A modified invasive weed

David C. Wyld et al. (Eds) : SAI, CDKP, ICAITA, NeCoM, SEAS, CMCA, ASUC, Signal - 2014
pp. 51–60, 2014. © CS & IT-CSCP 2014 DOI : 10.5121/csit.2014.41106
A MODIFIED INVASIVE WEED
OPTIMIZATION ALGORITHM FOR
MULTIOBJECTIVE FLEXIBLE JOB SHOP
SCHEDULING PROBLEMS
Souad Mekni and Besma Chaar Fayech
National School of Engineering of Tunis, Tunisia, LR-ACS-ENIT
msouadpop@gmail.com,Besma.fayechchaar@insat.rnu.tn
ABSTRACT
In this paper, a modified invasive weed optimization (IWO) algorithm is presented for
optimization of multiobjective flexible job shop scheduling problems (FJSSPs) with the criteria
to minimize the maximum completion time (makespan), the total workload of machines and the
workload of the critical machine. IWO is a bio-inspired metaheuristic that mimics the
ecological behaviour of weeds in colonizing and finding suitable place for growth and
reproduction. IWO is developed to solve continuous optimization problems that’s why the
heuristic rule the Smallest Position Value (SPV) is used to convert the continuous position
values to the discrete job sequences. The computational experiments show that the proposed
algorithm is highly competitive to the state-of-the-art methods in the literature since it is able to
find the optimal and best-known solutions on the instances studied.
KEYWORDS
Flexible job shop scheduling problem, Multiobjective optimization, Metaheuristics, Smallest
Position Value, Invasive Weed Optimization.
1. INTRODUCTION
Solving a NP-hard scheduling problem with only one objective is a difficult task. Adding more
objectives obviously makes this problem more difficult to solve. In fact, while in single objective
optimization the optimal solution is usually clearly defined, this does not hold for multiobjective
optimization problems. Instead of a single optimum, there is rather a set of good compromises
solutions, generally known as Pareto optimal solutions from which the decision maker will select
one. These solutions are optimal in the wider sense that no other solution in the search space is
superior when all objectives are considered. Recently, it was recognized that Invasive Weed
Optimization (IWO) was well suited to multiobjective optimization.
The invasive Weed Optimization algorithm developed by Mehrabian and Lucas [1] in 2006 is a
newly stochastic optimization approach inspired from a common phenomenon in agriculture:
colonization of invasive weeds. IWO is an appropriate competitor for other evolutionary
algorithms. In fact, it is simple and easy to understand and program. It has strong robustness and
fast global searching ability.

52 Computer Science & Information Technology (CS & IT)
Some of the distinctive properties of IWO in comparison with other numerical search algorithms
are the way of reproduction, spatial dispersal, and competitive exclusion. These properties are
presented in details in section 3. Section 2 introduces and formulates the flexible job shop
scheduling problem .The experiments are provided in section 4. Finally, brief conclusions and
future perspectives are discussed in section 5.
2. MATHEMATICAL FORMULATION
The problem of flexible job shop scheduling (FJSSP) belongs to the NP-hard family [2]. It
presents two difficulties. The first one is the assignment of each operation to a machine, and the
second one is the scheduling of this set of operations in order to optimize our criteria. The result
of a scheduling algorithm must be a schedule that contains the start times and allocation of
resources to each operation. The data, constraints and objective of our problem are as follows:
2.1. Data
• M represents a set of m machines. A machine is called ( 1,..., )kM k m= , each kM has a
Workload called kW .
• N represents a set of n jobs. A job is called ( 1,... )ij i n= , each job has a linear sequence
of in operations.
• ,i jO represents the operation number j of the job number i . The realization of each
operation ,i jO requires a machine kM and a processing time , ,i j kp . The starting time of
,i jO is ,i jt and the ending time is ,i jft .
2.2. Constraints
• Machines are independent of one another.
• A machine can be unavailable during the scheduling (case of machine breakdown).
• Jobs are independent of one another.
• In our work, we suppose that: each job ij can start at the date 0t = and the total number
of operations to perform is greater than the number of machines.
2.3. Criteria
We have to minimize 1Cr , 2Cr and 3Cr :
• The makespan: 1Cr
• The total workload of machines: 2Cr
• The workload of the most loaded machine: 3Cr
In this paper, the objective is to find a schedule which has a minimum makespan, a minimum
total workload of machines and a minimum workload of the critical machine. The sum of these
three objectives is taken as the objective function. To measure the quality of solutions found, we
use the lower bounds ( 1BCr for makespan, 2BCr for total machine workload, and 3BCr for
the workload of the most loaded machine) proposed in [3].

Computer Science & Information Technology (CS & IT) 53
3. INVASIVE WEED OPTIMIZATION ALGORITHM FOR FJSSP
The IWO algorithm was proposed by Mehrabian and Lucas [1] in 2006, and since then, it has
been successfully utilized in different practical optimization problems such as optimal positioning
of piezoelectric actuators [4], demanding a recommender system [5], Studying electricity market
dynamics [6], Design of an E shaped MIMO antenna [7] and encoding sequences for DNA
computing [8].
3.1. Invasive Weed Optimization Algorithm
A weed is any plant growing where it is not wanted. Weeds have shown very robust and adaptive
nature which turns them to undesirable plants in agriculture. A common belief in agronomy is
that “The Weeds Always Win”. The harder people try, the better they get [1]. Recently, many
studies are carried out with inspirations from ecological phenomena for developing optimization
techniques. The new algorithm that is motivated by a common phenomenon in agriculture is
colonization of invasive weeds. The flow chart of this algorithm is shown in Figure1 and the
details of IWO are addressed as follows:
Figure 1. Flow Chart of IWO
3.1.1. Initialization
A population of initial solutions (weeds) is randomly generated over the search space.

3.1.2. Evaluation
The fitness of each weed in the population is calculated.
3.1.3. Reproduction
Each weed in the population is allowed to produce seeds depending on its comparative fitness in
the population. In other words, a weed will produce seeds based on its fitness, the worst fitness
and the best fitness in the population. In such way, the increase of number of seeds produced is
linear. The number of seeds for each weed varies linearly between minS for the worst plant and
maxS for the best plant. Figure 2 illustrates the procedure of reproduction.
Figure 2. Procedure of reproduction
The equation for determining numWeed the number of seeds produced by each weed is presented in
equation (1):
min max min( ) worst
num
best worst
f f
Weed S S S
f f
−
= + −
−
Equation (1)
Where f is the fitness of the weed considered, worstf and bestf are respectively the worst and the
best fitness in the population. For better clarification, the application of equation (1) is shown in
Figure 3. In this figure, it is assumed that weed5 and weed1 are the worst and best weeds between
a population containing five weeds. So, the number of seeds around Weed5 is equal to minS and the
number of seeds around Weed1 is equal to maxS .

Figure 3. Schematic reproduction procedure for a problem with 5 weeds
3.1.4. Spatial Dispersal
This step ensures that the produced seeds will be generated around the parent weed, leading to a
local search around each plant. The generated seeds are randomly spread out around the parent
weeds according to a normal distribution with mean equal to zero and variance 2
σ . The standard
deviation of the seed dispersion σ decreases as a function of the number of iterationsiter . The
equation for determining the standard deviation for each generation is presented in equation (2):
max
max
( )
( )
( )
n
iter initial final finaln
iter iter
iter
σ σ σ σ
−
= − + Equation (2)
Where maxiter is the maximum number of iterations. iterσ is the standard deviation at the current
iteration and n is the nonlinear modulation index. Obviously, the value of σ defines the
exploration ability of the weeds. Therefore, as iter increases, the exploration ability of all weeds
is gradually reduced. At the end of the optimization process, the exploration ability has
diminished so much that every weed can only fine its position [9].
3.1.5. Competitive exclusion
After a number of iterations, the population reaches its maximum, and an elimination mechanism
is adopted: The seeds and their parents are ranked together and only those with better fitness can
survive and become reproductive. Others are being eliminated.
3.2. Weed representation of FJJSP
The original IWO is developed to solve continuous optimization problems, but it can not be
applied to discrete problems directly: individuals must be encoded appropriately to solve
scheduling problems. In this paper, we implement a coding that takes into account all the
constraints and the specifities of the problem. For the ( n jobs, m machines, O operations)
FJSSP, each plant is represented by four components: each component contains 2 O× number of
dimensions. Figure 5, Figure 6 and Figure 7 illustrate the solution representation of a weed
corresponding to (3 jobs, 5 machines, 8 operations) FJSSP described in Figure 4. The 1st
and
2nd
halves of the 1st
row of the weed (Figure 6 and Figure 7) represent operations as repetition of
jobs (Figure 5). For example ( 1J , 1J , 1J ) represents ( 1,1O , 1,2O , 1,3O ), ( 2J , 2J , 2J ) represents
( 2,1O , 2,2O , 2,3O ), and so on. The 2nd
row of (Figure 6 and Figure 7) represents weed’s position.
Each dimension of this row in Figure 6 maps one operation and each dimension of this row in

Figure 7 maps one machine. At this step, we use the Smallest Position Value (SPV) rule [10] to
find the permutation of jobs. The smallest component of the weed’s position in Figure 6 is -8
which corresponds to job number 1, thus 1J (or the first operation of 1J ) is scheduled first. The
second smallest component of the weed’s position is -5,2 which corresponds to job number 2,
therefore 2J (or the first operation of 2J )is the second in ordering, etc. The 2nd
row of Figure 6
contains a random number in the interval [0, ]m that indicates after being rounded to its nearest
integer the machine to which an operation is assigned during the course of IWO. The 3rd
row of
Figure 6 indicates the sequence of jobs in the ordering and the 3rd
row of Figure 7 indicates the
corresponding machines. Finally, the last row of Figure 6 indicates operations in the order and the
last row of Figure 7 indicates starting times. In conclusion, the weed itself presents a solution as it
shown in 3rd
and 4th
row of Figure 6 and Figure 7: First, the operation 1,1O of job 1J is executed
by the machine 1M at time 0t = , and then the operation 2,1O of job 2J is executed by the machine
1M at time 1t = , and so on.
1M 2M 3M 4M 5M
1,1O 1 9 3 7 5
1J 1,2O 3 5 2 6 4
1,3O 6 7 1 4 3
2,1O 1 4 5 3 8
2J 2,2O 2 8 4 9 3
2,3O 9 5 1 2 4
3J 3,1O 1 8 9 3 2
3,2O 5 9 2 4 3
Figure 4. Example of ( 3 J ,5 ,M 8 O ) FJSSP
Figure 5. Weed representation

Figure 6. The first half of the weed Figure 7. The second half of the weed
3.3. Pseudo-code of solving FJSSP by IWO algorithm
Begin{
• Initialize population of weeds, set parameters;
• Current_iteration=1;
While (Current_iteration< Max_iteration)do
{
• Compute the best and worst fitness in the population
• Compute the standard deviation std depending on iteration
For each weed w in the population W
{
• Compute the number of seeds for w depending on its
fitness
• Select the seeds from the feasible solutions around the
parent weed w in a neighborhood with normal distribution
having mean=0 and standard deviation=std;
• Add seeds produced to the population W
If (|W|>Max_SizePopulation)
{
• Sort the population W according to their fitness
• W=SelectBetter(weed,seed,Max_SizePopulation)
}End if
}End for
Current_iteration=Current_iteration+1;
}End while
}End
Figure 8. Pseudo code of IWO
4. EXPERIMENTAL RESULTS
Our approach is implemented in C++ on an Intel(R) Core(TM) i3 CPU M370@2,40 GHz
machine. The non deterministic nature of IWO algorithm makes it necessary to carry out multiple
runs on the same problem instance in order to obtain meaningful results. We run our algorithm
twenty times from different starting solutions and tested it on a number of instances from

literature. The convergence of IWO depends on the selection of three parameters: the initial
standard deviation initialσ , the final standard deviation finalσ and the non linear modulation
index n .The chosen parameters for IWO are given in table 1.
Table 1. Parameters of IWO.
Parameters values
Number of initial population 50
Maximum number of population 200
Maximum number of iterations: maxiter 5000
Maximum number of seeds 5
Minimum number of seeds 1
initialσ 10
finalσ 0,5
Non linear modulation index: n 3
To illustrate the effectiveness and performance of the algorithm used in this paper, we choose
different instances of the problem of flexible job shop scheduling problem taken from Kacem
[11]. Solutions in the literature to the instances presented in table 2 are presented in table3.
Table 2. Instances of Kacem.
Instances n(jobs) m(machines)
Instance 1 3 5
Instance 2 4 5
Instance 3 10 7
Instance 4 10 10
Instance 5 15 10
Instance 6 8 8
From table 3, we conclude that the obtained solutions are generally of a good quality. This is
noted while comparing them with the existing approaches in the literature (for example Xia
approach[12]) and also while comparing obtained values of the criteria with the computed lower
bounds [3]. In fact, for instance 1, instance 2 and instance 3 our value of makespan 1Cr is near
the lower bound 1BCr , our value of total machine workload 2Cr is near the lower bound 2BCr
and our value of the workload of the critical machine 3Cr is near the lower bound 3BCr .
For instance 4, instance 5 and instance 6 our values of criteria are near lower bounds and similar
or better (instance 4) than solutions found in [12].

Table 3. Solutions in Literature.
Instances Lower
Bounds
Xia et al
[12]
IWO
Instance 1
1BCr =4
2BCr =11
3BCr =2
-
-
-
1Cr =5
2Cr =13
3Cr =5
Instance 2
1BCr =11
2BCr =32
3BCr =6
-
-
-
1Cr =11
2Cr =32
3Cr =10
Instance 3
1BCr =11
2BCr =60
3BCr =8
-
-
-
1Cr =11
2Cr =61
3Cr =11
Instance 4
1BCr =7
2BCr =41
3BCr =4
1Cr =7
2Cr =44
3Cr =6
1Cr =7
2Cr =42
3Cr =6
Instance 5
1BCr =10
2BCr =91
3BCr =9
1Cr =12
2Cr =91
3Cr =11
1Cr =12
2Cr =91
3Cr =11
Instance 6
1BCr =12
2BCr =73
3BCr =9
1Cr =15
2Cr =75
3Cr =12
1Cr =14
2Cr =77
3Cr =12
5. CONCLUSIONS
In this paper, the performance of the Invasive Weed Optimization technique is investigated for
solving the multiobjective flexible job shop scheduling problem. The main highlighting features
in IWO are: it is simple and easy to understand and program and it has strong robustness and fast
global searching ability.
Experimental results are encouraging since that the proposed algorithm is able to find relevant
solutions minimizing makespan, total machine workload and the biggest machine workload on
the studied instances. A more comprehensive study on a large number of instances should be
made to test the efficiency of the proposed solution technique. Further investigation is needed to
fully reveal the ability of IWO in tackling scheduling problems and solving other optimization
problems. Future research should pay more attention to the hybridization of IWO and other
metaheuristics in order to benefit from advantages of each algorithm.
REFERENCES
[1] A R, Mehrabian. & C, Lucas, (2006) “A novel numerical optimization algorithm inspired from weed
colonization”, Ecological Informatics, Vol.1, pp355-366.
[2] M , Sakarovitch, (1984) “Optimisation combiantoire. Méthodes mathématiques et algorithmiques.
Hermann, Editeurs des sciences et des arts, Paris.

[3] R, Dupas, (2004) “ amelioration de performances des systems de production: apport des algorithms
évolutionnistes aux problems d’ordonnancement cycliques et flexibles, Habilitation , Artois
university.
[4] A R, Mehrabian & A, Yousefi-Koma (2007) “Optimal Positioning of Piezoelectric actuators on a
smart fin using bio-inspired algorithms”, Aerospace Science and technology, Vol 11, pp 174-182.
[5] H, Sepehri Rad & C, Lucas (2007) “ A recommender system based on invasive weed optimization
algorithm”, IEEE Congress on Evolutionary Computation, CEC 2007, pp 4297-4304.
[6] M, Sahaheri-Ardakani & M.Rshanaei & A, Rahimi-Kian & C, Lucas (2008) “ A study of electricity
market dynamics using invasive weed colonization optimization”, in Proc.IEEE Symp. Comput.Intell.
Games, pp 276-282.
[7] A, R, Mallahzadeh & S, Es’haghi & A, Alipour (2009) “Design of an E shaped MIMO antenna
using IWO algorithm for wireless application at 5.8 Ghz”, Progress in Electromagnetic Research,
PIER 90, pp 187-203.
[8] X, Zhang & Y,Wang & G, Cui & Y, Niu & J, Xu (2009) “Application of a novel IWO to the design
of encoding sequence for DNA Computing, Comput. Math. Appl. 57, pp 2001-2008.
[9] Z, D, Zaharis & C, Skeberis & T, D, Xenos (2012) “Improved antenna array adaptive beamforming
with low side lobe level using a novel adaptive invasive weed optimization method” , Progress in
Electromagnetics Research , Vol 124, pp 137-150.
[10] S, Mekni & B, Châar Fayéch & M, Ksouri (2010) “ TRIBES application to the flexible job shop
scheduling problem”, IMS 2010 10th IFAC Workshop on Intelligent Manufacturing Systems,
Lisbon, Portugal, July 1st -2nd 2010.
[11] I, Kacem & S, Hammadi & P, Borne (2002) “ Approach by localization and multiobjective
evolutionary optimization for flexible job shop scheduling problems. IEEE Trans Systems, Man and
Cybernetics, Vol 32,pp 245-276.
[12] W, Xia & Z, Wu (2005) “ An effective hybrid optimization approach for multiobjective flexible job
shop scheduling problems, Journal of Computers and Industrial Engineering, Vol 48,pp 409-425.
AUTHORS
Souad Mekni: received the diploma of Engineer in Computer Science from the Faculty of Science of
Tunis (Tunisia) in 2003 and the Master degree in Automatic and Signal Processing from the National
Engineering School of Tunis (Tunisia) in 2005. She is currently pursuing the Ph.D.degree in Electrical
Engineering at the National Engineering School of Tunis. Her research interests include production
scheduling, genetic algorithms, particle swarm optimization, multiobjective optimization, Invasive Weed
Optimization and artificial intelligence.
Besma Fayéch Chaâr: received the diploma of Engineer in Industrial Engineering from the National
Engineering School of Tunis (Tunisia) in 1999, the D.E.A degree and the Ph.D degree in Automatics and
Industrial Computing from the University of Lille (France), in 2000, 2003, respectively. Currently, she is
a teacher assistant in the Higher School of Sciences and Techniques of Tunis (Tunisia). Her research
interests include scheduling, genetic algorithms, transportation systems, multiagent systems and decision-
support systems.

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