Advertisements
Advertisements
Question
Refer to question 15. Determine the maximum distance that the man can travel.
Advertisements
Solution
Referring to the solution of Question No.15, we have
Maximise Z = x + y subject to the constraints
Let 2x + 3y = 120
| x | 0 | 60 |
| y | 40 | 0 |
Let 8x + 5y = 400
| x | 0 | 50 |
| y | 80 | 0 |
2x + 3y ≤ 120 ......(i)
8x + 5y ≤ 400 ......(ii)
x ≥ 0, y ≥ 0
On solving eq. (i) and (ii) we get
x = `300/7` and y = `80/7`
Here, OABC is the feasible region whose corner points are O(0, 0), A(50, 0), `"B"(300/7, 80/7)` and C(0, 40).
Let us evaluate the value of Z
| Corner points | Value of Z = x + y | |
| O(0, 0) | Z = 0 + 0 = 0 | |
| A(50, 0) | Z = 50 + 0 = 50 km | |
| `"B"(300/7, 80/7)` |
Z = `300/7 + 80/7` = `380/7` = 54.3 km |
← Maximum |
| C(0, 40) | Z = 0 + 40 = 40 km |
Hence, the maximum distance that the man can travel is `54 2/7` km at `(300/7, 80/7)`.
APPEARS IN
RELATED QUESTIONS
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:
Maximise Z = 3x + 2y
subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.
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.
A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced.
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 the function Z = 11x + 7y, subject to the constraints: x ≤ 3, y ≤ 2, x ≥ 0, y ≥ 0.
Minimise Z = 13x – 15y subject to the constraints: x + y ≤ 7, 2x – 3y + 6 ≥ 0, x ≥ 0, y ≥ 0
Feasible region (shaded) for a LPP is shown in Figure. Maximise Z = 5x + 7y.
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 quastion 12. What will be the minimum cost?
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 27. (Maximum value of Z + Minimum value of Z) is equal to ______.
In a LPP, the linear inequalities or restrictions on the variables are called ____________.
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 ______ value.
A feasible region of a system of linear inequalities is said to be ______ if it can be enclosed within a circle.
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?
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:
Objective function of a linear programming problem is ____________.
A linear programming problem is one that is concerned with ____________.
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 = 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 = 6x + 4y, subject to x ≤ 2, x + y ≤ 3, -2x + y ≤ 1, x ≥ 0, y ≥ 0.
