Spreadsheet modeling and decision analysis a practical introduction to business analytics 8th editio

Spreadsheet Modeling and Decision Analysis A Practical Introduction to Business Analytics 8th Edition Ragsdale Test Bank Visit to download the full and correct content document: https://testbankdeal.com/dow nload/spreadsheet-modeling-and-decision-analysis-a-practical-introduction-to-busines s-analytics-8th-edition-ragsdale-test-bank/

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Chapter 07 1. A constraint which cannot be violated is called a a. binding constraint. b. hard constraint. c. definite constraint. d. required constraint.

ANSWER: b 2. 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 ANSWER: d 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: c 4. 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: c 5. 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.

ANSWER: d 6. The RHS value of a goal constraint is referred to as the a. target value. b. constraint value. c. objective value. d. desired value.

ANSWER: a 7. The di+, di− variables are referred to as Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 a. objective variables. b. goal variables. c. target variables. d. deviational variables.

ANSWER: d 8. Which of the following are true regarding weights assigned to deviational variables? a. Larger weights are assigned to undesirable deviations from the respective goals. 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 weights must be nonzero.

ANSWER: a 9. The di+ variable indicates the amount by which each goal's target value is a. missed. b. underachieved. c. overachieved. d. overstated.

ANSWER: c 10. 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 ANSWER: a 11. 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+ ≤ 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 ANSWER: d 12. 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.

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Chapter 07 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 ANSWER: b 13. 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 ANSWER: a 14. Which of the following formulas is a deviation-minimizing objective function for a goal programming problem? a. b. c. d.

ANSWER: d 15. What is the meaning of the ti term in this objective function for a goal programming problem?

a. The time required for each decision variable. b. The percent of goal i met. c. The coefficient for the ith decision variable d. The target value for goal i.

ANSWER: d 16. What weight would be assigned to a neutral deviational variable? a. 0 b. 1 c. 10 d. 100 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 ANSWER: a 17. 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: b 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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A Problem Data Cost Coverage

B TV 20 10

C Radio 10 7

D

E

Goal Constraints Actual Amount +Under − Over = Goal Target Value

TV 0 0 0 0 6

Radio 0 0 0 0 12

Cost

Coverage

0 0 0 200

0 0 0 140

Percentage Deviation: Under Over

1 0

1 0

1 0

1 0

Weights Under Over Objective

0

18. 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) ANSWER: d 19. Refer to Exhibit 7.1. What formula goes in cell B9?

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Chapter 07 a. =SUM(B6:B8) b. =B6+B7-B8 c. =B6-B7+B8 d. =B10-B8 ANSWER: b 20. 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 ANSWER: c 21. 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 ANSWER: a 22. 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 ANSWER: a 23. 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 ANSWER: c 24. 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 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 25. 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.

ANSWER: c 26. 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.

ANSWER: b 27. 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.

ANSWER: c 28. 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.

ANSWER: d 29. MINIMAX solutions to multi-objective linear programming (MOLP) problems are a. dually optimal. b. Pareto optimal. c. suboptimal. d. maximally optimal.

ANSWER: b 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.

ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 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.

1 2 3 4 5 6 7 8 9 10 11

A Problem data Expected return Risk rating

B A 12% 0.50

C B 10% 0.20

D

Variables Amount invested Minimum required

A 0 $10,000

B 0 $15,000

Total 0 $150,000

Objectives: Average return Average risk

0 0

31. 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) ANSWER: a 32. 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$7 ANSWER: b 33. Refer to Exhibit 7.2. What formula goes in cell B11? a. =SUMPRODUCT(B2:C2,$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) ANSWER: c 34. Refer to Exhibit 7.2. Which cells are the changing cells in this model? a. $B$6:$C$6, $B$10:$B$11 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 b. $B$6:$C$6 c. $B$6:$D$6 d. $B$10:$B$11 ANSWER: b 35. 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 ANSWER: d 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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A Problem data Expected return Risk rating

B A 12% 0.50

C B 10% 0.20

D

Variables Amount invested Minimum required

A 0 $10,000

B 0 $15,000

Total 0 $150,000

Goals Average return Average risk

Actual 0 0

Objective:

0

Target 11.8% 0.22

Weights 1 1

E

Weighted % Deviation 0 0

36. 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) ANSWER: a 37. 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 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 b. D12, increase c. C12, increase d. D12, decrease ANSWER: b 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 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.

ANSWER: c 39. 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?

ANSWER: d1− = 5, d2+ = 6, d3− = 7, all others equal to zero.

40. Consider the following multi-objective linear programming problem (MOLP):

MAX: MAX: Subject to:

3 X1 + 4 X2 2 X1 + X2 6 X1 + 13 X2 ≤ 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?

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Chapter 07

ANSWER:

OBJ 1 OBJ 2 X1 X2 0 0 0 0 0 6 24 6 9 0 27 18 5.9 3.27 30.78 15.07 7.5 2 30.5 17 The solution to MAX 3 X1 + 4 X2 is (X1, X2) = (5.9, 3.27) and objective function value of 30.78.

The solution to MAX 2 X1 + X2 is (X1, X2) = (9, 0) and objective function value of 18.

41. 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.

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Chapter 07 Product A B Labor (hr/unit) 3 2 Material (ounces/unit) 1 2 Profit($/unit) 7 6 Management has developed the following set of goals

Available resources 150 200

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.

ANSWER: Let X1 = number of product 1 X2 = number of product 2 MIN: Subject to:

d1− + d1+ + d2 − + d2+ + d3− + d4+ + d5+ X1 + d1− − d1+ = 40 X2 + d2− − d2+ = 70 7 X1 + 6 X2 + d3− − d3+ = 400 3 X1 + 2 X2 + d4− − d4+ = 150 1 X1 + 2 X2 + d5− − d5+ = 200 Xi, di−, di+ ≥ 0 for all i

product 1 product 2 profit labor material

42. 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.

Product 1 2 Labor (hr/unit) 3 2 Material (ounces/unit) 1 2 Profit($/unit) 7 6 Management has developed the following set of goals

Available resources 150 200

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?

Let

X1 = number of product 1 X2 = number of product 2

MIN:

d1− + d1+ + d2− + d2+ + d3− + d4+ + d5+

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Chapter 07 X1 + d 1 − − d 1 + = 40 X2 + d 2 − − d 2 + = 70 7 X1 + 6 X2 + d3− − d3+ = 400 3 X1 + 2 X2 + d4− − d4+ = 150 1 X1 + 2 X2 + d5− − d5+ = 200 Xi, di−, di+ ≥ 0 for all i

Subject to:

product 1 product 2 profit labor material

(X1, X2) = (4, 69) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A Problem Data Labor Material Profit Goal Constraints Actual Amount + Under − Over = Goal Target Value

B A

C B

D

E

F

A

B

Labor

Material

Profit

Weights Under Over Objective

ANSWER:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A Problem Data Labor Material Profit Goal Constraints Actual Amount + Under − Over = Goal Target Value

B A 3 1 7 A 4 36 0 40 40

C B 2 2 6 B 69 1 0 70 70

D

E

F

Labor 150 0 0 150 150

Material 142 58 0 200 200

Profit 442 0 42 400 400

Weights Under Over

1 1

1 1

0 1

0 1

1 0

Objective

37

43. A company makes 2 products A and B from 2 resources. The products have the following resource requirements and Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 produce the accompanying profits. The available quantity of resources is also shown in the table.

Product 1 2 Labor (hr/unit) 3 2 Material (ounces/unit) 1 2 Profit($/unit) 7 6 Management has developed the following set of goals

Available resources 150 200

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 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.

Let

X1 = number of product 1 X2 = number of product 2

MIN: Subject to:

d1− + d1+ + d2− + d2+ + d3− + d4+ + d5+ X1 + d1− − d1+ = 40 X2 + d2− − d2+ = 70 7 X1 + 6 X2 + d3− − d3+ = 400 3 X1 + 2 X2 + d4− − d4+ = 150 1 X1 + 2 X2 + d5− − d5+ = 200 Xi, di−, di+ ≥ 0 for all i

product 1 product 2 profit labor material

(X1, X2) = (4, 69) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A Problem Data Labor Material Profit Goal Constraints Actual Amount + Under − Over = Goal Target Value

B A 3 1 7 A 4 36 0 40 40

C B 2 2 6 B 69 1 0 70 70

D

E

F

Labor 150 0 0 150 150

Material 142 58 0 200 200

Profit 442 0 42 400 400

Weights Under Over

1 1

1 1

0 1

0 1

1 0

Objective

37

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Chapter 07 ANSWER: Cell D6 E6 F6 B9 B16

Formula =SUMPRODUCT(B2:C2,$B$6:$C$6) =SUMPRODUCT(B3:C3, $B$6:$C$6) =SUMPRODUCT(B4:C4, $B$6:$C$6) =B6+B7-B8 =SUMPRODUCT(B7:F8,B13:F14)

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C9:F9

44. 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.

ANSWER: Let X1 = number of large trucks X2 = number of small trucks MIN: Subject to:

d1− + d1+ + d2− + d2+ + d3− + d3+ + d4− − d4+ X1 + d1− − d1+ = 10 X2 + d2− − d2+ = 15 10 X1 + 7 X2 + d3− − d3+ = 200 30 X1 + 20 X2 + d4− − d4+ = 600 Xi, di−, di+ ≥ 0 for all i

large trucks small trucks capacity cost

45. 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?

Let

X1 = number of large trucks X2 = number of small trucks

MIN: Subject to:

d1− + d1+ + d2− + d2+ + d3− + d3+ + d4− − d4+ X1 + d1− − d1+ = 10 X2 + d2− − d2+ = 15 10 X1 + 7 X2 + d3− − d3+ = 200 30 X1 + 20 X2 + d4− − d4+ = 600 Xi, di−, di+ ≥ 0 for all I

large trucks small trucks capacity cost

(X1, X2) = (10, 15) 1 2 3

A Problem Data Cost Capacity

B Large

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C Small

D

E

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Chapter 07 4 5 6 7 8 9 10 11 12 13 14 15 16

Goal Constraints Actual Amount + Under − Over = Goal Target Value

Large

Small

Cost

Capacity

Weights Under Over Objective

ANSWER:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A Problem Data Cost Capacity

B Large 10 30

C Small 7 20

D

E

Goal Constraints Actual Amount + Under − Over = Goal Target Value

Large 10 0 0 10 10

Small 15 0 0 15 15

Cost 205 0 5 200 200

Capacity 600 0 0 600 600

Weights Under Over

1 1

1 1

1 1

1 1

Objective

5

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.

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?

Let

X1 = number of large trucks X2 = number of small trucks

MIN: Subject to:

d1− + d1+ + d2− + d2+ + d3− + d3+ + d4− − d4+ X1 + d1− − d1+ = 10 X2 + d2− − d2+ = 15

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Chapter 07 10 X1 + 7 X2 + d3− − d3+ = 200 30 X1 + 20 X2 + d4− − d4+ = 600 Xi, di−, di+ ≥ 0 for all I

capacity cost

(X1, X2) = (10, 15) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A Problem Data Cost Capacity

B Large 10 30

C Small 7 20

D

E

Goal Constraints Actual Amount + Under − Over = Goal Target Value

Large 10 0 0 10 10

Small 15 0 0 15 15

Cost 205 0 5 200 200

Capacity 600 0 0 600 600

Weights Under Over

1 1

1 1

1 1

1 1

Objective

5

ANSWER: Cell D6 E6 B9 B16

Formula =SUMPRODUCT(B2:C2,$B$6:$C$6) =SUMPRODUCT(B3:C3, $B$6:$C$6) =B6+B7-B8 =SUMPRODUCT(B13:E14,B7:E8)

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C9:E9

47. 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.

Food 1 $1.00 15% 35% 50% 3000

Cost ($/pound) Fat Protein Carbohydrate Calories/pound Pounds of food 1 Pounds of food 2 Formulate the GP for this problem ANSWER: Let X1 = pounds of food 1

Food 2 $1.25 25% 40% 35% 2000

TARGET $1.15 300 pounds 370 pounds 400 pounds 2500 250 300

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Chapter 07 d1− + d1+ + d2+ + d3− + d4− + d5− + d5+ + d6− + d6+ + d7− + d7+ cost 1 X1 + 1.25 X2 + d1− − d1+ = 1.15 + fat .15 X1 + .25 X2 + d2− − d2 = 300 + protein .35 X1 + .40 X2 + d3− − d3 = 370 + carbohydrate .50 X1 + .35 X2 + d4− − d4 = 400 + calories 3000 X1 + 2000 X2 + d5− − d5 = 2500 + food 1 X1 + d6− − d6 = 250 + food 2 X2 + d7− − d7 = 300 food product required X1 + X2 = 1000 Xi, di−, di+ ≥ 0 for all i

MIN: Subject to:

48. A dietitian 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.

Food 1 1.00 15 35 50 3000 250

Cost ($/pound) % Fat % Protein % Carbohydrate Calories/pound Minimum pounds of food Formulate the MOLP for this problem.

ANSWER: Let X1 = pounds of food 1

Food 2 1.25 25 40 35 2000 300

Requirements Minimize Less than 30% At least 37% At least 40% Maximize

X2 = pounds of food 2 MIN: MAX: Subject to:

minimize cost or food product maximize the calories fat protein carbohydrate food 1 food 2 total food product required

1 X1 + 1.25 X2 3000 X1 + 2000 X2 .15 X1 + .25 X2 ≤ 300 .35 X1 + .40 X2 ≥ 370 .50 X1 + .35 X2 ≥ 400 X1 ≥ 250 X2 ≥ 300 X1 + X2 = 1000 Xi ≥ 0 for all i

49. A company needs to supply customers in 3 cities from its 3 warehouses. The supplies, demands and shipping costs are shown below.

Warehouse 1 2

1 34 70

Destination 2 60 40

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3 36 50

Supply 400 300 Page 17

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Chapter 07 3 56 40 Demand 500 300 The company has identified the following goals:

32 200

200

The company would like to come as close as possible to satisfying its customers demand.

Goal 2: It would also like to ensure that the cost is approximately $290,000.

Formulate a goal programming model of this problem.

ANSWER: Let Xij = number of units shipped from warehouse i to destination j Goal 1:

MIN: Subject to: X11 + X12 + X13 + d 1 − − d 1 + = 400 X21 + X22 + X23 + d 2 − − d 2 + = 300 X31 + X32 + X33 + d 3 − − d 3 + = 200 X11 + X21 + X31 + d 4 − − d 4 + = 500 X12 + X22 + X32 + d 5 − − d 5 + = 300 X13 + X23 + X33 + d 6 − − d 6 + = 200 34 X11 + 60 X12 + 36 X13 + 70 X21 + 40 X22 + 50 X23 + 56 X31 + 40 X32 + 32 X33 + d7− + d7+ = 290000 Xij, di−, di+ ≥ 0 for all i,j 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:

Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 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.

ANSWER: Let X1 = dollars in investment A X2 = dollars in investment B MAX: MIN: Subject to:

0.12 X1/50000 + 0.09 X2/50000 0.5 X1/50000 + 0.3 X2/50000 X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000 Xi ≥ 0 for all i

51. An investor wants to invest $50,000 in two mutual funds, A and B. The rates of return, risks and minimum investment Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 requirements for each fund are:

Fund Rate of return Risk Minimum investment A 12% .5 $20,000 B 9% .3 $10,000 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: MIN: Subject to:

0.12 X1/50000 + 0.09 X2/50000 0.5 X1/50000 + 0.3 X2/50000 X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000 Xi ≥ 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?

1 2 3 4 5 6 7 8 9 10 11

A Problem data Expected return Risk rating

B A

C B

D

Variables Amount invested Minimum required

A

B

Total

A Problem data Expected return Risk rating

B A 12% 0.50

C B 9% 0.20

D

Variables Amount invested Minimum required

A $20,000 $20,000

B $30,000 $10,000

Total $50,000 $50,000

Objectives: Average return Average risk

ANSWER:

1 2 3 4 5 6 7

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Chapter 07 8 9 10 11

Objectives: Average return Average risk

10.2% 0.32

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:

Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 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:

Let

X1 = dollars in investment A X2 = dollars in investment B

MAX: MIN: Subject to:

0.12 X1/50000 + 0.09 X2/50000 0.5 X1/50000 + 0.3 X2/50000 X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000 Xi ≥ 0 for all i 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.

1 2 3 4 5 6 7 8 9 10 11

A Problem data Expected return Risk rating

B A 12% 0.50

C B 9% 0.20

D

Variables Amount invested Minimum required

A $20,000 $20,000

B $30,000 $10,000

Total $50,000 $50,000

Objectives: Average return Average risk

10.2% 0.32

ANSWER: Cell D6 B10

Formula =B6+C6 =SUMPRODUCT(B2:C2,B6:C6)/D7

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Chapter 07 B11

=SUMPRODUCT(B3:C3,B6:C6)/D7

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:

Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 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.

ANSWER: Let X1 = dollars in investment A X2 = dollars in investment B MINIMIZE Subject to:

Q X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000

Xi ≥ 0 for all i, Q ≥ 0 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:

Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 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 Subject to:

Q X1 + X2 = 50000 X1 ≥ 20000

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Chapter 07 X2 ≥ 10000

Xi ≥ 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?

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A Problem data Expected return Risk rating

B A

C B

D

Variables Amount invested Minimum required

A

B

Total

Goals Average return Average risk

Actual

Target

Weights 1 1

E

Weighted % Deviation

Objective:

ANSWER:

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A Problem data Expected return Risk rating

B A 0.12 0.5

C B 0.09 0.3

D

Variables Amount invested Minimum required

A 15370 10000

B 34630 20000

Total 50000 50000

Goals Average return Average risk

Actual 0.09922 0.36148

Target 0.114 0.320

Weights 1 1

Objective:

0.12963

E

Weighted % Deviation 0.12963 0.12963

55. Robert Gardner runs a small, local-only delivery service. His fleet consists of three smaller panel trucks. He recently Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 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:

Truck Weight Capacity Box Capacity Cost per pound 1 800 pounds 5 $0.34 2 900 pounds 6 $0.42 3 700 pounds 4 $0.25 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.)

ANSWER: Y1 = weight loaded in truck 1; Y2 = weight loaded in truck 2; Y3 = weight loaded in truck 3; Xi,j = 0 if truck i not loaded with box j; 1 if truck i loaded with box j.

MIN 0.34Y1 + 0.42Y2 + 0.25Y3 + 50*(d1+ + d2+ + d3+) + 0.10*(d4− + d5− + d6−)*100 Subject to: Y1 = 210X11 + 160X12 + 320X13 + 90X14 + 110X15 + 70X16 + 410X17 + 260X18 + 170X19 + 240X1,10 + 80X1,11 + 180X1,12 Y2 = 210X21 + 160X22 + 320X23 + 90X24 + 110X25 + 70X26 + 410X27 + 260X28 + 170X29 + 240X2,10 + 80X2,11 + 180X2,12 Y3 = 210X31 + 160X32 + 320X33 + 90X34 + 110X35 + 70X36 + 410X37 + 260X38 + 170X39 + 240X3,10 + 80X3,11 + 180X3,12 Y1 ≤ 800 Y2 ≤ 900 Y3 ≤ 700 Y1 + d1− − d1+ = 767 Y2 + d2− − d2+ = 767 Y3 + d3− − d3+ = 767 X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 X13 + X23 + X33 = 1 X14 + X24 + X34 = 1 X15 + X25 + X35 = 1 X16 + X26 + X36 = 1 X17 + X27 + X37 = 1 X18 + X28 + X38 = 1 X19 + X29 + X39 = 1 X1,10 + X2,10 + X3,10 = 1 X1,11 + X2,11 + X3,11 = 1 X1,12 + X2,12 + X3,12 = 1 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 X11 + X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X1,10 + X1,11 + X1,12 + d4− − d4+ = 4 X21 + X22 + X23 + X24 + X25 + X26 + X27 + X28 + X29 + X2,10 + X2,11 + X2,12 + d5− − d5+ = 4 X31 + X32 + X33 + X34 + X35 + X36 + X37 + X38 + X39 + X3,10 + X3,11 + X3,12 + d6− − d6+ = 4 Xij ≥ 0, dk−, dk+ ≥ 0 for k = 1,2,3,4,5,6 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:

Truck 1 2 3

Weight Capacity 800 pounds 900 pounds 700 pounds

Box Capacity 5 6 4

Cost per pound $0.34 $0.42 $0.25

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; Xi,j = 0 if truck i not loaded with box j; 1 if truck i loaded with box j.

MIN 0.34Y1 + 0.42Y2 + 0.25Y3 + 50*(d1+ + d2+ + d3+) + 0.10*(d4− + d5− + d6−)*100 Subject to: Y1 = 210X11 + 160X12 + 320X13 + 90X14 + 110X15 + 70X16 + 410X17 + 260X18 + 170X19 + 240X1,10 + 80X1,11 + 180X1,12 Y2 = 210X21 + 160X22 + 320X23 + 90X24 + 110X25 + 70X26 + 410X27 + 260X28 + 170X29 + 240X2,10 + 80X2,11 + 180X2,12 Y3 = 210X31 + 160X32 + 320X33 + 90X34 + 110X35 + 70X36 + 410X37 + 260X38 + 170X39 + 240X3,10 + 80X3,11 + 180X3,12 Y1 ≤ 800 Y2 ≤ 900 Y3 ≤ 700 Y1 + d1− − d1+ = 767 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 Y2 + d2− − d2+ = 767 Y3 + d3− − d3+ = 767 X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 X13 + X23 + X33 = 1 X14 + X24 + X34 = 1 X15 + X25 + X35 = 1 X16 + X26 + X36 = 1 X17 + X27 + X37 = 1 X18 + X28 + X38 = 1 X19 + X29 + X39 = 1 X1,10 + X2,10 + X3,10 = 1 X1,11 + X2,11 + X3,11 = 1 X1,12 + X2,12 + X3,12 = 1 X11 + X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X1,10 + X1,11 + X1,12 + d4− − d4+ = 4 X21 + X22 + X23 + X24 + X25 + X26 + X27 + X28 + X29 + X2,10 + X2,11 + X2,12 + d5− − d5+ = 4 X31 + X32 + X33 + X34 + X35 + X36 + X37 + X38 + X39 + X3,10 + X3,11 + X3,12 + d6− − d6+ = 4 Xij ≥ 0, dk−, dk+ ≥ 0 for k = 1,2,3,4,5,6 Given the following spreadsheet solution of this integer goal programming formulation, answer the following questions.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A Cost Capacity 1 2 3 4 5 6 7 8 9 10 11 12 Total Items Under Over Actual Target Items Total Wt Under Over Actual Wt Target Wt

B $0.34 8.0 1 0 1 1 0 0 0 0 0 0 1 1 0 4 0 0 4 4 8 0.00 0.33 7.67 7.67

C $0.42 9.0 2 1 0 0 0 1 1 1 0 0 0 0 0 4 0 0 4 4 8 0.00 0.33 7.67 7.67

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D $0.25 7.0 3 0 0 0 1 0 0 0 1 1 0 0 1 4 0 0 4 4 7 0.67 0.00 7.67 7.67

E

F

G

Weight 2.1 1.6 3.2 0.9 1.1 0.7 4.1 2.6 1.7 2.4 0.8 1.8

Assigned 1 1 1 1 1 1 1 1 1 1 1 1

Required 1 1 1 1 1 1 1 1 1 1 1 1

Item Cost Wt. Cost

$50 $0.25

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Chapter 07 26

Cost

$7.8967

56. 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?

ANSWER: Trucks 1 and 2 are over the target weight by 33 pounds. Truck 3 is under the target weight by 67 pounds.

57. 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?

ANSWER: Nothing. Truck 3 is at its capacity of 700 pounds.

58. 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 Analytic Solver Platform (ASP)?

ANSWER: The solution obtained is the same regardless of scaling. In terms of pounds, the total cost would be multiplied by 100, to obtain $78,967.

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

ANSWER: =SUMPRODUCT(B21:D21, B1:D1) + F17*SUM(B18:D18) + F18*SUM(B22:D22) 60. 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?

ANSWER: Cell Formula Copied to B19 =B16 + B17 − B18 C19:E19 B24 =B21 + B22 − B23 C24:E24 61. 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 ANSWER: d 62. Goal programming (GP) is: a. iterative b. inaccurate c. static d. all of the above ANSWER: a 63. 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 ANSWER: a 64. A hard constraint Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 a. cannot be violated b. may be violated c. is always binding d. is always part of the feasible solution ANSWER: a 65. 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 ANSWER: a 66. 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 ANSWER: a 67. Deviational variables a. are added to constraints to indicate acceptable departures from the target values of their corresponding goals b. are negative c. are positive for underachievement only d. are positive for overachievement only ANSWER: a 68. 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 ANSWER: a 69. 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 ANSWER: a 70. Suppose that environmental and human variables are assigned the weight of zero. Then the "triple bottom line" approach reduces to: a. profit maximization Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 b. environmental issues c. HR objectives d. achieving social equilibrium ANSWER: a 71. 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 ANSWER: b 72. 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 ANSWER: a 73. 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 ANSWER: a 74. Goal Programming and Multiple Objective Optimization are not related a. True b. False ANSWER: False 75. Hard constraints can be violated, if necessary a. True b. False ANSWER: False 76. The second step in Goal Programming is defining the goals a. True b. False ANSWER: True 77. The deviational variables represent the amount by which the goal's target is underachieved a. True Copyright Cengage Learning. Powered by Cognero.

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Chapter 07 b. False ANSWER: False 78. In the goal programming problem, the weights, wi, attached to deviational variables must decay exponentially a. True b. False ANSWER: False 79. Trade-offs in goal programming can be made by modifying the weights to satisfy the priorities of a decision maker a. True b. False ANSWER: True 80. Linear programming problems cannot have multiple objectives a. True b. False ANSWER: False 81. In MOLP, a decision alternative is dominated if another alternative produces a better value of at least one objective without worsening the value of other objectives a. True b. False ANSWER: True

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