On path correlation and PERT bias

Abstract

Most studies of project time estimation assume that (a) activity times are mutually independent random variables; many also assume that (b) path completion times are mutually independent. In this paper, we subject the impact of both these assumptions to close scrutiny. Using tools from multivariate analysis, we make a theoretical study of the direction of the error in the classical PERT method of estimating mean project completion time when correlation is ignored. We also investigate the effect of activity dependence on the normality of path length via simulation.

Distributionally robust project crashing with partial or no correlation information

“…where and are the mean and SD of the original random duration of activity(i, j), and and are the minimal mean and SD of the duration of activity (i, j) that can be achieved by allocating resources to it. Further B is the amount of total cost budget, and c (1) ( ) and c (2) ( ) are the cost functions which have the following properties: (a) c (1) ( ) = c (2) ( ) = 0 which means the extra cost of activity (i, j) is 0 under the original mean duration and SD; and (b) c (1) ( ) is a decreasing function of ij and likewise c (2) ( ) is a decreasing function of ij . Note that it is possible in this formulation to just crash the means by forcing the SD to be fixed by setting = in the outer optimization problem.…”

“…In our setting,  is a network flow polytope and the projection can be computed by solving a least square problem that is efficiently solvable with standard convex quadratic programming solvers. Similarly, if we use a budgeted uncertainty set for and as given in Equation (3.4) where c (1) (⋅) and c (2) (⋅) are univariate piecewise linear or quadratic convex functions, then the projection on the set is computed by solving a convex quadratic program. For more general closed convex sets , the complexity of the projection operation would depend on the representation of the set.…”

“…where a (1) , a (2) , b (1) and b (2) are given nonnegative real numbers. These cost functions are chosen so that: (a) the costs are 0 under the normal mean and SD denoted by and ; and (b) the costs are convex decreasing functions of ij and i, j respectively.…”

“…We observe that when the budget is low, the deterministic model decreases the means of arcs (1,3) and (3,4) since they have been mistakenly identified as the critical arcs. On the other hand, the robust models recognize the importance of arc (1,2) and crash its mean. When the budget is abundant, we observe that the crashing decisions obtained by the CMM model is very close to those from SAA, while the deterministic solution does not have the flexibility to crash the arcs between nodes 2 and 4 appropriately.…”

“…1 For every activity, the means and the SDs of the original activity duration are generated by uniform distributions ∼ U(10, 20), ∼ U (6, 10), and the minimal values of mean and SD that can be obtained by crashing are ∼ U(5, 10), ∼ U(2, 6). The coefficients in the cost function (5.2) are chosen as follows a (1) ∼ U(1, 2), b (1) ∼ U(0, 1), a (1) ∼ U(1, 2) and b (2) ∼ U(0, 1). The amount of the cost budget is chosen as 1 4 ∑ (i,j) (c (1) ( ) + c (2) ( )).…”