A genetic algorithm approach for multi objective optimization of supply chain networks

A genetic algorithm approach for multi-objective
optimization of supply chain networks
Fulya Altiparmak a,*, Mitsuo Gen b
, Lin Lin b
, Turan Paksoy c
a
Department of Industrial Engineering, Gazi University, Turkey
b
Graduate School of Information, Production and Systems, Waseda University, Japan
c
Department of Industrial Engineering, Selcuk University, Turkey
Available online 22 August 2006
Abstract
Supply chain network (SCN) design is to provide an optimal platform for efficient and effective supply chain manage-
ment. It is an important and strategic operations management problem in supply chain management, and usually involves
multiple and conflicting objectives such as cost, service level, resource utilization, etc. This paper proposes a new solution
procedure based on genetic algorithms to find the set of Pareto-optimal solutions for multi-objective SCN design problem.
To deal with multi-objective and enable the decision maker for evaluating a greater number of alternative solutions, two
different weight approaches are implemented in the proposed solution procedure. An experimental study using actual data
from a company, which is a producer of plastic products in Turkey, is carried out into two stages. While the effects of
weight approaches on the performance of proposed solution procedure are investigated in the first stage, the proposed
solution procedure and simulated annealing are compared according to quality of Pareto-optimal solutions in the second
stage.
Ó 2006 Elsevier Ltd. All rights reserved.
Keywords: Supply chain network; Genetic algorithm; Multi-objective optimization
1. Introduction
A supply chain is a set of facilities, supplies, customers, products and methods of controlling inventory,
purchasing, and distribution. The chain links suppliers and customers, beginning with the production of
raw material by a supplier, and ending with the consumption of a product by the customer. In a supply chain,
the flow of goods between a supplier and customer passes through several stages, and each stage may consist
of many facilities (Sabri & Beamon, 2000). In recent years, the supply chain network (SCN) design problem
has been gaining importance due to increasing competitiveness introduced by the market globalization
(Thomas & Griffin, 1996). Firms are obliged to maintain high customer service levels while at the same time
0360-8352/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cie.2006.07.011
*
Corresponding author.
E-mail addresses: fulyaal@gazi.edu.tr (F. Altiparmak), gen@waseda.jp (M. Gen), lin@ruri.waseda.jp (L. Lin), tpaksoy@yahoo.com
(T. Paksoy).
Computers & Industrial Engineering 51 (2006) 197–216
www.elsevier.com/locate/dsw

they are forced to reduce cost and maintain profit margins. Traditionally, marketing, distribution, planning,
manufacturing, and purchasing organizations along the supply chain operated independently. These organi-
zations have their own objectives and these are often conflicting. But, there is a need for a mechanism through
which these different functions can be integrated together. Supply chain management (SCM) is a strategy
through which such integration can be achieved. Illustration of a supply chain network is shown in Fig. 1.
The network design problem is one of the most comprehensive strategic decision problems that need to be
optimized for long-term efficient operation of whole supply chain. It determines the number, location, capacity
and type of plants, warehouses, and distribution centers to be used. It also establishes distribution channels,
and the amount of materials and items to consume, produce, and ship from suppliers to customers. SCN
design problems cover wide range of formulations ranged from simple single product type to complex mul-
ti-product one, and from linear deterministic models to complex non-linear stochastic ones. In literature, there
are different studies dealing with the design problem of supply networks and these studies have been surveyed
by Vidal and Goetschalckx (1997), Beamon (1998), Erenguc, Simpson, and Vakharia (1999), and Pontran-
dolfo and Okogbaa (1999).
An important component in SCN design and analysis is the establishment of appropriate performance mea-
sures. A performance measure, or a set of performance measures, is used to determine efficiency and/or effec-
tiveness of an existing system, to compare alternative systems, and to design proposed systems. These
measures are categorized as qualitative and quantitative. Customer satisfaction, flexibility, and effective risk
management belong to qualitative performance measures. Quantitative performance measures are also cate-
gorized by: (1) objectives that are based directly on cost or profit such as cost minimization, sales maximiza-
tion, profit maximization, etc. and (2) objectives that are based on some measure of customer responsiveness
such as fill rate maximization, customer response time minimization, lead time minimization, etc. (Beamon,
1998). In traditional supply chain management, the focus of the integration of SCN is usually on single objec-
tive such as minimum cost or maximum profit. For example, Jayaraman and Pirkul (2001), Jayaraman and
Ross (2003), Yan, Yu, and Cheng (2003), Syam (2002), Syarif, Yun, and Gen (2002), Amiri (2006), Gen
Supplier
Plant
DC
Customer
C
C
C
C
C
C
C
C
C
C
C
C
S
S
S
P P
S
DC
P
DC
DC
DC
C
Supplier
Plant
DC
Customer
C
C
C
C
C
C
C
C
C
C
C
C
S
S
S
P P
S
DC
P
DC
DC
DC
C
Fig. 1. Illustration of a Supply Chain Network (ILOG).
198 F. Altiparmak et al. / Computers & Industrial Engineering 51 (2006) 197–216

and Syarif (2005), and Truong and Azadivar (2005) had considered total cost of supply chain as an objective
function in their studies. However, there are no design tasks that are single objective problems. The design/
planning/scheduling projects are usually involving trade-offs among different incompatible goals. Recently,
multi objective optimization of SCNs has been considered by different researchers in literature. Sabri and Bea-
mon (2000) developed an integrated multi-objective supply chain model for strategic and operational supply
chain planning under uncertainties of product, delivery and demand. While cost, fill rates, and flexibility were
considered as objectives, e-constraint method had been used as a solution methodology. Chan and Chung
(2004) proposed a multi-objective genetic optimization procedure for the order distribution problem in a
demand driven SCN. They considered minimization of total cost of the system, total delivery days and the
equity of the capacity utilization ratio for manufacturers as objectives. Chen and Lee (2004) developed a mul-
ti-product, multi-stage, and multi-period scheduling model for a multi-stage SCN with uncertain demands and
product prices. As objectives, fair profit distribution among all participants, safe inventory levels and maxi-
mum customer service levels, and robustness of decision to uncertain demands had been considered, and a
two-phased fuzzy decision-making method was proposed to solve the problem. Erol and Ferrell (2004) pro-
posed a model that assigning suppliers to warehouses and warehouses to customers. They used a multi-objec-
tive optimization modeling framework for minimizing cost and maximizing customer satisfaction. Guillen,
Mele, Bagajewicz, Espuna, and Puigjaner (2005) formulated the SCN design problem as a multi-objective sto-
chastic mixed integer linear programming model, which was solved by e-constraint method, and branch and
bound techniques. Objectives were SC profit over the time horizon and customer satisfaction level. Chan,
Chung, and Wadhwa (2004) developed a hybrid approach based on genetic algorithm and Analytic Hierarch
Process (AHP) for production and distribution problems in multi-factory supply chain models. Operating
cost, service level, and resources utilization had been considered as objectives in their study. The studies
reviewed above have found a Pareto-optimal solution or a restrictive set of Pareto-optimal solutions based
on their solution approaches for the problem. Our purpose in this paper is to present a solution methodology
to obtain all Pareto-optimal solutions for the SCN design problem and enable the decision maker for evalu-
ating a greater number of alternative solutions.
During the last decade, there has been a growing interest using genetic algorithms (GA) to solve a variety of
single and multi-objective problems in production and operations management that are combinatorial and NP
hard (Gen & Cheng, 2000; Dimopoulos & Zalzala, 2000; Aytug, Khouja, & Vergara, 2003). In this study, we
proposed a new approach based on GA for multi-objective optimization of SCNs which is one of the NP hard
problems. Three objectives were considered: (1) minimization of total cost comprised of fixed costs of plants
and distribution centers (DCs), inbound and outbound distribution costs, (2) maximization of customer ser-
vices that can be rendered to customers in terms of acceptable delivery time (coverage), and (3) maximization
of capacity utilization balance for DCs (i.e. equity on utilization ratios). The proposed GA was designed to
generate Pareto-optimal solutions considering two different weight approaches. To investigate the effectiveness
of the proposed GA, an experimental study using actual data from a company, which is a producer of plastic
products in Turkey, was carried out into two stages. While the effects of weight approaches on the perfor-
mance of proposed GA were investigated in the first stage, the proposed GA and multi-objective simulated
annealing (MO_SA) proposed by Ulungu, Teghem, Fortemps, and Tuyttens (1999) were compared according
to quality of Pareto-optimal solutions in the second stage.
The paper is organized as follows: In Section 2, multi-objective SCN design problem is formulated and dis-
cussed. Comprehensive explanation of the proposed GA is given in Section 3. Section 4 gives the computation-
al results to show the performance of the GA using actual data obtained from a company in Turkey. Finally,
concluding remarks are outlined and future research directions highlighted in Section 5.
2. Problem statement
The problem considered in this paper has been from a company which is one of the producers of plastic
products in Turkey. The company is planning to produce plastic profile which is used in buildings (vinyl
sidings, doors, windows, fences, etc.), pipelines and consumer materials. The main raw material of the plastic
profile is PVC. The company wishes to design of SCN for the product, i.e. select the suppliers, determine the
subsets of plants and DCs to be opened and design the distribution network strategy that will satisfy all
F. Altiparmak et al. / Computers & Industrial Engineering 51 (2006) 197–216 199

capacities and demand requirement for the product imposed by customers. The problem is a single-product,
multi-stage SCN design problem. Considering company managers’ objectives, we formulated the SCN design
problem as a multi-objective mixed-integer non-linear programming model. The objectives are minimization
of the total cost of supply chain, maximization of customer services that can be rendered to customers in terms
of acceptable delivery time (coverage), and maximization of capacity utilization balance for DCs (i.e. equity
on utilization ratios). The assumptions used in this problem are: (1) the number of customers and suppliers
and their demand and capacities are known, (2) the number of potential plants and DCs and their maximum
capacities are known, (3) customers are supplied product from a single DC. Fig. 2 presents a simple network
of three-stages in supply chain network.
The mathematical notation and formulation are as follows:
Indices: i is an index for customers (i 2 I). j is an index for DCs (j 2 J). k is an index for manufacturing
plants (k 2 K). s is an index for suppliers (s 2 S).
Model variables: bsk is the quantity of raw material shipped from supplier s to plant k. fkj is the quantity of
the product shipped from plant k to DC j. qji is the quantity of the product shipped from DC j to customer i.
zj ¼
1 if DC j is open
0 otherwise

; pk ¼
1 if plant k is open
0 otherwise

yji ¼
1 if DC j serves customer i
0 otherwise

Model parameters: Dk is the capacity of plant k. Wj is the annual throughput at DC j. sups is the capacity of
supplier s for raw material. di is the demand for the product at customer i. W is the maximum number of DCs.
P is the maximum number of plants. vj is the annual fixed cost for operating a DC j. gk is the annual fixed cost
for operating a plant k. cji is the unit transportation cost for the product from DC j to customer i. akj is the unit
transportation cost for the product from plant k to DC j. tsk is the unit transportation and purchasing cost for
raw material from supplier s to plant k. u is the utilization rate of raw material per unit of the product. hji is the
delivery time (in hours) from DC j to customer i. s is the maximum allowable delivery time (hours) from ware-
houses to customers. C(j) is the set of customers that can be reached from DC j in s hours, or
C(j) = {ijhji 6 s}. oD is the set of opened DCs, oP is the set of opened plants. r1 and r2 are the weights of plants
and DCs, respectively.
Objectives: f1 is the total cost of SCN. It includes the fixed costs of operating and opening plants and DCs,
the variable costs of transportation raw material from suppliers to plants and the transportation the product
from plants to customers through DCs. f2 is the total customer demand (in %) that can be delivered within the
stipulated access time s. f3 is the equity of the capacity utilization ratio for plants and DCs, and it is measured
1
1
21
2
j
i
s
K
J
I
S
1
Suppliers s∈S Plants k∈K DCs j∈J Customers i∈I
1st stage 2nd stage 3rd stage
k
……
……
……
……
tsk
t1k
tSK
akj
cji
3
a21
aKJ
c12
cJI
z1
zj
zJ
p1
p2
pk
pK
sup1
sups
supS
D1
D2
Dk
DK
W1
Wj
WJ
d1
d2
d3
di
dI
W
P
g1
g2
gk
gK
o1
oj
oJ
qji
q12
qJI
fkj
f21
fKJ
bsk
b1k
bSK
Fig. 2. A simple network of three-stages in supply chain network.
200 F. Altiparmak et al. / Computers Industrial Engineering 51 (2006) 197–216

by mean square error (MSE) of capacity utilization ratios. The smaller the value is, the closer the capacity
utilization ratio for every plant and DC is, thus ensuring the demands are fairly distributed among the opened
DCs and plants, and so it maximizes the capacity utilization balance.
minf1 ¼
X
k
gkpk þ
X
j
vjzj þ
X
s
X
k
tskbsk þ
X
k
X
j
akjfkj þ
X
j
X
i
cjiqji ð1Þ
maxf2 ¼
X
j2oD
X
i2CðjÞ
qji
!
=
X
i
di
!
ð2Þ
minf3 ¼ r1
X
k2oP
X
j2oD
fkj=Dk
!
À
X
k2oP
X
j2oD
fkj=
X
k2oP
Dk
! #2
=joPj
2
4
3
5
1=2
þ r2
X
j2oD
X
i
qji=W j
!
À
X
j2oD
X
i
qji=
X
j2oD
W j
! #2
=joDj
2
4
3
5
1=2
ð3Þ
s:t:
X
j
yji ¼ 1; 8i ð4Þ
X
i
diyji 6 W jzj; 8j ð5Þ
X
j
zj 6 W ð6Þ
qji ¼ diyji; 8i; j ð7Þ
X
k
fkj ¼
X
i
qji; 8j ð8Þ
X
k
bsk 6 sups; 8s ð9Þ
u
X
j
fkj 6
X
s
bsk; 8k ð10Þ
u
X
j
fkj 6 Dkpk; 8k ð11Þ
X
k
pk 6 P ð12Þ
zj ¼ f0; 1g; 8j ð13Þ
pk ¼ f0; 1g; 8k ð14Þ
yji ¼ f0; 1g; 8i; j ð15Þ
bsk P 0; 8s; k ð16Þ
fkj P 0; 8j; k ð17Þ
qji P 0; 8i; j ð18Þ
Eqs. (1)–(3) gives the objectives. While (1) deﬁnes the total cost of the SCN, (2) and (3) give the objectives
about customer service and equity of the capacity utilization ratio (i.e., capacity utilization balance), respec-
tively. Constraint (4) represents the unique assignment of a DC to a customer, (5) is the capacity constraint for
DCs, (6) limits the number of DCs that can be opened, (7) and (8) gives the satisfaction of customer and DCs
demands for the product, (9) describes the raw material supply restriction, (10) gives the supplier capacity con-
straint, (11) is the plant production capacity constraint, (12) limits the number of plants that are opened, (13)–
(15) impose the integrality restriction on the decision variables zj, pk, yij, (16)–(18) impose the non-negativity
restriction on decision variables bsk, fkj, qij. Since the third objective is nonlinear, the model given above is a
mixed-integer non-linear programming model.
F. Altiparmak et al. / Computers Industrial Engineering 51 (2006) 197–216 201

The multi-objective optimization problems as given above often contain many optimal solutions. These are
Pareto-optimal solutions. The set of Pareto-optimal solutions of a multi-objective optimization problem con-
sists of all decision vectors for which the corresponding objective vectors cannot be improved in a given
dimension without worsening another (Chankong Haimes, 1983). When a minimization problem and
two decision vectors X and Y are considered, the concept of Pareto optimality can be defined as follows: X
is said to dominate Y (also written as X 1 Y) iff:
fiðXÞ 6 fiðYÞ for all i 2 f1; 2; . . . ; mg and
fiðXÞ fiðYÞ for at least one i 2 f1; 2; . . . ; mg
All decision vectors, which are not dominated by another decision vector of a given set, are called non-dom-
inated as regards that set. There are various solution approaches for solving the multi-objective problem.
Among the most widely used techniques are sequential optimization, e-constraint method, weighting method,
goal programming, goal attainment, distance-based method and direction-based method. Recently, GA has
been successfully applied to obtain Pareto-optimal solutions for multi-objective optimization problems (Ceol-
lo, Van Veldhuizen, Lamont, 2002; Deb, 2001; Gen Cheng, 2000). GA deals simultaneously with a set of
possible solutions (population) instead of having to perform a series of separate runs in the case of the tradi-
tional mathematical programming techniques. This property increases its popularity on multi-objective opti-
mization. In this study, we also proposed a new GA approach to obtain Pareto-optimal solutions for SCN
design problem.
3. Proposed genetic algorithm
In this section, representation and genetic operators which were used in GA for multi-objective design of
SCN will be explained.
3.1. Representation
Representation is one of the important issues that affect the performance of GAs. Tree-based representa-
tion is known to be one way for representing network problems. Basically, there are three ways of encoding
tree: (1) edge-based encoding, (2) vertex-based encoding, and (3) edge-and-vertex encoding (Gen Cheng,
2000).
The first application of GAs to transportation/distribution problems was carried out by Michalewicz, Vign-
aux, and Hobbs (1991). They used matrix-based representation which belongs to edge-based encoding to rep-
resent transportation tree. When |K| and |J| are the number of sources and depots, respectively, the dimension
of matrix is |K|Æ|J|. Although it is a direct representation of the transportation tree, it needs not only excessive
memory on the computer environment, but also special genetic operators to obtain feasible solutions. Another
representation for transportation tree is Pru¨fer number. It belongs to vertex-based encoding and needs only
|K| + |J|-2 digits to represent a transportation tree with |K| sources and |J| depots. Although Pru¨fer number,
which was actually developed to encode of spanning trees, had been successfully applied to transportation
problem by Gen and Cheng (2000), it needs some repair mechanisms to obtain feasible solutions after classical
genetic operators.
In this study, to escape from these repair mechanisms in the search process of GA, we used priority-
based encoding developed by Gen and Cheng (2000). They had successfully applied this encoding to the
shortest path problem and the project scheduling problem. The first application of this encoding struc-
ture to a single product transportation problem was carried out by Gen, Altiparmak, and Lin (2006),
and its extension to design of multi-product, multi-stage SCN had been made by Altiparmak, Gen,
and Lin (2005). As it is known, a gene in a chromosome is characterized by two factors: locus, the posi-
tion of the gene within the structure of chromosome, and allele, the value the gene takes. In priority-
based encoding, the position of a gene is used to represent a node (source/depot in transportation net-
work), and the value is used to represent the priority of corresponding node for constructing a tree
among candidates.

For a transportation problem, a chromosome consists of priorities of sources and depots to obtain trans-
portation tree and its length is equal to total number of sources (|K|) and depots (|J|), i.e. |K| + |J|. The trans-
portation tree corresponding with a given chromosome is generated by sequential arc appending between
sources and depots. At each step, only one arc is added to tree selecting a source (depot) with the highest pri-
ority and connecting it to a depot (source) considering minimum cost. Fig. 3 represents a transportation tree
with 3 sources and 4 depots, its cost matrix and priority based encoding. The decoding algorithm of the pri-
ority-based encoding is given in Fig. 4. Table 1 gives trace table of the decoding procedure to obtain transpor-
tation tree in Fig. 3.
In SCN design problem, a chromosome consists of three segments. Each of the segments is used to obtain a
transportation tree of a stage on the supply chain, i.e. the rth segment of a chromosome matches the rth stage
on the SCN. We utilized two different encoding methods to design SCN. The priority-based encoding had
been used on the first and second stages. Since each customer for the product has to be assigned only one
DC on the last stage of our problem, integer encoding was used to define this situation. The length of the last
segment on a chromosome equals to number of customers on SCN. The position of a gene on the segment
represents a customer, and its value also represents the DC that corresponding customer will be assigned.
At the same time, gene values show that which DCs will be opened. The chromosome of SCN is decoded
on the backward direction. Firstly, transportation tree between opened DCs and customers is obtained with
decoding of the last segment of chromosome. In the second step, firstly, a decision about which plants will be
opened is given. Plants considering their priorities are opened consecutively until their number reaches to max-
imum number of plants to be opened or their total capacity is greater than or equal to total demand. After
that, second segment is decoded and transportation tree between opened DCs and opened plants is obtained.
Additionally, the amount of product which will be produced in opened plants and total amount of require-
ment for raw material on each plant are also determined in this stage. Lastly, transportation tree between sup-
pliers and opened plants is obtained with decoding of the first segment of chromosome. It is worthy note that
decisions about which DCs and plants will be opened are given during the decoding of the third and second
segments of the chromosome. If the number of opened DCs or plants is greater than their upper limit or their
capacities are not enough to meet customer demands, corresponding segments are repaired to obtain feasible
transportation tree for each stage of the SCN. Fig. 5 gives an illustration of a feasible chromosome for the
problem. In this example, we considered a SCN that has 3 suppliers, 3 plants, 4 DCs and 4 customers and
the upper limits of opened plants and DCs were taken as 3. It is important to note that since there is an unbal-
anced transportation problem on each stage of the SCN (i.e. total capacity of sources is greater than total
demands of depots for each stage) it is balanced by introducing a dummy depot on each segment of the chro-
mosome. Overall decoding procedure for priority-based encoding on SCN, decoding procedures for 3rd, 2nd,
and 1st stages, respectively, and repair algorithm, which is used when total capacity of opened DCs or plants is
not enough to meet customer demands, are given in Appendix A.
1
2
3
1
2
3
100
100
150
50
150
Source
Depot
4
50
50
100
50 50
50
4321
1243
5426
2561
3
2
1
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
5162473
4321321node ID l
100
50
priority v (l) :
Fig. 3. A sample of transportation tree and its encoding.

3.2. Evaluation
An important issue in multi-objective optimization is how to determine the fitness value of the chromosome
for survival. The fitness value of each individual reflects how good it is based upon its achievement of objec-
tives. In literature, there are different techniques to define fitness function (Gen Cheng, 2000). One of them,
also simplest approach, is weight-sum technique. Given m objective functions, fitness function is obtained by
combining the objective functions
evalðf Þ ¼
Xm
i¼1
wifi; ð19Þ
where wi is constant representing weight for fi, and
Pm
i¼1wi ¼ 1.
Fig. 4. Decoding algorithm for the priority-based encoding.
Table 1
Trace table of decoding procedure
Iteration v(k + j) a b k j gkj
0 [3 7 4 | 2 6 1 5] (100, 100, 150) (50, 150, 100, 50) 2 2 100
1 [3 0 4 | 2 6 1 5] (100, 0, 150) (50, 50, 100, 50) 3 2 50
2 [3 0 4 | 2 0 1 5] (100, 0, 100) (50, 0, 100, 50) 3 4 50
3 [3 0 4 | 2 0 1 0] (100, 0, 50) (50, 0, 100, 0) 3 3 50
4 [3 0 0 | 2 0 1 0] (100, 0, 0) (50, 0, 50, 0) 1 1 50
5 [3 0 0 | 0 0 1 0] (50, 0, 0) (0, 0, 50, 0) 1 3 50
6 [0 0 0 | 0 0 0 0] (0, 0, 0) (0, 0, 0, 0)

To determine the weight values, we adopted two approaches proposed by Murata, Ishibuchi, and Tanaka
(1996) and Zhou and Gen (1999). Approach 1 is based on random weight approach in which weights are ran-
domly determined for each step of evolutionary process (Murata et al., 1996). This approach explores the
entire solution space in order to avoid local optima and thus gives a uniform chance to search all possible
Pareto solutions along the Pareto frontier. In Approach 2, weights are determined based on the ideal point
generated in each evolutionary process (Zhou Gen, 1999). Fig. 6 illustrates these two strategies in the
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
461
253
134
3
2
1
321
t
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
6153
5364
3425
3
2
1
4321
a
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
2342
5634
1526
4253
4
3
2
1
4321
c
2
3
4
1Suppliers
Plants DCs
Customers
1
250
2
200
1
200
2
150 (150)
1
150 (150)
2
100
3
200 (150)
50
100
50
100
5 50 (Dummy)
4
1003
200 (150)
400 (Dummy)
50 (Dummy)5
43
250
50
100
50
100
50
150
150
50
150
100
200
50
200
54321543213214321321
33311172386541562473
v1(l) : 1st
stage v2(l) : 2nd
stage v3(l) : 3rd
stage
Fig. 5. An illustration of chromosome, transportation trees and transportation costs for each stage on SCN.
w2
w1
ideal point
))(),(( 1
2
1
min,1 xfxf
)(1 xf
)(2 xf
))(),(( 2
min,2
2
min,1 xfxf
Pareto Frontier
)(1 xf
)(2 xf
(1) Zhou and Gen’s strategy (2) Murata’s et al. Strategy
Fig. 6. Illustration of the weight strategies.

objective space. When weight-sum approach is used, firstly, objectives have to be normalized since they gen-
erally have different units. Normalization is carried out for each objective as follows:
f 0
i ¼
fi À fi;min
fi;max À fi;min
i ¼ 1; 2; . . . ; m; ð20Þ
where fi,min and fi,max are the minimum and maximum value of ith objective on the current generation,
respectively.
Approach 1. The weights in this approach are specified with Eq. (21) in each generation. After weights are
determined, the fitness value of each individual on a population is calculated using Eq. (19).
randomi $ Uð0; 1Þ
wi ¼ randomi=ðrandom1 þ Á Á Á þ randommÞ i ¼ 1; 2; . . . ; m: ð21Þ
Approach 2. Since the idea in this approach is to obtain Pareto-optimal solutions using ideal point generated
in each evolutionary process, the weight of each objective for an individual in current generation is determined
using Eq. (22).
w1 ¼
w0
1
w0
1 þ w0
2
; w2 ¼
w0
2
w0
1 þ w0
2
; ð22Þ
where
w0
1 ¼ f 0
1ðxÞ À f 0
1;minðxÞ; w0
2 ¼ f 0
2ðxÞ À f 0
2;minðxÞ
f 0
1;min and f 0
2;min are the minimum of f 0
1ðxÞ and f 0
2ðxÞ in the current population, i.e. they are ideal point in the
objective space whose value has been normalized.
In the rest of the paper, the proposed GA with Approaches 1 and 2 will be called as GA_A1 and GA_A2,
respectively.
3.3. Genetic operators
3.3.1. Crossover
The crossover is done to explore new solution space and crossover operator corresponds to exchanging
parts of strings between selected parents. We employed a segment-based crossover operator which was based
on uniform crossover. In this operator, each segment of offspring is randomly selected with equal chance
among the corresponding segments of parents. As it is seen in Fig. 7, crossover operator utilizes from a binary
mask. Its length is equal to number of stage in SCN. While ‘‘0’’ means that the first parent will transfer its
genetic materials to the offspring, ‘‘1’’ means that the offspring will take genetic materials from the second par-
ent for the corresponding segment. This crossover operator tends to preserve good gene segments of both
parents.
33311172386541562473
54321543213214321321
32231246753182376541
54321543213214321321
010
33311146753181562473
54321543213214321321
Parent 2
Binary Mask
Child
Parent 1
Fig. 7. An illustration of crossover operator.

3.3.2. Mutation
Similar to crossover, mutation is used to prevent the premature convergence and explore new solution
space. However, unlike crossover, mutation is usually done by modifying gene within a chromosome. As in
the crossover operator, we utilized from segment-based mutation as a mutation operator. In this operator,
firstly, a decision about which segments will be mutated is given with probability of 0.5 (i.e. using a binary
mask), and then selected segments are mutated. Since the chromosome consists of two different encoding
structure, mutation on the structures is also different. Swap operator is used for the first two segments in where
priority-based encoding is used. This operator selects two genes from the corresponding segment and exchang-
es their places. A conventional mutation operator is used for the last segment of the chromosome. In this oper-
ator, the value of randomly selected gene is replaced with new one which is selected between 1 and number of
DCs except to its current value. Fig. 8 gives an illustration of segment-based mutation operator. As it is seen
from figure, the first and third segments of the chromosomes are mutated.
3.4. Selection mechanism
In the proposed GA, initial population is randomly generated and Pareto-optimal set is created by non-
dominated solutions in the initial population. This set is updated by new individuals obtained with genetic
operators at every generation. As a selection mechanism, we adopted the (l + k) selection strategy. In this
strategy, l and k, respectively, represent the number of parents and offspring, which constitutes the evolving
pool in the current generation and competes for survival. After randomly selected two individuals from Par-
eto-optimal set as elite solutions are placed the population, the rest of population is filled by (l-2) different best
individuals selected from the evolving pool. If there are no (l-2) different individuals available, the vacant pool
of population is filled with randomly generated individuals. Additionally, we utilized from a diversification
strategy to increase the capability of proposed GA for reaching more Pareto-optimal solutions. This strategy
is based on the restart of genetic search. If the set of Pareto-optimal solutions has not been updated in the last
moves (number of generations/5, in our study), the population is reset. While the 10% of the population is
filled by non-dominated solutions which are randomly selected from the set of Pareto-optimal solutions, ran-
domly generated solutions are placed to the rest of population. If there are no enough non-dominated solu-
tions in the set of Pareto-optimal set to fill the 10% of the population, all non-dominated solutions are used in
the new population.
4. Performance evaluation of the algorithm
The proposed GA is tested with the actual data obtained from a company which is one of the producers of
plastic products in Turkey. In this section, after giving brief information about the company, computational
results, which are carried out into two stages, will be presented. While the effects of the weight-sum approaches
on the performance of GA are investigated in the first stage, the performances of GA and SA to obtain Pareto-
optimal solutions are comparatively examined in the second stage.
Based on the market research, the company is planning to produce plastic profile which is used in buildings
(vinyl sidings, doors, windows, fences, etc.), pipe lines and consumer materials. The market research shows
that the company can capture a portion of the national market. PVC is the main raw material of the product
33311146752181562473
54321543213214321321
Parent
101
33321146752181567423
54321543213214321321
Binary Mask
Child
Fig. 8. An illustration of mutation operator.

to be produced. In Turkey, this material is supplied from Petkim Co. in Izmir. But Petkim cannot afford to
supply all domestic demand. Thus, the company has to import this material from foreign suppliers in USA,
Belgium, France and Japan. The company intends to establish new plants. There are three potential locations
for the plants. These locations were determined depend on the some specific considerations. The first location
has been considered as Izmir, since the national supplier Petkim had been settled in there. The second is Istan-
bul, because customs and duties are paid, and vessels are entered in customhouses for all imported goods. The
last is Konya in where all other facilities of the company had been located. The company is planning to open
at most six DCs. Locations of DCs had been determined according to demand densities of 63 customer zones
to be served and access time from DCs to customer zones. The locations of DCs are Konya, Istanbul, Izmir,
Ankara, Trabzon, and Adana. The company intends to establish supply chain network that satisfying the
company objectives for the product. The company objectives, as given in mathematical model, are the mini-
mization of overall supply chain cost, maximization of customer services, i.e. the percentage of customer
demand that can be delivered within the stipulated access time s and the maximization of capacity utilization
balance for DCs (i.e. equity on utilization ratios). Table 2 gives information about suppliers’ capacities, and
capacity and fixed costs for plants and DCs. As it is seen from Table 2, fixed costs of plants are different from
each other, although their capacities are equal. Fixed cost of plants consists of expenditures such as hiring
costs of buildings and facilities; amortization of machines and tools; salaries of managers and guardians;
and insurance premiums. Although amortizations, fixed man-power and insurance cost are approximately
equal in Turkey, land and building costs depend on the developing and industrialization level of cities. Thus,
differences between fixed costs of plants come from this fact.
The company is planning to meet customer demands from DCs within half of day (i.e. 12 h). The scatter
diagram of the annual customer demand versus access time from the closest DCs is plotted to obtain infor-
mation about how large the customer demands are, and how far away they are located from DCs. When
Fig. 9 is examined, it is seen that the 93.4% of the customers have demands smaller than 20,000 packages
per year. Also, when the capacities of DCs are not taken in the consideration, it is possible to reach the
98.3% of the customers within 12 h.
Unit costs between suppliers and plants including purchasing and transportation costs change between
$707 and $775 per ton. Since the production costs do not exhibit any change for potential plant locations, unit
Table 2
Capacities and fixed costs for suppliers, plants, and DCs
Suppliers Capacity
(ton/year)
Plants Capacity
(package/year)
Fixed cost
(USD/year)
DCs Capacity
(package/year)
Fixed cost
(USD/year)
USA 10,000 Konya 640,000 440,000 Konya 200,000 70,000
Belgium 10,000 Istanbul 640,000 1,100,000 Istanbul 160,000 60,000
France 10,000 Izmir 640,000 720,000 Izmir 80,000 40,000
Japan 10,000 Ankara 120,000 50,000
Petkim 7200 Trabzon 80,000 40,000
Adana 120,000 50,000
0
20000
40000
60000
80000
100000
120000
0 5 10 15 20
Access time (hours)
customerdemand(package/year)
Fig. 9. Access time-demand distribution.

costs include only transportation costs between plants and DCs, and DCs and customers. The unit costs for
stage 2 and stage 3 of the SCN take a value between $0.18 and $3.56, $0.18, and $7.49 per package,
respectively.
4.1. Effects of weight-sum approaches on the performance of GA
In order to evaluate the performances of the GA_A1 and GA_A2 on SCN design problems with two-ob-
jective and three-objective, we considered three problems generated from original problem. They differ from
each other only according to selected objectives. While the first two problems include two objectives, the last
problem has three objectives, i.e. it is an original problem. The problems and their objective functions are list-
ed below:
Problem 1: minf1 and maxf2
Problem 2: minf1 and minf3
Problem 3: minf1, maxf2 and min f3
The proposed algorithm with two different evaluation approaches, GA_A1 and GA_A2, were coded with
C++ programming language and run on Pentium 4, 2.8 GHz clock pulse with 512 MB memory. GA_A1 and
GA_A2 run 10 times for each problem considering following parameters: population size = 400; crossover
rate = 0.5, mutation rate = 0.7, number of generation = 500. These parameters had been determined after pre-
liminary experiments. To evaluate the GA_A1 and GA_A2, we used two performance measures, which were
obtained over 10 runs. These are: (1) average number of Pareto-optimal solutions, and (2) average ratio of
Pareto-optimal solutions. The second performance measure was calculated following manner.
Let P1 and P2 be the sets of Pareto-optimal solutions obtained from one run of GA_A1, and GA_A2,
respectively, and P be the union of the sets of Pareto-optimal solutions (i.e., P = P1 [ P2) so that it includes
only non-dominated solutions. The ratio of Pareto-optimal solutions in Pi that are not dominated by any
other solutions in P is calculated using Eq. (23):
RPOSðPiÞ ¼
jPi À fX 2 Pij9Y 2 P : Y 0 Xgj
jPij
; ð23Þ
where Y 0 X means that the solution X is dominated by the solution Y. In (23), dominated solutions X by the
solutions Y in P are removed from the solution set Pi. The higher the ratio RPOS(Pi) is, the better the solution
set Pi is.
Experimental results are summarized in Table 3. As it is seen from table, while the average numbers of Par-
eto-optimal solutions are approximately equal on GA_A1 and GA_A2, GA_A1 outperforms the GA_A2 in
terms of average ratio of Pareto-optimal solutions for all problems. The average ratio of Pareto-optimal solu-
tions on GA_A1 changes between 52% and 78%. This ratio is between 51% and %70 on GA_A2. This result
suggests that GA_A1 tends to find higher quality solutions than GA_A2. It is expected result. Because the
proposed GA with Approach 1 randomly searches as many Pareto-optimal solutions as possible in the Pareto
frontier, while other (i.e. GA with Approach 2) only focuses on some areas on the Pareto frontier. Figs. 10–12
also support this result. These figures give the examples of Pareto-optimal solutions obtained by GA_A1 and
GA_A2 on a single run for each problem. As it is seen from these figures, most of the solutions generated by
Table 3
Comparison of GA_A1 and GA_A2
Average number of Pareto-optimal
solutions
Average ratio of Pareto-optimal
solutions
GA_A1 GA_A2 GA_A1 GA_A2
Problem 1 2.3 2.5 0.77 0.56
Problem 2 12.6 14.7 0.52 0.51
Problem 3 32.3 33.3 0.78 0.70

GA_A2 are dominated by the solutions obtained with GA_A1. Additionally, we investigated the effect of
diversification mechanism on the quality of Pareto-optimal solutions obtained by GA_A1 and GA_A2.
For this purpose, GA_A1 and GA_A2 without diversification mechanism were run 10 times for the third
problem. Table 4 gives the average number of Pareto-optimal solutions and average ratio of Pareto-optimal
solutions for GA_A1 and GA_A2 with diversification and without diversification. From Table 4, we can see
that while average numbers of Pareto-optimal solutions obtained by GA_A1 with and without diversification
are approximately equal, GA_A2 without diversification generates more Pareto-optimal solutions than its ver-
sion with diversification. Meanwhile, the 81% and 87% of Pareto-optimal solutions obtained by GA_A1 and
GA_A2 with diversification, respectively, are not dominated by GA_A1 and GA_A2 without diversification.
This result is an indicator that diversification mechanism increases the quality of Pareto-optimal solutions.
To give information about which plants and DCs are opened in Pareto-optimal solutions of original prob-
lem, we selected five solutions among the Pareto-optimal solutions because of the space limitation. Table 5
gives the objective function values, and locations of plants and DCs on the selected solutions. We could
Fig. 12. Pareto-optimal solutions of GA_A1 and GA_A2 for Problem 3.
78
80
82
84
86
88
90
92
19,15 19,2 19,25 19,3 19,35 19,4 19,45
Total cost (106
USD)Servicequality(%)
GA_A1
GA_A2
Fig. 10. Pareto-optimal solutions of GA_A1 and GA_A2 for Problem1.
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
Capacityutilizationratio
19,28 19,3 19,32 19,34 19,36 19,38
Total cost (106
USD)
GA_A1
GA_A2
Fig. 11. Pareto-optimal solutions of GA_A1 and GA_A2 for Problem 2.

not give the allocation of 63 customers to DCs, since it will consume more space. As it is seen from Table 5,
while the cost of SCN in the examples changes between $19.29 · 106
and $19.41 · 106
, service quality and
equity on utilization ratios take a value between 0.67 and 0.92, 0.001 and 0.113, respectively. There is a
trade-off between solutions. When the cost of SCN decreases, it is observed that there is a reduction on the
service quality, and equity on utilization ratios of the SCN. It is also important to note that when all Par-
eto-optimal solutions are examined, it is seen that one plant is opened on each solution, and its location in
the 99% of solutions is Konya (43%) or Izmir (56%). Also, we observe from Pareto-optimal solutions that
the number of opened DCs changes between three and five. While four DCs are opened in the 90% of solu-
tions, three and five DCs are opened in the 8% and 2% of solutions, respectively. Another important issue on
the Pareto-optimal solutions is the locations of DCs. While Konya, Istanbul, Izmir, and Ankara are selected in
the 60% of solutions, Konya, Istanbul, Ankara, and Adana are selected in the 33% of solutions.
4.2. Comparison of GA_A1 and MO_SA
To investigate the effectiveness of the GA_A1 for SCN design problem, the SA approach proposed by
Ulungu et al. (1999), called as MO_SA, was employed for the problem. Our purpose on selecting MO_SA
was that it was also based on weight-sum approach. This property provides us making a comparison of the
approaches on the same basis. We considered five problems on the comparison. While the first problem
was original problem, others were generated from the original problem by increasing the customer demands,
and the number of potential plants and DCs. The locations of additional plants were selected among the
developing cities in Turkey, and the locations of additional DCs were determined considering regional demand
Table 4
Comparison of GA_A1 and GA_A2 with diversification and without diversification for the problem 3
Average number of Pareto-optimal solutions Average ratio of Pareto-optimal solutions
GA_A1 with diversification 32.3 0.81
GA_A1 without diversification 33 0.44
GA_A2 with diversification 33.3 0.87
GA_A2 without diversification 37.7 0.54
Table 5
Examples for Pareto-optimal solutions (f1 · 106
)
Solutions (f1,f2,f3) Locations of opened plants Locations of opened DCs
1 (19.37, 0.92, 0.092) Konya Konya, _Istanbul, Trabzon, Ankara
2 (19.34, 0.76, 0.013) _Izmir Konya, _Istanbul, _Izmir, Ankara
3 (19.41, 0.91, 0.001) Konya Konya, _Istanbul, Trabzon, Ankara
4 (19.36, 0.81, 0.096) Konya Konya, _Istanbul, Ankara, Adana
5 (19.29, 0.67, 0.113) _Izmir Konya, _Istanbul, _Izmir, Ankara
Table 6
The size of new problems
Number of Plants,
|K|
Number of DCs,
|J|
Number of maximum
opened plants, P
Number of maximum
opened DCs, W
Problem 3 3 6 3 6
Problem 4 3 8 2 5
Problem 5 5 10 3 7
Problem 6 6 15 4 10
Problem 7 8 20 5 15

densities. Table 6 gives information about the number of potential plants and DCs, and number of maximum
opened plants and DCs in the new problems.
In MO_SA, a set of weight vectors are randomly generated, and the Pareto-optimal solutions are obtained
by running SA with predefined number of iterations for each random weight vector. SA starts with the ran-
domly generated solution, and a new solution in the each iteration is obtained by moving strategy. If the new
solution improves the current solution according to weight vectors or enters the set of Pareto-optimal solu-
tions, it is accepted as current solution; otherwise it is accepted with the probability of exp(ÀDs/T). We defined
Ds as [(eval(f0
) À eval(f))/eval(f)]*100 in where eval(f0
) and eval(f) were objective function values according to
weight vector for the new solution and current solution, respectively. Ds is a relative percent deviation of qual-
ity of the new solution from the current solution. When the MO_SA was implemented for SCN design prob-
lem, the encoding structure in GA was used to represent a solution, and the mutation operator in GA was
chosen as a moving strategy. The initial temperature was taken as 975 in which an inferior solution (inferior
by 50% relative to current solution) was accepted with a probability of 0.95. To make comparison of GA_A1
and MO_SA on the same basis, the number of solutions searched (NSS) was used as a stopping criterion, and
it depended on the problem size. In GA_A1, the population size and the number of generations were taken as
400 and NSS/400, respectively. In MO_SA, SA with the length of 100 iterations run for each of 400 different
randomly generated weight vectors. The reduction rate of temperature and the number of evaluated solutions
in each iteration of SA were 0.90 and NSS/(100*400), respectively. It is important to note that the MO_SA
was also coded with C++ programming language. GA_A1 and MO_SA run 10 times, and two performance
measures, mentioned in Section 4.1, were used to compare them. Table 7 summarizes experimental results. As
it is seen from table, MO_SA is inferior to the GA_A1 in terms of the average number of Pareto-optimal solu-
tions for all problems except to Problem 3. The comparison of GA_A1 and MO_SA with respect to average
ratio of Pareto-optimal solutions shows that while the average ratio of Pareto-optimal solutions on GA_A1 is
between 56% and 68%, it changes between 47% and 63% on MO_SA. These results suggest that GA_A1 tends
to find more solutions with higher quality than MO_SA.
Table 8 gives NSS and computation times on GA_A1 and MO_SA for each problem size. As it is seen from
the table that the computation times on GA_A1 and MO_SA increase based on problem size. Additionally,
Table 7
Comparison of GA_A1 and MO_SA
Average number of Pareto-optimal
solutions
Average ratio of Pareto-optimal
solutions
GA_A1 MO_SA GA_A1 MO_SA
Problem 3 32.3 36.7 0.58 0.54
Problem 4 59.6 45.1 0.56 0.47
Problem 5 55 55.2 0.65 0.55
Problem 6 42.9 38 0.64 0.63
Problem 7 53.9 48.3 0.68 0.62
Table 8
Number of solutions searched and CPU times for GA_A1 and MO_SA
Number of solutions searched (NSS) CPU times (min)
GA_A1 MO_SA
Problem 3 2 · 105
1.136 0.447
Problem 4 3 · 105
2.338 1.040
Problem 5 4 · 105
3.796 1.686
Problem 6 5 · 105
10.689 5.875
Problem 7 6 · 105
14.229 7.535

the computation times on GA_A1 are approximately two times higher than MO_SA on each problem.
Although the same stopping criterion had been used on the algorithms, this difference came from the fact that
the GA had some additional mechanisms such as selection mechanism and crossover, which were time
consuming.
5. Conclusion
In this paper, we presented mixed-integer non-linear programming model for multi-objective optimization
of SCN and a genetic algorithm (GA) approach to solve the problem which was met on a producer of the
plastic products in Turkey. Three objectives were considered: (1) minimization of total cost comprised of fixed
costs of plants and distribution centers (DCs), inbound and outbound distribution costs, (2) maximization of
customer services that can be rendered to customers in terms of acceptable delivery time (coverage), and (3)
maximization of capacity utilization balance for DCs (i.e. equity on utilization ratios). To deal with multi-ob-
jective and enable the decision maker to evaluate a greater number of alternative solutions, two different
weight approaches were implemented in the proposed GA. In order to evaluate the performances of the
GA with two different weight approach, called as GA_A1 and GA_A2, we considered three problems gener-
ated from original problem, which were different from each other according to selected objectives. Experimen-
tal results showed that while GA_A1 was capable to generate more Pareto-optimal solutions than GA_A2,
diversification mechanism was very effective on the quality of Pareto-optimal solutions. In addition,
GA_A1 was compared with the MO_SA using five problems which were generated from original problem.
This comparison showed that GA_A1 outperformed MO_SA according to not only average number of Par-
eto-optimal solutions but also quality of Pareto-optimal solutions. In future, new solution methodology based
on tabu search can be developed to obtain Pareto-optimal solutions for the multi-objective SCN design prob-
lem, and the effectiveness of GA_A1 according to this solution methodology can be investigated. Additionally,
uncertainty of costs and demands can be considered in the model and new solution methodologies including
uncertainty can be developed.
Acknowledgments
This research had been supported by The Matsumae International Foundation in Japan, while Dr. Fulya
Altiparmak was a visiting researcher at Graduate School of Information, Production and Systems, Waseda
University. Also this work was partly supported by Waseda University Grant for Special Research Projects
2004 and the Ministry of Education, Science and Culture, the Japanese Government: Grant-in-Aid for Scien-
tific Research (No. 17510138).
Appendix A. Decoding procedure of the chromosome for SCN
See Figs. 13–17.
procedure 2: decoding of chromosome for SCN
step 1 : find [qij] by procedure 3 ( 3rd
stage decoding);
step 2 : for j = 1 to |J|
if bj′= 0 then v2(j) ← 0;
step 3 : find [fjk] by procedure 4 ( 2nd
stage decoding);
step 4 : for k = 1 to |K|
if bk′= 0 then v1(j) ← 0;
step 5 : find [bsk] by procedure 5 ( 1st
stage decoding);
step 6 : calculate the value of objective functions (z1, z2, and z3) and stop.
Fig. 13. Decoding procedure for priority based encoding.

procedure 3 : 3rd
stage decoding
input : J : set of DCs, I : set of customers,
Wj : capacity of DC j, ∀j ∈ J,
di : demand for product of customer i, ∀i ∈ I
v3(i) : chromosome, ∀ i ∈ I
output : oD : set of opened DCs, cD : set of closed DCs
qij : the amount of product shipped from DC j to customer i
Wj′ : total customer demand for product on DC j
step 0 : zj ← 0, ∀j ∈ J, yij ← 0, ∀i ∈ I, ∀j ∈ J, oD←∅, cD ← J
qij ← 0, ∀i ∈ I, ∀j ∈ J; Wj′ ← 0, ∀j ∈ J
step 1 : determine the opened and closed DCs and find DC j serves customers i
{ } { })(,)(,1,1
to1for
33)()( 33
ivccivooyz
Ii
DDDDiiviv −=+=←←
=
step 2 : Calculate total capacity of opened DCs and total customer demand, and
number of opened DCs
∑=
j
jj zWcaptot _ , ∑=
i
iddemtot _ , ∑=
j
jD zo
step 3 : if WoD ≤ and tot_cap ≥ tot_dem then goto Step 5; else goto step 4.
step 4 : DOK ← oD, DCK ← cD, DOP← oP, DP← P,
d_tot_cap ← tot_cap, d_tot_dem ← tot_dem
call procedure 6,
obtain new v3(i) with reallocating the customers considering opened and closed DCs
and goto step 0.
step 5 : to1for Ii =
0,,, )()()()(*)(*)()( 3333333
←+′←′−←← iiiviviviiviviviiiv dqWWqWWdq
step 6 : if 0* jW for any j ∈ J , then select a customer from DC j and reallocate it to another
opened DC that satisfy 0* ≥jW for all j ∈ J , obtain new v3(i), and recalculate the
ijjj qWW ,,* ′
step 7 : output ijj qW ,' and return
Fig. 14. Decoding procedure for 3rd segment of the chromosome.
procedure 4 : 2nd
stage decoding
input : K : set of plants, J : set of DCs, P : maximum number of plants
W′j : total customer demand for product on DC j, ∀j ∈ J,
pjk : shipping cost of one unit of product from plant k to DC j,∀k∈K,∀j∈J,
v2((k+j)) : chromosome, ∀k ∈ K, ∀j ∈ J,
output : fkj : the amount of product shipped from plant k to DC j
D′k : total customer demand for product on plant k, ∀k ∈ K,
p(k) : priority of plant k for product, ∀k ∈ K,
NP : number of opened plants, oP : set of opened plants,
tot_cap : total capacity of opened plants
tot_dem : total demand of DCs
step 0 : qkl ← 0, ∀k ∈ K, ∀j ∈ J, Dk′ ← 0, ∀k ∈ K,
∑∑=
i j
ijqdemtot _
Fig. 15. Decoding procedure for 2nd segment of the chromosome.

procedure 5 : 1st
stage decoding
input : S : set of suppliers,
K : set of plants,
D′k : total customer demand for product on plant k, ∀k ∈ K,
tsk : unit transportation and purchasing cost of raw material from supplier s to
plant k ∀s ∈ S, ∀k ∈ K
u : utilization rate of raw material per unit of product ,
ak : the amount of raw material to produce the product on plant k
v1(s+k) : chromosome, ∀s ∈ S, ∀k ∈ K
output : bsk : the amount of raw material shipped from supplier s to plant k, ∀s ∈ S, ∀k ∈ K
step 0 : bsk ← 0, ∀s ∈ S, ∀k ∈ K
Calculate the amount of raw material to produce the product on plant k
uDa kk
′= , ∀k ∈ K
step 1 : Set the plants as depots and suppliers as sources and call Procedure 1 to obtain bsk
(i.e. transportation tree for the 1st stage of the SCN and return.
Fig. 16. Decoding procedure for 1st segment of the chromosome.
procedure 6: Repair algorithm
input: DOK : set of opened sources; DCK : set of closed sources;
DP : maximum number of sources; DOP : number of opened sources,
d_tot_cap : total capacity of opened sources
d_tot_dem : total requirement of depots
output: DOK : Set of opened sources
step 1. if DOP DP and d_tot_cap ≥ d_tot_dem then goto Step 2
If (DOP DP or DOP ≥ DP) and d_tot_cap d_tot_dem or then goto Step 3
step 2. obtain a set of sources (CS) from the DOK that closing a source in CS will also satisfy the
condition of d_tot_cap ≥ d_tot_dem.
repeat
if CS ≠ ∅, then close a source which is randomly selected from CS;
else close a randomly selected source from the set of opened sources.
until DOP ≤ DP.
Recalculate the tot_cap considering closed sources and return.
step 3. repeat
open a source which is randomly selected from DCK
until d_tot_cap ≥ d_tot_dem.
Recalculate the DOP considering opened sources and return.
Fig. 17. Repair algorithm.
step 1 : obtain p(k) from v2((k+j)), k ∈ K,
Kkkpkpd ∈∀← ),()(
step 2 : open the plants having high priorities until tot_cap ≥ tot_dem or NP ≥ P
{ }
{ }k,0)_(,1
,__,1
),(maxarg_
__
+←←+=
+=←
∈←
PPd
khpkhp
d
ookhppNPNP
Dcaptotcaptotp
Kkkpkhp
step 3 : set the priorities of the closed plants to 0 and keep the current priorities on
v2((k+j)) for the opened plants.
if oP ≤ P and tot_cap ≥ tot_dem then goto step 5; else goto step 4.
step 4 : DOK ← oP, DCK ← K - oP, DOP← NP, DP← P,
d_tot_cap ← tot_cap, d_tot_dem ← tot_dem
call procedure 6, and goto Step 3.
step 5 : set the DCs as depots and plants as sources, call Procedure 1 to obtain qkl
(i.e. transportation tree for the 2nd stage of the SCN), calculate Dk′ considering qkl and return.
Fig 15. (continued)

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