Scheduling and estimation techniques for transportation planning

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%?).