Operator workforce planning in the transit industry

0191-2607187$3X0+ .GU 0 19S7Pergamon Journals Ltd.

Trmpn. Res:A Vol. 21A. No. 2, Pp. 127-138,1987 PrInted,n Great Bnta~n.

OPERATOR

WORKFORCE PLANNING TRANSIT INDUSTRY

HARILAOS

N. Kou~~o~ou~os

Center for Transportation

IN THE

and NIGEL H. M. WILSON

Studies, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.

Abstract-Workforce planning in the transit industry has received increased attention in the last few years as management emphasis has shifted from expanding transit service to increasing service efficiency and effectiveness. This paper builds upon recent work on the optimal level of reserve personnel and extends the analysis approach to incorporate the probabilistic aspects of absenteeism by day of week and season, workforce attrition, and extra, non-scheduled work such as new operator on-street training and special events requiring extra service. It presents an integrated framework for operator workforce planning which includes three interrelated models: strategic, tactical and operational. A case study of workforce planning at the Massachusetts Bay Transportation Authority is used to demonstrate the potential of the general approach proposed in the paper.

force

INTRODUCI’ION Operator

workforce

of determining ator staffing

Still

today

has been most

operator

focused

planning transit

ad hoc staffing

In this paper,

in the transit

agencies

methods

framework

planning

is presented. to assist

both

number

of operators

erators

so that service

methodology

chastic work

nature

on

to determine

incorporates

the most

problem,

including

of both

operator

availability

to be performed,

and typical

cost. sto-

rule con-

paper

operator

is organized

workforce

in some sented

detail. including

tational

as follows.

planning

Then

problem

the various

the solution

requirements.

models

a case

the potential

The

operator

workforce

transit

industry

consists

defining

a hiring

program

of the following including

is pre-

operators

to be hired in each hiring cycle;

vacations

so that

the appropriate

available

throughout

so that

the appropriate

the year;

on each day of the week; during

each workday

changes

in service

allocating

in the

ing vacation

and

MacDorman,

1982).

workdays

without

work times

so that the appropriate

parent

work-

portant

127

an agency’s

at around

holidays

is

abil-

Absenteeism

compared

with most 10%

av-

exclud-

(MacDorman

and

In some cases the absolute

follows

predictable

patterns

Some

absences

of

of the year,

variation

level after accounting

day in advance,

prior

work

times the national

is also considerable

3. Deficiencies

service warning

required.

by day of week and season

variation.

with the in-

little

service.

to three running

one

major

minor

impairs

scheduled

erage-often

least

occur with

are high in transit

poral

size is

actual

While

and unscheduled

absenteeism

the expected

scheduling

generally occur

hirplans

before

are known.

can

of sched-

manpower

developed

the year

but there of

by the ser-

of lengthy

(but unpredictably)

variation

size is available

and allocating

planning

of a new timetable,

industries-two

decisions:

workforce

workforce

requirements

levels

from

the number

workforce

because

be

frequently

service

level of absence problem

be revised

the amount

must

ity to operate

DESCRIPTION planning

may

are determined

procedures,

2. Operator

the use of the methodology.

PROBLEM

the

However,

throughout

are pre-

study

va-

workdays

characteristics:

ing and training

frequently

and compubenefits

requirements

modifications

the

is described

methods

Finally,

sented which demonstrates

Initially

industry,

troduction

straints. The

transit

has the following

uled service.

and the

work

a long

rate),

annually,

times

vice plan which specifies

op-

the

will have

on the attrition

and work

se-

in the sequence.

decision

may be revised

quarterly

1. Work

important

of the

schedules

In the

within

at minimum

(based

implication

problem

the appropriate

is delivered

term

are made

of time commit-

daily.

A set of meth-

and to assign the available

decisions length

with each decision a hiring

revised

for oper-

managers

these

with a decreasing

So for example, cation

levels.

developed

characteristics

industry.

at each time of day. Typically

industry

ment associated

treat-

rely principally

ods has been

The

oper-

to set the appropriate

an integrated

agencies,

quentially

increas-

on the analytic

ator workforce transit

transit

Over the past few years,

of workforce

traditional

deals with the problem

the most cost-effective levels.

ing attention ment

planning

size is available

in the transit

around

for the temare known

while

others

at

occur

notice. in workforce

in reduced indicator

service

planning

become

reliability.

of the quality

An

of transit

apimser-

128

H. N. KOUTSOPOULOS and N. H. M. WUON vice, which is directly related to the workforce size, is the number of missed trips within a time period. Transit agencies are quite sensitive to this measure and only a very small percentage of missed trips is tolerable.

Given the importance of service reliability and the uncertainty about both the manpower available and the amount of work to be performed on a given day, transit agencies employ more operators than those actually scheduled for work. These extra operators are usually referred to as extraboard or cover operators, who, as with regular operators, may be either full- or part-time employees. Some of the extraboard uses may be known well in advance (for example, trippers and sporting events), others may be known only a day or so in advance, while still others may be completely unanticipated in terms of timing and location (for example, breakdowns and accidents). Some have a known pattern by week of timetable and season of the year so, for example, absences may be higher in summer because of good weather and in winter because of sickness than in spring and fall, and training of new operators may be concentrated in certain weeks of the timetable (Perin, 1984). Other extraboard needs will occur randomly. Despite the existence of extraboard operators, occasionally the assignments to be performed will exceed the available operators leading to overtime work and possibly resulting in missed trips. Total operating costs attributable directly to operator absenteeism and uncertain service requirements are estimated at $95 million annually nationwide with approximately 80% due to employing extra operators and 20% due to overtime (MacDorman et al., 1982, 1984). It is obvious that the determination of the most cost effective size of the extraboard force is a very important element of operator workforce planning. A large extraboard will result in low productivity since some operators who do not have any work to perform will still be paid for full days’ work based on typical labor contract provisions. On the other hand, a low number implies that a lot of overtimeat high marginal cost-has to be used. Moreover, there may be important side effects of too small an extraboard because high levels of overtime may cause even higher levels of absenteeism among operators. The optimal extraboard size should minimize the total expected costs of both overtime and extraboard compensation subject to a constraint on service reliability. Work rules and labor agreement provisions will strongly affect the optimal extraboard size (Chomitz and Lave, 1981). For example, if overtime premiums are small, then the extraboard should also be small because of the low cost of using more overtime. On the other hand, if unused extraboard operators are guaranteed less than a full day’s pay, then the minimum cost solution will shift towards more extraboard employees. Furthermore, work rules and la-

bor agreements usually restrict the number of parttime operators and the number of hours worked per day. Finally, in most agencies, special work rules exist which constrain the type of assignments performed by extraboard operators. Because of the patterns of absenteeism and work to be performed and the various uses of the extraboard, the tactical and operational management of the extra operators are other very important aspects of workforce planning. PRIOR

WORK

Despite the importance, and probably because of the complexity, of workforce planning in transit, until recently little research had been devoted to the topic, and practice in the industry could fairly be characterized as ad hoc and judgemental, with little benefit of analytic methods. Within the transit industry there is great diversity with respect to hiring, vacation scheduling, and extraboard practices, often with tradition and inertia seeming to play the strongest roles in determining practice within an agency. Perhaps the most common method to determine workforce size is a “rule of thumb” formula which relates the number of operator assignments to fulltime operator requirements (Porter etal., 1984). The number of operators is then determined by the number of assignments multiplied by this empirically determined factor. This factor theoretically reflects the effects of absenteeism, uncertainty in service requirements, attrition rates, promotions and vacation schedules. Not surprisingly, operating costs are very sensitive to the value of this factor, which is not derived analytically (Perin, 1984). The management of the available workforce is governed largely by judgement and experience with little analytical support (Perry and Long, 1984). While considerable literature exists on workforce planning in the manufacturing and production industries (McClain and Thomas, 1980), the methods that have been developed are not directly applicable to service systems in general, and transit systems in particular. The characteristic that makes transit systems unusual-besides complexity-is the immediacy of demand. Transit service has no shelf-life, unlike manufactured products, and thus sufficient manpower must be available at the required time, and in the required place, to provide the scheduled service. Recent analytic research on transit operator workforce planning is best exemplified by MacDorman’s analysisofextraboardsizing(MacDormanetal., 1982, 1984, 1985a, 1985b). In this method, a cost function is defined that incorporates for any extraboard size, the cost of overtime, the cost of unused extraboard operator time and the fixed fringe benefits associated with the extraboard. By approximating the cumulative distribution of absence time by a logit function, a simple analytic expression is derived which minimizes the total cost. While this method was the first

Operator workforce planning in the transit industry

serious attempt to determine the optimal extraboard size, it suffers from several significant shortcomings. First, the application of the method presented (MacDorman and MacDorman 1982) shows a highly aggregate analysis in which the absence distribution does not allow for systematic daily or seasonal variation, which are generally important (Perin, 1984). It is essential to perform a disaggregate analysis which is stratified to remove significant systematic differences in absence rates between days and between seasons. Second, by assuming that operators are always available to work overtime, and not recognizing the potential feedback effect of higher overtime creating more absenteeism, the suggested solution may violate a service reliability objective and not result in true long run minimum cost. As noted by MacDorman, under certain cost conditions, particularly in large systems which make use of part-time operators, the suggested “optimal solution” may be infeasible since the extraboard will be driven to zero or a negative number. Finally, the logit approximation is untested, and may be a poor approximation of the true absence rate slope in the vicinity of the purported optimum. While MacDorman’s work represents a good starting point for the development of more comprehensive workforce planning methods, it does not by itself constitute a full and effective treatment of the problem.

129

This multilevel structure is desirable for several reasons. It allows for incorporation of the stochastic elements of the problem and it defers detailed decisions until as late as possible when more information will be available. By decomposing the complex overall problem into smaller subproblems, it allows a range of procedures to be used at each level with varying computational and analytical requirements. It also meets the different needs of the various departments within the organization involved in manpower management. For example, to facilitate the personnel department’s recruitment process it is preferable to have a working plan with the expected number of employees to be hired at each hiring cycle over the next year. This provides personnel managers lead time to decide when to increase the size of the job applicant pool, which can be a lengthy process in light of affirmative action plans to which many agencies are committed. The models are interrelated since the optimal policy determined by the higher level model is transferred as input to the next level model. Short term scheduling has the support of the long term hiring decisions to assure that the manpower required in each time period is supplied at the least expected cost. Finally, the methodology is very flexible and the models at each level can be used independently. Strategic model

METHODOLOGY

From the previous description of the transit operator workforce planning problem, it is clear that the problem is complex. It is characterized by a very uncertain environment, complex work rules which restrict operator mobility, and the involvement of many departments such as service planning, scheduling, transportation and personnel-with different requirements and deadlines. However, there is a hierarchical structure which can be imposed on the problem which suggests a methodology incorporating three interrelated models: Strategic model: A weekly (or monthly)

optimization model operating over the period of a year. It suggests the most cost effective workforce size, hiring levels and vacation allocation for each operator type (full time and part time). The time unit for the analysis would be based on the frequency of hiring and the typical unit of incremental vacation award. Tactical model: A daily optimization model over the period of a timetable. It allocates days off for each day of the week and hence establishes the optimal number of operators for each day over several months when the schedule is relatively constant. Operational model: It assigns report times to those extraboard operators with no known assignment.

This is the level at which the most critical workforce planning decisions are made, including hiring levels and timing, vacation scheduling and extraboard design. Each of these three decision areas is related to the two others and a single procedure has been developed to assist in all three areas simultaneously. Critical input to the model at the strategic level includes: Schedule requirements by timetable; total vacation requirements by job classification; attrition rates by week (or month) based on historical and employee seniority data; movement between job classifications, and promotion policies; frequency of hiring; absence by week of year; required extra service by type and level of predictability; overtime premiums and availability of employees to work overtime; fixed fringe benefits by job classification; pertinent work rules and pay practices; design and utilization of existing extraboard; overtime by day of week and job classification; service reliability objective based on unavailable manpower alone. Given the above input, the output of the model is the simultaneous determination of the least cost values for:

1. Workforce size and consequently number of extraboard operators for each job classification for each week in the planning period. The extraboard is the basic mechanism within transit agencies for maintaining service reliability in the face of absenteeism and uncertain work

H. N. Kou~~o~ou~os

130

requirements, and thus is the heart of the model. Extraboard design involves determining how many employees are appropriate for this cover function by week of the year and by job classification for each garage (or rail line) in the system; 3 Hiring levels for each job classification. The -. model explicitly considers the various hiring alternatives the agency may have. Direct hiring and promotion from part time to full time operators are examples of these alternatives. 3. Allocation of vacation weeks during the planning period. The objective is to exploit the seasonal variation of both the service requirements and absenteeism. More operators will be on vacation during the low requirement weeks of the year than the peak periods. The summer timetable, for example, may require fewer operators because of reduced ridership associated with vacations among the general public and lack of school trips, and thus more operators can be scheduled for vacations. This problem is strikingly similar to the inventory control problem, with the inventory simply being the number of employees (beyond the schedule requirements) on the payroll, and the hiring level and interval corresponding to the order quantity and time between orders. Pursuing this analogy further, uncertainty is present in both problems, entering in exactly the same manner, and stockout costs in the inventory problem correspond to the overtime cost required to provide scheduled service if hiring levels have been too low. Given the above input, and output requirements the following formulation Pl of the strategic model is proposed: Pl. + g

Min:

[(F) + co $

c/y, + CPYP (c,hirf(i)

+ c,hirp(i))

xp(iDl]

E[Wxf(i),

subject to

xf(i)

= Ff(0) + y, +

J$(hirf(k)

k=,

- rf(k))

xp(i)

= PT(0)

+ y, + 2

k=,

+ vacp(i) Wo)

- p (xf(i)

+ Y, -

I

(hirp(k)

- rp(k)) xp(i)

- vucf(i)

- vacp(i)

+ v&(i))

P (Wo)

I II

5 0

+ Y,) 5 0

>

and N. H. M. WILSON

zvacf(i)

2

V,

N 2 vucp(i) 2 VP ,=,

2 hirf(i) ,=I

= 2 rf( i) t-1

2 hirp(i) ,=,

= 2

rp(i)

,=,

I

III

IV 1

/

{other policy constraints} where xf(i):

full-time

operators (FTOs) available in pe-

riod i; xp(i): part-time

operators (PTOs) available in pe-

riod i; hirf(i): hirp(i): vucf(i): riod i; vucp(i): riod i; IT(o):

full-time operators hired in period i; part-time operators hired in period i; full-time operator vacation weeks in pepart-time

operator vacation weeks in pe-

full-time operators, initial conditions; PT(o): part-time operators, initial conditions; yf: the difference in FTOs between the current and the optima1 steady state conditions; y,: the difference in PTOs between the current and the optima1 steady state conditions; rf(i): average FI’O attrition in period i; rp(i): average PTO attrition in period i; N: number of time periods per year; c,: long-term annual average costs per FTO (salary plus benefits); cp: long-term annual average costs per PTO (salary plus benefits); c,: long-term average compensation per hour of overtime; V,: lT0 vacation liability per year; VP: PTO vacation liability per year; E[ O(xf( i) , xp (i))] : expected overtime hours in period i. Formulation Pl assumes that there is 100% reliability, i.e. all open work is covered either on overtime or by extraboard operators, although it can be extended to incorporate more general reliability considerations. It is also assumed that the system is in steady state, i.e. work requirements at the end of the planning period are the same as at the start. The objective function represents total annual operating costs (salaries, benefits and overtime). Constraint set I determines the number of FTOs and PTOs available for work in each period, as a function of the hiring decisions and the vacation allocation. Constraint set II represents the work rule requirements with respect to the ratio of PTOs to FTOs. Constraint set III guarantees that vacation allocation

131

Operator workforce planning in the transit industry

satisfies the vacation liability for both FTOs and PTOs. Constraint set IV requires that the total hirings during the period under study should be equal to the total expected attrition, so that steady state conditions are maintained. Various management policies such as constant hiring levels or uniform allocation of vacations during each timetable, can be incorporated under the last constraint set. Problem Pl is a convex nonlinear program with linear constraints which can be solved efficiently by existing algorithms, for example the Frank-Wolfe algorithm. A special decomposition scheme, similar to Bender’s decomposition for mixed integer programs, has also been developed which solves the problem very efficiently (Koutsopoulos, 1986). An interesting aspect of problem Pl is that it has multiple solutions with respect to the number of operators hired and the vacation allocation during the year although the solution with respect to available operators xf(i) and xp(i) is unique. Since some of the solutions may be more desirable than others, the following two-step procedure is proposed to select the most desirable of the alternative optimal solutions: Step 1: Solve problem Pl. The outcome is the optimal operator levels xf(i)* and xp(i)* and one optimal hiring and vacation schedule. If this solution is not acceptable continue with step 2. Step 2: For the optimal values of xf( i)* and xp( i)* choose among the optimal schedules the one that satisfies whatever secondary objectives may exist. Such secondary objectives generally exist in any agency. To facilitate the discussion, the concept of the “ideal” hiring and vacation levels in period i is introduced: H,(i): ideal ITO hiring level in period i; H,(i): ideal PTO hiring level in period i; L,(i): ideal FTO vacation level in period i; L,(i): ideal PTO vacation level in period i.

Then the step 2 problem is to identify, among all the multiple solutions to the strategic model, the one that minimizes the difference between actual and “ideal” hiring and vacation levels. To identify this “best” optimal solution the following quadratic program is proposed.

P1.l.

Min: 2 [A,(H,(i) ,=I

- hirf(i))z

+ X,(H,(i)

- hirp(i))2

+ h,(L,(i)

- vacf(i))’

+ h,(L,(i)

- vncp(i))‘]

subject to Y, +

2 kf(k)

-

!f=l

vacf(i)

= xf(i)*

+

- IT(o)

2 +f(k),

k=,

y, + 2

hirp(k)

- vacp(i)

= xp(i)*

- IT(o)

k=,

+

k rp(kh

k=I

vucp(i)

- Pvucf(i)

= pxf(i)*

Y, - PY, = PFT(o) i

vucf(i)

- xp(i)*,

- PT(o),

= V,,

,=I

N

C vucp(i) i=*

= VP, N

N

2 W(i)

=

,=I

i

C rf(i), ,=*

hirp(i)

= 2

rp(i),

{other policy constraints}. The weights Ail are chosen so that at the optimal solution each term has approximately the same contribution. The values of the “ideal” vacation and hiring levels are predetermined so that they represent the secondary objectives. For example, if management is interested in minimizing the variability in the number of operators hired and vacation weeks allocated, the appropriate “ideal” levels are

H, = 2

rf(i)lN,

,=I

HP = i rp(i)lN, ,=I L, = V,IN, L, = V-,/N.

Apart from the obvious, direct application of the model to assist in setting hiring levels, allocating vacation weeks and sizing the workforce, because of its generality, it can also be valuable in analyzing various policy questions that may be of interest to a specific agency. For example, an agency may be used to hiring on a quarterly basis but is considering moving to a monthly hiring cycle; what will the implications be? Another agency may be under conflicting pressures either to maintain a constant hiring level per hiring cycle, or to maintain a constant workforce

132

H. N. Kourso~ou~os and N. H. M. Wn.so~

size. The model can be used to determine what the impacts of these policies would be and what each policy would cost above the optimal strategy. Various alternatives of how work for vacationing employees is covered, may also be evaluated. Frequently, vacation relief assignments require any employee selecting vacation relief to fill in for the scheduled activities of different vacationing employees throughout the timetable. In this case, a constant number of vacation spots will be available for selection in each week of a timetable. In other agencies, the work open due to vacations is covered directly from the extraboard. In this case, the number of vacation slots can be varied from week to week. Similarly, the effect on operating costs of the single day vacation option can be assessed. While the strategic level model would run with a one year (or longer) planning horizon, it would be re-run before each hiring decision to reduce the uncertainty on the best hiring level, given current information on actual attrition and best possible specification of the remaining inputs. Depending on needs, available data and computer resources, the model could be implemented at various levels of detail. In its simplest form, it is a direct extension of MacDorman et al., (1982, 1984) at a disaggregate level, recognizing explicitly the different job classifications. Benefits from applying the most detailed version of the model are likely to be highest for agencies that experience high levels of variability and for agencies in transition-for example, after major service changes. For agencies in a steady state condition, application of the simple version of the model should be adequate. Tactical model

While the model at the strategic level is concerned with workforce sizing, the models at both the tactical and the operational levels deal with the problem of making the most efficient utilization of the available workforce. At the strategic level, the optimal hiring sequence by job classification and allocation of vacation slots has been determined, defining the workforce size for each week. At the tactical level, the decisions concern the optimal extraboard size by day of week, and by job classification, given the restrictions on the available workforce and the work rule constraints on days off. Due to variations in expected absenteeism levels during the week, each day may have different requirements with respect to extraboard operators. Since minimization of expected overtime is the objective at this level, more operators, or equivalently fewer days off, will be allocated to the days with higher expected absenteeism (for example, Mondays and Fridays). The total number of days off during the week is directly determined by the number of available operators. Allocation of days off with the only objective being the satisfaction of work rules is a rather easy problem with multiple solutions and the proposed model chooses from among all the feasible

solutions the one which minimizes total expected overtime. The planning period is the timetable, and the day of week is the unit of analysis. The model can be used at the beginning of the timetable or as often as operators pick their days off. The formulation of the tactical model that follows corresponds to the case where the required reliability is 100% which implies that all trips are performed. P2.

Min: c

I=,

E[O(xf(i),

xp(i))]

subject to xf(i) = bf - uf(i), xp(i)

= bp - up(i),

uf(i) 2 2bf,

i ,=I 7

c

up(i) 2 2bp,

,=I

xf(i),xp(i),uf(i),uP(i)

2 0,

where uf(i):

number

of extraboard

FTOs having day i

number of extraboard

PTOs having day i

off; up(i):

off; xf(i): available extraboard FTOs on day i; xp(i): available extraboard PTOs on day i; bf: available extraboard FTO workforce in the

week under study; bp: available extraboard PTO workforce in the week under study; E[O(xf(i), xp(i))]: total expected overtime hours in day i. The form of the expected overtime function depends on the underlying distribution of open work. Problem P2 is convex with linear constraints and it can be solved efficiently by any appropriate algorithm (for example, by Frank-Wolfe). Besides the determination of appropriate extraboard levels for each day of the week, the tactical model can also be used to help the manager relocate operators temporarily from divisions with operator surplus to divisions with operator deficit (this last task, though, may also have an obvious or more easily obtainable solution). It can also be used to identify optimal days off for a marginal operator being added to the extraboard. Operational model The operational

level problem, which must be solved each day, is to design and schedule the daily assignments of extra operators to open runs. Because of the short time horizon (one day) there is much more information available than in the stra-

Operator workforce planning in the transit industry

tegic or tactical models. More specifically, absent operators belong to one of the following two categories: 1. Operators whose absences are known in advance (for example, operators on jury duty, military service, suspension or injured). 2. Operators who are unexpectedly absent. The operational level problem involves two distinct components: run-cutting to build efficient runs to cover known absences, and the assignment of report times to unassigned extraboard operators so as to cover unpredictable absences. It is at the operational level that all known-in-advance requirements (including trippers, extra service, attrition and known absences) are built into runs and assigned to the extraboard or to overtime. This problem is a small run-cutting problem with all the work rule complexity of the full run-cutting problem, but on a more manageable scale. Operators that remain, after the run-cutting of known open work has been performed, are utilized to cover work that may be open during the next day and the task is to assign report times so that operator availability best matches the expected needs. The determination of report times is sensitive to the probability density functions describing unanticipated requirements (whether caused by absence or extra service) by hour of day, the probable availability of employees for overtime work, and the work rules. It is also sensitive to the distribution of work throughout the day and to the number of available (i.e. unassigned) extraboard employees. While the solution of the first task is either trivial (direct assignment of open runs to operators) or well documented (run-cutting of the known open work), the solution to the second task is rather complicated. It is in essence a scheduling problem under uncertain demand. Exact formulation of the problem is computationally prohibitive because of interactions among operators who report in different time periods and open runs, which are combinatorial in nature. The modeling approach adopted is based on the profile of open work and operator availability as a function of the time of day. The operator availability profile is a function of the assigned report times and work rules while the open work profile is stochastic. Appropriate objectives at the operational level are the minimization of overtime, or unproductive extraboard time, or the number of missed trips or a combination of all the above. It turns out that with the proposed modeling approach, the above individual objectives are equivalent. By minimizing the overtime, the number of missed trips and the unproductive time are also minimized. Formulation P3 of the operational model that follows, assumes that extraboard operators work continuous assignments, but the formulation can be easily extended to represent the more general case of split assignments.

P3.

Min:

133

$ UO(rf(i),

xp(i))l

subject to

xf(i) = C yf(k), IrEIf

x&i> = 2 yf(k),

$,yf(k) =Nft i: YP(~) = N,, k=l

~f(k),yp(k)

integers,

where yf(i): full time extraboard operators reporting at start of time period i; yp(i): part time extraboard operators reporting at start of time period i; xf(i): full time extraboard operators available in time period i; xp(i): part time extraboard operators available in time period i; If(i): the set of report times for which an extraboard FTO who reports at time t E Zf(i) is still available at time i; Zp(i): the set of report times for which an extraboard PTO who reports at time t E Zp(i) is still available at time i; Nf: number of extraboard FTOs to be assigned report times; Np: number of extraboard PTOs to be assigned report times.

Formulation P3 requires that the decision variables, the number of operators reporting in each time period, are integers. Since algorithms to determine optimal integer solutions are generally computationally inefficient, a heuristic algorithm has been developed which gives good solutions to the operational model quite efficiently. According to. the heuristic, which is a variation of the greedy algorithm, the operating day is partitioned into periods appropriately small. Extraboard operators report in the beginning of these periods. The algorithm is iterative, with operators allocated successively one at a time. At each iteration the type of operator (full or part time), the type of assignment (continuous or split) and the time period of report are simultaneously determined based on the marginal reduction in total overtime (keeping the allocations already made fixed). A real-time operating problem also exists since the supervisor in the garage must assign required work to available extraboard employees or to overtime, must decide when and which trips should be dropped if manpower is unavailable, and when and

134

H. N. Kou~~o~ou~os

and N.

H. M. WILSON

Table 1. Strategic model results Case

1.

vacf(i)clOO vacp(i)Z 20

Overtime Increase

$118,853

Total Cost Increase

653,988

$191,239

where to provide extra service. If the work rules permit split cover assignments, he (or she) must also decide when and by how much to reassign available extraboard employees. These are all important decisions, but it is questionable whether, in the press of garage activity, there is room for a formal procedure to assist the decision-maker. Furthermore, if good decisions have been made at the operational level and therefore the report times have been correctly determined, incorrect real-time decisions should have minimal impact on operating costs. CASE STUDY

To demonstrate the potential of improved methods in operator workforce planning, the methods described in the preceding section have been applied to some of the workforce planning problems faced by the Massachusetts Bay Transportation Authority (MBTA). In this section, some of the results of this application are presented. Strategic model application The strategic model was applied to the entire MBTA

system using the 1986 service plan. The application of the model was made to demonstrate the importance of the simultaneous determination of workforce size, vacation allocation and hiring levels, and to evaluate the effects of constraints on operating costs. The increase in overtime and total operating costs was estimated when constraints were imposed on the distribution of vacation requirements (compared to the unconstrained base case). Two cases were examined. In the first case, upper bounds of 100 and 20 vacations per week were imposed on the distribution of FTO and PTO vacations, respectively. In case 2, vacation requirements for both FTOs and PTOs were uniformly distributed across the year. Table 1 shows the cost increases associated with each case compared with the base, unconstrained case. The effect and importance of using the allocation of vacations as a control variable is clear. In the base case, the vacation weeks were allocated by the model so that the total work requirements during the year are as uniform as possible. When constraints were imposed on the amount of vacation weeks allocated to each time period, the

peak requirements became higher than the base case and as a result, overtime increased substantially. The base case overtime cost was $1,383,730 and so the extra overtime represents increases of 9% and 32%, respectively. The increase in total operating costs was significantly smaller because the optimal workforce size was smaller in each case. The increase in overtime and reduction in workforce size was expected since the cost per operator per hour (salaries and benefits) was very similar to the cost of one overtime hour-around $20. Therefore, in the optimal solution it is preferable to use overtime when it is needed (to cover the extra peak requirements) instead of extraboard operators with their guaranteed annual salaries and benefits. Although the total cost increase is not very significant, the implications for service reliability should also be considered before any of the constrained solutions are employed. Since overtime increased substantially-by 16 h and 57 h per day, respectively, on the average for the entire MBTA system-sufficient manpower should be available to work all the required overtime hours. This also emphasizes the need, already mentioned, for constraints on the amount of overtime to be used in each period. Tactical model application

The situation analyzed corresponds to the spring 1985 timetable for one of the seven bus garages in the MBTA system to which 293 full time and 109 part time operators were assigned. The extraboard size was fixed at 2.5 full time and 13 part time operators and allocated among days of the week as shown in Table 2. This allocation is based on providing an extraboard size of about 10% of the scheduled runs on each day of the week, the recent MBTA policy on extraboard size. Table 2. Service requirements and operator allocation (Spring 1985)

135

Operator workforce planning in the transit industry Table 3. Absence hours by day of week Friday

Sat.

'Sunday

Mean(hrs) 259

Monday Tuesday 200

212

233

273

185

a4

36

31

36

30

52

24

25

Standard Deviation

Wednesday Thursday

Table 3 shows the mean and standard deviation of actual absence hours for each day of the week for the first nine weeks of the timetable. These figures clearly show the higher levels of absence (as a percent of scheduled runs) on Friday, Saturday, Sunday and Monday compared with mid-week, and the high levels of variability in absence hours on any day, but particularly on Fridays and Sundays. The tactical model was applied assuming that the distribution of total absence hours on any day is normal with the given mean and standard deviation. It was also assumed that part time operators could work only weekdays, replicating the practice at the MBTA in spring 1985. Table 4 compares the allocation of full time extraboard operators determined by the tactical model with the one actually used by the MBTA. As expected, the model allocates more of the fixed extraboard resource to the high absence days. Comparison of the expected overtime between the two solutions indicates a reduction in overtime of 42 hours per week or 24% of the actual overtime hours if the solution suggested by the short range model had been adopted. The difference is basically attributed to the fact that the actual uniform distribution of cover among weekdays did not take advantage of the systematic daily component of absenteeism. To validate the above results the overtime which would result from each extraboard allocation (model and actual) was determined directly using actual records of absence hours for each day of the nine week period. This eliminates the assumption that the absence hours on a particular day are normally distributed. Table 5 shows the difference in overtime for each day of the Spring 1985 timetable for both extraboard designs. In each cell of the matrix, the first number given is the overtime resulting from the model extraboard design while the numbers in parentheses are the overtime hours resulting from the extraboard design actually used. In both extraboard designs, overtime is estimated based on the optimal daily utilization of the available extraboard operators, i.e. daily report times match the absence patterns almost perfectly. Therefore, the resulting reduction in overtime is attributed exclusively to the

proper allocation of the available extraboard operators to each day of the week. This analysis suggests that a 29% reduction in overtime is possible as a result of changing the uniform allocation of cover to the one suggested by the tactical model which more closely reflects daily absenteeism patterns. This result demonstrates that the assumption of normality for the absence hour distribution should be acceptable. Table 5 also shows the great variation in amounts of overtime required on different days of the week with the actual extraboard design. In practice, this leads to poor service reliability, in particular on Fridays and Saturdays, when insufficient manpower is available to work all the required overtime hours. It should be emphasized that the reduction in overtime achieved by the tactical model occurs with no increase in the number of extraboard operators, rather by reallocating the extraboard operators to take advantage of the systematic daily variation in absence hours. Further analysis of MBTA absence patterns shows significant differences between seasons and between garages, in some cases resulting in deviations from the system-wide mean of more than 30%. This implies that substantial benefits can also be achieved by varying the size of the extraboard between garages and across seasons, rather than using the traditional figure of a flat 10%. These savings can be achieved by applying the tactical model to the full system, incorporating all garages for each timetable. Operational model application

The operational level model was applied to the same MBTA garage as the tactical level model case study, focusing on the determination of report times for the operators remaining after the known open runs have been assigned. Two applications of the model were made; first to compare current practice with the model results; and second to investigate the implications of alternative work rules on operating costs. The output of the operational model is an ordered set of report times for unassigned extraboard operators with an associated preference for part time

Table 4. Allocation of full time extraboard operators by day of week Monday Tuesday Wednesday Thursday Friday

7

Sat.

Sundav

Model

21

14

15

la

24

22

10

Actual

20

20

20

20

20

17

7

.

H. N. KO~T~OPOULOS and N. H. M. WILSON

136

Table 5. Overtime hours for model and actual extraboard design l-

Wed

O(O)

O(O)

O(O)

Thur

‘I

f!eek 4

#eek 5

Week

9

Week

18(26)

3(l)

34(42)

O(O)

ll(17)

6(O)

3(O)

O(O)

54(4)

12(O)

l(O)

O(O)

34(O)

51(11)

15(2)

9

Mean

O(O)

O(O)

O(O)

l8(2)

O(O)

35(19)

46(30)

14(7)

Fri

O(O)

O(O)

O(O)

O(7)

58(90)

O(24)

60(92)

21(38)

sat

O(35)

O(O)

3(43)

26(66)

1(41)

28(63)

16(56)

15(50)

SUfl

O(O)

O(16)

O(21)

41(61)

24(48)

8(32)

O(8)

ll(26)

Total

O(35)

ZO(26)

3(64)

llO(162:

83(179

39(185)

!25(201)

Numbers

in

( ) are estimates

of overtime

81(1ffl)j219(236)

associated

or full time operator for each report. Then the top report times are assigned to all available unassigned extraboard operators. To test the report times produced by the operational model, a detailed analysis of operator utilization was made for each day of the first nine weeks of the spring 1985 timetable for the selected MBTA garage, and utilization rates and resulting overtime were compared with the actual MBTA experience. Different report times were selected for each day of the week based on the expected pattern of absence that day. For example, although Mondays and Fridays had similar overall levels of absence (see Table 3) resulting in similar extraboard size, the patterns of absence over the day are quite different, with Monday having more morning absences and Friday having more afternoon absences. Accordingly, more early a.m. report times were provided by the operational model for Mondays than for Fridays. Another general observation on absence patterns over the day is that absence rates are significantly lower in the morning peak than later in the day. This may be due to the higher proportion of senior operators selecting morning runs and also tending to have lower levels of absenteeism. In the execution of the operational model a high weight was placed in the objective function on avoiding open work before 9 a.m., because of the importance of service reliability for the home-to-work commute and also because of the unlikelihood of having operators available to work overtime early in the morning. This results in a more than proportional assignment of report times in the early morning. Results of this detailed analysis showed that the

with

actual

extraboard

design.

operational model could reduce overtime by about 2530% compared with the actual report times used. In this case, unproductive extraboard operator time was also reduced by a similar percentage, increasing the overall utilization of the extraboard. The main difference between the report times actually used and those developed by the operational model was that too many reports were actually assigned in the early morning, resulting in a good deal of unproductive but paid time early in the day, and necessitating extra overtime later in the day. This seems to be an over-reaction to the concern for service reliability in the morning peak. To assess the implications of extraboard work rules on operating costs, the operational model was used to investigate two different scenarios: Scenario 1. Only continuous

pieces of work permitted for both full time and part time extraboard operators. Scenario 2. Split shifts permitted with two pieces of work spread over a 10 hour period. The impacts of these alternative work rules were investigated under two different levels of extraboard manpower with the results shown in Table 6. This analysis suggests a rather small reduction in both overtime and unassigned time of about 1 hour/day if two-piece extraboard assignments are possible. However, this is expected to underestimate the true savings because first, only a small subset of all possible two-piece assignments were considered and second, in the actual time-constrained decision context the likelihood of a poor initial decision on report

Table 6. Overtime and unproductive hours for two work rule scenarios

I-;1

19(140)

Unassi

ned

Overtime

Unassigned

137

Operator workforce planning in the transit industry 35 30 25 20 15 IO 5 I

0

!

40

I

I

60 80 100 Operator Hours Available

I

I

120

Fig. 1. Overtime as a function of available work hours.

times by the dispatcher is high. In the case of twopiece assignments, there is an opportunity to recover (at least partially) from these mistakes by reassigning the second piece of an early report, which would otherwise go unused, until later in the day. To investigate further the impact of the work rules on operating costs, the concept of the “ideal” operator profile is introduced. This is the result of allocating optimally the available extraboard operator

hours without being subject to work rule constraints. For example, constraints on the number of pieces of work, and their length are relaxed. The number of overtime hours required under these conditions is clearly a lower bound on all constrained solutions. Figure 1 compares the number of overtime hours under Scenarios 1 and 2 with this lower bound for different levels of extraboard manpower. The comparison indicates the generally small potential sav-

4 -.

(FTO=4,PTO=2)

04100AM

OSGOAM

02GOPM

Time

07:OOPM

12100Ak

of Day

lower bound -.-P-piece runs

i 040OAM

OSOOAM

02:OOPM 070OPM Time of Day

12:OOAM

Fig. 2. Extraboard operator availability by time of day.

138

H. N. Kowrso~ou~os

ings from using two-piece runs over a wide range of extraboard size. It also shows that the heuristic algorithm developed for the solution of the tactical model performs very well, closely approximating the lower bound solution. Figure 2 compares the extraboard operator availability by time of day for each scenario with the ideal operator availability profile. The inability to cover the peak requirement efficiently when one-piece assignments are used is very

characteristic and accounts for the additional benefits (albeit small) of allowing two-piece extraboard assignments. CONCLUSIONS

A methodology for operator workforce in the transit industry has been presented

planning

including the formulation of three models which address the two fundamental problems in workforce planning. For the first problem, the determination of the appropriate workforce size, a model is proposed which also incorporates related aspects of the problem including hiring decisions and vacation allocation. The second problem, the efficient utilization of the available manpower is addressed with two new models. The tactical model examines absenteeism on a day to day basis within a week and allocates the extraboard resources so as to minimize expected overtime. The operational model addresses the problem of the determination of report times for extraboard operators on a particular day. Application of the methodology to one MBTA bus garage provides results which are encouraging in terms of potential improvement in transit productivity. They demonstrate that the problem of operator workforce planning in transit systems is indeed amenable to analysis. The proposed methods are suitable for implementation and could lead to significant increases in both efficiency and effectiveness. Data needed as input to the model are available in almost all transit agencies but development of databases for workforce planning is an important

and N. H. M. WILSON

area for further work. Another promising research area is the development of guidelines which could be used by the dispatchers in real time. This is a very complicated aspect of the problem and would most likely require the use of simulation models to understand the dynamics of the real-time control issues involved.

REFERENCES

Baker H. S. and Schueftan D. (1980) Study of operator absenteeism and workers compensation trends in the urban mass transportation industry. UMTA Report UMTAPA-0050-80. Chomitz K. M. and Lave C. A. (1981) Part time labor, work rules and transit costs. UMTA Report UMTA-CA11-0018-81. Koutsopoulos H. N. (1986) A methodology for transit operator workforce planning. Unpublished Doctoral Dissertation, Department of Civil Engineering, M.I.T. McClain J. 0. and Thomas L. (1980) Operufions Munagement: Production of Goods and Services. Prentice Hall, New York. MacDorman L. C. and MacDorman J. C. (1982) The transit extraboard: Some opportunities for cost savings. Paper presented at the APTA Annual Meeting. MacDonnan J. C., Schwager D. S. and MacDorman L. C. (1984) Reducing the cost of absenteeism through workforce optimization. Paper presented at the APTA Western Conference. MacDorman L. C. (1985a) Extraboard management: Procedures and Tools. National Cooperative Transit Research and Development Program, Synthesis of Transit Practice Report No. 5. _ MacDorman and Assoc. (1985b) TOPDOG: Ooerator Productivity Diagnostic and Optimization Guidelines. Draft Users Guide. Perin C. (1984) The dynamics of vehicle operator absenteeism. Transportation Research Record 1002, 1-7. Perry J. L. and Long L. (1984) Extraboard scheduling, workers’ compensation, and operator stress in public transit: Research results and managerial implications. Transportation Research Record 1002, 21-28.

Porter B., Carter D. and MacDorman L. C. (1984) Operator availability management methods. U.S. DOT Report DOT-I-84-23. Wijngaard J. (1983) Aggregation in manpower planning. Management Science 29, 1427-1435.