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# Optimization Analysis Case Example

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### Optimization Analysis Case Example

1. 1. Case Pr esentation A study of OptimizationBy Bernie Karlowicz & Matt Osterstrom
2. 2. Case OverviewUnited Broadcast Network (UBN) • A commercial broadcasting network • Sells 15 & 30 second advertising slots on its T.V. shows • Bayside, Newsline, The Hour, Cops and Lawyers, The Judge, Friday Night Football, and ER Doctor • Develops detailed sales plans for customers, typically on monthly, bimonthly, 6-week, or quarterly basis. • Bases sales plans on performance scores for T.V. shows • Sets advertising rates based on performance scoresNanocom • Business software development firm • Purchaser of advertising slots from UBN in October and November, a 6 week segment • Advertising budget of \$600,000 • Wants to reach an older, more mature, upper-income audience that is likely to include many high-tech businesspeople.“The primary objective of both UBN and the advertiser is to develop a sales plan that will achieve the highest total performance score.”
3. 3. Case Overview (Cont…)Performance Scores explained • Based on several factors: 1. How well the show matches the desired audience demographics 2. Strength of the T.V. show 3. Historical ratings for the time slot 4. Competing shows in the same time slot 5. Performance of adjacent showsWeighting Factor •Performance scores and rates are multiplied by a weighting factor •The weighting factor is based on: • The week of the show • Total audience expected during that week. October November Week 3 4 1 2 3 4 Weight 1.1 1.2 1.2 1.4 1.4 1.6 Minimum 1 1 2 2 2 3 Maximum 4 5 5 5 5 5
4. 4. Case Overview (Cont…)T.V. shows • 7 shows, each with a 30 second and 15 second commercial slot • Base costs of commercial slots are variable per show • Base performance score of commercial slots are variable per show • Each Commercial option has a predefined inventory level Show Commercial Length Cost (\$1000s) Performance Score Available Inventory (seconds)Bayside 30 \$50.00 115.20 3  15 \$25.00 72.00 3The Hour 30 \$41.00 160.00 4  15 \$20.50 100.00 1Newsline 30 \$36.00 57.60 3  15 \$18.00 36.00 3Cops and Lawyers 30 \$45.00 136.00 4  15 \$27.50 85.00 2The Judge 30 \$52.00 100.80 2  15 \$26.00 63.00 2Friday Night Football 30 \$25.00 60.80 4  15 \$12.50 38.00 2ER Doctor 30 \$46.00 129.60 3  15 \$23.00 81.00 1
5. 5. Case Overview (Cont…)Sales plan timing • Nanocom requests a 6-week sales plan • Sales plan should cover the period beginning with the third week in October through November • This timing will include the November sweepsWith The Sales Plan involving a 6 week duration, for seven shows, withtwo different commercial lengths, and variable availability, it is inferred that decisions have to be made over time => Perfect for Optimization.
6. 6. Optimization – Overview Purpose: Attempt to develop the optimal solution to a problem, given a set of restrictions. Three Components to Optimization:Decision Variables Constraints • Restrictions to consider a) What decisions? • Restrictions placed on the b) What needs to be decided? decision Objective Function - The goal • Either Maximize or Minimize • A single linear function => Huh? • The sum of constants times decision variables ∑ (C * Xi) Question: What is Linear Programming?
7. 7. Linear Programming – Overview“Linear programming (LP) is a mathematical method for determining a way to achievethe best outcome (such as maximum profit or lowest cost) in a givenmathematical model for some list of requirements represented as linear equations.” –Wikipedia, 2010 Examples of linear equations: - Slope of a line: y = mx + b - Einstein’s Mass Equivalence: e = mc^2 - Pythagorean Theorem: a^2 + b^2 = c^2
8. 8. Case – Optimization AnalysisProvided data table, with assignment of Decision Variables: Show Commercial Cost (\$1000s) Performance Available Decision Length Score Inventory Variable (seconds) Bayside 30 \$50.00 115.20 3 X1   15 \$25.00 72.00 3 X2 The Hour 30 \$41.00 160.00 4 X3   15 \$20.50 100.00 1 X4 Newsline 30 \$36.00 57.60 3 X5   15 \$18.00 36.00 3 X6 Cops and  30 \$45.00 136.00 4 Lawyers X7   15 \$27.50 85.00 2 X8 The Judge 30 \$52.00 100.80 2 X9   15 \$26.00 63.00 2 X10 Friday Night  30 \$25.00 60.80 4 Football X11   15 \$12.50 38.00 2 X12 ER Doctor 30 \$46.00 129.60 3 X13   15 \$23.00 81.00 1 X14
9. 9. Case – Optimization Analysis (Cont…) Examining the Objective Function From the case: “The primary objective of both UBN and the advertiser is to develop a sales plan that will achieve the highest total performance score.” 115.20 * X1 +So we declare: 72.00 * X2 + 160.00 * X3 +P = Maximize Total Performance Score 100.00 * X4 + 57.60 * X5 +Si = Performance Scores 36.00 * X6 +Xi = Decision Variables (# of Advertizing Slots) OR… 136.00 * X7 + 85.00 * X8 +Giving: 100.80 * X9 + 63.00 * X10 + Pi = ∑ ( Si * Xi ) 60.80 * X11 + 38.00 * X12 + 129.60 * X13 + 81.00 * X14 P(Total)
10. 10. Case – Optimization Analysis (Cont…) Examining the Constraints First we note inventory constraints:Decision Show Comm AvailableVariable Length Inventory (secs) X1 Bayside 30 <= 3 X2   15 <= 3 X3 The Hour 30 <= 4 X4   15 <= 1 X5 Newsline 30 <= 3 X6   15 <= 3 X7 Cops and Lawyers 30 <= 4 X8   15 <= 2 X9 The Judge 30 <= 2 X10   15 <= 2 Friday Night  30 <= 4 X11 Football X12   15 <= 2 X13 ER Doctor 30 <= 3 X14   15 <= 1
11. 11. Case – Optimization Analysis (Cont…)Additional Constraints… Then we note the total budget constraint: Total Budget = \$600,000
12. 12. Case – Optimization Analysis (Cont…)Additional Constraints…We also note that “50% of the total number of advertisingslots it purchases to be on Newsline, The Hour, and FridayNight Football.”
13. 13. Case – Optimization Analysis (Cont…)Additional Constraints… “there should be a minimum and maximum number of advertising slots during each of the 6 weeks”
14. 14. Case – Optimization Analysis (Cont…)Additional Constraints… And finally:“Nanocom should have at most only one ad slot (either 15 or 30 seconds) per show per week”
15. 15. Case – Results Required detail tables for calculating either the Maximum Performance Score, or the Minimum Total Cost to consider the Weight Factor for a given show, in a given week. Performance Score per Week Cost per Week October November October November Week 3 Week 4 Week 1 Week 2 Week 3 Week 4 Week 3 Week 4 Week 1 Week 2 Week 3 Week 4Weight = 1.1 1.2 1.2 1.4 1.4 1.6 Weight = 1.1 1.2 1.2 1.4 1.4 1.6 X1 126.72 138.24 138.24 161.28 161.28 184.32 X1 \$55.00 \$60.00 \$60.00 \$70.00 \$70.00 \$80.00 X2 79.2 86.4 86.4 100.8 100.8 115.2 X2 \$27.50 \$30.00 \$30.00 \$35.00 \$35.00 \$40.00 X3 176 192 192 224 224 256 X3 \$45.10 \$49.20 \$49.20 \$57.40 \$57.40 \$65.60 X4 110 120 120 140 140 160 X4 \$22.55 \$24.60 \$24.60 \$28.70 \$28.70 \$32.80 X5 63.36 69.12 69.12 80.64 80.64 92.16 X5 \$39.60 \$43.20 \$43.20 \$50.40 \$50.40 \$57.60 X6 39.6 43.2 43.2 50.4 50.4 57.6 X6 \$19.80 \$21.60 \$21.60 \$25.20 \$25.20 \$28.80 X7 149.6 163.2 163.2 190.4 190.4 217.6 X7 \$49.50 \$54.00 \$54.00 \$63.00 \$63.00 \$72.00 X8 93.5 102 102 119 119 136 X8 \$30.25 \$33.00 \$33.00 \$38.50 \$38.50 \$44.00 X9 110.88 120.96 120.96 141.12 141.12 161.28 X9 \$57.20 \$62.40 \$62.40 \$72.80 \$72.80 \$83.20 X10 69.3 75.6 75.6 88.2 88.2 100.8 X10 \$28.60 \$31.20 \$31.20 \$36.40 \$36.40 \$41.60 X11 66.88 72.96 72.96 85.12 85.12 97.28 X11 \$27.50 \$30.00 \$30.00 \$35.00 \$35.00 \$40.00 X12 41.8 45.6 45.6 53.2 53.2 60.8 X12 \$13.75 \$15.00 \$15.00 \$17.50 \$17.50 \$20.00 X13 142.56 155.52 155.52 181.44 181.44 207.36 X13 \$50.60 \$55.20 \$55.20 \$64.40 \$64.40 \$73.60 X14 89.1 97.2 97.2 113.4 113.4 129.6 X14 \$25.30 \$27.60 \$27.60 \$32.20 \$32.20 \$36.80
16. 16. Case – Results (Cont…)Decision Show Comm Cost Performanc Available Inventory Actual Total Total CostVariable Length (\$1000s) e Score Inventory Inventory (secs) Actual Bayside 30 \$50.00 115.20 3 X1 0 0 0 0 0 0 0 \$0.00 15 \$25.00 72.00 3 X2 0 0 0 0 0 0 0 \$0.00 The Hour 30 \$41.00 160.00 4 X3 0 7.006E-11 1 1 1 1 4 \$229,600.00 15 \$20.50 100.00 1 X4 0 0 0 0 -2.703E-39 1 1 \$32,800.00 Newsline 30 \$36.00 57.60 3 X5 0 0 0 0 0 0 0 \$0.00 15 \$18.00 36.00 3 X6 0 0 0 0 0 0 0 \$0.00 Cops and 30 \$45.00 136.00 4 Lawyers X7 1 1 1 0.5068323 5.2848E-12 0 3.5068323 \$189,430.43 15 \$27.50 85.00 2 X8 0 0 0 0 1 1 2 \$82,500.00 The Judge 30 \$52.00 100.80 2 X9 0 0 0 0 0 0 0 \$0.00 15 \$26.00 63.00 2 X10 0 0 0 0 0 0 0 \$0.00 Friday Night 30 \$25.00 60.80 4 Football X11 0 0 0 0 0 0 0 \$0.00 15 \$12.50 38.00 2 X12 1.6184E-15 0 0 0.5068323 0 1 1.5068323 \$28,869.57 ER Doctor 30 \$46.00 129.60 3 X13 0 0 0 0 0 0 0 \$0.00 15 \$23.00 81.00 1 X14 0 0 0 0 0 1 1 \$36,800.00 TOTALS 315 447.5 1235 37 1 1 2 2.0136646 2 5 13.0136646 \$600,000.00 Cost (\$) \$49,500.00 \$54,000.00 \$103,200.00 \$98,200.00 \$95,900.00 \$199,200.00
17. 17. Case – Results (Cont…)We derive the Maximum Performance Score by calculating theSumproduct, with input arrays of: 1. All decision variables, for all time periods 2. All weighted performance scores from detail table previously mentioned.