Linear Programming Test

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Linear Programming Test - 5

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Question 1
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Let R be the feasible region for a linear programming problem,and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and

A
each of these occurs at a corner point (vertex) of R.

B
each of these occurs at themidpoints of the edges of R

C
each of these occurs at the centre of R.

D
each of these occurs at some points except corner points of R.

Solution

Let R be the feasible region for a linear programming problem,and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.
Question 2
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Minimize Z = 3x + 5y such that x + 3y ≥3, x + y ≥2, x, y ≥0.

A
Minimum Z = 7 at 3/2,1/2

B
Minimum Z = 8 at 3/2,1/2

C
Minimum Z = 9 at 3/2,1/2 110 V 60 Hz

D
Minimum Z = 10 at 3/2,1/2

Solution

Objective function is Z = 3x + 5 y ……………………(1).
The given constraints are : x + 3y ≥3, x + y ≥2, x, y ≥0 .
Question 3
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A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

A
8 Souvenir of types A and 27 of Souvenir of type B; Maximum profit = Rs 170

B
9 Souvenir of types A and 21 of Souvenir of type B; Maximum profit = Rs 163

C
8 Souvenir of types A and 20 of Souvenir of type B; Maximum profit = Rs 160

D
7 Souvenir of types A and 210 of Souvenir of type B; Maximum profit = Rs 161

Solution

Let number of souvenirs of type A = x
And number of souvenirs of type B = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 5x +6y , subject to the constraints : 5x +8y ≤200 and. 10x +8y ≤240 , x, y ≥0.

Here Z = 160 is maximum.
i.e. 8 Souvenir of types A and 20 of Souvenir of type B; Maximum profit = Rs 160.
Question 4
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Maximise the function Z = 11x + 7y, subject to the constraints: x ≤3, y ≤2,x ≥0, y ≥0.

A
49

B
50

C
47

D
48

Solution

We have , Maximise the function Z = 11x + 7y, subject to the constraints: x ≤3, y ≤2,x ≥0, y ≥0.
Question 5
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The feasible region for a LPP is shown in Figure. Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists

A
Minimum value = 2

B
Minimum value = 5

C
Minimum value = 3

D
Minimum value = 4

Solution

Corner Points(4,0)(2,1)(0,3)Corresponding value of Z 3 (Minimum)

From the shaded region, it is clear that feasible region is unbounded with the corner points A(4, 0, B(2, 1) and C(0, 3).

Also, we have Z=4x+y ,

Now, we see that 3 is the smallest value of Z at the corner point (0, 3). Note that here we see that the region is unbounded, therefore 3 may or may not be the minimum value of Z.

To decide this issue, we graph the inequality 4x+y <3 and check whether the resulting open half plan has no point in common with feasible region otherwise, Z has no minimum value.

From the shown graph above, it is clear that there is no point in common with feasible region and hence Z is minimum value of 3 at (0, 3).
Question 6
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A maximum or a minimum may not exist for a linear programming problem if

A
The feasible region is bounded

B
if the constraints are non linear

C
if the objective function is continuous

D
The feasible region is unbounded

Solution

A maximum or a minimum may not exist for a linear programming problem if The feasible region is unbounded .
Question 7
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Maximize Z = 3x + 2y subject to x + 2y ≤10, 3x + y ≤15, x, y ≥0.

A
Maximum Z = 20 at (4, 3)

B
Maximum Z = 22 at (4, 3)

C
Maximum Z = 24 at (4, 3)

D
Maximum Z = 18 at (4, 3)

Solution

Objective function is Z = 3x + 2 y ……………………(1).
The given constraints are : x + 2y ≤10, 3x + y ≤15, x, y ≥0.
Question 8
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A merchant plans to sell two types of personal computers –a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

A
200 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150000

B
260 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150400

C
220 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150020

D
240 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150040

Solution

Let number of desktop model computers = x
And number of portable model computers = y
Therefore , the above L.P.P. is given as :
Maximise , Z = 4500x +5000y , subject to the constraints : x +y ≤250 and 25000x +40000y ≤700000 .i.e. x +y ≤250 and 5x +8y ≤1400 , x, y ≥0.

Here Z = 1150000 is maximum.
i.e. 200 units of desktop model and 50 units of portable model; Maximum profit = Rs 1150000 .
Question 9
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Minimise Z = 13x –15y subject to the constraints : x + y ≤7, 2x –3y + 6 ≥0 , x ≥0, y ≥0.

A
–23

B
–32

C
–30

D
–34

Solution
Question 10
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In Figure, the feasible region (shaded) for a LPP is shown. Determine the maximum and minimum value of Z = x + 2y

A
Maximum = 7, minimum =

B
Maximum = 9, minimum =

C
Maximum = 8, minimum =

D
Maximum = 10, minimum =

Solution
Question 11
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In Corner point method for solving a linear programming problem the first step is to

A
Find the feasible region of the linear programming problem and determine its center points (vertices).

B
Find the infeasible region of the linear programming problem and determine its complement

C
Find the infeasible regions of the linear programming problem and determine theunion of the infeasible regions

D
Find the feasible region of the linear programming problem and determine its corner points (vertices).

Solution

In Corner point method for solving a linear programming problem the first step is : To find the feasible region of the linear programming problem and determine its corner points (vertices)
Question 12
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Minimize Z = x + 2y subject to 2x + y ≥3, x + 2y ≥6, x, y ≥0.

A
Minimum Z = 7 at all the points on the line segment joining the points (6, 0) and (0, 3).

B
Minimum Z = 6 at all the points on the line segment joining the points (6, 0) and (0, 3).

C
Minimum Z = 8 at all the points on the line segment joining the points (6, 0) and (0, 3).

D
Minimum Z = 9 at all the points (6, 0) and (0, 3).

Solution

Objective function is Z = x + 2y ……………………(1).
The given constraints are : 2x + y ≥3, x + 2y ≥6, x, y ≥0 .

Here , Z = 18 is minimum at (0, 3) and (6 , 0) .
Minimum Z = 6 at all the points on the line segment joining the points (6, 0) and (0, 3).
Question 13
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A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

A
Minimum cost = Rs 114

B
Minimum cost = Rs 134

C
Minimum cost = Rs 124

D
Minimum cost = Rs 104

Solution

Let number of units of food F1 = x
And number of units of food F2 = y
Therefore , the above L.P.P. is given as :
Minimise , Z = 4x +6y , subject to the constraints : 3 x + 6y ≥80, 4x + 3y ≥100, x,y ≥0.

Corner points Z =4x +6 y B(80/3 , 0) 320/3 D(24,4/3 ) 104 …………………(Min.) A(0,100/3) 200 Here Z = 104 is minimum. i.e. Minimum cost = Rs 104.
Question 14
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Maximize Z = 100x + 120y , subject to constraints 2x + 3y ≤30, 3x + y ≤17, x ≥0, y ≥0.

A
1260

B
1200

C
1300

D
1280

Solution

We have , Maximize Z = 100x + 120y , subject to constraints 2x + 3y ≤30, 3x + y ≤17, x ≥0, y ≥0.
Question 15
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Determine the minimum value of Z = 3x + 4y if the feasible region (shaded) for a LPP is shown in Figure above.

A
132

B
196

C
154

D
112