Deck 2: Linear Programming: Model Formulation and Graphical Solution

Question 1

A feasible solution violates at least one of the constraints.

Answer 1

False

Question 2

In linear programming models , objective functions can only be maximized.

Answer 2

False

Question 3

When using the graphical method, only one of the four quadrants of an xy-axis needs to be drawn.

Answer 3

True

Question 4

The equation 8xy = 32 satisfies the proportionality property of linear programming.

Question 5

Proportionality means the slope of a constraint is proportional to the slope of the objective function.

Question 6

Linear programming is a model consisting of linear relationships representing a firm's decisions given an objective and resource constraints.

Question 7

A constraint is a linear relationship representing a restriction on decision making.

Question 8

Typically, finding a corner point for the feasible region involves solving a set of three simultaneous equations.

Question 9

The terms in the objective function or constraints are multiplicative.

Question 10

All model parameters are assumed to be known with certainty.

Question 11

The feasible solution area contains infinite solutions to the linear program.

Question 12

A linear programming model consists of only decision variables and constraints.

Question 13

The values of decision variables are continuous or divisible.

Question 14

A parameter is a numerical value in the objective function and constraints.

Question 15

Linear programming models exhibit linearity among all constraint relationships and the objective function.

Question 16

All linear programming models exhibit a set of constraints.

Question 17

The objective function always consists of either maximizing or minimizing some value.

Question 18

The terms in the objective function or constraints are additive.

Question 19

The objective function is a linear relationship reflecting the objective of an operation.

Question 20

Objective functions in linear programs always minimize costs.

Question 21

A minimization model of a linear program contains only surplus variables.

Question 22

A ________ is a linear relationship representing a restriction on decision making.

Question 23

If at least one constraint in a linear programming model is violated, the solution is said to be ________.

Question 24

There is exactly one optimal solution point to a linear program.

Question 25

In the graphical approach, simultaneous equations may be used to solve for the optimal solution point.

Question 26

The following equation represents a resource constraint for a maximization problem: X + Y ≥ 20.

Question 27

A graphical solution is limited to solving linear programming problems with ________ decision variables

Question 28

The optimal solution for a graphical linear programming problem is the corner point that is the farthest from the origin.

Question 29

Multiple optimal solutions can occur when the objective function line is ________ to a constraint line.

Question 30

If the objective function is parallel to a constraint, the constraint is infeasible.

Question 31

Multiple optimal solutions occur when constraints are parallel to each other.

Question 32

The first step in formulating a linear programming model is to define the objective function

Question 33

Surplus variables are only associated with minimization problems.

Question 34

Slack variables are only associated with maximization problems.

Question 35

A manufacturer using linear programming to decide the best product mix to maximize profit typically has a(n) ________ constraint included in the model.

Question 36

The ________ solution area is an area bounded by the constraint equations.

Question 37

When a maximization problem is ________, the objective function can increase indefinitely without reaching a maximum value.

Question 38

The ________ is a linear relationship reflecting the objective of an operation.

Question 39

Graphical solutions to linear programming problems have an infinite number of possible objective function lines.

Question 40

________ are mathematical symbols representing levels of activity.

Question 41

Solve the following graphically:
Max z = 3x1 + 4x2
s.t.   x1 + 2x2 ?16
   2x1 + 3x2 ? 18
   x1 ? 2
   x2 ? 10
   x1, x2 ? 0
What are the optimal values of x1, x2, and z?

Question 42

Consider the following linear program:
MIN Z = 60A + 50B
s.t.  10A + 20B ? 200
   8A + 5B ? 80
   A ? 2
   B ? 5
Solve this linear program graphically and determine the optimal quantities of A, B, and the value of Z.

Question 43

The constraint 2X +XY violates the ________ property of linear programming.

Question 44

If the objective function is parallel to a constraint, the linear program could have ________.

Question 45

The management scientist constructed a linear program to help the alchemist maximize his gold production process. The computer model chugged away for a few minutes and returned an answer of infinite profit., which is what might be expected from a(n) ________ problem.

Question 46

In a constraint, the ________ variable represents unused resources.

Question 47

________ are at the endpoints of the constraint line segment that the objective function parallels.

Question 48

Consider the following minimization problem:
Min z = x1 + 2x2
   s.t. x1 + x2 ? 300
   2x1 + x2 ? 400
   2x1 + 5x2 ? 750
   x1, x2 ? 0
Which constraints are binding at the optimal solution? (x1 =250, x2 = 50)

Question 49

A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.
If this is a maximization, which extreme point is the optimal solution?

Question 50

________ is the difference between the left- and right-hand sides of a greater than or equal to constraint.

Question 51

The best feasible solution is ________.

Question 52

Corner points on the boundary of the feasible solution area are called ________ points.

Question 53

A linear programming problem that results in a solution that is ________ usually indicates that the linear program has been incorrectly formulated.

Question 54

Consider the following linear program:
MAX Z = 60A + 50B
s.t.  10A + 20B ? 200
   8A + 5B ? 80
   A ? 2
   B ? 5
Solve this linear program graphically and determine the optimal quantities of A, B, and the value of Z.

Question 55

Consider the following minimization problem:
Min z = x1 + 2x2
s.t.   x1 + x2 ? 300
    2x1 + x2 ? 400
    2x1 + 5x2 ? 750
    x1, x2 ? 0
What is the optimal solution?

Question 56

The ________ property of linear programming models indicates that the values of all the model parameters are known and are assumed to be constant.

Question 57

The ________ step in formulating a linear programming model is to define the decision variables.

Question 58

The ________ property of linear programming models indicates that the rate of change, or slope, of the objective function or a constraint is constant.

Question 59

The ________ property of linear programming models indicates that the decision variables cannot be restricted to integer values and can take on any fractional value.

Question 60

A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.
If this is a minimization, which extreme point is the optimal solution?

Question 61

Consider the following linear programming problem:
MIN Z =    3x1 + 2x2
Subject to:    2x1 + 3x2 ? 12
    5x1 + 8x2 ? 37
    x1, x2 ? 0
At the optimal solution point, the objective function value is 18. If the constraints are changed from greater than to less than constraints and the objective function is changed from minimize to maximize, what happens to the optimal solution? Demonstrate whether it falls at the same optimal point.

Question 62

A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.
What would the be the new slope of the objective function if multiple optimal solutions occurred along line segment AB?

Question 63

In a linear programming problem, a valid objective function can be represented as:

A) Max Z = 5xy
B) Max Z 5x2 + 2y2
C) Max 3x + 3y + 1/3 z
D) Min (x1 + x2) / x3

Question 64

Consider the following linear programming problem:
MIN Z =   & 2x1 + 3x2
Subject to:    x1 + 2x2 ? 20
    5x1 + x2 ? 40
    4x1 +6x2 ? 60
    x1 , x2 ? 0
What is the optimal solution?

Question 65

The minimization of cost or maximization of profit is the

A) constraint of operations management.
B) goal of management science.
C) objective of linear programming.
D) assumption of financiality.

Question 66

In a linear programming problem, the binding constraints for the optimal solution are:
5x1 + 3x2 ≤ 30
2x1 + 5x2 ≤ 20
Which of these objective functions will lead to the same optimal solution?

A) 2x1 + 1x2
B) 7x1 + 8x2
C) 80x1 + 60x2
D) 25x1 + 15x2

Question 67

A linear programming model consists of

A) decision variables.
B) an objective function.
C) constraints.
D) all of the above

Question 68

In a linear programming problem, the binding constraints for the optimal solution are:
5x1 + 3x2 ≤ 30
2x1 + 5x2 ≤ 20
As long as the slope of the objective function stays between ________ and ________, the current optimal solution point will remain optimal.

Question 69

Consider the following linear programming problem:
Max Z = $15x + $20y
Subject to:  8x + 5y ? 40
     0.4x + y ? 4
    x, y ? 0
Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is the amount of slack associated with the first constraint?

Question 70

Consider the following linear programming problem:
Max Z =    3x1 + 3x2
Subject to:    10x1 + 4x2 ? 60
     25x1 + 50x2 ? 200
    x1, x2 ? 0
Find the optimal profit and the values of x1 and x2 at the optimal solution.

Question 71

Given this set of constraints, for what objective function is the point x=5, y=3 in the feasible region?
s.t 3x + 6y ≤ 30
10x + 10y ≤ 60
10x + 15y ≤ 90

Question 72

Consider the following linear programming problem:
Max Z =    $3x + $9y
Subject to:    20x + 32y ? 1600
      4x + 2y ? 240
      y ? 40
      x, y ? 0
Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?

Question 73

Consider the following linear programming problem:
MIN Z =    3x1 + 2x2
Subject to:    2x1 + 3x2 ? 12
    5x1 + 8x2 ? 37
    x1, x2 ? 0
What is minimum cost and the value of x1 and x2 at the optimal solution?

Question 74

Which of the following could not be a linear programming problem constraint?

A) 1A + 2B ≠ 3
B) 1A + 2B = 3
C) 1A + 2B ≤ 3
D) 1A + 2B ≥ 3

Question 75

Consider the following linear programming problem:
MIN Z =    10x1 + 20x2
Subject to:    x1 + x2 ? 12
    2x1 + 5x2 ? 40
    x2 ? 13
    x1, x2 ? 0
At the optimal solution, what is the value of surplus associated with constraint 1 and constraint 3, respectively?

Question 76

The poultry farmer decided to make his own chicken scratch by combining alfalfa and corn in rail car quantities. A rail car of corn costs $400 and a rail car of alfalfa costs $200. The farmer's chickens have a minimum daily requirement of vitamin K (500 milligrams) and iron (400 milligrams), but it doesn't matter whether those elements come from corn, alfalfa, or some other grain. A unit of corn contains 150 milligrams of vitamin K and 75 milligrams of iron. A unit of alfalfa contains 250 milligrams of vitamin K and 50 milligrams of iron. Formulate the linear programming model for this situation.

Question 77

Consider the following linear programming problem:
MIN Z =    10x1 + 20x2
Subject to:    x1 + x2 ? 12
    2x1 + 5x2 ? 40
    x2 ? 13
    x1, x2 ? 0
What is minimum cost and the value of x1 and x2 at the optimal solution?

Question 78

Consider the following linear programming problem:
Max Z =    5x1 + 3x2
Subject to:    6x1 + 2x2 ? 18
      15x1 + 20x2 ? 60
      x1 , x2 ? 0
Find the optimal profit and the values of x1 and x2 at the optimal solution.

Question 79

Decision variables

A) measure the objective function.
B) measure how much or how many items to produce, purchase, hire, etc.
C) always exist for each constraint.
D) measure the values of each constraint.

Question 80

A company producing a standard line and a deluxe line of dishwashers has the following time requirements (in minutes) in departments where either model can be processed.
$$\begin{array} { | l | l | l | } \hline & \text { Standard } & { \text { Deluxe } } \\\hline \text { Stamping } & 3 & 6 \\\hline \text { Motor installation } & 10 & 10 \\\hline \text { Wiring } & 10 & 15 \\\hline\end{array}$$
The standard models contribute $20 each and the deluxe $30 each to profits. Because the company produces other items that share resources used to make the dishwashers, the stamping machine is available only 30 minutes per hour, on average. The motor installation production line has 60 minutes available each hour. There are two lines for wiring, so the time availability is 90 minutes per hour.
Let x = number of standard dishwashers produced per hour
y = number of deluxe dishwashers produced per hour
Write the formulation for this linear program.