 Marketing Application  Media Selection  Financial Application  Portfolio Selection  Financial Planning  Product Management Application  Product Scheduling  Data Envelopment Analysis  Revenue Management

LP Modeling Application For a particular application we begin with the problem scenario and data, then: 1) Define the decision variables 2) 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

 Helps marketing manager to allocate the advertising budget to various advertising media  News Paper  TV  Internet  Magazine  Radio

 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

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

We can use the graph of an LP to see what happens when: 1. An OFC changes, or 2. A RHS changes Recall the Flair Furniture problem ADVERTISING MEDIA # OF CUSTOMER REACHED COST PER ADVERTISMEN T MAX TIME AVAIALBLE PER MONTH EXPOSURE QUALITY UNITS DAY TIME TV(1 MIN) EVENING TV (30 SEC) DAILY NEWS PAPER SUNDAY NEWS PAPER RADIO 8 AM TO 5 PM NEWS 30 SEC

 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 ????

 Max 65DTV + 90ETV + 40DN + 60SN + 20R (Exposure quality )  Constraints  Availability of Media  Budget Constraint  Television Restriction

 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

 OBJ FUNCTION Value: 2370 (Exposure Quality unit)  Decision variable  Potential customers  ???? MEDIAFREQUENCY DTV10 ETV0 DN25 SN2 RADIO30

 dtvetvdnsn rRHS dual  Maximize  Constraint <=150  Constraint <=100  Constraint <=2516  Constraint <=40  Constraint <=3014  Constraint <=  Constraint >=10-25  Constraint <=  Constraint >=  Solution-> $2,370.

 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  1500DTV +3000ETV +400DN +1000SN +100R <=30,000 exposure quality unit ????  Dual price for constraint 7 is -25  DTV +ETV >=10 ???

 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

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.

Vitamin C Protein Iron Grain Units/lb Units/lb Units/lb Cost/lb

 Define the constraints Total weight of the mix is 8-ounces (.5 pounds): (1) x 1 + x 2 + x 3 + x 4 =.5 Total amount of Vitamin C in the mix is at least 6 units: (2) 9x x 2 + 8x x 4 > 6 Total amount of protein in the mix is at least 5 units: (3) 12x x x 3 + 8x 4 > 5 Total amount of iron in the mix is at least 5 units: (4) 14x x 3 + 7x 4 > 5 Nonnegativity of variables: x j > 0 for all j

OBJECTIVE FUNCTION VALUE = VARIABLE VALUE REDUCED COSTS X X X X 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.

 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

 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 ???? InvestmentProjected Rate of Return % Atlantic oil7.3% Pacific oil10.3% Midwest steel6.4% Huber Steel7.5% Govt Bonds4.5%

 Max 0.073A P M H G  1.A+P+M+H+G=  2.A+P <=50,000, M+H <= 50,000  3. G>=0.25(M + H) or G -0.25M H>=0  4. P<=0.60(A+P) or -0.60A +0.40P<=0

 Objective Function=8000 VariableValueReduced Cost A P M H G ConstraintSlack/surplusDual price InvestmentAmountExpected Annual Return A$20,000$1460 P$30,000$3090 H40,000$3000 G$10,000$450 Total$100000$8000 Overall Return ????

 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 value of optimal solution will increase by if one more dollar is invested.  A negative value for constrain 4 is which mean optimal solution get worse by if one unit on RHS of constrain is increased. What does this mean

 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%/

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

 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

 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

Input MeasureGeneralUniversityCountyState FTE Supply Expense Bed-days available ANNUAL SERVICES PROVIDED BY FOUR HOSPITALS Output MeasureGeneralUniversityCountyState Medicare patient days Non-Medicare patient days Nurses Trained Interns trained

 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.

 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

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

 Constraint 2:  Output for Composite Hospital >=Output for County Hospital  Medicare: 48.14wg wu wc ws >=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

 Constraint 3  Input for composite Hospital <=Resource available to Composite Hospital  FTE:285.20wg wu wc ws  Sup:123.80wg wu wc ws  Bed-dys:106.72wg+64.21wu wc ws  We need a value for RHS:  %tage of input values for county Hospital.

 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 ….

 FTE:285.20wg wu wc+210ws<=275.70E  SUP:123.80wg wu wc ws<= E  Beddays:106.72wg+64.21wu wc ws<= E  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,.

 Min E  Wg+wu+wc+ws=1  48.14wg wu wc ws >=36.72  43.10wg+27.11wu+45.98wc+54.46ws>=45.98  253wg+148wu+175wc+160ws >=175  41wg+27wu+23wc+84ws >=23  wg wu wc ws E <=0  wg wu wc ws E <=0  wg+64.21wu wc ws E <=0

VariableValueReduced cost E WG WU WC WS ConstraintSlack/Sur plus Dual Price 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

 E=0.905  Efficiency score of County Hospital is  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.