A Bi-Abjective Model for Optimizing Price, Warranty Length, and

A Bi-Abjective Model for Optimizing Price, Warranty Length, and

International Journal of Industrial Engineering & Production Management (2014)
July 2014, Volume 25, Number 2
pp. 131-142
http://IJIEPM.iust.ac.ir/
A Bi-Abjective Model for Optimizing Price, Warranty Length,
and Service Capacity Within Queuing Framework: Genetic
Algorithm and Fuzzy System
A. Mahmoudi & H. Shavandi*
Amin Mahmoudi, MSc, Faculty of Industrial and mechanical Engineering, Qazvin branch, Islamic Azad University, Qazvin, Iran
Hassan Shavandi, Associate Professor of Industrial Engineering, Sharif University of Technology, Tehran, Iran
Keywords
Bi-objective optimization,
Pricing, Queuing,
Fuzzy system,
Genetic algorithm,
Warranty,
1 ABSTRACT
In this paper, we have proposed a bi-objective model for pricingqueuing problem under fuzzy environment. The objectives are
maximizing the profit and minimizing the waiting time in system to
receive the service. Price, warranty length, and service capacity
decisions are analyzed for a seller with considering sale and service
channels. To formulate the demand function, a fuzzy system is
developed to estimate the demand value under price and warranty
length variables. Furthermore, a hybrid solution of genetic algorithm
and a fuzzy system is presented to solve the proposed model. At end,
numerical results are analyzed by solving sample problems.
© 2014 IUST Publication, IJIEPM. Vol. 25, No. 2, All Rights Reserved
*
Corresponding author. Hassan Shavandi
Email: Shavandi@sharif.edu
‫‪http://IJIEPM.iust.ac.ir/‬‬
‫‪ISSN: 2008-4870‬‬
‫*‬
‫سيستم فازي‪،‬‬
‫الگوريتم ژنتيک‪،‬‬
‫وارانتي‪،‬‬
‫] [‬
‫] [‬
‫] [ ] [‬
‫] [‬
‫امين محمودي‪ ,‬کارشناسي ارشد‪ ،‬دانشکده مهندسي صنايع و مکانيک‪ ،‬واحد‬
‫قزوين‪ ،‬دانشگاه آزاد اسالمي‪ ,‬قزوين‪ ،‬ايران‪amin.mahmoudi10@gmail.com ،‬‬
‫حسن شوندي‪ ،‬دانشيار دانشکده مهندسي‬
‫صنايع‪ ،‬دانشگاه صنعتي شريف‪Shavandi@sharif.edu ,‬‬
‫] [‬
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
L-p metric
-
M/M/1
[ ]
M/M/1
L-p metric
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫‪‬‬
‫] [‬
‫‪‬‬
‫‪Cr ‬‬
‫‪C ‬‬
‫‪D‬‬
‫‪‬‬
‫‪P‬‬
‫‪w‬‬
‫‪‬‬
‫زن يره امين مت رکز‬
‫دم ا‬
‫فرو‬
‫از‬
‫فروشنده‬
‫م‬
‫قي‬
‫و‬
‫ا ا‬
‫را‬
‫‪  . w .D‬‬
‫و دوره‬
‫ارانتي‬
‫مشتريان‬
‫کانا‬
‫دم ا‬
‫فرو‬
‫از‬
‫‪C r .‬‬
‫‪‬‬
‫‪C  .‬‬
‫کانا فرو‬
‫‪-‬‬
Z 1  P .D  C r .  C  .
( )
D  f ( p ,w )
M/M/1
[ ]
Z2 
M
L
FL
VH
M/M/1
1
( )
 
VL
FH
( x  a) (b  a); a  x  b

 X ( x)  1
xb
(c  x) (c  b); b  x  c

H
( )
Max :
Z 1  P .D  C r .  C  .
Min :
Z2 
1
 
( )
Subject to :
C .  C  .
P r
D
 
( x  a) (b  a); a  x  b

 X ( x)  1
b xc
(d  x) (d  c); c  x  d

( )
-
 X ( x)
1
If P is H and W is L Then D is L
b
a
x
c
 X ( x)
[ ]
-
1
a
b
c
x
d
y0 x0
MISO
Y ( y 0 )
X (x 0 )
h2 h1
 j  min(h1 , h2 )
-
( )
w
D
uj
uj
u
j
u0
2
Singleton Fuzzifier
Mamdani Implication
4 Centroid defuzzifier
5 Matching degree
3
1
Multi Input Single Output
P
J
u0 
u
j
 j
j 1
J

j
j 1
H
H
M
M
VL
VL
H
VH
M
H
L
VL
FL
VL
H
FH
M
VL
L
L
H
VH
H
VL
L
M
H
VH
VH
VL
M
H
H
M
VL
L
M
VH
H
H
L
L
VL
VL
VH
H
M
L
FL
L
VH
FH
H
L
FL
M
VH
FH
VH
L
L
H
VH
L
VL
M
M
VH
VH
M
L
M
-
-
-
M
M
M
X (x )
VL
1
X
min
L
X (x )
1
M
x2
x1
H
L
VH
x4
x3
‫د شد‬
VL
‫ان‬
Y ( y )
‫يم‬
‫يم‬
M
X
max
x5
VL
1
L
y2
y1
y min
Y ( y )
‫ا‬
H
VH
h1
‫شد‬
VL
1
‫و د‬
M
H
y3
y4
‫ان‬
L
‫و د‬
M
H
 Z (z )
VH
1
y5
FL
L
M
H
FH
z1
z2
z3
z4
z5
z6
0
y max
‫ا‬
VL
‫و ا د ود‬
 Z (z )
VH
VL
1
L
z7
z
Z max
‫ن‬
M
H
FH
u4
u5
u6
VH
j
Min
h2
FL
VH
u 'j
X min
x1
x2
x4
x3
‫يم‬
‫ام‬
‫د‬
x5
‫م‬
X
max
‫ي‬
y min
‫ان‬
y2
y1
‫و د‬
‫ام‬
y4
y3
‫د‬
‫م‬
y5
y max
0
u1
u2
u3
‫ي‬
L-p metric
-
MODM
[ ]
u7
z
U max
max : F (x )  f 1 (x ), f 2 (x ),..., f k (x )
Subject to: x  X
( )
X Rk
[ ]
K
MODM
L-p metric
GA
[ ] [ ]
GA
L-p metric
[ ]
k

Min : L  p    j
 j 1


-
Subject to:
t
 f (x * )  f (x ) 
j
 j

*


f
(
x
)
j


1
 t




( )
x X  R k
j
P
W

fj(x*)
j
1t  
µ
w
w p
p
Min: Z2
Max: Z1
Max: Z 2'  1 Z 2
b  ( Z 1, Z 2* )
L-p metric
1
Roulette Wheel
j
Min: Z2
a  ( Z 1* , Z 2 )
t
‫‪p‬‬
‫‪w‬‬
‫وليد‬
‫‪-‬‬
‫عي‬
‫اولي از ‪ , w ،p‬‬
‫‪ P, w‬‬
‫‪w P‬‬
‫مت ير اي ورودي‬
‫‪1‬‬
‫‪0‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪2‬‬
‫‪w2‬‬
‫‪P2‬‬
‫‪1‬‬
‫‪w1‬‬
‫‪P1‬‬
‫‪1‬‬
‫‪w2‬‬
‫‪P2‬‬
‫‪2‬‬
‫‪w1‬‬
‫‪P2‬‬
‫سيستم فازي‬
‫سيستم فازي‬
‫د‬
‫دد‬
‫‪‬‬
‫فازي ساز‬
‫ي اد‬
‫مو‬
‫فرآيند استنتا‬
‫فازي‬
‫ايگاه قوا د فازي‬
‫ير فازي ساز‬
‫مو‬
‫ي‬
‫ين ا ا‬
‫ارزيا ي کروموزو‬
‫ا وس‬
‫انت ا والدين‬
‫گر ا ع‬
‫گر ه‬
‫ير‬
‫‪p‬‬
‫‪w‬‬
‫‪‬‬
‫‪p‬‬
‫‪w p‬‬
‫‪w‬‬
‫‪‬‬
‫‪Tournament replacement‬‬
‫‪1‬‬
‫شر‬
‫و قف‬
‫ان ا شد‬
‫ا ع رازند ي‬
u0
-
L-p metric
 1  3 t=2   1 C   150 C r  40
2 1
‫يم‬
X (x )
VL
1
360
L
460
M
760
‫ان‬
Y ( y )
H
1554
1154
VH
1961
1
2041
VL
0.056
‫و د‬
L
0.44
2
M
H
4.12
6
U (u )
VH
1
8
0
9.7
VL
FL
L
M
16
166
333
500
H
FH
666
833
VH
u
980 1000
4
u0
4
u
j
 j

 j u j
j
j 1
u
j
j
j 1
P
D
W
P
H
M
L
FH
H
L
M
M
M
H
H
M
C r .  C  .
Z 1*  380071
D
 
Z 2*'  2020 ( Z 2*  0.000495)
p metric L-
Max: Z 2'  1 Z 2
Min: Z2
1
L-p metric
Min : L  p
Matlab
2
2 2
  *
 Z *'  Z '  
Z1  Z1 

2
2
 3  
  1 
 
*
 Z 2*'  
  Z 1 


Subject to:
Matlab
Z2
L-p metric
Z1

W
P
L-p metric
-
M/M/1
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