Cases of gun-related violence in the society often lead to diverse opinions among the citizens on the best
approach using which such issues could be resolved. The United States has experienced increased calls for
regulation of gun ownership, partly due to the increased cases of gun-related violence in the country.
Ultimately, scholars with interest in the subject have often made an effort to develop some form of
understanding on some of the possible measures floated as being possible solutions to cases of gun violence
in the society. Janet E.Rosenbaum set out to understand the impact of the policies on gun violence within an
analysis of the Israeli and Switzerland societies as the case study. The research therein was based on the
hypothesis that these countries have less strict gun laws, and encouragement of the residents therein to own
weapons has been influential in reducing cases of gun-related violence in the countries.
The scholar used primary research data, based on whose analysis it was clear that neither of the two
governments encourages gun ownership among the citizens. In Israel, a policy that ensured that soldiers
returning from war have limited to no access to firearms was instrumental in reducing the rates of suicide
among such war veterans (Rosenbaum, 2012). In any case, ownership of weapons in Israel comes following
sufficient explanation to the government on why a particular individual ought to be allowed to carry a gun.
Consequently, mandatory military service is not enough reason for improved security and limited cases of gunrelated violence. Alternative security measures that focus on individuals are the most appropriate measures to
reduce rates of gun violence to a significant extent.
Essentially, gun ownership in Israel requires a process greater than a background check on the individual
applying for ownership in addition to the necessary renewal of the license several times a year. The applicant
needs to convince the government enough of the necessity to own a weapon, and get back to the authorities
regularly to give an update of the situation that had led him or her to seeking gun ownership as a solution to the
issue in question. Ultimately, there are limited guns in the Israeli community in possession of the civilians. The
same case applies to Switzerland, with both countries having elaborate policies that discourage private gun
ownership, encouraging the citizens therein to seek assistance from the government whenever security is of
primary concern.
The primary solution to gun-related violence in the United States lies in total control of weapon access in the
society. The Second amendment has often been the primary source of reference based on which
conservatives seek to retain gun laws or increase access as a way of controlling cases of gun-related violence.
However, it would be appropriate to make appropriate interpretation of the law and use it in the context of the
modern society. At the time of the constitutional amendment, the weapons available were not as lethal as those
used and available in the contemporary world. If at all there should be a right to bear arms, there should be a
clear definition of the type of weapons that people can purchase. Otherwise, developing a policy that would
have young Americans participate in the military may not be effective enough in solving the problem as there
would be no guarantee of limited use once they have left the service.
This paragraph is very liberal Democrat and it may not be exactly what I believe, But you cannot take away
what the author has found in her research. I do believe that there needs to be stricter background checks, reapplication every couple of years, and that military service “weapon knowledge” will help lower the gun related
violence and injuries. Furthermore, the 2nd amendment needs to be protected but interpreted in a modern
sense. There is no need for fully automatic weapons, unless it is being used for military service. If you are
hunting, and you cannot hit your target with one shot or re-engaged second shot, you need to hit the range and
practice because that animal deserved to live and you did not deserve to eat. Maybe there needs to be
inspections for weapon registrations. We do it for our vehicles. Why not do it for our weapons. It is just an idea.
And maybe it will help pay for our ungodly debt.
(I do apologize, I had accidently posted the working copy during a hectic work day, work week. I have edited
what needed to be done to make it more complete.)
References
Rosenbaum, J. E. (2012). Gun utopias? Firearm access and ownership in Israel and Switzerland. Journal of
Public Health Policy, 33(1), 46-58.
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4
Quantitative Modeling Techniques
4.1
Introduction
In this book, quantitative models for a number of crucial issues in strategic
planning of reverse and closed-loop supply chains are presented. Depending on the decision-making situation, each model makes use of one or more
of the quantitative techniques introduced in this chapter. It must be noted
that only the basic concepts of each technique are presented here.
This chapter is organized as follows: Sections 4.2 through 4.19 introduce
the concepts of analytic hierarchy process (including eigen vector method),
analytic network process, fuzzy logic, extent analysis method, fuzzy multicriteria analysis method, quality function deployment, method of total preferences, linear physical programming, goal programming, technique for
order preference by similarity to ideal solution (TOPSIS), Borda’s choice rule,
expert systems, Bayesian updating, Taguchi loss function, Six Sigma, neural
networks, geographical information systems, and linear integer programming, respectively. Finally, section 4.20 gives some conclusions.
4.2
Analytic Hierarchy Process and Eigen Vector Method
The analytic hierarchy process (AHP) is a tool supported by simple mathematics that enables decision makers to explicitly weigh tangible and intangible criteria against each other for the purpose of resolving conflict or
setting priorities. The process has been formalized by Saaty [1] and is used
in a wide variety of problem areas, for example, siting landfills [2], evaluating employee performance [3], and selecting a doctoral program [4].
In a large number of cases (for example, [5]), the tangible and intangible
criteria are considered independent of each other; in other words, those criteria do not in turn depend upon subcriteria and so on. The AHP in such cases
is conducted in two steps: (1) weigh independent criteria, each of which can
compare two or more decision alternatives, using pair-wise judgments, and
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(2) compute the relative ranks of decision alternatives using pair-wise judgments with respect to each independent criterion.
1. Computation of relative weights of criteria: AHP enables a person to
make pair-wise judgments of importance between independent
criteria with respect to the scale shown in table 4.1. The resulting
matrix of comparative importance values is used to weigh the independent criteria by employing mathematical techniques like eigen
value, mean transformation, or row geometric mean. This step is
called the eigen vector method if an eigen vector is the employed
mathematical technique.
2. Computation of the relative ranks: Pair-wise judgments of importance
using the scale shown in table 4.1 are computed for the decision alternatives as well. These judgments are obtained with respect to each
independent criterion considered in step 1. The resulting matrix of
comparative importance values is used to rank the decision alternatives by employing mathematical techniques like eigen value, mean
transformation, or row geometric mean.
The degrees of consistency of pair-wise judgments in steps 1 and 2 are
measured using an index called the consistency ratio (CR). Perfect consistency implies a value of zero for CR. However, perfect consistency cannot be
demanded because, as human beings, we are often biased and inconsistent
in our subjective judgments. Therefore, it is considered acceptable if CR is
less than or equal to 0.1. For CR values greater than 0.1, the pair-wise judgments must be revised before the weights of criteria and the ranks of decision alternatives are computed. CR is computed using the formula
CR =
(λ max − n )
( n − 1)( R )
(4.1)
where λmax is the principal eigen value of the matrix of comparative importance values, n is the number of rows (or columns) in the matrix, and R is
Table 4.1
Scale for pair-wise judgments
Comparative
importance
Definition
1
Equally important
3
Moderately more important
5
Strongly important
7
Very strongly more important
9
Extremely more important
2, 4, 6, 8
Intermediate judgment values
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Table 4.2
Random index value for each n value
n
1
2
3
4
5
6
7
8
9
10
R
0
0
0.58
0.90
1.12
1.24
0.32
1.41
1.45
1.49
the random index for each n value that is greater than or equal to 1. Table 4.2
shows various R values for n values ranging from 1 to 10.
The AHP is illustrated in the form of a hierarchy of three levels, where the
first level contains the primary objective, the second level contains the independent criteria, and the last level contains the decision alternatives. Also, an
important feature of the AHP is that the tangible and intangible criteria in
the second level must be chosen in such a way that they can somehow help
the decision maker in comparing two or more decision alternatives.
4.3
Analytic Network Process
Analytic network process [6] (ANP) generalizes the AHP. AHP assumes independence among the criteria and subcriteria considered in the decision making, but real-life situations warrant against such assumption. ANP allows for
dependence within a set of criteria (inner dependence) as well as between
sets of criteria (outer dependence); therefore, ANP goes beyond AHP [7].
Whereas AHP assumes a unidirection hierarchical relationship among the
decision levels, ANP allows for a more complex relationship among decision
levels and attributes, as it does not require a strict hierarchical structure. The
looser network structure in ANP allows the representation of any decision
problem, irrespective of which criteria come first or which come next. Compared to AHP, ANP requires more calculations and requires a more careful
track of the pair-wise judgment matrices. ANP is used in a wide variety of
problem areas (for example, analyzing alternatives in reverse logistics for
end-of-life computers [7]; evaluating connection types in design for disassembly [8]; and modeling the metrics of lean, agile, and leagile supply chain
[9]). The steps involved in the ANP methodology are as follows:
Step 1: Model development and problem formulation: In this step, the decision problem is structured into its constituent components. The
relevant criteria, the subcriteria, and alternatives are chosen and
structured in the form of a control hierarchy.
Step 2: Pair-wise comparisons: In this step, the decision maker is asked
to carry out a series of pair-wise comparisons with respect to the
scale shown in table 4.1, where two main criteria are simultaneously
compared with respect to the problem objective, two subcriteria are
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simultaneously compared with respect to their main criteria, and
pair-wise comparisons are performed to address the interdependencies among the subcriteria. The relative matrix of comparative
importance values is then used to weigh the criteria using mathematical techniques like eigen vector, mean transformation, or row
geometric mean.
Step 3: Super matrix formulation: The super matrix allows for a resolution of interdependencies that exist among the subcriteria. It is a
partitioned matrix, where each submatrix is composed of a set of
relationships between and within the levels as represented by the
decision maker’s model. The super matrix M is made to converge to
obtain a long-term stable set of weights. For convergence, M must be
made column stochastic; that is done by raising M to the power of
2k+1, where k is an arbitrarily large number.
Step 4: Selection of the best alternative: The selection of the best alternative depends on the desirability index. The desirability index, DI, for
alternative i is defined as
J
DI =
Kj
∑∑ P A A S
j
j−1
D
kj
I
kj ikj
(4.2)
k=1
where Pj is the relative importance weight of main criterion j; AkjD
is the relative importance weight for subcriterion k of main criterion j for the dependency (D) relationships among subcriteria; AkjI is
the stabilized relative importance weight (determined by the super
matrix) for subcriterion k of main criterion j for interdependency
(I) relationships among subcriteria; and Sikj is the relative impact of
alternative i on subcriterion k of main criterion j.
4.4
Fuzzy Logic
Expressions such as “probably so,” “not very clear,” and “very likely,” which
are often heard in daily life, carry a touch of imprecision with them. This
imprecision or vagueness in human judgments is referred to as fuzziness in
the scientific literature. As the decision-making problem’s intensity grows,
this imprecision leads to results that can often be misleading, if the fuzziness
is not taken into account. Zadeh [10] first proposed fuzzy logic, after which
an increasing number of studies have dealt with fuzziness in problems by
applying fuzzy logic.
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Quantitative Modeling Techniques
When dealing with factors with uncertain or imprecise values, people use
linguistic values like high, low, good, and medium, to describe those factors.
For example, height may be a factor with an imprecise value, so its linguistic
value can be “very tall” or “very short.” Fuzzy logic is primarily concerned
with quantifying vagueness in human perceptions and thoughts. The transition from vagueness to quantification is performed by the application of
fuzzy logic, as shown in figure 4.1.
Zadeh proposed a membership function to deal with quantifying vagueness. Each quantified linguistic value is associated with a grade membership
value belonging to the interval [0, 1] by means of a membership function.
Thus, a fuzzy set can be defined as ∀x ∈ X , µA ( x ) ∈ [0,1] , where µA is the
degree of membership, ranging from 0 to 1, of a quantity x of the linguistic
value, A, over the universe of quantified linguistic values, X. X is essentially
a set of real numbers. The more x fits A, the larger the degree of membership of x. If a quantity has a degree of membership equal to 1, this reflects a
complete fitness between the quantity and the linguistic value. On the other
hand, if the degree of membership of a quantity is 0, then that quantity does
not belong to the linguistic value. The membership function looks like a typical cumulative probability function; however, the value of a membership
function represents the possibility of a fuzzy event, whereas the value of a
cumulative probability function represents the cumulative probability of a
statistical event.
A triangular fuzzy number (TFN) [11] is a fuzzy set with three parameters
(l, m, u), each representing a quantity of a linguistic value associated with a
degree of membership of either 0 or 1. Figure 4.2 shows a graphical depiction of a TFN. The parameters l, m, and u denote the smallest possible, most
promising, and largest possible quantities that describe the linguistic value.
Fuzzy Logic
Vagueness
Quantification
Figure 4.1
Application of fuzzy logic.
Degree of Membership
1.0
0.0
l
m
Parameters of TFN
Figure 4.2
Triangular fuzzy number.
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Each TFN, P, has linear representations on its left- and right-hand side such
that its membership function can be defined as
0,
x u
0,
(4.3)
For each quantity x increasing from l to m, the corresponding membership
function linearly increases from 0 to 1, and while x increases from m to u, the
corresponding membership function decreases linearly from 1 to 0.
The basic operations on TFNs are as follow [11, 12]; for example, P1 = (l, m,
u) and P2 = (x, y, z):
P1 + P2 = (l + x, m + y, u + z)
addition
(4.4)
P1 − P2 = (l − z, m − y, u − x )
subtraction
(4.5)
P1 × P2 = (l × x, m × y, u × z )
multiplication
(4.6)
division
(4.7)
P1 l m u
= , ,
P2 z y x
Defuzzification is a technique to convert a fuzzy number into a crisp real
number. There are several methods available for this purpose. The center-ofarea method [13] converts a fuzzy number P = (l, m, u) into a crisp number
Q, where
Q=
(u − l ) + (m − l )
+l
3
(4.8)
Defuzzification might be necessary in two situations: (1) when comparison
between two fuzzy numbers is difficult to perform, and (2) when a fuzzy
number to be operated on has negative parameters (in other words, we make
sure that upon performing an arithmetic operation on a TFN, we get a TFN
only; for example, squaring the TFN (–1, 0, 1) using equation (4.6) leads to (1, 0,
1), which is not a TFN, and hence we defuzzify (–1, 0, 1) before squaring it).
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4.5
Extent Analysis Method
Chang [14] proposed a new approach to handle situations that require use
of both fuzzy logic and AHP/ANP. First, TFNs (see section 4.4) are used for
pair-wise comparisons; then by using extent analysis method [15], the synthetic extent value of the pair-wise comparison is introduced, and by applying the principle of comparison of fuzzy numbers, the weight vectors with
respect to each element under a certain criterion can be computed. The steps
involved in the methodology are as follows.
Let X = {x1, x2, …, xn} be an object set and U = {u1, u2, …, um} be a goal set.
According to the extent analysis method, each object is taken and an extent
analysis for each goal, gi, is performed. Therefore, m extent analysis values
for each object can be obtained, with the following signs: M1gi, M2gi, …, Mmgi,
i = 1, 2, …, n, where all the Mjgi (j = 1, 2, …, m) are TFNs.
Step 1: The value of fuzzy synthetic extent with respect to the ith object
is defined as
m
Si =
∑
j=1
m
In order to obtain
∑M
j
gi
M ⊗
n
m
i=1
j=1
∑∑
j
gi
−1
M
j
gi
(4.9)
, perform the fuzzy addition operation of
j=1
m extent analysis values for a particular matrix such that
m
∑
j=1
M gij =
m
m
m
∑ ∑ ∑
lj ,
j=1
mj ,
j=1
j=1
u j
(4.10)
To obtain
n
m
i=1
j=1
∑∑
−1
M
,
j
gi
perform the fuzzy addition operation of M gij (j = 1, 2, …, m) values such
that
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n
m
∑∑
i=1
j=1
M gij =
n
n
n
∑ ∑ ∑
li ,
i=1
mi ,
i=1
i=1
ui
(4.11)
and then compute the inverse of the vector.
Step 2: The degree of possibility of M2 = (l2, m2, u2) ≥ M1 = (l1, m1, u1) is
expressed as
V (M2 ≥ M1)
= hgt (M1 ≥ M2)
= {1, if m2 ≥ m1; 0, if l1 ≥ u2; (l1 – u2)/((m2 – u2) – (m1 – l1))}
(4.12)
To compare M1 and M2, both V (M2 ≥ M1) and V (M1 ≥ M2) are required.
Step 3: The degree of possibility for a convex fuzzy number to be greater
than k convex fuzzy numbers Mi (i = 1, 2, …, k) can be defined as
V (M ≥ M1, M2, Mk)
= V [(M ≥ M1) and (M ≥ M2) and … and (M ≥ Mk)]
= min V (M ≥ Mi), i =1, 2, k
(4.13)
Let d’ (Ai) = min V (Si ≥ Sk), for k = 1, 2, n; k ≠ i. Then the weight vector
is given by
W’ = (d’ (A1), d’ (A2), d’ (An))T
(4.14)
Step 4: The weight vector obtained in step 3 is normalized to get the
normalized weights.
These steps are applied to deduce the weights of main criteria with respect
to the goal, subcriteria with respect to the main criteria, and alternatives
with respect to the main and subcriteria. The rest of the procedure is similar
to the traditional AHP/ANP.
4.6
Fuzzy Multicriteria Analysis Method
Multicriteria analysis problems require the decision maker to make qualitative assessments regarding the performance of the decision alternatives with
respect to each independent criterion and the relative importance of each
independent criterion with respect to the overall objective of the problem. As
a result, uncertain subjective data are present that make the decision-making
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Quantitative Modeling Techniques
process complex [16]. AHP enables a person to make pair-wise judgments of
importance between the independent criteria as well as the decision alternatives. However, traditional AHP is criticized for its unbalanced scale of judgment and failure to precisely handle the inherent uncertainty and vagueness
in carrying out pair-wise comparisons.
Deng [17] proposed a multicriteria analysis approach that extends Saaty’s AHP (see section 4.2) to deal with the imprecision and subjectiveness
in the pair-wise comparisons. TFNs (see section 4.4) are used for pair-wise
comparisons, and the concept of extent analysis method (see section 4.5)
is applied to solve the fuzzy reciprocal matrix for determining the criteria
importance and alternative performance. The α-cut concept is used to transform the fuzzy performance matrix representing the overall performance of
all alternatives with respect to each criterion into an interval performance
matrix. An overall performance index for each alternative across all criteria
that incorporates the decision maker’s attitude toward risk is obtained by
applying the concept of similarity to the ideal solution [18] using the vectormatching function.
The selection process starts with determining the criteria of importance
and performance of alternatives. By using TFNs, a fuzzy reciprocal matrix
for criteria importance (W) or alternative performance with respect to a specific criterion (Cj) can be determined as
a
11
a
W or C j = 21
...
a
k1
a12
...
a22
...
...
..
ak 2
...
a1k
a2 k
...
akk
(4.15)
where
1, 3, 5, 9, l < s ,
als = 1, l = s , l , s = 1, 2,...k ; k = m, or , n,
1 / a sl ,l > s
(4.16)
By applying the extent analysis method, the corresponding criteria weights
(wj) or alternative performance ratings (xij) with respect to a specific criterion
Cj can be determined as
k
∑
xij or w j =
s=1
als ÷
k
k
l=1
s=1
∑∑a
ls
(4.17)
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where i = 1, 2, …, n; j = 1, 2, …, m; and k = m or n depending on whether
the reciprocal judgment matrix is for assessing the performance ratings of
alternatives or weights of criteria involved. The decision matrix (X) and the
weight vector (W) can be respectively determined as
x11
x
X = 21
...
xn1
x12
x 22
...
xn 2
...
...
...
...
x1m
x 2m
...
xnm
W = (w1 , w 2 ,..., wm )
(4.18)
(4.19)
where xij represents the resultant fuzzy performance assessment of alternative Ai (i = 1, 2, n) with respect to criterion Cj, and wj is the resultant fuzzy
weight of criterion Cj (j = 1, 2, …, m) with respect to the overall goal of the
problem. A fuzzy performance matrix Z representing the overall performance of all alternatives with respect to each criterion is obtained by multiplying the weight vector by the decision matrix.
w1 x11
w x
Z = 1 21
...
w1 xn1
w 2 x12
w 2 x 22
....
w 2 xn 2
...
...
...
...
wm x1m
wm x 2m
...
wm xnm
(4.20)
An interval performance matrix [16] is derived by using an α-cut on the
performance matrix, where 0 ≤ α ≤ 1. The value of α represents the decision
maker’s degree of confidence in his or her fuzzy assessments regarding the
alternative ratings and criteria weights. The larger the value of α, the more
confident the decision maker is about the fuzzy assessments, viz., the assessments are closer to the most possible value a2 of the triangular fuzzy number
(a1, a2, a3).
z α , z α
11l 11r
...
Zα =
...
α α
z n1l , z n1r
z α , z α
12l 12r
...
...
z α , z α
n 2l n 2r
...
...
...
...
z α , z ε
1ml 1mr
...
..
...
z α , z α
nml nmr
(4.21)
An overall crisp performance matrix that incorporates the decision maker’s
attitude toward risk, using an optimism index λ (λ = 1 implies the decision
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maker has an optimistic view, 0 implies a pessimistic view, and 0.5 implies a
moderate view), is calculated.
z λ'
11α
'
...
z αλ =
...
'
z λ
n1α
'
λ
z12
α
...
...
...
...
...
'
z nλ2α
...
'
z1λmα
...
...
λ'
z nm
α
(4.22)
where
'
z ijλα = λz ijrα + (1 −λ )z ijlα ,λ ∈ [0,1]
(4.23)
A normalized performance matrix with respect to each criterion is calculated from equation (4.22).
zλ
11α
zλ
Zα = 21α
...
λ
z n1α
λ
z12
α
λ
z 22
α
....
z nλ2α
z1λmα
z 2λmα
...
λ
z nm
α
...
...
...
...
(4.24)
where
'
z ijλα = z ijλα ÷
n
∑ (z
λ' 2
ijα
)
i=1
(4.25)
Zeleny [18] introduced the concept of ideal solution in multiattribute decision analysis that was further extended by Hwang and Yoon [19], including
negative solution to avoid the worst decision outcome. In line with this concept, the positive- and negative-ideal solutions, respectively, can be determined by selecting maximum and minimum values across all alternatives
with respect to each criterion as follows:
Aαλ+ = (z1λα+ , z 2λα+ ,..., z mλ+α )
Aαλ− = (z1λα− , z 2λα− ,...., z mλ−α )
(4.26)
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where
z λjα+ = max(z λjα , z 2λ jα ,..., z njλ α )
z λjα− = min(z λjα , z 2λ jα ,...., z njλ α )
(4.27)
By applying the vector-matching function, the degree of similarity between
each alternative and the positive- and negative-ideal solutions can be calculated as
Siλα+ = Aiλα Aiλα+ / max( Aiλα Aiλα , Aαλ + Aαλ + )
Siλα− = Aiλα Aiλα− / max( Aiλα Aiλα , Aαλ − Aαλ − )
(4.28)
λ
where Aiλα = ( ziλ1α , ziλ2 α ,...., zim
α ) is the ith row of the overall performance matrix,
which represents the corresponding performance of alternative Ai with
respect to criterion Cj. The larger the value of Siλα+ , Siλα− , the higher the degree
of similarity between each alternative and the positive-ideal and negativeideal solutions [20]. A preferred alternative should have a higher degree of
similarity to the positive-ideal solution and a lower degree of similarity to
the negative-ideal solution. Hence, an overall performance index for each
alternative with the decision maker’s α level of confidence and λ degree of
optimism toward risk can be determined as
Pαλi = Siλα+ / (Siλα + Siλα− ), i −1, 2,…, n
(4.29)
The larger the performance index, the most preferred the alternative is.
4.7
Quality Function Deployment
Erol and Ferrell [21] define performance aspects as the features that the decision maker wishes to consider in the selection process and enablers as the
characteristics possessed by the alternatives, which can be used to satisfy the
performance aspects.
The absolute technical importance ratings (ATIRs), which measure how
effectively each enabler can satisfy all of the performance aspects, are computed by
I
ATIRj =
∑d R
i
i=1
ij
∀ j = 1, ..., J
(4.30)
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where di is the importance value of performance aspect i relative to the other
performance aspects, and Rij is the relationship score for performance aspect
i and enabler j. Because there is an ATIR for each enabler j, for the comparison of all enablers, it is normalized to form the relative technical importance
rating (RTIRj) as follows:
RTIRj =
ATIRj
∀ j = 1, ..., J
J
∑ ATIR
j
j=1
4.8
(4.31)
Method of Total Preferences
RTIRs (see section 4.7), together with additional human expert opinions, are
used to develop a single measure that reflects the rating of each alternative
as follows [21]:
J
TUPn =
∑ RTIR WA
j
nj
∀n
j=1
(4.32)
where TUPn is the total user preference for alternative n, and WAnj is the
(defuzzified) degree to which alternative n can deliver enabler j.
For the purpose of comparison of all alternatives, TUP of each alternative
is then normalized as follows:
NTUPn =
TUPn
N
∑ TUP
∀n
n
n=1
(4.33)
where NTUPn is the normalized total preference for alternative n, and N is
the total number of alternatives.
The alternative with the highest NTUP is considered the one with the
highest potential.
4.9 Linear Physical Programming
In the linear physical programming (LPP) method [22], four distinct classes
(1S, 2S, 3S, and 4S) are used to allow the decision maker to express his or her
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preferences for the value of each criterion (for decision making) in a more
detailed, quantitative, and qualitative way than when using a weight-based
method like analytic hierarchy process (see section 4.2). These classes are
defined as follows: smaller is better (1S), larger is better (2S), value is better
(3S), and range is better (4S). Figure 4.3 depicts these different classes.
The value of the pth criterion, gp, for evaluating the alternative of interest is
categorized according to the preference ranges shown on the horizontal axis.
Consider, for example, the case of class 1S. The preference ranges are:
zp
zp
z5
z5
zp
t-p5
t-p4
t-p3
t-p2
t-p1
g p(x)
Idea l
z2
z1
g p(x)
Desirable
t+p5
Tolerable
t+p4
Unacce ptable
t+p3
Highly
unde sirabl e
t+p2
Tolerable
t+p1
Highly
undesirable
z3
z2
z1
Unac ce ptable
z3
Unde sira ble
z4
Desira ble
z4
Unde sira ble
Cla ss- 2S
"Larger is Better"
Class-1S
"Smaller is Better"
Ideal
z5
Cla ss- 3S
"Value is Better"
z4
t+p5
g p(x)
Unac ce ptable
zp
t+p4
Highly
undesirabl e
t+p3
Undesirabl e
t+p2
Tolerable
tp1
Desira ble
t-p2
Idea l
t-p3
Desira ble
t-p4
Tolerable
t-p5
Unde sirabl e
Unac ceptable
z2
z1
Highly
unde sirable
z3
z5
Class-4S
"Range is Better"
z4
t+p4
t+p5
Unac ce ptable
t+p3
Highly
undesira ble
t+p2
Undesirable
t+p1
Tolerable
t-p1
Desirable
t-p2
Ideal
t-p3
Desirable
t-p4
Tolerable
t-p5
Highly
undesi rable
z2
z1
Undesirable
z3
Unac ce ptable
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50
g p(x)
Figure 4.3
Soft class functions for linear physical programming.
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Ideal range: g p ≤ t +p1
Desirable range: t +p1 ≤ g p ≤ t +p 2
Tolerable range: t +p 2 ≤ g p ≤ t +p 3
Undesirable range: t +p 3 ≤ g p ≤ t +p 4
Highly undesirable range: t +p 4 ≤ g p ≤ t +p5
Unacceptable range: g p ≥ t +p5
+
+
The quantities t p1
through t p5
represent the physically meaningful values
that quantify the preferences associated with the pth generic criterion. Consider, for example, the cost criterion for class 1S. The decision maker could
+
+
through t p5
in dollars as (10
specify a preference vector by identifying t p1
20 30 40 50). Thus, an alternative having a cost of $15 would lie in the desirable range, an alternative with a cost of $45 would lie in the highly undesirable range, and so on. We can accomplish this for a nonnumerical criterion
such as color as well by (1) specifying a numerical preference structure and
(2) quantitatively assigning each alternative a specific criterion value from
within a preference range (e.g., desirable, tolerable).
The class function, Zp, on the vertical axis in figure 4.3 is used to map the
criterion value, gp, into a real, positive, and dimensionless parameter (Zp is, in
fact, a piecewise linear function of gp). Such a mapping ensures that different
criteria values, with different physical meanings, are mapped to a common
scale. Consider class 1S again. If the value of a criterion, gp, is in the ideal
range, then the value of the class function is small (in fact, zero), whereas
+
, that is, in the unacceptable
if the value of the criterion is greater than t p5
range, then the value of the class function is very high. Class functions have
several important properties, including (1) that they are nonnegative, continuous, piecewise linear, and convex, and (2) that the value of the class function, Zp, at a given range intersection (say, desirable–tolerable) is the same for
all class types.
Basically, ranking of the alternatives is performed in four steps, as follows [23]:
Step 1: Identify criteria for evaluating each of the alternatives.
Step 2: Specify preferences for each criterion, based on one of the four classes
(see figure 4.3).
Step 3: Calculate incremental weights: Based on the preference structures
for the different criteria, the LPP weight algorithm [22] determines
incremental weights, ∆w +pr and ∆w −pr (used in step 4), that represent
the incremental slopes of the class functions, Zp. Here, r denotes the
range intersection.
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Step 4: Calculate total score for each alternative: The formula for the total
score, J, of the alternative of interest is constructed as a weighted
sum of deviations over all ranges (r = 2 to 5) and criteria (p = 1 to P),
as follows:
J=
P
5
p=1
r =2
∑ ∑(∆w
−
pr
d −pr + ∆w +pr d +pr )
(4.34)
where P represents the total number of criteria (each belonging to
one of the four classes in figure 4.3), ∆w +pr and ∆w −pr are the incre+
−
mental weights for the pth criterion, and d pr
and d pr
represent the
deviations of the pth criterion value of the alternative of interest
from the corresponding target values. An alternative with a lower
total score is more desirable than one with a higher total score.
The most significant advantage of using LPP is that no weights need to be
specified for the criteria for evaluation. The decision maker only needs to
specify a preference structure for each criterion, which has more physical
meaning than a physically meaningless weight that is arbitrarily assigned
to the criterion.
Note that there are no decision variables in the above ranking procedure.
LPP can be used in a problem consisting of decision variables as well, by
minimizing J in equation (4.34) and subjecting (if necessary) each criterion,
gp, to a constraint that falls into either one of the four classes (also called soft
classes) in figure 4.3 or one of the following four hard classes:
Class 1H: Must be smaller, i.e., gp ≤ tp,max
Class 2H: Must be larger, i.e., gp ≥ tp,min
Class 3H: Must be equal, i.e., gp = tp,val
Class 4H: Must be in range, i.e., tp,min ≤ gp ≤ tp,max
4.10
Goal Programming
Linear programming [24] assumes that the objectives of an organization can
be encompassed within a single objective function, such as maximizing the
total profit or minimizing the total cost. However, this assumption is not
always realistic, and there are several cases where the management focuses
on a variety of objectives simultaneously. Goal programming provides a way
of tackling such situations.
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Goal programming (GP; see [25]), generally applied to linear problems,
deals with the achievement of specific targets/goals. The basic approach
involves formulating an objective function for each objective and seeks a
solution that minimizes the sum (weighted sum in case of fuzzy goal programming) of the deviations of these objective functions from their respective goals. To this end, several criteria are to be considered in the problem
situation on hand. For each criterion, a target value is determined. Next,
the deviation variables are introduced, which may be positive or negative
(represented by ρk and ηk, respectively). The negative deviation variable,
ηk, represents the underachievement of the kth goal. Similarly, the positive deviation variable, ρk, represents the overachievement of the kth goal.
Finally, for each criterion, the desire to overachieve (minimize ηk) or underachieve (minimize ρk) or to satisfy the target value exactly (minimize ρk + ηk)
is articulated [26].
Goal programming problems can be categorized according to the type of
mathematical programming model. Another categorization is according to
how the goals compare in importance. In the case of preemptive goal programming, there is a hierarchy of priority levels for the goals, so the goals
of primary importance receive first attention and so forth. In case of nonpreemptive goal programming, all the goals are of roughly comparable importance [24].
In goal programming, it is necessary to specify aspiration levels for the
goals, and the overall deviation from the aspiration levels is minimized. In
most real-world scenarios, the aspiration levels and weights/importance levels of goals are imprecise in nature. In such situations, fuzzy GP comes in
handy, allowing the decision maker to obtain compromising results for multiple goals with varying aspiration levels. In fuzzy GP, the aspiration levels
are either in the “more is better” form or “less is better” form. [27]. A linear
membership function µi that represents goal fuzziness for the “more is better” form is expressed as [28]
1
G ( X ) − Li
µi = i
g i − Li
0
if Gi ( X ) ≥ g i
if L i ≤ Gi ( X ) ≤ g i
if Gi ( X )≤ Li
(4.35)
while a linear membership fraction µi that represents goal fuzziness for the
“less is better” form is expressed as
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1
if Gi ( X ) ≤ g i
U − Gi ( X )
µi = i
if g i ≤ Gi ( X ) ≤ U i
U i − g i
0
if Gi ( X )≥ U i
(4.36)
where gi, Li, and Ui are the aspiration level, lower tolerance limit, and upper
tolerrance limit respectively for the fuzzy goal Gi (X).
The simple form of the fuzzy GP problem with p fuzzy goals can be stated as:
p
Maximize
V (µ ) =
∑µ
i
(4.37)
i=1
subject to µ i =
Gi ( X ) − Li
U − Gi ( X )
or µ i = i
g i − Li
Ui − g i
(4.38)
AX ≤ b
(4.39)
µ i ≤1
(4.40)
X ,µi ≥0
(4.41)
where V(µ ) is the fuzzy achievement function or fuzzy decision function.
The objective is to obtain the µi value as close to 1 as possible. The weighted
additive model is widely used in GP and multiobjective optimization problems to reflect the relative importance of goals. In this approach, the decision maker assigns weights as coefficients of individual terms in the simple
additive fuzzy achievement function to reflect their relative importance. The
objective function for the weighted additive model is expressed as
p
V (µ ) =
Maximize
∑w µ
i i
i=1
(4.42)
where wi is the relative weight of the ith fuzzy goal. Fibonacci numbers are
used to assign weights to the goals. Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, etc.
(the next number is a result of the summation of the previous two numbers).
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The concept is applied by starting with numbers 1 and 2. For example, for
two goals, the weights would be in the ratio of 1:2, which approximately are
0.33 and 0.66. These weights are assigned to the two goals according to their
priority levels.
In some situations, the goals/objectives are not commensurable, or the goals
are such that unless a particular goal or subset of goals is achieved, other
goals should not be considered. In such situations, the weighting scheme is
not appropriate. The problem is divided into k subproblems, where k is the
number of priority levels. In the first subproblem, the fuzzy goals belonging to the first priority level will be considered and solved using the simple
additive model. At other priority levels, the membership values achieved at
earlier priority levels are added as additional constraints. In general, the ith
subproblem becomes:
Maximize
∑ (µ )
s pi
(4.43)
s
subject to µs =
Gs − Ls
g s − Ls
(4.44)
AX ≤ b
(4.45)
(µ ) pr = (µ * ) pr , r = 1, 2,…, j −1
(4.46)
µs ≤ 1
(4.47)
X, µi ≥ 0, i = 1,2,…, p
(4.48)
where (µs)pi refers to the membership functions of the goals in the ith priority
level and (µ*)pr is the achieved membership function value in the rth (r ≤ j – 1)
priority level.
4.11
Technique for Order Preference by
Similarity to Ideal Solution (TOPSIS)
The basic concept of the TOPSIS method [19] is that the rating of the alternative selected as the best from a set of different alternatives should have the
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shortest distance from the ideal solution and the greatest distance from the
negative-ideal solution in a geometrical (i.e., Euclidean) sense.
The TOPSIS method evaluates the following decision matrix, which refers
to m alternatives that are evaluated in terms of n criteria [29]:
Criteria
Alternatives
A1
A2
A3
.
.
.
.
Am
C1
w1
z11
z21
z31
…
…
…
…
zm1
C2
w2
z12
z22
z32
C3
w3
z13
z23
z33
…
…
…
…
…
Cn
wn
z1n
z2n
z3n
zm2
zm3
…
zmn
where Ai is the ith alternative, Cj is the jth criterion, wj is the weight (importance value) assigned to the jth criterion, and zij is the rating (for example, on
a scale of 1–10, the higher the rating, the better it is) of the ith alternative in
terms of the jth criterion.
The following steps are performed:
Step 1: Construct the normalized decision matrix. This step converts the
various dimensional measures of performance into nondimensional
attributes. An element rij of the normalized decision matrix R is calculated as follows:
rij =
z ij
∑
m
i=1
z ij2
(4.49)
Step 2: Construct the weighted normalized decision matrix. A set of weights
W = (w1, w2, …, wn) (such that ∑wj = 1), specified by the decision
maker, is used in conjunction with the normalized decision matrix
R to determine the weighted normalized matrix V defined by V =
(vij) = (rijwj).
Step 3: Determine the ideal and the negative-ideal solutions. The ideal (A*)
and the negative-ideal (A–) solutions are defined as follows:
{
A* = max vij
i
for i = 1, 2, 3, ....., m
= {p1, p2, p3, …, pn}
}
(4.50)
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{
A− = min vij
for i = 1, 2, 3, ....., m
i
}
= {q1, q2, q3, …, qn}
(4.51)
With respect to each criterion, the decision maker desires to choose
the alternative with the maximum rating (it is important to note that
this choice varies with the way he or she awards ratings to the alternatives). Obviously, A* indicates the most preferable (ideal) solution.
Similarly, A– indicates the least preferable (negative-ideal) solution.
Step 4: Calculate the separation distances. In this step, the concept of the
n-dimensional Euclidean distance is used to measure the separation
distances of the rating of each alternative from the ideal solution and
the negative-ideal solution. The corresponding formulae are
Si* =
∑ (v − p )
ij
2
j
for i = 1, 2, 3, ..., m
(4.52)
where Si* is the separation (in the Euclidean sense) of the rating of
alternative i from the ideal solution, and
Si− =
∑ (v − q )
ij
j
2
for i = 1, 2, 3, ..., m
(4.53)
where Si– is the separation (in the Euclidean sense) of the rating of
alternative i from the negative-ideal solution.
Step 5: Calculate the relative coefficient. The relative closeness coefficient for
alternative Ai with respect to the ideal solution A* is defined as follows:
Ci* =
Si−
Si* + Si−
(4.54)
Step 6: Rank the preference order. The best alternative can now be decided
according to the preference order of Ci*. It is the one with the rating that has the shortest distance to the ideal solution. The way the
alternatives are processed in the previous steps reveals that if an
alternative has the rating with the shortest distance to the ideal solution, then that rating is guaranteed to have the longest distance to the
negative-ideal solution. That means the higher the Ci*, the better the
alternative.
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4.12
Borda’s Choice Rule
Borda proposed a method in which marks of m – 1, m – 2, …, 1, 0 are assigned
to the best, second-best, …, worst alternatives, for each decision maker [30].
That means that a larger mark corresponds to greater preference. The Borda
score (maximized consensus mark) for each alternative is then determined
as the sum of the individual marks for that alternative, and the alternative
with the highest Borda score is declared the winner. That means that the different decision makers unanimously choose the alternative that obtains the
largest Borda score as the most preferred one.
4.13
Expert Systems
Expert systems are computer programs that can represent human expertise (knowledge) in a particular domain (area of expertise) and then use a
reasoning mechanism (applying logical deduction and induction processes)
to manipulate this knowledge in order to provide advice in this domain.
Although conventional computer programs also contain knowledge, their
main function is to retrieve information and carry out statistical analysis
and numerical calculations. They do not reason with this knowledge or
make inferences as to what actions to take or conclusions to reach. Thus,
what mainly distinguishes expert systems from conventional programs is
the capability to reason with knowledge. The main components of an expert
system are the following [31]:
• Knowledge base: This is where the knowledge is stored. Typically, this
consists of a set of rules of the form: if EVIDENCE, then HYPOTHESIS. The knowledge is written in the knowledge base using the
syntax of what is termed the knowledge representation language (e.g.,
Lisp and Prolog) of the system.
• Inference engine: This reasons with the knowledge resident in the
knowledge base using certain mechanisms.
• Reasoning mechanism: This traces the path or the knowledge steps
used to arrive at a conclusion and can relay it back to the user as
the justification for this conclusion. Examples of this mechanism are
deduction (cause + rule → effect), abduction (effect + rule → cause),
and induction (cause + effect → rule).
• Uncertainty modeling process: This aids the inference engine when
dealing with uncertainty.
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A shell is an expert system that is complete except for the knowledge base
[31]. Thus, a shell includes an inference engine, a user interface for programming, and a user interface for running the system. Typically, the programming interface comprises a specialized editor for creating rules in a
predetermined format and some debugging tools. The user of the shell enters
rules in a declarative fashion (if X, then Y) and ideally should not need to be
concerned with the working of the inference engine. Expert system shells are
easy to use and allow a simple expert system to be constructed quickly.
4.14
Bayesian Updating
Bayesian updating [32] is an uncertainty modeling technique that assumes
that it is possible for an expert in a domain to guess a probability to every
hypothesis or assertion in that domain and that this probability can be
updated in light of evidence for or against the hypothesis or assertion.
Suppose the probability of a hypothesis H is P(H). Then the formula for the
odds of that hypothesis, O(H), is given by
O(H) =
P(H)
1 – P(H)
(4.55)
A hypothesis that is absolutely certain, i.e., has a probability of 1, has infinite odds. In practice, limits are often set on odds values so that, for example,
if O(H) > 1,000, then H is true, and if O(H) < 0.01, then H is false.
The standard formula for updating the odds of hypothesis H, given that
evidence E is observed, is
O(H|E) = (A).O(H)
(4.56)
where O(H|E) is the odds of H, given the presence of evidence E, and A is the
affirms weight of E. The definition of A is
A=
P(E|H)
P(E|~H)
(4.57)
where P(E|H) is the probability of E, given that H is true, and P(E|~H) is the
probability of E, not given that H is true.
Bayesian updating assumes that the absence of supporting evidence is
equivalent to the presence of opposing evidence. The standard formula for
updating the odds of a hypothesis H, given that the evidence E is absent, is
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O(H|~E) = (D).O(H)
(4.58)
where O(H|~E) is the odds of H, given the absence of evidence E, and D is the
denies weight of E. The definition of D is
D=
P(~E|H)
1–P(E|H)
=
P(~E|~H) 1–P(E|~H)
(4.59)
If a given piece of evidence E has an affirms weight A that is greater than 1,
then its denies weight must be less than 1, and vice versa. Also, if A > 1 and D
< 1, then the presence of evidence E is supportive of hypothesis H. Similarly,
if A < 1 and D > 1, then the absence of E is supportive of H.
For example, while controlling a power station boiler, a rule “IF (temperature is high) and NOT (water level is low) THEN (pressure is high)” can also be
written as “IF (temperature is high—AFFIRMS A1, DENIES D1) AND (water
level is low—AFFIRMS A2, DENIES D2) THEN (pressure is high).” Here,
A1 =
P(Temperature is high | Pressure is high)
P(Temperature is high | ~Pressure is high))
D1 =
P(~Temperature is high | Pressure is high)
P(~Temperature is high | ~Pressure is higgh)
A2 =
D2 =
P(Water level is low | Pressure is high)
P(Water level is low | ~Pressure is high)
P(~Temperature is high | Pressure is high)
P(~Temperature is high | ~Pressure is higgh)
Sometimes, evidence is neither definitely present nor definitely absent. For
example, if one is diagnosing a TV set that is not functioning properly, it is
not definite if this is due to a malfunctioning picture tube. In such a case,
depending upon the value of the probability of the evidence P(E), the affirms
and denies weights are modified using the following formulae:
A' = [2.(A-1).P(E)]+2–A
(4.60)
D' = [2.(1–D).P(E)]+D
(4.61)
When P(E) is greater than 0.5, the affirms weight is used to calculate O(H|E),
and when P(E) is less than 0.5, the denies weight is used.
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If n statistically independent pieces of evidence are found that support or
oppose a hypothesis H, then the updating equations are given by
O(H|E1 &E 2 &E 3 ......En ) = (A1 ).(A 2 ).(A3 ).......(An ).O(H)
(4.62)
O(H| ∼ E1 & ∼ E 2 & ∼ E 3 ... ∼ En ) = (D1 ).(D2 ).(D3 ).......(Dn ).O(H)
(4.63)
and
Ai and Di are given by equations (4.64) and (4.65), respectively.
P(E i |H)
P(E i |~H)
(4.64)
P(~E i | H)
P(~E i |~H)
(4.65)
Ai =
Di =
4.15
Taguchi Loss Function
In traditional systems, the product is accepted if the product measurement
falls within the specification limits [33]. Otherwise, the product is rejected.
The quality losses occur only when the product deviates beyond the specification limits, thereby becoming unacceptable. These costs tend to be constant
and relate to the costs of bringing the product back into the specification
range. Taguchi [33] suggests a narrower view of characteristic acceptability
by indicating that any deviation from a characteristic’s target value results
in a loss. If a characteristic measurement is the same as the target value, the
loss is zero. Otherwise, the loss can be measured using a quadratic function,
after which actions are taken to reduce systematically the variation from the
target value.
There are three types of Taguchi loss functions: “target is best” (see
figure 4.4), “smaller is better” (see figure 4.5), and “larger is better” (see
figure 4.6).
If L(y) is the loss associated with a particular value of characteristic y, m is
the target value of the specification, and k is the loss coefficient whose value
is constant depending on the cost at the specification limits and width of the
specification, for the “target is best” type,
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(4.66)
For the “smaller is better” type,
L( y ) = k( y )2
(4.67)
L( y ) = k / ( y )2
(4.68)
For the “larger is better” type,
Lower Specification
Taguchi Loss
Taguchi Loss
m
Upper Specification
Figure 4.4
“Target is best” Taguchi loss.
Taguchi Loss
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L( y ) = k( y − m)2
Figure 4.5
“Smaller is better” Taguchi loss.
Upper Specification
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Taguchi Loss
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Lower Specification
Figure 4.6
“Larger is better” Taguchi loss.
4.16
Six Sigma
Statistical process control techniques help managers achieve and maintain a
process distribution that does not change in terms of its mean and variance
[34]. The control limits on the control charts signal when the mean or variability of the process changes. However, a process that is in statistical control may not be producing outputs according to their design specifications
because the control limits are based on the mean and variability of sampling
distribution, not the design specifications.
Process capability refers to the ability of the process to meet the design
specifications for an output. Design specifications are often expressed as a
target value ( τ ) and a tolerance (T). For example, the administrator of an
intensive care unit lab might have a target value for the turnaround time
of results to the attending physicians of 25 minutes and a tolerance of ± 5
minutes because of the need for speed under life-threatening conditions.
The tolerance gives an upper specification (U) of 30 minutes and a lower
specification (L) of 20 minutes. The lab process must be capable of providing the results of analyses within these specifications (see figure 4.7); otherwise, it will produce a certain proportion of “defects.”
Note that in most situations,
T=
U−L
U+L
τ=
2
2 ,
and
and hence it is assumed so here.
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L = 20 min
τ = 25 min
U = 30 min
Figure 4.7
Capable process.
Two essential quantitative measures to assess the capability of a process
are process capability ratio (Cp) and process capability index (Cpk).
4.16.1
Process Capability Ratio (Cp)
Assume that σ is the standard deviation of a process that produces a certain
dimension of interest for an output (good or service). This certain dimension
of interest will hereafter be called critical dimension. The process capability
ratio (Cp) is defined as
Cp =
U−L
6σ
(4.69)
The numerator represents the specification width and the denominator captures the total width of the 3σ limits of the process distribution. We consider
two examples, one for Cp = 1 and the other for Cp = 2.
If Cp = 1, the specification width is the same as the distribution width.
When the process mean ( ) is centered at (U+L)/2 without any shift from the
target value, τ , the probability that the actual critical dimension is within
the specification limits (assuming that the process distribution is normal)
is 0.9973 (2,700 ppm defect rate). Similarly, if Cp = 2, the specification width
is twice that of the distribution. When the process mean ( ) is centered at
(U+L)/2 without any shift from τ , the probability that the actual critical
dimension is within the specification limits is 0.999999998 (0.002 ppm defect
rate).
4.16.2
Process Capability Index (Cpk)
The process capability ratio (Cp) is enough to find out whether a process is
capable, only if is centered at (U+L)/2 without any shift from τ . For examEBSCO : eBook Collection (EBSCOhost) - printed on 4/10/2018 4:02 PM via AMERICAN PUBLIC UNIV SYSTEM
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ple, the lab process may have a good Cp value (i.e., more than the critical
value of, say, 1.5), but if
is closer to U, lengthy turnaround times may still
be generated. Likewise, if
is closer to L, very quick results may be generated. Thus, in order to check whether
is not far away from τ , there is a
need for an additional capability ratio, called the process capability index
(Cpk).
Cpk is defined in [35] as
C pk = C p (1 − k ) , where k =
τ −µ
T
(4.70)
This definition allows consideration of a mean shift, i.e., a shift of from τ . The
fraction k is the fraction of tolerance consumed by the mean shift. The Motorola
convention uses a one-sided mean shift of 1.5 σ . This is motivated by common physical phenomena such as tool wear. If Cp = 2 and Cpk = 1.5 (i.e., mean
shift consumes 25% of the tolerance), the probability that the actual critical
dimension is within the specification limits is 0.9999966 (i.e., 3.4 ppm defect
rate).
A process is said to be capable only if the process has good values (viz.,
more than the respective critical values) of both Cp and Cpk. If Cp is less than
the critical value, σ is too high. If Cpk is less than the critical value, either
is too close to U or L or σ is too high.
Six Sigma is an art of management that originated at Motorola in the early
1980s and is a business-driven, multifaceted approach to process improvement, cost reduction, and profit increase. Its fundamental principle is to
improve customer satisfaction by reducing defects in processes.
Traditionally, one needs both Cp and Cpk values in order to investigate
whether the process of interest is a Six Sigma process. We illustrate this by
calculating Cp and Cpk values (equations (4.69) and (4.70), respectively) for an
n Sigma process (where n is any positive real number; the higher the value
of n, the better the process is). We consider three different cases, viz., n = 3,
4.5, and 6. It must be noted that the mean shift in each case is allowed to be
up to 1.5 σ .
4.16.2.1
Three Sigma Process
See figure 4.8.
Cp =
U − L 6σ
=
=1
6σ
6σ
1.5σ
= 0.5
C pk = C p (1 − k) ≥ 11 −
3σ
Hence, if Cp = 1 and Cpk ≥ 0.5, it is considered a Three Sigma process.
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T= 3
T= 3
1.5
U= +3
L= -3
Figure 4.8
Three Sigma process.
4.16.2.2
4.5 Sigma Process
See figure 4.9.
Cp =
U − L 9σ
=
= 1.5
6σ
6σ
1.5σ
=1
C pk = C p (1 − k) ≥ 1.51 −
4.5σ
Hence, if Cp = 1.5 and Cpk ≥ 1, it is considered a 4.5 Sigma process.
4.16.2.3
Six Sigma Process
See figure 4.10.
Cp =
U − L 12 σ
=
=2
6σ
6σ
1.5σ
= 1.5
C pk = C p (1 − k) ≥ 21 −
6σ
Hence, if Cp = 2 and Cpk ≥ 1.5, it is considered a Six Sigma process.
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7
7
/
8
±
Figure 4.9
4.5 Sigma process.
7
7
/
8
Figure 4.10
Six Sigma process.
4.17
Neural Networks
A neural network is made up of simple processing units (called neurons)
combined in a parallel computer system following implicit instructions based
on recognizing patterns in data inputs from external sources [41]. Neural
networks can model complex relationships between inputs and outputs or
find patterns in data. Their usefulness derives from their ability to embody
inferential algorithms that alter the strengths or weights of the network connections to produce a desired significant flow. In this book, the following
equation [42] is used (see chapters 6 and 7) to calculate the weights (importance values) of evaluation criteria considered in a strategic planning issue.
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nH
∑
j
|Wv |=
nV
I vj
∑| I |
Oj
ij
i
nH
∑∑
i
nV
j
I vj
O
j
nV
| Iij |
i
∑
(4.71)
Here, the absolute value of Wv is the weight of the vth input node (evaluation criterion) upon the output node (rating of the decision alternative), nV is
the number of input nodes (evaluation criteria), nH is the number of hidden
nodes (can be any arbitrary number), Iij is the connection weight from the ith
input node to the jth hidden node, and Oj is the connection weight from the
jth hidden node to the output node. The connection weights [43] are obtained
upon training the respective neural network.
4.18
Geographical Information Systems
A geographical information system (GIS) is a computer system with a set of
processes for obtaining, managing, analyzing, and displaying data that have
been located geographically [37]. In a more generic sense, GIS is a tool that
allows users to create interactive queries (user-created searches), analyze the
spatial information, edit data, and map and present the results of all these
operations [38]. GISs have become powerful operation tools in the business
world.
In this book (see chapter 6), GIS is applied for displaying data. The motivation
for this application is a paper [39] that addresses the necessity for building
the strategic planning process “around a picture,” to make the chairperson
of the concerned supply chain company easily understand the “dense documents filled with numbers” and to convince him or her that it is important to
implement the proposed action. To this end, an excellent GIS-based business
mapping application, MapLand [40], is used to map the results obtained in
different phases of a model.
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4.19
Linear Integer Programming
Linear programming problems are about optimization of a linear objective
function, subject to linear equality and inequality constraints [36]. For example, in canonical form, a linear programming problem can be expressed as
Maximize CTX
subject to AX ≤ B and X ≥ 0
(objective function)
(constraints)
Here, X represents the vector of variables, whereas C and B are vectors of coefficients and A is the matrix of coefficients. If the variables of a linear programming problem are restricted to being integers, the programming is called linear
integer programming.
Linear integer programming can be applied to various fields. Most extensively, it is applied to business, economic, and engineering problems.
4.20
Conclusions
This chapter gave an introduction to various quantitative techniques
employed by strategic planning models presented in different chapters of
the book. Depending on the decision-making situation, each of the models
uses one or more of the techniques introduced in this chapter.
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AN: 238599 ; Pochampally, Kishore K., Gupta, Surendra M., Nukala, Satish.; Strategic Planning Models for
Reverse and Closed-Loop Supply Chains
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