EXERCISES
1.Explain the terminologies of linear programming model.
2.List and explain the assumptions of linear programming problems.
3.Define the following:
(a) Alternate optimum solution
(b) Unbounded solution
(c) Infeasible solution
(d) Degenerate solution
(e) Slack variable
(f) Surplus variable
(g) Artificial variable
(h) Basic variable
(i) Criterion value.
4.A factory is engaged in manufacturing two products A and B which involve lathe work,
grinding and assembling. The cutting, grinding and assembling times required for one unit of
A are 2, 1 and 1 hours respectively. The times required for one unit of B are 3, 1 and 3 hours
respectively. The profits on each unit of A and B are Rs. 2 and Rs. 3 respectively. Assuming
that 300 hours of lathe time, 300 hours of grinding time and 240 hours of assembling time are
available, pose a linear programming problem in terms of maximizing the profit on the items
manufactured.
[Ans. Max z = 2 X1+ 3 X2 s.t. 2 X1 + 3 X2 < 300, X1 + X2 < 300,X1+ 3 X2<240, X1, X2>0]
5. Upon completing the construction of his house, Mr. Sharma found out that 100 sq. ft. of
plywood scrap and 80 sq. ft. of white pine scrap were in a usable form for the construction of
tables and book-cases. It takes 16 sq. ft. of plywood and 8 sq. ft. of white pine to make a table, 12
sq. ft. of plywood and 16 sq. ft. of white pine are required to construct a book-case. By
selling the finished products to a local furniture store, Mr. Sharma can realize a profit of
Rs. 25 on each table and Rs. 20 on each book-case. How may he most profitably use the
leftover wood? Use graphical method to solve the problem.
[Ans. 4 tables, 3 book-cases; Max profit Rs. 160]
6. Food X contains 6 units of vitamin A and 7 units of vitamin B per gram and costs 12 paise per
gram. Food Y contains 8 units and 12 units of A and B per gram respectively and costs 20 paise
per gram. The daily requirements of vitamin A and B are at least 100 units and 120 units
respectively. Formulate the above as a linear programming problem. Use graphical method to
solve the problem.
[Ans. Min z = 0.12 X1 + 0.20 X2 s.t. 6 X1 + 8 X2 > 100, 7 X1 + 12 X2> 120, X1, X2> 0 ]
7. At a cattle breeding farm it is prescribed that the food ration for one animal must contain at
least 14, 22 and 1 units of nutrients/I, B and C respectively. Two different kinds of fodder are
available. Each unit weight of these two contains the following three nutrients.
Nutrient A
Nutrient B
Nutrient C
Fodder I
1
2
2
Fodder 2
1
3
1
It is given that the costs of unit quantity of fodders 1 and 2 are Rs. 3 and Rs. 2 respectively.
Pose a linear programming problem. [Ans. Min z = 3 X1, + 2 X2 s.t. 2 X1 + X2 > 14, 2 X1 + 3 X2 >
22,
X1 + X2> l; X1 >0, X2 >0]
Mr. X requires 10, 12 and 12 units of chemicals A, B and C for his garden. One jar of liquid
product contains 5, 2 and 1 units of A, B and C respectively. A dry product contains 1, 2 and
4 units of A, B and C per carton. If the liquid product sells for Rs. 3 per jar and the dry product
sells for Rs. 2 per carton, how many of each should he purchase to minimize the cost and meet
the requirements? Use graphical method to solve the problem.
[Ans. Min z = 3 X1 + 2 X2 s.t. 5 X1 + X2 > 10, 2 X1 + 2x2 > 12,X1 + 4 X2> 12; X1 >0, X2>0]
8.A firm manufactures two different products A and B. The selling price of one unit of A is Rs. 40
and that of B is Rs. 30. Both A and B need for their production three different ingredients C, D
and E. In order to produce one unit of A, the firm must use one unit of C, one unit of D and three
units of E. On the other hand, one unit of B needs for its production one unit of C, two units of D
and two units of E. The maximum amounts of C, D and E that can be made available are 50 units
of C, 80 units of D and 100 units of £ respectively. The firm wants to determine the quantities of
A and B to be produced in order to maximize the sale revenue. Formulate the above problem as a
linear programming problem. Use graphical method to solve the problem.
[Ans. Max z = 40 X1 + 30 X2 s.t. X1 + X2 < 50, X1 + 2 X2 < 80,3 X1 + 2 X2 < 100; X1 > 0, X2 > 0]
9. A company produces two types of leather belts, say, type A and type B. Belt A is of superior
quality and belt B is of inferior quality. Profits on the two types of belts are Rs. 4 and Rs. 3 per
belt respectively. Each belt of type A requires twice as much time as that for a belt of type B.
If all belts were of type B, the company could produce 1000 belts per day. But the supply of
leather is sufficient only for 800 belts per day. Belt A requires a fancy buckle and only 400
buckles are available for this per day. For belt of type B only 700 buckles are available per
day. How should the company manufacture the two types of belts so that the profit is
maximum? Use graphical method to solve the problem.
[Ans. Max z = 4 X1 + 3 X2 s.t. 2 X1 + X2 < 1000, X1 + X2 < 800, X1 < 400, X2 < 700; X1 > 0, X2> 0]
10. Draw the feasible space of the following inequations:
(i) 0 < X1 < 2, 0 < x2 < 3, X1 + x2 > 1
(ii) 3 X1 + 5x2 < 10, 5 X1 + 3x2 < 15, X1 > 0, x2 > 0
.(iii) X1 + 2x2 < 7, x, - x2 < 4, x, > 0, x2 > 0.
(iv) 3 X1 + 6x2 < 8, 5 X1 + x2 < 10, X1 > 0, x2 > 0.
11. Solve the following LP problem graphically:
Maximize Z = 20X1 + 80X2 subject to
4 X1+ 6X2 <=90
8X1 + 6X2 <=100
5X1 + 4X2 <=80
X1 and X2 >= 0
11. Solve the following LP problem graphically:
Minimize Z = 20 X1 + 10 X2
subject to
X1+ 2X2 < 40
3 X1 +X2 > 30
4 X1 + 3X2 > 60
X1 and X2 > 0
11. Solve the following LP problem graphically:
Maximize Z = 60 X1 + 90X2
subject to
X1 + 2X2 < 40
2 X1 + 3X2 < 90
X1 - X2 > 10
X1 and X2 > 0
12. Solve the following LP problem graphically:
Minimize Z = 45 X1 + 55X2
subject to
X1 + 2X2 < 30
2Xt + 3X2 < 80
X1-X2 > 8
X1 and X2 > 0
13. Solve the following LP problem graphically:
Maximize Z =3 X1+ 2X2 subject to
-2 X1 +3 X2 < 9
X1 - 5X2 > -20
X1 and X2 > = 0
Ali
Mansuri