Estimating future demand for surgical subspecialties and surgeons is an expensive, time-consuming, and politically charged process. Because OR allocations should only be increased tactically for those surgeons with above average CM/OR hour who also meet other criteria, demand functions need be estimated for only a small percentage of surgeons. Marketing research can be limited to a few subspecialties. Fewer surgeons must be consulted to solicit their expert opinions about future expansion. In addition, demand need not be predicted exactly, just estimated as a range. Thus, using demand information to make tactical decisions about OR design, equipment, and staffing becomes practical and straightforward. Conclusions are sometimes so obvious once surgeons have been excluded that the calculations in the Appendix are not needed, as shown for the case study hospital.
Whereas the usefulness of our method is a consequence of the ability to exclude most surgeons before estimating future OR workload a year in advance, the fundamental scientific advance is the consideration of OR allocation in two stages: tactical followed by operational. If the tactical decision is considered alone, surgeons with below average CM/OR hour cannot be excluded (8). Their exclusion is possible only because the two stages of OR allocation are considered together, and the surgeons have access to OR time during the second operational stage without regard to financial criteria. The end result is that the second operational stage can be far more important than the initial tactical stage in determining actual OR allocations. In fact, at the case study hospital, the appropriate tactical decision was to rely solely on the second operational stage for OR allocation to specific surgeons and subspecialties.
We recommend that implementation at facilities be performed by following the same ordered process that we used. Obtain data as described in the second paragraph of the “Case Study” section and then follow the steps through the last bullet point in Methods. The references provide the details. The relevant equations are given in the Appendix.
CM/OR hour may not be readily available for all surgeons. The case study hospital and professional group had a costing database that provided detailed financial information. Payer mix was incorporated automatically into the calculations through its effect on hospital or professional revenue. Even at institutions that lack such detailed accounting data, variable costs can be estimated sufficiently accurately for the purposes of tactical decision-making using the patients' OR times, hospital lengths of stay, ICU lengths of stay, and implant costs (9). Emphasizing the latter, implant cost accounting is essential (9).
Although the Lagrange relaxations take into account uncertainties in demand estimations, a limitation is that the sampling error of CM/OR (10) is not considered. Nevertheless, once quadratic programming has been used to eliminate outlier surgeons, uncertainties in the estimates for CM/OR hour contribute little to the statistical risk associated with determining overall CM because the portfolio of surgeons at a hospital is sufficiently diverse (i.e., many more than eight surgeons) (11).
For screening surgeons, future demand functions can be ranges in which the maximum possible demand is a proportional increase in current usage. More precise estimates can be obtained from hospital discharge data and demographic data using methods such as data envelopment analysis (5,6). However, these estimates must be modified and adjusted by administrators and surgeons based on specific factors unique to each hospital and region. When local experts at the study hospital were asked about demand distributions, they seemed to be speculating wildly, unable to do more than suggest possible values for maximum demand and guess the probability of reduced demand. For that reason, we assumed a uniform statistical distribution for the increase in demand for the surgeons included in the tactical analysis (Equation 10). This assumption does not affect our most important finding: When OR allocation is considered a two-stage process, demand data are needed for only a small subset of surgeons. Although the assumption affects the calculated values for initial OR allocations, subsequent operational decisions using actual workload (13,14) will correct and compensate for inaccuracies in the estimates of future demand.
This article’s relevance to any particular hospital depends on how the strategic decision is made to increase OR capacity. The methods are not appropriate if future use of new ORs has been predetermined by strategic decisions (e.g., local politics, educational needs, or directed gifts). The methods do not apply unless existing OR capacity is used fully. Finally, the method assumes that OR time is allocated in a two-stage process. If a hospital does not adjust allocations operationally based on OR efficiency, then these methods for tactical decision-making are inappropriate.
Only tactical increases, not reductions, in OR allocations were considered because reductions are rarely mandated via tactical mechanisms. If a strategic decision were made to reduce or eliminate a specific program, it is unlikely that fully used OR time would suddenly be reduced for that subspecialty. Instead, the hospital would stem further investment, possibly by choosing not to replace outdated equipment, limiting the number of implants purchased, or allowing the number of subspecialty surgeons to decrease through attrition. The subspecialty would decline gradually. As its OR workload decreases, operational processes would progressively reduce its OR allocations.
The allocation of OR time is a two-stage process. For the tactical stage, financial and operational data are integrated to identify a small subset of surgeons for whom future surgical demand must be estimated. These demand functions are used to determine initial OR allocations 1 yr in advance. When the ORs open, the second operational stage adjusts the allocations based on actual OR workload based on OR efficiency.
The true mean values for the total contribution margin and OR time of the sth surgeon’s elective cases are μ(CMs) and μ(ORs), respectively, s = 1, 2,…,N. For brevity, we denote the mean contribution margin per OR hour with
Prior (Qsp) and future (Qsf) OR times to be planned for each surgeon are related in total by
with αmoreOR representing the allowable proportional increase in overall OR resources.
Let Ts represent the contribution to the total contribution margin from OR time planned for the sth surgeon. Let E(Ts) be its expectation over the uncertain demand distribution. The objective is to choose
subject to the constraints
The latter constraint specifies that the tactical decision is used only to increase OR allocations.
Without loss of generality, we rank the surgeons in descending sequence of CM/OR hour:
where the weighted average contribution margin per OR hour
At the time of tactical decision-making, future demand for a surgeon’s services, ds, is unknown. However, we assume that it is bounded
Combining Equations 4 and 7,
To maximize financial gains, future OR allocations should exceed current OR time only for the first m surgeons, for whom CM/OR hour is above average (Equation 5). Thus, future demand needs to be estimated only for the first m surgeons.
When screening the m surgeons, time-consuming demand modeling can be replaced by estimating the maximal increase in demand as a proportional increase in demand (αmoresurg > 0). For example, α moresurg = 1.0 represents recruitment of another surgeon of the same subspecialty (8,11). Then, for initial screening,
provided future demand is not constrained by the surgeon’s lack of eligibility for increases in OR time because of a lack of ICU beds or other such resources. Then, instead, ds max = Qsp = Qsf.
With no or very limited distributional information available for increases in demand, a uniform distribution is assumed for the increase in demand:
To make the notation less cumbersome, we subsequently drop the subscript in Fs().
The contribution margin from OR time planned for sth surgeon is given by
If OR time allocated tactically exceeds actual demand, operational processes will fill the OR time without regard to finances, achieving a CM/OR hour equal to the average, R. The expected value equals
Substituting the uniform distribution from Equation (10),
Although Equation 12 and thus Equation 6 are maximized by setting Q s f = ds max, s = 1, 2, …, m, the constraint
may not be satisfied. The constraint is introduced through the Lagrange coefficient λ and the maximization of
The optimal values for Qsf are found by solving the system of m + 1 equations:
From Equation 14, the condition is satisfied by requiring that:
(i.e., the sum of initial OR allocations are a function of λ). Also
where as has vanished because the constant does not depend on Qsf. Setting ∂L/∂Q s f = 0 and rearranging terms twice,
Adding the conditions of Equation 8,
Because λ represents the value of CM/OR hour above which surgeons are eligible to receive increases in initial OR allocations, the smallest value is chosen satisfying Equation 15. We use the standard nonlinear GRG Solver Tool in Excel.
1. Dexter F, Macario A, O’Neill L. Scheduling surgical cases into overflow block time: computer simulation of the effects of scheduling strategies on operating room labor costs. Anesth Analg 2000;90:980–6.
2. Dexter F, Traub RD, Macario A. How to release allocated operating room time to increase efficiency. Predicting which surgical service will have the most under-utilized operating room time. Anesth Analg 2003;96:507–12.
3. Dexter F, Traub RD. How to schedule elective surgical cases into specific operating rooms to maximize the efficiency of use of operating room time. Anesth Analg 2002;94:933–42.
4. Dexter F, Macario A, Traub RD. Which algorithm for scheduling add-on elective cases maximizes operating room utilization? Use of bin packing algorithms and fuzzy constraints in operating room management. Anesthesiology 1999;91:1491–500.
5. Dexter F, O’Neill L. Data envelopment analysis to determine by how much hospitals can increase elective inpatient surgical workload for each specialty. Anesth Analg 2004;99:1492–500.
6. O’Neill L, Dexter F. Market capture of inpatient perioperative services using data envelopment analysis. Health Care Manag Sci 2004;7:263–73.
7. Macario A, Dexter F, Traub RD. Hospital profitability per hour of operating room time can vary among surgeons. Anesth Analg 2001;93:669–75.
8. Dexter F, Blake JT, Penning DH, Lubarsky DA. Calculating a potential increase in hospital margin for elective surgery by changing operating room time allocations or increasing nursing staffing to permit completion of more cases: A case study. Anesth Analg 2002;94:138–42.
9. Dexter F, Blake JT, Penning DH, et al. Use of linear programming to estimate impact of changes in a hospital’s operating room time allocation on perioperative variable costs. Anesthesiology 2002;96:718–24.
10. Dexter F, Lubarsky DA, Blake JT. Sampling error can significantly affect measured hospital financial performance of surgeons and resulting operating room time allocations. Anesth Analg 2002;95:184–8.
11. Dexter F, Ledolter H. Managing risk and expected financial return from selective expansion of operating room capacity: Mean-variance analysis of a hospital’s portfolio of surgeons. Anesth Analg 2003;97:190–5.
12. Kuo PC, Schroeder RA, Mahaffey S, Bollinger RR. Optimization of operating room allocation using linear programming techniques. J Am Coll Surg 2003;197:889–95.
13. Strum DP, Vargas LG, May JH. Surgical subspecialty block utilization and capacity planning: A minimal cost analysis model. Anesthesiology 1999;90:1176–85.
14. Epstein RH, Dexter F. Statistical power analysis to estimate how many months of data are required to identify operating room staffing solutions to reduce labor costs and increase productivity. Anesth Analg 2002;94:640–3.
15. Ragsdale CT. Spreadsheet modeling and decision analysis, a practical introduction to management science, 2nd edition. Cincinnati, Ohio: South-Western College Publishing, 1998;45–64, 128–141, 334–41.
16. Higle JL, Wallace SW. Sensitivity analysis and uncertainty in linear programming. Interfaces 2003;33:53–60.
17. Briggs AH, Mooney CZ, Wonderling DE. Constructing confidence intervals for cost-effectiveness ratios: An evaluation of parametric and non-parametric techniques using Monte Carlo simulation. Stat Med 1999;18:3245–62.
18. Dexter F, Macario A, Penning DH, Chung P. Development of an appropriate list of surgical procedures of a specified maximum anesthetic complexity to be performed at a new ambulatory surgery facility. Anesth Analg 2002;95:78–82.
19. Etzioni DA, Liu JH, Maggard MA, et al. Workload projections for surgical oncology: Will we need more surgeons? Ann Surg Oncol 2003;10:1112–7.