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1. 1. 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
2. 2. 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
3. 3. 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
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. 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
8. 8. 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
9. 9. 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
10. 10. 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
11. 11. 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
12. 12. 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
13. 13. 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
14. 14. 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
15. 15. 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 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
16. 16. 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. 15
17. 17. 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
18. 18. Case study 17
19. 19. 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
20. 20. 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
21. 21. The End