Morphological matrix quantisation

A variation on this morphological technique has been applied to a fuel system configuration design on fighter aircraft as described by Gavel et al. (2006, 2007), Gavel et al. (2008), Ölvander et al. (2009), and Svahn, (2006). In these accounts, matrix-based methods were employed to quantify the morphological matrix providing a solution which is characterised with a set of parameters such as system weight, cost, performance etc. The selection of the candidate solutions are modelled with decision variables and the selection and optimisation problem is formulated within a mathematical framework as described below. The quantification of the morphological matrix provides access to every potential solution which is described as either a physical or statistical model, or a combination of both. Therefore, using this approach the TPMs are quantified as metrics.

In the following example provided by Ölvander et al. (2009) the TPMs provided by Cost (C) and Weight (W) are important to the conceptual design of an aerospace vehicle.

As described by Ölvander et al. (2009) the morphological matrix X can be stated as n different functions with M potential solutions for each function, resulting in a matrix X as follows:

𝑋 = [

𝑋₁₁ ⋯ 𝑋_1𝑚

⋮ ⋱ ⋮

𝑋_𝑛1 ⋯ 𝑋_𝑛𝑚

] Equation 2

The concept, as reproduced here from Ölvander et al. (2009), relies on determining one solution to fulfil one function only. This can be expressed by letting xij equal 1, if solution j is selected to implement function i. Otherwise in the remaining cases, 0 applies. Therefore, in each row there can be only one element different from zero with this relationship implemented. This is shown below by the following:

∑^𝑚_𝑗=1𝑥_𝑖𝑗= 1, 𝑖 = 1 … . 𝑛 Equation 3 𝑥_𝑖𝑗 ∈ (0, 1)

For this matrix, system weight, W, is calculated by summing the weight of each solution as shown below.

𝑊 = ∑^𝑛_𝑖=1∑^𝑚_𝑗=1𝑤_𝑖𝑗𝑥_𝑖𝑗 Equation 4

Therefore, the weight of a specific solution, wij is calculated as function of the specific constraints implied by the system as well as the system specific parameters defined in the vector y defined below. However, weight is also a function of the chosen concept X. This allows for dependencies where for example the weight of one solution may be also dependent on other solutions. This is particularly relevant in the case of alternate fuel systems where the weight of the fuel tank may be dependent on the fuel state (liquid or gas) or battery type selected. That is a concept where the selected gaseous fuel may require a heavier and larger fuel tank to achieve a range requirement (which may be represented in y). Therefore, the weight of a particular solution could be determined according to the equation below.

𝑤_𝑖𝑗= 𝑤_𝑖𝑗(𝑋, 𝑦) Equation 5 Where X is the chosen concept, and

y is the specific system parameter vector

It should be noted that the above equations yield a non-linear expression for total weight and that the total weight meets the requirements in y.

As indicated by Ölvander et al. (2009), this system weight solution may be minimised as an optimisation problem within the following relationships:

𝑚𝑖𝑛 ∑^𝑛_𝑖=1∑^𝑚_𝑗=1𝑤_𝑖𝑗𝑥_𝑖𝑗

Such that

∑^𝑚_𝑗=1𝑥_𝑖𝑗= 1, 𝑖 = 1 … . 𝑛 Equation 6 𝑤_𝑖𝑗 = 𝑤_𝑖𝑗(𝑋, 𝑦)

𝑥_𝑖𝑗 ∈ (0, 1)

It should be noted that this approach provides numerous infeasible solutions, and therefore this method requires the introduction of a set of feasible solutions S, in which to explore for feasible solutions. In simple cases xab is incompatible with xcd could be expressed as:

𝑥_𝑎𝑏+ 𝑥_𝑐𝑑≤ 1 Equation 7

The relationship described as xef requires that both xgh and xij are within the concept and could be modelled as follows:

2𝑥_𝑒𝑓− 𝑥_𝑔ℎ− 𝑥_𝑖𝑗≤ 0 Equation 8

If there are only mi solutions for the function i, then the remaining elements xmi+1

– xm should be set to zero as described by the following:

𝑥_𝑖𝑗 = 0, 𝑖 = 1 … . 𝑛, 𝑗 = 𝑚_𝑖+1….𝑚 Equation 9

Furthermore, there may be many other system attributes that might need to be included when evaluating candidate solution concepts. Ölvander et al. (2009) illustrates this by inclusion of the cost attribute C, in the same manner as weight, W.

This can be expressed as the following function, where α1 and α2 are linear weightings for the objectives wij and cij.

𝑚𝑖𝑛 𝛼₁∑^𝑛_𝑖=1∑^𝑚_𝑗=1𝑤_𝑖𝑗𝑥_𝑖𝑗 + 𝑚𝑖𝑛 𝛼₂∑^𝑛_𝑖=1∑^𝑚_𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗 Equation 10 Such that

∑ 𝑥_𝑖𝑗

𝑚

𝑗=1

= 1, 𝑖 = 1 … . 𝑛

𝑤_𝑖𝑗 = 𝑤_𝑖𝑗(𝑦, 𝑋) (10)

𝑐_𝑖𝑗= 𝑐_𝑖𝑗(𝑦, 𝑋) (11)

𝑥_𝑖𝑗= 0, 𝑖 = 1 … . 𝑛, 𝑗 = 𝑚_𝑖+1….𝑚

𝑥_𝑖𝑗 ∈ (0, 1)

𝑋 ∈ 𝑆

An alternative formulation is obtained if one decision variable is used for each function. That is for each row in the decision matrix there is one variable that can be

optimised and applied to the integer values representing different solutions for that function. Thus, there are n decision variables which can take integer values from 1 to mi, where mi is the number of solutions for the function i, as shown by:

𝑥_𝑖 = {1, 2, … . , 𝑚_𝑖}, 𝑖 = 1, 2, … . , 𝑛 Equation 11 𝑋 = [𝑥₁, 𝑥₂, 𝑥₃… . 𝑥_𝑛]^𝑇

By adopting this approach, the formulation will have n integer variables instead of n.m binary variables. This is referred to as quantification of the morphological matrix.

The weight W of the concept is therefore calculated by summing up the weights of the functions. However, it can be observed that the weight required to determine function i is obviously a function of the adopted solution. That is wi = wi(xi). It will depend also the solution selections within the concept, that is wi = wi(x). Weight also is a function of external requirements y, so that wi = wi(x, y) as shown below:

𝑊 = ∑^𝑛_𝑖=1𝑤_𝑖 Equation 12 𝑤_𝑖 = 𝑤_𝑖(𝑥, 𝑦)

The lowest possible minimum weight can thus be determined by:

𝑚𝑖𝑛 ∑ 𝑤_𝑖

𝑛

𝑖=1

Such that

𝑤_𝑖 = 𝑤_𝑖(𝑥, 𝑦)

𝑥_𝑖 = {1, 2, … . , 𝑚_𝑖}, 𝑖 = 1, 2, … . , 𝑛

𝑋 = [𝑥₁, 𝑥₂, 𝑥₃… . 𝑥_𝑛]^𝑇

There may be characteristics of the system that need to be considered when optimising or evaluating the solution concepts. In other design studies, Ölvander et al.

(2009) includes other important design parameters such as electrical power consumption and compressed air consumption. Ölvander et al. (2009), indicates that electrical power consumption, pei and compressed air consumption, pairi may be handled much in the same way as weight. Similarly, in the context of alternate fuel

systems, other parameters may be included such as drag increment and cost of the modification.

The objectives can be therefore be aggregated to an overall objective function, as follows, where each objective is weighted by the α

𝑚𝑖𝑛𝛼₁∑ 𝑤_𝑖

Where ψ is the penalty function which is zero if the concept is within the feasible solution space and >0 if it is not. Ölvander et al. (2009) states that depending on the characteristics of the problem, and the optimisation method employed, then a binary or integer representation may be the best choice. The framework associated with this approach is provided in the next section.