Franca Rinaldi

A Network Design Problem for a Distribution System with Uncertain Demands

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.

A two-dimensional strip cutting problem with sequencing constraint

We study a strip cutting problem that arises in the production of corrugated cardboard. In this context, rectangular items of different sizes are obtained by machines, called corrugators, that cut strips of large dimensions according to particular schemes containing at most two types of items. Because of buffer restrictions, these schemes have to be sequenced in such a way that, at any moment, at most two types of items are in production and not completed yet (sequencing constraint). We show that the problem of finding a set of schemes of minimum trim loss that satisfies an assigned demand for each item size is strongly NP-hard, even if the sequencing constraint is relaxed. Then, we present two heuristics for the problem with the sequencing constraint, both based on a graph characterization of the feasible solutions. The first heuristic is a two-phase procedure based on a mixed integer linear programming model. The second heuristic follows a completely combinatorial approach and consists of solving a suitable sequence of minimum cost matching problems. For both procedures, an upper bound on the number of schemes (setups) is found. Finally, a computational study comparing the quality of the heuristic solutions with respect to an LP lower bound is reported.

Guaranteed cost control for multi-inventory systems with uncertain demand

In this paper we consider the problem of controlling a multi-inventory system in the presence of uncertain demand. The demand is unknown but bounded in an assigned compact set. The control input is assumed to be also constrained in a compact set. We consider an integral cost function of the buffer levels and we face the problem of minimizing the worst-case cost. We show that the optimal cost of a suitable auxiliary problem with no uncertainties is always an upper bound for the original problem. In the special case of minimum-time control, this upper bound is tight, namely its optimal cost is equal to the worst-case cost for the original system. Furthermore, the result is constructive, since the optimal control law can be explicitly computed.

A feedback strategy for periodic network flows

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.

A time-indexed LP-based approach for min-sum job-shop problems

In this paper we propose two time-indexed IP formulations for job-shop scheduling problems with a min-sum objective. The first model has variables associated to job scheduling patterns. The exponential number of variables calls for a column generation scheme which is carried out by a dynamic programming procedure. The second model is of network flow type with side constraints. This model can be strengthened by adding cutting inequalities of clique type. It turns out that the two models are equivalent, since the dual of the second formulation is equivalent to the compact dual of the first model. However, they require significantly different solution approaches and may behave differently in terms of computing time and memory usage. Good upper bounds are found by a heuristic procedure that randomly generates schedules from fractional solutions. These features allow for an effective pruning of the branch-and-bound tree and narrowing the gap between lower and upper bounds. However, the size of both models is critically affected by the time-indexed formulation which may heavily slow down the computation.

Stabilization of multi-inventory systems with uncertain demand and setups

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.

An exact algorithm for the min-cost network containment problem

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.

A dynamic game model for distribution problems with non-stochastic uncertainty

Abstract A discrete-time dynamic production/distribution problem for a commodity is considered, comprising several delivery points. At each time, only the global amount of the demand is known, whereas the demand at each delivery point is only bounded above. The system has both production and transportation capacity constraints. Capacities and demand bounds can vary with time. The problem is that of finding initial conditions from which control (i.e. production/transportation) strategies exist that can fulfill any demand using the available system capacities. A two-person dynamic game model on a network is formulated for this problem. It is shown that the required sets of initial conditions are 0-base polyhedra; a procedure to derive them is provided, and an upper bound on the number of required steps is derived.

Vulnerability and power on networks

Inspired by socio-political scenarios, like dictatorships, in which a minority of people exercise control over a majority of weakly interconnected individuals, we propose vulnerability and power measures defined on groups of actors of networks. We establish an unexpected connection between network vulnerability and graph regularizability. We use the Shapley value of coalition games to introduce fresh notions of vulnerability and power at node level defined in terms of the corresponding measures at group level. We investigate the computational complexity of computing the defined measures, both at group and node levels, and provide effective methods to quantify them. Finally we test vulnerability and power on both artificial and real networks.

Control policies for multi-inventory systems with uncertain demand and setups

We consider multi-inventory systems in the presence of unknown but bounded demand and uncertain setups. The control input is assumed to be constant in its operating regime and incur setup whenever a variation of this regime is required. Both the setup times and setup configurations are unknown. We provide necessary and sufficient stabilizability conditions which turn out to be the same for the case in which there are no setups. 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.