Optimization Analysis Case Example

Case PresentationA study of Optimization

By Bernie Karlowicz & Matt Osterstrom

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 scores

Nanocom• 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.”

Case Overview (Cont…)Performance Scores explained

• Based on several factors:1. How well the show matches the desired audience demographics2. Strength of the T.V. show3. Historical ratings for the time slot4. Competing shows in the same time slot5. Performance of adjacent shows

Weighting 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

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 (seconds)

Cost ($1000s) Performance Score Available Inventory

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

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 sweeps

With The Sales Plan involving a 6 week duration, for seven shows, with two different commercial lengths, and variable availability, it is inferred that decisions have to be made over time => Perfect for Optimization.

Optimization – Overview

Purpose: Attempt to develop the optimal solution to a problem, given a set of restrictions.

Decision Variables

a) What decisions?

b) What needs to be decided?

Question: What is Linear Programming?

Constraintsa) Restrictions to considerb) Restrictions placed on the

decision

Three Components to Optimization:

Objective Function - The goal

a) Either Maximize or Minimize

b) A single linear function => Huh?

c) The sum of constants times decision variables ∑ (C * Xi)

Linear Programming – Overview

“Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical 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

Case – Optimization Analysis

Provided data table, with assignment of Decision Variables:

Show Commercial Length

(seconds)

Cost ($1000s) Performance Score

Available Inventory

Decision Variable

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 Lawyers

30 $45.00 136.00 4X7

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 Football

30 $25.00 60.80 4X11

15 $12.50 38.00 2 X12

ER Doctor 30 $46.00 129.60 3 X13

15 $23.00 81.00 1 X14

Case – Optimization Analysis (Cont…)

Examining the Objective FunctionFrom 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 +

72.00 * X2 +

160.00 * X3 +

100.00 * X4 +

57.60 * X5 +

36.00 * X6 +

136.00 * X7 +

85.00 * X8 +

100.80 * X9 +

63.00 * X10 +

60.80 * X11 +

38.00 * X12 +

129.60 * X13 +

81.00 * X14

P(Total)

So we declare:

P = Maximize Total Performance Score

Si = Performance Scores

Xi = Decision Variables (# of Advertizing Slots)

Giving:

Pi = ∑ ( Si * Xi )

OR…


Examining the Constraints

First we note inventory constraints:

Decision Variable

Show Comm Length (secs)

Available Inventory

X1 Bayside 30 <= 3X2 15 <= 3X3 The Hour 30 <= 4X4 15 <= 1X5 Newsline 30 <= 3X6 15 <= 3X7 Cops and Lawyers 30 <= 4X8 15 <= 2X9 The Judge 30 <= 2X10 15 <= 2

X11


30 <= 4

X12 15 <= 2X13 ER Doctor 30 <= 3X14 15 <= 1


Additional Constraints…

Then we note the total budget constraint:Total Budget = $600,000



We also note that “50% of the total number of advertising slots it purchases to be on Newsline, The Hour, and Friday Night Football.”



“there should be a minimum and maximum number of advertising slots during each of the 6 weeks”



And finally: “Nanocom should have at most only one ad slot (either 15 or 30 seconds)

per show per week”

Case – Results

Performance Score per WeekOctober November

Week 3 Week 4 Week 1 Week 2 Week 3 Week 4Weight = 1.1 1.2 1.2 1.4 1.4 1.6

X1 126.72 138.24 138.24 161.28 161.28 184.32X2 79.2 86.4 86.4 100.8 100.8 115.2X3 176 192 192 224 224 256X4 110 120 120 140 140 160X5 63.36 69.12 69.12 80.64 80.64 92.16X6 39.6 43.2 43.2 50.4 50.4 57.6X7 149.6 163.2 163.2 190.4 190.4 217.6X8 93.5 102 102 119 119 136X9 110.88 120.96 120.96 141.12 141.12 161.28X10 69.3 75.6 75.6 88.2 88.2 100.8X11 66.88 72.96 72.96 85.12 85.12 97.28X12 41.8 45.6 45.6 53.2 53.2 60.8X13 142.56 155.52 155.52 181.44 181.44 207.36X14 89.1 97.2 97.2 113.4 113.4 129.6

Cost per WeekOctober November

Week 3 Week 4 Week 1 Week 2 Week 3 Week 4Weight = 1.1 1.2 1.2 1.4 1.4 1.6

X1 $55.00 $60.00 $60.00 $70.00 $70.00 $80.00X2 $27.50 $30.00 $30.00 $35.00 $35.00 $40.00X3 $45.10 $49.20 $49.20 $57.40 $57.40 $65.60X4 $22.55 $24.60 $24.60 $28.70 $28.70 $32.80X5 $39.60 $43.20 $43.20 $50.40 $50.40 $57.60X6 $19.80 $21.60 $21.60 $25.20 $25.20 $28.80X7 $49.50 $54.00 $54.00 $63.00 $63.00 $72.00X8 $30.25 $33.00 $33.00 $38.50 $38.50 $44.00X9 $57.20 $62.40 $62.40 $72.80 $72.80 $83.20X10 $28.60 $31.20 $31.20 $36.40 $36.40 $41.60X11 $27.50 $30.00 $30.00 $35.00 $35.00 $40.00X12 $13.75 $15.00 $15.00 $17.50 $17.50 $20.00X13 $50.60 $55.20 $55.20 $64.40 $64.40 $73.60X14 $25.30 $27.60 $27.60 $32.20 $32.20 $36.80

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.

Case – Results (Cont…)Decision Variable

Show Comm Length (secs)

Cost ($1000s)

Performance Score

Available Inventory

Inventory Actual Total Inventory

Actual

Total Cost

X1

Bayside 30 $50.00 115.20 3

0 0 0 0 0 0 0 $0.00

X2

15 $25.00 72.00 3

0 0 0 0 0 0 0 $0.00

X3

The Hour 30 $41.00 160.00 4

0 7.006E-11 1 1 1 1 4 $229,600.00

X4 15 $20.50 100.00 1

0 0 0 0 -2.703E-39 1 1 $32,800.00

X5

Newsline 30 $36.00 57.60 3

0 0 0 0 0 0 0 $0.00

X6

15 $18.00 36.00 3

0 0 0 0 0 0 0 $0.00

X7

Cops and Lawyers

30 $45.00 136.00 4

1 1 1 0.5068323 5.2848E-12 0 3.5068323 $189,430.43

X8

15 $27.50 85.00 2

0 0 0 0 1 1 2 $82,500.00

X9

The Judge 30 $52.00 100.80 2

0 0 0 0 0 0 0 $0.00

X10

15 $26.00 63.00 2

0 0 0 0 0 0 0 $0.00

X11


30 $25.00 60.80 4

0 0 0 0 0 0 0 $0.00

X12

15 $12.50 38.00 2

1.6184E-15 0 0 0.5068323 0 1 1.5068323 $28,869.57

X13

ER Doctor 30 $46.00 129.60 3

0 0 0 0 0 0 0 $0.00

X14

15 $23.00 81.00 1

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 $600,000.00Cost ($)

$49,500.00 $54,000.00 $103,200.00 $98,200.00 $95,900.00 $199,200.00

Case – Results (Cont…)

We derive the Maximum Performance Score by calculating the Sumproduct, with input arrays of:

1. All decision variables, for all time periods2. All weighted performance scores from detail table

previously mentioned.

Optimization Analysis Case Example

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