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Computers ind. Engng Vol. 35, Nos 3-4, pp. 415-418, 1998 © 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0360-8352(98)00122-3 0360-8352/98 $19.00 + 0.00

R E S O U R C E S M O O T H E N I N G IN R E P E T I T I V E P R O J E C T S M. H. Elwany L E. Korish M. Aly Barakat S. M. Hafez Professor

Professor

Associate Professor

Research Aasiztant

Faculty of Engineering- Alexandria University

ABSTRACT To manage construction projects properly, it is necessary to establish a project plan for resources utilization. This management function, usually, is carried out in two steps: resource allocation and resource usage smoothening. In this paper, a linear programming model for a single resource allocation and smoothening in repetitive construction project has been achieved. The linear programming model is based on one of the following objectives: 1. The resource requirements nearly follow a desirable resource histogram, Model M1. 2. Minimizing the number of changes in resource requirements, Model M2. © 1998 Published by Elsevier Science Ltd. All rights reserved.

Keywords: Construction management, resource leveling, repetitive construction projects. 1-1141 and M2 MODELS ASSUMPTIONS In the development of the linear programming models it is considered that; activities are assumed to be contInuous and resource requirements for each activity is assumed to be uniform along the activity duration.

2- M O D E L I N P U T The input data for all models are: Number of activities N, time domain, total and free floats, resource usage rate, project duration PD. (Korish, et al 1997), and the target resource profile configuration, for M1 Model. 3- F O R M U L A T I O N O F M1 M O D E L 3-1 Decision Variables The decision variables of M1 model are grouped into two basic categories: 1. The amount of activity time shifting, 8 k ; where ~ k < TFk . TF't

~k = ~ i ' x k ( i )

where;xk(i) ; xk(i) e(0,1).

(1)

I=l 71"F'A

And ~ X k ( i ) ~ 1

(2)

I=!

2. The deviation between the resource requirements V ( t ) and the target resource profile ,B(t) at each time unit t where ( t = 1,2.....PD), is y(t) ,where; y(t) =

p(t)

(3)

Variable, y(t) ,is the unrestricted and can have a negative value, y ( t ) , is expressed in terms of two non -negative variables as; y ( t ) = y'(t) - y"(t) where; y'(t),y"(t) > 0 (4) 3-2 Objective Function To minimize the deviations Y(O between the resource requirements, V(t) ,and the target resource, fl(t) ,profile the objective function Z is;

415

416

Selected papers from the 22nd ICC&IE Conference PD

Z = Min Et=lly(t)l

Z = Min E,.'~ {y'(t)+ y"(t)

}

(5)

3-3 M o d e l C o n s t r a i n t s

3-3-1 Network logic constraints The starting time for activity, k, should satisfy the following constraints (Korish, et aL - 1997)); TF'~

(6a)

E Xk (i) < 1 i=l TFk

TFat

(6b)

i.x, (i) - ~ i . x . (i) < FF,

i=1

Iffi l

3-3-2 Target resource profile constraints The deviation, y ( t ) ,between the resource required, V ( t ) , and the target resource level, f l ( t ) , are y ( t ) = ~ ( t ) - ~ ( t ) . By substituting for y(t) , V ( t ) - y ' ( t ) + y " ( t ) = fl ( t ) The required resource, V(t) at each unit time along the project duration is: RR k.(l-

rp,

Zt=~t_tt,) xt(i))

(7a)

: t l k < t < t 2 t a n d t < t3 k

k=l IV

E

: t l k < t < t 2 k a n d t > t3 k

RR k

k=I IV

~, (t)

t-tij-!

=

X

RR k .(1 - E ~=(,-tz~) k (l))

: t 2 k < t < t 4 k a n d t < t3 k

RRt " ( X re~ t=¢t-r2.) xk ( i))

: t 2 k < t < t 4 k a n d t > t3 k

k=l IV

2

k=!

0.0

: otherwise

(7b) 4- RATE OF RESOURCE CHANGE The rate of resource change, U(t),at time t is the difference between the resource requirements at unit time t and ( t - 1) where ( t = 2 ..... PD). U(t) = ~'(t) - ~'(t - 1) (8) The unrestricted variable, U(t) ,can be expressed in terms of two non negative variables: where ; U' (t),U" (t) > 0 (9) The value of V ( t - 1 ) can be obtained by substituting time t by ( t -1) in Equation (7b): U(t) = U'(t) -U"(t)

~

RRk'(1-

TFj Z i.~t-l-tt.x~k ( l ) )

: tlk < (t-

1)< t2 k and (t-l)

< t3 k

k-t IV

E

RRk

:tlk < (t-

kml IV ¢, ('t - z)

=

t-tit-2

Z RR,.(Zk=l

1)< t2 k and(t-I)>

:t2 k < (t-

1)< t4 k and(t-I)<

t3t t3 k

N

5". " " . . ( 2 k=l

0.0

""

:t2 k <(t-l)<

t4,

and(t-l)>

: otherwise

( 10-a,b,e,d,e)

t3 k

Selectedpapersfrom the 22ridICC&IEConference

417

4-1 Case o f t 1 k < t < t 2 t a n d t < t 3 k The resource requirements, g/ ( t ) ,at time unit t as ¢/ ( t ) = ~

I~R~.(1-

k ffi l

x~ (i) ) t ffi l - t l j

4-2 Case o f t l k < t < t 2 kand t > t 3 k The resource requirements, ~, ( t ) ,at time unit t is; N

~/(t)= Z

RRk

k=l

The considered time unit unit (t-l) should be less than t2k, the activity time domain t3 k > tlk. Hence, for all cases the time unit (t-l) will be greater than or equal tl k . The previous mathematical relation can be summarized as follows; t l k < ( t - 1) < t2 k 4-3 Case o f t 2 k < t < t 4 kand t < t 3 k The resource requirements V ( t ) at time unit t is; q/ ( t ) =

R R k. ( 1 k:]

x k(i) i:t-t2~

4-4 Case of t2 k < t < t4 k and t ~ t 3 , The resource requirements ~ ( t ) at time unit t can be calculated from the following Equation, ~, ( t ) = ~.

Rx..(1-

k:l

~,x,(O i=l-t2

)

i

It is clear that for all cases the time unit (t - 1) is equal or greater than t l , , and less than t4 k . 5- F O R M U L A T I O N O F M 2 M O D E L The objective function for this model is to smooth the resource profile. This objective was achieved by minimizing the number of drops in the resource requirements along the project duration. The mathematical model is composed and developed using F O R T R A N program (Korish, et a11997),. L I N D O is used to solve the model. 5-1 Decision Variables The decision variables in M2 model can be grouped into two categories: 1. The amount of activity shifting, ~ k ,beyond the early start time ES~o. 2. The rate of resource change, U ( t ) , at time t where ( t = 2 ..... P D ) , varies depending on activities shifting, 8 k , between Us~n ( t ) a n d Umax(t). The variable U(O is an integer and can be formulated as, u..(t)

U(t)=

~

i.O,(t)

,and O,(t)~(0,1)

(11)

t=U.~.(t)

where; 0 i ( t ) is an integer binary variable; and Umi.(t) and Umax(t) are the minimum and maximum change in resource requirements calculated from unit time t and (t - 1) where; t =2 ..... DP. To insure that only one variable of 0 i ( t ) is considered at a time, this requires the following constraint, Zv...(t)

i=U.J t )

Ot(t )

< 1

(12)

The variable Or(t) takes a value equal one, only when a drop in resource requirements occurs between time unit t and (t - 1) ,otherwise it takes a value equal zero.

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5-2 Objective Function To minimize the number of drops in the resource profile, the objective function Z will be; Z=Min

(13)

E,I"% ~i'=~(t~,) Or(t)

5-3 Model Constraints The model is subjected to the following constraints: 5-3-1 Network logic constraints To achieve the network logic, the following constraint previously illustrated in Section 4-3-1 should be considered. TF',

~x,(i)<_l l=l

TF,,

TF,

i.x,(i)- ~ i.x.(i) < FF, I ffil

I= l

5-3-2 Resource constraints The variable U(t) previously formulated using the resource requirements, ¥ ( t ) , in Equation 11 to 14 should be subjected to the following constraint, U. ( t )

U(t)= ~

LOt(t)

iffiU~,(t)

The number of constraints needed for M2 model depends on the project duration PD, number of paths in the network, and the number of activities, N , in the network. List of Notations

PD

Optimum project duration

PR,

Production ratefor activity k

Start node numberfor activity k

Q,

End node numberfor activity k

~t'

k k' kk

Activity code number

Rk

The set of activitiesfollows the activity k The set ofproject activities, which the end nodes Jk'=Maxj Total required resource

Number of last activity in each path

RR k

Thefollowing activity where kk ~ Qk

STk.~,)

m

Stage serial number m = 1,2...... S Number of activities in the network

S

D k

ES,(o

N

Specified duration of the activity k Early start timefor activity k in stage 1

STk( 0

Resource rate per unit timefor activity k Start time for the last activity in the network at last stage Number of stages in the project Starting timefor activity k atfirst stage

Selected References: I. Korish, L E., Elwany, M. 11., Barakat, M. A., and Hafez, S. M. (1997)" Time Cost Trade Offln Repetitive Projects" Alex. E q g . Journal, 36 (I), C39-C47. 2. Korish, L E., Elwany, M. EL, Barakat, M. A., and Hafez, S. M. (1997) "Resource Allocation in Repetltlve Projects" 6th PEDAC conference, 819- 830.

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