Bounds on Optimal Solution

While it is dicult to nd a closed form expression for the optimal class 2 booking limit, two simple approximations serve as lower and upper bounds respectively.

Lower Bound A lower bound can be derived by a variation on Littlewood's rule (Littlewood 1972) which provides the optimal class 2 booking limit if no overbooking is allowed. In this case, letb1(x2) =k−x2 and x2 =min(D2, b2) and write

V1(x2, k−x2) =pα2x2+pE[min(D1, k−x2)]. (4.11)

Using marginal analysis, we nd a variation of Littlewood's rule wherep2=pα2 andp1 =p.

With no overbooking, we accept thexth₂ patient if and only if

4V1(x2, k−x2) =pα2−pP(D1 > k−x2)≥0. (4.12)

We can guarantee a single optimum exists because P(D1 > k−x2) is increasing inx2. The

optimal policy if no overbooking is allowed is given by

blw₂ =max{x2 :P(D1> k−x2)≤α2}. (4.13)

The lower bound is therefore a special case of the of the original problem where Class 2 appointments attend with certainty but the clinic only receives the expected revenue pα2 < p.

In this case,b1(x2) =k−x2 and the clinic never overbooks becauseh > p. If Class 2 requests

have a positive probability of failing to attend, the clinic books at least as many Class 2 requests as in the case when all show up with certainty, and thusblw₂ is a lower-bound for the optimalb2.

Using this lower bound, we also gain insight into conditions for the optimality of a policy, such as same-day scheduling or pure-open access, which does not book class 2 requests, i.e.

also be superior to all policies which do not overbook which implies P(D1> k−1) > α2 or

similarlyP(D1≥k)> α2. However, this condition does not guarantee that a pure open access

policy will be optimal as later demonstrated numerically in Figure 5.

Upper Bound To derive an upper bound on Class 2 reservations, assume the clinic does not allow Class 1 patients to book, i.e. b1(x2) = 0. In this case, the marginal expected prot of

accepting an additional request given x2−1 have already been booked is:

4V0(x2,0)− 4V0(x2−1,0) =α2p−hα2P(Z2(x2−1)> k−1). (4.14)

Since the expected marginal prot is decreasing in x2, the clinic accepts Class 2 patients

until the expected marginal prot is less than zero or equivalently:

bub₂ =maxnx2:P(Z2(x2−1)> k−1)≤ p h o . (4.15) 4.3.5 Sensitivity Analysis

Since we cannot derive a closed-form expression for the optimal solution, we perform sensitivity analysis using computational search methods. We examine changes in the optimal prot and booking limits relative to parameter values reecting dierent clinical conditions, and we test the performance of the upper and lower bounds over these parameter values. We enumerate all feasible policies, and search over all policies where the clinic chooses the optimal Class 1 booking limit given a choice of the Class 2 booking limit. We assume Poisson distributions for the demand from each class with mean parameters λ1 and λ2 respectively. The base case

scenario for the sensitivity analysis uses parameter values: k= 24;α2 = 0.75;λ1 = 10;λ2 = 20;

p= 100; h= 150.

In Figure 4.4, we see that optimal prots are increasing in the Class 2 attendance rate, the mean Class 1 demand, and the revenue-to-cost ratio. In Figure 4.a we see that the optimal expected prot increases as the Class 2 attendance rate increases, and in Figure 4.b we nd that the optimal class 2 booking limit is generally decreasing in the Class 2 attendance rate with some minor jumps where it increases due to the discrete nature of the problem. This result agrees with our intuition that clinic prots should increase as both the no-show rate and resulting variability in service arrivals decrease. While the expected revenue from booking a Class 2 customer relative to a Class 1 customer increases with the Class 2 attendance rate, the clinic chooses to book fewer Class 2 patients as the eects of overbooking outweigh the trade-os between class allocations. In Figure 4.c and 4.d we see that the optimal prot is increasing in

λ1 and the optimal booking limit is decreasing inλ1. Since there is no cost for rejecting patient

requests, clinic prots should also increase when mean demand increases because the clinic cannot do any worse by receiving more requests. Intuitively, the clinic should allocate fewer slots to Class 2 patients when the probability of an additional Class 1 customer is higher. In Figures 4.e we see that the optimal prot is increasing in thep/hratio as the expected marginal

revenue of scheduling an additional patient increasingly outweighs the expected marginal cost. In Figure 4.f the optimal booking limit is increasing in the p/h ratio as the clinic is willing to

take on more risk.

Across all of the graphs in Figure 4 we see that for each individual parameter, when all other parameters are held constant, the gap between the upper and lower bound booking limit is either monotonically increasing or decreasing. In the case of sensitivity to the Class 2 attendance rate,

the lower bound value is increasing as the upper bound value is decreasing. For the other two sensitivity parameters, one bound stays constant (upper bound for Class 1 demand and lower bound forp/hratio) while the other converges with the optimal value. Likewise, for each

individual sensitivity parameter, the optimal policy tends to be closer to one bound at one end of the parameter range and closer to the other bound at the other end of the parameter range. When the Class 2 attendance rate is low, the optimal policy is closer to the upper bound; however, as the Class 2 attendance rate increases, both bounds converge toward the optimal solution with the lower bound showing a greater improvement. When the mean Class 1 demand,

λ1 is low, the optimal Class 2 booking limit is close to the upper bound and when the mean

Class 1 demand is high, the optimal Class 2 booking limit is close to the lower bound. When the p

h ratio is low, revenue per customer is low relative to cost, and the optimal Class 2 booking limit is close to the lower bound. When p

h is high, the optimal Class 2 booking limit is close to the upper bound since the clinic has more incentive to risk overtime cost. Further sensitivity analysis in Section 4.3 evaluates changes in policy performance with respect to changes in the parameters.

4.4 Numerical Study

We build a numerical study to determine the economic value of joint overbooking and capacity control in a clinic setting, test simple approximations, and provide insights for clinic managers across a variety of parameter settings. In the rst subsection, we describe the numerical study design which involves 198 scenarios from a wide range of model parameters and ten unique policies developed from our results, previous literature, and common practice. In the second subsection, we compare each policy's expected prot performance by looking at percent dif-

ference from optimal, percent improvement over rst-come-rst-serve, relative rankings, and counts of optimal and near-optimal scenarios for each policy. In the third subsection, we discuss conditions where a pure Open Access policy achieves optimal or near-optimal expected prot. In the fourth subsection 4.4, we investigate dierences in policy decision variables and other clinic performance measures. In the fth subsection, we analyze the sensitivity of expected prots to model parameters for all policies to test the robustness of our results. Finally, we provide managerial insights on when to use certain approximations over others.