ASSIGNMENT – 13 and 14
REVIEW
Problem – 1
An agricultural mill manufacture feed for cattle, sheep and chickens. This is done by
mixing the following main ingredients: corn, limestone, soybeans and fish meal. These
ingredients contain the following nutrients: vitamins, protein, calcium, and crude fat.
The contents of the nutrients in each kilogram of the ingredients are summarized in
the table.
Ingredients
Corn
Limestone
Soybeans
Fish meal
Vitamins
8
6
10
4
Nutrient
Proteins
Calcium
10
6
5
10
12
6
8
6
Crude Fat
8
6
6
9
The mill is contracted to produce 10, 6 and 8 (metric) tons of cattle feed, sheep feed,
and chicken feed. Because of the shortage, a limited amount of the ingredients is
available – namely, 6 tons of corn, 10 tons of limestone, 4 tons of soybeans, and 5 tons
of fish meal. The price per kilogram of these ingredients is respectively $0,20, $0,12,
$0,24 and $0,12. The minimal and maximal units of the various nutrients that are
permitted is summarized for a kilogram of the cattle feed, the sheep feed and the
chicken feed.
Problem – 2
A steel manufacturer produces four sizes of 1 beams: small, medium, large and extra
large. These beams can be produced on any one of three machine type: A, B and C.
The length in feet of the I beams that can be produced on the machines per hour are
summarized
Beam
Small
Medium
Large
Extra Large
A
300
250
200
100
Machine
B
600
400
350
200
C
800
700
600
300
Assume that each machine can be used up to 50hours per week and that the hourly
operating costs of these machines are respectively $30, $50 and $80. Further suppose
that 10.000, 8.000, 6.000 and 6,000 feet of the different size I beams are required
weekly. Formulate the machine scheduling problem as a linear programming
Problem – 3
Consider the following problem :
Maximize Z = 2X1 + 3X2
Subject to
X1 + X2 ≤ 2
4X1 + 6X2 ≤ 9
X1, X22 ≥ 0
a. Sketch the feasible region
b. Find two alternative optimal extreme (corner) points
c. Find a infinite class of optimal solution
Problem – 4
Consider the following problem :
Maximize Z = 2X1 + 3X2
Subject to
X1 + X2 ≤ 2
4X1 + 6X2 ≤ 9
X1, X22 ≥ 0
Find these two variables by using simplex method
Problem – 5
Consider the following problem :
Maximize Z = -X1 + 2X2
Subject to
3X1 + 4X2 = 12
2X1 - X2 ≤ 12
X 1, X 2 ≥ 0
Find these two variables by using Big-M method
Problem – 6
Consider the following problem :
Maximize Z = -X1 + 2X2
Subject to
3X1 + 4X2 = 12
2X1 - X2 ≤ 12
X 1, X 2 ≥ 0
Find these two variables by using 2-Phase method
Problem – 7
Solve the following problem by using Duality Theory
PRIMAL : Minimize Z = X1 + 4X2
– X4
4
Subject to -X1 + 2X2 – X3 + X4 = 2
2X1 + X2 – 2X3 - X4 ≤ 4
X1 – X3
+ X4 ≥ 2
X1, X2, X4 ≥ 0
X3 unrestricted
Problem – 8
Consider the following linear program
PRIMAL : Minimize Z = X1 + 5X2 + 3X3
Subject to
X1 + 2X2 + X3 = 3
2X1 - X2
=4
X1, X2, X3 ≥ 0
a) Write the associated dual problem
b) For each following objective function, find the optimum solution by using the
sensitivity analysis procedure :
a. Maximize Z = 2X1 + X2 + 4X3
b. Minimize Z = X1 - 2X2 + X3
Problem – 9
Find the starting solution in the following transportation problem by
a) Northwest-corner method
b) The least cost method
c) Vogel’s approximation method
Obtain the optimal solution by using the best starting solution
10
13
4
14
3
60
20
9
15
7
12
60
5
12
7
1
5
20
7
8
9
0
19
10
10
20
30
40
50
Problem – 10
Find the starting solution in the following transportation problem by
a) Northwest-corner method
b) The least cost method
c) Vogel’s approximation method
Obtain the optimal solution by using the best starting solution
5
3
7
9
5
1
2
5
6
10
0
4
2
0
15
20
10
15
15