th
19 International Conference on Production Research
SUNK COSTS / CRASH COSTS / TIME TRADE-OFFS: CONTRIBUTIONS FOR THE
PROJECT MANAGEMENT
M.B.F. Lima, A. Subramanian, G.R.S. Gomes, L.B. Silva
Department of Production Engineering, Federal University of Paraíba, Centro de Tecnologia – Campus I –
Bloco G – Cidade Universitária, Castelo Branco, CEP: 58051-970, João Pessoa, Paraíba, Brazil
Abstract:
This paper seeks to add contributions of the innovation and industrial Economics to more used techniques
of crashing, in the Projects Management domain. First, it presents the brute force method, improved for the
use of the MSProject. Second, it develops a linear program approaches for determining the earliest crash
completion time and for determining a least costly crash schedule, for the same home-building project used
in the brute force method. Third, it establishes costs laws, which allow inferring that the cost of a project
does not depend only on the production rate but depending also on the time were the first unit of production
will be available, on the global volume of production and on the project completion time. Concluding, it is
pointed out the importance of the project management concerning the search of dynamic and innovative
solutions to the more important productive problems of the emergent economies.
Keywords:
Project Management, Laws of Cost, Sunk Costs, Crash Cost.
1 INTRODUCTION
Crashing the activities of a project relates to the costevaluation of reducing the duration of those activities
which are in the critical path. After this evaluation, the
activities that correspond to the lowest cost for crashing
should be worked on. This means that the addition of
more financial resources, manpower (extra hours, for
example), materials or equipments, will cause an increase
in the project’s budget [1].
The second section of this paper presents the brute force
method which is used to crash a house construction
project, an example extracted from Ragsdale [2]. It is
important to emphasize that the MSProject speeds up
considerably the development of this method, as also
many applications of several others methods of planning,
programming and controlling the projects ([3], [4], [5]). It is
also important to point out that the technique mentioned
above is fully accomplished by the MSProject.
Section three presents some linear programming
approaches with a view to determine the least possible
time for a project’s completion; and to program the
project’s crashing that would implicate the least additional
cost, given a constraint imposed by contract’s condition,
for instance, a pre-determined completion time. Some
trade-off discussions concerning the crash costs, and
project’s duration are also carried on. According to [6],
trade-off, in economy, means the situation where there
are conflicts of choices, i.e., when some economic action
that aims to solve certain problem leads, inevitably, to
other ones.
Section four, deals with costs and products from a
temporal point of view, within the scope related to the
Industrial Economy and Innovation, by establishing socalled laws of cost that enable profound reflections
concerning the project’s crashing.
In conclusion, it is evident that, in the era of knowledge,
the role to be played by the Project Management in this
process of decision-making will be more and more
relevant.
2 THE BRUTE FORCE METHOD
This is the most known method, used for the crashing of a
project, approached by various authors, among them
Casarotto Filho [7].
The method is simple; however it takes much time and
allows the occurrence of mistakes, which makes it
inadequate to big projects. The steps of the brute force
method are:
1. It is chosen the activity with lesser daily cost of
crashing;
2. This activity is reduced to one time unit;
3. Table 1 is used to check the value of the daily
compression cost related to the activity chosen;
4. The start and finish times of the network are
recalculated. It is also checked if the reduction of one
time unit, in the duration of the accelerated activity,
has not modified the critical path;
5. Actions in the items 2, 3, and 4 are repeated until the
duration of maximum acceleration is obtained or until
a new critical path appears in the network;
6. In case of a new critical path appears in the network,
there might be two parallel critical paths. Thus, in
order to reduce the project duration in more one unit
of time, it is necessary the simultaneous compression
of the critical activities that have a lower daily
compression cost in each critical path. By doing this,
two parcels are being summed up to the project costs.
It is necessary to verify if the sum of its daily costs is
smaller than the marginal costs of another critical
activity not accelerated yet.
Table 1 and 2 summarizes the data related to the home
construction project, which is an example extracted from
Ragsdale [2]. Table 1 also brings the results obtained
after implementing the Brute Force Method. It can be
seen that the duration of the project was reduced to 28
days, leading to a additional crashing cost of $18.334, 00.
th
19 International Conference on Production Research
Table 1: Duration and normal and accelerated cost of the residence project
NORMAL
ACTIVITY
DESCRIPTION
A
B
D
Excavate
Foundation
Lay rough
plumbing
Frame
E
ACCELERATED
DAILY
CRASHING COST
Duration
(Days)
3
4
Cost
($)
5.000
12.000
Duration
(Days)
2
3
Cost
($)
6.000
15.000
3
3.000
2
3.500
500
10
20.000
6
25.000
1.250
Finish exterior
8
8.000
5
10.000
667
F
Install HVAC
4
11.000
3
12.000
1.000
G
H
I
J
K
Rough electrical
Sheet rock
Install cabinets
Paint
Final plumbing
6
8
5
5
4
3.500
5.000
8.000
4.000
7.000
4
5
3
2
2
4.500
6.500
9.500
5.500
8.500
500
500
750
500
750
L
Final electric
2
2.000
1
2.500
500
M
Install flooring
Total
4
46
10.000
98.500
2
28
12.000
116.834
1.000
C
1.000
3.000
Table 2: Activities for residence project and its predecessors
Activity
A
B
C
D
E
F
G
H
I
J
K
L
M
Predecessor
-
A
B
B
D
D
D
C, E, F, G
H
H
I
J
K, L
Variables:
Ti
Instant when the activity i starts
ti
Regular time of the activity i
Ci
Crashing time of the activity i
Model:
Min TM + t m − C m
(1)
Subject to:
TB − T A ≥ t A − C A
(2)
TH − TC ≥ t C − CC
(8)
T J − TH ≥ t H − C H
(13)
TC − TB ≥ t B − C B
(3)
TH − TE ≥ t E − C E
(9)
TK − TI ≥ t I − C I
(14)
TD − TB ≥ t B − C B
(4)
TH − TF ≥ t F − C F
(10)
TL − T J ≥ t J − C J
(15)
TE − TD ≥ t D − C D
(5)
TH − T J ≥ t G − C G
(11)
TM − TK ≥ t K − C K
(16)
TF − TD ≥ t D − C D
(6)
TI − TH ≥ t H − C H
(12)
TM − TL ≥ t L − C L
(17)
TG − TD ≥ t D − C D
(7)
Ci ≤ Maximum crashing time available for the activity i
Ti , C i ≥ 0
(18)
(19)
Chart 1: Linear Programming model
th
19 International Conference on Production Research
3
LINEAR PROGRAMMING MODELS RELATED TO
THE PROJECT CRASHING
3.1 Determining the earliest crash completion time
According to Ragsdale [2], the projects manager might
want to know the earliest date necessary to accelerate in
order to satisfy the deadline. The formulation of a linear
programming model to provide the solution to this
problem is presented as follows in chart 1.
The objective function (1) attempts to minimize the total
amount of time necessary to finish the project. Constraints
(2) – (17) make sure that a certain activity starts only after
its predecessors have been completed. Restriction (18)
guarantees that the crashing time of an activity does not
violate the established limit. Finally, expression (19) is
related to the nature of the variables.
It is observed in the implanted model that the project
duration of 28 days, corresponds to a crashing cost of $
19.000, 00, lightly different to that obtained with the brute
force method ($ 18.334, 00).
3.2 Including a time limit for concluding the project
It is important to point out that in some situations, the
time specified in the contract (Tp) can be smaller than the
normal execution time (Tn) of the project, in other words,
without crashing. Besides, a penalty value for each day
passed over the established date is also considered.
These characteristics lead the model to a closer proximity
to the reality of the projects.
However, conflicting problems come up concerning the
total cost and the project crashing time, i.e., it might occur
a trade-off between the elements mentioned. Thus, the
following question has to be answered: is it worth to
sufficiently crash the project in order to achieve the Tp or
just crash or not a certain amount of time and pay the
penalty?
It can be verified that the model presented previously
does not attend the conditions mentioned above. Thus,
that objective function (1) has been adapted in order to fit
the characteristics related to the project’s time, specified
by the contract, and the existence of a penalty P.
Therefore:
M
Min
∑CC ⋅ C − P(T − T )
i
i
e
p
(20)
i= A
where
CCi
is the crashing cost of the activity i, and Te is
the execution time of the project (where Te ≥ Tp).
The new objective function (20) tries to minimize the sum
of the crashing costs and at the same time adds a daily
penalty constant if Te > Tp. In other words, whenever the
Te increases by one the objective function increases by P.
The model was implemented using the software LINDO
6.1 and the following results were found:
Project’s time = 43 days;
Crashing Costs = $ 1500, 00;
Penalty = $ 1500, 00.
There are some situations where arises the necessity to
accomplish some specific activity in a certain time horizon
determined by a contract. Suppose that, in the house
building’s example, the activity D has to be completed
until the 16th day of the project. For that, the following
constraint has to be added.
Td + td − Cd ≤ 16
(21)
The results obtained after including the restriction (19)
were:
Project’s time = 42 days;
Crashing Costs = $ 2500, 00;
Penalty = $ 1000, 00.
3.3 Considerations
It can be observed that each one of the methods
presented in this paper, has distinct characteristics to
solve the same problem. It is not the objective of this work
to discuss which is the most efficient, but the most
effective according to the project’s manager reality.
Matta and Ashkenas [8], discuss the use of project plans,
chronograms and budgets by managers so as to reduce
the “execution risk”. This risk is related to the possibility
that the designated activities are not executed
appropriately.
The authors further observe that many decision-makers
neglect two other critical errors: “blank-space risk”, which
is associated with the fact that some activities may not
have been predicted and therefore are not included in the
original project; and the “integration risk” which relates to
the situation where activities that are very different are not
completed nearly simultaneously at the end of the project.
Thus, one can observe the great potential of the use of
the methods presented here as instruments of support for
Project Management and minimization of errors
mentioned above, each conserving their peculiarities.
The linear programming presents several advantages.
For example, it permits construction of flexible models
which take into account the specific characteristics of a
given project. Some of the disadvantages include the
need to have adequate know-how to build up the model,
knowledge of the mathematical programming techniques
as well as the related softwares used to implement the
models.
4
COSTS AND PRODUCTS IN A OPTICS ALONG
THE TIME: THE NEW CONCEPTION OF THE
NOTIONS OF COST AND PRODUCT
The costs and the products analyzed under a dimension
along the time, have a completely different conception
from the usual one. This other conception has, as a
specific feature, the fact that it inserts the capital theory in
the core of the firm theory; it allows defining laws or
propositions that are coherent with the empiric
observation which, in turn, most of the times, concern the
irregular systems of production, whose revenue is not
synchronized with the expense [9].
In the regular or permanent production regimen the
industrial company is technologically effective, in such a
way that the supporting costs during the relevant period
are, analytically imputable to the current production.
Basically, the incomes are synchronized with the
expenditures and the temporal dimensional is not
significant to the cost analysis. The companies that
operate on this regime consist mostly of industries where
operations in chain provides manufacturing cycles of
products and their lead times are denominated
elementary periods.
On the other hand, in the irregular or transitory regimen of
production, the company is not very effective
technologically since it is usually in a situation of
construction of a new production capacity where the funds
are not entirely used due, in some cases, to the problem
of learning a new technology. Here, the concept of funds
(in opposite to flows) corresponds to the elements that
th
19 International Conference on Production Research
participate in the production process without being
destroyed, for instance, the land, the physical capital
(buildings, machines and equipments) and the human
resources.
Therefore, in the irregular regimens there exists an
imbalance between the cost and production profiles that
do not allow the current costs to be analytically imputed in
the current production.
This fact would make the
temporal dimension consideration unequivocal.
In the context of such irregular regimen, a well defined
production function does not exist due to the fact that the
technology does not exist in principle and is in the
process of constitution. The adequate analytical
instrument seems to be a function of cost that, naturally,
is not a derivative of the function of production anymore
and, therefore, does not exclude situations of
technological efficacy. This cost function should help to
point out the role of time, in such a way that it is enrolled
in an analytical perspective which is related to the
temporal models of production.
4.1 The cost of production and modification of the
capital value
The way in which a temporal dimension can be introduced
consists in defining the costs, like a modification in the
company’s social capital value, resulting from a particular
operation, assuming the change in income is omitted in
the calculation of the variation of the social capital value.
An example to illustrate this definition is as follows:
presuming that at the beginning of an operation the actual
value of the assets is 100 in a company and will be 80 at
its conclusion, not taking into consideration the selling of
the products of this operation in the market; the actual
value of 80, where for example the updating rate is 6%
will be 75.45. The operational cost in the present capital
value is 25.43 (100 – 75.47).
A cost expression in social capital unit is inserted here
using a “funds” approach for company evaluation as
opposed to materialistic approach used in the present
case.
Roughly speaking, the “funds” capital approach is based
on the fact that, under a strategic point of view, the
detention of the fixed asset by the companies constitutes
for them a irreversible employment in relation to the future
benefits the shareholders will be expecting. This is true
especially in those cases where the detention of the
machines becomes imperious due to the difficulty in
replacing them in a short space of time by the market.
This is equally true in those cases where the human
resources are considered to be like collective ones.
The materialistic approach basically consists in the fact
that the capital is inassimilable in the immobilized assets,
such as depreciation funds which are money reserves
established by the companies and regulated by law. This
is overcome by the company’s depreciation rate of the
fixed asset (real estate, machines and equipments) and is
destined to renovate this asset.
A cost expression in social capital unit is in matter of fact
coherent with the idea that the capital is inassimilable in
immobilized assets, but constitutes a monetary value
which is solely the future beneficiary values the
shareholders will be expecting.
Each of these characteristics mentioned to designate a
production operation will affect the cost of production. In
the case of project construction of a single product, for
example a ship, x(t) will be equal to 1 unit/year, T and m
will coincide in time, due to the fact the formulas below
will not be of much use. In the small several residential
construction projects like the case presented in this
article, for example, T can be 45 days, x(t) will be 9
units/year and m will be equal to 40 days.
The production is an operation that flows along the time.
That is to say, it is like a program whose characteristics
are the following:
1. A production tax (x) which is generally the only aspect
considered in the economics standard analysis;
2. The total production volume (V) accumulated during the
production program;
3. The completion time of the production (m).
These three characteristics are summarized by the
following formula:
T +m
V =
∑ x(t )dt
(22)
T
Where V is the total production volume, x(t) is the
production tax in the t moment, T is the moment in which
the first product unit is given, and m is the time interval
during which the production is made possible.
Each of these characteristics mentioned to designate a
production operation will affect the cost of production. In
the case of project construction of a single product, for
example a ship, x(t) will be equal to 1 unit/year, T and m
will coincide in time, due to the fact the formulas below
will not be of much use. In the small several residential
construction projects like the case presented in this
article, for example, T can be 45 days, x(t) will be 9
units/year and m will be equal to 40 days.
4.2 Production costs out of the permanent system:
the laws of costs
On the basis of the precedent observations, Alchian [8]
elaborates a set of propositions on the manner by which
the costs are affected by a variation of these variables or
characteristics. Naturally, among the variables V, x, T e
m, only three of them are independent, being the fourth a
restriction contrarily to the permanent system where the
four characteristics are invariable (this justifies the facto
that only one of them is retained during the analysis). Let
C be the cost function, namely, of modification of the
social capital value, in a way that: C = F (V , x, T , m)
Proposition 1:
∂C
>0
∂x(t )
T = T0 ; V = V0
(23)
The value of the costs increase as the tax x, under which
a certain volume is produced, is higher, the period of
manufacturing of product m, is consequently reduced. For
example, in the residential project mentioned in this
article, let T remain as equal to 45 days, and V being fixed
as 11 residences, x(t) will change from 9units/year to
11units/year, thus m will be reduced from 40 days to 32
days, due to this the company has to support higher costs
for extra working hours.
Proposition 2:
∂ 2C
∂x 2
>0
T = T0 ; V = V0
(24)
The growing of the costs is a growing function in relation
to the production tax. That is, if the additional cost
incurred due to changing x(t) from 9 to 11 houses/year is
$8,000.00, the change of x(t) from 11 to 13 houses/year
will incur a cost greater than $8.000,00, with T = 45 days
(fixed) and fixing V = 13 units.
Proposition 3:
th
19 International Conference on Production Research
∂C
> 0 x = x0 ; T = T0
∂V
(25)
The cost increases with the production volume for data x
and T, consequently increasing the period of availability,
m. In the present case, if x(t) = 11 houses/year and T is
maintained constant in 45 days, the change from 11 to 13
units/year will clearly increase the cost.
Proposition 4:
∂C
V <0
∂V
T = T0
(26)
The growing of the costs diminishes as the production
volume increases. It is the phenomenon known as
economics of scale which can be obtained from an
experience curve as well as from the learning process
that the human resources acquire along time.
5 CONCLUSION
Speaking generally, one can assume that the Project
Management can be directed mainly to the solution of
production problems concerning innovative situations
found in organizations whose tasks present a high
innovation content level.
Additionally, according to Casarotto Filho [7], Projects
Management might considerably help organizations that
search for a better market position, providing a higher
significance to the competitive advantages to be
developed by the company along the time, such as: costs,
quality, speed (time), reliability, flexibility and innovations,
in a growing importance order concerning aspects of the
difficulties which the management of complex systems
are submitted [11].
It is also important to mention the trade-off relevance
between the project crashing costs, the completion time
and the sunk costs. This way, the linear programming
models presented in this paper are essential tools for
obtaining useful references to the planning, programming
and control of the productive systems. However, the
project crashing estimated cost hardly presents the same
value as the incurred actual cost.
The consideration of the costs laws might contribute for
the fulfilling of this gap, under the analysis of other
important variables, such as the global production volume
and the time in which the first unit of product will be
available for commercialization.
When dealing with the construction project of a single
item as in the case of ship production, these variables are
not of great importance. However, for the house project
mentioned in this article variables V and T can turn out to
be interesting.
Thus, the average cost of production program decreases
when the global production volume (V) and the rate of
production x(t) increase, maintaining T and m constant. In
the same manner, the cost of production program
decreases when the variable T increases, reducing in that
case the production rate x(t), provided all the remaining
variables being constant. This occurs as a corollary to
proposition 2. In effect, higher the T value, lower will be
the rate value to which the inputs are compared, therefore
also their prices, because the salesmen expenses are
lesser on applying proposition 2 to them and thus lesser
will be the cost of program production.
6 REFERENCES
[1] Menezes, Luís C. de M. Gestão De Projetos. Editora
Atlas, 2ª. Ed., São Paulo, 2003.
[2] Ragsdale, Cliff T. Spreadsheet Modeling and
Decision Analysis – A practical introduction to the
management
science.
Southwestern
college
publishing, 3a edição, New York, 2001.
[3] Vargas, Ricardo V. Microsoft Office Project 2003 conhecendo
a
principal
ferramenta
de
gerenciamento de projetos de mercado. Brasport,
Rio de Janeiro, 2004.
[4]
Writh,
Almir.
Planejando,
replanejando
e
controlando com msproject 2000. Editora Book
Express, 2ª. Ed., Rio de Janeiro, 2002.
[5] Figueiredo, Francisco C. de, e Figueiredo, Helio. MS
Project 98 utilização na Gerência de Projetos.
Editora Infobook, Rio de Janeiro, 1999.
[6] Sandroni,
Paulo.
Novíssimo
Dicionário
de
Economia. Editora Best Seller, 3ª. Ed., São Paulo,
1999.
[7] Casarotto Filho, N. Projeto de Negócio: Estratégias
e Estudos de Viabilidade. Editora Atlas, São Paulo,
2002.
[8] Matta, Nadim F. and Ashkenas, Ronald N. Gestão e
implementação de projetos. Editora Campus, Rio de
Janeiro, 2005, (p. 7 – 22).
[9] Gaffard, J.L. Economie Industrielle et de l'innovation,
Paris, Editora Dalloz, 1990, 470p.
[10] Alchian, A. Economic Forces at Work, Indianapolis,
Liberty Press, EUA, 1977.
[11] Lima, Márcio B. Da F. Groupware, uso das
Tecnologias da Informação e Organização do
Trabalho: Contribuições à Economia da Inovação.
Tese de Doutorado, UFSC, Florianópolis, 2002.