Master Linear Programming: Optimize Decision-Making in Industrial Engineering

Question 1

The graphical method for solving linear programming problems is applicable when:

Accepted Answer

There are two decision variables and one constraint.

Answer

There is only one decision variable.

Answer

The objective function is non-linear.

Answer

The feasible region is unbounded.

Question 2

The dual problem of a linear programming problem is:

Accepted Answer

Another linear programming problem.

Answer

A non-linear programming problem.

Answer

An integer programming problem.

Question 3

The objective function in a linear programming problem can be used to:

Accepted Answer

Both (a) and (b).

Answer

Maximize profits.

Answer

Minimize costs.

Question 4

The constraint boundary in a graphical representation of a linear programming problem is:

Accepted Answer

A straight line.

Answer

A curved line.

Answer

A parabolic curve.

Answer

A hyperbolic curve.

Question 5

The feasible region in a linear programming problem represents:

Accepted Answer

The set of all possible values of the decision variables that satisfy all the constraints.

Answer

The set of all optimal solutions.

Answer

The set of all non-negative values of the decision variables.

Question 6

Which of these is NOT a step in the graphical method for solving linear programming problems?

Accepted Answer

Apply the simplex method.

Answer

Plot the feasible region.

Answer

Identify the vertices of the feasible region.

Answer

Evaluate the objective function at each vertex.

Question 7

Which of these is **NOT** a valid assumption of the simplex method for solving linear programming problems?

Accepted Answer

Variables can be negative.

Answer

The objective function is linear.

Answer

The constraints are linear inequalities.

Answer

The feasible region is convex.

Question 8

In linear programming, sensitivity analysis is used to evaluate:

Accepted Answer

The model's resilience to changes in input parameters

Answer

The viability of alternative solutions

Answer

The complexity of the linear programming model

Answer

The necessity of solving the linear programming problem

Question 9

What is the primary purpose of a slack variable in linear programming?

Accepted Answer

To convert an inequality constraint into an equality constraint.

Answer

To convert an equality constraint into an inequality constraint.

Answer

To increase the number of variables in the model.

Answer

To decrease the number of constraints in the model.

Question 10

Which of the following methods is commonly used to solve linear programming problems with a small number of variables?

Accepted Answer

Graphical method

Answer

Simplex method

Answer

Interior point method

Answer

Branch and bound method

Question 11

What is the primary goal of sensitivity analysis in linear programming?

Accepted Answer

To determine how changes in input data affect the optimal solution.

Answer

To identify alternative optimal solutions.

Answer

To solve problems with non-linear constraints.

Answer

To handle problems with integer variables.

Question 12

What is the fundamental difference between a primal problem and its dual in linear programming?

Accepted Answer

The primal problem maximizes an objective function, while the dual problem minimizes it.

Answer

The primal problem minimizes an objective function, while the dual problem maximizes it.

Answer

The primal problem has equality constraints, while the dual problem has inequality constraints.

Answer

The primal problem has inequality constraints, while the dual problem has equality constraints.

Question 13

The primary objective of linear programming is to:

Accepted Answer

Optimize resource allocation

Answer

Maximize revenue

Answer

Minimize expenses

Answer

Predict future trends

Question 14

The graphical method for solving linear programming problems is most appropriate for:

Accepted Answer

Small-scale problems with a limited number of variables and constraints

Answer

Large-scale problems with hundreds or thousands of variables and constraints

Answer

Problems involving non-linear constraints

Answer

Problems with integer variables

Question 15

In linear programming, the dual problem is defined as:

Accepted Answer

A problem with the reversed objective function and constraints

Answer

A problem with the same objective function and constraints

Answer

A problem with the same objective function but different constraints

Answer

A problem with the objective function and constraints negated

Question 16

What is the graphical representation of a linear equation?

Accepted Answer

A straight line

Answer

A parabola

Answer

A circle

Answer

A hyperbola

Question 17

Which method is used to solve linear programming problems with two decision variables graphically?

Accepted Answer

Corner point method

Answer

Simplex method

Answer

Big M method

Answer

Artificial variable method

Question 18

What is the purpose of introducing surplus variables?

Accepted Answer

Convert inequality constraints to equality constraints

Answer

Maximize the objective function

Answer

Minimize the production cost

Answer

Determine the optimal solution

Question 19

What is the dual problem of a minimization linear programming problem?

Accepted Answer

A maximization linear programming problem

Answer

Another minimization linear programming problem

Answer

A quadratic programming problem

Answer

A nonlinear programming problem

Question 20

What is the value of the objective function at an optimal solution?

Accepted Answer

Depends on the problem

Answer

Always positive

Answer

Always negative

Answer

Always zero

Question 21

Which of the following is a feasible solution to a linear programming problem?

Accepted Answer

A solution that satisfies all constraints

Answer

A solution that maximizes the objective function

Answer

A solution that minimizes the production cost

Answer

A solution that uses all available resources

Question 22

What is the purpose of sensitivity analysis in linear programming?

Accepted Answer

To determine the impact of changes in input data on the optimal solution

Answer

To solve linear programming problems with a large number of variables

Answer

To identify the most profitable product to produce

Answer

To find the minimum production cost

Question 23

Linear programming is a valuable tool for optimization, but it has some limitations. A key limitation is that it can only handle continuous variables. How can this limitation be addressed?

Accepted Answer

Use integer programming or mixed-integer programming to incorporate integer variables.

Answer

Use nonlinear programming to incorporate non-linear functions.

Answer

Use decomposition techniques to break down large-scale problems into smaller ones.

Answer

Use stochastic programming or robust optimization to handle uncertainty or risk.

Question 24

Consider the linear programming problem:
Minimize Z = 2X + 3Y
Subject to:
X + 2Y ≤ 10
X - Y ≤ 5
X, Y ≥ 0
What is the graphical representation of the feasible region?

Accepted Answer

A closed and bounded region defined by the lines X + 2Y = 10, X - Y = 5, X = 0, and Y = 0.

Answer

An unbounded region that extends infinitely in all directions.

Answer

A region that is not closed and has no clearly defined boundaries.

Question 25

Which of the following is NOT an essential step in the simplex method for solving linear programming problems?

Accepted Answer

Finding the maximum value of the objective function.

Answer

Converting the problem into standard form.

Answer

Finding an initial basic feasible solution.

Question 26

What is the dual of the following linear programming problem?
Minimize Z = 2X + 3Y
Subject to:
X + 2Y ≥ 10
X - Y ≥ 5
X, Y ≥ 0

Accepted Answer

Maximize W = 10U + 5V
Subject to:
2U - V ≤ 2
U + V ≤ 3
U, V ≥ 0

Answer

Minimize W = 2U + 3V
Subject to:
U + 2V ≤ 10
U - V ≤ 5
U, V ≥ 0

Answer

Maximize Z = 10U + 5V
Subject to:
2U + V ≥ 2
U - V ≤ 3
U, V ≥ 0

Question 27

Which of the following is a practical application of linear programming in industrial engineering?

Accepted Answer

Optimizing production schedules to maximize profit while considering constraints such as resource availability and labor capacity.

Answer

Analyzing financial data for investment purposes.

Answer

Predicting weather patterns for agricultural planning.

Question 28

What is the definition of a feasible solution in linear programming?

Accepted Answer

A set of values for the decision variables that satisfy all the constraints of the problem and can be used to calculate the value of the objective function.

Answer

A solution that maximizes the objective function.

Answer

A solution that minimizes the objective function.

Question 29

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

Accepted Answer

The objective function and constraints are nonlinear.

Answer

The constraints can only be expressed as linear equations or inequalities.

Answer

The decision variables can take on fractional values.

Question 30

What is the fundamental purpose of a linear programming model?

Accepted Answer

To optimize a linear objective function subject to a set of linear constraints, typically with the goal of maximizing or minimizing a specific quantity.

Answer

To solve complex equations.

Answer

To predict future trends.

Question 31

Which of the following is NOT a benefit of using linear programming in industrial engineering?

Accepted Answer

Reduced customer complaints.

Answer

Reduced production costs.

Answer

Enhanced decision-making and resource allocation.

Answer

Improved productivity and efficiency.

Question 32

How does sensitivity analysis help make better decisions in linear programming?

Accepted Answer

It shows how changes to input values affect the optimal solution.

Answer

It eliminates the need to solve multiple linear programming models.

Answer

It finds other solutions that are just as good as the optimal one.

Answer

It makes sure the linear programming model is reliable.

Question 33

Which assumption of linear programming ensures that the decision variables can be any real number?

Accepted Answer

The decision variables take on continuous values.

Answer

The constraints are linear.

Answer

The objective function is linear.

Question 34

What is the primary objective of the simplex method in linear programming?

Accepted Answer

To determine the optimal solution to the problem.

Answer

To check the feasibility of the problem.

Answer

To identify a feasible solution to the problem.

Question 35

In the graphical method for solving linear programming problems, which step involves determining the points where the constraint lines intersect?

Accepted Answer

Finding the vertices of the feasible region.

Answer

Plotting the feasible region.

Answer

Evaluating the objective function at each corner point.

Question 36

Which of the following accurately describes the dual problem in linear programming?

Accepted Answer

It has the same constraints as the original problem, but the roles of the objective function and the constraints are swapped.

Answer

It is the same linear programming problem written in a different mathematical form.

Answer

It is a separate linear programming problem with a different objective function and constraints.

Question 37

What is the key function of slack variables in linear programming?

Accepted Answer

To transform inequality constraints into equations.

Answer

To ensure the boundedness of the feasible region.

Answer

To substitute for the original decision variables.

Question 38

What is the main purpose of the reduced cost value in the simplex method?

Accepted Answer

To identify the non-basic variable that should be brought into the basis.

Answer

To determine which basic variable should be removed from the basis.

Answer

To calculate the optimal value of the objective function.

Question 39

What is the main benefit of using a sensitivity report in linear programming?

Accepted Answer

It provides insights into how changes in input data affect the optimal solution.

Answer

It helps identify errors in the input data.

Answer

It improves the speed and efficiency of the simplex method.

Question 40

Which of the following represents the standard form of a linear programming model?

Accepted Answer

Maximize/Minimize Z = c1x1 + c2x2 + ... + cnxn subject to a11x1 + a12x2 + ... + a1nxn ≤/≥ b1, a21x1 + a22x2 + ... + a2nxn ≤/≥ b2, ..., am1x1 + am2x2 + ... + amnxn ≤/≥ bm, x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0

Answer

Maximize/Minimize Z = c1x1^2 + c2x2^2 + ... + cnxn^2 subject to a11x1 + a12x2 + ... + a1nxn ≤/≥ b1, a21x1 + a22x2 + ... + a2nxn ≤/≥ b2, ..., am1x1 + am2x2 + ... + amnxn ≤/≥ bm, x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0

Question 41

For a linear programming problem with two decision variables, which graphical method can be employed to find a solution?

Accepted Answer

Graphical Method

Answer

Simplex Method

Answer

Dual Method

Answer

Newton's Method

Question 42

In a linear programming model, what is the primary function of the objective function?

Accepted Answer

To establish the optimization goal, either maximizing or minimizing a specific linear combination of the decision variables.

Answer

To define limitations on the decision variables.

Question 43

Which of the following is not a limitation of the graphical method for solving linear programming problems?

Accepted Answer

It can be applied to problems with numerous decision variables.

Answer

Visualizing the feasible region can be challenging in higher dimensions.

Answer

It demands that all constraints are linear.

Question 44

In linear programming, what is the purpose of slack variables?

Accepted Answer

To transform inequality constraints into equality constraints, allowing the problem to be solved using the Simplex Method.

Answer

To include new decision variables in the model.

Question 45

In linear programming, which type of sensitivity analysis is used to assess changes in the optimal solution?

Accepted Answer

Ranging Analysis

Answer

Discriminant Analysis

Answer

Cluster Analysis

Answer

Regression Analysis

Question 46

What is the primary function of the Simplex Method in linear programming?

Accepted Answer

To iteratively find an optimal solution by moving from one vertex of the feasible region to another.

Answer

To graphically depict the feasible region of a linear programming problem.

Question 47

Which of the following is a fundamental assumption in linear programming?

Accepted Answer

The objective function and constraints must be linear.

Answer

The feasible region is bounded.

Answer

Decision variables are continuous.

Question 48

In linear programming, how does an optimal solution differ from a feasible solution?

Accepted Answer

A feasible solution meets all constraints but may not optimize the objective function, while an optimal solution is feasible and also optimizes the objective function.

Answer

A feasible solution is always an optimal solution.

Question 49

Which of the following is the correct structure of a linear programming objective function?

Accepted Answer

Maximize 3x + 5y

Answer

Maximize x^2 + y^2

Answer

Minimize (x - y)^3

Question 50

Which step involves identifying the variable that enters the basis in the simplex method for linear programming?

Accepted Answer

Determining the entering variable

Answer

Calculating the optimal solution

Answer

Updating the tableau

Answer

Finding the leaving variable

Question 51

Which of the following is a fundamental theorem in linear programming?

Accepted Answer

Fundamental Theorem of Linear Programming

Answer

Gauss's Theorem

Answer

Lagrange Multiplier Theorem

Question 52

In a linear programming problem with two variables, what is the graphical representation of the optimal solution?

Accepted Answer

The corner point(s) of the feasible region that maximizes the objective function

Answer

The point inside the feasible region with the highest objective function value

Answer

The midpoint of the feasible region

Answer

The point on the boundary of the feasible region with the highest objective function value

Question 53

What is the primary purpose of introducing slack variables in linear programming?

Accepted Answer

To convert inequality constraints into equalities, making the problem easier to solve

Answer

To identify infeasible solutions

Answer

To reduce the number of variables in the problem

Answer

To represent non-negative decision variables

Question 54

Which of the following is a characteristic of a feasible solution in linear programming?

Accepted Answer

Satisfies all constraints and has non-negative decision variables

Answer

Maximizes the objective function value

Answer

Is unique

Question 55

What is the significance of sensitivity analysis in linear programming?

Accepted Answer

To analyze the impact of changes in input parameters on the optimal solution and objective function value

Answer

To identify infeasible solutions

Answer

To find multiple optimal solutions