Simulation of stochastic activity networks using path control variates

Abstract

This article details several procedures for using path control variates to improve the accuracy of simulation‐based point and confidence‐interval estimators of the mean completion time of a stochastic activity network (SAN). Because each path control variate is the duration of the corresponding directed path in the network from the source to the sink, the vector of selected path controls has both a known mean and a known covariance matrix. This information is incorporated into estimation procedures for both normal and nonnormal responses. To evaluate the performance of these procedures experimentally, we examine the bias, variance, and mean square error of the controlled point estimators as well as the average half‐length and coverage probability of the corresponding confidence‐interval estimators for a set of SANs in which the following characteristics are systematically varied: (a) the size of the network (number of nodes and arcs); (b) the topology of the network; (c) the percentage of activities with exponentially distributed durations; and (d) the relative dominance of the critical path. The experimental results show that although large improvements in accuracy can be achieved with some of these procedures, the confidence‐interval estimators for normal responses may suffer serious loss of coverage probability in some applications.

A splitting scheme for control variates

“…In previous experimentation [4], we found that the assumption of joint normality (11) between the response and the controls becomes increasingly untenable as the relative dominance increases; and this results in serious degradation in confidenceinterval coverage. To assess the effect on confidence-interval coverage of departures from the normality assumption, we set the relative dominance at three broad levels (low, medium, and high); and to assess the effect of sample size n, we performed simulation experiments involving n = 48, 96, and 192 independent replications.…”

“…For each nondummy activity duration V i in a given network, the associated distribution was taken to be either (a) a normal distribution with a specified mean µ i and standard deviation σ i = µ i /4 whose tail was truncated below the value 0; or (b) an exponential distribution with a specified mean µ i . We chose the exponential distribution as the nonnormal alternative for reasons elaborated in [4]. For network 1 the set of activities with durations as in (a) was taken to be {(1,3), (2,6), (2,4), (8,11), (10,13), (12,18), (16,17) The following rule was used for selecting control variates.…”

Assessing the Quality of Convex Approximations for Two-Stage Totally Unimodular Integer Recourse Models

“…We use numerical experiments to evaluate the actual performance of the convex approximation, defined in (35), of the stochastic activity network investment problem. We consider a single stochastic activity network given in Elmaghraby (1977) and also used in Avramdidis et al (1991). Figure 3 depicts the precedence relations between the activities.…”

“…We assume that all random variables, ω i , are independently distributed and follow a truncated normal distribution. The mean, µ i , is the same as in Elmaghraby (1977) and Avramdidis et al (1991) and differs per activity i, whereas the standard deviation σ is the same for each arc, but varies over the experiments. Furthermore, we also vary the budget B and the upper bound, b, on the investments; the per unit investment costs are c i = 1 for every activity and every experiment.…”

Integrated Variance Reduction Strategies

“…To form a control-variate estimator, we use the same approach as in Avramidis, Bauer, and Wilson (1991). Ranking the directed r-s paths in decreasing order of expected duration, we let t l , t~, & be the first three such paths.…”