Decision making for multi‐objective problems: Mean and median metrics

When dealing with problems with more than two objectives, sophisticated multi‐objective optimization algorithms might be needed. Pareto optimization, which is based on the concept of dominated and non‐dominated solutions, is the most widely utilized method when comparing solutions within a multi‐objective setting. However, in the context of optimization, where three or more objectives are involved, the effectiveness of Pareto dominance approaches to drive the solutions to convergence is significantly compromised as more and more solutions tend to be non‐dominated by each other. This in turn reduces the selection pressure, especially for algorithms that rely on evolving a population of solutions such as evolutionary algorithms, particle swarm optimization, differential evolution, etc. The size of the non‐dominated set of trade‐off solutions can be quite large, rendering the decision‐making process difficult if not impossible. The size of the non‐dominated solution set increases exponentially with an increase in the number of objectives. This paper aims to expand a framework for coping with many/multi‐objective and multidisciplinary optimization problems through the introduction of a min‐max metric that behaves like a median measure that can locate the center of a data set. We compare this metric to the Chebyshev norm L_∞ metric that behaves like a mean measure in locating the center of a data set. The median metric is introduced in this paper for the first time, and unlike the mean metric is independent of the data normalization method. These metrics advocate balanced, natural, and minimum compromise solutions about all objectives. We also demonstrate and compare the behavior of the two metrics for a Tradespace case study involving more than 1200 CubeSat design alternatives identifying a manageable set of potential solutions for decision‐makers.

3][4][5] While in a single objective setting, the search for the preferred optimum solution is simple and more often is a single solution, for a MOP setting without any preference among the objectives, the result is a set of trade-off solutions. 6,7In practice, customers and other project stakeholders often are unclear of their needs until they are presented with some solutions for consideration.Therefore, finding a few promising solutions to be presented for customer and other stakeholder considerations is essential in any real-life setting.Since the end-users do not have clear preferred objectives in the early phases, it is not practical to rely on or associate any weights for the objectives.
MOP is the principal approach adopted to deal with such problems where the intent is on the identification of a set of non-dominated solutions.10] Farina and Amato 11 detailed the area increase for a non-dominated space about an increase in many objectives.This increase in search area challenges optimization for identifying non-dominated solutions nearby the Pareto front. 9A large Pareto-optimal set is a problem for effective decision-making, as the large set of Pareto solutions that can be quite spread along multiple dimensions (objectives) leaves the decision-maker perplexed as to what solution should be chosen. 12This perplexity is due to cognitive limitations of the mind, as our precision to choose the right solution dramatically diminishes as the number of alternatives increases.
The methodology of obtaining the preferred solutions is sometimes ascertained by an objective preference model. 13The limit for precise decision-making according to one's preference is commonly known to be limited to no more than five solutions 13,15 along with only two objectives.The issue of cognitive efficiency of simultaneously considering several solutions of interest is significant for organizations and settings where the authority of decision-making is allocated to different classes of professionals including modelers, designers, and analysts who analyze the problem and generate the solution space.Normally designers have a better intuitive understanding of the solution space topology.In response to this problem stemming from human cognitive limitation, different metrics within MOPs are introduced to select a handful of solutions from the Pareto set (of hundreds and thousands of solutions) for presentation to the authoritative decision-makers.Metrics can also assist with the convergence of evolutionary algorithms. 14 this paper, we try to address the design problems, particularly those at the early stages where it is preferred to treat all the objectives simultaneously and equally.Noting that deciding objectives and constraints prematurely may lead to many infeasible solutions and difficulties in reaching agreements, this paper proposes an approach to reach such an agreement using metrics that favor wellbalanced solutions.We propose a new metric that has the following attributes: 1. Simplicity, which means the metric is based on a straightforward and intuitive algorithm.
2. Independence from the search algorithm means the metric does not necessarily require non-dominated sorting as an initial step.It can, however, be applied to a non-dominated set with equal efficiency.
3. It leads to fewer solutions than the previous metrics, which is an ideal attribute from the decision-maker's perspective.4. It is independent of any normalisation/scalarization of data sets and does not require scaling to be applicable.
The organisation of the presented paper is as the following: Section 2 presents the background, and Section 3 presents the two mean-max metrics.Section 4 demonstrates the properties of the two metrics by using some standard 2-D and 3-D datasets.Section 5 presents a multidimensional satellite design problem and compares the application of the two metrics for that problem.We conclude with conclusions in Section 6 and some paths for future study.

BACKGROUND
Take note that Pareto solutions and non-dominated solutions are not the same.An algorithm can only attempt and deliver a set of ND solutions.There is no guarantee that they are Pareto solutions.Generally, a solution is considered as part of the non-dominated (ND) set when no other solution is available that can enhance one of the objectives with no disregard for any other objectives.Figure 1 shows the ND solutions for two objectives.The points represent all possible design alternatives for all objective functions of the system.The set of these points is also called the feasible region.In a minimisation case, the optimal solutions can be found on the furthest lower-left edge of the feasible computations required for analysis.There is also research on computational frameworks to facilitate multidisciplinary integrated analysis.
Nevertheless, there are still significant challenges and drawbacks concerning design optimisation, requiring more innovative approaches. 15e reality is that the chosen solution may not be the most favourable to the user. 16As Balling 18 described, the commonly used design process includes formulating design problems and obtaining models related to the analysis and implementation of optimisation algorithms.However, it has not satisfied the designers mainly due to problems not being formulated appropriately.A study by Shanteau 19 demonstrated that decision-makers often change the ratings when they are not content with the outcome of a rational decision-making process.Wilson and Schooler 17 showed that people do worse at some decision tasks when asked to analyse the reasons for their preferences or to evaluate all the attributes of their choices.Moreover, Pareto optimisation can lead to many solutions, rendering the decision-making process even more complex. 18Figure 2 demonstrates this fact.Each point refers to a solution space generated by random numbers using MATLAB.The size of the solutions is kept constant at n = 100 while the number of objectives changes from 2 to 12.We can see that as the number of objectives increases, the Pareto front solutions size also increases rapidly, to the extent that it might contain the entire solution space (all 100 solutions) for 10 or more objectives.This essentially renders Pareto-optimization useless.
5][26] While these efforts are worthwhile, they tend to ignore the fact that knee solutions might be quite undesirable in that they might be close to corners.Moreover, by increasing the number of objectives, the Pareto optimal solutions are subsequently increased, making it challenging to choose the best solution among the whole set. 27Other distance-based measures are undesirable because they are often used in low-dimensional cases depending upon the purpose of use.These limitations pose hardships for such measures to be applied as an all-purpose distance metric.Thus, it should only be utilized in specific cases.The methods that rely on the range, including but not limited to "sum of weighted objectives", "distance functions", and "min-max formulation", necessities understanding of the searched problem to enable the algorithm to find the desirable solutions. 28Contrarily, range-independent methods do not rely on the nature of the objectives or the considered problem because they function regardless of the effective range of each objective function. 29Therefore, a ranking method is characterized by two factors.The first factor is that the method needs to be not limited by individual factors such as the problem; it should be independent of the effective ranges of the objectives in individual applications (i.e., range-independent). 30The next section presents two min-max metrics, namely, mean and median metrics, that observe these characteristics.

MIN-MAX METRICS
In mathematical terms MOP is defined as 31 : where n is the number of scalar objective functions and x = (implying m different solutions), and the objective space mapping is objective functions where f i : Many MOP solvers try to determine a set of Pareto optimal solutions.The Pareto optimal set is defined as  = {z : z *  z, z * ∈ Z} which is defined by a set of objective vectors that are not dominated by any objective vector.Also, the lower bound is called the ideal objective vector, as: One of the most popular utility functions for decision-makers is the Chebyshev metric which can be calculated from the Minkowski distance as below 31 : and by assuming z * = (0, 0, … , 0) then, the Equation ( 2) can be as: by considering p = ∞, the Equation ( 3) can be written as: Minkowski distance is a generalized distance metric.The word "generalized" refers to the ability to alter the formula as mentioned above to compute the distance between two data points in various ways.
Therefore, by manipulating the p, the distance can be obtained in three different modes thereby, for exceptional cases: p = 1 : Manhattan metric, p = 2 : Euclidean metric, p = ∞ : Chebyshev metric.The Chebyshev distance uses the ideal point of MOP which is the smallest value in each objective.Thus, the Chebyshev metric (we refer to this metric as the Mean metric for reasons specified later) is calculated as: One important note about Chebyshev metric and other Minkowski distance-based measures is that for them to be effective and meaningful the feasible space must be normalized or scaled in a way that all the objectives' values are numerically within the same ranges and comparable.]32 We now present the Median metric that does not require normalisation of the objective space, based on a sorting function and the Chebyshev metric.The sorting function is defined as: For notational simplicity let's assume Then s j i is the transformed objective space by the sorting function, which can be represented by the matrix M p : where has n rows and m columns and has the indices of first n solutions regarding each objective.To find the optimum solution, the maximum solution of each row from M p will be chosen as: Then, we will sort the column of the matrix S m in ascending order based on their maximum value.Afterwards, we consider the 1st rank for the highest maximum values while the m-rank is for the lowest maximum.Thus, the rank vector (R) can be written as: Then the minimum values of the matrix S m according to their rank is the preferred solution (r 1 ).
We now further illustrate the two min-max metrics (mean and median) with the following simple example.The min-max method for both metrics is demonstrated in

MEAN AND MEDIAN METRICS PROPERTIES
The behaviours of the two metrics are compared in this section using various standard solution space topologies.A set of non-dominated  A preference region is defined as a subgroup of the non-dominated front.The location of the preference region is normally at convex bulges of the non-dominated front. 13The middle of a convex projection of the non-dominated front is known as a knee point. 13The knee, by definition, offers the greatest trade-off. 33For assessing the performance of the metrics in selecting the knee area the fronts obtained from DEB2DK, and DEB3DK were used.Figures 6 and 7 In Figure 6(D) where the distribution of the points is nonsymmetric, the knee point is on the left of the chosen region by the Median metric (blue region).Similarly, in Figure 7(D), the Median metric favours a region (blue solutions) that is not on the knee.However, from Figure 7 it is evident that neither Median nor Mean metrics necessarily pick the knee points, and the fact that the knee points and the Metrics preferred region are identical in Figure 6(A-C) is due to the fact that DEB2DK1B places the knee point at the centre of the solution points.
The importance of the median metric is in its ability to pinpoint significant parts of the Pareto Front and important decision points.While the mean metric and other distance-based metrics or polar metrics are used to find out about the significant parts of the Pareto Front, the median metric can single out parts of the Pareto Front that were so far not considered for example in evolutionary algorithms that rely on such metrics to find optimal solutions.We should also note that no single metric should be used to characterise the entire solution space and influence decision making.Rather a group of important metrics are required, and Median metric is certainly an important metric that should not be ignored.

EXPERIMENT RESULT AND ANALYSIS
The data of this case study and its description have been previously presented in ref. 34.We used the same case study to demonstrate the effectiveness of a completely different method proposed in this investigation.This section presents an example of CubeSat's design Tradespace, aiming to guide the reader using the described method datasets related to available designated designs. 35Each alternative is a coordinate in hyperspace that represents selected design parameters such as attributes (e.g., mass, volume, power consumption) or system parameters (e.g., total life cost, power distribution, deliver time, and reliability) that represent different design objectives. 36A suitably capable Tradespace exploration technique is essential to deal with the vast number of design alternatives and support systems engineering.

Case study
In this case study, a CubeSat is modelled as a structure consisting of six COTS components/subsystems.The subsystems are Communica-tion, Power, Solar Panel, ADCS, Command Data Handling, and Antenna, which are actual COTS components from CubeSat Shop. 34Table 3 shows the specifications for each COTS subsystem's alternative.For example, there are four options for the Communication subsystem, while the remaining five subsystems each have three options.An assumption is made that the COTS alternatives are compatible and easily accessible in the market.Choosing the most suitable combination of COTS alternatives that can satisfy the system requirement and limitations is a major challenge facing system designers.
The Tradespace (in Table 4) is, a combination of numbers, where n is the number of design alternatives that each of them includes multiple dimensions or variables.Each CubeSat architecture is defined as a set of board decision {k 1 , k 2 , k 3 , k 4 , k 5 , k 6 } that consists of six design variables, as follow: 1. k 1 , choose the k 1 th option for Communication subsystem, The requirement is to find design alternatives that satisfy the four objectives to minimise d 1 , d 2 , d 3 , d 4 .

Results
Firstly, we demonstrate the process to find the intersection via iterations.Since we have four objectives, that is, mass, volume, power, and cost, six 2-D objective spaces can be displayed in each iteration.The min-max method for all four objectives is illustrated in Figure 9 (combination of the subsystem of the intersection).Figure 9 shows the values

CONCLUSION
We addressed the issue of using a new metric for MOP with many objectives and a potentially large number of solutions on the Pareto front.There are common distance measures for decision-makers to identify a preferred solution of interest from a hundred to a thousand sets of solutions such as Euclidean distance, cosine similarity, Chebyshev distance etc. Recognizing these distance measures is essen-

TA B L E 4
Coordinates in the decision space and in the objective space.  in values is not entirely considered.When a recommender system is considered, as an example, the cosine similarity does not consider the difference in the rating scale. 37Another distance measure is called Manhattan distance, calculated as the distance of real-valued vectors, which does not work perfectly for high-dimensional data. 37Chebyshev distance is also known as the maximum distance between two vectors along any coordinated dimension.Although Chebyshev distance seems to work for high-dimensional data, it is appropriate for specific use cases, not all-purpose distance metrics.Minkowski distance is a metric employed in space where distances can be represented as a vector with a length.Nevertheless, Minkowski distance highly depends on the use-case and may not work effectively for high-dimensional data. 37other metric is called the Jaccard index, which is used to calculate the similarity and diversity of sample sets.However, the disadvantage of the Jaccard index is similar to Euclidean distance, which means it may not be suitable for a higher dataset.the mental workload of the decision-maker, and reaching a desirable decision quicker while ensuring the objectivity and soundness of the process.

F I G U R E 2
area.Such groups of ND solutions are known as Pareto-optimal solutions and are commonly referred to as Pareto Front, shown in the red points in Figure 1.It can be deduced that the Pareto optimal set associated with a selected problem can be obtained by the MOP algorithm subject to selecting a preferred solution amongst many possible solutions.Exiting efforts for designing sophisticated engineered systems have concentrated on algorithms and formulations used as a solution for optimising issues, and approximation techniques for reducing expenses related to F I G U R E 1 Concept of Pareto-optimality as the set of ND solutions.Pareto front size increases rapidly with the number of Tradespace objectives.

Figure 3 (
Figure 3(A, B), where parallel coordinate diagrams of SOI values (a) and single objective ranks (b) comprise the vertical axes.For both cases, the first solution that completely stays beneath the horizontal axes while it's moved upward is the preferred solution.This parallel coordinate representation shows that the number of preferred solutions according to each metric cannot be more than the number of objectives.

convex and nonconvex solutions are shown, respectively, in Figures 4 and 5 .
The preference of the solutions is colour coded, with the most preferred solution according to the matric as dark blue and the least preferred solutions with gold or yellow colour.The distribution and density of points in various segments of the solution space are changed to demonstrate the behaviour of the two metrics given the variation in densities of solutions.Solution spaces in Figures4(A, B) and 5(A, B) have uniform distributions, those in Figures4(C, D) and 5(C, D) have asymmetrical distributions, and those in Figures4(E, F) and 5(E, F) have nonuniform but symmetrical distributions.Both metrics behave relatively similarly where the distribution of solutions is symmetrical, with the more preferred solution lying precisely in the middle of the convex and concave curves.However, the metrics behave differently when the distributions are not symmetric, which is evident in F I G U R E 3 Demonstration of the max-min method with parallel coordinates plot for mean metric (A) and median metric (B).

Figures 4 (
Figures 4(C, D) and 5(C, D).The median metric in Figures 4(D) and 5(D) favors solutions, respectively, on the left and right.

Figure 8 (
Figure 8(C, D) show the front obtain from DEB3DK4B.

for
Tradespace exploration.Despite the volume limitation associated with CubeSat, most CubeSat architectures still offer multiple interacting subsystems and elements comparable to larger satellites.However, F I G U R E 4 A convex non-dominated Pareto front ranking by mean and median metrics.(A), (C), (E) show Mean metric rankings of solutions and (B), (D) and (F) show the ranking of the solutions based on the Median metric.F I G U R E 5 A concave non-dominated front-ranking by mean and median metrics.(A), (C), (E) shows the mean metric rankings of solutions and (B), (D) and (F) show the ranking of the solutions based on the median metric.F I U R E 6 Non-dominated solutions obtained from DEB2DK1B for the knee test problems.(A) and (C) show mean metric rankings of non-dominated solutions and (B) and (D) show the ranking of the non-dominated solutions based on the median metric.such subsystems that are smaller can be fully implemented at the board level.There are multiple standard Commercial-Off-The-Shelf (COTS) available versions of CubeSats, making it convenient to build and use straight away.A CubeSat bus may be considered a combined set of COTS parts, from which we can relatively quickly generate large

. k 2 , 3 F I G U R E 7 5 = 1 , 2 , 3 6. k 6 , 6 ∑ 6 ∑
choose the k 2 th option for Power subsystem, k 2 = 1,2,Non-dominated solutions obtained from DEB2DK4B for the knee test problems.(A) and (C) show mean metric rankings of non-dominated solutions and (B) and (D) show the ranking of the non-dominated solutions based on the median metric.3.k 3 , choose the k 3 th option for Solar panel, k 3 = 1,2,3 4. k 4 , choose the k 4 th option for ADCS subsystem, k 4 = 1,2,3,4 5. k 5 , choose the k 5 th option for Command Data Handling subsystem, k choose the k 6 th option in Antenna, k 6 = 1,2,3 We vary the six components in Table 4 to generate all possible combinations of these boards.This results in n = 4 × 3 × 3 × 4 × 3 × 3 = 1296 design alternatives.For each CubeSat design alternative, we record four intermediate variables (or performance variables/responses) of interest, as follows: 1. d 1 is the total mass M = n=1 m i.k 2. d 2 is the total volume V = 4 is the total power consumption P. Con. = n=1 p.con.i.k

F I U R E 8
Show the non-dominated fronts obtained from DEB3DK1B and DEB3DK4B.(A) and (B), respectively, show the mean and median rankings of DEB3DK1B non-dominated fronts.(C) and (D), respectively, show the mean and median rankings of DEB3DK4B. of the four objectives and the number of preferred solutions among 1296 solutions.

Figure 10
Figure 10 shows all the 2-D objective spaces, which indicates the number of solutions in the pairwise Pareto fronts acquired by mean,and median metrics problems.Each point represents a unique design alternative.We should note that the number of Pareto front solutions is 102 solutions when the four objectives are considered together in the search, some of which are highly undesirable solutions.Nonetheless, from these plots, no mapping information is available from decision space to object space.Then we try to find the intersection among the six objective spaces.

Figure 11 demonstrates
Figure 11 demonstrates the values and ranks of the four objectives and the number of preferred Pareto front solutions (102 solutions).Among all the obtained solutions shown in Figure11, we present the design of preferred solutions in Figure11(the thick lines are used to visualise the optimal design of value and rank, respectively).We obtain the design of 217 and 289 via the median and the design of 885 via the mean metric, where all solutions are non-dominated solutions that depend on their distribution.The 217, 289, and 885 designs are the preferred solutions for the four objectives shown in Table5shows the

Figure 12
Figure12shows all the objective spaces of preferred designs of 217, and 289 for the median and 885 for the mean metric in the pairwise Pareto fronts.For instance, according to Figure12, it is found that by considering mass-volume, volume-cost, volume-power consumption, and the mass-power consumption of 217 is lower among other designs of 102 solutions which is it verifies the 217 design is the preferred solution for decision-makers.

1 ⋮
Design alternative ntial to attain a preferred solution among a large set of objectives.For instance, Euclidean distance is one of the straightforward and popular distance measures known as the length of a segment connecting two points which is calculated from the cartesian coordinates when the magnitude of the vectors is important.However, this distance measure is appropriate only for low-dimensional data, and the data need to be normalized before using Euclidean distance.Cosine similarity measures the cosine of the angle of two vectors which is suitable for high-dimensional data.However, the magnitude of the vectors is not considered in Cosine similarity which indicates that the dissimilarityF I G U R E 9Objective values of all possible designs of four objectives (A) and objective ranks of all possible designs of four objectives (B).

F I U R E 1 0F I G U R E 1 1
Six objective spaces with Pareto front.Objective values of all possible designs of four objectives and 102 Pareto front solutions (A) and objective ranks of all possible designs of four objectives and 102 Pareto front solutions (B).

F I G U R E 1 2
The proposed median metric in this study works for both lowdimensional and high-dimensional spaces.Besides, it could be used for all-purpose distance metrics like Cosine similarity and Euclidean distance.The metric can be into various evolutionary algorithms where a uniform estimation of Pareto Front is desired such as the work discussed in ref.23.Simplicity is another characteristic of the presented new metric driven by a basic and intuitive algorithm.This method is also independent of the search algorithm, not initially relying upon non-dominated sorting as an essential prerequisite while it can still be applied to such a non-dominated set.More importantly, our proposed median metric leads to fewer solutions compared with previous metrics.This approach is suitable for decision-making in the early phases where objective preferences are not known, and all the objectives are treated as equally important.The future works include the incorporation of objective preferences into the Median metric.The general topic of preference modelling is discussed in refs.38-40.In a flow on paper, we will demonstrate the application of the median metric when the stakeholder preferences reflected in the objective preferences are known.The preference can be incorporated within the Median metric simply by skewing the moving axis in Figure3.The amount of skewness reflects the actual preference of one objective over the other.A possible extension to this work is the incorporation and the calculation of confidence intervals for Pareto Front when the objective functions contain uncertain parameters.41,42The confidence interval calculations are also useful when the Tradespace data are incomplete, in which case surrogate models established for example by State Vector Machines 43 might be used.To demonstrate the effectiveness of the presented method, a case study of CubeSat, including six COTS components (Communication, Power, Solar Panel, ADCS, Command Data Handling, and Antenna), is used.This approach attempts to extend the traditional Pareto optimisation method and allow designers to view more alternative Pareto-optimal solutions as well as similar designs to increase the chance the designers could reach an agreement on the understanding of the candidate system with non-preferred objectives.Thereby, additional benefits of working through a more streamlined alternative may include increasing the effectiveness of the design process, reducing Objective spaces of preferred design of 217.

Table 1
Assume a solution needs to be chosen from a set of 10 solutions points or Solutions of interest (SOIs) by minimising two objectives O1 and O2.Also, assume the values of 10 SOIs for each objective, as shown in Table1(a).Note that the value range for both objectives is between zero and one, thus comparable to one another in this example, meaning normalisation has already been performed in the objective space.In

Table 1 (
Values of SOIs for each objective and (b) and (c) values of SOIs for each objective and demonstration of median max-min method.
b). Figure2also demonstrates the L ∞ or MOP Mean metric.We should note that this metric is sensitive to the normalisation of SOI values.For example, if we stretch the O1 axis to include solution values from 0 to 1 by reducing the O1 values by 0.4 and then dividing them by 0.6, we arrive at the SOIs in Table 1(c), for which the L ∞ norm is S6.So, this metric is sensitive to the normalisation of solutions and ranges of the solution relative to single objectives.TA B L E 1 (a)TA B L E 2 (a) Ranks SOIs for each objective and demonstrates the median max-min method.(b) SOI correspondence to single objective ranks and demonstration of the intersection method of arriving at the median metric.

Table 2 (
a) shows the ranks of SOIs relative to single objectives for minimising SOI values.From this table, we can see that, for example, solution S1 is preferred to solution S2 regarding objective O1, S10 ranks 1 about objective O2, and S3 ranks 3 about objective O1.Now we find a solution that has the lowest maximum (aka min-max) rank for all objectives.To do this, we can scan the table from top to bottom and find the first solution (or a maximum of three solutions) present in all three columns.For example, we can see that S5 has a rank min-max rank of 6 with a ranks vector of [6 1].Thus, S5 ranks one according to this metric.Following a similar process, we see that S4 and S7 have the second rank.We refer to this metric as the MOP Median metric.Unlike the Mean metric, the Median metric is insensitive to the normalisation of SOI values.There is another way to arrive at the Median metric based on an intersection of a ranked set of solutions, as shown in Table2(a). 37 Obtained design from Pareto front solutions.
TA B L E 5