Two Hybrid Algorithms for Single-Machine Total

American Journal of Scientific Research ISSN 2301-2005 Issue 64(2012), pp. 22-29 © EuroJournals Publishing, Inc. 2012 http://www.eurojournals.com/ajsr.htm

Two Hybrid Algorithms for Single-Machine Total Weighted Tardiness Scheduling Problem with Sequence-Dependent Setup Hamidreza Feili Department of Industrial Engineering Alzahra University, Tehran, Iran E-mail: [email protected] Tel: 009802122494416 Hamidreza Haddad Department of Industrial Engineering Iran University of Science and Technology, Tehran, Iran E-mail: [email protected] Payam Ghanbari Department of Industrial Engineering Iran University of Science and Technology, Tehran, Iran E-mail: [email protected] Abstract This paper addresses Single-machine total weighted tardiness scheduling problem with dependent setup time. The problem, even in absence of setup time, is strongly NPhard and cannot be solved using common optimization methods in reasonable time. In this paper, first a mathematical model for solving the problem is presented and then, two metaheuristics; Genetic Algorithm (GA) and Simulated Annealing (SA), as well as a hill climbing heuristic are suggested. Each algorithm is examined individually and then two hybrid models are considered. The computational result shows that SA performs more efficient among the non-hybrid models while the hybrid of SA and Hill climbing is the best solution in general. Keywords: Hybrid algorithm; scheduling; single-machine; sequence-dependent setup;

1. Introduction This paper addresses the problem of sequencing N jobs on a single machine with sequence-dependent setup time when the aim is to find a schedule of jobs that minimizes the total weighted tardiness. This problem is denoted as 1/sij/∑WiTi in the literature. Lawler (1997) showed that this problem, even without setup time, is NP-hard and cannot be solved in reasonable time using common optimizing algorithms. Due to this fact, two groups of methods have been introduced to solve these kinds of problems include exact and near optimal solution methods. In field of exact algorithms these studies can be mentioned: Xiaochuan and Feng (2006) proposed a branch and bound for minimizing total tardiness with consider of dependent setup time and Baptiste et al (2009) presented a branch and bound to minimize total tardiness with arbitrary release dates. Ten et al (2000) solved the problem without considering

Two Hybrid Algorithms for Single-Machine Total Weighted Tardiness Scheduling Problem with Sequence-Dependent Setup

23

weight factor by using Simulated annealing SA, Genetic algorithm , Branch and Bound and a new heuristic method. According to their results, Branch and Bound can reach the global optima but is not desirable when the number of jobs passes 25 due to its unreasonable computational time. On the other hand, Genetic search was found efficient for a great number of jobs. On the other hand there are many researchers that investigated heuristic and metaheuristic methods to generate near optimal solutions. Lee and Pinedo (1997) offered a constructive method named as Apparent Tardiness Cost with Setups (ATCS) that is able to solve problem instances with the small size in a reasonable run time. However, the efficiency of model for the large-scaled instances is not proper. Holsenback (1999) presented a heuristic that solved the problem for 50 jobs. Feo (1996) used greedy method and considered the distinct penalty for any lateness. Valente (2008) offered a new beam search that its efficiency and its computational time were quite ideal for instances with low and medium size. Also, Sen (2003) discussed about the significance of weight factor of each job and showed that weight factor increases complexity of the problem exponentially. Furthermore, Cicirello (2003) accomplished a comprehensive research on tardiness problem and solved it using four different heuristics for 60 jobs. Allahverdi (1999) compared problems with dependent and independent setup time and concluded that although dependent setup time can liken the problem to real industry, but it causes more additional computations in solving process. Lin and Ying (2007) utilized three well known metaheuristics to solve the problem of total weighted tardiness with setup time including GA, TS and SA. The results were compared to the literature that validated the efficiency of proposed approaches. Tasgetiren et al (2009) offered a new algorithm named Discrete Differential Evolution (DDE) that was similar to the greedy approach. Der Chou (2009) presented an experienced learning GA for the problem without setup time and compared their results with popular J.E Beasley’s OR library. In addition, Anghinolfi and paolucci (2009) offered a discrete PSO and Rubin and Ragatz (1995) presented a GA for the problem of total tardiness and a Memetic algorithm is used by Franka et al (2001). The hybrid algorithms have used in recent years to solve the scheduling problems to minimize total tardiness without setup time and weight factor. Generally, these classes of algorithms are organized into two groups containing Low-level and High-level hybrids. In Low-level hybrids, first one heuristic or metaheuristic generates the initial solution and then the other algorithm tries to reform its performance whereas in High-level hybrids, one algorithm is known as the base and the other helps it for searching the neighborhood points without any changes on basic algorithm parameters. M’Hallah (2007) presented sets of high and Low-level hybrids based on dispatching rules, hill climbing, SA and a GA based evolutionary algorithm. Almeida and Centeno (1998) offered a compound algorithm using Tabu search, SA and Hill climbing to minimize total tardiness and Cheng et al (2010) presented a hybrid based ant colony and four elimination rules which is classified as a High-level hybrid. In this paper, the weight factor and dependent setup times are also considered and because of immense complexity of the mentioned problem, two metaheuristics include GA and SA methods and a hill climbing heuristic are utilized to generate the near optimal solution. In section 2, the problem is defined and its relative mathematical model is presented. In section 3, the solution methods are illustrated individually and finally the comprehensive computational results as well as the statistical analysis are presented in section 4.

2. Problem Formulation In this paper, the single machine scheduling problem with N jobs and sequence dependent setup times is studied. It is assumed that machine works with no idle, and all the jobs are ready for processing at time zero. Each job has a process time, a distinct due date and an importance weight factor that are

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Hamidreza Feili, Hamidreza Haddad and Payam Ghanbari

denoted by pi, di and wi respectively. The aim is to find a sequence of jobs that minimizes the total weighted of tardiness. To this aim the following mathematical model is suggested: n

 (W T

Min

i i

i 1

(1)

)

St:

x

1

(2)

x

1

(3)

ij

j

ij

i

C0  0

(4) n

C j  C j 1   xij Pi   xh , j 1 xij S h ,i i 1

n

Ti   c j xij  di

i

i

(5)

hi

(6)

j 1

Ti  0

i

(7)

xij  0,1 (8) Where xij equals 1 if job i is executed in priority j and 0 otherwise. Also Sh,i refers to a setup time required when job i is executed after job h and finally Cj is completion time of job in priority j. In this model, constraint (2) states that just one job could be situated in each priority while constraint (3) assures that each job lies just in one priority. Equations (4) and (5) show how completion time of jobs in each priority should be calculated and constraints (6) and (7) deal with the computation of tardiness times for each job. Finally, Constraint (8) indicates the binary limitation.

3. Solution Methods In order to solve the proposed model, two different hybrids, one based on SA and GA combination and the other based on SA and Hill climbing heuristic are applied. In each hybrid the initial solution is generated by the first algorithm and the other tries to reform the solution until its stopping condition is met. In remain part of this section, a brief description of each algorithm is given separately. 3.1. Genetic Algorithm

Holland at University of Michigan is a pioneer in introducing GA for the first time. GA is one of the most powerful metaheuristic methods that can be used as a useful tool to solve many optimization problems. The main idea of this approach is the use of natural genetic concepts and the effort is to code the optimization problem using chromosomes and gens. Equation (8) shows the fitness function that is used in this paper in which wi andTi demonstrate the weight factor and relative tardiness of job i respectively: 1 f (i )  (8)  Wi Ti Also roulette wheel mechanism is utilized for selecting suitable parents in each generation in which two operators are used to obtain the neighborhood points of the current solution. These two operators are as follows: In this paper, the probability of using mutation operator is considered equal to 0.3. For each generation, a random chromosome is selected using roulette wheel and then two random integer values are generated in [1-N] interval. Afterwards, the positions of generated numbers are changed in the sequence and the selected chromosome is replaced by this one in next generation. On the other hand, the probability of this operator is considered equal to 0.8. First two chromosomes are selected using

Two Hybrid Algorithms for Single-Machine Total Weighted Tardiness Scheduling Problem with Sequence-Dependent Setup

25

roulette wheel. Then a random number in [0-1] interval is generated. If the random number is lower than 0.5, then the first job will be selected from father chromosome and otherwise, the first job is selected from the mother and the selected job is located in a new chromosome (child chromosome). This process is iterated until all the jobs transformed into child chromosome. Afterwards, one of the parents is replaced by child randomly and the other would remain in the population in the next generation. This algorithm will be continuing till we reach 500th iteration while in each iteration, 30 chromosomes are generated randomly. 3.2. Simulated Annealing

Simulated annealing is among the most popular heuristic algorithms that have been employed extensively to solve lots of combinatorial optimization problems. Generally speaking, simulated annealing is a class of heuristic methods that performs a stochastic search through the solution space. The massive advantage of SA over classical local search methods is its ability to avoid getting trapped in local optima (1988). The main idea of this method arises from an analogy processes. It should be noted that SA procedures could be proportionately different in alternative problems but the basic steps of it could be described as follows: Start from a current solution x, while another solution y is generated by taking a stochastic step in some neighborhood of x. If the value of new solution increases, then y replaces x as the new current solution but if the value of the objective function does not improve, the new solution y is accepted with a probability which came from a significant formula1 in metallurgy process that is shown as below: (9) P ( E )  exp( E / T ) The value of T decreases in the course of iteration. Hence the probability of accepting worse solution will decrease too. In fact, when the algorithm is running in the last iterations, reduction in control parameter leads to maintain current solution and just accept the improving solutions. All the mentioned procedures enrich SA in the way that it would not be trapped in local optima. For more information about simulated annealing the reader is referred to the works by van Laarhoven and Aarts (1988), Aarts and Lenstra (1997). In this paper, the control parameter and cooling rate are considered equal to 1000 and 0.995 respectively. 3.3. Hill Climbing

In computer science, hill climbing is a mathematical optimization technique that is used just for local search and due to its simplicity of use this method was known as the first popular choice before. Although other advanced algorithms may be able to give more efficient results, in some situations hill climbing works just as well. Hill climbing starts with a random but not necessarily an appropriate solution, and iteratively makes small changes to the solution, each time improving it a little. When the algorithm cannot improve the best solution anymore, it terminates. Ideally, we expect the last solution be near optimal solution but there is no guarantee that hill climbing even come close to the optimal solution. Among different branches of this method, the simple hill climbing selects the first node which is closer to the current solution whereas the steepest ascent evaluates all the successors and then the closest point would be chosen. However, both forms will fail if they be trapped in the local maxima in the search space. In this paper, the place of each job is changed with its adjacent job and then objective function is calculated and if the value of objective function decreases, the change is applied. This algorithm first swaps the first job in the chain of sequence with the second job and will continue until all adjacent jobs are examined for a better objective function.

1

Boltzmann equation

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Hamidreza Feili, Hamidreza Haddad and Payam Ghanbari

4. Computational Results In this section, first the results achieved from solving the problem via each algorithm separately is presented and then the results obtained from hybrid methods are demonstrated. Also an appropriate statistical analysis is conducted and the relative results are shown afterwards. All the algorithms are coded via Visual Basic 6 on the system with CPU Core i7, 1.6 GHZ and 4 GB of RAM. In order to generate some random problems, the required values of parameters are generated according to Julai et al (2007) where processing times are selected from uniform distribution [1-100], weight factors from uniform distribution [1-10], setup times from uniform distribution [1-100] and finally due dates are generated from uniform distribution [0-ρ∑pi] in which ρ is a controlling parameter for due dates and take the values of 0.2, 0.4 and 0.8 throughout the research. In this survey, different scales of problems are generated that will cover small to large size of jobs. Each sample is solved 10 times and the best results achieved for each controlling parameter is depicted in Table 1. Table 1: N 10 25 50

Comparison between individual solution methods ρ 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.8

SA VOF 6767 37225 4092 31756 20797 12896 134057 77417 56309

GA Time (sec) 2 3 5

VOF 7202 37773 4562 48283 36703 32723 240588 182497 171803

Hill Time (sec) 15 29 77

VOF 9635 40376 7191 44393 40760 48875 267610 208877 155687

Time (sec) 0 1 3

Where N demonstrates the number of jobs and ρ shows the controlling parameter for due dates. Columns 3 till 8 corresponds the performance of proposed algorithms include value of objective function (VOF) and running time of algorithm respectively. According to the results illustrated in Table 1, SA offers the best solutions in all cases while Hill climbing operates the worst in all conditions with the exception of problem with N=25 and ρ=0.2. In order to draw a more clear contrast, the solutions achieved in the course of iterations for each of the three algorithms are illustrated in Fig. 1. As it could be seen, Hill climbing gives better results in comparison with GA method in the early iterations but loses its efficiency in the last iteration. Meanwhile, SA offers the best result that could stem from accepting unsuitable result throughout the solution procedure. Figure 1: Procedure of solution for each individual algorithm N= 25, ρ=0.6

Another probable reason that enables SA to outrank other methods could be its insensitivity to the initial solution. In fact, GA and Hill climbing are fully dependant to the initial solution and will be trapped in the local optimal if they start from unsuitable point. There are some ways to overcome this

Two Hybrid Algorithms for Single-Machine Total Weighted Tardiness Scheduling Problem with Sequence-Dependent Setup

27

dilemma. One is to use some simple priority rules like SPT and EDD for the purpose of creating the initial sequence and another is to develop some efficient hybrid algorithms which is applied it this paper. we proposed two hybrid algorithms that both start with SA and will follow by GA and Hill climbing separately. Table 2 shows the values of objective function when the hybrid algorithms are used. Table 2:

The comparison between hybrid algorithms N

ρ 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.8

10 25 50

SA+GA 6654 37225 4092 31463 20571 12626 134057 77417 56309

SA + Hill 6359 37225 4067 30466 17715 12137 133728 74985 56309

SA 6767 37225 4092 31756 20797 12896 134057 77417 56309

According to Table 2, the hybrid of SA and Hill are highly efficient and also gives better results in comparison with single methods. All the proposed hybrids have a stochastic approach to find the suitable solution, so it motivated us to give a statistical analysis for each algorithm. Each algorithm run for a typical instance runs 10 times and the results are shown in table 3. Table 3:

Examining the hybrids statistically

N

Algorithm SA+GA SA+hill SA+GA SA+hill SA+GA SA+hill

10 25 50

Best 37225 37225 18032 17308 138881 134080

Average 37646 37434 20042.2 18148.6 142733.6 139061.3

Worst 37969 37722 21981 19362 147482 146432

SD 319.8409 207.7677 1366.025 907.9671 2848.804 3986.411

SD/average 0.008496 0.00555 0.068157 0.05003 0.019959 0.028667

In Table 3. column 3,4 and5 show the best, average and the worst solution achieved. The two last columns also demonstrate the standard deviation and coefficient of variation respectively. For more comprehensive analysis on the obtained results, the 95% CI interval plots are provided for large scale of jobs that is depicted in Fig. 2,3 and 4: Figure 2: Comparison of the Figure 3: Comparison of the Figure 4: hybrids for N=50 and hybrids for N=50 and ρ=0.2 ρ=0.4 Interval Plot of SA+GA, SA+hill

95% CI for the Mean

95% CI for the Mean

145000

84000

144000

83000

143000

70000

68000

82000

142000

66000

140000

Data

81000

141000

Data

Data

Interval Plot of SA+GA, SA+hill

Interval Plot of SA+GA, SA+hill

95% CI for the Mean

Comparison of the hybrids for N=50 and ρ=0.8

80000

64000

79000

139000 138000

78000

137000

77000

136000

62000

60000

76000

SA+GA

SA+hill

SA+GA

SA+hill

SA+GA

SA+hill

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Hamidreza Feili, Hamidreza Haddad and Payam Ghanbari

According to the statistical examinations, it can be concluded that the hybrid of SA and hill climbing performs more efficient rather than hybrid SA and GA whereas its variation is more than of SA and GA hybrid.

5. Conclusion In this paper, the problem of single-machine total weighted tardiness scheduling problem with sequence-dependant setup has been studied. Since the problem was strongly NP-hard two metaheuristics based on GA and SA as well as a Hill climbing heuristic developed in order to obtain near optimal solution in reasonable time. Each method was examined individually and then they were considered as a hybrid. Since the hybrids have the stochastic approach to find the solutions, in order to give a more clear contrast a fruitful statistical analysis has been conducted. For further research, the user could adjust the parameter of the algorithms in a way that obtain better result or improve the basic proposed model and apply suggested hybrids for efficiently solving them.

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