Advertisements
Advertisements
प्रश्न
Solve the following Linear Programming Problems graphically:
Maximise Z = 3x + 2y
subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.
Advertisements
उत्तर
The system of constraints is
x + 2y ≤ 10 ....(i)
3x + y ≤ 15 ....(ii)
and x, y ≥ 0 ....(iii)
Let l1 : x + 2y = 10
l2 : 3x + y =15
The shaded region in the figure is the feasible region determined by the system of constraints (i) to (iii).
It is observed that the feasible region OCEB is bounded.
Thus, we use the Corner Point Method to determine the maximum value of Z.
We have: Z = 3x + 2y ....(iv)
The coordinates of O, C, E and B are (0, 0) (5, 0), (4, 3) (solving x + 2y = 10, 3x + y = 15) and (0, 5) respectively.
| Corner point | Corresponding values of Z |
| (0, 0) | 0 |
| (5, 0) | 15 |
| (4, 3) | 18 (Maximum) |
| (0, 5) | 10 |
Hence Zmax = 18 at (4, 3).
APPEARS IN
संबंधित प्रश्न
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.
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.
Maximise Z = 3x + 4y, subject to the constraints: x + y ≤ 1, x ≥ 0, y ≥ 0
Minimise Z = 13x – 15y subject to the constraints: x + y ≤ 7, 2x – 3y + 6 ≥ 0, x ≥ 0, y ≥ 0
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.
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.
Refer to question 13. Solve the linear programming problem and determine the maximum profit to the manufacturer
Maximise Z = x + y subject to x + 4y ≤ 8, 2x + 3y ≤ 12, 3x + y ≤ 9, x ≥ 0, y ≥ 0.
A manufacturer produces two Models of bikes-Model X and Model Y. Model X takes a 6 man-hours to make per unit, while Model Y takes 10 man-hours per unit. There is a total of 450 man-hour available per week. Handling and Marketing costs are Rs 2000 and Rs 1000 per unit for Models X and Y respectively. The total funds available for these purposes are Rs 80,000 per week. Profits per unit for Models X and Y are Rs 1000 and Rs 500, respectively. How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.
Refer to Question 27. Maximum of Z occurs at ______.
Refer to Question 30. Minimum value of F is ______.
Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F = 4x + 6y be the objective function. The Minimum value of F occurs at ______.
A corner point of a feasible region is a point in the region which is the ______ of two boundary lines.
Maximum value of the objective function Z = ax + by in a LPP always occurs at only one corner point of the feasible region.
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:
The maximum value of the object function Z = 5x + 10 y subject to the constraints x + 2y ≤ 120, x + y ≥ 60, x - 2y ≥ 0, x ≥ 0, y ≥ 0 is ____________.
Z = 7x + y, subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0. The minimum value of Z occurs at ____________.
In linear programming, optimal solution ____________.
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 = 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.
In the expression \[Z=250x+75y\], which quantities are the decision variables?
Which condition represents non-negative restrictions for two decision variables?
Which inequality is one of the given resource constraints in the formulation with objective function \[Z=250x+75y\]?
Linear Programming is a method of optimisation under which type of constraints?
What is the name of the quantity to be optimised in linear programming?
