Policy Performance

In this section, we compare expected prots of the policies on the basis of expected dierence from optimal, relative policy ranking, expected improvement over rst-come-rst-serve alloca- tion, and counts of optimality or near-optimality for each policy.

Table 4.1 provides ve summary statistics for the percentage dierence from optimal ex- pected prots for each of the nine policies across the 198 scenarios. The STOP policy performs very well but is the most dicult to compute. The median and mean percent errors are 0% and 0.09%, respectively, showing that the rst local maximum of the conditional prot function tends to perform well compared to the global maximum. The MR policy also performs well with a median percent error of 0.04% and mean of 0.29%. The MR policy is much easier to compute than the STOP policy; however, MR may still require nding multiple values ofb1(x2).

Given their computational simplicity, the LB and UB policies also perform well on average, but worst case scenarios are much worse than those of STOP and MR policies (19.60% and 26.87%, as opposed to 2.34% and 2.55% respectively). Across all scenarios, the average and standard deviation for the percent dierence from optimal is lower for UB than is for LB (1.58% vs. 2.24% and 2.33% vs. 3.49%, respectively). One possible explanation for why UB outperforms LB on average can be seen in Figure 4.4, where the prot curve plateaus after reaching its maximum for some scenarios. Therefore, a booking limit well above optimal may perform better than a booking limit slightly below optimal. For example, whenλ1 =λ2 = 18,α2 = 0.75, and

Table 4.1: Summary Statistics of Percent Dierence from Optimal by policy

% STOP MR LB UB ROA SC LW FCFS POA

Mean 0.09 0.29 2.24 1.58 3.86 3.95 4.38 13.63 23.00 Std. Dev. 0.37 0.51 3.49 2.33 4.11 4.90 5.55 10.80 24.57

Minimum 0 0 0 0.000 * 0 0.005 0 0.005 0

Maximum 2.34 2.55 19.60 26.87 16.43 26.87 26.39 38.59 74.21 Median 0 0.04 0.55 2.01 2.76 2.01 2.34 9.61 16.96 * UB never exactly matches optimal policy. Actual minimum dierence = 0.0000000227%

h= 200, the optimal policy sets the Class 2 booking limit at 16, whereas the LB policy chooses 9 and the UB policy chooses 32. Though the LB limit is only 7 lower than optimal and the UB limit is 16 higher than optimal, the UB policy has a higher expected prot (2246.75 vs. 2204.80). Table 4.4 reveals that while UB outperforms LB on average, it is not optimal for any scenario, though expected prot is close in 34 scenarios, while LB is optimal in 51 scenarios. Also, the worst case performance for the UB policy is worse than that for the LB policy, as seen in Table 4.1.

The performances of SC and LW provide some insight into the value of overbooking and the value of considering multiple classes in patient booking. The best policy which does not overbook, LW, has expected prots 4.38% less than optimal on average across all scenarios. Similarly, SC, a policy which assumes a single no-show rate, has expected prots 3.95% less than optimal on average across all scenarios. Using MR, a relatively simple joint overbooking and capacity control policy, expected prots average only 0.44% less than optimal. Thus, using a joint overbooking and capacity control policy means that expected prots are 3.5-4% closer to optimal as opposed to either one alone. Table 4.2 displays how policies perform compared to a FCFS allocation policy. This table provides further insight into the value gained from using either overbooking or capacity control alone and the additional value from integrating the two practices. By incorporating optimal capacity control without overbooking (LW), expected

Table 4.2: Summary Statistics of Percent Increase over FCFS by policy

% OPT STOP MR LB UB ROA SC LW

Mean 17.823 17.705 17.444 14.894 15.804 13.383 12.619 12.292 Std. Dev. 16.603 16.472 16.251 14.351 15.330 17.474 12.479 14.686 Minimum 0.005 0.005 0.005 0.005 0.002 -15.569 0 0.005 Maximum 62.832 62.832 62.832 62.832 57.312 62.832 56.662 62.832

Median 10.632 10.629 10.147 9.422 9.295 8.506 8.247 6.513

prots increase 12.3% above FCFS on average across all scenarios. By incorporating overbooking without capacity control, assuming a single weighted no-show rate, expected prots increase 12.6% above FCFS on average across all scenarios. All joint capacity control and overcooking policies outperform LW and SC with respect to increased expected prots over FCFS by 1.1-5.4% and 0.9% to 5.1%, respectively.

Summary statistics on the rank ordering of policies are compiled in Table 4.3. An interesting result is that neither SC nor FCFS is ever the best policy. In fact, the best rankings of these two policies are third and sixth, respectively. Another interesting result is that although the STOP policy is never more than 2.34% from optimal, its worst ranking is sixth. STOP tends to have a low ranking when the probability of Class 2 demand is signicantly high, since the optimal solution is likely to be a local optimum other than the rst. From further investigation, we de- termine that MR achieves its worst ranking of fourth in three scenarios where it is outperformed by STOP, UB, and SC respectively. These scenarios are characterized by moderately high mean Class 1 demand, low mean Class 2 demand, and high no-show rates. In these scenarios, the MR policy tends to underestimate the optimal Class 2 booking limit as it does not appropriately capture that the marginal expected overtime cost may actually be decreasing and a higher Class 2 booking limit may be preferable.

Table 4.3: Summary Statistics of Policy Rankings (1 highest expected prot, 9 lowest)

STOP MR LB UB ROA SC LW FCFS POA

Mean 1.268 1.970 3.495 3.980 5.404 5.783 4.490 8.136 6.975 Std. Dev. 1.101 0.860 1.930 2.038 2.198 1.736 2.584 0.859 2.867

Best Rank 1 1 1 1 1 3 1 6 1

Worst Rank 6 4 6 7 8 8 7 9 9

Median Rank 1 2 4 3 6 5 6 8 9

Table 4.4: Count of Scenarios (out of 198)

STOP MR LB UB ROA SC LW FCFS POA

Optimal 148 60 51 0 30 0 51 0 30

0.01% less than OPT 163 87 56 34 30 2 55 1 30

Best policy 185 73 64 3 30 0 64 0 30

number of scenarios each policy resulted in an expected prot less than 0.01% below optimal. The MR policy generalizes the LW policy and therefore dominates LW in performance. The simple approximations perform well, with LB less than 0.01% from optimal in 56 scenarios and UB less than 0.01% from optimal in 34 scenarios. Further investigation of results not captured in Table 4.3, reveals that both LB and UB are less than 0.01% from optimal in only two scenarios; meaning that in 88 distinct scenarios either LB or UB (or both) perform extremely well with expected prots less than only 0.01% from optimal. To provide further detail, for each scenario we let the Closer Bound Prot (CBP) denote the maximum expected prot achieved by either LB or UB. On average across all scenarios, we nd that CBP is only 0.30% dierent from optimal with a median of 0.03%. Thus, using LB and UB with discretion can be an eective policy. Of the two policies, LB (UB) should be implemented for high (low) values of λ1 and low (high)

values of p/h. While performance improves for both policies as the Class 2 attendance rate