Linear programming Maths - Linear programming

The corner points of the feasible region determined by the following system of linear inequalities:

2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5).

Let Z = px + qy, where p.q > 0.

Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is:

1. P = q
2. p = 2q
3. p = 3q
4. q = 3q

The objective function Z = 4x + 3y can be maximised subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6, x, y ≥ 0

1. at only one point
2. at two points only
3. at an infinite number of points
4. none of these

The value of objective function is maximum under linear constraints

1. at the centre of feasible region
2. at (0, 0)
3. at any vertex of feasible region
4. the vertex which is maximum distance from (0, 0)

The maximum value of Z = 4x + 3y subjected to the constraints 3x + 2y ≥ 160, 5x + 2y ≥ 200, x + 2y ≥ 80, x, y ≥ 0 is:

1. 320
2. 300
3. 230
4. none of these

A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1kg of food X and 1kg of food Y are given below:

	Vitamin A	Vitamin B	Vitamin C	Vitamin D
Food X	1	1	1	2
Food Y	2	1	3	1

One kg food X costs Rs. 5, whereas one kg of food Y costs Rs. 8.
Find the least cost of the mixture which will produce the desired diet.

A wholesale dealer deals in two kinds, A and B (say) of mixture of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The 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 nuts and 540 grams of hazel nuts. Mixture A costs Rs. 8 per kg. and mixture B costs Rs. 12 per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg. of each mixture which he should use to minimise the cost of the bag.

A manufacturer produces two types of steel trunks. He has two machines A and B. For completing, the first types of the trunk requires 3 hours on machine A and 3 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 2 hours on machine B. Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs. 30 and Rs. 25 per trunk of the first type and the second type respectively. How many trunks of each type must he make each day to make maximum profit?

To maintain one's health, a person must fulfil certain minimum daily requirements for the following three nutrients: calcium, protein and calories. The diet consists of only items I and II whose prices and nutrient contents are shown below:

	Food I	Food II	Minimum daily requirement
Calcium	10	4	20
Protein	5	6	20
Calories	2	6	12
Price	Rs. 0.60 per unit	Rs. 1.00 per unit

Find the combination of food items so that the cost may be minimum.

There are two types of fertilizers F₁ and F₂. F₁ consists of 10% nitrogen and 6% phosphoric acid and ​F₂ consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds the she needs atleast 14kg of nitrogen and 14kg of phosphoric acid for her crop. If F₁ costs Rs 6/kg and F₂ costs Rs 5/kg, determine how much of each type of fertilizer should be used so that the nutrient requirements are met at minimum cost. What is the minimum cost?