Assignment
1. Consider the project outlined in the table below.
|
Activity
|
Immediate Predecessor
|
Present Duration
|
Accelerated Duration
|
Present Cost
|
Accelerated Cost
|
|
A
|
-
|
10
|
9
|
11
|
15
|
|
B
|
-
|
15
|
13
|
20
|
25
|
|
C
|
A
|
10
|
6
|
9
|
20
|
|
D
|
A
|
20
|
18
|
25
|
30
|
|
E
|
C
|
15
|
10
|
20
|
35
|
|
F
|
B
|
17
|
15
|
20
|
30
|
|
G
|
B
|
12
|
10
|
15
|
25
|
|
H
|
D, F
|
9
|
8
|
12
|
18
|
|
I
|
G, H
|
7
|
6
|
10
|
15
|
a. Construct an activity network diagram
b. Disregarding the acceleration information, how long is the project expected to take to complete?
c. If the project duration was to be reduced by at least one unit of time what activity (or activities) would you crash?
d. Detail the steps you would take to crash the project to its minimum duration at the lowest cost
2. A mobile phone company services six geographical areas. The satellite distances in thousands of miles are given in the figure below. The company needs to determine the best quality message routes that should be established between each two areas in the network. Explain whether the company is facing shortest path problem or not.
3. School of Mathematical and Physical Sciences has three professors who each teach four courses per year. Each year, four sections of Algebra, OR, and Modelling must be offered. At least one section of each class must be offered during each semester (Semester 1 and 2). Each professor's time preference and preference for teaching various courses are given in the following table.
|
|
Professor 1
|
Professor 2
|
Professor 3
|
|
Semester 1 Preference
|
3
|
5
|
4
|
|
Semester 2 Preference
|
4
|
3
|
4
|
|
Algebra
|
6
|
4
|
5
|
|
OR
|
5
|
6
|
4
|
|
Modelling
|
4
|
5
|
6
|
The total satisfaction a professor earns teaching a class is the sum of the semester satisfaction. Thus, professor 1 derives a sati fiction of 3 + 6 =9 from teaching algebra during semester 1. Formulate an MCNFP that can be used to assign professors to courses so as to maximize the total satisfaction f the three professors.
4. During the next two months, Machineco must meet (on time) the demands for three types of products shown in the Table 1. Two machines are available to produce these products. Machine 1 can only produce products 1 and 2, and machine 2 can only produce products 2 and 3. Each machine can be used for up to 40 hours per month. Table 2 shows the time required to produce one unit of each product (independent of the type of machine); the cost of producing one unit of each product on each type of machine; and the cost of holding one unit of reach product in inventory for one month. Formulate an MCNFP that could be used to minimize the total cost for meeting all demands on time.
Table 1
|
Month
|
Product 1
|
Product 2
|
Product 3
|
|
1
|
50 units
|
70 units
|
80 units
|
|
2
|
60 units
|
90 units
|
120 units
|
Table 2
|
Product
|
Production Time (minutes)
|
Production costs ($)
|
Holding costs ($)
|
|
Machine 1
|
Machine 2
|
|
1
|
30
|
40
|
-
|
15
|
|
2
|
20
|
45
|
60
|
10
|
|
3
|
15
|
-
|
55
|
5
|
5. Televco produces TV picture tubes at three plants. Plant 1 can produce 50 tubes per week; plant 2, 100 tubes per week; and plant 3, 50 tubes per week. Tubes are shipped to three customers. The profit earned per tube depends on the site where the tube was produced and on the customer who purchases the tube based on the following table. Customer 1 is willing to purchase as many as 80 tubes per week; customer 2, as many as 90; and customer 3, as many as 100. Televco wants to find a shipping and production plan that will maximize the profits.
a. Formulate a balanced transportation problem that can be used to maximize Televco's profit.
b. Use the northwest corner method to find a bfs to the problem.
c. Use the transportation simplex to find an optimal to the problem.
d. Formulate the problem as a linear programming problem and solve them via Software. Compare the Software solution with your finding in (c).
|
From
|
To
|
|
Customer 1
|
Customer 2
|
Customer 3
|
|
Plant 1
|
75
|
60
|
69
|
|
Plant 2
|
79
|
73
|
68
|
|
Plant 3
|
85
|
76
|
70
|