Operational Research: An International Journal

Minimisation of total tardiness for identical parallel machine scheduling using geneticalgorithm

--Manuscript Draft--

Manuscript Number:

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

Corresponding Author SecondaryInformation:

Corresponding Author's Institution: Institute of Space Technology

Corresponding Author's SecondaryInstitution:

First Author: Imran Ali Chaudhry, PhD

First Author Secondary Information:

Order of Authors: Imran Ali Chaudhry, PhD

Abdul Munem Khan, PhD

Abid Ali Khan

Order of Authors Secondary Information:

Funding Information:

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.

Suggested Reviewers:

Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation

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: iman_chaudhry@yahoo.com

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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.

References

Anghinolfi D, Paolucci M (2007) Parallel machine total tardiness scheduling with a new

hybrid metaheuristic approach Comput Oper Res 34:3471-3490

doi:10.1016/j.cor.2006.02.009

Armentano VA, Yamashita DS (2000) Tabu search for scheduling on identical parallel

machines to minimize mean tardiness J Intell Manuf 11:453-460

doi:10.1023/A:1008918229511

Azizoglu M, Kirca O (1998) Tardiness minimization on parallel machines Int J Prod

Econ 55:163-168 doi:10.1016/S0925-5273(98)00034-6

Baker KR, Bertrand JWM (1982) A dynamic priority rule for scheduling against due-

dates J Oper Manag 3:37-42 doi:10.1016/0272-6963(82)90020-1

Barkhi R, Rolland E, Butler J, Fan W (2005) Decision support system induced guidance

for model formulation and solution Decis Support Syst 40:269-281

doi:10.1016/j.dss.2003.12.006

Bilge Ü, Kıraç F, Kurtulan M, Pekgün P (2004) A tabu search algorithm for parallel

machine total tardiness problem Comput Oper Res 31:397-414

doi:10.1016/S0305-0548(02)00198-3

Biskup D, Herrmann J, Gupta JND (2008) Scheduling identical parallel machines to

minimize total tardiness Int J Prod Econ 115:134-142

doi:10.1016/j.ijpe.2008.04.011

Briand LC, Feng J, Labiche Y (2002) Using genetic algorithms and coupling measures

to devise optimal integration test orders. Paper presented at the Proceedings of

the 14th international conference on Software engineering and knowledge

engineering, Ischia, Italy,

Chaudhry IA (2010) Minimizing flow time for the worker assignment problem in

identical parallel machine models using GA Int J Adv Manuf Technol 48:747-

760 doi:10.1007/s00170-009-2323-1

Chaudhry IA A Genetic Algorithm Approach for Process Planning and Scheduling in

Job Shop Environment. In: Proceedings of the World Congress on Engineering

2012, London, UK, 4-6 July 2012a. pp 1433-1437

Chaudhry IA (2012b) Job shop scheduling problem with alternative machines using

genetic algorithms J Cent South Univ 19:1322-1333 doi:10.1007/s11771-012-

1145-8

Chaudhry IA, Drake PR (2008) Minimizing flow-time variance in a single-machine

system using genetic algorithms Int J Adv Manuf Technol 39:355-366

doi:10.1007/s00170-007-1221-7

Chaudhry IA, Drake PR (2009) Minimizing total tardiness for the machine scheduling

and worker assignment problems in identical parallel machines using genetic

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

algorithms Int J Adv Manuf Technol 42:581-594 doi:10.1007/s00170-008-1617-

z

Chaudhry IA, Mahmood S No-wait flowshop scheduling using genetic algorithm. In:

Proceedings of the World Congress on Engineering 2012, London. UK, 4-6 July

2012. pp 1539-1542

Cheung S-O, Tong TK-L, Tam C-M (2002) Site pre-cast yard layout arrangement

through genetic algorithms Automat Constr 11:35-46 doi:10.1016/S0926-

5805(01)00044-9

Davis L Job Shop Scheduling with Genetic Algorithms. In: Proceedings of the 1st

International Conference on Genetic Algorithms, 1985. L. Erlbaum Associates

Inc., 657084, pp 136-140

Davis L (1991) Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York,

USA

Demirel T, Ozkir V, Demirel NC, Tasdelen B A genetic algorithm approach for

minimizing total tardiness in parallel machine scheduling problems. In:

Proceedings of the World Congress on Engineering 2011, London, UK, 6-8, July

2011. pp 1190-1193

Eusuff M, Ostfeld A, Lansey K (2000) An overview of HANDSS: Hula aggregated

numerical decision support system. In: Building Partnerships. American Society

of Civil Engineers, pp 1-6. doi:10.1061/40517(2000)30

Evolver: The genetic algorithm super solver V (1998). Palisade Corporation, New York,

USA

Fisher M (1976) A dual algorithm for the one-machine scheduling problem Math

Program 11:229-251 doi:10.1007/BF01580393

Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning.

Addison-Wesley Longman Publishing Co., Inc.,

Hayat N, Wirth A (1997) Genetic algorithms and machine scheduling with class setups

Int J Comput Eng Manage 5:10-23

He D, Grigoryan A (2002) Construction of double sampling s-control charts for agile

manufacturing Qual Reliab Eng Int 18:343-355 doi:10.1002/qre.466

Hegazy T, Ersahin T (2001) Simplified spreadsheet solutions II: Overall schedule

optimization J Constr Eng M 127:469-475 doi:10.1061/(ASCE)0733-

9364(2001)127:6(469)

Hegazy T, Kassab M (2003) Resource optimization using combined simulation and

genetic algorithms J Constr Eng M 129:698-705 doi:10.1061/(ASCE)0733-

9364(2003)129:6(698)

Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan

Press, Ann Arbor, MI

Hu P-C (2004) Minimising total tardiness for the worker assignment scheduling

problem in identical parallel-machine models Int J Adv Manuf Technol 23:383-

388 doi:10.1007/s00170-003-1716-9

Hu P-C (2006) Further study of minimizing total tardiness for the worker assignment

scheduling problem in the identical parallel-machine models Int J Adv Manuf

Technol 29:165-169 doi:10.1007/s00170-004-2487-7

Jeong SJ, Lim SJ, Kim KS (2006) Hybrid approach to production scheduling using

genetic algorithm and simulation Int J Adv Manuf Technol 28:129-136

doi:10.1007/s00170-004-2345-7

Nanvala H (2011) Use of genetic algorithm based approaches in scheduling of FMS: A

review Int J Eng Sci Techol 3:1936-1942

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Nassar K (2005) Evolutionary optimization of resource allocation in repetitive

construction schedules ITcon 10:265-273

Niu Q, Zhou T, Wang L (2010) A hybrid particle swarm optimization for parallel

machine total tardiness scheduling Int J Adv Manuf Technol 49:723-739

doi:10.1007/s00170-009-2426-8

Root GJ (1965) Scheduling with deadlines and loss functions on k parallel machines

Manage Sci 11:460-475

Ruiz R, Maroto C Flexible manufacturing in the ceramic tile industry. In: Proceedings

of Eighth International Workshop on Project Management and Scheduling,

Valencia, Spain, 2001. p 301−304

Sadegheih A (2007) Sequence optimization and design of allocation using GA and SA

Appl Math Comput 186:1723-1730 doi:10.1016/j.amc.2006.08.078

Saranga H, Kumar UD (2006) Optimization of aircraft maintenance/support

infrastructure using genetic algorithms - level of repair analysis Ann Oper Res

143:91-106 doi:10.1007/s10479-006-7374-1

Shim S-O, Kim Y-D Minimizing total tardiness in an identical-parallel machine

scheduling problem. In: Proceedings of the Fifth Asia Pacific Industrial

Engineering and Management Systems Conference, Gold Coast, Australia,

2004. pp 31.21.31-31.21.37

Shim S-O, Kim Y-D (2007) Scheduling on parallel identical machines to minimize total

tardiness Eur J Oper Res 177:135-146 doi:10.1016/j.ejor.2005.09.038

Shim S-O, Kim Y-D (2008) A branch and bound algorithm for an identical parallel

machine scheduling problem with a job splitting property Comput Oper Res

35:863-875 doi:10.1016/j.cor.2006.04.006

Shiue Y-R, Guh R-S (2006) Learning-based multi-pass adaptive scheduling for a

dynamic manufacturing cell environment Robot CIM-Int Manuf 22:203-216

doi:10.1016/j.rcim.2005.03.004

Shum Y-S, Gong D-C (2007) The application of genetic algorithm in the development

of preventive maintenance analytic model Int J Adv Manuf Technol 32:169-183

doi:10.1007/s00170-005-0314-4

Tanaka S, Araki M (2008) A branch-and-bound algorithm with Lagrangian relaxation to

minimize total tardiness on identical parallel machines Int J Prod Econ 113:446-

458 doi:10.1016/j.ijpe.2007.10.006

Wei M, Deng GL, Xu ZH, Gu XS (2012) Parallel machine tardiness scheduling based

on improved discrete differential evolution Adv Mat Res 459:266-270

doi:10.4028/www.scientific.net/AMR.459.266

Whitley D, Kauth K GENITOR: A different genetic algorithm. In: Proceedings of the

1988 Rocky Mountain Conference on Artificial Intelligence, 1988. pp 118-130

Yalaoui F (2012) Minimizing total tardiness in parallel-machine scheduling with release

dates Int J Appl Evol Computat 3:21-46 doi:10.4018/jaec.2012010102

Yalaoui F, Chu C (2002) Parallel machine scheduling to minimize total tardiness Int J

Prod Econ 76:265-279 doi:10.1016/S0925-5273(01)00175-X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65