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Minimisation of total tardiness for identical parallel machine scheduling using geneticalgorithm
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Full Title: Minimisation of total tardiness for identical parallel machine scheduling using geneticalgorithm
Article Type: Original Paper
Corresponding Author: Imran Ali Chaudhry, PhDInstitute of Space TechnologyPAKISTAN
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Corresponding Author's Institution: Institute of Space Technology
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First Author: Imran Ali Chaudhry, PhD
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Order of Authors: Imran Ali Chaudhry, PhD
Abdul Munem Khan, PhD
Abid Ali Khan
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Abstract: In the recent years parallel machine scheduling has received increased attention. Thispaper considers scheduling of jobs on a set of parallel machines where the objective isto minimize total tardiness. A spread sheet based genetic algorithm (GA) approach isproposed for the problem. The proposed approach is a domain independent generalpurpose approach which has been effectively used to solve this class of problem. Theperformance of GA is compared with branch & bound and particle swarm optimizationapproaches. Two set of problems having 20 and 25 jobs with number of parallelmachines equal to 2, 4, 6, 8 and 10 are solved with the proposed approach. Eachcombination of number of jobs and machines consist of 125 benchmark problems, thusa total for 2250 problems are solved. The results obtained by the proposed approachare comparable with two earlier approaches. It is also demonstrated how a simple GAcan be used to produce results that are comparable with problem specific approach.The proposed approach can also be used to optimise any objective function withoutchanging the GA routine.
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Minimisation of total tardiness for identical parallel machine scheduling using genetic
algorithm
Imran Ali Chaudhry
Institute of Space Technology
Sector H-12
1 Islamabad Highway
Islamabad
PAKISTAN
Abstract
In the recent years parallel machine scheduling has received increased attention. This
paper considers scheduling of jobs on a set of parallel machines where the objective is
to minimize total tardiness. A spread sheet based genetic algorithm (GA) approach is
proposed for the problem. The proposed approach is a domain independent general
purpose approach which has been effectively used to solve this class of problem. The
performance of GA is compared with branch & bound and particle swarm optimization
approaches. Two set of problems having 20 and 25 jobs with number of parallel
machines equal to 2, 4, 6, 8 and 10 are solved with the proposed approach. Each
combination of number of jobs and machines consist of 125 benchmark problems, thus
a total for 2250 problems are solved. The results obtained by the proposed approach
are comparable with two earlier approaches. It is also demonstrated how a simple GA
can be used to produce results that are comparable with problem specific approach.
The proposed approach can also be used to optimise any objective function without
changing the GA routine.
Keywords: Parallel machine scheduling; Genetic Algorithm (GA); Total tardiness; Scheduling
Corresponding Author:
Imran Ali Chaudhry
E-mail: [email protected]
Tel: +92-300-4279895
Title Page
Minimisation of total tardiness for identical parallel machine
scheduling using genetic algorithm
In the recent years parallel machine scheduling has received increased attention.
This paper considers scheduling of jobs on a set of parallel machines where the
objective is to minimize total tardiness. A spread sheet based genetic algorithm
(GA) approach is proposed for the problem. The proposed approach is a domain
independent general purpose approach which has been effectively used to solve
this class of problem. The performance of GA is compared with branch & bound
and particle swarm optimization approaches. Two set of problems having 20 and
25 jobs with number of parallel machines equal to 2, 4, 6, 8 and 10 are solved
with the proposed approach. Each combination of number of jobs and machines
consist of 125 benchmark problems, thus a total for 2250 problems are solved.
The results obtained by the proposed approach are comparable with two earlier
approaches. It is also demonstrated how a simple GA can be used to produce
results that are comparable with problem specific approach. The proposed
approach can also be used to optimise any objective function without changing
the GA routine.
Keywords: Parallel machine scheduling; Genetic Algorithm (GA); Total
tardiness; Scheduling
1. Introduction
Parallel machine scheduling problem is an important class of problem. It is a
generalization of the single machine scheduling problem. The parallel machines
scheduling problem can be classified as identical, uniform or unrelated parallel
machines scheduling problem. If the machines are identical, then all machines are
identical in terms of their speed and each and every job will take same amount of
processing time on each of the parallel machines. Uniform machines work at different
speeds, i.e., the processing time of each job differs by a constant factor for the
individual machines. If the machines are unrelated, then there is no relation between the
processing times of the jobs and the machines. This may be due to technological
differences of the machines, different features of the jobs, etc.
Parallel machine scheduling comes down to assigning each operation to one of the
machines and sequencing the operations assigned to the same machine. Many real life
problems can be modelled as parallel machine scheduling problem e.g. bank counters
can serve several customer at the same time; automotive assembly lines can assemble
several identical or non-identical car models; and a gas station can serve several cars
simultaneously.
Job tardiness is defined as the amount of completion time in excess of the due date. It is
one of the most common performance measures encountered in practical problem.
Minimizing total tardiness is one of the most important criteria in many manufacture
systems, especially in the current situation where competition is becoming more and
more intensive.
In this paper we consider scheduling of n jobs to be processed on m number of identical
parallel machines to minimize total tardiness of the jobs. The rest of the paper is
organized as follows: In section 2, literature review for identical parallel machine with
total tardiness as objective measure is presented. Problem and assumptions are defined
Blinded ManuscriptClick here to view linked References
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in section 3. Section 4 gives a brief introduction of GAs and the architecture of the
proposed approach. Experimental results are discussed in section 5 and finally the
paper is concluded in section 6.
2. Literature review: identical parallel machine with total tardiness as
objective measure
Due to its application in manufacturing, identical parallel machine total tardiness
problem has received wide attention from the researchers. Researchers have used wide
variety of algorithms, ranging from exact methods to hybrid algorithms. Root (1965)
considers scheduling of jobs on identical parallel machines in a single stage production
where all jobs have a common due date. Root presented a constructive method to
minimize the total tardiness. Azizoglu and Kirca (1998) proposed a branch and bound
algorithm combined with an efficient lower bounding scheme to minimize total
tardiness for n jobs to be processed on m identical parallel machines. The proposed
algorithm can find optimal solution for problems with up to 15 jobs.
Armentano and Yamashita (2000) presented a tabu search approach for scheduling jobs
on identical parallel machines with the objective of minimizing the mean tardiness.
Yalaoui and Chu (2002) have developed a branch-and-bound algorithm to minimize
total tardiness a set of jobs has to be scheduled, without pre-emption, on identical
parallel machines. They introduce some job dominance checks to improve their
algorithm and optimally solve problem instances with up to 15 jobs and three machines.
Bilge et al. (2004) also apply tabu search for scheduling a set of independent jobs with
sequence dependent setups on a set of uniform parallel machines such that total
tardiness is minimized. The authors show that the results obtained are considerably
better than those reported previously and constitute the best solutions known for the
benchmark problems as to date.
Hu (2004, 2006) extended the classical identical parallel machine scheduling problem
by including an additional dimension of worker assignment to machines. Hu suggested
that certain relationship exists between job processing time and the number of workers
assigned to the job. Therfore, the processing time is no longer a constant but is related
to the number of workers assigned to work on the job. Hu solves the problem in two
phases of job scheduling. The SES (SPT, EDD, SLACK) heuristic solves job
scheduling problem while, the largest marginal contribution (LMC) procedure is used to
minimise the total tardiness for the worker assignment phase.
Shim and Kim (2004; 2007) also developed branch and bound algorithm using
dominance rule (pruning rules) scheduling n independent jobs on m identical parallel
machines for the objective of minimizing total tardiness of the jobs. Anghinolfi and
Paolucci (2007) proposed a hybrid meta heuristic (HMH) approach that integrates
several features from tabu search (TS), simulated annealing (SA) and variable
neighbourhood search (VNS) by extending the problem by including non-zero ready
times and sequence dependent setups. Shim and Kim (2008) used a branch-and-bound
algorithm for solving a parallel machines scheduling problem with a job splitting
property. The author assumed that a job can be split into sub-jobs and these sub-jobs
can be processed independently on parallel machines.
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Tanaka and Araki (2008) combine Lagrangian relaxation with branch-and-bound to
minimize total tardiness on identical parallel machines. Authors show that optimal
solution can be obtained for instances with up to 25 jobs and any number of machines.
Biskup et al. (2008) considered the total tardiness minimization as performance measure
of a parallel machines scheduling problem, and then developed a new general heuristic
for finding optimal or near-optimal schedules. Chaudhry and Drake (2009) also
considered machine scheduling and worker assignment problems in identical parallel
machines to minimize total tardiness using genetic algorithms.
Niu et al. (2010) proposed hybrid particle swarm optimization (PSO) for identical
parallel machine scheduling with total tardiness as objective function. The authors
introduced a clonal selection algorithm (CSA) into PSO to enhance the performance.
Demirel et al. (2011) investigated parallel machine scheduling problem to minimize
total tardiness and propose a GA to solve the problem. Yalaoui (2012) considered
scheduling on n jobs on m identical parallel machines with job release times to
minimize total tardiness and propose an exact resolution, an Ant Colony Algorithm
(ACO), a Tabu Search (TS) method, a set of Heuristics based on priority rules and an
adapted Biskup et al. (2008) (BHG) method. Wei et al. (2012) proposed an improved
discrete differential evolution DDE_VND. The proposed algorithm hybridizes discrete
differential evolution (DDE) with variable neighbourhood descent (VND) to enhance its
local search ability. A modified due date (MDD) constructive heuristic (Baker and
Bertrand 1982) is employed to generate an initial solution in the algorithm to accelerate
the convergence of the algorithm.
3 Problem and assumptions
The problem considered in this paper can be formally described as scheduling n
independent jobs 𝑁 = 1, 2,…..…, 𝑛 on 𝑚 identical parallel machines 𝑀=1, 2,……,
𝑚 to minimize the total tardiness for jobs. A job is said to be tardy if its completion
time Ci is greater than its due date di and is measured by Ci - di. If a job is not tardy, its
tardiness is set to 0. Therefore, the tardiness of job i is given by max (Ci - di; 0).
The optimal solution satisfies following conditions:
(1) All jobs are ready at time zero.
(2) Each job has only one operation and a deterministic processing time.
(3) No job preemption (splitting is allowed);
(4) The number of jobs and machines is fixed.
(5) Each job has its own due date;
(6) Any machine can process any job.
(7) No machine may process more than one job at a time.
(8) Machine setup times are negligible.
(9) Transportation time between the machines is negligible.
(10) Jobs can wait for processing and each job has the same priority.
(11) All machines are always available, therefore machine breakdown is not
considered.
4 Genetic algorithms
Genetic Algorithms (GAs) are adaptive heuristic search algorithm based on the
evolutionary ideas of natural selection and genetics. GAs were first introduced by
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Holland (1975) at the University of Michigan. Genetic algorithms (GAs) are the main
paradigm of evolutionary computing. GAs are inspired by Darwin's theory about
evolution – the "survival of the fittest". In nature, competition among individuals for
scanty resources results in the fittest individuals dominating over the weaker ones.
GAs are the ways of solving problems by mimicking processes nature uses; i.e.,
selection, crossover, mutation and accepting, to evolve a solution to a problem. GAs
are therefore adaptive heuristic search based on the evolutionary ideas of natural
selection and genetics.
Since the late 1980s there has been a growing interest in genetic algorithms (GAs)
optimisation algorithms. As they are based on ‘evolutionary learning’, the technique
comes under the heading of artificial intelligence. Such algorithms have been used
widely for parameter optimisation, classification & learning and production scheduling.
A detailed comprehensive introduction to GAs is given in Goldberg (1989).
Applications of GAs are found in computational science, phylogenetic, economics,
engineering, chemistry, bioinformatics, manufacturing, physics and numerous other
fields. Detailed analysis of various application is given in Davis (1991).
Schedules can improve their ‘fitness’ by reproduction and survival of the fittest, in
exactly the same way as living creatures do. A key advantage of GAs is that they
provide a ‘general purpose’ schedule optimiser, with the peculiarities of any particular
application being accounted for in the calculation of the fitness of a schedule without
disturbing the logic of the standard GA. This means that it is a relatively straightforward
matter to adapt the software implementation of the method to meet the needs of a
particular application.
One of the earliest application of GA in scheduling has been reported by Davis (1985).
A recent review of GAs application in production scheduling has been presented by
Chaudhry and Drake (2008; 2009), Chaudhry (2010) and Nanvala (2011).
In this paper we use a commercial GA Evolver (Evolver: The genetic algorithm super
solver 1998) that functions as an add-in Microsoft Excel™ spreadsheet. Spreadsheet’s
built in functions are used to build the shop model with identical parallel machines.
Evolver generates a population of trial solutions corresponding to the population size.
The solution is continually improved by genetic algorithm in each successive trial.
Each possible solution becomes an independent "organism" that can "breed" with other
organisms. The spreadsheet model acts as an environment for the organisms,
determining which are "fit" enough to survive based on their results. The variables
(adjustable cells), the objective function and constraints are specified within the
spreadsheet environment. Figure 1 shows the spreadsheet-Evolver construction.
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Figure 1 Spreadsheet-Evolver construction.
Various benefits have been accrued from the proposed approach. The program runs in
the background thus freeing the user to work in the foreground. Furthermore, due to the
familiar layout of the spread sheet software helps the user to use the software easily and
carry out what-if analysis. Evolver™ has been used for various problems such as
scheduling (Chaudhry 2010; Chaudhry 2012a; Chaudhry 2012b; Chaudhry and Drake
2008; Chaudhry and Drake 2009; Chaudhry and Mahmood 2012; Hayat and Wirth
1997; Hegazy and Ersahin 2001; Jeong et al. 2006; Nassar 2005; Ruiz and Maroto
2001; Sadegheih 2007; Shiue and Guh 2006), maintenance (Saranga and Kumar 2006;
Shum and Gong 2007), resource optimization (Hegazy and Kassab 2003), construction
of double sampling s-control charts (He and Grigoryan 2002), optimal integration of test
orders in object-oriented systems (Briand et al. 2002), decision support systems (Barkhi
et al. 2005; Eusuff et al. 2000) and site precast layout arrangement (Cheung et al. 2002).
4.1 Genetic Algorithm Design
4.1.1 Chromosome Representation
In this paper single stage identical parallel machine scheduling problem has been
addressed. The chromosome for the problem consists of two parts: part 1 is the list of
jobs while part two has the machine associated to each job. An example of 6 jobs A-B-
C-D-E-F to be processed on 4 identical parallel machines is given in Fig 2. The
chromosome contains all the jobs and machines required for each of the job.
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Figure 2: Chromosome representation for single stage identical parallel machine model
The first six genes of the chromosome represent jobs while the next six genes are the
machines associated with each jobs. The chromosome would be read as: Job B is to be
processed on machine 4; job A on machine 3; job E on machine 1; job F on machine 1;
job D on machine 2 and job C on machine 3. For the first six genes the objective is to
find the right order or a sequence of jobs while for last six genes we need to find the
best combination of machine for each job such that total tardiness is minimised. Each
of the block is calculated at a different location within the spreadsheet and linked
together to calculate the objective function i.e., total tardiness.
4.1.2 Reproduction / selection
Evolver GA implements steady-state reproduction as has been used in GENITOR GA
(Whitley and Kauth 1988). Unlike generational replacement technique where all
parents in the population are replaced by child solutions, steady-state reproduction
replaces only one organism in each iteration. The advantage of using steady-state
reproduction is that all the genes are not lost, as is the case in generational replacement
where after replacement, many of the best individuals may not produce at all and their
genes may be lost. Steady-state reproduction is a better model of what happens in longer
lived species in nature. This allows parents to nurture and teach their offspring, but also
gives rise to competition between them (Whitley and Kauth 1988).
Objective function value, total tardiness in this case is used to measure the fitness of a
particular chromosome. In order to achieve a smoother probability curve, rank-based
mechanism is used to choose two parents. This is based on roulette wheel selection
where each parent has a slot proportional to its fitness. This prevents too quick
convergence of GA whereby preventing good organisms from completely dominating
the evolution from an early point.
4.1.3 Crossover operator
Crossover is an important element of GAs. Crossover is a process of taking more than
one parent solutions and producing a child solution from them.
For the first six genes in Fig 2, i.e., jobs order crossover developed by Davis (1991) is
used. The idea behind this operator is to preserve the relative order of the genes.
Figure 3 illustrates an order crossover. This operator is based on using a bit string (0-1)
template to determine which parent will contribute to make the child. This fills some
positions on offspring by copying the elements from the first parent P1 wherever the
binary template contains “1” in the same position as they appear in parent P1. The
elements from P1 associated with “0” in the template appear in the same order in the
offspring as they appear in second parent P2.
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Figure 3: Order Crossover
In Fig. 3, the elements associated with “1” in P1 are at position A, D, F, and G. These
elements are inherited by the offspring in the same positions. The remaining elements
associated with “0” are B, C, E, and H. These elements appear in the same order in the
offspring as they appear in P2.
For machine assignment i.e. genes 7-12 in Fig 2, uniform crossover is implemented.
Uniform crossover works better than one- and two-point crossovers (Davis 1991).
Figure 4 describes uniform crossover. The uniform crossover uses a fixed mixing ratio
between two parents. Uniform crossover enables the parent chromosomes to contribute
the gene level rather than the segment level.
Fig 4: Uniform crossover
Consider two parents in Fig. 4. Parent P1 has been coloured yellow while parent P2
green. A random mask is generated corresponding to the crossover rate. If the
crossover rate is 0.6, approximately five genes in the offspring will come from P1 while
three genes will come from P2. The offspring is produced by taking bit from P1 if the
corresponding mask bit is 1 or the bit from P2 if the corresponding bit is 0. The colour
of the child chromosome represents the mixing of genes if the crossover rate is 0.6.
For machine assignment (genes 7–12 in Fig. 2), we have a set of variables that are to be
adjusted and varied independently of one other. Uniform crossover is considered better
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for this type of situations as it preserves schema and can generate any schema from the
two parents.
4.1.4 Mutation operator
The purpose of the mutation is to ensure that diversity is maintained in the population.
It gives random movement about the search space, thus preventing the GA becoming
trapped in “blind corners” or “local optima” during the search. For the first six genes
that job ordering (genes 1-6 in Fig. 2), GA performs order-based mutation. In this
mutation, two tasks are selected at random and their positions are swapped. The
“mutation rate” determines the probability that mutation is applied after a crossover.
The number of swaps performed is increased or decreased proportionately to the
increase and decrease in the mutation rate setting.
For the next six genes (genes 7-6 in Fig. 2), mutation is performed by looking at each
variable individually. This is done by randomly generating a random number between 0
and 1 for each of the genes. If a gene gets a number that is less than or equal to the
mutation rate (for example, 0.06), then that gene is mutated. Mutating a variable
involves replacing it with a randomly generated value (within its valid min–max range).
5 Experimental results
In this section, the performance of the proposed GA approach is evaluated on a set of
benchmark instances proposed by Tanaka and Araki (2008) available online at
http://turbine.kuee.kyoto-u.ac.jp/~tanaka/index-e.html. Problem instances are generated
by the standard method by Fisher (1976). First, the integer processing times 𝑝𝑗(1≤ j ≤
n ) are generated by the uniform distributions in [1, 100]. Then, let P = ∑ 𝑝𝑗𝑛𝑗=1 and
the integer due dates 𝑑𝑗(1≤ j ≤ n ) are generated by the uniform distributions in [P(1 -
- R / 2 ) / m, P (1 - + R/2)/m]. The number of jobs n, the number of machines m , the
tardiness factor and the range of due dates R are changed by n = 20, 25; m = 2, 3, 4,
5, 6, 7, 8, 9, 10; = 0.2, 0.4, 0.6, 0.8, 1.0; and R = 0.2, 0.4, 0.6, 0.8, 1.0. For every
combination of n, m, and R, five problem instances are generated. Thus, for each
combination of n and m , 125 problem instances are generated. Hence a total of 2250
problems are solved. Optimal solutions for the data set are given by Tanaka and Araki
(2008) and available on web site.
The problems have been simulated on a Core 2 Duo 2.7 GHz computer having 1.5 GB
RAM. For each of the run, the following parameters have been used: population
size=65, crossover rate=0.65, mutation rate=0.001, and stopping criteria = 100,000
trials, which correspond to 2 min 35 s on a Core 2 Duo 2.7 GHz computer having 1.5
GB RAM. Same set of parameter setting as described earlier have been used for all GA
runs. The GA parameters have been selected based upon earlier findings by Chaudhry
and Drake (2008) where an empirical analysis to study the effect of crossover rate,
mutation rate, and the population size was carried out. The results demonstrate that GA
performance is insensitive to the crossover and mutation rate. Mutation rate possessed a
“range” over which its value is suitable rather than more precise value. Furthermore,
performance is fairly insensitive to population size provided the mutation rate is in the
“good” range.
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The performance of proposed GA approach is compared with branch-and-bound
algorithm with Lagrangian relaxation proposed by Tanaka and Araki (2008)Tanaka and
Araki [12] and a hybrid particle swarm optimization algorithm combining clonal
selection and PSO (CSPSO) proposed by Niu et al. (2010). Niu et al. compared the
performance of CSPSO with branch-and-bound by Tanaka and Araki only for 2nd and
4th instance each for m = 2, 4, 6, 8 and 10 for n = 20. Thus a total of 250 problems are
solved by Niu et al. The comparative performance of CSPSO and proposed GA
approach against optimal solutions are given in Table 1.
Table 1 Comparison of proposed GA approach with CSPSO for n = 20
No of
machines
Comparison of proposed GA with CSPSO Number of Optimal solutions
Total
problems Same Worse Better Niu et al. This GA
m = 2 50 46 2 2 48 48
m = 4 50 44 3 3 44 44
m = 6 50 48 1 1 49 48
m = 8 50 46 4 - 49 45
m = 10 50 48 - 2 47 49
Total 250 232 10 8 237 234
From Table 1 we can see that CSPSO found optimal solution for 237 problems (94.80%
optimal solutions) as compared to proposed GA which found optimal solution for 234
problems (93.60 % optimal solutions). We can deduce from the table that the
performance of the proposed approach is comparable with CSPSO. GA approach
proposed in this found worse solutions for 10 problems than CSPSO while for eight
problems proposed approach found better solutions than CSPSO.
Niu et al. (2010) solved only a small portion of benchmark problems given by Tanaka
and Araki (2008). All 1125 benchmark problems given by Tanaka and Araki for n = 20
were solved with proposed GA. Table 2 gives the comparative results of the two
approaches. The average, minimum and maximum times to reach the best solutions are
also given in Table 2.
Table 2 Comparison of proposed GA approach with optimal solutions (Tanaka and Araki
2008) for n = 20
No of
machines
Total
problems
Total
optimal
solution
found
% age
optimal
solutions
Avg
error
Time to reach best solution (secs)
Average Minimum Maximum
m = 2 125 121 96.80% 0.8132% 32 1 152
m = 3 125 121 96.80% 0.0185% 39 1 153
m = 4 125 113 90.40% 0.8137% 38 1 152
m = 5 125 112 89.60% 0.5196% 39 1 157
m = 6 125 116 92.80% 2.8320% 34 1 123
m = 7 125 116 92.80% 3.8294% 34 1 152
m = 8 125 119 95.20% 0.6271% 30 1 150
m = 9 125 120 96.00% 0.7519% 19 1 158
m = 10 125 120 96.00% 0.7407% 21 1 154
Total 1125 1058 94.04% 1.2162% 32 1 150
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From Table 2 we can see that the proposed GA was able to find optimal solution for
1058 problems (94.04%). The average, minimum and maximum times to find the best
solutions are 32, 1 and 150 secs respectively.
Tanaka and Araki also gives benchmark problems for n = 25. The same were also
attempted with the proposed GA approach. Table 3 gives the comparative results of
proposed approach and the optimal solutions. Proposed GA was able to find optimal
solutions for 840 problems (74.67% problems) out of a total of 1125 benchmark
problems.
Table 3 Comparison of proposed GA approach with optimal solutions (Tanaka and Araki
2008) for n = 25
No of
machines
Total
problems
Total
optimal
solution
found
% age
optimal
solutions
Avg
error
Time to reach best solution (secs)
Average Minimum Maximum
m = 2 125 108 86.40% 0.2765% 40 1 152
m = 3 125 90 72.00% 0.5013% 47 1 158
m = 4 125 81 64.80% 0.6424% 50 1 158
m = 5 125 72 57.60% 1.0993% 50 1 133
m = 6 125 85 68.00% 1.8320% 52 1 161
m = 7 125 94 75.20% 1.2100% 67 4 160
m = 8 125 96 76.80% 1.4552% 51 3 161
m = 9 125 99 79.20% 1.1942% 60 3 159
m = 10 125 115 92.00% 1.0224% 43 3 161
Total 1125 840 74.67% 1.0259% 51 2 156
From Table 3 we can see that the percentage of optimal solutions found was better for
higher number of parallel machines as compared to lower number of parallel machines.
The average, minimum and maximum times to find the best solutions are 51, 2 and 156
secs respectively.
Keeping in view the general purpose nature of the proposed approach and the
computational results, it can be concluded that the proposed approach generates high-
quality schedules in a timely fashion. Also the qualities of solutions found by the
proposed approach are comparable with previous approaches for the same problem.
6 Conclusions
In this study a general purpose GA is proposed for a class of scheduling problems to
minimize total tardiness on identical parallel machines. The GA has been implemented
in a spreadsheet environment, has been found to be easy to implement and customizable
to any condition without changing the GA routine, which makes it a domain-
independent approach. Furthermore, spreadsheet environment also enables carrying out
what-if analysis.
For n = 20, proposed GA approach found optimal solution for 93.60% of the problems
as compared to 94.80% for Niu et al. (2010). Similarly, the solutions found by
proposed approach are also comparable to branch and bound approach by Tanaka and
Araki (2008). The performance of proposed GA for n = 25, was worse as compared to
Tanaka and Araki, however being a general purpose approach, being able to be used for
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a wide class of problems, the results are considered to be comparable with previous
approaches. The proposed GA approach is truly general purpose as the spreadsheet
model is customizable to add more jobs or machines without changing the basic GA
routine.
The key advantage of GAs portrayed here is that they provide a general purpose
solution to the scheduling problem which is not problem-specific, with the peculiarities
of any particular scenario being accounted for in fitness function without disturbing the
logic of the standard optimization routine. The GA can be combined with a rule set to
eliminate undesirable schedules by capturing the expertise of the human scheduler.
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