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Due to NP-Hard nature of the Job Shop Scheduling Problems (JSP), exact methods fail to provide the optimal solutions in quite reasonable computational time. Due to this nature of the problem, so many heuristics and meta-heuristics have been proposed in the past to get optimal or near-optimal solutions for easy to tough JSP instances in lesser computational time compared to exact methods. One of such heuristics is genetic algorithm (GA). Representations in GA will have a direct impact on computational time it takes in providing optimal or near optimal solutions. Different representation schemes are possible in case of Job Scheduling Problems. These schemes in turn will have a higher impact on the performance of GA. It is intended to show through this paper, how these representations will perform, by a comparative analysis based on average deviation, evolution of solution over entire generations etc.

Scheduling is a decision-making process which deals with allocation of resources to tasks over given time-pe- riods and its goal is to optimize one or more objective functions. A scheduling problem is represented by triplet α/β/γ. α field describes machine environment; β field provides details of processing characteristics and con- straints and γ field describes the objective function to be minimized. Being essentially a combinatorial optimization problem, job shop scheduling has caught the attention of researchers in the last so many years for optimized performance. Combinatorial optimization problems can be classified as easy and hard. Problems which are poly- nomialy solvable with limited number of variables are treated easy and are called P. The notion polynomial solvable depends on the type of encoding. It is assumed that problems describing numerical data are binary encoded and the number of steps involved in solving these increases exponentially with increase in length of string and hence computational time will be enormously large and treated to be hard problems. Job scheduling problems belong to this category and are termed NP-Hard [

Extensive use of genetic algorithms to solve job shop scheduling problems can be seen through literature sur- vey [

Representations in GA environment applied so far in job shop scheduling can be classified into nine catego- ries as given by Cheng et al. (1996):

1) Operation based 2) Job based 3) Job pair relation based 4) Completion time based

5) Random keys 6) Preference list based 7) Priority rule based 8) Disjunctive graph based. 9) Machine based.

Nine categories mentioned above can be grouped into two basic encoding approaches—direct and indirect encoding. In direct approach, a Π_{j} schedule is encoded as a chromosome and genetic operators are used to evolve better individual ones. Categories 1 to 5 are examples of this category. In case of indirect approach, a sequence of decision preferences will be encoded into a chromosome. In this, encoding, genetic operators are applied to improve the ordering of various preferences and a Π_{j} schedule is then generated from the sequence of preferences. Categories 6 to 9 are examples of this category [

The rest of this paper is organized as follows: We will start with mathematical models with certain assump- tions that have been used in next section followed by the literature review on the different GA representations used in the case of JSP. Followed by review of GA representations, we will discuss regarding different GA op- erators frequently used by researchers and our own views on adding other operators not discussed so far. Now, we will analyze the experimental results conducted followed by the conclusion provided in the final part of this paper.

Since it is an important practical problem, some authors have formulated various JSP models based on different production situations and problem assumptions. The most common assumptions in case of JSP are:

1) A machine may process more than one job at a time;

2) No job may be processed by more than one machine at a time;

3) The sequence of machines which a job visits is completely specified and has a linear precedence structure;

4) Processing times are known. All the processing times are assumed to be integers;

5) Each job must be processed on each machine only once. There is no recirculation;

6) Set-up times are assumed zero;

7) Pre-emption is not allowed.

Let “J” represent a set of jobs and each job will be processed on a set of machines in a particular order. Let I = (1…..v) represent the operation indexes. The operation indexes are assigned such that for a job_{k} depending on the priority operation with higher or lower value is processed first. Let p_{i} be the processing time of ith operation, the job which it belongs to is j(i) and the machine on which ith operation car- ried is m(i).

Now the objective of scheduling process is to determine the start time st_{i} of an operation

and

Is the equation to satisfy the conflict of two jobs on the same machine at the same time.

Different total cost functions that can be studied are

The most common objective functions are the make span max

our objective function. Mannes’ [

In the disjunctive graph model, a disjunctive arc is defined between a pair of operations that share the ma- chine. Each disjunctive arc is assigned a binary decision variable such that selection on the value that variable defines the length and direction of each disjunctive arc. This is to the Mannes’ model. Very efficient algorithms like immediate selections and shifting bottleneck heuristics were proposed by Carlier [

A variable notation of the type

= 0...otherwise.

In ILP model was proposed by Bowman [^{th}’ position in the processing sequence on machine ‘m’

= 0...otherwise.

And. Mannes’ [

= 0...otherwise.

Darwin’s principle “survival of the fittest” can be used as a starting point in introducing evolutionary computa- tion. The problems of chaos, chance, non linear interactivities and temporality being solved by biological spe- cies are proved to be in equivalence with classic method of optimization [

Evolutionary computations techniques that contain algorithms based on evolutionary principles are used to search for an optimal or best possible solution for a given problem. In a search algorithm, number of possible solutions is available and the task is to find the best possible solution in a fixed amount of time. Traditional search algorithms randomly search (e.g. random walk) or heuristically search (e.g. gradient descent), explore one solution at a time in the search space to find best possible or optimal solution, which is computationally in- efficient as the search space grows in size. Whereas evolutionary algorithms from such traditional algorithms are population based. Evolutionary algorithm performs a directed efficient search by adaptation of successive gen- erations of a larger number of individuals. Genetic Algorithms is one such evolutionary algorithm in finding an optimal or near optimal solution to a problem. In a traditional genetic algorithm, the representation is bit length string. Its approach is to generate a set of random solutions from the existing solutions, so that there is an im- provement in the quality of solutions throughout the generations. This implementation is achieved through main GA operators’ viz. random selection of two solutions from individuals in the parent generation; performing crossover operation on these two solutions to generate two new child solutions. Crossover operation is per- formed by exchanging specific elements of the two solutions selected; and mutation operation is conducted on child solutions to further explore the search space for better solutions. Different variations in simple GA ap- proach can be found in literature survey to improve its search capabilities [

As explained above Cheng, Gen and Tsujimura [

In Priority Rule Based (PR) representation, a chromosome is represented as a string of (n − 1) entries (p_{1}, p_{2}…p_{n}) where n − 1 is the number of operations in the problem instance. An entry p_{1} represents a priority rule selected beforehand. Accordingly, a conflict in the i^{th} iteration of Giffler and Thompson algorithm [_{i}. It means an operation from the conflict set has to be selected by the p_{i} ties are broken randomly. In GA domain, a best set of priority rules should be selected. Here simple cros- sover yields feasible schedules.

In Random Keys Representation (RK) was first proposed by Bean [

In Operation based representation, each gene represents an operation. A chromosome contains as many genes as the number of operations. For example, an nx m JSP there will be nxm genes in the chromosome. Beirwirth proposed a technique “permutation with repetition” [

The Preference List based representation (PL) uses a string of operations for each machine instead of a single string for all operations which is a direct representation of processing sequence decision variables. Quite often violation of constraints is encountered which can be overcome by repair algorithm.

In the Machine based representation, [

In the Job based representation [

The reproduction and mutation operators applied to JSP model are generally adopted from Travelling Salesman Problem because of the similarity in representations. Reproduction operators are generally required in GA to conduct the neighborhood search and a mutation operator generally ensures that the solution is not trapped in local minima. The design of both operators is crucial for the success of GA. Among the reproduction operators reported in the literature, PMX (partially matched crossover) [

In general, the flow chart for GA can be represented as shown.

In our experiment, four representations are used viz. Operation based (OB), Job based (JB), Machine based (MB), Priority rule based (PR). All experiments are conducted with 50 generations and a population size of 1000. Mutation probability varies with 0.1 to 0.9 values dynamically and elite population size is 20%. Reproduction probability used in our experiment is 0.1 Parents in our experiment are selected from two groups sorted out based on fitness value (i.e. minimum make span). Each parent is selected from these groups probabilistically.

In our experimentation, GA is programmed with different reproduction and mutation operators’. Instead of selecting operators randomly as in [

. Results of benchmark instances under different representations

Problem | Size | No. of Operations | Best Known Solution | OB | OB | JB | JB | MB | MB | PR | PR |
---|---|---|---|---|---|---|---|---|---|---|---|

Best | Avg. | Best Avg. | Best | Avg. | Best | Avg. | |||||

mt06 | 6 × 6 | 36 | 55 | 55 | 64.889 | 55 | 65.712 | 55 | 61.822 | 55 | 66.648 |

mt10 | 10 × 10 | 100 | 971 | 989 | 1116.02 | 971 | 1100.9 | 992 | 1145.06 | 958 | 1100.42 |

mt20 | 5 × 20 | 100 | 1206 | 1220 | 1394.47 | 1206 | 1383.08 | 1245 | 1427.24 | 1242 | 1426.54 |

abz05 | 10 × 10 | 100 | 1259 | 1275 | 1394.79 | 1259 | 1386.12 | 1287 | 1409.87 | 1267 | 1390.94 |

abz06 | 10 × 10 | 100 | 971 | 958 | 1072.19 | 971 | 1075.97 | 996 | 1096.13 | 978 | 1080.3 |

abz07 | 15 × 20 | 300 | 742 | 734 | 821.16 | 742 | 804.892 | 751 | 817.937 | 730 | 807.128 |

abz08 | 15 × 20 | 300 | 758 | 751 | 833.362 | 758 | 825.982 | 763 | 838.59 | 755 | 826.954 |

abz09 | 15 × 20 | 300 | 752 | 784 | 877.468 | 752 | 849.838 | 773 | 873.541 | 764 | 859.258 |

car01 | 5 × 11 | 55 | 7038 | 7038 | 8747.84 | 7038 | 8694.01 | 7038 | 8707.83 | 7038 | 8782.28 |

car02 | 4 × 13 | 52 | 7376 | 7378 | 8788.38 | 7376 | 8738.23 | 7221 | 8817 | 7166 | 8881.94 |

car03 | 5 × 12 | 60 | 7725 | 7590 | 9219.19 | 7725 | 9195.36 | 7725 | 9293.86 | 7725 | 9272.51 |

car04 | 4 × 14 | 56 | 8072 | 8003 | 9620.16 | 8072 | 9452.62 | 8276 | 9697.21 | 8132 | 9643.3 |

car05 | 6 × 10 | 60 | 7835 | 7873 | 9207.14 | 7835 | 9130.26 | 7862 | 9251.68 | 7862 | 9407 |

car06 | 9 × 8 | 72 | 8505 | 8505 | 10017.7 | 8505 | 9886.82 | 8505 | 10229.5 | 8485 | 9830.33 |

car07 | 7 × 7 | 49 | 6558 | 6576 | 7673.64 | 6558 | 7782.76 | 6627 | 7751.89 | 6632 | 7738.75 |

car08 | 8 × 8 | 64 | 8407 | 8407 | 9436.29 | 8407 | 9500.61 | 8458 | 9470.57 | 8366 | 9470.64 |

la01 | 5 × 10 | 50 | 666 | 666 | 783.616 | 666 | 796.901 | 674 | 746.506 | 666 | 782.789 |

la02 | 5 × 10 | 50 | 655 | 665 | 748.122 | 655 | 774.664 | 660 | 745.747 | 667 | 757.79 |

la03 | 5 × 10 | 50 | 617 | 620 | 688.729 | 617 | 687.389 | 626 | 690.773 | 620 | 699.527 |

la04 | 5 × 10 | 50 | 607 | 595 | 695.259 | 607 | 690.822 | 619 | 699.926 | 602 | 688.268 |

la05 | 5 × 10 | 50 | 593 | 593 | 640.494 | 593 | 658.885 | 593 | 606.404 | 593 | 699.114 |

la06 | 5 × 15 | 75 | 926 | 926 | 1000.79 | 926 | 1021.85 | 926 | 958.039 | 926 | 1075.5 |

la07 | 5 × 15 | 75 | 890 | 890 | 998.253 | 890 | 1015.05 | 893 | 983.784 | 890 | 994.044 |

la08 | 5 × 15 | 75 | 863 | 863 | 981.109 | 863 | 985.795 | 863 | 959.264 | 863 | 995.054 |

la09 | 5 × 15 | 75 | 951 | 951 | 1051.15 | 951 | 1084.34 | 951 | 988.331 | 951 | 1167.81 |

la10 | 5 × 15 | 75 | 958 | 958 | 1017.01 | 958 | 1045.73 | 958 | 971.19 | 958 | 1089.97 |

la11 | 5 × 20 | 100 | 1222 | 1222 | 1308.89 | 1222 | 1334.96 | 1222 | 1264.12 | 1222 | 1389.85 |

la12 | 5 × 20 | 100 | 1039 | 1039 | 1132.34 | 1039 | 1157.8 | 1039 | 1104.58 | 1039 | 1226.58 |

la13 | 5 × 20 | 100 | 1150 | 1150 | 1248.7 | 1150 | 1278.51 | 1150 | 1191.37 | 1150 | 1314.64 |

la14 | 5 × 20 | 100 | 1292 | 1292 | 1320.59 | 1292 | 1348.95 | 1292 | 1295.72 | 1292 | 1388.82 |

la15 | 5 × 20 | 100 | 1207 | 1207 | 1336.66 | 1207 | 1352.88 | 1227 | 1368.04 | 1207 | 1352.41 |

la16 | 10 × 10 | 100 | 979 | 982 | 1083.26 | 979 | 1066.88 | 988 | 1088.68 | 987 | 1071.61 |

la17 | 10 × 10 | 100 | 797 | 793 | 890.389 | 797 | 885.073 | 832 | 905.275 | 807 | 888.012 |

la18 | 10 × 10 | 100 | 861 | 861 | 962.052 | 861 | 967.819 | 885 | 976.877 | 883 | 977.685 |

la19 | 10 × 10 | 100 | 875 | 875 | 970.966 | 875 | 972.302 | 899 | 983.686 | 877 | 976.925 |

la20 | 10 × 10 | 100 | 936 | 907 | 1022.37 | 936 | 1040.62 | 944 | 1039.52 | 914 | 1041.05 |

la21 | 10 × 15 | 150 | 1105 | 1098 | 1252.76 | 1105 | 1247.82 | 1115 | 1264.74 | 1111 | 1281.85 |

la22 | 10 × 15 | 150 | 972 | 988 | 1146.21 | 972 | 1125.68 | 1031 | 1161.32 | 990 | 1133.83 |

la23 | 10 × 15 | 150 | 1035 | 1045 | 1188.52 | 1035 | 1168.18 | 1037 | 1180.52 | 1068 | 1187.63 |

la24 | 10 × 15 | 150 | 1004 | 1006 | 1135.91 | 1004 | 1134.02 | 1029 | 1155.7 | 995 | 1149.69 |

la25 | 10 × 15 | 150 | 1040 | 1055 | 1177.18 | 1040 | 1170.37 | 1036 | 1175.69 | 1058 | 1178.36 |

la26 | 10 × 20 | 200 | 1269 | 1279 | 1457.86 | 1269 | 1424.04 | 1304 | 1466.35 | 1310 | 1446.12 |

la27 | 10 × 20 | 200 | 1341 | 1363 | 1529.94 | 1341 | 1500.85 | 1421 | 1539.77 | 1374 | 1538.38 |

la28 | 10 × 20 | 200 | 1301 | 1295 | 1454.95 | 1301 | 1456.94 | 1334 | 1463.16 | 1284 | 1453.35 |

la29 | 10 × 20 | 200 | 1274 | 1302 | 1441.65 | 1274 | 1416.42 | 1307 | 1429.08 | 1270 | 1425.34 |

la30 | 10 × 20 | 200 | 1418 | 1429 | 1576.32 | 1418 | 1554.81 | 1444 | 1592.45 | 1432 | 1591.74 |

la31 | 10 × 30 | 300 | 1784 | 1784 | 1927.56 | 1784 | 1938.51 | 1785 | 1934.69 | 1784 | 1933.39 |

la32 | 10 × 30 | 300 | 1850 | 1850 | 2019.59 | 1850 | 2029.95 | 1855 | 2024.57 | 1853 | 2031.31 |

la33 | 10 × 30 | 300 | 1719 | 1725 | 1890.07 | 1719 | 1873.95 | 1719 | 1871.39 | 1725 | 1883.41 |

la34 | 10 × 30 | 300 | 1757 | 1782 | 1942.11 | 1757 | 1916.99 | 1801 | 1941.67 | 1793 | 1941.16 |

la35 | 10 × 30 | 300 | 1890 | 1905 | 2090.49 | 1890 | 2079.74 | 1919 | 2116.75 | 1906 | 2097.97 |

la36 | 15 × 15 | 225 | 1348 | 1343 | 1515.99 | 1348 | 1492.07 | 1385 | 1519.31 | 1352 | 1498.9 |

la37 | 15 × 15 | 225 | 1486 | 1506 | 1674.69 | 1486 | 1651.36 | 1548 | 1698.19 | 1496 | 1687.76 |

la38 | 15 × 15 | 225 | 1319 | 1307 | 1455.03 | 1319 | 1474.01 | 1369 | 1494.87 | 1299 | 1486.33 |

la39 | 15 × 15 | 225 | 1316 | 1325 | 1486.54 | 1316 | 1479.86 | 1383 | 1519.21 | 1363 | 1508.05 |

la40 | 15 × 15 | 225 | 1296 | 1330 | 1469.25 | 1296 | 1460.38 | 1360 | 1485.68 | 1338 | 1492.6 |

orb01 | 10 × 10 | 100 | 1124 | 1130 | 1262.1 | 1124 | 1277.16 | 1150 | 1304.67 | 1126 | 1277.02 |

orb02 | 10 × 10 | 100 | 924 | 919 | 1047.74 | 924 | 1031.34 | 949 | 1045 | 931 | 1060.46 |

orb03 | 10 × 10 | 100 | 1067 | 1116 | 1254.65 | 1067 | 1232.93 | 1080 | 1262.65 | 1065 | 1230.21 |

orb04 | 10 × 10 | 100 | 1028 | 1055 | 1176.34 | 1028 | 1154.35 | 1075 | 1164.81 | 1053 | 1160.75 |

orb05 | 10 × 10 | 100 | 931 | 945 | 1075.54 | 931 | 1056.76 | 969 | 1109.28 | 931 | 1064.55 |

orb06 | 10 × 10 | 100 | 1046 | 1093 | 1248.37 | 1046 | 1213.47 | 1116 | 1272.66 | 1067 | 1233.12 |

orb07 | 10 × 10 | 100 | 419 | 415 | 469.22 | 419 | 467.724 | 425 | 472.424 | 418 | 470.205 |

orb08 | 10 × 10 | 100 | 928 | 940 | 1113.01 | 928 | 1075.88 | 969 | 1135.78 | 947 | 1102.29 |

orb09 | 10 × 10 | 100 | 949 | 953 | 1080.08 | 949 | 1059.32 | 958 | 1066.74 | 964 | 1079.28 |

orb10 | 10 × 10 | 100 | 977 | 989 | 1150.48 | 977 | 1125.03 | 991 | 1128.54 | 980 | 1122.7 |

The evolution process over 50 generations for the benchmark instance ABZ 5 for instance has been shown in

GA-flow chart

Deviations of different instances under different representations

Avg. deviation vs. GA representations

Evolution of ABZ5 under different representation

Evolution of CAR-07 under different representations

In our further study, we intend to use Job based representation In GA and with the aid of other techniques work to get optimum solutions in possible number of instances.