variance in demand and the utilization rate of the resources determine the waiting time. More variation and/or higher utilization rates give longer waiting times. Since lead times consist for a large part of waiting time, these two factors most probably also influence the length of the optimal planned lead time. For a set of representative supply chain structures we showed that this was indeed the case. With longer planned lead times, the flexibility in capacity allocation is higher. Additional flexibility gives lower safety stocks, but longer planned lead times also means more work-in-process. Hence, an important third factor which influence the optimal planned lead time is the holding costs structure. When using planned lead times, early produced items have to wait the remainder of their planned lead time. This seems contradictory, especially if these items are necessary to avoid or reduce backorders. Therefore we adapt the standard LP model in two ways. In the first model, items are made available for succeeding production steps directly after they are produced. And in the second model, produced items are only made available for succeeding production steps if they are needed to avoid or reduce backorders. Experiments showed that the first model does not improve the performance of the standard LP strategy. The advantages of planned lead times longer than one period are nullified by early availability of produced items. The second model indeed improves the performance of the standard LP strategy, but only when the planned lead times are optimal or longer. Comparing the introduced LP strategy with a so-called synchronized base stock policy under the assumption of infinite capacity, it turned out that the LP strategy is outperformed by the base stock policy. In order to obtain a better performance, we Summary 121 added linear allocation rules to the LP model. With these allocation rules shortages of child items are divided among the parent items using a predefined allocation fraction. A second way of balanced allocation of child items is obtained by replacing the linear objective function by a quadratic one. The results of a well-chosen set of experiments showed that although the synchronized base stock policy also outperforms the adjusted LP strategies, the difference in performance is small. Hence, the adjusted LP strategies are good alternatives for large, capacitated supply chain structures which cannot be solved by synchronized base stock policies. Comparing the model with linear allocation rules with the model with quadratic objective function, the preference is given to the latter model. This model does not only give the lowest inventory costs, it also has the shortest computation time. Furthermore, this model can easily be implemented and solved by existing software. Summarizing the main results of this thesis, we conclude that deterministic LP models can be used to solve the SCOP problem with stochastic demand by using the LP model in a rolling schedule concept. By using optimal planned lead times with multiperiod capacity allocation, early production during the planned lead times, and early availability of needed produced items before the end of the planned lead time, we can decrease the inventory costs. The costs can also be reduced by using allocation strategies to allocate shortages among parent items proportionally. Especially the results for the model with quadratic objective function are promising

Analysis of a two-echelon inventory system with two supply modes

In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We use results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and show that it is useful to decrease upstream stock levels. In cases with high demand uncertainty, high backlogging cost or long lead times, we conclude that dual-sourcing can lead to significant savings

Decision support system for vendor managed inventory supply chain:a case study

Vendor-managed inventory (VMI) is a widely used collaborative inventory management policy in which manufacturers manages the inventory of retailers and takes responsibility for making decisions related to the timing and extent of inventory replenishment. VMI partnerships help organisations to reduce demand variability, inventory holding and distribution costs. This study provides empirical evidence that significant economic benefits can be achieved with the use of a genetic algorithm (GA)-based decision support system (DSS) in a VMI supply chain. A two-stage serial supply chain in which retailers and their supplier are operating VMI in an uncertain demand environment is studied. Performance was measured in terms of cost, profit, stockouts and service levels. The results generated from GA-based model were compared to traditional alternatives. The study found that the GA-based approach outperformed traditional methods and its use can be economically justified in small- and medium-sized enterprises (SMEs)

Analysis of an assemble-to-order system with different review periods

We consider a single item assembled from two components. One of the components has a long leadtime, high holding cost and short review period as compared to the other one. We assume that net stocks are reviewed periodically, customer demand is stochastic and unsatisfied demand is back ordered. We analyze the system under two different policies and show how to determine the policy parameters minimizing average holding and backorder costs. First, we consider a pure base stock policy, where orders for each component are placed such that the inventory position is raised up to a given base stock level. In contrast to this, only the orders for one component follow this logic while the other orders are synchronized in case of a balanced base stock policy. Through mathematical analysis, we come up with the exact long-run average cost function and we show the optimality conditions for both policies. In a numerical study the policies are compared and the results suggest that the balanced base stock policy works better than the pure base stock policy under low service levels and when there is a big difference in the holding costs of the components

Near-optimal heuristics to set base stock levels in a two-echelon distribution network

We consider a continuous-review two-echelon distribution network with one central warehouse and multiple local stock points, each facing independent Poisson demand for one item. Demands are fulfilled from stock if possible and backordered otherwise. We assume base stock control with one-for-one replenishments and the goal is to minimize the inventory holding and backordering costs. Although this problem is widely studied, only enumerative procedures are known for the exact optimization. A number of heuristics exist, but they ??nd solutions that are far from optimal in some cases (over 20% error on realistic problem instances). We propose a heuristic that is computationally e??cient and ??nds solutions that are close to optimal: 0.1% error on average and less than 3.0% error at maximum on realistic problem instances in our computational experiment

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

An inventory control project in a major Danish company using compound renewal demand models

We describe the development of a framework to compute the optimal inventory policy for a large spare-parts’ distribution centre operation in the RA division of the Danfoss Group in Denmark. The RA division distributes spare parts worldwide for cooling and A/C systems. The warehouse logistics operation is highly automated. However, the procedures for estimating demands and the policies for the inventory control system that were in use at the beginning of the project did not fully match the sophisticated technological standard of the physical system. During the initial phase of the project development we focused on the fitting of suitable demand distributions for spare parts and on the estimation of demand parameters. Demand distributions were chosen from a class of compound renewal distributions. In the next phase, we designed models and algorithmic procedures for determining suitable inventory control variables based on the fitted demand distributions and a service level requirement stated in terms of an order fill rate. Finally, we validated the results of our models against the procedures that had been in use in the company. It was concluded that the new procedures were considerably more consistent with the actual demand processes and with the stated objectives for the distribution centre. We also initiated the implementation and integration of the new procedures into the company’s inventory management systemBase-stock policy; compound distribution; fill rate; inventory control; logistics; stochastic processes