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Engineering, Construction and Architectural Management 2000 7 1, 93103

Assessment of working capital requirements by fuzzy set

theory

V E L L A N K I S . S . K U M A R*, A W A D S . H A N N A & T E R E S A A D A M S

*Department of Civil Engineering, University College of Engineering (Autonomous), Osmania University, Hyderabad500

007, India; Department of Civil and Environmental Engineering, 2260 Engineering Hall, Madison, WI 53706, USA

Abstract The systematic assessment of working capital This article takes into consideration the uncertainty asso-

ciated with many of the project resource variables andrequirement in construction projects deals with the anal-

ysis of various quantitative and qualitative factors in these are reflected satisfactorily in the working capital

computations. A case study illustrates the application ofwhich information is subjective and based on uncer-

tainty. There exists an inherent difficulty in the classical the fuzzy set approach. The results of the case study

demonstrate the superiority of the fuzzy set approach toapproach to evaluate the impact of qualitative factors for

classical methods in the assessment of realistic workingthe assessment of working capital requirement. This

capital requirements for construction projects.paper presents a methodology to incorporate linguistic

variables into workable mathematical propositions for the Keywords fuzzy relation, fuzzy sets, membership func-

assessment of working capital using fuzzy set theory. tion, working capital

I N T R O D U C T I O N

Construction projects are generally complex and often

confronted by a number of operations and activities

with varying degrees of uncertainty. From the incep-

tion of the construction project, the project manager is

required to make numerous decisions that will deter-

mine the success or failure of the project both in

accomplishment of physical and in monetary terms.

One such decision is the working capital requirement

for the successful completion of the project.Conventionally, the first step in the estimation of

working capital is to forecast a total expenditure curve,

which is derived from a composition of constituent

expenditures, such as labour, equipment, maintenance

and material procurement advances. Then, the total

expenditure is superimposed with the anticipated cu-

mulative receipts of revenue, and the difference be-

tween the two curves is the working capital

requirement at any time during the project.

The realistic assessment of working capital in con-

struction projects involves usually quantitative re-

sources such as money, machinery, material and

labour, as well as qualitative features such as fluctuat-

ing market conditions, weather conditions, availability

and quality of materials, variation in material prices,

delay in receipts of revenue, technical problems, and

design changes.

In practice, experienced project engineers evaluate

the working capital requirement based on the past

experience in a similar situation and from cash flow

calculations. In some cases, experience is limited to

certain types of projects. Inexperienced project engi-

neers may overlook qualitative factors when assessing

the working capital requirement. Thus, there is a need

to incorporate qualitative factors in addition to quanti-

tative factors for the realistic assessment of working

capital requirements.

This article presents a methodology for the assess-

ment of working capital requirements by quantifying

the qualitative factors using fuzzy set theory. A casestudy is presented to demonstrate the use and practi-

cability of the fuzzy logic approach. In addition, the

approach described in this paper uncovers any gap in

experts thinking for optimum utilization of the avail-

able resources.

L I T E R A T U R E R E V I E W

Management of working capital in the construction

industry is an integral part of overall organization.

Working capital management involves rational deci-

sion making regarding the levels of inventory, receiv-

ables and cash balance to be maintained. In the

construction industry, working capital accounts for

about 60% of total investment.

Van Horne (1984) mentions that gross working

capital is an asset that converts back into cash within

an operating cycle, which is usually less than 1 year.

Major parts of gross working capital is financed by

2000 Blackwell Science Ltd93


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94 Kumar, V.S.S. et al.

current liabilities, which mature for repayment within

one operating cycle. He further defines the difference

between current assets and current liabilities as the net

working capital, which represents the ownership funds.

Archer et al. (1983) state the basic principle in working

capital management is minimization of the investment

on current assets without affecting the smooth runningof the organization. The rationale of this minimization

principle is that this investment by itself does not

generate profit, while the investment made on working

capital has a costexplicit or implicit.

Several authors emphasize that the maintenance of

working capital is intended to meet the current liabili-

ties promptly to ensure technical solvency of the firm.

In construction industry, in fact, at any given point of

time investment on working capital is more than the

investment on fixed assets. At the final stage of con-

struction, the entire unit can be viewed as inventory to

be sold to the owners. If the periodic cash inflows due

from an owner do not take place promptly, the need

for working capital will increase for the contractor.

Similarly, adverse inflationary impact also increases the

need for working capital.

Pilcher (1994) suggests that the capital invested on a

project is normally the initial expenditure for the pur-

chase of plant and other fixed assets together with the

associated working capital. He further explains that a

qualitative factor, for example fluctuating trading con-

ditions, would include a length of time over which

credit is allowed and received; accordingly, the amount

of stock or inventory carried influences the working

capital requirements.Similarly, the cost of the proposed construction pro-

ject along with the projects on hand also affect the

amount of working capital required. The gestation

period, which is affected by labour unrest, season,

nonavailability of construction material etc., is another

important factor that affects the quantum of working

capital. In fact, the longer the construction period, the

larger is likely to be the amount of working capital with

its own corresponding cost implications.

Desai (1997) suggests that proper management of

working capital synchronizes the cash receipts and

cash outlay so that a unit may function with cashreserves.

Ayyub & Haldar (1984) pioneered the use of fuzzy

set theory to evaluate the impact of qualitative factors

on the duration of construction project activities.

However, their study does not consider the effect of

qualitative factors on the working capital requirements

for a construction project.

The working capital requirements on account of the

relevant qualitative factors were not assessed so far;

this can be estimated through the application of fuzzy

set theory.

F A C T O R S A F F E C T I N G W O R K I N G

C A P I T A L R E Q U I R E M E N T S

The capital invested in construction projects is the

initial capital for the purchase of material and other

fixed assets plus associated working capital (Pilcher

1994). Working capital in this context is the require-

ment of the day-to-day finance until the completion of

the project. The working capital requirements vary

from one project to another and from one type unit to

another type.

In reality, the working capital for construction

projects is also influenced by qualitative factors as

shown in Fig. 1, in addition to quantitative factors. For

realistic assessment of working capital, variabilities ofthe qualitative factors have to be considered. For ex-

ample, fluctuating market conditions include a length

of time over which credit is allowed and received;

accordingly, the delay in receipts of revenue may con-

siderably influence the working capital requirements.

Similarly, the cost of a construction project relative to

other ongoing projects also affects the amount of work-

ing capital required. The down time that is affected by

labour unrest, season and nonavailability of construc-

tion materials is another important factor that affects

the quantum of working capital. In fact, the longer the

construction period, the greater will be the amount of

working capital with the corresponding costimplications.

The fuzzy logic approach presented here incorpo-

rates qualitative factors in the assessment of working

capital requirements. This methodology prompts the

Figure 1 Factors affecting the working capital in construction.

2000 Blackwell Science Ltd, Engineering, Construction and Architectural Management 7 1, 93103


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95Assessment of working capital requirements by fuzzy set theory

project engineer for information regarding each quali-

tative factor. In return, the methodology furnishes the

manager with a realistic working capital requirement

for each interval of the construction project.

F U Z Z Y S E T C O N C E P T

The fuzzy set concept is founded on the notion that

qualitative expressions usually involve the realm of

human perception, which are subject to a range of

interpretations. Although the values of these expres-

sions are inexact in quality, they are meaningful. Qual-

itative expressions usually consist of linguistic variables

and these are routinely used in construction projects.

These linguistic measures add to the overall uncer-

tainty in the final outcome of any decision process.

A linguistic variable is defined as a variable, the

values of which are expressed in words, phrases or

sentences in a given language (Ayyub & Haldar 1984).

The phrase performance of constructed facility, forexample, is a linguistic variable that can take the values

of excellent, good or low. The information expressed

in the phrase has a value that is not clearly defined.

The variabilities of the linguistic variables can be rep-

resented as fuzzy sets.

Fuzzy set theory in measuring qualitative criteria

A fuzzy set can be defined as a class of objects with

unclear boundaries in which transition from member-

ship to nonmembership is gradual. Since the transition

from member to nonmember appears gradual rather

than abrupt, the fuzzy set introduces vagueness byeliminating the sharp boundaries dividing members of

the set from nonmembers (Paek et al. 1993). The

qualitative factors that affect the working requirement

can be quantified by giving membership values to their

variabilities. This membership value is denoted by

vA(x), where x is an element of fuzzy set A. The

membership value vA(x) is the degree of belief with

which an element can be stated to belong to the set.

Therefore the fuzzy set:

A={x, vA (x)}, xA (1)

Each element, x, is associated with a membership

value vA(x), to the set A and is a real number. If A is

an ordinary, nonfuzzy, or crisp set, then the member-

ship function is given by:

1 if x belongs to A

vA(x)={ (2)

0 if x does not belong to A

In Equation (2), there are only two possibilities for

an element x, either being a member of A, i.e.,

vA(x)=1 or not being a member of A, i.e., vA(x)=0.

In this case, A has sharp boundaries. On the other

hand, if the membership function is allowed to take

values in the interval (0, 1), A is called a fuzzy set

(Lorterapong & Moselhi 1996).For example, the linguistic variable weather condi-

tions is interpreted using the following membership

function. Here, the linguistic variable is weather con-

ditions and the values of the linguistic variable are

poor, medium, good, etc. Dividing the range into

increments of 0.1, the following fuzzy subset poor

can be assigned for the variable weather conditions:

poor={x1=0.0/(vpoor(x1)=1.0), x2=0.1/

(vpoor(x2)=0.9), x3=0.2/(vpoor(x3)=0.5)} (3)

In Equation (3), the term x1=0.0/(vpoor(x1)=1.0),

indicates that if the value of the qualitative factorweather condition is 0.0, then the degree of belief

that the qualitative factor belongs to the, poor weather

conditions set is 1.0 (AbouRizk et al. 1994). Similarly,

for the value of weather condition 0.1, the degree of

belief is 0.9; and for the value of weather condition

0.2, the degree of belief is 0.5. For simplicity, Equa-

tion (3) can be written as in Equation (4):

poor=(0.0/1.0, 0.1/0.9, 0.2/0.5) (4)

These membership values are assigned based on sub-

jective judgement of experts.

In general, the membership values for variability ofqualitative factors that affect the working capital re-

quirements can be represented as in Equations (5)

(7).

pessimistic=(0.0/1.0, 0.1/0.9, 0.2/0.5) (5)

average=(0.3/0.2, 0.4/0.8, 0.5/1.0, 0.6/0.8, 0.7/

0.2) (6)

optimistic=(0.8/0.5, 0.9/0.9, 1.0/1.0) (7)

In some cases, a term set including linguistic values

such as pessimistic, average and optimistic may not besatisfactory in certain domains, and may instead re-

quire the use of linguistic values such as quite pes-

simistic and very optimistic. The membership values

for these variabilities can be defined as:

quite pessimistic=(pessimistic)1.25=(0.0/1.0, 0.1/

0.88, 0.2/0.42) (8)



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very optimistic=(optimistic)2=(0.8/0.25, 0.9/

0.81, 1.0/1.0) (9)

Equations (5) (9) can be applied to any of the

variabilities of linguistic variable in the project environ-

ment such as site conditions, technical problems,

weather conditions and design changes. For example,

if we say that the weight of the qualitative factor

weather condition is small (or low), then the pes-

simistic fuzzy set of Equation (5) can be applied by

treating the weight to vary continuously from 0 to 1.

In order to use fuzzy sets in practical problems such

as evaluating working capital requirements, some oper-

ational rules must be defined.

Assuming that A and B are two fuzzy sets with

membership functions of vA(x) and vB(x), then the

following operations can be defined on these sets. The

complement A0 , of a fuzzy set A with a membership

function vA(x) is defined as:

vA0(x)=1vA(x) (10)

The membership function for the union @ of two

fuzzy subsets A and B with membership functions

vA(x) and vB(x) is given in Equation (11).

vA@B(x)=max{vA(x), vB(x)} (11)

The membership function for the intersection S of

two fuzzy subsets A and B is given in Equation (12).

vASB(x)=min{vA(x), vB(x)} (12)

A fuzzy relation R or Cartesian product, AB,

between the two fuzzy subsets A (subset of universe X)

and B (subset of universe Y) is determined from the

following membership function in Equation (13).

vR(xi, yj)=vAB(xi, yj)=min{vA(xi), vB(yj)} (13)

A fuzzy relation is usually expressed in a matrix

form. Each element vR(xi, yj) is a membership value

for the ordered pair (xi, yj) and is a measure of

association between xi and yj. It is computed as the

minimum value of the membership values of vA(xi)

and vB(xj). For example, if A is a subset of weather

condition (X), and B is a subset of site condition (Y),

then vAB (xi, yj) is a measure of the associationbetween weather conditions and site conditions.

If R is fuzzy relation from X to Y, and T is a fuzzy

relation from Y to Z, the composition of R and T

(R o T) is a fuzzy relation between the fuzzy subsets X

and Z using the common fuzzy subset Y (Equation

(14)). The composition of two fuzzy relations is a

fuzzy relation.

vR o T(xi, zk)=max{min[vR(xi, yj), vT(yj, zk)]}

(14)

The preceding operational rules for fuzzy sets are

used in this paper (Ayyub & Haldar 1984).

M E T H O D O L O G Y

The applicable qualitative factors must be identified

for each interval of the project duration for the assess-

ment of the working capital requirement in the entire

project. Each of these factors has its own linguistic

values that can be converted into fuzzy sets by giving

suitable membership values. These membership values

are assigned based on subjective judgement. Each

membership value shows the degree of membership of

the corresponding element in the fuzzy set and is a real

number in the interval (0, 1).

Let the criterion Cij denote qualitative factors (i=1,

2,n) such as technical problems, design changes and

delay in receipts of revenue for each project interval

(j=1, 2,m), and Wij denotes a measurement of the

weight or importance of a criterion Cij. The impor-

tance of these criteria will depend on schedule, project

location, project characteristics, project managers

preferences and other factors. Here, the weight of a

criterion is also in terms of linguistic values.

The weight of qualitative factor, Wij, determines the

susceptibility of working capital requirement to each

criterion on each interval of the project duration. Here,

susceptibility (Sij) is a measure of sensitivity of the

working capital to the criterion i and project interval j.

The values of susceptibility also are in linguistic terms

and vary from zero to one. If the susceptibility ofinterval j to criterion i (Sij) equals to zero, the working

capital of interval j of the project is not affected by

criterion i.

The values of susceptibility and its affect vary with

different intervals of the project. If Sij equals to one,

the working capital of interval j is totally susceptible to

Cij. When Sij falls in the range of (0, 1), working

capital of interval j has some degree of susceptibility to

the criterion Cij (Chang et al. 1990). For example, if a

criterion called severity of rain exists, then by ones

judgement or expert advice, it is very reasonable to

believe that working capital of interval j (say, placing of

concrete) tends to be greatly affected by the rain.The methodology uses two fuzzy relations R and T.

Using Equation (13), the fuzzy relation R and T can

be determined. The fuzzy relation R is an association

of weight of the criterion (Wij) and susceptibility on

the working capital to the criterion during a particular

interval (Sij). The fuzzy relation T is an association

between susceptibility (Sij) and working capital (WCij).



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Table 1 Expenditure and revenue of the project

End of theEnd of the

Revenue Cum. exp.month Cum. exp.Expenditure RevenueCum. rev. Cum. rev.Expendituremonth

(3)(1) (4) (5) (6) (7) (8) (9) (10)(2)

0.0 0.6 0.0 7 3.0 3.71 13.0 11.80.6

0.0 1.0 0.0 8 1.60.4 2.6 14.6 14.42

0.83 0.9 1.8 0.9 9 0.7 1.7 15.3 16.1

2.24 1.0 4.0 1.9 10 0.7 0.9 16.0 17.0

2.3 7.0 4.2 11 0.43.0 0.4 16.4 17.45

3.9 10.0 8.1 12 0.66 1.6 17.0 19.03.0

Cum., cumulative; exp., expenditure; rev., revenue.

The stepwise procedure for assessing the working

capital requirement is as follows:

Step 1. Identify the relevant qualitative factors

(Cij) that affect each interval of the

total project duration.

Step 2. Tabulate the variabilities of the qualita-

tive factors for assessing working cap-

ital requirements by assigning weight

of importance (Wij), susceptibility

(Sij) and working capital (WCij).

Translate the variability of qualitativeStep 3.

factors into numerical measures by

using the membership values from

Equations (5)(9)

Compute fuzzy relation R between theStep 4.

weight (Wij) of the qualitative factor

and its susceptibility (Sij) through the

use of Equation (13).Compute fuzzy relation T between sus-Step 5.

ceptibility (Sij) and affect on working

capital (WCij) through the use of

Equation (13).

Establish fuzzy relations obtained inStep 6.

step 4 and 5. This is performed by

taking the largest membership value

in each column of the fuzzy relation

matrix.

Step 7. Formulate the fuzzy composition matrix

for all membership values for R and

T obtained in step 6, using Equation

(14) to arrive at the combinedinfluence.

Compute the working capital by choos-Step 8.

ing the decision subset, which maxi-

mizes the product of row summation.

The procedure is illustrated by the following case

study.

C A S E S T U D Y

An established construction company with cash on

hand of $1000000 is being awarded a contract to

construct warehouses and the total cost of the project

is $19000000. The construction work of the above

project is scheduled to be completed within12 months. From the second month until the end of

month 8, cumulative revenues lag behind the expendi-

tures. From the end of month 8 onward, there is an

increase in net revenue and at the end of the project

period the total revenue is $19000000. The expected

net profit for the project is $2000000. Table 1 shows

the projected monthly cash flow requirements of the

project.

To demonstrate the applicability of the fuzzy set

theory for the assessment of working capital require-

ment for the case study, the detailed procedure is

presented for the first interval of the project.

Analysis of working capital requirement by fuzzy

set theory

The first step in the assessment of working capital is to

identify the qualitative factors that affect the working

capital requirement. For the first interval, the relevant

qualitative factors are site conditions (poor, average),

weather conditions (good), cash flow (excellent) and

experience of an engineer (high). The weight for each

classification of these preceding factors as well as the

susceptibility on the requirement of working capital

estimation in linguistic terms are shown in Table 2.To translate the impact of qualitative factors for

i=15, the following fuzzy relations are calculated

based on Equations (5)(9). The relation R1 is defined

as: if the site conditions are poor then the susceptibility

is more for its great importance of the factor. There-

fore, the fuzzy relation Ri for poor site conditions is:

Site conditions (poor)



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Weight (great) Susceptibility (more)

0.8 0.9 1.0

0.8 0.50 0.500.50

0.900.500.9R1=W1S11 0.90

1.000.900.501.0

(15)

In the above relation, weight and susceptibility

are two different fuzzy sets. The membership values of

the second row of Equation (15), are evaluated as

follows from Equation (13).

vR1(0.9, 0.8)=min(0.90, 0.50)=0.50

From Equation (7), the membership values for the

variability of great weight and large susceptibility are

0.9 and 0.5. Then using Equation (13), the fuzzy

relation is obtained for the two fuzzy sets as the

minimum value of these two values is 0.5. The value

0.5 is an association for great weight and large suscep-tibility of the working capital for poor site conditions.

The second and third values of the second row are:

vR1(0.9, 0.9)=min(0.90, 0.90)=0.90

vR1(0.9, 1.0)=min(0.90, 1.00)=0.90

Similarly, the relations R2, R3, R4 and R5 are calcu-

lated from Table 2.

For site conditions (average)

Weight (small) Susceptibility (medium)

0.3 0.4 0.5 0.6

0.0 0. 20 0.80 1.00 0.80

R2=W21S21 0.1 0. 20 0.80 0.90 0.80

0.2 0. 20 0.50 0.50 0.50

(16)

For weather conditions (good)

Weight (medium) Susceptibility (small)

0.0 0.1 0.2

0.3 0.20 0.200.20

0.4R3=W31S31 0.500.800.80

0.5 1.0 0.90 0.50

0.6 0.80 0.80 0.50

(17)

For cash flow (excellent)

Weight (large) Susceptibility (small)

0.0 0.1 0.2

0.500.500.8 0.50

R4=W41S41 0.9 0.90 0.90 0.50

1.00 0.90 0.501.0

(18)

For engineers experience (high)

Weight (medium) Susceptibility (very small)

0.1 0.20.0

0.20 0.20 0.200.3

R5=W51S51 0.4 0.80 0.80 0.25

0.250.811.000.5

0.250.800.800.6

(19)

The total effect of all the factors is obtained by taking

the union of these five relations.

Total effect=R=(W11S11)@(W21S21)@(W31S31)@(W41S41)@(W51S51) (20)

Affect on working capitalSusceptibilityWeight (Wij)Qualitative factors (Cij)

(3)(2)(1) (4)

More1. Site conditions (poor) Very largeGreat

Medium MediumSmall2. Site conditions (average)

3. Weather conditions (good) SmallMedium Small

Very smallSmallLarge4. Cash flow (excellent)

Quite smallVery smallMedium5. Engineers experience (high)

Table 2

Qualitative

description of

weight and

susceptibility for

linguistic variables



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Therefore,

Susceptibility

0.1 0.2 0.3 0.4 0.5 0.60.0 0.7 0.8 0.9 1.0Weight

0.0 0.0 0.0 0.2 0.8 1.0 0.8 0.0 0.0 0.00.0 0.0

0.0 0.0 0.2 0.8 0.9 0.80.0 0.00.1 0.0 0.0 0.0

0.0 0.0 0.2 0.5 0.5 0.5 0.00.2 0.0 0.0 0.00.00.2 0.2 0.0 0.0 0.0 0.00.2 0.0 0.0 0.0 0.00.3

0.5 0.50.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00.8

0.5 0.5 0.0 0.0 0.0 0.00.9 0.0 0.0 0.0 0.0R=0.5

0.5 0.5 0.0 0.0 0.0 0.0 0.00.6 0.0 0.0 0.00.8

0.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.00.7

0.5 0.50.8 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.50.5

0.5 0.5 0.0 0.0 0.0 0.00.9 0.00.9 0.5 0.9 0.9

0.9 0.51.0 0.0 0.0 0.0 0.0 0.0 0.5 0.9 1.01.0

(21)

The fuzzy relations of R1, R2 R3, R4, and R5 are

projected in relation R using Equation (14) at appro-

priate positions. The other positions, in relation R are

given a membership value of 0.0, since there is no

qualitative factor affecting those positions. To establish

a fuzzy relation T between the fuzzy subsets of suscep-

tibility and working capital, the conditional statement

is as follows: if the susceptibility is more, its affect on

working capital will be very large. The membership

values for the variabilities of qualitative factors (affect

on working capital requirement) are given in the fol-

lowing Equation (22).

Very large=550000/0.25, 600000/0.81, 650 000/

1.00

Medium=550000/0.60, 600000/1.00, 650000/0.60

Small=550000/1.00, 600000/0.60, 650000/0.20

Very small=(small)2=550 000/1.00, 600000/

0.36, 650000/0.04

Quite small=(small)1.25=550000/1.00, 600000/

0.53, 650000/0.13 (22)

A range of the peak working capital requirement for

the present interval between $550 000 and $650 000 is

chosen based on the quantitative data and the experi-

ence of an engineer. If the range chosen is too small,

the effect of qualitative factors cannot be adequatelyrepresented. On the other hand, if the range is too

large, the qualitative factors will dominate the working

capital requirement. Hence, it is important to choose a

likely range for the working capital.

The maximum working capital for the chosen range

is $650 000. Since the affect on working capital re-

quirement is very large, the values nearer to $650 000

will assume higher membership values and large

membership values are applied for susceptibility factor.

Therefore, the fuzzy relations (T) from Table 2 are

as follows:

For site conditions (poor)

Susceptibility (more) Working capital (very large)

550 000 600 000 650 000

0.25 0.50 0.500.8

0.25 0.810.9 0.90T1=S11WC111.0 0.25 0.81 1.00

(23)

For site conditions (average)

Working capital (medium)Susceptibility (medium)550 00 600 000 650 000

0.3 0.20 0.20 0.20

T2=S21WC21 0.4 0.60 0.80 0.60

0.5 0.60 1.00 0.60

0.60 0.80 0.600.6

(24)

For weather conditions (good)

Working capital (small)Susceptibility (small) 550 000 600 000 650 000

0.0 1.00 0.60 0.20

0.90 0.600.1 0.20T3=S31WC310.50 0.50 0.200.2

(25)

For cash flow (excellent)

Working capital (very small)Susceptibility (small)

550 000 600 000 650 000

1.00 0.360.0 0.04

T4=S41WC41 0.1 0.90 0.36 0.04

0.50 0.36 0.040.2

(26)

For engineers experience (high)



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Working capital (quite small)Susceptibility

550 000 600 000 650 000(very small)

0.0 1.00 0.53 0.13

0.81 0.53 0.13T5=S51WC51 0.1

0.25 0.25 0.130.2

(27)

By taking the union of T1, T2, T3, T4 and T5, the

following relation is obtained:

Susceptibility Working capital

550 000 600 000 650 000

0.0 1.00 0.60 0.20

0.90 0.60 0.200.1

0.50 0.500.2 0.20

0.3 0.20 0.20 0.20

0.60 0.80 0.600.4

0.60 1.000.5 0.60T=0.6 0.60 0.80 0.60

0.7 0.00 0.00 0.00

0.25 0.500.8 0.50

0.9 0.25 0.81 0.90

1.0 0.25 0.81 1.00

(28)

Therefore, the working capital requirement is calcu-

lated by taking composition of R and T (R o T), from

Equation (14).

Weight Working capital

600 000 650 000 Sum550 000 Product

0.600.0 1.00 0.60 2.20 0.00

0.600.1 0.90 0.60 2.10 0.21

0.50 0.50 1.500.50 0.300.2

0.20 0.20 0.60 0.180.3 0.20

0.60 0.60 2.000.80 0.800.4

1.000.5 0.60 0.60 2.20 1.10

0.800.6 0.60 0.60 2.00 1.20

0.00 0.00 0.000.00 0.000.7

0.50 0.50 1.500.8 1.200.50

0.81 0.90 2.610.90 2.350.9

0.81 1.001.0 2.81 2.811.00

(29)

The fuzzy composition (R o T) considers the total

affect of the factors of Table 2 to assess the working

capital requirement. For example, the membership

value of 0.60 in the first row of Equation (29) is the

total effect of qualitative factors on working capital of

$550 000 for the weight of 0.0.

Now a row has to be chosen from the above equa-

tion, which maximizes the product of the row summa-

tion and the corresponding weight of the factor, to

calculate the working capital requirement for the inter-val. The last row of Equation (29) gives the maximum

value of this product for the case problem under

consideration. Therefore, the following subset is cho-

sen for the calculation of working capital requirement:

D=(550 000/1.00, 600000/0.81, 650000/1.00)

(30)

The probability mass function of the working capital

requirement for the interval can be calculated as fol-

lows (Zadeh 1968):

P (550 000)=1.00/(1.00+0.81+1.00)=0.356

P (600 000)=0.81/(1.00+0.81+1.00)=0.288

P (650 000)=1.00/1.00+0.81+1.00)=0.356

Therefore, the mean value of working capital re-

quirement is calculated as follows:

Working capital=5500000.356+6000000.288

+6500000.356=$600000

In addition to the mean working capital require-

ments, this methodology provides standard deviation

(|) and coefficient of variation (COV) for the interval

of the project as indicated below.

|2=55000020.356+60000020.288+6500002

0.356(600 000)2=1.78109 and

|=$42 190 with COV=42 190/600000=0.07

Similarly, for each interval in the case study, the

mean values of the working capital requirements were

calculated for the project and shown in Table 4. The

weight of qualitative factors and the adverse suscepti-

bility on working capital requirements are evaluated

subjectively based on experts judgement in relation of

performance to working capital.

DISCUSSION

Working capital is the requirement of the day-to-dayfinance until the completion of the project. It is essen-

tially determined by the difference between expendi-

ture assessed inclusive of next period expenditure and

revenue received at any given point of time. The peak

working capital requirement for the project is the

maximum value of the difference between cumulative

revenue of the previous month and the cumulative



9/11


Table 3 Working capital requirements of the project (conventional)

Working capital Net working End of theEnd of the Working capital Net working

month (conventional)capital capital(conventional) month

(2) (3) (4) (5) (6)(1)

0.6 71 4.9 1.20.6

1.0 1.0 8 2.8 0.22

3 1.8 0.9 9 0.9 +0.8

2.1 10 +0.13.1 +1.04

5.15 2.8 11 +0.6 +1.0

1.9 12 +0.4 +2.06 5.8

value represents the WC requirement and + value represents the surplus revenue at hand.

expenditure of the present month. The peak working

capital requirement for project can be noted to be

$5.8 million (Table 3). This can be met through the

availability of credit facility of the firm with the suppli-

ers in addition to the cash reserve at hand.

But in actual practice, due to inflow of revenue, the

Figure 2 Working capital requirements for project.



10/11


working capital requirement decreases for brief inter-

mittent intervals. In Fig. 2, the sloping lines represent

a gradual increase in the project requirements and

vertical lines represent monetary decreases in out-

standing working capital as and when the payments are

received. The discontinuous lines represent the work-

ing capital requirements in the absence of the expectedrevenue inflows.

Figure 3 represents the fuzzy-based working capital

computations and these are in general agreement with

the conventional evaluation as shown in Table 4.

However, in the absence of reliable data on the project

expenditure, the conventional approach cannot assess

the working capital requirements realistically. In such

cases, the fuzzy set approach can be applied with

advantage. This approach makes for explicit communi-

cation and incorporation of the effects of qualitative

factors on working capital assessment. Conventionally,

these relationships are known and retained with the

experts who may have intuitively used them, oftenunconsciously, to arrive at the assessment of working

capital requirements for construction projects. The

experts knowledge of these relationships is made ex-

plicit in the form of conditional statements (Table 2)

in order to apply the fuzzy set theory to other projects

systematically.

The application of fuzzy set theory to the assessment

of working capital was illustrated with a case studyincorporating qualitative inputs such as site conditions,

weather conditions and cash flow. Besides computing

the expected value of the working capital required,

fuzzy set analysis also provides the standard deviation

for this expected value. In projects of high priority to

the organization where a high degree of confidence in

the success of the project is called for, this standard

deviation can be used to establish additional require-

ments, which would increase the confidence in the

success of the project.

The methodology also provides the project manager

a greater insight and understanding to prepare the

working capital requirements that may affect the pro-ject implementation schedules.

Figure 3 Working capital requirements of project (fuzzy).



11/11


Table 4 Working capital requirements of the project

Working capitalWorking capital Working capitalEnd of theEnd of the Working capital

month(conventional)month (conventional)(fuzzy based)(fuzzy based)

(1) (6)(2) (3) (4) (5)

1 0.60 0.60 7 1.28 1.20

2 1.02 1.00 8 0.25 0.20

+0.80+0.8590.901.153102.102.05 +0.924 +1.00

5 2.92 2.80 11 +1.00 +1.00

6 1.90 1.90 12 +1.72 +2.00

value represents the WC requirement and + value represents the surplus revenue at hand.

C O N C L U S I O N S

In this paper, the fuzzy set theory applied to the

assessment of the working capital requirement is illus-

trated with a case study incorporating qualitative inputs

such as site conditions, weather conditions and cash

flow. This new methodology takes into consideration

the uncertainty associated with many of the project

resource variables and these are reflected satisfactorily

in the working capital computations.

Fuzzy set theory, under these circumstances, will be

useful to construction engineers in order to exercise

better control on cost allocation and financial planning.

Application of fuzzy set theory is a step towards the

elimination of bias or prejudice in the judgement of an

expert, since the steps leading to the judgement are

made explicit. This makes for explicit communication

and incorporation of the effects of qualitative factors for

the assessment of working capital requirement. Thishelps to uncover any gap in the experts thinking such

as in regard to qualitative factors, which may have not

been considered. This methodology can incorporate any

new information after completion of the project and for

multi-project planning.

R E F E R E N C E S

AbouRizk, S.M., Halpin, D.W. & Sawhney, A. (1994) Mod-

elling of uncertainty in construction simulation. In: Uncer-

tainty Modeling and Analysis: Theory and Applications, pp.

287304. Elseveir Science B.V., Netherlands.

Archer, S.H., Choate, G.M. & Rocette, G. (1983) Financial

Management. John Wiley and Sons, Inc., New York.

Ayyub, B.M. & Haldar, A. (1984) Project scheduling using

fuzzy set concept. Journal of Construction Engineering and

Management, 110, 189204.

Chang, T.C., Ibbs, C.W. & Crandall, K.C. (1990) Network

resource allocation with support of a fuzzy expert system.

Journal of Construction Engineering and Management, 116,

239259.

Desai, V. (1997) Project Management. Himalaya Publishing

House, Mumbai, India.

Lorterapong, P. & Moselhi, O. (1996) Project-network analysis

using fuzzy set theory. Journal of Construction Engineering and

Management, 122, 308318.

Paek, J.H., Lee, Y.W. & Ock, J.H. (1993) Pricing construction

risk: fuzzy set application. Journal of Construction Engineering

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Scientific, London.

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