Spreadsheet modeling and decision analysis 6th edition ragsdale test bank

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Chapter 8—Nonlinear Programming and Evolutionary Optimization MULTIPLE CHOICE 1. The main difference between LP and NLP problems is that NLPs will have a a. linear objective function and nonlinear or linear constraints. b. minimum of one nonlinear constraint or a nonlinear objective function. c. multilevel objective functions. d. multilevel objective function and nonlinear constraints.

ANS: B

PTS: 1

2. The optimal solution to a NLP problem can occur at a(n) I. II.

corner point. interior point.

a. b. c. d.

I is true II is true I and II are true neither I or II is true

ANS: C

PTS: 1

3. The optimal solution to a LP problem is always at I. II. III. a. b. c. d.

a corner point. an interior point the origin.

I is true II is true III is true I and III are true

ANS: A

PTS: 1

4. The GRG and Simplex algorithms are similar in that a. each algorithm process continues until there is no further improvement in the objective function. b. both algorithms take their starting solution from the spreadsheet. c. both return the globally optimal solution. d. both algorithms return a solution that satisfies at least one constraint at equality.

ANS: A

PTS: 1

5. The Risk Solver Platform (RSP) solution strategy for NLP problems is the GRG method. What does GRG stand for? a. Graphical Residual Gradient b. Generalized Reduced Gradient c. Goal Restricted Gradient d. Gradually Reduced Gradient ANS: B

PTS: 1

6. In the GRG algorithm the initial solution is called the a. originating point. b. insertion point. c. zero point. d. starting point.

ANS: D

PTS: 1

7. The GRG algorithm operates by a. moving in the direction of most rapid improvement in the objective function. b. choosing a search direction at random. c. searching directly for the optimum solution. d. moving in a clockwise direction.

ANS: A

PTS: 1

8. What is the search path for the following feasible solution space? The dashed line represents the objective function and the objective is to maximize the value of the objective function.

a. b. c. d.

A, D A, F, E, D A, B, C, D A, F, E

ANS: C

PTS: 1

9. In a maximization problem, the GRG algorithm's search for a feasible direction of improvement equates to ____ in the simplex algorithm for LP problems. a. an adjacent corner point b. a zero reduced cost c. a positive shadow price d. a positive reduced cost ANS: D

PTS: 1

10. The GRG algorithm terminates when it a. has completed 100 iterations. b. has reached the global optimal solution. c. when it detects no feasible direction for improvement. d. when it reaches the steepest gradient.

ANS: C

PTS: 1

11. In NLP a local optimum is best described as

a. b. c. d.

an interior point. a corner point. a point yielding no improving direction. a good starting point for subsequent searches.

ANS: C

PTS: 1

12. Which point or points are local optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function.

a. b. c. d.

D E F D, F

ANS: D

PTS: 1

13. Which point or points are global optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function.

a. b. c. d.

B D E F

ANS: B

PTS: 1

14. When using the GRG algorithm to solve NLPs one should try multiple starting points because a. If two different starting points return the same solution, that solution is optimal. b. the solution returned depends upon the starting point. c. the solution returned is always near the starting point. d. a random element of GRG requires multiple starting points.

ANS: B

PTS: 1

15. NLP problems which have slack in all the constraints a. are infeasible. b. are only at local optimal solutions. c. have zeros for all shadow prices. d. are at an interior point of the feasible region.

ANS: D

PTS: 1

16. If the "Standard LP/Quadratic Engine" option is chosen in the Risk Solver Platform (RSP) task pane, RSP a. will test for non-linearity in the model and apply GRG. b. will test for non-linearity in the model and apply Simplex. c. uses the B & B algorithm to solve the problem. d. will apply Simplex if the linearity test passes.

ANS: D

PTS: 1

17. The optimal trade-off between risk and return for a given portfolio problem can be summarized by the a. efficient frontier. b. investment frontier. c. portfolio boundary. d. variance boundary.

ANS: A

PTS: 1

18. The straight line (Euclidean) distance between two points (X1, Y1) and (X2, Y2) is calculated as a. X1 − Y1 + X2 − Y2 b. (X1 − X2)2 + (Y1 − Y2)2 c. d.

ANS: D

PTS: 1

19. What is the straight line (Euclidean) distance between the points (5,7) and (1, 11)? a. 2.83 b. 5.65 c. 8 d. 16 ANS: B

PTS: 1

20. The total annual cost for the economic order quantity model is a. purchase cost + ordering cost + holding cost. b. fixed ordering cost + holding cost. c. purchase cost + fixed ordering cost. d. unit cost + variable ordering cost + holding cost.

ANS: A

PTS: 1

21. Which of the following is not an assumption of an EOQ problem. a. Demand for a product is fairly constant. b. Inventory depletion rates vary non-linearly.

c. Each new order is delivered in full when the inventory level reaches 0. d. All are valid assumptions.

ANS: B

PTS: 1

22. A company has collected the following inventory data for an item. What is the total annual cost for this item?

Annual demand for the item Unit purchase cost for the item Fixed cost of placing an order Cost of holding one unit in inventory for a year Order quantity a. b. c. d.

D = 500 C = 10 S = 20 i = 30% Q = 50

275 450 5,275 5,450

ANS: C

PTS: 1

23. The Reduced Gradient is similar to which of these terms from linear programming? a. Shadow Price b. Allowable Decrease c. Allowable Increase d. Reduced Cost ANS: D

PTS: 1

24. The Lagrange Multiplier is similar to which of these terms from linear programming? a. Shadow Price b. Allowable Increase c. Allowable Decrease d. Reduced Cost ANS: A

PTS: 1

25. Why does the GRG algorithm not provide allowable increase or allowable decrease information with the Reduced Gradient and Lagrange multiplier information? a. Because all constraints have slack. b. Because the constraints may not be linear. c. Because the solution is not necessarily a global optimum. d. Because the solutions are not necessarily corner points.

ANS: B

PTS: 1

26. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of product A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of product B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.

Product Labor (hr/unit) Material (ounces/unit)

A 3 1

B 2 2

Available resources 150 200

Manufacturing cost($/unit) Demand (units)

25 20 − 0.10*P1

18 30 − 0.07*P2

Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs. The objective function for this problem is? a. MAX 25 X1 + 18 X2 b. MAX (20 − 0.10*P1) X1 + (30 − 0.07*P2) X2 c. MAX 20 P1 − 0.1 P12 + 30 P2 − 0.07 P22 − 1040 d. MAX 22.5 P1 − 0.1 P12 + 31.26 P2 − 0.07 P22 − 1040 ANS: D

PTS: 1

Exhibit 8.1 The following questions pertain to the problem and spreadsheet below.

A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.

Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)

A 3 1 25 20 − .10*P1

B 2 2 18 30 − .07*P2

Available resources 150 200

Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs A 1 2 3 4 5 6 7 8 9 10 11 12 13

B

C

Price Marginal Cost Profit Margin

A 112.50 25.00 87.50

B 223.29 18.00 205.29

Demand

8.75

14.37

Total Profit

3715.58

Constraints: Labor Material

3.00 1.00

2.00 2.00

D

E

Used 54.99 37.49

Available 150.00 200.00

27. Refer to Exhibit 8.1. What formula is used in cell B7 of the spreadsheet for this problem? a. = 30 − .07 * C3 b. = 20 − .1 * B3 c. = B3 − B4 d. = B5 * B7 + C5 * C7

ANS: B

PTS: 1

28. Refer to Exhibit 8.1. What formula is used in cell B9 of the spreadsheet for this problem? a. =B3*B7+C3*C7 b. =B5*B7+C5*C7 c. =(B5-B4)*B7+(C5-C4)*B7 d. =B3*B7+C3*B7+B4*B7+C4*B7 ANS: B

PTS: 1

29. Refer to Exhibit 8.1. What formula is used in cell D12 of the spreadsheet for this problem? a. =SUMPRODUCT(B3:C3, B12:C12) b. =SUMPRODUCT(B4:C4, B12:C12) c. =SUMPRODUCT(B5:C5, B12:C12) d. =SUMPRODUCT(B7:C7, B12:C12) ANS: D

PTS: 1

30. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 5% return but with minimum risk.

Let:

Pi = proportion of total funds invested in stock i i2 = variance of stock i ij = covariance between stocks i and j.

Ri = average return on stock i

What is the objective function for the NLP formulation of this problem? a. b. c. d.

and

ANS: B

PTS: 1

31. An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell G4 of the Excel spreadsheet?

1 2 3 4 5 6 7 8 9 10 11

A

B

Year 1 2 3 4 5 6 7 8

A 3.98% −1.51% 5.36% 4.98% 3.12% 5.58% 3.49% −2.37%

C

D

Annual Return B C 7.38% 8.76% 10.45% 6.66% 5.15% 6.55% 5.51% −1.58% 6.93% 8.43% 9.07% −4.32% 9.78% 9.65% 5.62% 8.96%

E

> > > > > > > > > > > >

12 13 14 15

9 10

5.92% 5.69%

6.28% 5.75%

−7.10% 10.96%

Average

3.42%

5.85%

6.03%

< < < < < < < < < < < < < < < <

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

a. b. c. d.

=COVAR(B4:B13) =VARP(B4:B13) =COVAR(B4:D13,$D$4:$B$13) =COVAR(B4:B13,$B$4:$B$13)

> > > >

F

G

H

I

A B C

A 0.00080 −0.00044 −0.00045

Covariance Matrix B −0.00044 0.00144 −0.00012

C −0.00045 −0.00012 0.00300

Portfolio

A 36.8%

B 40.6%

C 22.6%

Expected Return Required Return

5.00% 5.00%

Portfolio Variance

ANS: D

J

Total 100.0%

0.000274

PTS: 1

32. An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell H11 of the Excel spreadsheet?

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

A

B

Year 1 2 3 4 5 6 7 8 9 10

A 3.98% −1.51% 5.36% 4.98% 3.12% 5.58% 3.49% −2.37% 5.92% 5.69%

Annual Return B C 7.38% 8.76% 10.45% 6.66% 5.15% 6.55% 5.51% −1.58% 6.93% 8.43% 9.07% −4.32% 9.78% 9.65% 5.62% 8.96% 6.28% −7.10% 5.75% 10.96%

Average

3.42%

5.85%

F 1 2 3

C

D

E

6.03%

G

H

I

A

Covariance Matrix B

C

> > > > > > > > > > > > > > > > J

A B C

0.00080 −0.00044 −0.00045

−0.00044 0.00144 −0.00012

−0.00045 −0.00012 0.00300

Portfolio

A 36.8%

B 40.6%

C 22.6%

Expected Return Required Return

5.00% 5.00%

Portfolio Variance

0.000274

< < < < < < < < < < < <

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

a. b. c. d.

=SUMPRODUCT(B15:D15,G9:I9) =SUMPRODUCT(B15:D15,B4:D13) =SUMPRODUCT(B15:D15,I9) =SUMPRODUCT(B15:D15,G4:I6)

ANS: A

Total 100.0%

PTS: 1

33. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks.

It has four stores and their coordinates are listed in the accompanying spreadsheet. Which cell(s) in the spreadsheet represent the decision variables in the problem?

A 1 2 3 4 5 6 7 8 9

Warehouse Store 1 Store 2 Store 3 Store 4

a. b. c. d.

B3 B3:C3 B4:C7 D9

ANS: B

B

C

D

X-Coordinate 163.800 100 90 210 180

Y-Coordinate 101.000 130 60 80 110

Distance: 70.082 84.424 50.749 18.532

Total Distance:

223.787

PTS: 1

34. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks.

It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D4 of the spreadsheet?

A 1 2 3 4 5 6 7 8

Warehouse Store 1 Store 2 Store 3 Store 4

B

C

D

X-Coordinate 163.800 100 90 210 180

Y-Coordinate 101.000 130 60 80 110

Distance: 70.082 84.424 50.749 18.532

9

a. b. c. d.

Total Distance:

223.787

=(B4-$B$3)^2+(C4-$C$3)^2 =SQRT((B4-$B$3)+(C4-$C$3)) =SQRT((B4-$B$3)^2+(C4-$C$3)^2) =(B4-$B$3)+(C4-$C$3)

ANS: C

PTS: 1

35. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks.

It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D9 of the spreadsheet?

A

B

C

D

X-Coordinate 163.800 100 90 210 180

Y-Coordinate 101.000 130 60 80 110

Distance: 70.082 84.424 50.749 18.532

Total Distance:

223.787

1 2 3 4 5 6 7 8 9

Warehouse Store 1 Store 2 Store 3 Store 4

a. b. c. d.

=MIN(D4:D7)^2 =SUM(D4:D7) =SQRT(D4^2+D5^2+D6^2+D7^2) =SQRT(MMULT(B3:C3,D4:D7))

ANS: B

PTS: 1

36. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the variable cell in this problem?

1 2 3 4 5 6 7 8 9 10 11

a. b. c. d.

A

B

Annual Demand:

20,000

Cost per Unit: Ordering Cost: Carrying Cost:

$25 $30 25%

Order Quantity:

483.73

Total Cost:

B3 B7 B9 B11

$502,738.61

ANS: C

PTS: 1

37. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the objective cell in this problem?

1 2 3 4 5 6 7 8 9 10 11

a. b. c. d.

A

B

Annual Demand:

20,000

Cost per Unit: Ordering Cost: Carrying Cost:

$25 $30 25%

Order Quantity:

483.73

Total Cost:

$502,738.61

B3 B7 B9 B11

ANS: D

PTS: 1

38. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What formula goes in cell B11 in this problem?

1 2 3 4 5 6 7 8 9 10 11

a. b. c. d.

A

B

Annual Demand:

20,000

Cost per Unit: Ordering Cost: Carrying Cost:

$25 $30 25%

Order Quantity:

483.73

Total Cost:

$502,738.61

=B3*B7+B9/B3*B6+B9*2+B5*B7 =B3*B5+B3/B9*B6+B9/2*B5*B7 =SQRT(B3*B5+B3/B9*B6+B9/2*B5*B7) =B3*B3+B3/B9+B6*B9/2*B5*B7

ANS: B

PTS: 1

39. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What constraint would you impose on this problem to ensure that at least one order is placed per year?

1 2 3 4 5 6 7 8 9 10 11

a. b. c. d.

A

B

Annual Demand:

20,000

Cost per Unit: Ordering Cost: Carrying Cost:

$25 $30 25%

Order Quantity:

483.73

Total Cost:

$502,738.61

MAX B9 B9 = 1 B9  1 MIN B11

ANS: C

PTS: 1

40. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What formula goes in cell C7 of the spreadsheet?

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

a. b. c. d.

B

Amount Invested: Annual Return:

Year 1 2 3 4 5 6 7 8 9 10

C

D

E

F

Earnings After Taxes $699 $715 $729 $741 $753 $764 $774 $782 $789 $793

Premium Due $500 $534 $566 $593 $607 $635 $672 $695 $737 $793

Ending Balance $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207 $10,207

$9,000 10% Beginning Balance $9,000 $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207

=B7*(1+$C$3/4)^4-B7 =B7*(1+$C$3)-B7 =B7*($C$3/4)^4-B7 =B7*(1+$C$3/4)^4

Investment Earnings $971 $993 $1,012 $1,030 $1,046 $1,061 $1,075 $1,086 $1,096 $1,101

ANS: A

PTS: 1

41. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What constraint must be entered in the Risk Solver Platform (RSP) task pane?

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

a. b. c. d.

B

Amount Invested: Annual Return:

Year 1 2 3 4 5 6 7 8 9 10

C

D

E

F

Earnings After Taxes $699 $715 $729 $741 $753 $764 $774 $782 $789 $793

Premium Due $500 $534 $566 $593 $607 $635 $672 $695 $737 $793

Ending Balance $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207 $10,207

$9,000 10% Beginning Balance $9,000 $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207

Investment Earnings $971 $993 $1,012 $1,030 $1,046 $1,061 $1,075 $1,086 $1,096 $1,101

$D$7:$D$16  $E$7:$E$16 $C$7:$D$16  $E$7:$E$16 $C$7:$C$16  $E$7:$E$16 $D$7:$D$16  $E$7:$E$16

ANS: D

PTS: 1

42. How much must the objective function coefficient of the variable Pumpkin change before any Pumpkins are produced based on the following sensitivity report?

Changing Cells Cell $B$4 $C$4 $D$4

Name Corn Pumpkin Beans

Final Value 9.52 0 10.79

Reduced Gradient 0 −499.99 0

Final Value 200000 99 37777.78 29.84 8000

Lagrange Multiplier 0.016 12 0 0 3.49

Constraints Cell $E$8 $E$9 $E$10 $E$11 $E$12

Name Corn Pumpkin Beans Water Fertilizer

a. increase by 12

b. decrease by 12 c. increase by 499.99 d. decrease by 499.99 ANS: C

PTS: 1

43. How much are additional units of Labor worth based on the following sensitivity report?

Changing Cells Cell $B$4 $C$4

Name Number to make: X1 Number to make: X2

Final Value 9.42 1.71

Reduced Gradient 0 0

Final Value 42 132 24

Lagrange Multiplier 0 1.21 2.57

Constraints Cell $D$8 $D$9 $D$10 a. b. c. d.

Name Wood Labor Plywood

0 1.21 2.57 9.42

ANS: B

PTS: 1

PROBLEM 44. How many local maximum solutions are there on this graph of a function? Mark their locations on the graph.

ANS: There two local maximums, one at X = 13 and a second at X = 25.

PTS: 1

45. How many local minimum solutions are there on this graph of a function Mark their locations on the graph.

ANS: There are three local minima, X = 0, X = 20, and X = 30.

PTS: 1 46. Find the maximum solution on this graph of a function starting from X = 12. Mark its location on the graph.

ANS: The maximum solution is at X = 10.

PTS: 1 47. The Sweet Water beverage company is designing a new soft drink can. The designers wish to minimize the manufacturing cost of the can, a cost that is directly related to the amount of aluminum used in the can. The can must hold at least 350 ml (or cm3) of beverage, have a diameter between 3 and 7 cm, and have a height between 7 and 19 cm.

Formulate the NLP for Sweet Water.

ANS: Let

MIN Subject to:

D be the diameter of the can, H be the height of the can. DH + (/2)D2 (/4)D2H  350 3D7 7  H  19

Note: DH is the cylindrical surface area while each of the two ends has area (/4)D2 PTS: 1 48. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − .09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − .14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.

Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)

A 2 2 30 50 − 0.09*P1

B 4 8 20 30 − 0.14*P2

Available resources 150 220

Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs. Formulate the NLP for this company ANS: MAX: Subject to:

52.70 P1 − 0.09 P12 + 32.80 P2 − 0.14 P22 − 2100 X1 − 50 + 0.09P1 = 0 X2 − 30 + 0.14P2 = 0 2 X1 + 4 X2  150 2 X1 + 8 X2  220 90  P1, P2  140 X1, X2  0

PTS: 1 49. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − 0.09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − 0.14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.

Product Labor (hr/unit)

A 2

B 4

Available resources 150

Material (ounces/unit) Manufacturing cost($/unit) Demand (units)

2 30 50 − 0.09*P1

8 20 30 −14*P2

220

Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs. The NLP for the problem is: 52.70 P1 − 0.09 P12 + 32.80 P2 − 0.14 P22 − 2100 X1 − 50 + 0.09P 1 = 0 X2 − 30 + 0.14P2 = 0 2 X1 + 4 X2  150 2 X1 + 8 X2  220 90  P1, P2  140 X1, X2  0 and the solution (P1, P2) = (140.0, 117.14)

MAX: Subject to:

What values should go in cells B3:E18 of the spreadsheet for this problem?

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

B

C

A

B

D

E

Used

Available

D

E

Used 129.20

Available 150.00

Price Min Price Max Price Marginal Cost Profit Margin Demand Total Profit Constraints: Labor Material

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

B

C

Price Min Price Max Price Marginal Cost Profit Margin

A 140.00 90 140 30.00 110.00

B 117.14 90 140 20.00 97.14

Demand

37.40

13.60

Total Profit

5435.14

Constraints: Labor

2.00

4.00

15

Material

2.00

8.00

183.60

220.00

PTS: 1 50. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk.

The average return for the stocks is: Annual Return B 10.68%

A 10.72%

Average

C 11.87%

The covariance matrix for the stocks is: A 0.00009 −0.00009 −0.00011

A B C

B −0.00009 0.00032 −0.00007

C −0.00011 −0.00007 0.00122

Let: Pi = proportion of total funds invested in i, i = A, B, C Formulate the NLP for this problem.

ANS: MIN:

0.00009 P12 + 0.00032 P22 + 0.00122 P32 + 2 (−0.00009 P1P2 − 0.00011 P1P3 − 0.00007 P2 P3) P1 + P2 + P3 = 1 0.1072 P1 + 0.1068 P2 + 0.1187 P3  0.11 P1, P2, P3  0 P1, P2, P3  1

Subject to:

PTS: 1 51. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk.

Let: Pi = proportion of total funds invested in i, i = A, B, C The NLP for this problem is: 0.00009 P12 + 0.00032 P22 + 0.00122 P32 + 2 (−0.00009 P1P2 − 0.00011 P1P3 − 0.00007 P2P3) P1 + P2 + P3 = 1 0.1072 P1 + 0.1068 P2 + 0.1187 P3  0.11 P1, P2, P3  0 P1, P2, P3  1

MIN: Subject to:

What formulas should go in cells G4:J14 of the spreadsheet for this problem? NOTE: Formulas are not required in all of these cells.

A 1

B

C

D

E

> >

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

Year 1 2 3 4 5 6 7 8 9 10

A 9.73% 10.83% 12.14% 9.14% 11.16% 11.60% 11.06% 11.22% 9.25% 11.11%

Annual Return B 12.54% 9.52% 11.47% 13.72% 8.89% 10.72% 12.21% 8.56% 11.09% 8.10%

C 10.23% 16.10% 4.07% 12.93% 11.97% 12.03% 14.00% 16.28% 12.99% 8.06%

Average

10.72%

10.68%

11.87%

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

ANS: Cell B15 G4 G5 G6 H11 H14 J9

F

G

H

A B C

A 0.00009 −0.00009 −0.00011

Portfolio

A 63.7%

B 27.2%

Expected Return Required Return

11.00% 11.00%

Portfolio Variance

2.5312E-05

> > > > > > > > > > > > > > I

J

Covariance Matrix B C −0.00009 −0.00011 0.00032 −0.00007 0.00122 −0.00007 C 9.1%

Formula =AVERAGE(B4:B13) =COVAR(B4:B13,$B$4:$B$13) =COVAR(B4:B13,$C$4:$C$13) =COVAR(B4:B13,$D$4:$D$13) =SUMPRODUCT(B15:D15,G9:I9) =SUMPRODUCT(MMULT(G9:I9,G4:I6),G9:I9) =SUM(G9:I9)

Total 100.0%

Copied to C15:D15 H4:I4 H5:I5 H6:I6

PTS: 1 52. A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks.

It has four stores and their coordinates are listed in the below.

Store 1 Store 2 Store 3 Store 4

X-Coordinate 70 60 180 150

Y-Coordinate 160 90 90 120

Formulate the objective function for this problem. Let X and Y represent the X, Y coordinates of the new warehouse.

ANS: MIN:

PTS: 1 53. A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks.

It has four stores and their coordinates are listed in the accompanying spreadsheet. What formulas should go in cells D4:D9 of the spreadsheet for this problem?

1 2 3 4 5 6 7 8 9

A

B

C

D

Warehouse Store 1 Store 2 Store 3 Store 4

X-Coordinate 138.526 70 60 180 150

Y-Coordinate 116.227 160 90 90 120

Distance: 81.314 82.790 49.071 12.079

Total Distance:

225.253

ANS: Cell D4 D9

Formula =SQRT((B4-$B$3)^2+(C4-$C$3)^2) =SUM(D4:D7)

Copied to D5:D7

PTS: 1 54. A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It has four stores and their coordinates are listed in the below.

Store 1 Store 2 Store 3 Store 4

X-Coordinate 70 60 180 150

Y-Coordinate 160 90 90 120

Formulate the NLP for this problem. Let X and Y represent the X, Y coordinates of the new warehouse.

ANS: MIN: Subject to:

Q

PTS: 1 55. A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It has four stores and their coordinates are listed in the below.

X-Coordinate 70 60 180 150

Store 1 Store 2 Store 3 Store 4

Y-Coordinate 160 90 90 120

Let X and Y represent the X, Y coordinates of the new warehouse. The NLP for this problem and solution is the following.

MIN: Subject to:

Q

Solution is (X, Y) = (120.0, 117.4).

What values should go in cells B2:D9 of the spreadsheet for this problem?

A 1 2 3 4 5 6 7 8 9 10

B

C

D

X

Y

Distance

Warehouse Store 1 Store 2 Store 3 Store 4 Total Max

ANS: A 1 2 3 4 5 6 7 8 9 10

Warehouse Store 1 Store 2 Store 3 Store 4

B

C

D

120.0 X 70 60 180 150

117.4 Y 160 90 90 120 Total Max

Distance 65.85 65.85 65.85 30.14 227.698 65.854

PTS: 1 56. An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item.

Formulate the objective function for this problem. Let Q indicate the order quantity.

ANS: MIN: 5000 (50) + 5000/Q (75) + Q/2 (50)(.30) PTS: 1 57. An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item.

What values should go in cells B3:B11 of the spreadsheet for this problem if Q = 223.61?

A 1 2 3 4 5 6 7 8 9 10 11

B

Annual Demand: Cost per Unit: Ordering Cost: Carrying Cost: Order Quantity: Total Cost:

ANS:

1 2 3 4 5 6 7 8 9 10 11

A

B

Annual Demand:

5,000

Cost per Unit: Ordering Cost: Carrying Cost:

$50 $75 30%

Order Quantity:

223.61

Total Cost:

$253,354

PTS: 1 58. Calculate the annual inventory costs for the following data.

Order quantity Annual demand Unit purchase cost

= 400 units = 12,500 units = 50

Fixed cost of placing an order Percentage cost of holding one unit in inventory for a year

= 75 = 20%

ANS: $629,343.75 PTS: 1 59. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income.

What formulas should go in cells B7:F7 of the spreadsheet for this problem?

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

B

Amount Invested: Annual Return:

Year 1 2 3 4 5 6 7 8 9 10

ANS: Cell B7 B8 C7 D7 F7

C

D

E

F

Earnings After Taxes $1,689 $1,711 $1,730 $1,748 $1,765 $1,782 $1,798 $1,811 $1,822 $1,829

Premium Due $1,536 $1,570 $1,602 $1,629 $1,643 $1,671 $1,708 $1,731 $1,773 $1,829

Ending Balance $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994 $12,994

$12,000 18% Beginning Balance $12,000 $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994

Investment Earnings $2,346 $2,376 $2,403 $2,428 $2,452 $2,476 $2,497 $2,515 $2,531 $2,540

Formula =C2 =F7 =B7*(1+$C$3/4)^4-B7 =(1-.28)*C7 =B7+D7-E7

Copied to B9:B16 C8:C16 D8:D16 F8:F16

PTS: 1 60. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income.

If the Risk Solver Platform (RSP) is used, which are the Objective, Variables and Constraint cells in the spreadsheet for this problem?

A

B

C

D

E

F

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

Amount Invested: Annual Return: Beginning Balance $12,000 $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994

Year 1 2 3 4 5 6 7 8 9 10

ANS: Objective: Variables: Constraint:

$12,000 18% Investment Earnings $2,346 $2,376 $2,403 $2,428 $2,452 $2,476 $2,497 $2,515 $2,531 $2,540

Earnings After Taxes $1,689 $1,711 $1,730 $1,748 $1,765 $1,782 $1,798 $1,811 $1,822 $1,829

Premium Due $1,536 $1,570 $1,602 $1,629 $1,643 $1,671 $1,708 $1,731 $1,773 $1,829

Ending Balance $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994 $12,994

C3 C3 D7:D16E7:E16

PTS: 1 61. How much must the objective function coefficient of the variable X2 increase before any X2s are produced based on the following sensitivity report?

Changing Cells Cell $B$4 $C$4

Name Number to make: X1 Number to make: X2

Final Value 9.428 0

Reduced Gradient 0 −1.96

Final Value 42 132 24

Lagrange Multiplier 0 0.214 1.214

Constraints Cell $D$8 $D$9 $D$10

Name Used Used Used

ANS: 1.96 PTS: 1 62. How much are additional units of labor worth based on the following sensitivity report?

Changing Cells Cell $B$4 $C$4

Name Number to make: X1 Number to make: X2

Final Value 4 8.571

Reduced Gradient 0 0

Constraints Cell $D$8 $D$9 $D$10

Name Wood Labor Hinges

Final Value 84 4 8.571

Lagrange Multiplier 0.714 3.714 0

ANS: 3.714 PTS: 1 63. A construction company just purchased a 300  300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).

Formulate the NLP for the problem.

ANS: Let

MIN Subject to:

L = building length W = building width S = number of stories P = total parking area 0.60*10*X + 0.01*L*W S * L * W  60,000 L * W + P = 90,000 P  0.30 * S * L * W S * 10  65 0  L, W  300 0S6 P0

{total floor area} {total lot area} {min. parking area} {max. building height}

PTS: 1 Exhibit 8.2 The following questions pertain to the problem and spreadsheet below.

A construction company just purchased a 300  300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars).

The following is the NLP formulation for the problem.

Let

L = building length W = building width

S = number of stories P = total parking area MIN Subject to:

0.60*10*X + 0.01*L*W S * L * W  60,000 L * W + P = 90,000 P  0.30 * S * L * W S * 10  65 0  L, W  300 0S6 P0

{total floor area} {total lot area} {min. parking area} {max. building height}

The spreadsheet implementation of this formulation applies to the following questions.

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

B Length 108.39492 300

C Width 92.255243 300

D Stories 6 6

Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height

60,000 90,000 65,000 60

 =  

60,000 90,000 0 65

Cost Height Factor Area Factor

0.6 0.01

Total Height Floor Area Cost

60 10,000 $136.0

Max Values

E Parking 80,000

millions $

64. Refer to Exhibit 8.2. What values would you enter in the Risk Solver Platform (RSP) task pane for the above Excel spreadsheet?

Objective Cell: Variables Cells: Constraints Cells: ANS: Objective Cell: D13 Variables Cells: B2:E2 Constraints Cells: B5  D5 B6 = D6 B7  D7 B8  D8 PTS: 1 65. Refer to Exhibit 8.2. What formula would you place into cell B5 to calculate Total Floor Area?

ANS: =B2*C2*D2 PTS: 1 66. Refer to Exhibit 8.2. What formula would you place in cell B6 to calculate Total Lot Area?

ANS: =B2*C2+E2 PTS: 1 67. Refer to Exhibit 8.2. What formula would you place in cell D13 to calculate total cost?

ANS: =SUMPRODUCT(B11:B12,D11:D12) PTS: 1 68. Refer to Exhibit 8.2. The company wishes to have a relatively square building. Thus, they wish neither the building length nor the building width exceed the other by more than 25%. Add constraint(s) to enforce this design constraint.

ANS: −L + 0.75W  0 L − 1.25W  0 PTS: 1 PROJECT 69. Project 8.1 − Truck Company Expansion Kornfield Trucking handles private and commercial moves. They currently own 500 moving vans and employ 2000 full-time workers. Their trucks are used to pick-up and deliver office and household goods throughout the Eastern and Southeastern states. Kornfield mans each truck with three workers.

This allows driver swaps providing increased miles-covered-per-hour ratio while staying within safety requirements for individual driving time. A 3-person crew also reduces company reliance on local help. Local distributors and warehouses provide a pool of laborers for loading and unloading the moving trucks. While in the past this arrangement has worked well, the arrangement has soured recently as the temporary workers have demanded higher wages while produced less work. Despite their pay and benefits package, Kornfield still finds that the nature of the work (lifting and time on the road) makes for a high rate of turnover. Thus, Kornfield maintains an excess of workers/drivers, but no more than 3.75 workers per truck at any time.

The following information is available on the trucks in the Kornfield inventory and their options for new purchases.

New Truck Old Truck

Purchase Price $55,000 ---

Salvage Value --$25,000

Yearly Maintenance $ 750 $1500

Miles Per Year 100,000 100,000

Miles per Gallon* 12.5 mpg 10 mpg *assume $1.50/gal

The following information applies to Kornfield personnel actions.

New Worker Old Worker

Hiring Cost $1000 ---

Firing Cost --$1500

Salary $28,000 $35,000

Kornfield has an operating budget of $ 75M next year and wants to expand their operations. As a part of this expansion, they are considering options for their truck fleet and may purchase new trucks, sell old trucks to salvage or some combination of the two. Any salvage money received is rolled into the operating budget. Kornfield is also considering possible changes to their work force. Their current force has a fairly high average salary and their operating budget is not greatly affected by releasing current employees. On the other hand, newer employees carry a much lower average salary and do not tax the operating budget heavily in hiring and training costs.

The Cobb-Douglas production function is used to model the number of vehicle miles driven per year.

This function represents the quantity Kornfield management would like to maximize as they expand their operations. The general form of the Cobb-Douglas production function is the following:

where y is the output, each Xi represents an input and the letters represent constants. This function generalizes to fewer or a greater number of parameters than the three depicted above. The constants for the Kornfield Trucking production function are (a, b, c, d) = (9.1, 0.05, 0.40, 0.50).

Formulate the Kornfield Trucking problem as a non-linear programming problem. Implement the problem in Excel and use Risk Solver Platform (RSP) generalized reduced gradient (GRG) routing to obtain a solution to the problem. What is a recommended solution for Kornfield Trucking?

ANS: Define the following decision variables: X1 be the number of new trucks purchased; X2 be the number of old trucks salvaged; X3 be the number of new workers hired; X4 be the number of old workers fired; Y1 be the total trucks in expanded inventory; Y2 be the total workers in company; and Y3 be the total fuel consumed during the year (thousands of gallons).

Maximize: Subject to: Y1 − X1 + X2 = 500 Y2 − X3 + X4 = 2000 Y3 − 8X1 + 10X2 = 5000

Trucks Workers Fuel consumed (1000s of gals)

WC − $35000*(2000 − X4) − $1500X4 − $29000X3 = 0 MC − $750X1 − $1500(500 − X2) = 0 TC − $55000X1 + $25000X2 = 0 FC − $1500(500 − X2) − $1200X1 = 0

Worker cost Maintenance cost Truck costs Fuel costs per 10K miles

WC + MC + TC + 10*FC  $80,000,000

Budget constraint

Y2 − 3.75Y3  0

Worker/Truck ratio

Xi, Yj  0 The suggested solution is to purchase 647 new trucks while retaining all the old trucks. This requires 2299 new workers along with the current work force of 2000. This yields a total vehicle miles value of 37,081.31 vehicle miles with costs of $20,585,000 in truck costs, $37,915,000 in worker costs, $1,235,520 in maintenance costs, and $15,264,000 in fuel costs.

PTS: 1