Question

AZRB is a road bridge construction company. The company plan to build a major road bridge for its new project. Table shows the process sequence and other related data for the major road bridge development.

Table :Activities to construct the bridge, estimated time and crashing cost.

Activity	Predecessor	Time Estimates (Days)			Cost (RM per day)	Crash cost (RM per day)
		Optimistic	Most Likely	Pessimistic			Maximum crash time (days)
		a	m	b
A	-	65	73	90	100	150	6
B	A	20	27	40	80	120	6
C	A	46	50	66	110	165	10
D	B, C	33	40	50	80	120	4
E	B	15	22	35	90	135	5
F	D, E	50	71	80	130	195	3
G	D	25	40	85	100	150	5
H	F,G	15	30	45	90	145	7

1. Develop a network diagram for the above project. Determine the critical path and minimum project completion time.
2. Calculate the probability that the project can be completed in between 250 and 270 days.                                                                                                                        
3. If the project has to be shortened by five (5) days, determine which activity needs to be crashed and the additional cost involved. Cost per day as well as crash cost per day were based on expected time.

Answer 1

a.

Below are the calculations and network diagram.

We will use below approach for calculating critical path and completion time:

Earliest Start Time (ES) = the earliest time that an activity can begin

Earliest Finish Time (EF) = the earliest time that an activity can be completed

EF = ES + duration of activity        

Latest Start Time (LS) = the latest time that an activity can begin without lengthening the minimum project duration

Latest Finish Time (LF) = the earliest time than an activity can be completed without lengthening the minimum project duration

LF = LS + duration of activity   

Duration of the project = the difference between the maximum value of the "latest finish time" of projects and the minimum value of the "earliest start time" of projects

Project duration = max(LF) - min(ES)     

Slack = the amount of time that a project can be delayed without increasing the duration of the project

Slack = LF- LS or EF - ES

Critical path is the path with Total Slack = 0

Updated calculations are below.

b.

Probability that the project can be completed in between 250 and 270 days, we will calculate z statistic for the same

z= (X- mu)/ sigma

where mu is expected completion time

sigma is standard deviation of critical path

X = 250, 270

For X=250,

z= (X- mu)/ sigma = (250-266)/ 9.30 = -1.72

Looking at z table given below, we will find probability corresponding to z value of -1.72. Look at negative z value table on left side and value for row -1.7 and column 0.02. Probability value is 0.0427

For X=270,

z= (X- mu)/ sigma = (270-266)/ 9.30 = 0.43

Looking at z table given below, we will find probability corresponding to z value of 0.43. Look at positive z value table on right side and value for row 0.4 and column 0.03. Probability value is 0.6664.

Probability that the project can be completed in between 250 and 270 days,

P (-1.72 < Z < 0.43) = 0.6664 - 0.0427 =0.6237

Hence, probability that the project can be completed in between 250 and 270 days is 62.37%

c.

Possible paths in the network with expected time are:

• A-B-E-F-H: 74.50+28+23+69+30 = 224.50
• A-B-D-F-H: 74.50+28+40.5+69+30= 242
• A-C-D-F-H: 74.50+52+40.5+69+30= 266
• A-C-D-G-H: 74.50+52+40.5+45+30= 242

Now, in order to shorten project by 5 days we will crash activities on critical path with lowest crash cost per day.

Activity D has lowest crash of RM 120 per day and maximum crash time for D is 4 days. Hence, we will first crash activity D will be crashed by 4 days and crashing cost of RM 120*4 =480.

Next activity on critical path with lowest crash cost per day is activity H with crashing cost of RM 145 per day. So, we will crash activity H by one day and crashing cost of 480+145= RM 625.

Hence, activity D by 4 days and activity H by one day will be crashed to shorten project by 5 days. Total crashing cost = RM 625.

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