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(*Author for correspondence, E-mail: doina[email protected]). Key words: activity analysis, fuzzy logic, scheduling. Abstract. The paper focuses on how trip time ...
Transportation (2005) 32: 423–441

Ó Springer 2005

Modelling behavioural rules for daily activity scheduling using fuzzy logic DOINA OLARU1,* & BRETT SMITH2 1

Atmospheric Research, CSIRO, Australia; 2 Information Management and Marketing, University of Western Australia, Australia (*Author for correspondence, E-mail: [email protected])

Key words: activity analysis, fuzzy logic, scheduling Abstract. The paper focuses on how trip time variability affects re-scheduling of daily activities. A delay in a trip or an early arrival can contribute to changes in the timing, location of the next activity, and to the deletion/addition of some activities. We propose the idea of using fuzzy logic rules to ‘‘explain’’ the effect of variability in travel time on the benefits perceived by an individual with the changes, and to model different actions that the individuals take in order to re-establish the steadiness of the existing timetable. The fuzzy model is used to handle the imprecision of the data which is unstructured text. The results show that large deviations in trip duration are more likely to induce significant changes in the timetable whereas small deviations are either ignored or translated into modified timing of the next activity. In choosing an action, greater importance is assigned to the flexibility of the following activity, to the magnitude of the trip time saving/delay, and to the duration of the next activity. Time savings are not favoured unless they can be readily transferred into additional activity time allocated to the next activity or to a new activity. The fuzzy rules based system is capable of predicting satisfactorily the strategy of coping with uncertainty in travel times and the satisfaction sensed with the change.

1. Background Studies of travel behaviour responses to uncertainties have focused on the choice of alternatives for the trip in which the uncertainty is faced. Recent stated preference studies have included trip time variability as a key attribute of the trip profile presented to respondents (Noland et al. 1998; Ortu´zar & Iacobelli 1998; Bates et al. 2001; Hensher 2001). These studies measure behavioural responses like choosing an alternative route, mode, location, and changing the time of day. Such answers do not fully address the types of options available to the individuals, should they experience an unforeseen travel time saving or delay. In many instances the respondents must endure the unexpected circumstance of the trip and alter subsequent plans in their day’s activity schedule. Bates et al. (1987) found that the most likely response to an unexpected change in trip duration is acceptance (cited in Stern 1998: 178). Acceptance means that the driver does not react to congestion by trying a different route or breaking the

424 traffic rules, but compensates for time-loss by varying the duration and location of subsequent activities. What are the responses of individuals who experience unanticipated changes to their travel schedule? This question was put to activity-diary respondents, who were given a hypothetical scenario related to one trip they made during the survey day. The question was open-ended, allowing the respondents to discuss their coping strategy without having to comply with a set of alternatives imposed by the survey instrument. This type of question is often found at the end of detailed surveys and is worthy of exploration beyond the reporting of descriptive statistics. Fuzzy systems (FS) have been applied to cluster responses to an open ended question in marketing (Varki et al. 2000). The primary motivation in using FS was to organise the imprecise language used by the respondents. When faced with an expected delay or time saving, most individuals would attempt to re-establish their skeleton timetable (Kitamura & Fujii 1998) by lengthening or shortening the duration of the immediately affected activity. However, other responses include relocating subsequent activities, reordering of subsequent activities, adding or deleting an activity and reallocating tasks to other members of the household. We investigate the commonality of solutions provided by respondents – by accounting for the differing circumstances of each respondent – and explore the factors that affected the response. These factors are discussed in more detail in Section 2. They include: the degree of flexibility of the remaining scheduled activities (in terms of duration, frequency and location); the length of the delay or saving; the time of day when variation occurs; the duration of the next activity. The background to the modelling technique is presented in Section 3. This is important because this analysis tool is not widely familiar to researchers in travel behaviour and transportation. We have laid out the fundamentals of fuzzy logic as well as fuzzy membership functions, fuzzy operators and fuzzy rule based systems. The empirical setting is described in Section 4. In Section 5 the development of the model is outlined. The findings are presented in Section 6 and conclusions and further research in Section 7.

2. Activity scheduling: Elements considered in the fuzzy model The process of scheduling may be thought of as hierarchical, with mid-long term decisions about mobility being the first stage (Ben-Akiva & Bowman 1998; Papacostas & Prevedouros 2001). These long term decisions then impact on the daily (weekly) schedules of activity participation (Axhausen & Ga¨rling 1992; Ettema & Timmermans 1997; Arentze et al. 2000; Doherty et al. 2002).

425 Short term scheduling is done concomitantly by household members, accounting for shared resources and joint participation. In most instances there is no formal schedule, but individuals use a tacit awareness of the activity patterns of other members. If no formal schedule exists, the challenge for the modeler is to infer the unstated schedule from the actual activity patterns of the household. The analyst may view the activity patterns as an outcome of a skeletal timetable (Axhausen & Ga¨rling 1992; Kitamura & Fujii 1998; Doherty et al. 2002) drawn up by the individuals before the commencement of the day. This timetable schedules activities loosely around the windows of opportunity, as perceived by the individual. As more information becomes available – i.e., the size of each window of opportunity is known with a greater certainty – the individual adjusts this schedule accordingly.

2.1. Re-establishing schedules The above description of daily activity planning has implications on the use of utility maximising models for activity analysis. Typically, such models consider that individuals maximise utility by allocating time to home and out-of-home activities. Taken to its logical extreme, such models indicate that an individual would repeat the scheduling process each time there is a disruption to the existing timetable. However, the predominant response to the hypothetical ‘‘disorder’’ was an attempt to re-establish the existing timetable. This indicates that the utility/disutility of an uncertainty in travel time may be measured by the impact on the current schedule.

2.2. Nature and length of the activity A major consideration when researching behavioural responses to activity schedules is the type of activities undertaken by the individual (Golob & McNally 1995; Kitamura et al. 1996; Timmermans et al. 2002). It is reasonable to assume the response to an unexpected time delay or saving will be different depending upon whether the next activity is rigid or flexible. The flexibility of an activity is a function of its location, start/finishing time, and duration. Visiting a friend may be flexible in terms of start time and duration, but the location is fixed. Work and business activities are usually described as rigid, because they are often constrained by location and time. However, strict classification of activities into rigid or flexible classes is complicated because of the multi-dimensional aspect for the flexibility of an activity; added to those already given are: priority (business meetings may be rigid for all other dimensions, yet they may be deleted or postponed if another urgent matter

426 arises) and participation (i.e., is it essential that a particular household member conducts that activity?). Certain activities, such as work, appear under different grades of flexibility, due to the manner in which they may be conducted. Also, it is possible that each activity described does not fit perfectly within a grade. The potential for overlap is the motivation for using fuzzy sets to classify inputs to the model: fuzzy sets permit degrees of membership. Finally, activities that are ‘‘flexible’’ are often subject to exogenous constraints. For example, a concert at the theatre and a soccer match have a fixed start time and duration, and a delay in traffic will almost certainly mean a disutility is experienced, with no possibility of rescheduling. In these cases, the duration of the participation in the activity is reduced, and the benefits obtained from pursuing the activity vary accordingly (Kitamura & Supernak 1997). The scheduled duration of the next activity is important for two reasons. First, activities with long durations are more likely to be the principal activities and other activities are scheduled around them. It is not expected that the individual will remove these from schedule if a delay is experienced. Second, activities with long durations allow plenty of time to buffer the impact of a time savings/delay. They offer the chance to get back on schedule without seriously impacting on the next activity.

2.3. The length of the delay/savings The magnitude of the time saving/delay dictates the type of ‘‘corrective’’ action in the sense that very small changes are ignored or they are easily accommodated in the timetable by changing the starting time of the following activity; but significant modifications need a bit more thought. Large time savings allow possibilities of accessing more preferred destinations or including new activities. But significant time delays, arising within rigid schedules, are the most annoying, and they lead usually to activities being deleted from the timetable. Conversely, an unexpected time savings (a ‘‘good run’’ on a normally congested freeway) will also cause a disutility, when the savings do not allow the chance to insert a new activity into the schedule and waiting is disagreeable.

2.4. Time of day The moment when a disturbance occurs in the timetable will dictate on the course of action taken. A delay in the morning is very likely to knock on many more activities than a delay on the way home at the night. At the same time, the opportunities to solve the ‘‘conflict’’ in the timetable are more numerous

427 over the day, and the responses vary considerably. If a substantial change (delay) in the timetable appears late in the evening, the resolution is less elaborated and usually involves changing the timing of the next activity or deleting it altogether. We believe that this is not only due to the routine, but also affected by subjective sensations/feelings as fatigue, discomfort, boredom at the end of day. The model given below explores the factors that affect the type of responses and provides the potential action taken by individuals when facing changes in the travel time, as well as an index measure of the utility/disutility associated with the action.

3. Modelling background Rule-based systems model complex systems where way to represent knowledge is to use ‘‘IF-THEN’’ rules. Fuzzy rule-based systems (FRBS) offer a generalisation to the classic logic systems by addressing the imprecision and possible uncertainty of human knowledge. While traditional set theory defines set membership as a Boolean predicate, fuzzy sets allow us to represent the membership functions as a possibility distribution (Zadeh 1965; Kosko 1992). Fuzzy systems are based on degrees of membership. For example, a shopping destination is ‘‘flexible’’ (easily substituted with other destinations) or is ‘‘somewhat flexible’’ (accessing alternative destinations is possible but not preferred) and the degree of belonging to either of the categories is defined by membership functions. Fuzzy logic manages to model complex non-linear input-output relations as a synthesis of multiple simple input–output relations (fuzzy rules). The boundary of the rule areas is not sharp, but ‘‘fuzzy’’. The system output from one rule area to the next rule gradually changes. The difference between crisp and fuzzy rule-based systems is related to how the input space is partitioned (see Figure 1a and b). For instance, a small time delay is experienced before a scheduled shopping activity. In Figure 1a, if the shopping trip has flexible (b) Reschedule at convenient destination

Substitute destination

Length of delay

Length of delay

(a) Delete

Reduce time at destination

crisp rules

Rigidness of destination

Reschedule at convenient destination

Substitute destination

Delete

Reduce time at destination fuzzy rules

Figure 1. Rule partition of a two-dimensional input space.

Rigidness of destination

428 destination, the individual will choose a more convenient location. In Figure 1b,the rules share a ‘‘grey area’’. The individual may choose an alternate destination or choose to reduce the time taken when shopping. The resolution between the alternatives depends on other inputs (the diagram shows only two input dimensions) and on the shape of the membership functions.

3.1. Membership values for inputs In the example presented above, if the delay is 5 min (the vertical arrow in Figure 2), we might say that this is a small delay. However, it is possible the delay is negligible and it may be readily absorbed in the time spent at next activity. In Figure 2, the imprecision of the statements ‘‘the delay is small’’ and ‘‘the delay is negligible’’ are represented by the partitioning of all delays/savings into fuzzy sets or linguistic adjectives. The height of the graph describes the membership grade to a linguistic adjective. Strictly, a fuzzy set is a collection of ordered pairs A ¼ {x, mA(x)}, and mA(x) is the membership value for element x in set A. A normalised set (such as the one in Figure 2) is a set such that it contains at least one element with a membership value of 1. While not obligatory, normalised sets are the standard in fuzzy systems. Subsets are found in fuzzy systems and are useful to add strength (very) or ambiguity (somewhat or fairly) to a statement; these are known as linguistic hedges or modifiers. Conventionally, the square function is used to represent very and the cubed function is used to represent extremely. Also, the square root function is used to represent somewhat. They are particularly important as they allow non-linear functions for the fuzzy sets.

Figure 2. Adjectives for time savings/delays.

429 3.2. Fuzzy operators for combining inputs The two antecedents, size of delay and flexibility of location may be combined using fuzzy operators AND, OR, and NOT, and they are the operators used in this paper. Unlike classical set theory, there is not a definitive operator for each and different examples appear in the literature (for a review, see Klir & Folger 1988). We have chosen to use the basic formulations of these fuzzy operators: AND Intersection : mA\B ðxÞ ¼ min½mA ðxÞ; mB ðxÞ

ð1Þ

OR Union : mA[B ðxÞ ¼ max½mA ðxÞ; mB ðxÞ

ð2Þ

NOT Complement : mnot A ðxÞ ¼ 1  mA ðxÞ

ð3Þ

3.3. Fuzzy logic Where classical two-value predicate logic renders propositions as either true (T) or false (F), fuzzy logic deals in degrees of truth. Equally, it deals with degrees of falseness. As with classical logic, a fuzzy predicate is: x is A (e.g., 15 min is a ‘‘large_delay’’). However, unlike classical logic, the truth value for the proposition is itself a fuzzy set with varying degrees of membership. The simplest of truth sets is a one-to-one mapping of the membership function, for example, if msmall delay (5 min) = 0.63, then mT (5 min is a small delay) ¼ 0.63. Let us examine again the 5 min delay (as in Figure 2) experienced before a scheduled shopping activity, considered by a respondent as flexible (4 on the 1–5 scale, or membership degree of 0.4). If crisp sets are applied, the truth value for the proposition ‘‘the delay is small and the destination is flexible’’ is 1, and the strategy is ‘‘change destination’’. Applying fuzzy logic we have: mT (small delay and flexible activity) ¼ minðmsmall ð5Þ; mflexible ð4ÞÞ ¼ minð0:63; 0:4Þ ¼ 0:4: However, other truths may apply: mT (small delay and very flexible) ¼ min(0.63, 0.42) ¼ 0.16 and mT (negligible and flexible) = 0.17 are valid statements. It is natural to question which truth applies. To the logician it does not matter; these truths are membership values to the alternate truth sets. It is, however, important to the individual who must resolve the delay by adjusting his/her schedule, as well as to the modeler who is studying the responses. As stated at the beginning of Section 3 – fuzzy logic is a framework for modelling knowledge representation in an environment characterised by uncertainty and imprecision (Nguyen & Walker 2000). FRBS are used here to resolve strategies used by respondents to cope with unexpected changes in their daily schedules.

430

Knowledge Base Data Base

Real input x

Fuzzification interface

Rule Base

Inference system

Defuzzification interface

Real output y

Figure 3. Basic structure of Mamdani FRBS. Source: Cordon et al. (2001), p. 3.

3.4. Fuzzy rule-based system FRBS organise the input values, the linguistic labels (fuzzy sets), the rules (IF– THEN) and the rules consequences (output). The two types of FRBS for real values input and output are attributed to Mamdani and Takagi-Sugeno-Kang (Cordon et al. 2001). The fuzzy logic (FL) model used in this study is a Mamdani FRBS (Figure 3). The Knowledge Base stores the fuzzy rule semantics and the available knowledge about the problem in the form of fuzzy ‘‘IF–THEN’’ rules. The Data Base refers to the membership functions and scaling factors. The Rule Base holds a collection of ‘‘IF–THEN’’ linguistic rules built with AND and OR operators. For each rule, the antecedent (the rule’s premise) describes to what degree the rule applies, while the conclusion (the rule’s consequent) assigns a membership function to each of one or more output variables. The inference engine combines the input values and the knowledge base. Each input passes through the system, first undergoing a process of fuzzification (in the fuzzy interface), where its membership values are assigned. Next, a selection of appropriate rules is made, the rules are combined (inference system) and, finally, the fuzzy set is ‘‘restyled’’ in a crisp value (defuzzification interface) that constitutes the global output of the fuzzy system.

4. Empirical setting The survey was conducted in the capital city of Romania, Bucharest. The city’s population is over 2 million and it has a population density of 8600 inhabitants/km2 (NTS 2003). Since 1989 there has been a major shift towards car ownership and this has placed significant pressure on the limited road infrastructure. However, 80% of trips are still made by public transport. There are more than 2.5 million passenger trips per day on the public transport system, of which 85% are made on surface transport (trams, buses, trolleybuses, and light rail) and 15% are made on the underground. The frequency of services is

431 4–10 min during peak times, 5–10 min during off-peak periods and on the weekends, and 15 min over night (TRB 2003: 34). The question analysed in this study was part of a follow-up questionnaire to a travel diary data collection for academics and students from ten universities in Bucharest in November 1998. The 1441 questionnaires (89% response rate) recorded detailed trip and activity information for a typical weekday and additional socio-demographic variables. The respondents were then invited to participate in a second round of the survey. While 22% of the subjects agreed to participate, only 126 actually responded (Table 1). The respondents were asked to comment on how they would adjust their trip diary given a hypothetical change to the duration for one of their trips. For each of these respondents a varied length of saving/delay was given and this was applied randomly to a specified motorised trip from the sequence of their trips. The respondents were directed to discuss any subsequent alterations they would make to their day’s activity schedule and to rate the inconvenience (or benefit) experienced due to the change.

4.1. Descriptive statistics The average number of motorised trips taken by the students and academics is 3.1 trips during the working day, using an average of 1.7 transport modes. Only 11% of the trips are made by car and 63% are made on public transport. The average trip distance is 5.2 km and the average speed is 12.6 km/h. The trip details of the participants in the second stage of the survey were representative of the larger sample. While the second phase survey mechanism was an open-ended question, we did note a degree of consistency among the responses. Where possible, the respondents cushioned the impact by adjusting the timing and duration of the next activity – starting early and staying longer (trip time savings) or starting late and reducing the duration (trip time delay). However, other responses Table 1. Sample composition. Respondents

Female

Male

Total

Trip diary survey Academics Students Total

105 535 640

187 614 801

292 1149 1441

Follow-up survey Academics Students Total

19 37 56

28 42 70

47 79 126

432 included: deleting or adding an activity, changing the location, re-ordering the remaining activities, and consolidating remaining activities (trip chaining). In only three cases the respondents did choose alternative modes of transport for subsequent trips. Also we noted that substitution of activity location was an unpopular strategy; only five respondents adopted it. This supports Sch€ onfelder’s (2001) results, who found little spatial variability and 59% of all trip destinations within the household cluster. In Table 2 the frequencies of responses were recorded. Where responses were nearly exclusive to a range of travel delay/savings the size of the time variation is presented in brackets. Not surprisingly small variations in travel time cause minimal changes to the respondent’s daily activity schedule and large changes induce a more significant response when trying to get ‘‘back on track’’. The respondents also indicated the level of (dis)utility associated with having to alter their schedule. Understandably, time delays never raised the level of satisfaction for the change to a new activity schedule. Larger time delays had a greater impact on the respondent’s satisfaction. Large time savings were welcomed by the respondents. However, for some respondents small time savings were seen as an inconvenience. This occurred when the time saving was too small for the respondent to add a new activity and the nature of the subsequent activity was less flexible. In general respondents were happier when they were able to spend longer engaged in activities (transferring trip time to activity time) or when they could undertake more activities (having to delete an activity due to a time delay was most dissatisfying). Table 3 shows the frequencies of utility responses. 5. Development of the fuzzy rule based system The fuzzy logic model is used to generalise the individual’s decision rules, where the inputs are subjective. The development of the model can be partitioned into Table 2. Strategies to cope with changes in travel time. Course of action

Do nothing Change timing next activity (starting/finishing time) Change duration next activity Change location next activity Remove next activity Add new activity Change transport mode

Number cases (time saving)

Number cases (time delay)

Frequency (whole set)

Test set

10 (15 min) –

39 5 7 14

8 1 2 3

2 (>15 min)

3

0

433 Table 3. (Dis)utility indicator associated to the deviation of travel time Indicator of (dis)utility associated with time delay/saving Very happy (improves the satisfaction derived from the activity schedule) Happy No change Unhappy Very unhappy

Number cases (time saving) 39 (>10 min) 15 (5–20 min) 4(