A Fuzzy Multi-Objective Mixed Integer Programming Model for Multi Echelon
Supply Chain Network Design and Optimization
Turan PAKSOY*, Nimet YAPICI PEHLİVAN**
Eren ÖZCEYLAN***
*
Selçuk University, Department of Industrial Engineering, Campus, 42031, Konya, Turkey
(Tel: +90 332 223 20 40; e-mail: tpaksoy@yahoo.com)
**
Selçuk University, Department of Statistics, Campus, 42031, Konya, Turkey
(Tel: +90 332 223 13 48; e-mail: nimet@selcuk.edu.tr)
***
Selçuk University, Department of Industrial Engineering, Campus, 42031, Konya, Turkey
(Tel: +90 332 223 20 75; e-mail: eozceylan@selcuk.edu.tr)
Abstract: This paper applies a fuzzy multi-objective nonlinear programming model with piecewise
linear membership function to design a multi echelon supply chain network (SCN) by considering total
transportation costs and capacities of all echelons with fuzzy objectives. The model attempts to aim three
different fuzzy objective functions. The first one is minimizing the total transportation costs between all
echelons (suppliers, plants, distribution centers (DCs) and customers). Second one is minimizing of
holding and ordering costs in DCs and the last objective function is minimizing the unnecessary and
unused capacity of plants and DCs via decreasing variance of transported amounts between echelons. A
numerical example is solved to demonstrate the feasibility of applying the proposed model to given
problem, and also its advantages are discussed.
Keywords: Fuzzy modeling, optimization, network design, supply chain, nonlinear programming,
triangular fuzzy numbers.
1. INTRODUCTION
Supply chain management (SCM) has been a hot topic in
the management arena in the recent years. The term
“supply chain” (SC) conjures up images of products, or
supplies, moving from manufacturers to distributors to
retailers to customers, along a chain, in order to fulfill a
customer request (Gong et al. 2008). SCM explicitly
recognizes interdependencies and requires effective
relationship management between chains. The challenge in
global SCM is the development of decision-making
frameworks that accommodate diverse concerns of
multiple entities across the supply chain. Considerable
efforts have been expended in developing decision models
for SC problems (Narasimhan and Mahapatra, 2004).
Enterprises have to satisfy customers with a high service
level during standing high transportation, raw material and
distribution costs. In traditional SCs, purchasing,
production, distribution, planning and other logistics
functions are handled independently by decision makers
although SCs have different objectives. To overcome
global risks in related markets, decision makers are
obliged to fix a mechanism which different objective
functions
(minimizing
transportation/production,
backorder, holding, purchasing costs and maximizing
profit and customer service level etc.) can be integrated
together. Illustration of a SCN includes suppliers, plants,
DCs and customers in Fig. 1 (Syarif et al. 2002).
Fig. 1. Illustration of a SCN (Syarif et al. 2002)
The design of SCNs is a difficult task because of the
intrinsic complexity of the major subsystems of these
networks and the many interactions among these
subsystems, as well as external factors such as the
considerable multi objective functions (Gumus et al. 2009).
In the past, this complexity has forced much of the
research in this area to focus on individual components of
SCN. Recently, however, attention has increasingly been
placed on the performance, design, and analysis of the SC
as a whole.
SCs performance measures are categorized as qualitative
and quantitative. Customer satisfaction, flexibility, and
effective risk management belong to qualitative
performance measures. Quantitative performance measures
are also categorized by: (1) objectives that are based
directly on cost or profit such as cost minimization, sales
maximization, 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
(Altiparmak et al., 2006). However, the SCM design and
planning is usually involving trade-offs among different
goals.
In this study, we developed a mixed integer linear
programming model to design and optimize a supply chain
network via providing multi objective functions mentioned
above together. We considered three objectives for SCM
problem: (1) minimization of total transportation costs
between
suppliers-plants-distribution
centers
and
distribution costs between distribution centers and
customers, (2) minimization of holding and ordering costs
in DCs based EOQ (economic order quantity) and (3)
providing equity of the capacity utilization ratio of plants
and DCs. Much of the decision making in the real world
takes place in an environment in which the goals, the
constraints, and the consequences of the possible actions
are not known precisely. As in the same manner, SCs
operate in a somehow uncertain environment. Uncertainty
may be associated with target values of objectives,
external supply and customer demand etc. Supply chain
distribution network design models developed so far either
ignored uncertainty or consider it approximately through
the use of probability concepts (Selim and Ozkarahan,
2008). Because of the uncertain parameters, considering
this need, fuzzy set theory is applied to the model.
Additionally, to provide the DMs with a decision making,
we propose a novel fuzzy multi objective mixed integer
mathematical model. Through the solution approach, DMs
determine the preferred compromise solution.
The paper is further organized as follows: After the
introduction section, in Section 2, we described the
relevant literature about supply chain modeling and fuzzy
applications. In Section 3, the proposed model which is
fuzzy multi objective mixed integer linear programming
model and the model development are presented. We
tested the novel model with a numerical example and
discussed the results obtained by LINGO 12.0 (industrial
version) package programmer in Section 4. After
presenting the computational experiments, the paper is
finalized with conclusion section.
2. LITERATURE REVIEW
In real-world situations, most enterprises have only paid
attention
on
separately
optimizing
their
production/distribution planning decisions, but using these
decisions prevent possible improvement in decision
effectiveness. Hence, the issues of how to simultaneously
integrate manufacturing and distribution systems in a
supply chain with multi objectives have attracted
considerable interest from both practitioners and
academics (Liang and Cheng, 2009). Literature review
presents a review about supply chain modeling and fuzzy
applications in SC planning, respectively.
Cohen and Lee (1989) present a deterministic, mixed
integer, non-linear programming with economic order
quantity technique to develop a global supply chain plan.
Output of the model provides global resource deployment
policy for the plants, distribution centers and customer
zones. Pyke and Cohen (1993) develop a mathematical
programming model by using stochastic sub-models to
design an integrated supply chain that involves
manufacturers, warehouses and retailers. Due to not
considering multiple products and having only one plant,
one warehouse and one retailer, the model fails to
represent the complicated supply chain networks of the
real world. Ozdamar and Yazgac (1997) develop a
distribution/production
system
that
involves
a
manufacturer center and its warehouses. The paper tries to
minimize total costs such as inventory; transportation costs
etc. under production capacity and inventory equilibrium
constraints. Syarif et al. (2002) propose new algorithm
based genetic algorithm to design a multi stage supply
chain distribution network under capacity constraints for
each echelon. Although experimental results show that the
proposed algorithm can give better heuristic solutions than
traditional genetic algorithm, the comparison with the
performances of other meta-heuristic techniques (tabu
search, simulated annealing etc.) are remained unanswered.
Yan et al. (2003) try to contrive a network which involves
suppliers, manufacturers, distribution centers and
customers via a mixed integer programming under logic
and material requirements constraints. Chen and Lee
(2004) develop a multi-product, multi-stage, and multiperiod scheduling model to deal with multiple
incommensurable goals for a multi-echelon supply chain
network with uncertain market demands and product
prices. The supply chain scheduling model is constructed
as a mixed-integer nonlinear programming problem to
satisfy several conflict objectives, such as fair profit
distribution among all participants, safe inventory levels,
maximum customer service levels, and robustness of
decision to uncertain product demands, therein the
compromised preference levels on product prices from the
sellers and buyers point of view are simultaneously taken
into account. Gen and Syarif (2005) deal with a
production/distribution problem to determine an efficient
integration of production, distribution and inventory
system in order to minimize system wide costs while
satisfying all demand required. The study proposes a new
technique called spanning tree-based genetic algorithm. In
order to improve its efficiency, the proposed method is
hybridized with the fuzzy logic controller concept for autotuning the genetic algorithm parameters. The proposed
method is compared with traditional spanning tree-based
genetic algorithm approach. This comparison shows that
the proposed method gives better results. Lin et al. (2007)
compare flexible supply chains and traditional supply
chains with a hybrid genetic algorithm and mention
advantages of flexible ones. Azaron et al. (2008) develop a
multi-objective stochastic programming approach for
supply chain design under uncertainty. Demands, supplies,
processing, transportation, shortage and capacity
expansion costs are all considered as the uncertain
parameters. Their multi-objective model includes (i) the
minimization of the sum of current investment costs and
the expected future processing, transportation, shortage
and capacity expansion costs, (ii) the minimization of the
variance of the total cost and (iii) the minimization of the
financial risk or the probability of not meeting a certain
budget. You and Grossmann (2008) address the
optimization of supply chain design and planning under
responsive criterion and economic criterion with the
presence of demand uncertainty. By using a probabilistic
model for stock-out, the expected lead time is proposed as
the quantitative measure of supply chain responsiveness.
Farahani and Elahipanah (2008) develop a mixed integer
linear programming model with two objective functions:
minimizing costs, and minimizing the sum of backorders
and surpluses of products in all periods. They also consider
delivery lead times and capacity constraints in a multiperiod, multi-product and multi-channel network. Tuzkaya
and Önüt (2009) develop a model to minimize holding
inventory and penalty cost for suppliers, warehouse and
manufacturers based a holonic approach. The model is
restricted to only two-echelon and does not involve
determining location strategy of the supply chain.
Sourirajan et al. (2009) consider a two-stage supply chain
with a production facility that replenishes a single product
at retailers. The objective is to locate distribution centers in
the network such that the sum of facility location, pipeline
inventory, and safety stock costs is minimized. They use
genetic algorithms to solve the model and compare their
performance to that of a Lagrangian heuristic developed in
earlier work. Xu and Nozick (2009) formulate a two-stage
stochastic program and a solution procedure to optimize
supplier selection to hedge against disruptions. Their
model allow for the effective quantitative exploration of
the trade-off between cost and risks to support improved
decision-making in global supply chain design. Shin et al.
(2009) provide buying firms with a useful sourcing policy
decision tool to help them determine an optimum set of
suppliers when a number of sourcing alternatives exist.
They propose a probabilistic cost model in which
suppliers’ quality performance is measured by
inconformity of the end product measurements and
delivery performance is estimated based on the suppliers’
expected delivery earliness and tardiness.
In most real-world SCM problems, environment
coefficients and model parameters are frequently imprecise
/ fuzzy because some information is incomplete and/or
unavailable over the planning. Conventional LP and
special solution algorithms can not solve all fuzzy SCM
problems. Fuzzy set theory was developed by Zadeh in
1965, since then fuzzy set theory has been applied to the
fields of operations research (linear programming, nonlinear programming, multiple criteria decision making and
so on), management science, artificial intelligence/expert
system, statistics and many other fields. To formulate the
fuzzy/imprecise numbers, membership functions can be
used. Traditional mathematical programming techniques,
obviously, can not solve all fuzzy programming problems.
In practice, input data are usually fuzzy / imprecise
because of incomplete or non-obtainable information and
knowledge. Because of this, precise mathematics is not
sufficient to model a complex system. For solving decision
making problem and fuzzy linear programming problem,
various models are developed. In the last two decades,
multiple objective decision making techniques have been
applied to solve practical problems, such as academic
planning, production and manufacturing planning, location,
logistics, financial planning, portfolio selection, and so on
(Lai and Hwang, 1992).
Hu and Fang (1999) solve the problem of fuzzy
inequalities linear membership function by employing the
concepts of constraint surrogating and maximum entropy.
Vasant et al. (2005) propose a new fuzzy linear
programming based methodology using modified S-curve
membership function. Liang (2006) develops an
interactive fuzzy multi-objective linear programming
method for solving the fuzzy multi objective transportation
problems with piecewise linear membership function.
Peidro et al. (2007) propose a new mathematical
programming model for supply chain planning under
supply, process and demand uncertainty. Chang (2007)
proposes a new idea of how to formulate the binary
piecewise linear membership function. Alves and Climaco
(2007) review interactive methods for multi objective
integer and mixed integer programming. Liang (2008)
develops a fuzzy multi-objective linear programming
model with piecewise linear membership function to solve
integrated
multi-product
and
multi-time
period
production/distribution planning decisions problems with
fuzzy objectives. Liang and Cheng (2009) apply fuzzy sets
to multi-objective manufacturing/distribution planning
decision problems with multi-product and multi-time
period in supply chains by considering time value of
money for each of the operating categories. Peidro et al.
(2009) propose a new mathematical programming model
for supply chain planning under supply, process and
demand uncertainty. The model has been formulated as a
fuzzy mixed integer linear programming model where data
are ill-known and modeled by triangular fuzzy numbers.
Cao et al. (2010) develop stochastic chance constrained
mixed-integer nonlinear programming models to solve the
refinery short-term crude oil scheduling problem.
3. PROBLEM FORMULATION
In this section, a fuzzy multi-objective mixed integer
nonlinear programming model for multi echelon supply
chain network here can be described as follows.
3.1. Problem description, assumptions, and notations
Here, the constituted model represents three echelons,
multi supplier, multi manufacturer, multi DC, and multi
customer problem. Decision maker wishes to design of SC
network for the end product, select suppliers, determine
the manufacturers and DCs and design the distribution
network strategy that will satisfy all capacities and demand
requirement for the product imposed via customers. The
problem is a single-product, multi-stage SCN design
problem. Considering company managers’ objectives, we
formulated the SCN design problem as a fuzzy multiobjective mixed-integer non-linear programming model.
The objectives are minimization of the total cost of supply
chain, minimization holding and ordering costs in DCs,
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 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, for all i = 1,2…I
j is an index for DCs, for all j = 1,2…J
k is an index for plants, for all k = 1,2…K
s is an index for suppliers, for all s = 1,2…S
1,
if DC j is serves customer i
0,
otherwise.
yji
Model parameters;
is the fuzzy capacity of plant k
~
Dk
~
Sups
~
Wj
di
Cji
akj
tsk
Ch
S
is the fuzzy capacity of supplier s for raw material
is fuzzy distribution capacity of DC j
is the demand for the product at customer i
is the unit transportation cost for the product from
DC j to customer i
is the unit transportation cost for the product from
plant k to DC j
is the unit transportation and purchasing cost for the
raw material from supplier s to plant k
is the holding cost per year at DC j
is ordering cost to plant k from each of DCs
Objectives;
f1
is the total cost of SCN. It includes the variable
costs of transportation raw material from suppliers
to manufacturers and the transportation the product
from plants to customers through DCs.
f2
is annual holding and ordering cost of products in
DCs according to the economic order quantity
(EOQ) model.
f3
is the equity of the capacity utilization ratio for
manufacturers and DCs, and it is measured by mean
square error (MSE) of capacity utilization ratios.
The smaller value is, the closer the capacity
utilization ratio for every manufacturer and DC is,
thus ensuring the demand are fairly distributed
among the DCs and manufacturers, and so it
maximizes the capacity utilization balance.
Min f1 
t sk.bsk   akj. f kj   c ji. q ji
s
Fig 2. SCN of the proposed model
Model variables;
bsk
fkj
qji
is the quantity of raw material shipped from
supplier s to plant k
is the quantity of the product shipped from plant k
to DC j
is the quantity of the product shipped from DC j to
customer i
k
k





Min f 2 

j 



j
j
(1)
i
 2.S. f kj 

s. f kj ch . k

k
c
h




2

 2.S. f kj
(2)



k
ch
[ ( f kj / Dk )  ( f kj /  Dk ) ]
2
Min f3 
k
j
k
j
k
k
[ ( q ji /W j )  ( q ji / W i) ]

2
j
i
j
j
i
i
(3)
Subject To;
i
 y ji  1
(4)
j
~
 d i.y ji  W
j
j
(5)
q ji  d i . y ji
i, j
(6)
 f kj   q ji
j
(7)
bsk  Sups
s
(8)
 f kj  bsk
k
(9)
~
 f kj  D
k
k
(10)
y ji  0,1
i, j
(11)
bsk , f kj , q ji  0
i, j,k ,s
(12)
i
k
i
~
k
j
s


w1   t sk. bsk    a kj. f    c ji. q  
kj
ji
 s k

k j
j i
 

 2.S. f kj 
 
k

  s. f
ch .
kj

 
k
c
h
 
w 2  


 j   2.S . f
2
kj

  k

 

ch
 

2 

  [ ( f kj / D k )  (  f kj /  D k ) ] 
k
j
k j
k
w3 



k





[ ( q ji /W j )  ( q ji / W i) ]
2
j
j
The model is composed of three objective functions (1-3).
The first objective function (1) defines minimizing
shipment costs between suppliers, manufacturers, DCs and
customers. The second objective function is minimizing
holding and ordering costs in DCs using economic order
quantity model (2). Equation 3 (third objective) minimizes
equity of the capacity utilization ratio of manufacturers
and DCs.
Constraint (4) represents the unique assignment of a DC to
a customer, (5) is the fuzzy capacity constraint for DCs,
(6) and (7) gives the satisfaction of customer and DCs
demands for the product, (8) gives the fuzzy supplier
capacity constraint, (9) describes the raw material supply
restriction, (10) is fuzzy manufacturer production capacity
constraint. Finally, constraints (11) and (12) are integrality
constraints.
For convenience, in numerical example section, according
to Analytic Hierarchical Process (AHP) methodology,
fuzzy multi objective mixed integer nonlinear
programming problem is converted to a single objective
problem by calculating priorities of objectives as following,
Min g0 ( x)  w1 f1  w2 f 2  w3 f3
i
j
j
i
i
3.2. Model Development
Fuzzy nonlinear programming (FNLP) problem should be
transferred into an equivalent crisp nonlinear programming
(NLP) problem as follows;
~
Min g 0 ( x)
~
( Ax)  bi , i  1,2,..., m
(13)
x0
Then it can be solved by conventional solution techniques
or packages of NLP. Considering the Zimmermann’s
method with fuzzy constraints and fuzzy objective, the
problem is equivalent to;
Max 
 g 0 ( x)  
 i ( x)  
(14)
  [0,1]
x0
In the (14), S-curve membership function is used for fuzzy
nonlinear objective function and non-increasing linear
membership function is used for fuzzy constraints.
The S-curve membership function is a particular case of
logistic function with specific values of B, C and  . The
modified S-curve membership function proposed by
Vasant (2005) is given in (15) and depicted in Fig. 3.
1,
x  xa

0.999 , x  x a

 B
( x )  
, x a  x  xb
x
1  Ce

x  xb
0.001,
0,
x  xb

(15)
 is the degree of membership function. The values of B,
C and  are found as B=1, C=0.001001001 and
 =13.81350956 (Vasant, 2005).
(x)
1
0.999
0.5
0.001
x
0
xa
x0
xb
Fig. 3. The modified S-curve membership function
The non-increasing linear membership function proposed
by Verdegay is given by (16) and depicted in Fig. 4 (Lai
and Hwang, 1992).
1,
( Ax) i  bi

 [( Ax ) i  bi ]
 i ( x)  1 
, bi  ( Ax ) i  bi  p i
pi

0, ( Ax ) i  bi  p i

(16)
where; bi is right hand side of and pi is the tolerance of
the ith constraint.
 i (x)
1
bi
bi  pi
( Ax) i
Fig. 4. The membership functions for the fuzzy constraints
~
( Ax) i  bi
4. COMPUTER EXPERIMENT
In this section we present a numerical example to illustrate
the proposed model mentioned in previous section. The
application of the model is performed for a logical data
which was inspired from related cases in the real world.
The considered supply chain network includes five
suppliers which are located in different places, three
manufacturer plants, three distribution centers and four
customers (Fig. 2). The network is structured to supply
raw materials and transport products from suppliers to endusers is constituted from multi echelon and capacitated
elements of network considering minimizing the total
transportation costs between all echelons (suppliers,
manufacturers, distribution centers (DCs) and customers,
holding and ordering costs in DCs and unnecessary and
unused capacity of plants and DCs via decreasing variance
of transported amounts between echelons. Numerical data
used in example are given below, respectively. Table 1 and
2 gives the priorities of objectives obtained by Expert
Choice 11.5 program to find rate of purposes according to
AHP methodology.
Table 1. Relatives of Objective Functions (AHP)
f1
f2
f3
f1
1
2
3
f2
1/2
1
3/2
f3
1/3
2/3
1
1.83
3.67
5.5
Sum
Table 2. Normalized AHP Matrix
f1
f2
f1
0.545
0.545
f2
0.273
0.273
f3
0.182
0.182
f3
0.545
0.273
0.182
According to Table 2, weight of each objective function
is w1  0.542 , w2  0.273 and w3  0.182 , respectively.
Because of matrix consistency < 0.1, this matrix will be
accepted.
Number of Total Suppliers: 5
Number of Total Customers: 4
Number of Total Manufacturers: 3
Number of Total Distribution Centers: 3
S: 20tl; Ch: 1.5tl (tl: Turkish liras)
Table 3. Unit transportation costs values between
suppliers and manufacturers (TL)
Suppliers
Manufacturers
1
2
3
4
5
0.5
0.3
0.4
0.4
0.5
1
0.6
0.4
0.5
0.6
0.6
2
0.5
0.5
0.6
0.6
0.4
3
Table 4. Unit transportation costs values between
manufacturers and DCs (TL)
Manufacturers
DCs
1
2
3
1.4
1.1
1.1
1
1.1
1.2
0.8
2
1.3
1.4
0.9
3
Table 5. Unit transportation costs values between DCs and
Customers (TL)
DCs
Customers
1
2
3
0.9
0.9
0.7
1
0.7
0.6
0.6
2
0.8
0.5
0.7
3
0.6
0.9
0.8
4
Table 6. Fuzzy capacities of suppliers, manufacturers, DCs
and demands of customers (unit) (bi , bi  pi )
Suppliers
(5000, 5500)
(5500,6050)
(5250,5775)
(4750,5225)
(4500,4950)
1
2
3
4
5
Manufacturers
(7000,7700)
(6500,7150)
(6500,7150)
-----
DCs
(6300,6930)
(6700,7370)
(6000,6600)
-----
Customers
3100
3100
3100
3100
---
4.1 Computational analysis
For the proposed model, S-curve membership function of
the fuzzy objective g 0 ( x) is given as follows,


g 0 ( x)  g 0 ( x a )
1,

a
0.999 , g 0 ( x)  g 0 ( x )

1
( x )  
, g 0 ( x a )  g 0 ( x)  g 0 ( x b )
 g ( x) g ( x a ) 

13.81 0 b 0 a 
1  0.001e
 g0 ( x ) g0 ( x ) 

0.001,
g 0 ( x)  g 0 ( x b )

0,
g 0 ( x)  g 0 ( x b )
where: g 0 ( x a )  13678 .3 and g 0 ( x b )  18648 .7 are

1,
di y ji  6300

i

 [ di y ji  6300 ]

i ( x)  1  i
,
630


di y ji  6930
0,

i


 di y ji  6930
6300 
i

In the equation (8), nonincreasing linear membership
function for fuzzy capacity constraint of 1st supplier is
given as,
b
1,
 5000
sk

k

 [
 5000 ]
sk

k
i ( x)  1 
,
500


 5500
0,
sk

k

b
5000 
 bsk  5500
k
b
 i (x)
obtained as the value of minimizing and maximizing
g 0 ( x) .
1
 g 0 ( x)
5000
bsk
5500
k
1
0.999
In the equation (10), nonincreasing linear membership
function of 1st manufacturer’s fuzzy capacity constraint is
given as,
0.5
0.001
x
0
13678 .3 16163 .5 18648 .7
Fig. 5. The S-curve membership function
Nonincreasing linear membership function of 1 st DC’s
fuzzy capacity constraint in (5) is given as,
 i (x)
 f kj  7000
1,

j


 [

j
i ( x)  1 


0,

j


 f kj  7000 ]
700
,
7000 
 f kj  7700
j
 f kj  7700
 i (x)
1
1
6300
6930
 d i y ji
i
7000
7700
 f kj
j
According to output obtained by LINGO 12.0 package
program, results are given in Table 7.
Table 7. The results obtained by LINGO package program
Fuzzy Problem
Decision
Value
variable
2225
b1,1
5225
b4,3
3975
b5,1
975
b5,3
6200
f1,1
6200
f3,3
3100
q1,2
3100
q1,3
3100
q3,1
3100
q3,4
1
y1,2
1
y1,3
1
y3,1
1
y3,4
Crisp Problem
Decision
Value
variable
2000
b1,3
2428
b2,1
3072
b2,2
28
b3,2
372
b4,1
4500
b5,3
2800
f1,2
3100
f2,1
3400
f3,2
3100
f3,3
3100
q1,4
3100
q2,2
3100
q2,3
3100
q3,1
1
y1,4
1
y2,2
1
y2,3
1
y3,1
Total
Objective
16783.83 TL
Total
Objective
13678.27 TL
f1
f2
f3
30185 TL
1219.83 TL
0.6542
f1
f2
f3
30185 TL
1472.5 TL
0.4655
According to solving the crisp problem, decision maker
purchased raw materials from all suppliers. 2000 units
from first suppliers, 5500 units from second supplier, 28
units from third supplier, 372 units from fourth and 4500
units from fifth supplier, are transported to manufacturers.
2800 units which come from second and fourth supplier
are shipped to second DC from first manufacturer. Also
3100 units of product are transported from second
manufacturer to first DC. 3400 units to second DC and
3100 units to third DC totally 4500 units of product
shipped from third manufacturer. Supporting equation 4
constraint, each customer provided their demand only one
DC via providing a better balanced distribution. All
customers’ demand is supplied from DCs as 3100, 6200
and 3100 units respectively (Fig. 6). At three echelons, all
transportation costs and holding/ordering costs in DCs
(first and second objective functions) calculated about
13678.27 TL. Providing the third objective, the
unnecessary and unused capacity of plants and DCs are
minimized via decreasing variance of transported amounts
between second and third echelons. When we examined
the second and third echelons’ distribution, it’s seen that
the transportation between manufacturers-DCs-customers
come and go from 2800 units to 3400 units considering
balancing distribution.
When applying the fuzzy model to the computational
problem, the output is given in Table 7. According to the
fuzzy approach, the distribution follows like; 2225 units
from first supplier, 5225 units from fourth supplier, 4800
units of fifth suppliers are transported to the manufacturer
plants.
Fig. 6. Optimal distribution network
6200 units which come from first and fifth supplier are
shipped to first DC from first manufacturer plant. Also
6200 units of product are transported from third
manufacturer plant to third DC. Ensuring equation 4
constraint, each customer provided their demand only one
DC via providing a better balanced distribution. All
customers’ demand is supplied from DCs as 6200 units
from first and 6200 units from third DCs (Fig. 6). At three
echelons, all transportation costs and holding/ordering
costs in DCs (first and second objective functions)
calculated about 16783.83 TL. Supporting the third
objective, as crisp model, the unnecessary and unused
capacity of plants and DCs are minimized. When we
examined the second and third echelons’ distribution, it’s
seen that the transportation between manufacturers-DCscustomers steadied 6200 units. It is better than the crisp
model. As a result, although increased cost, fuzzy model
provides a better balanced distribution flows between
facilities.
5. CONCLUSION
In this study, a fuzzy mixed integer non-linear
programming model is developed to design a supply chain
network by combining three different objectives. We have
considered three objectives: (1) minimization of total
transportation cost of plants and distribution centers (DCs),
inbound and outbound distribution costs, (2) minimization
of holding and ordering costs via EOQ method (3)
maximization of capacity utilization balance for DCs (i.e.
equity on utilization ratios). We used the developed model
to determine from which suppliers, manufacturers, DCs
and how much amounts will be transported to answer
customers demand. We developed binary variables to
provide a DC for a customer. So we have prevented
unbalanced distributions between DCs and customers. For
the fuzzy nonlinear objective function, S-curve
membership function and non-increasing linear
membership function are used for the fuzzy nonlinear
objective function and fuzzy constraints, respectively. The
proposed model shows that fuzzy logic application could
increase the cost but it provides better balanced
distribution to satisfy customers.
As a future work, uncertainty of costs and demands can be
considered in the model and new solution methodologies
including uncertainty can be developed via fuzzy models
with piecewise membership function and various s-shaped
membership functions. Additionally, new solution
methodology based on tabu search or heuristic methods
can be developed to obtain new optimal solutions for the
multi-objective SCN design problem, and the effectiveness
of the solution methodology can be investigated.
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