mcgraw-hill/irwin modified for quan 6610 by dr. jim grayson optimization© the mcgraw-hill...

Optimization© The McGraw-Hill Companies, Inc., 2003

4.1McGraw-Hill/IrwinModified for Quan 6610 by Dr. Jim Grayson

Chapter 4 (Linear Programming: Formulation and Applications)

• Advertising-Mix Problem (Section 4.1) – Super Grain Corp | 4.2–4.5

• Resource Allocation Problems (Section 4.2)–Think-Big Capital Budgeting | 4.6–4.10

• Cost-Benefit-Trade-Off Problems (Section 4.3)–Union Airways | 4.11–4.15

• Distribution-Network Problems (Section 4.4)–Big M Co. | 4.16–4.20

• Student Exercises



Super Grain Corp. Advertising-Mix Problem

• Goal: Design the promotional campaign for Crunchy Start.

• The three most effective advertising media for this product are– Television commercials on Saturday morning programs for children.

– Advertisements in food and family-oriented magazines.

– Advertisements in Sunday supplements of major newspapers.

• The limited resources in the problem are– Advertising budget ($4 million).

– Planning budget ($1 million).

– TV commercial spots available (5).

• The objective will be measured in terms of the expected number of exposures.

Question: At what level should they advertise Crunchy Start in each of the three media?



Cost and Exposure Data

Costs

Cost CategoryEach

TV CommercialEach

Magazine AdEach

Sunday Ad

Ad Budget $300,000 $150,000 $100,000

Planning budget 90,000 30,000 40,000

Expected number of exposures

1,300,000 600,000 500,000



Spreadsheet Formulation

3456789101112131415

B C D E F G HTV Spots Magazine Ads SS Ads

Exposures per Ad 1,300 600 500(thousands)

Budget BudgetCost per Ad ($thousands) Spent Available

Ad Budget 300 150 100 4,000 <= 4,000Planning Budget 90 30 40 1,000 <= 1,000

Total ExposuresTV Spots Magazine Ads SS Ads (thousands)

Number of Ads 0 20 10 17,000<=

Max TV Spots 5



Algebraic Formulation

Let TV = Number of commercials for separate spots on televisionM = Number of advertisements in magazines.SS = Number of advertisements in Sunday supplements.

Maximize Exposure = 1,300TV + 600M + 500SSsubject to

Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand)Planning Cost: 90TV + 30M + 30SS ≤ 1,000 ($thousand)Number of TV Spots: TV ≤ 5

andTV ≥ 0, M ≥ 0, SS ≥ 0.



Think-Big Capital Budgeting Problem

• Think-Big Development Co. is a major investor in commercial real-estate development projects.

• They are considering three large construction projects– Construct a high-rise office building.

– Construct a hotel.

– Construct a shopping center.

• Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years.

Question: At what fraction should Think-Big invest in each of the three projects?



Financial Data for the Projects

Investment Capital Requirements

Year Office Building Hotel Shopping Center

0 $40 million $80 million $90 million

1 60 million 80 million 50 million



Net present value $45 million $70 million $50 million




345678910111213141516

B C D E F G HOffice Shopping

Building Hotel CenterNet Present Value 45 70 50

($millions) Cumulative CumulativeCapital Capital

Cumulative Capital Required ($millions) Spent AvailableNow 40 80 90 25 <= 25

End of Year 1 100 160 140 44.757 <= 45End of Year 2 190 240 160 60.583 <= 65End of Year 3 200 310 220 80 <= 80

Office Shopping Total NPVBuilding Hotel Center ($millions)

Participation Share 0.00% 16.50% 13.11% 18.11




Let OB = Participation share in the office building,H = Participation share in the hotel,SC = Participation share in the shopping center.

Maximize NPV = 45OB + 70H + 50SCsubject to

Total invested now: 40OB + 80H + 90SC ≤ 25 ($million)Total invested within 1 year: 100OB + 160H + 140SC ≤ 45 ($million)Total invested within 2 years: 190OB + 240H + 160SC ≤ 65 ($million)Total invested within 3 years: 200OB + 310H + 220SC ≤ 80 ($million)

andOB ≥ 0, H ≥ 0, SC ≥ 0.



Summary of Formulation Procedure for Resource-Allocation Problems

1. Identify the activities for the problem at hand.

2. Identify an appropriate overall measure of performance (commonly profit).

3. For each activity, estimate the contribution per unit of the activity to the overall measure of performance.

4. Identify the resources that must be allocated.

5. For each resource, identify the amount available and then the amount used per unit of each activity.

6. Enter the data in steps 3 and 5 into data cells.

7. Designate changing cells for displaying the decisions.

8. In the row for each resource, use SUMPRODUCT to calculate the total amount used. Enter ≤ and the amount available in two adjacent cells.

9. Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.



Union Airways Personnel Scheduling

• Union Airways is adding more flights to and from its hub airport and so needs to hire additional customer service agents.

• The five authorized eight-hour shifts are– Shift 1: 6:00 AM to 2:00 PM

– Shift 2: 8:00 AM to 4:00 PM

– Shift 3: Noon to 8:00 PM

– Shift 4: 4:00 PM to midnight

– Shift 5: 10:00 PM to 6:00 AM

Question: How many agents should be assigned to each shift?



Schedule Data

Time Periods Covered by Shift

Time Period 1 2 3 4 5

MinimumNumber of

Agents Needed

6 AM to 8 AM √ 48

8 AM to 10 AM √ √ 79

10 AM to noon √ √ 65

Noon to 2 PM √ √ √ 87

2 PM to 4 PM √ √ 64

4 PM to 6 PM √ √ 73

6 PM to 8 PM √ √ 82

8 PM to 10 PM √ 43

10 PM to midnight √ √ 52

Midnight to 6 AM √ 15

Daily cost per agent $170 $160 $175 $180 $195




3456789101112131415161718192021

B C D E F G H I J6am-2pm 8am-4pm Noon-8pm 4pm-midnight 10pm-6am

Shift Shift Shift Shift ShiftCost per Shift $170 $160 $175 $180 $195

Total MinimumTime Period Shift Works Time Period? (1=yes, 0=no) Working Needed

6am-8am 1 0 0 0 0 48 >= 488am-10am 1 1 0 0 0 79 >= 79

10am- 12pm 1 1 0 0 0 79 >= 6512pm-2pm 1 1 1 0 0 118 >= 872pm-4pm 0 1 1 0 0 70 >= 644pm-6pm 0 0 1 1 0 82 >= 736pm-8pm 0 0 1 1 0 82 >= 82

8pm-10pm 0 0 0 1 0 43 >= 4310pm-12am 0 0 0 1 1 58 >= 52

12am-6am 0 0 0 0 1 15 >= 15

6am-2pm 8am-4pm Noon-8pm 4pm-midnight 10pm-6amShift Shift Shift Shift Shift Total Cost

Number Working 48 31 39 43 15 $30,610




Let Si = Number working shift i (for i = 1 to 5),

Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5

subject toTotal agents 6AM–8AM: S1 ≥ 48Total agents 8AM–10AM: S1 + S2 ≥ 79Total agents 10AM–12PM: S1 + S2 ≥ 65Total agents 12PM–2PM: S1 + S2 + S3 ≥ 87Total agents 2PM–4PM: S2 + S3 ≥ 64Total agents 4PM–6PM: S3 + S4 ≥ 73Total agents 6PM–8PM: S3 + S4 ≥ 82Total agents 8PM–10PM: S4 ≥ 43Total agents 10PM–12AM: S4 + S5 ≥ 52Total agents 12AM–6AM: S5 ≥ 15

andSi ≥ 0 (for i = 1 to 5)



Summary of Formulation Procedure forCost-Benefit-Tradeoff Problems

1. Identify the activities for the problem at hand.

2. Identify an appropriate overall measure of performance (commonly cost).

3. For each activity, estimate the contribution per unit of the activity to the overall measure of performance.

4. Identify the benefits that must be achieved.

5. For each benefit, identify the minimum acceptable level and then the contribution of each activity to that benefit.

6. Enter the data in steps 3 and 5 into data cells.

7. Designate changing cells for displaying the decisions.

8. In the row for each benefit, use SUMPRODUCT to calculate the level achieved. Enter ≤ and the minimum acceptable level in two adjacent cells.

9. Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.



The Big M Distribution-Network Problem

• The Big M Company produces a variety of heavy duty machinery at two factories. One of its products is a large turret lathe.

• Orders have been received from three customers for the turret lathe.

Question: How many lathes should be shipped from each factory to each customer?



Some Data

Shipping Cost for Each Lathe

To Customer 1 Customer 2 Customer 3

From Output

Factory 1 $700 $900 $800 12 lathes

Factory 2 800 900 700 15 lathes

Order Size 10 lathes 8 lathes 9 lathes



The Distribution Network

F1

C2

C3

C1

F2

12 latheproduced

15 lathesproduced

10 lathesneeded

8 lathesneeded

9 lathesneeded

$700/lathe

$900/lathe

$800/lathe

$800/lathe $900/lathe

$700/lathe




3456789101112131415

B C D E F G HShipping Cost

(per Lathe) Customer 1 Customer 2 Customer 3Factory 1 $700 $900 $800Factory 2 $800 $900 $700

TotalShipped

Units Shipped Customer 1 Customer 2 Customer 3 Out OutputFactory 1 10 2 0 12 = 12Factory 2 0 6 9 15 = 15

Total To Customer 10 8 9= = = Total Cost

Order Size 10 8 9 $20,500




Let Sij = Number of lathes to ship from i to j (i = F1, F2; j = C1, C2, C3).

Minimize Cost = $700SF1-C1 + $900SF1-C2 + $800SF1-C3 + $800SF2-C1 + $900SF2-C2 + $700SF2-C3

subject toFactory 1: SF1-C1 + SF1-C2 + SF1-C3 = 12Factory 2: SF2-C1 + SF2-C2 + SF2-C3 = 15Customer 1: SF1-C1 + SF2-C1 = 10Customer 2: SF1-C2 + SF2-C2 = 8Customer 3: SF1-C3 + SF2-C3 = 9

andSij ≥ 0 (i = F1, F2; j = C1, C2, C3).



Types of Functional Constraints

Type Form* Typical Interpretation Main Usage

Resource constraint LHS ≤ RHSFor some resource, Amount used ≤ Amount available

Resource-allocation problems and mixed problems

Benefit constraint LHS ≥ RHSFor some benefit, Level achieved ≥ Minimum Acceptable

Cost-benefit-trade-off problems and mixed problems

Fixed-requirement constraint

LHS = RHSFor some quantity, Amount provided = Required amount

Distribution-network problems and mixed problems

* LHS = Left-hand side (a SUMPRODUCT function). RHS = Right-hand side (a constant).



Formulating an LP Spreadsheet Model

• Enter all of the data into the spreadsheet. Color code (blue).

• What decisions need to be made? Set aside a cell in the spreadsheet for each decision variable (changing cell). Color code (yellow with border).

• Write an equation for the objective in a cell. Color code (orange with heavy border).

• Put all three components (LHS, ≤/=/≥, RHS) of each constraint into three cells on the spreadsheet.

• Some Examples:– Production Planning

– Diet / Blending

– Workforce Scheduling

– Transportation / Distribution

– Assignment



Product Mix Exercise

Blue Ridge Hot Tubs manufactures and sells two models of hot tubs: the Aqua-Spa and the Hydro-Lux. Howie Jones, the owner and manager of the company needs to decide how many of each type of hot tub to produce during his next production cycle. Howie buys prefabricated fiberglass hot tub shells from a local supplier and adds the pump and tubing to the shells to create his hot tubs. (The supplier has the capacity to deliver as many hot tub shells as Howie needs.) Howie installs the same type of pump into both hot tubs. He will have only 200 pumps available during his next production cycle. From a manufacturing standpoint, the main difference between the two models of hot tubs is the amount of tubing and labor required. Each Aqua-Spa requires 9 hours of labor and 12 feet of tubing. Each Hydro-Lux requires 6 hours of labor and 16 feet of tubing. Howie expects to have 1,566 production labor hours and 2,880 feet of tubing available during the next production cycle. Howie earns a profit of $350 on each Aqua-Spa he sells and $300 on each Hydro-Lux he sells. He is confident that he can sell all the hot tubs he produces. The question is, how many Aqua-Spas and Hydro-Luxes should Howie produce if he wants to maximize his profits during the next production cycle?

Source: Ragsdale, Spreadsheet Modeling and Decision Analysis.



Elements Common to Every Problem

Decision variables:

number of aqua-spas (A) to produce and

number of hydro-luxes (H) to produce.

Objective function: Max: Profit = 350 A + 300 H

Constraints:

Pump 1A + 1H <= 200

Labor 9A + 6H <= 1566

Tubing 12A + 16H <= 2880



0 15 30 45 60 75 90 105120135150165180195210225240255270285300

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300AquaSpa

HydroLux

Payoff: 300.0 HydroLux + 350.0 AquaSpa = 0.0Optimal Decisions(HydroLux,AquaSpa): ( 0.0, 0.0)

Pump: 1.0HydroLux + 1.0AquaSpa <= 200.0

Graphical Solution Using Graphic LP Optimizer



0 15 30 45 60 75 90 105120135150165180195210225240255270285300

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HydroLux



: 6.0HydroLux + 9.0AquaSpa <= 1566.0




0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180

081624324048566472808896104112120128136144152160168

AquaSpa

HydroLux






Constraints



0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180

081624324048566472808896104112120128136144152160168

AquaSpa

HydroLux






Feasible Solution Space



0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180

081624324048566472808896104112120128136144152160168

AquaSpa

HydroLux

Payoff: 300.0 HydroLux + 350.0 AquaSpa = 66100.0

Optimal Decisions(HydroLux,AquaSpa): (78.0, 122.0)





Optimal Solution



Organizing Spreadsheet & Entering Formulas

D6. =SUMPRODUCT($B$5:$C$5,B6:C6)




Decision Variable Cells

Decision Variable Coefficients

Constraint Coefficients

Constraint RHS Formulas

Constraint RHS Limits



Tools | Solver



Sensitivity Analysis



A Classic Problem

See In class handout.

First, identify the decision variables, objective function and constraints.

Second, think about spreadsheet layout.

Third, implement and solve model.



In Class Exercise

End of chapter problem 4.6

mcgraw-hill/irwin modified for quan 6610 by dr. jim grayson optimization© the mcgraw-hill...

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