Identifying influential factors related to basing air refueling aircraft during wartime employment operations Open Access

This research examines several operational factors related to basing decisions for air refueling (AR) aircraft, commonly called tanker aircraft, within a notional theater of operations. AR is the transfer of fuel from an airborne tanker to a receiver aircraft, which are usually fighter, bomber or intelligence, surveillance and reconnaissance aircraft. Tanker aircraft used in the employment role can extend the range, payload or loiter time of receiver aircraft conducting wartime missions (Secretary of the Air Force, 2019a). During wartime operations, enemy threats in theater may require basing receiver aircraft at bases far from their fighting locations. These operational realities often result in tankers being the only means, via one or more AR events, for receiver aircraft to reach their fighting positions and return to home bases (U.S. Chairman of the Joint Chiefs of Staff, 2016).

Walton and Clark (2021) note modern military operations extensively use US Air Force tanker aircraft in the employment role, including 175 tankers for OPERATION Allied Force in Kosovo and 185 tankers for OPERATION Iraqi Freedom (OIF). Similarly, Kaplan and Rabadi (2012) note that studying AR in its employment role is more challenging than AR in its deployment role, which involves tankers escorting receiver aircraft from their bases (often in the Continental United States) to overseas bases. Readers interested in advanced modeling of AR in its deployment role are referred to the seminal work of Barnes et al. (2004). In recent large-scale operations, as is expected for future operations, no more than a dozen theater bases are used for basing US Air Force tankers. Hackler (2008) suggests tanker basing or tanker beddown, decisions affect other operational variables, including the amount of fuel available to offload to receivers. Determining the best tanker beddown strategy at theater bases is a complex problem, which requires balancing the preference to keep tankers close to expected AR locations (for higher fuel offloads to receivers) with the inherent congestion of aircraft at theater bases. In addition, tanker beddown decisions must consider the adequacy of base infrastructure in terms of runways and parking ramps, as well as the relative safety of the base given its proximity to enemy threats. The present research seeks answers to such questions, specifically how many tankers, and of which types, should be based in theater and where to base the tankers, to best meet AR employment demands? In addition, we examine which operational factors most affect AR-basing decisions for employment operations.

In this paper, we develop a novel mixed integer program (MIP) to optimize various operational considerations of AR within some notional theater of operations. The assignment of enough tankers to meet all receiver fuel onload requirements, i.e. ensuring AR effectiveness, is paramount in the present research. Therefore, our MIP formulation includes constraints ensuring receiver demands are met, and only then do we minimize the number of tankers via the MIP objective function. Although our optimization construct guarantees AR demands are met, the efficient assignment of tanker aircraft to receiver requirements is a doctrinal necessity (U.S. Chairman of the Joint Chiefs of Staff, 2019). Indeed, employing fewer tankers to meet AR demands promotes simplicity and economy of force in operational plans (Walton and Clark, 2021; Secretary of the Air Force, 2019a).

The key contributions of this paper are (1) presenting an integer program for tanker basing decisions and (2) demonstrating and quantifying the statistical significance of several operational factors associated with optimal tanker basing decisions according to the integer program. We studied three macro-level factors: the amount of fuel carried by the tanker aircraft, the flight time from tanker bases to AR locations and the receiver demand at AR locations. We selected these macro-level factors based on their expected influence on AR-basing decisions and the inherent variability of each factor in real-world applications, specifically that (1) allowable fuel loads change based on the specific tanker type as well as other operational factors (e.g. runway length, temperature), (2) the tanker flight time to an AR location is a direct consequence of tanker basing decisions and (3) receiver demands change based on the intensity of operations (e.g. peacetime, wartime). In addition, we studied three micro-level factors as they relate to basing two types of tankers at up to three tanker bases. The micro-level factors are the amount of parking space available, runway length and temperature at the base. These micro-level factors were selected because they were assessed as likely to affect how many, and which types, of tankers to beddown at specific bases, thereby answering our primary research questions. The research is important to AR analysts at the US Transportation Command (USTRANSCOM), who routinely model AR operations using mobility simulations. Although simulations can represent AR employment operations with a high degree of accuracy, USTRANSCOM analysts lack an optimization program to determine tanker beddown options at theater bases. More importantly, tanker analysts at the command lack a comprehensive understanding of the most influential operational factors with respect to basing tanker aircraft. The research described in this paper fills these gaps.

Recurring themes in the literature suggest several key factors are related to AR employment operations during wartime. Most recently, Altner et al. (2024) provided a comprehensive review of tanker operations using an MIP to generate operationally useful AR plans while incorporating several key AR factors, e.g. tanker fuel capacity, flight times between bases and AR locations and AR demands at specific times and locations. Additionally, Altner et al. (2024) included numerous planning constraints previously excluded from prior research on the topic, e.g. on-ground turnaround time, maximum sortie length, altitude restrictions on AR requests and airspace capacity limits. Altner et al. (2024) also benchmarked the quality of their produced AR plans against historical data, which showed the generated plans to be on average about 2.5% more fuel-efficient than human-generated plans. Similarly, Walton and Clark (2021) comprehensively summarized key factors related to AR basing decisions, including the total fuel tankers can deliver to AR locations from bases given tanker aircraft fuel capacities as well as tanker basing limitations, the flight times between tanker bases and receiver onload requirements, and the fluctuating AR demands during wartime given the dynamic nature of combat operations.

Based on these two recent AR publications, it seems reasonable that three factors (fuel load, flight time and demand) may be important predictors of the number of tankers required; however, researchers also suggest other factors (e.g. ramp space, runway length and temperature) directly affect the amount of fuel that can be delivered by tankers to receiver aircraft, and likewise will influence tanker beddown decisions. Additional literature related to each of these operational factors is reviewed in the following sub-sections.

Recent literature suggests the total amount of fuel tankers can deliver to AR locations depends on many factors, including the fuel capacity of the tanker, the number of tankers beddown at base ramps given tanker and ramp dimensions, and tanker-/base-specific factors such as runway length and base temperature. The Secretary of the Air Force (2022) noted robust infrastructure at theater bases was a prerequisite for supporting AR operations. Walton and Clark (2021) suggested tankers with small fuel loads were generally not viable in employment theaters, especially if tanker bases were far from AR locations. However, tankers with larger fuel capacities generally required more parking space on base ramps, which can be a significant constraint in employment theaters. In general, current tanker aircraft have a positive correlation between maximum fuel load and aircraft size (i.e. wingspan multiplied by length). Figure 1 shows this correlation for several current tanker aircraft, including the KC-135, KC-46 and A330-based Multi-Role Tanker Transport (MRTT).

Walton and Clark (2021) noted shorter runways and higher temperatures generally limit maximum takeoff weight, which translates into lower fuel loads. As an example, the fuel loads for KC-46 and KC-135 aircraft with a 12K′ runway at a temperature of 10 °C are 208K pounds (lbs) and 200K lbs, respectively. These fuel loads reflect the maximum fuel storage capacity of each tanker. Conversely, the fuel loads with a 8K′ runway at the same atmospheric conditions for KC-46 and KC-135 aircraft are 193K lbs and 171K lbs, respectively. Base temperature further affects allowable fuel loads. As an example, the fuel loads for KC-46 and KC-135 aircraft with a 8K′ runway at a temperature of 30 °C are 180K lbs and 160K lbs, respectively (U.S. Transportation Command, 2024). The interaction of these tanker-/base-specific factors adds complexity to real-world tanker aircraft basing decisions and, by extension, must be considered by USTRANSCOM analysts conducting AR modeling and simulation.

Another recurring theme in the literature related to AR employment operations suggests flight times between tanker bases and AR offload locations are a key factor in determining tanker requirements. Walton and Clark (2021) stressed that basing tankers farther from AR locations necessarily results in more fuel burned by the tanker and thus less fuel available for receivers. Similarly, Toydas and Saraç (2020) showed the distance from tanker bases to offload points affects the fuel savings possible when optimizing tanker assignments to receivers. Adding to the complexity is the fact that tanker and receiver bases closer to enemy threats are usually more vulnerable to enemy attack, which makes such bases less attractive from an aircraft survivability standpoint. Walton and Clark (2021) noted that a mix of tanker bases, some close and some far from AR demands, could improve operational flexibility and partially mitigate threats to tanker aircraft. Finally, evolving US Air Force operational concepts, such as Agile Combat Employment (ACE), have explored the benefits of theater combat and support aircraft dynamically moving between close and far bases to increase survivability while generating combat power (Gee and Nicastro, 2024; Secretary of the Air Force, 2022). Altogether, such tanker basing decisions directly impact associated tanker flight times to AR points, which has an inverse correlation with the amount of tanker fuel available for receivers. OPERATION Allied Force (Kosovo) showed the detrimental effects of having to base tankers far from AR locations. Begert (1999) highlighted wartime realities, including inadequate base infrastructure and various political constraints, which limited close-in tanker basing options and contributed to the relatively large tanker force of 175 aircraft to meet demands. The inherent relationship between tanker base locations, flight times to AR locations and directly impact on the amount of fuel available to receivers suggests flight time as a key operational factor.

Although the first air-to-air refueling occurred about 100 years ago, tanker aircraft were first used during combat operations, although only sparingly, during the Korean War (Lasley, 2018). The first significant use of AR during air combat operations was during Vietnam and OPERATIONs Desert Shield/Storm (Wallwork et al., 2009). Modern combat necessitates the use of AR to expand the force options available to a commander by increasing the range, payload and loiter time of receiver aircraft conducting various missions within the employment theater (Secretary of the Air Force, 2019a). There are two methods of transferring fuel during AR operations: boom or drogue. US Air Force receivers use the boom method and US Navy and Marine Corps receivers use the drogue method. The type of transfer required further complicates AR demands because not all tankers have the capability to conduct both boom and drogue operations on the same mission. Table 1 shows the prevalence of AR in recent wartime employment operations in terms of the number of tanker sorties, tanker aircraft flight hours and fuel offloaded to receiver aircraft.

Based on the amount of fuel offloaded to receivers during employment operations, daily AR demands could range from 1 to 6 M lbs of fuel offloaded during an average day, which is less than a peak demand day. Grant (2009) reported an average daily offload of about 3.4 M lbs in 2009 within the US Central Command during stability operations for OPERATIONs Enduring Freedom and Iraqi Freedom. Conversely, the first month of combat operations for OPERATION Iraqi Freedom, as reflected in Table 1, required an average daily AR demand of about 12.5 M lbs of fuel offloaded. The relatively wide range of AR demands during wartime operations, stated as an average over an extended timeframe or a peak during a short timeframe, suggests demand could be a key operational factor affecting the number of tankers required.

Analysts at USTRANSCOM studying AR tanker operations use various techniques ranging from low-fidelity, rough planning factors stated in Air Force Pamphlet 10-1,403, Air Mobility Planning Factors (Secretary of the Air Force, 2018) to high-fidelity AR simulations. Planning factor solutions are appropriate to answer relatively simple, isolated analysis questions, such as “how many tankers are needed to refuel six F-16 aircraft traveling 4,000 nautical miles (nm)?” Conversely, AR simulation solutions are necessary when developing feasible, but likely sub-optimal, assignment schedules for hundreds of tankers to thousands of receiver aircraft supporting large-scale wartime operations. The preceding techniques are well understood by USTRANSCOM tanker analysts; however, AR analysts are also interested in identifying general insights into optimal tanker-basing strategies, which is the focus of the present research. Specifically, we construct an MIP to provide optimal solutions of tanker aircraft to receiver demands subject to various operational constraints. We use several designs of experiment (DOEs) to provide structured analyses of the operational factors across a range of reasonable input values (Montgomery, 2005). Then, we use regression analysis to identify the relative importance of the factors and produce regression equations, which USTRANSCOM tanker analysts can use to predict the number of each tanker type to beddown at available theater bases.

Because AR optimization is inherently complex, we offer several simplifying assumptions to make the MIP tractable for our empirical analyses in this initial research effort. First, we do not consider onload requirements at AR locations on a timeline, nor do we consider specific receiver requirements, e.g. 2× F-16 aircraft each receiving 10K lbs of fuel. Instead, all AR demands represent an aggregate demand across all receivers for a single day at each AR location. Altner et al. (2024) similarly optimized AR operations for a single day to reduce problem complexity. Similar to the work of Rossillon (2015), we assume a negligible fuel transfer time from tankers to receivers at the AR location. This assumption is necessary because fuel transfer times vary based on specific receiver types, which are not represented in the present research. Next, we assume tankers are assigned to a single AR location during the day, i.e. tankers cannot “track jump” from one AR location to another. Rossillon (2015) made a similar assumption to reduce formulation complexity. Therefore, any tanker assigned to an AR location must only refuel receivers at that location, even for subsequent sorties during the day. As a direct consequence of the previous assumption, we do not allow AR aircraft to conduct boom and drogue offloads on the same missions or on the same day with subsequent missions, even if the tanker can perform both boom and drogue offloads on a single mission. Also, we assume tankers must return to their beddown base after each mission, i.e. tankers cannot switch bases. Next, some tankers can refuel other tankers in a force-extension role, i.e. tanker-on-tanker refueling, which may result in fewer tankers required to meet AR demands. However, this tanker capability is often limited to missions during execution based on numerous considerations (Walton and Clark, 2021) and therefore less appropriate for generating tanker beddown plans, as in the present research. Finally, we assume there are sufficient maintenance capabilities and aircrew available at each tanker base. Future researchers are encouraged to extend the present work by implementing the operational complexities noted above in subsequent analyses.

The indices for the tanker-basing formulation are defined as follows.

Next, we define the input data parameters as follows.

The MIP formulation requires several decision variables as follows.

x_k,r,i,h,j

Number of tankers of type k in configuration r to beddown at base i on ramp h to support AR location j (non-negative integer variable)

y_k,i,h

Represents if a tanker of type k is beddown at base i on ramp h, with y_k,i,h = 1 if x_k,r,i,h,j > 0 and y_k,i,h = 0 otherwise (binary variable)

z_k,r,i,h,j

Number of mission capable tankers of type k in configuration r at base i on ramp h to support AR location j (non-negative integer variable); the variable z_k,r,i,h,j is required to represent flyable tankers, because some of the physical tankers beddown at the base (x_k,r,i,h,j) are not flyable, primarily due to required maintenance inspections and repairs

We considered a single objective function to minimize the number of tankers beddown in theater, as in Equation (1).

The MIP constraints are provided in Equation (2) through Equation (10).

Equation (2) ensures the number of tankers of type k used does not exceed the inventory available. Similarly, Equation (3) ensures no more than the maximum number of tankers of type k can be parked at base i on ramp h. Equation (4) tracks each tanker used, based on a non-negative value of the primary decision variable x_k,r,i,h,j with BigM a positive value set at least as large as the expected number of tankers. Next, equation (5) ensures no more than one tanker type k can be based at a theater base i and ramp h. Equation (6) accounts for any tanker aircraft maintenance issues by allowing only flyable tanker aircraft to conduct missions. Equation (7) ensures all AR locations are supported with sufficient fuel by employment tankers, subject to the parameters fuel load, burn rate, flight time, minimum landing fuel and maximum sorties allowed from the tanker base to the assigned AR location. The remaining equations, (8)–(10), restrict the various decision and intermediate variables to the proper domains.

The preceding MIP formulation aligns with AR doctrine (Secretary of the Air Force, 2019a), specifically by ensuring receiver demands are met, which is guaranteed via equation (7) and by using assigned tankers as efficiently as possible, which is enforced in the objective function of Equation (1) by minimizing the number of tankers required.

USTRANSCOM tanker analysts are primarily interested in the number of tankers, by type, required to meet AR employment demands, as calculated in Equation (1) of the MIP objective function. Let N^k be the number of tankers of type k beddown in theater, as calculated in Equation (11).

Similarly, the total number of tankers required, of all types, is another useful metric. Therefore, let N^Tkrs be the number of tankers required to meet AR demands, as calculated in Equation (12).

In this paper, we use the MIP formulation to study two separate sets of operational problem factors with respect to the outcome measure N^Tkrs. Specifically, we analyze three macro-level factors (Fuel, FlightTime, Demand) and three micro-level factors (Ramp, Runway, Temp). Exploratory test solutions using fractional factorial designs suggested these factors and several factor interactions were likely important with respect to predicting N^Tkrs. Next, we consider how many levels to test for each factor in the two separate DOEs. Exploratory tests showed evidence of curvature in the outcome metric N^Tkrs, so we selected three-factor levels, encoded as (−1, 0, +1), for each suspected factor. As a result of these exploratory tests, we selected a 3³ full factorial design (FFD) with 27 runs for each DOE analysis, which allows a comprehensive analysis of the suspected factors and interactions (Montgomery, 2005).

We implement the MIP in the Python environment within Army Vantage, a data science platform created by Palantir that uses distributed systems technology. We use the following Python packages to solve the MIPs: NumPy (Harris et al., 2020), pandas (McKinney, 2010) and PuLP (Mitchell et al., 2011). The PuLP package uses the COIN–OR Branch and Cut Solver (Forrest, 2024).

For the statistical analysis, we use ordinary least squares linear regression to assess the relative influence of factor levels against the outcome metric (Montgomery, 2005). We use the regressors in the fitted model and their associated significance metrics (t-statistics and p-values) to assess the relative importance of the factors in predicting N^Tkrs. Finally, we use the Plotly package in R for plotting regression surfaces (Sievert, 2020).

This section provides the results of our empirical analysis. First, we present the macro-level analyses of the suspected factors Fuel, FlightTime and Demand. We also conduct several excursions to test the sensitivity of results to changes in the factor settings. Second, we provide micro-level analyses of the suspected tanker-/base-specific factors Ramp, Runway and Temp.

Let x^Fuel, x^FlightTime and x^Demand be the coded variables, each with (−1, 0, +1) settings, for the macro-level factors Fuel, FlightTime and Demand, respectively. In terms of the allowable Fuel settings, which are MIP inputs for fuel loads in lbs (fuel_k,i), we used the maximum tanker capacities of the current fleet of US Air Force tankers (KC-135, KC-46) to roughly set the mid-point at 210K lbs (see Figure 1). We set the low-point at 40K lbs less than the mid-point based on possible infrastructure limitations (i.e. short runway) and environmental factors (i.e. high temperatures), which reduce allowable fuel loads. Finally, we set the high-point at 40K lbs more than the mid-point based on the fuel capacity of the MRTT, which is another tanker used by several US Allied partners (see Figure 1). Together, these values reflect a reasonable range of fuel loads for this macro-level analysis. The subsequent micro-level factor analysis will more comprehensively explore the tanker-/base-specific interactions of runway length and base temperature on allowable fuel loads.

For the FlightTime settings, which are MIP inputs for the one-way flight time in hrs from a tanker base to an AR location (flight_k,i,j), we used operational data from historical tanker operations (see Table 1) to provide realistic domain bounds spanning the range of expected distances between bases and AR locations. Tanker flight times in OPERATION Desert Shield/Storm and OIF were relatively short with about a 2-h one-way flight time, which we assumed as the low-point setting. For the mid-point, we assumed a 4-h one-way flight time based on the average flight times in OPERATION Enduring Freedom. Next, we assumed a 6-h one-way flight time for the high-point, which roughly aligns with the extended flight times in OPERATION Allied Force (Kosovo) due to overflight restrictions. Next, we set the daily sortie rate (sortie_k,i,j) based on the selected one-way flight times. Assuming an aggressive 4-h turn-around time on the ground between missions for fueling and switching crews, sortie rates of 3.0, 2.0 and 1.0 are possible for flight times of 2-h, 4-h and 6-h, respectively.

For the Demand factor settings, which are MIP inputs for the daily AR demand (onload_j,r), we again leveraged data from historical tanker operations. Grant (2009) reported reasonable average daily AR demands ranging from 1M lbs to 3M lbs, which we assumed as the low-point and high-point settings, respectively. Then, we selected 2M lbs as the mid-point setting, but we acknowledge the selected range reflects an average daily demand instead of a peak demand. Also, most employment theaters will have dozens of AR locations, some requiring offloads via the boom method for US Air Force receivers and some requiring offloads via the drogue method for US Navy and Marine Corps receivers. However, for our analysis, we assume a single AR location (j = 1) with all receivers refueled via the boom method (r = boom).

Several other MIP parameter values are required for the macro-level analysis. We decided to hold these remaining parameters constant to focus on the three macro-level factors. First, we assumed a single base (i = 1) with relatively unconstrained ramp space on a single ramp, i.e. park_k,i,h = 500. We also assumed a relatively unconstrained tanker inventory, i.e. inventory_k = 500. Ferdowsi et al. (2020) made similar assumptions while determining the optimal placement of AR locations along the flight path of the receiver aircraft. Also, we assumed all beddown tankers can fly each day, i.e. the tankers have a 100% mission capable rate (capable_k = 1.0). Next, we assumed the single AR location can be serviced by the single base, i.e. feasible_j,i = 1. Then, we assume an hourly tanker fuel burn rate (burn_k,r) of 11,291 lbs/hour based on the KC-135 aircraft (Secretary of the Air Force, 2018), although the KC-46 burn rate is slightly less and the MRTT burn rate is slightly more than this rate. Finally, we assumed a reserve landing fuel (land_k) of 15K lbs based on operational averages, although numerous weather and geographical factors affect reserve fuels (Secretary of the Air Force, 2019b; Secretary of the Air Force, 2024).

Each MIP was solved to optimality in less than one minute of computation time. Table 2 provides the MIP solution results for each DOE run based on the combination of coded variables: x^Fuel, x^FlightTime, x^Demand.

Table 3 shows the regression results for predicting N^Tkrs based on the three macro-level factors as regressors. The regression model fit was highly significant (F-Statistic = 16.81; p-value <0.001) and accounted for about 80% of the variance in the solution data (Adjusted R² = 0.785). Also, we failed to detect non-normality of the residuals in the fitted model, which suggested robust results across the factor domains.

Figure 2 shows the fitted response surfaces from the regression equation predicting N^Tkrs based on the domain range of the three macro-level factors. The factors FlightTime and Fuel were placed on the x-axis and y-axis, respectively, because they were the most influential. The remaining factor, Demand, was overlaid as three response surfaces that predict N^Tkrs.

Next, we tested the sensitivity of the regression results to changes in the macro-level factor settings for Fuel and Demand. Testing the sensitivity of FlightTime was unnecessary because the range of factor settings (2-h to 6-h one-way flight times) covered expected flight times in most theaters of operation. Table 4 provides the statistical significance of each possible factor combination for the sensitivity analysis.

The previous analysis provided important insights about macro-level factors for a single tanker type beddown at a single theater base without considering the inherent complexities from tanker-/base-specific factors. For this micro-level analysis, we identified the influence of three suspected tanker-/base-specific factors on the outcome measure N^Tkrs. Let x^Ramp, x^Runway and x^Temp be the coded variables, each with (−1, 0, +1) settings, for the micro-level factors Ramp, Runway and Temp, respectively. In this sub-section, we describe the domain range for these settings and the remaining MIP parameters for two tanker types (k = {KC-135, KC-46}), which represent the current tanker types in the US Air Force.

For the domain range of the factor Ramp, we used three rectangular tanker ramp sizes based on sampling real-world tanker bases. The factor design settings (−1, 0, +1) correspond to small (1,750′ × 750′), medium (2,000′ × 750′) and large (2,250′ × 750′) ramps, respectively. To reflect real-world theaters, we assumed three tanker bases (i = {close, mid, far}) were available for basing tankers. The three bases aligned with the three FlightTime settings from the macro-level analysis, with the following one-way flight times from base i to the single AR location: 2-h from i = close, 4-h from i = mid and 6-h from i = far. For this analysis, we assumed each tanker type flies at the same speed, such that flight_KC-135,i,1 = flight_KC-46,i,1 for each base i. The associated round-trip flight times for these bases and expected ground times for current US Air Force tankers (six hours) suggested maximum sortie rates (sortie_i,j) of 2.0, 1.5 and 1.0 for the close, mid and far bases, respectively. To explore the dynamics of basing multiple tanker types on the same base, but on different ramps, we allowed two ramps (h = {1, 2}) at each tanker base with ramp dimensions as stated earlier. Table 5 shows the maximum number of tankers by type that can be parked on each ramp at each base. Parking estimates consider tanker aircraft dimensions, ramp dimensions and various safety restrictions (e.g. interior taxiway clearance, minimum distance between parked aircraft), as stated in basing guidance (Secretary of the Air Force, 2018).

The remaining two micro-level factors, Runway and Temp, together combined to affect allowable tanker fuel loads from each base (fuel_k,i) based on tanker performance characteristics in addition to the maximum fuel storage capacity of each tanker type, which is 200K lbs for the KC-135 and 208K lbs for the KC-46 (Secretary of the Air Force, 2018). For the range of Runway lengths in the micro-level analysis, we selected three runway lengths (8K′, 10K′ and 12K′) for the (−1, 0, +1) factor settings, respectively, which represented the expected range of tanker bases in most wartime scenarios. Next, for the range of Temp values, we used three temperatures (30 °C, 20 °C and 10 °C) for the (−1, 0, +1) factor settings, respectively, which represent the expected range in most wartime scenarios ranging from higher to lower temperatures.

Next, we determined the most appropriate AR demand (onload_j,r) for our micro-level analysis, which was held constant for each run to reduce complexity. Rather than provide an average daily AR demand, as in the macro-level analysis, we selected a stressing, peak-day AR demand based on recent historical events. Moseley (2003) reported the peak month during the start of OIF (19 March 2003–18 April 2003) consisted of ∼376.4 M lbs of fuel offload (see Table 1) or about 12.5 M lbs/day and USTRANSCOM analysts typically model stressing demands of ∼10M lbs/day. For this micro-level analysis, we selected 10M lbs of fuel as the peak daily AR demand.

The remaining MIP parameters for the micro-level analysis are set as follows. We assumed the single AR location could be serviced by each base, i.e. feasible_1,i = 1 for each base i = {close, mid, far}. Next, we assumed both tanker types offload fuel to receivers via the boom method (r = boom) for simplicity. Also, we assumed inventory was relatively unconstrained (inventory_KC-135 = inventory_KC-46 = 500). Then, we assumed burn_KC-135,boom = 11,291 lbs/hour and burn_KC-46,boom = 11K lbs/hour per published hourly fuel burn rates (Secretary of the Air Force, 2018). We also assumed land_KC-135 = 13K lbs and land_KC-46 = 20K lbs as the minimum landing fuel based on US Air Force flying operations manuals (Secretary of the Air Force, 2019b; Secretary of the Air Force, 2024). Finally, we assumed an 85% mission capable rate for both tanker types based on historical rates from OPERATION Allied Force (Begert, 1999) and OIF (Curtin, 2003; Moseley, 2003).

Each MIP was solved to optimality in less than one minute of computation time. Table 6 gives the MIP solution results for each DOE run based on a combination of coded variables: x^Ramp, x^Runway and x^Temp. The outcome measure N^Tkrs is reported along with the allowable fuel loads (each based on Runway and Temp factor settings) and basing solutions for each tanker type k = {KC-135, KC-46}.

Table 7 shows the regression results predicting N^Tkrs based on the three micro-level factors as regressors. The regression model fit was highly significant (F-Statistic = 58.83; p-value <0.001) and accounted for over 90% of the variance in the solution data (Adjusted R² = 0.940). As with the macro-level analysis, we failed to detect non-normality of the residuals in the fitted model via Shapiro–Wilk tests and concluded the results were consistent across the domain range of factors.

In this section, we discuss the results of our empirical analyses. To the best of our knowledge, the results presented here provide the most comprehensive, quantifiable relationship between the six operational factors studied and the number of tankers required for employment operations.

The macro-level DOE results confirmed each macro-level operational factor (FlightTime, Fuel and Demand) was an influential predictor of tanker-basing decisions. Based on the ranked t-statistics in Table 3, each macro-level factor was a statistically significant predictor of N^Tkrs with each p-value <0.01. The FlightTime regressor (β = 23.667) was the most influential with the largest t-statistic. In addition, the two-way interactions Fuel * FlightTime (β = −19.750) and FlightTime * Demand (β = 11.833), as well as the quadratic of FlightTime (β = 17.889), were likewise statistically significant with each p-value <0.02. The primacy of flight times in determining N^Tkrs was expected given the abundant literature citing the challenges of tankers beddown far from AR locations (Begert, 1999; Moseley, 2003; Curtin, 2004; MacDonald, 2005; Hackler, 2008; Grant, 2009; Walton and Clark, 2021). As for the importance of fuel loads, Walton and Clark (2021) suggested larger tanker fuel loads translated into fewer tankers needed to meet AR demands, which aligned with our macro-level results and the sign of the Fuel regressor β = −15.611. Finally, previous researchers noted modern air combat operations have a range of AR demands with the more tankers needed to meet peak demands (Lasley, 2018; Walton and Clark, 2021). The Demand regressor β = 11.278 confirmed this positive correlation with N^Tkrs.

Next, Table 4 shows the main factors, the two-way interaction Fuel * FlightTime and the quadratic of FlightTime were each statistically significant for the fuel and demand excursions. In addition, both fuel excursions and demand excursion #2 showed significant two-way interactions of Fuel * Demand and FlightTime * Demand as well as the three-way interaction of the factors. Together, the excursion results suggested relative consistency of the factors influencing the primary outcome measure N^Tkrs across a diverse range of fuel and demand settings.

The micro-level DOE results confirmed each micro-level operational factor (Ramp, Runway and Temp) was an influential predictor of tanker-basing decisions. Based on the ranked t-statistics in Table 6, each micro-level factor was a statistically significant predictor of N^Tkrs with each p-value <0.001. The regressor Runway (β = −9.111) was the most influential factor and the regressors Ramp (β = −4.890) and Temp (β = −3.000) were also significant. Additionally, the two-way interactions Runway * Temp (β = 2.833) and Runway * Ramp (β = 2.167) were likewise significant. In addition, the quadratic of Runway (β = 6.556) and quadratic of Ramp (β = 4.222) were also significant with each p-value <0.001.

Walton and Clark (2021) suggested access to suitable bases was likely the most influential factor affecting the number of tanker aircraft and associated fuel offload capabilities during employment operations. Similarly, the works of MacDonald (2005) and Hackler (2008) suggested base infrastructure considerations, including runway length and maximum-on-ground values for parking ramps, would influence tanker beddown decisions; however, both researchers considered the interaction of these factors beyond the scope of their respective studies. Indeed, the large number of tankers required for recent employment operations places a premium on securing suitable tanker basing. Gebicke (1993) noted the reason more air combat sorties were not conducted during OPERATIONs Desert Shield/Storm was due to ramps being saturated with combat aircraft and tankers. Similarly, Grant (2009) noted that during OPERATION Allied Force the employment tankers used up about 90% of available basing capacity with tankers reaching or nearly reaching, the maximum-on-ground limits given available parking spaces.

Several important insights were evident from the micro-level MIP solutions reported in Table 6. First, the furthest base (i = far) from the AR location was used only in Run 1, which represented the most stressed operational case with the least ramp space available, shortest runway length and highest temperature. In this extreme case, the KC-135 consumed both ramps at base i = close, while the KC-46 consumed both ramps at base i = mid and a fraction of a single ramp at base i = far. Run 2 and Run 3 showed the basing strategy with the same constrained ramps and runways, but with lower temperatures, was similar to Run 1 with the KC-135 assigned to the close base and the KC-46 assigned to the mid-range base. Next, Runs 4–6 showed that with runways at 10K′, the KC-46 achieved the maximum or nearly maximum (in the case of Run 4), allowable fuel loads. As a result, AR demands were met entirely with KC-46 tankers.

Interestingly, Runs 7–9 showed that the KC-135 consumed all ramps at the close base with KC-46 used at the mid-range base, presumably because the long runways set at 12K′ resulted in increased KC-135 fuel loads. This result was perhaps the most interesting of the micro-level analysis, because it suggested tanker basing strategies must consider tradeoffs between fuel loads and ramp space. Similarly, Walton and Clark (2021) suggested larger tanker fuel loads often come with larger-sized tankers, which consume more of the usually limited ramp space available in theater for tankers. Of note, assuming the maximum possible fuel loads with ideal basing infrastructure (see Figure 1), the KC-135 provides ∼96% of the fuel load of the KC-46, but the KC-135 takes up only ∼72% of the ramp space compared to the KC-46. Thus, Runs 7–9 confirmed the tradeoff between fuel loads and ramp space required when ramp space was constrained. However, the tradeoff between fuel loads and ramp space was less pronounced in the remaining runs, because no KC-135 tankers were beddown on any base ramp except for Run 18. The interesting insight from Run 18 was that the close base split its two ramps between the tanker types, with 22 KC-135 beddown on one ramp and 20 KC-46 beddown on the other ramp.

The main contribution of this paper is a quantitative analysis of several operational factors, which tanker analysts and planners have long suspected of influencing tanker aircraft basing decisions. The employment of tanker aircraft permits receiver aircraft to be based farther away from threats, carry more weapons or have more loiter time over the target. Such advantages for receiver aircraft are only possible by effectively employing tanker aircraft, which are also generally beddown at distant bases. Specifically, we identified the relative importance of three macro-level operational factors (FlightTime, Fuel and Demand) as well as three micro-level operational factors (Ramp, Runway and Temp). The MIP solutions and associated regression results from the macro-level analyses confirmed FlightTime as the dominant factor in determining the number of tankers required to meet AR demands; however, each factor studied was statistically significant along with several factor interactions. Furthermore, sensitivity analyses showed robust results spanning a wide range of Fuel and Demand values expected in real-world operations. Next, the three micro-level factors were also shown to be statistically significant predictors of required tanker fleet size. The micro-level analysis revealed important new insights about tanker fleet mix options between two current US Air Force tanker types. As a result of the present research, USTRANSCOM tanker analysts are now better prepared to analyze a diverse range of typical AR problems ranging from current, peacetime operations to future, wartime scenarios.

For this initial research effort, we made several simplifying assumptions to limit the problem complexity inherent in real-world AR operations. Specifically, we did not consider that receivers may need extended loiter times at the AR location or that conducting AR operations may require extra time to account for rendezvous or that some tankers may offload fuel to other tankers (acting as receivers) to increase their fuel available for offload to other receivers. The latter consideration is termed tanker-on-tanker refueling, which acts as a force extender mission. In addition, tanker aircraft routinely service multiple AR locations on the same mission, i.e. tanker aircraft can perform “track jumping” operations, which increases problem complexity. The interested reader is directed to the work of Altner et al. (2024), who incorporated track jumping (or airspace jumps) in their formulation using a time-expanded network model.

Also, we assumed perfectly efficient offloads, i.e. tankers offload all available fuel exceeding their minimum landing fuel (reserves). In practice, Gebicke (1993) noted tankers during OPERATION Desert Storm often returned to base with about 40% of their allowable fuel loads remaining. Although this “excess” tanker fuel could be used for emergency or air alert operations (Gebicke, 1993), future researchers may wish to account for typical, inefficient fuel offloads from tankers to AR locations.

Next, the present work should be extended by incorporating the effects of kinetic strikes against theater airbases within the range of enemy missiles. Such contested effects are likely influential factors related to tanker-basing decisions. The best tanker bases from a flight time standpoint are those closest to AR locations, which are often within enemy threat range (Walton and Clark, 2021). Along these lines, Goldfeld and Mason (2019) modeled diminished AR capacity due to tanker airbase attacks, which then resulted in reduced aircraft mission sorties in theater. Also, Brown et al. (2006) proposed several defender-attacker-defender optimization models to protect critical infrastructure against enemy attacks. In this case, tanker aircraft could be considered critical components of a theater’s fuel delivery supply chain. Future researchers may consider using similar approaches to capture the effects of enemy actions on tanker-basing decisions.

Finally, the macro- and micro-level operational factors studied in this research exhibit random fluctuations in practice. For example, flight times between tanker bases and AR locations may change due to winds or weather disturbances and AR demands at specific AR locations may change due to wartime requirements. Future researchers could incorporate these additional real-world AR considerations, including various stochastic elements as identified above, in follow-up AR research studies.