# 2011- Working Papers: Operations Research and Decisions

**Accounting for exact temporal behavior of costs in stochastic inventory systems**, 24 pp.

T. Avinadav and M. I. Henig

(Working Paper No. 4/2011)

Most periodic review inventory models do not attribute holding and shortage costs to their actual timing and therefore may miss optimality. This work shows how it is possible to overcome this deficiency in a general periodic review model. Exact formulas are presented to compute the optimal order quantity when inventory costs accrue in discrete or continuous times. The computational effort, not much different than that of the common periodic review models, involves some extra arithmetic. We introduce formulas to calculate the optimal policy for the Lévy process and especially for the Poisson, Wiener and gamma processes. The one for the Poisson process can be solved analytically while those for the Wiener and gamma processes can be approximated to the desired accuracy.

**The uncapacitated swapping problem on a line and on a circle**, 29 pp.

S. Anily and A. Pfeffer

(Working Paper No. 10/2011)

The *uncapacitated swapping problem* is defined by a graph consisting of *n* vertices, and *m* object types. Each vertex of the graph is associated with two object types: the one that it currently holds, and the one it requires. Each vertex holds or requires at most one unit of an object. The problem is balanced in the sense that for each object type, its total supply equals its total demand. A vehicle of unlimited capacity is assumed to ship the objects in order to fulfill the requirements of all vertices. The objective is to find a shortest route along which the vehicle can accomplish the rearrangement of the objects, given designated initial and terminal vertices. The uncapacitated swapping problem on a general graph, including a tree graph, is known to be *NP-Hard*. In this paper we show that for the line and circle graphs, the problem is polynomially solvable: we propose an *O**(**n**)*-time algorithm for a line and an *O**(**n ^{2}*

*)*-time algorithm for a circle.

**Homogeneous of degree one games are balanced with applications to service systems**, 23 pp.

S. Anily and M. Haviv

(Working Paper No. 11/2011)

A cooperative game with transferable utility is said to be *homogeneous of degree one* if for any integer m, the value of cloning *m* times all players at any given coalition, leads to *m* times the value of the original coalition. We show that this property coupled with sub-additivity, guarantee the non-emptyness of the core. A few examples for such games, which naturally emerge when servers in queueing systems cooperate, are presented.