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

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प्रश्न

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

पर्याय

  • True

  • False

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उत्तर

This statement is True.

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पाठ 12: Linear Programming - Solved Examples [पृष्ठ २४९]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12
पाठ 12 Linear Programming
Solved Examples | Q 11 | पृष्ठ २४९

संबंधित प्रश्‍न

Solve the following Linear Programming Problems graphically:

Maximise Z = 3x + 4y

subject to the constraints : x + y ≤ 4, x ≥ 0, y ≥ 0.


Solve the following Linear Programming Problems graphically:

Minimise Z = 3x + 5y

such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.


Solve the following Linear Programming Problems graphically:

Maximise Z = 3x + 2y

subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.


Show that the minimum of Z occurs at more than two points.

Minimise and Maximise Z = 5x + 10 y

subject to x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x, y ≥ 0.


Show that the minimum of Z occurs at more than two points.

Maximise Z = – x + 2y, Subject to the constraints:

x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.


A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional elements A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?


A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:

Type of toy Machines
I II III
A 12 18 6
B 6 0 9

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on each toy of type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

 


To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below:
 

  Food I
(per lb)
  Food II
(per lb)
    Minimum daily requirement
for the nutrient
 Calcium 10   5     20
Protein 5   4     20
 Calories 2   6     13
 Price (Rs) 60   100      


What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.


The minimum value of the objective function Z = ax + by in a linear programming problem always occurs at only one corner point of the feasible region


Determine the maximum value of Z = 11x + 7y subject to the constraints : 2x + y ≤ 6, x ≤ 2, x ≥ 0, y ≥ 0.


Maximise the function Z = 11x + 7y, subject to the constraints: x ≤ 3, y ≤ 2, x ≥ 0, y ≥ 0.


Determine the maximum value of Z = 3x + 4y if the feasible region (shaded) for a LPP is shown in Figure


A man rides his motorcycle at the speed of 50 km/hour. He has to spend Rs 2 per km on petrol. If he rides it at a faster speed of 80 km/hour, the petrol cost increases to Rs 3 per km. He has atmost Rs 120 to spend on petrol and one hour’s time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem


Refer to quastion 12. What will be the minimum cost?


The feasible solution for a LPP is shown in Figure. Let Z = 3x – 4y be the objective function. Minimum of Z occurs at ______.


The feasible region for an LPP is shown in the figure. Let F = 3x – 4y be the objective function. Maximum value of F is ______.


Refer to Question 32, Maximum of F – Minimum of F = ______.


In a LPP, the linear inequalities or restrictions on the variables are called ____________.


A feasible region of a system of linear inequalities is said to be ______ if it can be enclosed within a circle.


The feasible region for an LPP is always a ______ polygon.


If the feasible region for a LPP is unbounded, maximum or minimum of the objective function Z = ax + by may or may not exist.


In a LPP, the minimum value of the objective function Z = ax + by is always 0 if the origin is one of the corner point of the feasible region.


Based on the given shaded region as the feasible region in the graph, at which point(s) is the objective function Z = 3x + 9y maximum?


In the given graph, the feasible region for an LPP is shaded. The objective function Z = 2x – 3y will be minimum at:


For an objective function Z = ax + by, where a, b > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is:


In linear programming, optimal solution ____________.


In Corner point method for solving a linear programming problem, one finds the feasible region of the linear programming problem, determines its corner points, and evaluates the objective function Z = ax + by at each corner point. If M and m respectively be the largest and smallest values at corner points then ____________.


In Corner point method for solving a linear programming problem, one finds the feasible region of the linear programming problem, determines its corner points, and evaluates the objective function Z = ax + by at each corner point. Let M and m respectively be the largest and smallest values at corner points. In case feasible region is unbounded, M is the maximum value of the objective function if ____________.


If two corner points of the feasible region are both optimal solutions of the same type, i.e., both produce the same maximum or minimum.


In a LPP, the objective function is always ____________.


Maximize Z = 3x + 5y, subject to x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0.


Maximize Z = 4x + 6y, subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.


Maximize Z = 10×1 + 25×2, subject to 0 ≤ x1 ≤ 3, 0 ≤ x2 ≤ 3, x1 + x2 ≤ 5.


The feasible region for an LPP is shown shaded in the following figure. Minimum of Z = 4x + 3y occurs at the point.


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