NOAH as an Innovative Tool for Modeling the Use of Suburban Railways

Abstract

The paper presents an innovative method called the “Nest of Apes Heuristic” (NOAH) for modeling specific problems by combining technical aspects of transport systems with human decision-making. The method is inspired by nature. At the beginning of the paper, potential problems related to modeling a suburban rail system were presented. The literature review is supplemented with a short description of known heuristics. The basic terminology, procedures, and algorithm are then introduced in detail. The factors of the suburban rail system turn into “Monkeys”. Monkeys change their position in the nest, creating leaders and followers. This allows for the comparison of the factor sets in a real system. The case study area covers the vicinity of Wroclaw, the fourth largest city in Poland. Two experiments were conducted. The first takes into account the average values of the factors in order to observe the algorithm’s work and formulate the stopping criteria. The second is based on the current values of the factors. The purpose of this work was to evaluate these values and to assess the possibilities of changing them. The obtained results show that the new tool may be useful for modeling and analyzing such problems.

1. Introduction with the Literature Review

Suburban railways play or should play an important role in agglomerations as the main means of transport connecting the core with the surroundings. Significant numbers of lines, journeys, and seats can affect the choice of means of transport. This creates environmentally friendly travel. Modeling the use of suburban railways should take into account two main aspects: (a) rail operations and (b) cooperation in the transport system. A suburban railway works similarly to all other railways. It is slightly closer to metro systems due to high frequency, while being unlike long-distance rail due to higher stop density and lower speeds. Therefore, the typical problems of planning rail operations should be taken into account. There are numerous studies on these problems.

Table 1. List of publications analyzed in this paper concerning the considered problems and heuristic tools.

Source	Problem	Tool
Canca et al., 2017 [1]	Line planning	O
Yan et al., 2019 [2]		MILP
Hickish et al., 2020 [3]	Network optimization	GA
Schlechte et al., 2011 [4]		NH
Weik et al., 2020 [5]	Nodes (location, classification)	NH
Ahmed et al., 2020 [6]	Station location	GA
Dong et al., 2020 [7]		O
Qiannan et al., 2022 [8]		NH
Binder et al., 2021 [9]	Timetable creation and rescheduling	SA
Jamili and Pourseyed Aghaee, 2015 [10]		SA
Shang et al., 2018 [11]		NH
Wang et al., 2018 [12]		MILP
Jensen et al., 2016 [13]		NH
Yin et al., 2021 [14]		O
Zhang et al., 2018 [15]		O
Zhang et al., 2019a [16]		MILP
Zhang et al., 2019b [17]		NH
Zhang et al., 2022 [18]		O
Zhao et al., 2021 [19]		MILP
Whitbrook et al., 2018 [20]		MS
Zheng et al., 2014 [21]		BBO
Satoshi et al., 2011 [22]	Operations (delay management, train routing)	TS
Sama et al., 2017 [23]		SI
Oneto et al., 2017 [24]		O
Xiao et al., 2018 [25]	Demand side of the transport system	NH
Shen et al., 2016 [26]		O
Wu et al., 2013 [27]		O
Liu et al., 2019 [28]		ANN
Hansson et al., 2021 [30]	Bus systems and networks	NH
Wei et al., 2021 [31]		NH
Ngo et al., 2021 [32]	Ride-hailing services	NH
Behrends, 2017 [33]	Nodes, coordination	NH
Stead et al., 2019 [34]		NH
Yang et al., 2020 [35]		GA
Lianhua and Xingfang, 2022 [36]		SA
Chen et al., 2020 [37]	Autonomous Vehicles	O
Lopez and Farooq, 2020 [39]	Smart mobility data-markets	BC
Xavier and Xavier, 2016 [44]	Not related to transport systems	O
Martinelli and Teng, 1996 [45]	Railway operations	ANN
Fischetti and Monaci, 2016 [46]	Not related to transport systems	MILP
Teodorovic, 2008 [47]	Transportation engineering (general)	SI
Venter and S.-Sobieski, 2004 [48]	Optimization of a transport aircraft wing	SI
Steiger et al., 2016 [49]	Transport data	SOM
Cesme and Furth, 2014 [50]	Traffic signals	SOM
Hua et al., 2020 [51]	Intelligent Control in Heavy Haul Railway	BC
Kawaji et al., 2004, [52]	Not related to transport systems	O
Paradis et al., 2004, [53]	Not related to transport systems	O

Tool description: ANN = Artificial Neural Network, BBO = Biogeography-Based Optimization, BC = Blockchain Framework, GA = Genetic Algorithm, MILP = Mixed Integer Linear Programming, MS = Multiagent systems, NH = Not heuristic, O = other (heuristic), SA = Simulated Annealing, SI = Swarm Intelligence (incl. particle swarm intelligence, PSO, ant or bees colonies, ACO, BCO), SOM = Self Organized Maps, TS = Tabu Search.

contains the list of publications analyzed in this paper concerning the considered problems and heuristic tools. In [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], researchers address the problems related to railway modeling, and the authors of [25,26,27,28] consider the demand side of transport systems. It is worth noting that [29], discussed below, does not exist in the table because of its review character. In [30,31,32,33,34,35,36,37], researchers address the problems of integration in a suburban transport system. Sources [38,39] show directions for future research, and sources [40,41,42,43] contain important definitions (they were not added to the table). In [44,45,46,47,48,49,50,51,52,53], authors present different tools used to solve some problems (partially not as transportation problems, but with methodologies inspiring the method presented here). The tools are discussed in Section 2.

Sometimes the problems classified in Table 1 are more complex. Dong et al. [7] integrate the planning of train stops with the timetable, and Yan et al. [2] integrate the timetable with route planning. Wang et al. [12] and Zhao et al. [19] combine train timetables and rolling stock. Zhang et al. [16] integrate train timetables and track maintenance scheduling.

The above characterized examples illustrate the offer (supply side of the transport system). On the other hand, the result in the form of passenger flow (demand side of the transport system) was taken into account, inter alia, in: Xiao et al. [25], Shen et al. [26], Wu et al. [27], and Liu et al. [28]. These works include passenger flow as a direct result of modeling or simulation. In many studies, passenger flow is a factor influencing the modeled parameters, such as train schedules.

Rail is not the only means of transport in suburban areas. The railway is or should be one of several integrated components that work together at different levels. Access to rail should be improved by means of “complementary tools or means of transport”, forming a “delivery system” that includes local buses, private cars including car sharing, bicycles including rental, etc. It is important to optimize local systems, create nodes, and integrate tariffs and cost coordination. The importance of coordination studies is shown by a review by Liu et al. [29], who identified 135 papers on these topics. Further problems are related to the developing autonomy of vehicles. Examples of studies from recent years concerning cooperation in transport systems are presented in Table 1 ([25,26,27,28]).

Many parameters were taken into account in the models presented above. For example, Ahmed et al. [6] collect 27 input parameters, including: average travel speed, train headway, number of stations, spacing between stations, etc. The number of input parameters in Dong et al. [7] is 21 and includes, inter alia: the number of passengers arriving at the station, the number of trains, etc. A specific parameter is “passenger satisfaction” or “dissatisfaction” (Hickish et al. [3], Satoshi et al. [22], Stead et al. [34], Shen et al. [26]). Shen et al. [26] formulate nine elements creating passenger satisfaction: direction and guidance, cleanliness and comfort, speediness and convenience, safety and security, ticket service, equipment and facilities, staff service, information distribution, and convenient facilities for passengers.

Specific review studies (Liu et al. [29], Tang et al. [38]) formulate directions for future research. Integration with various planning activities is important. Data quality, data limitations or imperfections, uncertainty, and passenger behavior should be carefully considered. Modeling analyses should be more complex and include, inter alia, multi-objective optimization, multi-agent systems, and negotiations. Comprehensive and more flexible approaches will pay off.

All the parameters presented above affect the use of suburban railways. However, not only the “physical” ones (easy to identify and measure), such as speed or numbers, are important. Other parameters that are more difficult to identify and have a “psychological” aspect should also be taken into account. Lopez and Farooq [39] state that “transportation data are shared across multiple entities using heterogeneous mediums”. Such data vary on certain days (not only working days and holidays, but some typical working days may also differ in passenger flow depending on weather, accidents, and random factors). The influence of bounded rationality and unbounded uncertainty is significant (Khisty and Arslan [40]). Similar problems are discussed by Li et al. [41] and Wu et al. [27]. They wrote that the assumption of rational passenger behavior is not correct. Taking into account the behavior of passengers requires the use of advanced and unconventional tools in modeling. Tools inspired by nature and social behavior are called “artificial intelligence” (AI) or “heuristics” (in a broader sense, not just as an optimization tool).

The main research goal of this paper is to create a new algorithm (NOAH) not as an optimization tool but as a method of observation of selected datasets. The reason for this is the problems with the identification of the close set of important factors and with the collection and selection of the data. Known and used methods have other assumptions. The proposed algorithm allows us to find new and nonobvious connections between the factors (these are not correlations in the strictly mathematical sense). Assumptions to create an algorithm will be formulated after the presentation of the heuristics (Section 2). The rest of this paper is organized as follows: Section 3 presents the new algorithm, and Section 4 shows an example of its application (with the description of the case study area, Section 4.1; collection of factors, Section 4.2; and two experiments, Section 4.3 and Section 4.4). The last two sections contain a discussion and conclusions.

2. Heuristics as Inspirations from Nature

The term “heuristics” will be used here in a broader, philosophical sense, as defined by Kahneman [42]: “a heuristic is a mental shortcut that our brains use that allows us to make decisions quickly without having all the relevant information”. In more “technical” literature, this concept or tool is often referred to as “computational intelligence” or “artificial intelligence”. Regardless of the name, such tools are very popular and efficient in solving many problems, including modeling railways. Many tools developed in the last few decades can be considered “heuristics”. Tang et al. [38] identify 139 articles from the last decade on the use of heuristics in railway systems.

The third column in Table 1 presents tools used to solve the collected problems. Most of them are heuristics. An element inspired by nature, especially simulated human or animal behavior, is important. New developments in “metaheuristics” and their applications are presented by Lau et al. [43]. They evoke, among others, a new method called “flying elephants” (Xavier and Xavier [44]), which shows interesting and intriguing assumptions and solutions.

Some studies include more than one tool, including Yang et al. [35], who compared the effectiveness of GA and MINLP. The set in Table 1 contains only selected sources from a very large database. The selection focuses on methods dedicated to railway modeling or on tools that will be inspirations for the method formulated in this paper. Specifically, these are relatively new studies using PSO, SCO, BCO, SOM, blockchain, multiagent, or BBO methods. For example, Zheng et al. [21] used the earlier concept of Simon [54], biogeography-based optimization, to analyze emergency railway wagon scheduling. Similarly, Hua et al. [51] used the Nakamoto blockchain concept [55] for intelligent control on heavy haul railways.

Summarizing the above description, the conditions for a new model of suburban railway use are summarized below. Railways function in the transport system, and cooperation with other modes of transport is necessary. We may collect a large amount of data, but we do not know the significance (impact) of each individual piece of information. There are many factors that influence the use of suburban rail, and their impacts may vary from day to day. Passenger behavior (including the choice of means of transport) is not rational. We should consider bounded rationality and unbounded uncertainty. The modeled object (railway in the transport system) is variable. The “optimal” solution probably does not exist; rather, we are looking for an “acceptable” solution. An acceptable solution contains a set of factors that are realizable and make economic sense. The results from the model can support the decision-making process—for example, when choosing a specific option, planning system development, etc. It is desirable to use a dedicated metaheuristic in the new model. SOM, multiagent, and blockchain elements inspire certain assumptions about the new proposal. In particular, solutions based on animal or human behavior will be useful for creating a new modeling tool.

So, the new model (algorithm) should be allowed to compare different data with higher or lower complexity to show potential sets of them. It will be possible to analyze both the existing (observed) data as well as more theoretical values. The process of comparison should be flexible and based on partially random procedures. The assumptions collected above can be realized using a specific heuristic. A novel heuristic will be proposed based on the specific behaviors of monkeys.

3. The NOAH Concept Based on the Behavior of Monkeys

A novel tool created here and called “NOAH” (Nest of Apes Heuristic) is inspired by the social behavior of groups of monkeys. Numerous studies and publications have been devoted to groups of monkeys from different monkey species—such as diana (Decellieres et al. [56]), vervet (Gareta Garcia et al. [57]), capuchin (Leca et al. [58]), gelada (Miller et al. [59]), or colobus (Wikberg et al. [60])—which create various nests with specific social behaviors. Colobus monkeys create specific “social networks” based on interactions [60]. A visualization (model) of such a network is presented in Figure 1 (part b). Diana monkeys form specific relationships called “dear-enemy” or “nasty-neighbor” depending on the type of habitat [56]. Distributed leadership has been observed in the nests of white-faced capuchins [58]. All members can initiate a group movement, and many members recruit followers. Wild female vervets adapt their maneuvering to different pressures [57]. They are characterized by rapid social plasticity and flexible changes in care patterns (described by the authors as “Machiavellian-like”—this “human” analogy is important here). Miller et al. [59] identify leaders in gelada nests under the influence of out-of-group paternity. The behavior of such species has been compared with other primates and has been linked to human mating systems, including behaviors jealousy (Scelza et al. [61]) and reproductive strategies (Scelza et al. [62]). The implications presented by Miller et al. in [59] refer to the “weirdness” of various human populations described by Heinrich et al. [63]. WEIRD here is an acronym standing for western, educated, industrialized, rich, and democratic. The authors conclude that not all human groups can be characterized as above. Other classified groups have different social behaviors. Therefore, their description should assume specific and partially unknown parameters.

Hypothetically and in accordance with the heuristic methods described in Section 2, the behaviors presented above can be used in an algorithm (NOAH) that can describe not only groups of animals, but also technical systems containing parameters (factors) related to human behavior (like choice of transport means). Especially useful can be changeable leadership in the nest and the behaviors of followers. The parameters will be associated with “monkeys”—individuals in the nest that change their behavior (monkey position) according to specific procedures including leader creation, observations by followers, importance and hierarchy of individuals, dynamic changes in the nest, etc.

Changes to the nest will modify individuals (factors) before the algorithm stops. It will be possible to analyze and observe different sets of parameters (monkeys), their interactions, and their correlations. NOAH does not specify an optimal solution but shows possible datasets for comparison. It helps in choosing one or more. The operation of NOAH is very similar to the SOM (self-organized maps) concept, the stages of the blockchain, or the multiagent concept. A graphical representation of the exemplary methods is shown in Figure 1. Part (a) of this figure shows an SOM-like network, part (b) shows the colobus monkey nest described earlier, and part (c) shows the monkey nest and interactions between leaders (big black spots) and followers (little black spots) according to the NOAH concept.

Important for the application of NOAH in selected problems is the selection of parameters (factors) and their conversion into “monkeys”. Initially, the selection of parameters is made by an “expert” with the use of all of the available data. After the algorithm is stopped, the re-conversion procedure will follow. These elements will be described in Section 4 with a specific example. The basic and theoretical aspects of NOAH are presented here. Each monkey has a specific position in the nest that is variable. The monkey position values are limited to a range of 0 to 1 as defined by the procedures in NOAH.

A specific set of terms, parameters, and symbols used in NOAH is defined herein.

Nest (seat, habitat) is a set of individuals (representing factors in the model).

Position of the monkey in the nest, M_n, is a key variable in the algorithm. Each monkey changes its position in the nest, assuming the role of a leader or follower (representing variable values of factors and their importance in the model).

Steps (iterations) of changes in the nest, starting from zero, i = 0, are successive periods with a specific nest state (monkey position, i.e., factor values). The steps will continue until the socket is stable (will not change). See stopping criteria.

Importance of an individual, I_n, is a random variable indicating the subjective position of the monkey in the nest. The scope of this variable is determined by Formula (1).

$$I_n=random\left(0;1\right)$$

(1)

Hierarchy of an individual, H_n: This is a variable indicating a more objective position of the individual in the nest, assuming an actual value of M_n, a random importance I_n, a moderated followers coefficient L, and a number of steps i. The hierarchy is calculated using Formula (2).

$$H_n=\left|\frac{I_n-M_n}{L\cdot i}\right|$$

(2)

Followers coefficient (influence of leaders), L: This determines the time and efficiency of the algorithm (it should be precisely defined in accordance with the specification of the modeled problem). Its impact increases in subsequent iteration steps—see Formula (2). L values should oscillate around 1–3 (see example in Section 4).

Random hierarchy modifiers, R_n: These are used to modify the position of the monkey going to the next step of changes in the nest. The modifiers depend on the value of the hierarchy, taking into account the defined range of hierarchy modifiers according to Formula (3).

$$R_n=H_n\cdot random\left(R\min ;R\max \right)$$

(3)

Range of hierarchy modifiers, Rmin and Rmax: This also determines the runtime and efficiency of the algorithm; the Rmin value should be negative and the Rmax value positive (see example in Section 4).

NOAH works according to the algorithm shown in Figure 2. The position of the monkey in the next step is calculated using Formula (4). The position is changed taking into account the values of random hierarchy modifiers with the restrictions determined by Formula (5).

$$M_n^{i+1}=M_n^i+R_n^i$$

(4)

$$0\le M_n^{i+1}\le 1$$

(5)

The progressing steps create a changeable nest with individuals who change their positions. That means the changeable values of factors are considered in the model. The leaders (factors with higher importance) are identified, observed, and analyzed. The whole nest (set of all factors) can be analyzed too. The algorithm heads to the nest stability, which means reducing the changes in the monkey’s position during the steps. The tempo of such stabilization depends on the value of the followers coefficient and the range of hierarchy modifiers. However, specific stopping criteria are formulated. In each step the following “decisions measures” are calculated: M as the sum of all M_n, I as the sum of all I_n, H as the average from all H_n, and R as the average of all R_n. Consideration of these measures in the aspect of stopping criteria is shown in the example in Section 4.

The next steps of the algorithm create a changing nest with individuals of different positions. This refers to changes in the value of the factors included in the model. Leaders (factors of greater importance) are identified, monitored, and analyzed. The entire nest (set of all factors) can also be analyzed. The algorithm aims at nest stability, which means reducing the changes in the monkey’s position during steps. The pace of such stabilization depends on the value of the followers coefficient and the range of hierarchy modifiers. Specific stopping criteria have been formulated. At each step, the following “decision measures” are calculated: M as the sum of all M_n, I as the sum of all I_n, H as the average of all H_n, and R as the average of all R_n. The inclusion of these measures in terms of the stopping criteria is illustrated in the example in Section 4.

4. Application Example of NOAH

4.1. Case Study

The case study area is located near Wrocław, the fourth major city in Poland. Wrocław is the core of an agglomeration with approximately 1 million inhabitants. The Wrocław railway junction is large and forms the basis of the shape of the suburban railway system. This system is under construction, and several railway lines are being rebuilt or extended. In December 2021, the rebuilt line connecting Wrocław with Jelcz through the Czernica community was opened after 20 years without passenger transport. The occurrence of the reopening provides an opportunity for specific research, including testing of the NOAH algorithm. The use of NOAH allows for temporary changes to the rail network to be taken into account, e.g., the closure of a specific section of the line. The situation of the temporary closure of one railway connection makes it possible to observe all public transport connections between Wrocław and Jelcz using only one corridor.

Several factors and their collections are considered. Among them are new railroads, bus lines, and parking lots. These data were compared with the observed values of the number of passengers, etc. Figure 3 shows the map of the case study area, taking into account different types of railway lines (older ones in black, new ones in blue, and temporarily closed ones in red), two groups of bus lines (correlated with the railway line in green and competitive with red railways), target area in terms of the area of the Czernica commune, and car parks at railway stations in this area (park and ride, PR). Based on these conditions, 16 factors are formulated to be considered in the NOAH model.

4.2. Factors and Their Conversion

A set of 16 factors was used in this case study (

Table 2. The factors considered in the case study.

Factors
n	Group	Description
F1	positive	number of departures of trains (in a day)
F2	positive	number of passengers (in trains in a day)
F3	positive	number of people who get on the train in the chosen stations (one course)
F4 [%]	positive	percentage of seats taken up on the train (in a peak hour)
F5	positive	number of stops in the characteristic distance
F6	positive	number of residents in the “target area”
F7	positive	parking volume on PR
F8	positive	number of passengers who arrived at train from other forms of public transport (e.g., bus correlated in the timetable with trains)
F9	positive	number of departures in other correlated forms (in a day)
F10 [PLN]	negative	cost (the price of a typical ticket, single travel, adult)
F11	negative	number of departures in public transport competitive with the considered train line
F12 [minutes]	negative	travel time by train (in a peak hour)
F13 [minutes]	negative	travel time by competitive public transport (in a peak hour)
F14 [minutes]	negative	travel time by private cars (in a peak hour)
F15	negative	number of passengers on competitive public transport (in a day)
F16	negative	number of people who use private cars (in a day)

). Factors are divided into two groups: positive (nine factors) and negative (seven factors). Rising values of positive factors increase the total number of travelers (in all modes), and increasing values of negative factors reduce this number. This is a proposal used in this research considering specific assumptions (not defining the close set of factors and their role). For example, factors F2 and F3 are classified as “positive” according to an assumption of the offer presented by the rail operator having a higher number of places than the forecasted demand. In such an assumption, the trains will not become overcrowded. The higher number of passengers or people who get on the train will increase the use of suburban rail because of the creation of “good behavior” for new passengers. Conversion into monkeys and re-conversion differ depending on which group the factor belongs to. The set of factors adopted here is not complete in terms of all possible measures of the rail system, but it does contain exemplary, representative, and easy-to-collect data.

Assuming (or identifying) a range of all factors is a necessary and important element of NOAH. The minimum and maximum values of the selected factors are necessary to calculate the monkey position value (M_n). Monkey positions correspond with the factors collected in Table 2. Two experiments are carried out in this case study. The first (Section 4.3) tests the volatility of the factor values. The second (Section 4.4) takes the actual factor values and compares them with the NOAH results. The conversion from F to M only occurs in the second experiment, but the re-conversion from M to F applies to both experiments. The following conversion formulas (6)–(7) and re-conversion formulas (8)–(9) are used depending on the specificity of the positive (FP) or negative (FN) factors. The mathematical conversion formulas are used impose monkey position values in the range of 0 to 1.

$$M_n=\frac{F{P}_n-FP{\min }_n}{FP{\max }_n-FP{\min }_n}$$

(6)

$$M_n=\frac{FN{\max }_n-F{N}_n}{FN{\max }_n-FN{\min }_n}$$

(7)

$$F{P}_n=FP{\min }_n+\left(FP{\max }_n-FP{\min }_n\right)\cdot M_n$$

(8)

$$F{N}_n=FN{\max }_n-\left(FN{\max }_n-FN{\min }_n\right)\cdot M_n$$

(9)

where:

FPmin to FPmax is the range of positive factors;

FNmax to FNmin is the range of negative factors.

Data for both experiments are summarized in

Table 3. The data for both experiments.

Factor, Fn	min	max	Reference Value	Monkey’s Position
F1	0	20	10	0.500
F2	0	600	540	0.900
F3	0	100	60	0.600
F4 [%]	0	100	90	0.900
F5	6	21	15	0.600
F6	12,000	20,000	16,000	0.500
F7	0	100	80	0.800
F8	0	100	70	0.700
F9	0	10	4	0.400
F10 [PLN]	4.6	12	10.2	0.757
F11	0	20	16	0.200
F12 [minutes]	30	90	30	1000
F13 [minutes]	30	90	60	0.500
F14 [minutes]	30	90	45	0.750
F15	0	500	200	0.600
F16	2400	16,000	10,000	0.441

. 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.

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 (Rmin, Rmax) 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.

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. The results of the first experiment.

Mn	In	Hn	Rn	Fn
0.478	0.937	0.010	−0.003	10
0.538	0.083	0.009	0.003	323
0.566	0.333	0.005	0.000	57
0.440	0.675	0.005	−0.001	44
0.460	0.834	0.008	−0.002	13
0.513	0.523	0.000	0.000	16,101
0.545	0.403	0.003	0.001	54
0.516	0.858	0.007	0.000	52
0.538	0.752	0.004	0.001	5
0.482	0.800	0.007	0.000	8.4
0.478	0.584	0.002	0.000	10
0.491	0.729	0.005	0.001	61
0.506	0.513	0.000	0.000	60
0.500	0.295	0.004	−0.001	60
0.511	0.247	0.005	0.001	244
0.455	0.938	0.010	0.003	9811
M = 8.017	I = 9.504	H = 0.005	R = 0.000

. 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

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. The results of the second experiment.

Mn	In	Hn	Rn	Fn
0.525	0.039	0.010	0.001	10
0.862	0.708	0.003	−0.001	517
0.529	0.404	0.003	0.000	53
0.948	0.654	0.006	−0.001	95
0.560	0.308	0.005	0.000	14
0.397	0.809	0.009	−0.002	15,174
0.999	0.707	0.006	−0.001	100
0.699	0.689	0.000	0.000	70
0.413	0.074	0.007	−0.002	4
0.752	0.841	0.002	0.000	6.4
0.263	0.787	0.011	0.003	15
0.925	0.730	0.004	0.000	34
0.472	0.262	0.004	−0.001	62
0.755	0.901	0.003	0.001	45
0.607	0.087	0.011	0.003	196
0.447	0.078	0.008	0.000	9926
M = 10,153	I = 8078	H = 0.006	R = 0.000

. 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.

5. Discussion

Table 6. Set of values for all factors.

Fn
n	min	Reference	Exp. 1	Exp. 2	max	Short Description
F1	0	10	10	10	20	departures of trains in a day
F2	0	540	323	517	600	passengers on trains in a day
F3	0	60	57	53	100	people who get on the train
F4 [%]	0	90	44	95	100	percentage of seats taken up
F5	6	15	13	14	21	stops
F6	12,000	16,000	16,101	15,174	20,000	residents in the “target area”
F7	0	80	54	100	100	parking volume in PR
F8	0	70	52	70	100	passengers arrived
F9	0	4	5	4	10	departures in correlated forms
F10 [PLN]	4.6	10.2	8.4	6.4	12.0	cost
F11	0	16	10	15	20	competitive departures
F12 [minutes]	30	30	61	34	90	travel time by train
F13 [minutes]	30	60	60	62	90	travel time in competitive transport
F14 [minutes]	30	45	60	45	90	travel time in private cars
F15	0	200	244	196	500	passengers using competitive transport
F16	2400	10,000	9811	9926	16,000	people who use private cars

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” (F₄) can of course vary between 0 and 100%, and “parking volume on PR” (F₇) can vary between 0 and 100, the upper limit being the actual number of parking spaces in all of the considered locations. The “cost” (F₁₀) 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” (F₁₂–F₁₄) 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 (Rmin, Rmax). 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, Rmin = −0.3 and Rmax = 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.

The factors values obtained from the experiments should be analyzed as a complex set of parameters. In both experiments, sufficient numbers of trips (F₁) and stops (F₅) 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 (F₂). Comparing these values with the stable number of residents (F₆), there is a need to increase the role of the delivery system. The delivery system is represented here by factors including cars in PR (F₇) and passengers arriving from correlated forms of public transport (F₈). The travel cost (F₁₀) should be lower. The conditions tested above also show a lower number of journeys by means that compete with rail, both with private cars (F₁₆) and buses (F₁₅).

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 (F₆) or numbers of departures (F₁). 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.