. 187
5..1 EXPEDI TI NG A PRO J ECT
700
6
o
a
'o-'
600
E
500
17
20
19
18
(days)
Projectduration
Figure 6-6 Project cost vs. project
duration for sample crash problem.
to Crasha Project
Using Excel@
ffiiii::#;#lTlti.*?*:j,'i#,rffi
T"
llt
ttvolved inLcrashins.occasionallyrnurryur. i""otu"J'
of
how
use
we
demonstrate
quite
tedious.In this section,
become
6E"r*A;r]Fi"n
ivities to crash so
Excel@spreadsheetscan great
that a project is completed
To illustrate this, the data
;6GTTGG--tt.---into-t-hespreadsheetshown
in Table 6-2.Figure 6-7 shows the network diagram for this project based on the activa
do make one change to the data in-Tqble 6'1. Namely,
here can handle
not at all, we make this
5
>
E
change to simplify our ensuing discussion
At the top of the spreadsheet
and the
calcugiven in Table
by dividing the incremaximum number of
possible
the
by
mental cost of crashing the activity as much as
days the activity can be shortened.
In column H the maximum amount each activity can be crashed is calculated by
subtracting the crash duration from the normal duration. Column I corresponds to our
decision, namely, how much to crash each activity. Then based on the values entered
in column I, the cost of crashing each activity is calculated in column J. Finally, in column K the actual time to complete the activity is calculated by subtracting the amount
is entered
]:
node in the networ
you
occurs at ttme zero.
ttme each event occurs to ensure that the
ode 1 is excluded because we as'
see. we need to keep track of the
..relationships-in the network
r iw\fih'rt
C H A P T E R 6 / A L L O C A T I N G R E S O U R C E ST O TH E PR OJEC T
Table6.2
ple "crash" Problemin Table6-1 Transferred
to an Excel@spreadsheet
e{il
${r
6'
w
:(F6-E6y(C6-D6)
{copyto cellsGZ:G12}
=C6-D6 {copyto cellsH7:H12}
=I6*G6 {copyto celis l2}
J7:J
:C6-16 {copyto cellsK7:K12}
diagram are not violated.
we need to make sure that node 4 does not occur
A-'
until after node 2 occurs.
to
rate how Excel's@Solver can be used to
rmine which ac.
tivities to crash so that the entire project is completed withi
the minimum
costs.To begin, selectTools from the men., baiand then Solve#
,n. rr.*i rn".,u
that appears.The solver Parametersdialog box is now displayed(seeFigure6-8). The
Figure 6'7
problem.
AOA network of sample "crash"
. 189
6.1 EXPEDITIN G A PRO J ECT
Figure6,8 Excel's
Solver@loaded
with data and constraints from crash
problem.
refersto the cell in the spreadsheetthat we would like to either
Tareet Cell
S"t fr"g"fg[Ield
Set
mlnlmlze or maxrmrze.In our case,we would like tGGrE]F)t he total cost of completing the project, which is calculatedin cell B2. To specifythis, we- enter 82 in the Set
Target Cell field and then selectthe Min radio bRtton.
Next we tell Excel{what cells it can changelin orderto find the solutionwith the
minimumtotalprojectcffieadsheetshowninTab1e6.2,theva|.
;':"'5;xi::'?ifr
il'*Hi+Til'::il,T:'*1J
lfiii,:l;:l:',n:i"':H:"f
ure 6-8, these ranges were entered in the $y Changing Cells field.
onw eneedtoenteri sthel i mi tati onsorconstrai ntsasso.
ciated with this situation. Perhaps the most obvious constraint is that we want to complete rhe project within 20 days (cell B1). Since node 6 (cell B21) correspondsto the
event of the project being completed, we can specify this constraint as follows:
{-nzi=fl
Another important set of constraints is needed to make sure we don't crash an ac'
tivity more than the maximum number of days that it can be crashed. Constraints to
ensure this could be entered as follows:
16 < H6 (activity a)
I7 < H7 (activity b)
IB < HB (activityc)
19 < H9 (activity d)
I10 < Hl0 (activitye)
I11 < H11 (activityf)
ItZ = Hlz (activity g)
Altematively, by e ploying a shorthand approach that capitalizes on a spread'
sheet's ability to deal w h ranges, these seven constraints could be entered as a single
constraint as
<H6:HI7
16.112
Another ser of constraints is needed to make sure that the precedence relationships
specified in the network diagram are not violated. We do this by keeping track of the
19O o
C H A P T E R 6 / A L L O C A T I N G R E S O U R C E ST O T HE
PR OJEC T
event times of the nodes.For example,the eventtime
of Gaannot
occur unril
after activity a has beencompleted(assuming,tt"i rtr" pr":"ffit.,,
at time zero).The
time to complete activity a is its normal time lessth"
time it is crasheJ
Since cell 817 correspondsto the event time for .,od" j, ".o,r.riof
*"iffi"ri."lly
we could enter
this constraint asfollows,
a
Bl7>C6-t6
This constraint saysthat the event correspondingto
node 2 cannot occur until after activity a hasbeencompleted.
constraintt fo@ffi.rE$$could
be createdin a similarfashion.For example,
theconsrraintsfor(fi6Gj-6iiff?]-o-'.rldb.
B1B>C7_t7
B19>B17+CB_I8
The constraint for node 4 saysin effect that the event
correspondingto node 4 cannot
occur until after the evenr coffespondingto node 2 (cell
Bftj-*.,r^
plus the time it
takesto completeactivity c.
Moving on to node 5, nore that this node has.two_
arrowspointing to it. A node
with more than one arrow poinring to itrvill
riri.uirr, for each arrow.
"..!_g
Thus we need the following two constrai",, f".6iF"T)
B20>B17+C9_t9
B2O>81B+CiO_I10
This first constraint saysthat node 5 (cell B20) cannot
occur until after node Z has occurred(cell 817) plus the amount of time it takesro.o*pt.*
u.,irriry d. The second
constraint saysthat node 5 cannot occur until after node (cell
3
B1B) has o..rrrr"a ft,r,
thea.qmouEof-time
it takesto completeacrivrrye.
Nd9r
handledin a similarwayto node5 asfollows:
I
8 2 1> B 1 9 + C1 1 -t l1
B 2 I > 8 2 0 + Ct z -t I }
Finally, since it does not make senseto crash an
activi
nor does it make sense for a
4g5[to_oqcur at a tdleG
zero, we add conUsing Excel's@ shorthand ao-
16.112> 0
B 1 7 : B 2>1 0
In this examplewe assumethut th. iai"itioiii-b"
Gt
rr"ction of a day. If
th,a;theactivities
hadto t" .r"rr,.J "a-"l'uhot. dayor notat
:i
all,|::Y:1:"^"1?:."_T:
we couldeasilyaddadditionarconstraintsto the model "irr,o
ro r"R"., it ii;i#;:::'"'
To
enter theseconstraints,selectthe Add button i" ,rr.
sofJ"lt to th. Constraints
section of the Solver Parametersdialog tro*. rne entire
set oT.orrrtruints neededis as
follows:
(
I
I
I
I
nn =nt
< H6:HL2
16.112
817>C6- t6
nrs>c7-t7
8 1 9 > 8 1 7 + CB _ I B
. 1 91
EXPEDITING A PRO J ECT
II
I
I
i
gzo>B1z+c9-19
B20> BlB + C10- I10
BZr> B19+ Cl1 - I11
+ cr?.- rrz
nzr> B2o
> 0
16:112
I
(B 1 7 : B 2 1> 0
After entering theseconstraints,the Solver Parametersdialog box appearsas shown in
Figure6-8. Beforefinding the leastcostly way to crashthe projectdown to 20 days'seleJt the Qptions . .. brriton and click on the Assume Linear Model check box and
then click bK. No* to find the leastcost solution, selectthe Solve button in the Solver
parametersdialoe box. As is shown in Table 6-3, Excel@identified the samesolution
Tabfe 6-3
for CrashProblem,20-Daylolution and Cost
Excel@
Spreadsheet
(F6-E6V(C6-D6) {copyto cellsG7:G12}
-D6 {copyto cellsH7:H12}
:16*G6 [copyto cellsJ7Jl2]
=C6-16 {copyto cellsK7:K12)
192 o
C H A P T E R 6 / A L L O C A T I N G R E S O U R C E ST O T H E PR OJEC T
; 700
;
I 650
E ooo
'6'
o
F
500
450
Figure 6-9
Cost/duration graph
crashing
Project (
3:'.:rr,T
-.,-
that we obtained earlier using the manual approach.Specifically,the solution suggests
crashingactivity a by one day (cell 16).This resultsin completingthe projectby day 20
(cell 821) at a total projectcostof9430 (cell 82).
Having set up the spreadsheet,
we can now quickly and easilyevaluatethe cost of
:T
ffitilru::ili:il:;.Tt::,f;,':x.,n'L.':iTi;i:r#u,
ffi
#ryJ;ffii"a.g.
Fast-Trackinga Project
In addition to crashing a project in order to expedite it, a project may also
Used primarily in the construction industry, the term refers toane
ofa
f,
leted beforethe building phaseis started.Uzually designana phtt aie finished before
;
;i
-
-
----
--^---^-b
r^rqv!
!v
vsqrlvs
the building is started, soTetting them overlap reduces project duration-if the fact that
design and planning are incomplete does not result in a significant amount of rework
and ghange orders during the building phase.
a large proportion
number of change orders in fast-tracked construction projects is not significantly different
from that for similar projects that were not fast-tracked (Kurtulus and Narula, l9B2').
\7hen task durations are estimated, an assumption is made that task resources
are set at "normal" levels. This is the "standard practice" assumption. Traditionally, CPM project duration estimates also include a "crash" estimate together
with estimates of the crash time and the resourcesrequired to shorten the duration of project activities. By selectively choosing which activities to crash and
by how much, we can determine the minimum cost for all possible project completion times. Both manual and spreadsheetmethods are illusffated.
LoAptNG
ffind REsouRcE
From the first day on the job, the PM is concemed with resowrceloading.Resource loading refers to the amounts of specific resources that are scheduled for use on specific activities or projects at specific times. It usually takes the form of a list or table. Figure
6'10 is an MSP generated action plan and Gantt chart of a project aimed at producing a