Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems

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1 European Journal of Scientific Research ISSN X Vol.64 No.4 (2011), pp EuroJournals Publishing, Inc Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems R.Thamilselvan Assistant Professor, Department of Computer Science and Engineering Kongu Engineering College, Perundurai, Erode , Tamilnadu, India P.Balasubramanie Professor, Department of Computer Science and Engineering Kongu Engineering College, Perundurai, Erode , Tamilnadu, India Abstract In the manufacturing system, optimization of scheduling plays a vital role for effective utilization of resources. Efficient utilization of resources to improve the outcome of manufacturing system is the complex job shop scheduling. In recent years much attention has been given to solve the combinatorial optimization problems like Job Shop Scheduling using meta-heuristic approaches. This paper proposes a new approach to solve a Job Shop Scheduling Problem (JSSP). This paper introduces Order Preserving Multi Point Crossover (OPMPX), Unordered Subsequence Exchange Crossover (USXX) and Ordered Partially Mapped Crossover (OPMX) of Genetic Algorithm (GA). The above algorithms are experimented with the JSSP to analyze the performance. The objective of the proposed method is to minimize the makespan and flowtime with a variety of constraints. Job Scheduling is the process of completing jobs over a time with allocation of shared resources. The result and performance of the proposed work is tested with wellknown bench mark problems and also compared with the other conventional algorithms. Keywords: Genetic Algorithm, Job Shop Scheduling, Crossover, Makespan and Flowtime. 1. Introduction Optimization of scheduling can improve the production system and the effective utilization of resources. Scheduling deals with allocation of resources to perform a collection of tasks. One of the most difficult systems in scheduling is a job shop scheduling. The job shop scheduling problem can need enormous amount of backtracking. Obtaining exact solutions for such problem is computationally hard. Several algorithms have been used to solve the combinatorial optimization problem like job shop scheduling problem (JSSP). Meta-heuristics is used to solve with the computationally hard optimization problems. Meta-heuristics consist of a high level algorithm that guides the search using other particular methods. Mets-heuristics are used as a standalone approach for solving hard combinatorial optimization problems. Several algorithms have been tried to find the optimal solution for JSSP. Genetic Algorithm (GA) is one of the successful algorithms for order problem like JSSP and Travelling Salesman Problem

2 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems 539 etc. Because GA has successfully satisfied completeness, soundness non redundancy orders preservation. This paper presents an efficient GA for JSP to get the optimal solution. The crossover function is one of the major components in Genetic Algorithm and it creates major impact on GA. By using crossover generate a new schedule by mutually exchange the operations (genes) in the existing schedule. A simple GA uses 1-point crossover, 2-point crossover and uniform crossover etc. But we proposes a new powerful and efficient crossover algorithm like Order Preserving Multi Point Crossover (OPMPX), Unordered Subsequence Exchange Crossover (USXX) and Ordered Partially Mapped Crossover (OPMX) to effectively solve more difficult problems. This paper is organized as follows: Section 2 describes the problem and notation, Literature review is discussed in Section 3. Section 4 describes the Methodologies, Section 5 propose the new algorithm for crossover of GA. Results and Discussion is in Section 6. Section 7 concludes the paper. 2. Problem Description and Notation The nxm Job Shop Scheduling problem labeled by the symbol n, m, J, O, G and C max. It can be described by the finite set of n jobs J={j 0, j 1, j 2, j 3,..j n, j n+1 } (the operation 0 and n+1 has duration and represents the initial and final operations), each job consist of a chain of operations O={o 1,o 2,o 3,.o m }, Each operation has processing time {ʎ i1, ʎ i1, ʎ i1,. ʎ im }, finite set of m machines M={m 1, m 2, m 3.m m }, G is the matrix that represents the processing order of job in different machines and C max is the makespan that represents the completion time of the last operation in job shop. On O define A, a binary relation representing precedence between operations. If ( v, u) A then u has to be performed before v. A schedule is a function S : o IN { o} that for each operation u defines a start time S(u). A schedule S is feasible if u O : S ( u) 0 (1) u, v O, ( u, v) A : S( u) + λ( u) S( v) (2) u, v O, u v, M ( u) = M ( v) : S ( u) + λ ( u) S ( v) ors ( v) + λ( v) S( u) (3) The length of a schedule S is len ( S ) = maxv 0 ( S ( u) + λ ( u) ) (4) The goal is to find an optimal schedule, a feasible schedule of minimum length, min(len(s)). Table 1: Processing Time and Sequence for 4X3 problem Instance Job Operation Number and Processing Sequence Machine Assigned Processing Time Start Operation (Dummy) O 11 M 1 2 J1 O 12 M 2 3 O 13 M 3 4 O 21 M 3 4 J2 O 22 M 2 4 O 23 M 1 1 O 31 M 2 2 J3 O 32 M 3 2 O 33 M 1 3 O 41 M 1 3 J4 O 42 M 3 3 O 43 M 2 1 End Operation (Dummy) An instance of the JSS problem can be represented by means of a disjunctive graph G=(O, A, E).The vertices in O represent the operations, the conjunctive arcs in A represent the given precedence

3 540 R.Thamilselvan and P.Balasubramanie { } between the operations and the edge in E ( u, v) u, v O, u v, M ( u) M ( v) = = represent the machine capacity constraints. Each vertex u has a weight, equal to the processing time ʎ(u). Let us consider the bench mark problem of the JSSP with four jobs, each has three different operations and there are three different machines. Operation sequence, machine assignment and processing time are given in Table 1. Based on the above bench mark problem, we create a matrix G, in which rows represent the processing order of operation and the column represents the processing order of jobs. Also we create a matrix P, in which row i represents the processing time of J i for different operations. M1 M 2 M M 3 M 2 M G = M 2 M 3 M P = M1 M 3 M The processing time of operation i on machine j is represented by O ij. Let ʎ ij be the processing time of O ij in the relation O ij O ij C ij represents the completion of the operation O ij. So that the value C ij = C ik + ʎ ij represents the completion time of O ij. The main objective is to minimize of C max. It can be calculated as C = max ( C ) (5) max all Oij O ij The distinctive graph of the above bench mark job scheduling problem is shown in Figure 1, in which vertices are represents the operation. Precedence among the operation of the same job is represented by Conjunctive arc, which are doted directed lines. Precedence among the operation of different job is represented by Disjunctive arc, which are undirected solid lines. Two additional vertices S and E represented the start and end of the schedule. Figure 1: Illustration of disjunctive graph

4 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems 541 Figure 2: A Schedule of Gantt Chart for 4X3 problem Instance M1 O 11 O 41 O 23 O 33 Machine M2 O 31 O 12 O 22 O 43 M3 O 21 O 32 O 42 O 13 Time The Gantt Chart of the above bench mark job scheduling problem is shown in Figure 2. Gantt Chart is the simple graphical representation technique for job scheduling. It simply represents a graphical chart for display schedule; evaluate makespan, idle time, waiting time and machine utilization etc. 3. Literature Review Number of researchers has adopted GA technique for solving the job shop scheduling problem. Tabu search was first presented by Glover [2][3][4] (1986) and improved in the following years. The effectiveness of the technique in the job shop problem was examined by Taillard [1] (1994), Van Laarhoven et al. [5](1992), Barnes and Chambers (1995), Dell Amico and Trubian [8] (1993), Nowicki and Smutnicki [7] (1996) solve the job shop scheduling problem using fast tabu search and Genetic Algorithm with Penalty Function to solve JSSP by Liang Sun, Xiaochun Cheng, Yanchun Liang (2010) [22]. All algorithms demonstrated outstanding results comparing to simulated annealing and shifting bottleneck. Calderia et al. [12] presented Tabu-Hybrid using one of the representation for the JSSP called permutation with repetition (PWR) in which the order of operations within the permutation is interpreted as a sequence for building a schedule solution. Nowicki et al. [13] presented i-tsab algorithm which is an extension of their earlier landmark TSAM algorithm. In addition to the shortterm memory mechanism found in all implementations of the tabu search, i-tsab is characterized by the three algorithmic components: i) the highly restrictive move operator, ii) re-intensification of search around previously encountered high-quality solutions and iii) diversification of search via path relinking between high-quality solutions. i-tsab is largely deterministic. Christian Blum et al. [15] proposed MAX-MIN ant system in the hyper-cube framework approach to tackle the broad class of group shop scheduling problem instances. It probabilistically constructs solutions using the non delay guidance which employs the black-box local search procedures for improving the constructed solutions. Jun Zhang et al. [16] presented an investigation into the use of an ACS to optimize the JSSP. The main characteristics of this system are positive feedback, distributed computation, robustness and the use of a constructive greedy heuristic. The results have shown that the performance of the ACS for the JSSP largely depends on the parameter values and the number of the ants. V.P.Eswaramurthy and A.Tamilarasi [17] 2009 presented Hybridization of Ant Colony Optimization Strategies in Tabu Search for Solving Job Shop Scheduling Problem. Also the numbers of crossover strategies are proposed by researchers. Multi-Step Crossover method was introduced by Takeshi YAMADA and Ryohei NAKANO [18]. Ono.I, Yamamura.M and Kobayashi.S [19] [20] proposed a Job Based Order Crossover and Subsequence Exchange Crossover.

5 542 R.Thamilselvan and P.Balasubramanie 4. The Methodologies 4.1. Genetic Algorithm Figure 3: A standard genetic algorithm. Start Evaluate the fitness of each individual in the population Select the fittest solutions in the population Terminated Initialization of population Generate offspring solutions by genetic operators Stop Genetic algorithms are probabilistic Meta heuristic technique, which may be used to solve optimization problems. They are based on the genetic process of chromosome. Over many generations, natural populations evolve according to the principles of natural selection, i.e. survival of the fittest, first clearly stated by Charles Darwin in The Origin of Species. It starts with the initial solution called Population and it is filled with chromosome. Each element in chromosome is called gene. Job is represented by each gene in chromosome and the job sequence in a schedule based on the position of the gene. GA uses Crossover and Mutation operation to generate a new population. By crossover operation, GA generates the neighborhood to explore new feasible solution. A typical genetic algorithm is illustrated in Figure 3. It first creates an initial population consisting of randomly generated solutions. After applying genetic operators, namely selection, crossover and mutation, one after the other, new offspring are generated. Then the evaluation of the fitness of each individual in the population is conducted. The fittest individuals are selected to be carried over next generation. The above steps are repeated until the termination condition is satisfied. A GA is terminated after a certain number of iterations or if a certain level of fitness value has been reached. The construction of a genetic algorithm for the scheduling problem can be divided into four parts: the choice of representation of individual in the population; the determination of the fitness function; the design of genetic operators; the determination of probabilities controlling the genetic operators. The steps for basic genetic algorithm are Step 0: Start Step 1: Create initial random population P i randomly and set i=0 Step 2: Repeat 2.1 Evaluate the fitness function of each individual and select the fittest individual from the population that have: f j Pi < Pj = n f k = 1 k. 2.2 Select one individual based on fitness, perform reproduction then copy P i into P i Apply crossover to the element of P i to copy into P i Apply mutation to the element of P i to copy into P i+1.

6 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems Produce offspring or child until the population is filled up. 2.6 Evaluate the new population P i P i =P i+1 Step 3: Until the termination condition is true. Step 4: End 5. Proposed Algorithm 5.1. Genetic Algorithm Algorithm GA_Procedure gives steps for the proposed genetic algorithm and this algorithm introduces various reproduction strategies, which are described below. Different parameters are used in this algorithm. MAX represents the maximum number if iteration and NUM represent the total number of iteration. Current_Population procedure is used to create a current population. Evaluate_Current is used to evaluate the population; Pick_Population is used to select the best population. GA_Crossover is used to create the child chromosomes. We have proposed OPMPX,USXX and OPMX crossover strategies and it is applied with the genetic algorithm to find the optimal solution for job shop scheduling. Algorithm: GA_Procedure Step 1: /*Initialization*/ Initialize 0 to MAX Get the value for NUM Step 2: /*Generate initial population*/ Generate initial population using Current_Population() Step 3: /*Evaluate the chromosomes*/ Evaluate the fitness value of current chromosomes using Evaluate_Current() Step 4: Repeat For i=0 to NUM do For j=0 to 4 do pop_next[j]=pop_current[j] End for Sorting the given set of pop_current chromosomes. Step 4: /*Reproduction strategies*/ Apply any one Crossover strategies to get child chromosomes. Step 5: /*Mutation strategies*/ Apply Mutation() to mutate with low probability Step 6: /*Copy the chromosomes of pop_next to pop_current*/ For j=0 to 4 do pop_current[j]=pop_next[j] End for Increment MAX by 1 Step 7: /*Termination criteria*/ If MAX<NUM then Go to Step 4 Else Go to Step 8 Step 8: /*Output the solution*/ Stop

7 544 R.Thamilselvan and P.Balasubramanie Procedure Current_Population is used to generate the new population and the new population is stored in pop_current variable. Procedure: Current_Population() Inputs: RANDOM is a random number generated by random() function Output: Fitness value for pop_current Begin Assign VALUE and RANDOM /*Calculate fitness function*/ For i=0 to 4 do For j=0 to 6 do Create random value for RANDOM RANDOM=RANDOM%2 pop_current[i].bit[j]=random End for VALUE=Evaluate_Current(pop_current[i]) /*get the value of chromosome as integer*/ pop_current[i].fit=calculate_fitness(value) /*calculate the fitness value*/ End for Stop Procedure Evaluate_current() Input: pop_current[i] for i=0 to 4 Output: value of chromosomes as integer Begin z=pop_current.bit[0]*1 + pop_current.bit[1]*2 + pop_current.bit[2]*4 + pop_current.bit[3]*8 + pop_current.bit[4]*16 If pop_current.bit[5]==1 Then z=z*(-1) End if End Procedure Calcuate_Fitness() Inputs: VALUE ie value of chromosome as integer Output: objective function of chromosome Begin y= -(VALUE*VALUE)+5 End 5.2. Representation GA requires an appropriate chromosome (ie a collection of operations) to find a solution (Cheng et al. 1996). All the chromosomes must be generated during the evolutionary process for feasible solution. In a traditional JSSP consist of J jobs and M machines starting that JxM operation. The chromosome [g 1, g 2, g 3, g 4,, g JxM ] can represent a schedule of JxM operations. The chromosome could be generated based on sequence of operations. Once the basic schedule is generated, we need GA s crossover and mutation to generate a further schedule. The fundamental crossover of GA operates on two parent chromosome and generates two child chromosomes. This operation needs to present the job sequence characteristics. The representation of chromosome for JSSP is based on Gen et al (1994) and Cheng et al (1996). The chromosome is an ordered sequence of job/operations where each gene represents a one operation. Order of the operations represented in chromosome is the order of schedule. Let us consider an example of 4X3 job shop problem. Each job shop problem has constraints for scheduling the operation to the machine with processing time is shown in Table 1. For example J1 is processed in the order M1, M2 and M3 and J2 is processed in the order M3, M2 and M1 respectively and so on. The

8 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems 545 objective of the algorithm is to complete all the operations of a particulate job with minimum possible time. Also the processing of operations on machines taking into account of the precedence and processing time of operations. The main idea is how to represent the jobs in terms of sequence. In the relationship between the job scheduling and the chromosomes to represent the schedule. So that we can use the GA to find better job scheduling. For the above 4X3 job shop scheduling the chromosome such as [ ] may be formed and then change the order for the better schedule. In the given chromosome the genes 1 stands for J1, 2 stands for J2 and so on. The order of the operation corresponds to the relative position of the gene. For example the first gene 3 stands for first operation of J3, seventh gene 3 stands for the second operation of J3, second gene 4 stands for first operation of J4 and so on. The above scheduling chromosome is also represented as [O 31, O 41, O 11, O 21, O 12, O 42, O 32, O 43, O 13, O 22, O 33, O 23 ]. O ij stands for the j th operation of the job J i. For example O 31 stands for the first operation of J Crossover The crossover operator involves the swapping of genetic material (bit-values) between the two parent strings. Two parents produce two offspring. There is a chance that the chromosomes of the two parents are copied unmodified as offspring. There is a chance that the chromosomes of the two parents are randomly recombined (crossover) to form offspring. Generally the chance of crossover is between 0.6 and 1.0[21]. The following sections propose the new crossover algorithms for job shop scheduling Order Preserving Multi Point Crossover (OPMPX) We propose a new crossover strategy named as Order Preserving Multi Point Crossover (OPMPX) of GA. Given a sequence of operations as parent1 and parent2, OPMPX generates children1 and children2. The algorithm for OPMPX as follows. Step 1. Generate two random parent individual namely P1 and P2 with a sequence of all operations. Step 2. Choose more than one similar random crossover points from P1 and P2. Step 3. Split along those points from P1 and P2. Step 4. Exchange the alternate segments between P1 and P2 to create C1 and C2 if the operations are in P1 as well as in P2. Step 5. Starting from each exchangeable crossover point from P1 and P2, look for the elements of the same segment in P2 that are not in P1 and vice versa. Step 6. For each of these O i in P1 and O j in P2. Step 7. Copy each O i into C2 and each O j into C1 so as to maintain relative ordering. Figure 4: Order Preserving Multi Point Crossover (OPMPX)

9 546 R.Thamilselvan and P.Balasubramanie For example in Figure 4 there are two parent chromosomes are given namely P1 and P2. Select multiple crossover pointes. Move the operations of P1 and P2 from the first crossover point to second crossover point into C2 and C1 respectively, so as to maintain the same position. But the operation O 21 and O 41 are unable to exchange. So that O 21 is moved into the position of O 12 in C1 and O 41 is moved into the position of O 12 in C2. Similarly for all the crossover points Unordered Subsequence Exchange Crossover (USXX) We introduce one more cross over strategy named as Unordered Subsequence Exchange Crossover (USXX) that children inherit subsequences on each machine as far as possible from parents. Unordered Subsequence exchange crossover creates a new children s even the subsequence of parent1 is not in the same order in parent2. The algorithm for USXX is as follows. Step 1. Generate two random parent individual namely P1 and P2 with a sequence of all operations. Step 2. Generate two child individual namely C1 and C2. Step 3. Select random subset of operations (genes) from P1 and copy it into C1. Step 4. Starting from the first crossover point from P1, look for elements in P2 that have been copied as in the same order. Step 5. The remaining operations of P2 that are not in the subset can be filled in C1 so as to maintain their relative ordering. Step 6. If C1 is created then goto Step 3 to generate C2 analogously. For example in Figure 5 parent chromosome of M1 is [3, 4, 1, 2, 1, 4, 3, 4, 1, 2, 3, 2]. The selected sequence is [1, 2, 1]. It is the first operation of J1, first operation of J2 and the second operation of J1 respectively. Select the same operation from P2 even it is in different order. In a given sample first operation of J1 is in first gene, first operation of J2 is in fourth gene and the second operation of J1 is in eighth gene respectively. Copy the remaining operation of P2 in to C2 so as to maintain their relative ordering. Figure 5: Unordered Subsequence exchange crossover (USXX)

10 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems Ordered Partially Mapped Crossover (OPMX) This is the third crossover strategy named as Ordered Partially Mapped Crossover (OPMX) to generate children s from the parent chromosome. The procedure for OPMX is as follows. Step 1. Generate two random parent individual namely P1 and P2 with a sequence of all operations. Step 2. Generate two child individual namely C1 and C2. Step 3. Select random subset of operations (genes) from P1 and copy it into C1. Step 4. Starting from the first crossover point look for elements in that segment of P2 that have not been copied. Step 5. For each of these O i look in the subset to see what element O j has been copied in its place from P1 Step 6. Copy each O i in P2 into C1 so as to maintain their relative ordering from left to right. Step 7. The remaining operations of P2 that are not in the subset can be filled in C1 so as to maintain their relative ordering. Step 8. If C1 is created then goto Step 3 to generate C2 analogously. From the Figure 6, select a random set [O 21, O 12, O 42, O 32, O 43 ] of continuous operation from P1 and copy into C2, so as to maintain the same position. Look into the same set of continuous operations from P2 and find the elements that have not been copied. The elements are [O 41, O 22, O 23 ]. Copy those elements into C2, so as to maintain the relative ordering. Copy the remaining elements [O 11, O 31, O 13, O 33 ] that have not been copied into C2, so as to maintain the relative ordering. Figure 6: Ordered Partially Mapped Crossover (OPMX) 6. Results and Discussion To measure the effectiveness of the proposed algorithm, we consider the standard JSP test instances of Fischer and Thomson (1963) instances FT06, FT10, FT20, Lawrence (1984) instances from LA01 to LA20, D. Applegate, W. Cook (1991) instances ORB01-ORB10 and R.H. Storer, S.D. Wu, R. Vaccari (1992) instances SWV01-SWV10. The proposed algorithm is implemented using C++ programming language on windows platform with Intel Pentium E5800, 3.2 GHz and 2GB RAM. Table 2 summarizes the results such as problem name, problem size, optimal solution, Makespan and Relative error obtained by OPMPX, USXX and OPMX.. The performance of the proposed algorithm ids based on the relative percentage deviation (RPD) which is computed as

11 548 R.Thamilselvan and P.Balasubramanie Algosol Optsol Relative percentage deviation (RPD) = X 100 (6) Optsol Where Algo sol is the solution obtained by different existing and proposed algorithms and Opt sol is the optimal or best known solution. Genetic algorithm with USXX crossover strategy has succeeded in getting the best solution for the bench mark problem of job shop scheduling. USXX strategy has produced superior results when compared with other methods. Figure 7 shows the average makespan for the problem instance FT06, FT10 and FT20 using OPMPX, USXX and OPMX with GA. It clearly shows that the USXX produce an optimal makespan value compare to other crossover strategies. Also Figure 8 shows the average relative percentage deviation for the same problem instances using the above crossover strategies. Table 2: Results for instances by Fisher and Thomson (1963) Problem Name Problem Size Makespan Relative Error (%) Jobs (n) Machines (m) Optimal OPMPX USXX OPMX OPMPX USXX OPMX FT FT FT Average Figure 7: Average Makespan values by Different Crossover Strategies for FT06,FT10 and FT20 Figure 8: Average Relative Error values by Different Crossover Strategies for FT06,FT10 and FT20

12 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems 549 Figure 9 shows the typical runs of the problem instance FT20 with different crossover strategies of GA. The optimal value reached at the iteration 1350 using USXX. But for the other two crossovers the optimal is not reached until 1500 iterations. It clearly states that the USXX generate an optimal value faster than the other two approaches. Figure 9: The time evolutions of makespans for the FT20 (20 jobs and 5 machines) Table 3: Results for instances by Lawrence (1984) Problem Name Jobs (n) Problem Size Makespan Relative Error (%) Machines (m) Optimal OPMPX USXX OPMX OPMPX USXX OPMX LA LA LA LA LA LA LA LA LA LA LA LA LA LA LA LA LA LA LA LA Average Table 3 shows the makespan and relative error for the problem instance LA01 to LA20 using the different crossover strategies. Out of 20 problem instances USXX reach the optimal value for 16

13 550 R.Thamilselvan and P.Balasubramanie instances, OPMPX reach the optimal value for 6 and OPMX reach the optimal value for 9 instances respectively. It shows that the USXX is superior to the other two methods. Also the average makespan value is shown by the Figure 10 with USXX is better than the other two crossover approaches. Figure 10: Average Makespan values by Different Crossover Strategies for LA01-LA20 Figure 11: Average Relative Error values by Different Crossover Strategies for LA01-LA20 The typical online performance of LA20 is shown in Figure 12. In the given graph the OPMPX does not reach the optimal value until 1500 iterations. But the USXX reach the optimal solution at 1300 th iterations and OPMX reach at 1390 th iterations. It clearly shows that the USXX reach the optimal makespan value with less number of iterations.

14 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems 551 Figure 12: The time evolutions of makespan for the LA20 (10 jobs and 10 machines) Figure 13: Average Makespan values by Different Crossover Strategies for ORB01-ORB10 Table 4 shows the makespan and relative error for the problem instance ORB01 to ORB10 using the different crossover strategies. Out of 10 problem instances USXX reaches the optimal value for 8 instances, OPMPX reaches the optimal value for only one and OPMX reaches the optimal value for 2 instances respectively. It shows that the USXX is superior than the other two methods. Also the average makespan value is shown by the Figure 13 with USXX is better than the other two crossover approaches. Average makespan for optimal is 903, OPMPX is 917, 903 for USXX and 908 for OPMX. Table 5 shows the makespan for the problem instance SWV01 to SWV10 using the same crossover approaches. According to the table values, USXX produce the better result for this problem instances. Figure 15 and Figure 16 shows the average makespan value and relative error for SWV01 to SWV10 using the USXX, OPMPX and OPMX of GA.

15 552 R.Thamilselvan and P.Balasubramanie Table 4: Results for instances by Applegate and Cook (1991) Problem Size Makespan Relative Error (%) Problem Jobs Machines Name Optimal OPMPX USXX OPMX OPMPX USXX OPMX (n) (m) ORB ORB ORB ORB ORB ORB ORB ORB ORB ORB Average Figure 14: Average Relative Error values by Different Crossover Strategies for ORB01-ORB10 Table 5: Results for instances by Storer, Wu and Vaccari (1992) Problem Size Makespan Relative Error (%) Problem Machines Optimal Name Jobs (n) OPMPX USXX OPMX OPMPX USXX OPMX (m) UB LB SWV SWV SWV SWV SWV SWV SWV SWV SWV SWV Average

16 Analysis of Various Alternate Crossover Strategies for Genetic Algorithm to Solve Job Shop Scheduling Problems 553 Figure 15: Average Makespan values by Different Crossover Strategies for SWV01-SWV10 Figure 16: Average Relative Error values by Different Crossover Strategies for SWV01-SWV10 7. Conclusions Genetic Algorithm with various alternate crossover strategies for the job shop scheduling problem is presented. OPMPX, USXX and OPMX were analyzed to determine the effectiveness of genetic algorithm for job shop scheduling. The proposed algorithms are tested with the benchmark problem instance like FT06, FT10, FT20, LA01 - LA20, ORB01 - ORB10 and SWV01- SWV10. Crossover strategies of genetic algorithm play an important role to produce a quality of solution for job shop scheduling. We proposed three different algorithm to generate a new children s of chromosome. Experiment results of the proposed method were better when compare with other competing algorithms. The performance of USXX is better than OPMPX and OPMX for all the above said benchmark problems. References [1] Taillard E., Parallel taboo search techniques for the job shop scheduling problem, ORSA Journal on Computing, vol. 6(2), pp , 1994.

17 554 R.Thamilselvan and P.Balasubramanie [2] Glover, F., Future Paths for Integer Programming and Links to Artificial Intelligence, Computers and Operations Research, Vol. 13, pp , [3] Glover, F., Tabu Search - Part I, ORSA Journal on Computing, Vol. 1, pp , [4] Glover, F., Tabu Search - Part II, ORSA Journal on Computing, Vol. 2, pp.4-32, [5] Van Laarhoven P. J. M, Aarts E. H. L, Lenstra J. K, Job shop scheduling by simulated annealing, Operations Research, vol. 40(1), pp , [6] Barnes, J. W., Chambers, J. B., Solving the job shop scheduling problem using tabu search, IIE Transactions, vol. 27, pp , [7] Nowicki E, Smutnicki C.,A fast taboo search algorithm for the job-shop problem, Management Science, vol. 42(6), pp , [8] Dell'Amico M, Trubian M., Applying tabu search to the job-shop scheduling problem, Annals of Operations Research, vol. 41, pp , [9] Dorigo, M. and Gambardella, L. M., Ant colony System: A Cooperative Learning Approach to the Travelling Salesman Problem, IEEE Trans. On Evolutionary Computation, Vol. 1, No. 1, pp.53-66, [10] Dorigo, M., Maniezzo, V. and Colorni, A., The Ant System: Optimization by a colony of cooperating agents, IEEE Trans. Systems, Man, Cybernetics, Vol. 26, No. 2, pp.29-41, [11] Ferdinando Pezzella and Emanuela Merelli, A Tabu Search Method guided by Shifting Bottleneck for the Job Shop Scheduling Problem, European Journal of Operational Research, Elsevier, Vol. 120, pp , [12] Calderia, J. P., Melicio, F. and Rosa, A., Using a Hybrid Evolutionary-Taboo Algorithm to solve Job Shop Problem, ACM Symposium on Applied Computing, pp , [13] Nowicki, E. and Smutnicki, C., An Advanced Tabu Search Algorithm for the Job Shop Problem, Journal of Scheduling, Vol. 8, pp , [14] Nowicki, E. and Smutnicki, C., A fast tabu search algorithm for the job-shop problem, Management Science, Vol. 42, No. 6, pp , [15] Christian Blum and Michael Samples, An Ant Colony Optimization Algorithm for Shop Scheduling Problems, Journal of Mathematical Modeling and Algorithms, Kluwer Academic Publishers,Netherlands, Vol. 3, pp , [16] Jun, Zhang, Xiaomin, Hu, Tan, X., Zhong, J. H. and Huang, Q., Implementation of an Ant Colony Optimization technique for job shop scheduling problem, Transactions of the Institute of Measurement and Control, Vol. 8, No. 1, pp , [17] V.P.Eswaramurthy and A.Tamilarasi., Hybridization of Ant Colony Optimization Strategies in Tabu Search for Solving Job Shop Scheduling Problems, Vol.20, No. 2, pp ,2009. [18] Takeshi YAMADA and Ryohei NAKANO., A Genetic Algorithm with Multi-Step Crossover for Job Scheduling Problems, IEEE International Conference on Genetic Algorithms in Engineering Systems, pp , [19] Ono.I, Yamanura.M and Kobayashi.S., A Genetic Algorithm for Job Shop Scheduling Problem using Job Based Order Crossover, ICEC 96, pp [20] Ono.I, Yamanura.M and Kobayashi.S., An Efficient Genetic Algorithm for Job Shop Scheduling Problem, ICGA 95, pp [21] Man. K.F., Tan. K.S., Kwong. S: Genetic Algorithms Concepts and Design, Springer(1999) [22] Liang Sun, Xiaochun Cheng, Yanchun Liang., Solving Job Shop Scheduling Problem Using Genetic Algorithm with Penalty Function, International Journal of Intelligent Information Processing, Vol. 1, No. 2, pp , 2010.

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