Expansion of operating room (OR) capacity and

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1 Tactical Decision Making or Selective Exansion o Oerating Room Resources Incororating Financial Criteria and Uncertainty in Subsecialties Future Workloads Franklin Dexter, MD, PhD, Johannes Ledolter, PhD, and Ruth E. Wachtel, PhD, MBA Division o Management Consulting, Deartments o Anesthesia and Health Management & Policy, Deartment o Management Sciences, College o Business, and Deartment o Anesthesia, University o Iowa We considered the allocation o oerating room (OR) time at acilities where the strategic decision had been made to increase the number o ORs. Allocation occurs in two stages: a long-term tactical stage ollowed by short-term oerational stage. Tactical decisions, aroximately 1 yr in advance, detere what secialized equiment and exertise will be needed. Tactical decisions are based on estimates o uture OR workload or each subsecialty or surgeon. We show that grous o surgeons can be excluded rom consideration at this tactical stage (e.g., surgeons who need intensive care beds or those with below average contribution margins er OR hour). Lower and uer limits are estimated or the uture demand o OR time by the remaining surgeons. Thus, initial OR allocations can be accomlished with only artial inormation on uture OR workload. Once the new ORs oen, oerational decision-making based on OR eiciency is used to ill the OR time and adjust staing. Surgeons who were not allocated additional time at the tactical stage are rovided increased OR time through oerational adjustments based on their actual workload. In a case study rom a tertiary hosital, uture demand estimates were needed or only 15% o surgeons, illustrating the racticality o these methods or use in tactical OR allocation decisions. (Anesth Analg 2005;100: ) Exansion o oerating room (OR) caacity and lans or allocation o additional OR time are toics markedly aecting anesthesia roviders. Decisions directly aect anesthesia revenue, work hours, call requirements, and job security. Decisions aect the ability o academic deartments to educate residents and erorm clinical research. Decisions also aect anesthesia roviders indirectly through their inancial imact on the hosital and its caability to make other caital imrovements (e.g., urchase anesthesia inormation systems). We considered hositals at which the strategic decision had been made to increase the number o ORs because existing OR time was used ully and uture demand was exected to increase. Tactical decisions Franklin Dexter is the Director o the Division o Management Consulting within the Deartment o Anesthesia. He receives no unds ersonally other than his salary rom the State o Iowa, including no travel exenses or honoraria, and has tenure with no incentive rograms. Acceted or ublication October 21, Address corresondence and rerint requests to Franklin Dexter, MD, PhD, Division o Management Consulting, Deartment o Anesthesia, University o Iowa, Iowa City, IA Address to Franklin-Dexter@UIowa.edu. DOI: /01.AE D regarding how the additional OR time might be allocated are made roughly 1 yr in advance. I seciic subsecialties are to be targeted, the hosital must design and outit the new ORs and recruit sta with suicient exertise to meet the needs o those subsecialties. Alternatively, the hosital may decide to use the additional OR time as general urose overlow or services that have illed their allocated OR time (1,2) and have more cases to erorm (3,4). We describe a ractical method, using only limited inormation, or making tactical decisions regarding allocation o uture OR resources among surgical subsecialties. Such decisions require estimates or the uture demand or each subsecialty area (5,6). Assug the hosital wishes to target subsecialties with the highest otential or inancial growth (7 12), we show that OR allocation can be accomlished with demand inormation or only a small raction o surgeons, wherein surgeon is considered a surrogate or subsecialty (7,8,10 12). Furthermore, although demand estimates are aroximations, their accuracy is not crucial. Once the new ORs oen, oerational rocesses ill OR time that would otherwise be unused by adjusting staing, releasing allocated but unused OR time, and scheduling cases based on OR eiciency (1 4,13,14) by the International Anesthesia Research Society /05 Anesth Analg 2005;100:

2 1426 ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL. AESTH AALG TACTICAL OR MAAGEMET DECISIOS 2005;100: To illustrate these methods, data were taken rom a 28 OR tertiary hosital in the United States (US). The hosital had been exeriencing 3% surgical growth er year or the revious 5 years. Surgical intensive care unit (ICU) beds are usually ull. One third o ORs inish ater 5:30 m. The strategic decision was made to convert two storage rooms into two ORs and or two ORs to run to 9 m each workday, increasing caacity by 10%. The lanned time course or construction and recruitment o additional ersonnel was 1 yr. The hosital and medical grou s activity-based costing system was queried or all atients undergoing outatient or same day admit elective surgery in 2002 (11). Total contribution margin (CM) (hosital lus roessional), hours o OR time, and days o ICU care were calculated or each hysician who used OR time. The analysis was limited to the 122 hysicians who erormed at least 20 cases in This grou o hysicians, including a dermatologist, is henceorth reerred to as surgeons. Methods Financial considerations associated with allocation o OR time have been reviewed reviously (11): managerial cost accounting methods (9), rationale or selection o atients undergoing only elective surgery (7 9), use o surgeon as a surrogate or subsecialty (7 9), linear rogramg to include constraints such as limited regular ward or ICU beds (8,9), calculation o conidence intervals or CM er OR hour (CM/OR hour) or each surgeon (10), and quadratic rogramg to revent surious decisions rom outlier atients (11). Briely, CM/OR hour or each surgeon is detered by combining OR inormation systems data with hosital or roessional inancial data. CM is revenue (reimbursement) us variable costs. Proit is CM us ixed costs. To imize roit, a hosital should do more cases with a high CM/OR hour. Fixed costs are ignored because they do not change. Calculations were erormed using Excel XP s Solver tool (Microsot, Redmond, WA) (15). Allocation o additional OR time occurs as a twostage rocess. Tactical decisions made aroximately 1 yr in advance detere initial increases in allocations or each subsecialty. OR times are not reduced or any surgeon; they are increased or (mostly) let unchanged. Once the new ORs oen, oerational adjustments serve to ill available OR time and to match staing (i.e., OR allocations) to actual workload. Our novel aroach to the tactical rocess or increasing OR allocations is to use readily available inancial and oerational data irst to screen out grous o surgeons who should not receive increases in OR time. The advantage o eliating these surgeons rom consideration at this oint is that uture demand does not have to be estimated or these surgeons. Linear rogramg techniques or detering OR allocations can be used to eliate surgeons based on their corresonding need or other limited hosital resources that would not be available, such as regular ward, ICU, or ostanesthesia care unit time (8,9,16). Quadratic rogramg considers standard errors or each surgeon s CM/OR hour, and is used to eliate surgeons with uncertainties in CM/OR hour large enough to cause surious tactical decisions owing to outlier atients (10,11,15,17). Surgeons or whom the new ORs will be inherently unsuitable are eliated (18). For examle, the new ORs at the case study hosital are located at the tertiary surgical suite, ar rom the recovery and holding areas. They were deemed unsuitable or large-volume short outatient cases such as ediatric otolaryngology. Surgeons whose OR workload is already small will have little otential or substantive increases in workload. We required that doubling their workload would result in at least one extra case er week or two additional hours er week, whichever was larger. For these surgeons, even a large increase in relative demand would consume little additional OR time. The subsecialties reresented by these surgeons are unlikely to lay major roles in the tactical decisions associated with OR exansion. Additional OR time should not be allocated at this tactical stage to surgeons with below average CM/OR hour (7,10,11). Because the second oerational stage ills OR time without regard or CM/OR hour, thereby achieving the overall average CM/OR hour, allocating OR time tactically at a below average CM/OR hour would be disadvantageous inancially. The concet that surgeons can be excluded rom consideration or additional OR allocations at the tactical stage beore marketing data are obtained has not been recognized reviously. The concet is vital to the simlicity and racticality o our method. Because these surgeons will not have their OR allocations increased, their uture OR demand unctions can simly be ixed at their OR workload rom the receding year. All demand unctions are thus constrained demands, reresenting only that ortion o otential demand that can be met. Demand unctions are estimated or the remaining surgeons only. Seciically, results o data enveloment analysis (5,6) and other marketing data are used to redict the overall need or services in the uture and the extent to which nearby hositals may comete

3 AESTH AALG ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL ;100: TACTICAL OR MAAGEMET DECISIOS or the same atients, modiied to account or local actors such as the surgeons desires or exansion. Demand unctions do not have to be detered recisely. They can be seciied simly as ranges with uniorm distributions. Future OR allocations are then calculated (Aendix, Equation 16). Although many surgeons and subsecialties are excluded rom being allocated OR time at the irst tactical stage, they are not recluded rom receiving additional OR time during the second oerational stage. Surgeons and subsecialties ully using their allocated OR time and scheduling more cases can use much additional allocated OR time by having it released (2,3). Over a eriod o months, their OR allocations are increased to match staing to their actual OR workloads, thereby increasing OR eiciency (13). Surgeons excluded rom increased OR allocations tactically because o their need or ICU beds can schedule additional cases when ICU beds are available on short notice. Results Figure 1 shows the 122 surgeons rom the study hosital lotted according to CM/OR hour. CM/OR hour varied several-old among surgeons, consistent with results rom two other hositals (7,8), The average CM/OR hour among all surgeons was $1,773 (Equation 6), between reviously calculated values o $1,530 (8) and $1,864 (7), adjusted or the US Bureau o Labor Statistics national medical inlation rate. As the tactical goal is to imize inancial gains, allocations exceed current OR usage only or those surgeons (Equation 8) whose CM/OR hour is above the weighted average or all surgeons (Equation 3). Sixty-eight surgeons were excluded at the tactical stage rom receiving additional OR time because they did not satisy this criterion. At the tactical stage, 36 more surgeons were excluded rom receiving additional OR time or miscellaneous reasons: 15 surgeons because ICU caacity would not be increased suiciently, ive surgeons because o high uncertainties in CM/OR hour, seven surgeons based on the unsuitability o the ORs or their articular secialties, and nine surgeons who contribute negligibly to overall OR workload. Only 18 surgeons remained, reresenting just 15% o the original 122 surgeons. Achievable increases in demand or these surgeons were considered uniormly distributed within seciied ranges (Equation 10). The OR allocations at the tactical stage were detered (Equation 16) by imosing the constraint that total additional allocations cannot exceed the total additional OR time available. This constraint was introduced through the Lagrange coeicient (Aendix), reresenting the smallest value o CM/OR hour Figure 1. Contribution margins er oerating room (OR) hour sorted. Each symbol reresents one surgeon (n 122). Surgeon is used as a surrogate or subsecialty. Surgeons lotted with stars are those who were excluded rom consideration or exanded OR resources because their contribution margins er OR hour were less than the weighted average. Surgeons lotted with black squares were excluded by linear rogramg based on extensive intensive care unit usage. The surgeons lotted with circles were excluded by quadratic rogramg based on large uncertainty in estimated contribution margins er OR hour. Potential doubling o the surgeon s workload was considered equivalent to hiring another surgeon (8,11). Surgeons lotted with black triangles were excluded because their imum otential increases in OR workload were 2 h er week, even with doubling OR workload. The 18 surgeons included in the analysis were those marked with marks. above which surgeons were eligible to receive increases in OR allocations tactically. For the case study hosital, these calculations based on the Lagrange coeicient and CM/OR hour roved unnecessary. Data enveloment analysis (5,6) results were available or 11 surgeons in ive secialties. The hosital was already erorg as much surgery as exected in these ields based on the relationshis between secialty workloads, hosital characteristics, and demograhics throughout the region (5,6). There was no reason to exect OR workload in these secialties to increase at a rate exceeding the 1.8% annual growth rom oulation aging (19). Other marketing data or these 11 surgeons and secialties and seven additional surgeons in two additional secialties were rovided to local exerts and adistrators and discussed with the surgeons themselves. one o these surgeons and subsecialties aeared to have the otential or market exansion exceeding one extra case er week. Consequently, the decision was made not to target any subsecialty but to use the exanded OR resources or overlow time to be illed oerationally. Because these seciic tactical results may not aly to other hositals, we illustrate the calculation o initial (tactical) OR allocations. We created the samle data o Figure 2 by assug that the imum uture demand equals a simle 100% roortional increase

4 1428 ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL. AESTH AALG TACTICAL OR MAAGEMET DECISIOS 2005;100: Figure 2. Dierences in ercentage increases in oerating room (OR) resources (time) between the greedy algorithm and the Lagrange relaxation. Each oint reresents a single surgeon. To ermit visual comarison, all surgeons are considered to have imum ercentage increases o 100%. The 18 surgeons included are those marked with marks in Figure 1. The other 104 surgeons with zero increases are not lotted to revent clutter. The greedy algorithm rovides increased OR allocations to as ew surgeons as ossible because it assumes that uture demand and the ercentage increase are imal or each surgeon. The Lagrange relaxation assumes that demand is a random variable ranging between the imum increase o 0% and the imum increase o 100%. The data are rovided in Table 1. based on revious usage (Equation 9) (i.e., hiring another surgeon), just as we do to identiy surgeons with small workloads. Table 1 shows the above 18 surgeons who were not disqualiied. Two sets o OR allocation results are resented in Table 1 and Figure 2: a greedy algorithm in which as many surgeons as ossible are assigned the imum 100% increase (7) and our Lagrange relaxation in which the increase in demand or each surgeon is considered a random variable between 0% and 100% (Equation 16). These two methods are not equivalent (16) because the Lagrange relaxation considers uncertainties in knowledge o uture demand. Discussion Estimating uture demand or surgical subsecialties and surgeons is an exensive, time-consug, and olitically charged rocess. Because OR allocations should only be increased tactically or those surgeons with above average CM/OR hour who also meet other criteria, demand unctions need be estimated or only a small ercentage o surgeons. Marketing research can be limited to a ew subsecialties. Fewer surgeons must be consulted to solicit their exert oinions about uture exansion. In addition, demand need not be redicted exactly, just estimated as a range. Thus, using demand inormation to make tactical decisions about OR design, equiment, and staing becomes ractical and straightorward. Conclusions are sometimes so obvious once surgeons have been excluded that the calculations in the Aendix are not needed, as shown or the case study hosital. Whereas the useulness o our method is a consequence o the ability to exclude most surgeons beore estimating uture OR workload a year in advance, the undamental scientiic advance is the consideration o OR allocation in two stages: tactical ollowed by oerational. I the tactical decision is considered alone, surgeons with below average CM/OR hour cannot be excluded (8). Their exclusion is ossible only because the two stages o OR allocation are considered together, and the surgeons have access to OR time during the second oerational stage without regard to inancial criteria. The end result is that the second oerational stage can be ar more imortant than the initial tactical stage in detering actual OR allocations. In act, at the case study hosital, the aroriate tactical decision was to rely solely on the second oerational stage or OR allocation to seciic surgeons and subsecialties. The Aendix includes methods or calculating initial (tactical) values or OR allocations using a Lagrange relaxation. The greedy algorithm, linear rogramg, and quadratic rogramg have been used reviously or tactical OR allocation (7 12). In this aer, we considered OR allocation as a two-stage rocess o tactical decisions ollowed by oerational decisions, ermitting a large reduction in the number o surgeons or whom demand must be detered. We also incororated the additional inormation that uture demand or each surgeon is not truly a ixed value but a random variable. Whereas linear and quadratic rogramg are useul or excluding surgeons based on their imal otential increases in OR workload, they are not suitable or otimizing tactical OR allocations while simultaneously considering the ollowing oerational decision. The Lagrange relaxation is an aroriate method because it considers uncertainties in demand (16). We recommend that imlementation at acilities be erormed by ollowing the same ordered rocess that we used. Obtain data as described in the second aragrah o the Case Study section and then ollow the stes through the last bullet oint in Methods. The reerences rovide the details. The relevant equations are given in the Aendix. Limitations CM/OR hour may not be readily available or all surgeons. The case study hosital and roessional grou had a costing database that rovided detailed inancial inormation. Payer mix was incororated automatically into the calculations through its eect on hosital or roessional revenue. Even at institutions that lack such detailed accounting data, variable costs

5 AESTH AALG ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL ;100: TACTICAL OR MAAGEMET DECISIOS Table 1. Data Used to Create Figure 2 Surgeon classiied according to secialty Ratio o total contribution margin to total hours o OR time used Current average weekly use o OR time (h) Greedy Algorithm Lagrange Relaxation General $3, % 96% eurosurgery $2, % 93% General $2, % 92% Gynecology $2, % 91% Gynecology $2, % 90% Gynecology $2, % 89% Orthoedics $2, % 87% General $2, % 85% Orthoedics $2, % 82% Otolaryngology $2, % 80% General $2, % 78% Urology $2, % 76% Otolaryngology $1, % 39% Otolaryngology $1, % 31% Otolaryngology $1, % 12% Plastics $1, % 0% Orthoedics $1, % 0% General $1, % 0% OR oerating room. Another 104 surgeons were excluded and have 0% increases. For the 2 columns at the right, OR more 10%, more surg 1.0 (i.e., d s 2 Q s ). For the Lagrange calculations, R $1,773 and $ Readers can use the data to ensure they are using equation (16) correctly beore alying it to their own data. For the greedy algorithm, each surgeon s imum increase in demand could be set at 50%, equal to the average o a uniorm distribution between 0 and 1. Then, the total increases in OR allocations would equal 50% or all 18 surgeons, because the sum o the increases in OR allocations would be less than the available increase in OR time. can be estimated suiciently accurately or the uroses o tactical decision-making using the atients OR times, hosital lengths o stay, ICU lengths o stay, and imlant costs (9). Emhasizing the latter, imlant cost accounting is essential (9). Although the Lagrange relaxations take into account uncertainties in demand estimations, a limitation is that the samling error o CM/OR (10) is not considered. evertheless, once quadratic rogramg has been used to eliate outlier surgeons, uncertainties in the estimates or CM/OR hour contribute little to the statistical risk associated with detering overall CM because the ortolio o surgeons at a hosital is suiciently diverse (i.e., many more than eight surgeons) (11). For screening surgeons, uture demand unctions can be ranges in which the imum ossible demand is a roortional increase in current usage. More recise estimates can be obtained rom hosital discharge data and demograhic data using methods such as data enveloment analysis (5,6). However, these estimates must be modiied and adjusted by adistrators and surgeons based on seciic actors unique to each hosital and region. When local exerts at the study hosital were asked about demand distributions, they seemed to be seculating wildly, unable to do more than suggest ossible values or imum demand and guess the robability o reduced demand. For that reason, we assumed a uniorm statistical distribution or the increase in demand or the surgeons included in the tactical analysis (Equation 10). This assumtion does not aect our most imortant inding: When OR allocation is considered a two-stage rocess, demand data are needed or only a small subset o surgeons. Although the assumtion aects the calculated values or initial OR allocations, subsequent oerational decisions using actual workload (13,14) will correct and comensate or inaccuracies in the estimates o uture demand. This article s relevance to any articular hosital deends on how the strategic decision is made to increase OR caacity. The methods are not aroriate i uture use o new ORs has been redetered by strategic decisions (e.g., local olitics, educational needs, or directed gits). The methods do not aly unless existing OR caacity is used ully. Finally, the method assumes that OR time is allocated in a twostage rocess. I a hosital does not adjust allocations oerationally based on OR eiciency, then these methods or tactical decision-making are inaroriate. Only tactical increases, not reductions, in OR allocations were considered because reductions are rarely mandated via tactical mechanisms. I a strategic decision were made to reduce or eliate a seciic rogram, it is unlikely that ully used OR time would suddenly be reduced or that subsecialty. Instead, the hosital would stem urther investment, ossibly by choosing not to relace outdated equiment, limiting the number o imlants urchased, or allowing the number o subsecialty surgeons to decrease through

6 1430 ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL. AESTH AALG TACTICAL OR MAAGEMET DECISIOS 2005;100: attrition. The subsecialty would decline gradually. As its OR workload decreases, oerational rocesses would rogressively reduce its OR allocations. Summary The allocation o OR time is a two-stage rocess. For the tactical stage, inancial and oerational data are integrated to identiy a small subset o surgeons or whom uture surgical demand must be estimated. These demand unctions are used to detere initial OR allocations 1 yr in advance. When the ORs oen, the second oerational stage adjusts the allocations based on actual OR workload based on OR eiciency. Aendix The true mean values or the total contribution margin and OR time o the s th surgeon s elective cases are (CM s ) and (OR s ), resectively, s 1, 2,...,. For brevity, we denote the mean contribution margin er OR hour with c s CM s OR s. (1) Prior (Q s ) and uture (Q s ) OR times to be lanned or each surgeon are related in total by Q s s 1 (1 more ) OR s 1 Q s B OR, (2) with OR more reresenting the allowable roortional increase in overall OR resources. Let T s reresent the contribution to the total contribution margin rom OR time lanned or the s th surgeon. Let E(T s ) be its exectation over the uncertain demand distribution. The objective is to choose Q 1, Q 2,...,Q subject to the constraints Q s s 1 B OR s 1 E(T s ), (3) and Q s Q s, s 1, 2,...,. (4) The latter constraint seciies that the tactical decision is used only to increase OR allocations. Without loss o generality, we rank the surgeons in descending sequence o CM/OR hour: c 1 c 2 c m R c m 1 c, (5) where the weighted average contribution margin er OR hour c s Q s s 1 R. (6) s 1 At the time o tactical decision-making, uture demand or a surgeon s services, d s, is unknown. However, we assume that it is bounded Q s d s d s d s Combining Equations 4 and 7, d s Q s Q s d s, or s 1, 2,..., m Q s Q s, or s m 1,...,. To imize inancial gains, uture OR allocations should exceed current OR time only or the irst m surgeons, or whom CM/OR hour is above average (Equation 5). Thus, uture demand needs to be estimated only or the irst m surgeons. When screening the m surgeons, time-consug demand modeling can be relaced by estimating the imal increase in demand as a roortional increase in demand ( surg more 0). For examle, surg more 1.0 reresents recruitment o another surgeon o the same subsecialty (8,11). Then, or initial screening, (7) (8) d s 1 surg more Q s Q s Q s, (9) rovided uture demand is not constrained by the surgeon s lack o eligibility or increases in OR time because o a lack o ICU beds or other such resources. Then, instead, d s Q s Q s. With no or very limited distributional inormation available or increases in demand, a uniorm distribution is assumed or the increase in demand: F s Q s Q s d s d s d s F s d s d s d s d s d s or Q s d s d s. F s d s 1 or d s d s (10) To make the notation less cumbersome, we subsequently dro the subscrit in F s (). The contribution margin rom OR time lanned or s th surgeon is given by T s c sq s, Q s d s d s, c s d s R Q s d s, d s d s Q s or s 1, 2,..., m. (11)

7 AESTH AALG ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL ;100: TACTICAL OR MAAGEMET DECISIOS I OR time allocated tactically exceeds actual demand, oerational rocesses will ill the OR time without regard to inances, achieving a CM/OR hour equal to the average, R. The exected value equals E(T s ) (c s d s df(d s ) RQ s R) ds ds (c s R) ds d s df(d s ) df(d s ) RQ s ds d s df(d s ) c sq s df(d s ) c s Q s ds df(d s ) ds df(d s ). Substituting the uniorm distribution rom Equation (10), 1 E T s 2 d s d s R c s Q s 2 where a s (R c s ) 2 c s d s Rd s Q s a s, (12) ) (Q s ) 2 2(d s d Q s s d s df(d. s ) ds Although Equation 12 and thus Equation 6 are imized by setting Q s d s, s 1, 2,..., m, the constraint B OR B OR Q s s m 1 (13) may not be satisied. The constraint is introduced through the Lagrange coeicient and the imization o m m L(Q 1, Q 2,...,Q m ) E(T s ) B OR Q s. s 1 s 1 (14) The otimal values or Q s are ound by solving the system o m 1 equations: L Q s 0, L 0. s 1, 2,..., m From Equation 14, the condition is satisied by requiring that: m Q s ( ) B OR, (15) s 1 (i.e., the sum o initial OR allocations are a unction o ). Also L Q Q s R c s c s d s Rd s, s d s d s where a s has vanished because the constant does not deend on Q s. Setting L/ Q s 0 and rearranging terms twice, Q s d s d s Rd s c s d s R c s d s c s c s R d s d s. Adding the conditions o Equation 8, Q s Q s ; d s c s c s R d s d s d s, R c s, R c s m, B OR s 1 d s or s 1, 2,..., m. (16) Because reresents the value o CM/OR hour above which surgeons are eligible to receive increases in initial OR allocations, the smallest value is chosen satisying Equation 15. We use the standard nonlinear GRG Solver Tool in Excel. Reerences 1. Dexter F, Macario A, O eill L. Scheduling surgical cases into overlow block time: comuter simulation o the eects o scheduling strategies on oerating room labor costs. Anesth Analg 2000;90: Dexter F, Traub RD, Macario A. How to release allocated oerating room time to increase eiciency. Predicting which surgical service will have the most under-utilized oerating room time. Anesth Analg 2003;96: Dexter F, Traub RD. How to schedule elective surgical cases into seciic oerating rooms to imize the eiciency o use o oerating room time. Anesth Analg 2002;94: Dexter F, Macario A, Traub RD. Which algorithm or scheduling add-on elective cases imizes oerating room utilization? Use o bin acking algorithms and uzzy constraints in oerating room management. Anesthesiology 1999;91: Dexter F, O eill L. Data enveloment analysis to detere by how much hositals can increase elective inatient surgical workload or each secialty. Anesth Analg 2004;99: O eill L, Dexter F. Market cature o inatient erioerative services using data enveloment analysis. Health Care Manag Sci 2004;7: Macario A, Dexter F, Traub RD. Hosital roitability er hour o oerating room time can vary among surgeons. Anesth Analg 2001;93: Dexter F, Blake JT, Penning DH, Lubarsky DA. Calculating a otential increase in hosital margin or elective surgery by changing oerating room time allocations or increasing nursing staing to ermit comletion o more cases: A case study. Anesth Analg 2002;94:

8 1432 ECOOMICS, EDUCATIO, AD HEALTH SYSTEMS RESEARCH DEXTER ET AL. AESTH AALG TACTICAL OR MAAGEMET DECISIOS 2005;100: Dexter F, Blake JT, Penning DH, et al. Use o linear rogramg to estimate imact o changes in a hosital s oerating room time allocation on erioerative variable costs. Anesthesiology 2002;96: Dexter F, Lubarsky DA, Blake JT. Samling error can signiicantly aect measured hosital inancial erormance o surgeons and resulting oerating room time allocations. Anesth Analg 2002;95: Dexter F, Ledolter H. Managing risk and exected inancial return rom selective exansion o oerating room caacity: Mean-variance analysis o a hosital s ortolio o surgeons. Anesth Analg 2003;97: Kuo PC, Schroeder RA, Mahaey S, Bollinger RR. Otimization o oerating room allocation using linear rogramg techniques. J Am Coll Surg 2003;197: Strum DP, Vargas LG, May JH. Surgical subsecialty block utilization and caacity lanning: A imal cost analysis model. Anesthesiology 1999;90: Estein RH, Dexter F. Statistical ower analysis to estimate how many months o data are required to identiy oerating room staing solutions to reduce labor costs and increase roductivity. Anesth Analg 2002;94: Ragsdale CT. Sreadsheet modeling and decision analysis, a ractical introduction to management science, 2nd edition. Cincinnati, Ohio: South-Western College Publishing, 1998;45 64, , Higle JL, Wallace SW. Sensitivity analysis and uncertainty in linear rogramg. Interaces 2003;33: Briggs AH, Mooney CZ, Wonderling DE. Constructing conidence intervals or cost-eectiveness ratios: An evaluation o arametric and non-arametric techniques using Monte Carlo simulation. Stat Med 1999;18: Dexter F, Macario A, Penning DH, Chung P. Develoment o an aroriate list o surgical rocedures o a seciied imum anesthetic comlexity to be erormed at a new ambulatory surgery acility. Anesth Analg 2002;95: Etzioni DA, Liu JH, Maggard MA, et al. Workload rojections or surgical oncology: Will we need more surgeons? Ann Surg Oncol 2003;10: