ASAC 2008
Halifax, Nova Scotia
Anju Bajaj
S. S. Appadoo
C. R. Bector
University of Manitoba, Winnipeg, Canada
S. Chandra
Indian Institute of Technology, New Delhi, India
MEASURING PHYSICAL FITNESS AND CARDIOVASCULAR EFFICIENCY USING HARVARD
STEP TEST APPROACH UNDER FUZZY ENVIRONMENT4
One may not be able to measure exactly the various parameters involved
in computing physical fitness and cardiovascular efficiency of a human
being. Inexactness, in turn, may lead to inaccurate results and misleading
conclusions. In the present paper, using fuzzy arithmetic, we modify both
the Long Form and in Short Form approach associated with the Harvard
Step Test, under fuzzy environment. The advantage of fuzzy arithmetic is
that it takes into consideration the vagueness (inaccuracy or imprecision)
involved in the process of Harvard Step Test whereas the crisp arithmetic
approach used by other researchers ignores this aspect.
Keywords: Physical Fitness, Cardiovascular Efficiency, Harvard Step Test, Fuzzy Environment
1.
Introduction
A number of researchers have dealt with the measurement of physical fitness and cardiovascular
efficiency (for example, see [4]-[6], [10]-[14], [16], [20]-[23], [15], [19], [21], and [23]). In most of these
models it is assumed that the data associated with the parameters involved is crisp (precise). In 1943,
Brouha in her most celebrated paper [4], using easily available and inexpensive equipment and under
crisp environment, conducted a very simple field test, called Harvard Step Test (HST), for measuring
physical fitness (cardiovascular endurance) of a human being by using a mathematical formula for
determining an Index Number, called Physical Efficiency Index (PEI). Brouha et al. [5, 6] used the HST
in studying the physical fitness of college students and measuring physical fitness for hard muscular work
in adult young men. Brouha [4] in her paper states that when calculated if the PEI comes out to be; below
55 = poor physical condition; from 55 to 64 = low average; from 65 to 79 = high average; from 80 to 89 =
good; above 90 = excellent. Modifying the HST and PEI appropriately, Clark [11] in 1943 reported a
functional fitness test for finding the physical fitness of college women. Gallagher and Brouha [12]
discuss physical efficiency of boys aged 12-18 years where they modify the HST by reducing the duration
of the exercise from five minutes to four minutes and by reducing the height of the bench. To measure the
cardiovascular efficiency of girls and women, in 1963 Skubic and Hodgkins [23] modified Brouha’s
approach [4] by changing the height of the bench, pace of steps, duration of exercise, taking the pulse
count from 1.0 to 1.5 minute after exercise, and used it to measure the cardiovascular efficiency of girls
and women. Reducing the height of the bench and making other changes like adjusting the hand bar to
eye level of the subject, Cotton [9] modified the Brouha’s approach [4] for cardiovascular testing of
4
Acknowledgment: The authors are thankful to the anonymous referees for their useful suggestions and
comments.
129
school boys of Grade 4 by considering the duration of exercise as four minutes and varying the height of
the bench to 14, 16 or 20 inches depending upon the height of the subject. Ricci et al. [22] suggest that
arbitrary lowering of bench height for young subjects may not be justifiable.
It is important to point out here that none of the above mentioned researchers take into consideration the
errors that, while performing the experiments, can arise possibly due to the human judgments and the
uncertain environment. For example, numerous possibilities of errors exist in the collection of data and
selection of the size of each sample in which certain parameters are hard to measure and are subjectively
chosen. This suggests that use of fuzzy algebra may be a more desirable approach while dealing with
problems involving the uncertainties due to fuzzy environment. Fuzzy algebra is a simple and useful way
to propagate imprecision through a cascade of calculations and has often been used to model systems that
are hard to define precisely. As a methodology, it incorporates imprecision, subjective assessment, vague
data information, and sensitivity analysis into the model formulation, its analysis, and solution process.
Uncertainty and vagueness, when handled through the probability theory, sometimes encounter
difficulties where as it appears that fuzzy algebra may be a more supple technique that could provide
pragmatic answers to such problems. Therefore, in this paper we revisit the problem of physical
fitness/cardiovascular efficiency, considered by Brouha [4] and Brouha et al. [5, 6], under fuzzy
environment and demonstrate the use of the results by considering numerical examples. Another
advantage of fuzzy algebra approach is that, corresponding to every subject, we obtain the physical
efficiency index in the form of an interval. Approaches followed by other researchers (for example, [10][14], [16], [20]-[22], [11], [15], [19], [21], and [23]) can be modified on similar lines.
The present paper is divided in five sections. In Section 1 we provide the introduction and in Section 2 we
introduce the preliminaries, notation, and prerequisites. In Section 3, we introduce briefly the HST and
the concept of physical efficiency index (PEI) discussed by Brouha [4] and Brouha et al. [5], [6] under
crisp environment. In Section 4 we modify Brouha’s PEI [4] under fuzzy environment (call it fuzzy PEI)
and using simple concepts of fuzzy algebra (introduced in Section 2 and in [2], [3], [17], [18], [23], and
[24]) provide an approach to compute the fuzzy PEI (denoted by FPEI An advantage of the fuzzy
approach presented here is that it obtains the FPEI in the form of an interval. This is not the case if one
uses Brouha’s approach. In Section 5, we compute FPEI using numerical examples. Finally, we conclude
the contribution made and the results obtained in the paper.
2.
Preliminaries, notation and prerequisites
In this section we shall follow the notations and concepts as introduced in [2, 17, 18, and 25]. Let X be a
X be the set of real numbers.
subset of a universal set U, Rn X the space of n-tuples, and R
FUZZY SET [25]. A set A X is a fuzzy set if for every x
it is a set of ordered pairs (x, μA(x)). Then A is written as
A = {(x, μA(x)), x X }.
X, μ : X
M
R,
(2.1)
MEMBERSHIP FUNCTION. In (2.1), μA(x) is called the membership function or grade of membership,
or degree (level) of compatibility, or degree (level) of truth of x A that maps X to the membership
space M.
It is important to point out here that when M contains only two points 0 and 1 then A is nonfuzzy
(crisp) and μA(x) is identical to the characteristic function of a crisp set.
-LEVEL SET. A set
A = {x X : μA(x)
is called the -level set of the fuzzy set A.
}
(2.2)
130
STRONG -LEVEL SET. A set
A = {x X : μA(x) > }
is called the -level set of the fuzzy set A.
It may be pointed out here that in both (2.2) and (2.3), A is a crisp set.
-CUT OF A FUZZY SET A. A set A() given by
A() = {x X : μA(x) , is called the -cut of the fuzzy set A.
NOTE 1. In (2.4), if 1
2 then A(2)
(0, 1]}
(2.3)
(2.4)
A(1).
SUPPORT OF A FUZZY SET A. In (2.4) when = 0, then A is called the support of the fuzzy set A.
This means that
A(0) = {x X : μA(x) 0 }
(2.5)
CONVEX FUZZY SET. A fuzzy set A in Rn is called a convex fuzzy set if its -cuts A() are (crisp)
convex sets for all (0, 1].
THEOREM 2.1. A fuzzy set A in Rn is a convex fuzzy set if and only if for all x1 and x2 in Rn and
0 1,
μA[ x1+ (1 ) x2) ] = min (μA (x1), μA (x2))
NORMAL FUZZY SET. A fuzzy set A in Rn is said to be a normal fuzzy set if
max μA(x) = 1
x
FUZZY NUMBER. A fuzzy set A in R is called a fuzzy number if
(i)
A is normal,
(ii)
A is convex,
μA is upper semicontinuous, and
(iii)
(iv)
the support of A is bounded.
ARITHMETIC OF FUZZY NUMBERS. Using the notion of interval of confidence from Kaufmann
and Gupta [17, 18] and using (2.3) above, we write the -cut A() of a fuzzy number A in the form of an
interval as
where AL() AR().
(2.6)
A() = [AL(), AR()],
We now consider the arithmetic of fuzzy numbers based on the -cuts of fuzzy numbers as given in [2,
17, 18] and in (2.6). In this paper we shall make their extensive use. For this we consider two fuzzy
numbers A and B given by
A() = [AL(), AR()], and B() = [BL(), BR()].
(i)
(ii)
Addition of A() and B().
A() + B() = [AL(), AR()] + [BL(), BR()]
= [AL() + BL (), AR() + BR()]
(2.7)
Subtraction of A() and B().
A() B() = [AL(), AR()] [BL(), BR ()]
= [AL() BR(), AR() BL()]
(2.8)
131
(iii)
Multiplication of A() and B().
A() (.) B() = [AL(), AR()] (.) [BL(), BR()]
= [min (AL() (.) BL(), AL() (.) BR(), AR() (.) BL(), AR () (.) BR(),
max (AL() (.) BL(), AL() (.) BR(), AR() (.) BL(), AR() (.) BR()]
(2.9)
(iv)
Division of A() and B().
A() ( ) B()
= [min (AL()( )BR(), AL()( )BL(), AR()( )BR(), AR()( )BL(),
max (AL()( )BR(), AL()( )BL(), AR()( )BR(), AR()( )BL()].
(2.10)
(v)
When AL(), BR(), AR(), BL() 0,
A() ( ) B() = [AL()( )BR(), AR()(
In the sequel we shall write
A() ( ) B() = [AL()( )BR(), AR()(
as
)BL()],
(2.11)
)BL()]
(2.12)
=
SPECIAL FUZZY NUMBER AND THEIR CUTS
TRIANGULAR FUZZY NUMBER [2, 17, 18, 25]. A triplet A = (a1, a2, a3) is defined as a triangular
fuzzy number (TFN) if its membership function (m. f.) μA(x) is defined as
μA(x) =
(2.13)
In view of (2.6) and [2, 17, 18, 25] we characterize the TFN A = (a1, a2, a3) in terms of its -cut as
A() = [AL(), AR()]
= [a1 + (a2 a1), a3 + (a2 a3)]
Thus, we have
AL() = a1 + (a2 a1)
AR() = a3 + (a2 a3)
3.
[0, 1].
[0, 1].
[0, 1].
(2.14)
(2.15)
(2.16)
Physical efficiency index (PEI) under crisp environment
Using easily available and inexpensive equipment and under crisp environment, Brouha [4] and
Brouha et al. [5, 6] conducted HST, a very simple and promising field test, for measuring physical
efficiency (cardiovascular endurance) of a human being. In HST [4, 5, 6] the subject steps up and
132
down a 20 inch platform 30 times a minute for 5 minutes continuously unless he/she stops from
exhaustion before 5 minutes are over. At this point the observer records the duration of exercise. In
HST, the observer starts counting the time when the object starts the exercise.
According to Brouha [4] and Brouha et al. [5, 6], it is important for the observer to make sure that the
subject steps fully on the platform and takes a standing position at each step. Crouching is not
allowed. The subject must keep the pace accurately and if he/she falls behind because he/she is tired
the observer must stop him/her after he/she has been unable to keep pace for 10 to 15 seconds. The
pulse count is taken from 1 to 1.5, 2 to 2.5 and 3 to 3.5 minutes after the exercise is stopped. In this
case, the PEI is obtained as follows.
Let
d = length of exercise interval in seconds
ti = time interval in minutes after the exercise is stopped, i = 1, 2, 3.
where,
t1 = 1 minute to 1.5 minutes, t2 = 2 minutes to 2.5 minutes, and
t3 = 3 minutes to 3.5 minutes
pi = the pulse count during the interval ti ,
3.1
i = 1, 2, 3.
Full Form Formula
According to [4, 5, and 6] the Physical efficiency index (PEI) under crisp environment for those
subjects who complete 5 minutes duration of exercise is determined by using the following formula.
PEI =
3.2
(3.1)
Short Form Formula
The physical efficiency index (PEI) under crisp environment for those subjects who drop out
before they complete 5 minutes duration of exercise is determined as follows [4, 5, and 6]. Let
L = length of exercise interval in seconds, where L < 300 seconds, and
p = the pulse count during the interval 1 minute to 1.5 minutes after
the subject drops out before completing 5 minutes
Then, the (PEI) is given by
PEI =
(3.2)
According to [4, 5, 6] and [17], the physical efficiency index (PEI) under crisp environment for
those subjects whose complete 5 minutes duration of exercise is determined by using the following
formula.
133
4.
4.1
Physical efficiency index under fuzzy environment
Long Form Case
In this case we assume that d and each of pi , i = 1, 2, 3 are TFN’s. Thus, we take
d = (d1, d2, d3) = [d1 + (d2 d1), d3 + (d2 d3)]
dL() = d1 + (d2 d1)
dR() = d3 + (d2 d3)
And for i = 1, 2, 3
pi = (pi1, pi2, pi3) =
[0, 1],
(4.1)
[0, 1].
[0, 1].
(4.2)
(4.3)
[pi1 + (pi2 pi1), pi3 + (pi2 pi3)],
(4.4)
() = pi1 + (pi2 pi1),
[0, 1].
(4.5)
() = pi3 + (pi2 pi3),
[0, 1].
(4.6)
Then,
p1 + p2 + p3 =
=
[0, 1].
Now, using (4.2)(4.6) in conjunction with (2.7)(2.12) in (3.1), we obtain the cut of the Physical
Efficiency Index under Fuzzy Environment. Thus, denoting the Physical Efficiency Index under Fuzzy
Environment by FPEI and its cut by FPEI(), we have
FPEI() =
= [FL (), FR()]
[0, 1].
(4.8)
(4.9)
where,
F L( ) =
(4.10)
F R( ) =
(4.11)
From (4.7) we observe that the FPEI belongs to an interval [FL(), FR()] given by (4.8) in which the
lower limit is given by FL() [0, 1] and the upper limit is given by FR() [0, 1].
Setting = 0 in (4.7), we obtain the end points of the FPEI as
FL(0) =
,
and
FR (0) =
(4.12)
Setting = 1 in (4.7), we obtain the interior point of the FPEI as
134
FL(1) =
= FR (1)
(4.13)
From (4.12) and (4.13), the FPEI, whose -cut is given by (4.7), is given by
(4.14)
FPEI =
Using (4.14) we now make the following important observations.
Observation 4.1. If we assume the data for each subject to be crisp, then the values yielded by the interior
point (i.e. values corresponding to the value = 1) in FPEI coincide with the corresponding values in the
PEI. This means that the fuzzy algebra approach provides the physical efficiency index as an interval in
which the interior point is, as a special case, the crisp physical efficiency index.
Observation 4.2. In general the FPEI given by (4.14) is not a TFN. However, under certain conditions
described in Observation 3, we can approximate FPEI as a TFN. In that case we name the FPEI as
triangular fuzzy physical efficiency index (TFPEI).
Observation 4.3. We can draw the graphs of the membership function of FPEI and the membership
function of the approximated TFPEI by varying between 0 and 1. If each of the maximum of the left
divergence and the maximum of the right divergence turns out to be less that 3% , then we may treat FPEI
as a TFPEI [18].
Approximation of FPEI by a TFN
is a TFN. Then we have
Suppose we now assume that
TFPEI =
(4.15)
whose -cut is as follows.
T() = [
+(
) ,
+(
)
+(
)] (4.16)
where,
T L( ) =
(4.17)
135
T R( ) =
+(
)
(4.18)
As in [18], we define the left and the right divergence as follows.
Left Divergence Dleft . Dleft =
Dleft()=|
+(
) |
Dright = |TR() FR()|
Right Divergence Dright .
Dright () = |
|TL() - FL()|
+(
) |
We now state a theorem for the Full Form Case that will provide us the conditions under which we may
be able to approximate the FPEI by TFEPI..
THEOREM 1. Suppose the length of exercise interval in seconds is denoted by fuzzy number d = (d1, d2,
d3) as defined in (4.1) (4.3), and the pulse rates is given by a fuzzy numbers
pi = (pi1, pi2, pi3) , i = 1, 2, 3 as defined in (4.4) – (4.6). Then,
(i)
the maximum of Dleft () occurs at
=
D*left (
and
(ii)
,
) =
(4.19)
(4.20)
the maximum of Dright () occurs at
=
,
(4.21)
136
and
D*right (
) =
where in both (i) and (ii) 4.2
(4.22)
[0, 1].
Short Form Case
In this case, as in Long Form Case, we take L as a triangular fuzzy number
L = (l1, l2, l3),
whose cut is given by
[0, 1].
L() = [l1 + (l2 l1), l3 + (l2 l3)]
We take p also as a triangular fuzzy number given by
p = (p1, p2, p3),
whose cut is given by
p() = [p1 + (p2 p1), p3 + (p2 p3)]
[0, 1].
Then, analogous to (3.2) the Short Form Fuzzy Physical Efficiency Index (SFPEI) is given by
SFPEI() =
[0, 1].
(4.22)
Analogous to (4.14), we have the
(4.23)
SFPEI =
Treating SFPEI as a TFN we have,
TSFPEI =
(4.24)
whose cut is given by
TSFPEI () =
[0, 1]. (4.25)
Then, the cuts of the left divergence and the right divergence are defined as follows.
SDleft () = |
-
SDright () = |
-
|
|
Similar to the Theorem 1, we have the following Theorem 2.
THEOREM 2. Suppose the length of exercise interval in seconds is denoted by fuzzy number l = (l1, l2,
l3) as defined above, and the pulse rate is given by a fuzzy numbers p = (p1, p2, p3) as defined above.
Then for the short form case,
137
(ii)
the maximum of SDleft () occurs at
=
SD*left (
and
(ii)
,
) =
(4.27)
the maximum of SDright () occurs at
=
and
(4.26)
SD*right (
,
(4.28)
) =
where in both (i) and (ii) (4.29)
[0, 1].
5.
Collection of data and TFN format
We first describe the procedure for Long Form. The procedure for the Sort Form is a special case of the
procedure for the Long Form.
Step 1.
Determine an appropriate sample size n .
Data Collection and Analysis.
Step 2.
For the each subject in the sample size, collect the data on pulse rates pi , i = 1, 2, 3, both
for Long and Short Forms exactly on the lines as suggested by Brouha [4], and Brouha et
al. [5, 6].
Step 3.
Using Excel or otherwise, compute the standard deviation 1 from the column of p1 for
the whole sample size. Similarly, compute 2 and 3 for the columns o2 and p3.
To Compute Triangular Fuzzy Number (p11, p12, p13) corresponding to p1 (pulse count of a
subject corresponding to t1 we subtract and add 1 1 to yield
(p11, p12, p13) = (p1 1 1 , p1 , p1 + 11 )
for every element of the sample in the column of p1 to eliminate the imprecision or
subjective assessment from the data corresponding to the subjects. Thus, below for p1 and t1
we create an n 3 matrix whose columns are as follows. We fill the row corresponding to
Subject 1. Rows for other subjects are filled on similar lines.
Step 4.
Subject
Number
1
2
n
p11 = p1 1
pulse count 1
p12 = p1
pulse count
P13 = p1 + 1
pulse count + 1
pulse count n
pulse count
pulse count + n
138
It may be pointed here that we can also obtain the fuzzy elements (p11, p12, p13) from p1 by subtracting
from and adding to s 1 . This will yield
(p11, p12, p13) = (p1 s 1, p1 , p1 + s 1 ), s = 2, or 3
for all the subjects in the sample. However, in this case the TFN’s as compared with s = 1 will have broader
range in which case it may lose some precision. Similarly, we can obtain the TFN’s (p21, p22, p23) and (p31,
p32, p33).
6.
Numerical comparison of PEI and FPEI
Numerical Example 1 (Long Form case). To demonstrate as to how the method works we now
consider two numerical examples under fuzzy environment. Example 1 is for Long Form and Example 2
is for Short Form.
Example 1. Suppose we have a subject called n = 1 for which we have
(d1 , d2 , d3) = [295, 300, 305] in seconds,
(p11, p12 , p13) = (67.28, 70, 72.72), for ti = t1
(p21 , p22 , p23) = (64.80, 67, 69.20),
for ti = t2 and
for ti = t3
(p31 , p32 , p33) = (63.38, 65, 66.62),
Then, we have
FPEI() =
=
[0, 1].
Setting = 0, and = 1, we obtain
FPEI = (70.73, 74.26, 78.20)
Thus, we can say that in this case the FEPI will lie within 70 and 78 (taking the integer values), which
according to Brouha is classified as high average.
If we treat FPEI as a TFN approximately, then its -cut is given by
TFPEI = [70.73 + 3.53, 78.2 3.6 ]
[0, 1].
Thus, for the values of given below in [0, 1] we have the following Table.
Left
TEFPI
Right
TEFPI
0
70.
73
78.
20
.1
71.
08
77.
84
.2
71.
44
77.
48
.3
71.
79
77.
12
.4
72.
14
76.
76
.5
72.
50
76.
40
.6
72.
85
76.
04
.7
73.
20
75.
68
.8
73.
55
75.
32
.9
73.
90
74.
96
1
74.
60
74.
60
139
Thus for subject n = 1, the TFPEI lies between 70 and 78. Also, in this case, with every value of the
TFPEI we have the level of truth associated with it.
Numerical Example 2. As in Example 1, we take the duration of exercise as 4 minutes i.e. 240 seconds
such that
(l1 , l2 , l3)
= [235, 240, 245] in seconds, with -cut
= [235 + , 245 ]
We let
(p1 , p2 , p3)
[0, 1].
= [84, 90, 96]
Therefore, from (4.22), we have
SFPEI() =
[0, 1].
=
[0, 1].
=
[0, 1].
[0, 1].
= [44.55, 48.53, 53.08]
Treating SFPEI as a TFN we have,
TSFPEI =
=
whose cut is given by
TSFPEI () =
Thus, in this case we see that the SFPEI lies between 44 and 54, which according to Brouha is considered
as poor physical condition.
7.
8.
Conclusion
In the present paper we present Brouha’s PEI [4] under fuzzy environment and call it FPEI. Also, to
demonstrate the implementation of the results, we present two numerical examples. An advantage of the
fuzzy approach presented here is that it obtains the FPEI in the form of an interval. However, it may be
pointed out that using Brouha’s approach one gets PEI as a single point only.
140
References
[1]
Bajaj, Anju (2002), Effect of Dietary Pattern on Health and Motor Fitness of Punjab Players with
Special Reference to Sportive Events, Ph.D. Thesis, Department of Physiotherapy and Sports Sciences,
Punjabi University, Patiala, India.
[2]
Bector, C. R. and Chandra S. (2005), Fuzzy Mathematical Programming and Fuzzy Matrix Games,
Studies in Fuzziness and Soft Computing, Springer Verlag 23.
[3] Bojadziev, George and Bojadziev, Maria (1997), Fuzzy Logic for Business, Finance and Management,
World Scientific, New Jersey, U.S.A.
[4]
Brouha, Lucien (1943), “The Step Test: A Simple Method for Measuring Physical Fitness for Muscular
Work in Young Men”, Res.Quart. 14, pp. 31-36.
[5] Brouha, L., Graybiel, A., Heath, C.W. (1943), “The Step Test: A Simple Method for Measuring
Physical Fitness for Hard Muscular Work in Adult Man”, Rev.Canada Biol. 2, pp. 86-92.
[6] Brouha, L., Fradd, N.W. and Savage, B.M. (1944), “Studies in Physical Efficiency of College
Students”, Res. Quart. 15, pp. 211-224.
[7]
Buckley, James J., Eslami, Esfandiar, and Feuring, Thomas (2000), Fuzzy Mathematics in Economics
and Engineering, Physica-Verlag, A Springer-Verlag Company, Heidelberg, Germany.
[8] Chin-Teng Lin, Shing-Hong Liu, Jung Wan, and Zu-Chi Wen (2003), “ Reduction of Interference in
Oscillometric Arterial Blood Pressure Measurement using Fuzzy Logic Biomedical Engineering”, IEEE
Trans. 50 (4), pp. 432-441.
[9]
Cotton, Doyice J. (1971), “A Modified Step Test for Group Cardiovascular Testing”,
Res. Quart. 42, pp. 91-95.
[10] Clark, J. W. (1966), The Relationship of Initial Pulse Rate, Recovery Pulse Rate, Recovery Index and
Subjective Appraisal of Physical Condition After Various Deviations of Work, Master’s Thesis,
Louisiana State University, U.S.A.
[11] Clarke, H. L. (1943), “A Functional Physical Fitness Test for College Women”, J. Hlth. Phys. Educ. 14,
p. 358 (Abstract).
[12] Gallagher, J. Roswell, and Brouha, Lucien (1943), “A Simple Method for Testing the Physical Fitness
for Boys”, Res. Quart. 14, pp. 24-30.
[13] Gallagher, J. Roswell, and Brouha, Lucien (1944), “Physical Fitness -Its Evaluation and Significance”,
Journal of the American Medical Association 125, pp. 834-838.
[14] Hellebrandt, Frances A. (1946), “Contributions of Physical Education to Physical Fitness”, Rev. of.
Educational, Mental and Physical Health, 16 (5), pp. 457-460.
[15] Hodgkins, Jean and Skubic, Vera (1963), “Cardiovascular Efficiency Scores for Young Women in the
United States”, Res. Quart. 34, pp. 454-461.
[16] Keen, E.N., and Sloan, A. W. (1958),“Observations on Harvard Step Test” J. Appl. Physio. 13, p. 241.
141
[17] Kaufmann, A., and Gupta, M. M. (1985), Introduction to Fuzzy Arithmetic Theory and Applications,
Von Nostrand Reinhold Company, New York, U.S.A.
[18] Kaufmann, A., and Gupta, M. M. (1988), Mathematical Fuzzy Models in Engineering and Management
Sciences, North Holland, New York, U.S.A.
[19] Kurucz, Robert L. (1967), Construction of the Ohio State University Cardiovascular Fitness Test,
Doctoral Dissertation, Louisiana State University, U.S.A.
[20] McArdle, W. D., Pechar, G. S., Katch, F. I. and Magel, J.R. (1973), “Percentile Norms for a Valid Step
Test in College Women”, Res. Quart., 44, pp. 498-500.
[21] Nelson, J. K. and Johnson, B. L. (1988), Practical Measurement for Evaluation in Physical Education,
3rd Edition, Surjit Publication, Delhi, India.
[22] Ricci, B., Baldwin, K., Hakes, R., Fein, J., Sadowsky, D., Tuffs, S. and Wells, C. (1966), “Energy Cost
and Efficiency of Harvard Step Test Performance”, Inter. Zeits. Angew. Physio. Einnsch. Arbeits
Physio. 22, pp.125-130.
[23] Skubic, Vera and Hodkins, Jean (1963), “Cardiovascular Efficiency for Girls and Women”, Res. Quart.
34, pp.191-198.
[24] Zadeh, Lofti A. (1965), “Fuzzy Sets”, Information and Control, 8, pp. 338-353.
[25] Zimmermann, H. J. (2001), Fuzzy Sets Theory and Its Applications, 4th Edition, Kluwer Academic
Publishers, Nowell, MA, U.S.A.
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ASAC 2008
Halifax, Nova Scotia
Aim Solaiman
Weber Aircraft LP
Walid Abdul-Kader
University of Windsor
Ozhand Ganjavi
Laurentian University
AN INTEGRATED PRODUCTION COST MODEL WITH POSSIBLITY TO REDUCE
REWORK/SCRAP COST
This paper addresses the problem of reworking of defective items and
develops an integrated cost model so as to minimize the extra costs of
reworking/scraping of work pieces. Items to be reworked are the results
of quality problem. To improve a product’s quality, the selection of
process target is extremely important since it directly affects the process
defective rate, material cost, rework or scrap cost, and loss to customer
due to deviation of product from desired specification. The amount of
investment necessary to economically correct a defective process is still
an issue of research. This research is a contribution to this type of
problem. In addition to cost estimation, the integrated cost model focuses
on optimal tolerance, and optimal mean and variance of the output
characteristics. The symmetrical truncated loss function is used to
evaluate the cost of poor quality in a production system. Specifically, we
investigate the possible economic investment in a process improvement
to reduce its variance and shifting the process mean close to its target,
resulting in the reduction of waste like rework/scrap. Numerical
examples are presented to show all the steps involved and to verify the
proposed model. Utilizing this model, decision-makers can evaluate any
quality investment in order to achieve a significant financial return
Keywords: Economic investment; Rework/scrap costs; Process defects
1. Introduction
High quality and low cost are competitive strategies of many manufacturing companies. Waste reduction
is an important element of minimizing total cost of production (Raiman, 1990). Over the last two decades,
there have been numerous researches on process quality improvement and costs reduction. Most of these
researches focus on costs related to quality loss. In reality, a production process incurs several cost
including manufacturing costs, materials costs, quality loss cost, inspection costs, rework costs, and scrap
costs. Literature has shown that only a few attempts have been made to combine manufacturing and
quality related costs in a mathematical model. Our primary intent is to reduce the process variance as well
as shifting process mean closer to target, to reduce the amount of products that are prone to extra costs of
rework/repair. In this context, we are going to revisit the existing cost models with process quality based
on Taguchi’s quadratic loss function.
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