2019 Session 3 class notes (2).pptx - Lecture 3 MGMT 651...

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1 Lecture 3 MGMT 651 Linear Programming Applications Chapter 4
2 Possible Outcomes of a LP (Section 2.6) · A LP is either · Infeasible – there exists no solution which satisfies all constraints and optimizes the objective function · or, Unbounded – increase/decrease objective function as much as you like without violating any constraint · or, Has an Optimal Solution · Optimal values of decision variables · Optimal objective function value
3 Infeasible LP – An Example · minimize 4x 11 +7x 12 +7x 13 +x 14 +12x 21 +3x 22 +8x 23 +8x 24 +8x 31 +10x 32 +16x 33 +5x 34 Subject to · x 11 +x 12 +x 13 +x 14 =100 · x 21 +x 22 +x 23 +x 24 =200 · x 31 +x 32 +x 33 +x 34 =150 · x 11 +x 21 +x 31 =80 · x 12 +x 22 +x 32 =90 · x 13 +x 23 +x 33 =120 · x 14 +x 24 +x 34 =170 · x ij >=0, i=1,2,3; j=1,2,3,4 Total demand exceeds total supply Hence infeasible
4 Unbounded LP – An Example maximize 2x 1 + x 2 subject to - x 1 + x 2 1 x 1 - 2x 2 2 x 1 , x 2 0 x 2 can be increased indefinitely without violating any constraint => Objective function value can be increased indefinitely without violating any of the constraints
5 Multiple Optima – An Example Consider the following Linear Program maximize x 1 + 0.5 x 2 subject to 2x 1 + x 2 4 x 1 + 2x 2 3 x 1 , x 2 0 The optimal solution is: x 1 = 2, x 2 =0, objective function = 2 Is this solution also optimal: x 1 = 5/3, x 2 =2/3, objective function = 2? And what about x 1 = 0, x 2 =4, objective function = 2?
6 Marketing Application: Media Selection · Advertising budget for first month = $30000 · At least 10 TV commercials must be used · At least 50000 customers must be reached · Spend no more than $18000 on TV adverts · Determine optimal media selection plan Advertising Media # of potential customers reached Cost ($) per advertisement Max times available per month Exposure Quality Units Day TV 1000 1500 15 65 Evening TV 2000 3000 10 90 Daily newspaper 1500 400 25 40 Sunday newspaper 2500 1000 4 60 Radio 300 100 30 20
7 Media Selection Formulation · Step 1: Define decision variables · DTV = # of day time TV adverts · ETV = # of evening TV adverts · DN = # of daily newspaper adverts · SN = # of Sunday newspaper adverts · R = # of radio adverts · Step 2: Write the objective in terms of the decision variables · Maximize 65DTV+90ETV+40DN+60SN+20R · Step 3: Write the constraints in terms of the decision variables DTV <= 15 ETV <= 10 DN <= 25 SN <= 4 R <= 30 1500DTV + 3000ETV + 400DN + 1000SN + 100R <= 30000 DTV + ETV >= 10 1500DTV + 3000ETV <= 18000 1000DTV + 2000ETV + 1500DN + 2500SN + 300R >= 50000 Budget Customers reached TV Constraint s Availability of Media DTV, ETV, DN, SN, R >= 0
8 Optimal Solution Media Selection Problem using POM-QM
9 Optimal Solution – Media Selection Problem Using LINDO Exposure = 2370 units Variable Value DTV 10 ETV 0 DN 25 SN 2 R 30 Managerial Implications:

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