Question 1

A company makes two products, X₁ and X₂. They require at least 20 of each be produced. Which set of lower bound constraints reflect this requirement?

Accepted Answer

The lower bound constraints indicate the minimum value that each product should be produced. Therefore, we need to have a separate constraint for each product. Option A provides the correct constraints for each product separately, indicating that at least 20 units of each product should be produced. Option B is incorrect because it only provides a total lower bound constraint, not separate constraints for each product. Option C is incorrect because it indicates that the total number of products should be at least 40, which is not a requirement. Option D is incorrect because it includes an upper bound constraint, which is not necessary to meet the requirement of producing at least 20 units of each product.

Question 2

If a problem has infinite number solutions, the objective function

Accepted Answer

If a problem has infinite number of solutions, it means that the objective function is parallel to one of the binding constraints. This is because the slope of the objective function is the same as the slope of the binding constraint it is parallel to, and any point on that line would satisfy the same objective function value.

Question 3

Why do we study the graphical method of solving LP problems?

Accepted Answer

The graphical method is important in understanding the fundamentals of linear programming and how to approach solving LP problems. While the graphical method may be a useful tool in certain situations, it is not necessarily faster or better than computerized methods for solving LP problems. The main advantage of the graphical method is its accessibility and ease in visualizing LP problems.

Question 4

The constraints of an LP model define the

Accepted Answer

The constraints of an LP model define the feasible region, which is the set of all possible solutions that satisfy all the constraints. Therefore, choice A is the best answer. The other choices (B, C, and D) do not accurately describe the concept of constraints in LP modeling.

Question 5

When the objective function can increase without ever contacting a constraint the LP model is said to be

Accepted Answer

If the objective function can increase without ever contacting a constraint, the LP model is unbounded. This means that there is no optimal solution because the objective function can be made arbitrarily large without violating any of the constraints.

Question 6

Suppose that a constraint 2x₁+3x₂ ≥ 900 is binding. Then, a constraint 4x₁+6x₂ ≥ 600 is

Accepted Answer

If a constraint is binding, it means that it is already being fully utilized in the optimal solution. Adding another constraint with the same variables would not change the optimal solution or the feasible region. Therefore, it is redundant.

Question 7

A production optimization problem has 4 decision variables and a requirement that at least b₁ units of material 1 are consumed. Which of the following constraints reflects this fact?

Accepted Answer

The constraint should state that the consumption of material 1 must be greater than or equal to the specified requirement (b₁), which is reflected in option B. Option A states the consumption must be less than or equal to the requirement, which is incorrect. Option C states an equality, which is not necessary in this case, and option D states inequality, which does not reflect the requirement.

Question 8

Level curves are used when solving LP models using the graphical method. To what part of the model do level curves relate?

Accepted Answer

Level curves relate to the objective function in linear programming models. They represent different values of the objective function and are used to identify the optimal solution graphically by finding where these curves touch the feasible region defined by the constraints.

Question 9

The objective function for a LP model is 3 X₁ + 2 X₂. If X₁ = 20 and X₂ = 30, what is the value of the objective function?

Accepted Answer

The objective function is calculated as 3*20 + 2*30 = 60 + 60 = 120.

Question 10

A common objective in the product mix problem is

Accepted Answer

The common objective in the product mix problem is to maximize profit. The goal is to determine the combination of products to produce that will result in the highest overall profit for the company. Maximizing cost, minimizing production time, and maximizing production volume may be related factors to consider, but the ultimate goal is to maximize profit.

Question 11

In a mathematical formulation of an optimization problem, the objective function is written as z=2x₁+3x₂. Then:

Accepted Answer

x₁ is a decision variable because it is one of the variables that can be controlled to optimize the objective function.

Question 12

Which of the following special conditions in an LP model represent potential errors in the mathematical formulation?

Accepted Answer

The answer of Which of the following special conditions in...

Question 13

A production optimization problem has 4 decision variables and resource 1 limits how many of the 4 products can be produced. Which of the following constraints reflects this fact?

Accepted Answer

The constraint reflects a limit on the production of the 4 products due to resource 1, meaning the function of the decision variables must be less than or equal to the limit imposed by resource 1.

Question 14

A company uses 4 pounds of resource 1 to make each unit of X₁ and 3 pounds of resource 1 to make each unit of X₂. There are only 150 pounds of resource 1 available. Which of the following constraints reflects the relationship between X₁, X₂ and resource 1?

Accepted Answer

Each unit of X₁ requires 4 pounds of resource 1 while each unit of X₂ requires 3 pounds of resource 1. To find the relationship between the two variables and the available resource, we can use the inequality: 4X₁ + 3X₂ ≤ 150 This is because the company cannot use more than 150 pounds of resource 1. Therefore, B is the correct option.

Question 15

A mathematical programming application employed by a shipping company is most likely

Accepted Answer

A shipping company would most likely use mathematical programming for optimizing their routes and logistics in order to minimize costs and maximize efficiency.

Question 16

The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X₁ = number of product 1 produced in each batch X₂ = number of product 2 produced in each batch How many units of resource one (the first constraint) are used if the company produces 10 units of product 1 and 5 units of product 2?

Accepted Answer

The answer of The following linear programming problem has been...

Question 17

The constraint for resource 1 is 5 X₁ + 4 X₂ ≥ 200. If X₂ = 20, what it the minimum value for X₁?

Accepted Answer

5 X₁ + 4 X₂ ≥ 2005 X₁ + 4 (20) ≥ 2005 X₁ + 80 ≥ 2005 X₁ ≥ 120X₁ ≥ 24

Question 18

A common objective when manufacturing printed circuit boards is

Accepted Answer

The objective is to minimize the total distance the drill bit must be moved to increase efficiency and reduce manufacturing time.

Question 19

What most motivates a business to be concerned with efficient use of their resources?

Accepted Answer

Resources are limited and valuable, and efficient use of resources ensures that the business can operate sustainably and profitably in the long term. Inefficient use of resources can lead to waste, higher costs, and decreased productivity, all of which can harm the bottom line.

Question 20

The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X₁ = number of product 1 produced in each batch X₂ = number of product 2 produced in each batch How many units of resource 1 are consumed by each unit of product 1 produced?

Accepted Answer