LINEAR PROGRAMMING APPLICATIONS

Kashif Latif

Sumbal Babar

What is LP Applications

Most successful quantitative approach to decision making, also have been reported almost every industry. Application includes

Production Scheduling Media Selection Financial Planning Capital Budgeting Transportation Distribution System Design Staffing

What we’ll Cover

Marketing Applications Financial Applications Operations Management Applications

What is Marketing

Marketing is communicating the value of a product, service or brand to customers, for the purpose of promoting or selling that product, service, or brand.

Marketing Applications

Media Selection Marketing Research

Media Selection

Help marketing managers to allocate a fixed advertising budget to various advertising media. Media includes

Newspapers Magazines Radio Television Direct Mail

Objective

Objective of Media Selections includes

Maximize Reach Frequency Quality of Exposure

Restrictions

Company Policy Contract Requirements Media Availability

Relax-and-Enjoy Lake Development Corporation

Advertising Media

No. of Potential Customer Reached

Cost ($) per

Advertisement

Maximum Time

Available per

Month

Exposure Quality

Units

Daytime TV (1 min), station WKLA 1000 1500 15 65

Evening TV (30 sec), station WKLA 2000 3000 10 90

Daily Newspaper (full page), The Morning Journal

1500 400 25 40

Sunday Newspaper magazine (1/2 page color), The Sunday Press

2500 1000 4 60

Radio, 8:00 AM or 5:00 PM news (30 sec), station KNOP

300 100 30 20

Decision Variables

DTV = number of times daytime TV is used

ETV = number of time evening TV is used

DN = number of time daily newspaper is used

SN = number of time Sunday newspaper is used

R = number of times radio is used

Objective Function

With the objective of maximizing the total exposure quality units for the overall media selection plan, the objective function becomes

Max 65DTV+90ETV+40DN+60SN+20R

Formulate Constraints

DTV ≤ 15ETV ≤ 10

DN ≤ 25SN ≤ 4

R ≤ 30

Availability of Media

Continue…

1500DTV + 3000ETV + 400DN + 1000SN + 100R ≤ 30000 Budget

DTV + ETV ≥ 10

1500DTV + 3000ETV ≤ 18000

Television Restrictions

1000DTV + 2000ETV + 1500DN + 2500SN + 300R ≥ 50000 Customer Reached

DTV, ETV, DN, SN, R ≥ 0

Decision Variables

Decision Variable

sDTV ETV DNP SNP R

Adds 10 0 25 2 30

Advertising Plan

Media Frequency Budget ($)

Daytime TV 10 15,000

Daily Newspaper 25 10,000

Sunday Newspaper 2 2,000

Radio 30 3,000

Total 30,000

Results

Exposure Quality Units = 2,370Total Customers Reached =

61,500

Marketing Research

A research to learn about

Consumer Characteristics Attitudes Preferences

Marketing Research Firms

Specialized in marketing research for client organization. Services they offer includes:

Designing the Study Conducting Market Surveys Analyzing the Data Collected Providing Summary Reports &

Recommendations

Market Survey, Inc.

1. Interview at least 400 households with children.2. Interview at least 400 households without children.3. The total number of households interviewed during

the evening must be at least as great as the number of households interviewed during the day.

4. At least 40% of the interviews for households with children must be conducted during the evening.

5. At least 60% of the interviews for households without children must be conducted during the evening.

Previous Cost Estimations

Interview Cost

Household Day Evening

Children $20 $25

No Children $18 $20

Decision Variables

DC = the number of daytime interviews of households with childrenEC = the number of evening interviews of households with childrenDNC= the number of daytime interviews of households without childrenENC = the number of evening interviews of households without children

Objective Function

Using previous cost estimation, the object function would be

Min 20DC + 25EC + 18DNC + 20ENC

Formulate Constraints

1. DC + EC + DNC + ENC = 10002. DC + EC ≥ 4003. DNC + ENC ≥ 4004. EC + ENC ≥ DC + DNC

The usual format for linear programming model formulation places all decision variables on the left side of the inequality and a constant (possibly zero) on the right side. Thus, we rewrite this constraint as

4.– DC + EC – DNC + ENC ≥ 0

Formulate Constraints

5. EC ≥ 0.4(DC + EC) or -0.4DC + 0.6EC ≥ 0

6. ENC ≥ 0.6(DNC + ENC) or -0.6DNC + 0.4ENC ≥ 0

Nonnegativity Requirements

DC, EC, DNC, ENC ≥ 0

Interview Schedule

Number of Interviews

Household Day Evening Totals

Children 240 160 400

No Children 240 360 600

Totals 480 520 1000

Financial Application

In finance, linear programming can be applied in problem situations involving:

Capital BudgetingMake-or-Buy DecisionsAsset AllocationPortfolio SelectionFinancial Planning, and many more.

Financial Application Problems

Portfolio Selection Financial Planning

Portfolio Selection

Portfolio selection problems involve situations in which a financial manager

must select specific investments for example stocks and bonds from a variety of investment alternatives.

Objective Function

The objective function for portfolio selection problems usually is maximization of expected return or minimization of risk.

Constraints

The constraints usually reflect restrictions on the type of

Permissible InvestmentsState LawsCompany PolicyMaximum Permissible Risk, and so on.

Welte Mutual Funds, Inc.

Projected Rate of Return

Investment (%)

Atlantic Oil 7.3

Pacific Oil 10.3

Midwest Steel 6.4

Huber Steel 7.5

Government Bonds 4.5

Decision Variables

A = dollars invested in Atlantic OilP = dollars invested in Pacific OilM = dollars invested in

Midwest SteelH = dollars invested in Huber SteelG = dollars invested in government

bonds

Objective Function

Objective function for maximizing the total return for the portfolio is

Max 0.073A 0.103P 0.064M 0.075H 0.045G

Linear Programming Model

1. A + P + M + H + G = 100,0002. A + P ≤ 50,0003. M + H ≤ 50,0004. -0.25M - 0.25H + G ≥ 05. -0.6A + 0.4P ≤ 0

A, P, M, H, G ≥ 0

Optimal Portfolio Selection

Investment Amount ($) Expected Annual Return ($)

Atlantic Oil 20,000 1,460

Pacific Oil 30,000 3,090

Huber Steel 40,000 3,000

Government Bonds 10,000 450

Totals 100,000 8000

Expected Annual Return of $8000Overall Rate of Return = 8%

Financial Planning

Financial Planning is an ongoing process to help you make sensible decisions about money that can help you achieve your goals in life.

Hewlitt Corporation

Year 1 2 3 4 5 6 7 8

Cash Requirements 430 210 222 231 240 195 225 255

The cash requirements (in thousands of dollars) are due at the beginning of each year.

Government Bonds Investments

Bond Price ($) Rate (%) Years to Maturity

1 1150 8.875 5

2 1000 5.500 6

3 1350 11.750 7

Decision Variables

F = total dollars required to meet the retirement plan’s eight- year obligationB1 = units of bond 1 purchased at the beginning of year 1B2 = units of bond 2 purchased at the beginning of year 1B3 = units of bond 3 purchased at the beginning of year 1Si = amount placed in savings at the beginning of year i for

I = 1, . . . , 8

Objective Function

The objective function is to minimize the total dollars needed to meet the retirement plan’s eight-year obligation, or

Min F

General Form of Constraint

Funds available

at the beginning

of the year

-

Funds invested in bonds and

placed in

savings

-

Cash obligation for the

current

year

Constraint of Each Year

F - 1.15B1 - 1B2- 1.35B3 - S1 = 430 Year 10.08875B1+0.055B2+0.1175B3+1.04S1-S2 = 210 Year 20.08875B1+0.055B2+0.1175B3+1.04S2-S3 = 222 Year 30.08875B1+0.055B2+0.1175B3+1.04S3-S4 = 231 Year 40.08875B1+0.055B2+0.1175B3+1.04S4-S5 = 240 Year 51.08875B1+0.055B2+0.1175B3+1.04S5-S6 = 195 Year 6

1.055B2+0.1175B3+1.04S6-S7 = 225 Year 7 1.1175B3+1.04S7-S8 = 255 Year 8

Optimal Solution

Bond Units Purchased Investment Amount

1 B1=144.988 $1150(144.988)=$166,736

2 B2=187.856 $1000(187.856)=$187,856

3 B3=228.188 $1350(228.188)=$308,054

Operation Management Applications

Managing and directing the physical and/or technical functions of a firm or organization, particularly those relating to

DevelopmentProductionManufacturing

What we’ll cover

Make-or-Buy Decision Production Scheduling Workforce Assignment Blending Problems

Make-or-Buy Decision

It determine how much of each of several component parts a company should manufacture and how much it should purchase from an outside supplier.

Production Scheduling

Establish an efficient low-cost production schedule for one or more products over several time periods (weeks or months)

Advantages

Can help to smooth the demand signal Protects lead time and helps book future

deliveries Acts as a single communication tool to

the business Helps the Supply chain prioritize

requirement Helps stabilize production

Disadvantages

Complexity Cost Can be Skewed Lack of Flexibility

Workforce Assignment

Workforce assignment problems frequently occur when production managers must make decisions involving staffing requirements for a given planning period.

Blending Problems

Blending problems arise whenever a manager must decide how to blend two or more resources to produce one or more products.