Bc230202794

Mth601 Assignment 1

Question 1:
Using the data, draw the project network diagram showing all events and activities,
calculate the expected time and variance for each activity based on the time estimates, and
then determine the expected project completion time while also identifying the critical path
of the project.

Solution:

The expected time (te) and variance (σ2) are calculated using the formulas:

t o + 4 t m+ t p
t e=
6

( )
2
2 t p−t o
σ =
6

Activity to tm tp to σ
2

2
2+ 16+10 8
A 2 4 10 = 4.676 ( ) =1.78
6 6

2
3+20+9 6
B 3 5 9 =5.33 ( ) =1
6 6

2
1+ 12+ 5 4
C 1 3 5 = 3.00 ( ) = 0.44
6 6
 Activity to tm tp to σ
2

2
2+ 24+12 10
D 2 6 12 = 6.33 ( ) = 2.78
6 6

2
2+ 16+8 6
E 2 4 8 = 4.33 ( ) = 1.00
6 6

2
3+20+11 8
F 3 5 11 = 5.67 ( ) = 1.78
6 6

2
4+ 24+12 8
G 4 6 12 = 6.67 ( ) = 1.78
6 6

2
2+ 16+6 4
H 2 4 6 = 4.00 ( ) = 0.44
6 6

Path 1: 1 → 2 → 4 → 6 → 7: A → C → F → H
Duration: 4.67 + 3.00 + 5.67 + 4.00 = 17.34
Path 2: 1 → 2 → 5 → 6 → 7: A → E → G → H
Duration: 4.67 + 4.33 + 6.67 + 4.00 = 19.67

Path 3: 1 → 3 → 4 → 6 → 7: B → D → F → H
Duration: 5.33 + 6.33 + 5.67 + 4.00 = 21.33

As we know that longest path is the critical path. So,

Path 3: 1 → 3 → 4 → 6 → 7 is longest path with activities B → D → F → H and duration of
21.33.

Question 2:
A company has a constant demand of 24,000 units per year for one of its products. The cost
of placing each order is Rs. 200, and the holding cost per item is Rs. 2 per month. Assuming
there are no stockouts and that items are replenished instantly upon ordering, calculate the
Economic Order Quantity (EOQ), determine the time between two consecutive orders in
months, and find the number of orders the company should place annually to manage
inventory efficiently.

Solution:

Given Data:

 Annual demand (D) = 24,000 units

 Ordering cost (S) = Rs.200 per order
 Holding cost per item per month = Rs.2
 So, Annual holding cost per item (H) = 2×12=Rs.24 per unit per year.

To Find:

 Economic Order Quantity (EOQ)

 Number of orders
  The time between two consecutive orders in months

Step 1: Calculate EOQ

√ √
EOQ = 2 DS = 2 ×24000 × 200 = √ 400000 = 632.46
H 24

So, EOQ ≈ 632 units

Step 2: Calculate Number of Orders per Year

Number of Orders = EOQ = 632.46 ≈ 37.94 ≈ 38 orders/year

D 24000

Step 3: Time Between Two Orders (in Months)

12 12
Time between orders= = ≈ 0.316 months ≈ 9.5 days
Number of orders 38

Final Answers:

 EOQ ≈ 632 units

 Number of orders per year ≈ 38
 Time between orders ≈ 0.316 months or ~9.5 days

Question 3:

A company has a yearly demand of 15,000 units for a particular product. The setup cost
for each order is Rs. 500, and the holding cost is Rs. 2.5 per unit per year. The shortage cost
is Rs. 10 per unit per year, and the production rate is 30,000 units annually. Based on this
information, calculate the Economic Order Quantity and determine the Maximum
Inventory Level to help the company manage its inventory effectively.

Solution:

Given:

 Demand (D) = 15,000 units/year

 Setup cost (S) = Rs. 500 per order
 Holding cost (H) = Rs. 2.5 per unit/year
 Shortage cost (π) = Rs. 10 per unit/year
 Production rate (P) = 30,000 units/year

To Find:

 Economic Order Quantity (EOQ)

 Maximum Inventory Level

Step 1: EOQ with Shortages (EPQ Model with shortages)

√( ( )
2 DS π
EOQ= ×
H 1−
D
P ) π+H

√( ( )
2 ×15000 ×500 10
EOQ= ×
2.5 1−
15000
30000 )
10+ 2.5

EOQ=
√ 15000000
1.25
× 0.8

EOQ=√ 960000

EOQ=3098.39
So, EOQ ≈ 3098 units

Step 2: Maximum Inventory Level

Max Inventory=EOQ × 1− ( D
P)(
×
H
H +π )

Max Inventory=3098.39 × 1− ( 15000

30000)(
×
10
2.5+ 10 )
Max Inventory=3098.39 ×0.5 × 0. 8

Max Inventory=1,239.356 units

So, Max Inventory ≈ 1,239.36 units

Final Answers:

 Economic Order Quantity (EOQ) ≈ 3098 units

 Maximum Inventory Level ≈ 1,239.36 units