Evaluation of the OBP&SPSA Procedure

The revenue generated by the OBP&SPSA approach over all instances is 5.38% higher than the rev-enue calculated by the option-based procedure, 10.88% higher than the revrev-enue achieved by the FCFS approach, and 10.46% lower than the revenue generated in the ex post optimal solution.

The result of the option-based procedure is the initial solution of the OBP&SPSA algorithm. From this it follows that the solution of the option-based procedure is a lower bound for the OBP&SPSA result. An upper bound for the OBP&SPSA result is the ex post optimal solution.

The run-time of the OBP&SPSA procedure depends on the assumed demand and the defined capacity.

To achieve the result for one instance, the OBP&SPSA algorithm needs about two minutes in low de-mand settings considering the seat capacity to be 100 and up to four minutes in instances with a higher seat capacity and higher total demand.

The percentage gap (gap3) between the solution of the OBP&SPSA and the OBP approach is computed by:

gap3= OBP&SPSA−OBP

OBP ∗100.

The percentage gap (gap4) between the solution of the OBP&SPSA and the FCFS approach is computed by:

gap4= OBP&SPSA−FCFS

FCFS ∗100.

The percentage gap (gap5) between the ex post optimal solution and the solution of the OBP&SPSA is computed by:

gap5= ^{ex post}−OBP&SPSA OBP&SPSA ∗100.

Table 5.1 shows the achieved results aggregated over all instances in one capacity instance. The OBP&SPSA procedure determines better results in all three capacity settings than both the option-based procedure and the FCFS approach. The gap between the OBP&SPSA results and the ex post optimal solutions, however, can still be reduced. It can be noticed that gap3 decreases as the capacity scales up although the performance of the OBP&SPSA procedure improves as the capacity increases compared to the FCFS solutions. As the capacity grows, the results of the option-based method increase faster than the results of the OBP&SPSA procedure, which explains the shrinking of gap3.

Table 5.2 presents the revenue per seat in the capacity instances. Studying the revenue per seat cal-culated by the OBP&SPSA approach, we identify the same outcome as we identified while analyzing the results of the option-based procedure: The OBP&SPSA approach performs better in instances with higher capacity since the solution space expands when considering a higher seat capacity.

Capacity gap3 gap4 gap5 100 6.04 8.20 12.60 120 5.31 10.74 10.40 150 4.81 13.59 8.47

Table 5.1: OBP&SPSA – Results Aggregated over Demand, Revenue, and Price Instances

Capacity gap3/C gap4/C gap5/C

100 0.06 0.08 0.13

120 0.04 0.09 0.09

150 0.03 0.09 0.06

Table 5.2: OBP&SPSA – Results Aggregated over Demand, Revenue, and Price Instances Per Seat

5.3 Computational Study for Option-Based Procedure with SPSA We fixed the demand and aggregated the computed results over all revenue and price instances in a first observation in order to evaluate the estimated revenue values according to the demand variations.

Table 5.3 shows the aggregated results for the assumed capacity of 100, 120, and 150. The OBP&SPSA method improves the results of the option-based procedure in all demand settings and performs better than the FCFS approach in the instances with the demand being 120%, 130%, and 140% of the capac-ity. In low demand settings, the same effect as in the analysis of the option-based procedure appears:

The revenue generated by the FCFS method is higher than the revenue achieved by the OBP&SPSA ap-proach, although the gap between the OBP&SPSA method and the FCFS procedure decreased compared to the gap between the option-based method and the FCFS approach. The results of the OBP&SPSA procedure increase as the total demand scales up. Hence, in these cases (identical to the analysis of the option-based approach) the solution space increases and it is advantageous to reserve seat capacity for higher yielding classes through booking limits.

Capacity Demand in % gap3 gap4 gap5

100 110 3.49 -6.31 16.29

120 5.21 1.95 13.98

130 7.13 12.13 11.36

140 7.94 22.78 9.33

120 110 3.29 -5.10 13.80

120 4.85 6.12 10.91

130 5.87 15.78 9.10

140 7.23 26.17 7.81

150 110 2.93 -1.78 10.96

120 4.24 7.66 8.99

130 5.35 19.17 7.34

140 6.71 29.30 6.58

Table 5.3: OBP&SPSA – Results Aggregated over Revenue and Price Instances

To show the performance of the OBP&SPSA procedure in a low demand setting, we calculated the expected revenue for 13 demand instances assuming the total demand to be 90% of the capacity.

Capacity Demand in % gap3 gap4 gap5

100 90 5.17 -12.12 19.40

120 90 5.00 -11.01 17.27

150 90 4.64 -9.85 14.02

Table 5.4: OBP&SPSA – Results Aggregated over Revenue and Price Instances in Instances with Demand Intensity 0.9

Table 5.4 shows the results for the seat capacity being 100, 120, and 150. The average revenue of the OBP&SPSA approach over all demand, revenue, and price instances is much lower than the revenue generated by the FCFS approach and much lower than the ex post optimal solution. Although, the OBP&SPSA approach performs better than the option-based method in all revenue and price instances in which the demand is lower than the capacity, the results of the OBP&SPSA procedure are lower than the results of the FCFS approach in all these settings. The same effect could already be seen when comparing the results of the option-based procedure to the FCFS approach. Mentionable is the fact that the performance of the OBP&SPSA procedure is particularly bad in price instance d in which the sum of option price plus strike price is higher than the revenue achieved by selling a flight ticket in

the lower yielding booking classes of the operating carrier and the ticketing carrier (x+s > v21 and x+s > v22). In price instance d, the ticketing carrier only operates requests for the higher yielding booking class and the operating carrier rejects all requests for the second booking class as soon as the remaining capacity is equal to the number of options the ticketing carrier owns. If this point is reached in the booking process, the only booking classes in which requests are accepted are the operating carrier’s and ticketing carrier’s highest booking yielding class. And since the demand is low, especially for the higher yielding booking classes, most of the flight tickets for seats in the aircraft remain unsold which causes the poor performance of the OBP&SPSA procedure in price instance d.

In a second survey, we fixed the revenue and aggregated the computed results over all price and demand instances in order to evaluate the effect of revenue variation among the tested instances.

Table 5.5 contains the results for capacity 100, 120 and 150. The revenue gained from the OBP&SPSA approach is higher than the results gained from the option-based approach and the FCFS procedure in all capacity settings and revenue instances. The performance of the OBP&SPSA approach in the revenue instances 1, 2, and 3 is much better compared to the FCFS method. In the fourth revenue instance, the difference between the revenue gained by selling a flight ticket for the carriers’ higher yielding booking class and the revenue that the carriers achieve by selling a flight ticket for the lower yielding booking class is lower than in the other revenue settings. The conclusion is that the OBP&SPSA procedure is also most applicable if the protection of seats for higher yielding booking classes from the access of lower yielding booking classes achieves more revenue since the revenue for one sold flight ticket in the higher yielding booking class is remarkably higher than the revenue for one sold ticket in the lower yielding booking class.

Capacity Revenue Instance gap3 gap4 gap5

100 1 5.67 10.80 11.54

2 1.60 11.49 10.84

3 8.51 7.78 15.16

4 7.98 0.48 13.43

120 1 5.03 14.10 9.30

2 1.32 14.13 8.89

3 7.97 11.77 11.96

4 6.92 2.96 11.47

150 1 4.47 16.89 7.58

2 1.33 17.37 6.99

3 7.38 14.84 9.63

4 6.05 5.25 9.67

Table 5.5: OBP&SPSA – Results Aggregated over Demand and Price Instances

We fixed the option price and the strike price and aggregated the computed results over all revenue and demand instances in a third survey in order to evaluate the effect of the price variation among the tested instances.

In Table 5.6, the results for capacity 100, 120, and 150 are presented. The OBP&SPSA procedure accom-plished higher revenue results than the option-based approach in all price instances and higher revenue results than the FCFS procedure in the price instances a, b, and c. Similar to the performance of the option-based procedure, the OBP&SPSA approach obtains poor results especially in the price instance d although gap4 being not as small as in the option-based procedure analysis since the OBP&SPSA ap-proach enhances the results calculated by the option-based procedure.

5.3 Computational Study for Option-Based Procedure with SPSA

Table 5.6: OBP&SPSA – Results Aggregated over Demand and Revenue Instances

Procedure 1, which assigns the spare seats the ticketing carrier does not buy options for (if they exist) to a class of the operating carrier (described in Section 4.2.2), was also considered in the instances solved by the OBP&SPSA procedure. However, the result of the OBP&SPSA approach considering Procedure 1, averaged over all instances with seat capacity 120, is equal to the revenue gained from using Procedure 2, which solves the model of the operating carrier twice, not considering the additional class of the tick-eting carrier in calculating the booking limits for the operating carrier. The simultaneous perturbation stochastic approximation approach compensates in its 300 iterations lasting run the marginal differences in the initial values being the booking limits calculated by the two different procedures mentioned in Section 4.2.2.

We compared the performance of the OBP&SPSA procedure using standard nesting with the perfor-mance applying a theft nesting control for the seat capacity being 120. Exercising the two nesting con-trols results in two similar revenue outcomes aggregated over all instances (standard nesting performs 0.03% better than theft nesting). However, no declaration can be made about the comparative perfor-mance of the two nesting controls in general.

The performance of the simulation-based optimization procedure with buy-back opportunity of the operating carrier (OBP&SPSA+BB) is compared to the performance of the same procedure without back option (OBP&SPSA-BB). Compare Section 4.2.1 for the description of the operating carrier’s buy-back opportunity. The revenue generated by the OBP&SPSA+BB approach, averaged over all instances considering the capacity to be 120, is 2.04% higher than the revenue calculated by the OBP&SPSA-BB.

Similar to the performance of the option-based method with buy-back opportunity, the OBP&SPSA+BB procedure gains exactly the same revenue as the OBP&SPSA-BB approach in the revenue and price set-ting 4d. Only in 13 of the 832 instances studied, OBP&SPSA-BB performs barely better than OBP&SPSA+BB. This phenomenon occurs most likely in instances of revenue and price scenario 3a, combined with instances which describe a very low demand for the highest yielding booking class and a very high demand for the lowest yielding booking class of the operating carrier. The demand for the ticketing carrier’s booking classes is high in these instances. In revenue and price scenario 3a, the return for one sold flight ticket in one of the ticketing carrier’s booking classes is much higher than in one of the operating carrier’s booking classes and the option price plus the strike price is higher than the revenues gained by selling a flight ticket in the operating carrier’s lower yielding booking class. The high demand for the operating carrier’s second booking class causes that the state in the booking process is reached very fast in which the remaining seat capacity will be equal to the number of options the ticketing carrier holds. Henceforth, the operating carrier buys back options from the ticketing carrier if a request for the

first booking class of the operating carrier occurs. Since the total demand for the first booking class of the operating carrier is very small and since the ticketing carrier cannot fill the capacity the operating carrier took from the ticketing by buying back the options, a part of the capacity remains unsold. In this case, it is possible that the operating carrier buys back the options although the ticketing carrier could most likely sell the tickets and for a much higher price, which explains the inferior performance.

Finally, we evaluate the performance of the OBP&SPSA approach applied with the initial booking limit values coming from the EMSR-a+Options and EMSR-b+Options calculations introduced in Section 4.2.3 (which are referred to as OBP&SPSA+EMSR-a+Options method and OBP&SPSA+EMSR-b+Options procedure). We averaged the expected revenue of all instances with seat capacity 120 and gained an expected revenue from the OBP&SPSA+EMSR-b+Options procedure that is slightly higher than the revenue achieved by the OBP&SPSA+EMSR-a+Options approach. This outcome corresponds to the in-sight we gained comparing the procedures EMSR-a+Options and EMSR-b+Options not considering the simultaneous perturbation stochastic approximation. The results of the OBP&SPSA+EMSR-a+Options approach is 5.86% lower than the revenue calculated by the OBP&SPSA method with underlying DLP.

Considering the OBP&SPSA+EMSR-b+Options approach, we receive an expected revenue that is 4.29%

lower than the revenue achieved by the DLP underlying OBP&SPSA method observing all instances of the 120 seat capacity scenario.

Chapter 6