# Hybrid Priority-based Genetic Algorithm for Multi-stage Reverse

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```					IEMS Vol. 8, No. 1, pp. 14-21, March 2009.

Hybrid Priority-based Genetic Algorithm for
Multi-stage Reverse Logistics Network
Jeong-Eun Lee†
Graduate School of Information, Production and Systems Waseda University
Kitakyushu, 808-0135 Japan, E-mail: leeje@toki.waseda.jp

Mitsuo Gen
Graduate School of Information, Production and Systems Waseda University
Kitakyushu, 808-0135 Japan, E-amil: gen@waseda.jp

Kyong-Gu Rhee
Busan, 614-714 Korea, E-mail: rhee@deu.ac.kr

Received January 22 2008/Accepted September 14 2008 (Selected from APIEMS 2007)

Abstract. We formulate a mathematical model of remanufacturing system as multi-stage reverse Logistics
Network Problem (mrLNP) with minimizing the total costs for reverse logistics shipping cost and inventory
holding cost at disassembly centers and processing centers over finite planning horizons. For solving this
problem, in the 1st and the 2nd stages, we propose a Genetic Algorithm (GA) with priority-based encoding
method combined with a new crossover operator called as Weight Mapping Crossover (WMX). A heuristic
approach is applied in the 3rd stage where parts are transported from some processing centers to one
manufacturer. Computer simulations show the effectiveness and efficiency of our approach. In numerical
experiments, the results of the proposed method are better than pnGA (Prüfer number-based GA).

Keywords: Multi-stage Reverse Logistics Network Problem (mrLNP), Genetic Algorithm (GA), Priority-based
Encoding Method, Weight Mapping Crossover (WMX).

1. INTRODUCTION                                                  volved in product returns, source reduction, conserva-
tion, recycling, substitution, reuse, disposal, refurbish-
Many companies try to recover the residual value           ment, repair and remanufacturing (1992).
of their used products through remanufacturing process                Researches related to the reverse logistics are con-
for environmental concerns and saving costs. Product             ducting a lot of researches on various fields and subjects.
remanufacturing such as transforming used items into             In reuse logistics models, Kroon et al. (1995), they re-
marketable products through refurbishment, repair and            ported a case study concerning the design of a logistics
upgrading can also yield substantial cost benefits (2007).       system for reusable transportation packages. The authors
Therefore, until now, most of strategies of companies            proposed a mIP (mixed integer programming), closely
were focused on processes from making products to                related to a classical un-capacitated warehouse location
marketing. But, according as changing of the life cycle          model. Spengler et al. (1997) proposed a mIP model
of products including a process from collection of used          which based on the modified multi-level warehouse
products to remanufacturing, companies have to edit the          location problem. The model was solved by a modified
strategies with adjustment themselves to this condition.         Benders decomposition.
Reverse logistics is defined by REVLOG as “the                  In recycling models, Pati et al. (2006), they devel-
propose of planning, implementing and controlling flows          oped an approach based on a mixed integer goal pro-
of raw materials, in process inventory, and finished             gramming model to solve the problem. The model stu-
goods, from the point of use back to point recovery or           dies the inter-relationship between multiple objectives
point of proper disposal” (2004). In a broader sense,            of a recycled paper distribution network. This study
reverse logistics refers to the distribution activities in-      proposed reverse logistics network of remanufacturing

† : Corresponding Author
Hybrid Priority-based Genetic Algorithm for Multi-stage Reverse Logistics Network                            15

pro-cess. In remanufacturing models, Kim et al. (2006),                 ght Mapping Crossover (WMX), insertion mutation ope-
discussed a notion of remanufacturing systems in re-                    rator is adopted.
verse logistics environment. They proposed a general                          This paper is organized as follows: in Section 1, the
framework in view of supply planning and developed a                    problem is described in detail and the previous re-
ma-thematical model to optimize the supply planning func-               searches are reviewed; in Section 2, the mathematical
tion.                                                                   model of the reverse logistics network is introduced; in
Lee et al. (2008) proposed the multistage reverse                 Section 3, the priGA approach and heuristic approach
Logistics Network Problem (mrLNP) to minimize the                       are explained in order to solve this problem; in Section 4,
total cost which involves reverse logistics shipping cost               numerical experiments are presented to demonstrate the
and fixed cost of opening the disassembly centers and                   efficiency of the proposed method; finally, in Section 5,
processing centers.                                                     concluding remarks are outlined.
Recently, GAs have received considerable attention
regarding their potential as a novel approach to optimi-
zation problems and is often used to solve many real                    2. MATHEMATICAL FORMULATION
world problems, including the effective approaches on
the SPR problem, capacity and flow assignment, and the                       In this Section, we consider constituents, variables,
dynamic routing problem. It is noted that all of these                  and assumptions for formulating a multistage logistics
problem can be formulated as some sort of a combina-                    network model. In the remanufacturing process, after
torial optimization problem.                                            product recovery and disassembly part, the disassembled
In this paper, we consider a complex reverse logis-               parts are sent to processing centers. We then consider if
tics problem (rLP) including time period and inventory                  the state of these parts is the same as new products.
developing Lee et al. (2008). Furthermore, in disassem-                      The mathematical models in this analysis have the
bly process, the products are disassembled to several                   following assumptions:
different parts. In the practical reverse logistics prob-                    A1: We consider logistics network for treating sin-
lems, the different parts should be assigned different                             gle product.
processing processes based on the processing compe-                          A2: We consider the inventory factor at disassem-
tence of the processing center. For this reason, the chal-                         bly center and processing center over finite
lenges of our study are both in the mathematical formu-                            planning horizons.
lation and effective approach construction. In additional,                   A3: The demand of parts by manufacturer is known
multistage reverse logistics network problem (mrLNP)                               in advanced.
with minimum total reverse logistic shipping cost and                        A4: The maximum capacities about four echelons
inventory holding cost at disassembly center and                                   are known: returning centers, disassembly cen-
processing center over finite planning horizons has been                           ters, processing centers and manufacturer.
considered and new genetic algorithm approach is pro-                        A5: If the number of provided parts from
posed. And considering the priority-based encoding me-                             processing process is not enough for require-
thod, we propose a new crossover operator called Wei-                              ment of manufacturer, then manufacturer must

Supplier                    Processing                           Disassembly              Returning center i
center k                               center j                         1p
a1
cij xij(t)

cＳFm    xＳFm(t)                                                              b1                         2p
A
u1m   cｊｋm xjkm(t)                                              a2
1m                           B     1
B

cｋＤm                                                                   3p
a3
Manufacturer                                                           A           b2
A                                         2
2m                           B
dm                                  u2m                                                       4p
cｋＲm                                                                                a4
・・・

・・・

bj
cｋFm
Recycling                              km                                  j                          5p
A           ukm                      A
uＲ                                                                                                 a5
・・・

xkＲm(t)   xkFm(t)                                 B
Disposal
・・・

・・・

A
xkＤm(t)                                                                            ip
B
KM                             A         bJ
uKM                                                             ai
J
・・・

B
ckm2H y km2(t)                         cj1H y j1(t)
IP
aI

Figure 1. Multistage reverse logistics network model.
16                                      Jeong-Eun Lee·Mitsuo Gen·Kyong-Gu Rhee

buy parts from supplier.                                    xkDm(t): amount shipped from processing center k to
A6: If the number of provided parts from                                 disposal D in period t for part m
processing process exceeds in the requirement               xSFm(t): amount shipped from supplier S to manu-
of ma-nufacturer, then exceeded capacities are
distributed in order of recycling and disposal.                       facturer F in period t for part m
1
yj (t): inventory amount at disassembly center j in
The notations are defined as follows:                                   period t
Indices                                                                 2
ykm (t): inventory amount at processing center k in
i: index of returning center (i = 1, 2, …, I)
j: index of disassembly center (j = 1, 2, …, J)                           period t for part m
k: index of processing center (k = 1, 2, …, K)                  The mathematical model of the problem is:
m: index of part (m = 1, 2, …, M)
t: index of time period (t = 1, 2, …, T)                                T ⎡ I      J       J K M

Parameters                                                        min z = ∑ ⎢∑∑ cij xij (t) + ∑∑∑c jkm x jkm (t)
t =0 ⎣ i =1 j =1    j =1 k =1 m=1
I: number of returning centers
K     M                          K     M
J: number of disassembly centers
+∑∑ckFm xkFm (t) − ∑∑ckRm xkRm (t)
K: number of processing centers                                             k =1 m=1                          k =1 m=1
M: number of parts                                                           K     M                           M
T: planning horizons                                                      +∑∑ckDm xkDm (t) + ∑cSFm xSFm (t )
ai: capacity of returning center i                                          k =1 m=1                          m=1

bj: capacity of disassembly center j                                         J                    K   M  ⎤
+ ∑c1H y1j (t) + ∑∑ckm ykm (t)⎥
j
2H 2
(1)
ukm: capacity of processing center k for part m                             j =1           k =1 m=1      ⎦
uR: capacity of recycling                                                              K

uD: capacity of disposal                                        s.t. xSFm (t ) + ∑ xkFm (t ) ≥ d m ,                     ∀m, t              (2)
k =1
dm: demand of parts m in manufacturer F                               J

nm: the number of parts m from disassembling one                     ∑ x (t ) + y (t − 1) ≤ b ,
j =1
ij
1
j           j         ∀j , t                  (3)

unit of product                                                 K

cij: unit cost of transportation from returning center               ∑x
k =1
jkm   (t ) + ykm (t − 1) ≤ ukm ,
2
∀k , m, t         (4)
i to disassembly center j                                      xkRm (t ) ≤ uR , ∀k , m, t                                             (5)
cjkm: unit cost of transportation from disassembly
xij (t ), x jkm (t ), xkFm (t ), xkRm (t ), xkDm (t ), xSFm (t ) ≥ 0   (6)
center j to processing center k for part m
ckFm: unit cost of transportation from processing
The objective function (1) is to minimize total re-
center k to manufacturer F for part m             verse logistics, i.e. shipping cost and inventory holding
ckRm: unit cost of transportation from processing          cost. Equation (2) shows the demand of parts. Equations
center k to recycling R for part m                (3), (4) and (5) are shows constraints about capacity of
ckDm: unit cost of transportation from processing          the disassembly center, processing center and recycling
center k to disposal D for part m                 each other. Equation (6) shows the condition that all of
cSFm: unit cost of transportation from supplier S to       variables are non-negative number.
manufacturer F for part m
1H
cj : unit holding cost of inventory per period at
3. HYBRID GENETIC ALGORITHM
disassembly center j
2H
ckm : unit holding cost of inventory per period at
3.1 Representation and Initialization
processing center k for part m
Decision Variables                                                   We here adopt the priority-based encoding method
xij(t): amount shipped from returning center i to          developed Gen et al. (2006). Although this encoding
disassembly center j in period t                 had been successfully applied on shortest path problem
xjkm(t): amount shipped from disassembly center j          and project scheduling problem, the difference of our
to processing center k in period t for part m   approach comes from the facts of special decoding and
xkFm(t): amount shipped from processing center k to        encoding procedures for transportation trees. The prio-
rity-based encoding method is an indirect approach. In
manufacturer F in period t for part m
this method, a gene in chromosome contains two kinds
xkRm(t): amount shipped from processing center k to
of information: the locus, the position of the gene within
recycling R in period t for part m             the structure of a chromosome, and the allele, the value
Hybrid Priority-based Genetic Algorithm for Multi-stage Reverse Logistics Network                                                   17

the gene takes. The position of a gene is used to repre-                                     some.
sent a node (source or depot), and the value is used to
procedure 1.2 : 1st stage decoding
represent the priority of the node for constructing a tree                     input: I : number of returning centers
among candidates.                                                                    J : number of disassembly centers
For a transportation problem, a chromosome con-                                ai: capacity of returning center i, ∀i∈I ,
bj(t)=bj －yj1(t－1) : capacity of disassembly center j in period t, ∀j∈J ,
sists of priorities of sources and depots to obtain tran-                            cij: shipping cost of one unit product from i to j
sportation tree and its length is equal to total number of                           v1(i+j): chromosome, ∀i∈I, ∀j∈J ,
sources m and depots n, i.e. m + n. The transportation                         output: xij(t): the amount of shipment from i to j
step 0: xij(t)←0, ∀i∈I, ∀j∈J
tree corresponding with a given chromosome is genera-                          step 1: l ←argmax{v1(t), t∈I+J}; select a node
ted by sequential arc appending between sources and                            step 2: if l∈I, then i ← l; select a returning center
depots. At each step, only one arc is added to tree selec-                                     j* ← argmin{cil | v1(j)≠0, j∈J}; select a j with lowest cost
ting a source (depot) with the highest priority and con-                               else j ← l: select a disassembly process
i* ← argmin{cil | v1(i)≠0, i∈I}; select a i with lowest cost
necting it to a depot (source) considering minimum cost.                       step 3: xi* j* (t) ← min{ai*, bj*－ yj1(t－1) }; assign available amount of units
For mrLNP, we use two priority-based encodings                                    update the availabilities on i (ai* ) and j (bj*－ yj1(t－1) )
to represent the transportation trees on stages. This means                                  ai* = ai* － xi* j* (t), and bj* = (bj*－ yj1(t－1) ) － xi* j* (t)
step 4: if ai* = 0, then v1(i*) ←0
that each chromosome in the population consists of two                                  if (bj*－ yj1(t－1) ) = 0, then v1(I+j*) ←0
parts. While the first part (i.e. the first priority-based                     step 5: if v1(I+j) = 0 , ∀j∈J, output xij(t)
encoding) represents transportation tree between return                                else return step 1
centers and disassembly centers, the second part (i.e. the                    Figure 3. Decoding procedure for 1st stage of the chro-
second priority-based encoding) represents transporta-                                  mosome.
tion tree between disassembly centers and processing
centers.
The decoding procedure of 2nd stage priority-based
decoding and its trace table are given in Fig. 4. 2nd
procedure 1.1 : 1st stage encoding                                            stage encoding method is the same with in procedure 1.1
input: I : number of returning centers,                                       of 1st stage encoding.
J : number of disassembly centers
ai: capacity of returning center i, ∀i∈I                                     Fig. 5 gives an illustration of the 3rd stage of trans-
bj: capacity of disassembly center j, ∀j∈J                              portation between processing centers and a manufac-
cij: shipping cost of one unit product from i to j                      turer. The 3rd stage has three kinds of cases. In Case 1,
xij(t): the amount of shipment from i to j                              if the quantity of parts provided from the processing
output: v1(i+j): chromosome, ∀i∈I, ∀j∈J
step 1 : priority p ←(|I |+ |J |), v1(i+j) ←0, ∀i+j ∈|I |+ |J |;              centers is not enough for requirement of the manufac-
step 2 : (i*, j*) ← argmin{cil | xij(t)≠0 & (ai = xij(t)||, bj = xij(t)};     turer, parts for insufficient demand should be bought
step 3 : ai* = ai* - xi* j*(t), bj* = bj*- xi* j*(t);                         from a supplier. In Case 2, if the quantity of provided
step 4 : if ai* =0 then v1(i+j) ←p, p ←p-1;                                   parts from processing centers is the same requirement of
step 5 : if (ai* = 0, ∀i* ∈I ) & (bj* = 0, ∀j* ∈J ) then goto step 6;
else return step 1;
manufacturer, the parts privided from the processing
step 6 : for l = 1 to p do                                                    centers are distributed to the manufacturer. In Case 3, if
v1[i+j] ← l, t=random[1, (|I|+ |J|)]& v1[i+j] = 0;                   the quantity of parts provided from processing centers
step 7 : output encoding v1[i+j] , ∀ t ∈ |I|+ |J|                             exceeds the requirement of manufacturer, the exceeded
Figure 2. Encoding procedure for 1st stage of the chromo-                     parts should be recycled. When if the parts still remain,
procedure 2.2 : 2nd stage decoding
input: J : number of disassembly centers
K : number of processing centers
bj: capacity of disassembly center j, ∀j∈J ,                                J

ej(t) : shipping amount of disassembly center j, ∀j∈J , e j ( t ) = ∑1 x ij ( t ) + y ij ( t − 1)
1

j=
uk*m(t)=uk*m － ykm   2(t － 1): shipping amount of processing center k in period t for part m,∀k∈K ,

cｊｋm: unit cost of transportation from j to k for part m
v2(j+k): chromosome, ∀j∈J, ∀k∈K ,
output: xjkm(t): the amount of shipment from j to k for part m
step 0: xjkm(t): ←0, ∀j∈J, ∀k∈K
step 1: l ←argmax{v2(t), t∈J+K}; select a node
step 2: if l∈J, then j ← l; select a disassembly center a k with lowest cost
else k ← l: select a processing center
k* ← argmin{cjkm| v2(k)≠0, k∈K}; select a j with lowest cost
step 3: xj*k* m(t)← min{ej(t), , uk*m + ykm2(t － 1)}; assign available amount of units
update the availabilities on j (ej(t)) and k(uk*m + ykm2(t－1))
bj* = (ej(t)) － xj*k* m(t) and uk*m= (uk*m + ykm2(t－1)) － xj*k* m(t); update the availability
step 4: if bj* = 0, then v2(j*) =0
if (ej(t))= 0, then v2(J+k*) =0
step 5: if v2(J+k) = 0 , ∀k∈K, output xjkm(t),
else return step 1
Figure 4. Decoding procedure for 2nd stage of the chromosome.
18                                                    Jeong-Eun Lee·Mitsuo Gen·Kyong-Gu Rhee

Fig. 6 gives overall produce of proposed method.                                                    offspring by combining both chromosomes’ features. The
crossover is done to explore new solution space and cro-
3.2 Genetic Operators                                                                                    ssover operator corresponds to exchanging parts of st-
rings between selected parents.
Genetic operators mimic the process of heredity of                                                       Several crossover operators have been proposed for
gens to dreate new orrspring at each generation. Using                                                   permutation representation, such as partially mapping
the different genetic operators has very large influence                                                 crossover (PMX), order crossover (OX), position-based
on GA performance(1993). Therefore it is important to                                                    crossover (PX), cycle crossover (CX), Heuristic cros-
exmined different genetic operators.                                                                     sover, and so on. In this study, weight mapping crossov-
er (WMX) operator is used.
3.2.1 Crossover Operator                                                                                      WMX can be viewed as an extension of one-cut
Crossover is the main genetic operator. It operates                                                 point crossover for permutation representation. As one-
on two parents (chromosomes) at a time and generates                                                     point crossover, two chromosomes (parents) would be

procedure 3: 3rd stage of transportation between processing center and Manufacturer
input: K : number of processing centers,                                        K

ekm(t): shipping amount of processing center k for part m, e km ( t ) = ∑1 x jkm ( t ) + y km ( t − 1)
2

k=
uk*m(t): shipping amount of processing center k in period t , for part m, ∀k∈K ,
uR: capacity of recycling, uD: capacity of disposal, dm : demand of parts m in M
ckFm : unit cost of transportation from processing center k to manufacturer F for part m, ∀m
ckRm : unit cost of transportation from processing center k to recycling R for part m, ∀m
ckDm: unit cost of transportation from processing center k to disposal D for part m, ∀m
cSFm: unit cost of transportation from supplier S to manufacturer F for part m, ∀m
xjkm: the amount of shipment from j to k for part m , ∀m
output: xkFm(t): amount shipped from k to M for part m,                                   xkRm(t): amount shipped from k to R for part m
xkDm(t): amount shipped from k to D for part m,                                  xSFm(t): amount shipped from S to M for part m
step 0: Calculate total shipment from parts m.
K
e km ( t ) =   ∑x
k =1
jkm   ( t ) + y km ( t − 1)
2

step 1: for each part m, considering the follows cases:
case 1: dm> ekm(x) then goto step 2, case 2: dm= ekm(x) then goto step 3, case 3: dm< ekm(x) then goto step 4.
step 2 : if dm> ekm(x), distribute to the manufacturer provided parts ekm(x) from k,
then dm–ekm(x) = xSFm(t) , buy xSFm(t) from supplier.
step 3: if dm= ekm(x) distribute to the manufacturer provided parts ekm(x) from k.
step 4: if dm< ekm(x) distribute to the manufacturer with low cost among provided parts ekm(x) from processing center.
then if remaining parts ekm(x) – dm ≤ uR, distribute to the recycling parts ekm(x) – dm,
then if ekm(x) – dm> uR, distribute to the recycling parts uR and (ekm(x) – dm)– uR– uD distribute to the disposal.
step 5: output xkFm(t), xkRm(t), xkDm(t), xSFm(t)
Figure 5. Decoding procedure for 3rd stage.

procedure 4: Overall procedure
step 0: z ← 0 , z t ← 0 , y 1j ( t ) ← 0 , y km ( t ) ← 0 , ∀ t ;
2

step 1: for all t
find [xij(t)] by procedure 1
find [xjkm(t)] by procedure 2
find [xkFm(t)] [xkRm(t)] [xkDm(t)] [xSFm(t)] by procedure 3
step 2:calculate and output the total cost for all t:
:       T ⎡ I       J            J    K    M
z = ∑ ⎢ ∑ ∑ c ij x ij ( t ) + ∑ ∑ ∑ c jkm x jkm ( t )
t = 0 ⎣ i =1 j =1         j =1 k =1 m = 1
K     M                                K        M                          K    M                            M
+    ∑∑c
k =1 m =1
kFm   x kFm ( t ) −       ∑∑c
k =1 m =1
kRm   x kRm ( t ) +   ∑∑c
k =1 m =1
kDm   x kDm ( t ) +   ∑c
m =1
SFm   x SFm ( t )

J                         K      M                       ⎤
+    ∑c     1H
j     y 1j ( t ) +   ∑∑c             2H
km
2
y km ( t ) ⎥
j =1                        k =1 m =1                       ⎦
T
1
output z =
T
∑z
t =1
t

Figure 6. Overall procedure.
Hybrid Priority-based Genetic Algorithm for Multi-stage Reverse Logistics Network                  19

chose a random cut-point and generate the offspring by         better chance to get copied into the next generation. Also
using segment of own parent to the left of the cut-point,      the elitist method is employed to preserve the best chro-
then remapping the right segment that base on the wei-         mosome for the next generation.
ght of other parent of right segment.
In this study, we proposed a new crossover opera-
tor, weight mapping crossover (WMX).                           4. NUMERICAL EXPERIMENT
3.2.2 Mutation Operator                                              We used using pnGA (Prüfer number-based GA)
Mutation is a background operator which produces         proposed by Syarif and Gen (2003), to study the effec-
spontaneous random changes in various chromosomes.             tiveness of the developed GA with new encoding me-
Similar to crossover, mutation is done to prevent the          thod (priGA).
premature convergence and explores new solution space.               As it is seen in Table 2, the number of returning
However, unlike crossover, mutation is usually done by         centers changes between 4 and 30 on these problems,
modifying gene within a chromosome. We also investi-           number of disassembly centers and number of process-
gate the effects of two different mutation operators on        ing centers change between 3 and 15, and 3 and 20,
the performance of GA. Insert mutation is used for this        respectively. The data in test problems such as transpor-
purpose. Several mutation operators have been proposed         tation costs, demand of parts, capacitates of returning
for permutation representation, such as inversion, inser-      centers, disassembly processes, processing processes,
tion, displacement, and reciprocal exchange mutation.          recycles and manufacturer were also randomly genera-
In this study, insertion mutation has been adopted.      ted to provide realistic scenarios.
Insertion mutation selects a gene at random and inserts it           The parameters for the proposed GA approach are
in a random position.                                          set as follows:
• Population size: popSize = 10 Mutation probability: pM
3.2.3 Evaluation and selection                                    = 0.7
Evaluation aims to associate each individual with a      • Maximum generation: maxGen = 100 Crossover prob-
fitness value so that it can reflect the goodness of fit for      ability: pC = 0.7
an individual. This evaluation process intended to com-
pare one individual with other individuals in the popula-            Table 3 gives computational results for the priGA
tion. The choice of fitness function is also very critical     and pnGA, on four test problems. Stage 1 represents
because it has to accurately measure the desirability of       cost of product transportation from returning center to
the features described by the chromosome. The function         disassembly center. Stage 2 represents cost of disassem-
should be computationally efficient since it is used many      bled part transportation from disassembly center to pro-
times to evaluate each and every solution. In the pro-         cessing center and holding cost of inventory in disas-
posed algorithm, the fitness function has been taken as        sembly center. And stage 3 represents cost of processed
inverse of objective function.                                 part transportation from processing center to manufac-
The selection operator is intended to improve the        tuter, recyclig cost, disposal cost and holding cost of
average quality of the population by giving the high-          inventory in processing center.
quality chromosomes, i.e., a better chance to get copied             In priGA, one-cut point crossover and insertion
into the nest generation. The selection can be thought as      mutation operators were used as genetic operators and
the exploitation for the GA to guide the evolutionary          its rates were taken as 0.5. Each test problem is run by
process when we regard the genetic operation as the            10 times using GA approaches. As the results, priGA
exploration for the search in solution space. We employ        exhibits better performance than pnGA according to
roulette wheel selection with elitist strategy as a selec-     solution quality. This analysis indicates that the priGA is
tion mechanism.                                                superior to the pnGA.
In roulette wheel selection mechanism, the indi-
viduals on each generation are selected for survival into
the next generation according to a probability value pro-      5. CONCLUSIONS
portional to the ratio of individual fitness over total po-
pulation fitness; this means that average quality of the
population by giving the high-quality chromosomes a
20                                    Jeong-Eun Lee·Mitsuo Gen·Kyong-Gu Rhee

Table 3. Computational results with pnGA and priGA.
Problems       No. of        No. of shipping                                 pnGA                        priGA
Stage
No.       constraints       variables                           average          ACT        average           ACT
stage 1          6990           0.13        6990             0.13
stage 2          8470           0.12        8470             0.13
1             30                34
stage 3         10020             -         10020              -
total          25480             -         25480              -
stage 1         10790           0.18        10470            0.20
stage 2          9940           0.19        9720             0.19
2             48                71
stage 3         15100             -         15100              -
total          35830             -         35290              -
stage 1         16930           0.43        15730            0.45
stage 2         17110           0.40        16610            0.44
3            105               217
stage 3         22010             -         22010              -
total          56050             -         54350              -
stage 1         39265           0.84        37475            0.90
stage 1         42260           0.77        40435            0.77
4            168               556
stage 1         47535             -         47535              -
total         129060             -        125445              -
ACT: Average computation time as second.
In this paper, we address reverse Logistics Network           points about result in the case of real data are applied.
Problem for treating a remanufacturing problem which                    In the future, it is possible to investigate the per-
is one of the most important problems in the environ-              formance of the mrLNP on the large scale problems also
ment situation for the recovery of used products and               including real-data.
materials.
We formulated a mathematical model for the rLNP
by using priority-based genetic algorithm approach (priGA)         ACKMOWLEDGMENTS
and a heuristic approach. We also combined a new cro-
ssover operator, weight mapping crossover (WMX), with                  This work is partly supported by the Ministry of
insertion mutation in hybrid priGA.                                Education, Science and Culture, the Japanese Govern-
Numerical experiments demonstrated the efficiency             ment: Grant-in-Aid for Scientific Research (No. 17510138,
and effectiveness of the hybrid GA approach for solving            No.19700071, No. 20500143).
the mrLNP problem. Although memory requirement for
new representation is greater than pnGA, i.e., Prüfer
number-based GA, only 2 digits for each stage in trans-
portation problem, this representation shows very im-              REFERENCES
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Table 2. The size of test problems.
Problem          Time         Retuming         Disassembly         Processing          No. of        No. of shipping
No.          Period (t)     Centers (I)      Centers (J)         Centers (K)       Constraints       variables
1               3              4                 3                   3               30                 34
2               3              6                 5                   5               48                 71
3               3             15                 10                 12               105               217
4               3             30                 15                 20               168               556
Hybrid Priority-based Genetic Algorithm for Multi-stage Reverse Logistics Network                21

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