SAMPLE PROBLEM 3:

The table below shows the activities of a project and the durations of these activities. Some activities can be crashed and some can even be double crashed. Costs shown are in TL. Times shown are in days.

Assume that by allocating the proper amount of resources (costs) to an activity, an activity can be completed in any duration between its Normal Time and its fully crashed time.

ACTIVITY	Prerequisites	NORMAL		CRASH 1		CRASH 2
		Time	Cost	Time	Cost	Time	Cost
Q C J G A EE M B P X K L	C - C J,Q J M,A Q J,Q M P,B,G,EE EE G	3 4 5 6 7 1 6.5 5 3 5 2 3	8,000 4,500 6,000 6,000 10,000 3,000 7,250 5,000 3,500 15,000 1,000 5,500	2.5 3 4 5 5 - 4 - - 4 1 2	8,800 5,700 8,000 6,600 11,500 - 8,275 - - 18,000 1,200 6,000	- - - - 3 - 3 - - 3.5 - -	- - - - 15,100 - 9,000 - - 23,000 - -

The contractor company is expected to incur an overhead cost of 1500 YTL/day until Activity A is completed, and an overhead cost of 850 TL/day thereafter until the end of the project.

Also assume that the contractor has promised to complete this project in 19 days. If they can not finish the project in 19 days, they will incur a penalty charge of 1000 TL/day for every day they re late. But similarly, if they can complete the project faster than in 19 days, the cient will pay them a reward of 950 TL/day for every day they are earlier than promised.

1. Draw the Activities-On-Arcs network diagram.

2. Assuming no activities are crashed, calculate the project completion time. Show all your work (ES and LS times for nodes) in detail. Identify the critical path(s). Calculate the cost of the project in this case (without any crashing).

3. Now, assume all activities are fully crashed (i.e., crashed as much as possible), and calculate the project completion time. Show all your work (ES and LS times for nodes) in detail on a SEPARATE network diagram. Identify the critical path(s). Calculate the cost of the project in this case (with full crashing).

4. Now, formulate the optimal crashing strategy problem as a Linear Programming problem. In particular,

• List and define every decision variable precisely,
• Identify all your activity duration constraints and network constraints, and
• Identify the objective function to be minimized.