Deck 13: Introduction to Optimization Modeling

Question 1

Reduced costs indicate how much the objective coefficient of a decision variable that is currently 0 or at its upper bound must change before that the value of that variable changes.

Answer 1

True

Question 2

It is instructive to look at a graphical solution procedure for LP models with three or more decision variables.

Answer 2

False

Question 3

The proportionality property of LP models means that if the level of any activity is multiplied by a constant factor,then the contribution of this activity to the objective function,or to any of the constraints in which the activity is involved,is multiplied by the same factor.

Answer 3

True

Question 4

In general,the complete solution of a linear programming problem involves three stages: formulating the model,invoking Solver to find the optimal solution,and performing sensitivity analysis.

Question 5

If an LP model does have an unbounded solution,then we must have made a mistake - either we made an input error or we omitted one or more constraints.

Question 6

Proportionality,additivity,and divisibility are three important properties that LP models possess,which distinguish them from general mathematical programming models:.

Question 7

There is often more than one objective in linear programming problems

Question 8

Suppose the allowable increase and decrease for shadow price for a constraint are $25 (increase)and $10 (decrease).If the right hand side of that constraint were to increase by $10 the optimal value of the objective function would change.

Question 9

If a constraint has the equation $20 x + 10 y \leq 1000$ ,then the slope of the constraint line is function line is -2:

Question 10

When formulating a linear programming spreadsheet model,we specify the constraints in a Solver dialog box,since Excel does not show the constraints directly.

Question 11

Unboundedness refers to the situation in which the LP model has been formulated in such a way that the objective function is unbounded - that is,it can be made as large (for maximization problems)or as small (for minimization problems)as we like.

Question 12

All linear programming problems should have a unique solution,if they can be solved.

Question 13

All optimization problems include decision variables,an objective function,and constraints.

Question 14

When formulating a linear programming spreadsheet model,there is one target (objective)cell that contains the value of the objective function.

Question 15

In determining the optimal solution to a linear programming problem graphically,if the objective is to maximize the objective,we pull the objective function line down until it contacts the feasible region.

Question 16

Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase)and $10 (decrease).If the coefficient were to change from $50 to $60,the optimal value of the objective function would not change.

Question 17

Linear programming problems can always be formulated algebraically,but not always on spreadsheet.

Question 18

If a solution to an LP problem satisfies all of the constraints,then is must be feasible.

Question 19

Shadow prices are associated with nonbinding constraints,and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.

Question 20

When formulating a linear programming spreadsheet model,there is a set of designated cells that play the role of the decision variables.These are called the objective cells.

Question 21

There are two primary ways to formulate a linear programming problem,the traditional algebraic way and in spreadsheets.

Question 22

Infeasibility refers to the situation in which there are no feasible solutions to the LP model

Question 23

In using Excel to solve linear programming problems,the target cell represents the:

A) value of the objective function
B) constraints
C) decision variables
D) total cost of the model

Question 24

The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon,with lines forming all sides.

Question 25

If a constraint has the equation $3 x + 2 y \leq 60$ ,then the constraint line passes through the points (0,20)and (30,0):

Question 26

The optimal solution to any linear programming model is a corner point of a polygon.

Question 27

When the proportionality property of LP models is violated,then we generally must use non-linear optimization.

Question 28

Nonbinding constraints will always have slack,which is the difference between the two sides of the inequality in the constraint equation.

Question 29

The feasible region in all linear programming problems is bounded by:

A) corner points
B) hyperplanes
C) an objective line
D) all of these options

Question 30

The additivity property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.

Question 31

There are generally two steps in solving an optimization problem,model development and optimization.

Question 32

If the objective function has the equation $\min \{ 15 x + 25 y \}$ ,then the y-intercept of the objective function line is 40:

Question 33

If the objective function has the equation $\min \{ 4 x + 2 y \}$ ,then the slope of the objective function line is 2:

Question 34

The divisibility property of LP models simply means that we allow only integer levels of the activities.

Question 35

Consider the following linear programming problem:
Maximize $2 x _ { 1 } + 2 x _ { 2 }$
Subject to $4 x _ { 1 } + 3 x _ { 2 } \geq 12$
$- 2 x _ { 1 } - 3 x _ { 2 } \leq 6$
$x _ { 2 } \geq 2$
$x _ { 1 } , x _ { 2 } \geq 0$
The above linear programming problem:

A) has only one optimal solution
B) has more than one optimal solution
C) exhibits infeasibility
D) exhibits unboundedness

Question 36

Suppose a firm must at least meet minimum expected demands of 60 for product x and 80 of product y.An algebraic formulation of these constraints is:

A)
$x + y \geq 140$
B)
$60 x + 80 y \geq 140$
C)
$x + y = 140$
D)
$x \geq 60 , y \geq 80$

Question 37

It is often useful to perform sensitivity analysis to see how,or if,the optimal solution to a linear programming problem changes as we change one or more model inputs.

Question 38

The set of all values of the changing cells that satisfy all constraints,not including the nonnegativity constraints,is called the feasible region.

Question 39

It helps to ensure that Solver can find a solution to a linear programming problem if the model is well-scaled;that is,all of the numbers are of roughly the same magnitude.

Question 40

In an optimization model,there can only be one:

A) decision variable
B) constraint
C) objective function
D) shadow price

Question 41

When there is a problem with Solver being able to find a solution,many times it is an indication of a (n):

A) older version of Excel
B) nonlinear programming problem
C) problem that cannot be solved using linear programming
D) mistake in the formulation of the problem

Question 42

Related to sensitivity analysis in linear programming,when the profit increases with a unit increase in a resource,this change in profit is referred to as the:

A) add-in price
B) sensitivity price
C) shadow price
D) additional profit

Question 43

The solution of a linear programming problem using Microsoft Excel typically involves the following three stages:

A) formulating the problem,invoking Solver,and sensitivity analysis
B) formulating the problem,graphing the problem,and sensitivity analysis
C) the changing cells,the target cells,and the constraints
D) the inputs,the changing cells,and the outputs

Question 44

The most important solution method for linear programming problems is known as the:

A) spreadsheet method
B) solution mix method
C) complex method
D) simplex method

Question 45

Every linear programming problem involves optimizing a:

A) linear regression model subject to several linear constraints
B) linear function subject to several linear constraints
C) linear function subject to several non-linear constraints
D) non-linear function subject to several linear constraints

Question 46

The equation of the line representing the constraint $2 x + 4 y \leq 80$
Passes through the points:

A)
$( 20,0 ) \text { and } ( 0,40 )$
B)
$( 0,20 ) \text { and } ( 40,0 )$
C)
$( - 20,0 ) \text { and } ( 0 , - 40 )$
D)
$( 0 , - 20 ) \text { and } ( - 40,0 )$

Question 47

In linear programming we can use the shadow price to calculate increases or decreases in:

A) binding constraints
B) nonbinding constraints
C) values of the decision variables
D) the value of the objective function

Question 48

A linear programming problem with _____decision variable(s)can be solved by a graphical solution method.

A) 1
B) 2
C) 3
D) 4

Question 49

In most cases in solving linear programming problems,we want the decision variables to be:

A) equal to zero
B) nonnegative
C) nonpositive
D) All of these options

Question 50

The equation of the line representing the constraint $20 x + 10 y \leq 1000$
Is:

A)
$y \leq - 20 x + 1000$
B)
$y \leq 2 x + 100$
C)
$- y \leq 2 x + 100$
D)
$y \leq - 2 x + 100$

Question 51

Suppose a liquor store sells beer for a net profit of $1 per unit and wine for a net profit of $2 per unit.Let x equal the amount of beer sold and y equal the amount of wine sold.An algebraic formulation of the profit function is:

A)
$\min ( x + 2 y )$
B)
$\min ( 2 x + y )$
C)
$\max ( x + 2 y )$
D)
$\max ( 2 x + y )$

Question 52

Linear programming is a subset of a larger class of models called:

A) mathematical programming models
B) mathematical optimality models
C) linear regression models
D) linear simplex model

Question 53

Consider the following linear programming problem:
Maximize $5 x _ { 1 } + 5 x _ { 2 }$
Subject to $x _ { 1 } + 2 x _ { 2 } \leq 8$
$x _ { 1 } + x _ { 2 } \leq 6$
$x _ { 1 } , x _ { 2 } \geq 0$
The above linear programming problem:

A) has only one optimal solution
B) has more than one optimal solution
C) exhibits infeasibility
D) exhibits unboundedness

Question 54

The additivity property of linear programming implies that the contribution of any decision variable to the objective is of/on the levels of the other decision variables.

A) dependent
B) independent
C) conditional
D) the sum

Question 55

In using Excel to solve linear programming problems,the changing cells represent the:

A) value of the objective function
B) constraints
C) decision variables
D) total cost of the model

Question 56

Consider the following linear programming problem:
Maximize $2 x _ { 1 } + 4 x _ { 2 }$
Subject to $x _ { 1 } + x _ { 2 } \leq 5$
$- x _ { 1 } + x _ { 2 } \geq 8$
$x _ { 1 } , x _ { 2 } \geq 0$
The above linear programming problem:

A) has only one optimal solution
B) has more than one optimal solution
C) exhibits infeasibility
D) exhibits unboundedness

Question 57

Suppose a company sells two different products,x and y,for net profits of $5 per unit and $10 per unit,respectively.The slope of the line representing the objective function is:

A) 0.5
B) -0.5
C) 2
D) -2

Question 58

In linear programming,sensitivity analysis involves examining how sensitive the optimal solution is to changes in:

A) profit of variables in model
B) cost of variables in model
C) resources available
D) All of these options

Question 59

In some cases,a linear programming problem can be formulated such that the objective can become infinitely large (for a maximization problem)or infinitely small (for a minimization problem).This type of problem is said to be:

A) infeasible
B) inconsistent
C) unbounded
D) redundant

Question 60

The divisibility property of linear programming means that a solution can have both:

A) integer and noninteger levels of an activity
B) linear and nonlinear relationships
C) positive and negative values
D) revenue and cost information in the model

Question 61

Consider the following linear programming problem:
Minimize $5 x _ { 1 } + 6 x _ { 2 }$
Subject to $x _ { 1 } + 2 x _ { 2 } \geq 12$
$3 x _ { 1 } + 2 x _ { 2 } \geq 24$
$3 x _ { 1 } + x _ { 2 } \geq 15$
$x _ { 1 } , x _ { 2 } \geq 0$
The above linear programming problem:

A) has only one optimal solution
B) has more than one optimal solution
C) exhibits infeasibility
D) exhibits unboundedness

Question 62

The prototype linear programming problem is to select an optimal mix of products to produce to maximize profit.This type of problem is referred to as the:

A) product mix problem
B) production problem
C) product/process problem
D) product scheduling problem

Question 63

When using the graphical solution method to solve linear programming problems,the set of points that satisfy all constraints is called the:

A) optimal region
B) feasible region
C) constrained region
D) logical region

Question 64

All optimization problems have:

A) an objective function and decision variables
B) an objective function and constraints
C) decision variables and constraints
D) an objective function,decision variables and constraints
A company produces two products.Each product can be produced on either of two machines.The time (in hours)required to produce each product on each machine is shown below:
Machine 1
Machine 2
Product 1
5
4
Product 2
8
5
Each month,600 hours of time are available on each machine,and that customers are willing to buy up to the quantities of each product at the prices that are shown below:
​
Demands
Prices
Month 1
Month 2
Month 1
Month 2
Product 1
120
200
$60
$15
Product 2
150
130
$70
$35
​
The company's goal is to maximize the revenue obtained from selling units during the next two months.

Question 65

Linear programming models have three important properties.They are:

A) optimality,additivity and sensitivity
B) optimality,linearity and divisibility
C) divisibility,linearity and nonnegativity
D) proportionality,additivity and divisibility

Question 66

The term nonnegativity refers to the condition where:

A) the objective function cannot be less than zero
B) the decision variables cannot be less than zero
C) the right hand side of the constraints cannot be greater than zero
D) the reduced cost cannot be less than zero

Question 67

One of the things that you can do with linear programming and a spreadsheet model is to develop a user interface to make it easier for someone who is not an expert in using linear programming.The output can be a report that explains the optimal policy in non-technical terms.The type of system being described is called a (n):

A) expert system
B) decision support system
C) linear programming support system
D) production planning system

Question 68

If a manufacturing process takes 3 hours per unit of x and 5 hours per unit of y and a maximum of 100 hours of manufacturing process time are available,then an algebraic formulation of this constraint is:

A)
$3 x + 5 y \geq 100$
B)
$3 x - 5 y \leq 100$
C)
$3 x + 5 y \leq 100$
D)
$3 x - 5 y \geq 100$

Question 69

The optimal solution to any linear programming model is:

A) the maximum objective function line
B) the minimum objective function line
C) the corner point of a polygon
D) the maximum or minimum of a parabola

Question 70

What are the decision variables in this problem?
Western Chassis produces high-quality polished steel and aluminum sheeting and two lines of industrial chassis for the rack mounting of Internet routers,modems,and other telecommunications equipment.The contribution margin (contribution toward profit)for steel sheeting is $0.40 per pound and for aluminum sheeting is $0.60 per pound.Western earns $12 contribution on the sale of a Standard chassis rack and $15 contribution on a Deluxe chassis rack.During the next production cycle,Western can buy and use up to 25,800 pounds of raw unfinished steel either in sheeting or in chassis.Similarly,20,400 pounds of aluminum are available.One standard chassis rack requires 16 pounds of steel and 8 pounds of aluminum.A Deluxe chassis rack requires 12 pounds of each metal.The output of metal sheeting is restricted only by the capacity of the polisher.For the next production cycle,the polisher can handle any mix of the two metals up to 4,000 pounds of metal sheeting.Chassis manufacture can be restricted by either metal stamping or assembly operations;no polishing is required.During the cycle no more than 2,500 total chassis can be stamped,and there will be 920 hours of assembly time available.The assembly time required is 24 minutes for the Standard chassis rack and 36 minutes for the Deluxe chassis rack.Finally,market conditions limit the number of Standard chassis racks sold to no more than 1,200 Standard and no more than 1,000 Deluxe.Any quantities of metal sheeting can be sold.