Remarks on stochastic aspects on graphs

Abstract

For the description and solution of many problems in the area of operations research for which graph models have turned out to be an appropriate tool if problem data are fixed and known to enlarge the possibilities of application it is desirable to find corresponding formulations if variation of data has to be taken into consideration. With regard to this view-point problems including stochastic variation of data are of special interest and some efforts in combining stochastic and graphtheoretical aspects are cited in the references omitting, however, the large number of contributions using tools from reliability theory and stochastic programming(except [3] [15]). Of course, maximum flow and shortest path network formulations belong to the problems for which stochastic generalizations were considered (see e.g. [6][9],[25]), however, stochastic versions of project planning models seem to be mostly dealt with in this context. Defining a project to be a finite set of activities A = {a₁,…,a_n} ≠ ∅ with an irreflexive, asymmetric, transitive ordering relation σ ⊂ A×A (which allows representation as an adjacency-relation on the arc-set of a graph at least after introducing dummy activities, see e.g [1] [20] and nonnegative activity durations the problem is to determine the project duration which is yielded by maximizing over the sums of the durations of those activities which form maximal chains(with respect to σ). Assuming stochastic activity durations an intuitive idea is to replace the stochastic variables by their expected values and solve the resulting deterministic problem (using CPM), thus providing a rough estimate for the project duration mean. For this PERT-approach [18] several improvements have been suggested (see [4] [5] [10] [16] [22]). This paper gives a description of a class of estimation possibilities containig some estimators known from the literature as special cases.