Scheduling advertisements on a web page to maximize revenue

Scheduling advertisements on a
web page to maximize revenue
Speaker : Scott
Date : 17/6/2014
Subodha Kumar
Varghese S. Jacob
Cheeliah Sriskandarajah
173 (2006) 1067–1089
European Journal of Operational Research

Introduction
• The amount of users on the Internet is becoming stupendous.
• Advertisement revenue
 2003→$7.3 billion
 2002→$6 billion
 2006→$15.4 billion (prediction)
• Banner advertisements, major form
 The most common type, rectangular
• Limited space spawns the issue of maximizing revenue
• Three factors which will be considered
(1)time (2) number (3)size
• The problem belongs to a NP-Hard problem.
1

Problem description
• A set of n ads A = {𝐴1,…, 𝐴 𝑛} competing for space in a given planning horizon.
• Time fraction, access fraction, and ad geometry determines the expected number of
the impression of an ad.
 Time fraction, 𝑡𝑖, means the fraction of time for which 𝐴𝑖 is displayed.
 Access fraction, 𝑎𝑖, means
number of visitors who see 𝐴 𝑖
Total number of visitors
.
 Geometry is specified by 𝑙𝑖 which may represent the length of 𝐴𝑖.
 The width, W, of all ads is assumed to be the same.
• The length and the width of a rectangular slot are denoted as S and W, respectively.
• An instance, 𝐼1, is given by {(𝑎𝑖, 𝑡𝑖, 𝑙𝑖)|𝑎𝑖>0, 𝑡𝑖>0, 𝑙𝑖>0, 𝐴𝑖 ∈ 𝐴}.
 It can be transformed as 𝐼2 given by {(𝑠𝑖, 𝑤𝑖)|𝑠𝑖>0,𝑤𝑖>0, 𝐴𝑖 ∈ 𝐴}.
 𝑤 means frequency instead of W signified as the width of a slot previously.
• N represents the number of slots each having the size S.
2

Problem description
• The fullness of any slot j is 𝑓𝑗 = 𝐴 𝑗∈𝐵 𝑗
𝑠𝑖, 𝐵𝑗 ⊆ 𝐴
 max
𝑗
𝑓𝑗 ≤ 𝑆
• Three scenarios where 𝐼1 can be transformed as 𝐼2.
 Most accesses have very short duration.
 Most accesses have long duration
 Each ad has the same geometry and only one as is displayed at a time.
• A MAXSPACE problem
3

Related literature review
• Yager (1997), a general framework for the competitive selection
• Dreze and Zufryden (1997), intern.com Corp. (1998), Kohda and Endo (1996),
Marx (1996), Risdel et al. (1998), the issue of increasing of the effectiveness of
web ads.
• Aggarwel et al. (1998), the optimization of advertisements on webservers
• Adler et al. (2002), SUBSET-LSLF
4

Heuristic algorithm Integer programming formulation
max 𝑍 =
𝑗=1
𝑁
𝑖=1
𝑛
𝑠𝑖 𝑥𝑖𝑗
subject to
𝑖=1
𝑛
𝑠𝑖 𝑥𝑖𝑗 ≤ 𝑆 , 𝑗 = 1, 2, … , 𝑁
𝑗=1
𝑁
𝑥𝑖𝑗 = 𝑤𝑖 𝑦𝑖 , 𝑖 = 1, 2, … , 𝑁
𝑥𝑖𝑗 =
1 if ad 𝐴𝑖 is assigned to slot 𝑗.
0 otherwise
𝑦𝑖 =
1 if ad 𝐴𝑖 is selected.
0 otherwise
5

Heuristic algorithm SUBSET-LSLF
𝑠𝑖, 𝑤𝑖, i=1, 2,…,n.
N : number of slots
S : size of each slot
𝑠 = {𝐴𝑖|𝑠𝑖 = 𝑆}
𝑠 = {𝐴𝑖|𝑠𝑖 < 𝑆}
𝐵𝑠 =
𝐴 𝑖∈𝑠
𝑠𝑖 𝑤𝑖
𝐵 𝑠 =
𝐴 𝑖∈𝑠
𝑠𝑖 𝑤𝑖
If 𝐵𝑠 ≥ 𝐵 𝑠
Sort ads in 𝑠 with the order of frequency
Sort ads in 𝑠 by size
If 𝐵𝑠 < 𝐵 𝑠
Sort ads in 𝑠 by size
Sort ads in 𝑠 with the order of frequency
6

Heuristic algorithm Largest Size Most Fill (LSMF)
𝐶𝐿 = max
1
𝑁
𝑖=1
𝑛
𝑆𝑖 , max
1≤𝑖≤𝑛
𝑆𝑖
𝐶 𝑈 = 2𝐶𝐿
𝐶 =
𝐶𝐿(𝐼) + 𝐶𝑈(𝐼)
2
I = 1, 𝐾 = 10
If FFD(C) ≤ N
I=I+1
CU(I)=CL(I-1)
If I ≤ K, start from calculating C
ELSE
I=I+1
CU(I)=CU(I-1)
CL(I)=C
SUBSET-LSMF
7

Heuristic algorithm Largest Size Most Fill (LSMF)
1 SUBSET-LSMF();
2 If C ≤ S End;
3 Calculate the values of 𝐵𝑖 for all ads; Sort the ads by 𝐵𝑖, ⬆;
4 k=1; i=1; Schedule-={𝐵++𝑖(𝑠++𝑖)}; Discard(k) = 𝑠𝑖; SUBSET-LSMF();
5 If C ≤ S
6 If k = 1 End;
7 Else
8 Schedule+={Discard(k-1)}; SUBSET-LSMF();
9 If C > S Schedule-={Discard(k-1)}; Else k-=1;
10 Else
11 k+=1; Schedule-={𝐵++𝑖(𝑠++𝑖)}; Discard(k) = 𝑠𝑖; SUBSET-LSMF(); GOTO 5;
Algorithm LSMF
8

Heuristic algorithm A genetic algorithm
• For MAXSPACE problems, GA views sequences of ads as chromosomes.
• A simple GA is usually composed of three operations.
 Selection
 Crossover
 Mutation
• A design of experiments (DOE) approach was devised.
9

Heuristic algorithm A genetic algorithm
1 Assign ads to any slots with the principle
2 Calculate fitness value for each sequence; Sort all the sequences with descending order
3 Select ε for reproduction
4 k=0; Select 2 parents and cross them over; k+=1;
5 Mutate the children
6 Estimate the fitness of the children
7 If k<
𝑝𝑠
2
− 0.5 GOTO Line 4;
8 If i = 0 the overall best sequence = the current best sequence; GOTO Line 10;
9 If the overall best sequence < the current best sequence
10 i+=1; If i = 𝑛 𝑔𝑒𝑛, END; Else GOTO Line 2;
10

Heuristic algorithm Hybrid GA
• The whole processes are very much the same as the GA algorithm.
• The evaluation of fitness value are calculated three times with GA, LSMF, and
SUBSET-LSLF per sequence.
11

Computational studies
• 190 randomly generated problems with limitation
• Four algorithms were programmed in C.
• Parameterization of appropriate parameters for the GA algorithm
Set # No. of slots (N) Elite fraction (ε) Population size (ps)
Probability of
crossover (𝒑 𝒄)
Probability of
mutation (𝒑 𝒎)
1 10 0.25 75 0.95 0.10
2 25 0.25 75 0.75 0.05
3 50 0.25 75 0.60 0.05
4 75 0.25 200 0.75 0.01
5 100 0.25 200 0.75 0.01
12

Computational studies Comparison of results for
the small size problems
• 40 problems
Comparison of results for the small size problems with known optimal values
Prob.
Set #
No. of
slots (N)
Size of
each
slot (S)
%SUBSET-LSLF
gap
%LSMF gap %GA gap %Hybrid GA gap
Max Avg Min Max Avg Min Max Avg Min Max Avg Min
1 5 5 13.04 1.72 0 24 7 0 0 0 0 0 0 0
2 5 10 15.79 6.39 0 28 13 0 0 0 0 0 0 0
3 10 10 16.00 3.40 0 8 3.1 0 0 0 0 0 0 0
4 10 15 14.09 3.99 0 11.3 5.0 1.3 3.4 0.81 0 0 0 0
13

the small size problems
• 40 problems
Comparison of results for the small size problems with known optimal values
Prob.
Set #
No. of
slots (N)
Size of
each
slot (S)
%Imp in Avg %
gap of LSMF
Over
SUBSET-LSLF
%lmp in Avg %
gap of GA over
SUBSET-LSLF
%Imp in Avg %
gap of hybrid
GA over
SUBSET-LSLF
1 5 5 -306.9 100 100
2 5 10 -103.4 100 100
3 10 10 8.82 100 100
4 10 15 -25.3 79.7 100
14

the large size problems
• 150 problems
• The results generated from the three algorithms are compared to the upper bounds
calculated from CPLEX.
• For most of the test problems, GA and LSMF both provide improvements over
SUBSET-LSMF.
15

Case study
• The dataset was obtained by observing the ads on ValuePay’s pIggy Adbar.
• Ads on an Adbar will be updated periodically due to the characteristic of the function.
 Change every 20 seconds
 The planning horizon is 180
 Two banners, one is 468X60, the other is 120X60
• Assuming unit size = 12
• 33 different ads were displayed during the hour.
• For reaching the situation more closed to the practicality, 15 ads had been generated
randomly and added to the existing list.
 Four sets were generated.
• The price of an ad was determined by the CPM model.
• Total revenue： 𝐴 𝑖∈𝐴′ 𝑠𝑖 𝑤𝑖 1000
16

Conclusions and future research directions
• Growing business on the Internet
• The optimal utilization of space
• Efficient heuristics was designed.
• The LSMF was proposed and the hybrid GA was designed.
• The hybrid GA provided the optimal solutions for all the test problems.
• Revenue may increase within different situations
• Discussing the study with other emerging pricing models can be considered.
• Comparing different pricing models can be considered.
18

Comment
• Some similar symbols meaning different things bewilders people.
• In section 6, the authors said a phenomenon that usually there will be much more
ads competing for space by merely stating rather than providing some more
concrete evidence which can support the authors’ view.
• Many websites mentioned in the paper has changed their way of showing
webpages or even has been a wasteland, such as ValuePay’s Piggy.
19

Scheduling advertisements on a web page to maximize revenue

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