Linear Programming: Formulation

Linear programming is mainly used:

1. ato find the best way of using limited resources, such as the greatest profit or the least cost
2. bto solve a system of linear equations that has exactly one solution
3. cto predict the future values of a quantity from its past values
4. dto find the maximum of a non-linear function with no restrictions

Which of the following can be the objective function of a linear programming problem in the decision variables \(x\) and \(y\)?

1. a\(Z = 3x^{2} + 4y\)
2. b\(Z = 3xy\)
3. c\(Z = 3x + 4y\)
4. d\(Z = \dfrac{3}{x} + 4y\)

Match each term in List I with its meaning in List II:

1. a(i)-S, (ii)-R, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-S, (iii)-P, (iv)-Q
3. c(i)-R, (ii)-Q, (iii)-P, (iv)-S
4. d(i)-P, (ii)-S, (iii)-R, (iv)-Q

1. I. A linear programming problem may ask for either the maximum or the minimum value of its objective function.
2. II. In a diet problem, the decision variables are the quantities of the different foods to be used.

Which of the statements above is/are correct?

1. aOnly I
2. bOnly II
3. cNeither I nor II
4. dBoth I and II

A firm makes \(x\) units of item A and \(y\) units of item B. Each unit of A sells for ₹ 70 and costs ₹ 50 to make; each unit of B sells for ₹ 90 and costs ₹ 60 to make. If every unit made is sold, the objective function to be maximised (total profit) is:

1. a\(Z = 70x + 90y\)
2. b\(Z = 20x + 30y\)
3. c\(Z = 50x + 60y\)
4. d\(Z = 30x + 20y\)

Assertion (A): In a manufacturing problem, the constraints on available machine hours are usually of the type \(\le\).

Reason (R): In a manufacturing problem, the objective is usually to maximise the profit.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

A factory makes \(x\) units of product P and \(y\) units of product Q a week. Each unit of P needs 2 hours on machine M1 and 3 hours on machine M2; each unit of Q needs 1 hour on M1 and 2 hours on M2. M1 can run for at most 40 hours and M2 for at most 70 hours a week. The machine-time constraints are:

1. a\(2x + y \le 40\) and \(3x + 2y \le 70\)
2. b\(2x + 3y \le 40\) and \(x + 2y \le 70\)
3. c\(2x + y \le 70\) and \(3x + 2y \le 40\)
4. d\(2x + 3y \le 70\) and \(x + 2y \le 40\)

A garment unit makes \(x\) shirts and \(y\) trousers a day. The number of shirts must be at least twice the number of trousers. This condition is written as:

1. a\(y \ge 2x\)
2. b\(x \le 2y\)
3. c\(x \ge y + 2\)
4. d\(x \ge 2y\)

A carpenter makes \(x\) chairs and \(y\) tables in a week. Match each condition in List I with its inequality in List II:

1. a(i)-S, (ii)-Q, (iii)-R, (iv)-P
2. b(i)-R, (ii)-P, (iii)-S, (iv)-Q
3. c(i)-S, (ii)-P, (iii)-R, (iv)-Q
4. d(i)-R, (ii)-Q, (iii)-S, (iv)-P

Assertion (A): \(2x + 3y \le 12\) and \(x^{2} + y \le 9\) can both be constraints of a linear programming problem.

Reason (R): Every constraint of a linear programming problem must be a linear inequality or a linear equation.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

In a transportation problem, goods are sent from godowns to shops. The objective of such a problem is usually to:

1. aminimise the total cost of transporting the goods
2. bmaximise the number of trips made by the trucks
3. cminimise the number of shops that receive goods
4. dmaximise the stock left in the godowns

1. I. In this LPP, \(x\) and \(y\) are the decision variables.
2. II. Apart from the non-negative constraints, this LPP has two constraints.
3. III. The objective function of this LPP is \(x + y\).

Consider the LPP: maximise \(Z = 20x + 30y\) subject to \(x + y \le 10\), \(2x + 5y \le 40\), \(x \ge 0\), \(y \ge 0\). Which of the statements above is/are correct?

1. aOnly I
2. bOnly I and II
3. cOnly II and III
4. dI, II and III

A farmer can use at most 20 hectares of land for wheat and mustard. Wheat needs 3 days of labour per hectare and mustard needs 2 days per hectare, and at most 48 days of labour are available. The profit is ₹ 4,000 per hectare of wheat and ₹ 3,000 per hectare of mustard. If \(x\) hectares are used for wheat and \(y\) hectares for mustard, the farmer's problem is to maximise \(Z = 4000x + 3000y\) subject to:

1. a\(x + y \le 20\), \(2x + 3y \le 48\), \(x \ge 0\), \(y \ge 0\)
2. b\(x + y \le 48\), \(3x + 2y \le 20\), \(x \ge 0\), \(y \ge 0\)
3. c\(x + y \le 20\), \(3x + 2y \le 48\), \(x \ge 0\), \(y \ge 0\)
4. d\(x + y \ge 20\), \(3x + 2y \le 48\), \(x \ge 0\), \(y \ge 0\)

A dealer buys ceiling fans at ₹ 360 each and sewing machines at ₹ 240 each, and can invest at most ₹ 5,760. If he buys \(x\) fans and \(y\) sewing machines, the investment constraint in its simplest form is:

1. a\(3x + 2y \le 48\)
2. b\(2x + 3y \le 48\)
3. c\(3x + 2y \ge 48\)
4. d\(3x + 2y \le 5760\)

A potter makes \(x\) cups and \(y\) plates a day. A cup takes 30 minutes and a plate takes 1 hour on the wheel, and the wheel can be used for at most 8 hours a day. The wheel-time constraint is:

1. a\(30x + y \le 8\)
2. b\(x + 2y \le 8\)
3. c\(2x + y \le 16\)
4. d\(x + 2y \le 16\)

Assertion (A): The constraint \(\dfrac{x}{3} + \dfrac{y}{2} \le 4\) can be written as \(2x + 3y \le 24\).

Reason (R): Multiplying both sides of an inequality by a positive number reverses the inequality sign.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

A dietician mixes two foods P and Q. Each kg of P contains 4 units of vitamin A and 3 units of vitamin C; each kg of Q contains 2 units of vitamin A and 6 units of vitamin C. The mix must contain at least 20 units of vitamin A and at least 30 units of vitamin C. P costs ₹ 50 per kg and Q costs ₹ 70 per kg, and the dietician wants the cheapest such mix. If the mix uses \(x\) kg of P and \(y\) kg of Q, the LPP is:

1. aMinimise \(Z = 70x + 50y\) subject to \(4x + 2y \ge 20\), \(3x + 6y \ge 30\), \(x, y \ge 0\)
2. bMinimise \(Z = 50x + 70y\) subject to \(4x + 2y \ge 20\), \(3x + 6y \ge 30\), \(x, y \ge 0\)
3. cMinimise \(Z = 50x + 70y\) subject to \(4x + 2y \le 20\), \(3x + 6y \le 30\), \(x, y \ge 0\)
4. dMinimise \(Z = 50x + 70y\) subject to \(4x + 3y \ge 20\), \(2x + 6y \ge 30\), \(x, y \ge 0\)

1. I. One constraint of Asha's problem is \(x + y \le 50000\).
2. II. One constraint of Asha's problem is \(x \le 10000\).
3. III. The objective of Asha's problem is to maximise \(Z = 0.08x + 0.1y\).

Asha invests ₹ \(x\) in Scheme A and ₹ \(y\) in Scheme B. She has at most ₹ 50,000 to invest and wants at least ₹ 10,000 in Scheme A. Scheme A pays 8% and Scheme B pays 10% interest a year, and she wants the greatest yearly interest. Which of the statements above is/are correct?

1. aOnly I
2. bOnly I and II
3. cI, II and III
4. dOnly I and III

A company makes \(x\) units of product A and \(y\) units of product B. At least 25% of all the units it makes must be of product A. This condition is:

1. a\(x - 3y \ge 0\)
2. b\(4x - y \ge 0\)
3. c\(3x - y \ge 0\)
4. d\(x \ge 25\)

A trader has two godowns: P holds 60 bags of rice and Q holds 40 bags. Shop R needs 50 bags and shop S needs 30 bags. Let \(x\) bags go from P to R and \(y\) bags from P to S, and let Q send the rest of what each shop needs. The constraint that comes from the stock at Q is:

1. a\(x + y \ge 40\)
2. b\(x + y \le 40\)
3. c\(x + y \le 60\)
4. d\(x + y \ge 120\)