Abstract A network design problem which arises in the distribution of a public utility provided by several competitive suppliers is studied. The problem addressed is that of determining minimum‐cost (generalized) arc capacities in order to accommodate any demand between given source–sink pairs of nodes, where demands are assumed to fall within predetermined ranges. Feasible flows are initially considered as simply bounded by the usual arc capacity constraints. Then, more general linear constraints are introduced which may limit the weighted sum of the flows on some subsets of arcs. An exact cutting plane algorithm is presented for solving both of the above cases and some computational results are reported. © 2004 Wiley Periodicals, Inc.

In this paper, we consider different aspects of the problem of controlling a multi-inventory system in the presence of uncertain demand and setups. The demand is unknown but bounded in an assigned compact set. The control input is assumed to be constant in its operating regime and to incur setup whenever a variation of this regime is required. Both setup times and setup configurations are unknown. We provide necessary and sufficient stabilizability conditions which turn out to be the same in the case in which there are no setups. Stabilization can be achieved, provided that the planning horizon is large enough and a computable lower bound is given. We also face the problem of ultimately confining the state in an assigned constraint set and provide conditions on this set for the problem to be feasible. Furthermore, we consider the case in which the controls are quantized, as in the case of systems which work in a switching mode. Finally, we deal with the case in which multiple setups may happen during the planning horizon.

A class of production-distribution problems with unknown-but-bounded uncertain demand is considered. At each time, the demand is unknown, but each of its components is assumed to belong to an assigned interval. Furthermore, the system has production, transportation, and storage capacity constraints. The paper extends previous results to the case in which transportation delays are present. We show that the problem of finding a strategy which keeps bounded the storage levels reduces to that of finding a strategy for the associated instantaneous network, which is the network obtained by setting all the delays to zero in the original system. The state variables of the associated instantaneous network are the "inventory positions," given by the goods actually present in the warehouses plus the goods already ordered and leading to them.

We consider multiinventory production systems with control and state constraints dealing with unknown demand or supply levels. Unlike most contributions in the literature concerning this class of systems, we cope with uncertainties in an "unknown-but-bounded" fashion, in the sense that each unknown quantity may take any value in an assigned interval. For these situations, we perform a worst-case analysis. We show that a "smallest worst-case inventory level" exists, and it is associated to a steady-state control strategy. Then we consider the problem of driving the inventory levels to their smallest worst-case values. For this problem, we first give necessary and sufficient conditions, then we show that convergence occurs in a finite number of steps, and we give an upper bound for such a number.

We consider a dynamic network flow model for the control problem of a production-distribution system with periodic demand in the presence of storage and transportation capacity constraints. Unlike most papers dealing with control problems of dynamic networks, we derive a control strategy in feedback form. It is optimal in the sense that it involves, for any initial time, the set of all the initial states for which there exists a control strategy allowing the network variables (flows, storage levels) to remain in their constraint domain for all future times. The evaluation of such a strategy requires the previous computation of these maximal sets. We show that, due to the particular structure of the system, they are submodular polyhedra; in particular, they are finitely represented and the representation complexity is a priori known. Moreover, the evaluation of this sequence, even for the infinite horizon problem, can be performed in a finite number of steps that has a known upper bound depending on the dimension of the problem only. As a consequence of these results, the optimal strategy requires, at each step, to solve a submodular flow problem. The finite horizon problem is solved by the proposed method as a particular case. © 1996 John Wiley & Sons, Inc.