Question

18. Office Automation, Inc., developed a proposal for introducing a new computerized office system that will standardize the electronic archiving of invoices for a particular company. Contained in the proposal is a list of activities that must be accomplished to complete the new office system project. Use the following relevant information about the activities Crash Cost ($1000s) Activity Description Immediate NormalCrash Normal Predecessor Time TimeCost (weeks) (weeks ($1000s) 10 70 150 160 50 75 Plan needs Order equipment Install equipment Set up training lab Conduct training course Test system 30 120 100 40 50 6 10 6 10 60 a) b) Identify the critical path, and what is the expected project completion time? Assume that the company wants to complete the project in six months or 26 weeks, use the trial and error method to recommend crashing decisions that will meet the desired completion time at the least possible cost. What will be the total crash cost? And what will be the total cost for the project? Formulate the crash problem as a Linear Programming problem. Do not solve the LP problem c)

Answer 1

(a)

10 10 10 Project paths A-B-C-F A-D-E-F Duration 31 30

The critical path is the longest path i.e. A-B-C-F and the expected duration of the critical path (and that of the project) is 31 weeks.

(b)

Activity	Normal Time	Crash Time	Normal Cost	Crash Cost	Cost slope
A	10	8	30	70	20
B	8	6	120	150	15
C	10	7	100	160	20
D	7	6	40	50	10
E	10	8	50	75	12.5
F	3	3	60	-	-

Cost Slope = (Crash cost - Normal cost) / (Normal time - Crash time)

Step-Crash 'B' by 1 week 10 7 10 10 $400,000 $15,000 Normal cost Crash cost Total cost 30 30 A-B-C-F A-D-E-F $415,000 Project duration30

Step-2 Crash 'A' by 1 week 7 10 10 Normal cost $400,000 Crash Total cost $435,000 29 29 A-B-C-F A-D-E-F cost $35,000 Project duration 29

Step-3 Crash 'A' by 1 week 10 10 10 $400,000 $55,000 Normal cost Crash cost Total cost 28 28 A-B-C-F $455,000 A-D-E-F Project duration28

Step-4 Crash B' and D' each by 1 week 10 10 Project paths A-B-C-F A-D-E-F Duration 27 27 Normal cost $400,000 Crash Total cost $480,000 cost $80,000 Project duration 27

Final step Crash 'C' and 'E' each by 1 week 10 Normal cost $400,000 Crash cost $112,500 Total cost $512,500 26 26 A-B-C-F A-D-E-F Project duration 26

(b)

Let Tj be the start time of the activity-j and Cj be the crash weeks fro activity-j for j=A, B,..., F

Minimize Z = total cost = 400,000 + 20,000 CA + 15,000 CB + 20,000 CC + 10,000 CD + 12,500 CE

Subject to,

Precedence constraints:

TB - TA + CA >= 10
TD - TA + CA >= 10
TC - TB + CB >= 8
TE - TD + CD >= 7
TF - TC + CC >= 10
TF - TE + CE >= 10

Crash limit constraints:

CA <= 2
CB <= 2
CC <= 3
CD <= 1
CE <= 2
CF <= 0

Project completion constraint:

TF + 3 - CF <= 26 i.e. TF - CF <= 23

Tj, Cj >= 0 for j=A, B, ..., F