Tactical allocation and acceptance of multiple patient classes in the presence of no-shows

Abstract

Clinics that provide pediatric care are frequently confronted with family group appointment requests, where parents desire their children to be scheduled simultaneously or consecutively. This is potentially beneficial to the family by minimizing the number of trips to the provider’s office. However, offering prescheduled group appointments have the risk of reducing provider utilization, particularly if the entire group fails to meet their scheduled appointment. Similarly, reserving appointment slots for same day group appointment requests may also decrease utilization and impact profitability. This paper explores the impact of family group appointments on clinic performance in terms of provider utilization and profit. A finite-horizon, stochastic dynamic programming problem is presented to determine the optimal scheduling strategy given both individual and group appointment requests can be tactically accommodated via overbooking. On the basis of a computational study, we quantify the risk to clinic profitability and productivity resulting from the no-show behavior of prescheduled appointments. We also characterize the behavior of the optimal scheduling strategy as a function of prescheduled appointment allocations among the patient classes.

Appendix

Newsvendor solution to the allocation of slack capacity

We define the following notation before proceeding with the formal presentation of our results. Let s represent the slack capacity; d the demand for capacity (number of overbooked patients waiting for service); f(ɛ) the probability density function associated with the slack capacity; φ(·) is the cumulative distribution function; p₁ the penalty for excess capacity; and p₂ the penalty for insufficient capacity. Since prescheduled appointments will never exceed the physician’s capacity, we can examine the additional profit obtained via overbooking as a function of the slack capacity and the demand for capacity. The expected profit is defined as follows.

The first two terms represent the expected revenue obtained as a function of the number of patients overbooked and the slack capacity. The third term represents the expected penalty due to excess capacity. The fourth term represents the expected penalty due to insufficient capacity.

The expected profit function can be simplified to the following expression.

where μ is the expected slack capacity E[ɛ].

Setting G′(d)=0 yields the following.

The solution of the equation above maximizes the profit function since G(d) is concave.

A similar argument can be made using marginal analysis when the slack capacity is defined by a discrete distribution. In particular, the number of patients to overbook is the largest number d satisfying

Optimal number to overbook in the last decision epoch

The terminal period is a one-period problem with no decision. Using marginal analysis, we analyze the behavior of the profit function g(s, d).

Case 1.:

If s⩽d Profit will be rs−p₂(d−s)+(r−p₂)(d−s)=rd−2p₂d+2p₂s The change in profit from overbooking one additional patient increases since

Case 2.:

If s⩾d, Profit will be r(d+1)−p₁(s−d+1))=rd+r−p₁s+p₁d−p₁=(r+p₁)d+(r+p₁)−p₁s

The expected change in profit from overbooking one additional patient is always increasing since

As a result, it is better to end the period with some patients overbooked. In our problem since we have an overbooking limit policy variable, the number of patients to overbook is bounded and thus the expected profit is bounded. The value of d is decided in the prior period. In time period t=T−1, (assuming no decision is made in the terminal period) the selection of the number to overbook is made in advance of time T based on the slack capacity in period T.