Deck 7: Goal Programming and Multiple Objective Optimization

Question 1

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.

Refer to Exhibit 7.1. Which of the following is a constraint specified to Risk Solver Platform (RSP) 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

Answer 1

C

Question 2

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.

Answer 2

C

Question 3

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.

Answer 3

C

Question 4

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.

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 5

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 6

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

A) d₁^- = 4, d₁⁺ = 0
B) d₁^- = 0, d₁⁺ = 4
C) d₁^-= 4, d₁⁺ = 4
D) d₁^- = 8, d₁⁺ = 0

Question 7

The d_i⁺ variable indicates the amount by which each goal's target value is

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

Question 8

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) d₁^- = 0, d₁⁺ = 10
B) d₁^? = 10, d₁⁺ = 0
C) d₁^- = 5, d₁⁺ = 5
D) d₁^- = 6, d₁⁺ = 0

Question 9

What weight would be assigned to a neutral deviational variable?

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

Question 10

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 11

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.

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

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

Question 12

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 13

A constraint which cannot be violated is called a

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

Question 14

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.

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

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

Question 15

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.

Question 16

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 17

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

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 18

The d_i⁺, d_i^-variables are referred to as

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

Question 19

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

A) 3X₁ + 2 X₂ + d₁^  d₁⁺ = 10
B) 3X₁ + 2 X₂ + d₁^ + d₁⁺ = 10
C) 3X₁ + 2 X₂  d₁^  d₁⁺  10
D) 3X₁ + 2 X₂ + d₁^  d₁⁺  10

Question 20

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 X₁ + 4 X₂ + d₁^-- d₁⁺ $\le$ 36
B) 3 X₁ + 4 X₂ -d₁^- - d₁⁺ = 36
C) 3 X₁ + 4 X₂ + d₁^- + d₁⁺ = 36
D) 3 X₁ + 4 X₂ + d₁^- -d₁⁺ = 36

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.

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 22

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.

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 23

The decision maker has expressed concern with Goal 1, budget, achievement. He indicated that future candidate solutions should stay under budget. How can you modify your goal programming model to accommodate this change?

A) Make budget a hard constraint in the model.
B) Give d₁⁺ an extremely large weight in the objective function.
C) Remove d₁⁺ from the goal constraint.
D) All of these.

Question 24

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.
Management has developed the following set of goals
Formulate a goal programming model of this problem.

Question 25

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 40 X₁ + 20 X₂ + d₄^ + d₄⁺ = 300
To accommodate this requirement?

A) Increase the RHS value from 300 to 350.
B) Replace the constraint with 40 X₁ + 20 X₂  350.
C) Do not modify the constraint, add a constraint d₄⁺  50.
D) Do not modify the constraint, add a constraint d₄⁺ = 50.

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.
$$\begin{array} { | c | l | c | c | c | } \hline & { \text { A } } & \mathrm { B } & \mathrm { C } & \mathrm { D } \\\hline 1 & \text { Problem data } & \mathrm { A } & \mathrm { B } & \\\hline 2 & \text { Expected retun } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm { A } & \mathrm { B } & \text { Tatal } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & \text { Objectives: } & & & \\\hline 10 & \text { Average return } & 0 & & \\\hline 11 & \text { Average risk } & 0 & & \\\hline\end{array}$$

-Refer to Exhibit 7.2. What Risk Solver Platform (RSP) 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 27

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 28

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.

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 29

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.
$$\begin{array} { | c | l | c | c | c | } \hline & { \text { A } } & \mathrm { B } & \mathrm { C } & \mathrm { D } \\\hline 1 & \text { Problem data } & \mathrm { A } & \mathrm { B } & \\\hline 2 & \text { Expected retun } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm { A } & \mathrm { B } & \text { Tatal } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & \text { Objectives: } & & & \\\hline 10 & \text { Average return } & 0 & & \\\hline 11 & \text { Average risk } & 0 & & \\\hline\end{array}$$

-Refer to Exhibit 7.2. Which cell(s) is(are) 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 30

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.
$$\begin{array} { | c | l | c | c | c | } \hline & { \text { A } } & \mathrm { B } & \mathrm { C } & \mathrm { D } \\\hline 1 & \text { Problem data } & \mathrm { A } & \mathrm { B } & \\\hline 2 & \text { Expected retun } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm { A } & \mathrm { B } & \text { Tatal } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & \text { Objectives: } & & & \\\hline 10 & \text { Average return } & 0 & & \\\hline 11 & \text { Average risk } & 0 & & \\\hline\end{array}$$

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

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

Question 31

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 32

Consider the following MOLP:
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 33

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) MIN d₃⁺
B) d₃^  1000
C) d₃⁺ = 1000
D) d₃⁺  1000

Question 34

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.

Refer to Exhibit 7.1. Which cell(s) is(are) the objective cell(s) in this model?

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

Question 35

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 36

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.
$$\begin{array} { | c | l | c | c | c | } \hline & { \text { A } } & \mathrm { B } & \mathrm { C } & \mathrm { D } \\\hline 1 & \text { Problem data } & \mathrm { A } & \mathrm { B } & \\\hline 2 & \text { Expected retun } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm { A } & \mathrm { B } & \text { Tatal } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & \text { Objectives: } & & & \\\hline 10 & \text { Average return } & 0 & & \\\hline 11 & \text { Average risk } & 0 & & \\\hline\end{array}$$

-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 37

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

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

Question 38

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 39

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 40

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.
$$\begin{array} { | c | l | c | c | c | } \hline & { \text { A } } & \mathrm { B } & \mathrm { C } & \mathrm { D } \\\hline 1 & \text { Problem data } & \mathrm { A } & \mathrm { B } & \\\hline 2 & \text { Expected retun } & 12 \% & 10 \% & \\\hline 3 & \text { Risk rating } & 0.50 & 0.20 & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \mathrm { A } & \mathrm { B } & \text { Tatal } \\\hline 6 & \text { Amount invested } & 0 & 0 & 0 \\\hline 7 & \text { Minimum required } & \$ 10,000 & \$ 15,000 & \$ 150,000 \\\hline 8 & & & & \\\hline 9 & \text { Objectives: } & & & \\\hline 10 & \text { Average return } & 0 & & \\\hline 11 & \text { Average risk } & 0 & & \\\hline\end{array}$$

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

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

Question 41

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.
Formulate the integer goal programming problem for Robert. (Hint: objective function involves decision and deviation variables.)

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 in truck 3;
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 43

A company needs to supply customers in 3 cities from its 3 warehouses. The supplies, demands and shipping costs are shown below.
The company has identified the following goals:
Formulate a goal programming model of this problem.

Question 44

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.
Management has developed the following set of goals
Based on the following GP formulation of the problem, and the associated optimal solution, what formulas should go in cells D6:F6, B9:F9, and B16 of the following Excel spreadsheet? NOTE: Formulas are not required in all of these cells.

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 in truck 3;
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 46

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 in truck 3;
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 47

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.
Formulate a goal programming model with a MINIMAX objective function.

Question 48

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 49

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.
Formulate the GP for this problem

Question 50

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.
Management has developed the following set of goals
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 51

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 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.

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:
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).
Based on this solution, what values should go in cells B2:D11 of the spreadsheet?

Question 53

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 in truck 3;
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 54

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.
Formulate the MOLP for this investor.

Question 55

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 goal programming formulation, associated solution, and spreadsheet model, what formulas should go in cells D6:E6, B9:E9, and B16 of the spreadsheet?

Question 56

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 57

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 must contain at least 250 pounds of Food 1 and 300 pounds of Food 2.
Formulate the MOLP for this problem.

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 in truck 3;
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. Based on the integer goal programming formulation, the associated solution, and spreadsheet model, what formulas should go in cells B19:E19 and B24:E24 of the spreadsheet?

Question 59

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.
with solution (X₁, X₂) = (15,370, 34,630).
What values should go in cells B2:D14 of the spreadsheet?