MIXED 0−1 GOAL PROGRAMMING APPROACH TO INTERVAL-VALUED BILEVEL PROGRAMMING PROBLEMS VIA BIO-INSPIRED COMPUTATIONAL ALGORITHM

Rupak Bhattacharyya et al. (Eds) : ACER 2013,
pp. 47–58, 2013. © CS & IT-CSCP 2013 DOI : 10.5121/csit.2013.3205
MIXED 0−1 GOAL PROGRAMMING APPROACH
TO INTERVAL-VALUED BILEVEL
PROGRAMMING PROBLEMS VIA BIO-INSPIRED
COMPUTATIONAL ALGORITHM
Debjani Chakraborti1
and Papun Biswas2
1
Department of Mathematics, Narula Institute of Technology, Kolkata, W.B,
India
debjani_333@yahoo.co.in
2
Department of Electrical Engineering, JIS College of Engineering, Kalyani,
W.B, India
papunbiswas@yahoo.com
ABSTRACT
This paper presents how the mixed 0-1 programming in the framework of goal programming
(GP) can be used to solve interval-valued fractional bilevel programming (IVFBLP) problems
by employing genetic algorithm (GA) in a hierarchical decision making system. In the model
formulation of the problem, a goal achievement function for minimizing the lower-bounds of the
necessary regret intervals defined for the target intervals of achieving the goals and thereby
arriving at a compromise decision is constructed by using both the aspects of ‘minsum’ and
‘minmax’ approaches in GP. In the decision process, an GA scheme is employed for execution
of the problems at the two stages, target interval specification and optimal decision
determination, for distribution of decision powers to the decision makers (DMs) in the order of
hierarchy. A numerical example is provided to illustrate the potential use of the approach.
KEYWORDS
Fractional bilevel programming, Goal programming, Genetic algorithm, Interval-valued GP,
Mixed 0-1 programming.
1. INTRODUCTION
The bilevel programming (BLP) approaches to hierarchical decision problems have been studied
widely since Candler and Townsley [4] demonstrated the use of BLP to large hierarchical
decision making and planning organizations.
In a BLP problem (BLPP), the two DMs located at the two different hierarchical levels control a
vector of decision variables and each is interested in optimizing his/ her own benefit in the
decision making horizon. Here, in actual practice, it has been realized that the cooperation
between the DMs and a motivation to sacrifice the individual decision are needed for survival and
sustainable growth of an organization.
In such a context, BLPPs as well as multilevel programming problems (MLPPs) as an extension
of BLPPs have been studied [2, 3, 5, 8, and 22] deeply in the past. The fuzzy programming (FP)
approaches [7,19] to decentralized hierarchical decision problems have also been investigated

48 Computer Science & Information Technology (CS & IT)
from the point of view of potential use to different real-life decision problem like traffic control,
economic system, warfare, network design, conflict resolution, and others.
The GAs [9, 17] as prominent tools to optimization of multiobjective decision making (MODM)
problems [15, 16] have also been introduced to solve BLPPs [12, 18]. The GA based fuzzy goal
programming (FGP) approaches [20, 24] to linear as well fractional BLPPs and MLPPs have
been studied by Pal et al. [29, 30] in the recent past. However, the extensive study in this area is
at an early stage.
Now, in most of the real-life decision situations, it is to be observed that, although the FP as well
FGP as an extension of conventional GP [1] have been successfully implemented to different
areas in the field of MODM, the DMs are often faced with the problem of assigning fuzzy
aspiration levels to the objectives due to highly ambiguous in nature of them regarding
achievement of their targets in an inexact decision making environment.
To overcome the above difficulty, interval programming (IP) approaches [10] have appeared as a
prominent tool for solving decision problems with interval parameter values. The methodological
aspects of IP studied in the past have been surveyed by Olivera et al. [25] in 2007. The
methodological aspect of IP has also been studied by Pal et al. [26, 27] in the past. The GA based
approach to IP problems with fractional criteria has been reported in [26] in the past. The
potential use of IP to a real-life problem has also been investigated by Pal et al. [31] in the recent
past.
However, methodological extension of IP is still at an early stage. Further, the use of an IP
approach to hierarchical decision problems is yet to circulate in the literature.
In this article, a fractional bilevel programming problem having the characteristics of IP with
interval coefficients in the objectives of both the DMs is considered. In the GP model formulation
of the problem, the target intervals of the objectives of the DMs as well the control vector of the
upper-level DM (the leader) are determined first by defining the best and worst objective values
of the leader and the lower-level DM (the follower) with the use of an GA scheme. The interval-
valued objectives of the DMs and the control vector of the leader are then transformed into the
standard goals in GP by using the notion of interval arithmetic technique in IP. In the goal
achievement function, minimization of both the under- and over-deviational variables associated
with the aspired goal levels under the framework of both the minsum [11] and minmax [1]
approaches in GP is taken into consideration for minimizing the necessary lower-bounds of the
regret intervals defined for the target intervals from the optimistic point of view of arriving at a
compromise decision for both the DMs and overall benefit of the organization.
In the solution process, the formulated model is transformed into a mixed 0-1 GP formulation to
overcome the combinatorial hard in nature of the executable problem. The proposed GA scheme
is finally employed to reach a satisfactory decision on the basis of the weights of importance of
achieving the goals.
The potential use of the approach is illustrated by a numerical example.
Now, formulation of the IVFBLP problem is presented in the following Section 2.
2. PROBLEM FORMULATION
Let X=(x1,x2,…,xn) be the vector of decision variables involved with the two hierarchical decision
systems. Then, let Fk and Xk be the objective function and the decision vector, respectively, of the
k-th objective Fk, k=1,2; where .X}2,1kX{ k
k
==U

Computer Science & Information Technology (CS & IT) 49
Then, the fractional BLPP with interval coefficients in a hierarchical decision structure can be
presented as:
Find )X,X(X 21 so as to:
],[X]d,d[X]d,d[
],[X]c,c[X]c,c[
)X,X(FMax U
1
L
12
U
12
L
121
U
11
L
11
U
1
L
12
U
12
L
121
U
11
L
11
211
X1 ββ++
αα++
= (Leader’s problem) (1)
and, for given 21 X,X solves
],[X]d,d[X]d,d[
],[X]c,c[X]c,c[
)X,X(FMax U
2
L
22
U
22
L
221
U
21
L
21
U
2
L
22
U
22
L
221
U
21
L
21
212
X2 ββ++
αα++
= (Follower’s problem) (2)
subject to,






≥





≤
≥
+=∈ 0X,bXAXA|)X,X(SX 221121 , (3)
where mn
Rb,RX ∈∈ , and ]c,c[ U
k
L
k ll , ]d,d[ U
k
L
k ll (k, l =1,2) are the interval coefficient vectors,
U
k
L
k α,α , U
k
L
k β,β , (k=1,2) are constants, L and U stands for the lower and upper bounds,
respectively, of the defined intervals. 21 AandA are constant matrices and b is a constant vector.
It is assumed that the feasible region )Φ(S ≠ is a convex set.
Again, it is customary to assume that 0]β,[β]Xd,[d]Xd,[d U
k
L
k2
U
k2
L
k21
U
k1
L
k1 >++ .
Now, using the interval arithmetic operation rule [14], the interval-valued objectives in (1) and
(2) can be successively expressed as [27]:
XdXd
XcXc
,
XdXd
XcXc
)X,X(FMax L
12
L
121
L
11
U
12
U
121
U
11
U
12
U
121
U
11
L
12
L
121
L
11
211
X1 







β++
α++
β++
α++
= = )],X,X(T),X,X(T[ 21U121L1 (say) (4)
XdXd
XcXc
,
XdXd
XcXc
)X,X(FMax L
22
L
221
L
21
U
22
U
221
U
21
U
22
U
221
U
21
L
22
L
221
L
21
212
X2 







β++
α++
β++
α++
= = )],X,X(T),X,X(T[ 21U221L2 (say) (5)
Now, to determine the target intervals of the expressions in (4) and (5) and thereby formulating
the GP model of the problem, an GA scheme adopted in the solution search process is presented
in the Section 3.
3. DESIGN OF GA SCHEME
The two major operational activities in a GA approach are selection and crossover. In the present
GA search process, the fitter Codon selection [23] and two-point crossover [9] used is defined as
follows:
(i) fitter Codon selection:
The cordons are parts of a binary coded chromosome in a population. In a conventional GA
method to optimization problems in [9, 13, 17], Roulette-wheel scheme studied in [9] is used for
selection of parents. In the fitter Codon selection scheme in [23, 28] a comparison of the selected
strings with the given string lengths are used to determine the fitter one. In the present GA
approach, the cordons selection is made by considering the portion of a string from its most
significant bit to the bit position with a specified length. In such a selection scheme, the
consideration of full string length of a chromosome as well as conversion to its binary value is not

required in the selection process. This process substantially reduces the computational load in the
selection search process.
(ii) two-point crossover:
In the conventional GA approach [13] single-point crossover is considered. In the present GA
scheme, two-point crossover in [9] is considered. The merit of choosing this type of operation is
that a completely new population from the initial population is generated in a less number of
iterations in contrast to the single-point crossover.
The algorithmic steps of the GA method are presented in the following Section 3.1
3.1 Steps of the Proposed GA
Step 1. Representation and Initialization
Let VP denotes the binary coded representation of chromosome in a population as
VP = { n21 x,...,x,x }P, where ‘n’ denotes the length of a chromosome, and P = 1, 2,..., pop_size,
represents the population size, and where pop_size chromosomes are randomly initialized in its
search domain.
Step 2. Fitness function
The fitness value of each chromosome is determined by the value of an objective function. The
fitness function is defined as:
eval (VP) = PK )F( , k = 1, 2; P = 1,2, . . ., pop_size.
The best chromosome for the best and least value of the objective function is determined as:
V* = max {eval (VP) | P = 1, 2, pop_size},
And V* = min {eval (VP) | P = 1, 2, pop_size}, respectively.
Step 3. Selection
The fitter Codon selection scheme is used in the proposed GA. Here, chromosomes are likely to
be selected from the population depending on their fitness score. The merit of the fitter Codon
selection is to reach a solution with predefined level of fitness.
For instance, the following four chromosomes in a population are considered.
(i) 111010 000
(ii) 111101 010
(iii) 010111010
(iv) 101010010
Here, cordons are selected from the stand point of maximum occurrence of dominant values of
the most significant bits, where Codon length is defined by the number of bits from most
significant bit position bit to the position of first non-matching bit in the selected pair. It is
observed here that the chromosomes in (i) and (ii) are the fitter with Codon length 4 in

comparison to the others in (iii) and (iv). Again, it is to be followed that the chromosome in (ii) is
fitter than the chromosome in (i). It is clear from the above that the decimal equivalents of
chromosomes are not required here for selection of one with better fitness score. The merit of the
use of this selection scheme to different problems has been studied [29] in the recent past.
Step 4. Crossover
The probability of crossover is defined by the parameter Pc. Here in a two-point crossover genetic
system, the mating chromosomes interchange their middle portion in the process of reproduction.
Again, a chromosome is selected as a parent, if for two defined random number r, r1 ∈ [0, 1]; r,
r1 < Pc with r + r1 < 1 is satisfied.
In the selection of two parents, another random number r2 is defined such that r2=1-r- r1. Then,
two parents V1, V2 ∈ S yield two offspring as:
1
1
V = ( r + r2). V1 + r1 V2, 1
2
V = r1.V1 + ( r + r2). V2, where 1
1
V , 1
2
V ∈S
Step 5. Mutation
The parameter Pm is conventionally defined as the probability of mutation. The mutation
operation is performed on a bit-by-bit basis, where for a random number r ∈ [0, 1], a
chromosome is selected for mutation provided that r < Pm.
Step 6. Termination
The execution process terminates when the generated best chromosome is reported after a certain
number of generations as the decision in the genetic search process.
Now, formulation of the interval-valued goals of the problem is presented in the Section 4.
4. INTERVAL-VALUED GOAL FORMULATION
To formulate the GP model of the BLPP, the target intervals for both the objectives F1 and F2 and
the decision vector 1X controlled by the leader are to be defined in the decision making
environment.
4.1 Determination of Target Interval
To determine the target intervals, the best and worst solutions of the objectives are to be
determined first, and that can be obtained using the GA scheme by defining the parameter values
of it.
Let the individual best and least solutions of the leader be
)T;X,X( *
U1
b
2
b
1
ll
and )T;X,X( *
L1
w
2
w
1
ll
, respectively,
where )
2
X,
1
X(TMax
U1S)2X,1X(U1T
∈
=∗
,
and )X,(XT 21L1
L1
T
(
Min
S)2X,1X ∈
=∗
.

Similarly, the individual best and worst solutions of the follower can be obtained as
)T;X,X( U2
fb
2
fb
1
∗
and )T;X,X( *
L2
fw
2
fw
1 , respectively,
Where )X,(XT 21U2
U2
TMax
S)2X,1X( ∈
=∗
,
And )X,(XT 21L2
L2
T
(
Min
S)2X,1X ∈
=∗
.
Now, in the decision making context, it is reasonably assumed that both the leader and follower
are motivated to cooperate each other and each is willing to sacrifice his/ her own benefit up to a
certain level for a gain of the other from the view point of survival as well as sustainable growth
of the organization.
From the above view point, the target intervals of the objective kF can be determined as:
[ ] 2,1k,t,t U
k
L
k =
Where ,2,1k,TttT *
kU
U
k
L
k
*
kL =≤≤≤ and the consideration of which depends on the decision
making situation.
Then, the expressions in (4) and (5) with the target intervals can be presented as:
[ ] ]t,t[)X,X(T),X,X(T U
1
L
121U121L1 = , (Leader’s problem) (6)
[ ] ]t,t[)X,X(T),X,X(T U
2
L
221U221L2 = (Follower’s problem) (7)
Again, since the leader has a higher power of making decision, relaxation on the best decision
b
1Xl
up to a certain level as the lower tolerance limit should be considered by the leader for
searching of a better decision by the follower. Let )XXX(X b
11
w
11
llll
<< be the lower tolerance
limit of the decision vector 1
X controlled by the leader.
Now, using the concept of mid-point arithmetic of IP [14] the interval objective of the control
vector 1
X can be obtained as:
]X,X[X b
111
ll
= (8)
Now, the standard goal representation of the defined objectives in the GP formulation is
discussed in the following Section 4.2.
4.2. Standard Goal Representation of Interval-Valued Goal
To formulate the GP model of the problem, the objectives in (6), (7) and (8) are to be transformed
into the standard goals by introducing the target intervals and the under- and over- deviational
variables to each of them.
The standard goal expressions of the objectives are successively obtained as
,tdd)X,X(T L
1L1L121L1 =−+ +−
(9)

and ;tdd)X,X(T U
1U1U121U1 =−+ +−
(Leader’s problem) (10)
,tdd)X,X(T L
2L2L221L2 =−+ +−
(11)
and ,tdd)X,X(T U
2U2U221U2 =−+ +−
(Follower’s problem) (12)
Where 2,1k,0)d,d( kUkL
=≥−−
represent under-deviational variables and 2,1k,0)d,d( kUkL
=≥++
represent over-deviational variables, respectively, associated with the respective goal expressions.
Again, the goal expressions for the control vector 1X are obtained as:
,XddX 1LL1
l
=−+ +−
(13)
And .XddX b
1UU1
l
=−+ +−
(14)
Where, 0)d,d(and)d,d( UULL ≥+−+−
represent the vectors of under- and over- deviational variables,
respectively, and where the dimension of each of them depends on 1X .
Now, formulation of the GP model of the problem is presented in the Section 5.
5. GP MODEL FORMULATION
In a decision making situation, the aim of each of the DMs, is to achieve the goal values within
the specified ranges by means of minimizing the necessary regrets in terms of the deviational
variables involved in the decision situation.
In the present decision situation, the goal achievement function is termed as the regret function,
since the regret intervals defined for goal achievement within the specified target intervals are to
be minimized to the extent possible in the decision making environment.
Now, in the field of interval programming, both the aspects of GP, minsum GP [11] for
minimizing the sum of the weighted unwanted deviational variables as well as minmax GP [1] for
minimizing the maximum of the deviations, are simultaneously taken into account as a convex
combination of them to reach a satisfactory solution within the specified target intervals of the
goals.
Here, from the optimistic point of view of both the DMs, minimization of the necessary regrets
involved with the regret intervals )2,1k(),d,d(),d,d( kUkLkUkL
=−++−
, and )d,d(),d,d( ULUL
−++−
are taken into
consideration.
Now, for model simplification let it is assumed that )nn(n 11
< be the number of decision
variables involved with the control vector X1.
Then, the GP model of the problem under consideration takes the form:
Find X(X1, X2) so as to
Minimize Z=






++++∧∧∧∧++++−−−−++++






++++∧∧∧∧++++ −−−−++++++++−−−−
++++∈∈∈∈
++++
====
++++
====
−−−−−−−−++++++++++++++++−−−−−−−−
∑∑∑∑ ∑∑∑∑ )dd(}dd{max)1()dwdw()dwdw( iUiLiUiL
2ni
2n
1i
2n
1i
iUiUiLiLiUiUiLiL
1
1 1
λλ (15)

and satisfy the goal constraints defined in (9) – (14), subject to the system constraint in (3),
where Z represents the regret function for goal achievement; )),2n(,...,2,1i(d,d,d,d 1iUiLiUiL +=−++−
are successively renamed for ),2,1k(d,d,d,d kUkLkUkL =−++−
and n1 components of each of the
−++−
ULUL d,d,d,d , and where 0)w,w,w,w( iUiLiUiL >−++−
with 1)wwww(
i
_
iUiLiUiL =+++∑
++−
denote the
numerical weights of importance of achieving the goals within the respective target intervals, and
;10 <λ< ∧∧∧∧ stands for min operator.
Now, it is to be followed that the function Z in (15) is non-convex in nature in the field of
combinational optimization.
The mixed 0-1 programming technique as the most widely used [10] and the simplest version of
solving such problems is introduced here to solve the problem.
5.1. Mixed 0-1 Programming to GP Model
Introducing the variable }1,0{∈iz , (either 0 or 1), 2))(n1,2,....,(k 1 += the regret function Z in
(15) can be recast as:
Minimize Z = ,V)1()z1()dwdw(z)dwdw( i
2n
1i
2n
1i
iUiUiLiLiiUiUiLiL
1 1
λ−+






−+++λ ∑ ∑
+
=
+
=
−−++++−−
(16)
Where
V)}dd()dd{(max iUiLiUiL
2ni 1
====






++++∧∧∧∧++++ −−−−++++++++−−−−
++++∈∈∈∈
(17)
Now, the expression in (17) in general format can be represented as:
V)d(d)d(d iUiLiUiL ≤≤≤≤++++∧∧∧∧++++ −−−−++++++++−−−−
, i = 1, 2, (n1+2) (18)
Then, incorporating the variable zi defined above, the relational expression in (18) can be
presented in its equivalent form as:
,V)z(1)d(dz)d(d iiUiLiiUiL ≤−+++ −++−
where }1,0{∈iz ; i = 1, 2,..,(n1+2) (19)
Finally, the executable GP model appears as:
Minimize Z = ,V)1()z1()dwdw(z)dwdw( i
2n
1i
2n
1i
iUiUiLiLiiUiUiLiL
1 1
λ−+






−+++λ ∑ ∑
+
=
+
=
−−++++−−
(20)
subject to the constraints sets defined in (15) and (19) Now, since GA is a goal satisfier [9] rather
than optimizer, the defined GA scheme can be employed here to minimize the regret function ‘Z’
in (20) and thereby to reach a satisfactory decision by minimizing the regrets of both the DMs.
Here, the fitness function appears as:
Eval (VP) = (Z) P, P = 1, 2, pop_size.
The best chromosome V*
with highest fitness score at a generation is determined as:

}.pop_size1,2....,P)min{eval(VV P
*
========
To illustrate the proposed approach, a numerical example is solved.
6. NUMERICAL EXAMPLE
Let the two decision variables 1x and 2x are under the control of the leader and follower,
respectively.
Then, the FBLPP with interval coefficients can be presented as:
Find X (x1, x2) so as to
]3,3[x]7,3[x]5,4[
]8,7[x]11,5[x]2,1[
)x,x(FMax
21
21
211
x1 ++
++
= , (Leader’s problem) (21)
And, for given x1 , x2 solves
]6,5[x]4,2[x]7,6[
x]2,1[x]4,3[
)x,x(FMax
21
21
212
x2 ++
+
= , (Follower’s problem) (22)
subject to
x1 + x2 ≤ 6, -x1 + x2 ≤ 3,
x1 + 2x2 ≥ 2, x1 ≤4,
x1, x2 ≥ 0 . (23)
Now, following the procedure, the Leader’s objective in interval form appear as
,
3x3x4
8x11x2
,
3x7x5
7x5x
21
21
21
21






++++++++
++++++++
++++++++
++++++++
And that of the Follower takes the form:






++++++++
++++
++++++++
++++
5x2x6
x2x4
,
6x4x7
xx3
21
21
21
21
.
To solve the problem by employing the proposed GA scheme, the following genetic parameter
values are found effective in the solution search process:
The probability of crossover Pc = 0.8,
Probability of mutation Pm = 0.07,
Population size =100,
Chromosome length =30.
The GA is implemented using the Programming Language C. The execution is done in an Intel
Pentium IV with 2.66 GHz. Clock-pulse and 1 GB RAM.
The Leader’s best and worst solutions are obtained as:
)41.3;3,0()T;X,X( *
U1
b
2
b
1 ====ll
And )47.0;0,4()T;X,X( *
L1
w
2
w
1 ====
ll
, respectively.

The Follower’s best and worst solutions are found as
)65.0;5.4,5.1()T;X,X( *
U2
bf
2
bf
1 ====
and )1.0;1,0()T;X,X( *
L2
wf
2
wf
1 = , respectively.
Now, following the procedure, the interval objectives with target intervals can be obtained as:
]41.3,47.0[
3x3x4
8x11x2
,
3x7x5
7x5x
21
21
21
21
=





++
++
++
++
, (Leader’s problem)
]65.0,1.0[
5x2x6
x2x4
,
6x4x7
xx3
21
21
21
21
=





++
+
++
+
, (Follower’s problem)
Again, the decision variable x1 with its target interval appears as:
[1, 1] x1 = [0,1.5].
Then, the goals in standard GP formulation are obtained as:
,47.0dd
3x7x5
7x5x
L1L1
21
21
=−+
++
++ +−
,41.3dd
3x3x4
8x11x2
U1U1
21
21
=−+
++
++ +−
,1.0dd
6x4x7
xx3
L2L2
21
21
=−+
++
+ +−
,65.0dd
5x2x6
x2x4
U2U2
21
21
=−+
++
+ +−
,0ddx L3L31 =−+
+−
5.1ddx U3U31 =−+ +−
(24)
Using the expressions of Z in (20) and following the procedure, the executable GP model in the
form of mixed 0-1 programming problem appears as:
Find X(x1, x2) so as to
Minimize Z= ,V)1()}z1()dwdw(z)dwdw{( i
3
1i
iUiUiLiLiiUiUiLiL λ−+






−+++λ ∑=
−−++++−−
(25)
And satisfy the goal expressions in (24)
subject to, ,V)z1)(dd(z)dd( iiUiLiiUiL ≤−+++ −++−
i=1,2,3 ; }1,0{zi ∈ , and the system constraints
in (23).
Now, for simplicity, introducing the equal weights w1 = w2 = w3 = 1/3 for goal achievement and
taking ,5.0=λ the problem is solved by employing the GA scheme, where the function Z
defined in (25) appears here as the fitness function.
The resulting decision is obtained as:
).0656.2,0()x,x( 21 =
The achieved objective function values in the interval-valued form are:
],45.0,14.0[Zand]34.3,99.0[Z 21 ==

The result shows that a satisfactory decision is achieved here from the view point of distributing
the proper decision powers to both the DMs in the decision making context.
Note: It may be noted that if the crisp coefficients, instead of interval coefficients, are involved
with the objectives, then using the mid-point arithmetic rule [14] in IP, the problem can easily be
solved under the framework of the proposed approach.
Again, it is worth mentioning that the computational complexity arising out of the fractional
objectives [6] as well as the computational load involved with the use of conventional linearization
approach [21] does not occur here due to the use of the proposed GA based solution approach.
7. CONCLUSION
The main advantage of using the proposed IP approach to the FBLPP is that the ambiguity of
assigning the fixed objective values as necessarily introduced to the other MODM approaches
(deterministic/ fuzzy) do not involve here due to consideration of the goals in the interval-valued
form for their achievement. Here, on the basis of the needs and desires of the DMs, the objective
values can be achieved within the specified intervals according to the given ranges of the input
parameter values of the coefficients in the decision making environment. Further, the approach is
flexible enough to set up the interval parameter values based on the needs of an organization in
the hierarchical decision system. The proposed approach can be extended to MLPPs with
multiplicity of objectives in a large hierarchical decision making organization, which is a problem
for future study. However, it is expected that the proposed approach may open up many new
looks into the field of practical hierarchical decision problems for sustainable growth of an
organization in the current competitive world for survival.
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