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Optimizing Bus Services with Variable Directional and
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Temporal Demand Using Genetic Algorithm
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Qu Hezhou1, Chien Steven I-Jy2, Liu Xiaobo3*, Zhang Peitong4, and Bladikas Athanassios5
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School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China; Sichuan Province Key Laboratory of Comprehensive Transportation, Chengdu, 610031, China College of Automobile, Chang'an University, Xi'an, 710064, China; John A. Reif Jr. Department of Civil and Environmental Engineering, New Jersey Institute of Technology, Newark, 07102-1982, USA School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China (*Corresponding Author) Phone: (+86)-28-87600750, e-mail:
[email protected] School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China Mechanical & Industrial Engineering, New Jersey Institute of Technology, Newark, 07102-1982, USA
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Abstract
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As a major mode choice of commuters for daily travel, bus transit plays an important
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role in many urban and metropolitan areas. This study proposes a mathematical model to
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optimize bus service by minimizing total cost and considering a temporally and directionally
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variable demand. An integrated bus service, consisting of all-stop and stop-skipping services
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is proposed and optimized subject to directional frequency conservation, capacity and
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operable fleet size constraints. Since the research problem is a combinatorial optimization
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problem, a genetic algorithm is developed to search for the optimal result in a large solution
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space. The model was successfully implemented on a bus transit route in the City of
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Chengdu, China, and the optimal solution proved to be better than the original operation in
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terms of total cost. The sensitivity of model parameters to some key attributes/variables is
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analyzed and discussed to explore further the potential of accruing additional benefits or
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avoiding some of the drawbacks of stop-skipping services.
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Keywords:
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Bus transit; Cost; Travel time; Service patterns; Optimization; Genetic algorithm
Introduction
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As a major mode choice of commuters’ daily travel, public transit is expected to offer
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sufficient capacity to serve as many passengers as possible in an effective way so that the
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level of service may be sufficiently high, but at a reasonable cost. Improving bus service and
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maintaining efficient system performance have been critical concerns of transit suppliers.
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One of the major drawbacks of bus service is frequent stops, which increase travel time and
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make the service unattractive. Stop-skipping (or express) service, allowing some buses to
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skip certain stops, can reduce passenger as well as vehicle travel time. However, all-stop
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service expands the service coverage and offers passengers in light demand areas a travel
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choice they did not have before. Thus, the success of integrating stop-skipping and all-stop
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services may reduce the cost to passengers and the transit provider. There are already fairly
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successful systems operated in the US (i.e., subways systems in New York City, NY and
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Boston, MA and bus transit systems in Newark, NJ and Madison, WI).
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As indicated in previous studies (Ercolano [1], Casello and Hellinga [2]), passenger
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travel time can be significantly reduced by introducing limited-stop (or called stop-skipping)
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service into a traditional all-stop service. El-Geneidy and Surprenant-Legault [3] evaluated
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limited-stop service which saved vehicle running time and user travel time. The stop-skipping
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operation has been successful in increasing average speed while maintaining high service
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frequency and it is superior to the all-stop service in terms of reducing passenger travel time
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(Vuchic [4]). While investigating the schedule of stop-skipping service in a congested urban
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setting during peak periods, Niu [5] formulated a nonlinear programming model to minimize
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the overall waiting times and the in-vehicle traveler costs subjected to a limited number of
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vehicles and using a bi-level genetic algorithm.
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While analyzing transit service quality and system efficiency, several studies (Tzeng
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and Shiau [6], Chien [7], Zhao and Zeng [8], Hafezi and Ismail [9], Verbas et al [10])
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indicated that the service level may be improved at a reasonable operator cost by employing
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optimized service strategies. Furth and Day [11] investigated three service strategies for
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planning bus operations, including short-turn, deadheading and express services. It was found
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that short-turn service is preferable when high-demand exists over a bus route segment,
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because it can reduce fleet size as well as operating cost. Short-turn and deadheading service
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patterns have been researched extensively (Furth [12], Ceder [13], Ceder and Stern [14],
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Furth [15]). Ulusoy et al [16] developed a model to optimize the integration of all-stop, short-
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turn and express services. The travel demand was assumed even on both directional services.
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The objective total cost was minimized subjected to frequency conservation and operable
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fleet size. To reduce total cost, Tirachini et al [17] introduced short-turn service on high
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demand segments, so that the negative impact of spatial demand concentration could be
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alleviated. Furthermore, Cortés et al [18] minimized total cost considering short turning and
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deadheading for a single transit line, and determined the optimal bus frequencies, vehicle
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capacities, and the stations where the strategy begins and ends.
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Some studies focused on the integration of stop-skipping and all-stop services. Fu et
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al [19] investigated the performance of a service strategy that allows a lead vehicle to
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perform all-stop service and the following one express service. A nonlinear integer model
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was developed to minimize the total cost incurred by both operator and passengers, while the
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optimal schedule was found by an exhaustive search method. Chien et al [20] developed an
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analytical model to minimize total cost for a general rail transit route with a many-to-many
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demand pattern. The integrated service patterns and frequencies were optimized subject to a
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set of constraints to ensure frequency conservation and operable fleet size. Assuming
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symmetric Origin-Destination (OD) passenger demand, Ulusoy and Chien [21] developed a
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model to optimize an integrated service (i.e., express and short-turn services, and all-stop
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services) which yielded the minimum total cost operation for a bus transit route with many
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stops considering transfer demand elasticity. Transfer demand at stops was determined by a
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logit model. A Genetic Algorithm was used to search for the optimal solution.
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Chiraphadhanakul and Barnhart [22] optimized a limited-stop service frequency in parallel
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with the local service of a transit route, to obtain the maximum users' welfare.
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Many studies related to optimizing transit services focused on minimizing users’
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travel time and system operating costs on an hourly basis. Previous studies (Chien et al [20],
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Leiva et al [23], Larrain et al [24]) concentrated on smaller networks in which the stopping
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patterns of express services are pre-defined. In reality, a bus route with many stops might
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traverse areas with different land uses and traffic conditions, which increases the difficulty of
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formulating the model and the computation time to search for optimal service patterns. Since
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the research problem discussed here is NP-hard, an efficient algorithm or heuristic is needed
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to search for the optimal solution. In addition, the ridership might vary temporally and
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spatially.
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The objective of this study is to minimize the total cost of an integrated system (i.e.
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all-stop and stop-skipping) by optimizing the frequencies subject to directional frequency
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conservation, capacity and operable fleet size constraints. Unlike previous studies, the
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objective total cost function is developed to adapt to demand that varies by direction and over
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time. A genetic algorithm is developed to search for the optimal solution. In this study, a
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transfer penalty, and different values for the various components of users’ time (i.e., in-
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vehicle, transfer, and wait time) are considered. Unlike previous models, the proposed
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method optimizes stop-skipping services of a bus route with uneven demand. Thus, the
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optimized service frequencies vary temporally and skipped stops vary directionally and
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temporally. The needed data were collected from a real-world bus route, and are used to
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demonstrate the model’s efficiency and applicability. The sensitivity of model parameters to
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some key attributes/variables is analyzed and discussed to explore in detail the potential
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benefits of a stop-skipping service.
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Model formulation
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A mathematical model, consisting of an objective function and a set of constraints, is
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developed and discussed in this section. To minimize the objective total cost function (the
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sum of user and operator costs), decision variables, including bus frequencies, and stops
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served by stop-skipping service, shall be optimized subject to directional service frequency
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conservation, capacity, and fleet size constraints. The model parameters and variables utilized
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for formulating the proposed model are defined and summarized in Table 1. The assumptions
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considered in this study are as follows:
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1. A general L-kilometer long bus route with N stops (See Figure 1) is given. End terminals
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are served by both all-stop and stop-skipping services.
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{ 120 121
Fig. 1 A general bus route and associated services
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2. The frequencies of the outbound and inbound services are the same. The end stations
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have sufficient space for storing the required buses and there is no deadheading.
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3. The stop-skipping vehicle trips will be evenly spaced within the study time period. For
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instance, an integrated service with frequency of 2 stop-skipping buses and 6 all-stop
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buses per hour, the pattern would be dispatching 1 stop-skipping bus follow by 3 all-stop
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buses every half an hour.
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4. The Origin-Destination (OD) demand is constant within a time period, but may vary by
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direction and time period. Passenger arrivals at stops are random, and the average
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wait/transfer time is a fraction (i.e. 0.5) of the service headway (the inverse of frequency).
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5. The choice of a transfer location for passengers is determined considering the shortest
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travel time. The probability of more than one transfer per trip is negligible, and the single
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bus fare without transfer cost for passengers transferring from one service to another is
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considered in the model.
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6. The average delay time of entering and leaving a bus stop is given. The dwell time at a
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stop linearly increases as the number of boarding and alighting passenger increases, and
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the average bus running speed is constant within a time period, but may vary over
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different time periods. Table 1 Variable Definitions and Baseline Values
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Baseline Values
Parameters
Units
aAsd /bAsd
pass
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aEsd /bEsd
pass
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b CI /CR /CW CO /CU c Dijk d F fAd /fEd G i/j/s k L ls N O QAsd /QEsd qij qAij /qEij r TAd /TEd tADsd /tEDsd tba tl to tIijk /tRijk /tWijk uk V x/y αR /αW δ λid
$/bus-hr $/hr $/hr spc/bus pass/hr vehicles km km pass/hr pass/hr pass/hr min hr hr sec/pass hr/stop min hr km/hr % $/pass-hr
15 90 99 500 21 34 500 5 4 5 0.5 0.9 0.3/0.3/0.3 1.3/2/2 1.5
ijk
rS /rC /rM θI /θR /θW I/ R/ W
Descriptions Average alighting/boarding passengers per bus vehicle at stop s, direction d with all-stop service Average alighting/boarding passengers per bus vehicle at stop s, direction d with stop-skipping service Average bus operating cost User in-vehicle/transfer/wait cost Operator/User cost Bus capacity Demand from stop i to j, travel choice k Direction of service (1: Outbound; and 2:Inbound) Maximum operable fleet size Frequency of all-stop/stop-skipping service, direction d Number of iteration for GA Indices of stops (i, j, s ∈ S) Index of travel choice (k ∈K) ; K is a set of travel choices Bus route length Distance of link s (from stop s to s+1) Total number of stops along the bus route Size of population in GA Demand on link s, direction d with all-stop/stop-skipping service Demand from stop i to j Demand from stop i to j using all-stop/ stop-skipping service Total transfer penalty per transfer Bus travel time on direction d for all-stop/stop-skipping service Dwell time at stop s, direction d (all-stop /stop-skipping) Average boarding/alighting time per passenger Average delay per stop Layover time at the end stop In-vehicle/Transfer/Wait time from stop i to j, travel choice k Disutility of travel choice k Average bus speed Index of transfer stop (x, y ∈ X; X is a set of transfer stops) Ratio of the average transfer/wait time to service headway Load factor Availability stop-skipping service at stop i, direction d Percentage of demand from stop i to j with travel choice k Selection/crossover/mutation ratio in GA Demand parameter associated with in-vehicle/transfer/wait time Value of users’ In-vehicle/Transfer/Wait time
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Demand estimation
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This study considers that demand uses a specific service based on its availability at
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stop i in direction d (1: Outbound and 2: Inbound), denoted by the binary variables λid, and
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the weighted travel time (or disutility). This variable is equal to 1 if stop-skipping service is
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available; otherwise, it is 0. Thus,
1; λid = 0;
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Served by stop-skipping service ∀i, d Skipped by stop-skipping service
(1)
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A passenger may begin his/her journey with either all-stop or stop-skipping service
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depending on the service availability at bus stop and travel time (i.e. the sum of wait, transfer
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and in-vehicle times). Thus, the demand from stop i to j denoted as qij may have four
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alternatives (k ∈ K = {1, 2, 3, 4}), as illustrated in Figure 2. The transfer location is chosen by
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a passenger considering his/her shortest travel time. Hence, demand Dijk from stop i to j with travel choice k can be determined by a fraction of qij. Thus, Dijk = qijψ ijk
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∀i , j
(2)
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where
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choice k, as shown in Equation 3. The available choices for passengers originating at stop i
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and traveling to stop j vary with the availability of services at both stops, as shown in Table 2.
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Table 2 Travel choices vs. service availability at stops
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can be estimated by a logit model using a disutility function (i.e., uk) of travel
Alternatives for travel choice λid
1 0
λjd 1
0
k=1, 4 k= 3, 4
k= 2, 4 k= 4
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Situation 1
Both origin stop i and destination stop j are served by stop-skipping (λid=1; λjd=1) and
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all-stop services. Thus, there are two travel choices that a passenger can make (k=1 and 4).
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Situation 2
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The stop-skipping service does serve origin stop i but not destination stop j (λid=1;
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λjd=0). Some passengers may begin with the stop-skipping service and then transfer to the all-
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stop service (k=2), while others may take the all-stop service to stop j without transfer (k=4).
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Situation 3
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The stop-skipping service does not serve origin stop i but does serve destination stop j
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(λid=0; λjd=1). Therefore, some passengers will begin their journey with the all-stop service,
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and then transfer to the stop-skipping service at a downstream stop (k=3), while others will
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not transfer (k=4).
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Situation 4
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The stop-skipping service does not serve origin stop i and destination stop j (λid=0; λjd=0). All passengers from stop i to j have to use the all-stop service without transfer.
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Fig. 2 Travel choices for passengers from stop i to j Hence, the proportions of demand (
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ijk)
that will use the four choices illustrated in
Figure 2 can be calculated as follows:
ψ ijk
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e−( u1 ) − (u1 ) −(u4 ) λid λ jd +e e − ( u2 ) e λ (1 − λ jd ) −( u2 ) −( u4 ) id = e +e e( −u3 ) (1 − λid )λ jd ( − u3 ) ( − u4 ) e +e 1 − (ψ ij1 +ψ ij 2 +ψ ij 3 )
∀i, j; k = 1 ∀i, j; k = 2
(3)
∀i, j; k = 3 ∀i, j; k = 4
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where uk is assumed to be the weighted sum of wait, transfer (if any), and in-vehicle time.
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Thus,
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u k = θW tWijk + θ I t Iijk + θ R t Rijk
∀i , j , k
(4)
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where θW , θR and θI represent the sensitivity of demand to wait (tWijk), transfer (tRijk), and in-
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vehicle time (tIijk), respectively.
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Objective function
The objective function considered in this paper minimizes the total cost (TC) and is
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defined as the sum of user CU and operator CO costs, as shown in Equation 5. min TC = min ( CU + CO )
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(5)
User cost (CU)
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Considering the full path of a bus trip, the user cost is determined by including the
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time required for access, wait, transfer, in-vehicle travel and egress. Since the stop locations
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are given, the user access/egress time is constant and will not affect the optimal solution.
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Hence, the access/egress costs are omitted. The user cost, denoted as CU, is the sum of wait,
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transfer, and in-vehicle costs. Thus, N
i =1
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where
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respectively.
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N
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CU = ∑∑∑ Dijk (tWijk µW + tRijk µR + tIijk µI )
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W,
R
and
I
(6)
j =1 k =1
represent the value of user’s wait, transfer and in-vehicle time,
Wait cost is incurred by passengers waiting for buses at origin station i. tWijk is the average initial wait time, a fraction αW of headway, which is the inverse of frequency. Thus,
tWijk
αW f Ed = αW f Ad
∀i, j , d
k = 1, 2 (7)
∀i, j, d
12
k = 3, 4
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where fEd and fAd represent the frequencies of stop-skipping and all-stop services in direction
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d, respectively.
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In this study, a passenger waits for the bus to transfer without extra transfer walking
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time and no transfer fare. Thus, tRijk represents the transfer time, which is the sum of average
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transfer waiting time (a fraction αR of the pickup bus headway) and the total transfer penalty
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per transfer (r). There are several factors that affect passenger decisions to transfer, and
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include walking distance, waiting time, transfer fare, seat availability, service reliability,
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pedestrian environment, personal attributes etc. (Algers et al [25], Han [26], Guo and Wilson
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[27 and 28]). In previous studies, the effects of all the factors were incorporated into a single
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total transfer penalty. Note that as passengers' travel choice k is 1 and 4, they can reach their
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destination stops without transfer. The transfer time can be formulated as
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tRijk
αR f +r / 60 = Ad α R +r / 60 f Ed
∀i, j; k = 2 (8)
∀i, j; k = 3
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In-vehicle cost is incurred by the in-vehicle time for passengers. The in-vehicle time
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of passenger from station i to j with travel choice k, denoted as tIijk, includes the travelling
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time along the bus route, the average delay and dwell time at each stop. For k=1and k=4,
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passengers begin and end their travel with stop-skipping and all-stop service respectively, and
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there is no transferring. Thus,
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t Iijk
j −1 ls ( + (t EDs +1,d + tl )λs +1,d ) ∑ s =i V = i −1 ∑ ( ls + (t ED + tl )λsd ) sd s = j V
t Iijk
j −1 ls ( + t ADs+1,d + tl ) ∑ s =i V = i −1 ∑ ( ls + t AD + tl ) sd s = j V
i < j; d = 1; k = 1
(9) i > j; d = 2; k = 1 i < j; d = 1; k = 4
(10) i > j; d = 2; k = 4
Considering that passengers may use the combination of all-stop and stop-skipping services when k is equal to 2 or 3, the in-vehicle time is formulated as follows:
tIix1 +tI xj 4 tIijk = tIiy 4 +tI yj1
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∀i, j; k = 2 ∀i, j; k = 3
(11)
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where tIix1 (tIyj1) and tIxj4 (tIiy4) can be obtained using Equations 9 and10. It is worth noting that
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x and y (x, y ∈ X) represent transfer stops which can be identified based on the shortest travel
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time. V is the average bus speed; ls represents the spacing between bus stops s and s+1; tl is
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the average delay per stop resulting from deceleration and acceleration while a bus entering
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to and leaving from a stop; tEDsd and tADsd represent the dwell time at stop s in direction d of
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the stop-skipping and all-stop service, respectively, which can be derived by the average
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boarding/alighting time per passenger multiplied by the sum of boarding passengers and
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alighting passengers. Thus,
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tEDsd = tba (bEsd + aEsd )/3600
∀s, d
(12)
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t ADsd = tba (bAsd + a Asd )/3600
∀s , d
(13)
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For stop-skipping service, the average boarding and alighting passengers per bus at
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stop s in direction d are denoted as bEsd and aEsd, respectively, and can be calculated by
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Equations 14 and 15 based on demand qEij and frequency fEd. Thus,
bEsd
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aEsd
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N qEsj ∑ = j = s +1 f Ed 0
s < j; d = 1
(14)
s = N; d = 1
s −1 qEis ∑ = i =1 f Ed 0
i < s; d = 1
(15)
s = 1; d = 1
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Similarly, the average boarding and alighting passengers per vehicle and the
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corresponding dwell time at stop s in direction d using all-stop service can be derived as
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Equations 16-17.
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bAsd
aAsd
N q Asj ∑ = j = s +1 f Ad 0
s < j; d = 1
(16)
s = N; d = 1
s −1 qAis ∑ = i =1 f Ad 0
i < s; d = 1
(17)
s = 1; d = 1
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where qEij/qAij represents the demand from stop i to j using the stop-skipping/all-stop service
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and can be derived from demand Dijk, the stops served by stop-skipping and the transfer stops
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x and y (see Figure 2). There are three travel choices (k=1, 2 and 3) for stop-skipping service
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passengers. Note that, in categories 2 and 3, passengers using the stop-skipping service would
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alight at transfer stop x and board at transfer stop y, respectively. Thus, qEij can be calculated
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as q Eij = Dij1 + Dix 2 +D yj 3
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(18)
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There are three travel choices (k=2, 3 and 4) for passengers who want to use an all-
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stop service. Note that, in categories 2 and 3, passengers using all-stop service would board at
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transfer stop x and alight at transfer stop y, respectively. Thus, qAij can be calculated as q Aij = Dxj 2 + Diy 3 + Dij 4
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(19)
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Note that at the last stop in both directions there are no passengers boarding the bus,
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and there are no passengers alighting off the bus for the first stop in both directions. The
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inbound traffic (direction 2) can be similarly formulated.
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Operator cost (CO)
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The operator cost is incurred by operating buses, which is the sum of the product of
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vehicle one-way travel time, vehicle frequency and average vehicle per hour operating cost
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(b). Thus, the operator cost is 2
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CO = ∑ (TEd f Ed + TAd f Ad )b
∀d
(20)
d =1
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TEd and TAd are defined as the vehicle one-way travel time for stop-skipping and all-
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stop service in direction d, respectively, including the sum of bus moving time along the
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route, dwell time at each stop and layover time at the end stop (to). Thus,
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N −1 ls ( + (tEDs+1,d + tl )λs +1, d )+to /60 ∑ s =1 V TEd = N −1 ( ls +(t + t )λ )+t /60 ∑ EDsd l sd o s =1 V
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N −1 ls ( + t ADs+1,d + tl ) +to /60 ∑ s =1 V TAd = N −1 ( ls + t + t ) +t /60 ∑ ADsd l o s =1 V
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264
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d = 1; ∀s (21)
d = 2; ∀s d = 1; ∀s (22) d = 2; ∀s
Model constraints
Considering that vehicle operations should be realistic, directional frequency
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conservation, capacity and fleet size constraints are formulated and discussed below.
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Frequency conservation constraint
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The frequency conservation constraint is designed to ensure that the number of vehicles dispatched from both end stops will be the same. Thus,
f E1 = f E 2 and
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f A1 = f A 2
(23)
Capacity constraint
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Equation 24 describes the capacity constraints needed to provide sufficient vehicle
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service capacity for all-stop and stop-skipping services to satisfy the available demand. For
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stop-skipping and all-stop service, the maximum demand on link s (link connects stops s to
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s+1) in direction d is defined as Max{QEsd} and Max{QAsd}, respectively. Thus,
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Max {QEsd , QAsd } ≤ {cδ f Ed , cδ f Ad }
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for all s, d
(24)
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where δ is a load factor appropriate for the level of service, and c represents the vehicle capacity. QEsd and QAsd can be calculated from the stop i to j demand that uses stop-skipping
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and all-stop services, and denoted as qEij and qAij respectively. Equations 25 and 26 can be
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used to write the outbound (direction 1) constraints. The inbound (direction 2) constraints can
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be formulated similarly. s
N
QEsd = ∑ ∑ qEij
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i < j; d = 1; ∀s
(25)
i < j; d = 1; ∀s
(26)
i =1 j = s +1 s
N
QAsd = ∑ ∑ q Aij
283
i =1 j = s +1
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Fleet size constraint
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The limited operable fleet size denoted as F must be greater than or equal to the sum
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of the fleets for all-stop and stop-skipping services. It is noted that fleet size F is the product
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of vehicle travel time and its service frequency. Thus, 2
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F ≥ ∑ (TEd f Ed + TAd f Ad ) ∀d
(27)
d =1
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It is worth noting that the total cost formulated in Equation 5 can deal with
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heterogeneous demand distribution over the study bus route. With that the demand might be
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uneven for both directional services. Considering that the demand may vary temporally (i.e.
292
over different time periods of a day), the proposed model may take demand of different time
293
periods as input and output the corresponding optimal services which minimize the total cost.
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Solution algorithm
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As discussed earlier, the study problem is a combinatorial optimization problem. An
296
efficient solution algorithm is needed to quickly search for the optimal result in a large
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solution space. In previous studies (Ulusoy et al [16], Fu et al [19]) that considered small
298
scale problems, the Exhaustive Search Algorithm (ESA) was used as the solution approach.
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However, a real bus route has a greater number of stops leading to a large number of
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combination possibilities for stop-skipping services. Owing to increasing computation time,
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ESA may not be the best method to optimize the solution for such a bus route. To optimize a
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large scale combinatorial problem, the genetic algorithm (GA) has been regarded as an
303
effective solution approach (Chien et al [20], Ulusoy and Chien [29], Holland [30], Goldberg
304
[31]). GA is a powerful and broadly used tool to perform a stochastic search in large solution
305
spaces (Gen and Cheng [32]), such as transit route network design (Fan and Machemehl [33]),
306
rail transit routes optimization (Jha et al [34]), urban planning (Balling et al [35]), transit
307
scheduling optimization (Chakraborty et al [36]), and bus frequency optimization (Ulusoy
308
and Chien [29]; Yu et al [37]). Therefore, to reduce the computational time consumed by
309
conventional techniques, a GA based approach is developed to search for the optimal solution.
310
311
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To use the genetic algorithm, each individual in the population for a time period is represented by an encoded solution called chromosome, which is represented as follows:
{f
Ad
, f Ed , λi1 , λi 2 }
19
(28)
313
The chromosome consists of three parts of genes encoded by a binary vector as shown
314
in Figure 3. In Part 1 the all-stop and stop-skipping service frequencies are represented by an
315
integer string consisting of a series of cells, while in Parts 2 and 3 the binary vector indicates
316
for each direction whether a stop is served or skipped. In Part 1 of the chromosome, the 1st
317
and 2nd six-cell strings represent all-stop and stop-skipping service frequencies, respectively,
318
which can be decoded based on the combinations of 0 and 1. The number of digits in Parts 2
319
and 3 are identical, representing the fact that the number of stops is equal for both directions.
320
λid=1 denotes that stop i is served by stop-skipping service, and λid=0 otherwise. Considering
321
that both end stops must be served by stop-skipping service, the variables λ1d and λnd are set as
322
1 before executing the optimization processes. Thus, there are N-2 genes in both Parts 2 and 3
323
(N is the total number of stops along the bus route). Similarly, for special neighborhoods or
324
CBD areas needed stop-skipping service, the corresponding genes associated with those
325
location can be set as 1.
326
f Ad f Ed λi 1 λi 2 6 47 4 8 678 6474 8 6474 8 001010101000 001......010010......110 1442443 14243 14243 Part1
327
Part 2
Part 3
Fig. 3 The encoding illustration of a chromosome
328
Three operators are included in GA executing processes: selection, crossover and
329
mutation. Elitist selection is generally used to preserve the best chromosome in the new
330
generation. Crossover is used for two randomly selected parent chromosomes to produce two
331
better offspring by swapping corresponding segments of the parents, which intends to inherit
20
332
good genes from parent chromosomes. The mutation operator is called mutation ratio which
333
introduces genetic diversity into the population. The elitist selection, the two-point crossover
334
operation and the two-point mutation which have been fully tested (Ulusoy and Chien [21])
335
are used in each generation until a terminating criteria is yielded.
336
In this study, the minimum total cost TC represents the best solution and the
337
maximum fitness of the chromosomes. To ensure that the solutions always satisfy directional
338
frequency conservation, capacity and fleet size constraints, an enormous penalty is introduced
339
so that the infeasible solutions will be excluded. Thus, the objective function can be
340
reformulated as follows:
341
342
343
m = 0; TC = TC ( Z ) + m m = Value of penalty;
if Z is a feasible solution if Z is not a feasible solution
(29)
Solution approach
A step-to-step procedure using the developed GA to search for the optimal solution
344
(i.e., service frequencies and skipped stops) is discussed below and illustrated in Figure 4.
345
Step 1: Initialize the parameters for the GA such as population size, number of total
346
generations, selection ratio (rS), crossover ratio (rC), mutation ratio (rM), termination
347
rule, etc.
348
349
Step 2: Population initialization. Generate the initial group of solutions randomly and regard the initial group of solutions as the 1st generation.
21
350
Step 3: Decode the binary string into real numbers for each corresponding chromosome to
351
obtain the service frequencies and the served or skipped stops by stop-skipping
352
service in the outbound and inbound directions.
353
354
355
356
357
358
359
360
Step 4: Use the capacity and fleet size constraints as determined by Equations 23, 24 and 27 to check whether each solution satisfies the constraints. Step 5: Calculate the objective function (total cost) value for each chromosome in the solutions by using Equation 5. Step 6: Sort the chromosomes in the population in ascending order by their objective function values. Step 7: Implement the elitist mechanism in the selection process, and use the crossover and mutation operations to reproduce the new solutions.
361
Step 8: Use the constraints to check whether the new solutions satisfy the constraints. Re-
362
evaluate the new solutions in terms of total cost and apply the constraint handling
363
method via Equation 29 to discard the infeasible solutions. Copy the remaining
364
solutions to the next generation.
365
Step 9: Update the next generation and best solution.
366
Step 10: Check if the stop criteria (i.e., maximum iterations) is satisfied. Otherwise, go to
367
368
Step 3. Step 11: Terminate the GA search and output the best solution.
22
369 370
371
Fig. 4 Flow Chart of Genetic algorithm Case study
372
The studied bus Route 56 in Chengdu (China), shown in Figure 5, is approximately
373
21-km long providing all-stop service at 34 stops in a heavy-demand corridor. The service
374
hours are segmented into 5 time periods that represent EA (Early Morning: 6:00 am ~
375
7:00am), AM (Peak Morning: 7:00 am ~ 9:00 am), MD (Middle Day: 9:00 am: ~ 5:00pm),
376
PM (Peak Afternoon: 5:00 pm ~ 7:00 pm) and NT (Night: 7:00 pm ~ 11:00 pm). The services
377
in the AM and PM periods were of concern to the agency. Therefore, the associated
378
passenger Origin-Destination (OD) demand during the AM and PM periods of weekdays was
379
collected by a 3-week survey and the results are presented in Figure 6.
23
Outbound
10
18
Inbound
380
Fig. 5 The Study Route 56 in Chengdu
381 600
AM
Passenger (pass/h)
400 200 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
200 StopID
400
382
Board Outbound Board Inbound
Alight Outbound Alight Inbound
SumBA Outbound SumBA Inbound
600 600
PM Passenger (pass/h)
400 200 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
200 400
383 384
600
StopID Board Outbound Board Inbound
Alight Outbound Alight Inbound
SumBA Outbound SumBA Inbound
Fig. 6 Hourly passenger demand distribution (AM and PM)
24
385
The values of model parameters were provided by the operating agency. The average
386
bus operating speeds in the AM and PM periods are 30 km/hr and 25 km/hr (excluding dwell
387
time and average delay at bus stop), respectively. A load factor of 0.9 was used to represent
388
an acceptable level of service, making the effective capacity of a bus 81 passengers. Based on
389
the survey at bus stops, the average delay of entering and leaving the stop in the AM and PM
390
periods was 50s and 55s, respectively. The values of other model parameters were presented
391
in Table 1. The optimization process was programmed using the MATLAB software. The
392
distance between adjacent stops, OD demand matrix and formulations were entered into the
393
program. Figure 7 shows the value of objective function versus number of iterations during
394
the optimized process. It is notable that the total cost reduced quickly at the beginning 100
395
iterations and then a converged result was found around 200 iterations.
396 397
398
Fig. 7 Total cost vs. iteration number (AM and PM) Original vs. optimal operations
399
The costs incurred by passengers and vehicles under the original and two scenario
400
operations (S-I: All-stop service only and S-II: integrated all-stop and stop-skipping service)
25
401
are summarized in Table 3 where the optimal service frequencies for both all-stop and stop-
402
skipping services in the AM and PM periods are included.
403
Original operation
404
Based on the original operation, the frequency of all-stop service is 22 and 26
405
buses/hr in the AM and PM periods, respectively. In the AM period, the associated total cost
406
is 5,483 $/hr, consisting of 4,421 and 1,062 $/hr user and operator costs, respectively. In the
407
PM period, the associated total cost is 6,938 $/hr, consisting of 5,575 $/hr user cost and 1,363
408
$/hr operator cost.
409
Scenario I (S-I): All-stop only service
410
The optimal frequency of all-stop service in the AM period is 21 buses/hr and
411
achieves a minimum cost of 5,445 $/hr, and 23 buses/hr in the PM period that achieves a
412
minimum cost of 6,808 $/hr. Compared to the original service, nearly 38 $/hr and 130 $/hr
413
can be saved in the AM and PM periods, respectively. The reduced cost is a result of a
414
reduced service frequency, which leads to a higher vehicle utilization rate and lower fleet size
415
and operator cost, albeit the slight increase in waiting cost.
416
417
418 419
26
Table 3 Optimal Solutions for Various Scenarios (AM and PM)
420
AM Parameters
PM
Units Original
S-I
S-II
Original
S-I
S-II
fAd
buses/hr
22
21
14
26
23
16
fEd
buses/hr
0
0
7
0
0
8
F
buses
71
68
60
91
81
75
CW
$/hr
206
216
377
210
237
423
CR
$/hr
0
0
78
0
0
93
CI
$/hr
4,215
4,215
3,810
5,365
5,365
4,891
CU
$/hr
4,421
4,431
4,265
5,575
5,602
5,407
CO
$/hr
1,062
1,014
890
1,363
1,206
1,123
TC
$/hr
5,483
5,445
5,155
6,938
6,808
6,530
421
Note: “frequency = 0” means no service
422
Scenario II (S-II): Integrated all-stop and stop-skipping service
423
The optimal service frequencies for all-stop and stop-skipping services are 14 and 7
424
vehicles per hour, respectively in the AM period and 16 and 8 vehicles per hour, respectively
425
in the PM period (Figure 8). For the outbound stop-skipping service in the AM period, stops
426
1, 2, 14, 15, 16, 17, 18 and 34 are served, while in the inbound direction, the served stops
427
include 1, 5, 6, 7, 8, 9, 10, 25, 28, 29, 30, 31, and 34. For the outbound PM period service,
428
stops 1, 6, 7, 10, 11, 15, 16, 21, 22, 23 and 34 are served by the stop-skipping service, while
429
in the inbound, the served stops include 1, 5, 7, 10, 14, 15, 25, 28, 29, 30, 33 and 34.
430
Compared to the original service, nearly 328 $/hr and 408 $/hr can be saved in the AM and
431
PM periods, respectively. Implementing integrated service may reduce the in-vehicle time,
432
vehicle running time per trip, and fleet size, albeit the slight increase in wait and transfer
27
433
times. It is worth noting that with integrated stop-skipping and all-stop service, bus average
434
speed increases by 32.1% in the AM period and 26.2% in the PM period.
435
Fig. 8 Total cost vs frequency (AM and PM)
436
437
438
439
Sensitivity analysis
In this section, the sensitivity of model parameters to some key attributes/variables is analyzed and discussed based on the data for the AM period.
440
The sensitivity of the user’s value of time (the same value for wait, transfer and in-
441
vehicle time) to the optimized service frequencies is illustrated in Figure 9-a. It was found
442
that increasing the user’s value of time leads to increasing frequencies for both all-stop and
443
stop-skipping services so that the users' wait and transfer costs can be reduced. However, if
444
the user’s value of time less than 6 $/pass-hr, the optimal frequency of stop-skipping service
445
increases faster than that of all-stop service, because the in-vehicle time of a passenger using
446
the stop-skipping service is shorter than that of a passenger using the all-stop service, which
447
induces more passengers to use the stop-skipping service. It was also found that if the user’s
28
value of time exceeds 7 $/pass-hr, the all-stop and stop-skipping service frequencies remain
449
constant due to the constraint of fleet size. 24
50000
20
Total Cost ($/hr)
60000
40000
16
30000
12
20000
8 Total Cost All-stop frequency Stop-skipping frequency
10000 0
10000
25
8000
20
6000
15
4000
10 Total Cost All-stop frequency Stop-skipping frequency
0 1
5
Frequency (buses/hr)
Fig. 9-a Minimized total cost and optimized frequency vs. value of time
2000
452 453
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 User's value of time ($/pass-hr)
Total Cost ($/hr)
450 451
4
Frequency(buses/hr)
448
0
2 3 4 5 6 7 8 9 10 11 12 13 14 15 User's value of wait and transfer time ($/pass-hr)
Fig. 9-b Minimized total cost and optimized frequency vs. value of wait/transfer time
454
The effects of the user’s value of wait and transfer time on the minimum total cost and
455
optimized frequencies are illustrated in Figure 9-b. In this analysis, the in-vehicle time
456
remains the same at 1.5 $/pass-hr and the users' values of wait and transfer times vary. It was
457
found that increasing the user’s value of wait and transfer time leads to increasing frequencies
458
for both all-stop and stop-skipping services so that the expensive wait cost can be reduced.
459
However, the rate of increase in stop-skipping service frequency is greater than that of the all-
29
460
stop service. If the user’s value of time exceeds 9 $/pass-hr, the optimal frequencies of the
461
all-stop and stop-skipping service remains constant because of the fleet size constraint.
462
The impact of the transfer penalty on transfer demand and optimized frequencies is
463
shown in Figure 10. It was found that increasing the transfer penalty results in decreasing
464
transfer demand. Hence, the frequency of the stop-skipping service is reduced, while the
465
frequency of the all-stop service increases. Total Transfer Pass All-stop frequency Stop-skipping frequency
466 467
Transfer Passengers (pass/hr)
960
25 20
720
15
480
10
240
5
0
Frequency (buses/hr)
1200
0 0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Transfer Penalty （min/transfer）
Fig. 10 Transfer passengers and optimized frequency vs. transfer penalty
468
Figure 11 indicates that as bus capacity increases, the optimized frequencies of all-
469
stop and stop-skipping services decrease and lead to a higher total cost. Increasing bus
470
capacity may reduce the service frequency and increases wait time. The frequency of all-stop
471
service decreases faster than that of the stop-skipping service. If the bus capacity is between
472
100 and 120 spaces/bus, the optimal frequency of the stop-skipping service remains constant
473
and the frequency of the all-stop service is reduced. The same variation of frequencies exists
474
between the stop-skipping and all-stop services when the bus capacity is between 130 and
30
476
operating cost increases due to the larger bus size. 6000
30
5000
25
4000
User Cost 20 Operator Cost Total Cost 15 All-stop frequency Stop-skipping frequency 10
3000 2000 1000
5
0
Frequency(buses/hr)
150 spaces/bus and more. However, the operator cost increases slightly as the average bus
Cost ($/hr)
475
0 50
60
70
477
80 90 100 110 120 130 140 150 Bus Cpacity (spc/bus)
478
Fig. 11 Costs and optimized frequency vs. bus capacity
479
The sensitivity of optimized service frequencies and costs on the bus operating cost is
480
presented in Figure 12. As the bus operating cost increases from 5 to 25 $/bus-hr, the
481
optimized stop-skipping service frequencies decrease. However, the all service frequencies
482
remain the same when the bus operating cost is over 25 $/bus-hr due to the capacity
483
constraint. User Cost Total Cost Stop-skipping frequency
7200
Operator Cost All-stop frequency
16 12
3600
8
1800
4
Cost ($/hr)
5400
0
0 5
484 485
20
10
15 20 25 30 35 40 Bus Operating Cost ($/bus-hr)
45
50
Fig. 12 Cost and frequency vs. average bus operating cost
31
Frequency (buses/hr)
9000
486
Conclusions
487
Considering a temporally and directionally variable demand, this study developed a
488
mathematical model to optimize integrated bus services (all-stop and stop-skipping) and
489
determined the associated service frequencies which minimize the total cost subject to
490
capacity and fleet size constraints. The study problem is a combinatorial optimization
491
problem. Because of the number of stops on the route, a large number of stop combinations
492
served by stop-skipping service are candidates for implementation. To search for an optimal
493
solution quicker, a genetic algorithm (GA) was developed. As demonstrated in the case study,
494
the proposed method offers a practical approach to design an efficient transit service, which
495
improves system performance and generates savings for Route 56 in Chengdu, China. In this
496
study, the modeling approach proposed is very flexible, and can be utilized to optimize a
497
generalized transit route when the OD demand and the route/stop locations are available.
498
Transit agencies may easily adopt the developed model and solution algorithm with minor
499
modification to estimate user and operator costs.
500
501
A sensitivity analysis investigated the impacts of changes in some key model attributes/variables and generated the following insights:
502
1. Compared with the conventional pure all-stop service, stop-skipping service is an
503
effective strategy to improve transit service quality and operating efficiency. User’s in-
504
vehicle cost and operator cost can be reduced, although the waiting cost may slightly increase.
32
505
Because of increased travel speed, the benefit of reduced in-vehicle time and fleet size is
506
sufficient to compensate for the increased waiting and transfer costs.
507
2. When all components of the user’s value of time or only the value of wait and
508
transfer time are increased, operator cost increases because the frequencies for both bus
509
services increase in to reduce the expensive wait and transfer user costs. It was found that
510
increasing the transfer penalty leads to decreased transfer demand and increased total cost.
511
512
3. As bus capacity increases, the optimized service frequencies decrease, leading to a higher user as well as operator cost due to the higher average operating cost of bigger buses.
513
4. Increasing bus operating costs tend to decrease frequencies of both all-stop and
514
stop-skipping services. However, if the operator cost continuously increases, the optimal
515
frequencies of all-stop and stop-skipping services and user cost remain constant due to the
516
capacity constraint.
517
Since the highest demand section in both directions of the studied route is nearly
518
equal, in this case equal frequencies per direction were considered. Designing unequal
519
integrated service frequencies for each travel direction may be beneficial in cases where the
520
demand between the two directions is quite unbalanced, and this can be an immediate
521
extension of this study. Including other strategies such as deadheading should be explored
522
also to improve potentially the system’s performance. In addition, optimal dispatching
523
solutions of all-stop and stop-skipping services at the starting terminal could be developed to
33
524
coordinate transfer and analyze passenger waiting time in depth. Considering that the bus fare
525
may affect the passenger trip choice, the relation among the demand, passenger trip cost and
526
bus fare can be established as soon as the bus fare configuration information is collected,
527
which can be an immediate extension of this study.
528
Acknowledgements
529
This work is supported by Sichuan Province Key Laboratory of Comprehensive
530
Transportation (Project No.: B01B1203) and the Southwest Jiaotong University (Project No.:
531
SWJTU09BR141), P. R. China.
532
34
533
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