CBSE Class 12 Mathematics Linear Programming Notes

TOPIC 11

LINEAR PROGRAMMING

KEY CONCEPTS

Linear Programming Problem : Linear programming problem is one that is concerned with finding the optimal value ( maximum or minimum value ) of a linear function of several variables called objective function

Feasible Region:Feasible region is the region which is common to all the linear constraints (linear inequalities )

Important LPP are

1. Diet Problems

2. Manufacturing Problems

3. Transportation Problems

Steps for solving a LPP

Solving linear programming problem using Corner Point Method. The method comprises of the following steps:

1. Convert the word problem into mathematical formulation by using given constraints.

2. Solve the linear inequations formed in step 1 and plot the graph.

3. Find the feasible region of the linear programming problem and determine its corner points either by inspection or by solving the two

equations of the lines intersecting at that point.

4. Evaluate the objective function Z = ax + by at each corner point. Let M and m, respectively denote the largest and smallest values of these points.

5. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z.

(ii) In case, the feasible region is unbounded, we have:

6. (a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value.

 (b) Similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value.

Problems of LPP

1. A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs.5 and Rs.4 per unit respectively. One unit of the food A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of the food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost, but it must satisfy the requirements of the sick person. Form the question as LPP and solve it graphically. Explain the importance of balanced diet.

Hint : Let x units of food A, y units of food B are mixed Z = 5x + 4y 200x+100y ≥ 4000, 40x + 40y ≥ 1400, x ≥ 0, y ≥ 0 2m Correct graph 2m

Points (50,0), (20,15), (5,30), (0, 40) and minimum cost is at (5, 30) 2m

2. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.Which computer would you prefer to buy?

Solution:-Let the merchant stock x desktop models and y portable models. Therefore, x ≥ 0 and y ≥ 0

The cost of a desktop model is Rs 25000 and of a portable model is Rs 4000. However, the merchant can invest a maximum of Rs 70 lakhs.

The monthly demand of computers will not exceed 250 units.

The profit on a desktop model is Rs 4500 and the profit on a portable model is Rs 5000.

Total profit, Z = 4500x + 5000y

Thus, the mathematical formulation of the given problem is

subject to the constraints,

The feasible region determined by the system of constraints is as follows

The corner points are A (250, 0), B (200, 50), and C (0, 175).

The values of Z at these corner points are as follows.

The maximum value of Z is 1150000 at (200, 50).

Thus, the merchant should stock 200 desktop models and 50 portable models to get the maximum profit of Rs 1150000.

3. A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1costs Rs. 4 per unit and F2costsRs. 6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets

the minimal nutritional requirements.

Solution:-

. Minimize Z = 4x + 6y

Cost will be minimum when 24 units of F1& 4/3 units of F2 will be mixed and minimum cost will be Rs 104

4. One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming  hat there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solvegraphically.

Solution:-Let x and y be the no. of cakes of type I & II respectively. Then according to question 

20 cakes of first kind and 10 cakes of 2nd kind should be made to get max. numbers of cake. 

5: A firm has to transport 1200 packages using large vans,which can carry 200 packages each& small vans which can carry 80 packages each.The cost of engaging each large van is Rs 400 7 each small van is Rs 200. Not more than 3000 is to be spent on the job& the no. of large vans cannot exceed the no. of small vans. Formulate the problem as CPP, given that the objective is to minimize the cost.

Let x large vans and y small vans are engaged

Then LPP is

To minimize Z = 400 x + 200y

Subject to constraints

x≥o, y≥ 0

400x+200y≤3000 =>2x+y ≤15

200x+80y≥1200 =>5x+2y ≥30

and x ≤ y

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