Scheduling advertisements on a web page to maximize revenue

Speaker : ScottDate : 17/6/2014

Subodha KumarVarghese S. JacobCheeliah Sriskandarajah

173 (2006) 1067–1089European 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 = {,…, } 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, , means the fraction of time for which is displayed. Access fraction, , means . Geometry is specified by which may represent the length of .

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, , is given by {(, , )|>0, >0, >0, }. It can be transformed as given by {(, )|>0,>0, }.

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 ,

• Three scenarios where can be transformed as . 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

subject to

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Heuristic algorithm SUBSET-LSLF

, , i=1, 2,…,n.

N : number of slots

S : size of each slot

If Sort ads in with the order of frequencySort ads in by size

If Sort ads in by sizeSort ads in with the order of frequency

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Heuristic algorithm Largest Size Most Fill (LSMF)

,

If FFD(C) ≤ NI=I+1CU(I)=CL(I-1)If I ≤ K, start from calculating C

ELSE I=I+1CU(I)=CU(I-1)CL(I)=C

SUBSET-LSMF

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Heuristic algorithm Largest Size Most Fill (LSMF)1　 SUBSET-LSMF();

2　 If C ≤ S　 End;

3　 Calculate the values of for all ads; Sort the ads by , ;⬆

4　 k=1; i=1; Schedule-={}; Discard(k) = ; 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-={}; Discard(k) = ; SUBSET-LSMF(); GOTO 5;

Algorithm LSMF

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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.

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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< 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 = , 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.

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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 ()

Probability of mutation ()

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

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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-LSLFgap

%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

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)

%Imp in Avg %gap of LSMFOverSUBSET-LSLF

%l\mp 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

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Computational studies Comparison of results for 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.

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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：

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Case study

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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.

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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.

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The End