A proposed integrated methodology for evaluation and adjustment of Balance of Performance (BoP) schemes in Grand Touring (GT) racecar championships

Abstract

The concept of Balance of Performance (BoP) has been extensively applied to various forms of motorsports, and especially championships involving Grand Touring (GT) racecars. The organizing bodies of these championships have devised schemes restricting various technical features of the participating racecars, in order to achieve competition parity and subsequently avoid dominance of certain racecars. BoP schemes mainly focus on adjusting engine horsepower and racecar weight, additionally including provisions for restrictions pertinent to aerodynamics, suspension, fuel tank capacity, etc. However, the methodology used by the organizing bodies of motorsports championships in order to evaluate and adjust the BoP schemes is not so transparent to competitors and other interested parts (e.g., racecar manufacturers). In fact, no motorsports organizing body applying the BoP concept to GT racecars, has disclosed details associated with imposing, evaluating, and adjusting the BoP scheme for each championship season. In the current paper, an integrated methodology deploying statistical analysis and Data Envelopment Analysis (DEA) is proposed, for the purpose of evaluating and adjusting BoP schemes in any motorsports championship, primarily focused on GT racecars. The scientifically justified framework along with the simulated application resulting in a performance differentiation range of around 0.40 to 0.45%, comprise strong arguments for the validity and effectiveness of the proposed methodology. In this context, implementation of the proposed methodology by motorsports organizing bodies could potentially render the process of evaluating and adjusting BoP schemes more objective and consequently transparent.

1 Introduction

Motorsports and motor racing in general, comprise a peculiar form of competitive sports. In traditional sports like athletics, soccer, basketball etc., athletes define the winning potential. In the case of motorsports, winning potential is primarily dependent on the race vehicle/driver combination, with the race vehicle comprising the major factor for maintaining competitive capability. Race vehicle manufacturers and motorsports contenders, i.e. factory teams or customer teams, constantly seek to obtain competitive advantage by incorporating improved or novel systems, which are the product of research and development process.

In order to ensure competitive parity among contenders in various motorsports championships, organizing bodies devise and subsequently impose specific regulations. These regulations are primarily technologically-oriented, i.e. aim to restrict technological features associated with race vehicle performance within certain thresholds. In motorsports series with worldwide popularity, organizing bodies tend to change the set of regulations on a year-by-year basis, mainly for the purpose of deterring the dominance of certain competitors and subsequently retain or increase the grid of contenders.

The major concept devised by organizing bodies of motorsports championships and especially those associated with the so-called Grand Touring (henceforth GT) racecars, in order to achieve parity among contenders, is the one widely known as “Balance of Performance” (henceforth BoP). Although the concept was initially named “Equivalence of Technology” (EoT), the “Balance of Performance” term is currently adopted by all organizing bodies related to GT racecars (Cotton 2019).

The topic of BoP in motorsports besides its technological aspects, has become a kind of political debate between organizing bodies and contenders and among contenders (Cotton 2016). However, BoP enables organizing bodies to maintain sustainability of a racing series, which is directly associated with a significant number of contenders (grid size), several renowned manufacturers, rationally-sized budget for team owners/racecar manufacturers, along with a dedicated fan base (Raymond 2019).

Organizing bodies of motorsports championships involving GT racecars, publicly disclose the racecars’ technical regulations. However, when it comes to the adopted BoP schemes, only power and weight restrictions are reported. In addition, organizing bodies do not openly disclose the statistical or other mathematical methods applied for defining the deficit or surplus in performance between/among racecars, along with the justification for imposed advantage or handicap. Hence, it can be deduced that BoP schemes are characterized by a lack of transparency (Raymond 2021a).

Despite the fact that racecar performance equalization has gained increased attention in the field of motosports, the subject of evaluating and adjusting BoP schemes in motorsports championships has not been stressed by any scholar. After thorough search by the authors in every scientific research database, no piece of research or study was found, dealing with racecar performance evaluation and adjustment within the framework of a BoP scheme.

Taking into account the above, the current paper comprises a novel attempt to devise a methodology/algorithm based on scientific tools, in order to achieve the following:

• Objectively evaluate BoP schemes imposed by motorsports championship organizing bodies, through detection of performance differences between/among racecars.

• Objectively quantify the advantage/handicap applied to any racecar ascertained to respectively demonstrate inferior/superior performance, when compared to the performance of all other racecars participating in the specific racing series.

• Set the foundations for creating a transparent methodology, which could be adopted by organizing bodies involved with motorsports, and especially those involved with GT racecar championships. Hence, current research effort simultaneously aims to stimulate organizing bodies to publicize these aspects and to motivate other researchers to devise alternative methodologies pertinent to BoP scheme evaluation and adjustment.

The structure of the rest of the paper is as follows: Sect. Balance of Performance (BoP) concept & relevant literature initially describes the concept of Balance of Performance (BoP) in motorsports, subsequently presenting the existing literature relevant to BoP. Section BoP Schemes in GT Racecar Championships provides information pertinent to the BoP schemes adopted by major GT racecar championships across the world, additionally providing research motivation background. Section Proposed Algorithm for BoP Scheme Evaluation & Adjustment is comprised by six subsections. The first, describes the basic concept of the algorithm devised for BoP scheme evaluation and adjustment, while the subsequent five analyze the discrete algorithm steps. In Sect. Simulated Algorithm Application, the devised algorithm is applied to a simulated environment, i.e. a racecar simulator which adopts a BoP scheme and incorporates real-world GT racecars and racetracks. Finally, Sect. Conclusions & Proposals for Future Research makes a summary of the concluding remarks, additionally discussing some proposals for future research.

2 Balance of Performance (BoP) concept & relevant literature

2.1 BoP definition & framework

As previously reported, scholar scientific research dealing with the subject of BoP in motorsports is virtually non-existent. In this context, alternative sources were utilized in order to obtain relevant information, namely how BoP is defined and which elements constitute the BoP framework. As a matter of fact, main literature source is the Racecar Engineering monthly magazine, which is highly specialized in topics pertinent to racecar technology and motorsports series. Other important source is internet and more specifically the webpages of organizing bodies of motorsports series like FIA (Fédération Internationale de l’Automobile), IMSA (International Motorsports Association), and SRO (Stéphane Ratel Organization).

After consulting the aforementioned sources, the following statements were found to best represent the fundamental framework of BoP:

• “… equalize disparate production cars to provide a level field in the interest of competition.” (Divila 2014).

• “… to put the cars in each class on as level a technical field as possible.” (Autoweek 2019).

• “… equalize the machinery part of the playing field, to ensure all competitors have an equal opportunity to stand on the top step of the podium when the race is over.” (Raymond 2020b).

• “… an adjustment method to maintain competitive equivalency between cars within each class, and between classes.” (IMSA 2022b).

Taking into account the previous statements, a more comprehensive definition of BoP can be phrased as follows:

The scheme which is adopted for ensuring that any substantial competitive advantage/disadvantage among racecars is eliminated, so that race wins and podium finishes across the racing season are distributed to as many contenders as possible, thus avoiding dominance of a single or extremely reduced number of competitors.

Besides setting forth the definition of BoP, it is of paramount importance to describe its framework. Due to the advanced technology utilized in motorsports, the task of achieving parity among the competing racecars is deemed quite elaborate, especially when dealing with GT racecars, which incorporate a fairly high diversity in terms of engine/tramsmission layout and external design form (e.g. supercar, sportscar, grand tourer coupe etc.). However, the BoP framework is definitely simpler, mainly because the technical regulations stipulated by the organizing bodies, ensure a high level of commonality regarding technical features among competing racecars.

The framework of BoP lies in the fact that the organizing body has to engineer an entire grid of racecars, in order to ensure a level competition field (Raymond 2020a). However, the organizing body should essentially focus on each racecar which possibly outperforms or is outperformed by any other racecar. According to Raymond (2020c), the BoP variables and consequently the fundamental BoP framework constituents are the following racecar features:

• Racecar Mass: Taking into account Newton’s Second Law, the capability of a vehicle to accelerate and decelerate is inversely impacted by its mass. Hence, acceleration and braking performance are impaired when mass increases.

• Power/Torque Output: The engine comprises the propulsion system of a racecar and any vehicle in general. Its output magnitude determines the acceleration and maximum speed that a vehicle is able to attain. Power and torque represent different measures and subsequently different effects on vehicle dynamics. Power represents the capacity to produce work over time, while torque represents the rotational force at a given point in time. Regarding their effects on vehicle dynamics, available power is associated vehicle’s maximum speed, while torque is associated with the vehicle’s acceleration capability (Hossay 2020).

• Minimum Ride Height: Organizing bodies usually restrict racecars to a prescribed ride height, due to the fact that ride height affects center of gravity and more importantly total downforce and drag. As a general rule, increasing ride height reduces available downforce and subsequently the forces generated by tires on the racetrack.

• Aerodynamic Elements: These elements are usually wings, dive planes, splitters etc., which can be attached to or removed from the racecar. For elements like wings, organizing bodies stipulate the range of permissible angles, given the fact that increasing wing angles increases downforce and also increases drag.

• Fuel Capacity: The distance that can be covered by a racecar for a specific fuel consumption rate, is directly related to its fuel tank capacity. Consequently, competitors demand equality in terms of how far they can go on a full tank of fuel.

2.2 BoP relevant literature

As stressed in the previous subsection, the literature dealing with the subject of BoP in racecars is scarce. In essence, the only source of specialized articles pertinent to BoP in motorsports is the Racecar Engineering magazine. Hence, in the current subsection we shall refer to articles from the particular magazine, written by motorsports experts, in a sequential timeline. As a matter of fact, the magazine issues that were searched, have been published from February 2011 till November 2022. However, the first article focusing on BoP is the one of Divila (2014) (Racecar Engineering, April 2014), albeit concurrently using the term “Equivalence of Technology” (EoT), in order to describe the effort of motorsports organizing bodies to equalize racecar performance. In the particular article, the author delves into the framework and technology developed by the organizers of the Le Mans 24-h race, aiming to restrict fuel consumption and subsequently equalize racecar performance.

The article of Wright (2014) (Racecar Engineering, July 2014), also uses the term “Equivalence of Technology” along with the term “Balance of Performance”. Its main focus is BoP testing framework and the technology used, in the World Endurance Championship (WEC) and the Le Mans 24-h race, which are organized by FIA-ACO (Fédération Internationale de l’Automobile – Automobile Club de l’Ouest). Subsequently, the article of Collins (2015) (Racecar Engineering, May 2015) describes GT racecar homologation process implemented by organizing bodies, mainly focusing on GT3 category racecars and the associated technical aspects affected by the process.

From year 2016 to year 2019, Racecar Engineering magazine published articles dedicated to BoP, authored by Andrew Cotton. The first, published in August, 2016 (Cotton 2016), analyzes the BoP regulatory framework implemented by three different organizing bodies, i.e. FIA-ACO, SRO, and IMSA. Moreover, the specific article descibes the concepts and technological means used for evaluating and adjusting the BoP schemes imposed by the aforementioned organizing bodies. The second article, published in June, 2017 (Cotton 2017) describes the BoP scheme implemented by FIA-ACO for WEC. The author states that FIA-ACO applies a specially developed algorithm for evaluating and adjusting the BoP scheme, however the details of the algorithm are not disclosed. The third article, published in July, 2019 (Cotton 2019) focuses again on the BoP scheme implemented by FIA-ACO in the framework of WEC. Apart from general description of the technical aspects, the author also refers to the policy aspects associated with BoP in the particular championship. Moreover, it should be noted that both the terms “Equivalence of Technology” and “Balance of Perfomance” are used in the article. After year 2019, there is another article by Andrew Cotton in the March 2021 issue of Racecar Engineering (Cotton 2021), which deals with BoP in WEC, focusing on the automated BoP system developed and adopted by FIA-ACO in GTE (Grand Touring Endurance) racecars. However, the author does not disclose any methodological details of the specific automated system.

Another author extensively dealing with the topic of BoP in the Racecar Engineering magazine, is Scott Raymond. In his first article in December 2019 issue (Raymond 2019), he attempts to justify the necessity of a BoP scheme in motorsports. In addition, author cites politics, subjective observations and objective analysis as the main factors associated with BoP decision framework. Raymond’s subsequent article published in March 2020 issue of Racecar Engineering magazine (Raymond 2020a), describes the homologation processes implemented by organizing bodies of GT racecar championships (FIA, IMSA, SRO), mainly focusing on the concept named “performance windows”, which are defined as windows around coupled parameters, i.e. acceleration versus drag, downforce versus drag etc. Also, author refers to the impact of tire manufacturer/compound type selection in the framework of BoP, citing specific examples of organizing bodies either allowing variability in tire manufacturer/compound type selection or imposing usage of tires from a single manufacturer. In addition, author introduces the concept of “reference car”, i.e. the racecar which is used to comparatively evaluate the performance of all other racecars taking part in a racing series.

The special topic of “sandbagging” is analyzed by Scott Raymond in the July 2020 issue of Racecar Engineering magazine (Raymond 2020b). The author defines “sandbagging” as the process of masking a racecar’s performance capabilities, in order to hide its true potential. Alternatively, the term “performance management” is introduced, which includes driver performance management, car performance management, and engine performance management. With respect to driver performance management, any good driver can manipulate inputs (throttle, brakes, and other adjustable components) to make the racecar appear slow. In an analogous way, car performance management is associated with deliberately choosing setups for tires (inflation pressure, compound type), ride height, aerodynamic devices, and suspension, which degrade racecar performance artificially. In addition, the author states that achieving 0.3% of lap time difference among participating racecars, designates elimination of “sandbagging” techniques.

The article of Scott Raymond published in the November 2020 issue of Racecar Engineering magazine (Raymond 2020c), deals with the fundamental physics associated with the BoP process. The author states that general BoP variables are mass, power output, minimum ride height, aerodynamic elements, fuel capacity, and to some extent tires. Subsequently, the author proceeds with performing a concise analysis of each aforementioned BoP variable from a physics perspective, along with its implications on racecar performance. The subsequent article in the February 2021 issue of Racecar Engineering magazine (Raymond 2021a), describes the guidelines asssociated with racecar performance evaluation via race event analysis. The author additionally refers to the exogenous factors that may affect racecar performance, while stressing the usefulness of simulation for assessing the performance potential of a racecar. The ensuing article in the August 2021 issue of Racecar Engineering magazine (Raymond 2021b) descibes the features of the BoP scheme imposed by the organizing body of the year 2021 Nürburgring 24-h race, with the main point being the fact that the different driver line-ups (i.e. professional drivers intermixed with amateur drivers) add uncertainty to the ultimate performance capability of some racecars.

The article published in the May 2022 issue of Racecar Engineering magazine (Raymond 2022a) introduces the novel concept of “raceability” into the framework of BoP. The author defines raceability as the ability of cars to race each other, while he stresses that it additionally refers to a racecar’s ability to deal with traffic between different racecar classes. Moreover, the author stresses the need to focus on balancing racecars over a stint (i.e. bundle of laps between successive pitstops for refueling and/or tire change), instead of focusing on individual lap times. In addition, the importance of racecar homologation process is cited as the foundation of an objective BoP scheme. In the most recent article of Scott Raymond, published in the November 2022 issue of Racecar Engineering magazine (Raymond 2022b), the author draws conclusions pertinent to the dimension of “raceability” within the BoP framework, with the most prominent being: (i) the importance of racecar homologation process to the BoP framework, along with data analysis and decision making, (ii) the necessity of BoP, in order to avoid excessive money spending by manufacturers and competing teams.

3 BoP schemes in GT racecar championships

3.1 General

In the current section, we shall proceed with the description of the BoP schemes adopted by organizers of major GT racecar championships across the globe. More specifically, we shall initially refer to the racecar classes that take part in each championship and subsequently describe the general framework of each organizing body in terms of creating/imposing and sequentially adjusting/modifying the BoP scheme, if any racecar performance differences emerge during races.

3.2 World Endurance Championship – 24 Hours of Le Mans

The World Endurance Championship (WEC) is organized by FIA-ACO, which mainly includes 6-hour endurance races. The first part of defining the BoP scheme is racecar homologation. The racecar manufacturer must initially submit a complete datasheet to FIA-ACO, containing the vehicle features and subsequently the engine performance is evaluated using race fuel on a dynamometer. The second part of the BoP scheme is racecar aerodynamic testing in a wind tunnel. After performing aeroodynamic testing and confirming that the racecar aerodynamic features (especially downforce) fall within the prescribed limits set by FIA-ACO, the positions/angles of the various aerodynamic devices are recorded and set to a specific range of variation, thus restricting the maximum top speed and cornering ability to a pretty narrow value range. After the aforementioned sequence, racecar configuration is frozen to incorporate the mechanical and aerodynamical features determined during the homologation process.

Subsequently, the racecar is permitted to enter the championship. During the course of the championship, racecar performance is evaluated using telemetry data acquired by a variety of sensors fitted on all competing vehicles. Before each race, scrutineering via vehicle scanning takes place, in order to verify that the prerequisites pertinent to dimensions and weight are met. Top speed and cornering speeds are closely monitored on track, while the race is in progress, in order to verify whether any deviation exists with respect to the prescribed aerodynamic features. In addition, acceleration and deceleration are monitored via specialized sensors.

In order to evaluate the BoP scheme, FIA-ACO takes into account for each racecar model, the fastest average lap time of the fastest racecar of the specific model and for each year of homologation (e.g. base model of 2016, evolution of model of 2017 etc.). Additionally, average attained times are calculated for individual track sectors, for the purpose of evaluating cornering speeds, acceleration, and braking. In order to ratify the acquired data as acceptable, the following prerequisites must hold:

• The racecar has covered sufficient distance, i.e. at least 60% of the race distance.

• The racecar has not suffered any significant damage that degrades its performance.

• Weather conditions are not abnormal, i.e. no existence of heavy rain, snow, hail, strong winds etc.

The acquired data are fed into an algorithm, which is available to all teams, in order to elicit any significant performance differences. Although the details of the aforementioned algorithm have not been revealed by FIA-ACO, it is claimed that it has been developed in cooperation with the racecar manufacturers. (Cotton 2017, 2021; FIA World Endurance Championship 2017, 2021; Raymond 2022a).

3.3 IMSA Sportscar Championship

The technical regulations stipulated by IMSA for the Sportscar Championship include specific rules for enforcing BoP, i.e. Attachment 2 of the applicable regulations. The basic elements of the BoP scheme implemented by IMSA are engine power, racecar weight, and fuel tank capacity. Any racecar model intending to participate in the racing series, must complete aerodynamic testing in accordance with FIA rules, at least 30 days before the inaugurating racing event of the season.

For evaluating racecar performance, IMSA fundamentally adopts the terms “Demonstrated Performance” and “Model Representative Car” (MRC). The first term refers to the verified data generated in the course of a racing event, by a racecar which adheres to the IMSA technical regulations, while the second term refers to the racecar selected by IMSA Technical Committee to represent the racecar model and/or manufacturer in analysis. Regarding adjustments in the BoP scheme, the IMSA Technical Committee shall determine the need to make any adjustment after a single or multiple races, taking into account the demonstrated performance of each MRC. With respect to weight adjustments, these are rounded to the nearest 5 Kg increment, while any power adjustment must be tested in advance and then cleared for approval by the IMSA Technical Committee. Also, retrieved data may be used for adjusting the fuel tank capacity of any racecar.

The evaluation of the acquired data from official timing and data loggers takes place immediately after the end of the racing event, while the IMSA Technical Committee proceeds with publishing Technical Bulletins stipulating adjustments at least 7 days before the next racing event, after discussions with teams and/or racecar manufacturers. However, IMSA does not give any further details pertinent to the methodology and/or algorithm implemented, for the purpose of evaluating and adjusting the BoP scheme in the course of the championship series (Autoweek 2019; IMSA 2021, 2022b).

3.4 GT World Challenge

The organizing body is Stéphane Ratel Organization (SRO), which is also the organizing body of many other GT racecar championships across the globe. Current paper shall focus on the GT World Challenge Europe championship, due to the availability of published specialized regulations pertinent to BoP. In specific, year 2021 regulations were retrieved from the website of the particular championship, given that year 2022 regulations available in its website do not contain any special section dealing with the details of BoP scheme. Additionally, several pieces of information regarding the BoP framework of GT World Challenge Europe, were obtained from articles published in Racecar Engineering magazine.

The main racecar class competing in GT World Challenge Europe is GT3. The specific racecar class has become a worldwide standard. As of year 2022, championships organized by SRO include nearly 70% of the GT3 racecars worldwide (Cleeren 2021).

The GT World Challenge Europe and the other racing championships organized by SRO, do differentiate significantly from FIA WEC and IMSA Sportscar Championship regarding the BoP framework. First of all, before the start of the championship season, BoP testing sessions are held in specific racetracks. All racecar manufacturers aiming to compete in the championship, must first make available for BoP testing at least one of their racecars. Subsequently, the racecar undergoes wind tunnel aerodynamic testing. After the racecar obtains clearance with respect to its aerodynamic features, it undergoes testing in Paul Ricard circuit by professional racecar drivers appointed by SRO, with testing data acquisition performed by specially-developed data loggers. Testing at Paul Ricard takes place in two different circuit configurations, namely a fast one with few low-speed turns and long straights and a slow one with many low-speed turns and short straights.

The aforementioned BoP testing process takes 4 days for each racecar, while besides the lap times, sector times are also recorded. In some cases, additional BoP testing can be conducted in the course of the championship season, between racing events, for the purpose of verifying any performance difference. It is worth noting that BoP testing and subsequently participation in every championship is done using tires from the same manufacturer, thus reducing any performance differentiation induced by variations in rubber compound composition.

Another unique feature of the BoP scheme adopted by GT World Challenge Europe and other championships organized by SRO, is the segmentation of racetracks in 4 categories, with each category having different BoP settings. The racetrack categories are the following:

• A: High-speed, lower downforce.

• B: High-speed, medium downforce.

• C: Mid-speed, higher downforce.

• D: Low-speed, medium downforce.

The individual BoP settings for each racetrack category are established by analyzing the data acquired during testing at Paul Ricard circuit, along with performing simulation using specially-developed software. Stéphane Ratel, who is the head of SRO, names the specific concept “variable BoP”, while he also stresses that “a well done BoP is to balance cars, not to balance a team or drivers”. In a similar way to FIA-ACO and IMSA, any further details pertinent to the methodology and/or algorithm implemented by SRO in order to evaluate and adjust the BoP scheme have not been disclosed (Cotton 2016, 2022; RACB 2021).

4 Proposed algorithm for BoP scheme evaluation & adjustment

4.1 Algorithm basic concept

The development of the proposed algorithm was executed taking into account the fundamental algorithm theory of Knuth (1997), thus incorporating the following discrete steps:

• Initial Racecar Testing (1st Step): All the models/types of racecars participating in a motorsports championship, are initially tested implementing a well-defined set of rules, in order to obtain objective figures of each racecar’s performance potential. The set of rules implemented within the particular step aims to ensure that a ceteris paribus testing framework is adopted.

• Designation of Statistically Significant Performance Differences (2nd Step): After completion of the racecar testing, it is imperative to designate any significant performance differences between any racecars. In order to detect those differences, a proper statistical analysis method is required, primarily featuring increased power in pairwise comparisons.

• Selection of Racecar Requiring Performance Adjustment (3rd Step): After designating the statistically significant performance differences, we define which racecar demonstrating superior or inferior performance – when compared to all other participating racecars – has top priority for undergoing performance adjustment, in order to eliminate the aforementioned differences.

• Selected Racecar Power/Weight Adjustment Theoretical Quantification (4th Step): The target performance of the racecar selected in the previous step is equalized to a properly defined “reference racecar”. Similar to the BoP concepts of FIA, IMSA, and SRO, power and weight are modified in order to adjust performance. The theoretical adjusted values of power and weight for achieving the target performance are computed by deploying Data Envelopment Analysis in an inverse way.

• Selected Racecar Re-Testing (5th Step): After obtaining the theoretical adjusted values from the previous step, we re-test the selected racecar implementing the exactly same framework adopted in the first step.

Following completion of the re-testing session, we shall verify the elimination of any pre-existing performance differences by reiterating the second step. If those performance differences continue to exist, we repeat the sequence of Steps 3, 4, 5, and 2.

4.2 Initial Racecar Testing (1st Step)

The specific step is essential in terms of defining how each participating racecar stands compared to every other competing racecar, before the championship series’ inauguration. After consulting the available literature in order to devise a suitable testing framework, it was concluded that the most proper is the approach of SRO described in subsection 3.4. More specifically, all participating racecars individually undergo testing in two different racetrack configurations, i.e. high-speed and mid-speed, by a single professional test driver or a cadre of professional test drivers with similar experience/expertise, who are appointed by the organizing body (Cotton 2016).

Apart from the fundamental elements of racetrack configurations and cadre of test drivers, racecar testing framework should embed the following prerequisites in order to achieve ceteris paribus, i.e. other things being equal:

• Track Temperature/Weather Conditions: Racecar testing must be undertaken on dry racetrack, within a specified ambient temperature range. Precipitation negatively affects racecar traction and grip, while a narrow ambient temperature range ensures equal air density, thus ensuring engine performance is virtually the same across the testing sessions. Moreover, narrow ambient temperature range ensures uniform tire behavior.

• Tires: Using same tires, namely same brand and same compound (soft, medium, hard), ensures that the racecar/racetrack interface is virtually the same for all racecars undergoing testing. In this way, suspension architecture/setup incorporated within each individual racecar shall be the decisive factor for attaining grip and traction. In order to extract maximum racecar performance, soft compound tires are the recommended option, despite the fact they wear faster than medium and hard compound tires (Asher 2022).

• Fuel Tank Capacity: As stressed in the previous section, organizing bodies adopt differentiated approaches regarding the aspect of fuel tank capacity. For the purpose of achieving racecar parity during testing, it is preferable to impose same fuel tank capacity for all racecars undergoing testing. As a matter of fact, it would be more proper to impose fixed fuel tank capacity for all racecars across the season, in order to force racecar manufacturers to develop engines with similar fuel consumption figures and consequently similar power output figures, which can also lead to racetrack performance parity.

As stressed by Raymond (2022a), BoP schemes adopted by organizing bodies mainly aim to balance individual lap times, thus neglecting to balance racecar performance over a stint. The term “stint” refers to a group of consecutive laps executed between successive refueling pit stops. In the course of a stint, a racecar has to deal with two important factors affecting performance. The one is onboard fuel, i.e. in the start of the stint fuel weight is degrading performance, while in the final stage of the stint as onboard fuel decreases racecar weight is reduced, thus enabling faster lap times. The other factor in the course of a stint is tire wear, which degrades traction and grip, thus negating the reduced fuel weight advantage.

Taking into account the concept of raceability, the racecar testing framework should be designed accordingly, in order to evaluate the racecar performance over the course of a stint. Hence, racecar testing should be executed in a stint format, i.e. running from full allowable fuel to minimum feasible fuel. Given the average individual race durations and the available pre-season duration, the authors’ recommendation is for every racecar to execute 3 stints of testing. In essence, our approach pertinent to racecar testing framework, is to focus on evaluating the performance potential of every individual racecar by incorporating all these prerequisites, which ensure the highest feasible level of objectivity.

4.3 Designation of Statistically Significant Performance Differences (2nd Step)

After executing the initial testing of every individual racecar adhering to the prerequisites reported above, it is important to determine any performance differences, which affect the competitive parity between racecars. The scientific way to designate any performance differences is to conduct pairwise comparisons using statistical testing, in order to deduce whether the differences of means of the testing lap times are statistically significant.

In order to designate the most proper statistical test for conducting reliable multiple pairwise comparisons, literature dealing with statistical testing was studied. Before describing the most suitable statistical testing method for multiple pairwise comparisons, we should initially refer to the terms related to the potential of the used statistical test for conducting multiple pairwise comparisons, i.e. power, average power, any-pair power, and all-pairs power.

Power of a statistical test refers to its ability to detect a difference, when there is one. In other words, power represents the probability of correctly detecting a difference between the means of two groups. Subsequently, average power is the proportion of true differences designated. Any-pair power is the probability of rejecting at least one null hypothesis in the correct direction, when conducting a set of comparisons. The any-pair power is nearly identical to the family-wise error rate, which is essentially the probability of erroneously rejecting at least one null hypothesis. Finally, all-pairs power represents the probability of rejecting all false null hypotheses of pairs of means, each in the correct direction. Hence, all-pairs power is a stricter measure compared to any-pair power (Ramsey and Ramsey 2009; Sauder and DeMars 2019).

Fundamentally, racecar testing lap times are data which do not follow a specific probability distribution. Hence, we should not apply parametric statistical tests, thus resorting to nonparametric statistical tests (García et al. 2010). Other significant peculiarities which must be taken into account, in order to select the proper statistical test, are the unequal variances of the recorded lap times and the equal sample size, i.e. each racecar has executed the same number of laps.

Given the features of the testing lap times’ dataset and the existing literature pertinent to statistical tests for multiple pairwise comparisons, we have concluded that the most suitable statistical test for the second step of the algorithm is the Games-Howell statistical test. The justification for the selection of the specific statistical test is outlined as follows (Hsiung and Olejnik 1994; Ramsey and Ramsey 2009; Ramsey et al. 2010; Sauder and DeMars 2019):

• It adequately controls the Type I error, i.e. rejection of null hypothesis for pairs of groups with identical means, in any condition of equal/unequal sample size and variance.

• The any-pair power is superior to other statistical tests dealing with multiple pairwise comparisons.

Regarding the case of the current paper which involves equal sample size, unequal variance, and quite large sample size, the Games-Howell statistical test is designated by Sauder and DeMars (2019) as the one achieving superior performance compared to all other tests in terms of any-pair power, along with sufficiently restricting Type I error.

4.4 Selection of Racecar Requiring Performance Adjustment (3rd Step)

The particular step deals with designating which racecar has significantly superior/inferior performance compared to all other racecars, thus needing to get handicap/advantage in order to eliminate the significant performance excess/deficit. In other words, it fundamentally aims to answer the question “Is the specific racecar too fast or part of the competition is too slow?”

However, if there are several (i.e. more than one) racecars demonstrating statistically significant performance differences compared to all others, we have to devise a method defining the priority of imposing handicap/advantage to any overperforming/underperforming racecar. In this context, we count the instances of statistically significant superior/inferior pairwise performance differences for each racecar. In order to visualize the particular concept, we construct Tables 1 and 2, which represent two different cases of testing lap times’ comparison for 7 racecars, utilizing the Games-Howell statistical test.

Table 1 Example for designating significant performance differences (faster racecar)

In Table 1, we observe that racecar Number 4 has recorded 2 instances of statistically significant superior performance (compared to racecar Number 6 and Number 3), while racecars Number 6 and Number 3 have both recorded 1 instance of statistically significant inferior performance (when compared to racecar Number 4). Hence, we deduce that we must impose handicap on racecar Number 4.

Table 2 Example for designating significant performance differences (slower racecar)

In Table 2, we observe the reverse condition, namely racecars Number 4 and Number 5 have both recorded 1 instance of statistically significant superior performance (compared to racecar Number 3), while racecar Number 3 has recorded 2 instances of statistically significant inferior performance (compared to racecars Number 4 and Number 5). In a similar manner to Table 1, we deduce that we must enhance performance of racecar Number 3. Hence, algorithm’s 3rd step is completed, thus answering the question whether a racecar is fast or slow compared to the competition.

4.5 Selected Racecar Power/Weight Adjustment Theoretical Quantification (4th Step)

The fundamental question that emerges within the context of current step is the following: “How much handicap/advantage should we impose on the faster/slower racecar, in order to eliminate the condition of statistically significant performance differences?”. At this point, we shall refer to the term “reference car”, which has been described by Raymond (2020a) as the racecar undergoing minimal changes in the course of a racing series. In addition, organizing bodies evaluate the measured performance of all other participating racecars relative to the reference car. However, the above methodologies for selecting the reference car are deemed quite ambiguous from a quantitative standpoint.

Our approach for determining the handicap/advantage imposed on a racecar in order to eliminate any statistically significant excessive/insufficient performance, starts by establishing the terms “upper tier reference car” and “lower tier reference car”. Upper tier reference car is defined as the racecar with the highest performance figures among racecars with zero instances of statistically significant performance differences. Regarding the example in Table 1, upper tier reference car is racecar Number 5, while for the example in Table 2, upper tier reference car is racecar Number 2. In an analogous way, lower tier reference car is defined as the racecar with the lowest performance figures among racecars with zero instances of statistically significant performance differences. For the example in Table 1, lower tier reference car is racecar Number 7, while for the example in Table 2, lower tier reference car is racecar Number 6.

After defining the upper and lower tier cars, we implement the following course of action:

• For the example in Table 1, we shall prioritize imposing handicap on racecar Number 4, in order to match the performance attained by racecar Number 5 (upper tier reference car).

• For the example in Table 2, we shall prioritize imposing advantage on racecar Number 3, in order to match the performance attained by racecar Number 6 (lower tier reference car).

From a technical standpoint, our approach attempts to eliminate statistically significant changes without incorporating major adjustments in power and/or weight figures, in a similar way to FIA-ACO for WEC. According to Cotton (2017), FIA-ACO implements limits of 10 kW and 20 kg on power and weight adjustments respectively, in order to avoid any performance pendulum effect.

Following determination of the revised performance figures that should be attained by a racecar in order to eliminate any statistically significant performance differences, we shall attempt to theoretically quantify the magnitude of handicap/advantage imposed on a racecar to achieve the revised performance figures. Given the features of the BoP schemes reported in Sect. 3 of the current paper, we shall confine adjustments to engine power and racecar curb weight. From a technical perspective, the particular approach of exclusively adjusting engine power and/or racecar curb weight is preferable, due to the fact that it does not incorporate extensive technical interventions. Moreover, the interaction of engine power and weight with other racecar internal/external elements is quite straightforward, thus avoiding any excessive increase/reduction in overall performance by applying modest changes in power and/or weight.

The mathematical tool deployed for quantifying the theoretical magnitude of engine power and/or weight changes is Data Envelopment Analysis (DEA), albeit in a manner deviating from the vast majority of DEA applications in various scientific fields dealing with production systems. In essence, DEA applies mathematical programming techniques and models, for the purpose of evaluating relative performance of peer units called Decision Making Units (DMUs). It uses selected performance measures/metrics, which are classified as “inputs” and “outputs”. The optimal solution of the mathematical models/techniques incorporated in various forms of DEA, fundamentally represents best-practice, i.e. the optimal combination of resources (inputs) for a given production level (output) (Zhu 2014).

In our case, racecars do not comprise a typical production system like factories, banks, terminals etc. However, they constitute integrated engineering systems, incorporating several interconnected subsystems/components with certain inputs/outputs. In a similar way to a production system, an engineering system utilizes resources (inputs) for the purpose of providing outputs (e.g. electrical power by a nuclear powerplant). With respect to racecars, utilized resources are represented by the materials used for its construction and the fuel consumed for the engine operation, while outputs are represented by acceleration, top speed, lap times etc.

As previously stressed, 4th step aims to theoretically quantify the magnitude of handicap/advantage imposed on a racecar in order to achieve the revised performance figures, while DEA computes relative efficiency. Hence, we shall deploy DEA in an inverse way, i.e. given the efficiency of a racecar, how much shall we increase/reduce power and/or weight (inputs) in order to achieve the prescribed revised performance level (output).

The concept of inverse DEA was first introduced by the research of Wei et al. (2000), which is outlined as follows: Assuming that the DMU maintains its efficiency score when compared to a fixed set of DMUs, how much more/less input it would require in order to produce a certain higher/lower level of output? The subsequent research of Jahanshahloo et al. (2004) alternatively outlines the inverse DEA concept as the determination of best possible output for a given input level, assuming that the optimal objective value (i.e. efficiency) of the initial DEA model remains unchanged. In addition, the recent research of Amin et al. (2019) states that inverse DEA models determine the required inputs and outputs for a given efficiency target. Regarding the scope of the current paper, it can be assumed that it is closer to the inverse DEA concept proposed by Wei et al. (2000).

After referring to the theoretical foundations of our approach for the determination of handicap/advantage imposed on a racecar in order to bring its performance to a certain level, we shall construct the DEA model that properly suits our case. Initially, we shall determine the inputs and outputs which best correspond to the functional aspects of a racecar. After consulting relevant technical literature, these are the following:

4.5.1 Inputs

• Nominal Maximum Engine Power: Fundamentally, power represents the rate of work production. For the case of a racecar and any vehicle in general, every constructor refers to the nominal maximum engine power figure. From a physics perspective, the magnitude of nominal maximum engine power proportionally affects the top speed that is attainable by a vehicle. Hence, given constant drag coefficient, providing more engine power to a racecar results in higher attainable top speed, thus assuming that engine power is a desirable input.

• Nominal Maximum Engine Torque: The specific measure basically represents the potential of the engine to transfer tractive force to the driving wheels and consequently the potential to accelerate the vehicle. Depending on the architecture of the engine, available torque across engine RPM (revolutions per minute) range and maximum torque varies, consequently resulting in differentiations in acceleration capability. Given that increasing torque results in higher acceleration, engine torque is a desirable input.

• Curb Weight: The specific measure is defined as the weight of the racecar without any fuel and driver onboard. As previously stressed, organizing bodies adopt curb weight in their BoP schemes. From a physics perspective, weight negatively affects the performance of any vehicle. Increased weight is translated into high inertia and consequently decreased acceleration/braking capability. In addition, increased weight is translated into higher friction coefficient for the tires, thus inflicting increased resistance and consequently increased tire wear rate. Hence, weight comprises an undesirable input.

4.5.2 Outputs

• Total Required Time: The particular measure is essentially the total time each racecar records, for completing the required testing session. In our case, each car is tested for 3 stints, with each stint comprising of a standard number of laps, depending on fuel consumption, which is mainly dictated by the racetrack configuration. We have opted for the total required time instead of mean lap time, given that racecar performance over a stint is more representative of a racecar’s potential. Given that the range of mean lap time differences lies within 0.3–0.5%, it is preferable from a computational standpoint to adopt total required time instead of mean lap time. As a matter of fact, total time differences are estimated to be 2-digit (usually 10 to 20 s), while even in lengthy racetracks the mean lap times differences are estimated to be around 0.6 to 1.0 s for a racetrack with 200 s average lap time. Given that superior racecar performance is represented by lower total time, the specific output is undesirable.

Following the determination of the assigned inputs and outputs, we shall construct the DEA model formulation, taking into account the features of a racecar as an engineering system and the inherent attributes of the inputs/outputs. After extensive search of the literature related to DEA, the model deemed more suitable for our case is the non-oriented Slacks-Based Measure DEA model initially suggested by Tone (2001) and subsequently refined by Tone (2010) and Tone (2011), additionally incorporating Variable Returns to Scale (VRS). The formulation of the specific DEA model is written as follows:

$$\:{\rho\:}_{IO}^{*}={\text{m}\text{i}\text{n}}_{\lambda\:,{s}^{+}{s}^{-}}\:\frac{1-\left(\frac{1}{m}\right)\sum\:_{i=1}^{m}\frac{{s}_{i}^{-}}{{x}_{io}}}{1+\left(\frac{1}{s}\right)\sum\:_{r=1}^{s}\frac{{s}_{r}^{+}}{{y}_{ro}}}\:\:\:$$

s.t.

$\:\:\:\:\:\:\:\sum\:_{j=1}^{n}{\lambda\:}_{j}{x}_{ij}+{s}_{i}^{-}={x}_{io}\:(i=\text{1,2},\dots\:,\:m)$$\:\:\:\:\:\:\:\sum\:_{j=1}^{n}{\lambda\:}_{j}{y}_{rj}-\:{s}_{r}^{+}=\:{y}_{ro}\:(\:r=\text{1,2},\dots\:,\:s)$

$$\:\:\:\:\:\:\:\sum\:_{j=1}^{n}{\lambda\:}_{j}=1,$$

$$\:\:\:\:\:\:\:\:{s}_{i}^{-}\:,\:{s}_{r}^{+}\:\ge\:0\:(\:i=\:1,\:2,\:\dots\:,\:m;\:\:\:\:r=\text{1,2},\:\dots\:,\:s)$$

$$\:\:\:\:\:\:\:{\lambda\:}_{j}\:\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:(j=\text{1,2},\:\dots\:,\:n)$$

Where:

• $\:{(\rho\:}_{IO}^{*})$ is the efficiency

• (n) is the number of DMUs under evaluation.

• (m) is the number of inputs of each DMU.

• (s) is the number of outputs of each DMU.

• (x_ij) is the i-th input of the j-th DMU.

• (x_io) is the i-th input of the DMU_o under evaluation.

• $\:\left({s}_{i}^{-}\right)$ is the slack of the i-th input of the DMU_o under evaluation.

• (y_rj) is the r-th output of the j-th DMU.

• (y_ro) is the r-th output of the DMU_o under evaluation.

• $\:\left({s}_{r}^{+}\right)$ is the slack of the r-th output of the DMU_o under evaluation.

• ($\:{\lambda\:}_{j})$ are the intensity variables

However, due to the fact that racecar curb weight comprises an undesirable input and total time comprises an undesirable output we shall resort to some transformations, in order to conform to the fundamental prerequisite of the implemented DEA model, i.e. all inputs and outputs being desirable. With respect to the output (total time), we shall apply the transformation suggested by Halkos and Petrou (2019), namely subtracting each recorded value from a positive number significantly greater in value. Hence, if $\:\left({y}_{ro}\right)$ is the the value of each recorded total time, then it shall be transformed as $\:{u}_{ro}=\:-\:{y}_{ro}+\beta\:$, where (β) is a positive number substantially greater than $\:\left({y}_{ro}\right)$, e.g., nearly two times the value of ($\:{y}_{ro})$.

The transformation devised for dealing with the undesirable input of curb weight is converting both maximum engine nominal power and torque into indexed measures, i.e. dividing both with the curb weight and subsequently multiplying with 1,000. The resulting inputs are nominal maximum engine power per tonne and nominal maximum engine torque per tonne. The justification for alternatively implementing the aforementioned indexed measures is presented below:

• Maximum Acceleration: Taking into account the Newton’s second law to any vehicle under accelerated motion on a level surface, we get the following equation (Stone and Ball 2004):

$$\:m\times\:\gamma\:=F-(R+D)$$

where (m) is the mass of the vehicle, (γ) the acceleration, (F) the tractive force, (R) the rolling resistance, (D) the aerodynamic drag at a given speed.

Hence, solving with respect to acceleration, we get: $\:\gamma\:=\frac{F}{m}-\left(\:\frac{R+D}{m}\right)$

Given that tractive force (F) is directly proportional to the engine torque, consequently acceleration is directly dependent on engine torque to vehicle weight ratio. Hence, maximum acceleration for any vehicle speed is directly proportional to the fraction of maximum nominal engine torque divided by vehicle mass.

• Maximum Speed: Assuming again motion on a level surface, at any attained speed the following equation holds (Sun and Zhu 2015):

$\:{P}_{ENG}=\:{P}_{DRAG}+\:{P}_{ROLL\:}<=>\:{P}_{ENG}=\left({F}_{DRAG}\times\:v\:\right)+(R\times\:v$).

Where (P_ENG) is the power provided by the engine, (P_DRAG) is the power consumed to counterbalance drag, (P_ROLL) is the power consumed to counterbalance rolling resistance, (F_DRAG) is the aerodynamic drag force, (R) is the rolling resistance, and (v) is the velocity.

Regarding the rolling resistance, the following equation holds: R=(K_R)x(W)=(K_R)x(m)x(g), where (K_R) is the pavement friction coefficient, (m) the mass of the vehicle, and (g) the gravitational acceleration.

Hence, the maximum attainable speed is expressed as below:

$$\:{v}_{max}=\:\frac{{P}_{ENG\left(max\right)}}{{F}_{DRAG\left(max\right)}+({K}_{R\:}\times\:m)}$$

Taking into account the previous equations, it is deduced that maximum attainable acceleration and maximum attainable speed of a racecar are inversely proportional to its weight, thus designating that the alternative measures of nominal maximum engine power per tonne and nominal maximum engine torque per tonne adequately incorporate the negative effect of the racecar’s weight.

Regarding the selection of the DEA model for the purpose of assessing efficiency of racecars, the justification is the following:

• Being non-radial model, it excludes the assumption of proportional changes in inputs and outputs, concurrently dealing directly with slacks (Tone 2017). The specific feature fits properly to engineering systems, taking into account that processes and interactions within these systems are not always linear.

• It is non-oriented, namely aims to both minimize inputs and maximize outputs (Tone 2017). Given the fact that engineering system designers apply optimization techniques simultaneously reducing utilized resources and maximize performance, the non-oriented approach is definitely proper for the case of racecars.

• With respect to additionally incorporating the VRS (Variable Returns to Scale) assumption, it should be noted that VRS DEA models calculate pure technical efficiency (Ramanathan 2003), i.e. the efficiency of a production process converting inputs into outputs, regardless of the operations’ scale and the associated prices/costs (Avkiran 2006). Consequently, the technical efficiency is the type of efficiency applicable to engineering systems, where resource utilization (inputs) and performance (outputs) are inherently of technical texture.

It should be stressed that the DEA model of Tone (2001, 2010, 2011) has not been included in research efforts pertinent to inverse DEA, taking into account the most focused relevant research (Wei et al. 2000; Jahanshahloo et al. 2004; Lim 2016; Zhang and Cui 2020; Chen and Wang 2021).

In order to deal with the aformentioned limitation, we shall implement a “trial & error” approach, i.e. for a target total required time (output) we shall compute efficiency for different combinations of revised inputs and compare it to the initial efficiency. The lowest difference of the computed revised efficiency compared to initial effiiency shall be the decisive criterion for implementing a specific combination of revised inputs. For the purpose of making the above process comprehensible, we shall construct the following example, which is numerically represented by Tables 3 and 4.

Table 3 Example data for input adjustment of racecar demonstrating statistically significant performance (faster racecar)

As previously stressed, Number 5 racecar comprises the upper-tier reference car, hence the target value for racecar Number 6 regarding total required time shall be 6,283.432 s. Subsequently, we shall impose combinations of handicaps (inceased curb weight/reduced maximum power) and recalculate efficiency, as shown in Table 4. At this point, it should be stressed that power handicaps are imposed in 5 BHP decrements, while curb weight handicaps are imposed in 10 kg increments. Moreover, it is important to underline that for every engine setting a certain power curve and a certain torque curve corresponds. With this in mind, we shall conform with the handicap/advantage concept of BoP schemes described in Chap. 3, which include power and weight adjustment (Fig 1).

The approximate theoretical magnitude of power and weight handicaps imposed on racecar Number 6 shall be determined by the recalculated efficiency scores. In order to monitor the value proximity of each recalculated efficiency score, we shall include a column containing efficiency delta, i.e. the initial efficiency subtracted by the recalculated efficiency. Hence, the selected revised power and weight figures for racecar Number 6, shall be the combination which results to the smallest efficiency delta.

Table 4 Power/weight adjustment and recalculated efficiency scores for racecar number 6 (target total time: 6,283.432 s)

Taking into account the data of Table 4, it is deduced that implementing the adjustment coded as “P0 + W1”, is the less severe handicap to impose on racecar Number 6, in order to achieve the revised target total time of 6,283.432 s and simultaneously retain its efficiency score. Hence, assuming that efficiency score remains the same, the recommended adjustment in order to achieve the aforementioned target total time is the one represented by code “P0 + W1”, with the primary criterion being that it encompasses the less severe handicap and consequently incorporates less risk for incurring a pendulum effect on racecar performance.

4.6 Selected Racecar Re-testing (5th Step)

After defining the approximate theoretical magnitude of imposed handicap, in order to achieve the revised total time and subsequently eliminate the pre-existing statistically significant performance difference, it is imperative to actually re-test the racecar (Number 6 for the case of the preceding example). Of course, the repeated testing shall follow the very same concept implemented for initial testing, i.e. the racecar driven ideally by the same test driver (or alternatively by a test driver of similar capability) under similar track conditions, using same tires and same fuel tank capacity.

Further modifications to the racecar except for power and/or weight are not recommended, in order to avoid disturbing the functionality and the interconnection among various systems/subsystems, which is directly related to lap time production capability and consequently to racecar efficiency. For the case of weight adjustment, it is crucial for the organizing body not only to dictate the magnitude of adjustment, but to additionally designate the racecar areas that shall receive extra weight or have weight removed, for the purpose of maintaining the pre-exising center of gravity.

With respect to the algorithm reiteration after racecar re-testing, 2nd step shall be repeated in order to verify that the statistically significant performance difference of the re-tested racecar has been eliminated. After successive reiterations from 2nd step to 5th step, we shall terminate the particular cyclical process when all statistically significant performance differences are eliminated, thus achieving a fair BoP scheme for the specific grid of racecars.

Fig. 1

(Source: Own elaboration)

Schematic diagram of devised algorithm.

5 Simulated Algorithm Application

5.1 Simulation Framework

In many cases, motorsports organizing bodies alternatively select to impose handicap/advantage on a racecar and skip actual testing by utilizing simulation methods, as stressed by the relevant literature. With this in mind, the authors resorted to utilizing a racecar simulator. The selected racecar simulator incorporates a great degree of realism in vehicle dynamics, while it hosts many virtual online races and championships on a worldwide scale. Moreover, major criterion for choosing the specific racecar simulator, is the fact that it implements a BoP scheme for online races and championships, thus comprising ideal choice for evaluating the algorithm.

The first element of the simulation framework is the cadre of racecars that shall constitute the grid and subsequently the subjects of performance evaluation/adjustment. The racecar category selected is GT3, taking into account that nearly all GT racecar championships across the world include this category, while even championships not currently including GT3 racecars (e.g., FIA WEC) are planning to incorporate GT3 racecars as the primary racecar format (Cotton 2022; Kilbey 2021). Hence, 14 racecars conforming to the GT3 racecar format were selected. In order to avoid any inference pertinent to brand preference and affiliation, we renamed the 14 racecars using words from the aviation phonetic alphabet, i.e. Alpha, Bravo, Charlie, Delta, Echo, Foxtrot, Golf, Hotel, India, Juliet, Kilo, Lima, Mike, and November.

With respect to racecar testing, the following prerequisites were implemented, for the purpose of attaining ceteris paribus testing conditions:

• A single driver (i.e. one of the authors) exclusively executed testing and re-testing of the racecars. At the time of testing, the appointed test driver possessed driving experience in the specific racecar simulator of over 240,000 km, with online race performance rated within top 7% worldwide. Hence, it is deduced that the appointed test driver has sufficient capability/skills to test the cadre of racecars near to their technical limits for prolonged time periods.

• Testing (and re-testing) sessions were conducted in same track conditions, namely same temperature and completely dry.

• Tires were all of the same brand/compound. More specifically, soft compound tires were utilized, given that they provide better grip/traction compared to medium and hard compound tires.

• Fuel tank capacity was specified to 100 L for all racecars.

• Tire wear and fuel consumption rate were assigned equal to unit (defined “1x” in the racecar simulator interface), namely corresponding to real-world figures, as provided by the racecar simulation software.

• Each stint started from the exit of the pit paddock, with the racecar being initially stationary.

• Number of laps within each stint was fixed for all racecars, determined by the racecar with the highest recorded fuel consumption rate. For example, if the less fuel-efficient racecar could make 10 complete laps with 100 L of fuel, then all racecars were tested for 3 stints, each comprising of 10 complete laps.

• All racecars undergoing testing had exactly the same downforce values, as provided by the simulator software.

• Damage due to crash or exit from the racetrack was set to “Heavy”, i.e. any improper contact with barriers or any other object affects the driveability of the racecar. If any damage was inflicted during the course of the stint, the effort was aborted and subsequently re-initiated.

• Racetrack limits’ violation setting was set to “Strong”, i.e. any instance that the racecar is assessed to ignore the track limits and results into an unfair lap time advantage, a severe time penalty is imposed. If any such penalty was imposed in the course of a stint, the effort was aborted and the stint was started from the beginning.

• Minor racecar slides resulting into minimal time losses, were not assumed to be rejection criterion for a particular stint. On the other hand, discrete exits or slides which can add over 1 s to the lap time, are deemed as criteria for rejecting a stint.

• Moreover, the Traction Control System (TCS) setting was the same for all tested cars. The extensive previous driving experience, designated setting “2 out of 5” (“0” no TCS intervention and “5” maximum TCS intervention), which has proven to be the best combination of minimal TCS intervention and safety for avoiding inadvertent loss of racecar control.

The racetracks selected for conducting the racecar testing, in order to evaluate the BoP scheme implemented by the racecar simulator, are the 24-hour configuration of the Nürburgring racetrack and the Le Mans 24 Hours racetrack (alternatively known as Circuit de la Sarthe). The former is 25.378 km in length, designated as a medium-speed and high technicality racetrack. The latter is 13.629 km in length, designated as high-speed and reduced technicality racetrack (RacingCircuits.Info 2022).

At this point, it is important to stress that SRO would implement different BoP schemes for the aforementioned racetracks, while the selected racecar simulator implements unified BoP scheme. Hence, current research effort shall additionally draw conclusions whether the SRO concept of implementing different BoP schemes for different categories of racetracks, is preferable to the unified BoP scheme adopted by the racecar simulator. The maximum nominal horsepower/torque and the racecar curb weight figures of the 14 racecars under consideration, in the framework of BoP scheme implemented by the racecar simulator, are depicted in Table 5.

Table 5 Racecar Simulator BoP Scheme features

5.2 Nürburgring 24 Hours racetrack

Each racecar was tested for a total of 18 laps (3 stints, each comprising of 6 laps), which results in nearly 457 km of covered distance and nearly 2.5 h of testing the racecar to its technical limits. The descriptive statistics of the recorded lap times for each tested racecar are contained in Table 6, while the individual lap times are contained in Appendix B, Part A (Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14).

Table 6 Nürburgring 24 h Racetrack Testing Lap Times Descriptive Statistics (units in seconds)

After completing the 1st step of the devised algorithm, we proceed with the 2nd step, i.e. perform pairwise comparisons with respect to recorded testing lap times, for all possible racecar pairs, using the Games-Howell statistical test. The results of the pairwise comparison statistical tests for the lap times recorded in the framework of the initial racecar testing, are analytically depicted in Appendix D, Table 1. Taking into account the ascertained statistically significant performance differences, we construct Table 7.

Table 7 Dominant/Outperformed instances after initial testing (Nürburgring 24 h racetrack)

With respect to the 3rd step of the algorithm, we shall initially designate which racecar is excessively fast or slow compared to the competition. By observing Table 7, we can deduce that racecar November recorded the highest number of instances being dominant, namely the instances its performance superiority is statistically significant compared to other racecars. Hence, November racecar shall be the first to undergo power and/or weight adjustment in order to eliminate the statistically significant performance differences.

In the framework of the 4th step of the algorithm, i.e. defining the power and/or weight handicap in order to eliminate the statistically significant performance differences of Table 7, the performance of November racecar shall be reduced to the level achieved by Kilo racecar (upper tier reference racecar). The numerical values of DEA model inputs and output are contained in Table 8. At this point, it should be stressed that the product of the sum of inputs and outputs multiplied by number 3 is 9 (3 by 3), which is less than the number of DMU’s (i.e. 14). According to Avkiran (2006), this condition constitutes a fundamental prerequisite for ensuring adequate discriminatory power for the applied DEA model.

Regarding the column named “Transformed Total Required Time (seconds)”, the transformation is executed by abstracting the actual total time from number 18,000, hence $\:{u}_{ro}=\:-\:{y}_{ro}+\beta\:=\:-\:{y}_{ro}+\text{18,000}$. As previously stressed, the specific transformation is necessary in order to incorporate total required time in the DEA model as desirable output.

The efficiency scores along with the resulting slack and lambda values, are contained in Table 1 of Appendix C. The efficient DMUs (efficiency score equal to unit) are Alpha, Mike, and November racecars, i.e. 3 out of 14 or approximately 21.42% of the DMUs. Hence, we can deduce that discriminatory power is satisfactory, given that the number of efficient DMUs does not exceed one-third of the sample (Avkiran 2006).

Table 8 Nürburgring 24 h racetrack DEA inputs & output

In order to quantify the power and/or weight handicap imposed on November racecar, for the purpose of reducing its performance to the level achieved by Kilo racecar (upper tier reference racecar), we shall recalculate efficiency with revised (handicapped) power/torque and weight values. At this point, it is important to stress that the racecar simulator enables power and weight increase/decrease in percentile increments/decrements. Given the particular feature, the theoretical magnitude of imposed handicap shall be determined applying the specific percentile increment/decrement pattern. The results of revised November racecar efficiency scores for different power/torque and weight handicaps, while assuming 9,029.027 s as the fixed output value (transformed value), are contained in Table 1 of Appendix E. Data contained in Table 1 of Appendix E, designate that a 2% decrease in power and a 3% increase in curb weight comprises the less severe handicap for November racecar, in order to attain the performance level of Kilo racecar, with efficiency remaining the same.

After defining the magnitude of handicap required for reducing the November racecar performance to the level attained by Kilo racecar, we shall proceed with the 5th step of the algorithm, i.e. re-test the November racecar with its power/torque set to 544 BHP/65.4 kgfm and its weight set to 1,317 kg. Re-testing session lap times of November racecar are contained in Table 29 of Appendix B. The revised total required time is equal to 8,981.947 s, i.e. a transformed value of 9,018.053 s. Hence, the deviation of attained total required time compared to target total required time is (8,981.947-8,970.973)/(8,970.973) = 0.001223 or 0.1223%.

We reiterate the 2nd step of the algorithm, i.e. perform the Games-Howell statistical test for all possible pairs. The results are depicted in Table 2 of Appendix D, with the major observation being the elimination of all previous statistically significant differences, when November racecar is compared to Delta, Echo, Foxtrot, Golf, Hotel, and India racecars. The instances of statistically significant performance differences are shown in Table 9 (3rd step of the algorithm).

Table 9 Dominant/Outperformed instances after November Racecar Re-testing (Nürburgring 24 h racetrack, 544 BHP, 64.4 kgfm, 1,317 kg)*

The data of Table 9 designate the requirement to handicap Mike racecar. Once more, Kilo racecar is the upper tier reference racecar, hence we shall bring the performance of Mike racecar to the level attained by Kilo racecar. In the framework of algorithm’s 4th step, we recalculate the efficiency of the racecars under investigation using the revised numerical values of November racecar. Efficiency scores, slack values, and lambda values, are contained in Table 2 of Appendix C. The efficient DMUs are Alpha and Mike (2 out of 14), thus deducing that discriminatory power is satisfactory in accordance with Avkiran (2006).

For the purpose of quantifying the theoretical power and/or weight handicap that should be imposed on Mike racecar, in order to bring its performance to the level attained by Kilo racecar (upper tier reference racecar), we recalculate efficiency with revised (handicapped) power/torque and weight values. The results of revised efficiency scores for different power/torque and weight handicaps, while assuming 9,029.027 s as the transformed output value target, are shown in Table 2 of Appendix E.

Observation of data contained in Table 2 of Appendix E, indicates that the less severe handicap from a BHP per tonne standpoint is a 1% power decrease, while the less severe handicap from a torque per tonne standpoint is a 1% weight increase. We opt for the 1% weight increase, due to the fact that it is less complex from a technical view. As a matter of fact, it is far less time-consuming and costly to add some kilos of weight, than modifying an engine to lose 5 BHP of maximum nominal performance.

We proceed with the 5th step of the algorithm, i.e. re-testing of Mike racecar with unchanged power/torque and its weight set to 1,339 kg. The lap times recorded by Mike racecar in the framework of re-testing session are contained in Table 30 of Appendix B. Total required time attained in re-testing session is 8,970.392 s, namely with a transformed value of 9,029.608 s. Subsequently, the deviation of attained total required time compared to target total required time is (8,970.392-8,970.973)/(8,970.973) = -6.476 × 10^− 5 or -0.00647%, i.e. the revised total required time is almost identical to the target time.

As with the case of November racecar, we reiterate the 2nd step of the algorithm (Games-Howell test for all possible racecar pairs). Obtained results are depicted in Table 3 of Appendix D, which clearly demonstrate the elimination of all statistically significant differences of Mike racecar when compared to Delta, Echo, and India racecars. Given that Bravo racecar continues to exhibit statistically significant performance differences, we again proceed with the 3rd step of the algorithm, thus constructing the following table.

Table 10 Dominant/Outperformed instances after Mike Racecar Re-testing (Nürburgring 24 h racetrack, 556 BHP, 65.7 kgfm, 1,339 kg)*

Table 10 data observation dictates that a handicap should be imposed on Bravo racecar. This time, Mike racecar is assigned as the upper tier reference racecar, which means that we shall bring the performance of Bravo racecar to the level achieved by Mike racecar. We proceed with the 4th step of the algorithm, by initially recalculating the efficiency of the racecars under investigation using the revised numerical values of November and Mike racecars. The attained efficiency scores, along with the associated slack and lambda values are depicted in Table 3 of Appendix C. The efficient DMUs are Alpha, Bravo, and Mike (3 out of 14), thus assuming that discriminatory power is satisfactory taking into account Avkiran (2006).

The power and/or weight handicap that should be imposed on Bravo racecar, for the purpose of bringing its performance to the same level as Mike racecar (upper tier reference racecar), is determined by recalculating efficiency using revised (handicapped) power/torque and weight values. The results of revised efficiency scores for different power/torque and weight handicaps, while assuming 9,029.608 s as the transformed output value target, are shown in Table 3 of Appendix E.

Data contained in Table 3 of Appendix E, designate as less severe handicap for Bravo racecar to achieve performance identical to Mike racecar, while simultaneously retaining previous efficiency, the 1% power decrease and 2% weight increase. Other handicap combinations, i.e. 2% power decrease/1% weight increase and 2% power decrease/2% weight increase, are definitely more severe, taking into account the associated BHP per tonne and kgfm per tonne figures.

Subsequent re-testing of Bravo racecar conforming with all prerequisites of subsection 5.1, is executed in the framework of the algorithm’s 5th step. Lap times recorded by Bravo racecar are contained in Table 31 of Appendix B. Total required time attained in re-testing session is 8,971.862 s, which corresponds to a transformed value of 9,028.138 s. The deviation of the attained total required time compared to the target total required time is (8,971.862-8,970.392)/(8,970.392) = + 1.639 × 10^− 4 or + 0.01639%, which designates that the revised total required time is almost identical to the target time.

Table 11 Dominant/Outperformed instances after Bravo Racecar Re-testing (Nürburgring 24 h racetrack, 558 BHP, 66.4 kgfm, 1,351 kg)*

Reiteration of the algorithm’s 2nd step (pairwise comparisons deploying the Games-Howell test for all possible racecar pairs) draws the results contained in Table 4 of Appendix D. Observation of the statistical tests results and Table 11, confirms that all previous statistically significant performance differences have been totally eliminated, thus managing to create a fair BoP scheme, at least from a statistical standpoint. Comparing the highest value of total required time to the lowest value of total required time, we get a ratio equal to 9,010.175/8,970.392 = 1.0044349, namely a 0.44349% performance difference, when the slowest racecar is compared to the fastest one.

5.3 Le Mans 24 Hours Racetrack

For the specific racetrack, we implemented exactly the same racecar testing framework as Nürburgring 24 Hours racetrack. Taking into account the fuel consumption features of all racecars under investigation, each racecar was tested for a total of 36 laps, namely 3 stints, each comprising of 12 laps. Hence, the distance covered by each racecar nearly equates nearly 490 km, which is deemed adequate for the purpose of drawing an objective performance potential outlook. The descriptive statistics of the lap times recorded by each tested racecar in the initial testing (1st step of the algorithm), are contained in Table 12, while the individual lap times are contained in Part B of Appendix B (Tables 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 and 28).

Table 12 Le Mans 24 h Racetrack Testing Lap Times Descriptive Statistics (units in seconds)

With respect to the 2nd step of the algorithm, we perform the Games-Howell statistical test for all possible racecar pairs, using the recorded lap times of the initial testing session. The results of the pairwise comparisons are contained in Table 5 of Appendix D, being utilized to subsequently construct Table 13.

Observation of the last column of Table 13, leads us to deduce that Echo racecar is confidently characterized as demonstrating inferior performance when compared to all other racecars. The 11 instances of statistically significant inferior performance clearly designate the requirement to adjust power and/or weight of Echo racecar in an advantageous way (3rd step of the algorithm). Taking into account the data of Table 13, Alpha racecar shall comprise the reference racecar, i.e. Echo racecar should attain the performance level of the Alpha racecar, in order to eliminate the statistically significant performance inferiority. Hence, power and/or weight adjustments of Echo racecar (4th step of the algorithm) shall be defined with the specific assumption.

Table 13 Dominant/Outperformed instances after initial testing (Le Mans 24 h racetrack)

The inputs and outputs of the DEA model formulation for the case of Le Mans 24 Hours racetrack are depicted in Table 14. The inputs remain the same, given that engine nominal maximum power and torque per tonne are not altered, as racecar simulator implements unified BoP scheme. Regarding the output, which corresponds to the column named “Transformed Total Required Time (sec)”, the transformation results by abstracting the actual total required time from the number 18,000, as in the case of the Nürburgring 24 Hours racetrack.

The obtained efficiency scores along with the slack and lambda values, are contained in Table 5 of Appendix C. The efficient DMUs (efficiency score equal to unit) are Alpha, Kilo, Lima, and Mike racecars, namely 4 out of 14 or approximately 28.57% of the DMUs are efficient. Subsequently, we can deduce that discriminatory power is satisfactory, taking into account that the number of efficient DMUs does not exceed one-third of the sample (Avkiran 2006).

Table 14 Le Mans 24 h racetrack DEA inputs & output

The quantification of theoretical power and/or weight advantage that should be provided to Echo racecar, in order to enhance its performance to the level achieved by Alpha racecar (lower tier racecar), shall be performed by initially recalculating efficiency with revised power/torque and weight figures. In a similar manner to Nürburgring 24 Hours racetrack, the selection of the revised power/torque and weight figures shall be made by adhering to the criteria of achieving the same efficiency with least possible increase/decrease to power/torque and/or weight respectively. The revised efficiency scores with a target value for transformed total required time equal to 9,014.754 s, for different combinations of power/torque increase and weight reduction are shown in Table 4 of Appendix E.

Given that minimum efficiency delta is achieved by adopting the initial power/1% weight decrease combination, which simultaneously comprises the least advantageous in terms of BHP per tonne and kgfm per tonne, we proceed with its selection. Moreover, as stressed for the case of Mike racecar in Nürburgring 24 Hours racetrack, opting for weight reduction instead of power increase, is less complex from a technical standpoint and of course less costly. Hence, we shall proceed with re-testing of the Echo racecar (5th step of the algorithm) implementing unchanged power/torque and 1% weight decrease, along with adhering to the prerequisites stipulated by subsection 5.1.

The lap times recorded in the framework of the re-testing session are contained in Table 32 of Appendix B. The total required time is equal to 8,666.874 s, i.e. a transformed value of 9,333.126 s. The deviation of the attained total time compared to the target total time is (8,666.874-8,682.840)/(8,682.840) = -0.0018388 or -0.18388%. After the re-testing session, we reiterate the 2nd step of the algorithm (Games-Howell statistical test for all possible racecar pairs) thus obtaining the results contained in Table 6 of Appendix D. The specific results designate the elimination of all previous statistically significant performance differences, subsequently leading to the termination of the algorithm sequence. Hence, we have managed to shape a fair BoP scheme, given that no statistically significant performance difference is existent, as depicted in Table 15. In terms of slowest and fastest racecar comparison, the attained ratio of biggest to smallest total required time is (8,682.840/8,648.347) = 1.003988, or 0.3988% overall performance difference.

Table 15 Dominant/Outperformed instances after Echo Racecar Re-testing (Le Mans 24 h racetrack, 528 BHP, 56.4 kgfm, 1,273 kg)*

6 Conclusions & Proposals for Future Research

Given that no previous scientific research has dealt with BoP in motorsports, the current paper comprises a starting point for this special topic. Additionally, the tendency of motorsports organizing bodies implementing BoP to avoid disclosure of any detail related to the BoP process, renders the whole subject quite challenging, taking into account the aspects of competitiveness and scientific objectivity. Hence, current research effort fundamentally attempts to set the foundations for objectively evaluating and adjusting any BoP scheme, by deploying proven scientific tools.

The algorithm that has been devised for the evaluation and adjustment of BoP schemes is deemed scientifically sound, given the solidity of the mathematical methods incorporated within its steps. Additionally, the results obtained in the framework of the simulated application of the algorithm, comprise a tangible confirmation of its effectiveness to objectively evaluate and adjust BoP schemes. Hence, the devised algorithm has proven its potential to sufficiently deal with the overlooked topic of BoP in motorsports. Moreover, the discrete steps are quite straightforward in terms of computational complexity, thus rendering its execution quite seamless.

With respect to limitations, it could be stated that the “trial-and-error” approach in the 4th step, due to lack of inverse DEA model incorporating non-radial/non-oriented features and slacks-based measures, adds some computational complexity. In addition, the inherent inability of the adopted DEA model to directly deal with undesirable inputs and outputs, is also adding computational complexity, despite the fact that gives quite accurate adjusted power/weight values.

Taking into account the results of the simulated algorithm application extracted in Sect. 5, the following conclusions are drawn:

• The unified BoP scheme, i.e. same BoP scheme for all racetracks included within a championship calendar, embodies a great amount of bias regarding the demonstrated performance of racecars. As it is observed in Tables 6 and 12 of the paper, the performance of racecars in two different types of tracks (mid speed, high technicality for Nürburgring 24 Hours racetrack and high speed, reduced technicality for Le Mans 24 Hours racetrack) is definitely differentiated between these tracks. Hence, it can be confidently deduced that the SRO approach for segmentation of racetracks in distinct categories is the preferred one, when it comes to defining the BoP scheme before the start of a motorsports championship season and the necessary adjustments while the championship is in progress. Consequently, the approach implemented by other organizing bodies, where adjustments are determined by largely focusing only on the results of previous race(s), without taking into account the racetrack physical configuration, is deemed to encompass reduced subjectivity in terms of detecting performance differences.

• The concept of assigning racecar testing to test driver(s), as implemented by SRO, is deemed as far more proper in order to assess the performance potential of every racecar. Someone can argue that digital telemetry via data loggers (installed on racecars) is sufficient to define any performance inequalities, however the experience and subsequently the driving perception of a seasoned professional test driver cannot be substituted.

• The finalized adjusted BoP schemes, which result after implementation of the devised algorithm, demonstrate a reduced range of racecar performance differentiation. In specific, the adjusted BoP scheme for Nürburgring 24 Hours racetrack has a 0.44349% total time difference when comparing the fastest to the slowest racecar, while the adjusted BoP scheme for Le Mans 24 Hours racetrack has a 0.3988% total time difference when comparing the fastest to the slowest racecar. Despite the fact that the difference of the fastest racecar compared to the slowest racecar is slightly greater than the 0.3% threshold dictated by Raymond (2022b), the elimination of statistically significant performance differences within the framework of testing stints is more solid in scientific terms, given the power of the Games-Howell statistical test and the incorporation of the raceability factor by testing the racecars in stints.

• The adoption of the “upper tier reference racecar” and “lower tier reference racecar” has proven more proper than the “Model Representative Car” adopted by IMSA, in terms of adjusting power and/or weight features of racecars within a fairly reduced numerical value range. If a sole “reference racecar” was adopted in the framework of the devised algorithm, then fundamental difficulties would arise. First and foremost, adjustments of power and/or weight would be definitely of greater magnitude, thus creating a pendulum effect on racecar performance. Secondly, many more racecar re-testing sessions would be required, as all racecars’ performance should lie within a narrower performance range.

Apart from the computational aspects of the devised algorithm, it is of paramount importance to stress that any motorsports organizing body which would use the algorithm and openly inform the contenders about its details, could definitely increase trust and significantly reduce any existing tensions among participating racing teams over the issue of racecar performance. This enhanced transparency would attract additional contenders and subsequently lead to the enlargement of the championship series in geographical terms (more racetracks included in the championship calendar) and financial terms (more participation fees, sponsors, advertised companies etc.). In any case, it is deemed necessary for motorsports organizing bodies to consider implementation of any scientifically sound research on the topic of BoP, in order to render the competition rulebook more transparent and subsequently maintain a sufficient grid size.

Regarding proposals for future research, main statement is that any subsequent research effort dealing with the topic of BoP in motorsports, should concentrate on methodology pertinent to detection of performance differences and the quantification of adjustments. As a matter of fact, proposing alternative statistical tests with increased any-pair power and DEA models which could directly deal with undesirable inputs/outputs and possess sufficient discriminatory power, would be the way for future research to enhance the devised algorithm. In addition, future research could deploy super-efficiency DEA or even SFA (Stochastic Frontier Analysis) for quantitative determinination of the power/weight adjustments.

Finally, it should be pointed out that the interdisciplinary texture of the subject of BoP in motorsports, creates the potential for collaboration among researchers from various scientific fields. In specific, scientists dealing with engineering, statistics, and operational research can cooperate and contribute accordingly, in order to produce relevant research that will propose enhanced methodologies regarding BoP in motorsports. Ideally, endorsement of such research by motorsports organizing bodies could improve their policies over the BoP issue, thus augmenting viability and marketability of their championship series.