Linear Programming quiz

In an LPP, the objective function is always:

The optimal value of the objective function in a bounded feasible region occurs at:

The feasible region of an LPP is always:

The corner points of the feasible region for x + y <= 4, x >= 0, y >= 0 are:

The maximum value of Z = 3x + 4y at corner points (0,0), (4,0), (0,4) is:

Non-negativity constraints mean:

If the feasible region is empty, the LPP has:

If Z has the same maximum value at two corner points, then:

The point (2, 3) lies in the region x + y <= 6:

In an unbounded feasible region, the minimum of Z found at a corner exists only if:

Write the general form of a linear programming problem in two variables. (2 marks)

Find the corner points of the region x + y <= 6, x <= 4, x >= 0, y >= 0. (2 marks)

What are decision variables? Give an example from a manufacturing problem. (2 marks)

Why do we include x >= 0 and y >= 0 in real-life problems? (2 marks)

What is meant by an unbounded feasible region? Draw a rough sketch. (2 marks)

Check whether the point (3, 2) satisfies 2x + 3y <= 12 and x - y >= 0. (2 marks)

What is the meaning of an infeasible LPP? Give one example of a pair of contradictory constraints. (2 marks)

Formulate: a tailor has 16 m of cloth; a shirt needs 2 m and a trouser 4 m, with profit Rs 50 and Rs 80. Write the objective function and constraint. (2 marks)

At corner points (0, 0), (5, 0), (3, 4), (0, 5) find the maximum of Z = 2x + y. (2 marks)

Why is linear programming called linear? (2 marks)

Maximise Z = 3x + 2y subject to x + 2y <= 10, 3x + y <= 15, x >= 0, y >= 0. (3 marks)

Minimise Z = 200x + 500y subject to x + 2y >= 10, 3x + 4y <= 24, x >= 0, y >= 0. (3 marks)

Maximise Z = 4x + y subject to x + y <= 50, 3x + y <= 90, x >= 0, y >= 0. (3 marks)

Find the maximum value of Z = 2x + 3y subject to x + y <= 6, x <= 4, x >= 0, y >= 0. (3 marks)

The corner points of a feasible region are (0, 10), (5, 5), (15, 15) and (0, 20). Find the maximum and minimum values of Z = 3x + 9y and state where they occur. (3 marks)

Read the passage and answer the questions. A cottage industry makes pedestal lamps and wooden shades. A lamp needs 2 hours on a grinding machine and 3 hours on a sprayer; a shade needs 1 hour on the grinder and 2 hours on the sprayer. The grinder is available for 12 hours and the sprayer for 20 hours a day. Profit is Rs 5 per lamp and Rs 3 per shade. (i) Formulate the LPP. (ii) Find the corner points of the feasible region. (iii) Find the number of lamps and shades for maximum profit. (5 marks)

Read the passage and answer the questions. A dietician mixes two foods. Food X costs Rs 4 per unit and Food Y costs Rs 6 per unit. The mixture must give at least 10 units of protein and 8 units of iron. One unit of X has 1 unit of protein and 2 units of iron; one unit of Y has 2 units of protein and 1 unit of iron. (i) Formulate the LPP to minimise cost. (ii) Find the corner points. (iii) Find the minimum cost and check that it exists. (5 marks)

Read the passage and answer the questions. A student draws the constraints x + y >= 8 and 3x + 5y <= 15 with x, y >= 0 and finds no shaded common region. (i) What does this mean? (ii) Verify by testing the corner points of 3x + 5y <= 15. (iii) What conclusion can be drawn about the LPP? (5 marks)

Read the passage and answer the questions. The corner points of a bounded feasible region are (0, 0), (5, 0), (6, 5), (6, 8), (4, 10) and (0, 8). (i) Find Z = 3x - 4y at each corner. (ii) Find the maximum value of Z. (iii) Find the minimum value of Z. (5 marks)

Read the passage and answer the questions. For an LPP with objective function Z = px + qy, where p and q are positive, the maximum value occurs at both corner points (3, 4) and (0, 5). (i) Write the equation relating p and q. (ii) Find q in terms of p. (iii) What can you say about the other points on the segment joining them? (5 marks)

Minimise Z = x + 2y subject to 2x + y >= 3, x + 2y >= 6, x >= 0, y >= 0 graphically. Show that the minimum occurs at more than one point. (6 marks)

A company makes two models of a product. Model P needs 9 labour hours for fabricating and 1 hour for finishing; model Q needs 12 hours and 3 hours. At most 180 fabricating hours and 30 finishing hours are available weekly. Profit is Rs 8,000 per model P and Rs 12,000 per model Q. Find the production for maximum profit graphically. (6 marks)

Show graphically that the LPP minimise Z = 3x + 2y subject to x + y >= 8, 3x + 5y <= 15, x >= 0, y >= 0 has no feasible solution. (6 marks)

Food F1 costs Rs 4 per unit and food F2 Rs 6 per unit. One unit of F1 has 3 units of vitamin A and 4 units of minerals; one unit of F2 has 6 units of vitamin A and 3 units of minerals. The diet needs at least 80 units of vitamin A and 100 units of minerals. Find the minimum cost of the diet graphically. (6 marks)

A farmer has 50 hectares of land for wheat and rice, with profits of Rs 10,500 and Rs 9,000 per hectare. Herbicide needed is 20 litres per hectare for wheat and 10 litres per hectare for rice, and at most 800 litres may be used. Find the allocation of land for maximum profit graphically. (6 marks)