Chapter 3 1. 3-2 Super Grain Corp. Advertising-Mix Problem Goal: Design the promotional campaign for...

Chapter 3

1

3-2

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?

3-3

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

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Spreadsheet Formulation

3456789

101112131415

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

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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 ($thousand)Planning Cost: 90TV + 30M + 40SS ≤ 1,000 ($thousand)Number of TV Spots: TV ≤ 5

andTV ≥ 0, M ≥ 0, SS ≥ 0.

3-6

The TBA Airlines Problem• TBA Airlines is a small regional company that specializes in short flights in small

airplanes.• The company has been doing well and has decided to expand its operations.• The basic issue facing management is whether to purchase more small airplanes to

add some new short flights, or start moving into the national market by purchasing some large airplanes, or both.

Question: How many airplanes of each type should be purchased to maximize their total net annual profit?

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Data for the TBA Airlines Problem

SmallAirplane

LargeAirplane

CapitalAvailable

Net annual profit per airplane $1 million $5 million

Purchase cost per airplane 5 million 50 million $100 million

Maximum purchase quantity 2 —

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Violates Divisibility Assumption of LP• Divisibility Assumption of Linear Programming: Decision variables in a linear

programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values.

• Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated.

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Spreadsheet Model

3456789

1011121314

B C D E F GSmall Airplane Large Airplane

Unit Profit ($millions) 1 5

Capital CapitalSpent Available

Capital ($millions) 5 50 100 <= 100

Total ProfitSmall Airplane Large Airplane ($millions)

Units Produced 0 2 10<=

Maximum Small Airplanes 2

Capital Per Unit Produced

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Integer Programming FormulationLet S = Number of small airplanes to purchase

L = Number of large airplanes to purchase

Maximize Profit = S + 5L ($millions)

subject to

Capital Available: 5S + 50L ≤ 100 ($millions)

Max Small Planes: S ≤ 2

and

S ≥ 0, L ≥ 0

S, L are integers.

3-11

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?

3-12

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

3-13


3456789

10111213141516

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

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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.

3-15

Template for Resource-Allocation Problems

Activities

Unit Profit profit per unit of activityResources Resources

Used Available

SUMPRODUCTresource used per unit of activity (resource used per unit,

changing cells)

Total ProfitLevel of Activity changing cells SUMPRODUCT(profit per unit, changing cells)

<=

Const

rain

ts

3-16

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.

3-17

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?

3-18

Schedule DataTime 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

3-19


3456789

101112131415161718192021

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

3-20

Algebraic FormulationLet 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)

3-21

Template for Cost-Benefit Tradoff Problems

Activities

Unit Cost cost per unit of activityBenefit Benefit

Achieved Needed

SUMPRODUCTbenefit achieved per unit of activity (benefit per unit,

changing cells)

Total CostLevel of Activity changing cells SUMPRODUCT(cost per unit, changing cells)

>=

Const

rain

ts

3-22

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.

3-23

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

Transportation problems and mixed problems

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

3-24

Continuing the Super Grain Case Study

• David and Claire conclude that the spreadsheet model needs to be expanded to incorporate some additional considerations.

• In particular, they feel that two audiences should be targeted — young children and parents of young children.

• Two new goals

– The advertising should be seen by at least five million young children.

– The advertising should be seen by at least five million parents of young children.

• Furthermore, exactly $1,490,000 should be allocated for cents-off coupons.

3-25

Benefit and Fixed-Requirement Data

Number Reached in Target Category (millions)

EachTV Commercial

EachMagazine Ad

EachSunday Ad

MinimumAcceptable

Level

Young children 1.2 0.1 0 5

Parents of young children 0.5 0.2 0.2 5

Contribution Toward Required Amount

EachTV Commercial

EachMagazine Ad

EachSunday Ad

RequiredAmount

Coupon redemption 0 $40,000 $120,000 $1,490,000

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3456789

101112131415161718192021

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

Exposures per Ad 1,300 600 500(thousands)

Cost per Ad ($thousands) Budget Spent Budget AvailableAd Budget 300 150 100 3,775 <= 4,000

Planning Budget 90 30 40 1,000 <= 1,000

Number Reached per Ad (millions) Total Reached Minimum AcceptableYoung Children 1.2 0.1 0 5 >= 5

Parents of Young Children 0.5 0.2 0.2 5.85 >= 5

TV Spots Magazine Ads SS Ads Total Redeemed Required AmountCoupon Redemption per Ad 0 40 120 1,490 = 1,490

($thousands)Total Exposures

TV Spots Magazine Ads SS Ads (thousands)Number of Ads 3 14 7.75 16,175

<=Maximum TV Spots 5

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Algebraic FormulationLet TV = Number of commercials for separate spots on television

M = 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

Young children: 1.2TV + 0.1M ≥ 5 (millions)Parents: 0.5TV + 0.2M + 0.2SS ≥ 5 (millions)

Coupons: 40M + 120SS = 1,490 ($thousand)

andTV ≥ 0, M ≥ 0, SS ≥ 0.

3-28

Template for Mixed ProblemsActivities

Unit Profit or Cost profit/cost per unit of activityResources Resources

Used Available

SUMPRODUCTresource used per unit of activity (resource used per unit,

changing cells)

Benefit BenefitAchieved Needed

SUMPRODUCTbenefit achieved per unit of activity (benefit per unit,

changing cells)

Total Profit or CostLevel of Activity changing cells

Const

rain

ts

SUMPRODUCT(profit/cost per unit, changing cells)

<=

>=

=

3-29

LP Example #2 (Diet Problem)A prison is trying to decide what to feed its prisoners. They would like to offer some combination of milk, beans, and oranges. Their goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. The cost and nutritional contents of each food, along with the minimum nutritional requirements are shown below.

Milk(gallons)

NavyBeans(cups)

Oranges(large Calif.Valencia)

MinimumDaily

Requirement

Niacin (mg) 3.2 4.9 0.8 13.0

Thiamin (mg) 1.12 1.3 0.19 1.5

Vitamin C (mg) 32 0 93 45

Cost ($) 2.00 0.20 0.25

Question: What should the diet for each prisoner be?

3-30


Let x1 = gallons of milk per prisoner,x2 = cups of beans per prisoner,x3 = number of oranges per prisoner.

Minimize Cost = $2.00x1 + $0.20x2 + $0.25x3

subject toNiacin: 3.2x1 + 4.9x2 + 0.8x3 ≥ 13 mgThiamin: 1.12x1 + 1.3x2 + 0.19x3 ≥ 1.5 mgVitamin C: 32x1 + 93x3 ≥ 45 mg

andx1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

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345678910111213

B C D E F G HMilk Beans(gal.) (cups) Oranges

Cost $2.00 $0.20 $0.25Minimum

Total RequirementNiacin (mg) 3.2 4.9 0.8 13 >= 13

Thiamin (mg) 1.12 1.3 0.19 3.438 >= 1.5Vitamin C (mg) 32 0 93 45 >= 45

Quantity Total Cost(per prisoner) 0 2.574 0.484 $0.64

Nutritional Contents (mg)

3-32

George Dantzig’s Diet• Stigler (1945) “The Cost of Subsistence”

– heuristic solution. Cost = $39.93.• Dantzig invents the simplex method (1947)

– Stigler’s problem “solved” in 120 man days. Cost = $39.69.

• Dantzig goes on a diet (early 1950’s), applies diet model:

– ≤ 1,500 calories– objective: maximize (weight minus water content)– 500 food types

• Initial solutions had problems

– 500 gallons of vinegar– 200 bouillon cubes

For more details, see July-Aug 1990 Interfaces article “The Diet Problem”

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Least-Cost Menu Planning Models in Food Systems Management

• Used in many institutions with feeding programs: hospitals, nursing homes, schools, prisons, etc.

• Menu planning often extends to a sequence of meals or a cycle.• Variety important (separation constraints).• Preference ratings (related to service frequency).• Side constraints (color, categories, etc.)• Generally models have reduced cost about 10%, met nutritional requirements

better, and increased customer satisfaction compared to traditional methods.• USDA uses these models to plan food stamp allotment.For more details, see Sept-Oct 1992 Interfaces article “The Evolution of the Diet Model in Managing Food Systems”

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