Question 1

The Sensitivity Report typically analyses the impact of simultaneous changes in the objective function coefficients and right-hand side constraint values on the optimal solution.

Accepted Answer

The Sensitivity Report analyzes the impact of changes in the objective function coefficients or right-hand side constraint values individually, not simultaneously, on the optimal solution.

Question 2

The slack values in the Answer Report can refer to either slack or surplus values based on the type of the inequality.

Accepted Answer

The slack values in the Answer Report can refer to either slack or surplus values depending on whether the inequality is a less than or a greater than constraint. For less than constraints, the slack value indicates the amount by which the left-hand side of the constraint can be increased before the constraint becomes binding. For greater than constraints, the slack value indicates the amount by which the left-hand side of the constraint can be decreased before the constraint becomes binding. In both cases, the same numerical value can be either a slack or a surplus, depending on the direction of the inequality.

Question 3

A given constraint has 40 unused hours.The shadow price for this resource would most likely be zero.

Accepted Answer

If there are unused hours, it means the resource is not fully utilized, so the shadow price would most likely be zero.

Question 4

As long as the objective function coefficient for a given variable increases within its allowable increase,the current optimal solution remains the same

Accepted Answer

This statement is true. If the objective function coefficient for a given variable increases within its allowable increase, the current optimal solution remains the same. This is because the increased coefficient increases the contribution of that variable to the objective function, making it more attractive for the solver to increase its value, but only up to the point where it hits its upper bound. Since the upper bound remains constant, the optimal solution will not change unless another variable becomes more attractive to increase.

Question 5

When the Allowable Increase or Allowable Decrease columns for the objective function coefficient of a variable has a value of zero in the Adjustable Cells table,this indicates the presence of a unique solution.

Accepted Answer

The Allowable Increase or Allowable Decrease columns for the objective function coefficient of a variable indicate how much you can increase or decrease the coefficient before the current optimal solution changes, not the uniqueness of the solution.

Question 6

Use this information to answer the following questions.
A company can ship its product from any of its three factories,F1,F2,and F3,to any of its retail outlets,R1,R2,and R3.The capacity,demand,and shipping cost information is provided as follows:
 $\begin{array}{ll}
\text { Demand (units) } & \text { Capacity (units) } \
\text { R1:300 } & \text { F1:250 } \
\text { R2: } 500 & \text { F2: } 350 \
\text { R3:200 } & \text { F3:400 }
\end{array}$

$\begin{array}{l}
\text { Shipping Cost/unit (\$) }\
\begin{array}{lll}
&\mathrm{R} 1 & \mathrm{R} 2& \mathrm{R} 3 \
F1&1&3&2\
F2&3 & 4 & 2 \
F3&2 & 2 & 3
\end{array}
\end{array}$
 
The company wants to come up with an optimal shipping strategy that will allow it to minimize its total shipping cost.
 
-Use the Sensitivity Report to answer the following questions:
a.Will any of the retail outlets experience any shortages in meeting their demand requirements if we implement the optimal solution?
b.Will any of the factories have any remaining quantities of the product if we implement the optimal solution?
c.What is the total minimal shipping cost?

Accepted Answer

The answer of Use this information to answer the following...

Question 7

Use this information to answer the following questions.
An insurance company has three secretaries,A,B,and C that each is capable of processing four different types of insurance claims.The amount of time required by each secretary to process a particular type of a claim is summarized in the following table.
  
On a typical week,the insurance firm has 5 type 1 claims,4 type 2 claims,2 type 3 claims,and 3 type 4 claims.Each secretary works a maximum of 40 hours per week.The office manager wants to know how many of each type of an insurance claim should be processed by each secretary to minimize the total processing time.
 \(\begin{array}{ccccccc}
\hline \text { Cell } & \text { Name } & \begin{array}{c}
\text { Final } \
\text { Value }
\end{array} & \begin{array}{c}
\text { Reduced } \
\text { Cost }
\end{array} & \begin{array}{c}
\text { Objective } \
\text { Coefficient }
\end{array} & \begin{array}{c}
\text { Allowable } \
\text { Increase }
\end{array} & \begin{array}{c}
\text { Allowable } \
\text { Decrease }
\end{array} \
\hline \text { SC\$13 } & \text { A1 } & 5 & 0 & 2 & 1 & 1 \mathrm{E}+30 \
\hline \text { SD\$13 } & \text { A2 } & 0 & 1 & 3 & 1 \mathrm{E}+30 & 1 \
\hline\text { SE\$13 } & \text { A3 } & 0 & 1 & 2 & 1 \mathrm{E}+30 & 1 \
\hline \text { SF\$13 } & \text { A4 } & 0 & 3 & 4 & 1 \mathrm{E}+30 & 3 \
\hline \text { SC\$14 } & \text { B1 } & 0 & 2 & 4 & 1 \mathrm{E}+30 & 2 \
\hline\text { SD\$14 } & \text { B2 } & 0 & 3 & 5 & 1 \mathrm{E}+30 & 3 \
\hline \text { SE\$14 } & \text { B3 } & 0 & 2 & 3 & 1 \mathrm{E}+30 & 2 \
\hline \text { SF\$14 } & \text { B4 } & 3 & 0 & 1 & 3 & 1 \mathrm{E}+30 \
\hline\text { SC\$15 } & \text { C1 } & 0 & 1 & 3 & 1 \mathrm{E}+30 & 1 \
\hline\text { SD\$15 } & \text { C2 } & 4 & 0 & 2 & 1 & 1 \mathrm{E}+30 \
\hline\text { SE\$15 } & \text { C3 } & 2 & 0 & 1 & 1 & 1 \mathrm{E}+30 \
\hline\text { SF\$15 } & \text { C4 } & 0 & 4 & 5 & 1 \mathrm{E}+30 & 4 \
\hline
\end{array}\)

\[\begin{array}{l}
\text { Constraints }\
\begin{array} { l l c c c c c } 
\hline \text { Cell } & \text { Name } & \begin{array} { c } 
\text { Final } \
\text { Value }
\end{array} & \begin{array} { c } 
\text { Shadow } \
\text { Price }
\end{array} & \begin{array} { c } 
\text { Constraint } \
\text { R.H. Side }
\end{array} & \begin{array} { c } 
\text { Allowable } \
\text { Increase }
\end{array} & \begin{array} { c } 
\text { Allowable } \
\text { Decrease }
\end{array} \
\hline \text { \$G\$13 } & \text { A constraint } & 10 & 0 & 40 & 1 \mathrm { E } + 30 & 30 \
\hline \text { \$G\$14 } & \text { B constraint } & 3 & 0 & 40 & 1 \mathrm { E } + 30 & 37 \
\hline \text { \$G\$15 } & \text { C constraint } & 10 & 0 & 40 & 1 \mathrm { E } + 30 & 30 \
\hline \text { \$C\$16 } & \text { Claim Type 1 } & 5 & 2 & 5 & 15 & 5 \
\hline \text { \$D\$16 } & \text { Claim Type 2 } & 4 & 2 & 4 & 15 & 4 \
\hline \text { \$E\$16 } & \text { Claim Type 3 } & 2 & 1 & 2 & 30 & 2 \
\hline \text { \$F\$16 } & \text { Claim Type 4 } & 3 & 1 & 3 & 37 & 3 \
\hline
\end{array}
\end{array}\]
-Use the Sensitivity Report to answer the following questions:
a.Suppose that the number of type 1 claims increases to 6 and the number of type 2 claims decreases to 3.What impact would this simultaneous change have on the current optimal objective function value?
b.What is the impact on the current solution and the objective function value if secretary A takes 3 hours to process claim 1 and secretary B takes 5 hours to process claim 1?

Accepted Answer

The answer of Use this information to answer the following...

Question 8

The information provided in the Sensitivity Report assumes that we are considering a change to only a single input data value.

Accepted Answer

The Sensitivity Report in linear programming and optimization analysis provides information based on the assumption that changes are made to one input data value at a time, allowing for the assessment of how sensitive the optimal solution is to changes in the parameters.

Question 9

A constraint right-hand side value is increased within its allowable increase value.The shadow price for this constraint will still be valid.

Accepted Answer

The shadow price represents the increase in the objective function value per unit increase in the right-hand side value of a constraint, assuming all other variables remain fixed. As long as the allowable increase value for the constraint is not exceeded, the shadow price will still be valid even if the right-hand side value is increased within its allowable range.

Question 10

Use this information to answer the following questions.
An insurance company has three secretaries,A,B,and C that each is capable of processing four different types of insurance claims.The amount of time required by each secretary to process a particular type of a claim is summarized in the following table.
  
On a typical week,the insurance firm has 5 type 1 claims,4 type 2 claims,2 type 3 claims,and 3 type 4 claims.Each secretary works a maximum of 40 hours per week.The office manager wants to know how many of each type of an insurance claim should be processed by each secretary to minimize the total processing time.
 \(\begin{array}{ccccccc}
\hline \text { Cell } & \text { Name } & \begin{array}{c}
\text { Final } \
\text { Value }
\end{array} & \begin{array}{c}
\text { Reduced } \
\text { Cost }
\end{array} & \begin{array}{c}
\text { Objective } \
\text { Coefficient }
\end{array} & \begin{array}{c}
\text { Allowable } \
\text { Increase }
\end{array} & \begin{array}{c}
\text { Allowable } \
\text { Decrease }
\end{array} \
\hline \text { SC\$13 } & \text { A1 } & 5 & 0 & 2 & 1 & 1 \mathrm{E}+30 \
\hline \text { SD\$13 } & \text { A2 } & 0 & 1 & 3 & 1 \mathrm{E}+30 & 1 \
\hline\text { SE\$13 } & \text { A3 } & 0 & 1 & 2 & 1 \mathrm{E}+30 & 1 \
\hline \text { SF\$13 } & \text { A4 } & 0 & 3 & 4 & 1 \mathrm{E}+30 & 3 \
\hline \text { SC\$14 } & \text { B1 } & 0 & 2 & 4 & 1 \mathrm{E}+30 & 2 \
\hline\text { SD\$14 } & \text { B2 } & 0 & 3 & 5 & 1 \mathrm{E}+30 & 3 \
\hline \text { SE\$14 } & \text { B3 } & 0 & 2 & 3 & 1 \mathrm{E}+30 & 2 \
\hline \text { SF\$14 } & \text { B4 } & 3 & 0 & 1 & 3 & 1 \mathrm{E}+30 \
\hline\text { SC\$15 } & \text { C1 } & 0 & 1 & 3 & 1 \mathrm{E}+30 & 1 \
\hline\text { SD\$15 } & \text { C2 } & 4 & 0 & 2 & 1 & 1 \mathrm{E}+30 \
\hline\text { SE\$15 } & \text { C3 } & 2 & 0 & 1 & 1 & 1 \mathrm{E}+30 \
\hline\text { SF\$15 } & \text { C4 } & 0 & 4 & 5 & 1 \mathrm{E}+30 & 4 \
\hline
\end{array}\)

\[\begin{array}{l}
\text { Constraints }\
\begin{array} { l l c c c c c } 
\hline \text { Cell } & \text { Name } & \begin{array} { c } 
\text { Final } \
\text { Value }
\end{array} & \begin{array} { c } 
\text { Shadow } \
\text { Price }
\end{array} & \begin{array} { c } 
\text { Constraint } \
\text { R.H. Side }
\end{array} & \begin{array} { c } 
\text { Allowable } \
\text { Increase }
\end{array} & \begin{array} { c } 
\text { Allowable } \
\text { Decrease }
\end{array} \
\hline \text { \$G\$13 } & \text { A constraint } & 10 & 0 & 40 & 1 \mathrm { E } + 30 & 30 \
\hline \text { \$G\$14 } & \text { B constraint } & 3 & 0 & 40 & 1 \mathrm { E } + 30 & 37 \
\hline \text { \$G\$15 } & \text { C constraint } & 10 & 0 & 40 & 1 \mathrm { E } + 30 & 30 \
\hline \text { \$C\$16 } & \text { Claim Type 1 } & 5 & 2 & 5 & 15 & 5 \
\hline \text { \$D\$16 } & \text { Claim Type 2 } & 4 & 2 & 4 & 15 & 4 \
\hline \text { \$E\$16 } & \text { Claim Type 3 } & 2 & 1 & 2 & 30 & 2 \
\hline \text { \$F\$16 } & \text { Claim Type 4 } & 3 & 1 & 3 & 37 & 3 \
\hline
\end{array}
\end{array}\]
-Use the Sensitivity Report to answer the following questions:
a.Which constraints are binding?
b.Suppose that secretary A has a total of 45 weekly hours.What impact would this have on the current optimal solution?
c.What is the total number of unused weekly hours for secretary B?
d.Suppose that the number of type 4 claims increases to 4? What impact would this have on the current optimal solution?

Accepted Answer

The answer of Use this information to answer the following...

Question 11

The Sensitivity Report has two parts: one part deals with objective function coefficient changes,and the other part deals with constraint coefficient changes.

Accepted Answer

The answer of The Sensitivity Report has two parts: one...

Question 12

Use this information to answer the following questions.
An investment company currently has $1 million available for investment in five different stocks.The company wants to maximize the interest earned over the next year.The five investment possibilities along with the expected interest earned are shown below.To manage risk,the investment firm wishes to have at least 35% of the investment in stocks A and B.Furthermore,no more than 15% of the investment may be in stock E.
 $$\begin{array} { l c } 
\text { Investment } & \text { Expected Interest Earned (\%) } \
\text { Stock A } & 7 \
\text { Stock B } & 9 \
\text { Stock C } & 8 \
\text { Stock D } & 10 \
\text { Stock E } & 11
\end{array}$$ 
 
-Use the Sensitivity Report to answer the following questions:
a.What would be the impact on the optimal allocation if the expected interest earned on stock A decreases to 6%?
b.What would be the impact on the optimal allocation if the expected interest earned on stock A increases to 10%?
c.What should the minimal expected interest earned for stock C be before it would be desirable to invest in this particular stock?
d.What would be the impact on the optimal allocation and the objective function value if the expected interest earned on stock B decreases by 1%?

Accepted Answer

The answer of Use this information to answer the following...

Question 13

Use this information to answer the following questions.
A company can ship its product from any of its three factories,F1,F2,and F3,to any of its retail outlets,R1,R2,and R3.The capacity,demand,and shipping cost information is provided as follows:
 $\begin{array}{ll}
\text { Demand (units) } & \text { Capacity (units) } \
\text { R1:300 } & \text { F1:250 } \
\text { R2: } 500 & \text { F2: } 350 \
\text { R3:200 } & \text { F3:400 }
\end{array}$

$\begin{array}{l}
\text { Shipping Cost/unit (\$) }\
\begin{array}{lll}
&\mathrm{R} 1 & \mathrm{R} 2& \mathrm{R} 3 \
F1&1&3&2\
F2&3 & 4 & 2 \
F3&2 & 2 & 3
\end{array}
\end{array}$
 
The company wants to come up with an optimal shipping strategy that will allow it to minimize its total shipping cost.
 
-Use the Sensitivity Report to answer the following questions:
a.Suppose that the shipping cost per unit for variable X1 (i.e. ,route F1-R1)increases to $2.00.What impact will this have on the current optimal solution and the objective function value?
b.Suppose that the shipping cost per unit for variable X6 (i.e. ,route F2-R3)increases by $2.00 to $5.00.What impact will this have on the current optimal solution and the objective function value?
c.Suppose that the factory constraints and the retail outlet constraints become "≤".What would the optimal solution and the objective function coefficient be? (Hint: The answer to this question is not given in the Sensitivity Report. )

Accepted Answer

The answer of Use this information to answer the following...

Question 14

A binding constraint means that the constraint is exactly satisfied,and its left-hand side does not equal its right-hand side.

Accepted Answer

A binding constraint means that it is at its maximum or minimum value, but its left-hand side must be equal to its right-hand side.

Question 15

Use this information to answer the following questions.
An investment company currently has $1 million available for investment in five different stocks.The company wants to maximize the interest earned over the next year.The five investment possibilities along with the expected interest earned are shown below.To manage risk,the investment firm wishes to have at least 35% of the investment in stocks A and B.Furthermore,no more than 15% of the investment may be in stock E.
 $$\begin{array} { l c } 
\text { Investment } & \text { Expected Interest Earned (\%) } \
\text { Stock A } & 7 \
\text { Stock B } & 9 \
\text { Stock C } & 8 \
\text { Stock D } & 10 \
\text { Stock E } & 11
\end{array}$$ 
 
-Use the Sensitivity Report to answer the following questions:
a.What is the optimal total expected interest earned for next year?
b.What is the dollar amount that should be invested in each stock?
c.Which constraints are binding? Which constraints are not binding?
d.Is the solution to the problem unique or are there alternate optimal solutions?
e.Does the optimal solution call for investing the entire $1,000,000?

Accepted Answer

The answer of Use this information to answer the following...

Question 16

The slack of a constraint indicates the amount of a used resource.

Accepted Answer

The slack of a constraint is the amount of time a task or project can be delayed without delaying the overall end date. It is not related to the amount of a used resource.

Question 17

Use this information to answer the following questions.
An investment company currently has $1 million available for investment in five different stocks.The company wants to maximize the interest earned over the next year.The five investment possibilities along with the expected interest earned are shown below.To manage risk,the investment firm wishes to have at least 35% of the investment in stocks A and B.Furthermore,no more than 15% of the investment may be in stock E.
 $$\begin{array} { l c } 
\text { Investment } & \text { Expected Interest Earned (\%) } \
\text { Stock A } & 7 \
\text { Stock B } & 9 \
\text { Stock C } & 8 \
\text { Stock D } & 10 \
\text { Stock E } & 11
\end{array}$$ 
 
-Use the Sensitivity Report to answer the following questions:
a.Suppose that the amount of money available for investment increases by $50,000.What impact would this have on the current optimal objective function value?
b.Suppose that total investment in stocks A and B must be at least 40% of the total amount available for investment (i.e. ,$400,000).What impact would this have on the current optimal objective function value?
c.Suppose that the total investment in stocks A and be must be at least 30% of the total amount available for investment.What impact would this have on the current optimal objective function value?
d.Assume that no more than 30% of the investment may be in stock E.What impact would this have on the current optimal objective function value?

Accepted Answer

The answer of Use this information to answer the following...

Question 18

Use this information to answer the following questions.
An insurance company has three secretaries,A,B,and C that each is capable of processing four different types of insurance claims.The amount of time required by each secretary to process a particular type of a claim is summarized in the following table.
  
On a typical week,the insurance firm has 5 type 1 claims,4 type 2 claims,2 type 3 claims,and 3 type 4 claims.Each secretary works a maximum of 40 hours per week.The office manager wants to know how many of each type of an insurance claim should be processed by each secretary to minimize the total processing time.
 \(\begin{array}{ccccccc}
\hline \text { Cell } & \text { Name } & \begin{array}{c}
\text { Final } \
\text { Value }
\end{array} & \begin{array}{c}
\text { Reduced } \
\text { Cost }
\end{array} & \begin{array}{c}
\text { Objective } \
\text { Coefficient }
\end{array} & \begin{array}{c}
\text { Allowable } \
\text { Increase }
\end{array} & \begin{array}{c}
\text { Allowable } \
\text { Decrease }
\end{array} \
\hline \text { SC\$13 } & \text { A1 } & 5 & 0 & 2 & 1 & 1 \mathrm{E}+30 \
\hline \text { SD\$13 } & \text { A2 } & 0 & 1 & 3 & 1 \mathrm{E}+30 & 1 \
\hline\text { SE\$13 } & \text { A3 } & 0 & 1 & 2 & 1 \mathrm{E}+30 & 1 \
\hline \text { SF\$13 } & \text { A4 } & 0 & 3 & 4 & 1 \mathrm{E}+30 & 3 \
\hline \text { SC\$14 } & \text { B1 } & 0 & 2 & 4 & 1 \mathrm{E}+30 & 2 \
\hline\text { SD\$14 } & \text { B2 } & 0 & 3 & 5 & 1 \mathrm{E}+30 & 3 \
\hline \text { SE\$14 } & \text { B3 } & 0 & 2 & 3 & 1 \mathrm{E}+30 & 2 \
\hline \text { SF\$14 } & \text { B4 } & 3 & 0 & 1 & 3 & 1 \mathrm{E}+30 \
\hline\text { SC\$15 } & \text { C1 } & 0 & 1 & 3 & 1 \mathrm{E}+30 & 1 \
\hline\text { SD\$15 } & \text { C2 } & 4 & 0 & 2 & 1 & 1 \mathrm{E}+30 \
\hline\text { SE\$15 } & \text { C3 } & 2 & 0 & 1 & 1 & 1 \mathrm{E}+30 \
\hline\text { SF\$15 } & \text { C4 } & 0 & 4 & 5 & 1 \mathrm{E}+30 & 4 \
\hline
\end{array}\)

\[\begin{array}{l}
\text { Constraints }\
\begin{array} { l l c c c c c } 
\hline \text { Cell } & \text { Name } & \begin{array} { c } 
\text { Final } \
\text { Value }
\end{array} & \begin{array} { c } 
\text { Shadow } \
\text { Price }
\end{array} & \begin{array} { c } 
\text { Constraint } \
\text { R.H. Side }
\end{array} & \begin{array} { c } 
\text { Allowable } \
\text { Increase }
\end{array} & \begin{array} { c } 
\text { Allowable } \
\text { Decrease }
\end{array} \
\hline \text { \$G\$13 } & \text { A constraint } & 10 & 0 & 40 & 1 \mathrm { E } + 30 & 30 \
\hline \text { \$G\$14 } & \text { B constraint } & 3 & 0 & 40 & 1 \mathrm { E } + 30 & 37 \
\hline \text { \$G\$15 } & \text { C constraint } & 10 & 0 & 40 & 1 \mathrm { E } + 30 & 30 \
\hline \text { \$C\$16 } & \text { Claim Type 1 } & 5 & 2 & 5 & 15 & 5 \
\hline \text { \$D\$16 } & \text { Claim Type 2 } & 4 & 2 & 4 & 15 & 4 \
\hline \text { \$E\$16 } & \text { Claim Type 3 } & 2 & 1 & 2 & 30 & 2 \
\hline \text { \$F\$16 } & \text { Claim Type 4 } & 3 & 1 & 3 & 37 & 3 \
\hline
\end{array}
\end{array}\]
-Use the Sensitivity Report to answer the following questions:
a.What is the total minimal processing time in hours?
b.Suppose that secretary A can process a type 2 claim in two hours rather than three hours.How would this impact the current optimal solution?
c.Assume that secretary C can process a type 4 claim in 6 hours rather than 5 hours.How would this impact the current optimal solution?
d.Currently,secretary C is not processing any type 4 claims.Suppose that we force secretary four to process one type 4 claim.What impact would this have on the optimal solution?

Accepted Answer

The answer of Use this information to answer the following...

Question 19

Surplus is typically associated with "≤" constraints.

Accepted Answer

Surplus is typically associated with ">=" constraints, while slack is associated with "<=" constraints.

Question 20

The Reduced Cost may be viewed as the difference between the marginal contribution of a variable and the marginal value of the resources it consumes.

Accepted Answer

This is a correct definition of reduced cost. Marginal contribution refers to the increase in objective function value when one unit of the variable is added while holding all other variables constant. Marginal value of resources refers to the decrease in objective function value when one unit of the resource is removed while holding all other variables constant. The difference between the two yields the reduced cost of the variable.

Quiz 4: Linear Programming Sensitivity Analysis

The Sensitivity Report typically analyses the impact of simultaneous changes in the objective function coefficients and right-hand side constraint values on the optimal solution.

The slack values in the Answer Report can refer to either slack or surplus values based on the type of the inequality.

A given constraint has 40 unused hours.The shadow price for this resource would most likely be zero.

As long as the objective function coefficient for a given variable increases within its allowable increase,the current optimal solution remains the same

When the Allowable Increase or Allowable Decrease columns for the objective function coefficient of a variable has a value of zero in the Adjustable Cells table,this indicates the presence of a unique solution.

The information provided in the Sensitivity Report assumes that we are considering a change to only a single input data value.

A constraint right-hand side value is increased within its allowable increase value.The shadow price for this constraint will still be valid.

The Sensitivity Report has two parts: one part deals with objective function coefficient changes,and the other part deals with constraint coefficient changes.

A binding constraint means that the constraint is exactly satisfied,and its left-hand side does not equal its right-hand side.

The slack of a constraint indicates the amount of a used resource.

Surplus is typically associated with "≤" constraints.

The Reduced Cost may be viewed as the difference between the marginal contribution of a variable and the marginal value of the resources it consumes.