Parametric Cost Prediction Models for Light General Aviation Aircraft in the Early Design Phase

The global forecast for the aircraft industry is set to rise due to the growing adoption of light and ultralight aircraft in the military, commercial, and civil arenas. According to a report from MarketsAndMarkets [1], the CAGR (compound annual growth rate) is currently 5.9%. Thus, the light aircraft market is planned to rise from \$7.5 to \$11.9 billion between 2022 and 2030. This report classified the light and ultralight aircraft market according to Region (North America, Europe, Asia Pacific, and rest of the world), System (airframes, avionics, cabin interior, and aircraft systems), Flight operation (CTOL and VTOL), Material (aluminium, composites, titanium, and others), Propulsion (conventional fuel and electric-hybrid), Technology (manned and unmanned), End Use (civil & commercial, and military), and Aircraft Type (light and ultralight). However, new opportunities will arise that require a competitive advantage with this growth. The utilization of new materials and technologies would inevitably mean new designs that address technical, legal, ecological, and operational aspects. The growth in the light aviation area inevitably means many new designs that are fit for specific purposes. The greatest challenge that befalls the designer, is to determine the sensitivity of key design parameters that affect not only the performance but the overall cost of the aircraft. The overall cost affects the direct operating cost (DOC) and the seat mile cost (SMC) which in turn are key financial metrics that determine the viability of the design for commercial exploitation. Additionally, the maximum take off weight (MTOW) which is a function of wing size, fuselage size, geometric layout, aerodynamic and propulsive aspects, determines the cost. Each design aspect ultimately results in the utilization of resources such as Material, Manpower, Equipment, Services, Facilities, and Time, resulting in costs that can be direct or indirect.

In many business and engineering decisions, including the aircraft industry, cost estimation is challenging since it needs to factor in various expenses such as materials, labor quantities, sales, floor space, utilities, and overheads. These expenses are utilized as inputs to analysis like decision tree analysis or Monte Carlo simulation [2, 3]. Therefore, the cost estimation must be accurate to avoid making wrong decisions or overrunning the total developed cost [4]. Reliable cost estimation may also improve the chances of getting external funding - a vital point for start-ups. To tackle this issue, the authors developed a research method to investigate potential cost models for GA aircraft that can serve as a guideline for, e.g., start-ups and research works. More specifically, Shahriar et al. [5] presented a comprehensive summary of all the cost-estimating techniques in design, development, manufacturing, and production cycles. Although parametric models exist, usually not of high fidelity. DAPCA-IV was developed by RAND Corporation in the late ’60s and is based on empirical relations where information on the airframe, engine, avionics, labor, and material is processed. Furthermore, the DAPCA-IV program requires information that is generally not available during early design phases.

The analogous cost estimation method is the most common approach for estimating a new product cost. “Similar products have similar costs” is the basis for this technique [6, 7]. The benefit of using the available past cost data is obtaining a valuable estimate with minimal time [8, 9]. References [10-12] offer additional research on the analogous cost estimation method. Parametric cost estimation is another widely used method to estimate a new product cost. Its main principle is to generate what is known as “cost estimation relationships”. Furthermore, a single model may have several relationships [13-15]. The engineering build-up method is another approach for cost-estimating a new product. In this approach, the part design details, the process interactions, and a deep process understanding should be available [16, 17]. More information is available in the references [18-20]. The commonly used method's strengths and weaknesses and related applications are presented in the reference [21], point to note is that software development, data-driven risk analysis, sensitivity analysis, long-range planning, architectural studies, cross-checking, design-to-cost trade studies are cited as the main advantages in parametric methods. The models developed later on in this work are particularly suited to performing sensitivity analysis, and design-to-cost trade studies. These methods are significantly important in the Preliminary and detailed phases of development [22].

Various parameters, such as time constraints, availability of data, level of detail required, and adequacy of project/program definition, are used to decide the type of cost estimation method. However, in selecting a methodology, the analyst should note that the cost-estimating task predicts upcoming costs by extrapolating the existing schedule data and historical costs. Figure 1 shows the relative significance of each method throughout the product development phases [22], Phase A (Conceptual), Phases B and C (Preliminary), and Phase D (Detailed).

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Figure 1. Relative significance of cost-estimating methods during the aircraft design phases [22]

For the preliminary design phase (phases B and C), it is clear that the parametric method has the highest relative significance. Therefore, it is recommended to use it where data and samples are lacking. Moreover, the aircraft design process is complicated, more often than not iterative, and often has a long development cycle due to many uncertainties and often due to the incorporation of new technologies, which directly impact the overall costs. Consequently, the prediction of aircraft cost is not only tricky but also a complex task. Accurate cost predictions at an early design stage allow the design drivers' impact to be evaluated and avoid the risk of bad decision-making or overrunning the total developed cost [4]. It is true for estimating the cost of both commercial and light aircraft [23].

However, parametric cost estimation is a widely used method to estimate a new product's cost in the early stages [24]. Its main principle is to generate what is known as “cost estimation relationships”. These relationships correlate the product cost with a number of its parameters, usually identified as “cost drivers”. For instance, the product size can be a parameter; the manufacturing costs will increase as the size increases. Thus, cost and size can be mathematically correlated through statistical analysis. The mathematical relationship (the model) can have a number of parameters or design variables like weight, size, density, thickness, etc. Note that these parameters (cost drivers) should greatly impact the cost changes. Further, a single model may have several relationships [14, 23, 25].

Conversely, most textbooks and design references use a simple linear regression model, defining the cost as a function of the MTOW. This approach is commonly adopted, with a prediction error of around 10%. Thus, the main contribution of this paper is to develop more accurate cost estimation models (with prediction error less than ±5%) for light general aviation aircraft (MTOW less than 2000 kg). In addition, the linearity of the models facilitates the process of investigating the impact of each design variable on the aircraft cost. Thus, these models help, simplify, and accelerate the decision-making process.

Lappas and Bozoudis [31] used fuselage length, empty weight, MTOW, SFC, max speed, and ceiling parameters in their paper to predict cost per flying hour. The choice of the variables is logical, as the variables chosen affect the capital costs and recurrent costs. For example, wing area affects many aspects, principally the weight, therefore more material has to be used, as well as increasing lift and drag, to obtain the impact of many performance aspects. Noticeably, the choice of the variables and their relative significance and impact on the desired objective function is not obvious and dependent on the experience and wider knowledge base of the aircraft designers. The TLAR (top level aircraft requirements) allows few design choices to be made and performance and operational requirements impact the other choices, which are often honed by trade-off analysis, for which parametric is essential.

Having accurate and up-to-date data for obtaining cost estimation models is the first and foremost requirement. The knowledge of the Cost in design passes B and C is critically important. Any model of any significant fidelity must only use variables that are known in these phases. Later in the section we outline such variables. Data from retired or even old aircraft is discarded as it is based on old technologies, manufacturing methods, and materials. Therefore, only aircraft that are in production and also in service are included in this paper.

Moreover, only piston-engine aircraft with the same features and design characteristics are considered. Although there is a growing trend toward all-electric aircraft, internal combustion engines will continue to be used and designed for the foreseeable future. The sampled data (aircraft) used in modeling and testing are classified into two categories, namely the tricycle (category A) and taildragger (category B). The aircraft used to build the cost-estimation models are listed in Table 1 for category A and Table 2 for category B.

Table 1. The prices and maximum takeoff weights of category A

Table 2. The prices and maximum takeoff weights of category B

From the list above, it can be noted that both conventional and composite aircraft have been included. All electric aircraft are relatively new, with just a few certified; hence, they have been excluded. So, the design variables need to be generic, specifically, those that are not a function of technology.

In order to establish the cost models, a typical set of variables has been chosen [30]. In this work, the selected parameters critically affect almost all design and cost-estimation aspects and are evaluated, estimated, or known at the conceptual design phase. The selected variables have by far the most impact on the overall cost and are described in the following:

1. W_to – The maximum take-off weight – is the primary parameter and must be considered during all design phases. Its importance impacts endurance, range, wing loading, thrust loading, and all aspects of aircraft performance (including take-off, climb, cruise, turning, etc.). Various techniques are available for estimating this parameter in the conceptual design phase.

2. PAX – Number of passengers (Number of Seats) – is one of the crucial requirements and has a bearing on fuselage length and/or the diameter of the fuselage. This requirement is generally available at the conceptual phase. As the number of seats increases, the aircraft's structural weight goes up, and hence, the aircraft cost increases. The increased fuselage length increases the tail arm, which affects static and dynamic stability, requiring larger volume coefficients, leading to larger empennage areas, and, eventually, the cost indirectly affects the PAX parameter. Since the occupants' weight is significant compared to the empty weight, it is likely to be significant.

3. R – Aircraft range – is largely dependent on the amount of fuel that can be carried and computed by the Breguet Range Equation, which requires the knowledge of including initial and final weights (W_o and W_f), drag and lift (D and L) characteristics, specific fuel consumption (SFC), and cruise speed (V_c). For example, increasing the range requires a robust, lightweight structure, a higher L/D ratio, and efficient engines. All of these improvements will result in increased costs.

4. F – Fuel capacity – is directly proportional to aircraft range, limited by the maximum take-off weight and performance. It also affects the wing geometry and fuselage design due to volume storage requirements.

5. V_c – Cruise speed – directly impacts the stage or mission time. Increasing the cruise speed involves obtaining increased power. This situation translates into larger engines, which implies heavier engines, and in turn, the wing loading (structure) and aircraft cost go up.

6. P – Engine power – A certain excess power is needed to obtain good performance. This case limits the permitted total drag, affecting the aerodynamic design and, to a large extent the structural design, as well.

7. L_f – Aircraft length – mainly depends on the seating arrangement, the number of passengers, and fore and after body requirements.

8. b – Wing span affects wing loading and aspect ratio selection.

9. S – Aircraft wing area – depends on wing loading obtained from the constraint analysis that requires several simultaneous constraints to be met.

In applying multiple linear regression (MLR) properly, it should be noted that the number of independent and dependent parameters must be less than one than the number of aircraft in the specified category under consideration. However, this work uses nine design (independent) parameters listed in section three and one independent (aircraft cost) parameter. Thus, to include all these parameters in constructing the cost models, the number of aircraft in each category should be more than ten. Otherwise, correlation analysis is applied first to reduce the number of independent parameters by removing parameters with the lowest correlation coefficients. Starting with Category A, there are 11 aircraft, and thus there was no need to apply correlation analysis. After performing the first pass of MLR, the fuel parameter with the biggest p-value was eliminated. The aircraft length was removed in the second pass, while the span parameter was removed in the third. The fourth pass showed that all parameters had a value less than 0.05, and the process was stopped. Thus, the mathematical equation of the cost-estimation model (category A) is:

$\begin{aligned} C_{\text {est. } A[p-\text { value }]} & =16184.74 \times S+195 \times R \\ & +1514 \times V_c+2622.74 \times P \\ & +156908 \times P A X \\ & -860.82 \times W_{\text {to }}-384648\end{aligned}$                    (9)

Logically all the retained variables affect aircraft cost, while investigating category B showed that the number of sampled data is ten aircraft. Thus, only eight design parameters should be considered in performing the MLR. Correlation analysis should be performed to decide which parameters should be eliminated. The correlation matrix is obtained in Table 4.

The wing span (b) has the lowest correlation coefficient value and is removed in the first pass. In subsequent passes, more variables are dropped, resulting in a reduced mathematical model of category B, which is:

$\begin{aligned}\left.C_{\text {est. }[[p-v a l u e}\right] & =1123.798 \times F-235.5 \times R \\ & +2896.625 \times V_c+254.8 \times W_{\text {to }} \\ & +82245.46 \times S \\ & -1959.83 \times P-1614.28\end{aligned}$                   (10)

The aircraft in these categories are covered by the CAA requirements for experimental aircraft, known as e-Conditions [38]. The empirical models will benefit small-scale aircraft designers and manufacturers up to a maximum take-off weight of 2000 kg for accurately estimating the aircraft cost quickly.

Table 8. The design parameter values of the additional aircraft for case studies

Table 9. Cost prediction accuracy of the models for the test aircraft

Designing a new aircraft mostly starts with some mission requirements, which impact by many design choices. The design choices not only affect the MTOW of the aircraft, but also the cost. In this paper major design variables available in the preliminary design phase are used to predict the cost. The Power loading (P/W) and wing loading (W/S) are the key entities in satisfying the constraints, which could be takeoff distance, climb rate, turning, gliding or cruise performance. Changing the weight or the wing area or the power can greatly affect the overall cost.

This work introduced the multi-linear regression analysis along with the p-value analysis as a parametric cost estimation approach to estimate the cost of light general aviation aircraft. The aircraft that are still “in production” and “in service” are used to develop the empirical models and are classified into two categories based on their landing gear configuration. Moreover, nine parameters that significantly affect the aircraft design, and its ultimate cost, are considered. Equations (9) and (10) can be used to predict the cost. The findings showed that the developed models could effectively estimate the aircraft cost at early design stages with a prediction error of less than ± 5%., but in most cases the prediction error is less than ± 3%. Thus, giving the designers an accurate method for cost prediction, which will enable either venture capital or bank loans to be secured against an accurate cost forecast. Moreover, among the available cost estimation methods, this accuracy of the presented models is the best and offers the possibility to perform parametric studies to highlight the cost sensitivity to the changes in key design variables.