Policy Sensitivity Analysis

In this subsection, we analyze the sensitivity of the expected prot of the policies to changes in parameters. We are interested in the consistency of our revenue-based model results with those of authors who have used cost-based models, in order to establish generality across dierent clinic objectives and situations.

Tables 4.7 provides sensitivity analysis of how the expected prot changes with respect to the Class 2 attendance rate as measured by the average percent dierent from optimal across scenarios. All of the policies except for ROA and POA show an improvement as the attendance rate increases. This agrees with intuition that if advanced scheduling is allowed, high no-show rates imply higher risk for overtime costs. The lower bound policies, such as LB and LW, show a greater relative improvement than the upper bound policies, such as UB and SC, with respect to the Class 2 attendance rate. LB prots are less than 0.3% from optimal on average when α2 = 0.9. Consistent with results found by other authors (Qu et al. 2007, LaGanga and

Lawrence 2007, Robinson and Chen 2010), the performance of POA and the performance of ROA improve as the no-show rate increases. Across dierent values of mean demand and cost parameters, POA performs much worse than simple alternatives even when the attendance rate

Table 4.7: Avg. Percent Dierence from Optimal by Class 2 Attendance Rate

% STOP MR LB UB ROA SC LW FCFS POA

α2 = 0.6 0.219 0.579 4.471 1.913 2.777 6.090 7.244 23.120 21.563

α2 = 0.75 0.045 0.233 1.949 1.738 3.721 4.057 4.551 13.483 22.984

α2 = 0.9 0.002 0.054 0.285 1.096 5.089 1.713 1.332 4.272 24.461

is quite low. Even FCFS strongly outperforms POA and slightly outperforms ROA, when the attendance rate is high.

Table 4.8summarizes how the mean percent dierence from optimal expected prot changes with respect to changes in mean Class 1 demand. As earlier results indicate, lower bound policies (LB, LW) show improved performance as mean Class 1 demand increases, while upper bound policies (UB, SC) show decreased performance. This result is illustrated in 4.5 where the optimal Class 2 booking limit is close to the upper bound when mean Class 1 demand is low and close to the lower bound when it is high. The MR results are interesting in that performance worsens as mean Class 1 demand increases; however when mean Class 1 demand is very high, the policy performs extremely well with an average of 0% dierence from optimal. POA shows dramatic improvement in performance as mean Class 1 demand increases. Alternatively, ROA results are mixed as the average dierence rst increases, then decreases. This is likely due to the fact that at rst demand is small enough that the chance of booking up to capacity is very small. As demand increases, the risk of booking some Class 2 patients and booking up to capacity in Class 1 patients increases. Once the probability of booking up to capacity in Class 1 patients is high enough, the ROA policy becomes a pure open access policy and does not allow any advance bookings. These results support a nding from Qu et al. (2007) that nearly all appointments will be held open if Class 1 demand is signicantly higher than capacity. Other authors, such as Liu et al. (2010) and Green and Savin (2008) nd that high demand leads to poor performance in open access because they explicitly model the impact of denying patients

Table 4.8: Avg. Percent Dierence from Optimal by Mean Class 1 Demand

% STOP MR LB UB ROA SC LW FCFS POA

λ1 = 6 0.000 0.012 5.242 0.117 2.381 0.451 10.386 12.704 70.511

λ1 = 12 0.002 0.187 3.990 0.328 5.617 1.557 8.241 13.166 44.116

λ1 = 18 0.005 0.380 2.256 0.696 7.893 2.816 4.829 12.328 19.760

λ1 = 24 0.092 0.733 1.241 1.688 3.159 4.877 1.598 13.604 3.921

λ1 = 30 0.292 0.000 0.000 4.246 0.021 8.185 0.000 15.862 0.021

Table 4.9: Avg. Percent Dierence from Optimal by Mean Class 2 Demand

% STOP MR LB UB ROA SC LW FCFS POA

λ2 = 6 0.114 0.154 0.393 0.679 2.433 1.292 0.997 3.327 6.120

λ2 = 12 0.097 0.245 1.044 1.038 3.512 2.174 2.569 7.934 15.877

λ2 = 18 0.080 0.281 1.948 1.343 3.787 3.297 4.253 12.838 27.484

λ2 = 24 0.082 0.337 3.099 1.932 4.347 5.157 5.883 18.258 28.522

λ2 = 30 0.082 0.364 3.717 2.450 4.591 6.426 6.512 20.511 28.832

service. Our model assumes Class 1 requests leave without cost if rejected.

Sensitivity analysis for the eect of mean Class 2 demand on percent dierence from optimal expected prot is shown in Table 4.9. Since Class 2 patients have a higher no-show rate, the percent dierence from optimal increases on average for all policies except for STOP. The performance for STOP improves slightly because the likelihood that the rst local maximum will be the global maximum is higher asλ2, the mean demand for Class 2, increases. Whenλ2

is extremely low, LB outperforms UB on average; however, asλ2 increases, UB outperforms LB

on average. Likewise, λ2, the LW policy outperforms the SC policy on average, meaning that

if the clinic can implement only capacity control or overbooking, it prefers capacity control. As

λ2 increases, the SC policy outperforms the LW policy on average and the clinic would prefer

single class overbooking to capacity control with no overbooking. Since POA does not allow any Class 2 patients to book, its performance clearly deteriorates as λ2 increases, because it incurs

a higher opportunity cost by not allowing Class 2 patients.

Table 4.10: Avg. Percent Dierence from Optimal byp/h, (p= 100)

% STOP MR LB UB ROA SC LW FCFS POA

p/h= 0.25 0.057 0.181 1.450 2.706 5.950 5.680 3.030 12.415 22.025

p/h= 0.5 0.175 0.322 2.284 1.467 3.720 3.858 4.425 13.671 23.026

p/h= 0.75 0.034 0.363 2.972 0.574 1.917 2.322 5.671 14.789 23.957

ratio of revenue and cost parameters. As the overtime cost, h, increases, the lower bound

policies, LB and LW, perform better, where the opposite is true for the upper bound policies, UB and SC. This result is expected because the upper bound policies are exposing the clinic to more overtime risk and perform better when the overtime cost parameter is lower. This result provides insight into the sensitivity analyses in earlier sections and Figure 4.5. The performance of MR policy improves as h decreases. Likewise, the performance of POA increases slightly as h increases. POA does not overbook, so changes in expected prots are not due to changes in h, given the same values for other parameters. The improvement in performance is more an

indication of how the optimal policy is changing and the decrease in the opportunity cost for POA of not booking Class 2. Alternatively, the performance of ROA decreases as h increases

because ROA is a joint overbooking and capacity control policy which allows some overbooking unless POA is optimal. Both policies are similar in that they oer all Class 1 patients an appointment (up to capacity), so we might expect similar behavior with regard to the changes

h. It is also interesting because the result agrees with Robinson and Chen (2010) that open

access performs better when there are small overtime surcharges.