Master Linear Programming: Enhance Your Optimization Skills

Question 1

A company produces two types of products, A and B. Each unit of product A requires 2 hours of labor and 1 unit of raw material, while each unit of product B requires 3 hours of labor and 2 units of raw material. The company has 120 hours of labor and 60 units of raw material available. The profit per unit of product A is $5, and the profit per unit of product B is $8. Which of the following is the correct formulation of the objective function in a linear programming model to maximize the total profit?

Accepted Answer

Maximize Z = 5A + 8B

Answer

Maximize Z = 5A + 8B + 120 + 60

Answer

Maximize Z = 120A + 60B

Answer

Maximize Z = 2A + 3B

Question 2

Which of the following is NOT a characteristic of a feasible solution in a linear programming problem?

Accepted Answer

It must be the optimal solution.

Answer

It must be a combination of decision variables that satisfies the problem's requirements.

Answer

It must satisfy all constraints.

Answer

It must be a point within the feasible region.

Question 3

In the graphical solution of a linear programming problem, the feasible region is the area...

Accepted Answer

bounded by the constraints.

Answer

where the objective function is maximized.

Answer

where the objective function is minimized.

Answer

where all decision variables are positive.

Question 4

Which of the following methods is used to solve linear programming problems with more than two variables?

Accepted Answer

Simplex method

Answer

Graphical method

Answer

Substitution method

Answer

Trial and error method

Question 5

A constraint in a linear programming problem represents...

Accepted Answer

a limitation on the available resources.

Answer

the objective function to be optimized.

Answer

the optimal solution to the problem.

Answer

the feasible region.

Question 6

Which of the following is NOT an assumption of linear programming?

Accepted Answer

Non-linear relationships between variables.

Answer

Additivity of contributions.

Answer

Divisibility of resources.

Answer

Proportionality of resources.

Question 7

The corner point theorem states that the optimal solution to a linear programming problem occurs...

Accepted Answer

at one of the corner points of the feasible region.

Answer

on a line segment connecting two corner points.

Answer

at the center of the feasible region.

Question 8

A company is trying to decide how many units of two products, X and Y, to produce. The profit per unit of product X is $10 and the profit per unit of product Y is $15. The production of each product requires certain resources, and the company has limited resources available. Which of the following is the correct objective function for this linear programming problem?

Accepted Answer

Maximize Z = 10X + 15Y

Answer

Maximize Z = 15X + 10Y

Answer

Maximize Z = X + Y

Answer

Maximize Z = 10X - 15Y

Question 9

Which of the following is NOT a characteristic of a linear programming problem?

Accepted Answer

Nonlinear objective function

Answer

Feasible region defined by linear constraints

Answer

Objective function to be maximized or minimized

Question 10

Consider the linear programming problem: maximize z = 2x + 3y, subject to the constraints 2x + y ≤ 12, x + y ≤ 8, x ≥ 0, and y ≥ 0. What is the maximum value of z?

Accepted Answer

36

Answer

24

Answer

40

Answer

30

Question 11

Which of the following is a method for solving linear programming problems with more than two variables?

Accepted Answer

Simplex method

Answer

Graphical method

Answer

Dynamic programming

Question 12

Consider the linear programming problem: minimize c = 5x + 3y, subject to the constraints x + y ≥ 6, x ≤ 4, and y ≤ 5. What is the optimal solution?

Accepted Answer

x = 4, y = 2

Answer

x = 6, y = 0

Answer

x = 0, y = 6

Answer

x = 2, y = 4

Question 13

Which of the following is a limitation of linear programming?

Accepted Answer

Assumes linearity of objective function and constraints

Answer

Provides optimal solutions

Answer

Can handle large-scale problems

Question 14

Consider the linear programming problem: maximize z = 3x + 4y, subject to the constraints x + 2y ≤ 10, 3x + y ≤ 12, and x ≥ 0, y ≥ 0. Which of the following is the dual of this problem?

Accepted Answer

minimize w = 10u + 12v, subject to the constraints u + 3v ≥ 3, 2u + v ≥ 4, u ≥ 0, v ≥ 0

Answer

minimize z = 3u + 4v, subject to the constraints u + 2v ≤ 10, 3u + v ≤ 12, u ≥ 0, v ≥ 0

Answer

maximize w = 10u + 12v, subject to the constraints u + 3v ≤ 3, 2u + v ≤ 4, u ≥ 0, v ≥ 0

Question 15

Which of the following is an application of linear programming in transportation?

Accepted Answer

Minimizing the cost of shipping goods between warehouses and customers

Answer

Assigning employees to shifts

Answer

Scheduling flights for an airline

Question 16

Consider the linear programming problem: maximize z = 2x + 3y, subject to the constraints x + y ≤ 12, 2x + y ≤ 14, and x ≥ 0, y ≥ 0. What is the corner point solution?

Accepted Answer

(x = 4, y = 8)

Answer

(x = 10, y = 2)

Answer

(x = 6, y = 6)

Answer

(x = 8, y = 4)

Question 17

Which of the following is a benefit of using linear programming to solve optimization problems?

Accepted Answer

Provides a systematic approach to solving complex problems

Answer

Can model non-linear relationships

Answer

Always guarantees a globally optimal solution

Question 18

Consider the linear programming problem: minimize c = 2x + 5y, subject to the constraints 3x + 2y ≥ 12, 2x + 3y ≥ 15, and x ≥ 0, y ≥ 0. What is the minimum value of c?

Accepted Answer

25

Answer

35

Answer

30

Answer

20

Question 19

Which of the following is the standard form of a linear programming problem?

Accepted Answer

Maximize/Minimize cTx subject to Ax ≤ b, x ≥ 0

Answer

Minimize/Maximize cTx subject to Ax ≥ b, x ≤ 0

Answer

Maximize/Minimize cTx subject to Ax ≥ b, x ≥ 0

Answer

Minimize/Maximize cTx subject to Ax ≤ b, x ≤ 0

Question 20

What is the graphical method of solving linear programming problems best suited for?

Accepted Answer

Solving 2-variable linear programming problems with few constraints

Answer

Solving 3-variable linear programming problems with complex constraints

Answer

Solving nonlinear programming problems with quadratic objectives

Question 21

Which of the following is NOT a condition for the simplex method to find an optimal solution?

Accepted Answer

All variables must be basic variables.

Answer

The reduced cost of the entering variable must be negative.

Answer

The leaving variable must have a positive pivot.

Question 22

What is the dual problem of the following linear programming problem: Maximize z = 2x + 3y subject to x + y ≤ 6, 2x + y ≤ 8, x ≥ 0, y ≥ 0?

Accepted Answer

Minimize w = 6u + 8v subject to u + 2v ≥ 2, u + v ≥ 3, u, v ≥ 0

Answer

Minimize z = 2x + 3y subject to x + y ≥ 6, 2x + y ≥ 8, x ≤ 0, y ≤ 0

Answer

Maximize w = 6u + 8v subject to u + 2v ≤ 2, u + v ≤ 3, u, v ≤ 0

Question 23

Which of the following is an advantage of using linear programming to solve optimization problems?

Accepted Answer

Provides a systematic and efficient method for finding optimal solutions.

Answer

Can be easily solved by hand for problems with a large number of variables.

Answer

Guarantees a globally optimal solution even for nonlinear problems.

Question 24

What is the economic interpretation of the dual variables in linear programming?

Accepted Answer

Shadow prices, representing the additional value of relaxing the corresponding constraint by one unit.

Answer

Lagrange multipliers, representing the sensitivity of the objective function to changes in the constraints.

Question 25

Which of the following is a necessary condition for a linear programming problem to have an unbounded solution?

Accepted Answer

There exists a feasible solution and a direction along which the objective function can be improved without violating any constraints.

Answer

The feasible region is empty.

Question 26

What is the purpose of sensitivity analysis in linear programming?

Accepted Answer

To determine how changes in the input parameters (e.g., coefficients, constraints) affect the optimal solution.

Answer

To find a new optimal solution if the original problem is infeasible.

Question 27

Which of the following is an application of linear programming in finance?

Accepted Answer

Portfolio optimization

Answer

Image processing

Answer

Natural language processing

Answer

Numerical integration

Question 28

What is the purpose of the Big M method in linear programming?

Accepted Answer

To convert a linear programming problem with inequality constraints into an equivalent problem with only equality constraints.

Answer

To find a feasible solution to an infeasible linear programming problem.