Deck 18: Simplex-Based Sensitivity Analysis and Duality

Question 1

Given the simplex tableau for the optimal primal solution

A)the values of the dual variables can be found from the c_j -z_j values of the slack/surplus variable columns.
B)the values of the dual surplus variables can be found from the c_j - z_j values of the primal decision variable columns.
C)the value of the dual objective function will be the same as the objective function value for the primal problem.
D)each of the above is true.

Answer 1

each of the above is true.

Question 2

As long as the objective function coefficient remains within the range of optimality, the variable values will not change although the value of the objective function could.

Answer 2

True

Question 3

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

A)the slack value.
B)the dual price.
C)never negative.
D)the 100% rule.

Answer 3

B

Question 4

The range of optimality is calculated by considering changes in the c_j - z_j value of the variable in question.

Question 5

For an objective function coefficient change outside the range of optimality, explain how to calculate the new optimal solution. Must you return to the (revised) initial tableau?

Question 6

A one-sided range of optimality

A)always occurs for non-basic variables.
B)always occurs for basic variables.
C)indicates changes in more than one coefficient.
D)indicates changes in a slack variable's coefficient.

Question 7

If the dual price for b₁ is 2.7, the range of feasibility is 20 $\le$ b₁ $\le$ 50, and the original value of b₁ was 30, which of the following is true?

A)There currently is no slack in the first constraint.
B)We would be willing to pay up to $2.70 per unit for up to 20 more units of resource 1.
C)If only 25 units of resource 1 were available, profit would drop by $13.50.
D)Each of the above is true.

Question 8

A linear programming problem with the objective function 3x₁ + 8x₂ has the optimal solution x₁ = 5, x₂ = 6. If c₂ decreases by 2 and the range of optimality shows 5 $\le$ c₂ $\le$ 12, the value of Z

A)will decrease by 12.
B)will decrease by 2.
C)will not change.
D)cannot be determined from this information.

Question 9

The dual price is the improvement in value of the optimal solution per unit increase in the value of the right-hand side associated with a linear programming problem.

Question 10

The range of feasibility indicates right-hand side values for which

A)the value of the objective function will not change.
B)the values of the decision variables will not change.
C)those variables which are in the basis will not change.
D)more simplex iterations must be performed.

Question 11

If the simplex tableau is from a maximization converted from a minimization, the signs and directions of the inequalities that give the objective function ranges will need to be adjusted to apply to the original coefficients.

Question 12

The dual variable represents

A)the marginal value of the constraint
B)the right-hand side value of the constraint
C)the artificial variable
D)the technical coefficient of the constraint

Question 13

The ranges for which the right-hand side values are valid are the same as the ranges over which the dual prices are valid.

Question 14

The entries in the associated slack column of the final tableau indicate the changes in the values of the current basic variables corresponding to a one-unit increase in the right-hand side.

Question 15

The range of optimality is useful only for basic variables.

Question 16

The dual price for an equality constraint is the z_j value for its artificial variable.

Question 17

There is a dual price associated with each decision variable.

Question 18

The range of optimality for a basic variable defines the objective function coefficient values for which the variable will remain part of the current optimal basic feasible solution.

Question 19

For the basic feasible solution to remain optimal

A)all c_j -z_j values must remain $\le$ 0.
B)no objective function coefficients are allowed to change.
C)the value of the objective function must not change.
D)each of the above is true.

Question 20

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

A)dual analysis.
B)optimality analysis.
C)ranging analysis.
D)sensitivity analysis.

Question 21

Given the following linear programming problem
Max Z
0.5x₁ + 6x₂ + 5x₃
s.t.
4x₁ + 6x₂ + 3x₃ $\le$ 24
1x₁ + 1.5x₂ + 3x₃ $\le$ 12
3x₁ + x₂ $\le$ 12
and the final tableau is $$\begin{array} { c c | c c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\\text { Basis } & \mathrm { c } _ { \mathrm { B } } & .5 & 6 & 5 & 0 & 0 & 0 & \\\hline \mathrm { x } _ { 2 } & 6 & 1 & 1 & 0 & .22 & - .22 & 0 & 2.67 \\\mathrm { x } _ { 3 } & 5 & 0 & 0 & 1 & .11 & - .44 & 0 & 2.67 \\\mathrm {~s} _ { 3 } & 0 & 2.33 & 0 & 0 & - .22 & .22 & 1 & 9.33 \\\hline & \mathrm { z } _ { \mathrm { j } } & 4 & 6 & 5 & .77 & .88 & 0 & 29.33 \\& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & .5 & 0 & 0 & - .77 & - .88 & 0 &\end{array}$$
a.Find the range of optimality for c_1, c_2, c_3, c_4, c_5, and c₆.
b.Find the range of feasibility for b₁, b₂, and b₃.

Question 22

Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

Question 23

For the following linear programming problem
Max Z
-2x₁ + x₂ - x₃
s.t.
2x₁ + x₂ $\le$ 7
1x₁ + x₂ + x₃ $\ge$ 4
the final tableau is $$\begin{array} { c c | c c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm { a } _ { 2 } & \\\text { Basis } & \mathrm { C } _ { \mathrm { B } } & - 2 & 1 & - 1 & 0 & 0 & - \mathrm { M } & \\\hline \mathrm { s } _ { 2 } & 0 & 1 & 0 & - 1 & 1 & 1 & - 1 & 3 \\\mathrm { x } _ { 2 } & 1 & 2 & 1 & 0 & 1 & 0 & 0 & 7 \\\hline & z _ { j } & 2 & 1 & 0 & 1 & 0 & 0 & 7 \\& \mathrm { C } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & - 4 & 0 & 1 & - 1 & 0 & - \mathrm { M } &\end{array}$$
a.Find the range of optimality for c₁, c₂ , c₃.c₄, c₅ , and c₆.
b.Find the range of feasibility for b₁, and b₂.

Question 24

Explain why the z_j value for a slack variable is the dual price.

Question 25

Write the dual of the following problem
Min Z
= 2x₁ -3x₂ + 5x₃
s.t.
-3x₁ + 2x₂ + 5x₃ $\ge$ 7
2x₁ -x₃ $\ge$ 5
4x ₂ + 3x₃ $\ge$ 8.

Question 26

The primal problem is
Min
2x₁ + 5x₂ + 4x₃
s.t.
x₁ + 3x₂ + 3x₃ $\ge$ 30
3x₁ + 7x₂ + 5x₃ $\ge$ 70
x₁, x₂, x₃ $\ge$ 0
The final tableau for its dual problem is $$\begin{array} { c c | c c c c c | c } & & \mathrm { u } _ { 1 } & \mathrm { u } _ { 2 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\\text { Basis } & \mathrm { c } _ { \mathrm { B } } & 30 & 70 & 0 & 0 & 0 & \\\hline \mathrm { u } _ { 2 } & 70 & 0 & 1 & 3 / 4 & 0 & - 1 / 4 & 1 / 2 \\\mathrm {~s} _ { 2 } & 0 & 0 & 0 & - 3 / 2 & 1 & - 1 / 2 & 0 \\\mathrm { u } _ { 1 } & 30 & 1 & 0 & - 5 / 4 & 0 & 3 / 4 & 1 / 2 \\\hline & z _ { \mathrm { j } } & 30 & 70 & 15 & 0 & 5 & 50 \\& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & 0 & - 15 & 0 & - 5 &\end{array}$$ Give the complete solution to the primal problem.

Question 27

Explain the simplex tableau location of the dual constraint for each type of constraint.

Question 28

For this optimal simplex tableau the original right-hand sides were 100 and 90. The problem was a maximization.
a.What would the new solution be if there had been 150 units available in the first constraint?
b.What would the new solution be if there had been 70 units available in the second constraint?

Question 29

The linear programming problem
Max
6x₁ + 2x₂ + 3x₃ + 4x₄
s.t.
x₁ + x₂ + x₃ + x₄ $\le$ 100
4x₁ + x₂ + x₃ + x₄ $\le$ 160
3x₁ + x₂ + 2x₃ + 3x₄ $\le$ 240
x₁, x₂, x ₃, x₄ $\ge$ 0
has the final tableau $$\begin{array} { c c | c c c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm { x } _ { 4 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\\text { Basis } & \mathrm { c } _ { \mathrm { B } } & 6 & 2 & 3 & 4 & 0 & 0 & 0 & \\\hline \mathrm { x } _ { 2 } & 2 & 0 & 1 & 1 / 2 & 0 & 3 / 2 & 0 & - 1 / 2 & 30 \\\mathrm { x } _ { 1 } & 6 & 1 & 0 & 0 & 0 & - 1 / 3 & 1 / 3 & 0 & 20 \\\mathrm { x } _ { 4 } & 4 & 0 & 0 & 1 / 2 & 1 & - 1 / 6 & - 1 / 3 & 1 / 2 & 50 \\\hline & \mathrm { z } _ { \mathrm { j } } & 6 & 2 & 3 & 4 & 2.33 & .67 & 1 & 380 \\& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & 0 & 0 & 0 & - 2.33 & - .67 & - 1 &\end{array}$$ Fill in the table below to show what you would have found if you had used The Management Scientist to solve this problem.
LINEAR PROGRAMMING PROBLEM
MAX
6X1+2X2+3X3+4X4
S.T.
1) 1X1 + 1X2 + 1X3 + 1X4 < 100
2) 4X1 + 1X2 + 1X3 + 1X4 < 160
3) 3X1 + 1X2 + 2X3 + 3X4 < 240
OPTIMAL SOLUTION Objective Function Value $=$
$\begin{array}{ccc}\text { Variable } & \text { Value } & \text { Reduced Cost } \\X 1 & - &- \\X 2 & - &- \\X 3 & - &- \\X 4 & - &-\end{array}$

$\begin{array}{ccc}\text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\1 & - & - \\2 & - & - \\3 & - & -\end{array}$
$\text { OBJECTIVE COEFFICIENT RANGES }$

$\begin{array}{cccc}\text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\mathrm{X} 1 & - & -&- \\\mathrm{X} 2 & - & -&- \\\mathrm{X} 3 & - & -&- \\\mathrm{X} 4 & - & -&-\end{array}$
$\text { RIGHT HAND SIDE RANGES }$

$\begin{array}{cccc}\text { Constraint }& \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\1 & - & -&-\\2 & - & -&-\\3 & - & -&-\\\end{array}$

Question 30

When sensitivity calculations yield several potential upper bounds and several lower bounds, how is the range determined?

Question 31

For this optimal simplex tableau, the right-hand sides for the two original $\ge$ constraints were 300 and 250. The problem was a minimization. $$\begin{array} { c c | c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \\\text { Basis } & \mathrm { C } _ { \mathrm { B } } & - 100 & - 110 & - 95 & 0 & 0 & \\\hline \mathrm { x } _ { 3 } & - 95 & 0 & - 1.5 & 1 & 1 & - 1.5 & 75 \\\mathrm { x } _ { 1 } & - 100 & 1 & 2 & 0 & - 1 & 1 & 50 \\\hline & \mathrm { z } _ { \mathrm { j } } & - 100 & - 57.5 & - 95 & 5 & 42.5 & - 12125 \\& \mathrm { C } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & - 52.5 & 0 & - 5 & - 42.5 &\end{array}$$
a.What would the new solution be if the right-hand side value in the first constraint had been 325?
b.What would the new solution be if the right-hand side value for the second constraint had been 220?

Question 32

Write the dual to the following problem.
Min
12x₁ + 15x₂ + 20x₃ + 18x₄
s.t.
x₁ + x₂ + x₃ + x₄ $\ge$ 50
3x₁ + 4x₃ $\ge$ 60
2x₂ + x₃ - 2x₄ $\le$ 10
x₁, x₂, x₃, x₄ $\ge$ 0

Question 33

Given the following linear programming problem
Max
10x₁ + 12x₂
s.t.
1x₁ + 2x₂ $\ge$ 40
5x₁ + 8x₂ $\le$ 160
1x₁ + 1x₂ $\le$ 40
x₁, x₂ $\ge$ 0
the final tableau is $$\begin{array} { c c | c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\\text { Basis } & \mathrm { c } _ { \mathrm { B } } & 10 & 12 & 0 & 0 & 0 & \\\hline \mathrm { x } _ { 2 } & 12 & 0 & 1 & - 2.5 & - .5 & 0 & 20 \\\mathrm { x } _ { 1 } & 10 & 1 & 0 & 4 & 1 & 0 & 0 \\\mathrm {~s} _ { 3 } & 0 & 0 & 0 & - 1.5 & - 5 & 1 & 20 \\\hline & \mathrm { z } _ { \mathrm { j } } & 10 & 12 & 10 & 4 & 0 & 240 \\& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & 0 & - 10 & - 4 & 0 &\end{array}$$
a.Find the range of optimality for c₁ and c₂.
b.Find the range of feasibility for b₁, b₂, and b₃.
c.Find the dual prices.

Question 34

Creative Kitchen Tools manufactures a wide line of gourmet cooking tools from stainless steel. For the coming production period, there is demand of 1200 for 8 quart stock pots, and unlimited demand for 3 quart mixing bowls and large slotted spoons. In the following model, the three variables measure the number of pots, bowls, and spoons to make. The objective function measures profit. Constraint 1 measures steel, constraint 2 measures manufacturing time, constraint 3 measures finishing time, and constraint 4 measures the stock pot demand.
Max
5x₁ + 3x₂ + 6x₃
s.t.
3x₁ + 1x₂ + 2x₃ $\le$ 15000
4x₁ + 4x₂ + 5x₃ $\le$ 18000
2x₁ + 1x₂ + 2x₃ $\le$ 10000
x₁ $\le$ 1200
x₁, x₂, x₃ $\ge$ 0
The final tableau is: $$\begin{array} { c c | c c c ccc c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 }& \mathrm {~s} _ { 4 } \\\text { Basis } & \mathrm { C } _ { \mathrm { B } } &5&3&6&0&0&0&0& \\\hline \mathrm {~s} _ { 1 } &0&0&-2&-1.75&0&-.75&0&0&1500\\ \mathrm {~s} _ { 4 }&0&0&1&1.25&0&.25&0&1&3300\\ \mathrm {~s} _ { 3 }&0&0&-1&-.5&0&-.5&1&0&1000\\ \mathrm { x } _ { 1 } &5&1&1&1.25&0&.25&0&0&4500\\\hline& \mathrm { z } _ { j }&5&5&6.25&0&1.25&0&0&22500\\&\mathrm{C}_{\mathrm{j}}-z_{\mathrm{j}}&0&-2&-.25&0&-1.25&0&0&\end{array}$$
a.Calculate the range of optimality for c₁, c₂, and c₃.
b.Calculate the range of feasibility for b₁, b₂, b₃, and b₄.
c.Suppose that the inventory records were incorrect and the company really has only 14000 units of steel.What effect will this have on your solution?
d.Suppose that a cost increase will change the profit on the pots to $4.62.What effect will this have on your solution?
e.Assume that the cost of time in production and finishing is relevant.Would you be willing to pay a $1.00 premium over the normal cost for 1000 more hours in the production department? What would this do to your solution?