1. Case Pr esentation
A study of Optimization
By Bernie Karlowicz & Matt Osterstrom
2. Case Overview
United 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.”
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 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
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 3
The Hour 30 $41.00 160.00 4
15 $20.50 100.00 1
Newsline 30 $36.00 57.60 3
15 $18.00 36.00 3
Cops and Lawyers 30 $45.00 136.00 4
15 $27.50 85.00 2
The Judge 30 $52.00 100.80 2
15 $26.00 63.00 2
Friday Night Football 30 $25.00 60.80 4
15 $12.50 38.00 2
ER Doctor 30 $46.00 129.60 3
15 $23.00 81.00 1
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 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.
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. 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
8. Case – Optimization Analysis
Provided 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. 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. Case – Optimization Analysis (Cont…)
Examining the Constraints
First we note inventory constraints:
Decision Show Comm Available
Variable 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. Case – Optimization Analysis (Cont…)
Additional Constraints…
Then we note the total budget constraint:
Total Budget = $600,000
12. Case – Optimization Analysis (Cont…)
Additional Constraints…
We also note that “50% of the total number of advertising
slots it purchases to be on Newsline, The Hour, and Friday
Night Football.”
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. 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”
17. Case – Results (Cont…)
We derive the Maximum Performance Score by calculating the
Sumproduct, with input arrays of:
1. All decision variables, for all time periods
2. All weighted performance scores from detail table
previously mentioned.
