A review on non traditional algorithms for job shop scheduling

International Journal of Production Technology and Management TECHNOLOGY–AND
INTERNATIONAL JOURNAL OF PRODUCTION (IJPTM), ISSN 0976 6383
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ISSN 0976- 6383 (Print)
ISSN 0976 - 6391 (Online)
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A REVIEW ON NON TRADITIONAL ALGORITHMS FOR JOB SHOP
SCHEDULING
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

A great deal of research has been focused on solving the job-shop problem, over the
last fifty years, resulting in a wide variety of approaches. Recently, much effort has been
concentrated on hybrid methods to solve job shop scheduling problem. JSSP is stated as a NP
Hard problem [36, 37] so that as a single technique cannot solve this stubborn problem. As a
result much effort has recently been concentrated on techniques that combine the specific
methods and a meta-strategy which guides the search out of local optima. These approaches
currently provide the best results. Such hybrid techniques are known as iterated local search
algorithms or meta-heuristics. In this paper we seek to assess the work done in the job-shop
domain by providing a review of many of the techniques used. The impact of the major
contributions is indicated by applying these techniques to a set of standard benchmark
problems. It is established that methods such as Tabu Search, Genetic Algorithms, Simulated
Annealing should be considered complementary rather than competitive. In addition this
work suggests guide-lines on features that should be incorporated to create a good job shop
scheduling system. Finally the possible direction for future work is highlighted so that current
barriers within job shop scheduling problem may be surmounted as we approach the 21st
Century.

Key Words: Job shop, scheduling, review, exact, approximation algorithms.

1. INTRODUCTION

Research in scheduling theory has evolved over the past fifty years and has been the
subject of much significant literature with techniques ranging from unrefined dispatching
rules to highly sophisticate parallel branch and bound algorithms and bottleneck based
heuristics. Not surprisingly, approaches have been formulated from a diverse spectrum of
researchers ranging from management scientists to production workers. However with the
advent of new methodologies, such as neural networks and evolutionary computation,

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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383

researchers from fields such as biology, genetics and neurophysiology have also become regular contributors to
scheduling theory emphasizing the multidisciplinary nature of this field.
One of the most popular models in scheduling theory is that of the job-shop, as it is considered to be a
good representation of the general domain and has earned a reputation for being notoriously difficult to solve. It
is probably the most studied and well developed model in deterministic scheduling theory, serving as a
comparative test-bed for different solution techniques, old and new and as it is also strongly motivated by
practical requirements it is clearly worth understanding.
Jain and Meeran (1999) provided a concise overview of JSPs over the last few decades and highlighted
the main techniques. The JSP is the most difficult class of combinational optimization. Garey, Johnson, and
Sethi (1976) demonstrated that JSPs are non-deterministic polynomial-time hard (NP-hard); hence we cannot
find an exact solution in a reasonable computation time. The single objective JSP has attracted wide research
attention. Most studies of single-objective JSPs result in a schedule to minimize the time required to complete
all jobs, i.e., to minimize the makespan. Many approximate methods have been developed to overcome the
limitations of exact enumeration techniques. These approximate approaches include simulated annealing (SA)
(Lourenço, 1995), tabu search (Nowicki & Smutnicki, 1996; Pezzella & Merelli, 2000; Sun, Batta, & Lin, 1995)
and genetic algorithms (GA) (Bean, 1994; Gonçalves, Mendes, & Resende, 2005; Kobayashi, Ono, &
Yamamura, 1995; Wang & Zheng, 2001). However, real world production systems require simultaneous
achievement of multiple objective requirements. This means that the academic concentration of objectives in the
JSP must been extended from single to multiple. Recent related JSP research with multiple objectives is
summarized as below. Ponnambalam, Ramkumar, and Jawahar (2001) has offered a multi-objective GA to
derive optimal machine-wise priority dispatching rules for resolving job-shop problems with objective functions
that consider minimization of makespan, total tardiness, and total machine idle time. Ponnambalam’s multi-
objective genetic algorithm (MOGA) has been tested with various published benchmarks, and is capable of
providing optimal or near-optimal solutions. A Pareto front provides a set of best solutions to determine the
tradeoffs between the various objects, and good parameter settings and appropriate representations can enhance
the behavior of an evolution algorithm. Esquivel, Ferrero, and Gallard (2002) studied the influence of distinct
parameter combinations as well as different chromosome representations.
Initial results showed that:
(i) larger numbers of generations favour the building of a Pareto front because the search process does not
stagnate, even though it may be rather slow,
(ii) Multi-recombination helps to speed the search and to find a larger set size when seeking the Pareto optimal
set, and
(iii) operation-based representation is better than priority-list and job-based representation selected for contrast
under recombination methods.
The Pareto archived simulated annealing (PASA) method, a meta-heuristic procedure based on the SA
algorithm, was developed by Suresh and Mohanasndaram (2006) to find non-dominated solution sets for the JSP
with the objectives of minimizing the makespan and the mean flow time of jobs. The superior performance of
the PASA can be attributed to the mechanism it uses to accept the candidate solution. Candido, Khator, and
Barcia (1998) addressed JSPs with numbers of more realistic constraints, such as jobs with several subassembly
levels, alternative processing plans for parts and alternative resources of operations, and the requirement for
multiple resources to process an operation. The robust procedure worked well in all problem instances and
proved to be a promising tool for solving more realistic JSPs. Lei and Wu (2006) first designed a crowding-
measure-based multi-objective evolutionary algorithm (CMOEA) makes use of the crowding-measure to adjust
the external population and assign different fitness for individuals. Compared to the strength Pareto evolutionary
algorithm, CMOEA performs well in job-shop scheduling with two objectives including minimization of
makespan and total tardiness.

2. OBJECTIVES OF SCHEDULING

The scheduling is made to meet specific objectives. The objectives are decided upon the situation,
market demands, company demands and the customer’s satisfaction. There are two types for the scheduling
objectives:
(i) Minimizing the makespan
(ii) Due date based cost minimization
The objectives considered under the minimizing the makespan are,
(a) Minimize machine idle time
(b) Minimize the in process inventory costs
(c) Finish each job as soon as possible

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(d) Finish the last job as soon as possible
The objectives considered under the due date based cost minimization are,
(a) Minimize the cost due to not meeting the due dates
(b) Minimize the maximum lateness of any job
(c) Minimize the total tardiness
(d) Minimize the number of late jobs

Fig 1: Different algorithms for JSSP

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3. SCHEDULING TECHNIQUES

There are number of optimization and approximation techniques are used for
scheduling of job shop scheduling problem. The techniques are generally,
(i) Traditional Techniques
• Traditional Techniques are also called as Optimization Techniques. These
techniques are slow and guarantee of global convergence as long as problems are
small.
Mathematical programming (Linear Programming, Integer programming, Goal
Programming, Dynamic Programming, Transportation, Network, Branch-and-
Bound, Cutting Plane / Column Generation Method, Mixed Integer Linear
programming, Surrogate Duality), Enumerate Procedure Decomposition
(Lagrangian Relaxation) and Efficient Methods.
(ii) Non Traditional Techniques
• Non Traditional Techniques are also called as Approximation Methods. These
methods are very fast but they do not guarantee for optimal solutions.
Constructive Methods(priority dispatch rules, composite dispatching rules),
Insertion Algorithms (Bottleneck based heuristics, Shifting Bottleneck
Procedure(SBP)), Evolutionary Programs(Genetic Algorithm, Particle Swarm
Optimization), Local Search Techniques(Ants Colony Optimization,
Simulated Annealing, adaptive Search, Tabu Search, problem Space Methods
like Problem & Heuristic Space and GRASP), Iterative Methods((Artificial
Intelligence Techniques(Constraint Satisfacton (CSPs)),(Expert Systems),
(Artificial Neural Network(Hopfield Networks and Back-Error
propagation(BEP))), Heuristics Procedure, Beam-Search, and Hybrid
Techniques.

3.1. Literature Review on JSSP Scheduling
Many researchers have been focusing on scheduling during the last few decades. A
number of approaches have been developed and employed for solving various problems of
Job Shop Scheduling considering various objectives. The following sections discuss the
literature available in scheduling using various traditional and non-traditional optimization
techniques.

3.1. Review on Job Shop Scheduling using Non Traditional Optimization Techniques

Table:1 Methodologies to solve the Job Shop scheduling Problem:
S.no Method Author 1 Author 2
Approximation Fisher and Rinnooy Kan(1988) Blazewicz et al.(1996)
Algorithms
(I) Constructive Methods
(1) Jackson(1955) Smith(1956)
Priority Dispatch Rules Giffer and Thomson(1960) Fisher and Thompson(1963)
Crowston et al.(1963) Jeremiah et al.(1964)
Gere(1966) Moore(1968)
Panwalkar and Iskander(1977) Blackstone et al.(1982)
Lawrence(1984) Haupt(1989)
Chang et al.(1966) Sabuncuoglu and Bayiz(1997)

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(2) Werner and Winkler(1995)
Insertion Algorithms
(i) Bottleneck based Alvehus(1997)
heuristics
(ii) Shifting Adams et al.(1988) Applegate and Cook(1991)
Bottleneck Dauzere-Peres and lasserre (1993) Balas et al.(1995)
Procedure(SBP) Balas and Vazacopoulos(1998) Holtsclaw and Uzsoy(1996)
Demirkol et al.(1997)
(II) Iterative Methods
(1) Artificial intelligence Glover and Greenberg(1989)
(i) Constrained Eerschler et al.(1976) Fox(1987)
Satisfaction(CSPs) Sadeh(1991) Caseau and Laburthe(1994,95)
Nuijten and Aarts(1994,96) Harvey(1995)
Harvey and Ginsberg(1995) Sadeh et al.(1995)
Baptiste and Le Pape(1995) Baptiste et al.(1995)
Pesch and Tetzlaff(1996) Sadeh and Fox(1996)
Cheng and Smith(1997) Nuijten and Le Pape(1998)
(2) Neural Networks Wang and Brunn(1995) Jain and Meerun(1998)

(i) Hopfield Networks Foo and Takefuji(1988a-c) Zhou et al.(1990,91)
Van Hulle(1991) Lo and Bravian(1993)
Willems and Rooda(1994) Satake et al.(1990,91)
Foo et al.(1994,1995) Sabuncuoglu and Gurgun(1996)
(ii) Back Error Dagli et al.(1991) Watanabe et al.(1993)
Propagation(BEP) Cedimoglu(1993) Sim et al.(1994)
Kim et al.(1995) Dagli & Sittisathanchai(1995)
(3) Expert Systems Alexander(1987) Kusiak and Chen(1988)
Biegel and wink(1989) Charalambous and Hindi(1991)
Shakhlevich et al.(1996) Sotskov(1996)
(III) Local Search Evans(1987) Vaessens(1995)
Methods Vaessens et al.(1995,96) Arts and Lenstra(1997)
Mattfeld et al.(2000)
(1) Problem Space Methods
(i) Problem & Storer et al.(1992,95)
Heuristic Space

(ii) GRASP Resende(1997)
(2)Genetic Local Search Aarts et al.(1991,94) Pesch(1993)
Della Croce et al.(1994) Dorndorf and Pesch(1995)
Mattfeld(1996) Yamada and Nakano(1995b,1996b,c)
Imen Essafi, Yazid Mati, Stéphane Dauzère-Pérès(2008)
(3)Ant Optimization Colorni et al.(1995,96)
Colorni, M. Dorigo et V. Maniezzo(1991)
S. Goss, S. Aron, J.-L. Deneubourg et J.-M. Pasteels
(4)Reinsertion Methods Werner and Winker(1995)
(1)
Threshold Algorithm Aarts et al(1991,94)
Threshold

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(i) Improvement Aarts et al(1991,94) Storer et al.(1992)
(ii) Simulated Annealing Matsuo et al.(1988) Yan Laarhoven et al.(1988,92)
Aarts et al(1991,94) Yamada et al.(1994)
Sadeh and Nakakuki(1996) Yamada and Nakano(1995a,1996a)
Kolonko (2001)
(2)
Threshold Acceptance Aarts et al.(1991,1994)
(i) Large step Lourenco (1993,1995) Lourenco & Zwijnenburg(1996)
Optimization Brucker et al.(1996a,1997a)
(3)Tabu Search Taillard(1989,1994) Dell’Amico and Trubian(1993)
Barnes and Chambers(1995) Sun et al.(1995)
Nowioki and Smutnioki(96) Ton Eikelder et al.(1997)
Thomson(1997)
Alain Hertz(1996) Marino Widmer(1996)
(IV) Evolutionary Algorithms
(1) Genetic Algorithm Davis(1985) Falkenauer and Bouffouix(1991)
Nakano and Yamada(1991) Tamaki and Nishikawa(1992)
Yamada and Nakano(1992) Davidor et al.(1993)
Ross et al.(1993) Mattfeld et al.(1994)
Della Croce et al.(1995) Kobayashi et al.(1996)
Norman and Bean(1995) Bierwirth(1995)
Bierwirth et al.(1996) Cheng et al.(1996)
Shi(1996)
(2) Particle Swarm Tsung-Lieh Lin, Shi-Jinn Chen, Ray-Shine Run, Rong-Jian Chen,
Optimization Horng, Tzong-Wann Kao, Jui-Lin Lai, I-Hong Kuo
Yuan-Hsin Xinyu Shao, Peigen Li, Liang Gao(2009)
Guohui Zhang(2009) Xingsheng Gu(2008)
Qun Niu, Bin Jiao, (2008) Cheng-Yu Hsu(2006)
D.Y. Sha(2006)
Deming Lei(2008) Zhiming Wu(2005)
Weijun Xia(2005) Hsing-Hung Lin(2009)
D.Y. Sha (2009) Ren-qian ZHANG, Guo-ping XIA(2007)
Kun FAN (2007) Davide Anghinolfi (2009)
Massimo Paolucci(2009) G. Baharian Khoshkhou(2009)
M. Rabbani, M. Aramoon
Bajestani(2009)

(3)Memetic Algorithm Jin-hui Yang, Liang Sun(2009) Yun Qian(2009)
Heow Pueh Lee(2009) Yan-chun Liang(2009)
Lacomme, Nikolay Tchernev Anthony Caumond, Philippe (2008)
Nima Safaei, Farrokh Sassani Reza Tavakkoli-Moghaddam(2009)
Mohsen Sadegh Amalnik Jalil Layegh, Fariborz Jolai, (2009)
(5)Immune Algorithm A. Bagheri, M. Zandieh(2009) I. Mahdavi, M. Yazdani(2009)
(6)Hybrid Algorithms
(i) Hybrid PSO Dongyun Wang (2008) Liping Liu(2008)
Peng-Yeng Yin, Shiuh-Sheng Yu, Pei-Pei Wang, Yi-Te Wang(2007)
Dušan Teodorović(2008)
(ii) Hybrid GA Jie Gao, Mitsuo Gen(2007) Linyan Sun, Xiaohui Zhao(2007)
Hong Zhou, Waiman Cheung Lawrence C. Leung(2009)

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4. OPTIMIZATION AND APPROXIMATION TECHNIQUES

Various tools have been used in the field of Job Shop Scheduling. The methods used for JSP
are generally divided into two broad categories: traditional approaches and nontraditional approaches.
The traditional approaches are divided into two categories:
(i) Theoretical research dealing with optimization procedures
(ii) Experimental research dealing with dispatching rules
The nontraditional approaches are used recently are robust for the JSP scheduling problems, as the
problems are NP-hard. This section analyzes some traditional and nontraditional approaches that have
been implemented on JSP Scheduling.

4.1. Non – Traditional Techniques
4.1.1. Genetic Algorithm
Genetic Algorithms have been widely studied, experimented and applied in many fields of
engineering. GA is stochastic search procedure for combinatorial optimization problems
based on the mechanism of natural selection and natural genetics [71]. These use the idea of
survival of the fittest by progressively accepting better solutions to the problems. The element
and mechanism of genetic algorithm are representation, population, evaluation, selection,
operator and parameter. The algorithm starts with a randomly generated initial set of
population called chromosomes that represent the solution of problem. These are evaluated
for the fitness function and the selected according to their fitness value. Many selection
procedures are based on the fitness value of the individuals of current generation. This
selection alone cannot introduce any new individuals into population in the need of search
large space. The operators such as crossover and mutation are works on it.
GA coding scheme
As GA works on coding of parameters, the feasible job sequences ( the parameter of
the considered problems) are coded in two different ways.
(i) Pheno style coding
(ii) Binary coding
Genetic Operations
Reproduction
Rank order selection method is used for reproduction. The individuals in the
population are ranked according to fitness, and the expected value of individual depends on
its rank rather than on its absolute fitness. Ranking avoids giving the largest share of
offspring to a small group of highly fit individuals, and thus reduces the selection pressure
when the fitness variance is high, thus keeping up selection pressure when the fitness
variance is low. Each individual in the population is ranked in increasing order of fitness
from 1 to N. The expected value of each individual ‘i’ in the population at time ‘t’ is given by
(Max-Min)(Rank(i,t)-1)
Expected value (i,t) = Min + --------------------------------
N-1
Where, N= Population Size, Min.=0.4 and Max.=0.6
After calculating the expected value of each rank, reproduction is performed using Monte
Carlo simulation by employing random numbers.
Cross over
The strings in the mating pool formed after reproductions are used in the crossover operation.
The possibility of cross over is used for this work. With phenol type coding scheme, two
strings are selected at random and crossed at random site. Since the mating pool contains
strings at random, we pick of pairs of strings from the top of the list. When two strings are

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chosen for crossover, first a coin is flipped with a probability Pc=0.6 to check whether a
crossover is desired or not. If outcome of the coin flipping is true, the crossover is performed;
otherwise the strings are directly placed in the intermediate population for subsequent genetic
operation.
Crossover site is chosen by creating a random number between 1 and 20. For example
if the random number is 10, the strings are crossed after 10th position. The current sequence
after 10th position the first string in that pair, is rearranged according to the next string.
Similarly the sequence after the 10th position of the 2nd string in that pair is rearranged
according to the first sequence.
(E.g.) pair of strings before crossover: Crossover = 10th position
5 4 6 9 10 15 19 20 1 17 / 7 3 11 13 18 2 8 12 14 16
1 20 15 9 19 6 10 5 4 6 / 3 11 2 14 12 16 7 8 18 13
Pair of strings after crossover:
5 4 6 9 10 15 19 20 1 17 / 3 11 2 14 12 16 7 8 18 13
1 20 15 9 19 6 10 5 4 6 / 7 3 11 13 18 2 8 12 14 16
Mutation
The possibility of mutation is determined by a probability from 0.01 to 0.1. In this
work, bit wise mutation is used. With the phenol type scheme, two sites are selected by
generating two numbers of random numbers between 1 1nd 20. If random numbers generated
are 8 and 15, then the corresponding job numbers in these positions are exchanged and the
new sequence is obtained.
(E.g.) string before mutation:
5 4 6 9 10 15 19 20 1 17 7 3 11 13 18 2 8 12 14 16
String after mutation:
5 4 6 9 10 15 2 20 1 17 7 3 11 13 18 19 8 12 14 16
Algorithm:
Step 1: Generate random population of n chromosomes(suitable solution for problem)
Step 2: Evaluate the fitness f(x) of each chromosome x in the population.
Step 3: Create a new population by repeating following steps until the new population is
complete
Step 4: Select two parent chromosomes from a population according to their fitness (the
better fitness, the bigger chance to be selected)
Step 5: With a crossover probability crossover the parents to form a new offspring(children).
If no crossover was performed, offspring is an exact copy of parents.
Step 6: with a mutation probability mutate new offspring at each locus(position in
chromosome)
Step 7: place new offspring in a new population
Step 8: use new generated population for a further run of algorithm.
Step 9: if the end condition is satisfied, stop, and returns the best solution in current
population and Go to step 2.

4.1.2. Particle Swarm Optimization
One of the latest evolutionary techniques for unconstrained continuous optimization is
particle swarm optimization (PSO) proposed by Kennedy and Eberhart (1995), inspired by
social behavior of bird flocking or fish schooling.
PSO learned from these scenarios are used to solve the optimization problems. In
PSO, each single solution is a “bird” in the search space.[37],[38],& [39]. We call it
“particle”. All of particles have fitness values, which are evaluated by the fitness function to
be optimized, and have velocities, which direct the flying of the particles. The particles are
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“flown” through the problem space by following the current optimum particles.PSO is
initialized with a group of random particles (solutions) and then searches for optima by
updating generations. In each iteration, each particle is updated by following two “best”
values. The first one is the best solution (fitness) it has achieved so far. (The fitness value is
also stored.) this value is called ‘pbest’. Another “best” value that is tracked by the particle
swarm optimizer is the best value, obtained so far by any particle in the population. This best
value is a global best and called ‘gbest’.

Algorithm :
Step 1: initialize a population on n particles randomly.
Step 2: Calculate fitness value for each particle. If the fitness value is better than the best
fitness value (pbest) in history. Set current value as the new pbest.
Step 3: Choose particle with the best fitness value of all the particles as the gbest.
Step 4: for each particle, calculate particle velocity according to the equation
V[]=c1*rand()*(pbest[]-present[])+c2*rand()*(gbest[]-present[])
Where present[]=present[]+v[]
V[] is the particle velocity, present[] is the current particle(solution),
rand() is random functions in the range [0,1].
c1,c2 are learning factors=0.5.
step 5: particle velocities on each dimension are clamped to a maximum velocity Vmax. If the
sum of acceleration would cause the velocity on that dimension to exceed Vmax (specified by
user), the velocity on the dimension is limited to Vmax.
Step 6: Terminate if maximum of iterations is reached. Else, goto Step2
The original PSO was designed for a continuous solution space. In original PSO we can’t
modify the position representation, particle velocity, and particle movement. Then another
heuristic is made to modify the above parameters, called Hybrid PSO. So they work better
with combinational optimization problems. This Hybrid PSO was designed for a discrete
solution space.

4.1.3. Ant Colony Optimization
The ant system is a new kind of co-operative search algorithm inspired by the behaviour of
colonies of real ants. The blind ants are able to find astonishing good solutions to shortest
path problems between food sources and their home colony [72]to[80]. The medium used to
communicate information among individuals regarding paths, and decide where to go, was
the pheromone trails. A moving ant lays some pheromone on the path they move, thus
marking the path by the substance. While an isolated ant moves essentially at random, it can
encounter a previously laid trail and device with high probability to follow it, and also
reinforcing the trail with its own pheromone. The collective that emerges in a form of
autocatalytic behaviour where the more the ants following a trail, the more attractive that trail
becomes for being followed.
The path of ants
There is a path along which ants are walking from nest to the food source and vice
versa. If a sudden obstacle appears and the path is cut off, the choice is influenced by the
intensity of the pheromone trails left by proceeding ants. On the shorter path more
pheromone is laid down.
The below figure shows the behaviour of ants when faced with an obstacle in its search path.

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Figure: path traced by ants without and with obstacle

AN FS

O
B
S FS
AN
T
A
C
L
E

Ants Colony algorithm can be applied for the continuous function optimization problems.
Here, the domain has to be divided into a specific number of R randomly distributed regions.
These regions are indeed the trail solutions and act as local stations for the ants to move and
explore. The fitness of these regions are first evaluated and stored on the basis of fitness.
Totally a population of ants explores these regions; the updating of the region is done locally
and globally with the local search and global search mechanism respectively. The distribution
of local and global ants is illustrated in the below figure.

Algorithm:
Step 1: fix the evaporation rate and no. of runs.
Step 2: while (number of runs is less than required).
Step 3: Initialize pheromone values.
Step 4: Call random no. generation function.
Step 5: Generate group of ants with difference paths(sequences).
Step 6: Call the function for calculating COF.
Step 7: Sort the COF values in ascending order.
Step 8: For best routes update pheromone level.
Step 9: Repeat 4,5,6,7&8 till obtaining required no. of runs.
Step 10: Print the best routes and the COF values.
Step 11: Change evaporation rate and no. of runs for next trail.

4.1.4. Simulated Annealing Algorithm
The simulated annealing algorithm resembles the cooling process of molten metals
through annealing. At high temperature, the atoms in the molten metal can move freely with
respect to each another. But, as the temperature is reduced, the movement of the atoms gets
reduced. The atoms start to get ordered and finally form crystals having the minimum
possible energy. However, the formation of the crystal depends on the cooling rate. If the
temperature is reduced at a fast rate, the crystalline state may not be achieved at all; instead
the system may end up in a polycrystalline state, which may have a higher energy state than
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the crystalline state. Therefore, to achieve the absolute minimum state, the temperature needs
to be reduced at a slow rate. The process of slow cooling is known as annealing in
metallurgical parlance. SA simulates this process of slow cooling of molten to achieve the
minimum function value in a minimization problem. The cooling phenomenon is simulated
by controlling a temperature – like parameter introduced with the concept of the Boltzmann
probability distribution. [83]to [87].
Algorithm:
Step 1: Choose an initial point X^(o), a termination criterion ∈. Set T as a sufficiently high
value number of iterations to be performed at a particular temperature n, and set t=0.
Step 2: Calculate the neighbouring point X^(t+1) = N(x^(t)). Usually, a random point in the
neighbourhood is created.
Step 3: If ∆E= E(x^(t+1))-E(x^(t))<0, set t=t+1; Else create a random number (r) in the range
(0,1). If r≤exp(∆E/T), set t=t+1; Else go to Step 2.
Step 4: if (x^(t+1)-x^(t))< ∈ and T is small, Terminate. Else go to Step 2.

4.1.5. Memetic algorithm
Memetic algorithm is a combination of a population based global search and the
heuristic local search made by each of the individual. Memetic algorithms (MA) are
evolutionary algorithms where, two non-traditional techniques are combined to form an
algorithm. Combining global optimization approaches has in fact been recognized as a
powerful algorithmic paradigm for evolutionary computing, as it consists of a separate local
search process to refine individuals. In particular the relative advantage of MA over EA is
quite consistent on complex search spaces. Each algorithm is designed to obtain global
optimum solution. The output of the first iteration of the algorithm is obtained and is given to
the second algorithm in the same iteration. Memetic algorithm is a population based
approach. The flowchart of the Memetic algorithm illustrated in below figure.

Algorithm:
Step 1: Initialize the population randomly. Set iteration k=0
Step 2: Implement algorithm 1 in the population
Step 3: Implement algorithm 2 on the population obtained from step 2
Step 4: Calculate the fitness and find the best solution.
Step 5: If termination criteria is not achieved, increment k=k+1 and go to step 2.

4.1.6. Tabu search
The basic idea of Tabu search (Glover 1989, 1990) is to explore the search space of
all feasible scheduling solutions by a sequence of moves. A move from one schedule to
another schedule is made by evaluating all candidates and choosing the best available, just
like gradient-based techniques. Some moves are classified as tabu (i.e., they are forbidden)
because they either trap the search at a local optimum, or they lead to cycling (repeating part
of the search). These moves are put onto something called the Tabu List, which is built up
from the history of moves used during the search. These tabu moves force exploration of the
search space until the old solution area (e.g., local optimum) is left behind. Another key
element is that of freeing the search by a short term memory function that provides “strategic
forgetting”.
Tabu search methods have been evolving to more advanced frameworks that includes longer
term memory mechanisms. These advanced frameworks are sometimes referred as Adaptive
Memory Programming (AMP, Glover 1996). Tabu search methods have been applied

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successfully to scheduling problems and as solvers of mixed integer programming problems.
Nowicki and Smutnicki (Glover 1996) implemented tabu search methods for job shop and
flow shop scheduling problems. Vaessens (Glover 1996) showed that tabu search methods (in
specific job shop scheduling cases) are superior over other approaches such as simulated
annealing, genetic algorithms, and neural networks.

CONCLUSION

Since job shop scheduling problems fall into the class of NP-complete problems, they
are among the most difficult to formulate and solve. Operations Research analysts and
engineers have been pursuing solutions to these problems for more than 35 years, with
varying degrees of success.
While they are difficult to solve, job shop scheduling problems are among the most
important because they impact the ability of manufacturers to meet customer demands and
make a profit. They also impact the ability of autonomous systems to optimize their
operations, the deployment of intelligent systems, and the optimizations of communications
systems. For this reason, operations research analysts and engineers will continue this pursuit
well into the next coming centuries.
Limitations of traditional techniques:
In the traditional approaches the following limitations are observed:
• Most traditional techniques that give optimal solutions apply only to problems of very
small size.
• These techniques require excessive computation time and are not practical for use on
a daily basis.
• The traditional techniques are not efficient in handling multiple objectives.
• The convergence to an optimal solution depends on the chosen initial random
solution.
• These techniques start with a single point and are not efficient when practical search
space is too large.
• The results tend to stick with local optima and follow a deterministic rule.
• Most scheduling problems are NP-hard, and this degrades the performance of
traditional operations research techniques and also modelling the method is a difficult
task.
• In branch and bound, the number of decision nodes are numerous depending upon the
problem size and thus requiring large computations.
• The performance of heuristic is suitable as long as the operating characteristics and
objectives of the system remain the same.
These limitations urge the researchers to implement nontraditional optimization techniques in
the domain.
Merits of Non Traditional Techniques
The following strengths of the nontraditional techniques are observed over the traditional
techniques.
• The nontraditional techniques yield a global optimal solution
• The techniques use a population of points during search.
• Initial populations are generated randomly which enable to explore the search space.
• The techniques efficiently explore the new combinations with available knowledge to
find a new generation.
• The objective functions are used rather than their derivatives.
72


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