Advertisements
Advertisements
प्रश्न
Solve the following linear programming problems by graphical method.
Maximize Z = 40x1 + 50x2 subject to constraints 3x1 + x2 ≤ 9; x1 + 2x2 ≤ 8 and x1, x2 ≥ 0.
Advertisements
उत्तर
Given that 3x1 + x2 ≤ 9
Let 3x1 + x2 = 9
| x1 | 0 | 3 |
| x2 | 9 | 0 |
Also given that x1 + 2x2 ≤ 8]
Let x1 + 2x2 = 8
| x1 | 0 | 8 |
| x2 | 4 | 0 |
3x1 + x2 = 9 ………(1)
x1 + 2x2 = 8 ……..(2)
6x1 + 2x2 = 18 ……..(3) [Multiply by 2 for eq. (1)]
− 5x1 = − 10
x1 = 2
x1 = 2 substitute in (1)
3(2) + x2 = 9
x2 = 3
The feasible region satisfying all the conditions is OABC.
The co-ordinates of the corner points are O(0, 0), A(3, 0), B(2, 3), C(0, 4)
| Corner points | Z = 40x1 + 50x2 |
| O(0, 0) | 0 |
| A(3, 0) | 120 |
| B(2, 3) | 40 × 2 + 50 × 3 = 80 + 150 = 230 |
| C(0, 4) | 200 |
The maximum value of Z occurs at B(2, 3).
∴ The optimal solution is x1 = 2, x2 = 3 and Zmax = 230
APPEARS IN
संबंधित प्रश्न
A company manufactures two types of chemicals Aand B. Each chemical requires two types of raw material P and Q. The table below shows number of units of P and Q required to manufacture one unit of A and one unit of B and the total availability of P and Q.
| Chemical→ | A | B | Availability |
| Raw Material ↓ | |||
| P | 3 | 2 | 120 |
| Q | 2 | 5 | 160 |
The company gets profits of ₹ 350 and ₹ 400 by selling one unit of A and one unit of B respectively. (Assume that the entire production of A and B can be sold). How many units of the chemicals A and B should be manufactured so that the company gets a maximum profit? Formulate the problem as LPP to maximize profit.
A doctor has prescribed two different units of foods A and B to form a weekly diet for a sick person. The minimum requirements of fats, carbohydrates and proteins are 18, 28, 14 units respectively. One unit of food A has 4 units of fat, 14 units of carbohydrates and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is ₹ 4.5 per unit and that of food B is ₹ 3.5 per unit. Form the LPP, so that the sick person’s diet meets the requirements at a minimum cost.
In a cattle breeding firm, it is prescribed that the food ration for one animal must contain 14, 22, and 1 unit of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit weight of these two contains the following amounts of these three nutrients:
| Nutrient\Fodder | Fodder 1 | Fodder2 |
| Nutrient A | 2 | 1 |
| Nutrient B | 2 | 3 |
| Nutrient C | 1 | 1 |
The cost of fodder 1 is ₹ 3 per unit and that of fodder ₹ 2 per unit. Formulate the L.P.P. to minimize the cost.
Solve the following L.P.P. by graphical method:
Maximize: Z = 4x + 6y
Subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.
Choose the correct alternative :
Feasible region; the set of points which satify.
Fill in the blank :
A dish washing machine holds up to 40 pieces of large crockery (x) This constraint is given by_______.
Choose the correct alternative:
Z = 9x + 13y subjected to constraints 2x + 3y ≤ 18, 2x + y ≤ 10, 0 ≤ x, y was found to be maximum at the point
A company produces two types of pens A and B. Pen A is of superior quality and pen B is of lower quality. Profits on pens A and B are ₹ 5 and ₹ 3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. The supply of raw material is sufficient only for 1000 pens per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B, only 700 clips are available per day. Formulate this problem as a linear programming problem.
A firm manufactures pills in two sizes A and B. Size A contains 2 mgs of aspirin, 5 mgs of bicarbonate and 1 mg of codeine. Size B contains 1 mg. of aspirin, 8 mgs. of bicarbonate and 6 mgs. of codeine. It is found by users that it requires at least 12 mgs. of aspirin, 74 mgs. of bicarbonate and 24 mgs. of codeine for providing immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LLP.
Solve the following linear programming problem graphically.
Maximize Z = 3x1 + 5x2 subject to the constraints: x1 + x2 ≤ 6, x1 ≤ 4; x2 ≤ 5, and x1, x2 ≥ 0.
