Deck 17: the Simplex Solution Method

Question 1

In using the simplex method, the number of basic variables is equal to the number of constraints.

Answer 1

True

Question 2

Row operations are used to solve simultaneous equations where equations are multiplied by constants and added or subtracted from each other.

Answer 2

True

Question 3

A basic feasible solution satisfies the model constraints and has the same number of variables with negative values as there are constraints.

Answer 3

False

Question 4

The simplex method can be used to solve quadratic programming problems.

Question 5

A basic feasible solution satisfies the model constraints and has the same number of variables with non-negative values as there are constraints.

Question 6

The simplex method is a general mathematical solution technique for solving linear programming problems.

Question 7

The first step in solving a linear programming model manually with the simplex method is to convert the model into standard form.

Question 8

The simplex method cannot be used to solve quadratic programming problems.

Question 9

The simplex method moves from one better solution to another until the best one is found, and then it stops.

Question 10

The basic feasible solution in the initial simplex tableau is the origin where all decision variables equal zero.

Question 11

The simplex method does not guarantee an integer solution.

Question 12

At the initial basic feasible solution at the origin, only slack variables have a value greater than zero.

Question 13

The mathematical steps in the simplex method replicate the process in graphical analysis of moving from one extreme point on the solution boundary to another.

Question 14

The last step in solving a linear programming model manually with the simplex method is to convert the model into standard form.

Question 15

In solving a linear programming problem with simplex method, the number of basic variables is the same as the number of constraints in the original problem

Question 16

Slack variables are added to constraints and represent unused resources.

Question 17

Artificial variables are added to constraints and represent unused resources.

Question 18

In the simplex method, the model is put into the form of a table, and then a number of mathematical steps are performed on the table.

Question 19

At the initial basic feasible solution at the origin, only slack variables have a value greater than 1.

Question 20

The simplex method is a general mathematical solution technique for solving nonlinear programming problems.

Question 21

Multiple optimal solutions cannot be determined from the simplex method.

Question 22

The variable with the largest positive cj - zj is the ________ variable.

Question 23

The theoretical limit on the number of decision variables that can be handled by the simplex method is 50.

Question 24

A(n) ________ maximization linear programming problem has an artificial variable in the final simplex tableau where all cj - zj values are less than or equal to zero.

Question 25

The dual form of a linear program is used to determine how much one should pay for additional resources.

Question 26

Whereas the maximization primal model has ≤ constraints, the ________ dual model has ≥ constraints.

Question 27

In using the simplex method, ________ optimal solutions are identified by cj - zj = 0 for a non-basic variable.

Question 28

If the primal problem has three constraints, then the corresponding dual problem will have three ________.

Question 29

The first step in solving a linear programming model manually with the simplex method is to convert the model into ________ form.

Question 30

When solving a linear programming problem, a decision variable that leaves the basis in one iteration of the simplex method can return to the basis on a later iteration.

Question 31

The ________ column is the column corresponding to the entering variable.

Question 32

A change in the objective function coefficient of a basic variable cannot change the value of zj for a non-basic variable in the final simplex tableau.

Question 33

The quantity values on the right-hand side of the primal inequality constraints are the ________ coefficients in the dual.

Question 34

A primal maximization model with ≤ constraints converts to a ________ minimization model with constraints.

Question 35

The ________ variable allows for an initial basic feasible solution, but it has no meaning. Therefore, after we get the simplex tableau started, they are discarded in later iterations.

Question 36

Final tableaus cannot be used to conduct sensitivity analysis.

Question 37

In solving a minimization problem, artificial variables are assigned a ________ in the objective function to eliminate them from the final solution.

Question 38

The ________ values are computed by multiplying the cj column values by the variable column values and summing.

Question 39

________ variables are added to constraints and represent unused resources.

Question 40

The ________ values are contribution to profit for each variable.

Question 41

Given the following linear programming problem:
$\begin{array} { l l } \operatorname { maximize } & 4 x _ { 1 } + 3 x _ { 2 } \\\text { subject to } & 4 x _ { 1 } + 3 x _ { 2 } \leq 23 \\& 5 x _ { 1 } - x _ { 2 } \leq 5 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$
What are the basic variables in the initial tableau?

Question 42

To determine the sensitivity range for the coefficient of a variable in the objective function, calculations are performed such that all values in the cj - zj row are ________.

Question 43

________ in linear programming is when a basic variable takes on a value of zero (i.e., a zero in the right-hand side of the constraints of the tableau).

Question 44

Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the optimal value for the objective function?

Question 45

Given the following linear programming problem:
$\begin{array} { l l } \operatorname { maximize } & \mathrm { Z } = \$ 100 x _ { 1 } + 80 x _ { 2 } \\\text { subject to } & x _ { 1 } + 2 x _ { 2 } \leq 40 \\& 3 x _ { 1 } + x _ { 2 } \leq 60 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$
Using the simplex method, what is the value for S₁ in the final basic feasible solution?

Question 46

The ________ form of a linear program is used to determine how much one should pay for additional resources.

Question 47

Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the optimal value for X1?

Question 48

Given the following linear programming problem:
$\begin{array} { l l } \operatorname { maximize } & 4 x _ { 1 } + 3 x _ { 2 } \\\text { subject to } & 4 x _ { 1 } + 3 x _ { 2 } \leq 23 \\& 5 x _ { 1 } - x _ { 2 } \leq 5 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$
What are the Cj values for the basic variables?

Question 49

A(n) ________ problem can be identified in the simplex procedure when it is not possible to select a pivot row.

Question 50

Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the value of X1 in the final tableau?

Question 51

In a ________ problem, artificial variables are assigned a very high cost.

Question 52

Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the optimal value for X2?

Question 53

Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the value of x2 in the final tableau?

Question 54

Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the optimal value of this objective function?

Question 55

Solve the following problem using the simplex method.
$\begin{array} { l } \text { Minimize } \mathrm { Z } = 3 x _ { 1 } + 4 x _ { 2 } + 8 x _ { 3 } \\\text { Subject to: } \quad 2 x _ { 1 } + x _ { 2 } \geq 6 \\      x _ { 2 } + 2 x _ { 3 } \geq 4 \\       x _ { 1 } , x _ { 2 } \geq 0 \\\end{array}$

Question 56

Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
How many iterations did we have to perform before reaching the final tableau?

Question 57

Given the following linear programming problem:
$\begin{array} { l l } \operatorname { maximize } & \mathrm { Z } = \$ 100 x _ { 1 } + 80 x _ { 2 } \\\text { subject to } & x _ { 1 } + 2 x _ { 2 } \leq 40 \\& 3 x _ { 1 } + x _ { 2 } \leq 60 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$
Using the simplex method, what is the value for S₂ in the optimal tableau?

Question 58

Given the following linear programming problem:
$\begin{array} { l l } \operatorname { maximize } & 4 x _ { 1 } + 3 x _ { 2 } \\\text { subject to } & 4 x _ { 1 } + 3 x _ { 2 } \leq 23 \\& 5 x _ { 1 } - x _ { 2 } \leq 5 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$
What is the (Ci- Zi) value for S2 at the initial solution?

Question 59

Solve the following problem using the simplex method.
$\begin{array} { l } \text { Minimize } \mathrm { Z } = 2 x _ { 1 } + 6 x _ { 2 } \\\text { Subject to: } \quad 2 x _ { 1 } + 4 x _ { 2 } \leq 12 \\\qquad 3 x _ { 1 } + 2 x _ { 2 } \geq 9 \\x _ { 1 } , x _ { 2 } \geq 0\end{array}$

Question 60

Given the following linear programming problem:

$\begin{array} { l l } \operatorname { maximize } & 4 x _ { 1 } + 3 x _ { 2 } \\\text { subject to } & 4 x _ { 1 } + 3 x _ { 2 } \leq 23 \\& 5 x _ { 1 } - x _ { 2 } \leq 5 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$
What is the (Cj - Zj) value for S1 at the initial solution?

Question 61

Slack variables are added to ________ constraints and represent unused resources.

A) ≤
B) <
C) ≥
D) >
E) =

Question 62

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-To what value can the profit on red nail polish drop before the solution would change?

Question 63

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-By how much will the second marketing restriction be exceeded?

Question 64

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-By how much can the amount of space decrease before there is a change in the profit?

Question 65

The basic feasible solution in the initial simplex tableau is the origin where all decision variables equal:

A) 0
B) 1
C) -1
D) 1 or -1

Question 66

At the initial basic feasible solution at the origin, only slack variables have a value greater than:

A) 0
B) 1
C) -1
D) 1 or -1

Question 67

Consider the following linear programming problem and the corresponding final tableau.
MAX Z = 3x1 + 5x2
s.t. x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≥ 18
What is the sensitivity range for the first constraint?

Question 68

You are offered the chance to obtain more space. The offer is for 15 units and the total price is 1500. What should you do?

Question 69

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-By how much can the profit on green nail polish increase before the solution would change?

Question 70

Consider the following linear programming problem and the corresponding final tableau.
MAX Z = 3x1 + 5x2
s.t. x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≥ 18
What is the shadow price for each constraint?

Question 71

The ________ step in solving a linear programming model manually with the simplex method is to convert the model into standard form.

A) first
B) second
C) last
D) only

Question 72

Row operations are used to solve simultaneous equations where equations are ________ by constants and added to or subtracted from each other.

A) converted
B) restrained
C) divided
D) multiplied

Question 73

The simplex method is a general mathematical solution technique for solving ________ programming problems.

A) integer
B) non-linear
C) linear
D) A, B, and C

Question 74

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-What is the profit?

Question 75

At the initial basic feasible solution at the origin, only ________ variables have a value greater than zero.

A) linear
B) slack
C) non-linear
D) integer

Question 76

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-How much time will be used?

Question 77

The simplex method ________ be used to solve quadratic programming problems.

A) can
B) cannot
C) may
D) should

Question 78

Write the dual form of the following linear program.
MAX Z = 3x1 + 5x2
s.t. x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≥ 18

Question 79

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$

-How much space will be left unused?

Question 80

Consider the following linear programming problem:
$\begin{array} { l l } \text { MAX } & \mathrm { Z } = 10 x _ { 1 } + 30 x _ { 2 } \\\text { s.t. } & 4 x _ { 1 } + 6 x _ { 2 } \leq 12 \\& 8 x _ { 1 } + 4 x _ { 2 } \leq 16\end{array}$

Use the two tables below to create the initial tableau and perform 1 pivot.