Part 3

CPM provided the project manager with the option of adding resources to selected activities to reduce project completion time as earliest as possible. However, added resources (such as more workers, overtime, and so on) generally increase project costs, so the decision to reduce activity times must take into consideration the additional cost involved. In effect, this construction manager must take a decision that involves trading reduced activity time for additional project cost.

Because, current completion time within 270 days and only less 75% sure about to completing. We can shorten selected activity times. This shortening of activity times, which usually can be achieved by adding resources, is referred to as crashing. So, we will want to identify the activities that cost the least to crash. Then crash those activities only the amount necessary to meet desired project completion time.

We could use linear programming to solve the network crashing problem. With CPM, we know that when an activity start at its earliest start time. In this case, we may have

Finish time > Earliest start time + Activity time

Because we do not know ahead of time whether an activity will start at its earliest start time, we use the following

Finish time ≥ Earliest start time + Activity time

Consider activity A, which has expected time of 30 days. Let FA finish time for activity, A, and CA amount of time activity A is crashed. If we assume that the project begins at time 0, the earliest start time for activity A is 0. Because the time for activity A is reduced by the amount of time that activity A is crashed, the finish time for activity A must satisfy the relationship

FA ≥ 0 + (30 – CA)

Moving CA to the left side,

FA +CA ≥ 30

In general, Let we define:

Ci = amount of crash time used for activity i

Fi = earliest finish time for activity i

If we follow the same approach that we used for activity A, the constraint corresponding to the finish time for activity C is

FC + CC - FA ≥ 65

As you notice, 2 final activity lead directly to the Finish node of a project network, We suggest creating an additional variable, FIN, which indicates the finish or completion time for the entire project. The fact that the project cannot be finished until both activities L and F are completed can be modeled by the two constraints

FIN ≥ FF

FIN ≥ FL

Moving to the left,

FIN - FF ≥ 0

FIN - FL ≥ 0

The constraint that the project must be finished by time T can be added as FIN ≤ T. In this case, the manager has proposed T equal to 240 and 250.

As with all linear programming, we add the usual non-negativity requirements for the decision variables.

The total project cost for normal completion time is fixed. We can minimize the total project cost by minimizing the total crashing costs. Thus, the linear programming objective function becomes

Min 1500 CA + 3500 CB + 4000 CC + 1900 CD + 9500 CE + 0 CF + 2500 CG + 2000 CH + 2000 CI + 6000 CJ + 0 CK + 4500 CL

Thus, to determine the optimal crashing for each of the activities, we must solve along with the constraints below.

Constraints RHSFA+CA ≥ 30FB+CB-FA ≥ 60

FC+CC-FA ≥ 65FD+CD-FC ≥ 55FE+CE-FB ≥ 30FF+CF-FE ≥ 1FG+CG-FE ≥ 30FG+CG-FD ≥ 30FH+CH-FG ≥ 20FI+CI-FH ≥ 30FJ+CJ-FH ≥ 10FK+CK-FJ ≥ 1FL+CL-FI ≥ 30FL+CL-FK ≥ 30FIN-FF ≥ 0FIN-FL ≥ 0FIN ≤ 240CA ≤ 10CB ≤ 40CC ≤ 15CD ≤ 25CE ≤ 5CF ≤ 0CG ≤ 5CH ≤ 10CI ≤ 10CJ ≤ 2CK ≤ 0CL ≤ 10

The linear programming solution from Excel Solver provides the optimal solution for finishing time within 240 days of crashing activity by crashing activity A by 10 days and activity D by 10 days. In this case study, Hill construction would have to shorten Bonding, insurance, tax structuring by 10 days. Besides that, getting more resources to shorten activity D, which upgrading walkways, stairwells and elevators by 10 days. The minimum crashing cost of completion within 240 days is $34000.

Activity Days CrashCrash cost

Amount($)

A 20.00 10 1500 15000B 60.00 0 3500 0C 65.00 0 4000 0D 45.00 10 1900 19000E 30.00 0 9500 0

F 1.00 0 0 0G 30.00 0 2500 0H 20.00 0 2000 0I 30.00 0 2000 0J 10.00 0 6000 0K 1.00 0 0 0L 30.00 0 4500 0

Total crash cost 34000

The linear programming solution from Excel Solver provides the optimal solution for finishing time within 240 days of crashing activity by crashing activity A by 10 days. With crashing cost of $15000.

Activity Days CrashCrash cost Amount

A 20.00 10 1500 15000B 60.00 0 3500 0C 65.00 0 4000 0D 55.00 0 1900 0E 30.00 0 9500 0F 1.00 0 0 0G 30.00 0 2500 0H 20.00 0 2000 0I 30.00 0 2000 0J 10.00 0 6000 0K 1.00 0 0 0L 30.00 0 4500 0

Total crash cost 15000