Advertisements
Advertisements
प्रश्न
In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in milligrams per tablet) are given as below:
| Tablets | Iron | Calcium | Vitamin |
| X | 6 | 3 | 2 |
| Y | 2 | 3 | 4 |
The person needs atleast 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamin. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?
उत्तर
Let there be x units of tablet X and y units of tablet Y
So, according to the given information, we have
6x + 2y ≥ 18
⇒ 3x + y ≥ 9 ......(i)
| x | 0 | 3 |
| y | 9 | 0 |
3x + 3y ≥ 21
⇒ x + y ≥ 7 ......(ii)
| x | 0 | 3 |
| y | 9 | 0 |
2x + 4y ≥ 16
⇒ x + 2y ≥ 8 ......(iii)
| x | 0 | 8 |
| y | 4 | 0 |
x ≥ 0, y ≥ 0 ......(iv)
The price of each table of X type is ` 2 and that of y is ` 1.
So, the required LPP is
Minimise Z = 2x + y subject to the constraints
3x + y ≥ 9, x + y ≥ 7, x + 2y ≥ 8, x ≥ 0, y ≥ 0
On solving (ii) and (iii) we get
x = 6 and y = 1
On solving (i) and (ii) we get
x = 1 and y = 6
From the graph, we see that the feasible region ABCD is unbounded whose corner points are A(8, 0), B(6, 1), C(1, 6) and D(0, 9).
Let us evaluate the value of Z
| Corner points | Value of Z = 2x + y | |
| A(8, 0) | Z = 2(8) + 0 = 16 | |
| B(6, 1) | Z = 2(6) + 1 = 13 | |
| C(1, 6) | Z = 2(1) + 6 = 8 | ← Minimum |
| D(0, 9) | Z = 2(0) + 9 = 9 |
Here, we see that 8 is the minimum value of Z at (1, 6) but the feasible region is unbounded.
So, 8 may or may not be the minimum value of Z.
To confirm it, we will draw a graph of inequality 2x + y < 8 and check if it has a common point.
We see from the graph that there is no common point on the line.
Hence, the minimum value of Z is 8 at (1, 6).
Tablet X = 1
Tablet Y = 6.
APPEARS IN
संबंधित प्रश्न
Solve the following Linear Programming Problems graphically:
Maximise Z = 5x + 3y
subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0
Solve the following Linear Programming Problems graphically:
Minimise Z = x + 2y
subject to 2x + y ≥ 3, x + 2y ≥ 6, 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.
Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?
A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of one kg food is given below:
| Food | Vitamin A | Vitamin B | Vitamin C |
| X | 1 | 2 | 3 |
| Y | 2 | 2 | 1 |
One kg of food X costs Rs 16 and one kg of food Y costs Rs 20. Find the least cost of the mixture which will produce the required diet?
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.
The feasible region for a LPP is shown in Figure. Find the minimum value of Z = 11x + 7y
Refer to Exercise 7 above. Find the maximum value of Z.
Refer to question 13. Solve the linear programming problem and determine the maximum profit to the manufacturer
Refer to question 14. How many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit.
Refer to question 15. Determine the maximum distance that the man can travel.
Maximise Z = x + y subject to x + 4y ≤ 8, 2x + 3y ≤ 12, 3x + y ≤ 9, x ≥ 0, y ≥ 0.
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 30. Minimum value of F is ______.
A corner point of a feasible region is a point in the region which is the ______ of two boundary lines.
The feasible region for an LPP is always a ______ polygon.
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:
A linear programming problem is as follows:
Minimize Z = 30x + 50y
Subject to the constraints: 3x + 5y ≥ 15, 2x + 3y ≤ 18, x ≥ 0, y ≥ 0
In the feasible region, the minimum value of Z occurs at:
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. 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 ____________.
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 the feasible region is unbounded, m is the minimum value of the objective function.
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.
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 = 7x + 11y, subject to 3x + 5y ≤ 26, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0.
Maximize Z = 10×1 + 25×2, subject to 0 ≤ x1 ≤ 3, 0 ≤ x2 ≤ 3, x1 + x2 ≤ 5.
