3.1 Conceptual model

The conceptual model presented in Fig. 2 outlines the workflow. Initially, a dataset was generated with careful consideration of the parameters depicted in the figure. This facilitated the creation of both the dataset and test problems. Subsequently, the mathematical models for the SOO and MOO models were developed using the GAMS program. All result files were saved, and statistical analyses were conducted using the IBM SPSS 22 software.

Figure 2.

Conceptual model.

3.2 Mathematical model

MOO models were developed to optimise the sequencing of arrival aircraft by considering the transit passenger numbers in each aircraft. Also, the SOO model was formed by combining two objectives into one equation.

Separation minima between arrival-arrival and departure-departure are given in Table 1. All separation durations are given in seconds and were used in the mathematical model as parameters csarr(o1,o2) and csdep(o1,o2).

Equations 1 and 2 present the objective functions in the mathematical model. Equation 1 focuses on minimising arrival delay of arrival flights that have more transit passengers than threshold value while Equation 2 consists of air and ground delay of all other aircraft. Equation 3 presents the combination of the objective functions for the SOO model. Equations 4–6 represent this combination for MOO models. For these formulations and detailed explanation, please refer to [Reference Kasimbeyli, Ozturk, Kasimbeyli, Yalcin and Erdem25].

(1) \begin{align}{f_1} = {\rm{\;}}\mathop \sum \limits_{i|i \in I} {\rm{a}}{{\rm{d}}_i},\end{align}

\begin{align*}{\rm{transi}}{{\rm{t}}_i} \gt avetransit,{\rm{\;}}fmod{e_i} = 1{\rm{\;}}\end{align*}

(2) \begin{align} & \quad \quad \quad \quad \quad \quad {f_2} = \mathop \sum \limits_{i1|i1 \in I} {\rm{a}}{{\rm{d}}_{i1}} + \mathop \sum \limits_{i2|i2 \in I} {\rm{g}}{{\rm{d}}_{i2}}{\rm{\;}}, \\ \nonumber&i1 \ne i2,{\rm{\;transi}}{{\rm{t}}_{i1}} \le avetransit,{\rm{\;}}fmod{e_{i1}} = 1,{\rm{\;}}fmod{e_{i2}} = 2\end{align}

Table 1.

Arrival and departure separation minima (sec.)

(3) \begin{align}\min\nolimits_{SOO}\left( {\mathop \sum \limits_{n = 1}^2 {f_n}} \right)\end{align}

(4) \begin{align}{\min\nolimits_{WSS}}\left( {\mathop \sum \limits_{n = 1}^2 {w_n}\ {\rm{*}}\ {f_n}} \right)\end{align}

(5) \begin{align}{\min\nolimits_{CONIC}}\left( {\mathop \sum \limits_{n = 1}^2 {w_n}\ {\rm{*}}\ ({f_n} - {a_n}) + \alpha\ \ {\rm{*}}\ \ \mathop \sum \limits_{n = 1}^2 \left| {{f_n} - {a_n}} \right|} \right)\end{align}

(6) \begin{gather} {\min\nolimits_{EPSILON}}{\rm{\;}}\left( {{f_1}} \right)\nonumber\\[9pt]\mathrm{s.t.}\ {f_2} \le eps \end{gather}

Constraint 7 calculates the entry times of the arrival aircraft into the TMA. Constraint 8 gives the passing times of each aircraft over waypoints by considering holding and margin delays. The calculation of the flight duration between two waypoints utilises the distance and speed information published in the AIP. Constraints 9 and 10 are responsible for monitoring the separations between aircraft flying towards the same waypoint.

(7) \begin{gather}{t_{i,w}} = e{t_i}{\rm{\;}}, \nonumber\\[8pt]w = e{p_i},{\rm{\;}}fmod{e_i} = 1,{\rm{\;}}\forall i \in I,{\rm{\;}}\forall w \in W \nonumber \end{gather}

(8) \begin{gather}{t_{i,w2}} = {t_{i,w1}} + ({d_{w1,w2}}/{v_{w1,w2}})\ {\rm{*}}\ ct + {h_{w1}}\ {\rm{*}}\ h{d_{i,w1}} + m{d_{i,w1,w2}}{\rm{\;}},\nonumber\\[8pt]w1 \ne w2,{\rm{\;}}{r_{p,w1}} = 1,{\rm{\;}}p = e{p_i},{\rm{\;}}se{g_{w1,w2}} = 1,fmod{e_i} = 1,{\rm{\;}}\forall i \in I,\\[8pt]\forall \!\left({w1,w2} \right) \in W,\forall p \in P \nonumber\end{gather}

(9) \begin{gather}{t_{i2,w}} - {t_{i1,w}} \geq sep - M\ {\rm{*}}\ \left( {1 - {x_{i1,i2}}} \right){\rm{\;}}, \nonumber\\[8pt]i1 \ne i2,{\rm{\;}}{r_{p1,w}} = 1,{r_{p2,w}} = 1,{\rm{\;}}p1 = e{p_{i1}},{\rm{\;}}p2 = e{p_{i2}},{\rm{\;}}fmod{e_{i1}} = 1,{\rm{\;}}fmod{e_{i2}} = 1\nonumber\\[8pt]\forall \!\left( {i1,i2} \right) \in I,{\rm{\;}}\forall w \in W,\forall\! \left( {p1,p2} \right) \in P \end{gather}

(10) \begin{gather}{t_{i1,w}} - {t_{i2,w}} \geq sep - M\ {\rm{*}}\ \left( {{x_{i1,i2}}} \right){\rm{\;}},\nonumber\\[8pt]i1 \ne i2,{\rm{\;}}{r_{p1,w}} = 1,{r_{p2,w}} = 1,{\rm{\;}}p1 = e{p_{i1}},{\rm{\;}}p2 = e{p_{i2}},{\rm{\;}}fmod{e_{i1}} = 1,{\rm{\;}}fmod{e_{i2}} = 1\nonumber\\[8pt]\forall \!\left( {i1,i2} \right) \in I,{\rm{\;}}\forall w \in W,\forall\! \left( {p1,p2} \right) \in P \end{gather}

Constraint 11 calculates the holding delay times for aircraft that enter a holding pattern. The division operation within this constraint ensures that, in the presence of a holding pattern, the delay time for each pattern is based on the minhd and its multiples. This formulation is employed to enhance both the smoothness of the holding pattern and its realism. Constraints 12 and 13 count holding instructions and constrain the total number of them, respectively. Constraint 14 limits the total holding pattern duration for each aircraft. Constraint 15 guarantees that each waypoint can only be used for a holding pattern up to the allowed maximum number. Constraint 16 includes a set of constraints to ensure that, if an aircraft is assumed to experience margin delay, minmd is guaranteed. If such a manoeuver is not required, the margin delay time for these aircraft is set to zero. Additionally, it limits the margin delay of each aircraft to a maximum duration of maxmd between waypoints. This set of constraints is presented in grouped form for readability. Constraints 17 and 18 count margin delay instructions and constrain the total number of them, respectively. Constraints 19 limits total margin delay for each aircraft.

(11) \begin{gather}{n_{i,w}} = ({h_w}\ {\rm{*}}\ h{d_{i,w}})/minhd{\rm{\;\;}}, \nonumber\\[8pt]{r_{p,w}} = 1,{\rm{\;}}p = e{p_i},{\rm{\;}}fmod{e_i} = 1,{\rm{\;}}\forall i \in I,{\rm{\;}}\forall w \in W,\forall p \in P\end{gather}

(12) \begin{gather}crmHL{D_i} = \mathop \sum \limits_{(w|w \in W)} \mathop \sum \limits_{(p|p \in P)} {n_{i,w}}{\rm{\;}}, \nonumber\\[8pt]{r_{p,w}} = 1,{\rm{\;}}p = e{p_i},fmod{e_i} = 1,{\rm{\;}}\forall i \in I\end{gather}

(13) \begin{gather} crmHL{D_i} \le maxhld{\rm{\;}}, \nonumber\\[8pt]fmod{e_i} = 1,\ \forall i \in I\end{gather}

(14) \begin{gather}\mathop \sum \limits_{(w|w \in W)} \mathop \sum \limits_{(p|p \in P)} {h_w}\ {\rm{*}}\ h{d_{i,w}} \le maxhlddur{\rm{\;}}, \nonumber\\[8pt]{r_{p,w}} = 1,{\rm{\;}}p = e{p_i},fmod{e_i} = 1,{\rm{\;}}\forall i \in I\end{gather}

(15) \begin{gather}\mathop \sum \limits_{(i|i \in I)} \mathop \sum \limits_{(p|p \in P)} {n_{i,w}} \le maxhldcount{\rm{\;}}, \nonumber\\[8pt]{r_{p,w}} = 1,{\rm{\;}}p = e{p_i},fmod{e_i} = 1,{h_w} = 1,{\rm{\;}}\forall w \in W\end{gather}

\begin{gather*}m{d_{i,w1,w2}} \geq 0 - M\ {\rm{*}}\ {y_{i,w1,w2}} \\[8pt]m{d_{i,w1,w2}} \le 0 + M\ {\rm{*}}\ {y_{i,w1,w2}} \\[8pt]m{d_{i,w1,w2}} \geq minmd - M\ {\rm{*}}\ \left( {1 - {y_{i,w1,w2}}} \right)\end{gather*}

(16) \begin{gather}m{d_{i,w1,w2}} \le maxmd{\rm{\;\;}},\nonumber\\[8pt]w1 \ne w2,{\rm{\;}}{r_{p,w1}} = 1,{\rm{\;}}p = e{p_i},{\rm{\;}}se{g_{w1,w2}} = 1,fmod{e_i} = 1,{\rm{\;}}\forall i \in I,{\rm{\;}}\forall\! \left( {w1,w2} \right) \in W,\forall p \in P\end{gather}

(17) \begin{gather}crmMCT{R_i} = \mathop \sum \limits_{(w1|w1 \in W)} \mathop \sum \limits_{(w2|w2 \in W)} \mathop \sum \limits_{(p|p \in P)} {y_{i,w1,w2}}{\rm{\;}},{\rm{\;}}\nonumber\\[8pt]w1 \ne w2,{\rm{\;}}{r_{p,w1}} = 1,{\rm{\;}}p = e{p_i},{\rm{\;}}se{g_{w1,w2}} = 1,fmod{e_i} = 1,{\rm{\;}}\forall i \in I\end{gather}

(18) \begin{gather}crmMCT{R_i} \le maxmctr{\rm{\;}}, \nonumber\\[8pt]fmod{e_i} = 1,\ \forall i \in I\end{gather}

(19) \begin{gather}\mathop \sum \limits_{(w1|w1 \in W)} \mathop \sum \limits_{(w2|w2 \in W)} \mathop \sum \limits_{(p|p \in P)} m{d_{i,w1,w2}} \le maxmctrdur{\rm{\;}},{\rm{\;}} \nonumber\\[8pt]w1 \ne w2,{\rm{\;}}{r_{p,w1}} = 1,{\rm{\;}}p = e{p_i},{\rm{\;}}se{g_{w1,w2}} = 1,fmod{e_i} = 1,{\rm{\;}}\forall i \in I\end{gather}

Constraint 20 provides the entry times of the aircraft into the PMS. Constraint 21 calculates the passing time for each aircraft over the merge point. This calculation considers the time aircraft spend on the sequencing leg and the flight duration from the sequencing leg to the merge point. Constraints 22 and 23 ensure that the separation between consecutive arrival aircraft adheres to the time intervals specified in Table 1. Constraint 24 constrains the maximum flight duration on the sequencing legs.

(20) \begin{gather}t{p_i} = {t_{i,w}}{\rm{\;}}, \nonumber\\[8pt]w = pm{s_p},{\rm{\;}}p = e{p_i},{\rm{\;}}fmod{e_i} = 1,\ \forall i \in I,{\rm{\;}}\forall w \in W,\forall p \in P\end{gather}

(21) \begin{gather}mp{t_i} = t{p_i} + durseqmpt + durseqle{g_i}{\rm{\;}}, \nonumber\\[8pt]fmod{e_i} = 1,{\rm{\;}}\forall i \in I \end{gather}

(22) \begin{gather}mp{t_{i2}} - mp{t_{i1}} \geq csar{r_{o1,o2}} - M\ {\rm{*}}\ \left( {1 - y{2_{i1,i2}}} \right){\rm{\;}}, \nonumber\\[8pt]i1 \ne i2,{\rm{\;}}o1 = ca{t_{i1,{\rm{\;}}}}o2 = ca{t_{i2,{\rm{\;}}}}\ fmod{e_{i1}} = 1,{\rm{\;}}fmod{e_{i2}} = 1,\ \forall \left( {i1,i2} \right) \in I,{\rm{\;}}\forall \left( {o1,o2} \right) \in O\end{gather}

(23) \begin{gather}mp{t_{i1}} - mp{t_{i2}} \geq csar{r_{o1,o2}} - M\ {\rm{*}}\ y{2_{i1,i2}}{\rm{\;}}, \nonumber\\[8pt]i1 \ne i2,{\rm{\;}}o1 = ca{t_{i2,{\rm{\;}}}}o2 = ca{t_{i1,{\rm{\;}}}}\ fmod{e_{i1}} = 1,{\rm{\;}}fmod{e_{i2}} = 1, \forall \left( {i1,i2} \right) \in I,{\rm{\;}}\forall \left( {o1,o2} \right) \in O\end{gather}

(24) \begin{gather}durseqle{g_i} \le durseq{\rm{\;}}, \nonumber\\[8pt]fmod{e_i} = 1,\ \forall i \in I\end{gather}

Constraint 25 calculates the departure times of aircraft on the ground by considering the ground delay time. Constraints 26 and 27 are used to ensure the separations between landing and departing aircraft by sep2 (60 sec.) and sep3 (75 sec.), respectively. In these equations, ICAO Doc 8168 was used to determine the final approach speed ( $vfa{f_o}$ ) for each aircraft category [26]. An approach speed of 200 knots is applied between the initial and final approach fix. Additionally, for heavy traffic, a final approach speed of 150 knots is applied while for medium traffic, a speed of 130 knots is used. Constraints 28 and 29 ensure that the separation between consecutive departure aircraft adheres to the time intervals specified in Table 1. Constraint 30 constrains the maximum delay duration on the ground. Constraint 31 calculates the total airborne delay duration for each aircraft. This calculation considers the time spent on the sequencing legs, holding delays and margin delays. Constraint 32 presents the sign constraints of the model.

(25) \begin{gather}deptim{e_i} = e{t_i} + g{d_i}\, \nonumber\\[8pt]fmod{e_i} = 2,\ \forall i \in I\end{gather}

(26) \begin{gather}deptim{e_{i2}} - \left( {mp{t_{i1}} + durmptfaf + \left( {\frac{{dist}}{{vfa{f_o}}}\ {\rm{*}}\ ct} \right)} \right) \geq sep2 - M\ {\rm{*}}\ \left( {1 - y{3_{i1,i2}}} \right){\rm{\;}}, \nonumber\\[8pt]i1 \ne i2, \ fmod{e_{i1}} = 1, fmod{e_{i2}} = 2, \forall \left( {i1,i2} \right) \in I \end{gather}

(27) \begin{gather}\left( {mp{t_{i1}} + durmptfaf + \left( {\frac{{dist}}{{vfa{f_o}}}\ {\rm{*}}\ ct} \right)} \right) - deptim{e_{i2}} \geq sep3 - M\ {\rm{*}}\ y{3_{i1,i2}}{\rm{\;}}, \nonumber\\[8pt]i1 \ne i2,\ fmod{e_{i1}} = 1, fmod{e_{i2}} = 2, \forall \left( {i1,i2} \right) \in I \end{gather}

(28) \begin{gather}deptim{e_{i2}} - deptim{e_{i1}} \geq csde{p_{o1,o2}} - M\ {\rm{*}}\ \left( {1 - y{4_{i1,i2}}} \right){\rm{\;}}, \nonumber\\[8pt]i1 \ne i2,{\rm{\;}}o1 = ca{t_{i1,{\rm{\;}}}}o2 = ca{t_{i2,{\rm{\;}}}}\ fmod{e_{i1}} = 2,{\rm{\;}}fmod{e_{i2}} = 2, \forall \left( {i1,i2} \right) \in I,{\rm{\;}}\forall \left( {o1,o2} \right) \in O\end{gather}

(29) \begin{gather}deptim{e_{i1}} - deptim{e_{i2}} \geq csde{p_{o1,o2}} - M\ {\rm{*}}\ y{4_{i1,i2}}{\rm{\;}}, \nonumber\\[8pt]i1 \ne i2,{\rm{\;}}o1 = ca{t_{i2,{\rm{\;}}}}o2 = ca{t_{i1,{\rm{\;}}}}\ fmod{e_{i1}} = 2,{\rm{\;}}fmod{e_{i2}} = 2, \forall \left( {i1,i2} \right) \in I,{\rm{\;}}\forall \left( {o1,o2} \right) \in O\end{gather}

(30) \begin{gather}g{d_i} \le maxgd{\rm{\;}}, \nonumber\\[8pt]fmod{e_i} = 2,\ \forall i \in I\end{gather}

(31) \begin{gather}a{d_i} = durseqle{g_i} + \mathop \sum \limits_{(w1|w1 \in W)} \mathop \sum \limits_{(w2|w2 \in W)} \mathop \sum \limits_{(p|p \in P)} m{d_{i,w1,w2}}{\rm{\;}} + {\rm{\;}}\mathop \sum \limits_{(w|w \in W)} \mathop \sum \limits_{(p|p \in P)} {h_w}\ {\rm{*}}\ h{d_{i,w}} \nonumber\\[8pt]w1 \ne w2,{r_{p,w1}} = 1,{\rm{\;}}{r_{p,w}} = 1,{\rm{\;}}p = e{p_i},{\rm{\;}}se{g_{w1,w2}} = 1,fmod{e_i} = 1,{\rm{\;}}\forall i \in I\end{gather}

(32) \begin{gather}{x_{i1,i2}}, {y_{i,w1,w2}}, y{2_{i1,i2}}, y{3_{i1,i2}}, y{4_{i1,i2}} \in {\rm{\;}}\left\{ {0,{\rm{\;}}1} \right\} \nonumber\\[8pt]{n_{i,w}},{\rm{\;}}crmHL{D_i},crmMCT{R_i} \in {Z^ + } \nonumber\\[8pt]{\rm{all\;remaining\;variables}} \geq 0 \end{gather}

4.1 Numerical results

The comparison between the average delay of HTP and LTP flights is shown in Table 2. It presents the average total delays, in seconds, for flights labeled as HTP and LTP in the results of three different MOO models and the SOO model. There are 50 iterations for each MOO model, constituting a total of three distinct sub-problems across these 50 iterations. To elucidate, consider the model WSS-50H-50M-1: it undergoes 50 iterations to obtain Pareto-optimal points. Additionally, there exist two more sub-problems denoted as WSS-50H-50M-2 and WSS-50H-50M-3. Consequently, the average delay is computed by aggregating all delays for HTP and LTP, and subsequently dividing this cumulative value by 150 (representing the product of 50 iterations and 3 sub-problems). This calculation serves to portray the overall performance of each model across different sub-problems. To enhance clarity, direct representation is preferred over percentages in this table, and single objective results were separated as HTP and LTP as well after obtaining the results from the model.

Table 2.

Average delays of HTP and LTP flights (sec.)

When looking at delay times on an average basis, it is evident that HTP flights under the category distribution of 30–70 exhibited the lowest delay times in the CS model. It can be observed that this model has proven to be more effective in reducing delay times for HTP flights compared to other models. However, due to the trade-off approach, it has affected the delay times for LTP aircraft, but it has not significantly deviated from the results obtained from the other models. The SOO model has the lowest LTP delay time, at 139.25 seconds, in the 100M. However, its HTP delay time is higher compared to CS, WSS and EC models in the same category distribution. Figure 3 provides a comparison between MOO models and the SOO model based on the average delay times given in Table 2.

Figure 3.

Comparison of models in terms of HTP and LTP delays.

The percentages provided in Fig. 3 encompass the comparison of the results between the SOO model and the respective MOO models based on the averages for the given category distributions. Positive values indicate that the SOO model produced better results while negative values indicate that the corresponding MOO model achieved superior results. As can be observed from the figure, the CS model outperformed the SOO model by −22.91%, −24.93%, −22.58% and −16.73% for 50H-50M, 70H-30M, 30H-70M and 100M category distribution on HTP delay, respectively. On the other hand, when considering the LTP delay times, the WSS model yielded better results when compared to the other MOO models across all categories. But, for the same delay, the SOO model achieved superior results compared to the MOO models across all category distributions. For 100M distribution, EC delivered the best result for HTP delay among the MOO models while the opposite was true for the LTP delay.

In Fig. 4, average delay times for each distribution are provided for the MOO and the SOO models. As evident from this figure, the model has scarcely utilised holding delays in any solution model or category distribution. In the 50-50 distribution, while the SOO model and WSS model nearly equally employ margin delays, the CS model proves more efficient for sequence delay in the same distribution. In the 70-30 distribution, the CS model continues to yield better results for sequence delay, while for ground delay, the SOO model exhibits a slight but favourable differentiation from the others. For margin delay, the WSS model achieved superior result compared to the MOO models in the 30-70 distribution. This model also outperforms the other MOO models in terms of holding delays across all category distributions. In the 100M distribution, ground delay was needed less compared to the others. The main reason for this is that the required separation time for medium traffics is shorter than for mixed traffics.

Figure 4.

Type of delays.

Multi-objective optimisation inherently provides decision-makers with various solutions, catering to the complexities of conflicting objectives. The number of pareto-optimal points obtained from each MOO model is presented in Fig. 5. The number of pareto-optimal points is provided for each distribution in the figure, along with their sub-problems (t1-t3), for each MOO model. These numbers were found by counting the strong pareto dominances obtained for each outcome. The EC model generally produced a greater number of pareto results than the other models.

Figure 5.

Total number of pareto optimal points.

Figure 6 depicts the pareto-optimal fronts generated by the MOO models for the results of a randomly selected test problem in this study. In this figure, the discrepancy in the number of pareto-optimal points among the MOO models may indicate the effectiveness of each model in exploring the solution space. Specifically, the EC model, with 8 pareto-optimal points, seems to be the most effective in finding various trade-off solutions compared to the other models for the related category distribution and test problem. The greater number of pareto-optimal points in the EC model could suggest that it provides a wider range of feasible solutions that balance the competing objectives, which can be valuable when decision-makers seek a variety of options.

Figure 6.

Pareto optimal fronts of each MOO model (a: 70H-30M; test no: 1, b: 100M; test no: 1).

4.2 Emission insights in optimisation results

In the context of optimising the solution, the primary goal lies in managing trade-offs between the objective functions which are related to aircraft delay in this study. Despite the absence of a direct optimisation goal for emissions, their quantification provides a critical insight regarding the environmental implications of the solutions.

The formula, Equation 33 [27], provided below has been utilised for the analysis of emission values within this study. TIM, FF, EI, Ne and E stand for time in mode, fuel flow, emission index, the number of engines and total emissions of each pollutant, respectively. The indices i, j and k are used to denote the pollutant, aircraft and aircraft operation mode, respectively. The FF and EI values of each aircraft type used in the study can be seen in Table 3 [28]. In the absence of a distinct holding pattern index in the database, the APP index was applied for emission calculations during the holding phase if any pattern exists. These values were employed uniformly for each test problems.

Table 3.

Emission index and fuel flow coefficients

All emission calculations have been conducted based on holding patterns, margin delay instructions, time spent on the sequencing leg and ground delays, excluding emission calculations for routine flight phases for each aircraft. The primary reason for this is to integrate emission calculations more cohesively with the objective functions of the research. This approach aims to align the calculation methodology with the overarching goal of the study.

(33) \begin{align}{E_{ij}} = \sum TI{M_{jk}}\ {\rm{*}}\ F{F_{jk}}\ {\rm{*}}\ E{I_{ijk}}\ {\rm{*}}\ N{e_j}\end{align}

Emission values calculated based on the MOO and the SOO models are represented in Fig. 7. In the figure, average emission values for HC, CO and NOx are provided for each MOO and the SOO model. While models generally exhibit relatively little variation for HC, distinctions are evident for CO and NOx emissions. In a 50H-50M distribution, the WSS model may be considered better choice for CO among the MOO models, whereas for NOx in the same distribution, the CS model yields a lower emission value. In a 70H-30M distribution, the SOO model achieves the best values for CO while the CS model once again attains superior emission values for NOx compared to the other models. However, when examining the 30H-70M distribution, the dynamics shift slightly. While the WSS model is more effective for CO among the MOO models, the SOO model yields superior results for the NOx pollutant. In the 100M distribution, the SOO and CS delivered better results than the others for the CO and the NOx pollutant, respectively.

Figure 7.

Emission values for HC, CO and NOx.

4.3 Statistical analysis

Statistical analyses were carried out to compare the differences between the means of the MOO models’ results. In this study, three distinct MOO models constitute the independent variables while the variables HTP and LTP delay serve as the dependent variables. The objective is to ascertain whether these three different models employed for problem-solving exhibit statistically significant differences from one another. The results of the three test problems for each category distribution were combined and subjected to statistical testing according to their respective distributions. To decide on the appropriate statistical method, an initial step involved testing whether the variables (HTP and LTP) exhibited a normal distribution. This test was conducted using IBM SPSS 22. Prior to the analysis, missing data were examined, and outliers were identified and excluded. Subsequently, a test for normality was performed with the interpretation relying on the Skewness and Kurtosis values. When these values fall within the range of −1 to +1, it is considered that the variable values follow at least approximately normal distribution and some parametric tests such as the one-way ANOVA are quite robust to show meaningful results under this condition [Reference Leech, Barrett and Morgan29].

As a result of the normality test, it was seen that the Skewness and Kurtosis values for each analysis were located between −1 and +1. Hence, it was assumed that the dependent variables in each analysis followed a normal distribution, allowing for the applicability of parametric tests to compare the mean values of the dependent variables. With both the assumption of normality and the presence of four independent variables, the choice was made to employ the one-way ANOVA test for the analyses. This test is a statistical technique employed to assess the differences among the means of three or more groups [Reference Büyüköztürk30]. It is utilised to determine whether there are significant variations in the dependent variable concerning the categorical independent variable. The results of this test is given in Table 4.

Table 4.

ANOVA and Welch test

In accordance with the Levene statistic, it is observed that the variance homogeneity assumption ( ${p_{value}}$  ≥ .05) is met for the LTP delay in the 70H-30M distribution and HTP delay in the 100M distribution, whereas it is not satisfied for the other distributions. Consequently, the ANOVA test has been employed for the group with homogeneous variances while the Welch test has been utilised for the remaining groups. All results marked with an asterisk (*) indicate that the MOO models differ from each other in a statistically significant way ( ${p_{value}}$  < .05) in corresponding dependent variables. Furthermore, post hoc tests were conducted.

The Scheffe test was employed as a post hoc analysis in cases where the variance homogeneity assumption ( ${p_{value}}$  ≥ .05) was met in the Levene statistic. On the contrary, the Games-Howell test was applied in other instances. The results of this are summarised in Table 5, highlighting only the statistically significant differences. For the mean difference values calculated positively according to the (I–J) operation in the Mean Difference column of Table 5, the mean delay values of variables in column I are greater than the mean values of the variables in column J.

Table 5.

Post hoc tests

Upon examining the significant differences, it can be generally stated that the WSS and CS models provided lower delay values than the EC model. As an example, the largest difference occurred in the dependent variable LTP delay for the 70H-30M distribution, specifically between EC and WSS. In this case, the average LTP delay value obtained with EC is 288.02 seconds higher than that obtained with WSS. The smallest difference (13.14 seconds) was observed in the HTP delay variable for the 30H-70M distribution, again between WSS and CS. In this case, the average value obtained with WSS. is higher than that obtained with CS.

In conclusion, the one-way ANOVA and post hoc tests conducted in this section provided valuable insights into the differences and trends within the results. The results have elucidated the significant differences between the MOO models as revealed by the post hoc comparisons in Table 5.