Article Contents
Article Contents

# Exact and heuristic methods for personalized display advertising in virtual reality platforms

• * Corresponding author: Kemal Kilic

Mathematics Subject Classification: Primary: 90C40; Secondary: 90C39.

 Citation:

• Figure 1.  The sample mean revenues of the six algorithms for varying number of replications in Experiment 111

Figure 2.  Computed expected revenues (ER) and the sample mean revenues (SMR) obtained for various L = 1/h values for the finite difference algorithm in Experiment #111

Figure 3.  Computed expected revenues determined at each iteration (that is, $n \mapsto U_n$) for the value iteration algorithms with different resolution parameter values in Experiment # 111

Figure 4.  Computed expected revenues determined by the value iteration algorithm after 40 iterations for different resolution parameter values in Experiment #111

Figure 5.  The computational time in days for the value iteration algorithm with iteration number = 40, for different resolution parameter values in Experiment 111

Figure 6.  The expected revenues determined by the value iteration algorithm with iteration number = 40, for different resolution parameter values in Experiment 111

Figure 7.  The sample mean revenues (SMRs) determined by the finite difference algorithm for different h-value in Experiment 111

Table 1.  Parameters for numerical experiments

 Problem Specific Parameters Algorithm Specific Parameters Problem Size $h$ value Initial States Iteration Number Transition Rates Resolution (i.e., Step Length in Time) $\beta$-probabilities Exposure Payment Matrix Min./Max. Display Constraint Min./Max. Payment Constraint

Table 2.  Experimental Conditions

 Experiment # Maximum Display Minimum Payment Maximum Payment 111 5 10 40 112 5 10 70 121 5 30 40 122 5 30 70 211 8 10 40 212 8 10 70 221 8 30 40 222 8 30 70

Table 3.  Revenue performance of the algorithms for different experimental conditions

 Heuristics Finite Difference Value Iteration Exp.# A B C Random SMR ER SMR ER 111 32.25 45.46 42.24 28.83 45.93 45.57 45.90 44.57 112 32.69 45.99 42.24 28.99 46.47 46.01 46.49 45.01 121 13.28 14.86 9.54 9.70 30.75 30.46 30.79 29.61 122 13.82 15.40 9.54 9.86 33.71 33.38 33.64 32.35 211 33.81 49.55 49.13 29.71 49.63 49.27 49.62 48.16 212 35.28 49.85 49.34 30.11 49.89 49.55 49.91 48.46 221 15.66 31.44 30.75 11.52 34.80 34.36 34.85 33.17 222 17.00 31.75 30.96 11.92 39.94 39.90 39.90 38.55

Figures(7)

Tables(3)