Journal of Applied Operational Research (2012) 4(1), 2–13
© 2012 Tadbir Operational Research Group Ltd. All rights reserved. www.tadbir.ca
ISSN 1735-8523 (Print), ISSN 1927-0089 (Online)
Using a genetic algorithm to optimize the total
cost for a location-routing-inventory problem
in a supply chain with risk pooling
Fatemeh Forouzanfar 1 and Reza Tavakkoli-Moghaddam 2,*
1
2
South Tehran Branch, Islamic Azad University, Tehran, Iran
College of Engineering, University of Tehran, Tehran, Iran
Abstract. This paper addresses a problem of designing a multi-echelon supply chain with the single sourcing type and the
related inventory systems. We also presents a novel mathematical model considering the risk-pooling, lead time, multi-echelon
inventory under the demand uncertainty, routing of vehicles from distribution centers (DCs) to customers in order to give
services in a stochastic supply chain system, simultaneously. This problem is formulated as a mix integer non-linear
programming model. The aim of this model is to determine, the number of the located distribution centers, their locations
and capacity levels, to allocate customers to distribution centers and distribution centers to suppliers optimally. In addition,
it is also to determine the net lead time of distribution centers and the inventory control decisions. In addition, it is to determine
the service time of every distribution centers and routing decisions. All these are done in a way that the total system cost is
minimized. The GAMS software is used to solve the presented model for small-size problems. The given problem belongs
to the class of NP-hard ones. Hence, to solve the large-sized instances, a genetic algorithm is used. The sensitivity analysis
has been performed to investigate the impact of effective parameters on the final solutions.
Keywords: supply chain; inventory control; risk-pooling; uncertainty; capacity levels
* Received October 2011. Accepted January 2012
Introduction
Nowadays, the competition in business environment is increasing considerably. The life cycle of products is becoming
shorter, customer demands are getting more uncertain, and the lead time on their service is getting very effective.
The demand’s variety can be recognized as one of the important sources of uncertainty in a supply chain (Gupta
et al. 2000; Park et al. 2010). Risk pooling is a strategy to redesign the supply chain, the production process or
the product to either reduce the uncertainty the firm faces or hedge uncertainty so that the firm is in a better position to
mitigate the consequence of uncertainty. The proposed risk-pooling strategy and centralizing the inventory
at distribution centers are considered as one of the effective ways to manage such a demand uncertainty for
achieving appropriate service levels to customers. Due to the increasing pressure for remaining competition in the
global market place, optimizing inventories across the supply chain has become a major challenge for the process
of industries to reduce the costs and improve the customer service. This challenge requires integrating inventory
management in a supply chain network design (You and Grossmann, 2009). The lead time is one of the effective
* Correspondence: Reza Tavakkoli-Moghaddam, Department of
Industrial Engineering, College of Engineering, University of Tehran,
P.O. Box 11155-4563 Tehran, Iran. E-mail: tavakoli@ut.ac.ir
F Forouzanfar and R Tavakkoli-Moghaddam
3
factors in the safety stock levels due to the customer demand uncertainty (Park et al. 2010). Surely, the lower level
for product, is considered as an additional value that can gain a long term or short term competitive benefit in the
market. In the recent decades, the topic of multi-depot heterogeneous vehicle routing problem (MDHVRP) is
proposed to increase the productivity and efficiency of transportation systems, in which this model leads to the
least cost function by minimizing the number of vehicles (Bettinelli et al. 2011).
One of the important factors of the total productivity and profitability of a supply chain is to consider its distribution
network, which can be used to achieve the various supply chain objectives. Designing a distribution network consists
of three sub-problems; namely, location-allocation, vehicle routing, and inventory control. In the literature, there
are some research studies amalgamating two of the above sub-problems, such as location-routing, inventory-routing,
and location-inventory problems (Ahmadi Javid and Azad, 2010). These three sub-problems of a distribution
network design are considered in few papers simultaneously. However, in this paper for the first time, we present
a model to concurrently optimize the location, allocation, capacity, lead time, inventory, and routing decisions
with risk-pooling in a stochastic supply chain system. Location-routing-inventory problems are laid in the nondeterministic polynomial-hard category, due to the intrinsic complexity of calculations. Therefore, solving large-sized
problems is not possible by linear programming using ordinary operational research software in a reasonable
time. A genetic algorithm (GA) has unique characteristics compared to other meta-heuristic methods. The following
advantages have been added in the revised paper (Goldberge, 1989).
 A genetic algorithm (GA) works with the coding of the parameter set, not the parameters themselves.
 It is a population-based solution, not a single point.
 It uses probabilistic transition rules, not deterministic rules.
 It trades-off between exploration and exploitation.
 It works on the number of variables at the same time.
 It is capable of working with any kinds of the objective functions and constraints in linear and/or non-linear
forms within any solution space (discrete or continuous).
Consequently, it is applied in this paper with respect to the model complexity. It is a bio-inspired algorithm
taken from the nature and it is also one of the most popular meta-heuristics, which is applied in many optimization
problems with different functions. The previous information is derived and used in searching for promising solutions
within the solution space. Furthermore, it has been utilized for solving supply chain problems extensively. For
further study, readers may refer to Wang et al. 2011; Kannan et al. 2010; Yun et al. 2009; Zegordi et al. 2010 and
Chang et al. 2010.
Problem formulation
The mathematical model of the considered problem, minimizes the fixed cost of locating the opened distribution
centers, the safety stock costs of distribution center by considering uncertainty in customer’s demand, inventory
ordering and holding costs and also vehicles routing beginning from a distribution center (DC) with the aim of
replying and covering to the devoted customer’s demands to that DC by considering the risk-pooling. The important
assumptions in this paper are as follows.
1) One kind of product is involved (Paksoy and Chang, 2010).
2) Each distribution center j is assumed to follow a (Qi, Rj) inventory policy (Ahmadi Javid and Azad,
2010).
3) A single sourcing strategy is considered in the whole supply chain (Park et al., 2010).
4) It is considered that the customers’ demands after reaching to retailer are independent and follows a normal
distribution (Park et al., 2010; Ahmadi Javid and Azad, 2010).
5) Each plant can give any kinds of services in any amount of demands to the related devoted distribution
centers.
6) We consider different capacity levels for each distribution center, and finally one capacity for each of
them is selected.
7) Each DC with the limited capacity carries on-hand inventory to satisfy demands from customer demand
zones as well as safety stock to deal with the mutability of the customer demands at customer demand
zones to attain risk-pooling profits (Park et al., 2010).
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Journal of Applied Operational Research Vol. 4, No. 1
8) All customers must be served.
9) The number of available vehicles for each type and the number of allowed routes for each DC are limited
(Bettinelli et al., 2011)
10) To determine all the feasible routes, the following factors are taken into account:
- Each customer must be visited by only one vehicle.
- Each route begins at a DC and ends at the same DC.
- The sum of the demands of the customers served in each route must not exceed the capacity of the
associated vehicle.
- Each of the distribution centers and the vehicles has the various, limited and determined capacity
(Bettinelli et al., 2011; Marinakis and Marinaki, 2010).
Model formulation
Following are the notations introduced for the mathematical description of the proposed model.
Indices
I
J
K
Nj
V
 jv
Set of plants indexed by i
Set of candidate DC locations indexed by j
Set of customer demand zones indexed by k
Set of capacity levels available to DCj (j  J)
Set of vehicles
Set of all feasible routes using a vehicle of type v (v ∊V ) from DCj (j  J)
Parameters
n
Fj
Yearly fixed cost for opening and operating distribution center j with capacity level n ( n  N j , j  J )
j
Safety stock factor of DCj (j  J)
hj
Unit inventory holding cost at DCj (j  J), (annually)
k
 2k
Mean demand at customer demand zone k
Aj
bj
n
t ij
Variance of demand at customer demand zone k
Fixed inventory ordering cost at DCj
Capacity with level n for DCj
Order processing time of DCj if it is served by plant i; including material handling time of DCj, transportation
time from plant i to DCj, and inventory review period.
Si
Service time of plant i
Cost of each demand unit in route r (these costs include the fixed cost of vehicle plus the transportation
cost of each demand unit in route r. the mentioned transportation cost for each demand unit is not related to customer
demand zone and it is considered fixed for all locations in each route r.
v
Number of available vehicles of each type v
Cr
gj
Number of routes associated with each distribution center j
Binary coefficients
Pkr
1 if and only if customer k is visited by route r; and 0, otherwise
F Forouzanfar and R Tavakkoli-Moghaddam
5
Decision variables
Unj
1 if distribution center j is opened with capacity level n; and 0, otherwise
X ji
1 if distribution center j is served by plant i; and 0, otherwise
Z jk
1 if customer k is assigned to distribution center j; and 0, otherwise
Xr
Lj
1 if and only if route r is selected; and 0, otherwise
j
Service time of DCj
Qj
Order size at distribution center j
Net lead time of DCj
The problem formulation is as follows.
F
Minimize
j J n N
  
U j n   j h j
j J
j

j J v V k K r  jv
L 
n
j
j
k K
2
k
Z jk   
i I j J k K
k Z jk A j
QjX
ji
 
i I j J
h jQ j X
2
ji

Pkr X rC r
k
s.t.
U
nN j

k K
1
n
j
Z jk 
k
Q X
iI
j
ji
(1)
j  J
b
nN j
n
j
Uj
L 
 j
kK
j
2
k
(2)
j  J
n
Z jk 
L j   ( S i  t ij ) X ji   j
b
nN j
n
j
j  J
Uj
n
j  J
(3)
(4)
iI
X
ji
Z
jk
1
k  K
(6)
X
ji
1
j  J
(7)
U
j
X
ji
iI
jJ
iI
nN j
iI

U
nN j
n
 Z jk
 Z jk
  P
kr
jJ vV r jv
j  J
n
j
j  J , k  K
j  J , k  K
X r 1
k  K
(5)
(8)
(9)
(10)
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Journal of Applied Operational Research Vol. 4, No. 1
 X
r
 v
v V
(11)
 X
r
 gj
j  J
(12)
jJ r jv
vV r jv
  Z
vV r jv k K
U nj  0,1
X ji  0,1
jk
Pkr X r  1
(13)
j  J
j  J , n  N j
(14)
j  J , i  I
Z jk  0,1
j  J , k  K
X r  0,1
r 

jv
jJ ,vV
Lj  0
j  J
j 0
j  J
Qj  0
j  J
(15)
This model minimizes the total expected cost consisting of the fixed cost for opening distribution centers with
a certain capacity level, the expected annual inventory cost, and the annual routing cost. Constraint (1) ensures
that each distribution center can be assigned to only one capacity level. Constraints (2) and (3) are the capacity
constraints associated with the distribution centers. Constraint (4) imposes limits on the minimum amount of net
lead time of DCj. Constraints (5) states that if the distribution center j with n capacity is opened, it is serviced by a
plant. Constraints (6) and (7) represent the single-sourcing constraints for each customer demand zone and each
DC, respectively. Constraint (8) ensures that if the distribution center j is allocated to the customer k, that center
should certainly be established by a determined capacity level. Constraint (9) makes sure that if the distribution
center j gives the service to the customer k, that center must get services from a plant. Constraint (10) is standard
set covering constraints, modeling assumption 6. Constraints (11) and (12) impose limits on the maximum number
of available vehicles of each type and maximum number of permitted routes for each DC, modeling assumption
7. Constraint (13) implies that there is at least one customer in one selected route. Constraint (14) enforces the
integrality restrictions on the binary variables. Finally, Constraint (15) enforces the non-negativity restrictions on
the other decision variables.
Solution methodology
Some different small-sized problems have been solved by the conventional branch-and-bound imbedded in the
GAMS (General Algebraic Modeling System) software in order to consider the feasibility and validity of the presented
mathematical model in small-sized problems. To solve large-sized problems, a genetic algorithm (GA) is proposed.
Chromosome definition
It is obvious that the solution representation is the base of any meta-heuristic approach. Four one-dimensional
matrices are used to demonstrate the solution. The first matrix is the 1×m one (m is the number of the distribution
centers) and denotes that each distribution center is established with its capacity levels. Each array of the matrix
is corresponding to a number between 1 and Nj, as shown in Figure 1.
Nj1
Fig. 1. Presentation of the first matrix
Nj2
Nj3
….
Njm
F Forouzanfar and R Tavakkoli-Moghaddam
7
The second matrix is the 1×m and represents the assignment of the distribution centers to the plants. Each array
of the matrix is associated with a number between 1 and n (n is the number of plants) as shown in Figure 2.
I1
I2
I3
….
Im
Fig. 2. Presentation the second matrix
The third matrix is the 1×k one (k is the number of the customers) that shows the customers allocation to the
distributors. The associated arrays are determined by a number between 1 and m, as shown in Figure 3.
J1
J2
J3
….
Jk
Fig. 3. Presentation of the third matrix
The fourth matrix is the 1×r one (r is the number of the routes) that shows the selected routes, as shown in
Figure 4. Each array of the matrix belongs to the numbers 0 or 1. It can be understood that the given route has
been used if the i-th cell of the matrix is 1; otherwise, it is 0.
A1
A2
A3
….
Ar
Fig. 4. Presentation of the fourth matrix
Establishing an initial population
The first step is to generate an initial population from the chromosomes once so that each one indicates to a specific
solution. The required feasible solutions are generated randomly in this section.
Fitness function
The fitness function is similar to considered objective function. As the chromosomes are formed and modified,
the objective function value is calculated for each one to justify it.
Sampling mechanism
The sampling mechanism pertains how the chromosomes are chosen with respect to the sampling space. The
bi-tournament approach is applied in this paper, in which the best solution is selected from the population and
then the next optimal solution is selected from the rest.
Crossover operator
In this paper, we use a two-point crossover operator for all the four matrices. However, it should be noted that the
obtained solutions from the crossover operator may be infeasible. Thus, they must be transformed into feasible
solutions by modifying practices. In this operator, two random indices are generated in the interval of 1 and the
length of the matrix for each one. The first and second offspring are also generated as follows.
 The first part of the first parent + the second part of the second parent + the third part of the first parent.
 The first part of the second parent + the second part of the first parent + the third part of the second parent.
Mutation operator
Variable Neighborhood Search (VNS) is used in this paper for the mutation structure. The VNS structure applies
four Neighborhood Search Structures (NSS). These four structures are used in the framework of VNS and the
entire structure can be demonstrated by Figure 5. The pseudo-code of our VNS is as follows.
8
Journal of Applied Operational Research Vol. 4, No. 1
{for each input particle
K=1
While the stopping criterion is met do
New particle=Apply NSS type k (Input particle)
If new particle is better than input particle then
K=1
Input particle= new particle;
Else
K=k+1
If k=5 then
K=1
Endif
Endif
Endwhile
}
Fig. 5. Shows the VNS algorithm
Strategy in dealing with the constraints
The mutation operator is designed in a way that no infeasible solution can be generated. Just, the crossover
operator may lead to infeasible solutions. Since new solutions are generated during the algorithm implementation,
a specific procedure has been deployed to check whether the constraints are satisfied by the given solution or not.
Hence, if it is necessary, the feasible solutions remain and infeasible solutions can be transformed into feasible
ones. The mentioned procedure tries to transform the solution into an acceptable one, whenever one or more
constraints are dissatisfied by the obtained solution.
Stopping criterion
The algorithm is terminated when it cannot find a new solution anymore, or in other words the objective function
values do not change.
Design of experiments
To investigate the validity and feasibility of the proposed mathematical model, different small-sized problems are
solved by the conventional branch-and-bound (B&B) solver in the GAMS software. In order to do that, ten
random instances are taken into account. Afterwards, the results are compared with those of the GA to validate
the approach. The obtained results of the GA are compared with the exact ones of GAMS, as shown in Table 1.
The parameters are set with respect to the following intervals.












Establishment cost of distribution centers by different capacity levels ~ U[650-5500]
Stock level for each distribution center ~ U[0.2, 2.3]
Annual inventory holding cost ~ U[4, 15]
Average demand of each customer ~U[1, 11]
Variance of the customers’ demands ~U[0, 3]
Capacity level of the distribution centers ~U[1, 30]
Processing time of each plant ~U[1, 4]
Service time of distribution centers ~U[1, 5]
Fixed ordering cost in each distribution center ~U[15, 60]
Cost of each demand unit in route r ~U[100, 650]
Available vehicles type v ~U[2, 3]
Number of possible routes related to distributor j ~U[1, r]
F Forouzanfar and R Tavakkoli-Moghaddam
9
Table 1. Obtained results of small-sized problems for the GA and GAMS.
Problem
2×2×3
2×2×3
2×2×5
2×2×2
3×3×3
3×3×4
3×3×5
3×3×6
2×2×3
2×2×2
GAMS
GA
Gap (%)
OFV
CPU time (sec.)
OFV
CPU time (sec.)
517670
697750
1740100
154480
780490
2719700
3368100
3250500
726950
447130
1.438
1.391
4.5
0.468
17.656
17.985
138.593
1005.703
2.406
1.641
517670
697750
1740100
154480
780490
2719700
3368100
3250500
726950
447130
0.011
0.013
0.002
0.027
0.01
0.011
0.004
0.017
0.021
0.012
0
0
0
0
0
0
0
0
0
0
The results indicate that the obtained objective function values for the GA and GAMS are the same in small-sized
instances. However, the CPU times of the GA are less than GAMS. Given the assumptions and parameters, 30
random instances are considered for medium and large-sized problems. The results are presented in Table 2 in
terms of the CPU time and the objective function value (OFV). Each instance is solved for five times by the GA
and the average value is illustrated in the tables.
Table 2. Results obtained by the proposed GA.
Medium-sized problems
Large-sized problems
Problem No.
OFV
CPU time (sec.)
Problem No.
OFV
CPU time (sec.)
10×10×15
10×10×20
10×10×25
10×10×30
10×15×15
10×15×20
10×15×25
10×15×30
10×20×25
10×20×30
10×20×40
10×25×30
10×25×40
10×25×50
10×30×40
10×30×50
10×30×60
15×15×20
15×15×25
15×15×30
15×20×25
15×20×30
15×20×40
15×25×30
15×25×40
15×25×50
15×30×30
15×30×40
15×30×50
15×30×60
1542905
1700312
2419531
2569362
3149300
3573900
3747400
4001100
4127800
4462900
4683200
5999300
5219400
5670000
6464700
6298800
6700000
4643900
3800000
4098900
5098900
5210000
5329100
5765400
5032000
5099900
6765900
7098700
7379900
7100200
3
2.8
2.825
2.925
2.9
2.9
3.475
3.525
4.725
5.5
4.975
4.95
4.975
5.525
6.05
6.3
7.275
2.825
3.075
3.2
3.9
3.825
4.225
4.3
5.5
5.575
4.975
5.94
6.4525
7.475
40×50×50
40×50×60
40×50×70
40×50×80
40×50×90
40×50×100
50×70×50
50×70×60
50×70×70
50×70×80
50×70×90
50×70×100
50×70×120
50×70×140
50×80×90
50×80×100
50×80×120
50×80×140
50×80×160
50×90×100
50×90×120
50×90×140
50×90×160
50×90×180
50×100×110
50×100×120
50×100×140
50×100×160
50×100×180
50×100×200
10929000
11211000
12748000
12789000
13734000
14007000
16988000
17765400
16997000
16371000
17342000
18756000
15023000
16917000
17600000
17008000
18432900
17699000
19809000
20112000
22259000
21977000
20033000
21098000
24989000
24396000
24164000
24621000
26791000
28866000
10.6825
11.1
12.8
15.275
17.475
20.1
14.35
19.125
22.775
27.575
34.975
34.825
47.625
52.75
36.95
44
56.125
70.3
89.675
52.75
49.975
72.675
92.725
122.3
72.775
69.25
92.2
125.15
150.1
181.775
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Journal of Applied Operational Research Vol. 4, No. 1
In order to show the proper performance of the GA for the given problems, the following assumptions should
be taken into account.




Rates of the mutation and crossover operators are assumed to be 0.1 and 0.8, respectively.
Local-iteration is assumed equal to 5.
Population size is assumed to be 100.
Initial population is generated randomly.
The parameters are set with respect to the following intervals.












Etablishment cost of distribution centers by different capacity ~ U[1,40]
Stock level for each distribution center ~ U[0, 3]
Annual inventory holding cost ~ U[1, 40]
Average demand of each customer ~U[1, 100]
Variance of the customers’ demands ~U[1, 4]
Capacity level of the distribution centers ~U[1000, 2500]
Processing time of each plant ~U[1, 40]
Service time of distribution centers ~U[1, 40]
Fixed ordering cost in each distribution center ~U[1, 40]
Cost of each demand unit in route r ~U[1, 100]
Available vehicles type v ~U[1, 10]
Number of possible routes related to distributor j ~U[1, r]
Sensitivity analysis
The effective rates of mutation and crossover and the efficient population size are given below. The rate of the
mutation is equal to 0.1, and the value of the crossover rate is considered equal to 0.6, 0.7 and 0.8. The three
levels can be seen in Figure 6, so that each level shows the combination of the rate of two operators (Naderi et
al., 2009).
a1: rate 0.6 for crossover and 0.1 for mutation.
a2: rate 0.7 for crossover and 0.1 for mutation.
a3: rate 0.8 for crossover and 0.1 for mutation.
The vertical axis shows the value of a criterion, namely relative percentage deviation (RPD), which is
calculated by:
RPD 
Algsol  min sol
 100
min sol
(16)
where the Algsol is the objective function value for each problem by combining the parameters and Min sol is
the minimum objective function value in all the combinations. In fact, each instance is run by each of the three
combinations and the RPD criterion is calculated for each one. The results are presented in Figure 6. In this figure, it is obvious that the best crossover rate is 0.8.
In addition, the crossover rate is equal to 0.8 and the mutation rates are set to 0.1 and 0.2. Two levels are seen
in this figure, so that each one is a combination of the rate of two operators given below.
a1: the rates 0.8 for crossover and 0.1 for mutation.
a2: the rates 0.8 for crossover and 0.2 for mutation.
In fact, each problem is carried out by each of two possible combinations and the RPD criterion is calculated
for each problem. It is concluded that the best mutation rate is 0.1 as depicted in Figure 7.
F Forouzanfar and R Tavakkoli-Moghaddam
11
analysis for crossover rate
95% CI for the Mean
3.0
2.5
Data
2.0
1.5
1.0
0.5
0.0
a1
a2
a3
RPD
Fig. 6. RPD diagram with respect to combinations of the fixed mutation rates and different Crossover rates
analysis for mutation rate
95% CI for the Mean
3.0
2.5
Data
2.0
1.5
1.0
0.5
a1
a2
RPD
Fig. 7. RPD diagram with respect to combinations of the fixed Crossover rates and different mutation rates
Figure 8 pertains to the population size. Four different sizes (i.e., 30, 50, 100 and 200) are considered as shown
below:
a: shows size 30
b: shows size 50
c: shows size 100
d: shows size 200
Likewise, the vertical axis shows the RPD criterion.
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Journal of Applied Operational Research Vol. 4, No. 1
where the Algsol is the objective function for each level of the population size for each problem, which is obtained
by the algorithm and Minsol is the minimum calculated value among all the considered size levels for each problem.
As it is observed from Figure 8, the most efficient and reliable population size is 100.
analysis for popsize
95% CI for the Mean
3.0
2.5
Data
2.0
1.5
1.0
0.5
0.0
a
b
c
d
RPD
Fig. 8. RPD diagram regarding the population size
Conclusions
In this paper, a new mathematical model for designing the multi-echelon supply chain has been presented by
considering the inventory under uncertain demands, risk-pooling, lead time and vehicles routing. This model has
been formulated for the first time as a location-inventory-routing problem with a risk-pooling strategy in a
multi-echelon supply chain. Feasibility of the developed model was checked by presenting small-sized random
instances and solving them by commercial optimization software. Then, the results obtained from the GA were
compared to the exact ones of GAMS in small-sized instances in order to validating the GA. The results showed
that the CPU times were less for the GA in comparison with those of GAMS. A number of medium and large-sized
problems were solved by the proposed GA because of the NP-hardness of the given problems. Some future studies
are as follows: considering each parameter as a fuzzy, time windows, multi-period planning and solving the presented
model by the use of heuristic or other meta-heuristic algorithms.
References
Ahmadi Javid, A., Azad, N., 2010. Incorporating location,
routing and inventory decisions in supply chain network design. Transportation Research Part E: Logistics
and Transportation Review 46, 582-597.
Bettinelli, A., Ceselli, A., Righini, G., 2011. A branch-andcut-and-price algorithm for the multi-depot heterogeneous vehicle routing problem with time windows.
Transportation Research - Part C: Emerging Technologies 19, 723-740.
Chang, Y.-H., 2010. Adopting co-evolution and constraintsatisfaction concept on genetic algorithms to solve
supply chain network design problems., Expert Systems
with Applications 37, 6919-6930.
Goldberge, D.E., Genetic algorithms in search, optimization
and machine learning, Addison-Wesley Publishing, 1989.
Gupta, A., Maranas, C.D., McDonald, C.M. 2000. Mid-term
supply chain planning under demand uncertainty:
F Forouzanfar and R Tavakkoli-Moghaddam
Customer demand satisfaction and inventory management.
Computers & Chemical Engineering 24, 2613-2621.
Kannan, G., Sasikumar, P., Devika, K., 2010. A genetic algorithm approach for solving a closed loop supply
chain model: A case of battery recycling. Applied
Mathematical Modeling 34, 655-670.
Marinakis, Y., Marinaki, M., 2010. A hybrid geneticparticle swarm optimization algorithm for the vehicle
routing problem. Expert Systems with Applications 37,
1446-1455.
Naderi, B., Khalili, M., Tavakkoli-Moghaddam, R., 2009. A
hybrid artificial immune algorithm for a realistic variant of job shops to minimize the total completion time.
Computers & Industrial Engineering 56, 1494-1501.
Paksoy, T., Chang, C.T., 2010. Revised multi-choice goal
programming for multi-period, multi-stage inventory
controlled supply chain model with popup stores in
Guerrilla marketing. Applied Mathematical Modeling
34, 3586-3598.
Park, S., Lee, T.E., Sung, C.S., 2010. A three-level supply
chain network design model with risk-pooling and lead
times. Transportation Research - Part E: Logistics and
Transportation Review 46, 563-581.
Wang, K.-J., Makond, B., Liu, S.-Y., 2011. Location and
allocation decisions in a two-echelon supply chain with
stochastic demand – A genetic-algorithm based solution. Expert Systems with Applications 38, 6125-6131.
You, F., Grossmann, I.E., 2009. Optimal design of largescale supply chain with multi-echelon inventory and
risk pooling under demand uncertainty. Computer Aided Chemical Engineering 26, 991-996.
Yun, Y., Moon, C., Kim, D., 2009. Hybrid genetic algorithm with adaptive local search scheme for solving
multistage-based supply chain problems. Computers &
Industrial Engineering 56, 821-838.
Zegordi, S.H., Kamal Abadi, I.N., Beheshti Nia, M.A., 2010.
A novel genetic algorithm for solving production and
transportation scheduling in a two-stage supply chain.
Computers & Industrial Engineering 58, 373-381.
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