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Chapter 4: Linear Programming Applications

Chapter 4: Linear Programming Applications. Marketing Application Media Selection Financial Application Portfolio Selection Financial Planning Product Management Application Product Scheduling Data Envelopment Analysis Revenue Management.

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Chapter 4: Linear Programming Applications

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  1. Chapter 4:Linear ProgrammingApplications

  2. Marketing Application • Media Selection • Financial Application • Portfolio Selection • Financial Planning • Product Management Application • Product Scheduling • Data Envelopment Analysis • Revenue Management Linear Programming (LP) Can Be Used for Many Managerial Decisions:

  3. LP Modeling Application For a particular application we begin with the problem scenario and data, then: • Define the decision variables • Formulate the LP model using the decision variables • Write the objective function equation • Write each of the constraint equations • Implement the Model using QM or MS

  4. Helps marketing manager to allocate the advertising budget to various advertising media • News Paper • TV • Internet • Magazine • Radio Media selection application

  5. A Construction Company wants to advertise his new project and hired an advertising company. • The advertising budget for first month campaign is $30,000 • Other Restrictions: • At least 10 television commercial must be used • At least 50,000 potential customer must be reached • No more than $18000 may be spent on TV advertisement • Need to recommend an advertising selection media plan Media selection

  6. PLAN DECISION CRETERIA EXPOSURE QUALITY It is a measure of the relative value of advertisement in each of media. It is measured in term of an exposure quality unit. Potential customers Reached Media Selection

  7. We can use the graph of an LP to see what happens when: • An OFC changes, or • A RHS changes Recall the Flair Furniture problem Media selection

  8. DTV : # of Day time TV is used ETV: # of times evening TV is used DN: # of times daily news paper used SN: # of time Sunday news paper is used R: # of time Radio is used Advertising plan with DTV =65 DTV Quality unit Advertising plan with ETV =90 DTV Quality unit Advertising plan with DN =40 DTV Quality unit Objective Function ???? Decision Variables

  9. Max 65DTV + 90ETV + 40DN + 60SN + 20R (Exposure quality ) • Constraints • Availability of Media • Budget Constraint • Television Restriction Objective function

  10. Availability of Media • DTV <=15 • ETV <=10 • DN<=25 • SN<=4 • R<=30 • Budget constraints • 1500DTV +3000ETV +400DN +1000SN +100R <=30,000 • Television Restriction • DTV +ETV >=10 • 1500DTV +3000ETV<=18000 • 1000DTV+2000ETV+1500DN +2500SN +300R >=50,000

  11. OBJ FUNCTION Value: 2370 (Exposure Quality unit) Decision variable Potential customers ???? OPTIMAL sOLUTION

  12. dtvetvdn sn r RHS dual Maximize 65 90 40 60 20 Constraint 1 1 0 0 0 0 <= 150 Constraint 2 0 1 0 0 0 <= 100 Constraint 3 0 0 1 0 0 <= 2516 Constraint 4 0 0 0 1 0 <= 40 Constraint 5 0 0 0 0 1 <= 3014 Constraint 6 1500 3000 400 1000 100 <= 300000.06 Constraint 7 1 1 0 0 0 >= 10-25 Constraint 8 1500 3000 0 0 0 <= 18000 0 Constraint 9 1000 2000 1500 2500 300 >= 50000 0 Solution-> 10 0 25 1.999999 30 $2,370.

  13. Dual Price for constraint 3 is 16 ???? • (DN >=25) exposure quality unit ???? • Dual price for constraint 5 is 14 • (R <=30) exposure quality unit ???? • Dual price for constraint 6 is 0.060 • 1500DTV +3000ETV +400DN +1000SN +100R <=30,000 exposure quality unit ???? • Dual price for constraint 7 is -25 • DTV +ETV >=10 ??? discussion

  14. Reducing the TV commercial by 1 will increase the quality unit by 25 this means The reducing the requirement having at least 10 TV commercial should be reduced

  15. Blending Problem Frederick's Feed Company receives four raw grains from which it blends its dry pet food. The pet food advertises that each 8-ounce can meets the minimum daily requirements for vitamin C, protein and iron. The cost of each raw grain as well as the vitamin C, protein, and iron units per pound of each grain are summarized on the next slide. Frederick's is interested in producing the 8-ounce mixture at minimum cost while meeting the minimum daily requirements of 6 units of vitamin C, 5 units of protein, and 5 units of iron.

  16. Blending Problem Vitamin C Protein Iron Grain Units/lb Units/lb Units/lb Cost/lb 1 9 12 0 .75 2 16 10 14 .90 3 8 10 15 .80 4 10 8 7 .70

  17. Blending Problem • Define the constraints Total weight of the mix is 8-ounces (.5 pounds): (1) x1 + x2 + x3 + x4 = .5 Total amount of Vitamin C in the mix is at least 6 units: (2) 9x1 + 16x2 + 8x3 + 10x4 > 6 Total amount of protein in the mix is at least 5 units: (3) 12x1 + 10x2 + 10x3 + 8x4 > 5 Total amount of iron in the mix is at least 5 units: (4) 14x2 + 15x3 + 7x4 > 5 Nonnegativity of variables: xj> 0 for all j

  18. Blending Problem OBJECTIVE FUNCTION VALUE = 0.406 VARIABLEVALUEREDUCED COSTS X1 0.099 0.000 X2 0.213 0.000 X3 0.088 0.000 X4 0.099 0.000 Thus, the optimal blend is about .10 lb. of grain 1, .21 lb. of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. The mixture costs Frederick’s 40.6 cents.

  19. Portfolio Selection 1.A company wants to invest $100,000 either in oil, steel or govt industry with following guidelines: 2.Neither industry (oil or steel ) should receive more than $50,000 3.Govt bonds should be at least 25% of the steel industry investment 4.The investment in pacific oil cannot be more than 60% of total oil industry. What portfolio recommendations investments and amount should be made for available $100,000 Financial application s

  20. Decision Variables A = $ invested in Atlantic Oil P= $ invested in Pacific Oil M= $ invested in Midwest Steel H = $ invested in Huber Steel G = $ invested in govt bonds Objective function ????

  21. Max 0.073A + 0.103P + 0.064M + 0.075H + 0.045G 1.A+P+M+H+G=100000 2.A+P <=50,000, M+H <= 50,000 3. G>=0.25(M + H) or G -0.25M -0.25 H>=0 4. P<=0.60(A+P) or -0.60A +0.40P<=0 Constraints & obj function

  22. Objective Function=8000 Solution Overall Return ????

  23. Dual price for constraint 3 is zero increase in steel industry maximum will not improve the optimal solution hence it is not binding constraint., Others are binding constraint as dual prices are zero For constrain 1 0.069 value of optimal solution will increase by 0.069 if one more dollar is invested. A negative value for constrain 4 is -0.024 which mean optimal solution get worse by 0.024 if one unit on RHS of constrain is increased. What does this mean Discussion

  24. If one more dollar is invested in govt bonds the total return will decrease by $0.024 Why??? Marginal Return by constraint 1 is 6.9% Average Return is 8% Rate of return on govt bond is 4.5%/ Discussion

  25. Associated reduced cost for M=0.011 tells Obj function coefficient of for midwest steel should be increase by 0.011 before considering it to be advisable alternative. With such increase 0.064 +0.011 =0.075 making this as desirable as Huber steel investment. Discussion

  26. It is an application of the linear programming model used to measure the relative efficiency of the operating units with same goal and objectives. • Fast Food Chain • Target inefficient outlets that should be targeted for further study • Relative efficiency of the Hospital, banks ,courts and so on Data envelopment Analysis

  27. General Hospital; University Hospital County Hospital; State Hospital Input Measure # of full time equivalent (FTE) nonphysician personnel Amount spent on supplies # of bed-days available Output Measures Patient-days of service under Medicare Patient-days of service notunder Medicare # of nurses trained # of interns trained Evualating Performance of Hospital

  28. Annual Resource consumed by 4 Hospital ANNUAL SERVICES PROVIDED BY FOUR HOSPITALS

  29. Construct a hypothetical composite Hospital Output & inputs of composite hospital is determined by computing the average weight of corresponding output & input of four hospitals. Constraint Requirement All output of the Composite hospital should be greater than or equal to outputs of County Hospital If composite output produce same or more output with relatively less input as compared to county hospital than composite hospital is more efficient and county hospital will be considered as inefficient. Relative Efficiency of County Hospital

  30. Wg= weight applied to inputs and output for general hospital Wu = weight applied to input & output for University Hospital Wc=weight applied to input & output for County Hospital Ws = weight applied to input and outputs for state hospital

  31. Constraint 1 Wg+ wu + wc + ws=1 Output of Composite Hospital Medicare: 48.14wg + 34.62wu + 36.72wc+ 33.16ws Non-Medicare:43.10wg+27.11wu+45.98wc+54.46ws Nurses:253wg+148wu+175wc+160ws Interns:41wg+27wu+23wc+84ws Output constraints

  32. Constraint 2: Output for Composite Hospital >=Output for County Hospital Medicare: 48.14wg + 34.62wu + 36.72wc+ 33.16ws >=36.72 Non-Medicare:43.10wg+27.11wu+45.98wc+54.46ws>=45.98 Nurses:253wg+148wu+175wc+160ws >=175 Interns:41wg+27wu+23wc+84ws >=23 Output constraints

  33. Constraint 3 Input for composite Hospital <=Resource available to Composite Hospital FTE:285.20wg+162.30wu+275.70wc+210.40ws Sup:123.80wg+128.70wu+348.50wc+154.10ws Bed-dys:106.72wg+64.21wu+104.10wc+104.04ws We need a value for RHS: %tage of input values for county Hospital.

  34. E= Fraction of County Hospital ‘s input available to composite hospital Resources to Composite Hospital= E*Resources to County Hospital If E=1 then ??? If E> 1 then Composite Hospital would acquire more resources than county If E <1 …. Input Constraints

  35. FTE:285.20wg+162.30wu+275.70wc+210ws<=275.70E SUP:123.80wg+128.70wu+348.50wc+154.10ws<=348.50E Beddays:106.72wg+64.21wu+104.10wc+104.04ws<=104.10E If E=1 composite hospital=county hospital there is no evidence county hospital is inefficient If E <1 composite hospital require less input to obtain output achieved by county hospital hence county hospital is more inefficient,. Input constraints

  36. Min E Wg+wu+wc+ws=1 48.14wg + 34.62wu + 36.72wc+ 33.16ws >=36.72 43.10wg+27.11wu+45.98wc+54.46ws>=45.98 253wg+148wu+175wc+160ws >=175 41wg+27wu+23wc+84ws >=23 285.20wg+162.30wu+275.70wc+210.40ws-275.70E <=0 123.80wg+128.70wu+348.50wc+154.10ws-348.50E <=0 106.72wg+64.21wu+104.10wc+104.04ws-104.10E <=0 Model

  37. Optimal Solution Composite Hospital as much of as each output as County Hospital (constrain 2-5) but provides 1.6 more trained nurses and 37 more interim. Contraint 6 and 7 are for input which means that Composite hospital used less than 90.5 of resources of FTE and supplies

  38. E=0.905 Efficiency score of County Hospital is 0.905 Composite hospital need 90.5% of resources to produce the same output of County Hospital hence it is efficient than county hospital. and county hospital is relatively inefficient Wg=0.212;Wu=0.26;Ws=0.527. Discussion

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