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3 Figure 2(b). Note that the choice of which slot to display in a time interval can be made by cycling through a deterministic permutation of N slots. Adler et al. (2001) provide a heuristic algorithm (called SUBSET-LSLF) for the MAXSPACE problem. Kumar et al. (2001) developed a hybrid genetic algorithm (GA) integrating SUBSET-LSLF with GA. RESEARCH METHODOLOGY BEING USED Since the MAXSPACE problem is NP-Hard (Dawande et al. 2001), it is unlikely that the problem can be solved by an efficient optimal algorithm (Garey and Johnson 1979) and we need to develop heuristic algorithms. We select the ads from the set of available ads and schedule them in such a way that the space utilization is maximized. The schedule provides details of the identity of ads that appear together and the time in which interval they appear. Figure 2. An Example to Illustrate MAXSPACE Problem Integer Programming Formulation max where ; We used the CPLEX Linear Optimizer to solve the problem. CPLEX fails to find the optimal solution for large size problems with N > 25 as the memory space is exceeded. However, CPLEX provides an upper bound for all the problems. The performances of the heuristic methods are evaluated by comparing with these upper bounds. LSMF We develop an algorithm, called LSMF, in which we first determine the maximum slot fullness. If the maximum slot fullness obtained is less than or equal to the available space for each slot (S), then we have a feasible solution, otherwise we discard some of the ads Twenty-Second International Conference on Information Systems 463

5 Step 11. If k <((ps/2) - 0.5), go to Step 8. Step 12. If i = 0, set the overall best sequence = current best sequence and go to step 14. Step 13. If the overall best sequence is better than the current best sequence, replace the current worst sequence with the overall best sequence; else, set the overall best sequence = current best sequence. Step 14. Set i = i + 1. If i = n gen, terminate; else, go to Step 5. CURRENT STATUS OF THE PROJECT For computational studies, we use 150 randomly generated problems. Problems are generated in such a way that i A s i w i = N * S for any test problem and all the ads fit into the given N slots. Here, A is the set of ads generated for the test problem. Therefore, the optimal values are known and for any test problem it is equal to N * S. These problems are generated for 15 different combinations of N and S. The size s i of ad A i in any test problem is generated randomly between S/3 and 2S/3, where S is the size of each slot for that problem. It is found that the problems generated with these limits on the value of s i are more difficult for the proposed heuristic to solve than randomly generated problems without any limits on the value of s i. A statistical experimental framework is used to discover the most appropriate values of the GA parameters. The experimental layout is full-factorial design (Montgomery 1996). This experimental layout used three levels (possible values) each of ε, p c and p m. For small size problems, two levels of ps are used and for larger size problems, three levels of ps are used. Comparison of Results In order to evaluate their relative performance, SUBSET-LSLF, LSMF, and the GA are coded in C and executed on a SunOS 5.6 system. The solutions produced by these algorithms are compared with the optimal solution. We find that CPLEX fails to find the optimal solution for problems with N > 25 as the memory space is exceeded. CPLEX gives upper bounds for all of the problems. In order to evaluate the performance of algorithms for the test problems with unknown optimal values, we will use the upper bounds obtained by CPLEX. First the pure heuristics without any hybridization are compared. Table 1 shows the results. Each row in this table is the result for a different combination of N and S and there are 15 rows. For all the test problems, the values of N and S are shown in the table. Percentage SUBSET-LSLF gap in the table is the percentage deviation of SUBSET-LSLF from the optimal solution. It is calculated as follows: (Optimal Solution) (SUBSET LSLF) Percentage SUBSET LSLF Gap = 100% Optimal Solution We have generated 10 problems for each combination of N and S. So, Max, Avg and Min columns show the maximum, average and minimum values of percentage gaps respectively, out of these 10 problems in the set. Similarly, GA percentage gap is the percentage deviation of GA from the optimal solution and LSMF gap is the percentage deviation of LSMF from the optimal solution. We see that the average percentage GA gap is 0% for small size problems (N = 10), which means that the GA achieves the optimal solution for all these problems. For medium and large size problems, this average percentage gap is in the range of 0.2% to 2.3%. Table 1 shows that the average percentage LSMF gap is very small for all the problems (in the range of 0% to 1.75%). Percentage improvement in average percentage gap of LSMF over SUBSET-LSLF is calculated as follows: (Avg %SUBSET LSLF Gap) (Avg % LSMF Percentage Improvement in Avg %Gap = 100% (Avg %SUBSET LSLF Gap) We find that the LSMF outperforms SUBSET-LSLF for all the test problems and the improvement is in the range of 50% to 100%. Percentage improvement in average percentage gap of LSMF over GA is calculated similarly. We find that the LSMF outperforms GA for all the medium and large size test problems (N = 50, 75, and 100) and improvement is in the range of 67% to 100%. For some of the small size problems, GA performs better than LSMF Twenty-Second International Conference on Information Systems 465

6 Table 1. Comparison of Test Problems Results The last three columns indicate the average CPU times taken by SUBSET-LSLF, GA and LSMF respectively. Each row gives the average of CPU times taken by 10 problems in that problem set. We see that CPU times taken by SUBSET-LSLF and LSMF are much lower than that of GA. We can say here that SUBSET-LSLF and LSMF are computationally efficient because they are specially designed heuristics, which carefully and cleverly exploit the structure of these problems. To improve the results further, we develop hybrid GA-LSLF and hybrid GA-LSMF by combining GA with SUBSET-LSLF and LSMF respectively. The results are shown in Table 2. Columns in Table 2 are similar to the columns in Table 1. We find that the average percentage gap for GA-LSLF is very small (in the range of 0% to 0.78%). The average percentage gap is 0% for all the test problems in the case of GA-LSMF and thus the improvement over SUBSET-LSLF and GA-LSLF is 100%. This indicates that the GA-LSMF provides optimal solutions for all 150 test problems. It has to be noted that the algorithm takes advantage of both GA search process and problem specific information by LSMF to provide the optimal solution. Note that the average CPU times taken by GA-LSMF are much lower than the other GAs, especially for large size test problems. In this case the GA starts with a very good solution obtained by LSMF and converges very fast. FUTURE RESEARCH AND PROPOSAL FOR CONFERENCE PRESENTATION We are currently working on testing these algorithms on a more diverse set of test problems for which the optimal values are not known. Research is also being conducted to improve the LSMF algorithm and its hybridization with GA. We have collected some real-world data and are analyzing the performance of these algorithms on this data. Using this data, we will illustrate how even a small improvement in the schedule can significantly impact the revenue generated from the ads Twenty-Second International Conference on Information Systems

7 Table 2. Comparison of Test Problems Results with Hybrid Gas References Adler, M., Gibbons, P. B., and Matias, Y. Scheduling Space-Sharing for Internet Advertising, Journal of Scheduling, 2001 (forthcoming). Coffman, Jr., E. G., Garey, M. R., and Johnson, D. S. An Application of Bin-Packing to Multiprocessor Scheduling, SIAM Journal of Computing, (7:1), February 1978, pp Dawande, M., Kumar, S., and Sriskandarajah, C. Performance Bounds of Algorithms for Scheduling Advertisements on a Web Page, Working Paper. The University of Texas at Dallas, Garey, M. R., and Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, Kumar, S., Jacob, V. S., and Sriskandarajah, C. Scheduling Advertisements on a Web Page to Maximize Space Utilization, Working Paper, The University of Texas at Dallas, McCandless, M. Web Advertising, IEEE Intelligent Systems, May/June, 1998, pp Montgomery, D. C. Design and Analysis of Experiments, 4 th edition, John Wiley & Sons, New York, Novak, T. P., and Hoffman, D. L. New Metrics for New Media: Toward the Development of Web Measurement Standards, World Wide Web Journal (W3J) (3:1), Winter Rewick, J. Choices, Choices: A Look at the Pros and Cons of Various Types of Web Advertising, The Wall Street Journal, April 23, 2001, p. R Twenty-Second International Conference on Information Systems 467

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