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208 Deolinda M. L. D. Rasteiro and António J. B. Anjoization of Bellman first-in-first-out labelling-correcting algorithm used to the determine theshortest path in directed networks with deterministic parameters associated to each arc.The most common methods used to solve the classical deterministic shortest path problemare the label-correcting methods. In this approach to each node i ∈ N , ((N , A) is a probabilisticnetwork where N = {v 1 , . . . , v n } is the set of nodes and A = {a 1 , . . . , a m } ⊆ N × N is theset of arcs) is assigned a label (usually the distance or cost to the destination node) and putinto a queue. Then we scan through the nodes in the queue and update the labels, if necessary.After a node is scanned it will be removed from the queue, and then some nodes maybe inserted or repositioned in the queue. The process is repeated until the queue is empty orall labels are correct. There are several implementations of this methods including the firstin-first-outalgorithm of Bellman [3]. A review of these methods, which basic difference is theway how the queue is manipulated, can be found in Ahuja et al [1].We present computational results, for networks with 100 up to 10000 nodes and densities 2,5 and 10, which prove that our approach is very efficient in terms of memory and time.2. Problem DefinitionIn the stochastic shortest path problem a directed probabilistic network (N , A) is given whereeach arc (i, j) ∈ A is associated to the real random variable X ij which is called the randomparameter of the arc (i, j) ∈ A. We assume that the real random variables X ij have discretedistributions and are independent. The variables X ij are sometimes referred as cost, time ordistance.The set of outcomes of X ij will be denoted by S Xij ={d 1 ij , d2 ij , . . . , dr ij}. We will assumethat the dimension of S Xij , i.e, r is always a finite value. The probability of X ij assume thevalue d l ij is denoted by pl ij .If an appropriate utility measure is assigned to each possible consequence and the expectedvalue of the utility measure of each alternative is calculated, then the best action is to considerthe alternative with the highest expected utility (which can be the smallest expected value).Different axioms that imply the existence of utilities with the property that expected utility isan appropriate guide to consistent decision making are presented in [5, 7–10]. The choice ofan adequate utility function for a specific type of problem can be taken using direct methodspresented in Keeney and Raiffa’s book.The utility of the arc (i, j) ∈ A in the optimal path is measured calculating the minimumof the real random variables X iw where w is such that the arc (i, w) ∈ A, i.e, w belongs to theforward star 1 of i. Thus U((i, j)) = minw∈F(i) X iw.Associated to the path p, we define the real random variableX p = ∑minX iww∈F(i)(i,j)∈prepresenting the random parameter of the loopless path p ∈ P.With the objective of determine the optimal path, we consider a real function U : P −→ IR,called utility function, such that for each loopless path p, U(p) depends on the random variablesassociated to the arcs of p and is defined as⎛U(p) = E ⎝ ∑minX iww∈F(i)(i,j)∈p⎞⎠ .1 The forward star of node i is the set formed by the terminal nodes of its outgoing arcs.