SMDA6e Chapter 03

Chapter 3—Modeling and Solving LP Problems in a Spreadsheet

MULTIPLE CHOICE

1. An LP problem with a feasible region will have

a. an optimal solution at some interior point. b. an optimal solution at some extreme point. c. an optimal solution only at the origin. d. an optimal solution at two interior points.

ANS: B PTS: 1

2. Microsoft Excel, Quattro Pro and Lotus 1-2-3 contain built-in optimizers called

a. what-if engines. b. calculators. c. solvers. d. risk analyzers.

ANS: C PTS: 1

3. Which type of spreadsheet cell represents the objective function in an LP model?

a. Objective cell b. Changing variable cell c. Constraint cell d. Constant cell

ANS: A PTS: 1

4. Which type of spreadsheet cell represents the decision variables in an LP model?

a. Target or set cell b. Variable cell c. Constraint cell d. Constant cell

ANS: B PTS: 1

5. Which type of spreadsheet cell represents the left hand sides (LHS) formulas in an LP model?

a. Target or set cell b. Changing variable cell c. Constraint cell d. Constant cell

ANS: C PTS: 1

6. The constraints X1 ≥ 0 and X2 ≥ 0 are referred to as

a. positivity constraints. b. optimality conditions. c. left hand sides. d. nonnegativity conditions.

ANS: D PTS: 1

7. In the Risk Solver Platform (RSP) dialog box simple upper and lower bounds for decision variables

are specified by

a. referring directly to the decision variable cells in the Constraints-Bound area. b. requiring the addition of the bounds above and below the variable cells. c. resolving the problem with the bounds added. d. incorporating the bounds in the objective function.

ANS: A PTS: 1

8. The built-in Solver in Excel is found under which tab on the ribbon?

a. Tools b. Insert c. Data d. Window

ANS: C PTS: 1

9. Which tab in the Risk Solver Platform (RSP) task pane is used to define an optimization problem?

a. Guess b. Model c. Change d. Delete

ANS: B PTS: 1

10. Spreadsheet modeling is an acquired skill because

a. there is generally only one correct way to build a model. b. the spreadsheet is free-form providing many modeling options. c. using Risk Solver Platform (RSP) requires lots of experience. d. spreadsheets are not very easy to use.

ANS: B PTS: 1

11. Models which are setup in an intuitively appealing, logical layout tend to be the most

a. Reliable b. Modifiable c. Auditable d. Organized

ANS: C PTS: 1

12. The English-reading eye scans

a. Right to left b. Bottom to top c. Left to Bottom d. Left to right

ANS: D PTS: 1

13. Numeric constants should be

a. embedded in formulas. b. placed in individual cells c. placed in separate workbooks. d. entered manually every time a model is solved.

ANS: B PTS: 1

14. The "Analyze Without Solving" tool in Risk Solver Platform (RSP) is useful for

a. verifying the equations in a spreadsheet model. b. toggling between absolute and relative cell referencing. c. executing the Excel spreadsheet layout Wizard. d. naming cells and cell ranges for easier modifiability.

ANS: A PTS: 1

15. The "Objective Value of" option in the Risk Solver Platform (RSP) task pane may be used to

a. find a solution at a maximum value. b. find a solution at a minimum value. c. find a solution for a specific objective function value. d. returns the best feasible solution.

ANS: C PTS: 1

16. The "Objective Sense" option in the Risk Solver Platform (RSP) task pane may be used to

a. return a heuristic solution to the problem. b. tell the Solver what value it should seek for your optimization objective. c. determine the value of the objective based on specified decision variable cells. d. always works correctly.

ANS: B PTS: 1

17. What action is required to make Risk Solver Platform (RSP) solve a specified problem?

a. Type go in cell A1. b. Click the "Optimize" button on the RSP Ribbon, or the green arrow "Solve" in the Task

Pane. c. Click the Close button in the RSP Parameters dialog box. d. Click the Guess button in the RSP Parameters dialog box.

ANS: B PTS: 1

18. What does the Excel =SUMPRODUCT(A1:A5,C6;C10) command do?

a. Sums each range and multiplies the sums. b. Sum each pair of cells and multiples each sum. c. Multiplies the contents of cells containing the =SUM() command. d. Multiplies each pair of cells and sums each product.

ANS: D PTS: 1

19. What command is used to add the contents of cells A1, A2 and A3?

a. =A1+A2+A3 b. =ADD(A1:A3) c. =TOTAL(A1:A3) d. =PRODUCT(A1:A3)

ANS: A PTS: 1

20. Which command is equivalent to =SUMPRODUCT(A1:A3,B1:B3)?

a. =SUM(PRODUCT((A1:A3,B1:B3)) b. =PRODUCT(SUM((A1:A3,B1:B3)) c. =PRODUCT(A1+B1,A2+B2,A3+B3)) d. =A1*B1+A2*B2+A3*B3

ANS: D PTS: 1

21. Problems which have only integer solutions are called a. discrete programming problems b. integer programming problems c. discrete programming problems d. infeasible programming problems

ANS: B PTS: 1

22. What is the significance of an absolute cell reference in Excel?

a. The cell reference will not change if the formula containing the reference is copied to another location

b. The cell will always contain the absolute value of any number entered into it c. The cell reference changes if the formula containing the reference is copied to another

location d. It is the only formula used to refer to a cell on another spreadsheet

ANS: A PTS: 1

23. How many decision variables are there in a transportation problem which has 5 supply points and 4

demand points? a. 4 b. 5 c. 9 d. 20

ANS: D PTS: 1

24. How many constraints are there in a transportation problem which has 5 supply points and 4 demand

points? (ignore the non-negativity constraints) a. 4 b. 5 c. 9 d. 20

ANS: C PTS: 1

25. A heuristic solution is

a. used by Risk Solver Platform (RSP) when the Guess button is used. b. guaranteed to produce an optimal solution. c. used by Risk Solver Platform (RSP) if Standard GRG Nonlinear method is selected. d. a rule-of-thumb for making decisions.

ANS: D PTS: 1

26. Scaling problems

a. can cause Risk Solver Platform (RSP) to consider a linear problem as nonlinear. b. can cause problems in accuracy of solutions returned. c. are caused by small numbers and large numbers used in the same problem. d. all of these.

ANS: D PTS: 1

27. Which of the following describes Data Envelopment Analysis (DEA).

a. DEA finds the most effective company among some set of companies. b. DEA determines if a company is converting inputs to outputs as effectively as possible. c. DEA determines how effective a company converts inputs to outputs compared to other

companies. d. DEA compares how effective a company converts inputs to outputs compared to a

benchmark composite of all companies.

ANS: C PTS: 1

28. Data Envelopment Analysis (DEA) is an LP-based methodology in which weighted sums of inputs and

outputs are calculated and a. the constraints capture the maximum effectiveness of each unit. b. the objective is to maximize every units output. c. the constraints ensure the sum of the weighted outputs is one. d. the objective for each unit is to maximize the weighted sum of its outputs.

ANS: D PTS: 1

29. Using Data Envelopment Analysis (DEA) for an inefficient unit, a more efficient composite unit can

be found by a. Solving its DEA problem and retrieving the weights from the answer report. b. Solving its DEA problem and examining those units whose final value is non-zero. c. Solving its DEA problem and using the resulting shadow prices as composite weights. d. Solving its DEA problem and using the positive resulting shadow prices as composite

weights.

ANS: C PTS: 1

Exhibit 3.1 The following questions are based on this problem and accompanying Excel windows. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced. Let X1 = Number of Beds to produce X2 = Number of Desks to produce The LP model for the problem is MAX: 30 X1 + 40 X2 Subject to: 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for desks) X1, X2 ≥ 0

A B C D E 1 Jones Furniture 2 3 Beds Desks 4 Number to make: Total Profit: 5 Unit profit: 30 40 6 7 Constraints: Used Available 8 Carpentry 6 4 36

9 Varnishing 4 8 40 10 Desk demand 1 8

30. Refer to Exhibit 3.1. What formula should be entered in cell E5 in the accompanying Excel

spreadsheet to compute total profit? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B8:C8,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10)

ANS: A PTS: 1

31. Refer to Exhibit 3.1. What formula should be entered in cell D8 in the accompanying Excel

spreadsheet to compute the amount of carpentry used? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B8:C8,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10)

ANS: B PTS: 1

32. Refer to Exhibit 3.1. Which cells should be changing cells in this problem?

a. B4:C4 b. E5 c. D8:D10 d. E8:E10

ANS: A PTS: 1

33. Refer to Exhibit 3.1. Which cells should be the constraint cells in this problem?

a. B4:C4 b. E5 c. D8:D10 d. E8:E10

ANS: C PTS: 1

34. Refer to Exhibit 3.1. Which of the following statements represent the carpentry, varnishing and limited

demand for desks constraints? a. B4:C4 ≤ B5:C5 b. E5 ≤ 0 c. D8:D10 ≤ E8:E10 d. E8:E10 ≤ D8:D10

ANS: C PTS: 1

Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table.

Computer Model

Profit per Model ($)

Maximum demand for

product

Wiring Hours

Required

Assembly Hours

Required

Inspection Hours

Required Plain 30 80 .4 .5 .2 Fancy 40 90 .5 .4 .3

Hours Available 50 50 22 Let X1 = Number of Plain computers to produce X2 = Number of Fancy computers to produce MAX: 30 X1 + 40 X2 Subject to: .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0

A B C D E 1 Byte Computer Company 2 3 Plain Fancy 4 Number to make: Total Profit: 5 Unit profit: 30 40 6 7 Constraints: Used Available 8 Wiring 0.4 0.5 50 9 Assembly 0.5 0.4 50

10 Inspection 0.2 0.3 22 11 Plain Demand 1 80 12 Fancy Demand 1 90


spreadsheet to compute total profit? a. =B4*C4+B5*C5 b. =SUMPRODUCT(B4:C4,B5:C5) c. =SUM(B5:C5) d. =SUM(E8:E10)

ANS: B PTS: 1


spreadsheet to compute the amount of wiring used? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B8:C8,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10)

ANS: B PTS: 1


a. B4:C4 b. E5

c. D8:D10 d. E8:E10

ANS: A PTS: 1


a. B4:C4 b. E5 c. D8:D12 d. E8:E12

ANS: C PTS: 1

39. Refer to Exhibit 3.2. Which of the following statements will represent the constraint for just assembly

hours? a. B4:C4 ≤ B5:C5 b. D9 ≤ E9 c. D8:D10 ≤ E8:E10 d. E8:E10 ≤ D8:D10

ANS: B PTS: 1

Exhibit 3.3 The following questions are based on this problem and accompanying Excel windows. Jack's distillery blends scotches for local bars and saloons. One of his customers has requested a special blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products, call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50 1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B. The customer also specified that the blend have an alcohol content of at least 85%. Product A contains 95% alcohol while product B contains 78%. The blend is sold for $12.50 per bottle. Product A costs $7 per liter and product B costs $3 per liter. The company wants to determine the blend that will meet the customer's requirements and maximize profit. Let X1 = Number of liters of product A in total blend delivered X2 = Number of liters of product B in total blend delivered MIN: 7 X1 + 3 X2 Subject to: X1 + X2 = 1.5 * 50 (Total liters of mix) X1 ≥ 0.48 * 1.5 * 50 (X1 minimum) X2 ≥ 0.5 * 50 (X2 minimum) .0.95 X1 + 0.78 X2 ≥ 0.85 * 1.5 * 50 (85% alcohol minimum) X1, X2 ≥ 0

A B C D E 1 Jacks' Distillery 2 3 A B 4 Liters to use Total Cost: 5 Unit cost: 10.5 4.5 6 7 Constraints: Supplied Requirement

8 Total Liters 1 1 75 9 A required 1 36

10 B required 1 25 11 85% alcohol 0.95 0.78 63.75


spreadsheet to compute total cost? a. =B4*C4+B5*C5 b. =SUMPRODUCT(B4:C4,B5:C5) c. =SUM(B5:C5) d. =SUM(E8:E10)

ANS: B PTS: 1


spreadsheet to compute the total liters of alcohol supplied? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B11:C11,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10)

ANS: B PTS: 1


a. B4:C4 b. E5 c. D8:D10 d. E8:E10

ANS: A PTS: 1


a. B4:C4 b. E5 c. D8:D11 d. E8:E10

ANS: C PTS: 1

44. Refer to Exhibit 3.3. Which of the following statements could represent a constraint in this problem?

a. B4:C4 ≤ B5:C5 b. E5 ≤ 0 c. D8 = E8 d. E8:E11 ≤ D8:D11

ANS: C PTS: 1

Exhibit 3.4 The following questions are based on this problem and accompanying Excel windows.

A financial planner wants to design a portfolio of investments for a client. The client has $300,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 25% of the money in any one investment, at least one third should be invested in long-term bonds which mature in seven or more years, and no more than 25% of the total money should be invested in C or D since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.

Investment Return Years to Maturity Rating A 6.45% 9 1-Excellent B 7.10% 8 2-Very Good C 8.20% 5 4-Fair D 9.00% 8 3-Good

Let X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D MAX: .0645 X1 + .071 X2 + .082 X3 + .09 X4 Subject to: X1 + X2 + X3 + X4 ≤ 300000 X1 ≤ 75000 X2 ≤ 75000 X3 ≤ 75000 X4 ≤ 75000 X1 + X2 + X4 ≥ 100000 X3 + X4 ≤ 75000 X1, X2, X3, X4 ≥ 0

A B C D > 1 Amount Maximum > 2 Bond Invested 25.0% Return > 3 A $0 $75,000 6.45% > 4 B $0 $75,000 7.10% > 5 C $0 $75,000 8.20% > 6 D $0 $75,000 9.00% > 7 Total Invested: $0 Total: $0 > 8 Total Available: $300,000 >

< E F G H < 1 Years to 7+ years? Good or worse? < 2 Maturity (1-yes, 0-no) Rating (1-yes, 0-no) < 3 9 1 1-Excellent 0 < 4 8 1 2-Very Good 0 < 5 5 0 4-Fair 1 < 6 8 1 3-Good 1 < 7 Total: $0 Total: $0 < 8 Required: $100,000 Allowed: $75,000

45. Refer to Exhibit 3.4. What formula should be entered in cell B7 in the accompanying Excel

spreadsheet to compute total dollars invested? a. =ADD(B3:B6) b. =SUM(B3:B6)

c. =TOTAL(B3:B6) d. =TALLY(B3:B6)

ANS: B PTS: 1


spreadsheet to compute the total return? a. =B7*SUM(D3:D6) b. =SUMPRODUCT(B3:B6,D3:D6) c. =SUM(B3:B6) d. =SUMPRODUCT(B3:E3,B6:E6)

ANS: B PTS: 1

47. Refer to Exhibit 3.4. Which cells are changing cells in the accompanying Excel spreadsheet?

a. B3:B6 b. B7:I7 c. C7 d. E7

ANS: A PTS: 1

Exhibit 3.5 The following questions are based on this problem and accompanying Excel windows. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost is stable but demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand. Quarterly inventory holding cost is 3% of the cost of production. The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month. There are currently 5,000 units in inventory. The company wants to produce at no less than one half of its maximum capacity in any quarter. Quarter 1 2 3 4 Unit Production Cost $ 300 $ 300 $ 300 $ 300 Units Demanded 2,000 9,000 12,000 11,000 Maximum Production 8,000 7,000 8,000 9,000 Let Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i MIN: 300 P1 + 300 P2 + 300 P3 + 300 P4 + 9(B1 + B2)/2 + 9(B2 + B3)/2 + 9(B3 + B4)/2 + 9(B4 + B5)/2 Subject to: 4000 ≤ P1 ≤ 8000 3500 ≤ P2 ≤ 7000 4000 ≤ P3 ≤ 8000 4500 ≤ P4 ≤ 9000 3000 ≤ B1 + P1 − 2000 ≤ 12000 3000 ≤ B2 + P2 − 9000 ≤ 12000 3000 ≤ B3 + P3 − 12000 ≤ 12000 3000 ≤ B4 + P4 − 11000 ≤ 12000 B2 = B1 + P1 − 2000

B3 = B2 + P2 − 9000 B4 = B3 + P3 − 12000 B5 = B4 + P4 − 11000 Pi, Bi ≥ 0

A B C D E F 1 Quarter 2 1 2 3 4 3 Beginning Inventory 5,000 11,000 9,000 5,000 4 Units Produced 8,000 7,000 8,000 9,000 5 Units Demanded 2,000 9,000 12,000 11,000 6 Ending Inventory 11,000 9,000 5,000 3,000 7 8 Minimum Production 4,000 3,500 4,000 4,500 9 Maximum Production 8,000 7,000 8,000 9,000

10 11 Minimum Inventory 3,000 3,000 3,000 3,000 12 Maximum Inventory 12,000 12,000 12,000 12,000 13 14 Unit Production Cost $300 $300 $300 $300 15 Unit Carrying Cost 3.0% $9.00 $9.00 $9.00 $9.00 16 17 Quarterly Production Cost $2,400,000 $2,100,000 $2,400,000 $2,700,000 18 Quarterly Carrying Cost $72,000 $90,000 $63,000 $36,000 19 20 Total Cost $9,861,000

48. Refer to Exhibit 3.5. What formula should be entered in cell C6 in the accompanying Excel

spreadsheet to compute ending inventory? a. =C3-C4+C5 b. =C3+C4-C5 c. =C3-(C4-C5) d. =C5-C4-C3

ANS: B PTS: 1

49. Refer to Exhibit 3.5. What formula should be entered in cell C18 in the accompanying Excel

spreadsheet to compute the quarterly carrying costs? a. =C15*C3+C6 b. =C15*(C3+C6) c. =C15*C3/2 d. =C15*(C3+C6)/2

ANS: D PTS: 1

50. Refer to Exhibit 3.5. Which cells are changing cells in the accompanying Excel spreadsheet?

a. C4:F4 b. C9:F9 c. F20 d. C12:F12

ANS: A PTS: 1

51. Refer to Exhibit 3.5. What formula could be entered in cell F20 in the accompanying Excel

spreadsheet to compute the Total Cost for all four quarters?

a. SUMPRODUCT($C$4:$F$4,C17:F17) b. SUM(C17:F17) + SUM(C18:F18) c. PRODUCT(SUM(C14:F15,C17:F18) d. SUMPRODUCT(C4:F4,C14:F14) + SUMPRODUCT(C6:F6,C15:F15)

ANS: B PTS: 1

PROBLEM

52. You have been given the following linear programming model and Excel spreadsheet to solve this

problem. What numbers should be entered into cells B5:C5 and B8:C10 to implement this model? MAX: 2 X1 + 7 X2 Subject to: 5 X1 + 9 X2 ≤ 90 9 X1 + 8 X2 ≤ 144 X2 ≤ 8 X1, X2 ≥ 0

A B C D E 1 2 3 X1 X2 4 Number to make: OBJ. FN. VALUE 5 Unit profit: 6 7 Constraints: Used Available 8 1 90 9 2 144

10 3 8

ANS:

A B C D E 1 2 3 X1 X2 4 Number to make: OBJ. FN. VALUE 5 Unit profit: 2 7 6 7 Constraints: Used Available 8 1 5 9 90 9 2 9 8 144

10 3 0 1 8

PTS: 1


problem. What formulas should be entered into cells E5 and D8:D10 to implement this model? MAX: 2 X1 + 7 X2 Subject to: 5 X1 + 9 X2 ≤ 90 9 X1 + 8 X2 ≤ 144 X2 ≤ 8

X1, X2 ≥ 0


10 3 0 1 8

ANS: Cell Formula Copied to E5 =SUMPRODUCT(B4:C4,B5:C5) D8 =SUMPRODUCT($B$4:$C$4,B8:C8) D9:D10

PTS: 1


problem. What cell references would you enter in the Risk Solver Platform (RSP) task pane for the following? Objective Cell: Variables Cells: Constraints Cells: MAX: 8 X1 + 5 X2 Subject to: 3 X1 + 5 X2 = 54 11 X1 + 10 X2 ≤ 144 X1 ≥12 X1, X2 ≥ 0


10 3 1 0 12

ANS: Objective Cell: $E$5 Variables Cells: $B$4:$C$4

Constraints Cells: $B$4:$C$4 ≥ 0 $D$8 = $E$8 $D$9 ≤ $E$9 $D$10 ≥ $E$10 (or $B$4 ≥ $E$10)

PTS: 1


problem. What numbers should be entered into cells B5:C5 and B8:C10 to implement this model? MAX: 4 X1 + 3 X2 Subject to: 6 X1 + 7 X2 ≤ 84 X1 ≤ 10 X2 ≤ 8 X1, X2 ≥ 0


10 3 8

ANS:


10 3 0 1 8

PTS: 1


problem. What formulas should be entered into cells E5 and D8:D10 to implement this model? MAX: 4 X1 + 3 X2 Subject to: 6 X1 + 7 X2 ≤ 84 X1 ≤ 10 X2 ≤ 8 X1, X2 ≥ 0

A B C D E

1 2 3 X1 X2 4 Number to make: OBJ. FN. VALUE 5 Unit profit: 4 3 6 7 Constraints: Used Available 8 1 6 7 84 9 2 1 0 10

10 3 0 1 8


PTS: 1


problem. What cell references would you enter in the Risk Solver Platform (RSP) task pane for the following? Objective Cell: Variables Cells: Constraints Cells: MAX: 12 X1 + 9 X2 Subject to: 9 X1 + 10.5 X2 ≤ 126 X1 ≥ 5 X2 ≥ 6 X1, X2 ≥ 0

A B C D E 1 2 3 X1 X2 4 Number to make: OBJ. FN. VALUE 5 Unit profit: 12 9 6 7 Constraints: Used Available 8 1 9 10.5 126 9 2 1 0 5

10 3 0 1 6

ANS: Objective Cell: $E$5 Variables Cells: $B$4:$C$4 Constraints Cells:

$B$4:$C$4 ≥ 0 $D$8 ≤ $E$8 $D$9:$D$10 ≥ $E$9:$E$10

PTS: 1


problem. What numbers should be entered into cells B5:C5 and B8:C10 to implement this model? MIN: 8 X1 + 3 X2 Subject to: X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0


10 3 60

ANS:


10 3 3 5 60

PTS: 1


problem. What formulas should be entered into cells E5 and D8:D10 to implement this model? MIN: 8 X1 + 3 X2 Subject to: X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0

A B C D E

1 2 3 X1 X2 4 Number to make: OBJ. FN. VALUE 5 Unit profit: 8 3 6 7 Constraints: Used Available 8 1 1 8 9 2 8 5 80

10 3 3 5 60


PTS: 1


problem. What cell references would you enter in the Risk Solver Platform (RSP) task pane for the following? Objective Cell: Variables Cells: Constraints Cells: MIN: 8 X1 + 3 X2 Subject to: X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0

A B C D E 1 2 3 X1 X2 4 Number to make: OBJ. FN. VALUE 5 Unit profit: 6 7 Constraints: Used Available 8 1 1 8 9 2 8 5 80

10 3 3 5 60

ANS: Objective Cell: $E$5 Variables Cells: $B$4:$C$4 Constraints Cells: $B$4:$C$4 ≥ 0

$D$8:$D$10 ≥ $E$8:$E$10

PTS: 1

61. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of

Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available. The following table summarizes the data for the problem. Crop

Profit per Acre ($)

Yield per Acre (lb)

Maximum Demand (lb)

Irrigation (acre ft)

Fertilizer (pounds/acre)

Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 Formulate the LP for this problem.

ANS: Let X1 = aces of corn X2 = acres of pumpkin X3 = acres of beans MAX: 2100X1 + 900X2 + 1050X3 Subject to: 21X1 ≤ 200 10X2 ≤ 180 3.5X3 ≤ 80 X1 + X2 + X3 ≤ 20 2X1 + 3X2 + 1X3 ≤ 50 5X1 + 4X2 + 3X3 ≤ 80 X1, X2, X3 ≥ 0

PTS: 1


Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available. The following table summarizes the data for the problem. Crop

Profit per Acre ($)

Yield per Acre (lb)

Maximum Demand (lb)



Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 Enter the numbers in the appropriate cells of ranges B12:D12 and E8:F12 in the Excel spreadsheet to solve this problem based on the following formulation. Let X1 = aces of corn X2 = acres of pumpkin X3 = acres of beans

MAX: 2100X1 + 900X2 + 1050X3 Subject to: 21X1 ≤ 200 10X2 ≤ 180 3.5X3 ≤ 80 X1 + X2 + X3 ≤ 20 2X1 + 3X2 + 1X3 ≤ 50 5X1 + 4X2 + 3X3 ≤ 80 X1, X2, X3 ≥ 0

A B C D E F 1 Farm Planning Problem 2 3 Corn Pumpkin Beans 4 Acres to plant Total Profit: 5 Profit per acre 6 7 Constraints: Used Available 8 Corn demand 9 Pumpkin demand

10 Bean demand 11 Water 12 Fertilizer

ANS:

A B C D E F 1 Farm Planning Problem 2 3 Corn Pumpkin Beans 4 Acres to plant Total Profit: 5 Profit per acre 2100 900 1050 6 7 Constraints: Used Available 8 Corn demand 21000 200000 9 Pumpkin demand 10000 180000

10 Bean demand 3500 80000 11 Water 2 3 1 50 12 Fertilizer 500 400 300 8000

PTS: 1


Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. The following table summarizes the data for the problem. Crop

Profit per Acre ($)

Yield per Acre (lb)

Maximum Demand (lb)



Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 What are the key formulas for this Excel spreadsheet implementation of the following formulation?

Let X1 = aces of corn X2 = acres of pumpkin X3 = acres of beans MAX: 2100X1 + 900X2 + 1050X3 Subject to: 21X1 ≤ 200 10X2 ≤ 180 3.5X3 ≤ 80 2X1 + 3X2 + 1X3 ≤ 50 5X1 + 4X2 + 3X3 ≤ 80 X1, X2, X3 ≥ 0

A B C D E F 1 Farm Planning Problem 2 3 Corn Pumpkin Beans 4 Acres to plant Total Profit: 5 Profit per acre 2100 900 1050 6 7 Constraints: Used Available 8 Corn demand 21000 200000 9 Pumpkin demand 10000 180000

10 Bean demand 3500 80000 11 Water 2 3 1 50 12 Fertilizer 500 400 300 8000

ANS: Cell Formula Copied to F5 =SUMPRODUCT(B4:D4,B5:D5) E8 =SUMPRODUCT($B$4:$D$4,B8:D8) E:E12

PTS: 1

64. A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must

work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table. Time period Required # of Nurses Wage ($/hr) 12 am − 4 am 20 15 4 am − 8 am 30 16 8 am − 12 pm 40 13 12 pm − 4 pm 50 13 4 pm − 8 pm 40 14 8 pm − 12 am 30 15 Formulate the LP for this problem.

ANS: Let Xi = number of nurses working in time period i; i = 1,6 MIN: 1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 Subject to: 1X1 + 1X2 ≥ 30

1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0

PTS: 1


work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table. Time period Required # of Nurses Wage ($/hr) 12 am − 4 am 20 15 4 am − 8 am 30 16 8 am − 12 pm 40 13 12 pm − 4 pm 50 13 4 pm − 8 pm 40 14 8 pm − 12 am 30 15 Enter the numbers in the appropriate cells of ranges B6:G11 and B13:G13 in the Excel spreadsheet to solve this problem based on the following formulation. Let Xi = number of nurses working in time period i; i = 1,6 MIN: 1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 Subject to: 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0

A B C D E F G H I 1 Nurse Hiring 2 3 Hours for each shift 4 Mid 4am 8am Noon 4pm 8pm Nurses Wages per 5 Shift 4am 8am Noon 4pm 8pm Mid Scheduled Nurse 6 1 $15 7 2 $16 8 3 $13 9 4 $13

10 5 $14 11 6 $15 12 Available: Total Wages: 13 Required:

ANS:

A B C D E F G H I

1 Nurse Hiring 2 3 Hours for each shift 4 Mid 4am 8am Noon 4pm 8pm Nurses Wages per 5 Shift 4am 8am Noon 4pm 8pm Mid Scheduled Nurse 6 1 1 1 0 0 0 0 $15 7 2 0 1 1 0 0 0 $16 8 3 0 0 1 1 0 0 $13 9 4 0 0 0 1 1 0 $13

10 5 0 0 0 0 1 1 $14 11 6 1 0 0 0 0 1 $15 12 Available: Total Wages: 13 Required: 20 30 40 50 40 30

PTS: 1


work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table. Time period Required # of Nurses Wage ($/hr) 12 am − 4 am 20 15 4 am − 8 am 30 16 8 am − 12 pm 40 13 12 pm − 4 pm 50 13 4 pm − 8 pm 40 14 8 pm − 12 am 30 15 What are the key formulas for this Excel spreadsheet implementation of the following formulation? Let Xi = number of nurses working in time period i; i = 1,6 MIN: 1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 Subject to: 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0

A B C D E F G H I 1 Nurse Hiring 2 3 Hours for each shift 4 Mid 4am 8am Noon 4pm 8pm Nurses Wages per 5 Shift 4am 8am Noon 4pm 8pm Mid Scheduled Nurse 6 1 $15 7 2 $16 8 3 $13

9 4 $13 10 5 $14 11 6 $15 12 Available: Total Wages: 13 Required:

ANS: Cell Formula Copied to I12 =SUMPRODUCT(H6:H11,I6:I11) B12 =SUMPRODUCT(B6:B11,$H$6:$H$11) C12:G12

PTS: 1


work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table. Time period Required # of Nurses Wage ($/hr) 12 am − 4 am 20 15 4 am − 8 am 30 16 8 am − 12 pm 40 13 12 pm − 4 pm 50 13 4 pm − 8 pm 40 14 8 pm − 12 am 30 15 What values would you enter in the Risk Solver Platform (RSP) task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem? Objective Cell: Variables Cells: Constraints Cells: Let Xi = number of nurses working in time period i; i = 1,6 MIN: 1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 Subject to: 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0

A B C D E F G H I 1 Nurse Hiring 2 3 Hours for each shift 4 Mid 4am 8am Noon 4pm 8pm Nurses Wages per 5 Shift 4am 8am Noon 4pm 8pm Mid Scheduled Nurse 6 1 1 1 0 0 0 0 $15

7 2 0 1 1 0 0 0 $16 8 3 0 0 1 1 0 0 $13 9 4 0 0 0 1 1 0 $13

10 5 0 0 0 0 1 1 $14 11 6 1 0 0 0 0 1 $15 12 Available: Total Wages: 13 Required: 20 30 40 50 40 30

ANS: Objective Cell: $I$12 Variables Cells: $H$6:$H$11 Constraints Cells: $H$6:$H$11 ≥ 0 $B$12:$G$12 ≥ $B$13:$G$13

PTS: 1

68. A company needs to purchase several new machines to meet its future production needs. It can

purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. Formulate the LP for this problem.

ANS: Let Xi = number of machines of type i purchased MAX: 200X1 + 250X2 + 300X3 Subject to: 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0

PTS: 1


purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. Enter the numbers in the appropriate cells of range B5:F10 in the Excel spreadsheet to solve this problem based on the following formulation. Let Xi = number of machines of type i purchased MAX: 200X1 + 250X2 + 300X3

Subject to: 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0

A B C D E F 1 Capital Expansion 2 3 Machine Types 4 Machine 1 Machine 2 Machine 3 5 Number to buy Total Output: 6 Machine output 7 8 Requirements: Used Available 9 Square feet

10 Cost

ANS:

A B C D E F 1 Capital Expansion 2 3 Machine Types 4 Machine 1 Machine 2 Machine 3 5 Number to buy Total Output: 6 Machine output 200 250 300 7 8 Requirements: Used Available 9 Square feet 2,000 3,000 5,000 20,000

10 Cost 80,000 50,000 40,000 500,000

PTS: 1


purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. What are the key formulas for this Excel spreadsheet implementation of the following formulation? Let Xi = number of machines of type i purchased MAX: 200X1 + 250X2 + 300X3 Subject to: 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0

A B C D E F G 1 Machine A Machine B Machine C 2 Number to Buy 3

4 Production Possible 200 250 350 Total: 5 6 Resources Hours Required Used Available: 7 Floor Space Req'd 2 3 4 20 8 Assemble 80 50 40 500

ANS: Cell Formula Copied to G4 =SUMPRODUCT(B2:D2,B4:D4) E7 =SUMPRODUCT($B$2:$D$2,B7:D7) E8

PTS: 1


purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. What values would you enter in the Risk Solver Platform (RSP) task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem? Objective Cell: Variables Cells: Constraints Cells: Let Xi = number of machines of type i purchased MAX: 200X1 + 250X2 + 300X3 Subject to: 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0

A B C D E F 1 Capital Expansion 2 3 Machine Types 4 Machine 1 Machine 2 Machine 3 5 Number to buy Total Output: 6 Machine output 200 250 300 7 8 Requirements: Used Available 9 Square feet 2,000 3,000 5,000 20,000

10 Cost 80,000 50,000 40,000 500,000

ANS: Objective Cell: $F$6 Variables Cells:

$B$5:$D$5 Constraints Cells: $B$5:$D$5 ≥ 0 $E$9:$E$10 ≤ $F$9:$F$10

PTS: 1

72. State Farm Supply has just received an order for 10,000 pounds of chicken feed. The farmer has

specified certain that the feed meet minimum requirements for Protein, Carbohydrate, Fat and Vitamins. State Farm can blend four different feeds to produce the required mix. The farmer would like to pay the lowest possible price for the feed. The data for the problem is summarized in the following table.

State Farm Supply Percent of Nutrient in: Minimum Nutrient Feed 1 Feed 2 Feed 3 Feed 4 Req'd Amt Protein 15 20 30 15 18 Carbohydrate 20 10 10 15 12 Fat 20 30 15 20 20 Vitamin 1 1.50 0.75 0.50 1 Cost/1,000 lbs $500 $600 $550 $450 Formulate the LP for this problem.

ANS: Let Xi = pounds of feed i used in mixture MIN: .5X1 + .6X2 + .55X3 + .45X4 Subject to: .15X1 + .20X2 + .3X3 + .15X4 ≥ 1800 .20X1 + .10X2 + .1X3 + .15X4 ≥ 1200 .20X1 + .30X2 + .15X3 + .20X4 ≥ 2000 .01X1 + .015X2 + .0075X3 + .005X4 ≥ 100 1X1 + 1X2 + 1X3 + 1X4 = 10000 Xi ≥ 0

PTS: 1

73. A paper mill has received an order for rolls of paper. The customer wants 400 12" wide rolls, 300 18"

rolls and 200 24" rolls. The company has 40" wide rolls of paper which it can slit to the appropriate width. The company wants to minimize the number of rolls it must use to fill the order. Formulate the LP for this problem.

ANS: Define the following cutting patterns. Number of widths in roll

Cutting pattern 12" 18" 24" 1 3 0 0 2 1 1 0 3 1 0 1 4 0 2 0 400 300 200

Let Xi = number of rolls cut in pattern i MIN: 1X1 + 1X2 + 1X3 + 1X4 Subject to: 3X1 + 1X2 + 1X3 ≥ 400 1X2 + 2X4 ≥ 300 1X3 ≥ 200 Xi ≥ 0

PTS: 1

74. Pete's Plastics manufactures plastic at plants in Miami, St. Louis and Cleveland. Pete needs to ship

plastic to customers in Pittsburgh, Atlanta and Chicago. He wants to minimize the cost of shipping the plastic from his plants to his customers. The data for the problem is summarized in the following table. Distance From Plants to Customers Plant Pittsburgh Atlanta Chicago Supply Miami 1200 700 1300 30 St. Louis 700 550 300 40 Cleveland 125 675 350 50 Demand 40 60 20 Formulate the LP for this problem.

ANS: Let Xij = tons shipped from plant i to customer j (i and j = 1, 2, 3) MIN: 1200X11 + 700X12 + 1300X13 + 700X21 + 550X22 + 300X23 + 125X31 + 675X32 + 350X33 Subject to: X11 + X12 + X13 = 30 X21 + X22 + X23 = 40 X31 + X32 + X33 = 50 X11 + X21 + X31 ≥ 40 X12 + X22 + X32 ≥ 60 X13 + X23 + X33 ≥ 20 Xij ≥ 0

PTS: 1

75. A financial planner wants to design a portfolio of investments for a client. The client has $400,000 to

invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 30% of the money in any one investment, at least one half should be invested in long-term bonds which mature in six or more years, and no more than 40% of the total money should be invested in B or C since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.

Investment Return Years to Maturity Rating A 6.45% 6 1-Excellent B 8.5% 5 3-Good C 9.00% 8 4-Fair D 7.75% 4 2-Very Good

Formulate the LP for this problem.

ANS: Let X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D MAX: .0645 X1 + .085 X2 + .090 X3 + .0775 X4 Subject to: X1 + X2 + X3 + X4 ≤ 400000 X1 ≤ 120000 X2 ≤ 120000 X3 ≤ 120000 X4 ≤ 120000 X1 + X3 ≥ 200000 X2 + X3 ≤ 160000 X1, X2, X3, X4 ≥ 0

PTS: 1




Let X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D MAX: .0645 X1 + .085 X2 + .090 X3 + .0775 X4 Subject to: X1 + X2 + X3 + X4 ≤ 400000 X1 ≤ 120000 X2 ≤ 120000 X3 ≤ 120000 X4 ≤ 120000 X1 + X3 ≥ 200000 X2 + X3 ≤ 160000 X1, X2, X3, X4 ≥ 0

A B C D > 1 Amount Maximum > 2 Bond Invested 30.0% Return >

3 A $0 $120,000 6.45% > 4 B $0 $120,000 8.5% > 5 C $0 $120,000 9.00% > 6 D $0 $120,000 7.75% > 7 Total Invested: $0 Total: $0 > 8 Total Available: $400,000 >

< E F G H < 1 Years to 6+ years? Good or worse? < 2 Maturity (1-yes, 0-no) Rating (1-yes, 0-no) < 3 6 1 1-Excellent 0 < 4 5 0 3-Good 0 < 5 8 1 4-Fair 1 < 6 4 0 2-Very Good 1 < 7 Total: $0 Total: $0 < 8 Required: $200,000 Allowed: $160,000 What values would you enter in the Risk Solver Platform (RSP) task pane for the following cells for this Excel spreadsheet implementation of this problem? Objective Cell: Variables Cells: Constraints Cells:

ANS: Objective Cell: D7 Variables Cells: B3:B6 Constraints Cells: B3:B6 ≤ C3:C6 B3:B6 ≥ 0 B7 ≤ B8 F7 ≥ F8 H7 ≤ H8

PTS: 1




Let X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D MAX: .0645 X1 + .085 X2 + .090 X3 + .0775 X4 Subject to: X1 + X2 + X3 + X4 ≤ 400000 X1 ≤ 120000 X2 ≤ 120000 X3 ≤ 120000 X4 ≤ 120000 X1 + X3 ≥ 200000 X2 + X3 ≤ 160000 X1, X2, X3, X4 ≥ 0

A B C D > 1 Amount Maximum > 2 Bond Invested 30.0% Return > 3 A $0 $120,000 6.45% > 4 B $0 $120,000 8.5% > 5 C $0 $120,000 9.00% > 6 D $0 $120,000 7.75% > 7 Total Invested: $0 Total: $0 > 8 Total Available: $400,000 >

< E F G H < 1 Years to 6+ years? Good or worse? < 2 Maturity (1-yes, 0-no) Rating (1-yes, 0-no) < 3 6 1 1-Excellent 0 < 4 5 0 3-Good 0 < 5 8 1 4-Fair 1 < 6 4 0 2-Very Good 1 < 7 Total: $0 Total: $0 < 8 Required: $200,000 Allowed: $160,000 What formulas are required for the following cells in the Excel spreadsheet implementation of the formulation? B7 D7 F7 H7

ANS: B7 =SUM(B3:B6) D7 =SUMPRODUCT($B$3:$B$6,D3:D6) F7 =SUMPRODUCT($B$3:$B$6,F3:F6) H7 =SUMPRODUCT($B$3:$B$6,H3:H6)

PTS: 1

78. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost, demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 100 units and management wants to keep at least 50 units on hand. Quarterly inventory holding cost is 4% of the cost of production. There are currently 50 units in inventory. The company wants to produce at no less than one half of its maximum capacity in any quarter. Quarter 1 2 3 4 Unit Production Cost $55 $50 $50 $45 Units Demanded 100 150 180 120 Maximum Production 150 150 160 130 Holding cost $2.2 $2 $2 $1.8 Let Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i MIN: 55 P1 + 50 P 2 + 50 P3 + 45 P4 + 2.2 (B1 + B2)/2 + 2 (B2 + B3)/2 + 2 (B3 + B4)/2 + 1.8 (B4 + B5)/2 Subject to: 75 ≤ P1 ≤ 150 75 ≤ P2 ≤ 150 80 ≤ P3 ≤ 160 65 ≤ P4 ≤ 130 50 ≤ B1 + P1 − 100 ≤ 100 50 ≤ B2 + P2 − 150 ≤ 100 50 ≤ B3 + P3 − 180 ≤ 100 50 ≤ B4 + P4 − 120 ≤ 100 B2 = B1 + P1 − 100 B3 = B2 + P2 − 150 B4 = B3 + P3 − 180 B5 = B4 + P4 − 120 Pi, Bi ≥ 0

A B C D E F 1 Quarter 2 1 2 3 4 3 Beginning Inventory 50 70 70 50 4 Units Produced 120 150 160 120 5 Units Demanded 100 150 180 120 6 Ending Inventory 70 70 50 50 7 8 Minimum Production 75 75 80 65 9 Maximum Production 150 150 160 130

10 11 Minimum Inventory 50 50 50 50 12 Maximum Inventory 100 100 100 100 13 14 Unit Production Cost $55 $50 $50 $45 15 Unit Carrying Cost 4.0% $2.20 $2.00 $2.00 $1.80 16 17 Monthly Production Cost $6,600 $7,500 $8,000 $5,400 18 Monthly Carrying Cost $132 $140 $120 $90 19

20 Total Cost $27,982 What formulas are required for cells D3, D6, D8, D15, D17 and D18 in the Excel spreadsheet implementation of the formulation?

ANS: D3 =C6 D6 =D3+D4-D5 D8 =D9/2 D15 =$B$15*D1 D17 =D14*D4) D18 =D15*(D3+D6)/2)

PTS: 1

79. A grain store has six types of grain, each varying in cost, quality, and nutritional content. Periodically,

excess inventory of these grains are consolidated into two local products, Feed-M-All and Supreme-Feed. Feed-M-All sells for $6.50 for a 10-pound bag while Supreme-Feed sells for $8.50 for a 10-pound bag. These feeds are advertised as having the following nutritional content: Grain Minimum Protein Minimum Fat Maximum Carbohydrates Feed-M-All 16% 18% 10% Supreme-Feed 18% 18% 9% The component grains have the following content characteristics: Grain Cost/10 lbs Quality Protein Fat Carbohydrates Pounds Avail.

A $4.75 4 15% 10% 10% 90 B $4.00 2 20% 20% 8% 120 C $3.75 1 10% 25% 5% 150 D $4.25 3 15% 20% 10% 125 E $4.50 3 20% 20% 10% 85 F $5.00 4 25% 15% 12% 165

Targets for Feed-M-All are a cost of $ 4.35 per 10-pound bag, a quality rating of 2.25, along with the minimum percentages of protein and fat, and the maximum percentage of carbohydrates. Similar targets are set for Supreme-Feed with cost set at $ 4.60 and quality at 2.45. There must be at least a 70%-30% mix among these two local feeds. Formulate an LP model for this product mix problem.

ANS: Let Xij = amount of grain i in feed j where i = A, B, C, D, E, F and j = 1(Feed-M-All), 2(Supreme-Feed) Yj = total amount of feed j produced MAX: $6.50Y1 + $8.50Y2 Subject to: Y1 = X11 + X21 + X31 + X41 + X51 + X61 Define Yj values Y2 = X12 + X22 + X32 + X42 + X52 + X62

X11 + X12 ≤ 90 Grain availability X21 + X22 ≤ 120

X31 + X32 ≤ 150 X41 + X42 ≤ 125 X51 + X52 ≤ 85 X61 + X62 ≤ 165 220.5 ≤ Y1 ≤ 514.5 Mix requirements 220.5 ≤ Y2 ≤ 514.5 4X11 + 2X21 + X31 + 3X41 + 3X51 + 4X61 ≥ 2.25Y1 Quality targets 4X12 + 2X22 + X32 + 3X42 + 3X52 + 4X62 ≥ 2.45Y2

4.75X11 + 4X21 + 3.75X31 + 4.25X41 + 4.5X51 + 5X61 ≤ 4.35Y1 Cost targets 4.75X12 + 4X22 + 3.75X32 + 4.25X42 + 4.5X52 + 5X62 ≤ 4.60Y2

10X11 + 20X21 + 10X31 + 15X41 + 20X51 + 25X61 ≥ 16Y1 Protein targets 10X12 + 20X22 + 10X32 + 15X42 + 20X52 + 25X62 ≥ 18Y2

10X11 + 20X21 + 25X31 + 20X41 + 20X51 + 15X61 ≥ 18Y1 Fat targets 10X12 + 20X22 + 25X32 + 20X42 + 20X52 + 15X62 ≥ 18Y2

10X11 + 8X21 + 5X31 + 10X41 + 10X51 + 12X61 ≤ 10Y1 Carbohydrate targets 10X12 + 8X22 + 5X32 + 10X42 + 10X52 + 12X62 ≤ 9Y2

Xij ≥ 0 for all i and j, Yj ≥ 0 for all j.

PTS: 1

80. Carlton construction is supplying building materials for a new mall construction project in Kansas.

Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):

Week Trucking Limits Railway Limits Air Cargo Limits 1 45 60 15 2 50 55 10 3 55 45 5

Costs ($ per 1000 tons) $200 $140 $400 Formulate an LP model for this logistics problem.

ANS: Let Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3 WLij = weekly limit of mode i in week j (as provided in above table) MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33) Subject to: Xij ≤ WLij for all i and j Weekly limits by mode X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks X11 + X21 + X31 ≥ 120 Total at end of first week X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement

X31 + X32 + X33 ≤ 0.15*250 Air mix limit Xij ≥ 0 for all i and j

PTS: 1




Costs ($ per 1000 tons) $200 $140 $400 The following is the LP model for this logistics problem. Let Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3 WLij = weekly limit of mode i in week j (as provided in above table) MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33) Subject to: Xij ≤ WLij for all i and j Weekly limits by mode X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks X11 + X21 + X31 ≥ 120 Total at end of first week X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement X31 + X32 + X33 ≤ 0.15*250 Air mix limit Xij ≥ 0 for all i and j A B C D E F

1 Costs $200.00 $140.00 $500.00 2 by Truck by Rail by Air Totals Required 3 Week 1 45 60 15 120 120 4 Week 2 50 55 0 225 200 5 Week 3 13 12 0 250 250 6 Shipped by 108 127 15 7 Percentage 45% 40% 15% 8 Total Limit 108 100 37.5 9

10 Total Cost $46,880.00 11 Weekly Limits 12 Truck Rail Air 13 Week 1 45 60 15 14 Week 2 50 55 10

15 Week 3 55 45 5 What formula goes in cells F10, E3, E4, E5, and B6 of this Excel spreadsheet?

ANS: F10 =SUMPRODUCT($B$1:$D$1,$B$6:$D$6) E3 =SUM($B$3:$D$3) E4 =SUM($B$4:$D$4) E5 =SUM($B$5:$D$5) B6 =SUM($B$3:$B$5)

PTS: 1




Costs ($ per 1000 tons) $200 $140 $400 The following is the LP model for this logistics problem. Let Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3 WLij = weekly limit of mode i in week j (as provided in above table) MIN: 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33) Subject to: Xij ≤ WLij for all i and j Weekly limits by mode X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 Total at end of three weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of two weeks X11 + X21 + X31 ≥ 120 Total at end of first week X11 + X12 + X13 ≥ 0.45*250 Truck mix requirement X21 + X22 + X23 ≥ 0.40*250 Rail mix requirement X31 + X32 + X33 ≤ 0.15*250 Air mix limit Xij ≥ 0 for all i and j A B C D E F

1 Costs $200.00 $140.00 $500.00 2 by Truck by Rail by Air Totals Required 3 Week 1 45 60 15 120 120 4 Week 2 50 55 0 225 200 5 Week 3 13 12 0 250 250

6 Shipped by 108 127 15 7 Percentage 45% 40% 15% 8 Total Limit 108 100 37.5 9

10 Total Cost $46,880.00 11 Weekly Limits 12 Truck Rail Air 13 Week 1 45 60 15 14 Week 2 50 55 10 15 Week 3 55 45 5 What values would you enter in the Risk Solver Platform (RSP) task pane for the cells in this Excel spreadsheet implementation of this problem? Objective Cell: Variables Cells: Constraints Cells:

ANS: Objective Cell: F10 Variables Cells: B3:D5 Constraints Cells: B3:D3 ≤ B13:D13 E3:E5 ≥ F3:F5 B6:C6 ≥ B8:C8 D6 ≤ D8

PTS: 1

83. Robert Hope received a welcome surprise in this management science class; the instructor has decided

to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: • homework can account for up to 25% of the grade, but must be at least 5% of the grade; • the project can account for up to 25% of the grade, but must be at least 5% of the grade; • the mid-term and final must each account for between 10% and 40% of the grade but

cannot account for more than 70% of the grade when the percentages are combined; and • the project and final exam grades may not collectively constitute more than 50% of the

grade. Formulate an LP model for Robert to maximize his numerical grade.

ANS: Let W1 = weight assigned to homework W2 = weight assigned to the project W3 = weight assigned to the mid-term W4 = weight assigned to the final

MAX: 75W1 + 94W2 + 85W3 + 92W4 Subject to: W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40

PTS: 1

84. Robert Hope received a welcome surprise in this management science class; the instructor has decided

to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: • homework can account for up to 25% of the grade, but must be at least 5% of the grade; • the project can account for up to 25% of the grade, but must be at least 5% of the grade; • the mid-term and final must each account for between 10% and 40% of the grade but

cannot account for more than 70% of the grade when the percentages are combined; and • the project and final exam grades may not collectively constitute more than 50% of the

grade. The following LP model allows Robert to maximize his numerical grade. Let W1 = weight assigned to homework W2 = weight assigned to the project W3 = weight assigned to the mid-term W4 = weight assigned to the final MAX: 75W1 + 94W2 + 85W3 + 92W4 Subject to: W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40 A B C D E

1 Percentage Limits 2 Grade to grade Lower Upper 3 Mid Term 85 0.40 0.10 0.40 4 Final 92 0.25 0.10 0.40 5 Project 94 0.25 0.05 0.25 6 Homework 75 0.10 0.05 0.25 7 1.00 8 100% 1.00 9

10 Grade 88.00 11 Total Limit 12 Both Exams 0.65 0.70 13 Final & Project 0.5 0.50

What values would you enter in the Risk Solver Platform (RSP) task pane for the cells in this Excel spreadsheet implementation of this problem? Objective Cell: Variables Cells: Constraints Cells:

ANS: Objective Cell: C10 Variables Cells: C3:C6 Constraints Cells: C3:C6 ≤ D3:D6 C3:C6 ≥ E3:E6 C7 = C8 B12 ≤ C12 B13 ≥ C13

PTS: 1

85. The hospital administrators at New Hope, County General, and City East recently received notice of an

impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months. Hospital New Hope County General City East Input Measures Bed days unused (1000s) 83.0 105.0 104.1 Supply expense ($1000s) 123.8 162.3 154.0 Full-time staff 225.0 200.0 231.0 Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 Customer satisfaction 98.0 88.0 83.0 a. Formulate a DEA LP model to evaluate the efficiency of City East. b. Implement a spreadsheet model for this problem and compute the DEA efficiency for each

facility. Which facilities are efficient?

ANS: a. Let wi = weight assigned to output j, j = 1, ...,4 vi = weight assigned to input i, i = 1,...,3 MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4

Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0 b. A B C D E > 1 Patient Asst > 2 Days Nurses on Cust > 3 Hospital (1000s) Qual. Staff Sat. > 4 New Hope 105.10 253.00 125.0 98 > 5 Cnty. General 71.00 92.00 45.0 88 > 6 City East 82.70 175.00 65.0 83 > 7 > 8 Weights 0.002009 0 0 0.00778 > 9 > 10 UNIT 3 > 11 Output 0.812259 > 12 Input 1 > < F G H I J K < 1 Bed-Days Supply Full < 2 Unused Expense Time Wgt. Wgt. < 3 (1000s) ($1000s) Staff Output Input Diff < 4 83.00 123.80 225.00 97% 97% 0.0000 < 5 105.00 162.30 200.00 83% 87% −0.0381 < 6 104.10 154.00 231.00 81% 100% −0.1877 < 7 < 8 0 0 0.004329 < 9 < 10 < 11 < 12 Results: DEA Unit Efficiency New Hope 1.0000 County General 0.9297 City East 0.8123 New Hope is an efficient facility.

PTS: 1


impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months. Hospital

New Hope County General City East Input Measures Bed days unused (1000s) 83.0 105.0 104.1 Supply expense ($1000s) 123.8 162.3 154.0 Full-time staff 225.0 200.0 231.0 Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 Customer satisfaction 98.0 88.0 83.0 Based on the following formulation, is City East efficient? If not, what input and output values should they aspire to in order to become efficient? Let wi = weight assigned to output j, j = 1, ..., 4 vi = weight assigned to input i, i = 1,...,3 MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0

ANS: No, City East is not efficient. The following shows that 78.69% of New Hope input and outputs produces a composite unit with outputs greater than or equal to those of City East requiring less input than City East. --- Outputs --- > Patient Asst > Days Nurses on Cust > Unit (1000s) Qual. Staff Sat. > New Hope 105.10 253.00 125.0 98 > County General 71.00 92.00 45.0 88 > City East 82.70 175.00 65.0 83 > > Comp Vals 82.7 199.1 98.4 77.1 > < --- Inputs --- < Bed-Days Supply Full < Unused Expense Time Composite < Unit (1000s) ($1000s) Staff Weight < New Hope 83.00 123.80 225.00 0.7869 < County General 105.00 162.30 200.00 0 < City East 104.10 154.00 231.00 0 < < Comp Vals 65.3 97.4 177.0 Note, however, the drop in customer satisfaction. City East will not want to aspire to that particular level. These composite values will make City East efficient.

A B C D E > 1 Patient Asst > 2 Days Nurses on Cust > 3 Hospital (1000s) Qual. Staff Sat. > 4 New Hope 105.10 253.00 125.0 98 > 5 Cnty. General 71.00 92.00 45.0 88 > 6 City East 82.70 199.0 98.4 83 > 7 > 8 Weights 0.004279 0 0 0.007784 > 9 >

10 UNIT 3 > 11 Output 1 > 12 Input 1 > < F G H I J K < 1 Bed-Days Supply Full < 2 Unused Expense Time Wgt. Wgt. < 3 (1000s) ($1000s) Staff Output Input Diff < 4 83.00 123.80 225.00 121% 127% −0.0586 < 5 105.00 162.30 200.00 99% 113% −0.1411 < 6 65.3 97.4 177 100% 100% 0.0000 < 7 < 8 0 0 0.005649 < 9 < 10 < 11 < 12

PTS: 1


impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months. Hospital New Hope County General City East Input Measures Bed days unused (1000s) 83.0 105.0 104.1 Supply expense ($1000s) 123.8 162.3 154.0 Full-time staff 225.0 200.0 231.0 Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 Customer satisfaction 98.0 88.0 83.0 Enter the numbers in the appropriate cells of ranges B4:H6 in the Excel spreadsheet to solve this problem based on the following formulation. Let wi = weight assigned to output j, j = 1, ..., 4 vi = weight assigned to input i, i = 1,...,3

MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0

A B C D E > 1 Patient Asst > 2 Days Nurses on Cust > 3 Hospital (1000s) Qual. Staff Sat. > 4 New Hope > 5 Cnty. General > 6 City East > 7 > 8 Weights 0 0 0 0 > 9 >

10 UNIT 3 > 11 Output 0.81 > 12 Input 1.0 > < F G H I J K < 1 Bed-Days Supply Full < 2 Unused Expense Time Wgt. Wgt. < 3 (1000s) ($1000s) Staff Output Input Diff < 4 97% 97% 0.0000 < 5 83% 87% −0.0381 < 6 81% 100% −0.1877 < 7 < 8 0 0 0 < 9 < 10 < 11 < 12

ANS:

A B C D E > 1 Patient Asst > 2 Days Nurses on Cust > 3 Hospital (1000s) Qual. Staff Sat. > 4 New Hope 105.10 253.00 125.0 98 > 5 Cnty. General 71.00 92.00 45.0 88 > 6 City East 82.70 175.00 65.0 83 > 7 > 8 Weights 0.002009 0 0 0.00778 > 9 >

10 UNIT 3 > 11 Output 0.812259 > 12 Input 1 > < F G H I J K

< 1 Bed-Days Supply Full < 2 Unused Expense Time Wgt. Wgt. < 3 (1000s) ($1000s) Staff Output Input Diff < 4 83.00 123.80 225.00 97% 97% 0.0000 < 5 105.00 162.30 200.00 83% 87% −0.0381 < 6 104.10 154.00 231.00 81% 100% −0.1877 < 7 < 8 0 0 0.004329 < 9 < 10 < 11 < 12

PTS: 1


impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months. Hospital New Hope County General City East Input Measures Bed days unused (1000s) 83.0 105.0 104.1 Supply expense ($1000s) 123.8 162.3 154.0 Full-time staff 225.0 200.0 231.0 Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 satisfaction 98.0 88.0 83.0 What are the key formulas for this Excel spreadsheet implementation of the following formulation? Let wi = weight assigned to output j, j = 1, ..., 4 vi = weight assigned to input i, i = 1,...,3 MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0

A B C D E > 1 Patient Asst > 2 Days Nurses on Cust > 3 Hospital (1000s) Qual. Staff Sat. > 4 New Hope 105.10 253.00 125.0 98 > 5 Cnty. General 71.00 92.00 45.0 88 >

6 City East 82.70 175.00 65.0 83 > 7 > 8 Weights 0.002009 0 0 0.00778 > 9 >

10 UNIT 3 > 11 Output 0.812259 > 12 Input 1 > < F G H I J K < 1 Bed-Days Supply Full < 2 Unused Expense Time Wgt. Wgt. < 3 (1000s) ($1000s) Staff Output Input Diff < 4 83.00 123.80 225.00 97% 97% 0.0000 < 5 105.00 162.30 200.00 83% 87% −0.0381 < 6 104.10 154.00 231.00 81% 100% −0.1877 < 7 < 8 0 0 0.004329 < 9 < 10 < 11 < 12

ANS: Cell Formula Copied to I4 =SUMPRODUCT($B$8:$E$8,B4:E4) I5:I6 J4 =SUMPRODUCT($F$8:$H$8,F4:H4) J5:J6 K4 =I4-J4 K5:K6 B11 =INDEX(I4:I6,B10,1) B12 =INDEX(J4:J6, B10,1)

PTS: 1

PROJECT

89. Project 3.1 − The Diet Problem: Ordering Meals from McDonald's

Based on: Robert A. Bosch, "Big Mac Attack: The Diet Problem revisited, Eating at McDonald's," OR/MS Today, August 1993, pp 30-31. Tina Simpson is a new fourth-grade teacher at Forest Ridge Elementary. The first teacher workshop for the upcoming school year is next Monday and by majority vote, McDonald's was selected as the food of choice. As the new person, Tina is tasked with developing the meal for the workshop. McDonald's has graciously offered to deliver whatever food Tina decides to order, along with a variety of condiments applicable to whatever is ordered. Rather than offer a menu choice, Tina has decided to simply order the same meal for each person in the workshop. To get started, Tina took a trip to McDonald's and obtained their published information on the nutritional content of their food. That data is summarized in the table below. Price Protein Fat Sodium > Menu Item ($) Calories (grams) (mg) > Hamburger 0.59 255 12 9 490 > McLean Dlx 1.79 320 22 10 670 >

Big Mac 1.65 500 25 26 890 > Small Fries 0.68 220 3 12 110 > McNuggets 1.56 270 20 15 580 > Honey 0.00 45 0 0 0 > Chef Salad 2.69 170 17 9 400 > Garden Salad 1.96 50 4 2 70 > Egg McMuffin 1.36 280 18 11 710 > Wheaties 1.09 90 2 1 220 > Yogurt Cone 0.63 105 4 1 80 > Milk 0.56 110 9 2 130 > Orange Juice 0.88 80 1 0 0 > Grapefruit juice 0.68 80 1 0 0 > Apple Juice 0.68 90 0 0 5 > Prices recorded August, 1991 in Oberlin Ohio > < Vit A Vit C Vit B1 Vit B2 Niacin Calcium Iron < Menu Item % U.S. Recommended Daily Allowance (RDA) < Hamburger 4 4 20 10 20 10 15 < McLean Dlx 10 10 25 20 35 15 20 < Big Mac 6 2 30 25 35 25 20 < Small Fries * 15 10 * 10 * 2 < McNuggets * * 8 8 40 * 6 < Honey * * * * * * * < Chef Salad 100 35 20 15 20 15 8 < Garden Salad 90 35 6 6 2 4 8 < Egg McMuffin 10 * 30 20 20 25 15 < Wheaties 20 20 20 20 20 2 20 < Yogurt Cone 2 * 2 10 2 10 * < Milk 10 4 8 30 * 30 * < Orange Juice * 120 10 * * * * < Grapefruit juice * 100 4 2 2 * * < Apple Juice * 2 2 * * * 4 < Prices recorded August, 1991 in Oberlin Ohio < * Contains less than 2% of the U.S. RDA of these nutrients Tina wants the meal to be nutritionally complete. The National Research Council publishes their Recommended Daily Allowances. In this publication, they contend that a diet (in this case the meal) should provide at least 100 percent of the U.S. RDA of numerous nutrients. The specific amount of the RDA depends on such factors as age, weight and gender. In addition, the council recommends daily sodium and cholesterol intakes be kept to at most 2.4 grams of sodium and 300 milligrams of cholesterol. Further, at most 30 percent of the calories consumed should come from fat, and at most 10 percent from saturated fat. Each gram of fat contains 9 calories. Based on the above information, Tina wants to design a least-cost meal that provides at least 100% of the U.S. RDA of vitamins A, C, B1, B2, niacin, calcium, and iron; supplies at least 55 grams of protein; contains at most 3 grams of sodium; and contains at most 30 percent of its calories from fat. Only those foods list in the table above are available for the meal. Formulate the LP model for Tina's problem. Develop a spreadsheet model of the problem and use Excel Solver to determine the least-cost meal that meets all the stated requirements. What is the recommended meal? Is this meal reasonable? If not, modify the model to obtain what you believe to be a reasonable meal that meets the stated requirements.

ANS: Answer not provided.

PTS: 1

SMDA6e Chapter 03

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