Pergamon Computers ind. Engng Vol. 29, No. 1---4, pp. 473--476, 1995

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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OPTIMIZATION AND HEURISTIC MODELS TO INTEGRATE

PROJECT TASK AND MANPOWER SCHEDULING*

James Bailey, Phl)

Arizona State University

HeshamAlfare$, PhD

King FahdUniverslty of Petroleum and Minerals, Saudl Arabia

win Yuan Lin, PhD

ProvldenceUniversity, Tiawan China

ABSTRACT

The objective of project scheduling

is to determine start dates and the labor

resources assigned to each activity in

order to complete a project o n time. By

moving start dates within available slack

times and altering labor levels, the daily

labor-demand profile can be changed. The

objective of personnel scheduling is to

determine how many of each feasible work-

day tour are required to satisfy a given

labor-demaand profile while minimizing the

cost of labor plus overhead. Integrating

these two problems permits the simultaneous

determination of start dates, labor levels

and tours for a mlnimum-cost and on-time

schedule. In this paper, single and

multiple resource optimization models and

heuristic solution procedures to solve the

integrated problem are presented. The

heuristic procedure outperformed the non-

integrated two-step scheduling procedure by

re4hacing the cost of labor and overhead and

performed nearly as well as t h e

optimization procedure.

* The research reported here was

accon~lished as part of the PhD

requirements for Drs. Alfares (1) and Lin

( 2 ) .

INTRODUCTION

At present, labor a n d project

scheduling are accomplished in two

sequential steps. A project scheduling

algorithm is used to determine the start

time, duration, and labor level for each

activity which esstablishes a daily labor-

~ d profile for labor. A manpower

scheduling algorithm is then used to

determine the number of workers assigned

needed to satisfy that a~nd profile. This

practice is clearly less than optimal.

When each task starts, how much labor is

assigned, and how long it is active affects

beth numbers of people to be scheduled.

Therefore, when to determine the minimum

cost schedule one should integrate the task

and labor scheduling problems.

AN INTEGRATED OPTIMIZATION MODEL

Consider first the single labor

class problem. The objective of the

integrated model is to minimize the sum of

overhead plus labor costs. There are five

sets of constraints. The first set of

constraints establish unique task start

dates for each task. The second set of

constraints require that the start date of

any task be later than the co~91etion date

of all its immediate predecessors. The

third set of constraints requires that the

number of people assigned to work any given

day will satisfy the labor M~-~nd of all

tasks scheduled for that day. This is the

constraint that integrates the personnel

scheduling and the project scheduling

models. The fourth set of constraints

establishes the project duration time as

the oom%Dletion of the last task fzca the

set of tasks with no successors. A last

set of constraints limits the total work

force size in any given week to some

473

474

prescribed number. For more, the reader is

refered to Alfaras(l} and Lin (2).

NWK7 Minimize Z Z Ci*Ywl + OH*PT (1)

w-I i-i

Subject to:

LSjt Z Z Xjdt=l, J=l,2 .... , NACT (2)

d~$j VteTj LSpt LBjt

Z Z(d+t)*Xpd t <= Z Z d*Xjdt , ~pePj (3)

d~Sp VteTp d~Sj VtgTj

NACT ~n(d,LSjt) 7 NWK

Z Z Z MANjt*Xjq t <= Z Z aawid*Y w (4)

j-1V~Tjq~max(d-t+l,ESj) i-i w-1

LSjt Z Z (d+t-l)*Xjd t <= PT, VJeL (5)

d~sj VteTj

7

Z Ywi <= SIZEw' w=l,2,..., NWK (6)

i=l

Where :

C l Ywi

OH PT

xjd t

Tj

ESj

LSjt

N~'T

P

pJ

t

d

MANjt

~ j

aawid

L

SIZEw

VteT:)

= Weekly cost of tour i.

= Number of tours of type i in week w.

= Daily overhead cost.

= Pro~ect duration in days.

= Project due date in weeks.

= 1 if task J with duration time t is

started on day d. = 0 otherwlse.

= The set of activity duration times

for activity J-

= Earliest possible start date for

activity 3-

= Latest possible start date for

activity ~ if it has a duration of t.

= The number of tasks in the project.

= An iE~-~Liate predecessor of task J,

vpepj. = Set of immediate predecessors for

task 3.

= Duration days of predecessor p.

= Start day of predecessor p.

= Daily requlredmanpower for task J if

its duration is t. MANJt*t>~TMANJ

= Total man-days of effort requlredby

task 3.

= 1 if day d is a work day for shift i

in week w. = 0 otherwise.

-- Pro~ect duration time.

= set of all tasks which have no

predecessors.

= Maximum size of the work force in

week w.

is read, for all values of t in

the set Tj.

17th International Conference on Computers and Industrial Engineering

An extension of this model to the

multiple labor catagory case is quite

simple. The objection function needs to be

expanded to include a sum over K multiple

labor catagories. The decision variable

must be expanded to include each labor

catagory. Likewise, there must be one

manpower constraint for each catagory. It

should be noted that these additions do

increase an already large integer

progr~-,-4 ~g model. The need for a good

heuristic is more apparent as the number of

activities and number of labor categories

increases.

HEURISTIC SOLUTION PROCEDURE

Each week has a finite set of

feasible project schedules defined by all

feasible combinations of tasks in the

availale-task set. Each project schedule

defines a labor-demand profile which can be

used to find an optimal solution to the

personnel scheduling problem. Call the

solution to any week' s project/personnel

scheduling problem a feasible state for

that week. In a backwards dynamic

progr a~mtlng terminology the optimal

schedule for the present and future weeks

can be found when all feasible states at

the end of the previous week are known,

without regard to how one got to those

states.

The problem is that there may be

many feasible project schedules. A

bounding procedure is needed to reduce the

size of this set. Two logical bounds can

be employed. First, to be considered

further, the daily labor demand should lle

between two previously established limits

so that labor expenditure is reasonably

level during the project. Two bounds are

determined by the average daily labor

required to complete the project between

the critical path time (TL) and due date

time (TU) . A second bounding procedure

also eE~loys a limit on the daily staff

size. Baker (3) presented a procedure for

establishing the minimum staff size needed

to complete the project on time. To be

considered further, the work force

requirement must be either equal to or one

greater than the Baker minimum.

17th International Conference on Computers and Industrial Engineering 475

HIF~tlSTZC OPTIMAL

JOB LAB-COHT

• 8AV'8 • 8AV'8

1 19.89 18.82

2 17.97 17.97

3 16.14 16.14

4 18.22 19.16

5 1 7 . 3 5 1 7 . 3 5

6 1 8 . 0 0 18.00

7 20.53 20.53

8 17.46 17.46

9 1 8 . 4 6 1 8 . 4 6

i0 19.12 19.12

HELmXHTIC OPTIMAL BEURIHTIC OPTIMAL

LAB-COHT TOT-COHTTOT-COST JOB LAB-COST LAB-COBT TOT-COST TOT-CO8T

% SAV'8 % BAV'8 NL~B]~% 8AV'8 % 8AV'8 % 8AV'8 % 8AV'8

10.98 12.20 II 14.83 17.37 7.23 8.96

14.26 14.26 12 17.37 17.37 7.62 9.60

12.72 12.72 13 25.81 26.34 10.80 11.18

14.47 15.30 14 9.01 9.03 5.30 5.81

10.17 10.17 15 9.78 9.78 3,89 3.89

12.07 12.07 16 10.48 10.48 4,90 4.90

13.80 13.80 17 10.63 12.03 5,10 6.13

11.83 11.83 18 4.53 5.39 1.40 1.97

9 . 3 8 9 . 3 8 19 4 . 0 9 5 . 0 9 1 . 3 0 1 . 9 9

1 1 . 2 7 1 1 . 2 7 20 7 . 1 0 9 . 1 8 3 . 6 9 5 . 1 9

R]P~kIATIC OPTIMAL

Aver's 14.83 15.25 8.61 9.13

Table I: Percent 8avlnqs in Total Cost and Labor Cost for the Heuristic and

Optimal Procedures over the Traditional Two-step Heuristic

P ~ r e for the 20 ProblemJ.

Table 2:

JOB LAB-COST TOT-COST JOB LAB-CO8T TOT-COST

% SAV'8 % SAV'S NU~mER % SAV'S % SAV'S

I 29.06 15.50 25 19.07 7.41

2 0.00 0.00 26 4.71 2.32

3 6.45 3.22 27 12.08 6.72

4 3.34 1.55 28 18.10 5.72

5 13.04 7.00 29 7.13 3.99

6 15.37 7.00 30 7.81 2.10

7 25.51 12.26 31 11.73 6.53

8 12.13 5.40 32 16.17 4.18

9 3.19 1.46 33 22.86 10.23

10 22.30 9.78 34 8.80 1.31

11 26.50 12.49 35 26.19 11.60

12 12.03 5.14 36 21.49 5.30

13 2.03 0.86 37 8.78 4.60

14 1.09 0.36 38 15.59 8,10

15 12.17 5.55 39 15.77 16.10

16 0.00 0.00 40 10.57 5.67

17 42.56 19.36 41 6.67 3.31

18 2.61 1.06 42 0.44 0.16

19 2 2 . 6 9 9 . 5 8 43 1 2 . 3 2 5 . 9 0

20 2 1 . 3 8 7 . 4 4 44 7 . 8 5 1 . 2 3

21 1 1 . 3 6 4 . 6 3 45 1 2 . 3 7 5 . 9 8

22 1 1 . 7 3 4 . 2 9 46 3 . 8 5 1 . 3 9

23 1 0 . 1 2 3 . 7 6 47 1 7 . 9 9 9 . 0 9

24 8 . 9 0 3 . 4 7 48 9 . 0 8 0 . 0 0

Aver. 12.71 5.65

P e r c e n t S a v i n g s i n T o t a l C o s t a n d L a b o r C o s t f o r t h e O p t i m a l P r o c e d u r e s

o v e r t h e T r a d i t i o n a l T w o - s t e p H e u r i s t i c P r o c e d u r e f o r t h e 48 P r o b l m .

476

For each surviving project

schedule, the LP personnel scheduling

formulation is run. If a solution is not

found with this model, the project schedule

is discarded. The best of all feasible

solutions is then carried on to the next

earlier week using a recurson equation.

IMPERICAL COMPARISON OF PROCEDURES

17th International Conference on Computers and Industrial Engineering

The optimal and heuristic

procedures presented here were compared to

the traditional two-step scheduling

approach using 20 test problems. The two

step approach employed the Elmaghraby (4)

resource constrained project scheduling and

Baker (5) personnel scheduling procedures.

For test problem detail, see (1) and (2).

The results of the 20 test runs for

the single labor class case are displayed

in Table 1 which contains the percentage

i~)rovement of the optimal and heuristic

procedures over the traditional two-step

procedure. Note that on average, the ILP

optimal procedure yielded a 9.1 percent

reduction in total project cost and the

heuristic performed almost as well,

yielding an 8.6 percent reduction. In the

test cases, labor productivity for the

heuristic procedure averaged 98 percent, an

increase of 14 percent over the traditional

procedure. Note also that the heuristic

yielded an optimal solution in half the

test cases and averaged only 0.6 percent

higher costs relative to the optimal

procedure. Finally, the heuristic

procedure required 58 percent fewer CPU

seconds than did the optimal integer

programming procedure.

The percent saving in total and

labor for the integrated optimization

formulation over the two step procedure for

a two labor class case are shown in Table

2. In this case, 48 test problems were

used. A 5.65 percent savings in total cost

and a 12.71 percent savings in labor were

accomplished using the integrated

procedure.

CONCLUS IONS

The task and personnel scheduling

problems have been integrated in a model

which appears to offer substantial savings

in the overall cost of a project. That

savings comes as a result of selecting a

start time and labor level for each task

that reduces labor cost. The integer

prograualng optimization formulation is,

however, too large for realistic problems.

A very efficient heuristic approach based

on dynamic programming is presented and

tested.

The heuristic solution procedure

yielded an 8.6 percent reduction in total

cost as compared to the traditional two-

step procedure, when only one labor class

was considered. In the multiple labor

class case, the similar savings were 12.71%

in labor and 5.65% overall cost.

The primary conclusion is that

integrating the solutions of the project

and personnel scheduling problem is both

feasible and wise. Two reasonable follow-

up efforts would be to explore refinements

to the heuristic procedure presented here

and to program user friendly decision

support tools for this scheduling activity.

Bibliography

1.Alfares, H.K., Integrated Project

Activities and Manpower Scheduling,

Unpublished PhD dissertation, Arizona

State University, Tempe,1991.

2.Lin, W.Y., A Model of the Integrated

Multiple Resource Project and Personnel

Scheduling Problem, Unpublished PhD

dissertation, Arizona State University,

Terape, 1993

3. Baker, K.R. , "Scheduling a Full-time

Workforce to Meet cyclic Staffing

Requirements," Management Science,

Vol.20, No.12, (1974)

4.Elmaghraby, S.E., Activity Networks:

Project Planning and Control by Network

Models, John Wiley & Sons, New York,

(1977)

5.Baker, K.R. , "Workforce Allocation in

Cyclical Scheduling Problems: A survey,"

Operation Research Quarterly, Vol. 27,

No.l, (1976)