A NETWORK DESIGN PROBLEM FOR A DISTRIBUTION SYSTEM WITH UNCERTAIN DEMANDS ∗ FRANCO BLANCHINI † , FRANCA RINALDI † , AND WALTER UKOVICH ‡ SIAM J. OPTIM. c  1997 Society for Industrial and Applied Mathematics Vol. 7, No. 2, pp. 560–578, May 1997 017 Abstract. A class of production–distribution planning problems with nonstochastic uncertain demands is modeled as a dynamic game between two players who control flows on a network with node and arc capacity constraints. Simple conditions are derived for determining which player wins the game. These conditions are then used to design a minimum cost network with the property that its feasible control strategies are allowed to meet the demand without violating the capacity constraints. Key words. dynamic networks, dynamic games, network design AMS subject classifications. 90B05, 90B06, 90B10, 90B15, 90B30, 90C60, 90D43, 90D50 PII. S1052623494266262 1. Introduction. Many important problems concerning production, transporta- tion, and distribution of goods can be addressed by network models in which nodes represent storage capabilities and arcs represent production units or transportation links. Basically, such problems consist in determining a strategy to decide arc flows in order to ship the commodity from some nodes to other nodes of the network in order to satisfy a certain demand. The literature on this subject is very extensive and we refer the reader to several textbooks (among the most recent ones, see, for instance, [1], [6], [12], [19], [20], and [28]). In particular, dynamic network problems have received great attention. In this case, flow values, storage levels, and demands are time-varying quantities. A typical problem concerning this kind of model consists of planning the commodity flow and storage at each time in order to minimize transportation and stocking costs. For an extensive survey of these topics, see [2]. If the demand is known in the assigned time horizon, the dynamic flow problem can be handled via the well-known time- expanded network method (see, again, [2] and [31]). Unfortunately, the demand is often unknown and this fact has led to the use of stochastic methods (see, for instance, [5], [30]) to handle problems of this kind. However, the stochastic approach to the control of dynamic networks requires stochastic information which can be unavailable in some cases. In this paper, uncertainties are modeled in a different way. Production and de- mand are assumed to have a known range of allowed values, but no knowledge is given on which allowed values will actually be taken. These unknown-but-bounded specifications for uncertainties are quite realistic in several situations. In general, upper and lower bounds for production and demand can be inferred from historical data or decision makers’ experience much more easily and with much more confidence than empirical probability distributions for the same quantities. Sometimes, they are a consequence of a particular operational condition or a technological characteristic * Received by the editors April 18, 1994; accepted for publication (in revised form) November 1, 1995. http://www.siam.org/journals/siopt/7-2/26626.html † Dipartimento di Matematica ed Informatica, University of Udine, Italy (blanchini@uniud.it, rinaldi@dimi.uniud.it). These authors were supported by C.N.R. under research grant 94.00543.CT11. ‡ Dipartimento di Elettrotecnica, Elettronica ed Informatica, University of Trieste, Italy (ukovich@univ.trieste.it). This author was supported by C.N.R. under contract 94.01472.PF74. 560