SCHEDULING AND ESTIMATION TECHNIQUES FOR TRANSPORTATION PLANNING LAWRENCE D. BODIN* University
of Maryland,
College
Park.
MD 20740. U.S.A
DONALD B. ROSENFIELD+ Arthur D. Little, Inc.,
Acorn
Park, Cambridge,
MA 01140.
USA
and ANDY S. Brookhaven
National
Laboratories,
KYDES~ Upton,
NY
11973. U.S.A
Abstract-The Urban Transportation Planning System (UTPS) is a computer system developed by the Urban Mass Transportation Administration for the purpose of assisting planners in the analysis of proposed mass transit systems. While UTPS provides many indicators of system performance, the estimates of operating cost have not been satisfactory. The authors have developed an alternative model for estimating the operating cost of mass transit systems. This model is designed primarily for road-based systems although both rail and road systems can be analyzed. Recognizing the relationship of operating costs with scheduling complexities, the authors also developed procedures to determine line schedules and vehicle schedules, and to estimate manpower requirements for the proposed transit system. These procedures, which will be integrated in future versions of UTPS, are the subject of this paper.
The Office of Planning (UMTA)
Methods and Support of the Urban Mass Transportation
has been developing
the Urban Transportation
Planning
years, The goal in the design of UTPS has been the availability allows planners long-range
to study and analyze
planning
environment.
multi-modal
Operating
mass transit system, UTPS provides
transportation
the number
of passengers
many indicators
on each line.
of a computer
of the performance
The system
the estimate of operating
OPERATING
COST =
cost. Until
to
of the proposed
of a proposed transit
specific zones in a transit region and is primarily
although both rail and road systems can be analyzed. A characteristic of the proposed transit system that is not provided estimate of the system operating
system that
systems in an intermediate
on a specific set of characteristics
system such as the number of passengers going between
Administration
System (UTPS) over several
the study described
for road-based
systems
in UTPS is an accurate
in this paper was carried out,
cost in UTPS was found by:
COST IN SERVICE
VEHICLE
MILE
x(N0.
OF IN SERVICE
MILES)
(I) *Lawrence 0. Bodin is a Professor in the College of Business and Management at the University of Maryland. He was formerly Associate Professor in the College of Urban and Policy Sciences at the State University of New York at Stony Brook. He is the author of numerous articles in scheduling and routing. tDonald B. Rosenfield is a consultant in Operations Research at Arthur D. Little, Inc. and a part-time lecturer at the Sloan School of Management at M.I.T. His areas of interest include production and inventory logistics. scheduling.and applications of probability. He holds a S.B. in Mathematics, S.M. in Operations Research and Electrical Engineer degree, all from M.I.T.. and a Ph.D in Operations Research from Stanford University. He was formerly a faculty member at the State University of New York at Stony Brook. He has authored several articles in probability applications and logistics. tAndy S. Kydes is Associate Mathematician and head of Energy Models Groupat the Brookhaven National Laboratory. He was formerly a visiting Assistant Professor at the College of Urban and Policy Sciences at the State University of New York at Stony Brook. He has authored several papers in the areas of energy and transportation.
16
L. D. BODW et al.
Models such as this are known as unit cost models, and relate levels of cost to levels of a causal factor where the causalfactor is some physical characteristic of the system (total vehicle miles. total vehicle hours, number of passengers, etc.). Some of these cost models partition the operating costs into categories (cost for operator wages, costs for supervisory personnel. costs for maintenance, costs for fuel, etc.) and then find the total operating cost by adding up the costs found in each category. Although such models may give accurate cost estimates in a particular situation, these models do not take into account enough of the significant characteristics (manpower and vehicle schedules) of the system in estimating system operating cost to be used as general cost models. For example, most cost models never utilize an accurate estimate for the number of operators in cost computations, and the wages paid to the operators make up a significant portion of the operating cost of any transportation system. As another example, maintenance costs, which can also be a critical cost, should be based on accurate estimates of numbers of vehicles as well as vehicle mileage. In order to develop accurate cost estimates, and in recognition of the fact thar cost depends on number of vehicles and number of crews, procedures were developed to determine the number of vehicles and to estimate the number of crews needed to service the proposed transit system. Estimates of crews and vehicles, of course, are also important to the planner in their own right. These procedures along with the cost model outlined in [41 have separately been coded to UTPS specifications and a significant portion will be implemented in future versions of UTPS. In this paper we describe the analytical techniques used in the development of the procedures. I. INTRODUCTION
A planner typically specifies a daily transit system and a pattern of transfer demands between lines in the system. Two of the problems that have to be addressed by the transit planner include: (a) What are the vehicle and manpower levels necessary to service the proposed transit system? (b) What is the estimate of operating cost for the proposed transit system? The procedures for solving the problems in (a) are developed in this paper. A new and unique cost model based on the solutions to (a) has been designed and is described in [4]. 3ecause of the problems with unit cost models, the authors developed methods for estimating the operating cost of transit systems based on: (I) Determining accurate estimates of crew and vehicle requirements over the day for the system. (2) Estimating operating cost based on the characteristics of the resulting vehicle scheduies and manpower estimates. We have found that as well as being accurate, this approach is fast computationally for long and intermediate range planning. Efficiency is very important because a planner may want to try alternative transit system designs while analyzing the needs of the region. Thus, for example. since weekly and seasonal variations can affect comparisons of different systems, the models described can be used to evaluate various workload scenarios in a long and intermediate range planning scheme. For operational planning, actual crew schedules are needed. In meeting our objectives, the research led naturally to four closely related componentsthe line scheduling component, the vehicle scheduling component, the crew estimation component, and the cost estimation component. The highlights of the first three components are found in this paper. The cost estimation component was described in [4] and is briefly described in this paper. Although some of the techniques have been previously developed, the integration of the techniques for transportation planning problems was a unique aspect of our research. Assume that a transit system is specified in terms of the stops along each line plus headways for each time period of the day. The line scheduling component was designed to convert these line specifications into a feasible timetable (or line schedule). A iine is the specification of a roufe and the times between stops on the route that a vehicle is to follow. A line can bz run many times during a time period (part of a day). A run on a line is one traversal of the line from the beginning to end by a vehicle. The headway of a line is the time between runs on a line. and
2-l
Schedulingand estimationtechniquec for transportation planning
is assumed
to be a constant
the determination components normally
in any time period.
of the line
are independent
different
schedules,
(The
while
of this assumption).
for the different
constant
the
vehicle
headway
assumption
scheduling
and crew
The headways
time periods
for any line in the system
over the day. In a line schedule.
the day when each run on each line is to begin and end is specified. for determining
vehicle
setting
line
up the
description
scheduling
service
schedules dules
schedules
and estimating
is the
of the line scheduling
The vehicle feasibly
schedules
part
Furthermore,
of
the
component.
the
of the Dilworth day
are
characteristics
Input
to the vehicle
(b) deadheading
time
between
of
total
the minimum The
chain
to the
of the
vehicle
schedules
scheduling
component
the end points
proposed which
system.
make
Input
up the workload
user. In the manpower specifications) procedure
consists
which
of the system,
estimation
component,
are
are sufficient
to cover
used to determine
Wagner[
the crew
5, a description
cost based on the vehicle of the implementation
schedules
size estimates
The line scheduling proposed
transportation
and crew
estimation
description
data is specified major
components.
Input
criterion
which
we call
rates between
line within
from
each
in scheduling,
total passenger
time also indicates
problems.
For example, the planner
can effect vehicle downtown
require
district
time
presented.
4.
operating
A description
over the day for all lines in the component
scheduling
consists
order),
of (1) a
the time to each of
on the line, and (2) a list of all
each pair of intersecting
Two
and, hence,
This lines
lines. This
starting require
the locations
times
in order
additional
gives
the
rise to a con-
such criteria The
minimization
of
pairs of lines may
Synchronization
of lines
a line from a suburb connecting to synchronize
crews
if
are not
of line synchronization
time between
in his system.
For example,
waiting
maintaining
are said to be synchronized
transfer
headways
criterion
considerations.
to the planner
the total
while
in time. Although
they are important
requirements.
is to minimize
to another
period.
of large passenger
earlier
line
point closely
the appropriate
and manpower
lines may
lines in the central
indications
to change
one
synchronization”.
on both lines reach the intersection transfer
of system
as input to the vehicle
the line schedule
in transferring “line
and
in Section
of the day.
the only ones involved
prompt
is needed
to the line scheduling
used in determining
of each
The
of Veinott
COMPONENT
stop of the line, and the headway
encounter
type
are determined.
an estimate
a line schedule
demand
by the
by crew
is described
is briefly
the
5.
SCHEDULING
determines
and the transfer
headway
sideration
estimates
The line schedule
for each time period
passengers
vehicles
system.
the origin
lines that intersect
constant
component
down
on procedures
of the lines (lists of stops on the lines in the appropriate
these stops from
The
and manpower
is
component
as determined
(broken
operators)
and
Output
scheduling
for the system
used to derive
estimation
in Section 3. requirements for
the vehicle
(based
cost
information.
specifications
telephone
issues is also given in Section
LINE
in the
sche-
component.
of (a) the line schedule,
size estimates
the workload
of the procedure
2. THE
time
crew
I31 and later by Segal[ I I] for scheduling
In Section
used
to
vehicle
Vehicle
estimation
is described work force
and (b) shift
these
procedure[6].
consists
runs from
in
The
necessary
in deriving
of the lines and garage
of (a) aggregated
used time.
of vehicles
used
manpower
actual vehicle schedules. The vehicle scheduling component The manpower estimation component estimates the
transfer
2. number
decomposition
as input
is the basis
objective
passenger
procedure
used
The
are
the times over
A line schedule
requirements.
is given in Section
describes
all runs in the line schedule.
is an adaptation
over
minimization
component
component
crew
is used in estimation
properly
with
to the
and vehicles.
The approach taken is heuristic. Although the problem can be formulated as a mixed integer program, full optimization would be too time-consuming with little extra usefulness for the planning problem. The heuristic, on the other hand, gives a line schedule with a reduced total transfer time, synchronizes the most important set of lines, and is computationally fast. For lines not synchronized optimally, finite waits are insured. We omit the fine details of the heuristic in this paper, with any heuristic, details,
the reader
there are different is referred
to [2].
variations
but note some of the key aspects.
for different
cases. For a full discussion
As
of the
28
L. D. BODWrl
ui
The heuristic is based on the UTPS assumption that headway5 are constant in each time period. The major part of the heuristic is the determination of schedules for each time period, Determination of the daily schedule is simply the combination of schedules for different time periods. This may involve adjustments such as additions or deletions of runs on the boundary between two time periods. These adjustments, which are not described in this paper, make up the interface procedure in [2j. For a given time period, the heuristic determines the starting times for the first run on each line in the period using the criterion of minimization of total passenger transfer time. Knowing the starting time for the first run on a line, the starting times for all runs on the line in the period are also known, since by assumption the headways on each line are constant during the period. For example, if line i has a headway of d; and fi + & is the starting time of the first run on line i, subsequent runs on this line start at times fi f 23:. ti + 34, etc. Thus, it is assumed that &hereare N lines with constant known headways Ai,. . . , AN and it is desired to determine the starting times ti,. . , , fN for the N lines that exist in the period. By changing any of the starting times t;, the time at which passengers reach intersection points changes. Therefore, the amount of time that passengers spend at the intersection points to transfer from one tine to another changes also. To derive the passenger waiting function for a given time period, assume that afl M components of transfer demand have been enumerated. A component of transfer demand consists of an origin line, a destination line; an intersection node, and a transfer demand rate. The transfer demand rate is a specification of a given number of passengers per unit time desiring to transfer from the origin line to destination line at the given intersection point. For example, three persons~minute may wish to transfer from line Four to line seven in the time period. The demand rate is determined from the load characteristics of the system by another program in UTPS. For the jth component, denote the origin line as O(j) and the destination line as d(j). Further, let the starting times for these two lines be tmj, and fdii) and the headways be Jwj, and Ad(j). Finally (see Fig. I) let 4 be the time required for the vehicle on the origin line to reach the given intersection point from the first stop on its run; and let Sj be the time that the vehicle on the destination line requires to reach the given intersection point from the first stop on its run. A passenger riding on the mth run on line O(j) will board a vehicle which leaves the origin node at time tMij+mhoci, for some integer value of m (m is known). This vehicle will reach the intersection node at time foci) -t m&j, f q = ft. Time fdf,)+ Khd
node before
It is assumed him;
this passenger
that the passenger
arrives.
rides the first vehicle
on the destination
line available
to
i.e. K is the smallest integer such that f? z f,. Thus, K is the smallest integer satisfying:
\
_________Origfn
node Desftnatbn ilne
Fig. 1. Passenger transfer.
Transfer
node
Schedulingand estimationtechniquesfor transportationplanning
19
Using the above as a base, a detailed mathematical analysis can be carried out to determine the total passenger waiting time during a time period. Under the assumption that the length of the time period is proportional to ail the Ai, then H is given by the following expression.
where Di = Demand per time period for transfer between route i and j, and g.c.f. (a. 6) = greatest common factor of Q and b. Therefore, the line scheduling problem for any time period consists of determining starting times t,, . . . , fN to minimize (2). The expression (2) consists of non-linear periodic terms with step increases and linear decreases, The heuristic developed for solving (2) simply orders the terms by the amplitudes of the steps in each term. Successive terms are minimized until ail starting times are determined. The idea behind the heuristic is the assignment of the most important terms in (2) to a basis for the solution and the insurance of finite waits for other transfer points. An advantage of such an approach is computational speed. If the length of the time period is not a multiple of ail the headways, the objective function is a slight variation of (2) and a modified heuristic has been developed to handle this. Details of this heuristic are in[2]. 3. THE
VEHICLE
SCHEDULING
COMPONENT
The objectives of the vehicle scheduling component are to determine the minimum number of vehicles needed to cover all the runs in the line schedule and to generate actual vehicle schedules for the fleet. The line schedule can be specified by the planner independent of the line scheduling component: hence, the vehicle scheduling and manpower estimation components can be run independently. The procedure used is an adaptation of the ~i~worth Chain Decomposition~&~ and is a special case of the problem described by Dantzig and Fuikerson in [5]. The chain decomposition operates on a network description H = [M, B] of the vehicle scheduling problem. (M is the set of nodes and B is the set of directed branches.) The ith node of H, miyrepresents the ith run from the line schedule. INI, the number of elements in h4, is equal to the number of runs in the line schedule. A branch &K= (mj, mj)eB and directed from mieM to m,&f is said to exist in H if it is possible for a single vehicle to service run j after completing run i. More specifically, let (i) dii = Start time of run j -end time of run i. (ii) Ai/ = crew layover time. The crew layover time is normally fixed in the crew’s contract. The layover time (which can be a function of the lengths of the runs and other factors) gives the crew scheduled relief time at the beginning or the end of each run. (iii) a = minimum time required between runs. A branch bK = (mi, mi)eB if (i) di/ 2 CY+ A;j; (ii) It is possible to deadhead from the last stop of run i to the first stop of run j in no more than d+ - Aii minutes. The vehicle scheduling problem then becomes the following: find the minimum number of chains which can cover ail of the nodes in the directed network H. The ordering of the nodes on a chain found by the Diiworth Chain Decomposition procedure gives the sequence of runs on a vehicle schedule. In the Dilworth procedure, the network H is converted into a bipartite network G = [S, T, A] where S = M and 7’ = M. In addition, for every fmi, mi) E B define branch (xi, yi) E A where XiE S and yi E T. Each node in S is connected to a super source and each node in T is connected to a super sink. The capacities on all branches connected to the super source or the super sink is one. A maximum flow problem is solved on the network G. CAOR Vol. 8. No, L-C
30
L. D. BODIK et al.
Let
f(x, y) be the flow
on branch
(x, .v), x E S, y E
in the optimal
T
solution.
Let
C =
u(x,
y) E A: f(x. y) = 1). Th e b ranches in C are used to determine the minimum number of chains in H.By the Konig-Egervary theorem [6], the minimum number of chains in the network
H
equals
IS/ -ICI.
After
the maximum
vehicle
schedules.
node T. (A vehicle (S;, q)
from
branch
found
procedure
schedule
in C equals
continues
the branches
in C where
is to begin with the run represented (Sj,
the index
in which
of indices
are found
in this manner
As an example, and 1-4).
Tk) E C.Note
the runs
corresponding
T,) is found
associated
until all branches
network
with
nodes
T2), (S2,
modifications can have
G in the Dilworth
procedure
flow solution
T2), (S2, T5), (S3,
computer
preaggregation
stage,
partial
specified
parameter In this way,
time
If
dij -A,
B satisfies BP,
orderly
S,.
form
H
Additional
a
chains
of chains source
is two (e.g.
and super
sink
in Fig. 4.
and
H
then
service
are
hundreds
of thousands
produce
stages;
In a partial
prepares
input
stay with their
since as few crews
a
vehicle
(/3 is a user
for
vehicles
of
a useable
up into two
formed.
often
a small transit
by more than p minutes
crews
can be simulated
problem
Even
therefore
was broken
preaggregation
of tasks for which
stage works the
The
and
schedules
are separated
to a practical
procedure.
can have
storage
T4) or runs 3 and 4.
manpower as much as
as possible
change
day.
conditions
is then utilized
vehicle
which
as 20 minutes). clusters
their work
The preaggregation
procedure
such
and produces
in
super
procedure
of runs and the network
to reduce
no two runs are connected
branch
in C. This
T4).
and this was the case with the Dilworth
schedule,
during
number
with
T5) or runs I, 2, and 5. The second chain is (S3,
thousands
In order
possible.
schedule.
is displayed
procedure for planning purposes, the Dilworth procedure preaggregation stage and a final vehicle scheduling stage.
vehicles
found
in the set C are covered.
The number of chains is (S( -ICI = 5 - 3 = 2. As with many applications, the use of a theoretical
estimation
branch
in C are the following:
One chain is (Sl,
In the
branch
node on the first
m;.mj,mk...mp.m, in vehicle
in Fig. 2. The minimum
C = (St,
branches.
j on the f
such that there is no flow out of branch
to tasks for a feasible
let H be as denoted
The
The branches
system
by node S;). Eliminate
the index
the
so that there is no flow into
j on the S node on the second (S,,
is given in Fig. 3. The maximum
involves
in C are used to determine
S; is found
of branches
a chain
added,
is found.
a branch
until a branch
sequence
l-2-5
solution
7;) be
C and find the branch
The sequence
define
flow
Let (Si.
same
as follows.
A network
(i) and (ii) mentioned crew
may
not
to find the minimum
H = [M, B] is created
before
service number
plus the condition
both
run
of chains
M.
Fig. 2. Example of task network
H.
where
i and run j. The needed
every
that dij - Aij 5 p.
to cover
Dilworth
the nodes in
Scheduling and estimation techniques for transportation
Fig. 3. Corresponding In the second
together schedules locations
without found
stage, the aggregated any constraints
bipartite network G with unit capacities
runs from
on the maximum
in this stage are augmented
for each vehicle
are chosen
planning
the first vehicle time
between
with deadheading
to minimize
deadhead
scheduling
stage are joined
runs (B = 1440).
to and from
The
vehicle
the garages.
Garage
time for each vehicle.
A vehicle
is
cc@\-__: ,’
0s,
Fig. 4. Maximum
solution of unit flows.
Sink
32
L. D. Boom et al.
assigned to return to his garage during the day if the time between runs on its schedule becomes greater than a specified parameter. Because of dimension and size problems, it is often advantageous to split the problem into two stages. For example, if a transit system consists of more than 1000runs, the runs can be ordered with respect to starting times and can be processed in sequence in batches of 1000runs in the first stage of the procedure. Stage 2 can then be carried out over the final collection of partial vehicle schedules. The approach taken in the vehicle scheduling component results in the type of task clustering planners would like to see in their schedules. The two stages in the vehicle scheduling component also allow for larger vehicle scheduling problems to be solved. To test this procedure, a 650 run (or node) system was broken down as follows: Stage I: Solve a 650 node chain decomposition problem with a limited number of adjacencies. Stage 2: Solve a 75 node decomposition problem. Although there is a very minor approximation to the overall optimum, using the two stage approach in contrast to solving one 650 node problem involves considerably less time and computer storage. The final vehicle schedules for a 650 run transit system were found in about 24 minutes on a 32 K PDP IO computer (equivalent to l/2 minute on the IBM 370/155). This particular example was for a mid-western city which utilizes 51 vehicles for 25 lines and 650 runs. The procedure in this paper required 49 vehicles for the same line structure. This comparison illustrates that this approach can derive a reasonable estimate of the fleet size along with realistic vehicle schedules. 4. THE
MANPOWER
ESTIMATION
COMPONENT
The manpower estimation component estimates the number of crews necessary to cover all of the required runs in the line scheduling component. The file which is input to the manpower estimation component consists of the partial vehicle schedules determined in the first stage of the vehicle scheduling component. This file lists the sets of runs which have been aggregated together in order to be serviced by a crew. Furthermore, each partial vehicle schedule is updated in the vehicle scheduling component to take into account the times that crews spend in going to and from the garage. The parameter /? in the vehicle scheduling component also controls the manpower estimation component. If a delay between tasks is greater than /3, then the tasks are not aggregated in stage one of the vehicle scheduling component and the delay does not have to be covered by a crew. The manpower estimation component is based on a procedure based on a generalized capacity planning problem of Veinott and Wagner[ 131and later used by Segal[l I] for scheduling telephone operators. This procedure estimates the number of crews necessary to cover the workload as specified by the partial vehicle schedules created in the first stage of the vehicle scheduling component. This approach does not give feasible work schedules but does yield an accurate estimate for the number of crews required. It furthermore gives a breakdown on the types of workers (full-time, split shift and trippers) and starting and ending times for each of the scheduled shift segments. The first step in the procedure is to form a histogram (called the demand histogram) of the number of crews required to cover all the partial vehicle schedules. To construct this histogram from the input to the file described above, the time of day is broken down into ten minute intervals (although this could be varied) where it is assumed that any run falling into any part of these smal! time intervals requires a crew to service it for the entire time interval. For example, a particular schedule starting at 7:09 generates a crew requirement from 7:OOto 7:lO. Experimental evidence indicates that the choice of a 10 minute time interval over the day is a reasonable parameter setting for deriving accurate crew size estimates. A typical demand histogram would look like the one in Fig. 5. The problem is now converted into a network G = [M, B] that can be solved using a minimum cost flow problem. Let L be the length of the time interval used in the demand histogram and let the number of daily time intervals P = 1400/L (10 and 144 respectively in the current software system). Two cases exist, one when the line schedule is non-cyclical (under 24
Scheduling and estimation techniques for transportation
33
planning
-
Time of doy
Fig. 5. Typical demand histogram.
hours)
and the other
case, it is assumed
when the line schedule
that there is a portion
any lines are scheduled service
to ofirate.
is cyclical
In this case,
runs on both sides of this portion
non-cyclical
case by cutting
24 hour demand
histogram
day in question following The
the demand evolves
(24 hours duration).
of the day (generally
it is assumed
of the day. The
histogram
after
that
cyclical
at midnight.
In the non-cyclical
midnight)
when
the same
will
case is converted
Therefore,
a slightly
since trips from the garage can begin before
and some runs concluding
no runs on
crew
not
into the
longer than
midnight
of the
trips to the garage can be ended after midnight
of the
day. forward
branches
in G correspond
to the
work
requirements
in the
various
time
intervals. Specifically, for 1 5 n I p - 1, branches going from node n to node n + 1 have a lower bound on flow equal to the number of crews required corresponding to node n. These requirements branches
are found
is infinite
from
that there must be an amount The reverse flow
branches
which
a crew
for example,
accounted
for in the partial
vehicle
specification
from 9:OO AM assumes
works
two
shift
shift
530
1:OOPM and one from 530 to 9:OOPM. from combinations of shorter shifts.
segments.
(These
provide
is a set of consecutive
breaks
(layover
for long shift
by the following
the manpower segments.
to 930
segments
on these
branches
is
and
lunch
the time
times
are
segments
can be
length
of shift
input to this procedure.
on which shift
on flow
breaks
characteristics:
crew cost (including differential report to work on this shift
of two shorter
bound
idea of the forward
segment
without
schedules
to 130 PM and from
the crew
shift
that lunch
is designated
the alternatives
shifts are combinations
to work
upper
The
to the allowable
it is assumed
segment (8 hours, 4 hours, etc.), times of day when crews can form
The
the workload.
An allowable
is assumed
Thus,
specifications
histogram.
of flow to “cover”
branches.)
included).
.4 shift segment
demand
in G correspond
for the requirement
intervals
the
and the cost per unit of flow is zero.
PM, each
Thus,
and overtime), segment. The
estimation if a crew
first and last shift segment
is to be based. is to work
Split
a split shift
then the manpower
estimation
component
of 4 hours
one from
9:OO AM
It is assumed
length;
that feasible
Assume a shift segment (of k time intervals duration) n and ends at the beginning of time interval n + k. Then
to
split shifts can be derived
starts at the beginning of time interval a branch is drawn in the network from
node n + k to node n. The lower bound on flow on this branch is 0, the upper bound on flow on this branch is x and the cost/unit flow equals the cost for one crew working this shift segment. All potential manner.
shift
segment
specifications
are represented
as branches
in the network
in this
34
L. D. BODINet al.
After all branches are represented in G, a minimum cost flow problem is solved using the out-of-kilter algorithm[6]. In this solution, the flow from node n to node n + 1 represents the number of crews available in time interval n. The number of crews of shift segment duration k starting in time interval n is represented by the flow on the branch from node n f k to node n. As an example. suppose that one hour time intervals are stipulated and that the demand histogram is given in Fig. 6. Suppose that the two allowable work shift specifications are 6 hours at $35 and 3 hours at $20. Then the network requires a lower bound on flow of 4 from the first to second nodes (8:00-9:00), second to third nodes and sixth to seventh nodes. Branches (3,4), (4,5) and (5,6) have a lower bound on flow of 6. Shifts at a cost of $20 are branches reversed from node n + 3 to node n, 4 5 n 17 (e .g. node 7-4) and the one possible reverse branch for six-hour shifts goes from node 7 to node I. The entire network is illustrated in Fig. 7. The (_Y,u/c) notation on the branches denotes a lower bound of flow 2, and upper bound of flow u and a cost/unit flow c. Node I represents a time interval beginning at 8:00, node 2 represents a time interval beginning a 9:00, etc. The optimal flow is illustrated in Fig. 8. The solution stipulates that 4 full-timers are needed who start
their
day at time
(node 3). The user can specify particular number. crews
time
(node
a certain
of day by setting
I) and 2 trippers
number
of crews
the lower
bound
It has been
found
derives
demand
histogram.
number
of minutes
of shift
segments
maximum
at 8:00 AM,
to work
overtime
to half the length
has a length
maximum
overtime
If only full-time specify
should
10 minute
estimates
in an assumed
segments
planner
with
has a length
equal
allowance crews
utilize
full-time
equal
a great
work
shift
reverse
a minimum
at IO:00
segment branch
number
to half
at a
to this
of full-time
(e.g. 20-30
are allowed
deal of flexibility
of shift
to cover
workday
a second
class
plus an assumed
a third class of shift segments and the fourth
of the full-time
the
in length equal to the
(e.g. 8 hours);
class of default
workday
has a shift
plus an assumed
minutes).
and the same cost/crew
of shift
test cases indicate
types
needed
of the crew
workday
the length
use of four of crews
of the full-time
to I hour);
of the full-time
the
number
has a shift segment
workday
(e.g. 40 minutes
two classes
intervals,
to the length
segments
should
similar
Similarly,
applies
if only full-time
solution
the types
above
but
crews (no overtime)
of day
Fig. 6. Demand histogram for example
improve
model are utilized.
of alternative
could be allowed.
Time
the day, then the
does not greatly
of the type in the default
in simulating
over
to the first two cited
be utilized.
that the derived
more than four classes of shift segments however,
time
shift segments
then a single shift segment
Runs on several
a particular
require
for the minimum
his own cost and length of shift segment.
are permitted,
who start
4:00 PM and midnight.
One class of crew
allowable
equal
that accurate
are needed
of the appropriate
This situation could occur, e.g. if regulations to begin work
specifications
length
890
shift
when
There is,
segments
that
Scheduling and estimation techniques for transportation
IO, w/201
to,-/20)
IO,-120)
Fig. 7. Network
As an example
of the results obtained
planning
Q-/201
for example
for the manpower
estimation
component,
we tested a
transit system mentioned before with approximately 25 lines, 650 runs, 51 vehicles, and 69 drivers. We obtained the sets of results shown in Table I for the manpower estimation component. intervals
In the table, of work)
length of shift, As noted approximation timetable
the results
turned
if this timetable
daily work
be covered specifies
full-time
crews
is either
530
out to be relatively
were adopted
the runs from
schedule).
by a worker,
only amounts
gives an accurate
time
a full-time
intervals
insensitive
Each backward but there
of work
estimate
for use. Operationally,
the original
crew
of work).
(>30
Except
to the parameter
however,
which
flow in the solution
is no specification
of the number
of time intervals
timetable
and not specific
The above results were found (equivalent to l/2 minute on the IBM number
crew
(each
time for the
settings.
previously, the solution found in the manpower estimation component is an to the optimal crew configuration which would be required to service the
not give the planner actual
an effective
or two part-time
of crews
found will
is to service
represents
a portion
of which
work.
task breakdowns.
in about 370/155).
the solution
each crew
The
input
Nevertheless,
and is also computationally
(i.e. an
of work to histogram
the procedure efficient.
24 minutes on a PDPIO 32K computer The efficiency is primarily dependent on the
used and not on the number
of lines in the transit
system.
5. IMPLEMENTATION
The importance use by planners, UTPS.
These
This paper
model
has already Perhaps
directly
the
described
vehicle
scheduling
analyzed
some
directed
toward
sample
how the procedures
developed
scheduling
this analysis
based
against
estimation
existing
compare
cost
The authors
that the authors
estimation were
model envision
of
of UTPS. with operational
based on the components
systems
components
on the existing
and the ways
of UTPS.
and manpower
for possible
to the specifications
versions
issue is how a cost model
with the cost model
manpower
results
is its development
the future
has not been calibrated and
in this paper
each of the procedures within
a more important
uses the vehicle
though the cost model
coded
will be integrated
in this paper compares
that
described
computer
coded procedures
characteristics. described
of the methodology The authors
also coded
a cost
components.
in the same manner calibrated,
of UTPS.
This
Althat
the authors section
is
the usage of the system.
The UTPS cost model was described by equation (1). Note that the equation is only an approximate estimate of system costs. Historically, of course, system operating costs (as well
Fig. 8. Optimal flow for example.
L. D. Ehmn ef al.
36
Table1. case 1
2
3
4
5
6
of
Length of Shift ---
Cost of Shift
Drivers
48
40
59
24
20
30
48
40
35
52
48
9
24
:0
37
26
24
11
48
40
33
<9
42
6
50
44
0
51
46
0
52
48
7
24
20
27
25
22
9
26
24
7
NO.
44
40
41
48
46
5
22
20
26
24
23
30
48
40
49
52
r6
9
21
21
10
21
23
16
48
40
36
SZ
46
11
24
21
29
26
24
12
Effective Full-Time
Crew
Size
74
68
67-l/2
14
71
67-l/2
as capital costs) increase as in-service vehicle miles increase. Current systems vary considerably in cost per vehicle mile, however. In a survey conducted by the authors[2, Appendix F], Jacksonville, New York, Minneapolis and Cleveland, for example, varied in cost between $1.05 and $2.70 per vehicle mile in 1975. (Fixed guideway systems have a great deal more variability.) Although a planner might be able to account for local factors, there is still a large range over which the estimate might be in error and this error will be compounded over the length of the planning period. In addition, estimates for capital investments for I I cities[2, Appendix F] range between one vehicle per 20 thousand and one vehicle per 42 thousand vehicle miles. With a new cost model based on the outputs of the vehicle scheduling and manpower estimation components, the accuracies are greatly improved. Capital costs are directly proportional to the number of estimated vehicles and operating costs are broken down into fifteen categories. Each category is either a FARE category, and aggregation of FARE categories, or a part of a FARE category. The FARE system, which was developed by Arthur Anderson & Company[l], will be required to become the standard system for accounting and evaluation of all U.S. transit systems receiving federal aid. The fifteen cost categories in our model were designed to be familiar to transportation planners. These I5 categories are: (1) Operator salaries. (2) Fringe benefits and other salaries for revenue vehicle operators. (3) Fuel, lubricants, and power, including fuel taxes for revenue vehicles. (4) Tires and tubes for revenue vehicles. (5) Leases and licensing of revenue vehicles. (6) Transportation operations.
Scheduling and estimation techniques for transportation
(7) Servicing
revenue
vehicles.
(8) Inspection
and maintenance
(9) Vandalism
repairs
(10)
Fuel, service,
(I 1) Ticketing (12) (14)
of revenue
of revenue
inspection
and maintenance including
and maintenance
maintenance
Scheduling
and general
of power
of workers
0 0
Number Number
of peak vehicles. of vehicle miles broken
Specifically, Maintenance these
operator
are
on total
miles
costs can be based follows
Given
and other
improve
accuracy
only
on number
directly
the accuracies
on lengths (not
from
the vehicle
presented
variables within
but output
from
scheduling
for the latter
and available and causative
costs, however,
It is important
developing the transit loadings. on the Additional
factors,
scheduling
of
estimation
cost accuracy
depending
component
should be based on future
work
force
and a summary
a unique
aspect
is
on vehicle
should
greatly
costs of an operator,
of the improvement
in the system.
other advantages such as costs per changes, etc. These attributes have
statistics,
of system
of a planning
involving
and experiments
described
various
operating
characteristics.
package.
these models
experimentation
He would then develop of
lines
estimation variations continual
and
components would
in developing
costs broken
of these reports
can be used by the planner. crew
down
by
The plots for vehicles
Copies
input scenarios.
a set of possible
variations
headways.
each
would
result from
experimentation
tions can be performed Table 1. The interactive prove
and manpower
This process
size estimates
are in
Planning
is similar earlier.
to
After
the various inputs required by UTPS such as demand patterns and a description of network, the user would execute a battery of programs to determine the network
numbers
manpower involve
trials
stratified.
Determination
involve several reports. These include summary in service and deadheading, numbers of workers
to note the ways in which
continual
and leaving.
speed
two components,
give an indication
by type, summary
and manpower represent Refs. [2] and (31.
are
miles.
may still be variations
the vehicle
The coded versions of the procedures statistics on waiting time, plots of vehicles required
and
UTPS.
Present
category
miles)
and vehicle
In addition to more improved costs, the system provides category, the ability to measure the sensitivity of workrule not yet been available to transportation planners.
the various
one through
in our cost model for
of shift and time of reporting in-service
of vehicles
It should be noted that the true comparison fuel and vehicles.
involves
will be categories
down by speed.
depend
greatly improved over the figures noted above. For the particular case of maintenance, there types
system
or characteristics
by type.
salaries
based
characteristics
components.
in any transit causal factors
include:
Number
costs
facilities. administration.
cost categories
0
Fuel
vehicles. of equipment.
administration.
four plus seven and eight. The underlying these categories
of service
maintenance
and maintenance
(15) General function. The dominant operating
vehicles.
vehicles.
and fare collection
Operation
(13) Other
37
planning
For
be executed
various
of the base network
variation,
the
and system
crew size assumptions.
and evaluation.
vehicle
with variations scheduling
costs would
and
be evaluated.
The process
would
thus
If the data bases are on file, these evalua-
interactively. The process is illustrated by the variations scenario used in a long and intermediate range planning
presented in mode should
most effective.
Acknowledgements-The authors wish to express their gratitude to Dr. Robert B. Dial, Director of the Office of Planning Methods and Support of the Urban Mass Transportation Administration, and his staff for their direction and assistance, Dr. Adelbert Roark for his technical advice and guidance, Paul Dempsey for his computer programming, and Jonh Bennett of Peat, Marwick and Mitchell and Mike James for their advice on cost model development, This research was sponsored under Grant NO. NY-I I-0012 from the Urban Mass Transportation Administration. REFERENCES I. Arthur Anderson and Co., Reporting System Instrucfions. Vol. II of Project FARE Task IV Report (1973). 2. L. Bodin and D. Rosenfield, Estimation of the operating cost of mass transit systems. Rep. No. WAHCUPS-&WA-I-
38
3. 4. 5. 6. 7. 8. 9. IO. It.
I!. 13.
L. D. BODIH et al. 76, College of Urban and Policy Sciences, State University of New York, Stony Brook, New York (1976). (Available through the National Technical Information Service (NTIS)). L. Bodin. D. Rosenfield and A. Kydes, UCOST: A planning, scheduling and costing system. 1. L’rban Analysis 5, 47-69 (1978). L. Bodin. D. Rosenfield. A. Kydes and A. Roark, Operating cost model for transit based on direct systems characteristics. Transpn Res. Rec. 654, 38-30 (1977). G. Dantzig and D. R. Fulkerson. Minimizing the number of tankers to meet a fixed schedule. Vacal Res. Logistics Q. I. 217-Z! (1954). L. R. Ford, Jr. and D. R. Fulkerson. I%H~S in Networks. Princeton University Press. Princeton, New Jersey (I%?). W. Gavin and A. L. Roark. WMAl’X Bus Operating COSI Model. Memorandum Rep. NO. 10. Subtask 2.b.4, Transit Technical Studies. Wilbur Smith and Associates, Washington, DC. (1974). W. C. Gilman and Company and Allen Voorhees and Associates. Revenues and Operating Costs, prepared for Washington Metropolitan Transit Authority (1971). J. H. Miller and J. C. Rea. A Comparison of Cost Models for Urban Transit. Pennsylvania Transportation and Traffic Safety Center. (1973). R. P. Roess. Operating cost models for urban public transportation and their USC in analysis. Transpn Res. Rec. 490. Transportation Research Board (1974). M. Segal. The operator-scheduling problem: A newwork flow approach. Ops Res. 12. pp. 808-823 (1974). Urban Mass Transportation Administration, llTPS Reference &fanva/. Urban Mass Transportation Administration, Washington. D.C. (1975). A. Veinott and H. Wagner, Optimum capacity scheduling, I and II. Ops Res. lo(4) (I%?).