Advertisements
Advertisements
Question
A wholesale dealer deals in two kinds of mixtures A and B of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew and 180 grams of hazel nuts. A dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew and 540 grams of hazel nuts. Mixture A costs ₹ 8 and B costs ₹ 12 per kg. How many kgs of each mixture should he use to minimize the cost of the kgs
Solution
Let the dealer use x kg of mixture A and y kg of mixture B.
Since x and y cannot be negative, x ≥ 0, y ≥ 0
Mixture A costs ₹ 8 and Mixture B costs ₹ 12 per kg.
∴ Total cost = Z = 8x + 12y
We construct a table with the constraints of Almond, Cashew and Hazelnut as follows:
| Mixture A | Mixture B | Least value | |
| Almond | 60 | 30 | 240 |
| Cashew | 30 | 60 | 300 |
| Hazelnut | 30 | 180 | 540 |
From the table, the constraints are
60x + 30y ≥ 240
30x + 60y ≥ 300
30x + 180y ≥ 540
∴ Given problem can be formulated as follows:
Minimize Z = 8x + 12y
Subject to 60x + 30y ≥ 240
30x + 60y ≥ 300
30x + 180y ≥ 540, x ≥ 0, y ≥ 0
To draw the feasible region, construct table as follows:
| Inequality | 60x +30y ≥ 240 | 30x+ 60y ≥ 300 | 30x+ 180y ≥ 540 |
| Corresponding equation (of line) | 60x+ 30y = 240 | 30x + 60y = 300 | 30x+ 180y = 540 |
| Intersection of line with X-axis | (4, 0) | (10, 0) | (18, 0) |
| Intersection of line with Y-axis | (0, 8) | (0, 5) | (0, 3) |
| Region | Non-origin side | Non-origin side | Non-origin side |
Shaded portion XABCDY is the feasible region, whose vertices are A(18, 0), B, C and D(0, 8).
B is the point of intersection of the lines 30x + 180y = 540 and 30x + 60y = 300.
Solving the above equations, we get
x = 6, y = 2
∴ B ≡ (6, 2)
C is the point of intersection of the lines 60x + 30y = 240 and 30x + 60y = 300.
Solving the above equations, we get
x = 2, y = 4
∴ C ≡ (2, 4)
Here, the objective function is
Z = 8x + 12y
∴ Z at A(18, 0) = 8(18) + 12(0)
= 144
Z at B(6, 2) = 8(6) + 12(2)
= 48 + 24
= 72
Z at C(2, 4) = 8(2) + 12(4)
= 16 + 48
= 64
Z at D(0, 8) = 8(0) + 12(8)
= 96
∴ Z has minimum value 64 at x = 2 and y = 4.
∴ 2 kgs of mixture A and 4 kgs of mixture B should be used to minimize the cost of the kgs.
APPEARS IN
RELATED QUESTIONS
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
| Product A | Product B | Weekly capacity | |
| Department 1 | 3 | 2 | 130 |
| Department 2 | 4 | 6 | 260 |
| Selling price per unit | ₹ 25 | ₹ 30 | |
| Labour cost per unit | ₹ 16 | ₹ 20 | |
| Raw material cost per unit | ₹ 4 | ₹ 4 |
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
Solve the following L.P.P. by graphical method:
Minimize: Z = 6x + 2y subject to x + 2y ≥ 3, x + 4y ≥ 4, 3x + y ≥ 3, x ≥ 0, y ≥ 0.
Choose the correct alternative :
The maximum value of z = 5x + 3y. subject to the constraints
Fill in the blank :
The region represented by the in equations x ≤ 0, y ≤ 0 lines in _______ quadrants.
State whether the following is True or False :
The region represented by the inqualities x ≤ 0, y ≤ 0 lies in first quadrant.
Choose the correct alternative:
If LPP has optimal solution at two point, then
Choose the correct alternative:
The maximum value of Z = 3x + 5y subjected to the constraints x + y ≤ 2, 4x + 3y ≤ 12, x ≥ 0, y ≥ 0 is
State whether the following statement is True or False:
The maximum value of Z = 5x + 3y subjected to constraints 3x + y ≤ 12, 2x + 3y ≤ 18, 0 ≤ x, y is 20
State whether the following statement is True or False:
If LPP has two optimal solutions, then the LPP has infinitely many solutions
State whether the following statement is True or False:
A convex set includes the points but not the segment joining the points
State whether the following statement is True or False:
Of all the points of feasible region, the optimal value is obtained at the boundary of the feasible region
State whether the following statement is True or False:
The point (6, 4) does not belong to the feasible region bounded by 8x + 5y ≤ 60, 4x + 5y ≤ 40, 0 ≤ x, y
State whether the following statement is True or False:
The graphical solution set of the inequations 0 ≤ y, x ≥ 0 lies in second quadrant
The feasible region represented by the inequations x ≥ 0, y ≤ 0 lies in ______ quadrant.
If the feasible region is bounded by the inequations 2x + 3y ≤ 12, 2x + y ≤ 8, 0 ≤ x, 0 ≤ y, then point (5, 4) is a ______ of the feasible region
Smita is a diet conscious house wife, wishes to ensure certain minimum intake of vitamins A, B and C for the family. The minimum daily needs of vitamins A, B, and C for the family are 30, 20, and 16 units respectively. For the supply of the minimum vitamin requirements Smita relies on 2 types of foods F1 and F2. F1 provides 7, 5 and 2 units of A, B, C vitamins per 10 grams and F2 provides 2, 4 and 8 units of A, B and C vitamins per 10 grams. F1 costs ₹ 3 and F2 costs ₹ 2 per 10 grams. How many grams of each F1 and F2 should buy every day to keep her food bill minimum
A chemist has a compound to be made using 3 basic elements X, Y, Z so that it has at least 10 litres of X, 12 litres of Y and 20 litres of Z. He makes this compound by mixing two compounds (I) and (II). Each unit compound (I) had 4 litres of X, 3 litres of Y. Each unit compound (II) had 1 litre of X, 2 litres of Y and 4 litres of Z. The unit costs of compounds (I) and (II) are ₹ 400 and ₹ 600 respectively. Find the number of units of each compound to be produced so as to minimize the cost
Maximize Z = 2x + 3y subject to constraints
x + 4y ≤ 8, 3x + 2y ≤ 14, x ≥ 0, y ≥ 0.
Minimize Z = 24x + 40y subject to constraints
6x + 8y ≥ 96, 7x + 12y ≥ 168, x ≥ 0, y ≥ 0
