Design

#### Project Management

Critical Path

Delays

 Project Management - Delays

Critical path analysis provides only two specific schedules, the early schedule and the late schedule. For the early schedule each activity starts at the early start time determined by the analysis. For the late schedule, each activity starts at the late start time.

The early schedule is shown below. The early start through the late finish columns are calculated with critical path analysis. The scheduled start and scheduled finish columns provide the start and finish times associated with a specific schedule. The early start and the scheduled start columns are the same indicating that each activity starts at its earliest possible time. The activities with the minimum slack values are critical. They are marked with the red bars in the critical column.

The late schedule is shown below. The scheduled finish column is the same as the late finish column. Each activity finishes at its latest possible time in order to complete the project by its due time (59). Review of the critical column shows that every activity is critical. When an activity finishes at the latest possible time, any increase in the activity time will cause the project duration to increase beyond the due time.

Activity Delay

To provide schedules different than the early and late schedules we define activity planned delay as the difference between the scheduled start time and the earliest start time for the activity. To simplify the discussion we use delay to refer to planned delay. Again we use node as synonym for activity in the following.

Although we use the same notation for early and late times as for critical path analysis, we use different equations to compute these quantities as shown below. In addition to its usefulness for defining schedules the planned delay for an activity has relevance for practical planning.

Early Start

The early start time for a node is the time when all its predecessors have finished.

Early Start Time = Max(Early Finish Time for all predecessors)

Including delay, the formulas for early start, early finish, scheduled start and scheduled finish times are below. We include a delay for node 1 representing a delay in the start of the project. The finish time for node n, the end node, must be less than or equal to the due time.

We evaluate these formulas in order of increasing node index.

We illustrate the formulas using columns from the project worksheet. We show only columns that are directly relevant to this calculation. Recall that nodes that have no predecessors shown on the table are immediately preceded by the Start node. Nodes that have no successors shown in the table are immediate predecessors to the End node. The due time is 59 in this analysis. We have arbitrarily specified delay for some of the activities.

Some of the steps of the process of solving the equations are shown in the table. Of the nodes shown, nodes 9 and 16 involve two predecessors. Except for the start node, the others have only one.

 Node Name 1 Start ES(1)=0 EF(1)=ES(1)+d(1)+t(1)=0 2 A ES(2)=EF(1)=0 EF(2)=ES(2)+d(2)+t(2) EF(2)=0+1+12=13 3 B ES(3)=EF(1)=0 EF(3)=ES(3)+d(3)+t(3) EF(3)=0+15+1.75=15.25 4 C ES(4)=EF(1)=0 EF(4)=ES(4)+d(4)+t(4) EF(4)=0+0+30=30 5 D ES(5)=EF(2)=13 EF(5)=ES(5)+d(5)+t(5) EF(5)=13+0+24=37 6 E ES(6)=EF(3)=15.25 EF(6)=ES(6)+d(6)+t(6) EF(6)=15.25+12+8=35.25 7 F ES(7)=EF(4)=30 EF(7)=ES(7)+d(7)+t(7) EF(7)=30+11+11.33=52.33 8 G ES(8)=EF(4)=30 EF(8)=ES(8)+d(8)+t(8)=40 EF(8)=30+10+10=50 9 H ES(9)=Max(EF(5),EF(6)) ES(9)=Max(37,35.25)=37 EF(9)=ES(9)+d(9)+t(9) EF(9)==37+0+6=43 ... ... 16 End ES(16)=Max(EF(14),EF(15)) ES(16)=Max(59,55.52)=59 EF(16)=ES(16)+d(16)+t(16) EF(16)=59+0+0=59

Latest Finish

The latest finish time for a node is based on the logical requirement that a node must end before any of its successors may be begin.

Latest Finish Time = Min(Latest Start Time for all successors)

The latest start time is the latest finish time, less the delay time and less the activity time for the node.

We describe these relationships with mathematical notation below.

The latest finish times and latest start times are computed in the reverse order of the node indices. First we set the latest finish time for the End node to the due time for the project. This must be no less than the earliest finish time for the project. The latest finish time for a node must be the minimum of all the latest start times of all successor nodes. We can compute the latest finish time for a node because the latest start times for all successor nodes have already been computed. We illustrate the formulas using columns from the project worksheet. Recall that although the form does not show it, the End node has as predecessors all nodes with no successors. In the example, the predecessors of the End node are M and N. For the example, the due time is 59.

Some of the steps of the process are shown in the table.

 Node Name 16 End LF(16)=59 LS(16)=LF(16)-d(16)-t(16) LS(16)=59-0-0=59 15 N LF(15)=LS(16)=59 LS(15)=LF(15)-d(15)-t(15) LS(15)=59-0-1.02=57.98 14 M LF(14)=LS(16)=59 LS(14)=LF(14)-d(14)-t(14) LS(14)=59-0-4.5=54.5 13 L LF(13)=Min(LS(14),LS(15)) LF(13)=Min(54.5,57.98)=54.5 LS(13)=LF(13)-d(13)- t(13) LS(13)=54.5-0-1.5=53 12 K LF(12)=Min(LS(14),LS(15)) LF(12)=Min(54.5,57.98)=54.5 LS(12)=LF(12)-d(12)- t(12) LS(12)=54.5-1-1.5=52 11 J LF(11)=LS(13)=53 LS(11)=LF(11)-d(11)- t(11) LS(11)=53-1-9=43 ... ... 1 Start LF(1)=Min(LS(2),LS(3),LS(4)) LF(16)=Min(0,1.75,0.67)=0 LS(1)=LF(1)-d(1)- t(1) LS(1)=0-0-0=0

Slack Time

The slack time is the difference between late and early start times. This is the same definition as for critical path analysis, but the calculation of early and late start times now depends on delay.

The figure below shows the results shown previously and the slack column.

The slack value is the increase the amount of time an activity can be delayed without delaying the completion of the project. When the early start time and late start time are equal, the activity cannot be delayed, so this activity is said to be critical. Activities with positive slack are not critical because they can be delayed without affecting the project duration.

It should be emphasized that the slack of an activity depends on the delays of other activities in the project. We can see this by showing the slack and delay columns for several selections of delays. At the left we see the slacks with all 0 delays, the early start schedule. When we increase the delay for J, the slacks for all the activities on the critical path are reduced. Part of the project slack has been used up by the planned delay. Now, increasing the delay of D to 1 further reduces the slack along the critical path to 0. In the last figure we delay I by 4. This activity is not on the critical path, but the slack for both I and K are reduced since I is a predecessor of K.

Planned Delays

Planned delays have importance for scheduling studies because every possible schedule can be derived by some selection of delay values. We will search for an optimal schedule by varying the values of the activity delays.

Delays also have importance in practice. Since scheduling is done without exact knowledge of how long activities will take, a schedule with no delays is almost sure to fail. A planned delay allows time between when the activity may start and when it is scheduled. Thus if some predecessor activities take longer than estimated, the delay can simply be reduced and the activity can start at the scheduled time. Delays serve the purpose of decoupling the effects of variability in much the same way that inventory protects a production system from the variability of demand.

Operations Management / Industrial Engineering
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by Paul A. Jensen