Linear Programming Application
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This document discusses various applications of linear programming (LP) in areas such as marketing, finance, and operations management. In marketing, LP can be used for media selection and marketing research problems. It provides examples of using LP to determine an optimal advertising plan that maximizes exposure quality within a budget. In finance, LP is used for portfolio selection and financial planning problems. Operations management applications covered include make-or-buy decisions, production scheduling, workforce assignment, and blending problems.
Linear Programming Application
- 2. 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
- 3. What we’ll Cover Marketing Applications Financial Applications Operations Management Applications
- 4. 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.
- 5. Marketing Applications Media Selection Marketing Research
- 6. Media Selection Help marketing managers to allocate a fixed advertising budget to various advertising media. Media includes Newspapers Magazines Radio Television Direct Mail
- 7. Objective Objective of Media Selections includes Maximize Reach Frequency Quality of Exposure
- 8. Restrictions Company Policy Contract Requirements Media Availability
- 9. Relax-and-Enjoy Lake Development Corporation Advertising Media No. of Potential Customer Reached Cost ($) per Advertiseme nt Maximum Time Available per Month Exposure Quality Units Daytime TV (1 min), station WKLA 1000 1500 15 65 Evening TV (30 sec), station WKLA 2000 3000 10 90 Daily Newspaper (full page), The Morning Journal 1500 400 25 40 Sunday Newspaper magazine (1/2 page color), The Sunday Press 2500 1000 4 60 Radio, 8:00 AM or 5:00 PM news (30 sec), station KNOP 300 100 30 20
- 10. Decision Variables DTV = number of times daytime TV is used ETV = number of time evening TV is used DN = number of time daily newspaper is used SN = number of time Sunday newspaper is used R = number of times radio is used
- 11. Objective Function With the objective of maximizing the total exposure quality units for the overall media selection plan, the objective function becomes Max 65DTV+90ETV+40DN+60SN+20R
- 12. Formulate Constraints DTV ≤ 15 ETV ≤ 10 DN ≤ 25 SN ≤ 4 R ≤ 30 Availability of Media
- 13. Continue… 1500DTV + 3000ETV + 400DN + 1000SN + 100R ≤ 30000 Budget DTV + ETV ≥ 10 1500DTV + 3000ETV ≤ 18000 Television Restrictions 1000DTV + 2000ETV + 1500DN + 2500SN + 300R ≥ 50000 Customer Reached DTV, ETV, DN, SN, R ≥ 0
- 14. Decision Variables Decision Variables DTV ETV DNP SNP R Adds 10 0 25 2 30
- 15. Advertising Plan Media Frequency Budget ($) Daytime TV 10 15,000 Daily Newspaper 25 10,000 Sunday Newspaper 2 2,000 Radio 30 3,000 Total 30,000
- 16. Results Exposure Quality Units = 2,370 Total Customers Reached = 61,500
- 17. Marketing Research A research to learn about Consumer Characteristics Attitudes Preferences
- 18. 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
- 19. 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
- 20. Previous Cost Estimations Interview Cost Household Day Evening Children $20 $25 No Children $18 $20
- 21. Decision Variables DC = the number of daytime interviews of households with children EC = the number of evening interviews of households with children DNC = the number of daytime interviews of households without children ENC = the number of evening interviews of households without children
- 22. Objective Function Using previous cost estimation, the object function would be Min 20DC + 25EC + 18DNC + 20ENC
- 23. Formulate Constraints 1. DC + EC + DNC + ENC = 1000 2. DC + EC ≥ 400 3. DNC + ENC ≥ 400 4. 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
- 24. 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
- 25. Interview Schedule Number of Interviews Household Day Evening Totals Children 240 160 400 No Children 240 360 600 Totals 480 520 1000
- 26. Financial Application In finance, linear programming can be applied in problem situations involving: Capital Budgeting Make-or-Buy Decisions Asset Allocation Portfolio Selection Financial Planning, and many more.
- 27. Financial Application Problems Portfolio Selection Financial Planning
- 28. 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.
- 29. Objective Function The objective function for portfolio selection problems usually is maximization of expected return or minimization of risk.
- 30. Constraints The constraints usually reflect restrictions on the type of Permissible Investments State Laws Company Policy Maximum Permissible Risk, and so on.
- 31. 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
- 32. Decision Variables A = dollars invested in Atlantic Oil P = dollars invested in Pacific Oil M = dollars invested in Midwest Steel H = dollars invested in Huber Steel G = dollars invested in government bonds
- 33. Objective Function Objective function for maximizing the total return for the portfolio is Max 0.073A 0.103P 0.064M 0.075H 0.045G
- 34. Linear Programming Model 1. A + P + M + H + G = 100,000 2. A + P ≤ 50,000 3. M + H ≤ 50,000 4. -0.25M - 0.25H + G ≥ 0 5. -0.6A + 0.4P ≤ 0 A, P, M, H, G ≥ 0
- 35. 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 $8000 Overall Rate of Return = 8%
- 36. Financial Planning Financial Planning is an ongoing process to help you make sensible decisions about money that can help you achieve your goals in life.
- 37. Hewlitt Corporation Year 1 2 3 4 5 6 7 8 Cash Requirements 430 210 222 231 240 195 225 255 The cash requirements (in thousands of dollars) are due at the beginning of each year.
- 38. Government Bonds Investments Bond Price ($) Rate (%) Years to Maturity 1 1150 8.875 5 2 1000 5.500 6 3 1350 11.750 7
- 39. Decision Variables F = total dollars required to meet the retirement plan’s eight- year obligation B1 = units of bond 1 purchased at the beginning of year 1 B2 = units of bond 2 purchased at the beginning of year 1 B3 = units of bond 3 purchased at the beginning of year 1 Si = amount placed in savings at the beginning of year i for I = 1, . . . , 8
- 40. Objective Function The objective function is to minimize the total dollars needed to meet the retirement plan’s eight-year obligation, or Min F
- 41. 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
- 42. Constraint of Each Year F - 1.15B1 - 1B2- 1.35B3 - S1 = 430 Year 1 0.08875B1+0.055B2+0.1175B3+1.04S1-S2 = 210 Year 2 0.08875B1+0.055B2+0.1175B3+1.04S2-S3 = 222 Year 3 0.08875B1+0.055B2+0.1175B3+1.04S3-S4 = 231 Year 4 0.08875B1+0.055B2+0.1175B3+1.04S4-S5 = 240 Year 5 1.08875B1+0.055B2+0.1175B3+1.04S5-S6 = 195 Year 6
- 43. 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
- 44. Operation Management Applications Managing and directing the physical and/or technical functions of a firm or organization, particularly those relating to Development Production Manufacturing
- 45. What we’ll cover Make-or-Buy Decision Production Scheduling Workforce Assignment Blending Problems
- 46. 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.
- 47. Production Scheduling Establish an efficient low-cost production schedule for one or more products over several time periods (weeks or months)
- 48. 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
- 49. Disadvantages Complexity Cost Can be Skewed Lack of Flexibility
- 50. Workforce Assignment Workforce assignment problems frequently occur when production managers must make decisions involving staffing requirements for a given planning period.
- 51. Blending Problems Blending problems arise whenever a manager must decide how to blend two or more resources to produce one or more products.