Data Model Chap 4

Linear Programming Applicationsin Marketing, Finance and Operations

• Marketing Applications

• Financial Applications

• Operations Management Applications

• Blending Problems

LP applications• Care must be taken to ensure that the Linear

Programming model accurately reflects the real problem.

• Always review your formulation thoroughly before attempting to solve the model.

• One application of linear programming in marketing is media selection.

• LP can be used to help marketing managers allocate a fixed budget to various advertising media.

• The objective is to maximize reach, frequency, and quality of exposure.

• Restrictions on the allowable allocation usually arise during consideration of company policy, contract requirements, and media availability.

Marketing ApplicationsMarketing Applications

Media SelectionMedia Selection

SMM Company recently developed a new SMM Company recently developed a new instantinstantsalad machine, has $282,000 to spend on salad machine, has $282,000 to spend on advertising. The product is to be initially test advertising. The product is to be initially test marketed in the Dallasmarketed in the Dallasarea. The money is to be spent onarea. The money is to be spent ona TV advertising blitz during onea TV advertising blitz during oneweekend (Friday, Saturday, andweekend (Friday, Saturday, andSunday) in November.Sunday) in November. The three options availableThe three options availableare: daytime advertising,are: daytime advertising,evening news advertising, andevening news advertising, andSunday game-time advertising. A mixture of one-Sunday game-time advertising. A mixture of one-minute TV spots is desired. minute TV spots is desired.


Estimated AudienceEstimated AudienceAd TypeAd Type Reached With Each AdReached With Each Ad Cost Cost Per AdPer Ad Daytime Daytime 3,000 3,000 $5,000 $5,000 Evening News Evening News 4,000 4,000 $7,000$7,000Sunday Game Sunday Game 75,000 75,000 $100,000$100,000

SMM wants to take out at least one ad of each SMM wants to take out at least one ad of each type (daytime, evening-news, and game-time). type (daytime, evening-news, and game-time). Further, there are only two game-time ad spots Further, there are only two game-time ad spots available. There are ten daytime spots and six available. There are ten daytime spots and six evening news spots available daily. SMM wants evening news spots available daily. SMM wants to have at least 5 ads per day, but spend no to have at least 5 ads per day, but spend no more than $50,000 on Friday and no more than more than $50,000 on Friday and no more than $75,000 on Saturday.$75,000 on Saturday.


Solution SummarySolution Summary

Total new audience reached = Total new audience reached = 199,000 199,000

Number of daytime ads on Friday Number of daytime ads on Friday = 8= 8 Number of daytime ads on Saturday Number of daytime ads on Saturday = 5= 5 Number of daytime ads on Sunday Number of daytime ads on Sunday = 2= 2 Number of evening ads on Friday Number of evening ads on Friday = 0= 0 Number of evening ads on Saturday Number of evening ads on Saturday = 0= 0 Number of evening ads on Sunday Number of evening ads on Sunday = 1= 1 Number of game-time ads on SundayNumber of game-time ads on Sunday = 2= 2

Notes and comments

• The above media selection model uses new audience reached as the objective function and places a constraint on expenditure on the ads.

• An alternative formulation could be to use the exposure quality as the objective function and place an additional constraint on customers reached.

• The model in such a case would require subjective evaluations of the exposure quality for the media alternatives. Judgement is an acceptable way of obtaining input for an LP model.

Financial ApplicationsFinancial Applications

LP can be used in financial decision-making LP can be used in financial decision-making that involves capital budgeting, make-or-buy, that involves capital budgeting, make-or-buy, asset allocation, portfolio selection, financial asset allocation, portfolio selection, financial planning, and more.planning, and more.

Portfolio selectionPortfolio selection problems involve choosing problems involve choosing specific investments – for example, stocks and specific investments – for example, stocks and bonds – from a variety of investment bonds – from a variety of investment alternatives.alternatives.

This type of problem is faced by managers of This type of problem is faced by managers of banks, mutual funds, and insurance banks, mutual funds, and insurance companies.companies.

The objective function usually is maximization The objective function usually is maximization of expected return or minimization of risk.of expected return or minimization of risk.

Portfolio Selection

Winslow Savings has $20 million available

for investment. It wishes to invest

over the next four months in

such a way that it will maximize

the total interest earned over the four

month period as well as have at least

$10 million available at the start of the fifth month for a high rise building venture in which it will be participating.

Portfolio SelectionFor the time being, Winslow wishes to invest

only in 2-month government bonds (earning 2% over

the 2-month period) and 3-month construction loans

(earning 6% over the 3-month period). Each of these

is available each month for investment. Funds not

invested in these two investments are liquid and earn

3/4 of 1% per month when invested locally.

Portfolio SelectionFormulate a linear program that will

help Winslow Savings determine how to invest over the next four months if at no time does it wish to have more than $8 million in either government bonds or

construction loans.

Portfolio SelectionPortfolio Selection

The Management ScientistThe Management Scientist SolutionSolution

Objective Function Value = Objective Function Value = 1429213.79871429213.7987

VariableVariable ValueValue Reduced CostsReduced Costs

GG1 1 8000000.0000 8000000.0000 0.00000.0000

GG2 2 0.0000 0.0000 0.00000.0000

GG3 3 5108613.9228 5108613.9228 0.00000.0000

GG4 4 2891386.0772 2891386.0772 0.00000.0000

CC1 1 8000000.0000 8000000.0000 0.00000.0000

CC2 2 0.0000 0.0000 0.04530.0453

CC3 3 0.0000 0.0000 0.00760.0076

CC4 4 8000000.0000 8000000.0000 0.00000.0000

LL1 1 4000000.0000 4000000.0000 0.00000.0000

LL2 2 4030000.0000 4030000.0000 0.00000.0000

LL3 3 7111611.0772 7111611.0772 0.00000.0000

LL4 4 4753562.0831 4753562.0831 0.00000.0000

Notes and comments• The optimal solution to the Winslow problem indicates

that $ 5108613.92285108613.9228 is to be spent on the government bonds in the third month. If government bonds are selling for $100 per bond, we would have to purchase exactly 51086.139 bonds.the difficulty of purchasing fractional bonds is usually handled by purchasing the largest possible integer number of bonds with the allotted funds(e.g. 51086 bonds). This will guarantee that the budget constraint is not violated.

• Financial portfolio theory stresses obtaining a proper balance between risk and return.

• In the above example, we explicitly considered return in the objective function. Risk is controlled by choosing constraints that ensure diversity among different stocks and a balance between government bonds and the construction loans.

Operations Management ApplicationsOperations Management Applications

LP can be used in operations management to LP can be used in operations management to aid in decision-making about product mix, aid in decision-making about product mix, production scheduling, staffing, inventory production scheduling, staffing, inventory control, capacity planning, and other issues.control, capacity planning, and other issues.

An important application of LP is multi-periodAn important application of LP is multi-period planning such as planning such as production schedulingproduction scheduling..

Usually the objective is to establish an Usually the objective is to establish an efficient, low-cost production schedule for one efficient, low-cost production schedule for one or more products over several time periods.or more products over several time periods.

Typical constraints include limitations on Typical constraints include limitations on production capacity, labor capacity, storage production capacity, labor capacity, storage space, and more.space, and more.

Chip Hoose is the owner of Hoose Custom Chip Hoose is the owner of Hoose Custom Wheels. Chip has just received orders for 1,000 Wheels. Chip has just received orders for 1,000 standard wheelsstandard wheelsand 1,250 deluxe wheels next monthand 1,250 deluxe wheels next monthand for 800 standard and 1,500 deluxeand for 800 standard and 1,500 deluxethe following month. All orders mustthe following month. All orders mustbe filled.be filled.

Production SchedulingProduction Scheduling

The cost of making standard wheelsThe cost of making standard wheelsis $10 and deluxe wheels is $16. Over-is $10 and deluxe wheels is $16. Over-time rates are 50% higher. There aretime rates are 50% higher. There are1,000 hours of regular time and 500 hours of 1,000 hours of regular time and 500 hours of overtimeovertimeavailable each month. It takes .5 hour to make available each month. It takes .5 hour to make aastandard wheel and .6 hour to make a deluxe standard wheel and .6 hour to make a deluxe wheel.wheel.The cost of storing a wheel from one month to The cost of storing a wheel from one month to the next is $2.the next is $2.


We want to determine the regular-time and We want to determine the regular-time and overtimeovertime production quantities in each month for production quantities in each month for standard andstandard and deluxe wheels.deluxe wheels.

Month 1Month 1 Month 2Month 2WheelWheel Reg. TimeReg. Time OvertimeOvertime Reg. TimeReg. Time OvertimeOvertimeStandard Standard SRSR11 SOSO11 SRSR22 SOSO22 Deluxe Deluxe DRDR11 DODO11 DRDR22 DODO22

Define the Decision Define the Decision VariablesVariables


We also want to determine the inventory We also want to determine the inventory quantitiesquantities for standard and deluxe wheels.for standard and deluxe wheels.

SISI = number of standard wheels held in = number of standard wheels held in inventory from month 1 to month inventory from month 1 to month 22 DIDI = number of deluxe wheels held in = number of deluxe wheels held in inventory from month 1 to month inventory from month 1 to month 22

Define the Decision Define the Decision VariablesVariables

Objective Function Value = 67500.000Objective Function Value = 67500.000

VariableVariable ValueValue Reduced CostReduced Cost

SRSR1 1 500.000 500.000 0.000 0.000

SOSO1 500.000 1 500.000 0.0000.000

SRSR2 2 200.000 200.000 0.000 0.000 SOSO2 600.000 2 600.000 0.0000.000 DRDR1 1 1250.000 1250.000 0.000 0.000 DODO1 1 0.000 0.000 2.0002.000 DRDR2 1500.000 2 1500.000 0.0000.000 DODO2 2 0.000 0.000 2.0002.000 SISI 0.000 0.000 2.0002.000 DIDI 0.000 0.000 2.0002.000

The Management ScientistThe Management Scientist SolutionSolution


Thus, the recommended production schedule is:Thus, the recommended production schedule is:

Month 1Month 1 Month 2Month 2

Reg. TimeReg. Time OvertimeOvertime Reg. TimeReg. Time OvertimeOvertimeStandard Standard 500 500 500 500 200 600 200 600 Deluxe Deluxe 1250 1250 0 0 1500 0 1500 0

No wheels are stored and the minimum total cost No wheels are stored and the minimum total cost is $67,500.is $67,500.

Solution summarySolution summary


Product MixFloataway Tours has $420,000 that can be

used to purchase new rental boats for hire during the summer. The boats can be purchased from two different manufacturers.

Floataway Tours would like to purchase at least 50 boats and would like to purchase the same number from Sleekboat as from Racer to maintain goodwill. At the same time, Floataway Tours wishes to have a total seating

capacity of at least 200.

Formulate this problem as a linear program.

Maximum Expected

Boat Builder Cost Seating Daily Profit

Speedhawk Sleekboat $6000 3 $ 70

Silverbird Sleekboat $7000 5 $ 80

Catman Racer $5000 2 $ 50

Classy Racer $9000 6 $110

Product Mix

• Solution Summary– Purchase 28 Speedhawks from Sleekboat.– Purchase 28 Classy’s from Racer.– Total expected daily profit is $5,040.00.– The minimum number of boats was exceeded by 6

(surplus for constraint #2).– The minimum seating capacity was exceeded by 52

(surplus for constraint #4).

Product Mix

Diet problemDiet problem

Ferdinand Feed Company receives four Ferdinand Feed Company receives four rawraw

grains from which it blends its dry pet food. grains from which it blends its dry pet food. The petThe pet

food advertises that each 8-ounce packetfood advertises that each 8-ounce packet

meets the minimum daily requirementsmeets the minimum daily requirements

for vitamin C, protein and iron. Thefor vitamin C, protein and iron. The

cost of each raw grain as well as thecost of each raw grain as well as the

vitamin C, protein, and iron units pervitamin C, protein, and iron units per

pound of each grain are summarized onpound of each grain are summarized on

the next slide. the next slide.


Vitamin C Protein Iron Vitamin C Protein Iron

Grain Units/lb Units/lb Units/lb Grain Units/lb Units/lb Units/lb Cost/lbCost/lb

1 9 1 9 12 12 0 0 .75 .75

2 16 2 16 10 10 14 14 .90 .90

3 83 8 10 10 15 15 .80 .80

4 10 4 10 8 8 7 7 .70 .70

Ferdinand is interested in producing the 8-Ferdinand is interested in producing the 8-ounceounce

mixture at minimum cost while meeting the mixture at minimum cost while meeting the minimumminimum

daily requirements of 6 units of vitamin C, 5 daily requirements of 6 units of vitamin C, 5 units ofunits of

protein, and 5 units of iron.protein, and 5 units of iron.

The Management ScientistThe Management Scientist Output Output

OBJECTIVE FUNCTION VALUE = 0.406OBJECTIVE FUNCTION VALUE = 0.406

VARIABLEVARIABLE VALUEVALUE REDUCED COSTSREDUCED COSTS X1 X1 0.099 0.099 0.0000.000 X2 X2 0.213 0.213 0.0000.000 X3 X3 0.088 0.088 0.0000.000 X4 X4 0.099 0.099 0.0000.000

Thus, the optimal blend is about .10 lb. of grain Thus, the optimal blend is about .10 lb. of grain 1, .21 lb.1, .21 lb.

of grain 2, .09 lb. of grain 3, and .10 lb. of grain of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. The4. The

mixture costs Frederick’s 40.6 cents.mixture costs Frederick’s 40.6 cents.


Data Model Chap 4

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