C-diagrams, shifts and solidarity values

In the theory of cooperative games so called dividends of a coalition $S$ are considered, which are defined as $\frac{c_S}{|S|}$. The costs $c_S$ form a c-diagram. On these c-diagrams several types of shifts are defined and analysed. Different solution concepts and their properties are related to shifts. We introduce reward games and fine games as components of a cooperative game. Some solution concepts for applications are analysed in terms of c-diagrams, as well as the solidarity concept. \u

A simple dual ascent algorithm for the multilevel facility location problem

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. \u

An approximation algorithm for a facility location problem with stochastic demands

In this article we propose, for any $\epsilon>0$, a $2(1+\epsilon)$-approximation algorithm for a facility location problem with stochastic demands. This problem can be described as follows. There are a number of locations, where facilities may be opened and a number of demand points, where requests for items arise at random. The requests are sent to open facilities. At the open facilities, inventory is kept such that arriving requests find a zero inventory with (at most) some pre-specified probability. After constant times, the inventory is replenished to a fixed order up to level. The time interval between consecutive replenishments is called a reorder period. The problem is where to locate the facilities and how to assign the demand points to facilities at minimal cost per reorder period such that the above mentioned quality of service is insured. The incurred costs are the expected transportation costs from the demand points to the facilities, the operating costs (opening costs) of the facilities and the investment in inventory (inventory costs). \u

A multiple-choice knapsack based algorithm for CDMA downlink rate differentiation under uplink coverage restrictions

This paper presents an analytical model 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 downlink rates, is maximized. \u

The standard set game of a cooperative game

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 $\tau-$value or the core can be considered in the context of the standard set games. \u

A combinatorial approximation algorithm for CDMA downlink rate allocation

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

A combinatorial approximation algorithm for CDMA downlink rate allocation

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

An approximation algorithm for the 2-level uncapacitated facility location

We present an approximation algorithm for the maximization version of the two level uncapacitated facility location problem achieving a performance guarantee of $0.47.$ The main idea is to reduce the problem to a special case of MAX SAT, for which an approximation algorithm based on randomized rounding is presented

Locating repair shops in a stochastic environment

In this paper, we consider a repair shop location problem with uncertainties in demand. New 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, an 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 we focus on, is finding locations for the local repair shops, deciding their capacity, i.e., number of servers and base stock levels, such that the total expected cost is minimized and the fraction of customers that can leave the local shops without waiting is above some specified level. We assume that the central repair shop is already opened, but that the repair capacity still has to be set. The costs we consider are the costs for keeping the repair shops operational, for the transport of items and for the inventory. For this problem, a local search heuristic is proposed and experimental results are presented