The document proposes genetic algorithm approaches to solve the job shop scheduling problem (JSSP). It introduces new crossover and mutation operators designed based on the problem characteristics. The crossover combines operation orders of different machines from two parents. The mutation permutes successive operations on the same machine that are part of the critical path. Experimental results test the approach over 10 runs with a population size of 100, crossover probability of 0.7 and mutation probability of 0.1.

1. Submitted to:
Prof. Dr. N. SELVARAJ
Department of Mechanical Engineering
National Institute of Technology Warangal
Prepared by:
Sanjeev Singh Yadav
Roll no. 133565
M.Tech 1st year (cim)

2.  Each job has m operations that must be processed
at m machines.
 The operations of a given job have to processed in
a given order.
 The objective is to determine the schedule which
minimizes the makespan -The time required to
complete all the jobs.
2
Job Operation routing (processing time )
1 1(3) 2(3) 3(3)
2 1(2) 3(3) 2(4)
3 2(3) 1(2) 3(1)

3.  Genetic algorithms have been tried to solve the
job-shop scheduling.
 However, the simple genetic algorithm is with a
slow convergent speed and is easy to converge
prematurely.
 But, the crossover and mutation operators
◦ not sufficiently made use of the characteristics of the
problem structure.

4.  Hence, in this paper, To sufficiently use the
information of the problem structure, a new
crossover and mutation operators based on the
characteristics of the job shop problem were
designed
 The proposed genetic operators are explained
using disjunctive graph theory model

5.  Given an instance of JSSP, it is associate with a
disjunctive graph G = (V, A, E)
 with V being the set of nodes (operations )
 A the set of conjunctive directed arcs
 E the set of disjunctive undirected arcs (edges)

6.  V = {0,1,.., N, N +1} , where {0} and {N +1} are
special nodes which identify the start and
completion of the overall jobs
 A = {(i, j) : operation i is an immediate predecessor
of operation j in the chain of job }
 E = { (i, j): operation i and operation j are
processed on the same machine , i, j ∈V }.
 For each vertex i∈V , a weight di is associated, and
di is the duration of the operation i .
 d is 0 for node 0 and N+1

7.  If length of a path is defined as the sum of the
weights of the vertices in the path, solving the job
shop scheduling problem corresponds to finding
an acyclic orientation of G so that the length of the
longest path between 0 and N +1 (critical path) is
minimized.

8.  In this representation, the chromosome consists of
n*m genes.
 i.e each job will appear m times exactly.
 E.x (3-job and 3 machine problem ) a
chromosome is given as [2 1 3 1 2 1 2 3 3].
◦ So, 1 represents the job 1, 2 represents the job 2 and 3
represents the job 3.
◦ Because each job consists of three operations, it occurs
exactly three times in the chromosome.

9. [2 1 3 1 2 1 2 3 3]

10.  The fitness function is the function of the
objectives function and defined as
 And the selection probability is

11.  It is driven be the belief that the good gene
characteristics preservation and the feasibility
are the most important criteria to design
crossover operation in JSSP.
 In this paper, a new crossover operator based on
the characteristic of the JSSP itself was designed.
The offspring generated can keep the good
characteristics of the problem structure and
satisfy the feasibility.

12.  Suppose , there are two parents: parent 1 and
parent 2
Parent 1.
Parent 2.

13.  Divide the machine numbers into two
complementary sets, such as {1, 3} and {2}.
 Combine the operation orders of machines {1, 3} in
the parent 1 and the operation orders of machine
{2} in the parent 2 to form child 1.
 Similarly, Combine the operation orders of machine
{2} in the parent 1 and the operation orders of
machine {1, 3} in the parent 2 to form child 2.

14. Child 1.
Child2.

15.  Given an individual chromosome, mutation
generates the child by the following procedure:
 Step 1. Calculate/specify the critical path of this
individual.
 Step 2. Permuting two successive operations v and
w assigned to the same machine with probability of
pm and for which the arc (v, w) is on a critical path
in that individual.

16.  For example: the graph of the parent 1is and the
critical path of the parent 1 is 0-1-8-9-10.
 Then we know that the operations 1 and 8 are
assigned to the same machine 1.
 Permuting two successive operations 1 and 8
assigned to the same machine with probability of
pm and get the child 1 as shown below.

17.  Experimental results
◦ Population size 100
◦ Cross over probability 0.7
◦ Mutation probability 0.1
◦ 10 independent runs for each test

18. 18