Paul Parker Donald McKinnon Production Scheduling Spring 13 Agenda Problem Overview Our Solution Analysis Conclusion Extensions 2 Problem Overview 1 Description Schedule shows of equal duration such that the sum of the viewers over all periods is greatest ID: 178696
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Slide1
Broadcast Programming
Paul Parker
Donald McKinnon
Production Scheduling Spring ‘13Slide2
Agenda
Problem Overview
Our Solution Analysis Conclusion Extensions
2Slide3
Problem Overview (1)
Description
: Schedule shows of equal duration such that the sum of the viewers over all periods is greatest.
Given: The number of viewers a shows garners is dependent on
its time period and neighbors.
Example
: Schedule 48 different 30 minute shows in a 24 hour period. Maximize the sum of viewers over all periods, where number of viewers of the show in period 35 depends on the 35th period and the shows in periods 34 and 36.
3Slide4
Problem Overview (2)
Upper Bound
:
The sum of the maximum number of viewers any show can have over each time slot (typically infeasible to schedule).
NP Hard:
Broadcast
Programming reduces KnapsackAssumptionsAll jobs (shows) have the same duration.
Unscheduled jobs
gain no
viewers.Jobs can only occur in 1 period in a schedule (no reruns).
4Slide5
Our Solution: Ratio Based S
wapping
Objective
: Given n shows,
m
periods, and
function(ni-1,n
i
,n
i+1
,m
j
) = vi,j, viewers for show ni in period mj, max Σj (vi,j), where i=1…n, and j=1…m.Algorithm:Randomly schedule shows until all m time slots have a distinct show, or all shows are scheduled.Determine the show S with the smallest ratio of its viewers in its current period, P, to its average viewers in all periods of the current schedule: vS,P ÷ AVG(all vS,j) where j=1…m. If S can be swapped with another period such that the sum of the viewers over all periods increases, then swap S such that the greatest increase in viewers results and go to Step 2. Otherwise, STOP and output the objective value of the current schedule.
5Slide6
Analysis – Function 1
Method
Max Objective Value
Average Objective
Value
Average Number
of Swaps
Our
Heuristic
(5 trials)
8677832215Random Scheduling (1,000,000 trials)72974816-*Simulations done in MATLAB or JavaVi,j = 100 * [1 + sine(Si,j-1 * Si,j * Si,j+1)], where i=1…n, and j=1…mSlide7
Analysis – Function
2
Method
Max Objective Value
Average Objective
Value
Average Number
of Swaps
Our
Heuristic
(10,000 trials)10757293Random Scheduling (10,000 trials)978665-*Simulations done in MATLAB or JavaVi,j = Si,j-1 * Si,j * Si,j+1, where i=1…n, and j=1…mSlide8
Analysis – Function
3
Method
Max Objective Value
Average Objective
Value
Average Number
of Swaps
Our
Heuristic
(1,000 trials)338718393Random Scheduling (10,000 trials)30901557-*Simulations done in MATLAB or JavaVi,j = j * (Si,j-1 * Si,j * Si,j+1), where i=1…n, and j=1…mSlide9
Conclusion
Our Heuristic:
Consistency Average objective value is good (above average)
Efficiency
Good objective values result after few swaps Scalability
Yields high objective values given different viewership functionsSlide10
Algorithmic Extensions Considered
D
erive initial schedule as follows
Schedule the show and its neighbors that will garner the most viewers in the remaining time slot until there are no empty time slots remaining.Expanded search after finding local maximum
When a maximum objective value is achieved, swap the two jobs in the time slots having the least viewers and resolve this instance, disallowing the algorithm to re-swap these two jobs immediately.
Advertisements
Model advertisements for shows as shorter jobs and schedule them (not necessarily adjacent to the show they promote) to maximize viewership.
10Slide11
Heuristic for Initial S
chedule
C
onfigurationResultsInconclusivePossible Implication:
11
Objective Value
Initial
Final
Heuristic < Random
Heuristic >
Random
*Simulations done in MATLAB or JavaSlide12
Thank You