Patrizia Beraldi

Designing robust emergency medical service via stochastic programming

Abstract This paper addresses the problem of designing robust emergency medical services. Under this respect, the main issue to consider is the inherent uncertainty which characterizes real life situations. Several approaches can be used to design robust mathematical models which are able to hedge uncertain conditions. We are using here the stochastic programming framework and, in particular, the probabilistic paradigm. More specifically, we develop a stochastic programming model with probabilistic constraints aimed to solve both the location and the dimensioning problems, i.e. where service sites must be located and how many emergency vehicles must be assigned to each site, in order to achieve a reliable level of service and minimize the overall costs. In doing so, we consider the randomness of the system as far as the demand of emergency service is concerned. The numerical results, which have been collected on a large set of test problems, demonstrate the validity of the proposed model, particularly in dealing with the trade-off between quality of service and costs management.

A branch and bound method for stochastic integer problems under probabilistic constraints

Stochastic integer programming problems under probabilistic constraints are considered. Deterministic equivalent formulations of the original problem are obtained by using p-efficient points of the distribution function of the right hand side vector. A branch and bound solution method is proposed based on a partial enumeration of the set of these points. The numerical experience with the probabilistic lot-sizing problem shows the potential of the solution approach and the efficiency of the algorithms implemented.

Rolling-horizon and fix-and-relax heuristics for the parallel machine lot-sizing and scheduling problem with sequence-dependent set-up costs

In this paper we develop new rolling-horizon and fix-and-relax heuristics for the identical parallel machine lot-sizing and scheduling problem with sequence-dependent set-up costs. Unlike previous papers, our procedures are based on a compact formulation relying on the hypotheses of identical machines. This feature makes our approach suitable for large-scale applications (with hundreds of machines) arising in the textile and fiberglass industries. Moreover, our procedures are shown to provide a feasible solution for any feasible instance. Comparisons with lower bounds provided by a truncated branch-and-bound show that the gap between the best heuristic solution and the lower bound never exceeds 3%.

The Probabilistic Set-Covering Problem

In a probabilistic set-covering problem the right-hand side is a random binary vector and the covering constraint has to be satisfied with some prescribed probability. We analyze the structure of the set of probabilistically efficient points of binary random vectors, develop methods for their enumeration, and propose specialized branch-and-bound algorithms for probabilistic set-covering problems.

Optimal capacity allocation in multi-auction electricity markets under uncertainty

The advent of competitive markets confronts each producer with the problem of optimally allocating his energy/capacity so as to maximize his profits. The multiplicity of auctions in electricity markets and the non-trivial constraints imposed by technical and bidding rules make the problem of crucial importance and difficult to model and solve. Further difficulties are represented by the dynamic and stochastic natures that characterize the decision process. We formulate the problem as a multi-stage mixed-integer stochastic optimization model under the assumption that the seller is a price taker. We validate the effectiveness of the proposed model on a representative test problem.

A heuristic approach for resource constrained project scheduling with uncertain activity durations

In this paper, we address the resource constrained project scheduling problem with uncertain activity durations. Project activities are assumed to have known deterministic renewable resource requirements and uncertain durations, described by independent random variables with a known probability distribution function. To tackle the problem solution we propose a heuristic method which relies on a stage wise decomposition of the problem and on the use of joint probabilistic constraints.

A two-stage stochastic programming model for electric energy producers

The bilateral contract selection and bids definition constitute a strategic issue for electric energy producers that operate in competitive markets, as the liberalized electricity ones. In this paper we propose a two-stage stochastic integer programming model for the integrated optimization of power production and trading which include a specific measure accounting for risk management. We solve the model by means of a novel enumerative solution approach that exploits the particular problem structure. Finally, we report some preliminary computational experiments.

Parallel algorithms to solve two-stage stochastic linear programs with robustness constraints

Abstract In this paper we present a parallel method for solving two-stage stochastic linear programs with restricted recourse. The mathematical model considered here can be used to represent several real-world applications, including financial and production planning problems, for which significant changes in the recourse solutions should be avoided because of their difficulty to be implemented. Our parallel method is based on a primal-dual path-following interior point algorithm, and exploits fruitfully the dual block-angular structure of the constraint matrix and the special block structure of the matrices involved in the restricted recourse model. We describe and discuss both message-passing and shared-memory implementations and we present the numerical results collected on the Origin2000.

An exact approach for solving integer problems under probabilistic constraints with random technology matrix

This paper addresses integer programming problems under probabilistic constraints involving discrete distributions. Such problems can be reformulated as large scale integer problems with knapsack constraints. For their solution we propose a specialized Branch and Bound approach where the feasible solutions of the knapsack constraint are used as partitioning rules of the feasible domain. The numerical experience carried out on a set covering problem with random covering matrix shows the validity of the solution approach and the efficiency of the implemented algorithm.

Beam search heuristic to solve stochastic integer problems under probabilistic constraints

Abstract This paper proposes a Beam Search heuristic strategy to solve stochastic integer programming problems under probabilistic constraints. Beam Search is an adaptation of the classical Branch and Bound method in which at any level of the search tree only the most promising nodes are kept for further exploration, whereas the remaining are pruned out permanently. The proposed algorithm has been compared with the Branch and Bound method. The numerical results collected on the probabilistic set covering problem show that the Beam Search technique is very efficient and appears to be a promising tool to solve difficult stochastic integer problems under probabilistic constraints.