Deck 7: Goal Programming and Multiple Objective Optimization

Question 1

Which of the following is false regarding a goal constraint?

A)A goal constraint allows us to determine how close a given solution comes to achieving a goal.
B)A goal constraint will always contain two deviational variables.
C)Deviation variables are non-negative.
D)If two deviation variables are used in a constraint at least one will have a value of zero.

Answer 1

B

Question 2

Exhibit 7.1
The following questions are based on the problem below.
A company wants to advertise on TV and radio.The company wants to produce about 6 TV ads and 12 radio ads.Each TV ad costs $20,000 and is viewed by 10 million people.Radio ads cost $10,000 and are heard by 7 million people.The company wants to reach about 140 million people,and spend about $200,000 for all the ads.The problem has been set up in the following Excel spreadsheet.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { TV } & \text { Radio } & & \\\hline 2 & \text { Cost } & 20 & 10 & & \\\hline 3 & \text { Coverage } & 10 & 7 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { TV } & \text { Radio } & \text { Cost } & \text { Coverage } \\\hline 6 & \text { Actual Amount } & 0 & 0 & & \\\hline 7 & \text { +Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 0 & 0 \\\hline 9 & \text { F Goal } & 0 & 0 & 0 & 0 \\\hline 10 & \text { Target Value } & 6 & 12 & 200 & 140 \\\hline 11 & & & & & \\\hline 12 & \text { Percentage Deviation: } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 0 & 0 & 0 & 0 \\\hline 15 & & \\\hline 16 & \text { Weights } & \\\hline 17 & \text { Under } & \\\hline 18 & \text { Over } & \\\hline 19 & & \\\hline 20 & \text { Objective } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.1.What formula goes in cell B9?

A)=SUMB6:B8)
B)=B6+B7-B8
C)=B6-B7+B8
D)=B10-B8

Answer 2

=B6+B7-B8

Question 3

What is the soft constraint form of the following hard constraint?
3X₁ + 2 X₂ ? 10

A) $3 \mathrm{X}_{1}+2 \mathrm{X}_{2}+\mathrm{d}_{1}^{-}-\mathrm{d}_{1}^{+}=10$
B) $3 \mathrm{X}_{1}+2 \mathrm{X}_{2}+\mathrm{d}_{1}+\mathrm{d}_{1}^{+}=10$
C) $3 \mathrm{X}_{1}+2 \mathrm{X}_{2}-\mathrm{d}_{1}^{-}-\mathrm{d}_{1}^{+} \leq 10$
D) $3 \mathrm{X}_{1}+2 \mathrm{X}_{2}+\mathrm{d}_{1}^{-}-\mathrm{d}_{1}^{+} \geq 10$

Answer 3

$3 \mathrm{X}_{1}+2 \mathrm{X}_{2}+\mathrm{d}_{1}^{-}-\mathrm{d}_{1}^{+}=10$

Question 4

What is the meaning of the t_i term in this objective function for a goal programming problem?
$\operatorname{MIN} \sum \frac{1}{t_{i}}\left(d_{i}^{-}+d_{i}^{+}\right)^{2}$
$\operatorname{MIN} \sum \frac{1}{t_{i}}\left(d_{i}^{-}+d_{i}^{+}\right)^{2}$

A)The time required for each decision variable.
B)The percent of goal i met.
C)The coefficient for the i^th decision variable
D)The target value for goal i.

Question 5

Decision-making problems which can be stated as a collection of desired objectives are known as what type of problem?

A)A non-linear programming problem.
B)An unconstrained programming problem.
C)A goal programming problem.
D)An integer programming problem.

Question 6

Suppose that the first goal in a GP problem is to make 3 X₁ + 4 X₂ approximately equal to 36.Using the deviational variables d₁^? and d₁⁺,the following constraint can be used to express this goal.
3 X₁ + 4 X₂ + d₁^? ? d₁⁺ = 36
If we obtain a solution where X₁ = 6 and X₂ = 2,what values do the deviational variables assume?

A) $\mathrm{d}_{1}^{-}=0, \mathrm{~d}_{1}^{+}=10$
B) $\mathrm{d}_{1}^{-}=10, \mathrm{~d}_{1}^{+}=0$
C) $\mathrm{d}_{1}^{-}=5, \mathrm{~d}_{1}^{+}=5$
D) $\mathrm{d}_{1}^{-}=6, \mathrm{~d}_{1}^{+}=0$

Question 7

Exhibit 7.1
The following questions are based on the problem below.
A company wants to advertise on TV and radio.The company wants to produce about 6 TV ads and 12 radio ads.Each TV ad costs $20,000 and is viewed by 10 million people.Radio ads cost $10,000 and are heard by 7 million people.The company wants to reach about 140 million people,and spend about $200,000 for all the ads.The problem has been set up in the following Excel spreadsheet.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { TV } & \text { Radio } & & \\\hline 2 & \text { Cost } & 20 & 10 & & \\\hline 3 & \text { Coverage } & 10 & 7 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { TV } & \text { Radio } & \text { Cost } & \text { Coverage } \\\hline 6 & \text { Actual Amount } & 0 & 0 & & \\\hline 7 & \text { +Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 0 & 0 \\\hline 9 & \text { F Goal } & 0 & 0 & 0 & 0 \\\hline 10 & \text { Target Value } & 6 & 12 & 200 & 140 \\\hline 11 & & & & & \\\hline 12 & \text { Percentage Deviation: } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 0 & 0 & 0 & 0 \\\hline 15 & & \\\hline 16 & \text { Weights } & \\\hline 17 & \text { Under } & \\\hline 18 & \text { Over } & \\\hline 19 & & \\\hline 20 & \text { Objective } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.1.Which of the following is a constraint specified to Analytic Solver Platform for this model?

A)$B$9:$E$9=$B$6:$E$6
B)$B$9:$E$9<$B$10:$E$10
C)$B$9:$E$9=$B$10:$E$10
D)$B$9:$E$9>$B$10:$E$10

Question 8

The d⁺
Variable indicates the amount by which each goal's target value is

A)missed.
B)underachieved.
C)overachieved.
D)overstated.

Question 9

The RHS value of a goal constraint is referred to as the

A)target value.
B)constraint value.
C)objective value.
D)desired value.

Question 10

Exhibit 7.1
The following questions are based on the problem below.
A company wants to advertise on TV and radio.The company wants to produce about 6 TV ads and 12 radio ads.Each TV ad costs $20,000 and is viewed by 10 million people.Radio ads cost $10,000 and are heard by 7 million people.The company wants to reach about 140 million people,and spend about $200,000 for all the ads.The problem has been set up in the following Excel spreadsheet.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { TV } & \text { Radio } & & \\\hline 2 & \text { Cost } & 20 & 10 & & \\\hline 3 & \text { Coverage } & 10 & 7 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { TV } & \text { Radio } & \text { Cost } & \text { Coverage } \\\hline 6 & \text { Actual Amount } & 0 & 0 & & \\\hline 7 & \text { +Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 0 & 0 \\\hline 9 & \text { F Goal } & 0 & 0 & 0 & 0 \\\hline 10 & \text { Target Value } & 6 & 12 & 200 & 140 \\\hline 11 & & & & & \\\hline 12 & \text { Percentage Deviation: } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 0 & 0 & 0 & 0 \\\hline 15 & & \\\hline 16 & \text { Weights } & \\\hline 17 & \text { Under } & \\\hline 18 & \text { Over } & \\\hline 19 & & \\\hline 20 & \text { Objective } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.1.Which cells are the variable cells in this model?

A)$B$6:$C$6,$B$7:$E$8
B)$B$6:$C$6
C)$B$9:$E$9
D)$B$6:$E$8

Question 11

A constraint which represents a target value for a problem is called a

A)fuzzy constraint.
B)vague constraint.
C)preference constraint
D)soft constraint

Question 12

Suppose that X₁ equals 4.What are the values for d₁⁺ and d₁^? in the following constraint?
X₁ + d₁^?? d₁⁺ = 8

A) $\mathrm{d}_{1}^{-}=4, \mathrm{~d}_{1}^{+}=0$
B) $\mathrm{d}_{1}^{-}=0, \mathrm{~d}_{1}^{+}=4$
C) $\mathrm{d}_{1}^{-}=4, \mathrm{~d}_{1}^{+}=4$
D) $\mathrm{d}_{1}^{-}=8, \mathrm{~d}_{1}^{+}=0$

Question 13

The d_i⁺,d_i⁻ variables are referred to as

A)objective variables.
B)goal variables.
C)target variables.
D)deviational variables.

Question 14

Suppose that the first goal in a GP problem is to make 3 X₁ + 4 X₂ approximately equal to 36.Using the deviational variables d₁^? and d₁⁺,what constraint can be used to express this goal?

A) $3 \mathrm{X}_{1}+4 \mathrm{X}_{2}+\mathrm{d}_{1}^{-}-\mathrm{d}_{1}^{+} \leq 36$
B) $3 \mathrm{x}_{1}+4 \mathrm{X}_{2}-\mathrm{d}_{1}^{-}-\mathrm{d}_{1}^{+}=36$
C) $3 \mathrm{X}_{1}+4 \mathrm{X}_{2}+\mathrm{d}_{1}^{-}+\mathrm{d}_{1}^{+}=30$
D) $3 \mathrm{X}_{1}+4 \mathrm{X}_{2}+\mathrm{d}_{1}^{--} \mathrm{d}_{1}^{+}=36$

Question 15

Which of the following are true regarding weights assigned to deviational variables?

A)The weights assigned can be negative.
B)The weights assigned must sum to one.
C)The weight assigned to the deviation under a particular goal must be the same as the weight assigned to the deviation above that particular goal.
D)All of these are false.

Question 16

Exhibit 7.1
The following questions are based on the problem below.
A company wants to advertise on TV and radio.The company wants to produce about 6 TV ads and 12 radio ads.Each TV ad costs $20,000 and is viewed by 10 million people.Radio ads cost $10,000 and are heard by 7 million people.The company wants to reach about 140 million people,and spend about $200,000 for all the ads.The problem has been set up in the following Excel spreadsheet.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { TV } & \text { Radio } & & \\\hline 2 & \text { Cost } & 20 & 10 & & \\\hline 3 & \text { Coverage } & 10 & 7 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { TV } & \text { Radio } & \text { Cost } & \text { Coverage } \\\hline 6 & \text { Actual Amount } & 0 & 0 & & \\\hline 7 & \text { +Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 0 & 0 \\\hline 9 & \text { F Goal } & 0 & 0 & 0 & 0 \\\hline 10 & \text { Target Value } & 6 & 12 & 200 & 140 \\\hline 11 & & & & & \\\hline 12 & \text { Percentage Deviation: } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 0 & 0 & 0 & 0 \\\hline 15 & & \\\hline 16 & \text { Weights } & \\\hline 17 & \text { Under } & \\\hline 18 & \text { Over } & \\\hline 19 & & \\\hline 20 & \text { Objective } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.1.What formula goes in cell D6?

A)=SUMPRODUCTB2:B3,B6:B7)
B)=B2*C2+B6*C6
C)=SUMPRODUCTB2:C2,B10:C10)
D)=SUMPRODUCTB2:C2,B6:C6)

Question 17

Goal programming differs from linear programming or integer linear programming is that

A)goal programming provides for multiple objectives.
B)goal programming excludes hard constraints.
C)with goal programming we iterate until an acceptable solution is obtained.
D)goal programming requires fewer variables.

Question 18

What weight would be assigned to a neutral deviational variable?

A)0
B)1
C)10
D)100

Question 19

Which of the following is true regarding goal programming?

A)The objective function is not useful when comparing goal programming solutions.
B)We can place upper bounds on any of the deviation variables.
C)A preemptive goal program involves deviations with arbitrarily large weights.
D)All of these are true.

Question 20

A constraint which cannot be violated is called a

A)binding constraint.
B)hard constraint.
C)definite constraint.
D)required constraint.

Question 21

Exhibit 7.1
The following questions are based on the problem below.
A company wants to advertise on TV and radio.The company wants to produce about 6 TV ads and 12 radio ads.Each TV ad costs $20,000 and is viewed by 10 million people.Radio ads cost $10,000 and are heard by 7 million people.The company wants to reach about 140 million people,and spend about $200,000 for all the ads.The problem has been set up in the following Excel spreadsheet.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { TV } & \text { Radio } & & \\\hline 2 & \text { Cost } & 20 & 10 & & \\\hline 3 & \text { Coverage } & 10 & 7 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { TV } & \text { Radio } & \text { Cost } & \text { Coverage } \\\hline 6 & \text { Actual Amount } & 0 & 0 & & \\\hline 7 & \text { +Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 0 & 0 \\\hline 9 & \text { F Goal } & 0 & 0 & 0 & 0 \\\hline 10 & \text { Target Value } & 6 & 12 & 200 & 140 \\\hline 11 & & & & & \\\hline 12 & \text { Percentage Deviation: } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 0 & 0 & 0 & 0 \\\hline 15 & & \\\hline 16 & \text { Weights } & \\\hline 17 & \text { Under } & \\\hline 18 & \text { Over } & \\\hline 19 & & \\\hline 20 & \text { Objective } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.1.Which cells)isare)the objective cells)in this model?

A)$B$20
B)$D$6
C)$E$6
D)$B$13:$E$14,$B$9:$E$9

Question 22

Exhibit 7.2
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following multi-objective linear programming MOLP)has been solved in Excel.
$\begin{array}{|c|l|c|c|c|} \hline& {\text { A }} & \text { B } & \text { C } & \text { D } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm{A} & \mathrm{B} & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & \\\hline 9 & \text { Objectives: } & \\\hline 10 & \text { Average return } & 0 \\\hline 11 & \text { Average risk } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.2.What formula goes in cell B11?

A)=SUMPRODUCTB2:C2,$B$6:$C$6)/$D$7
B)=B2*C2+B3*C3
C)=SUMPRODUCTB3:C3,$B$6:$C$6)/$D$7
D)=SUMPRODUCTB3:C3,$B$6:$C$6)

Question 23

Given the following goal constraints
5 X₁ + 6 X₂ + 7 X₃ + d₁⁻ − d₁⁺ = 87
3 X₁ + X₂ + 4 X₃ + d₂⁻ − d₂⁺ = 37
7 X₁ + 3 X₂ + 2 X₃ + d₃⁻ − d₃⁺ = 72
and solution X₁,X₂,X₃)= 7,2,5),what values do the deviational variables assume?

Question 24

Exhibit 7.1
The following questions are based on the problem below.
A company wants to advertise on TV and radio.The company wants to produce about 6 TV ads and 12 radio ads.Each TV ad costs $20,000 and is viewed by 10 million people.Radio ads cost $10,000 and are heard by 7 million people.The company wants to reach about 140 million people,and spend about $200,000 for all the ads.The problem has been set up in the following Excel spreadsheet.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { TV } & \text { Radio } & & \\\hline 2 & \text { Cost } & 20 & 10 & & \\\hline 3 & \text { Coverage } & 10 & 7 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { TV } & \text { Radio } & \text { Cost } & \text { Coverage } \\\hline 6 & \text { Actual Amount } & 0 & 0 & & \\\hline 7 & \text { +Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 0 & 0 \\\hline 9 & \text { F Goal } & 0 & 0 & 0 & 0 \\\hline 10 & \text { Target Value } & 6 & 12 & 200 & 140 \\\hline 11 & & & & & \\\hline 12 & \text { Percentage Deviation: } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 0 & 0 & 0 & 0 \\\hline 15 & & \\\hline 16 & \text { Weights } & \\\hline 17 & \text { Under } & \\\hline 18 & \text { Over } & \\\hline 19 & & \\\hline 20 & \text { Objective } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.1.If the company is very concerned about going over the $200,000 budget,which cell value should change and how should it change?

A)D13,increase
B)D13,decrease
C)D14,increase
D)D14,decrease

Question 25

The primary benefit of a MINIMAX objective function is

A)it yields any feasible solution by changing the weights.
B)it is limited to all corner points.
C)it yields a larger variety of solutions than generally available using an LP method.
D)it makes many of the deviational variables equal to zero.

Question 26

Exhibit 7.2
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following multi-objective linear programming MOLP)has been solved in Excel.
$\begin{array}{|c|l|c|c|c|} \hline& {\text { A }} & \text { B } & \text { C } & \text { D } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm{A} & \mathrm{B} & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & \\\hline 9 & \text { Objectives: } & \\\hline 10 & \text { Average return } & 0 \\\hline 11 & \text { Average risk } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.2.Which cells)isare)the target cells in this model?

A)$B$6:$C$6,$B$10:$B$11
B)$B$6:$C$6
C)$B$6:$D$6
D)$B$10:$B$11

Question 27

Exhibit 7.2
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following multi-objective linear programming MOLP)has been solved in Excel.
$\begin{array}{|c|l|c|c|c|} \hline& {\text { A }} & \text { B } & \text { C } & \text { D } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm{A} & \mathrm{B} & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & \\\hline 9 & \text { Objectives: } & \\\hline 10 & \text { Average return } & 0 \\\hline 11 & \text { Average risk } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.2.What Analytic Solver Platform constraint involves cells $B$6:$C$6?

A)$B$6:$C$6=$B$7:$C$7
B)$B$6:$C$6?$B$7:$C$7
C)$B$6:$C$6?$B$7:$C$7
D)$B$6:$C$6=$D$7

Question 28

A manager wants to ensure that he does not exceed his budget by more than $1000 in a goal programming problem.If the budget constraint is the third constraint in the goal programming problem which of the following formulas will best ensure that the manager's objective is met?

A) $\text { MIN } d_{3}^{+}$
B) $\mathrm{d}_{3}^{-} \geq 1000$
C) $\mathrm{d}_{3}{ }^{+}=1000$
D) $\mathrm{d}_{3}{ }^{+} \leq 1000$

Question 29

A company makes 2 products A and B from 2 resources.The products have the following resource requirements and produce the accompanying profits.The available quantity of resources is also shown in the table.
$\begin{array}{rllr}\text { Product } & 1 & 2 & \text { Available resources } \\\hline \text { Labor hr/unit) } & 3 & 2 & 150 \\\text { Material ounces/unit) } & 1 & 2 & 200 \\\text { Profit\$/unit) } & 7 & 6 &\end{array}$

Management has developed the following set of goals
Goal 1: Produce approximately 40 units of product 1.
Goal 2: Produce approximately 70 units of product 2.Goal 3: Achieve a profit over $400.
Goal 4: Consume less than 150 hours of labor Goal 5: Consume less than 200 ounces of material
Based on this GP formulation of the problem and the associated optimal integer solution what values should go in cells B2:F16 of the following Excel spreadsheet?

Question 30

If no other feasible solution to a multi-objective linear programming MOLP)problem allows an increase in any objective without decreasing at least one other objective,the solution is said to be

A)dually optimal.
B)Pareto optimal.
C)suboptimal.
D)maximally optimal.

Question 31

An optimization technique useful for solving problems with more than one objective function is

A)dual programming.
B)sensitivity analysis.
C)multi-objective linear programming.
D)goal programming.

Question 32

MINIMAX solutions to multi-objective linear programming MOLP)problems are

A)dually optimal.
B)Pareto optimal.
C)suboptimal.
D)maximally optimal.

Question 33

Exhibit 7.3
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following minimax formulation of the problem has been solved in Excel.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \text { A } & \text { B } & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & & & & & \text { Weighted } \\\hline 10 & \text { Goals } & \text { Actual } & \text { Target } & \text { Weights } & \% \text { Deviation } \\\hline 11 & \text { Average return } & 0 & 11.8 \% & 1 & 0 \\\hline 12 & \text { Average risk } & 0 & 0.22 & 1 & 0 \\\hline 13 & & & & & \\\hline 14 & \text { Objective: } & 0 & & &\\\hline\end{array}$

-Refer to Exhibit 7.3.What formula goes in cell E11?

A)=D11*C11?B11)/C11
B)=C11?B11)/C11
C)=D11*C11
D)=D11*C11?B11)

Question 34

Consider the following multi-objective linear programming problem MOLP):
MAX: ^{3 X}₁ ^{+ 4 X}₂
MAX: ^{2 X}₁ ^{+ X}₂
Subject to: ^{6 X}₁ ^{+ 13 X}₂ ^{? 78}
12 X₁ + 9 X₂ ? 108
8 X₁ + 10 X₂ ? 80 X₁,X₂ ? 0
Graph the feasible region for this problem and compute the value of each objective at each extreme point.What are the solutions to each of the component LPs?

Question 35

A company makes 2 products A and B from 2 resources,labor and material.The products have the following resource requirements and produce the accompanying profits.The available quantity of resources is also shown in the table.
$$\begin{array} { l r r c } \text { Praduct } & \text { A } & \text { B } & \text { Available resources } \\\hline \text { Labor hr/unit) } & 3 & 2 & 150 \\\text { Material ounces/unit) } & 1 & 2 & 200\\\text { Profit\$/unit)}&7&6\end{array}$$
Management has developed the following set of goals
Goal 1: Produce approximately 40 units of product 1.
Goal 2: Produce approximately 70 units of product 2.Goal 3: Achieve a profit over $400.
Goal 4: Consume less than 150 hours of labor Goal 5: Consume less than 200 ounces of material
Formulate a goal programming model of this problem.

Question 36

Exhibit 7.3
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following minimax formulation of the problem has been solved in Excel.
$\begin{array}{|c|l|c|c|c|c|} \hline&{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \text { A } & \text { B } & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & & & & & \text { Weighted } \\\hline 10 & \text { Goals } & \text { Actual } & \text { Target } & \text { Weights } & \% \text { Deviation } \\\hline 11 & \text { Average return } & 0 & 11.8 \% & 1 & 0 \\\hline 12 & \text { Average risk } & 0 & 0.22 & 1 & 0 \\\hline 13 & & & & & \\\hline 14 & \text { Objective: } & 0 & & &\\\hline\end{array}$

-Refer to Exhibit 7.3.Which value should the investor change,and in what direction,if he wants to reduce the risk of the portfolio?

A)D11,increase
B)D12,increase
C)C12,increase
D)D12,decrease

Question 37

Exhibit 7.2
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following multi-objective linear programming MOLP)has been solved in Excel.
$\begin{array}{|c|l|c|c|c|} \hline& {\text { A }} & \text { B } & \text { C } & \text { D } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm{A} & \mathrm{B} & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & \\\hline 9 & \text { Objectives: } & \\\hline 10 & \text { Average return } & 0 \\\hline 11 & \text { Average risk } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.2.What formula goes in cell B10?

A)=SUMPRODUCTB2:C2,$B$6:$C$6)/$D$7
B)=B2*C2+B3*C3
C)=SUMPRODUCTB3:C3,$B$6:$C$6)/$D$7
D)=SUMPRODUCTB2:C2,$B$6:$C$6)

Question 38

Goal programming solution feedback indicates that the d₄⁺ level of 50 should not be exceeded in future solution iterations.How should you modify your goal constraint to accommodate this requirement?
40 X₁ + 20 X₂ + d₄^? + d₄⁺ = 300

A) $\text { Increase the RHS value from } 300 \text { to } 350 \text {. }$
B) $\text { Replace the constraint with } 40 \mathrm{X}_{1}+20 \mathrm{X}_{2} \leq 350$
C) $\text { Do not modify the constraint, add a constraint } \mathrm{d}_{4}^{+} \leq 50$
D) $\text { Do not modify the constraint, add a constraint } d_{4}+=50$

Question 39

Exhibit 7.2
The following questions are based on the problem below.
An investor has $150,000 to invest in investments A and B.Investment A requires a $10,000 minimum investment,pays a return of 12% and has a risk factor of .50.Investment B requires a $15,000 minimum investment,pays a return of 10% and has a risk factor of .20.The investor wants to maximize the return while minimizing the risk of the portfolio.The following multi-objective linear programming MOLP)has been solved in Excel.
$\begin{array}{|c|l|c|c|c|} \hline& {\text { A }} & \text { B } & \text { C } & \text { D } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & \\\hline 2 & \text { Expected return } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm{A} & \mathrm{B} & \text { Total } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & \\\hline 9 & \text { Objectives: } & \\\hline 10 & \text { Average return } & 0 \\\hline 11 & \text { Average risk } & 0 \\\hline\end{array}$

-Refer to Exhibit 7.2.Which cells are the changing cells in this model?

A)$B$6:$C$6,$B$10:$B$11
B)$B$6:$C$6
C)$B$6:$D$6
D)$B$10:$B$11

Question 40

The MINIMAX objective

A)yields the smallest possible deviations.
B)minimizes the maximum deviation from any goal.
C)chooses the deviation which has the largest value.
D)maximizes the minimum value of goal attainment.

Question 41

A MINIMAX objective function in goal programming GP):

A)is used to minimize the maximum deviation from a goal
B)is captured in a decision table
C)is estimated by trial-and-error
D)often produces an infeasible solution

Question 42

Exhibit 7.4
The following questions are based on the problem below.
Robert Gardner runs a small,local-only delivery service.His fleet consists of three smaller panel trucks.He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers.The box weights are: 210,160,320,90,110,70,410,260,170,240,80 and 180 for boxes 1 through 12,respectively.Since each truck differs each truck has different load capacities as given below:
Robert would like each truck equally loaded,both in terms of number of boxes and in terms of total weight,while minimizing his shipping costs.Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight.
The following integer goal programming formulation applies to his problem.
Y₁ = weight loaded in truck 1;Y₂ = weight loaded in truck 2;Y₃ = weight loaded intruck3;X_i,j = 0 if truck i not loaded with box j;1 if truck i loaded with box j.
Given the following spreadsheet solution of this integer goal programming formulation,answer the following questions.

Refer to Exhibit 7.4.What formulas should go in cell E26 of the spreadsheet?

Question 43

A company wants to purchase large and small delivery trucks.The company wants to purchase about 10 large and 15 small trucks.Each large truck costs $30,000 and has a 10 ton capacity.Each small truck costs $20,000 and has a 7 ton capacity.The company wants to have about 200 tons of capacity and spend about $600,000.
Based on the following formulation and associated integer solution,what values should go in cells B2:E16 of the spreadsheet?

Question 44

An investor wants to invest $50,000 in two mutual funds,A and B.The rates of return,risks and minimum investment requirements for each fund are:
$\begin{array}{lrrr}&&&\text { Minimum }\\\text { Fund } & \text { Rate of return } & \text { Risk }&\text { investment }\\\hline\mathrm{A} & 12 \% & 0.5 & \$ 20,000 \\\mathrm{~B} & 9 \% & 0.3 & \$ 10.000\end{array}$
Note that a low Risk rating means a less risky investment.The investor can invest to maximize the expected rate of return or minimize risk.Any money beyond the minimum investment requirements can be invested in either fund.
Formulate the MOLP for this investor.

Question 45

Exhibit 7.4
The following questions are based on the problem below.
Robert Gardner runs a small,local-only delivery service.His fleet consists of three smaller panel trucks.He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers.The box weights are: 210,160,320,90,110,70,410,260,170,240,80 and 180 for boxes 1 through 12,respectively.Since each truck differs each truck has different load capacities as given below:
Robert would like each truck equally loaded,both in terms of number of boxes and in terms of total weight,while minimizing his shipping costs.Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight.
The following integer goal programming formulation applies to his problem.
Y₁ = weight loaded in truck 1;Y₂ = weight loaded in truck 2;Y₃ = weight loaded intruck3;X_i,j = 0 if truck i not loaded with box j;1 if truck i loaded with box j.
Given the following spreadsheet solution of this integer goal programming formulation,answer the following questions.

Refer to Exhibit 7.4.The solution indicates Truck 3 is under the target weight by 67 pounds.What if anything can be done to this model to provide a solution in which Truck 3 is closer to the target weight?

Question 46

A company wants to purchase large and small delivery trucks.The company wants to purchase about 10 large and 15 small trucks.Each large truck costs $30,000 and has a 10 ton capacity.Each small truck costs $20,000 and has a 7 ton capacity.The company wants to have about 200 tons of capacity and spend about $600,000.
Formulate a goal programming model of this problem.

Question 47

One major advantage of goal programming GP)is that the technique:

A)allows a decision-maker to jointly examine several objectives
B)multiple objectives can be assigned different weights depending on their relative importance
C)can focus on a single objective,if necessary
D)all of the above

Question 48

A soft constraint

A)represents a target a decision-maker would like to achieve
B)is always tight
C)cannot be violated
D)typically represents a single goal

Question 49

Goal programming problems

A)typically include a set of multiple goals
B)cannot include hard constraints
C)consist of soft constraints only
D)must have a single objective function

Question 50

An investor wants to invest $50,000 in two mutual funds,A and B.The rates of return,risks and minimum investment requirements for each fund are:

Note that a low Risk rating means a less risky investment.The investor can invest to maximize the expected rate of return or minimize risk.Any money beyond the minimum investment requirements can be invested in either fund.
The following is the multi-objective linear programming MOLP)formulation for this problem:
Let ^X₁ ^{= dollars in investment A} X₂ = dollars in investment B
MAX: ^{0.12 X}₁^{/50000 + 0.09 X}₂^/50000
MIN: ^{0.5 X}₁^{/50000 + 0.3 X}₂^/50000
Subject to: ^X₁ ^{+ X}₂ ^{= 50000}
X₁ ≥ 20000
X₂ ≥ 10000
X_i ≥ 0 for all i
The solution for the second LP is X₁,X₂)= 20,000,30,000).
Based on this solution,what values should go in cells B2:D11 of the spreadsheet?

Question 51

Deviational variables

A)are added to constraints to indicate acceptable departures from the target vaoues of their corresponding goals
B)are negative
C)are positive for underachievement only
D)are positive for overachievement only

Question 52

An investor wants to invest $50,000 in two mutual funds,A and B.The rates of return,risks and minimum investment requirements for each fund are:
$\begin{array}{lrrr}&&&\text { Minimum }\\\text { Fund } & \text { Rate of return } & \text { Risk }&\text { investment }\\\hline\mathrm{A} & 12 \% & 0.5 & \$ 20,000 \\\mathrm{~B} & 9 \% & 0.3 & \$ 10.000\end{array}$
Note that a low Risk rating means a less risky investment.The investor wants to maximize the expected rate of return while minimizing his risk.Any money beyond the minimum investment requirements can be invested in either fund.The investor has found that the maximum possible expected rate of return is 11.4% and the minimum possible risk is 0.32.
Formulate a goal programming model with a MINIMAX objective function.

Question 53

An investor wants to invest $50,000 in two mutual funds,A and B.The rates of return,risks and minimum investment requirements for each fund are:

Note that a low Risk rating means a less risky investment.The investor wants to maximize the expected rate of return while minimizing his risk.Any money beyond the minimum investment requirements can be invested in either fund.The investor has found that the maximum possible expected rate of return is 11.4% and the minimum possible risk is 0.32.
The following Excel spreadsheet has been created to solve a goal programming problem with a MINIMAX objective based on the following goal programming formulation with MINIMAX objective and corresponding solution.
MINIMIZE Q
Subject to: ^X₁ ^{+ X}₂ ^{= 50000}
X₁ ≥ 20000
X₂ ≥ 10000




X_i ≥ 0 for all i,Q ≥ 0
with solution X₁,X₂)= 15,370,34,630).
What values should go in cells B2:D14 of the spreadsheet?

Question 54

Exhibit 7.4
The following questions are based on the problem below.
Robert Gardner runs a small,local-only delivery service.His fleet consists of three smaller panel trucks.He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers.The box weights are: 210,160,320,90,110,70,410,260,170,240,80 and 180 for boxes 1 through 12,respectively.Since each truck differs each truck has different load capacities as given below:
Robert would like each truck equally loaded,both in terms of number of boxes and in terms of total weight,while minimizing his shipping costs.Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight.
The following integer goal programming formulation applies to his problem.
Y₁ = weight loaded in truck 1;Y₂ = weight loaded in truck 2;Y₃ = weight loaded intruck3;X_i,j = 0 if truck i not loaded with box j;1 if truck i loaded with box j.
Given the following spreadsheet solution of this integer goal programming formulation,answer the following questions.

Refer to Exhibit 7.4.The spreadsheet model has scaled all the weights from pounds into 100s pounds.How does this scaling effect the solution obtained using the Risk Solver Platform RSP)?

Question 55

Goal programming GP)is:

A)iterative
B)inaccurate
C)static
D)all of the above

Question 56

An investor wants to invest $50,000 in two mutual funds,A and B.The rates of return,risks and minimum investment requirements for each fund are:
$\begin{array}{lrrr}&&&\text { Minimum }\\\text { Fund } & \text { Rate of return } & \text { Risk }&\text { investment }\\\hline\mathrm{A} & 12 \% & 0.5 & \$ 20,000 \\\mathrm{~B} & 9 \% & 0.3 & \$ 10.000\end{array}$
Note that a low Risk rating means a less risky investment.The investor can invest to maximize the expected rate of return or minimize risk.Any money beyond the minimum investment requirements can be invested in either fund.
The following is the MOLP formulation for this problem:




The solution for the second LP is X₁,X₂)= 20,000,30,000).
What formulas should go in cells B2:D11 of the spreadsheet? NOTE: Formulas are not required in all of these cells.
$\begin{array}{|c|c|c|c|c|}\hline & \text { A } & \text { B } & \text { C } & \text { D } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & \\\hline 2 & \text { Expected return } & 12 \% & 9 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4\\\hline 5 & \text { Variables } & \text { A } & \text { B } & \text { Total } \\\hline 6 & \text { Amount invested } & \$ 20,000 & \$ 30,000 & \$ 50,000 \\\hline 7 & \text { Minimum required } & \$ 20,000 & \$ 10,000 & \$ 50,000 \\\hline 8 & & & & \\\hline 9 & \text { Objectives: } & \\\hline 10 & \text { Average return } & 10.2 \% \\\hline 11 & \text { Average risk } & 0.32 \\\hline\end{array}$

Question 57

Goal programming GP)is typically

A)a minimization problem of the sum of weighted percentage deviations
B)a maximization problem of positive deviations only
C)a minimization problem of negative deviations only
D)a maximization problem of continuous goals

Question 58

Exhibit 7.4
The following questions are based on the problem below.
Robert Gardner runs a small,local-only delivery service.His fleet consists of three smaller panel trucks.He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers.The box weights are: 210,160,320,90,110,70,410,260,170,240,80 and 180 for boxes 1 through 12,respectively.Since each truck differs each truck has different load capacities as given below:
Robert would like each truck equally loaded,both in terms of number of boxes and in terms of total weight,while minimizing his shipping costs.Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight.
The following integer goal programming formulation applies to his problem.
Y₁ = weight loaded in truck 1;Y₂ = weight loaded in truck 2;Y₃ = weight loaded intruck3;X_i,j = 0 if truck i not loaded with box j;1 if truck i loaded with box j.
Given the following spreadsheet solution of this integer goal programming formulation,answer the following questions.

Refer to Exhibit 7.4.Given the solution indicated in the spreadsheet,which trucks,if any,are under an equal weight
amount,and which trucks are over an equal weight amount?

Question 59

A dietician wants to formulate a low cost,high calorie food product for a customer.The following information is available about the 2 ingredients which can be combined to make the food.The customer wants 1000 pounds of the food product and it should contain 250 pounds of Food 1 and 300 pounds of Food 2.The final cost of the blend should be about $1.15 and contain about 2500 calories per pound.The percent of fat,protein,carbohydrate in each food is summarized below with the target values for the goals.The dietician would prefer the food product be low in fat while also high in protein and carbohydrates.
$\begin{array}{lrrr} & \text { Food 1 } & \text { Food 2 } & \text { TARGET } \\\hline \text { Cost } \$ / \text { pound) } & \$ 1.00 & \$ 1.25 & \$ 1.15 \\\text { Fat } & 15 \% & 25 \% & 300 \text { pounds } \\\text { Protein } & 35 \% & 40 \% & 370 \text { pounds } \\\text { Carbohydrate } & 50 \% & 35 \% & 400 \text { pounds } \\\text { Calories/pound } & 3000 & 2000 & 2500 \\\text { Pounds of food 1 } & & & 250 \\\text { Pounds of food 2 } & & & 300\end{array}$

Formulate the GP for this problem

Question 60

A hard constraint

A)cannot be violated
B)may be violated
C)is always binding
D)is always part of the feasible solution

Question 61

In the "triple bottom line" the term "people" refers to:

A)social responsibility issues
B)environmental issues
C)financial objectives
D)all of the above

Question 62

Suppose that environmental and human variables are assigned the weight of zero.Then the "triple bottom line" approach reduces to:

A)profit maximization
B)environmental issues
C)HR objectives
D)achieving social equilibrium

Question 63

Multi-objective linear programming MOLP)provide

A)a way to analyze LP problems with multiple conflicting objectives
B)a way to incorporate soft constraints
C)a way to incorporate hard constraints
D)a simple way to solve the problem as a relaxed LP

Question 64

Suppose that profit and human variables are assigned the weight of zero.Then the "triple bottom line" approach reduces to:

A)profit maximization only
B)environmental considerations only
C)HR objectives only
D)achieving social happiness

Question 65

The "triple bottom line" incorporates multiple objective decision-making by:

A)simultaneously considering profit,people and planet
B)environmental issues only
C)financial objectives only
D)wealth redistribution in the society