Preemptive expediting to improve project due date performance

Abstract

Project expediting is often viewed as a corrective action taken in response to prior scheduling errors and is usually applied in the later stages of projects when it appears that predefined due dates will not be met. However, large-scale projects with uncertain activity durations tend to have numerous probabilistic network paths with complex interactions that require some level of expediting to ensure successful scheduling outcomes. Because potential expediting options are consumed over time, delaying expediting efforts until the later stages of projects is likely to result in higher expediting costs or poorer due date performance. This research introduces a preemptive expediting approach that evaluates the probability of completion before the due date throughout the life of a project and selects expediting options per a prespecified probability tolerance or expediting budget. An experiment demonstrates the benefits of this preemptive approach.

Appendix

In each of the experiments the O ₀ matrix is compiled from a randomly generated adjacency matrix per the network type factor settings and the flow diagram in Figure A1 using the following definitions:

E _i :

=set of candidate predecessors for activity i

P _i :

=set of selected predecessors for activity i

f _ik :

∼U(0,1) for activity i in sample k

d _i :

=selection determinant for activity i

n _i :

=goal number of predecessors for activity i

p _ij :

=position of selected candidate for activity i in iteration j

N(i):

=number of activities

N(E _i ):

=number of elements in E _i

N(P _i ):

=number of elements in P _i

Figure A1

Random path generation procedure.

The goal number of predecessors for each activity is randomly determined by a selection determinant (d _i ) that is continually adjusted in order to generate a network based on the network type factor level (d ₀), as follows:

The purpose of Eq. (6) is to handle the competing goals of generating randomly interesting networks and allowing d _i to converge to d ₀. This is accomplished programmatically by defining different probabilities based on the value of d _i . For example, P(n _i <d ₀)=0.95 when d _i <d ₀ and P(n _i >d ₀)=0.95 when d _i >d ₀. Then the predecessors for each activity are randomly determined by iteratively selecting from the set of candidates (sorted in numerical order) as follows:

At each iteration the selected candidate is added to the set of selected predecessors (P _i ) and the set of candidate predecessors (E _i ) is reduced by the selected candidate and all predecessors and successors of the selected candidate per the flow diagram in Figure A1. After completing a number of practice runs with a range of network type factor levels (d ₀) to view the relationship between the factor levels and the resultant networks, levels of 1.05 (sparse) and 1.40 (dense) were chosen for this study because they covered a wide range of network structures.

For demonstration purposes, consider the five-activity adjacency matrix depicted in Table A1. This matrix, which shows the predecessors (In) as the rows and the successors (Out) as the columns for each activity, is initialized with all zeros per step 1 in Figure A1. Defining activity indices in order of precedence similar to Fulkerson (1962) requires only that the portion of the matrix below the diagonal be completed and results in an initial activity (the start event) having no predecessors. Then the process proceeds as follows for each activity:

• 1 E ₁={0}, d ₁=d ₀

  

  Candidate predecessor of activity 1 is activity 0 and the selection determinant (d ₁) is set to the network density factor level (d ₀). Activity 1 is randomly determined to have 1 predecessor and activity 0 in position 0 is selected from the set of candidates.

• 2 E ₂={0,1}, d ₂=2d ₀−1

  

  Candidate predecessors of activity 2 are equal to the set E ₂ and the selection determinant (d ₂) is updated per the previous period's results. Activity 2 is randomly determined to have 1 predecessor and activity 0 in position 0 is randomly selected from the set of candidates.

• 3 E ₃={0,1,2}, d ₃=2d ₀−1

  

  Candidate predecessors of activity 3 are equal to the set E ₃ and the selection determinant (d ₃) is updated per the previous errors. Activity 3 is randomly determined to have 2 predecessors. Activity 2 in position 2 is randomly selected from the set of candidates in the first iteration and activity 1 in position 0 is selected in the second iteration.

• 4
  

  Candidate predecessors of activity 4 are equal to the set E ₄ and the selection determinant (d ₄) is updated per the previous values. Activity 4 is randomly determined to have 2 predecessors. Activity 3 in position 3 is randomly selected from the set of candidates in the first iteration. The procedure terminates after this iteration because E ₄ becomes a null set.

• 5
  

  Candidate predecessors of activity 5 are equal to the set E ₅ and the selection determinant (d ₅) is updated per the previous values. Activity 5 is randomly determined to have 1 predecessor. Activity 2 in position 2 is randomly selected from the set of candidates.

Table 3 Example adjacency matrix

Tracing the predecessor values in the adjacency matrix results in paths 0-1-3-4, 0-2-3-4 and 0-2-5.