A coordination mechanism with fair cost allocation for divergent multi-echelon inventory systems

This paper is concerned with the coordination of inventory control in divergent multiechelon inventory systems under periodic review and decentralized control. All the installations track echelon inventories. Under decentralized control the installations will decide upon replenishment policies that minimize their individual inventory costs. In general these policies do not coincide with the optimal policies of the system under centralized control. Hence, the total cost under decentralized control is larger than under centralized control.\ud To remove this cost inefficiency, a simple coordination mechanism is presented that is initiated by the most downstream installations. The upstream installation increases its base stock level while the downstream installation compensates the upstream one for increased costs and provides it with additional side payments. We show that this mechanism coordinates the system; the global optimal policy of the system is the unique Nash equilibrium of the corresponding strategic game. Furthermore, the mechanism results in a fair allocation of the costs; all installations enjoy cost savings

Cost sharing of cooperating queues in a Jackson network

We consider networks of queues in which the independent operators of individual queues may cooperate to reduce the amount of waiting. More specifically, we focus on Jackson networks in which the total capacity of the servers can be redistributed over all queues in any desired way. If we associate a cost to waiting that is linear in the queue lengths, it is known how the operators should share the available service capacity to minimize the long run total cost. We answer the question whether or not (the operators of) the individual queues will indeed cooperate in this way, and if so, how they will share the cost in the new situation. One of the results is an explicit cost allocation that is beneficial for all operators. The approach used also works for other cost functions, such as the server utilization

Supply chain collaboration

In the past, research in operations management focused on single-firm analysis. Its goal was to provide managers in practice with suitable tools to improve the performance of their firm by calculating optimal inventory quantities, among others. Nowadays, business decisions are dominated by the globalization of markets and increased competition among firms. Further, more and more products reach the customer through supply chains that are composed of independent firms. Following these trends, research in operations management has shifted its focus from single-firm analysis to multi-firm analysis, in particular to improving the efficiency and performance of supply chains under decentralized control. The main characteristics of such chains are that the firms in the chain are independent actors who try to optimize their individual objectives, and that the decisions taken by a firm do also affect the performance of the other parties in the supply chain. These interactions among firms’ decisions ask for alignment and coordination of actions. Therefore, game theory, the study of situations of cooperation or conflict among heterogenous actors, is very well suited to deal with these interactions. This has been recognized by researchers in the field, since there are an ever increasing number of papers that applies tools, methods and models from game theory to supply chain problems

Efficiency of repeated network interactions

In this paper we consider a network with interactions by two users. Each of them repeatedly issues download requests on the network. These requests may be unsuccessful due to congestion or non-congestion related errors. A user decides when to cancel a request (that is, what his impatience threshold is) and how long to wait before reissuing his request after cancellation of the previous request (that is, what his waiting time will be). This pair of impatience threshold and waiting time is his strategy. If a customer decides not to wait but to reissue his request immediately, that is, he sets his waiting time to zero, then he is said to use a so-called restart strategy. The goal of the user is to maximize the number of successful requests over a given time span.\ud We study optimal strategies for the users in a game-theoretic framework. We find that in case congestion is the only cause of unsuccessful requests then each of the users will be very patient and any waiting time is optimal. Hence, restart strategies are among the optimal strategies. Second, in case non-congestion related errors may occur, users will also set large impatience times, but now they will set waiting times to zero; in other words: they immediately reissue an unsuccessful download. In this case all optimal strategies are restart strategies. Hence, in both cases restart strategies are among the optimal strategies. Finally, implementing social optimal strategies instead of individual optimal ones cannot improve the efficiency of the network usage

Coordination mechanisms for inventory control in three-echelon serial and distribution systems

This paper is concerned with the coordination of inventory control in three-echelon serial and distribution systems under decentralized control. All installations in these supply chains track echelon inventories. Under decentralized control the installations will decide upon base stock levels that minimize their own inventory costs. In general these levels do not coincide with the optimal base stock levels in the global optimum of the chain under centralized control. Hence, the total cost under decentralized control is larger than under centralized control. \ud To remove this cost inefficiency, two simple coordination mechanisms are presented: one for serial systems and one for distribution systems. Both mechanisms are initiated by the most downstream installation(s). The upstream installation increases its base stock level while the downstream installation compensates the upstream one for the increase of costs and provides it with a part of its gain from coordination. It is shown that both coordination mechanisms result in the global optimum of the chain being the unique Nash equilibrium of the corresponding strategic game. Furthermore, all installations agree upon the use of these mechanisms because they result in lower costs per installation. The practical implementation of these mechanisms is discussed

Convexity in stochastic cooperative situations

This paper introduces a new model concerning cooperative situations in which the payoffs are modeled by random variables. We analyze these situations by means of cooperative games with random payoffs. Special attention is paid to three types of convexity, namely coalitional-merge, individual-merge and marginal convexity. The relations between these types are studied and in particular, as opposed to their deterministic counterparts for TU games, we show that these three types of convexity are not equivalent. However, all types imply that the core of the game is nonempty. Sufficient conditions on the preferences are derived such that the Shapley value, defined as the average of the marginal vectors, is an element of the core of a convex game

Cooperation in stochastic inventory models with continuous review

Consider multiple companies that continuously review their inventories and face Poisson demand. We study cooperation strategies for these companies and analyse if there exist allocations of the joint cost such that any company has lower costs than on its own; such allocations are called stable cost allocations. We start with two companies that jointly place an order for replenishment if their joint inventory position reaches a certain reorder level. This strategy leads to a simple expression of the joint costs. However, these costs exceed the costs for non-cooperating companies. Therefore, we examine another cooperation strategy. Namely, the companies reorder as soon as one of them reaches its reorder level. This latter strategy has lower costs than for non-cooperating companies. Numerical experiments show that the gametheoretical distribution rule — a cost allocation in which the companies share the procurement cost and each pays its own holding cost — is a stable cost allocation. These results also hold for situations with multiple companies

Stratified breast cancer follow-up using a continuous state partially observable Markov decision process

Frequency and duration of follow-up for breast cancer patients is still under discussion. Currently, in the Netherlands follow-up consists of annual mammography for the first five years after treatment and does not depend on the personal risk of developing a locoregional recurrence or a second primary tumor. The aim of this study is to gain insight in how to allocate resources for optimal and personalized follow-up. We formulate a discrete-time Partially Observable Markov Decision Process (POMDP) over a finite horizon with both discrete and continuous states, in which the size of the tumor is modeled as a continuous state. Transition probabilities are obtained from data of the Netherlands Cancer Registry. We show that the optimal value function of the POMDP is piecewise linear and convex and provide an alternative representation for it. Under some reasonable conditions on the dynamics of the POMDP, the optimal value function can be obtained from the parameters of the underlying probability distributions only. Finally, we present results for a stratification of the patients based on their age to show how this model can be applied in practice