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Hybrid Algorithm for Rough Multi-level Multi-objective Decision Making Problems
Shereen Fathy El-Feky1, Tarek H.M. Abou-El-Enien2*
1 Teaching Assistant at Faculty of Computer Science, Department of Computer Science, Modern Science and Arts University, Giza
12613, Egypt
2 Department of Operations Research & Decision Support, Faculty of Computers & Information, Cairo University, Giza 12613,
Egypt
Corresponding Author Email: t.hanafy@fci-cu.edu.eg
https://doi.org/10.18280/isi.240101
ABSTRACT
Received: 28 October 2018
Accepted: 12 January 2019
The purpose of this paper is to generate compromise solutions for the multi-level multi-
objective decision making (MLMODM) problems with rough parameters in the objective
functions (RMLMODM) based on TOPSIS method and "Lower & Upper” approximations
method. We introduce a computational hybrid algorithm for solving RMLMODM problems.
Also, we solved illustrative numerical example and compared the solution of the proposed
algorithm with the solution of Global Criterion (GC) method. The engineers and the scientists
can apply the introduced hybrid algorithm to various practical RMLMODM problems to obtain
numerical solutions.
Keywords:
compromise programming, rough
programming, TOPSIS method, global
criterion method, multi-objective
programming, multi-level programming
1. INTRODUCTION
Rough set theory is an important mathematical tool for
dealing with the description of vague objects. Rough set
methodology has been introduced by Pawlak (1991), [1].
Linear optimization problem which is considered where
some or all of its coefficients in the objective function and/or
constraints are rough intervals is introduced by Hamzehee et
al. [2].
Various hybrid algorithms for solving several kinds of
multi-objective optimization problems based on TOPSIS
approach are presented in [3].
S. F. El-Feky and T.H.M. Abou-El-Enien, [4], develop a
methodology to find compromise solutions for rough multi-
objective optimization problems.
In the following section, the formulation of RMLMODM
problems is given. A new hybrid algorithm based on the
TOPSIS method [5-7], and "Lower & Upper” approximations
method [1-2], for solving RMLMODM problems is proposed
in section (3). For the sake of illustration, we present an
example and compared the solution of the example by the
proposed algorithm with the ideal solutions and the solution of
global criterion (GC) method.
2. PROBLEM DEFINITION
Consider the following linear multi-level multi-objective
decision making (LMLMODM) problem with rough
parameters in the objective functions:
[]
where,
[]
where,
[]
where,
where,
[]
subject to (1)
where
, (2)
m: The number of constraints,
n: The number of variables,
h: The number of levels,
Ingenierie des Systemes d'Information
Vol. 24, No. 1, February, 2019, pp. 1-17
Journal homepage: http://iieta.org/Journals/isi
1
k: The number of objective functions,
: jth level decision maker, i=1,2,…,h,
: The number of objective functions of the ,
i=1,2,….,h,
,
: The number of variables of the , i=1,2,….,h,
Real constants coefficients of the objective
functions.
: An m-dimensional column vector of right-hand sides of
constraints
: Ancoefficient matrix,
: The set of all real numbers,
X: An n-dimensional column vector of variables,
Xi: An -dimensional column vector of variables for the ith
level, i=1,2,…,h,
={1,2,…..,n},
,
are rough interval coefficients of
the objective functions
.
Using the upper and lower approximation method, [2], the
multi-level multi-objective decision making problems with
rough parameters in the objective functions (RMLMODM)
can be transformed to the following four deterministic linear
multi-objective decision making (LMODM) problems :
(Lower interval coefficients-Lower interval)
[]
where,
[]
where,
[]
where,
where,
[]
subject to (3)
(Upper interval coefficients-Lower interval)
[]
where,
[]
where,
[]
where,
[]
subject to (4)
(Lower interval coefficients -Upper interval)
[]
where,
[]
where,
[]
where,
where,
[]
2
subject to (5)
(Upper interval coefficients -Upper interval)
[]
where,
[]
where,
[]
[]
subject to (6)
where
, (7)
, (8)
, (9)
, (10)
3. HYBRID ALGORITHM FOR RMLMODEM
A modified version of TOPSIS method, [3, 5], is introduced
to find compromise solutions, [8-10], for the RMLMODM
problems. Modified equations for the distance function
equation from the positive ideal solution (PIS) and the distance
function equation from the negative ideal solution (NIS) are
introduced. Thus, we present the following hybrid algorithm
based on a modified version of TOPSIS method and the upper
and lower approximation method to generate compromise
solutions, [4], for RMLMODM problems.
Algorithm:
Phase (1):
Step 1:
Let h = the number of the levels of the RMLMODM
problem (1). Set i=1, "The 1st level".
Step 2:
Use the "Lower & Upper” approximations method to
transform the RMLMODM Problem (1) into the four
deterministic LMLMODM problems (3)-(6).
Step 3:
Construct the positive ideal solution (PIS) payoff tables, [4],
of the following problems, for i=1:
[]
subject to (11)
subject to (12)
subject to (13)
subject to (14)
and obtain the PIS:
,
,
and
,
Step 4:
Construct the negative ideal solution (NIS) payoff tables,
[4], of the problems (11-14) for i=1 and obtain the NIS:
,
and
,
Step 5:
Let where
and ,
.
Step 6:
Construct distance functions
and
,
[4], by the above steps (3, 4 & 5):
3
(15)
And
(16)
Construct distance functions
and
by the above steps (3, 4 & 5):
(17)
and
(18)
Construct distance functions
and
by the above steps (3, 4 & 5):
(19)
and
(20)
Construct distance functions
and
by the above steps (3, 4 & 5):
(21)
and
(22)
Step 7:
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives, [4], using the
distance functions
and
:
subject to (23)
where
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives using the distance
functions
and
:
subject to (24)
where
=1,2,…..,∞.
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives using the distance
functions
and
:
subject to (25)
where
=1,2,…..,∞.
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives using the distance
functions
and
:
4
subject to (26)
where
Step 8:
Construct PIS Payoff table for problem (23) and obtain:
where
.
Construct PIS Payoff table for problem (24) and obtain:
Construct PIS Payoff table for problem (21) and obtain:
Construct PIS Payoff table for problem (22) and obtain:
Step 9:
Construct the following satisfactory level model, [11-13],
(for finite value of p) for problem (23):
subject to (27)
,
,
Construct the following satisfactory level model (for finite
value of p) for problem (24):
subject to (28)
,
,
Construct the following satisfactory level model (for finite
value of p) for problem (25):
subject to (29)
,
,
Construct the following satisfactory level model (for finite
value of p) for problem (26):
subject to (30)
,
5
,
Step 10:
Solve problems (27-30) to obtain the satisfactory levels
, and for the
compromise solutions , and
. If the is satisfied with the solutions, then go
to step (11). Otherwise, go to step (5).
Step 11:
Ask theto select the maximum acceptable negative
and positive tolerance (relaxation) values, [14, 15]:
on the decision vectors:
and
Set i = i+1, go to the next phase.
Phase (2):
Step 12:
Set i=2, “The 2nd level".
Construct the PIS payoff table of problems (11-14) for i=2,
and obtain the PIS:
,
,
and
,
Step 13:
Construct the negative ideal solution (NIS) payoff tables of
the problems (11-14) for i=2 and obtain the NIS:
,
and
,
Step 14:
Let where
and ,
.
Step 15:
Construct distance functions
and
:
(31)
And
(32)
Construct distance functions
and
:
(33)
and
(34)
Construct distance functions
and
:
(35)
and
(36)
Construct distance functions
and
:
(37)
6
and
(38)
Step 16:
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives, using the distance
functions
and
:
subject to (39)
where
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives using the distance
functions
and
:
subject to (40)
where
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives using the distance
functions
and
:
subject to (41)
where
Construct the following bi-objective problem with two
commensurable (but conflicting) objectives using the distance
functions
and
:
subject to (42)
where
Step 17:
Construct PIS Payoff table for problem (39) and obtain:
where
.
Construct PIS Payoff table for problem (40) and obtain:
Construct PIS Payoff table for problem (41) and obtain:
Construct PIS Payoff table for problem (42) and obtain:
Step 18:
Construct the following satisfactory level model, [11-13],
(for finite value of p) for problem (39):
7
subject to (43)
,
,
,
.
Construct the following satisfactory level model (for finite
value of p) for problem (40):
subject to (44)
,
,
,
.
Construct the following satisfactory level model (for finite
value of p) for problem (41):
subject to (45)
,
,
,
,
.
Construct the following satisfactory level model (for finite
value of p) for problem (42):
subject to (46)
,
,
,
,
Step 19:
Solve problems (43-468) to obtain the satisfactory levels
, and for the
compromise solutions , and
. If the is satisfied with the solutions, then go
to step (20). Otherwise, go to step (14).
Step 20:
Ask theto select the maximum acceptable negative
and positive tolerance (relaxation) values:
on the decision vectors:
8
and
Set i = i+1, if i h goes to the next phase. Otherwise, stop.
4. ILLUSTRATIVE NUMERICAL EXAMPLE
Consider the following linear multi-level multi-objective
decision making (LMLMODM) problem with rough
parameters in the objective functions
First Level:
Second Level:
Third Level:
Subject to:
Solution:
Phase (1):
First Level:
+ 3++2
++1
Second Level:
+ +5
+ +3
Third Level:
2++4
+6
Subject to
First Level:
+ 5++3
++3
Second Level:
+ ++6
+ +4
Third Level:
2++5
+7
Subject to
First Level:
+ 2++1
+
Second Level:
+ +3
+ +2
Third Level:
2++3
+5
Subject to
First Level:
+ 7++4
++5
Second Level:
+ 7
+ + 6
9
Third Level:
2++6
+8
Subject to
(First Level):
Obtain PIS and NIS payoff tables for problem
Table 1. PIS payoff table for problem
62.54
10.38
0.77
0
10.38
0.77
0
PIS:=(45.84615, 62.53846)
Table 2. NIS payoff table for problem
1
0
0
0
11.6
0
3.2
0
NIS: =(2,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ==0.5 and p=2
Table 3. PIS payoff table of problem (11) when p=2
45.9
62.5
10.
4
0.
8
0
0
45.85
62.5
10.
4
0.
8
0
= (,) ,
= (0, ).
Now, it is easy to compute step (10):
Subject to
The maximum “satisfactory level” (=0.9447279E-
07) is achieved for the solution =10.38462,
=0.7692308,= zero.
(Second Level)
Obtain PIS and NIS payoff tables for problem
Table 4. PIS payoff table for problem
-
0.2667
6.3333
3.2
0
10
0
0
PIS:= (, 13)
Table 5. NIS payoff table for problem
0
0
0
0
21
0
3.2
0
NIS: = (5,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ====0.25 and p=2,
Table 6. PIS payoff table of problem (11) when p=2
29.62
11.1
10.
4
0.7
7
0
0.04
29.62
11.1
10.
4
0.7
7
0
=( ,),
=(0.04301006758,
).
Now, it is easy to compute step (10):
10
Subject to
,
The maximum “satisfactory level” (=0.7886892E-
07) is achieved for the solution =10.38462,
=0.7692308,= zero.
Level):
Obtain PIS and NIS payoff tables for problem
Table 7. PIS payoff table for problem
10
0
0
10.38
0.77
0
PIS:= (, )
Table 8. NIS payoff table for problem Table 8. NIS
payoff table for problem
12.4
0
3.2
0
5.67
0
0
1.67
NIS: = (,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ======1/6 and p=2,
Table 9. PIS payoff table of problem (11) when p=2
23.23
59.46
10.
4
0.7
7
0
0.03
23.23
59.46
10.
4
0.7
7
0
= ( ),
=(0.02925962351,
).
Now, it is easy to compute step (10):
Subject to
,
The maximum “satisfactory level” (=0.6082756E-
07) is achieved for the solution =10.38462,
=0.7692308,=zero.
(First Level):
Obtain PIS and NIS payoff tables for problem
Table 10. PIS payoff table for problem
74.92
10.38
0.77
0
10.38
0.77
0
PIS: =(69.15385, 74.92308)
Table 11. NIS payoff table for problem
3
0
0
0
19
0
3.2
0
NIS: =(3,)
Next, construct equation and obtain the following equations:
11
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ==0.5 and p=2,
Table 12. PIS payoff table of problem (11) when p=2
69.1
5
74.9
2
10.3
8
0.7
7
0
0
69.1
5
74.9
2
10.3
8
0.7
7
0
= (,) ,
= (0, ).
Now, it is easy to compute step (10):
Subject to
The maximum “satisfactory level”
(=0.1536454E-01) is achieved for the solution
=10.38462, =0.7692308,= zero.
(Second Level):
Obtain PIS and NIS payoff tables for problem
Table 13. PIS payoff table for problem
3.56
7.4
2.44
0.64
10
0
0
PIS: = (, 14)
Table 14. NIS payoff table for problem
4
0
0
0
25.2
0
3.2
0
NIS: = (6,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ====0.25 and p=2,
Table 15. PIS payoff table of problem (11) when p=2
39.
2
3.6
7.4
2.4
4
0.6
4
0.1
31.
4
11.
3
10.
4
0.7
7
0
= (,),
= (0.06597257621,
).
Now, it is easy to compute step (10):
Subject to
,
The maximum “satisfactory level” ( =zero) is
achieved for the solution =10.38462,
=0.7692308,= 0.2121311e-03.
Level):
Obtain PIS and NIS payoff tables for problem
Table 16. PIS payoff table for problem
10.38
0.77
0
10.38
0.77
0
PIS: = (, )
12
Table 17. NIS payoff table for problem
19.8
0
3.2
0
6.67
0
0
1.67
NIS: = (,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ======1/6 and p=2,
Table 18. PIS payoff table of problem (11) when p=2
55.4
62
10.
4
0.7
7
0
0.04
55.4
62
10.
4
0.7
7
0
= (),
= (0.04398171747,
).
Now, it is easy to compute step (10):
Subject to
,
The maximum “satisfactory level” ( =zero) is
achieved for the solution =10.38462,
=0.7692308,= zero.
(First Level):
Obtain PIS and NIS payoff tables for problem
Table 19. PIS payoff table for problem
51.15
10.38
0.77
0
10.38
0.77
0
PIS:= (23.30769, 51.15385)
Table 20. NIS payoff table for problem
0
0
0
0
7.4
0
3.2
0
NIS: = (1,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ==0.5 and p=2,
Table 21. PIS payoff table of problem (11) when p=2
23.31
51.2
10.
4
0.
8
0
0
23.31
51.2
10.
4
0.
8
0
= (,) ,
= (0, ).
Now, it is easy to compute step (10):
Subject to
The maximum “satisfactory level”
13
(=0.1536454E-01) is achieved for the solution
=10.38462, =0.7692308,= zero.
(Second Level):
Obtain PIS and NIS payoff tables for problem
Table 22. PIS payoff table for problem
10.85
10.38
0.77
0
10
0
0
PIS: = (, 12)
Table 23. NIS payoff table for problem
2
0
0
0
12.6
0
3.2
0
NIS: = (3,)
Next, construct equation and obtain the following equations:
=
=
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ====and p=2,
Table 24. PIS payoff table of problem (11) when p=2
10.9
10.
4
0.
8
0
0.03
10.9
10.
4
0.
8
0
= (,),
= (0.02462469512,
).
Now, it is easy to compute step (10):
Subject to
The maximum “satisfactory level” (=3561380) is
achieved for the solution =10.38462,
=0.7692308,= zero.
Level):
Obtain PIS and NIS payoff tables for problem
Table 25. PIS payoff table for problem
10
0
0
10.38
0.77
0
PIS: = (, )
Table 26. NIS payoff table for problem
11.4
0
3.2
0
4.67
0
zero
1.67
NIS: =(,)
Next, construct equation and obtain the following equations:
=
=
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ======1/6 and p=2,
14
Table 27. PIS payoff table of problem (11) when p=2
13
55
10
0
0
0.02
11.85
58.46
10.
4
0.7
7
0
= ( ),
= (0.01658311422,
).
Now, it is easy to compute step (10):
Subject to
,
The maximum “satisfactory level” ( =zero) is
achieved for the solution =10, = zero,= zero.
(First Level):
Obtain PIS and NIS payoff tables for problem
Table 28. PIS payoff table for problem
97.69
10.4
0.77
0
10.4
0.77
0
PIS:= (113.2308, 97.69231)
Table 29. NIS payoff table for problem
5
0
0
0
26.4
0
3.2
0
NIS: = (4,1.8)
Next, construct equation and obtain the following equations:
=
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ==0.5 and p=2.
Table 30. PIS payoff table of problem (11) when p=2
113.23
97.69
10.
4
0.
8
0
0
113.23
97.691
10.
4
0.
8
0
= ,,
=(zero,
).
Now, it is easy to compute step (10):
Subject to
The maximum “satisfactory level” ( =zero) is
achieved for the solution =10.38462,
=0.7692308,= zero.
(Second Level):
Obtain PIS and NIS payoff tables for problem
Table 31. PIS payoff table for problem
-6.87
6.3
3.2
0
0
0
1.67
PIS:=(, )
Table 32. NIS payoff table for problem
17.67
0
0
1.67
32.6
0
3.2
0
NIS: = (-13,)
Next, construct equation and obtain the following equations:
15
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ====0.25 and p=2,
Table 33. PIS payoff table of problem (11) when p=2
-6.87
6.3
3.
2
0
0.04
11.77
10.
4
0.
8
0
=( ,),
=(0.03948343031,
).
Now, it is easy to compute step (10):
Subject to
The maximum “satisfactory level” ( =zero) is
achieved for the solution =10.38462,
=0.7692308,= zero.
Level):
Obtain PIS and NIS payoff tables for problem
Table 34. PIS payoff table for problem
10.4
0.8
0
10.4
0.8
0
PIS:=(, 66.07692)
Table 35. NIS payoff table for problem
33.6
0
3.2
0
7.67
0
0
1.67
NIS: =(,)
Next, construct equation and obtain the following equations:
Thus, problem (11) is obtained. In order to get numerical
solutions, assume that ======1/6 and p=2,
Table 36. PIS payoff table of problem (11) when p=2
87.54
66.08
10.
4
0.7
7
0
0.05
87.54
66.08
10.
4
0.7
7
0
= () ,
= (0.04546254747, ).
Now, it is easy to compute step (10):
Subject to
,
The maximum “satisfactory level”
(=0.2253846E-06) is achieved for the solution
=10.38461,
=0.7692309,
= 0.9999998E-07.
Note:
Numbers at Table (1) to Table (36) are approximated.
Table (37) presents a comparison among the proposed
TOPSIS method, Global Criterion (GC) Method and the ideal
objective vector (IOV). In general, the proposed TOPSIS
algorithm is a good method to generate compromise solutions
(at p=2).
16
Table 37. Comparison among the proposed algorithm, the
GC method and the vector of ideal solutions
Objective
Proposed
TOPSIS
Algorithm
method
(p=2)
Global
Criterion
(GC)
Method
Ideal Objective
Vector
PIS
NIS
23.230778
5.6678893
24
-2.4
59.465616
1.0033662
59.4615
4.3
55.384638
13
55.38462
-1.4
62.0090232
15
62
5.333
13
3
13
-3.4
55
11
58.4615
3.3333
87.538418
13
87.5384
-0.4
66.076897
20
66.0769
6.3333
5. CONCLUSIONS
This paper extended TOPSIS approach to find compromise
solutions for the multi-level multi-objective decision making
problems with rough parameters in the objective functions
(RMLMODM). A new hybrid algorithm based on modified
TOPSIS method and the "Lower & Upper” approximations
method for solving RMLMODM problems is proposed. Also,
an illustrative numerical example is solved and compared the
compromise solutions of the proposed algorithm with the
vector of ideal solutions and the traditional global criterion
method. The engineers and the scientists can apply the
introduced hybrid algorithm to various practical RMLMODM
problems to obtain numerical solutions
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