An Evolutionary Algorithmic Approach for Single Machine Early Tardy Scheduling Problem

Most of the real world scheduling problems incorporates Just-In-Time production philosophy which leads to a growing interest in the development of various nature inspired metaheuristic algorithms. Single Machine Early Tardy scheduling problem (SMETP) is one such problem in which jobs have to be scheduled on a single machine against a restrictive common due date parameter and this problem is strongly a NP-hard combinatorial optimization problem. As job sizes vary from 10 to 1000, problems of larger job sizes cannot be solved by exact algorithms. Hence in this research study, we propose genetic algorithm with variations in local search to find an optimal schedule which jointly minimizes the summation of earliness and tardiness cost penalties of ‘n’ jobs from a common due date by satisfying the three SMETP scheduling properties. The performance of this evolutionary algorithm is validated on the 280 benchmark instances proposed by Biskup and Feldmann for various job sizes and the results show that genetic algorithm works well for smaller job sizes.


INTRODUCTION
Sequencing and scheduling are decision making processes which play a crucial role in manufacturing and production industries.Scheduling jobs on a Single machine against a restrictive common due date to Minimize Early and Tardy Penalties (SMETP) has been studied by many researchers for nearly 3 decades with the principles of Just-In-Time (JIT) inventory management.SMETP problem belongs to a class of scheduling problems formally defined as 1/d/ (# α i E i + β i T i. .Each job has three characteristics namely processing time, earliness penalty cost and tardiness penalty cost associated with it.In this study, we focus on single machine restricted common due date problems where some jobs may be completed before the common due date which is referred to as earliness penalty and after the common due date is referred to as tardiness penalty respectively.Execution of jobs before the common due date may lead to storage of products in the industries, whereas execution of jobs after the common due date may lead to loss of reputation of customer's goodwill.Hence the objective is to find an optimal schedule which exactly finishes execution of jobs on the common due date to jointly minimize the summation of earliness and tardiness penalties costs.
Common due date scheduling problems are categorized into restrictive and unrestrictive ones.In the case of unrestricted common due date scheduling problems, all jobs complete its execution before the common due date.Hence there is no challenge in optimizing algorithms.In our research study, we focus on restrictive single machine common due date scheduling problems.
In this study, we address the issues of minimizing error offset for the evolutionary algorithms, applying local improvement methods in the algorithms to reduce the computation time and when to terminate the algorithm.

Problem formulation and properties of SMETP:
Restrictive common due date scheduling problem is formulated as follows: • 'n' jobs are available for processing at time zero, which have to be processed on a single machine.• Each of the jobs needs exactly one operation.No pre-emption of jobs is allowed.• The processing times p j of the jobs 1...n are deterministic, common due date 'd' is computed as: where h is the restrictive common due date parameter.
• Completion time C j of each job is computed by: (# .p j -1 + p j • A job is referred to as early if its completion time falls below the common due date and the earliness penalty of job j is given by E j = max (0, d -C j ).• A job is tardy if its processing time ends after common due date and the tardiness penalty is computed by T j = max (C j -d, 0) respectively for all jobs j = 1..n.
The objective in our study is to find an optimal schedule σ which jointly minimizes the earliness and tardiness penalties of all jobs closer to the due date is given as: For the restricted SMETP, an optimal schedule should satisfy the following three optimality properties.
Property 1: No idle times are inserted between consecutive jobs (Cheng and Kahlbacher, 1991).
Property 2: The optimal schedule is 'V' shaped around the common due date.But a straddling job may exist, i.e., a job whose execution starts before and finishes after the due date, (Baker and Scudder, 1989).
Property 3: Either the processing time of first job starts at time zero or one job is completed at the due date in the optimal schedule (Hoogeveen and van de Velde, 1991).Due to the complexity of SMETP, branch and bound techniques and several nature inspired metaheuristic algorithms like genetic algorithms, simulated annealing, tabu search, differential evolution, artificial bee colony optimization and hybrid evolutionary algorithms are addressed by various researchers to tackle this problem.

LITERATURE REVIEW
Many researchers have studied about the nature of SMETP which is strongly proved to be a NP hard combinatorial optimization problem.Baker and Scudder review the literature on scheduling models with Early Tardy penalties in 1989.The researchers pointed out that the single-machine scheduling problem with a restricted common due date has never been addressed in the literature.By that time, Hall et al. (1991) proved that this problem is NP-hard.Due to its complexity, many authors addressed this problem using nature inspired metaheuristic methods and compared their results with state-of-the-art metaheuristics.Lee and Kim (1995) developed a parallel genetic algorithm, while James (1997) used Tabu search approach to address this.Biskup and Feldmann (2001) presented 280 benchmarks for the restrictive common due-date problem with general earliness and tardiness penalties.Feldmann and Biskup (2003) studied the restricted E/T problem postponing the schedule by applying different metaheuristics: Evolutionary Search (ES), Simulated Annealing (SA) and Threshold Accepting (TA).Lin et al. (2007) used a sequential exchange approach while Liao and Cheng (2007) proposed a variable neighborhood search for minimizing single machine weighted earliness and tardiness with common due date.Nearchou (2008) used Differential Evolution algorithm.Le and Hong (2013) developed a hybrid metaheuristic Permutation-based Harmony search algorithm by incorporating Variable Neighborhood Search (PHVNS) and demonstrated that their algorithm shows high competitiveness by comparing with some state-of-the-art metaheuristics.

MATERIALS AND METHODS
Proposed algorithm: Genetic algorithm with local improvement for SMETP: John Holland's invention of Genetic algorithm is a population-based metaheuristic evolutionary algorithm that evolves from one population of chromosomes to a new population by natural evolution such as reproduction, crossover and mutation and follows Charles Darwin's "Survival of the fittest".

Sequence representation:
We use permutation encoding mechanism to represent a sequence of jobs.A sequence is mapped into a chromosome with the alleles assuming different and non-negative integer values in the (1..n) interval.For a 5 jobs problem, the complete sequence is represented as ((1) (2) (3) (4) (5)) where [i] is the position of the i th job in the sequence.The objective function is to find a sequence σ which jointly minimizes the sum of early and tardy cost penalties for single machine restrictive common due date scheduling problems.

Initial population generation and fitness evaluation:
Jobs are scheduled according to p/α heuristic for jobs that complete before the common due date and p/β heuristic for jobs that complete after the common due date respectively.This is used to generate the first individual in the initial population.The remaining sequences in initial population are generated by constructive heuristics which places [i] job in all possible combinations.

Reproduction of chromosomes:
We have used roulette wheel selection strategy to select the chromosome with minimum fitness value to evolve from current population to the new population.

Ordered crossover:
We have implemented ordered crossover operator for the mating parents which are selected randomly from the mating pool.Two crossover points for the mating parents are randomly generated to determine the range for crossover.The length of the crossover is in the range with Lower Limit (LL) (1, n-1) Offspring is generated by retaining the elements of the parent that falls within the crossover range and inheriting the remaining elements from the parent in the order in which they appear in that parent beginning from the first position following the second crossover point and the elements are skipped if they are already present in the newly generated offspring.

Mutation:
The resultant offspring represents the sequence σ and the value of the total early and tardy penalties z (σ) is calculated by using Eq.(1).We have implemented sliding mutation strategy followed by pair-wise random swap mutation.Two alleles in a parent are selected based on two randomly generated positions.The allele in position 2 is shifted to the allele in position 1 and the allele in position 1 is shifted right by 1 place and follows the sane order of alleles in the parent and results in new offspring (Fig. 2).
The newly generated offspring again undergoes pair-wise random swap mutation and generates a new offspring σ1.Two alleles in a parent are selected randomly and their positions are swapped and result in a new offspring (Fig. 3).
The fitness function z (σ1) is computed and checked with the fitness value of z (σ).If z (σ1) < z (σ), the newly generated offspring is added to the population set and the sequence with fitness value z (σ) is removed from the population set by applying elitism replacement strategy.If there is no improvement after several generations, the original offspring is added to the population set.
Mutation with local improvement: Each resultant offspring of job size 'n' generated after mutation operation again undergoes adjacent pair-wise swap mutation and yields new sequences (Fig. 4).
The fitness function is computed for all new offsprings and the offspring which returns minimum fitness value is added to the new population set.
Finally out of n * n offsprings, 'n' offsprings are added to the new population set which form the chromosomes for the next generation.This undergoes roulette wheel selection, ordered crossover and 4. Retain the offspring with the minimum fitness value.

Add this offspring to the new population set
Termination criteria: We have generated 2n + n 2 individuals for all job sizes 'n' in each generation and the best 3n individuals are added to the population set.We run our genetic algorithm by fixing the number of generations as 1000.

Materials:
The benchmark instances of restricted single-machine common due date problems are proposed by Biskup and Feldmann (2001) on job sizes n = 10, n = 20, n = 50, n = 100, n = 200, n = 500 and n = 1000.The common due date d is calculated by d = round (SUM_P * h) where round (X) gives the biggest integer which is smaller than or equal to X; Sum_P denotes the sum of the processing times of the n jobs and the parameter h is used to calculate more or less restrictive common due dates.For the following 280 benchmarks we used h = 0.2, h = 0.4, h = 0.6 and h = 0.8.The instances are available at http:// people.brunel.ac.uk/~mastjjb/jeb/orlib/schinfo.html.

RESULTS AND DISCUSSION
Experiments conducted: The developed evolutionary algorithm is implemented in Java language on a computer with 2.27 GHz Intel (R) Core i5 CPU and 3 GB RAM with main memory, running Windows 8.1 operating system with Java NetBeans IDE 8.2.We solved 280 benchmark instances for the different sizes n = 10, n = 20, n = 50, n = 100, n = 200, n = 500 and n = 1000 with h = 0.2, h = 0.4, h = 0.6 and h = 0.8.For all the 280 benchmark problems, parameters and its values chosen for our study are listed in Table 1.

Case I:
The benchmark instances considered from OR-Library by J E Beasley has to schedule 10 jobs in a single machine for the values of common restrictive due date parameter 'h' taking h = 0.2, 0.4, 0.6 and 0.8.* indicates optimal objective function values.In the Table 2, the attributes UB represents known Upperbound function value, COST GA represents the fitness cost value obtained by job scheduling.The deviation in cost percentage is computed as % error = ((COST GA-UB) /UB) *100 and the results are tabulated and given below.

Case III:
The benchmark instances considered have to schedule 50 jobs in a single machine for the values of common restrictive due date parameter 'h' taking h = 0.2, 0.4, 0.6 and 0.8, respectively.
The proposed algorithm generated better results for 50 jobs and the results are listed in Table 6 and 7.

Case IV:
The benchmark instances considered have to schedule 100 jobs in a single machine for the values of common restrictive due date parameter 'h' taking h = 0.2, 0.4, 0.6 and 0.8, respectively.
The proposed algorithm has minimized the early tardy penalty costs for all instances for h = 0.2 and 0.4 and shown in Table 8 while Table 9 shows % error deviation to be high for h = 0.8.

Case V:
The benchmark instances considered have to schedule 200 jobs in a single machine for the values of common restrictive due date parameter 'h' taking h = 0.2, 0.4, 0.6 and 0.8, respectively.
The proposed algorithm has minimized the early tardy penalty costs for all instances for h = 0.2 and some instances for h = 0.4 and the results are tabulated in Table 10.
Results shown in Table 11 shows that % error deviation is high for larger values of h.

Case VI:
The benchmark instances considered have to schedule 500 jobs in a single machine for the values of common restrictive due date parameter 'h' taking h = 0.2, 0.4, 0.6 and 0.8, respectively.
We can infer from Table 12 and 13 that % error obtained is high for 500 job size for all values of common due date restrictive parameter 'h'.

CONCLUSION
In this study, we have carried out several experiments to determine the best crossover rate.Results show clearly that genetic algorithm works well for smaller job sizes 10, 20, 50, 100 and 200, respectively for restrictive common due date parameter h = 0.2 and 0.4 but fails to work for larger job sizes 500 and 1000 for larger values of h.This algorithm can be further extended by incorporating local search techniques to minimize the penalty cost as well as error value.

Table 1 :
GA parameters and values