*Mn* <sup>=</sup> *FPn* <sup>−</sup> *FP*min*<sup>n</sup> FP*max*<sup>n</sup>* − *FP*min*<sup>n</sup>*

(6)

$$M\_n = \frac{FN \text{max}\_{\text{n}} - FN\_{\text{n}}}{FN \text{max}\_{\text{n}} - FN \text{min}\_{\text{n}}} \tag{7}$$

*FPn* = *FP*min*<sup>n</sup>* + (*FP*max*<sup>n</sup>* − *FP*min*n*) · *Mn* (8)

$$FN\_{\text{ll}} = FN \text{max}\_{\text{ll}} - \left(FN \text{max}\_{\text{ll}} - FN \text{min}\_{\text{ll}}\right) \cdot M\_{\text{ll}} \tag{9}$$

where:

*FP*min to *FP*max is the range of positive factors;

*FN*max to *FN*min is the range of negative factors.

Data for both experiments are summarized in Table 3. The "reference value" of a factor and the corresponding monkey position value show the actual (current) data. All factor values are commented on in Section 5.




**Table 3.** *Cont.*

#### *4.3. Experiment 1, Testing the Changeability of Factors' Values*

In this experiment, the initial values of all monkey positions are midway between the minimum and maximum, which is 0.500. The followers coefficient (*L*) is 1.5, and the range of hierarchy modifiers (*R*min, *R*max) are −0.3 and 0.3. These values were selected based on testing various algorithm stop criteria. This aspect is commented on in Section 5. The effect of the NOAH algorithm after 32 steps is shown in Figure 4.

**Figure 4.** Changes in decision measures during the steps in Experiment 1.

Because the initial monkey position value assumption is 0.5 and the number of monkeys considered is 16, the starting value of the first decision measure (*M*) is 8.0. The value of this measure is changed in the subsequent steps of the algorithm in accordance with the procedures described in Section 3. These changes should be observed in order to formulate the algorithm stopping criteria. One such criterion could be the difference between the *M* values in the following steps. When this difference reaches a predetermined minimum, the algorithm stops. We observe "stabilization in the nest".

The second decision measure is the sum of importance values (*I*). Because each individual's value for importance is random, the measure *I* varies from step to step but fluctuates

around an average of 8.0. Observing the changes in the value of *I* makes sense in order to formulate the second algorithm stopping criterion. If you change this value significantly, you can stop the algorithm or ignore this particular step. Here we see "a remarkable change of subjective meaning in the nest".

The average value of the hierarchy (*H*) is the third decision measure. This value drops to zero. Achieving the assumed minimum value may terminate the algorithm's work. A similar situation takes place in the next decision measure—the average value of the random hierarchy modifier (*R*). It is possible to use only one decision measure and only one stop criterion, but the observation of all of them increases the set of analyzed solutions. In this experiment, stabilization in the socket was achieved after 32 steps, and the algorithm was completed. The values of all NOAH parameters for all monkeys and the decision measures in the final step are presented in Table 4. This table also shows the recalculated values of the analyzed factors as a set characterizing the final step of the algorithm. The interpretation of the coefficient values is presented in Section 5


**Table 4.** The results of the first experiment.

#### *4.4. Experiment 2, Considering the Real Data*

In this experiment, the initial values of the monkeys are calculated according to actual (reference) factor values (see Table 3). The values of the followers coefficient and hierarchy modifier ranges are the same as in Experiment 1. The effect of the NOAH algorithm after 32 steps is shown in Figure 5. The same four decision measures and stopping criteria were also used as in Experiment 1. The values of all NOAH parameters for all monkeys and decision measures in the final step are presented in Table 5. This table also shows the re-converted values of the analyzed factors as a set characterizing the final step of the algorithm. The interpretation of the factor values is presented in Section 5.

**Table 5.** The results of the second experiment.



**Table 5.** *Cont.*

**Figure 5.** Changes in decision measures during the steps in Experiment 2.

#### **5. Discussion**

Table 6 summarizes the important data for all factors. The minimum, maximum, reference (actual), and final values from both experiments are shown. The reference values correspond to the observed (measured) situations. Measurements and data collection were performed in the spring of 2022. The minimum and maximum values are calculated or assumed according to physical conditions or other possibilities. For example, "percentage of spaces occupied" (*F*4) can of course vary between 0 and 100%, and "parking volume on PR" (*F*7) can vary between 0 and 100, the upper limit being the actual number of parking spaces in all of the considered locations. The "cost" (*F*10) could vary from PLN 4.6 to 12.0, which results from the comparison of various fees in the considered journeys between the origin and the destination; "travel times" (*F*12–*F*14) oscillate between values obtained from measurements or calculated taking into account the possible speeds.

The values obtained in the experiments depend on the values of the parameters adopted in the algorithm: the followers coefficient (*L*) and the range of hierarchy modifiers (*R*min, *R*max). These values may vary depending on the specifics of the problem under consideration (e.g., depending on a number of factors) and should be tested in the experimental phase of the research. Finally, the values were selected—*L* = 1.5, *R*min = −0.3 and *R*max = 0.3—as the best according to the decision measure change process. The impact of

the abovementioned parameters on the results of NOAH requires further research and will be explored in the future.


**Table 6.** Set of values for all factors.

The factors values obtained from the experiments should be analyzed as a complex set of parameters. In both experiments, sufficient numbers of trips (*F*1) and stops (*F*5) were identified in relation to the values of the other factors. These values led to an intuitive decrease in the number of passengers during the day (*F*2). Comparing these values with the stable number of residents (*F*6), there is a need to increase the role of the delivery system. The delivery system is represented here by factors including cars in PR (*F*7) and passengers arriving from correlated forms of public transport (*F*8). The travel cost (*F*10) should be lower. The conditions tested above also show a lower number of journeys by means that compete with rail, both with private cars (*F*16) and buses (*F*15).

Some results differ in both experiments. Because the first experiment starts with the "average" values of the factors, the results are close to the middle between the minimum and the maximum. This shows the potential NOAH effect but is not of practical use. The results of the second experiment are more realistic and allow you to judge the real conditions. The maintenance of a clear difference between the travel time of rail travel and those of competitive forms is particularly visible. The results show the possibilities of further use of the constructed methodology and algorithm. For example, it is possible to test other numbers of residents (*F*6) or numbers of departures (*F*1). It is also possible to test different values of the minimum and maximum factors. They should allow the modeling of various conditions in the suburban rail system and the observation of correlations between all factors. Initial values of the abovementioned parameters should be carefully collected. This limitation requires further work to be overcome.

The obtained results correspond with actual topics involved in the interdisciplinary field of sustainable urban planning and transport development. Significantly, connection with heuristics used in analyses of transportation and railway systems [64,65] occurred. This aspect shows the potential of the proposed algorithm and methodology to be developed in other studies not only connected with railway systems and not only in transportation engineering.

#### **6. Conclusions**

The parameters describing a suburban rail system have different characteristics. The set of possible factors is not precisely defined, and the data collection process has specific problems. Some data are incomplete, while others contain errors. The influence of each selected factor and its importance are not fully understood. Therefore, modeling and analysis of suburban rail systems requires specific methods that also take into account human behavior. Here, heuristics as a holistic tool can help in the analysis of possible correlations between factors, as well as be helpful in the decision-making process. The novel method presented here allows using an open set of data. The factors could be different and somewhat "chaotic" at first sight (as in the presented experiments). This is the key innovation in analyses of "technical systems" like the suburban railways considered here.

The main advantage of the proposed method is the creation of the possibility to observe various sets of data and their interactions without precise knowledge about the influence of a specific factor on the result. The data (describing the transportation systems) depend partially on human decisions. For example, the choice of the mode of transport could be a reaction to the behaviors of other people (neighbors, social media groups). So, an individual can observe and copy leaders as a follower. The use of the specific transport means influencing the number of passengers or parking volume in PR has some uncertainties and could be modeled using heuristics.

NOAH, which is inspired by the behavior of groups of monkeys, especially taking into account dynamic changes in leadership, is likely to be useful in specific technical problems where physical parameters (such as the number of departures or stops) are compared with human decisions moderated by travel time, prices, etc. The basic definitions, the procedures, the algorithm, and the potential application of NOAH are illustrated in simple examples with a relatively small set of factors. The NOAH algorithm modifies the values of factors in a heuristic way and shows possible solutions that could be introduced in reality. This method created originally by the author is quite similar to swarm intelligence algorithms (like PSO and ACO) but contains new elements based on specific behaviors of monkeys which were described in Section 3.

Strict comparison of this new method with others is not possible because of the other goals of those methods. The heuristics search mainly for optimal solutions. NOAH can compare factors, not showing the best result. This is an intentional assumption representing the difficulty of evaluation of data by an individual person. Thus, the evaluation of the effectiveness of the proposed method is difficult, especially in the present stage of research. The study introduced in one of the Polish agglomerations will be continued and should formulate remarks to modify selected elements of the transport system (e.g., number of departures or cost) with the observation of changes in other factors. When the obtained results are similar to the model, NOAH will be effective.

The presented material introduces the new algorithm by showing its procedures that can allow for its use in similar problems. This could test obtained assumptions and improve procedures in the future. Future works should contain influence analyses of the parameters adopted in the algorithm (the followers coefficient and the range of hierarchy modifiers), other (broader) sets of factors, and comprised studies in other areas. The NOAH method could be also used for other problems where incomplete and different data make observations of technical systems difficult.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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