A Hybrid Pareto Based Multi Objective Evolutionary Algorithm for a Partial Flexible Open-Shop Scheduling Problem

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. II (Jan. - Feb. 2017), PP 32-38
www.iosrjournals.org
DOI: 10.9790/5728-1301023238 www.iosrjournals.org 32 | Page
A Hybrid Pareto Based Multi Objective Evolutionary Algorithm
for a Partial Flexible Open-Shop Scheduling Problem
N. Jananeeswari1*
Dr. S. Jayakumar2
Dr. M. Nagamani3
1Research Scholar of Mathematics, Aringar Anna government Arts college, Cheyyar, Tiruvannamalai, India.
2Assistant Professor of Mathematics, Aringar Anna government Arts college, Cheyyar, Tiruvannamalai, India,
3Associate Professor of Mathematics, Global Institute of Engineering & Technology, Vellore, Tamilnadu, India,
Abstract: In this paper, A Partial flexible, open-shop scheduling problem (FOSP) is a combinatorial
optimization problem. This work, with the objective of optimizing the makespan of an FOSP uses a hybrid
Pareto based optimization (HPBO) approach. The problems are tested on Taillard’s benchmark problems. The
consequences of Nawaz, Encore and Ham (NEH) meta heuristic are introduced to the HPBO to direct the
search into a quality space. Variable neighbourhood search for (VNS) is employed to overcome the early
convergence of the HPBO and helps in global search. The results are compared with standalone HPBO,
traditional meta heuristics and the Taillard’s upper bounds. Five problem locate are taken from Taillard’s
benchmark problems and are solved for various problem sizes. Thus a total of 35 problems is given to explain.
The experimental results show that the solution quality of FOSP can be improved if the search is directed in a
quality space in light of the proposed LHPBO approach (LHPBO-NEH-VNS).
Keywords: Minimize the make span, Open shop scheduling, Hybrid Pareto based, Variable neighbourhood
search, Evolutionary algorithms, Partial flexible, Local search.
I. Introduction
Scheduling is the assignment of resources (e.g., machines) to tasks (e.g., jobs) in order to ensure the
completion these tasks in a reasonable measure of time. In an open shop, the technological constraints demand
that the jobs pass between the machines in the same order; i.e., If J1 must be prepared on machine Mk before
machine Mi then the same is true for all jobs. A permutation schedule is one on which every machine processes
the jobs in the same order; i.e., if on a machine M1 job Ji is processed before Jk, then the same is valid for all
machines [1]. The objective is to find an arrangement of jobs that minimizes the make span. I.e., the time
required to finish every one of the jobs. This problem is observed to be NP-hard [2].The meta heuristics can be
partitioned into either helpful heuristics or improvement meta heuristics, the previous are heuristics that
assemble a partial feasible schedule from scratch and the last are heuristics that attempt to improve a previously
generated schedule by typically applying some form of specific problem knowledge.
The multi objective of this work is to discover a permutation and combination of jobs that minimizes
the make span. The present review explores the further improvement in the solution quality, if the populace in
the improvement phase is provided with better starting solutions. The work depicts the percentage improvement
that the proposed technique is able to obtain when contrasted with some of existing meta heuristics because of
the improvisation in the search space of HPBO. This approach can be considered as the critical contribution of
this study. This work goes for directing the search space of HPBO into certain quality. Then, VNS is employed
to make the solution jump out of the local optimal solution. This approach is named as HPBO -NEH-VNS heuristic. The
solution thus obtained is compared with the results obtained using the HPBO and the upper limits of Taillard’s [3]
benchmark problems. The approach proves that outcomes approach the optimum faster when the search space is
nearer to the optimum.The remains of the paper are sorted out as follows. Section-2 discusses the literature
survey. The methodology is presented in Section-3. The results and discussions are made in Section-4.
Conclusions and further future scope are given in the Section-5
II. Literature Survey
2.1 Meta-Heuristics
Many constructive meta heuristic methodologies have been proposed in the literature. Johnson’s
algorithm is the soonest known meta heuristic for the permutation Partial flexible open shop problems. The two
machine open shop problem with the objective of minimizing make span is otherwise called Johnson’s [4]
problems. An optimal sequence is found by following a meta heuristic of finding the minimum machining time
and allotting the job to the machine in a particular order is adopted. The literature is replete with customary meta
heuristics [5-7] and nontraditional meta heuristics (Tabu search ([8-10]); Simulated annealing [11-14].
Evolutionary algorithm [15-17], Ant colony optimization [18] to give some examples. Contrary to constructive
heuristics, improvement heuristics begin from an already built schedule and attempt to improve it by some given
methodology. Many improvement heuristic approaches likewise have been proposed in the research [19-21]

A Hybrid Pareto Based Multi Objective Evolutionary Algorithm for a Partial Flexible Open-Shop
2.2 Hybrid Pareto based Evolutionary Algorithms
Duda [22] assesses a Hybrid Pareto based Evolutionary algorithm with variable neighbourhood search
against a standalone VNS algorithm. The results of the conducted experiments demonstrate that hybridization of
a meta-heuristic with local search algorithms may not continually bring additional performance advantage. Ali
and Fawaz [23] address the open shop scheduling problem with respect a due date-based best performance
measure, i.e., maximum lateness. Lian et al. [24] proposes a novel Hybrid Pareto based Evolutionary algorithm
to minimize the make span of an open shop. Kuo et al. [25] propose an open shop scheduling algorithm based
on Hybrid Pareto based Evolutionary algorithm model. Liu et al. [26] propose a Hybrid Pareto based
Evolutionary algorithm for open shop scheduling with stochastic handling time. Chen [27] proposes an
improved Hybrid Pareto based Evolutionary algorithm for open shop scheduling. Sha and Lin [28] propose a
HPBO demonstrate for multi-objective partial flexible open shop scheduling problem. Zhang et al. [29] provide
a hybrid interchange two phases Hybrid Pareto based Evolutionary algorithm for open shop scheduling
problems. Deb [30] talks about the evolutionary approaches used in solving the multi-objective optimization
problems.
III. Methodology
The objective of the work is to minimize the make span of the Partial flexible open shop scheduling
problems. This is attempted by developing a HPBO approach. 5 problem sets are taken from “Taillard” [2]
benchmark problem and are solved for different sizes. A total of 35 problems is illuminated. The results of
HPBO-NEH-VNS are compared with the upper bound (UB) of Taillard’s benchmark and also compared with
the results of CDS, NEH, HPBO, HPBO-NEH, and HPBO-VNS.The Taillards benchmark problem is initially
solved using NEH meta heuristics and a sequence is obtained. The HPBO requires a populace of sequences to be
initialized. In this problem, the HPBO is introduced with fifteen randomly generated populations. The sequence
that is obtained by NEH meta heuristic is instated as one of the initial population. This HPBO is referred to as
HPBO instated with NEH (HPO-NEH). This initialization is done with the trust that the search space is closer to
the optimal solution. The results obtained by the HPBO-NEH are further improved with the VNS. The VNS
keeps the early convergence of HPBO and jump out of locally optimal solution. This approach is named as
HPBO-NEH-VNS. Hansen et al. [31] express that Variable neighbourhood search (VNS) is a meta-heuristic, or
a structure for building meta heuristics, based upon deliberate changes of neighbourhoods both in drop phase, to
find a local minimum, and in perturbation phase to emerge from the relating valley.
3.1 HPBO algorithm for FOSP
A hybrid Pareto based optimization (HPBO), roused by the motion of an Evolutionary searching for
based solution, was produced by Kennedy and Eberhart [32] for optimization of continuous nonlinear functions.
To find the optimal solution each evolutionary called a scheduling adjusts its searching direction according to
two variables, its own best past involvement (pbest) and the best experience of all other members (gbest). The
system is introduced with a population of random solutions and searches for optimal solution by updating
generations. In HPBO potential solutions called Evolution, fly through the problem space by following the
current optimum evolution. HPBO introduced the behaviour of bird flocking. In HPBO each single solution is a
“Hybrid evolutionary” in the search space. We call it “Pareto optimal”. Every one of the particles has fitness
value and velocities which direct the flying of the particles. The evolutionary through the problem space by
following the current optimum evolution. Every particle is updated by following two “best” values. The first
value is the best solution, it has accomplished so far known as “pbest”. Second value is the best value obtained
so far by any evolutionary is the populace called “gbest”. After finding the two best values the particle updates
its speed and position with the following equations.
Let some job i  ni 1 is to be scheduled on machine j  mj 1 in the same technological order with
criteria to be optimized as minimization of make span C*
max under no-wait. Let ti j be the time of processing of
the job i on the machine j, Ti be the sum total of processing times corresponding of job i on m machines, D( p, q)
be the minimum delay of the first job p on the first machine after the job p is completed under no-wait constraint
and can be calculated by Reddi and Ramamoorthy (1972) equation as
mkttttt
ttttttttttqpD
k
j
jq
k
j
jpmqqq
mppppqqppqp





 2),0,max()0),..........(
.........),.......,(,max(),(
1
1
,
2
,1,2,1,
,4,2.2,2,1,3,2,1,2,
The calculation for C*
max are as follows
For i=1,2,3,…..,n and j=1,2,3,……,m
1,1
*
1,1 tC 

jijijiji
ii
jjj
tCCC
niiiDtt
mjttC
,
*
),1(
*
)1(,
*
,
1,1),1(
*
i,1
,1)1(,1
*
,1
),max(
,......4,3,2),,1(C
,........,4,3,2,






and the make span under no-wait is
*
,
*
max mnCC 
The proposed algorithm has the following assumptions:
 All jobs and machines are available one’s disposal at the start of the processing.
 Jobs pre-emption is not permitted.
 The machines are accessible all through the processing and never breakdown.
 All processing times of the machines are deterministic and understood.
 Each job is prepared through each of the machine precisely once.
 Each machine can perform only one single undertaking at a time.
 A job is not accessible to the next machine until and unless processing on the present machine is completed.
 The processing time of jobs includes the setup times on machines or otherwise can be disregarded.
Algorithm 1: Pareto Based Evolutionary Algorithm
Step 1: Generate Initial Population (P)
Step 2: Evaluate (P)
Step 3: While termination conditions aren't met do if same
Step 4: Fitness values occur in P then
Step 5: Apply Immigrate
Step 6: End if
Step 7: Selection (P1)
Step 8: Crossover (P2)
Step 9: Mutations (P3)
Step 10: ILS (P4)
Step 11: Pairwise Comparison (P1, P2)
Step 12: End while.
3.2 Local Search hybrid Pareto based optimization (LHPBO)
Although EAs have been applied to solve many optimization problems effectively, they are also known
for lack of the ability of exploitation. The EA combines the HEA and local search to gather their advantages to
improve the search capacity. In the proposed LEA, we do local search to the best NLS individuals in the
population of EA every TLS generations. The local search procedure depends on the framework of variable
neighborhood descent (VND) algorithm. We utilize three neighborhood functions, all focused on critical
operations. The Critical operations are the operations on the critical path, which is the longest path on the
disjunctive graph representation of a schedule. In the event that more than one critical path exists, we select one
randomly. Table 1 gives the pseudo code the principal neighborhood (N1) follows the neighborhood proposed.
It swaps the initial two and last two operations in the critical blocks. For the primary (last) critical block, only
the last (initial) two operations are swapped. The second neighborhood (N2) chooses one random critical
operation and one random qualified machine. Then, the operation is inserted into every single possible position
on the machine. The third neighborhood (N3) is an extension of N2. It inserts the selected operation into every
possible position on all qualified machines. We apply the three neighborhood functions one by one utilizing the
steepest descent algorithm until the neighborhood ideal is reached. The local search goes at minimizing the
make span, and thus the fitness function WS (N1) here is a straight weighted summation.
Pseudo Code of the Local Search Procedure
Local Search (best Individual x)
Begin
y* = x
For t = 1 to TSH
If t = 1 Then y = x
Else y = Shaking(x)
For n = 1 to 3
y’= Steepest Descent (y, Nn, nE)
If WS(y)≤ WS(y) Then y = y’
If WS(y)≤ WS(y*) Then y=y*

End
End
x* = y
End
We allowed accept equal-fitness neighboring solution for at most n E times continuously. Shaking of a solution
is accomplished by inserting a random critical operation to a random position on a random qualified machine.
IV. Results And Discussions
The objective of this work is to build a HPBO model to solve the Partial flexible open shop scheduling
problems for minimizing the make span; Five problem set are taken from Taillard benchmark problem and are
solved. The problems are 20 jobs, 5, 10 and 20 machines; 50 jobs, 5, 10 and 20 machines and 100 jobs, 5
machines. The proposed HPBO-NEH-VNS clearly gives better results than other methods such as CDS, NEH,
HPBO, HPBO-NEH, and HPBO-VNS. This is possible because the search space of HPBO-NEH-VNS is in the
area of superior solution. The search space of HPBO is initialized with the results of NEH meta heuristic; this
ensures that the solution space is of certain quality. The VNS helps in the solution of HPBO jumping out of the
local optimal solution. Thus, the interchange of VNS has helped the HPBO to come out of the pre convergence.
In smaller size problems the average difference in make span between upper bound and HPBO-NEH-VNS is
less and this difference increases with increasing number of machines. Average difference in make span
between HPBO-NEH-VNS and upper bound for 20 Job 5 machines is 12.5, 20 Job 10 machines is 21 and 20 Job
20 machines is 32.
Table I shows the results of the five instances of 20 × 5, 20 × 10 and 20 × 20 problems that are explained using
the various methods. Make span of 20 jobs, 5, 10 and 20 machines.
SL No. Problem
size
CDS NEH HPBO HPBO-
NEH
HPBO-
VNS
HPBO-
NEHVNS
UB
1 20 × 5 1390 1286 1297 1286 1320 1278 1278
2 20 × 5 1424 1365 1373 1365 1368 1365 1359
3 20 × 5 1249 1159 1124 1132 1164 1100 1080
4 20 × 5 1418 1325 1364 1325 1349 1309 1293
5 20 × 5 1324 1305 1250 1263 1277 1250 1236
6 20 × 10 1757 1680 1660 1635 1673 1586 1582
7 20 × 10 1854 1729 1727 1722 1747 1684 1659
8 20 × 10 1645 1556 1517 1556 1588 1521 1496
9 20 × 10 1547 1439 1434 1419 1514 1399 1378
10 20 × 10 1558 1502 1492 1502 1537 1450 1419
11 20 × 20 2559 2410 2365 2386 2446 2330 2297
12 20 × 20 2285 2150 2177 2148 2195 2110 2100
13 20 × 20 2565 2411 2387 2399 2508 2342 2326
14 20 × 20 2432 2262 2304 2251 2410 2248 2223
15 20 × 20 2506 2397 2358 2388 2470 2302 2291
Table II shows the make span of 50 jobs and 5, 10 and 20 machines problem size. The proposed
HPBO-NEH-VNS clearly give better results than other methods. The make span obtained by HPBO-NEH-VNS
is nearly equal (within 7%) to the UB of the Taillard’s benchmark problems and in some cases it is equal to the
UB. (for 50 × 5, SL No. 1). Similar to the last problem set, the average difference between upper bound and
HPBO-NEH-VNS increases with increasing number of machines. (For 50 Job 5 machines is 6.1; 50 job 10
machines is 52.0; 50 Job 20 machines is 136.2.
Table II. Make span of 50 jobs, 5, 10 and 20 machines.
SL No. Problem
size
CDS NEH HPBO HPBO-
NEH
HPBO-
NEH-VNS
UB
1 50 × 5 2816 2732 2729 2729 2724 2724
2 50 × 5 3032 2843 2906 2843 2843 2834
3 50 × 5 2703 2640 2676 2620 2631 2620
4 50 × 5 2884 2782 2824 2782 2761 2750
5 50 × 5 3038 2868 2873 2864 2864 2863
6 50 × 10 3421 3135 3240 3134 3059 3025
7 50 × 10 3246 3032 3093 3025 2934 2892
8 50 × 10 3280 2986 3139 2965 2932 2864
9 50 × 10 3392 3198 3236 3172 3115 3064
10 50 × 10 3375 3160 3186 3115 3052 2986
11 50 × 20 4328 4082 4192 4061 4010 3875
12 50 × 20 4216 3921 4066 3918 3864 3714
13 50 × 20 4189 3927 3981 3907 3808 3668

14 50 × 20 4280 3969 4067 3956 3844 3752
15 50 × 20 4121 3835 3998 3832 3815 3634
Refers to upper bound of Taillard’s benchmark problems.
Make span values of 100 jobs and 5, 10 and 20 machines are shown in Table 3. For the 100 jobs
problem as well the proposed HPBO-NEH-VNS method gives better results when compared with other
methods. In certain instances (For 100 × 5, SL No. 1), the HPBO-NEH-VNS is able to give the make span
values equivalent to the upper bound of the Taillard’s benchmark problems Table III.
Table III. Make span of 100 jobs, 5 machines.
SL No. Problem
size
CDS NEH HPBO HPBO-
NEH
HPBO-
NEH VNS
UB
1 100 × 5 5592 5518 5527 5519 5492 5492
2 100 × 5 5563 5348 5327 5300 5290 5268
3 100 × 5 5492 5218 5253 5213 5213 5174
4 100 × 5 5273 5023 5078 5023 5023 5014
5 100 × 5 5461 5265 5323 5266 5252 5250
Consistently, the HPBO-NEH-VNS is giving best results than the other various methods taken for
comparison. Hence, it can be stated that with a better solution search space provided for the HPBO, the HPBO
with the help of VNS is able to avoid early convergence. It can also be induced that the search is able to
converge faster nature of the initial quality of the search space. In order to compare computational time required,
various methodologies are coded in matlab and run on a 3 G B RAM, 2.99 GHz, Dual Core system with
Windows XP operating system. The CPU time for different size problems are shown in Table-4 Computational
time.
Table IV. CPU time in seconds
SL No. Problem
size
CDS NEH HPBO HPBO-
NEH
HPBO-
VNS
HPBO- NEH
VNS
1 20 × 5 0.0156 0.0156 1.29 1.32 7.96 7.96
2 20 × 10 0.0156 0.0156 1.58 1.50 35.82 36.68
3 20 × 20 0.0157 0.0314 1.96 1.92 206.92 197.84
4 50 × 5 0.0156 0.0156 1.96 1.96 13.08 13.06
5 50 × 10 0.0156 0.0156 2.26 2.26 60.62 60.45
6 50 × 20 0.0156 0.0782 2.95 2.95 332.14 330.84
7 100 × 5 0.0314 0.0156 3.08 3.06 22.50 22.10
It can be observed that computational time for HPBO NEH VNS approach is slightly lesser than HPBO VNS for
larger problem sizes.
Table V. % Improvement of makespan over
SL No. Problem size CDS NEH HPBO HPBO NEH HPBO VNS
1 20 × 5 8.08 0.00 0.85 0.00 2.64
2 20 × 5 4.32 0.00 0.58 0.00 0.21
3 20 × 5 13.54 5.36 2.27 2.90 5.81
4 20 × 5 8.32 1.22 4.20 1.22 3.05
5 20 × 5 5.84 4.40 0.00 1.04 2.16
6 20 × 10 10.78 5.92 4.66 3.08 5.48
7 20 × 10 10.09 2.67 2.55 2.25 3.74
8 20 × 10 8.15 2.36 0.26 2.30 4.40
9 20 × 10 10.57 2.85 2.50 1.42 8.22
10 20 × 10 7.44 3.58 2.89 3.58 6.06
11 20 × 20 9.26 2.90 0.98 1.87 4.44
12 20 × 20 87.73 1.36 2.64 1.27 3.48
13 20 × 20 8.82 2.29 1.27 1.78 6.44
14 20 × 20 8.27 0.62 2.49 0.13 7.20
15 20 × 20 7.73 3.05 1.37 2.66 6.23
Table V shows the percentage improvement of results of HPBO VNS NEH for a 20 job problem size,
when compared with other methods used in this work.Improvement of makespan using HPBO NEH VNS. It can
be observed that HPBO do better than HPBO-VNS. HPBO when initialized with the results of NEH does better
than both HPBO and HPBO-VNS algorithms. The VNS takes the HPBO to early convergence. But when the
same solution is initialized with best quality solution space, obtained from the constructive meta heuristic
(NEH), it is able to perform better solution.

Table VI shows the comparison of experimental results from review literature The results show the LHPBO-
NEH-VNS is providing results that are comparable to the Kuo et al. [25] LHPBO and in one instance (20 × 10)
it is able to provide better result. This indicates the LHPBO-NEH-VNS model is worth exploring.
Table VI Comparison of results.
Problem size Makespan
UB Kuo et al. LHPBO NEH-LHPBO –VNS
20 × 5 1276 1276 1276
20 × 10 1582 1586.3 1586
20 × 20 2296 2306 2310
50 × 5 2723 2723 2723
Though the results show the proposed meta heuristic performs superior certain HPBO models, the rate
deviation from the optimum still persists. VNS has an inalienable characteristic that is not taking the results to
the optimum. But, the objective of the work is to prove that if the initial solution space is given with quality
solutions, the HPBO can be utilized for obtaining speedier results.
V. Conclusions
A Hybrid Pareto based optimization algorithm named HPBO-NEH-VNS is used to illuminate the
FOSP with the goal of minimizing makespan. The research demonstrates that the HPBO instated with a superior
solution space obtained from the useful meta heuristic (NEH), can give ability to provide solutions that are
better than some of the existing meta heuristics and takes the solution to early convergence of results. The
outcomes show that the percentage deviation of the makespan increments with the increasing machine, proving
that the unpredictability of the partial flexible open shop scheduling problems depends on the expanding
machine size. The results additionally show that the percentage deviation of the makespan with respect to the
expanding job is not significant.
Acknowledgments
The authors wish to thank the unknown analysts for giving their important proposals in improving the quality of
the best paper.
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A Hybrid Pareto Based Multi Objective Evolutionary Algorithm for a Partial Flexible Open-Shop Scheduling Problem