Nomenclature

b

wingspan, m

F

fuel capacity, L

L _f

aircraft length, m

MTOW

maximum take-off weight, kg

(Note: In this paper, weight is expressed in kilograms as a mass-equivalent for consistency with industry practice)

P

engine power, kW

PAX

number of passengers (including crew)

R

range, km

S

wing area, m²

V _c

cruise speed, km/h

${W_{to}}$

actual maximum take-off weight, kg

${W_{to.est\;\left[ {cat.\;\;x} \right]}}$

maximum take-off weight estimate, where suffix x denotes aircraft category type (A, B, or C), kg

2.0 Theoretical background

Traditional approaches for estimating the MTOW of aircraft have long served as foundational approaches in the aircraft industry. These methodologies typically rely on established aircraft design principles, engineering calculations and historical data analysis. As a result, the traditional methods of estimating MTOW encompass well-established practices, including adherence to standard guidelines, utilisation of empirical formulas, reference to payload-range diagrams and consultation of performance charts [Reference Basgall, Liu, Cassady and Anemaat6]. The breakdown weight-estimate technique, which decomposes the MTOW into components such as empty weight, fuel weight and payload weight, is widely used in aircraft design [Reference Lee and Chatterji7] and is extensively explained in several textbooks [Reference Raymer8, Reference Kundu, Price and Riordan9]. Each component in this technique is determined using analytical methods, historical data and design configurations, employing a wide range of assigned/estimated design variables. More specifically, this technique plays a critical role in ensuring that the aircraft’s structure, systems and propulsion are properly balanced to meet the predetermined weight requirements.

On the other hand, the parametric estimation approach implies developing and applying formulae that illustrate the relationship between the target (dependent) parameter (e.g. weight, cost, volume, power, etc.) and several source (independent) parameters that should be carried out, continued and aged. In other words, parametric estimation depends on simulated models, which are statistical-based equations. The mathematical approach to estimating weight is known as parametric weight estimation (PME), like the parametric cost estimation presented in Ref. [Reference Dean10]. It replaces the target parameter with weight instead of cost. This paper employed non-weight parameters (design variables) to estimate the MTOW of light aircraft. Multiple linear regression, combined with p-value analysis, is used to develop a weight estimation model with the fewest design variables for light aircraft with an MTOW of less than 5,700 kg.

2.1 Multiple linear regression (MLR)

The linear relationship between a particular independent variable and its response is known as ordinary (simple) linear regression (OLR (SLR)), while for multiple independent variables and a single response, it is known as multiple linear regression (MLR) [Reference Chatterjee and Simonoff11]. For instance, if there are n independent variables, then an MLR model obtains the formula:

(1) \begin{align}\;y = {\beta _0} + {\beta _1}{x_1} + {\beta _2}{x_2} + \ldots + {\beta _n}{x_n} + \varepsilon \end{align}

where the j ^th independent variable is x _j ; the response variable is y; the error term is $\varepsilon$ ; and $\beta_{j}$ is the mean impact on y of a one-unit increment in x _j , keeping all other independent variables unchanging. The values of $\beta_{0},\ldots,\beta_{n}$ are obtained using the least squares method to reduce the sum of squared residuals (SSR). This method relies on matrix algebra to find the $\beta$ coefficient estimates:

(2) \begin{align}SSR = \sum {\left( {{y_j} - {{\hat y}_j}} \right)^2}\end{align}

where the actual response value for the j ^th sample and the predicted response value based on the MLR model is ${\hat y_j}$ . Note that evaluating these coefficient estimates requires matrix algebra. Luckily, statistical software such as Excel and SPSS can calculate these coefficients. In this work, MS Excel was used.

2.1.1 Interpreting the MLR output

Let us have a simple example to understand the MLR output [Reference Bazdaric, Sverko, Salaric, Martinovic and Lucijanic12]. Suppose we have a response variable, take-off distance (m), and two independent variables: aircraft take-off weight (kg) and headwind speed (knots). Table 1 lists the values of the two independent variables and their relevant responses for ten aircraft samples.

Table 1.

The ten values of the two independent variables and the response

The MLR output might look like Fig. 1, generated in Excel.

Figure 1.

A sample of MLR output in Excel.

From Fig. 1, the mathematical MLR model can be formulated as:

(3) \begin{align}takeoff\,distance = 849.1245 + 0.1198 \times takeoff\,weight - 9.7545 \times headwind\end{align}

The coefficients can be interpreted as:

1. (a) Increasing the independent variable take-off weight by one kilogram is coupled with a 0.1198 m increase in take-off distance, assuming the headwind speed is kept unchanged.

2. (b) Increasing the headwind speed by one knot is coupled with a 9.7545 m decrease in take-off distance, assuming the take-off is unchanged.

3. (c) The model can predict the take-off distance of an aircraft if the aircraft’s take-off weight and headwind speed are known. For instance, if an aircraft weighing 2,200 kg with a 10 knots headwind speed, then the predicted take-off distance is:

   (4) \begin{align}takeoff\,distance = 849.1245 + 0.1198 \times 2200 - 9.7545 \times 10 = 1015.43\,m\end{align}

The following is the rest of the MLR output interpretation:

1. (a) R-square: It is defined as the coefficient of determination. It can also be described in terms of the descriptive variables as the ratio of the variation in the response variable. From Fig. 1, 88.15% of the deviation in take-off distance is explained by take-off weight and headwind speed. The resulting R-squared value (88.15%) indicates a strong explanatory power.

2. (b) Standard error: It is the mean distance between the regression line and the observed values. From the above example, the mean distance of the observed values from the regression line is 30.99 m. It represents the ‘typical prediction error’.

3. (c) F: For the regression model, it is the overall F-statistic, and it is computed as (regression mean squares [MS]divided by residual MS).

4. (d) Significance F: It shows that the regression model is statistically significant. It represents the p-value accompanying the overall F-statistic. In other words, it explains if two descriptive (independent) variables jointly have a statistically significant association with the dependent (response) variable. Note that the p-value is under 0.002 in the above example, which denotes that the variables take-off weight and headwind speed jointly have a statistically significant association with the take-off distance variable

5. (e) Coefficient p-value: The p-value of each independent variable indicates if it is statistically significant or not. The take-off weight variable has p = 0.0015, indicating it is statistically significant, whereas the headwind speed variable is more statistically significant, with p = 0.0008. Thus, both independent variables meaningfully contribute to the model.

2.1.2 Evaluating the fitness of the MLR model

In general, two metrics are commonly used in aerospace engineering to evaluate the fitness of an MLR model: R-squared and Standard error.

1. (a) R-squared: It is the change percentage in the dependent (response) variable that the independent variables can explain. It has a value between 0 and 1. A value of 0 means the independent variable cannot explain the dependent variable. In comparison, a value of 1 means that the dependent variable is completely explained by the independent variable without error. Thus, as the R-squared value approaches 1, the model fits the data with error approaching zero. In the above example, R-squared indicates the percentage of variation in take-off distance explained by the independent variables. However, 0.88 shows strong predictive ability.

2. (b) Standard error: This metric, which often denotes S, gives an idea (in terms of units) of how accurate the model predictions will be. Thus, it is more useful than the R-squared metric. And, in turn, as the standard error becomes smaller, the better the model fits the data. In the above example, a standard error of 30.99 m means the model can typically predict take-off distances within ±30.99 m of the actual values, which is useful for runway planning.

3.0 Selection of aircraft weight-modelling variables

In their paper, Lappas and Bozoudis [Reference Lappas and Bozoudis15] calculated the cost per flying hour using MTOW, maximum speed, ceiling, SFC, empty weight and fuselage length, which were chosen logically because they affect recurrent and capital costs. For instance, wing area affects many attributes, including weight (which requires additional material), as well as lift and drag, influencing various performance aspects. Notably, the broader knowledge base and experience of aircraft designers play a crucial role in shaping the preferred objective function, its relative importance and the selection of variables. Operation and performance requirements influence a few choices when considering the top-level aircraft requirements (TLAR), while the other design choices are allowed. These choices are frequently made using trade-off analysis; therefore, a parametric approach is fundamental.

Up-to-date and accurate data are the primary requirements for developing weight estimation models. Keeping an eye on MTOW in all design phases is crucial. Models with high trustworthiness should utilise only known variables during the early phase. These variables are summarised later in this section. Data from old or retired aircraft is excluded because it is based on outdated materials, manufacturing methods and technologies. This paper only includes aircraft currently ‘in service’ and/or ‘in production’.

Additionally, only piston-engine aircraft with identical design characteristics and features are considered. All-electric aircraft are becoming more popular, but for the foreseeable future, internal combustion engines will still be employed in design and production. Three categories, single-engine tricycle (Category A), double-engine tricycle (Category B) and single-engine taildragger (Category C), are utilised to classify the sampled data (aircraft) used in modelling and testing. The aircraft mentioned in Table 2 are utilised to establish the weight-predicting models.

Table 2.

The aircraft categories with their MTOW

In exploring the aircraft listed in Table 2, it is noted that both conventional and composite aircraft are represented. In contrast, because there are only a handful of certified electric aircraft, all of which are very new, they are also omitted. Therefore, the design variables must be general; more specifically, they must not be technology-specific.

However, a typical collection of variables has been selected to build the weight-predicting models [Reference Al-Shamma and Ali16]. The chosen variables used in this study were determined, predicted or known at the conceptual design stage and have a significant impact on practically all design and weight-prediction aspects. These design variables include aircraft length (L _f ), wing area (S), wingspan (b) and number of passengers (PAX) from the geometry section; aircraft cruise speed (V _c ) and range (R) from the performance section; as well as engine power (P) and fuel capacity (F) from the propulsion section.

3.1 Multicollinearity problem

Multicollinearity in regression arises when two or more independent variables are highly correlated, thereby violating the assumption of independence among predictors. One desirable attribute of regression is the isolation and quantification of the individual effect of each independent variable on the dependent variable, associated with a one-unit change in each independent variable whilst holding all other independent variables constant. When multicollinearity is present, it becomes difficult to interpret the results because the independent variables are highly correlated.

Multicollinearity gives rise to two principal issues. First, it inflates the variance of the estimated coefficients, reducing their precision and, consequently, diminishing the model’s statistical power. This leads to unreliable p-values and undermines the identification of statistically significant independent variables. Second, the estimated coefficients become unstable and highly sensitive to small changes in the model, producing substantial fluctuations in coefficient magnitude and even sign, reflecting an inherent lack of robustness in the estimates [Reference Hayes and Scott17].

Whether multicollinearity needs to be addressed depends on how severe it is and on what the model is being used for. Several considerations, as outlined in Ref. [Reference Forest18], are worth examining. As the degree of multicollinearity increases, its detrimental effects become more pronounced; however, at moderate levels, corrective action may not be necessary. Moreover, multicollinearity primarily affects those variables that are strongly correlated, and if such variables are not of substantive interest, intervention may be unnecessary. Importantly, multicollinearity does not materially affect overall quality-of-fit measures or predictive accuracy; rather, its effect is confined to the estimation of individual coefficients.

Diagnostic evaluation of multicollinearity typically involves examining pairwise correlations among independent variables and computing the variance inflation factor (VIF). These techniques are widely implemented in statistical software packages, including Microsoft Excel. A VIF value of 1 indicates no correlation between the independent variables under consideration and the remaining independent variables. Values between 1 and 5 suggest moderate correlation that is generally acceptable, whereas values exceeding 5 indicate potentially problematic multicollinearity. In such cases, coefficient estimates are likely to be inaccurate, and p-values unreliable, and phenomena such as sign reversals and inflated standard errors may arise, as stated before.

The VIF can be calculated in Excel by exploiting its definition rather than relying on a built-in function. For each independent variable ${X_i}$ , a separate (auxiliary) regression is performed in which ${X_i}\;$ is treated as the dependent variable, and all remaining independent variables are used as independent variables. Using Excel’s Data Analysis ToolPak, this regression yields an ${R^2}$ value, which indicates how well ${X_i}\;$ is explained by the other independent variables. The VIF is then computed using the standard relationship [Reference Shrestha19]:

(5) \begin{align}VI{F_i} = \frac{1}{{1 - R_i^2}}\end{align}

This process is repeated for each independent variable in the model. A VIF value of 1 indicates no multicollinearity, while values between 1 and 5 suggest moderate correlation that is often acceptable in practice. Values exceeding 5 – and particularly above 10 – indicate increasingly severe multicollinearity, which leads to inflated standard errors, unstable coefficient estimates and unreliable statistical inference. Although Excel requires this iterative approach, the method is entirely consistent with standard statistical practice [Reference Montgomery, Peck and Vining20], which provides formal guidance on diagnosing and interpreting multicollinearity in regression models.

4.0 The proposed approach and model development

The data used to establish the empirical models have been extracted from the web, manufacturers’ technical specifications, pilot operating handbooks, FAA-type certificates, manufacturer performance charts, flight test reports, aviation databases and aircraft design textbooks, to name a few. Data from these sources have been compiled in Table 1 for the MTOW and in Table 2 for the main design variables.

This work developed three models to estimate the aircraft MTOW using multiple linear regression, in conjunction with p-value analysis, which is used to reduce the set of independent variables to a minimum while maintaining high estimation accuracy. This reduction in variables in the resulting model may speed up decision-making, highlight the importance of parametric analysis and make it easier to estimate aircraft MTOW. The p-values for the regression model coefficients can also be used to assess the significance of each design variable. In other words, if the p-value is high, the parameter significance is low, and vice versa. It is inversely proportional to the p-value.

To perform the MLR, several statistical platforms and software are available on the market. This work used the most popular software, MS Excel. It has several built-in data analysis tools, as part of the Analysis ToolPak, such as correlation, regression and t-test. To access the regression tool, select the Data tab from the home page, then select the Data analysis tab from the Analyse section. Finally, select the regression tool from the opening list window.

Note that successful use of MLR requires that the total number of dependent and independent parameters be less than the number of aircraft in the category, by at least 1. Thus, no more than 14 independent variables can be chosen. Alternatively, correlation analysis is used first to minimise the number of independent parameters by eliminating those with the lowest correlation coefficients. In Section 3, the total number of design variables (independent parameters) is 8, plus the dependent parameter (MTOW variable in this study). Moreover, all categories (A, B, and C) have 15 aircraft. Therefore, there is no need to apply the correlation analysis to minimise the number of independent parameters.

Table 3.

The input values of the design variables

Based on the aircraft in Table 2, Table 3 provides the input values of the design variables for each aircraft in the three aircraft categories. Starting with Category A, the mathematical model of the estimated MTOW can be extracted from the MLR output as mentioned in Section 2.1:

(6) \begin{align}{W_{to.est\;\left[ {cat.\;\;A} \right]}} = 120.9809 \times PAX + 97.22259 \times {L_f} + 8.990606 \times b - 16.2701 \times S + 0.571669 \times P \nonumber\\ + 1.317788 \times {V_c} + 0.067629 \times R + 0.661598 \times F - 564.382\end{align}

Similarly, the mathematical models for Categories B and C are:

(7) \begin{align}{W_{to.est\;\left[ {cat.\;\;B} \right]}} = 51.77997 \times PAX + 221.1475 \times {L_f} - 7.04112 \times b + 37.58779 \times S + 2.923553 \times P \nonumber\\ + 1.05773 \times {V_c} + 0.151307 \times R - 0.33279 \times F - 1600.58\end{align}

(8) \begin{align}{W_{to.est\;\left[ {cat.\;\;C} \right]}} = 21.45537 \times PAX - 25.1091 \times {L_f} + 57.37207 \times b + 5.397891 \times S + 6.395582 \times P \nonumber\\ + 0.44584 \times {V_c} + 0.09982 \times R + 0.46689 \times F - 430.84\end{align}

Note that the MLR analysis considered all the independent variables. To reduce the number of independent variables to the minimum possible while keeping the error accuracy within the desired range, p-value analysis is considered. This analysis performs the MLR analysis in an iterative procedure. At the end of each iteration, the design variable with the highest p-value is removed from the regression parameter list. The iterative procedure ends once all p-values fall below 0.05. The final model is obtained using the remaining variables.

Thus, in considering Category A, the P parameter, which had the highest p-value, was eliminated after the first MLR execution, and the b parameter was eliminated after the second. Hence, the final mathematical formula is:

(9) \begin{align}{W_{to.est\;\left[ {cat.\;\;A} \right]}} = 0.721811 \times F + 111.4565 \times {L_f} + 1.504516 \times {V_c} + 133.9247 \times PAX \nonumber\\ + 0.081801 \times R - 20.1199 \times S - 573.377\end{align}

Similarly, in Category B, the b and F parameters were eliminated. The final mathematical formula is:

(10) \begin{align}{W_{to.est\;\left[ {cat.\;\;B} \right]}} = 176.3897 \times {L_f} + 33.60172 \times S + 2.054176 \times P + 1.046836 \times {V_c} \nonumber\\ + 0.172823 \times R + 69.59588 \times PAX - 1360.57\end{align}

And, in Category C, the S, L _f , F and V _c parameters were eliminated. The final mathematical formula is:

(11) \begin{align}{W_{to.est\;\left[ {cat.\;\;C} \right]}} = 50.75259 \times b + 5.888885 \times P + 19.50779 \times PAX - 0.07986 \times R - 365.99\end{align}

In Category A, it should be noted that the wing area coefficient has a negative sign (Equation (9)), whereas the variable itself has a strong relationship with the MTOW. The wing area variable also has strong correlations with PAX, length, engine power and fuel capacity. Thus, it works here as a suppressor variable. It effectively removes from the model the part of the independent variable that does not relate to the outcome. Physically, as the wing area increases, the lift force increases, and the climb stage time is reduced. Hence, fuel consumption decreases, and the total fuel capacity is reduced. Thus, the MTOW is decreased. Similarly, in Category C, the range variable has a strong relationship with the cruise speed variable (see Equation (11)), and it also acts as a suppressor variable.

Note that the E-Conditions of the CAA requirements for experimental aircraft [21] cover the aircraft whose MTOW is less than 2,000 kg. Most aircraft in Categories A and C are below this limit. In general, models (Equations (6), (7) and (8)) or shorter ones (Equations (9), (10) and (11)) can help predict MTOW for aircraft specifically under 6,000 kg.

6.0 Case studies

Evaluating the proposed method requires testing additional aircraft to certify their suitability. In Categories A and B, five aircraft were employed (not used in building the estimation models), and a further four aircraft were utilised for Category C due to the limited number of available aircraft. Moreover, the conditions still ‘in service’ and/or ‘in production’ are also applied. Table 10 presents these aircraft and their design parameter values.

Table 10.

The design parameter values (case studies aircraft)

For all categories, Equations (6)–(11) were used to estimate the aircraft MTOW. Figure 4 presents the actual and estimated MTOW, using MLR and MLR+p-value techniques. It is clearly demonstrated that the estimated findings from both techniques are very close to the actual MTOWs.

Figure 4.

The actual and estimated MTOW for the tested aircraft of all categories.

Consequently, to determine the accuracy of the tested aircraft’s error precisely, the difference (Error) and the percentage difference (Error %) were also calculated. Figure 5 shows the error accuracy of each tested aircraft for all categories. The graphs showed that the error was less than 5% across all categories for both the long and short formulae.

Figure 5.

The tested aircraft error accuracy % for all categories.

Figure 6.

The impact of the variables for the full and short models of Category A.

However, according to many textbooks, early in the design process, it is acceptable for the estimated MTOW to be within 10% of the target value. The developed models (Equations (6)–(8) and (9)–(11)) enable designers to estimate the MTOW of new designs with an error accuracy of less than 5%. The weight estimates obtained with the test aircraft confirm the applicability of the developed models and facilitate parametric analyses based on key design variables.

7.0 Parametric studies

However, an impartial evaluation is required to evaluate the sensitivity of the created models. Because of the model’s linearity, MTOW estimation is easily done. Additionally, it makes parametric studies easier by allowing the design parameters to be used to determine the sensitivity and effect of changes. Every parameter coefficient has an effect. The largest coefficient corresponds to the greatest effect, and vice versa. Furthermore, a crucial factor in identifying the direction of proportionality is the sign of the coefficient. If the sign is positive, the predicted MTOW is directly proportional; if the sign is negative, inverse proportionality is indicated. For example, the Category A short model (Equation (9)) contains all positive (directly proportional) coefficients except the S parameter (F, L _f , V _c , PAX and R). Furthermore, since the PAX parameter has the largest coefficient, it has the greatest impact. While adding 1 m to the aircraft length would raise the MTOW by just 111 kg, reducing one passenger would result in 133 kg of MTOW savings. Logically, adding 1 litre of fuel will increase MTOW by 0.7 kg.

In considering the MLR model (Equation (10)) for Category B, the aircraft length has the largest coefficient, since adding 1 m will increase the MTOW by 221 kg, whereas decreasing the wing area by 1 m² will save 37.5 kg. The range parameter has the lowest coefficient value and the least influence (see Fig. 7).

Figure 7.

The impact of the variables for the full and short models of Category B.

Finally, the short model (Equation (11)) of Category C has only four parameters: b, P, PAX and R. The Span coefficient is the highest, and increasing it by 1 m will increase the MTOW by 50 kg. Similar to Category B, the Range coefficient has the lowest impact on the MTOW (see Fig. 8).

Figure 8.

The impact of the variables for the full and short models of Category C.