Information Sciences 168 (2004) 77–94
www.elsevier.com/locate/ins
Measuring flexibility of computer
integrated manufacturing systems using
fuzzy cash flow analysis
Cengiz Kahraman a, Ahmet Beskese
a
c
b,*
, Da Ruan
c
Department of Industrial Engineering, Istanbul Technical University, 34367 Macka,
Istanbul, Turkey
b
Department of Industrial Engineering, University of Bahcesehir, 34538,
Bahcesehir, Istanbul, Turkey
Fuel Research Unit, Belgian Nuclear Research Centre (SCK CEN), Boeretang 200,
2400 Mol, Belgium
Received 16 June 2003; received in revised form 29 August 2003; accepted 10 November 2003
Abstract
There exist a number of methods proposed in the literature to quantify manufacturing flexibility in monetary terms and to use a financial evaluation model with a
decision criterion based on present worth. However, most of these methods are unable
to handle problems with incomplete and uncertain data. To obtain a sensible result in
quantifying the manufacturing flexibility in computer integrated manufacturing systems, this paper proposes some fuzzy models based on fuzzy present worth. The fuzzy
models based on present worth are basically engineering economics decision models in
which the uncertain cash flows and discount rates are specified as triangular fuzzy
numbers. To build such a model, fuzzy present worth formulas of the manufacturing
flexibility elements are formed. Flexibility for continuous improvement, flexibility for
trouble control, flexibility for work force control, and flexibility for work-in-process
control are quantified by using fuzzy present worth analysis. Formulas for both inflationary and non-inflationary conditions are derived. Using these formulas, more reliable
results can be obtained especially for such a concept like flexibility that is described in
many intangible dimensions. These models allow experts’ linguistic predicates about
computer integrated manufacturing systems.
2003 Elsevier Inc. All rights reserved.
*
Corresponding author. Tel.: +90-212-669-6523/1251; fax: +90-212-669-4398.
E-mail address: [email protected] (A. Beskese).
0020-0255/$ - see front matter 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.ins.2003.11.004
78
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
Keywords: Fuzzy cash flow analysis; Computer integrated manufacturing systems;
Fuzzy numbers; Fuzzy intervals; Manufacturing flexibility
1. Introduction
Computer integrated manufacturing (CIM) systems can be viewed as a total
system, which provides an automatic link between product design, manufacturing engineering, and the factory floor. Effective implementation of advanced
manufacturing technologies through a CIM system is the cornerstone of factory modernization. CIM systems provide many important benefits such as
greater process flexibility, reduced inventory, reduced floor space, faster response to shifts in market demand, lower lead times, and a longer useful life of
equipment over successive generations of products. Manufacturing flexibility
may be defined as the ability to cope with changing circumstances or instability
caused by the environment. Flexibility is emerging as one of the key competitive strengths in today’s manufacturing systems, since it provides a critical
measure of total manufacturing performance. Hence, measuring the flexibility
of manufacturing systems is important to operations managers engaged in
decision making on strategic issues related to flexibility. Flexibility is widely
recognized as a multidimensional attribute. Most of the studies reported in the
literature have focused on measuring separate dimensions independently and
also have tended to be non-financial in nature. As a result, many of these
measures have only limited application in strategic decision making.
There exist various methods to measure the value of flexibility by using
surrogate measures to represent the intangible parts. However, managers still
prefer the methods by which all intangible parts of flexibility can be quantified
as far as it is possible in monetary terms. This preference leads the decisionmakers into an uncertain economic decision environment of the contemporary
business world, where an expert’s knowledge about the cash flow information
usually consists of a lot of vagueness instead of randomness. For example, to
describe a sales profit that may be implicitly forecasted from past incomplete
information, linguistic descriptions like ‘‘around one million’’ are often used.
The major contribution of the fuzzy approach used in this paper is its capability of representing vague knowledge in such an economic environment. In
real word applications, precise data concerning flexibility factors of CIMs are
not available or very hard to be extracted. In addition, decision-makers prefer
natural language expressions rather than sharp numerical values in assessing
flexibility parameters. So, manufacturing flexibility is an inherently fuzzy notion, which can be measured by the synthesis of its constituents. Fuzzy logic
offers a systematic base in dealing with situations, which are ambiguous or not
well defined. Indeed, the uncertainty in expressions such as ‘‘low flexibility’’ or
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
79
‘‘high utilization,’’ which are frequently encountered in the flexibility literature,
is fuzziness.
Tsourveloudis and Phillis [29] are the only researchers using a fuzzy logic
framework to measure manufacturing flexibility. They propose a fuzzy rulebased method that handles imprecise data and knowledge about a production
system. However, there is no paper published in refereed international journals
aimed at measuring flexibility in terms of fuzzy cash flows.
With the rapid expansion of CIM and flexible manufacturing systems
(FMS), there are hundreds of studies on the definition, classification, and
quantification of manufacturing flexibility and on decision-making tools for
investment analysis of flexible automation. Nonetheless production managers
of many firms, under pressure to make automated manufacturing systems more
flexible, are finding that there is still a considerable gap between the benefits
promised by the financial evaluation tools and the benefits realized in practice.
As a value-added to the literature on the topic, this paper aims at providing
practitioners with a fuzzy point of view to the traditional cash flow analysis
method for dealing quantitatively with imprecision or uncertainty and at
obtaining a fuzzy flexibility quantification from this point of view which will
close this gap considerably. Fuzzy intervals will be used to represent the cash
flows and the other non-monetary parameters of flexibility. Beskese et al. [1,2]
provide a basis for this study.
The paper is organized as follows: Section 2 gives a literature review for the
quantification of manufacturing flexibility. Section 3 includes a general
knowledge about fuzzy present worth analysis, two applications of this analysis
for measuring manufacturing flexibility, and a numerical example. Section 4
concludes the current research results and future plan of the study reported
in the paper.
2. Quantification of manufacturing flexibility
The economic justification of CIM has received increased attention in recent
years. Arbitrarily high hurdle rates, comparison with the status quo, and
insufficient benefit analysis are cited as major efforts in application. However,
traditional methods for the economic justification of new manufacturing
technologies fail to include benefits such as better quality, greater flexibility and
reduced work-in-process, since these benefits are difficult to measure and
therefore hard to quantify. Application of traditional capital budgeting
methods, for example, does not fully account for the benefits arising from
increased flexibility. Some special tools must be used to facilitate a thorough
analysis of tangible and intangible benefits.
In the literature, many researchers have tried to quantify the manufacturing
flexibility and to integrate it with decision-making tools. Their methods of
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C. Kahraman et al. / Information Sciences 168 (2004) 77–94
quantifying the flexibility can be classified into three groups: (1) there are many
intangible parts of flexibility, so it should be considered as blackbox [11,18,21];
(2) the intangible parts of flexibility that cannot be quantified in monetary
terms can be measured by a surrogate value [8,27,30,32]; and (3) all intangible
parts of flexibility can be quantified as far as it is possible in monetary terms
[12,25,26,28].
Pyoun and Choi [25] distinguish the flexibility that is inherent in the manufacturing system from the flexibility that the user can attain after implementation. They introduce the concepts of potential flexibility and realizable
flexibility to obtain a deeper insight into flexibility and to give consideration to
all possible aspects of user experience and capability. Potential flexibility is the
flexibility inherent in a manufacturing system from the system manufacturer’s
point of view, before it is implemented and operated by the user. Realizable
flexibility is the flexibility that will be realized by operating the manufacturing system using both potential flexibility and the users’ engineering and
management capability. Pyoun and Choi [25] classify potential flexibility into
four elements and realizable flexibility into 11 elements. They propose a systematic procedure for quantifying realizable flexibility in monetary terms and
use a financial evaluation model with a decision criterion based on present
worth.
Although such efforts are highly appreciated, one should be aware that there
are some difficulties to quantify manufacturing flexibility by only using crisp
methodologies. Some of them can be mentioned as: there is no effective
method for the synthesis of the functional parameters affecting each type of
flexibility and the flexibility types, which are observed on different hierarchical
levels and are crucial in the determination of manufacturing flexibility, there
is no correspondence between flexibility and the physical characteristics of the
production system, and certain operational characteristics have contradictory
effects on flexibility. To overcome such difficulties, this paper proposes a fuzzy
flexibility quantification method using fuzzy present worth analysis based on
fuzzy interval arithmetic.
3. Fuzzy present worth analysis in flexibility measurement
3.1. Fuzzy present worth analysis
Most economic decision problems involve the uncertainty feature of cash
flow modeling. If sufficient objective data is available, probability theory is
commonly used in modeling cash flows and performing decision analysis.
Unfortunately, decision-makers rarely have enough information to perform
the decision analysis, since probabilities can never be known with certainty and
the economic decision is attributable to many uncertain derivations. In this
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
81
µp (x)
1.0
f1 (.)
f2 (.)
y
0.0
a
a+(b-a) y
c
b
X
c+(b-c) y
Fig. 1. A triangular fuzzy number, Pe ¼ ða; b; cÞ.
situation, most decision-makers rely on experts’ knowledge in modeling cash
flows.
In the literature, triangular and trapezoidal fuzzy numbers that are the
special forms of LR-type fuzzy numbers are usually used to capture the
vagueness of the parameters related to the topic. The arithmetic operations of
these types of fuzzy numbers can be found in [33]. In this paper, triangular
fuzzy numbers (TFNs) will be used to consider the fuzziness of the flexibility
parameters. A TFN is designated as Pe ¼ ða; b; cÞ. It is graphically depicted in
Fig. 1 in which f1 ðÞ is the left side, and f2 ðÞ is the right side representation of
the TFN.
To deal quantitatively with imprecision or uncertainty, fuzzy set theory is
primarily concerned with vagueness in human thoughts and perceptions. As an
alternative to conventional cash flow models where cash flows are defined as
either crisp numbers or risky probability distributions, Chiu and Park [7]
propose an engineering economics decision model in which the uncertain cash
flows and discount rates are specified as triangular fuzzy numbers. They
examine deviation between exact present worth (PW) and its approximate form
(PWA) and perform the fuzzy project selection by applying different dominance rules as shown in Eqs. (1) and (2) respectively. The result of the exact
present worth is also a fuzzy number with a non-linear membership function. It
is in complex non-linear representations that require tedious computational
effort [7]. For the reason of simplicity, a TFN can be used as an approximate
form of the complex (exact) present worth formula in Eq. (1).
"
!
lðyÞ
lðyÞ
N
X
maxfFt ; 0g
minfFt ; 0g
PW ¼
þ Qt
;
Qt
rðyÞ
lðyÞ
t¼0
t0 ¼0 ð1 þ Rt0 Þ
t0 ¼0 ð1 þ Rt0 Þ
!#
rðyÞ
rðyÞ
N
X
maxfFt ; 0g
minfFt ; 0g
þ Qt
ð1Þ
Qt
lðyÞ
rðyÞ
t¼0
t0 ¼0 ð1 þ Rt0 Þ
t0 ¼0 ð1 þ Rt0 Þ
lðyÞ
rðyÞ
where Ft is the left side representation, Ft is the right side representation of
lðyÞ
rðyÞ
the fuzzy cash flow Fe at time t, and Rt0 is the left side representation, Rt0 is
82
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
e at time t0 . N is a crisp
the right side representation of the fuzzy interest rate R
number denoting the project life.
lðyÞ
When the degree of membership (y) in Eq. (1) is equal to 0, Ft ¼ ft0 ,
rðyÞ
lðyÞ
rðyÞ
Ft ¼ ft2 , Rt0 ¼ rt0 , and Rt0 ¼ rt2 . When the degree of membership (y) in Eq.
lðyÞ
rðyÞ
lðyÞ
rðyÞ
(1) is equal to 1, Ft ¼ Ft ¼ ft1 , and Rt0 ¼ Rt0 ¼ rt1 . Substituting these to
the exact present worth formula, the approximate form of the present worth
formula can be derived as in Eq. (2). PWA is represented using its three
parameters and it is easier to implement because they are in linear representations.
X
N N
X
maxfft0 ; 0g
minfft0 ; 0g
ft1
PWA ¼
þ Qt
;
;
Qt
Qt
0
0
ð1
þ
r
Þ
ð1
þ
r
Þ
ð1
þ rt 0 1 Þ
t2
t0
t0 ¼0
t0 ¼0
t0 ¼0
t¼0
t¼0
!
N X
maxfft2 ; 0g
minfft2 ; 0g
Qt
þ Qt
ð2Þ
t0 ¼0 ð1 þ rt0 0 Þ
t0 ¼0 ð1 þ rt0 2 Þ
t¼0
Chiu and Park [7] compute the maximum deviation as a measure of the fitness
between PW and PWA. They use very small increments of y as the measurement method instead of derivative method since the latter is difficult to calculate. Using a simulation software, they calculate the deviations for different
ranges of cash flows and discount rates, and find out that the deviations are not
significant unless the confident width of discount rate is larger than an absolute
range of ±4%. In the real world applications, when the discount rates are
usually estimated within the width of ±4%, PWA can be used in project
analysis. The deviations of PW and PWA are depicted in Fig. 2.
In literature, some other approaches to calculate fuzzy PW can be found.
For instance, Buckley [4] forms the membership function for the fuzzy present
worth, P f
W N , as in Eq. (3)
h
i
W N Þ=pwN 2 ; pwN 2 =fN 2 ðyjP f
W N Þ; pwN 3
lðxjP f
W N Þ ¼ pwN 1 ; fN 1 ðyjP f
ð3Þ
where N is the crisp useful life of the project, pwNi is the least, the most, and the
largest possible values of P f
W N respectively for i ¼ 1; 2; 3, pwN 1 < pwN 2 <
W N Þ is a continuous monotone increasing function of y ¼
pwN 3 , f1 ðyjP f
W N Þ ¼ pwN 1 and f1 ð1jP f
W N Þ ¼ pwN 2 , and
lðxjP f
W N Þ for 0 6 y 6 1 with f1 ð0jP f
f
f2 ð0jP W N Þ is a continuous monotone decreasing function of y for 0 6 y 6 1
W N Þ ¼ pwN 3 and f2 ð1jP f
W N Þ ¼ pwN 2 .
with f2 ð0jP f
lðxjP f
W N Þ is determined by
h
i
W N Þ ¼ fi ðyj Fe Þ 1 þ fk ðyj~rÞN
fNi ðyjP f
ð4Þ
for i ¼ 1; 2 where k ¼ i for negative Fe and k ¼ 3 i for positive fuzzy future
worth, Fe [4]. In Eq. (4), the fuzzy interest rate (~r) is assumed to be kept con-
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
83
PW
µ (PW)
PWA
1.0
dl
y
dr
l
0.0
a
b
c
PW
Fig. 2. Deviation between PW and PWA.
stant whereas it is a parameter changing from year to year in Chiu and Park’s
[7] formula in Eq. (1). In other words, ~r in Eq. (4) represents the average annual
fuzzy interest rate along N years.
e is fuzzy. If Fe is the final amount in the
Assume that the project life N
account, then its membership function is defined by [4]
lðxj Fe Þ ¼ supðhÞ;
ð5Þ
CðxÞ
where
e Þ; lðvj~rÞ; lðwj N
e ÞÞ
h ¼ minðlðujP W
w
CðxÞ ¼ fðu; v; wÞ j uð1 þ vÞ ¼ xg
ð6Þ
ð7Þ
The definition of Fe is simply the extension principle applied to Eq. (8) [4].
N
PWð1 þ rÞ ¼ F
ð8Þ
For some relevant publications on fuzzy capital budgeting techniques, the
reader is referred to [3,5,9,13,15–17,19,22].
3.2. Application of fuzzy present worth analysis for measuring flexibility
In the literature, some publications can be found on measuring flexibility
using crisp present worth analysis. A significant contribution to this research
area by Pyoun and Choi [25], and the fuzzification of this model are given in
the following sub-sections.
3.2.1. Pyoun and Choi’s quantification of flexibility value
Pyoun and Choi [25] classify realizable flexibility into three categories: (1)
the internal control group, (2) the investment policy group, and (3) the marketing adaptation group. The flexibility value is the sum of the values in these
three categories. All of the formulas of the elements of these three groups
are based on present worth. Therefore, we have decided to develop fuzzy
present worth formulas for the elements of the internal control group only, to
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C. Kahraman et al. / Information Sciences 168 (2004) 77–94
constitute an example. Similar transformations can be applied easily to the
elements of the other two groups. The flexibility formulas of the internal
control group are given in Table 1. In the formulas in Table 1, t is the annual
income tax rate, r is the interest rate and N is the project life. Pyoun and Choi
[25] use the flexibility value (FV) in their financial evaluation model as shown
in Eq. (9).
PWðrÞ ¼
N
X
(
n¼1
þ
ð1 tÞRn
In
ð1 tÞOn
tDn
n n þ
n
ð1 þ rÞ
ð1 þ rÞ
ð1 þ rÞ
ð1 þ rÞn1
tBN
ð1 tÞS
þ
þ FV
ð1 þ rÞN ð1 þ rÞN
)
ð9Þ
where t is the income tax rate, N is the project life, R revenue, I investment
cost, O operating cost, D depreciation cost, B book value, and S salvage
value.
In this paper, only the fuzzy flexibility formulas in the internal control group
are obtained since the other modifications are similar. The internal control
group is categorized by the user’s integrated operational capability to cope with
their internal changes and is calculated as the sum of four elements, flexibility
Table 1
Flexibility formulas of internal control group
Internal control group
Flexibility for continuous
improvement (CI)
Flexibility for trouble control
(TC)
Flexibility for workforce
control (WC)
Work-in-process control (IP)
Flexibility formula
P
sn cn
CI ¼ ð1 tÞ Nn¼1 ð1þrÞ
n
where sn is the average number of parts revisions, and cn is the
average cost required for alteration of tooling and software in
the nth year P
Tn bn
TC ¼ ð1 tÞ Nn¼1 ð1þrÞ
n
where TC ¼ 0 if Ca ¼ 0 and, Tn ¼ ed minfCa Mn ; Qn g c2 , Tn is
the average increased production revenue per breakdown in the
nth year, after subtracting off the rescheduling cost, bn is the
average number of breakdowns in the nth year, e is the average
revenue per item, d is the average period required for repair per
breakdown, Ca is the average manufacturing capacity available
during breakdown (% of Mn ), Mn is the maximum manufacturing capacity at the beginning of the nth year, Qn is the fuzzy
average demand in the nth year, and c2 is the rescheduling cost
P /Jn Wn
WC ¼ ð1 tÞ Nn¼1 ð1þrÞ
n
where / is the average benefit per job, Jn is the average number
of different jobs, Wn is the average number of operators
P
ucv Vn
IP ¼ ð1 tÞ Nn¼1 ð1þrÞ
n
where u is the average revenue per item, cv is the inventory
carrying-cost rate, and Vn is the average work in process in the
N th year
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
85
for continuous improvement, flexibility for trouble control, flexibility for
workforce control, and flexibility for work-in-process control [25].
Pyoun and Choi [25] do not consider that some deviations in the estimates of
flexibility parameters, such as the average number of parts revisions in the nth
year (sn ), may occur. Our aim in this paper is to adapt their formulation of
realizable flexibility to the uncertainty of real-world conditions by applying
Fuzzy Sets Theory. So here, in this paper, the terms sn , cn , and r are assumed to
be fuzzy parameters because in a CIM environment, the decision-makers
usually have imprecise data about these three. The income tax rate, t, is defined
by the government and usually is a fixed value (kept unchanged) for a very long
time. So, it is assumed to be a crisp value. The project life, N , is also assumed to
be a crisp number here in the formulation. However, the fuzziness for this
parameter will be incorporated in the numerical example for the practical
purposes.
We may define the parameters that Pyoun and Choi used in their paper [25]
as triangular fuzzy numbers such as ~sn ¼ ðsn0 ; sn1 ; sn2 Þ. We may also define the
discount rate during time n such as ~rn ¼ ðrn0 ; rn1 ; rn2 Þ. The indices 0, 1, and 2 in
the representations of ~sn and ~rn stand for the smallest possible value, the most
possible value, and the largest possible value, respectively. The exact form of
the fuzzy present worth of the flexibility for continuous improvement (P f
WoCI)
under non-inflationary conditions can be calculated as follows:
"
#
N
N
lðyÞ
rðyÞ
X
X
slðyÞ
srðyÞ
n cn
n cn
f
P WoCI ¼ ð1 tÞ
; ð1 tÞ
Qn
Qn
rðyÞ
lðyÞ
n¼1
n¼1
n0 ¼1 ð1 þ Rn0 Þ
n0 ¼1 ð1 þ Rn0 Þ
ð10Þ
Under inflationary conditions, it is very important to take the inflation rate
into consideration. Hence, the average cost required for alteration of tooling
and software (cn ), which is the only monetary parameter in Eq. (10) changes
from year to year. In this case, the present worth formula for geometric cash
flows, which is frequently used in Engineering Economics, can be used [23].
Using Buckley’s notation and assuming that c1 will increase at a fuzzy constant
rate (~
g), and ~s is fixed along N years, Eq. (10) will change to,
"
#
N
N
1
f
ðyj~
r
Þ
f
ðyj~
g
Þ
k
k
fNi ðyjP f
W N Þ ¼ ð1 tÞfi ðyj~sÞc1
ð11Þ
fk ðyj~rÞ fk ðyj~
gÞ
Other monetary parameters in the formulas of the following elements of
internal control group are Tn for trouble control, / for workforce control, and
u for work-in-process control. Similar modifications derived by inflationary
conditions are applied to Pyoun and Choi’s formulas in Eqs. (12), (19) and (21)
to yield with formulas in Buckley’s notations in Eqs. (18), (20) and (22),
respectively. Please note that, to achieve a more precise present worth under
86
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
inflationary conditions forcing us to deal with geometric growth rates, we
substitute the arithmetic average values / with /n , and u with un for
n ¼ 1; 2; . . . ; N , in Eqs. (20) and (22), respectively.
The second element of the internal control group is flexibility for trouble
control (TC). It is the user’s capability to adapt the manufacturing system to
handle a breakdown of the system. This flexibility is applicable only when the
breakdown period is long enough to reschedule the manufacturing system and
the user also needs to keep producing during this period with reduced capacity.
The terms Tn , bn , r, e, d, Ca , Mn , Qn , and c2 are assumed to be fuzzy
parameters. Only t and N are assumed to be crisp numbers for the reasons
mentioned in the fuzzification of CI above. Considering these, flexibility for
trouble control P f
WoTC is measured by
"
#
N
N
lðaÞ lðaÞ
rðaÞ rðaÞ
X
X
T
b
T
b
n
n
n
n
Pf
WoTC ¼ ð1 tÞ
; ð1 tÞ
Qn
Qn
rðaÞ
lðaÞ
ð1
þ
R
Þ
ð1
þ
Rn0 Þ
0
0
0
n¼1
n¼1
n ¼1
n ¼1
n
ð12Þ
where Ten is obtained by using the following equation:
ea M
e n Þ ~c2
e n; Q
Ten ¼ ~e d~ minð C
ð13Þ
In Eq. (13), it is necessary to use a ranking method to compare the fuzzy
ea M
e n . There are many ranking methods in the literature
e n and Q
numbers, C
and these methods may give different ranking results. Some of them are
Chang’s [6], Chiu and Park’s [7], Dubois and Prade’s [10], Jain’s [14], Kaufmann and Gupta’s [20], Liou and Wang’s [24], and Yager’s [31] methods. For
example, Liou and Wang [24] propose the total integral value method with an
e be a fuzzy number with left membership
index of optimism x 2 ½0; 1. Let A
function f L and right membership function f R . Then the total integral value is
eA
eA
defined as
e ¼ xER ð AÞ
e þ ð1 xÞEL ð AÞ
e
Ex ð AÞ
where
e ¼
ER ð AÞ
Z
b
a
and
e ¼
EL ð AÞ
ð14Þ
Z
R
xfe
ðxÞ dx
A
ð15Þ
L
xfe
ðxÞ dx
A
ð16Þ
d
c
e ¼ ða; b; cÞ,
where 1 < a 6 b 6 c 6 d < þ1. For a triangular fuzzy number, A
the total integral value is obtained by
e ¼ 1½xða þ bÞ þ ð1 xÞðb þ cÞ
ð17Þ
Ex ð AÞ
2
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
87
Under inflationary conditions, assuming that T1 will increase at a fuzzy constant rate (~
g) and ~
b is fixed along N years, Eq. (12) becomes
"
#
N
N
1
f
ðyj~
r
Þ
f
ðyj~
g
Þ
k
k
~ 1
W N Þ ¼ ð1 tÞfi ðyjbÞT
fNi ðyjP f
ð18Þ
fk ðyj~rÞ fk ðyj~
gÞ
The third element of the internal control group is flexibility for workforce
control (WC). It is the user’s capability to manage the size and the technical
and managerial capability of the workforce required for operation of the
manufacturing system. This flexibility is measured by the size and the technical
and managerial capability of the required workforce and by the average benefit
that the user can attain per job.
The terms /, Jn , Wn , and r are assumed to be fuzzy parameters. Only t and N
are assumed to be crisp numbers for the reasons defined in the fuzzification of
CI above. Considering these, the fuzzy present worth of flexibility for workforce control (P f
WoWC) can be formulized as in Eq. (19).
"
#
N
N
lðaÞ lðaÞ
rðaÞ rðaÞ
lðaÞ
rðaÞ
X
X
/
J
W
/
J
W
n
n
n
n
Pf
WoWC ¼ ð1 tÞ
; ð1 tÞ
Qn
Qn
rðaÞ
lðaÞ
ð1
þ
R
Þ
ð1
þ
R
0
0
0
n¼1
n¼1
n
n0 Þ
n ¼1
n ¼1
ð19Þ
Under inflationary conditions, assuming that /1 will increase at a fuzzy cone are fixed along N years, Eq. (19) becomes
stant rate (~
g) and J~ and W
"
#
N
N
1
f
ðyj~
r
Þ
f
ðyj~
g
Þ
k
k
f
e Þ/1
fNi ðyjP W N Þ ¼ ð1 tÞfi ðyj e
J Þfi ðyj W
ð20Þ
fk ðyj~rÞ fk ðyj~
gÞ
The fourth element of the internal control group is work-in-process control
(IP). It is the user’s capability to minimize the work in process required for
operation of the manufacturing system. This flexibility is measured by the
average amount of work in process required for operation of the manufacturing system and the inventory carrying-cost rate.
The terms u, Cv , Vn , and r are assumed to be fuzzy parameters. Only t and N
are assumed to be crisp numbers for the reasons defined in the fuzzification
of CI above. Considering these, the fuzzy present worth of flexibility for workin-process control (P f
WoIP) can be formulized as in Eq. (21).
"
#
N
N
lðaÞ lðaÞ lðaÞ
rðaÞ rðaÞ rðaÞ
X
X
u
c
V
u
c
V
v
n
v
n
Pf
WoIP ¼ ð1 tÞ
; ð1 tÞ
Qn
Qn
rðaÞ
lðaÞ
n¼1
n¼1
n0 ¼1 ð1 þ Rn0 Þ
n0 ¼1 ð1 þ Rn0 Þ
ð21Þ
Under inflationary conditions, assuming that u will increase at a fuzzy constant
rate (~
g) and Ve and ~c are fixed along N years, Eq. (21) becomes
88
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
"
e Þu1
fNi ðyjP f
W N Þ ¼ ð1 tÞfi ðyj Ve Þfi ðyj W
N
1 fk ðyj~rÞ fk ðyj~
gÞ
fk ðyj~rÞ fk ðyj~
gÞ
N
#
ð22Þ
3.2.2. Numerical example
EGEY, a Turkish Motors Company, produces transmission gearboxes for
cars. The fuzzy average numbers of part revisions ð~sn Þ, the fuzzy maximum
e n Þ, and the fuzzy
e n Þ, the fuzzy average demand ð Q
manufacturing capacities ð M
~
average number of breakdowns ðbn Þ in the years 2003–2008 are expected to be
as in Table 2. However, the analysis period is not certain since the company is
not sure that it will produce these products for the next six years. The proe years where lðni j N
e Þ is given in Table 3. The fuzzy
duction will continue for N
interest rate is expected to be fixed as (8%, 10%, 12%) along this period while
the fuzzy average cost required for alteration of tooling and software is also
expected to be fixed as (20, 30, 50) (·$1000). The fuzzy average manufacturing
capacity available during breakdown is (85%, 90%, 95%), the fuzzy average
revenue per item is $(6.0, 6.5, 7.0) (·$10), the fuzzy average period required for
repair per breakdown is (0.015, 0.032, 0.038) years, the fuzzy rescheduling cost
e years. The
is $(180, 210, 240) and they are all assumed to be fixed through N
crisp annual income tax rate is 35%.
Using Eq. (10), the exact fuzzy present worth of the flexibility for continuous
improvement is calculated as follows:
8
2 1þy
3
2þy
2þy
<
þ ð1:120:02yÞ
2 þ
1:120:02y
ð1:120:02yÞ3
5;
Pf
WoCI ¼ 0:65 ð20 þ 10yÞ 4
3þy
3þy
3
:
þ ð1:120:02yÞ
þ ð1:120:02yÞ
4 þ
6
ð1:120:02yÞ5
2
39
4y
2
3
=
þ ð1:08þ0:02yÞ
2 þ
ð1:08þ0:02yÞ
ð1:08þ0:02yÞ3
5
0:65 ð50 20yÞ 4
4y
5y
4
;
þ ð1:08þ0:02yÞ
þ ð1:08þ0:02yÞ
4 þ
6
ð1:08þ0:02yÞ5
Since the discount rate is estimated within the width of ±4%, P f
WA can be used
in the problem under consideration [7]. To obtain the approximate form of
Table 2
e n , and ~bn
e n, Q
Fuzzy expected values for e
Sn, M
e
e
Year, n
Sn
Mn
2003
2004
2005
2006
2007
2008
(1,
(2,
(2,
(3,
(3,
(3,
2,
3,
3,
3,
4,
4,
2)
3)
4)
4)
4)
5)
(3600,
(3900,
(4000,
(4300,
(4500,
(4600,
3800,
4000,
4200,
4500,
4700,
4800,
4000)
4200)
4300)
4600)
4800)
4900)
en
Q
(3100,
(3500,
(3600,
(3700,
(4000,
(4100,
~bn
3400,
3650,
3700,
3900,
4200,
4300,
3900)
3900)
4000)
4400)
4600)
4700)
(3,
(4,
(4,
(5,
(6,
(7,
4,
5,
5,
6,
7,
8,
5)
6)
7)
7)
7)
8)
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
89
Table 3
Possibility table for the project life
Ni
4
5
6
eÞ
lðNi j N
0.7
1.0
0.8
Pf
WoCI, for n ¼ 4, 5, and 6, y will be assigned a value of 0 and 1 in the left side
representation, and then a value of 0 in the right side representation in the
equation above. Thus, the results in Table 4 are obtained.
After obtaining both the exact and approximate equations for P f
WoCI, the
values of these two functions were calculated by using very small increments of
0.001 on y axis. Spearman correlation coefficient was used to decide whether
the approximate form can be used instead of the exact form, or not. The
coefficient was calculated as 0.9984 for the left sides, whereas it was 0.9993 for
the right sides. Since these coefficients are close enough to 1.00, it was proven
for our example that the approximate form can be used instead of the exact
form.
In Fig. 3, the approximate fuzzy present worth of the flexibility for continuous improvement is shown by its TFNs.
The numeric results of the fuzzy flexibility for trouble control are given in
the following:
ea M
e 1 ¼ ð3100;
e 1 ¼ ð3060; 3420; 3800Þ and Q
For the year 2003, C
3400; 3900Þ. Using Liou and Wang’s ranking method [24], for a moderately
ea M
e 1 Þ ¼ 3425 and
optimistic decision-maker, x ¼ 0:5, the minimum of E0:5 ð C
e 1 Þ ¼ 3450 is 3425. So we select C
ea M
e 1 value and use this value in Eq.
E0:5 ð Q
(13). By calculating all Ten ’s in the same way, we obtain
Te1 ¼ ð2514; 6904; 9928Þ;
Te3 ¼ ð3000; 7486; 10460Þ;
Te5 ¼ ð3203; 8588; 11950Þ;
Te2 ¼ ð2744; 7278; 10433Þ
Te4 ¼ ð3090; 8602; 11524Þ
Te6 ¼ ð3279; 8776; 12202Þ
By using these values, the exact fuzzy present worth of flexibility for trouble
control is obtained as
Table 4
Pf
WoCI values for n ¼ 4; 5; 6
Life, n
4
5
6
fn;i ðyjP f
WoCIn Þ
y ¼ 0 and i ¼ 1
y ¼ 1, i ¼ 1 or i ¼ 2
y ¼ 0 and i ¼ 2
$75,625.67
$97,755.32
$117,513.93
$167,709.86
$216,141.72
$260,170.69
$342,527.79
$431,003.61
$533,406.17
90
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
µ PW~ oCI
1.0
0.8
0.7
533,406.17
431,003.61
342,527.79
260,170.69
216,141.72
167,709.86
97,755.32
117,513.93
75,625.67
~
PWoCI
Fig. 3. The approximate fuzzy present worth of the flexibility for continuous improvement shown
by its TFNs.
8
3 9
2
ð4390yþ2514Þð3þyÞ
>
>
þ ð4534yþ2744Þð4þyÞ
>
>
2
1:120:02y
>
ð1:120:02yÞ
>
>
7 >
6
>
>
ð4486yþ3000Þð4þyÞ
ð5512yþ3090Þð5þyÞ
>
>
7
6 þ
>
>
þ
3
4
>
>
7
6
ð1:120:02yÞ
ð1:120:02yÞ
> 0:65 6
;
>
>
7 >
>
>
ð5385yþ3203Þð6þyÞ
ð5497yþ3279Þð7þyÞ
>
>
5
4
>
>
þ ð1:120:02yÞ5 þ ð1:120:02yÞ6
>
>
>
>
=
<
3
2
f
P WoTC ¼
ð99283024yÞð5yÞ
ð104333155yÞð6yÞ
>
>
þ ð1:08þ0:02yÞ2
>
>
>
7>
6 1:08þ0:02y
>
>
>
>
ð104602974yÞð72yÞ
ð115242922yÞð7yÞ
>
>
þ ð1:08þ0:02Þ3
þ ð1:08þ0:02yÞ4 7
>
> 0:65 6
7
6
>
>
>
>
7
6
>
>
ð119503362yÞ7
ð112023426yÞ8
>
>
5
4
>
>
þ
>
>
5 þ
6
>
>
ð1:08þ0:02yÞ
ð1:08þ0:02yÞ
>
>
;
:
To obtain the approximate form of P f
WoTC, y will be assigned a value of 0 and
1 in the left side representation, and then a value of 0 in the right side representation in the equation above. Thus, the least possible value, the most possible value, and the largest possible value of the fuzzy present worth of the
flexibility for trouble control are calculated as $33,844.19, $118,552.69, and
217,048.22 for n ¼ 4; $44,749, $155,880, and $273,979 for n ¼ 5, and 56,377.70,
190,994.79, and 330,452.28 for n ¼ 6, respectively. In Fig. 4, the approximate
fuzzy present worth of the flexibility for trouble control is shown by its TFNs.
The FV in Eq. (9) will be the sum of the results which are obtained for the
internal control group, the investment policy group, and the marketing
adaptation group. Here, in this numerical example, we include only two of the
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
91
µ PW~oTC
1.0
0.8
330,452.28
273,979.00
217,048.22
190,994.79
155,880.00
118,552.69
44,749.00
56,377.70
33,844.19
0.7
~
PWoTC
Fig. 4. The approximate fuzzy present worth of the flexibility for trouble control shown by its
TFNs.
elements of the internal control group, which are P f
WoCI and P f
WoTC, just to
show the way how the total FV amount is calculated. Thus, the result of the
summation of these two elements is shown in Fig. 5.
Using Fig. 5, the possibility of any FV can be found easily. In case of having
incomplete information, this possibility distribution provides the decisionmaker with a detailed information source. For example, the most possible
value in Fig. 5 is $372,021.72. If the decision-maker wants to know the interval
of flexibility values having a possibility larger than 0.90, he/she should make
the calculations below:
1. Derive the functions for the area above the desired possibility value (0.90 for
our example) using the bold lines in Fig. 5.
x1 ¼ 142; 504:32 þ 229; 517:4y
x2 ¼ 704; 982:61 332:960:89y
2. Substitute the desired possibility value into these equations and find the
extreme points of the interval.
x1 ¼ 142; 504:32 þ 229; 517:4 0:90 ¼ 349; 069:98
x2 ¼ 704; 982:61 332:960:89 0:90 ¼ 405; 317:81
According to the results of the calculations above, the interval of flexibility
values having a possibility larger than 0.90 is (349,069.98; 405,317.81).
92
C. Kahraman et al. / Information Sciences 168 (2004) 77–94
µ PW~oFV
1.0
0.8
863,858.45
704,982.61
559,576.01
451,165.48
372,021.72
286,262.55
142,504.32
173,891.63
109,469.86
0.7
~
P W oFV
Fig. 5. The approximate fuzzy present worth of the flexibility value shown by its TFNs.
4. Conclusion
In this paper, the fuzzy present worth formulas for optimal level of flexibility
and the flexibility elements in the internal control group, which is categorized
by the user’s integrated operational capability to cope with their internal
changes, are obtained. Using these formulas, as shown in the given numeric
example, more informative results of flexibility measures can be achieved. The
flexibility modeling using triangular fuzzy numbers allows experts’ linguistic
predicate about computer integrated manufacturing systems. When there are
more than one alternative of CIM systems to compare in terms of flexibility,
using the fuzzy present worth formulas of several types of flexibility and
applying some dominance rules on triangular fuzzy numbers, the fuzzy flexibility evaluation process is completed. This study involves discrete compounding for calculating the fuzzy present worth of flexibility. Further research
can be aimed at including continuous compounding for the same calculation.
Also, trapezoidal fuzzy numbers can be used instead of TFNs when the decision-maker uses ‘‘between’’ instead of ‘‘around’’ to express his/her estimations
about the flexibility parameters.
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