The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty

The urban transport planning process has four main activities: Network design, Timetable construction, Vehicle scheduling and Crew scheduling; each activity has subactivities. In this paper the authors work with the subactivities of timetable construction: minimal frequency calculation and departure time scheduling. The authors propose to solve both subactivities in an integrated way. The developed mathematical model allows multi-period planning and it can also be used for multimodal urban transportation systems. The authors consider demand uncertainty and the authors employ fuzzy programming to solve the problem. The authors formulate the urban transportation timetabling construction problem as a bi-objective problem: to minimize the total operational cost and to maximize the number of multi-period synchronizations. Finally, the authors implemented the SAUGMECON method to solve the problem.

1. Introduction. The urban transport planning problem is very complex, this is the reason Desaulniers & Hickman [9], propose to divide the whole urban transport planning problem into: strategical, tactical, operational, and during operations, real-time control. Strategic planning problems concern long-term decisions such as the design of transit routes and networks. Most of these problems fall into the category of network design problems and require solving passenger assignment problems. Tactical planning problems concern the decisions related to the service offered to the public, namely the frequencies of service and the timetables. Operational planning problems relate to how the operations should be conducted to offer the proposed service at minimum cost. The problem the authors tackle in this paper is about tactical planning; this kind of planning is solved generally on a seasonal basis or sceneries [26]. It is important to solve this problems because there is an urgent need to reduce traffic congestion and improve the quality of life [22].
The urban transport planning process has four main activities: 1) Network design, 2) Timetable construction, 3) Vehicle scheduling and 4) Crew scheduling, each one of these has subactivities and the output of each activity or subactivity is the input for the next one [5]. In that way, timetable development has frequency calculation and departures time assignment and they are usually executed in sequence by planners, which have been found by [3] to provide suboptimal solutions than the integrated approach. Figure 1. Transport planning process [5] The problem being addressed in this research is about the timetable construction problem (second activity of Fig. 1). This problem consists of: (i) the minimal frequency of departures of units (maximum quantity of departures), and (ii) departure time of each unit into a given planning horizon (See Fig. 2).
The integration of minimal frequency calculation and departures times assignment, besides making this problem harder to solve from a computationally perspective than solve each subactivity sequentially [13], makes the problem harder to model because the number of departures is unknown at the time when the departures times should be assigned. This is not the case when both subactivities are solved sequentially, in this case the minimal frequency is calculated first then it is used as input for the assignment of departures times. So far, the authors have found no paper where minimal frequency calculation and departures times assignments are solved in an integrated way.
Multi-period scheduling and multi-period synchronizations also make the problem harder to solve and to model, thus more decision variables, and indexes and constraints are included in the model. Also the authors have not found any previous paper where multi-period synchronizations are addressed.
In what follows, the notable characteristics of the problem are synthesized, the authors group those into typical (1-4 of the next list), present in most of the papers reviewed; recently incorporated (5 and 6 of the following list), present in most recent papers; and desirable (7 and 8 of the list), not found in any of the reviewed papers: 1. Minimal frequency determination depends of headways definition, also departure times depends of minimal frequency (Column 1 of Table 1). For example, [12], [26] and [2] focus on the problem of frequency setting and their objective is to minimize the total cost, with another author working on the problem of minimal frequency: [24] who also combines the frequency setting problem with timetables. 2. There are certain nodes of the transportation system network called transfer nodes. At these nodes, passengers change route. A transfer will occur when the involved routes synchronize to allow passengers to do the transfer within a fixed time window [14](Column 2 of Table 1). For example, Eranki [10] proposes a model whose objective is to maximize the number of synchronization subjects to several policy constraints. She added a special characteristic, allowing synchronizations within a window time. Ibarra-Rojas & Ríos-Solís [13] formulate the timetabling problem with the objective of maximizing the number of synchronizations to facilitate passenger transfers and avoid bus bunching along the network. Zhang et al. [28] Besides proposing a multimodal model, they consider two kind of transfers: one is between the same mode; the other one is between different modes. 3. There are certain nodes of the transportation system networks called bunching nodes. At these nodes typically, there is bunching among the vehicles. This bunching should be controlled (Column 3 of Table 1). In the literature, some authors including bunching nodes are [13]. 4. The operational cost of a trip can be approximately calculated as i) a variable cost per kilometer of the route, that could include driver expenses, maintenance of the vehicle, fuel and other typical expenses; and ii) a fixed cost related to the organization that manage the urban transportation system network (Column 4 of Table 1). The cost is the most popular objective function ( [7], [12], [2], [15], [6], [30]). 5. The urban transportation system network is typically composed by more than one transportation mode, each one with its own regulation structure and requirements (Column 5 of Table 1). Liu et al. [15] propose a model where the passengers, to get their destination can alternate between different modes (buses and subways). They minimize the total travel time and total travel cost, where transfer time includes, the walking time, the waiting time and the stopping time. Zhang et al. [28] proposed a super-network approach where the networks for different modalities are integrated in a single network. This model provides a multi-modal routing system. Also, Wang et al. [27] proposed an integrated, region-urban, multimodal transportation model. The model introduces a time allocation step, suggests better approaches in steps such as trip distribution and mode choice, and improves the connections among each mode. 6. Although demand is unknown at the time of planning, generally demand variation within a time period follows certain patterns which depends on the time of the day, the day of the week or the season of the year, among other less frequent scenarios (Column 6 of Table 1). Authors like [25], [19], [7], [21], [12], [2] consider the stochastic nature of some parameters like demand and travel time; Chen et al. [7] address the problem of computing multiple headways and take into account stochastic demand and travel time; while [21] and [24] work with bus routing-timetables and frequency setting-timetables respectively. Both consider uncertainty at some parameters and their approaches apply between cities and only work-for buses. 7. Typically, the planning horizon is fixed a priori. Generally, demands vary significantly through the planning horizon. Due to this, the schedulers divide the planning horizon into multiple periods within them. The demand varies a little (fluctuates around a central value), but demand varies considerably between two consecutive periods (Column 7 of Table 1). Successfully addressing this issue also implies tackling multi-period synchronizations. Only one previous work address multi-period scheduling (see Fig. 4) Ibarra-Rojas [13], although multi-period synchronizations are not modeled. 8. The integration of frequency calculations and departures-scheduling is a desirable characteristic of any mathematical model and solution-oriented approach for this problem (Column 8 of Table 1). Otherwise suboptimal solutions are suboptimal timetables [3]. Some authors take into account two or more activities of the urban transport system like, [6], [30], and [23]. Chakroborty [6] works with the network design (transit routing) and timetable (scheduling) problem, but in a sequential way. His objective is to minimize the total transfer time and the total initial waiting time-the stopping time of a bus at a stop. Zhao & Zeng [30] present a metaheuristic method for optimizing transit networks that includes network design (route network design) and timetables (vehicle headway and timetable assignment). Their objective is to minimize the passenger cost function. Szeto & Wu [23] aim to reduce the number of transfers and the total travel time (including in-vehicle travel time and waiting time) of users by solving the bus network design (route design) and timetable (frequency setting) problem simultaneously.
It is worth mentioning that the authors also have not found a published paper about urban transport planning where all issues (1) -(8) are considered for developing models or solutions methods for that problem. This affirmation is supported by [8], where presented is an exhaustive literature review of papers dealing with timetable and frequency calculation.  It is also worth mentioning that, the proposed model can be applied for monomodal transportation systems and multimodal transportation systems without any extra work needed to make the switch from one mode to another. However, the most important feature is multimodal synchronizations, to avoid bunching between different transport systems, allowing effective transfers, since in transfer nodes demand can be distributed between different transport modes, additionally to define policies for operations cost, service level, etc.
We mentioned earlier the problem being addressed in this paper is related to tactical planning, therefore the authors do not pay too much attention to the service level because it makes more sense to do so when the authors are dealing with the operational planning. However, Avila et al. consider a minimal service level at the frequency calculation due to the methods the authors employed.
The paper is organized as follows: In Section 2 the mathematical model is presented. Also, the preprocessing and the assumptions made. In Section 3, the authors show and analyze the results of numerical experiments. Finally, in Section 4 conclusions and future work are mentioned. 2. Mathematical model.

2.1.
Modeling approach. In this section, the authors will refer briefly to the original approach they follow in this paper for representing the main elements of the problem being addressed: the modeling of decision variables (departures, synchronizations) and constraints (synchronizations and headways policies). What makes a difference with other approaches found in the reviewed literature are the characteristics (1) to (8) mentioned in the introduction, which are fully implemented in their proposed model.
Authors assume to know an average value around which demand fluctuates on the node for each route in each period and this behavior can be represented as a triangular, fuzzy number. Since authors are interested in the local behavior, there is no problem in following the fuzzy approach.
2.2. Departures assignment by time intervals. Two approaches are proposed in [5] for assignment of departures in the timetable construction activity: (i)Even headways and (ii) Even loads work well as a guide for planners, but in real situations where some variability in the system parameters is expected, there is a need for more flexible approaches for assigning the departures. Here the authors present an original approach that offers flexibility, and reduces the size of decision variables and associated constraints in a significant way.
Here Avila et al. assume that the planners are capable of supplying maximum and minimum headway for each route in each period of the scheduling horizon (H max and H min ). Departures are modelled as decision variables in the optimization model; typically, departures are arranged as in Fig. 5(a) where each time unit is considered as a variable ( [5], [30]). What the authors found inconvenient in their case in the first place was that the number of departures for each route in each time period was not known in advance. Also, this was due to the high dimensionality of the decision variable arrays, and high number of constraints derived from the headway requirements.
To avoid the issues previously mentioned, the authors propose to represent departures as in Fig Although with this representation, the authors need to introduce another variable (α), the dimensionality of the decision variable representing the departures is noticeably reduced in regard to other models presented for timetable construction ( [5], [30]). Meanwhile, the number of constraints increase because of the introduction of the α being negligible. N is an upper bound of the departures in a time period T v , and is calculated as . However, N v is also the maximum number of departures that can be scheduled in time period T v .

Headways policies.
It is common that an assignation of departures in an urban transportation system obeys certain headway policies. The headways can be provided by the decision makers or calculated. Among the most common headway polices are: to divide the period length by the frequency [5] and to provide the headways polices for first departure, for last departure and for two consecutive departures [10], [13].
For a certain route i of a certain time period v: (A) a typical policy for the first departure is that this departure should take place between the beginning of the period and the maximum headway (H v maxi ) [10]. (B) two consecutive departures must to be assigned in such a way that they should be separated by at least the minimum headway (H v mini ) and at most the maximum headway (H v maxi ) [10]. (C) for the last departure a typical policy is to ensure that this departure should   In what follows, the authors formally defined those policies by interval ranges. The range of departures time intervals for a given route i and a period v will be represented by the index set  Let RN v i ∈ RD v i be the range of time intervals (p v i and q v i , time interval where a departure is assigned) for the scheduling of two consecutive departures of a route i and period v is given by the index set That is mcp v i and msp v i are the farthest and the nearest previous intervals where the departure p v i was assigned in relation to the actual interval where departure q v i was assigned. In other words, here defined are the time intervals where a consecutive departure of a route in a period can depart.  Let RL v i ∈ RD v i be the range of time intervals for the scheduling of the last departure of a route i and period v is given by the index set . and it is the farthest interval (from N v i ) where the last departure must be assigned. In other words, here defined are the time intervals where the last departure of a route in a period can depart.
Multimodal and multi-period synchronizations for transfers or to control bunching. In their formulation transfer, synchronization is required at: a single transfer node (b), transfer between two close nodes (c) (see Fig. 3). Synchronizations are also modeled as decision variables, where each synchronization has an O-D pair of route, and a pair of nodes (where the origin node is the same destination node when the authors are facing a single node synchronization), and also the time scheduling period where the synchronization should take place.
All those properties of a synchronization require at least 5 indices, which in practice yield a huge amount of decision variables representing synchronizations and related constraints, the reason why it is very important to reduce the total amount of variables representing synchronizations. But, for the sake of fine control over synchronizations Avila et al. introduce two further indices: the index of the departure of origin and the index of the departure of destination, so the authors employ seven indices for the variable synchronization.
Synchronizations at transfer, or bunching nodes are directional. This means when a pair of routes synchronize, the transfer can occur from one route to another and vice-versa, then two synchronizations should be counted. The authors consider a round trip as the same route (when distinguished between origin and destination), then the characterization of the period brings the weighing of waiting time naturally by demand as it seeks to minimize the cost among other objectives.
2.5. Fuzzy programming. It is usual that the coefficients of a real linear programming (LP) problem where human estimation is used, are inexact because of either some lack of precision or some vagueness about the data being used in the problem [4]. On the other hand, the decision maker may feel more comfortable in specifying vague over crisp data. In that case, the theoretical support provided by the fuzzy numbers may be very appropriate to model the problem under consideration [4].
In the stochastic programming approach, uncertainty is modelled through discrete or continuous random variables. On the other hand, fuzzy programming considers uncertain parameters as fuzzy numbers and uncertain constraints are treated as fuzzy sets [20].
Here are different types of fuzzy numbers: among them trapezoidal and triangular, for example. In the model presented in the next section, the authors employ triangular fuzzy numbers. A fuzzy numberã in R is said to be a triangular fuzzy number if real numbers exist there s and l, r ≥ 0, such that (1).
A fuzzy linear problem is considered partially fuzzy, if some parameters are fuzzy or completely fuzzy, if all parameters are fuzzy. When there is uncertainty only (partially fuzzy) in constraint coefficients, there are methods that can help us to transform the fuzzy linear problem into a crisp problem by solving a fuzzy number ranking problem. The problem of ranking fuzzy numbers has been extensively studied in the literature [4]. An important ranking method is the k-preference method, which has been used recently by Perez et al. [18], [17] and it has given good results.
In this paper, the k-preference method is implemented to compare two fuzzy numbersã= (a, a, a) ≤b=(b, b, b) implies [4]: According to k-preference method where k is a confidence level, the confidence that the decision maker has about a parameter.
In this paper, fuzzy programming is employed instead of stochastic programming because it is considering the demand within periods that have an almost stable behavior around certain value. With fuzzy programming the variability or fuzziness around that value is represented.
2.6. Preprocessing. The first preprocessing procedure the authors applied is maybe the simplest, but it has turned out to be the more effective. The domain reduction applied to decision variables, especially in the variable that determines the departure time. As the authors divide the scheduling horizon into intervals, then the domain for these variables is delimited by the end of the intervals. Avila et al. take into consideration the properties of the problem previously described; they developed the following procedures to preprocess the data and reduce the size of the valuation of a model instance (to reduce the number of decision variables and constraints).
2.6.1. Algorithm to determine frequency method. Ceder [5] proposes four methods to determine the frequency, and he divided them into two groups: max load methods and load profile methods.
• Max load methods. Method I: Satisfies the demand of the maximum load point during the day and Method II: Satisfies the maximum load point of a time period. • Load profile methods. Method III: Guarantees that the node with the maximum load will not have overcrowding and Method IV: To control the possible overcrowding situations. This method sets a percentage of the route with overcrowding. With these methods they can estimate bounds for the frequency of a route. In order to determine which frequency method to use, Ceder [5] proposes an algorithm to select the most appropriate. In the flowchart of the Figure 10 this process is presented. First, they need a ride-check passenger count, then they construct the load profile (ρ) for each period, that is the total of passenger-kilometres divided into the length of the route by the maximum load. If ρ is less or equal to 0.5, then they calculate the frequency with method 3 and method 4 (different percentages). The results of method 3 are considered as lower bound and they use method 4 with the selected percentage. If ρ is greater than 0.5 then they compare method 1 and method 2 with a χ 2 test. If the value obtained for method 1 is equal to method 2 then they use method 1, otherwise they use method 2.
By incorporating these frequency calculation methods, the authors guarantee that important characteristics mentioned by Ceder are taken into account for the timetable construction. Avila et al. approach these frequencies as lower bounds  In the problem that is being investigated here, unlike the problem of Ibarra-Rojas et al., Avila et al. consider multi-period synchronizations, and also the possibility that synchronizations can occur between two different nodes of the system (but also they allow single node synchronization). Also it is possible to synchronize different transport modes.
In the following, they define the procedure: 1. For a given period, v they select a synchronization node, for each pair of routes that should synchronize: they identify the origin route of the transfer (O i ) and the destination route (D j ). 2. For each pair, (O i , D j ) they determine the departures p i and q j of the period v and the previous periods that arrive at the synchronization node of the period v. 3. For each departure of each set, p i they determine which departures of the set q j can synchronize the transfer between both, considering the maximum transfer time between both nodes and the holding time at the node of the vehicle unit of the route j.
In this way, the authors restrict the set of variables even more that represent the possible synchronizations, as well as the constraints of synchronizations. In the following proposition, they show this result formally.  3 ) is based on the maximum between the load point ( P v maxi ) for a route, a period and the average of passengers-kilometer ( P as v i ). This method guarantees that the node with the maximum load will not present overcrowding. Frequency method IV (M C v 4 ) is similar to method III, but this method does not exceed a percentage of the length of the route (β v i ). This method sets a level of service restricting the overcrowding to a portion of the route.

P.ÁVILA-TORRES, F. LÓPEZ-IRARRAGORRI, R. CABALLERO AND Y. RÍOS-SOLÍS
Remember, the authors are considering triangular fuzzy numbers so they need the central value (F C v i , (3)), the upper value (F C v i ,(4)) and the lower value (F C v i ,(5)). They used basic operations to transform fuzzy numbers into crisp numbers.
The minimum frequency (F reM in v i ) that their model has to satisfy is the maximum between the total number of departures (N v ), the frequency obtained with one of the frequency methods (F C v i ) and a basic frequency required (f v mri ), calculated as . The frequency obtained with the frequency methods and the frequency required are fuzzy numbers, because the demand is present in them, which converts the minimum frequency also fuzzy, the reason the central value, the upper value (F reM in v i ) and the lower value (F reM in v i ) are needed.
Assumptions. Here the authors present the assumptions of the problem addressed in this work: • Periods are created according to the fluctuation of demand; two consecutive periods have different demand in mean over the nodes for all routes. Inside each period, demand has a stable behavior. • Headways (minimum and maximum) do not change within a period for a route. • Vehicle units have the same capacity (at this moment, this is unimportant).
• The vehicle fleet is enough to perform the proposed planning schedule.
• Synchronization and bunching nodes are fixed by the planner.
• All passengers desire to transfer to the nearest vehicle unit.
• Demand does not change significantly in each period. • Demand is unknown, but it can be estimated for each period.
• Period lengths must be enough to allow the schedule of needed departures for satisfying demand.
The most popular objective functions are: minimize cost, minimize time (waiting time, total travel time, etc.) and maximize synchronizations [8], among others. This model consists of 2 objective functions, the first objective function (6) minimizes the total operation cost. The authors are considering a fixed cost (F ixedCost v i ) and a variable cost (V ariableCost v i ) which is affected by the length of the route (L i ) in kilometers multiplied by the number of departures in a route in a period ( p∈N v X v ip ). The second function (7) is to maximize the number of synchronizations between two bus routes with departure times in the same period or different periods (Y v ijkupq ).
Constraints (8 -9) guarantee that if there is no departure in the period v for the route i in the segment p, then it is not assigned a departure time. Also with these constraints, if the variable X v ip is 1, then α v ip will take a value less or equal to the upper bound of the interval p (8). With the next equation; if X v ip is 1, then α v ip will take a value greater or equal to the lower bound of the interval p (9). In this way, they limit the value that α v ip takes if there is a trip in the interval p. These constraints are related to the intervals to represent the departures.
In constraints (10-11) the sum of all schedule departures for a route i ( p∈N v X v ip ) has to be greater or equal to lower and upper bound of the minimum frequency (F reM in v i and F reM in v i ), considering the confidence level (σ) of the decision maker.
Constraints (12) to (21) are referring to headways constraints, which are depicted in Fig. 6. All these constraints are related to the properties of headway policies.
Constraint (12 -13) represents the first departure for the first period. If there is not assigned a trip in the first possible intervals, that means there is not a departure from the first interval to the second-last interval (tf v i ), in other words tf v i c=1 X v ic is equal 0, then the first departure is assigned in the last possible interval (tc v i ) and the departure time α v i(tc v i ) has to be less or equal to the maximum headway.
464 P.ÁVILA-TORRES, F. LÓPEZ-IRARRAGORRI, R. CABALLERO AND Y. RÍOS-SOLÍS Constraint (14)(15) represents the first departure for all periods greater than 1. The interval for the first possible departure since second period is defined (f v i ), also the last possible interval according to the previous period is defined (l v−1 i ), then the difference between the last departure of the previous period and the first departure of the current period (α v ) has to be greater or equal than the maximum of the minimum headways and less or equal than the minimum of the maximum headways (headways of current and previous period). In the case that max( Constraint (16) is for the consecutive departures, here indicates that the departure time must be between the minimum and maximum headway. If there is no a trip from the second-last interval to the first possible interval ( ic =0) then the next departure time will be assigned in the last possible interval (mcp v i ) and the difference of time between the next departure and the previous departure ) has to be less or equal to the maximum headway. Constraint 17 means, if the next trip is assigned to the first possible interval (msp v i ), then they just verify that the difference of time between the next departure and the departure assigned in the first possible interval (α v ip − α v i(msp v i ) ) is less or equal to the maximum headway. Also, if the next trip is assigned to the first possible interval (msp v i ), then they just verify that the difference of time between this two trips (α v ip − α v i(msp v i ) ) is greater or equal to the minimum headway (18). From the last possible interval to the first possible interval ( ic ) has to be at least one departure assigned (19).
For the last departure (20)(21) If there is not a trip assigned from the second-last possible interval to the maximum number of intervals ( ic = 0) then last departure time will be assigned to the last possible interval, and the time has to be greater or equal to the end of the period minus a desired time (T v f in − γ v i ). (20) and at least one departure must be assigned (21).
When the variable Y v ijkupq is equal to 1, then the difference between the departure time (α v ip ), travel time (t v ik ) and transfer time (δ v ijk ) of the origin route and the departure time (α v * jq ), travel time (t v ju ) and holding time (s v jk ) of the destination route, has to be greater or equal to the minimum time window (W v mini , constraint 22) and less or equal to the maximum time window (W v maxi , constraint 23). With the next equation they guarantee that there are two trips for which there is a synchronization (Constraint 24). These constraints are related to the synchronization and bus bunching policies.
3. Numerical experiments. The main objective of this experiment is to analyze the influence of instances factors (routes, periods, nodes, density, and headway) on the objective functions; also analyze the influence of the confidence level, uncertainty (fuzziness) and level of demand on the cost. The impact of these factors are not considered on synchronizations because the demand has no influence on this objective function.
The authors generated 32 instances. The generator was coded with R. The parameters that form an instance are: a fixed and a variable cost per route and period, the number of passengers at each node, the travel time between nodes, the distance between two nodes, the number of passengers who wish to transfer between routes, besides others. But there are 5 parameters that define an instance and they gave special attention to them: the number of routes, the number of periods, the number of nodes, the number of synchronization nodes and the range of headways. The authors established two levels for each one of these 5 important parameters and they combined them. The authors show the levels of these parameters in Table 2. 5-10 5-20 priority to decision variables during the execution. Also, they passed an initial solution from one run to another.
They found a very low correlation level between cost and synchronization, they are practically independent. There is a correlation between the execution time and cost; and execution time and synchronization, there is also a correlation between cost and periods and routes ( Figure 12).
The factor which establishes a higher variation on execution time is the quantity of periods, followed by routes and nodes, then confidence level, follow by density, headways, demand, and finally fuzziness is the less influence factor in relation to the execution time ( Figure 13). That means that the size of the instance has a greater effect than synchronizations or fuzziness parameter of level of demand.
Also, the authors analyzed the cost behavior in relation to confidence, fuzziness and level of demand; the characteristics of the instance selected are: 20 routes, 2 periods, 10 nodes per route; 12 points of synchronization per period and a range of 15 minutes between minimum and maximum headway. In Figure 14 the authors  In the same figure, the results obtained indicate that to greater confidence better results (100, 110, 101 and 111), in fact, the best results are obtained when the confidence level is high, the fuzziness is low and the level of demand is low (100), followed by high confidence level, high fuzziness and low demand (110). A midpoint is given by low confidence, low fuzziness and low demand (000). The worst results are obtained when the confidence level is low, the fuzziness is high and there is a high level of demand (011).
It should be noted that the authors established a gap of 5% to obtained results; here they presented results for one instance but similar results are obtained with the other instances. The authors identified another case, the instance they presented to show this behavior has: 20 routes, 2 periods, 150 nodes per route; 12 synchronization points per period and a range of 5 minutes between the minimum and maximum headway. In Figure 15 the results indicate that the results are better in those cases were the confidence was high (100, 110, 101 and 111) and the results obtained when the confidence is low are the worst.
It is important to notice the difference between both cases presented here, because it is evident that the fuzzy effect over the demand has a bigger impact when the instance is bigger too, especially when the number of nodes is bigger, because this implies a bigger demand; this is the case presented in Figure 15 where you can see more defined the different combinations of factors (Confidence, Uncertain and Demand). In the other case, the one presented in Figure 14, the impact over the demand it is not as evident than Figure 15. With these two cases, Avila et al. show the impact of the use of the fuzzy methodology in this problem.
Besides, Avila et al. analyzed the cost behavior in relation to the characteristics of the instances and they find out the most influential factors in the cost are: periods, routes and nodes. Also, when the range between the minimum and maximum headway is wider the cost increases. Furthermore, when the instance has more  (Instance 24) synchronization points, the cost also increases due to it trying to generate a bigger number of trips to satisfy the synchronizations objective. See Fig. 16.  Here the authors presented a mathematical model for the frequency and timetable integrated problem considering demand uncertainty. The proposed model can be used for multimodal transportation systems; it can be used for buses, subway, trains. The authors minimized the operation cost and they maximized the number of synchronizations (same or different period). The authors implemented the SAugmecon to solve the problem and they employed fuzzy programming. This method turned out to be suitable to obtained Pareto fronts with the objectives they used. The authors tested the model with instances which were generated randomly.
They tested the influence of different elements of the instances like periods, routes, nodes, synchronization points and the range of headways. They designed an experiment with three factors: confidence level, fuzziness and level of demand. The execution time will be higher when the number of synchronizations is high and the number of ships to schedule is also high. But the number of periods is the main factor in the variation of execution time.
In the future, the authors will incorporate uncertain of travel time and they would like to experiment with other ranking methods for fuzzy numbers, like the second index of Yager.