7
MANAGEMENT USES OF THE PHYSICIAN STAFFING METHODOLOGY

The previous three chapters have demonstrated how VA physician requirements can be computed from several different perspectives, then reconciled to produce staffing recommendations. The discussion of how the VA decision maker might put these results to use proceeds at two levels. In the first—management philosophy and strategy—there are two main issues to consider. In implementing any selected variant of the Reconciliation Strategy (see chapter 6), what is the most appropriate role of decision makers at VA Central Office vis à vis those at the VAMCs? And, how does the nature of the methodology itself influence this? The second issue is the degree to which analytical models for physician staffing, such as those developed here, should become one part of a larger decision support system for resource management in the VA.

Finally, specific examples are presented showing how the VA decision maker can apply components of the physician staffing methodology to ask certain ''what if'' questions important to resource management. In the process, the committee introduces the analytical machinery to address an issue raised frequently in the course of the study. Can physician requirements be determined through application of the empirically based models in ways that actively promote, or at least protect, the quality of care? The answer, in principle, is yes. An example at the end of this chapter shows how this might work.

VA DECISION MAKERS IN CENTRAL OFFICE AND THE VAMCs: PROMOTING A DIALOGUE

Each VAMC's overall budget and personnel ceiling are approved by VA Central Office. But within these two management parameters, the number of



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Physician Staffing for the VA: Volume I 7 MANAGEMENT USES OF THE PHYSICIAN STAFFING METHODOLOGY The previous three chapters have demonstrated how VA physician requirements can be computed from several different perspectives, then reconciled to produce staffing recommendations. The discussion of how the VA decision maker might put these results to use proceeds at two levels. In the first—management philosophy and strategy—there are two main issues to consider. In implementing any selected variant of the Reconciliation Strategy (see chapter 6), what is the most appropriate role of decision makers at VA Central Office vis à vis those at the VAMCs? And, how does the nature of the methodology itself influence this? The second issue is the degree to which analytical models for physician staffing, such as those developed here, should become one part of a larger decision support system for resource management in the VA. Finally, specific examples are presented showing how the VA decision maker can apply components of the physician staffing methodology to ask certain ''what if'' questions important to resource management. In the process, the committee introduces the analytical machinery to address an issue raised frequently in the course of the study. Can physician requirements be determined through application of the empirically based models in ways that actively promote, or at least protect, the quality of care? The answer, in principle, is yes. An example at the end of this chapter shows how this might work. VA DECISION MAKERS IN CENTRAL OFFICE AND THE VAMCs: PROMOTING A DIALOGUE Each VAMC's overall budget and personnel ceiling are approved by VA Central Office. But within these two management parameters, the number of

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Physician Staffing for the VA: Volume I physician Full-Time-Equivalent Employees (FTEE), by specialty, is determined largely at the VAMC level, with many local factors—including short-term personnel constraints—influencing the specification of physician staffing, by specialty. By its very structure and logic, the Reconciliation Strategy implies that the allocation of physician FTEE across the system would be more centrally directed than is currently the case. Within a given specialty or program area, all facilities would be judged by the same criteria. There is the presumption that facilities with similar mission-related demands would be prescribed similar physician FTEE levels. The committee was not asked to consider the budgetary costs of meeting VA physician requirements or how, if at all, the methodology could or should be linked to the budget process. However, the committee can envision a resource management policy in which the portion of the VAMC budget allocated to staff physicians is established in accordance with the FTEE targets (and intermediate targets) derived through applications of the Reconciliation Strategy. The committee does believe that the likelihood of the physician staffing methodology influencing VA physician staffing is substantially greater if the methodology is made an integral part of the budget process at the facility level. Therefore, the committee recommends that the VA take steps to achieve this integration concurrently with the implementation of the methodology. For the Reconciliation Strategy to be implemented successfully and to be improved over time, however, there must be strong channels of communication between Central Office and each VAMC. And the dialogue must be an active, two-way interchange. There are two reasons why this is crucial. First, the acceptability of specific physician staffing levels—and of the methodology that produced them—is likely to be greater if they emerge from a process that genuinely engages the local facility. This does not mean that every facility director or chief of staff would be in agreement with every staffing target (or intermediate target) finally set. But local decision makers would have had the opportunity to present information judged relevant to the decision. Moreover, the criteria and reasoning behind that decision would be clear, precisely because it arises from a clearly defined decision process—the Reconciliation Strategy. Second, any broadly applicable methodology for determining VA physician requirements—no matter how sound the statistics or thoughtful the expert judgment—will necessarily use models that are simplifications of reality. A model, by definition, cannot incorporate every factor that could influence the number of physicians required at every VAMC. In addition, certain variables that the VA decision maker might wish to include may have to be omitted simply because the data are missing or inadequate. Errors may occur in the measurement of some variables in the model in ways known to the local VAMC, but not apparent to the decision maker in VA Central Office.

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Physician Staffing for the VA: Volume I Hence, a second fundamental purpose of the dialogue between Central Office and the VAMC is to ferret out and evaluate context-specific, "local" information that reduces the likelihood of model error. Put differently, the dialogue should improve the likelihood that each VAMC is treated fairly in the execution of the Reconciliation Strategy. How might this dialogue work in practice? Applying the Reconciliation Strategy, decision makers in Central Office would derive, for all physician specialties and program areas at a given VAMC, FTEE targets and intermediate targets (see chapter 6). Whatever differences exist between actual and targeted staffing would be communicated to the facility, along with information describing how the targets were computed. The facility would be expected to respond. If it agreed with the recommendations, there is little more to debate (except perhaps where the funds will be obtained to meet proposed staffing levels). But, on occasion, the facility may take exception to the targets; if so, it might wish to introduce supporting evidence not generally available at Central Office. For example, non-VA consulting physicians might contribute significantly to productivity on the inpatient medicine patient care area (PCA); their exclusion from that PCA's production function (PF) means that the model might understate the staff physician FTEE required to meet workload if these consultants were to be curtailed. Similar "omitted variable" biases could arise if a nonphysician practitioner, such as the physician assistant, is a significant contributor to workload. It could also arise if capital equipment or physical space factors not included in the model are relevant to productivity in the medicine PCA. This VAMC might be aware of certain data measurement problems that skew the medicine PCA staffing figures in its Cost Distribution Report (CDR). Or it might wish to contend that patients in this PCA are more severely ill than its overall weighted work unit (WWU) score indicates because the diagnosis-related-group (DRG) scheme (the clinical basis for the WWUs) does not discriminate adequately among these patients. In each case, the VAMC could provide the supplementary information required for improving Central Office's understanding of the facts. There also may be occasions when the facility would request new (typically additional) physician staffing levels in a specialty or program area as part of a proposed expansion of services or in response to other local conditions. The Reconciliation Strategy could be applied to generate evidence either supporting, or failing to support, the facility's request. There are already precedents in the VA for policy decisions being influenced by a dialogue between Central Office and decision makers in the field; one of the more prominent examples arose in the context of Medical District Initiated Program Planning (MEDIPP) (Veterans Health Services and Research Administration, 1989). Preliminary hospital bed projections for each of (what used to be) 27 districts were transmitted from planners in Central Office to

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Physician Staffing for the VA: Volume I district planners and directors, who could either accept the allocations or appeal them by bringing to bear relevant VAMC-specific data and other pertinent information. Because the proposed dialogue between Central Office and the VAMCs is oriented around the interpretation and evaluation of formal models for staffing, it is appropriate to reflect briefly on the general role of models in health care resource management. USE OF MODELS IN MANAGEMENT DECISION MAKING All management activities use models of some type in defining the scope of the problem under discussion and refining the elements of the decision making process. From the balance sheet of the accountant showing the sources and uses of funds to the map of the field commander showing the disposition of friendly and enemy forces, models of countless varieties and applications are used everywhere in administrative and management activities. So ubiquitous are implicit models or rules of thumb that many managers use them extensively without ever consciously realizing it. It brings to mind the story of Monsieur Jourdain in Moliere's Bourgeois Gentilhomme (Miller and Starr, 1960) who discovers, to his complete amazement, that he has been speaking prose for 40 years without realizing it! Model building is not a totally new way of looking at a familiar problem, in this case, physician staffing. Rather, model building seeks to make explicit and to quantify the relationships between elements in the real world and to improve one's understanding of real-world phenomena. When the abstraction of reality that is used in management decision making consists only of words and some loosely related numbers, the resultant management decisions may carry with them some of the same quality of fuzziness. As the process of model building becomes more explicit, the assumptions used in the abstractions of reality become better defined. Also, since no model can capture all the richness of the reality upon which it is based, the process of modeling forces the manager to define the boundaries of the abstraction. Models compress data and relationships that exist in life into manageable representations of reality that can be explored and manipulated more easily. The model, then, is always less complex and less complete than reality; but, to be useful, it must be sufficiently complete to represent those elements of the real world under investigation. Unfortunately, as the basis of a model becomes less familiar and, in particular, derives from statistical analyses, the model sometimes takes on a mystique that inhibits its appropriate utilization by management or, in the extreme, precludes even a rational evaluation of its merits. In the remainder of this chapter, an attempt is made to remove some of this mystique. In particular, ways are suggested in which components of the

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Physician Staffing for the VA: Volume I physician staffing methodology might profitably be used in management decision making. Some suggestions are also made on how management might ensure that the models presented are valid representations of reality. To be used effectively in management decision making, the physician staffing methodology must be adaptable to changing circumstances and to the availability of new data. Further, the usefulness of any management model will depend on whether it can be readily understood and presented to decision makers in a "user-friendly" fashion. No model, regardless of how carefully it is crafted, can ever provide the final and definitive answer to a management problem. Models must be seen as advisory to the decision process. Indeed, this is an undergirding precept of the committee's proposed Reconciliation Strategy. THE PHYSICIAN STAFFING METHODOLOGY AS AN COMPONENT OF AN VA DECISION SUPPORT SYSTEM In the early days of management science modeling, the analysis of data and the solution of complex mathematical models were so time-consuming that results could be developed only offline and presented to management as "the solution." The process left management with little opportunity to interact with the models, to ask "what if" questions, or to posit entirely different formulations. Recent advances in the speed and availability of computing new solution methods and software for the creative presentation of data have all revolutionized the way management science models are developed and used. The integration of these models into management decision making has been facilitated by the development of decision support systems. As generally used, the term "decision support system" means a computer-based system that assists the decision process by providing the manager with timely and relevant information so that the effects of alternative resource allocations can be understood and easily communicated. A decision support system typically has three components: a comprehensive data base, a high-level data-base manager or information processing software, and a set of appropriate decision models. It is the availability of decision models that differentiates the decision support system from a conventional management information system. A fourth element sometimes implied is the availability of user-friendly inputs and outputs. These typically take the form of menu-driven software perhaps with lightpen or touch-screen menu selection for input and the easy availability of presentation-quality graphic displays for outputs. These displays (charts, tables, three-dimensional plots) show the relationships between variables in the "what if" questions in a way that can be rapidly seen, understood, and communicated. Figure 7.1 depicts the relationship of these elements.

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Physician Staffing for the VA: Volume I A management information system alone can be used to produce standard reports of operations or special reports on request. The decision support system goes beyond that to allow managers to pose complex "what if" questions, to explore the interrelationships of changes in one part of the system to others, and to forecast system performance under a variety of future alternative scenarios. Models for exploring these "what if" questions can take a variety of forms, but they have one common characteristic. Each is a compact representation of the relationships thought to exist in the real world. In the present study, each of the modeling components of the physician staffing methodology is such a compact representation. To function properly within a VA decision support system, the physician staffing methodology would need to be backed up with accurate data bases on actual staffing for physicians and direct and indirect support personnel, availability of residents and fellows by specialty and postgraduate year of training, current workload levels, current and planned programs affecting physician requirements, facility characteristics, and so forth, as detailed in chapters 4 through 6. In a comprehensively defined decision support system, the decision maker would also want the capability to explore the fiscal implications of physician staffing decisions vis à vis decisions about other types of personnel, capital equipment, and facilities. Although these nonphysician factors are important elements in a decision support system, they lie outside the scope of this study and are not analyzed in the examples that follow. APPLYING THE METHODOLOGY TO RESOURCE MANAGEMENT QUESTIONS This section illustrates how components of the physician staffing methodology can be further analyzed to help the VA decision maker better understand the implications of alternative management decisions. Sensitivity Analysis Sensitivity analysis permits management, through manipulation of the methodology's component models, to explore the following generic question: How does a key system output (e.g., patient workload) change in response to systematic variation in one or more inputs (e.g., physicians) or productivity-influencing factors (e.g., affiliation status)? Because the analysis is conducted within a model of the process, rather than the process itself, a broad range of alternatives may be explored quickly and cheaply with no disruption of the patient care process, or of administrative and clinical relationships. The ability

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Physician Staffing for the VA: Volume I to perform sensitivity analyses quickly and easily is an essential attribute of a decision support system. Sensitivity analysis can also help the decision maker validate a model and better understand its implications. As management poses questions of the staffing methodology (e.g., the physician FTEE required to staff the psychiatry service at VAMC I in FY 1991), sensitivity analysis can indicate whether the answers are implausible or counterintuitive. If the methodology's recommendations are consistently at variance with management's prior expectations, it is possible that some important relationship was misspecified in the component models, that inappropriate input data were used in exercising the models, or that some important real-world constraints have been ignored. In any event, sensitivity analysis likewise can be used to identify changes that must be made in the models or their application to improve validity. A pair of examples drawn from the current study illustrates the process of sensitivity analysis and also provides a vehicle for noting both its benefits and its limitations. Both the production function (PF) and the inverse production function (IPF) variants of the Empirically Based Physician Staffing Models (EBPSM) provide a means of studying the relationship between physician staffing and patient workload (typically measured in WWUs). Exploring the PF Variant Consider again the estimated PF equation for the inpatient medicine PCA, as presented in chapter 4 (but with t-statistics now omitted): To proceed with the sensitivity analysis, some VAMC is selected and plausible values are assigned to the right-hand-side variables; in Equation 4.11, all except HGROUP6 involve inputs into a production process that yields workload (W), expressed here in terms of WWUs. Then, one or more inputs are varied from their initial values and the workload response can be observed. That is, the sensitivity of output to specific changes in input usage can be examined. (A preview of such analyses was presented in chapter 4 when the coefficient estimates in Equation 4.11 were being interpreted.) Suppose the initial variable values are as follows: MED_MD = 9.00, SUR_MD = 0.25, PSY_MD = 0.20, NEU_MD = 0.50, RESIDENTS =

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Physician Staffing for the VA: Volume I 20.00, FELLOWS = 15.00, SUPPORT/MD = 12.00, SOCW = 4.50, and HGROUP6 = 0 (i.e., this inpatient medicine PCA is assumed not to be located in a large psychiatric hospital). Given these hypothetical values, the derived predicted value of workload is 5,045 WWUs/yr. Assume, for illustration, that two additional full-time internists are added (i.e., MED_MD is increased from 9.00 to 11.00), while support staff are incremented so as to preserve the current level of support-staff intensity—that is, maintain SUPPORT/MD at 12.00.1 Then, all else equal, predicted workload increases to approximately 5,335 WWUs/yr. The comparatively small increment in predicted workload is attributable to the combined effect of diminishing marginal productivity (as captured in the squared term in Equation 4.11) and the negative interaction effect with FELLOWS, which worsens as MED_MD increases (for any given value of FELLOWS). This type of analysis can easily be extended to consider changes in several other inputs at once (in addition to SUPPORT) or to trade off increases in one input for decreases in another. If there were dollar costs available to associate with each input change, the net change in workload for each proposed change in budgetary allocation could be estimated. If such sensitivity analyses were to be performed in a functioning decision support system, the response of output to specified input changes could be displayed graphically in real time to permit rapid visualization and interpretation. To illustrate, the graph of workload as a function of internist FTEE in the inpatient medicine PCA of a hypothetical facility is shown in Figure 7.2; as above, it is assumed that all other variables in Equation 4.11, including SUPPORT/MD, are held fixed. The figure clearly indicates that successive increases in internist FTEE cause workload to increase, but at a decreasing rate, all else equal. The graph also illustrates the danger of projecting too far beyond the range of the data on which the model is based. Figure 7.2 implies that if internist FTEE is increased beyond about 12, the marginal output per additional internist becomes negative. The real world probably does not behave this way. This is a result of the model providing the best statistical fit within the range of the original data. It also illustrates the value of having readily available graphic displays to supplement tabular summaries of the data. This statistically derived PF indicates most precisely the relationships between inputs and outputs when the prediction is made at the sample mean 1    This would be accomplished as follows. The assumption that SUPPORT/MD = 12.00 at baseline implies that SUPPORT = 119.40 FREE, since MD (the sum of all direct-care FREE for hands-on physicians in the PCA) is 9.00 + 0.25 + 0.20 + 0.50 = 9.95. That is, 119.40/9.95 = 12.00. When MED_MD becomes 11.00, MD increases to 11.95, and SUPPORT must rise to 143.40 FTEE for SUPPORT/MD to remain 12.00.

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Physician Staffing for the VA: Volume I values of the inputs. When an attempt is made (as it surely will be) to ask the model to predict output for input values departing significantly from those observed in the sample, the statistical confidence in the prediction diminishes (see Kmenta, 1986). In general, if radical changes in scale are contemplated, such models are ill-equipped to provide management with accurate insights on the anticipated outcome. Also, the model assumes that the technology of care underlying the input-output relationships found in the original sample will not change significantly over time. Similar cautions apply to the expert judgment models. The decision maker's confidence in prescriptions derived through the SADI or the DSE diminishes as the forecasts extend beyond the scale of operations, organizational structures, and technologies familiar to the expert judges. The examples above represent but one type of application of sensitivity analysis. Within the EBPSM framework, many other interesting questions can be asked, some more statistically complex than above. Note, for example, that the parameters of the PF and the IPF are estimated with uncertainty; each reported coefficient estimate is, in fact, the mean of an estimated distribution of possible values. Computer simulations can be conducted to investigate the sensitivity of workload production (in the PF) or physician requirements (in the IPF) to random variations in these coefficient values around the estimated means. Sensitivity analysis could be applied equally well within the expert judgment approaches to staffing. In applications of the SADI, the sensitivity of physician requirements for PCA-related activities to variations in task performance times could be examined. In another type of query involving the SADI (or the DSE), one could study the effect on physician FTEE requirements of alternative rules for combining the task-time estimates of individual experts to derive group consensus estimates. Yet another application, quite simple but powerful, would be to chart how the FTEE recommendations emerging from the Reconciliation Strategy vary with the weighting parameters b, c, and d, as defined in Equation 6.1. Exploring the IPF Variant Consider next the estimated IPF equation for surgeons, as presented in chapter 4 (again, t-statistics are omitted):

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Physician Staffing for the VA: Volume I The salient sensitivity analysis question now is: How does the surgeon FTEE required for patient care and resident education at the VAMC vary as the amount of surgical workload is systematically altered? Let the hypothetical baseline values of the right-hand-side variables be: SURWWU = 0.65; SURCAPWWU = 150; and HGROUP6 = 0. (For computational reasons, workload in the IPFs is deflated by the constant multiplier 10,000.) The effect of changes in surgery inpatient workload on surgeon requirements is summarized in Figure 7.3. In this case, there is evidence of diminishing marginal productivity for surgeons, but it is not so visually obvious as in the medicine example; a close examination of Figure 7.3 reveals that the relationship between SURWWU and SUR-MD' is not linear, but slightly concave. Thus, as inpatient workload increases, surgeon FTEE for patient care and resident education must increase, at a slightly increasing rate. As with the PF equation, caution must be exercised about predicting physician FTEE for workload values that differ significantly from those in the sample that is used to estimate the equation. In particular, as each right-hand-side variable in Equation 4.26 departs from its sample mean value, the prediction error on SUR-MD' increases. Correspondingly, the 95 percent prediction interval widens (see chapter 4), and statistical confidence in the prediction is reduced (Kmenta, 1986). Such sensitivity analyses provide useful insights about input-output relationships at a given VAMC. However, they are not geared to deal with a second type of management issue (outlier analysis) of considerable importance to the VA decision maker if the Reconciliation Strategy is to be implemented as advocated earlier. Outlier Analysis: Comparing Actual Versus Model-Predicted Values for Physician Ftee and Patient Workload One potentially important aspect of the dialogue envisioned between Central Office and the individual VAMC is a careful scrutiny by all parties of the facility's actual performance, along several possible dimensions, in comparison with the performance predicted from components of the physician staffing methodology. Two important, and related, dimensions are physician FTEE levels and workload productivity. If there is little difference between actual and predicted performance in a particular area, the facility is operating according to expectations, and further inquiry typically would not be indicated. If there is a significant difference between actual and predicted—that is, if the facility is an "outlier"—the reasons should be explored.

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Physician Staffing for the VA: Volume I Such gaps do not necessarily indicate that the VAMC is managing its resources poorly. There may be good justification for why physician FTEE or workload production is higher or lower than expected, based on one or more of the staffing models. Or there may not be. Through the two-way dialogue, these points can be put forward, discussed, and resolved. To illustrate how the physician staffing methodology can inform this discussion, an actual-versus-predicted analysis using both variants of the EBPSM is conducted. Specifically, IPFs are used for medicine, surgery, and psychiatry specialties to predict the total amount of physician FTEE for patient care and resident education expected at two actual facilities, VAMC II and VAMC III, in FY 1989. These predictions are then compared with the corresponding actual FTEE reported by the facility, and the percentage difference is computed. In parallel, PFs are used for the medicine, surgery, and psychiatry inpatient PCAs to predict the workload volume expected in these PCAs at VAMCs II and III in FY 1989. (Recall from the PFs reported in chapter 4 that workload is measured in WWUs in the inpatient medicine and surgery equations and in bed-days of care in the inpatient psychiatry equation.) The workload predictions are compared with the corresponding actual WWUs generated in FY 1989, and the percentage difference is computed. These calculations are summarized in Table 7.1. The data are displayed so that the percentage difference in FTEE for each specialty at a facility is paired with the percentage difference in workload production for the PCA in which it is arguably the ''dominant'' physician specialty. Thus, the internist is assumed to be the dominant physician in the inpatient medicine PCA, and so on. The focus is first on psychiatry staffing at VAMC III, a medium-sized unaffiliated facility. When VAMC III's actual FY 1989 values for workload and other variables are inserted in the psychiatry IPF, it can be shown that the expected FTEE level for patient care and resident education is 12.84. The facility's CDR indicates that 8.47 psychiatrist FTEE were allocated to these purposes in FY 1989. The percentage difference is thus [(8.47-12.84)/12.84] × 100 = -34.0. In both surgery and psychiatry, a common pattern arises at both facilities. For a given total workload (not necessarily just inpatient), each IPF indicates that actual staffing is below what is expected for a VA facility with its attributes (e.g., affiliation status). Likewise, for a given FTEE distribution of physician and nonphysician personnel (not just in the dominant specialty), each PF indicates that actual workload productivity in the PCA is greater than expected for a VA facility with its attributes. Although such a parallel pattern is not logically required, it does appear plausible. On the other hand, the pattern does not arise in medicine. This could be explained by historical staffing patterns that have not been adjusted to reflect actual workload. The main point is that such analyses focus and facilitate the inquiry about the appropriateness of current staffing levels. Questions naturally are raised,

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Physician Staffing for the VA: Volume I information is introduced in response, and the Reconciliation Strategy can function as intended. The actual-versus-predicted analysis of Table 7.1 can be extended clearly to encompass all physician specialties, PCAs, and also the expert judgment approaches to deriving predicted FTEE. Choosing an Optimal Specialty Mix of VA Physicians Through Linear Programming The examples considered thus far have been developed in an unconstrained environment. That is, only the relationships between specified inputs and outputs have been considered without taking into account factors that impinge on the availability and productivity of inputs. Each input costs money, takes up space, requires supervision or support services, and so on. If these realities are not considered, the solution developed from the model might make perfect statistical sense, but violate real-world fiscal or operational constraints. In the day-to-day delivery of VA medical care, there are implicit bounds (upper and lower) on the relative proportions in which various providers are combined to meet mission-related activities in the PCAs. For example, residents require supervision, which is typically less than 1 to 1 but more than 1 to 30. In the inpatient medicine PCA, consultations are made by physicians from most other services, but rarely is the implied FTEE close to that contributed by the medicine service itself. In the delivery of VA medical care as envisioned in the SADI and the DSE, there are prescriptively determined (though implicit) bounds on the ratios between physician specialties and between physicians and other types of personnel. Moreover, in some instances, the judgment may be that some minimum level of physician FTEE, in one or more specialties, is required to promote the quality of care. One methodology that can be used to represent these constraints explicitly—and ensure that all are considered simultaneously while deriving an optimum allocation of resources—is linear programming (Dorfman et al., 1958). Three simple applications are presented below to illustrate further how components of the physician staffing methodology might fit into the decision support framework. With respect to Figure 7.1, such mathematical programming models represent one member in the set of possible models that fit into the third box. The VA decision maker could call upon the models as needed. The following three linear programming problems focus on the ambulatory medicine PCA. The VAMC modeled here represents no particular facility, but would be representative of many large affiliated VAMCs (assume RAM Group 5) with a busy ambulatory medicine PCA.

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Physician Staffing for the VA: Volume I Problem 1 Suppose a decision maker at the VAMC wishes to determine what combination of staff physician FTEE, by specialty, will minimize the total annual physician-related salary cost of providing care in the ambulatory medicine PCA as the number of assigned residents is varied from zero to some previously agreed upon maximum, while satisfying the following constraints: (a) provider FTEE will be adequate, in sum, to meet the workload projected for the PCA, and (b) the FTEE ratios between types of attending physicians will not vary beyond the upper and lower bounds of similar ratios observed across the VA system. To operationalize the output constraint specified in (a), the PF for the ambulatory medicine PCA (Equation 4.17) is used. Specifically, FTEE levels entered into this PF must be large enough, and in the proper mix, so that the ambulatory workload levels that these inputs are expected to produce equal or exceed the projected ambulatory workload. For this example and the variants that follow, the projected workload is assumed to be 3,859,312 capitation weighted work units (CAPWWUs). To effectively use the linear programming method of solving constrained optimization problems, both the objective function (in this case, the sum of the weighted salaries) and the constraints (in this case, the PF and the staffing ratios) must be linear functions of the decision variables (i.e., contain no quadratic or higher-order terms, no interaction terms, and so on). When these conditions are not met, one generally must resort to a nonlinear programming solution technique. In the case at hand, Equation 4.17 does contain a quadratic (squared) term, but it enters in such an uncomplicated manner that the equation can be readily approximated by a piecewise linear function with little loss in solution speed and accuracy. Using systemwide representative salaries for MED_MD, OTHER_MD, and RESIDENTS, and assuming (for simplicity only) that there are no rehabilitation medicine physicians available for this PCA at the facility, the linear programming problem can be stated formally as follows:

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Physician Staffing for the VA: Volume I The objective function to be minimized is simply a general statement of the salary costs of these physicians defined as relevant to the problem. The left-hand-side of the production constraint, (a), is that part of the ambulatory medicine PCA PF pertaining to the decision variables in the linear programming; the remaining terms of this PF (in general, those not pertaining to a decision variable) are collected on the right-hand side of the constraint. Referring to Equation 4.17, RMS_MD has been assumed to be zero; the facility is in RAM Group 5, so that HGROUP(3+5) = 1; and thus HGROUP6 = 0, as will the interaction term involving it. Hence, the right-hand side of (a) is computed in this instance as Regarding the inequalities in (b), the first constrains the ratio of MED_MD to OTHER_MD to be ± 20 percent of the typical such ratio in similar VAMCs. The other constraint is designed to ensure adequate staff physician supervision of residents. Both constraints are, of course, illustrative. To demonstrate the effect on staffing and cost of systematically varying the number of residents, these relationships are used but the number of residents is constrained to be less than a given upper bound, which starts at zero and stops when RESIDENTS passes the point of positive marginal productivity. The result of this analysis is shown in Table 7.2 and displayed graphically in Figure 7.4. In each case, the staffing shown for the MED_MD and OTHER_MD is that which results in the minimum salary cost, given that the number of residents is constrained to be no less than 0, 1, 2, etc. In this ambulatory medicine PCA, residents closely substitute for staff physicians, resulting in diminishing salary costs until the point is reached where the addition of more residents no longer justifies their salary expense. Clearly, in an actual application with these characteristics, one would not add more than six residents unless it was felt that the teaching mission or some other benefit not captured in the salary minimization objective justified this additional expense. Problem 2 As noted in chapter 2, the VA Office of Quality Management is developing statistical models relating measurable outcome indicators of the quality of care to various structure and process characteristics of the VA system. In due course, the relationship between physician FTEE and quality indicators will likely be examined. Suppose, for example, that such analyses were to indicate that for high-quality care, the number of residents should never exceed the number of staff

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Physician Staffing for the VA: Volume I physicians assigned to the ambulatory medicine PCA. In this case, the constraint on resident supervision in (b) must be adjusted so that Under these more constrained conditions the result of the linear programming analysis is: The salary cost is $588,427 which is, of course, lower than the four-resident solution but higher than the five-resident solution. Although these tradeoffs are easy to see in this rather simple example (which is one reason why it was chosen), they are less obvious in cases where many more variables are involved and literally an infinite number of solutions are feasible, but only one (or a very small number) of "best" solutions exist. Problem 3 As a final example, suppose that staffing policy at the VAMC requires that this PCA must have at least 0.500 OTHER_MD assigned. Let the number of residents achieve its optimum (i.e., cost minimizing) level. Under these conditions the optimum solution is: The cost of this solution is $548,761, higher than the optimum solution in which OTHER_MDs is smaller. In fact, any deviation from the optimum six-resident solution presented earlier will lead to higher costs. The only way to lower costs is to staff at a level less than required to produce the projected CAPWWU output. This would clearly violate the original workload target requirement. An interesting alternative linear programming formulation of this problem (not presented here) is to recast the question as: What is the maximum output obtainable within a given budget constraint? In that case, the roles of the objective function and the output constraint above are reversed, but the analysis proceeds similarly.

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Physician Staffing for the VA: Volume I These few examples illustrate the variety of ways that the empirically based models can be utilized within a decision support framework to assist management in the rapid evaluation of alternative staffing configurations. Combined with the expert judgment approaches, these systems (once fully developed) should lead to a better understanding of the budgetary and organizational consequences of staffing decisions. Although not illustrated here, sensitivity analysis and other decision management techniques can be applied likewise to the expert judgment staffing models. If both the empirically based and expert judgment models are fully integrated into a comprehensive VA decision support system, it would be possible to derive a better understanding of the budgetary and organizational consequences of alternative staffing proposals. REFERENCES Dorfman, R., Samuelson, P.A., and Solow, R.M. 1958. Linear Programming and Economic Analysis. New York: McGraw-Hill Book Company. Kmenta, J. 1986. Elements of Econometrics. New York: Macmillan Publishing Co. Miller, David W., and Starr, Martin K. 1960. Executive Decisions and Operations Research. Englewood Cliffs, N.J.: Prentice-Hall, Inc. Veterans Health Services and Research Administration. 1989. Manual M-9, "MEDIPP." Washington, D.C.: Department of Veterans Affairs.

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Physician Staffing for the VA: Volume I TABLE 7.1 Comparisons of Actual Versus Model-Predicted Values of Physician Staffing and Workload for Three Specialties at Two Actual VAMCs % Departure from Predicted Value = [(Actual Value-Predicted Value)/Predicted Value] × 100 Physician Specialties VAMC Physician FTEE for Patient Care and Resident Education1 (%) Weighted Work Units (WWUs) for the Specialty's Dominant Inpatient PCA2 (%) Medicine II 1.2 8.2   III 44.9 -0.3 Surgery II -3.8 9.0   III -13.9 12.4 Psychiatry II -15.4 26.5   III -34.0 103.3 1 From the Inverse Production Function. 2 From the Production Function.

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Physician Staffing for the VA: Volume I TABLE 7.2 Optimal Staff Physician FTEE and Corresponding Total Salary Cost for a Hypothetical Ambulatory Medicine PCA as the Number of Assigned Residents is Varied RESIDENTS MED_MD OTHER_MED Salary Cost 0 11.968 0.767 $1,128,939.00 1 9.873 0.633 961,442.00 2 8.078 0.518 822,169.00 3 6.582 0.422 711,120.00 4 5.386 0.345 628,298.00 5 4.488 0.288 573,692.00 6 3.890 0.249 547,313.00 7 3.890 0.249 577,381.00

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Physician Staffing for the VA: Volume I FIGURE 7.1 Elements of a Decision Support System

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Physician Staffing for the VA: Volume I FIGURE 7.2 Nonlinear Relationship between Internist FTEE for Patient Care and Medicine Service Workload, as Derived from the Inpatient Medicine Production Function

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Physician Staffing for the VA: Volume I FIGURE 7.3 Impact of Surgery Inpatient Workload on Surgeon Requirements for Patient Care and Resident Education, as Derived from the Surgery Inverse Production Function

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Physician Staffing for the VA: Volume I FIGURE 7.4 Impact of Variations in Resident FTEE on Physician Salary Cost in the Ambulatory Medicine PCA of a Hypothetical Large Affiliated VAMC