Linear Programming Test

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Linear Programming Test - 3

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Question 1
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In linear programming feasible region (or solution region) for the problem is

A
The common region determined by all the constraints including the non –negative constraints x ⩾ 0, y ⩾ 0

B
The common region determined by all the x ⩾ 0 and the objective function

C
The common region determined by all the objective functions including the non –negative constraints x ⩾ 0, y ⩾ 0

D
The common region determined by all the x ⩾ 0, y ⩾ 0 and the objective function

Solution

In linear programming feasible region (or solution region) for the problem is given by The common region determined by all the constraints including the non –negative constraints x ⩾0, y ⩾ 0
Question 2
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In a LPP if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same

A
Minimum value

B
Mean value

C
Maximum value

D
Upper limit value

Solution

In a LPP if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same maximum value .
Question 3
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A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman ’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman ’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman ’s time. What number of rackets and bats must be made if the factory is to work at full capacity?

A
5 tennis rackets and 15 cricket bats

B
4 tennis rackets and 12 cricket bats

C
4 tennis rackets and 14 cricket bats

D
3 tennis rackets and 13 cricket bats

Solution

Let number of rackets made = x
And number of bats made = y
Therefore , the above L.P.P. is given as :
Maximise , Z = x +y , subject to the constraints : 1.5x +3y ≤42 and. 3x +y ≤24, i.e.0.5x + y ≤14 i.e. x +2y ≤28 and 3x +y ≤24 , x, y ≥0.

Here Z = 16 is maximum. i.e Maximum number of rackets = 4 and number of bats = 12.
Question 4
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A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden? should the delivery be scheduled in order that the transportation cost is minimum?

A
45 bags of brand P and 105 bags of brand Q; Minimum amount of nitrogen = 480 kg.

B
40 bags of brand P and 100 bags of brand Q; Minimum amount of nitrogen = 470 kg.

C
50 bags of brand P and 100 bags of brand Q; Minimum amount of nitrogen = 490 kg.

D
47 bags of brand P and 107 bags of brand Q; Minimum amount of nitrogen = 499 kg.

Solution

Let the number of bags used for fertilizer of brand P = x And the number of bags used for fertilizer of brand Q = y . Here , Z = 3x + 3.5y subject to constraints : :1.5 x +2 y ≤310, x + 2y ≥240, 3x + 1.5y ≥270 , x,y ≥0

Here Z = 470 is minimum i.e. 40 bags of brand P and 100 bags of brand Q; Minimum amount of nitrogen = 470 kg.
Question 5
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In a LPP, the linear inequalities or restrictions on the variables are called

A
Inequalities

B
Linear constraints

C
Constraints

D
Limits

Solution

In a LPP, the linear inequalities or restrictions on the variables are called Linear constraints
Question 6
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In linear programming infeasible solutions

A
fall on the x = 0 plane

B
fall inside the feasible region

C
fall outside the feasible region

D
fall inside the a regular polygon

Solution

In linear programming infeasible solutions fall outside the feasible region .
Question 7
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In linear programming problems the function whose maxima or minima are to be found is called

A
Delta function

B
Selection function

C
Subjective function

D
Objective function

Solution

In linear programming problems the function whose maxima or minima are to be found is called Objective function .
Question 8
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A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman ’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman ’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman ’s time. If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity?

A
Maximum profit = Rs 220

B
Maximum profit = Rs 210

C
Maximum profit = Rs 200

D
Maximum profit = Rs 230

Solution

Let number of rackets made = x
And number of bats made = y
Therefore , the above L.P.P. is given as :
Maximise , Z = x +y , subject to the constraints : 1.5x +3y ≤42 and. 3x +y ≤24, i.e.0.5x + y ≤14 i.e. x +2y ≤28 and 3x +y ≤24 , x, y ≥0.

Here Z = 16 is maximum. i.e Maximum number of rackets = 4 and number of bats = 12.
Here , profit function is P = 20x + 10y
Profit is maximum at x = 4 and y = 12 .
Therefore , maximum profit = 20(4) + 10 ( 12) = 200.i.e. Rs.200.
Question 9
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A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What isthe maximum amount of nitrogen added?

A
150 bags of brand P and 50 bags of brand Q; Maximum amount of nitrogen = 625 kg

B
160 bags of brand P and 52 bags of brand Q; Maximum amount of nitrogen = 635 kg

C
140 bags of brand P and 50 bags of brand Q; Maximum amount of nitrogen = 595 kg

D
145 bags of brand P and 55 bags of brand Q; Maximum amount of nitrogen = 555 kg

Solution

Let the number of bags used for fertilizer of brand P = x And the number of bags used for fertilizer of brand Q = y . Here , Z = 3x + 3.5y subject to constraints : :1.5 x +2 y ≤310, x + 2y ≥240, 3x + 1.5y ≥270 , x,y ≥0

Here Z = 595 is maximum i.e. 140 bags of brand P and 50 bags of brand Q; Maximum amount of nitrogen = 595 kg .
Question 10
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Determine the maximum value of Z = 3x + 4y if the feasible region (shaded) for a LPP is shown in Figure above.

A
216

B
206

C
226

D
196

Solution
Question 11
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In linear programming, optimal solution

A
satisfies all the constraints only

B
is not unique

C
satisfies all the constraints as well as the objective function

D
maximizes the objective function only

Solution

In linear programming, optimal solution satisfies all the constraints as well as the objective function .
Question 12
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In linear programming problems the optimum solution

A
satisfies a set of piecewise –linear inequalities (called constraints)

B
satisfies a set of linear inequalities (called linear constraints)

C
satisfies a set of quadratic inequalities (calledconstraints)

D
satisfies a set of cubic inequalities (calledconstraints)

Solution

In linear programming problems the optimum solution satisfies a set of linear inequalities (called linear constraints) .
Question 13
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A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?

A
3 packages of nuts and 3 packages of bolts; Maximum profit = Rs 73.50.

B
4 packages of nuts and 4 packages of bolts; Maximum profit = Rs 76.50.

C
3 packages of nuts and 4 packages of bolts; Maximum profit = Rs 74.50.

D
4 packages of nuts and 3 packages of bolts; Maximum profit = Rs 75.50.

Solution

Let number of packages of nuts produced = x
And number of packages of bolts produced = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 17.50x +7y , subject to the constraints : x +3y ≤12 and. 3x +y ≤12, x, y ≥0.

i.e 3 packages of nuts and 3 packages of bolts;
Maximum profit = Rs 73.50.
Question 14
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A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

A
830 dolls of type A and 430 dolls of type B; Maximum profit = Rs 16300

B
800 dolls of type A and 400 dolls of type B; Maximum profit = Rs 16000

C
840 dolls of type A and 404 dolls of type B; Maximum profit = Rs 16500

D
820 dolls of type A and 420 dolls of type B; Maximum profit = Rs 16200

Solution

Here , Maximise Z = 12x + 16y , subject to constraints : : x + y ≤1200, x - 2y ≥0, x - 3y ≤600 , x,y ≥0

Here Z = 16000 is maximum.i.e. 800 dolls of type A and 400 dolls of type B; Maximum profit = Rs 16000 .
Question 15
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Feasible region (shaded) for a LPP is shown in Figure. Maximize Z = 5x + 7y.

A
49

B
45

C
43

D
47

Solution