Download - Linear Programming Application
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.