## Question

### Solution

Correct option is

x = 6, y = 12 & Z = 240

Converting the given in-equations into equations, we obtain the following equations:

x + 2y = 40, 3x + y = 30, 4x + 3y = 60, x = 0 and y = 0.

Region represented by x + 2y ≤ 40: The line x + 2y = 40 meets the coordinates axes at A1 (40, 0) and B1 (0, 20) respectively. Join these points to obtain the line x + 2y = 40. Clearly, (0, 0) satisfies the in-equationx + 2y ≤ 40. So, the region in xy-plane that contains the origin represents the solution set of the given in-equation.

Region represented by 3x + y ≥ 30: The line 3x + y = 30 meets X and Yaxes at A2 (10, 0) and B2 (0, 30) respectively. Join these points to obtain this line. We find that O (0, 0) does not satisfy the inequation 3x + y ≥ 30. So that region in xy-plane which does no contain the origin is the solution set of this inequation.

Region represented by 4x + 3y ≥ 60: The line 4x + 3y = 60 meets X and Yaxes at A3 (15, 0) and B1 (0, 20) respectively. Join these points to obtain the line 4x + 3y = 60. We observe that O (0, 0) does not satisfy the inequation 4x + 3y ≥ 60. So, the region not containing the origin in xy-plane represents the solution set of the given inequation. Region represented by x≥ 0, y ≥ 0: Clearly, the region represented by x ≥ 0 and y ≥ 0 is the first quadrant in xy-plane.

The shaded region A3A1 QP in fig. represents the common region of the regions represented by the above inequations. This region expresents the feasible region of the given LPP.

The coordinates of the corner-points of the shaded feasible are A3 (15, 0)A1 (40, 0), Q (4, 18) and P (6, 12). These points have been obtained by solving the equations of the corresponding intersecting lines, simultaneously.

The values of the objective function at these points are given in the following table:

 Point (x, y) Value of the objective function               Z = 20x + 10y A3 (15, 0) Z = 20 × 15 + 10 × 0 = 300 A1 (40, 0) Z = 20 × 40 + 10 × 0 = 800 Q (4, 18) Z = 20 × 4 + 10 × 18 = 260 P (6, 12) Z = 20 × 6 + 10 × 12 = 240

Clearly, Z is minimum at P (6, 12). Hence, x = 6, y = 12 is the optimal solution of the given LPP. The optimal value of Z is 240.

#### SIMILAR QUESTIONS

Q1

A company makes two kinds of leather belts, A and B. Belt A is high quality belt, and B is of lower quality. The respective profits are Rs 4 and Rs 3 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (bothA and B combined). Belt A requires a fancy buckle, and only 400 buckles per day are available. There are only 700 buckles available for belt B. What should be the daily production of each type of belt? Formulate the problem as a LPP.

Q2

A dietician whishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of Vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C while food ‘II’ contains 1 unit per kg of vitamin A and 2 units per kg of vitamin C. It costs Rs 5.00 per kg to purchase food ‘I’ and Rs 7.00 per kg to produce food ‘II’. Formulate the above linear programming problem to minimize the cost of such a mixture.

Q3

A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available: F1 which costs Rs 2 per unit, and F2 which costs Rs 4 per unit. A unit of food F1 contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein; a unit of food F2 contains 25 units of carbohydrate, 10 units of fat, and 20 unit of protein. Find the minimum cost for a diet consists of a mixture of these two foods and also meets the minimum nutrition requirements. Formulate the problem as a linear programming problem.

Q4

The objective of a diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited to milk, beaf and eggs, and to vitamins ABC. The number of milligrams of each of these vitamins contained within a unit of each food is given below:

 Vitamin Litre of milk Kg of beaf Dozen of eggs Minimum daily requirements A B C 1 100 10 1 10 100 10 10 10 1 mg 50 mg 10 mg Cost Rs 1.00 Rs 1.10 Re 0.50

What is the linear programming formulation for this problem?

Q5

There is a factory located at each of the two places P and Q. From these locations, a certain commodity is delivered to each of the three depots situated at AB and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are 8 and 6 units respectively. The cost of transportation per unit is given below.

 To From Cost (in Rs) A B C P Q 16 10 10 12 15 10

How many units should be transported from each factory to each in order that the transportation cost is minimum. Formulate the above as a linear programming problem.

Q6

A brick manufacturer has two depots, A and B, with stocks of 30,000 and 20,000 bricks respectively. He receives orders from three builders PQand R for 15,000, 20,000 and 15,000 bricks respectively. The cost in Rs of transporting 1000 bricks to the builders from the depots are given below:

 From To P Q R A B 40 20 20 60 30 40

How should the manufacturer fulfil the orders so as to keep the cost of transportation minimum?

Formulate the above linear programming problem.

Q7

A company is making two products A and B. The cost of producing one unit of products A and B are Rs 60 and Rs 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit product A requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LLP.

Q8

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 Rs 25 Rs 30 Labour cost per unit Rs 16 Rs 20 Raw material cost per unit Rs 4 Rs 4

The problem is to determine the number of units of produce each product so as to maximize total contribution to profit. Formulate this as a LLP.

Q9

Solve the following LPP graphically:

Maximize    Z  = 5x + 3y

Subject to

3x + 5y ≤ 15

5x + 2y ≤ 10

And,    xy ≥ 0.

Q10

Solve the following LPP graphically:

Minimize and Maximize Z = 5x + 2y

Subject to –2x – 3y ≤ – 6

x – 2y ≤ 2

3x + 2y ≤ 12

–3x + 2y ≤ 3

xy ≥ 0