AN INVASIVE WEED OPTIMIZATION _IWO_ APPROACH FOR MULTI-OBJECTIVE JOB SHOP SCHEDULING PROBLEMS _JSSPs_

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AN INVASIVE WEED OPTIMIZATION _IWO_ APPROACH FOR MULTI-OBJECTIVE JOB SHOP SCHEDULING PROBLEMS _JSSPs_ Powered By Docstoc
					International Journal of Mechanical Engineering OF MECHANICAL 0976 – 6340(Print),
INTERNATIONAL JOURNAL and Technology (IJMET), ISSN ENGINEERING
ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME
                            AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 3, Issue 3, September - December (2012), pp. 627-637
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     AN INVASIVE WEED OPTIMIZATION (IWO) APPROACH FOR
    MULTI-OBJECTIVE JOB SHOP SCHEDULING PROBLEMS (JSSPs)

                             Hymavathi Madivada1, C.S.P. Rao2
1
    (Research Scholar, Department of Mechanical Engineering, National Institute of Technology
               – Warangal, Warangal – 506004, India, hyma.madivada07@gmail.com)
      2
        (Professor, Department of Mechanical Engineering, National Institute of Technology –
     Warangal, Warangal – 506004, India, csp_rao@rediffmail.com, csp_rao63@yahoo.com)

ABSTRACT

       In this paper, a new meta-heuristic solution approach for Multi-objective Job Shop
Scheduling Problems (JSSP) is presented. The proposed algorithm makes use of Mehrabian &
Lucas’s heuristic ‘Invasive Weed Optimization’ (IWO) in generating optimal schedules. For
performance evaluation of solutions in a Multi-objective scenario, a concept called’ Fuzzy
dominance’ has been employed. The results obtained from our study have shown that the
proposed algorithm can be used as a new alternative solution technique for finding good
solutions to the complex Multi-objective Job Shop Scheduling problems.

Keywords:Invasive Weed Optimization, Job Shop scheduling, Metahueristics, Multi-
Objective Optimization

1. INTRODUCTION

1.1. Job Shop Scheduling
        Scheduling may be viewed as an optimization process where limited resources are
allocated over time among both parallel and sequential activities.It comes into picture when
certain activities are to be carried out with limited resources. In a Job-shop Scheduling
Problem (JSSP) activities refer to the operations on jobs and resources refer to machine
hours. It is a NP-hard problem. In many cases, the combination of goals and resources
exponentially increases the search space, and thus the generation of good schedule is difficult
because we have a very large combinatorial search space. Several problems in various
industrial environments are combinatorial. This is the case for numerous scheduling and
planning problems. Generally, it is extremely difficult to solve this type of problems in their
general form, as it comprises several concurrent goals and several resources which must be
allocated to lead to our goals, which are to maximize the utilization of individuals and/or machines
and minimize the time required to complete the entire process being scheduled. Therefore, the exact

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methods such as the branch and bound method, dynamic programming and constraint logic
programming need a lot of time to find an optimal solution. So, we expect to find not necessarily an
optimal solution, but a good one to solve the problem. Realistically, we are satisfied by obtaining a
good solution near the optimal one. New search techniques such as genetic algorithms, simulated
annealing or tabu search[2] are able to lead to our objective i.e. to find near-optimal solutions for a
wide range of combinatorial optimization problems.
        The task of production scheduling consists in the temporal planning of the processing of a
given set of orders. The processing of an order corresponds to the production of a particular product.
It is accomplished by the execution of a set of operations in a predefined sequence on certain
resources, subject to several constraints. The result of scheduling is a schedule showing the temporal
assignment of operations of orders to the resources to be used. Each operation can be performed by
some machines with different processing times. The difficulty is to find a good assignment of an
operation to a machine in order to obtain a schedule which minimizes the total elapsed time (make-
span).
1.2. Multi-Objective Job Shop Scheduling Problem
        Multi Objective Job Shop Scheduling Problem deals with sequencing the operations so that
given set of objectives are achieved. One cannot identify a single solution that simultaneously
optimizes each objective. While searching for solutions, one reaches points such that, when
attempting to improve an objective further, other objectives suffer as a result. A tentative solution is
called non-dominated, Pareto optimal, or Pareto efficient if it cannot be eliminated from consideration
by replacing it with another solution which improves an objective without worsening another one.
Finding such non-dominated solutions, and quantifying the trade-offs in satisfying the different
objectives, is the goal when setting up and solving a multi objective optimization problem.
In the present work an attempt has been made to find the solution to Multi Objective Job Shop
Scheduling the objectives considered being
     • Minimizing Make-span, make-span being the maximum completion time of all jobs or the
        time taken to complete the last job on the last machine in the schedule.
     • Minimizing Tardiness, tardiness being the lateness of the job if it fails to meet its due-date,
        and zero otherwise.
     • Minimizing Mean Flow-time which measures the average response of the schedule to
        individual demands of jobs for service.

2. A REVIEW OF INVASIVE WEED OPTIMIZATION (IWO)
         Invasive weed optimization (lWO), first designed and developed by Mehrabian and Lucas, is
a relatively novel numerical stochastic optimization algorithm inspired from colonization of invasive
weeds. The algorithm is simple but has shown to be effective in converging to optimal solution by
employing basic properties, e.g. seeding, growth and competition, in a weed colony. A weed is any
plant growing where it is not wanted; any tree, vine, shrub or herb may qualify as a weed, in any
specified geographical area, depending on the situation. Weeds have shown a very robust and
adaptive nature that renders them undesirable plants in agriculture. In D-dimensional search space, a
weed which represents a potential solution of the objective function is represented by W = (w1, w2,·· ,
wm) . Firstly, M weeds, called a population of plants, are initialized with random growth position, and
then each weed produces seeds depending on its fitness and the colony's lowest fitness and highest
fitness to simulate the natural survival of the fittest process. The number of seeds each plant produce
increases linearly from minimum possible seed production to its maximum. The generated seeds are
being distribution randomly in the search area by normal distribution with mean equal to zero and a
variance parameter decreasing over the number of iteration. By setting the mean equal to zero, the
seeds are distributed randomly such that they locate near to the parent plant and by decreasing the
variance over time, the fitter plants are grouped together and inappropriate plants are eliminated over
times. The whole process is depicted in Fig 1 shown below [1] . The model and simulation of the
colonizing behavior of weeds in order as a novel optimization algorithm, with some basic properties
of the colonization process is presented in the following steps given below.

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2.1 Initialization
         A finite number of seeds are being dispread over the D-dimensional problem space with
random positions (initializing a population)




                 Fig 1: Flowchart for Invasive Weed Optimization Algorithm

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2.2 Reproduction
       Every seed grows to a flowering plant and produces seeds depending on their fitness
(reproduction): a member of the colony of weeds is allowed to produce seeds depending on
its own and the lowest and highest fitness values of the colony, where the number of seeds
each plant produces increases linearly from a possible minimum to its maximum; in other
words, a plant will produce seeds based on its fitness and the lowest and highest fitness
values of the colony to ensure that the increase is linear. This step adds a significant property
to the search algorithm. In evolutionary algorithms that are adopted to solve optimization
problems, intuitively, the infeasible individuals are not allowed to be reproduced, and feasible
individuals could be thought to be the ones with better fitness values than infeasible
individuals, although it is possible that some of the infeasible individuals carry more useful
information than feasible individuals during the evolution process; in the reproduction
method, this chance is given to infeasible individuals to survive and reproduce similar to the
mechanisms that occur in nature
2.3 Spatial Dispersion
       The produced seeds are being randomly dispread over the search area and grow to
new plants (spatial dispersal): the generated seeds are being randomly distributed over the D-
dimensional search space by normally distributed random numbers with a mean equal to zero
but with a varying variance of Witer. Thus, seeds will be randomly distributed such that they
abide near the parent plant. The SD of the random function is reduced from a previously
defined initial value Wini to a final value Wfin in every step (generation). In simulations, a
nonlinear alteration has shown satisfactory performance.
     =                              +     ………..(1)

Where itermax is the maximum number of iterations, Witer the SD at the present time step
and n the nonlinear modulation index. This alteration ensures that the probability of dropping
a seed in a distant area decreases nonlinearly at each time step, which results in grouping
fitter plants and the elimination of inappropriate plants
2.4 Competitive-Exclusion
        This process continues until the maximum number of plants is attained by fast
reproduction. At this stage, only the plants with higher fitness can survive and produce seeds,
whereas others are eliminated (competitive exclusion). In this process, after the maximum
number of weeds in a colony is reached, each weed is allowed to produce seeds, spread them
over the search area, and find their position and rank together with their parents. Next, weeds
with lower fitness values are eliminated in order to attain the maximum allowable population
in a colony. The course continues until the maximum iterations are reached and hopefully the
plant with the best fitness is the closest to the optimal solution. . It is worth mentioning that
the IWO has some distinctive properties when compared with the traditional GA and other
numerical search algorithms, such as reproduction, spatial dispersal and competitive
exclusion. In addition, no genetic operators are employed in the proposed algorithm, which
makes it more dissimilar from the GA.
         In this way the algorithm mimics the ecological process of weed colonization and has
been proved to be a powerful tool for finding out competitive solutions. It is capable of
solving general multi-dimensional, linear and non-linear optimization problems with
appreciable efficiency. It has been shown to be statistically significant than some state of art
existing evolutionary algorithms.

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3. IMPLEMENTATION OF IWOFOR SOLVING MULTI-OBJECTIVE JSSP

3.1 Weed representation of JSSP solution
        The solution to JSSP is a schedule of operation for jobs. In the present work IWO is
used to find the optimum schedule[2]. A weed represents feasible schedule in this case .This
is similar to a chromosome representing feasible schedule in case of genetic algorithm. In the
present work a direct approach “Operation based representation” is employed. Following is
the brief description of the representation. The schedule is represented in the form of a string
as shown in figure 3.1. The representation encodes a schedule as a sequence of operations,
and each character of the string stands for one operation. All operations for a job are
represented by the same symbol, the job number and they are interpreted according to the
order of their occurrence in the sequence in which they appear in the solution string. In this
case the string is called weed and the algorithm IWO is used to evolve these weeds to
discover potential schedules. For example, for a three-job-three-machine problem the
representation would be as shown below.
                                  3 2 1 2 1 3 3 1 2

3.2 Inputs for the problem
For solving JSSP using IWO, the following inputs are required:
    • Number of jobs
    • Number of Machines
    • Machine order for all the jobs
    • Processing Times of all the operations
    • Due dates for all the jobs
    • Number of iterations to be carried out
    • Maximum population allowable
    • Initial and Final standard deviations of seeds from parent weed
    • Nonlinear modulation index for finding out standard deviation in each iteration.

3.3 Step-wise implementation of IWO for JSSP
        This section gives a detailed explanation of implementation of the algorithm. The
actual algorithm has been slightly modified for enhancing the performance. INVASIVE
WEED OPTIMIZATION adopts the colonization process of weeds for finding optimum
solutions efficiently. In this case weed is a potential solution and the algorithm helps the
weeds to evolve and generate better population thus giving rise to fitter weeds which
represent competitive schedules. In the present work the algorithm has been programmed
using matlab. Several bench mark problems have been tested using make span and tardiness
as objectives.

3.3.1 Initialization
        In the original algorithm initialization is random. ‘n’ number of weeds are generated
randomly. The modification is that initially 10 weeds i.e 10 different schedules are generated
using dispatching rules which present a significant optimization capacity. Since the choice of
the initial population has a high impact on the speed of evolution and the quality of final
results, the solution scenario would be focused on generating initial population using priority
dispatching rules. Priority dispatching rules are actually the most widely used for solving
JSSP where all the operations available to be scheduled are assigned a priority .The operation

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with the highest priority is chosen to be sequenced. A priority dispatching rule is a simple
mathematical formula that, based on some processing parameters , specifies the priority of
operations to be executed. 10 initial schedules i.e. weeds are generated using 10 commonly
used priority dispatching rules given in the table below.

          EXPRESSION                               DESCRIPTION
  Shortest Processing Time (SPT)   The job with shortest time on machines selected.
                                   pi≤ pi+1≤ pi+2≤………≤ pn
  Longest Processing Time(LPT)     The job with longest time on machines
                                   selected.pi≥ pi+1≥ pi+2≥……≥pn
  Minimum      Slack    Time   Per Time remaining until the due date – Processing
  Operation(MINSOP)                timeremaining
  Minimum Due Date(MINDD)          The job with earliest due date is processed first
                                   Di≤ Di+1≤ Di+2≤…………….≤ Dn
  Critical Ratio(CR)                 Remaining due date/Remaining processing time
  Most work remaining (MWKR)         Select the operation associated with the job of the
                                     most work remaining to be processed
  Least work remaining(LWKR)          Select the operation associated with the job of
                                     the least work remaining to be processed
  Shortest   remaining   Minimum Min(processing tine remaining- minimum
  Processing Time(SRMPT)             processing time)
  Longest    remaining   Maximum Max(processing tine remaining- maximum
  Processing Time(LRMPT)             processing time)
  RANDOM(random selection)           Select the next job to be processed randomly.
                         Table 1: Priority Dispatching Rules

3.3.2 Performance Evaluation
        The initial population is made to reproduce depending on the fitness of the individual.
The present problem is a Multi- Objective optimization , the objectives being minimizing
make-span, minimizing tardiness and minimizing flow-time. So the fitness of the individual
has to be evaluated considering all the three aspects.The individuals are ranked based on their
fitness values(all the three). This ranking is done based on concept called Fuzzy-Pareto-
Dominance.[3] The ranking scheme assigns dominance degrees to any set of vectors in a
scale-independent, non-symmetric and set-dependent manner. Based on such a ranking
scheme, the fitness values of a population can be replaced by the computed ranking values
representing the ”dominating strength” of an individual against all other individuals in the
population.
        The three scheduling objectives used in this implementation are (1)make-span of the
sequence (2) mean-flow time of the jobs, and (3)the mean tardiness of jobs.

3.3.2.1. Make-Span Module (Fitness 1)
         In Scheduling literature, make-span is defined as the maximum completion time of all
jobs, or the time taken to complete the last job on the last machine in the schedule-assuming
that the processing of the first job began at time 0. Make-span is denoted by cmax and
computed as        cmax =max {Fj} , where Fj is the flow time for job j(the total time taken by
job j from the instant of its release to the shop to the time its processing by the last machine is
over).

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3.3.2.2. Mean Tardiness Module (Fitness 2)
        The lateness of a job measures the conformity of the schedule to that job’s due date.
Lateness is defined as the amount of time by which the completion time of a job exceeds its
due-date. Mathematically
Lj=Cj-dj
Lj: Lateness of job j
Cj: Completion time of job j
dj: Due date of job j.
Tardiness of job is the lateness if it is positive else it is zero.
Mean tardiness(T) is defined as the average of tardiness of all jobs.
T = (sum of tardiness of all jobs) / number of jobs

3.3.2.3.Mean Flow Time Module(Fitness 3)
         The mean flow time measures the average response of the schedule to individual
demands of jobs for service. Mathematically, mean flow time is the average of the flow times
of all jobs.
Mean flow time = sum of flow times of all jobs / number of jobs

3.3.2.4. Fuzzification Of Pareto Dominance And Ranking
        In multiobjective optimization, the optimization goal is given by more than one
objective to be extreme. Formally, given a domain as subset of Rn, there are assigned m
functions f1(x1, xn) . . . fm(x1. . . xn). Usually, there is not a single optimum but rather the
so-called Pareto set of non-dominated solutions.
        Evolutionary Computation (EC) has been shown to be a powerful technique for multi-
objective optimization (EMO - Evolutionary Multi-Objective Optimization). This
biologically inspired methodology offers both flexibility in goal specification and good
performance in multimodal, nonlinear search spaces. If we want to solve a highly complex
multi-objective optimization problem, we might select one of the best ranked evolutionary
approaches reviewed in the literature, like NSGA-II and hopefully start reaching good results
quickly.[11] However, all these algorithms need dominated individuals in the population, to
perform the corresponding genetic operators. For a higher number of objectives, this might
become a problem, since the probability of having a dominated individual in the population
will rapidly go to zero.
        The need for a revision of the Pareto dominance relation for also handling a larger
number of objectives was already pointed out in a few studies, esp. given by Farina and
Amato. There, we also find the suggestion to use fuzzy-membership degrees for the degree of
a point belonging to the Pareto set (so called fuzzy optimality). Authors design their revised
dominance measure in a way that the approach to the Pareto front can be registered more
early in the search. The approach was shown to work successfully in the domain of more than
two objectives. It came out that the use of fuzzy concepts is fruitful in this regard

The fuzzification of Pareto dominance relation can be written then as follows:
It is said that vector a dominates vector b by degree µa with:
                                  µa (a,b) =                   ,
It is said that the vector a is dominated by vector b by degree µp with:
                                   µp (a,b) =


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Here i denote the dimension of vector, in the present case ‘i’ is 3 as we have considered 3
objectives.The definitions differ in denominator and thus are not symmetric. “Dominating by
degree µ” and being dominated by degree µ have different fuzzy values.
For a Pareto dominating b,
                µa (a,b) = 1 and µp (b,a) = 1 but µp (a,b) < 1 and   µa (b,a) < 1
We use these dominance degrees to rank a set M of multi-variate data(vectors).The vectors in
this case are that sets of three fitness values of the schedules in the population.The vector
here is (fitness1 ,fitness2 ,fitness3). Consider two weeds or schedules a and b µp(a,b) = s/p.
Where s = min (fitness1a, fitness1b) ×min (fitness2a, fitness2b) ×min (fitness3a, fitness3b)
        p = fitness1b×fitness2b× fitness3b

       Each element of M(set of schedules) is assigned the maximum degree of being
dominated by any other element in M, and the elements of M are sorted in increasing order of
the value of dominance degree. After sorting the population is ranked sequentally. So the
individuals having good rank(lower) are fitter members in the population. With the help of
ranking scheme performance evaluation is carried out.

3.3.3 Reproduction
         The number of seeds generated by the parent weed depends upon the parent’s fitness,
the maximum fitness of the population and the minimum fitness. For all the weeds generated
in the first phase number of seeds for each weed is found out using the following formula.

N=max
Max=max no of seeds a weed can have
w =worst rank
b =best rank,
 i =fitness of the weed considered

The seed has similarities with the parent. The seed differs from the parent weed by a standard
deviation the value of which reduces from Sin to Sfin as iterations progress thus ensuring
convergence. At each iteration the value of standard deviation(s) is calculated as follows

Sf =
Sin =1       (assumption)
Sfin=0.001 (assumption)
f = ( Sin - Sfin)×Sf
S = f + Sfin
iter = maximum number of iterations for each run
n = non-linear index (in this case it is assumed to be 3)
i = the iteration for which the standard deviation is being calculated

Initially a 10×o (o is no of operations) matrix of random numbers is generated. Each row of
the matrix corresponds to a weed (schedule) generated in the first step. Let ‘x’ be the matrix
of random numbers and ‘p’ be the matrix of corresponding schedules(each row of the matrix
is a schedule coded in operation-based mode) generated using priority dispatching rules.
From the initial set of schedules seeds are generated,the number of seeds depending on the
rank of the parent weed.
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3.3.4 Competitive Exclusion
        This process continues until the maximum number of plants is attained by fast
reproduction. At this stage, only the plants with higher fitness can survive and produce seeds,
whereas others are eliminated (competitive exclusion). In this process, after the maximum
number of weeds in a colony is reached, each weed is allowed to produce seeds, spread them
over the search area, and find their position and rank together with their parents. Next, weeds
with lower fitness values are eliminated in order to attain the maximum allowable population
in a colony. The course continues until the maximum iterations are reached and hopefully the
plant with the best fitness is the closest to the optimal solution. . It is worth mentioning that
the IWO has some distinctive properties when compared with the traditional GA and other
numerical search algorithms, such as reproduction, spatial dispersal and competitive
exclusion. In addition, no genetic operators are employed in the proposed algorithm, which
makes it more dissimilar from the GA.
   The initial weeds generated and their seeds are together compared for their fitness.All are
sorted in descending order of their fitness. Only the first ‘m’ solutions are considered and are
allowed to proceed to the next generation, ‘m’ being the maximum allowable population
which is an assumption depending upon the problem complexity.
Maximum allowable population = 20×no of operations (assumption in this case)

3.3.5 Stopping Criteria
The stopping criteria is one of the following
           • Maximum number of iterations
           • The population’s worst and best fitness becomes equal
The cycle of steps explained is carried out until one of the stopping criteria is met

4. RESULTS AND DISCUSSIONS

       In this section, the execution of the program is presented by using an example
problem. The problem is a 5 machine and 10 job problems [2]. The input to the program is as
follows

S.No     Due date      Machine Order Matrix                   Processing Time Matrix
Job1     37            1       5      4       2       3       13      16     19         7    14
Job2     74            4      5       2      1        3      19       7      13         17   19
Job3     111           3      2       5      4        1      19       18     16         18   19
Job4     148           1      4       5      2        3      14       15     10         13   17
Job5     185           1      2       5      4        3      8        8      19         7    9
Job6     222           3      2       5      4        1      16       15     20         18   10
Job7     259           2      4       3      1        5      14       17     18         5    20
Job8     296           1      2       4      5        3      8        6      9          20   7
Job9     333           5      4       3      2        1      16       13     9          16   12
Job10    370           2      1       3      5        4      12       19     9          6    7
                              Table 2: Input data for the problem
1-Best Make-span
2-Best Mean Flow Time

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   S.NO                                     IWO                                        NSGA-II
     1                          165          184        169               190            198       193
     2                          143          128        136              153.2          140.9     142.4
                            Table 3: Results obtained for the above problem using IWO



                                 Comparison of results of IWO and
                                             NSGA
                                160
               Mean Flow Time




                                150
                                140
                                130                                                       MOIWO
                                120
                                                                                          NSGA
                                      160     170      180         190           200
                                                    Make Span

                                      Fig2: Comparision of results of IWO and NSGA

        In view of the results obtained by implementing IWO to solve JSSP (MOIWO), it
appears that IWO is efficient. Fuzzy dominance applied for decision making in Multi-
objective scenario yielded promising results. The algorithm has been improved by changing
its solution coding method and hybridizing with priority dispatching rules leading to fast
convergence. The algorithm’s performance has been compared to that of standard Non-
dominated- Search- Genetic- Algorithm (NSGA-2) with the help of some bench-mark
problems and has been found to be superior to the latter.

5. CONCLUSION

        In the present work the algorithm has been programmed for JSSP using mat-lab.
Bench mark problems have been tested using make span and tardiness as objectives using
IWO algorithm and the results are compared with the best known ones. It is concluded that
the application of IWO to Multi Objective JSSP is a new area to be explored for competitive
solutions

REFERENCES

Journal Papers
[1] Siddharth Pal, Anniruddha Basak and Swagatam Das “Linear Antenna Array Synthesis
    with Invasive Weed Optimization”,International Conference of Soft Computing and
    Pattern recognition, 2009
[2] Edurado Fernandez, Edy Lopez, Segio Bernal “Evolutionary multi-objective optimization
    using a Fuzzy-based Dominance Concept”
[3] Prithwish Chakraborty, Gaurab Ghosh Roy “On Population Variance and Explorative
    Power of Invasive Weed Optimization ” World congress on nature and biologicaaly
    inspired computing, 2009

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ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME

[4] Ritwik Giri ,Aritra Chowdhury, Arnob Ghosh “A Modified Invasive Weed Optimization
    for training of feed forward neural networks”
[5] M.Ramezani Ghalenoei, H.Hajmirsadeghi, C.Lucas “Discrete Invasive Weed
    Optimization and its application to Co-operative multi-task assignment of UAVs” Comin
    prac. 48th IEEE conference on Decision and Control, Dec 2009,in Press
[6] Rafal Zdonek, Tomasz Ignor “UMTS base station location planning with Invasive Weed
    Optimization”
[7] Takeshi Yamada and Ryohei Nakano “ Genetic Algorithn for Job-shop scheduling
    problem”Proceedings of Modern Hueristic for Decision support, pp.March 1997, Pages
    67-81
[8] Takeshi Yamada and Ryohei Nakano “ Job shop scheduling ”Job Shop Scheduling,
    pp.IEEE Control engineering Services 55,pages 134-160
[9] S.Q. Liu, H.L. Ong, K.M. Ng “Applying Tabu search to Job Shop Scheduling Problem ”
    Annals of Operations research 41, 1993, Pages 231-252

Books
[10] Tapan P. Bagehi “Multi-objective Scheduling By genetic Algorithms”

Chapters in Books
[11] Khald Mesghouni, Pierre Borne “Evolutionary Algorith For Job shop scheduling”
   Int.j.appl.Math.Comput.Sci.vol.14,2004,Pages 91-103
[12] Hossein Hajimirsadeghi, Amin Ghazanfari “Co-operative co-evolutionary Invasive
        Weed Optimization and its application to Nash Equilibrium search in Electricity
   Markets”




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