In this article, I describe a mathematical modeling approach to determine the optimal structure (dollars, space) for allocating resource packages when recruiting new faculty, based on expected financial returns from those faculty. Although I use the University of Arizona College of Medicine as an illustrative case study (the model was applied there from 2005 to 2008), this model is a simple and flexible approach that can be adopted by other medical schools irrespective of the magnitude of the resources allocated

Construction of new university facilities for biomedical research accelerated markedly over the last decade. This acceleration was spurred, in large part, by the doubling of the budget of the National Institutes of Health between 1998 and 2002. In general, facilities were constructed with the intent to expand the size of the research base, with the expectation that additional faculty would be recruited to occupy the incremental space rather than to replace outdated buildings.^{1} An intrinsic assumption was that sponsored research support from federal sources would provide the financial safety net for the expansion, through a continually increasing federal allocation. Concomitantly, new facilities provided an avenue to recruit junior faculty, to repopulate the graying biomedical research base.^{2} An assumption that has held for four decades is that most junior faculty will obtain sufficient sponsored research support during a three- to four-year period after their recruitment to meet their institution’s expectations. Even if not successful in doing so, other revenue streams (clinical revenues, philanthropy, state support, tuition) have been available to cross-subsidize and/or stabilize the situation.

However, a different dynamic has been in play since 2003. The flattening in federal research support, especially relative to the number of applications, coupled with declining clinical margins and local/state government support, have undermined the financial safety net. Institutions are competing with one another to recruit from a relatively fixed number of funded investigators, in a form of zero-sum game.^{1,3} This situation has driven up the size of recruitment packages, through simple supply and demand, as well as through the recruitment process itself, which I have previously suggested has characteristics of an auction.^{4} The latter often leads to the “winner’s curse,” in which the winning bidder (e.g., the institution that successfully recruits a faculty member) ultimately concludes that it paid too much for the winning bid (e.g., recruitment package, which typically includes commitments for salary, start-up costs, equipment, additional positions, administrative support, and both research and office/administrative space).

What are the options for dealing with the above situation, realizing that new facilities do exist? The one nearly immutable fact is that the facilities cannot be left unoccupied. In other words, institutions must develop strategies to deal with the current realities while simultaneously advocating a return to a more stable research funding climate.^{5} Discretionary institutional resources must be used as efficiently as possible to maximize the benefit. I describe herein a model that I have developed for addressing this issue. The model allows hypotheses to be tested about the optimal structure of recruitment packages, by varying their configuration and rapidly determining the consequences on predicted financial return to central administration.

#### The Model

##### Optimizing resource allocations

In creating the strategy described below, I made three assumptions:

* There is a maximum amount of space, and a maximum dollar amount, to allocate from central administration over a defined period of time.

* For each faculty member who is recruited, there will be a specified commitment of dollars and space from central administration. The aggregate of these two commitments, for all recruited faculty, will define the total central commitment, which cannot exceed the maximum available for allocation.

* For each faculty member who is recruited, there will be an expected revenue generation that will accrue to central administration over a defined period of time. The aggregate of the totals, for all recruited faculty, will define the total financial return to college of medicine (COM) central administration during that period.

All of these parameters are linked in linear fashion and can be modeled mathematically (see details in the Appendix). For any series of inputs, constituting the dollars and space committed per faculty member by rank, the number of faculty recruited at each rank is directly related to a series of outputs, which include the associated aggregate amount of space and dollars committed and the expected financial returns. A computer program can then be applied to optimize (typically to maximize) a particular output, as inputs are varied, to a high level of precision. For the optimizations illustrated herein, to fill the available space, the expected aggregate financial return to COM central administration was maximized as the number of faculty (at various ranks recruited to a combination of departments and centers) was varied.

In the particular situation discussed below, the model was developed to guide resource allocation from central administration to department heads and center directors (e.g., allocations for “faculty positions”), under the following circumstances:

* Space for allocation to departments and centers was not necessarily contiguous, because it was assigned by research theme.

* Faculty positions were committed to unit heads in half packages (i.e., half of the dollars and space needed to constitute a full recruitment package), with the expectation that unit directors would partner with one another to create a “full” package. The rationale for, and implementation of, this approach have been described previously.^{6,7} The results of applying this model (described below) are reported in full positions, for simplicity of presentation.

Although the above dynamics influenced the details of how the model was constructed and applied at the University of Arizona College of Medicine (UACOM), they did not influence the conclusions, which are generic. Similarly, the numeric values are specific to the UACOM and are used to demonstrate the application of the model in an actual setting. They can and should be modified as appropriate for any particular institution.

The model assumes that total and net revenue generation by existing faculty will not be influenced by whether they move to the new space. Hence, the financial and organizational consequences of space assignment to existing faculty are held constant, and they need not be optimized.

##### Fixed inputs

##### Amount and configuration of space available for allocation.

Available incremental research space was enumerated by modules, consisting of laboratory space and associated support space. The net square feet associated with each module varied depending on the building and laboratory space configuration (see the Appendix for details). Appropriate adjustments were made to allow allocation of modules in noninteger values. A total of 112 incremental modules were available for allocation. In the model, all incremental space for allocation was assigned programmatically. As described in detail previously,^{6} this approach was adopted to facilitate programmatic research, through a recruitment strategy that strongly encourages units to pair with one another.

##### Available financial resources, over time.

Financial resources applicable to recruitment and support of research faculty (from college, university, practice plan, hospital, development, and other sources) were estimated for the duration of the modeling period. This was necessary to constrain projected costs by available resources. For each revenue source, the maximum percentage of the available dollars accessible to a unit or program was entered as an input. Examples:

* Practice plan and hospital funds were assigned for use exclusively in recruiting faculty to programs involving clinical department faculty.

* Funds from philanthropy were committed only to those units appropriate to receive revenues from existing or anticipated endowments under the control of COM central administration.

* Commitments from other colleges or research institutes were designated only for known or anticipated shared programmatic recruitments.

Resources from university administration were designated only for already approved initiatives.

##### Expected financial returns, over time.

Expected financial returns per faculty member, by rank, were expressed as net present value (NPV) for positive cash flows (NPV[+])—see Table 1—as described previously,^{4,8} using data from the UACOM.^{9} The NPV tool explicitly accounts for the time-value of money, allows the financial value of projects to be compared irrespective of their magnitude or duration, and permits addition or subtraction of projects.

The total NPV for any project is equivalent to the sum of the NPV for negative cash flows (NPV[−]), derived from expenses for the recruitment package, and the NPV for positive cash flows (NPV[+]), derived from revenues generated by the faculty member after hire. Because NPV[−] and NPV[+] can be added, the two parameters can be assessed separately. Because the model is for purposes of future projections, estimated NPV[+] values were used, because actual positive cash flows will be known only in the future.

To calculate NPV[−] for a given faculty member, all cash allocations from central administration to support the individual’s recruitment are “discounted” back to the present (Time 0) to reflect their value at Time 0. The magnitude of the discount depends on the value for the discount rate (described elsewhere^{8}) and the number of discounted periods. For example, the present value of $250,000 paid out at the start of a four-year recruitment package is ($250,000), because it is considered Time 0 and is not discounted. (Note: negative values are presented within parentheses.) By contrast, $150,000 paid out at the start of the second year (e.g., after one period is complete), and using a discount rate of 3%, has a present value of ($150,000)/(1.00 + 0.03) = ($145,630), $100,000 paid out at the start of the third year has a present value of ($100,000)/(1.00 + 0.03)^{2} = ($94,260), and $50,000 paid out at the start of the fourth year has a present value of ($50,000)/(1.00 + 0.03)^{3} = ($45,760). The total NPV[−] for the four-year recruitment package is the sum of the four values, or ($535,650). The NPV[+] is calculated using the same logic, by discounting all projected future positive cash flows accruing to central administration back to time 0. As above, total NPV[+] for a given number of periods is the sum of the NPV[+] for each period.

Although NPV[+] varies substantially when laboratory and nonlaboratory faculty, and/or when tenure-track and non-tenure-track faculty, are compared, the optimization model was applied solely to tenure-track, laboratory-based faculty for the first six years after their recruitment, in the analysis presented herein. Of great importance, only the numeric value, and not the derivation of the value, dictates the results of optimization. I have shown the data in the form of NPV primarily to link these results to results in previous publications.^{4,8,9}

##### Hypothesis testing

The two major variables under control are allocation of space and dollars for recruitment. Using the model, I tested the following hypothesis regarding those variables: *Revenue generation to central administration will be optimal if space and dollar allocation to individual faculty are scaled based on expected financial returns from those faculty*.

I tested this hypothesis in two sequential steps. In the first step, I held the dollar amount committed per position constant, irrespective of rank, and varied the process for space allocation. In the second step, the dollar commitment per recruit was scaled proportional to the expected financial returns by rank, while also varying the allocation of modules. By so doing, all possible combinations were tested and were directly compared to determine the optimal configuration.

The average total central cash allocation per faculty position was set at $550,000, dispersed across four years, as described further in the Appendix. Rather than total dollars expended for recruitment, this was expressed as NPV[−] = ($536,500), using a discount rate of 3%, and calculated as described above. This value is an arbitrary amount, and the conclusions reached from the modeling are independent of the absolute value.

##### Step one: Varying space allocation, by rank, holding dollar allocation constant.

Three different scenarios for space allocation were compared (see Table 2):

* The ratio of modules allocated by rank was determined on the basis of expected sponsored research funding. Values for predicted annual support for all sponsored research for newly recruited faculty (assistant, associate, and professor tenure track) at the UACOM during the first six years after their recruitment^{9} were divided by the college benchmark of $400 total costs per net square foot. The quotient gave an expected value for the appropriate space allocation of 2.5, 2.1, and 1.7 modules for professors, associate professors, and assistant professors, respectively (see Table 2, column A). These noninteger values are easily accommodated with modular space assigned at a unit level.

* A fixed, arbitrary ratio of modules was allocated on the basis of faculty rank. For the modeling shown, I used a ratio of 3.0:2.0:1.0 for professors, associate professors, and assistant professors, which would be considered a “typical” approach (see Table 2, column B).

* The number of modules allocated was not differentiated by rank but was, instead, set at an average of 2.1 per faculty (see Table 2, column C).

Using the inputs in column A, and maximizing NPV[+], the optimal configuration is to recruit a total of 55.5 faculty: 14.5 professors, 14.5 associate professors, and 25.5 assistant professors. In total, these faculty will occupy 110 of the available 112 modules. The aggregate NPV[+] from this cadre of faculty, during a six-year period, is $7.765 million. Strikingly, the NPV[−] is ($30,601) million, and the total NPV (sum of NPV[+] and NPV[−]) is ($22.836) million. These large negative values are discussed further below. More detail from the modeling specific to the UACOM, but not affecting the general conclusions described here, is provided in the Appendix.

Using the inputs in Table 2, column B, the optimal configuration was to recruit almost exclusively assistant professors, resulting in a higher total number of recruits—94—and a higher NPV[+], of $11.527 million. However, the recruiting costs increased dramatically, to more than ($51 million), such that the total NPV is nearly ($40 million). All of these results derive directly from the proportionally lower space commitment to assistant professors, allowing more recruits, but with higher recruitment expenses relative to revenues generated.

The inputs in column C gave the optimal result of the three scenarios tested. The NPV[+] was higher, and the total NPV was less negative, as a result of recruiting more professors relative to other ranks. This result derives from the proportionately greater revenue generated by professors relative to space allocated.

Column D shows the optimal result when the constraint in columns A through C, requiring a minimum of 25% of modules to assistant professors, is removed. Consistent with the logic in the above paragraph, essentially all of the positions are allocated to professors, with an NPV[+] exceeding $9 million and the smallest negative total NPV of all scenarios. Although this might be the optimal outcome mathematically, this is neither a practical nor desirable approach, and it is shown only for purposes of illustration.

##### Step two: Varying space and dollar allocation, by rank.

In the second step (columns E, F, and G), the dollar commitment per recruit was scaled proportional to the expected financial returns by rank, as determined previously. Under these circumstances, the allocations were ($635,000) for professors, ($537,000) for associate professors, and ($437,000) for assistant professors.

Column E shows the results when space allocation is also scaled according to the expected financial returns. Framed differently, this is the model that I hypothesized would provide the optimal central NPV[+]. Although only slightly less favorable than the optimal value shown in column C, and not likely to have a recognizable impact during implementation, it is nonetheless surprising, and it refutes my primary hypothesis.

Columns F and G show results for space allocated in a 3.0:2.0:1.0 ratio, and in a constant amount, respectively. The results in column F are qualitatively similar to those in Column B, with predominantly assistant professors recruited at a module allocation ratio of 3.0:2.0:1.0. Results in column G are qualitatively similar to, but less optimal financially, than those in column C, with predominantly professors recruited at a constant allocation of 2.1 modules/faculty member.

#### Testing Other Variables

##### Sensitivity testing

The impact of altering select input variables over a wide range of values was determined. I describe results for (1) varying the minimum percentage of modules assigned to assistant professors, and (2) varying the ratio of modules allocated by rank.

##### Varying the minimum percentage of modules assigned to assistant professors.

The minimum percentage of modules assigned to assistant professors was varied from 10% to 40% (Figure 1A–C). As expected, the number of professors decreased, and the number of assistant professors increased (Figure 1A), with a nearly linear decline in NPV[+] (Figure 1B) and total NPV (Figure 1C). As previously shown (Table 2, Column D), having no assistant professors is the optimal result when the minimum is 0%.

##### Varying the ratio of modules allocated by rank.

The ratio of modules by rank (professor:associate professor:assistant professor) was varied from the baseline of 2.1 for all ranks, by incremental increases of 0.1 for professors, and incremental decreases of 0.1 for assistant professors (Figure 2A–C). For ratios between 2.1:2.1:2.1 and 2.4:2.1:1.8, the optimal result was preferential recruitment of professors (Figure 2A). At ratios of 2.5:2.1:1.7 and higher, the optimum was to recruit more associate professors and a progressively larger fraction of assistant professors. This result follows directly from the numeric ratio of NPV[+]/modules. Once this number becomes higher for assistant professors than for professors (which occurs at 2.5:2.1:1.7), the preferential result is to recruit assistant professors. NPV[+] (Figure 2B) decreased linearly until a ratio of 2.5:2.1:1.7 was reached, primarily as a consequence of fewer professor positions (and total positions) allocated; then, it rose sharply in association with the linear increase in assistant professors (and total) positions. Total NPV (Figure 2C) became slightly less negative as the module ratio widened to 2.4:2.1:1.8, consistent with fewer total positions allocated, but then it dropped sharply as a consequence of the higher recruiting costs relative to revenue generated for the increasing number of assistant professor positions allocated.

The optimization model was then used to probe other variables, including the parameter optimized, the period of observation, and the discount rate.

##### Maximizing the number of faculty recruited at a specific rank.

Shown in Table 3 are results maximizing either the number of assistant professors (column A), associate professors (column B), or professors (column C). The same numbers of assistant, associate, and professor positions are allocated (as dictated by the space constraints), but the financial parameters vary on the basis of the expected NPV[+].

##### Maximizing total NPV.

The optimal financial circumstance for central administration is to maximize total NPV. As shown in Table 3, column D, doing so suggests that only 15.5 assistant professors be recruited. This number derives from the constraint that a minimum of 25% of all modules be assigned to assistant professors. (In the model, the 25% minimum is applied to individual units, rather than in the aggregate. As a consequence, 32.5 modules are committed, rather than the 28 that would be expected based on taking 25% of the 112 modules available.) Nonetheless, total NPV is still negative. If the constraint to distribute 25% of modules to assistant professors is removed, and total NPV is maximized, the optimal result is to recruit no faculty (column E).

##### Increasing the observation period to 10 years.

In comparison with the observation period of six years (Table 2, column C, text), increasing the observation period to 10 years distributes positions more equivalently across ranks (Table 3, column F). This reflects the relative increase in NPV[+] from assistant professors and associate professors, compared with professors, with longer modeling periods (Table 1). Removing the constraint to hire 25% assistant professors (column G) has less impact on the distribution than doing so with a six-year horizon.

##### Varying the discount rate.

A discount rate of 3% was used for optimizations reported herein, on the basis of considerations described earlier.^{8} The consequences of varying the discount rate are illustrated in Table 4. Effects on the input values are illustrated in the top half of Table 4. The higher the discount rate, the lower the values for NPV[+] and the less negative the values for NPV[−]. The consequences are substantially greater for NPV[+] than for NPV[−], and for the 10-year period in comparison with the 6-year period. This is because the positive cash flows progressively increase over the entire time frame, and the relative effect of the discount rate becomes progressively greater at later time points. In contrast, the effect of discount rate on NPV[−] is not influenced by the time period (6 years versus 10 years), because all negative cash flows occur by year four.

Effects of varying the discount rate on optimization results are illustrated in the bottom half of Table 4. For the six-year period, the slight drop in NPV[+] with increasing discount rate is almost precisely matched by the slightly less negative NPV[−], such that total NPV remains relatively constant. For a 10-year observation period, the consequences are more substantial, with a decrease of nearly 20% in NPV[+] and an NPV[−] more negative by greater than 12%. This is for the reason described in the previous paragraph.

#### Implementation

I have used these results to justify providing to unit heads a single, standard recruitment package for all new positions. Although the number of positions varies by unit (see Appendix for UACOM modeling), dependent on local factors, the positions per se are not distinguished by rank, dollars per position, or amount of space per position. This does not mean that the same commitments are ultimately made to all newly recruited faculty, nor that the distribution of faculty recruited will precisely mimic results from modeling. Rather, unit heads are empowered to subdivide the resources provided to optimally meet the specific recruitment needs and desires for their unit. In essence, the unit heads optimize the outputs by distributing resources (varying the inputs) for positions as best suits their needs and recruiting requirements.

One response to the above is that optimization modeling was not necessary to reach these conclusions; for instance, it is commonplace in academic health centers to provide a block of space and a total dollar commitment to unit heads, to use at their discretion. As indicated earlier, the critical differences in the situation described here are (1) the location of the space to be occupied by recruits to a particular unit is not identified a priori but, instead, is determined on the basis of the programmatic theme into which the recruit best fits, and (2) faculty are jointly recruited between units. Both are features of the intent to drive programmatically organized research, although they create anxiety for unit heads, who are concerned about their ability to identify partners or about having sufficient space to accommodate their faculty recruits. Particularly for the latter concern, modeling was important to align available resources with commitments. At the level of central administration, this process also creates a level playing field, decreases any initial inclination to customize the recruitment package on the basis of the requests or credentials of a specific recruit, and minimizes the “winner’s curse.”

The analysis I have presented here necessarily sets boundaries and constraints, to provide clarity in the optimization process. I explored central administration expenses and revenues only. Revenues and expenses accruing to individual units, which are managed in a more heterogeneous fashion, cannot be readily optimized using a single model. It is both feasible and reasonable for individual units to apply this same approach to their faculty recruitment planning.

A striking observation, derivative from the central purpose and findings of this article, is the large negative value for total NPV at six years. This does not mean that the institution will be in a deficit at the end of those periods. That is true for two reasons: (1) an explicit constraint is that dollar commitments cannot exceed available funds, and (2) numbers for NPV[−] and total NPV apply only to central college administration. Regarding the second of these reasons, total sponsored research dollars—direct + indirect (i.e., facilities and administration)— and total clinical collections are each 25-fold higher than the values for NPV[+] per faculty member. The average laboratory-based tenure-track assistant professor will therefore generate $3 million of total revenues, and the average laboratory-based professor will generate $4.3 million of total revenues, during the first six years of their respective appointments. Although these numbers will be highly variable, depending on clinical responsibilities, they indicate that recruitment of 50 to 55 faculty will generate hundreds of millions of dollars of total revenues. These revenues will be used to offset both recruitment costs and ongoing operations. Nonetheless, the large negative total NPV does mean that recruiting costs are disproportionately borne by COM central administration. Framed differently, even during a period of 10 years, COM central administration will recover far less than $0.50 on the dollar, especially when considering additional recruitment costs not illustrated. Even recognizing that the institution as a whole may recover as little as $0.80 on the dollar from F&A revenues,^{10} the brunt of the costs is borne by the COM. Appropriate distribution of F&A revenues at the institution is needed to more equitably balance the costs and benefits associated with faculty expansion.

In summary, allocation to unit heads of homogeneous recruitment packages for new recruits, independent of rank or expected revenue generation by those recruits, optimizes projected positive cash flows to COM central administration. As stated at the beginning of this article, this model is a simple and flexible approach that can be adopted irrespective of the magnitude of the resources allocated.

#### Acknowledgments

The author thanks Steven Wormsley, Tucson, Arizona, for assistance with this work.