Characterization of the nucleolus for a class of n-person games

An inductive procedure is given for finding the nucleolus of an n-person game in which all coalitions with less than n-1 players are totally defeated. It is shown that, for such a game, one of three things may occur: (a) all players receive the same amount; (b) each player receives his quota, plus a certain constant (which may be positive, nerative, or zero); (c) the weakest player receives one half his quota, and the other players divide the remaining profit according to the nucleolus of a similar (n-1)-person game. It is also shown that the nucleolus of such a game yields directly the nucleolus of each derived game. An example is worked out in detail.     

Callable products for an implicit time model with dependent demands

Capacity providers such as airlines and hotels have traditionally increased revenues by practicing market segmentation and revenue management, enabling them to sell the same capacity pool to different consumers at different prices. Callable products can enhance profits and improve consumers' welfare by allowing the firm to broker capacity between consumers with different willingness to pay. A consumer who buys a callable product gives the capacity provider the right to recall capacity at a prespecified recall price. This article studies callable products in the context of the model most commonly used in industry, which handles time implicitly imposing fewer restrictions on the nature of randomness compared to the Poisson arrival process favored in academia. In the implicit time model, capacity providers set booking limits to protect capacity for future high-fare demand. Our numerical study identifies conditions where callable products result in significant gains in profits.     

Oligopolistic contracting: Channel coordination under competition

We study contracts between a single retailer and multiple suppliers of two substitutable products, where suppliers have fixed capacities and present the retailer cost contracts for their supplies. After observing the contracts, the retailer decides how much capacity to purchase from each supplier, to maximize profits from the purchased capacity from the suppliers plus his possessed inventory (endowment). This is modeled as a noncooperative, nonzero‐sum game, where suppliers, or principals, move simultaneously as leaders and the retailer, the common agent, is the sole follower. We are interested in the form of the contracts in equilibrium, their effect on the total supply chain profit, and how the profit is split between the suppliers and the retailer. Under mild assumptions, we characterize the set of all equilibrium contracts and discuss all‐unit and marginal‐unit quantity discounts as special cases. We also show that the supply chain is coordinated in equilibrium with a unique profit split between the retailer and the suppliers. Each supplier's profit is equal to the marginal contribution of her capacity to supply chain profits in equilibrium. The retailer's profit is equal to the total revenue collected from the market minus the payments to the suppliers and the associated sales costs.     

Economic lot scheduling of fully loaded processes with external setups

Without restricting the class of permissible schedules, we derive optimal schedules for economic lot scheduling problems that are fully loaded, have external setups, and have only two products. The fully loaded condition accurately represents certain types of bottlenecks. We show that the optimal schedule must have the Wagner-Whitin property. We also develop a measure of aggregate inventory, derive an optimal steady-state aggregate inventory policy, and provide conditions under which the aggregate inventory level of an optimal schedule must approach a steady state. By restricting the class of permissible schedules to rotation cycle schedules, we extend these results to more than two products.     

The economic lot scheduling problem with finite backorder costs

We are concerned with the problem of scheduling m items, facing constant demand rates, on a single facility to minimize the long-run average holding, backorder, and setup costs. The inventory holding and backlogging costs are charged at a linear time weighted rate. We develop a lower bound on the cost of all feasible schedules and extend recent developments in the economic lot scheduling problem, via time-varying lot sizes, to find optimal or near-optimal cyclic schedules. The resulting schedules are used elsewhere as target schedules when demands are random. © 1992 John Wiley & Sons, Inc.     

Managing waiting times of backordered demands in single‐stage (Q,r) inventory systems

We present a service constrained (Q, r) model that minimizes expected holding and ordering costs subject to an upper bound on the expected waiting time of demands that are actually backordered. We show that, after optimizing over r, the average cost is quasiconvex in Q for logconcave continuous lead time demand distributions. For logconcave discrete lead time demand distributions we find a single‐pass efficient algorithm based on a novel search stopping criterion. The algorithm also allows for bounds on the variability of the service measure. A brief numerical study indicates how the bounds on service impact the optimal average cost and the optimal (Q, r) choice. The discrete case algorithm can be readily adapted to provide a single pass algorithm for the traditional model that bounds the expected waiting time of all demands (backordered or not). © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 557–573, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10028     

Political games

A modification of the Shapley value is suggested, which takes into account the fact that (due to personal affinities among the players) certain coalitions are more easily formed than others. This is done by assigning to each player a point in space, and looking at the distances between pairs of points. The method seems to be especially applicable to voting games among political parties (in, e. g., parliaments), and, for such games, gives a value which is considerably easier to compute than the usual Shapley value. Some examples are considered.     

Strategic investment to reduce setup times in the economic lot scheduling problem

This article considers the Economic Lot Scheduling Problem where setup times and costs can be reduced by an initial investment that is amortized over time. The objective is to determine a multiple-item single facility cyclic schedule to minimize the long run average holding and setup costs plus the amortized investment. We develop a lower bound on the long run average inventory carrying and setup costs as a function of the setup times, and show that this lower bound is increasing concave on the setup times when the out-of-pocket setup costs are zero or proportional to the setup times. We then develop a model that may be helpful in deciding the magnitude and the distribution of a one-time investment in reducing the setup times when the investment is amortized over time. Numerical results based on randomly generated problems, and on Bomberger's ten item problem indicate that significant overall savings are possible for highly utilized facilities. Most of the savings are due to a significant reduction in the long run average holding cost. © 1995 John Wiley & Sons, Inc.     

Cost allocation and communication

This article considers a special class of cost allocation problems, where the communication possibilities among the agents are restricted. Integral formulas are derived for two allocation rules: The Myerson value and the position value. © 1993 John Wiley & Sons, Inc.     

When is a base stock policy optimal in recovering disrupted cyclic schedules?

The extended economic lot scheduling problem (EELSP) is concerned with scheduling the production of a set of items in a single facility to minimize the long-run average holding, backlogging, and setup costs. Given an efficient cyclic production schedule for the EELSP, called the target schedule, we consider the problem of how to schedule production after a single schedule disruption. We propose a base stock policy, characterized by a base stock vector, that prescribes producing an item until its inventory level reaches the peak inventory of the target schedule corresponding to the item's position in the production sequence. We show that the base stock policy is always successful in recovering the target schedule. Moreover, the base stock policy recovers the target schedule at minimal excess over average cost whenever the backorder costs are proportional to the processing times. This condition holds, for example, when the value of the items is proportional to their processing times, and a common inventory carrying cost and a common service level is used for all the items. Alternatively, the proportionality condition holds if the inventory manager is willing to select the service levels from a certain set that is large enough to guarantee any minimal level of service, and then uses the imputed values for the backorder costs. When the proportionality condition holds we provide a closed-form expression for the total relevant excess over average cost of recovering the target schedule. We assess the performance of the base stock policy when the proportionality condition does not hold through a numerical study, and suggest some heuristic uses of the base stock policy. © 1994 John Wiley & Sons, Inc.