In this paper we consider a repair shop location problem with uncertainties in demand. New local repair shops have to be opened at a number of locations. At these local repair shops, customers arrive with broken, but repairable, items. Customers go to the nearest open repair shop. Since they want to leave as soon as possible, a (small) inventory of working items is kept at the repair shops. A customer immediately receives a working item from stock, provided that the stock is not empty. If a stockout occurs, the customer has to wait for a working item. The broken items are repaired in the shop and then put in stock. Sometimes, however, a broken item cannot be fixed at the local repair shop, and it has to be sent to a central repair shop. At the central repair shop the same policy with inventory and repair is used.The problem that we focus on, is not only finding locations for the local repair shops, but also minimizing the stock levels at the shops, such that the fraction of customers that can leave the local shops without waiting (the so called fill rate), is above a prespecified level. We assume that the central repair shop is already opened, but that the repair capacity still has to be set. The local repair shops can be opened at a number of locations, which may have different repair capacities.The goal is to minimize the total cost, that is the total cost for keeping the local shops operational, for the transport of items and for the inventory. For this minimizing problem, a local search heuristic with respect to the open locations, repair capacities and inventory levels is presented.

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We present a simple dual ascent method for the multilevel facility location problem which finds a solution within 6 times the optimum for the uncapacitated case and within 12 times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique.

This paper presents a combinatorial algorithm for downlink rate allocation in Code Division Multiple Access (CDMA) mobile networks. By discretizing the coverage area into small segments, the transmit power requirements are characterized via a matrix representation that separates user and system characteristics. We obtain a closed-form analytical expression for the so-called Perron-Frobenius eigenvalue of that matrix, which provides a quick assessment of the feasibility of the power assignment for a given downlink rate allocation. Based on the Perron-Frobenius eigenvalue, we reduce the downlink rate allocation problem to a set of multiple-choice knapsack problems. The solution of these problems provides an approximation of the optimal downlink rate allocation and cell borders for which the system throughput, expressed in terms of utility functions of the users, is maximized.

We show that for every cooperative game a corresponding set game can be defined, called the standard set game.Values for set games can be applied to this standard game and determine allocations for the cooperative game. On the other hand, notions for cooperative games, like the Shapley value, the τ −value or the core can be considered in the context of the standard set games.