1. business and industrial
2. manufacturing

The 2012 International Symposium on Semiconductor Manufacturing Intelligence
Hybrid Sampling Strategy-based Multiobjective Evolutionary
Algorithm for Process Planning and Scheduling Problem
Wenqiang Zhang1, Mitsuo Gen2 and Jungbok Jo3*
Graduate School of IPS, Waseda University, Fukuoka, Japan
2
Fuzzy Logic Systems Institute (FLSI), Fukuoka, Japan and
Dept. of Industrial & Management Eng., Hanyang University, Korea
3
Division of Computer and Information Engineering, Dongseo University, Korea
1
*
[email protected]
Abstract
Process planning and scheduling (PPS) is a most important,
practical but very intractable problem in manufacturing systems.
Many research works use multiobjective evolutionary algorithm
(MOEA) to solve such problems; however, they cannot achieve
satisfactory results in both quality and speed. This paper proposes
a hybrid sampling strategy-based evolutionary algorithm
(HSS-EA) to deal with PPS problem. HSS-EA tactfully unites the
advantages of vector evaluated genetic algorithm (VEGA) and a
sampling strategy according to Pareto dominating and dominated
relationship-based fitness function (PDDR-FF). The sampling
strategy of VEGA prefers the edge region of the Pareto front and
PDDR-FF-based sampling strategy has the tendency converging
toward the central area of the Pareto front. These two mechanisms
preserve both the convergence rate and the distribution
performance. Numerical comparisons show that the convergence
performance of HSS-EA is better than NSGA-II and SPEA2 while
the distribution performance is slightly better or equivalent, and
the efficiency is obviously better than they are.
Keywords:
evolutionary
algorithm,
hybrid
sampling,
multiobjective optimization, process planning and scheduling
1. Introduction
Process planning and scheduling (PPS) is to process a set of
prismatic parts into completed products effectively and
economically in a manufacturing system (MS). A prismatic part to
be produced is generally described by features. For each feature,
one or more corresponding operations are determined according to
its feature geometry and available machining resources. Each
operation includes the selection of machines, tools and tool access
direction (TAD). There are precedence relationship constraints
among operations according to the geometrical and technological
consideration. Process planning is the determination of an optimal
process plan, i.e. operations and their sequences, within the
precedence relationship constraints and manufacturing resource
(machine and tool) constraints. The scheduling is the allocation of
the resources in the shop over time to manufacture the various
parts [1]. PPS becomes more important for the effective allocation
and utilization of resources in the MS. However, seeking an
optimal solution rapidly and effectively from all of the
permutations, combinations of all of the tasks, manufacturing
resources according to specified criteria is very difficult for
decision maker.
Duos to its importance, practicality and difficulty, in the past
decade, many research works have addressed the PPS problem.
These approaches include simulated annealing (SA) algorithm [1],
[2], tabu search algorithm [3], agent-based approach [4], [5],
particle swarm optimization (PSO) algorithm [6], [7] and genetic
algorithm (GA) [8-12].
Unfortunately, most of the above previous researchers
concern single objective optimization rather than multiobjective
optimization problems (MOOPs). In the real world, almost every
problem involves optimization of several incomparable or
incommensurable objectives simultaneously.
Multiobjective evolutionary algorithms (MOEAs) have been
recognized to be well-suited for solving MOOPs [13],[14]. As the
first one of MOEAs, vector evaluated genetic algorithm (VEGA)
divides the population into m sub-populations (m is number of
objectives), each of which evolves toward a single objective [15].
The benefit of VEGA is the strong ability to converge to the edge
region of the Pareto front by its simple sampling strategy and less
time complexity [16]. However, the qualities (especially, diversity)
of VEGA are not good because of the selection bias. Therefore,
VEGA cannot be used to address multiobjective PPS problems in
practical situation.
As two classical MOEAs, NSGA-II [17] and SPEA2 [18]
have been proven to be able to get better quality in solving
MOOPs. NSGA-II can get the better quality owing to its Pareto
ranking and crowding distance mechanism. SPEA2 depends on
raw fitness assignment mechanism and density mechanism.
Unfortunately, these two algorithms both need much CPU time.
These drawbacks of NSGA-II and SPEA2 cause that they cannot
be applied to solve multiobjective PPS problems online.
Ho et al. propose a generalized Pareto-based
scale-independent fitness function (GPSI-FF) to solve the large
parameter optimization problem [19]. The GPSI-FF can obviously
speed up the convergence rate [20], especially around the central
area of the Pareto front. However, the lack of diversity
preservation mechanism causes that GPSI-FF also cannot be
chosen to deal with multiobjective PPS problems. Moreover, the
difference between nondominated and dominated individuals can
be decreased by GPSI-FF values. This disadvantage causes that
more dominated individuals could be held in archive (external
population) while nondominated ones would be removed from
archive. It reduces the performance of archive, unless an enough
large size of its archive is set to store a sufficient number of
individuals.
To increase the quality both of convergence and distribution,
and to reduce the computation time, a new hybrid sampling
strategy-based evolutionary algorithm (HSS-EA) is proposed to
solve the multiobjective PPS problem.
For covering the drawback of GPSI-FF, a new Pareto
dominating and dominated relationship-based fitness function
(PDDR-FF) is proposed to evaluate the individuals. It gives the
sensible difference values between the nondominated and
dominated individuals. Moreover, ones with different numbers of
dominating are also set as different fitness values even though
they are all nondominated individuals. The individuals locating
around the central region of Pareto font will have smaller values
than the edge points.
The proposed method deliberately unites the two different
mechanisms. One is the sampling strategy of VEGA with a
preference for the edge region of the Pareto front, and the other is
the sampling strategy of PDDR-FF with tendency converging
toward the central area of the Pareto front. These two mechanisms
not only preserve the convergence rate, but also guarantee the
better
distribution
performance.
Moreover,
some
problem-dependent crossover, mutation and local search methods
are also used to improve the performance of the algorithm.
The paper is organized as follows: Section 2 describes the PPS
problem description and formulates the mathematical model;
Section 3 presents the detailed HSS-EA approach; Section 4 gives
Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem
a discussion and analysis of numerical experiments results; finally,
the conclusion and future work are given in Section 5.
2. PPS Problem
2.1 Problem Description
The illustration of the PPS problem in a MS is shown in Fig.
1. In Fig. 1, there are a set of parts (four parts) to be processed by
a number of machines (four machines) with a number of tools and
different TADs. Each part has several operations (four parts have
4, 3, 3, and 4 operations respectively), and each operation can be
performed on more than one suitable machines with different
processing times. Table 1 shows the operation information of the
4 parts. Each column describes the part ID, operation ID,
successors, operation name, TAD candidates, machine candidates,
tool candidates and machining time, respectively.
part1
part2
o12
o11
o14
o42
o33
o11
o41
o43
o42
o44
o32
o23
o44
part4
o31
o21
o13
o21
part3
o22
o12
o31
o33
o32
o41
o43
o22
o23
o13
o14
o21
Machine m1
o14
o12
Machine m2
o42
Machine m3
o44
o33
Machine m4
o41
o11
o32
o43
o23
o13
Makespan
o22
Idle time and
change time
o31
0
Time
Figure 1: Illustration of PPS problem
Table 1: Operation information of part 1 to part 4
Part-ID Op-ID Successor Operations
Part 1
Part 2
Part 3
Part 4
makespan and minimizing variation of workload for each machine
are important and frequently-used. Reducing makespan means
that the MS can get higher producing efficiency in a limited
period. Decreasing variation of machine workload can balance the
utilization of machines. The mathematical model of them is
expressed in the following notations.
Indices
i, k : indices of part, (i, k = 1, 2, , I ) .
j, h : indices of operation for part i , ( j ,h=1,2, ,J i ) .
m : index of machine, (m = 1, 2, , M ) .
l : index of tool, (l = 1, 2, , L) .
d : index of TAD, (d = 1, 2, , D) .
Decision variables
if oij is performed by m,
M 1,
xmij
=
0,
otherwise

1, if oij is performed by l ,
xlijT = 
0, otherwise
1, if oij is performed by d ,
D
xdij
=
0, otherwise
1, if oij is performed directly before okh ,
yijkh = 
0, otherwise
Parameters
I : number of parts.
J i : number of operations for part
M : number of machines.
L : number of tools.
D : number of TADs.
Oi :set of operations for part
oij : the j-th operation of part
TAD
candidates
Machine
candidates
Tool
Machining time
candidates (time unit)
mm : the m-th machine.
tl : the l-th tool.
o11
o12, o13
Milling
+z
m2, m3, m4
t6 , t7, t8
40, 40, 30
o12
o14
Milling
-z
m2, m3, m4
t6 , t7, t8
40, 40, 30
o13
o14
Milling
+x
m2, m3, m4
t6 , t7, t8
20, 20, 15
o14
--
Drilling
+z, -z
m1, m2, m3, m4 t2
12, 10, 10, 7.5
o21
o22, o23
Drilling
+z, -z
m1, m2, m3, m4 t1
12, 10, 10, 7.5
o22
--
Milling
-x, +y, -y, -z m2, m3, m4
t12
20, 20, 15
o23
--
Milling
+y
m2, m3, m4
t5, t6, t11
18, 18, 13.5
o31
o32
Milling
+z
m2, m3, m4
t6 , t7, t8
20, 20, 15
o32
--
Milling
-z
m2, m3, m4
t6 , t7, t8
20, 20, 15
o33
o32
Milling
+x, -x, +y, -z m2, m3, m4
t6 , t7, t8
15, 15, 11.25
o41
o43
Milling
-y
m2, m3
t6 , t9
12,15
o42
o44
Milling
-y
m2, m3
t9, t10
21,18
o43
--
Milling
-z
m2, m3
t3
18,25
o44
--
Milling
+x, -x
m2, m3
t1 , t3
27,25
i.
i , Oi ={oij | j=1,2, ,J i } .
i.
ad : the d-th TAD.
M ij : set of machines that can process oij .
Am : set of operations that can be processed on machine m.
rijh : precedence constraints. rijh = 1 , if oij is predecessor of
oih ; 0, otherwise.
M
: machining time of oij by machine m.
tmij
MC
t : machine change time. It is needed when two adjacent
operations belong to 1) different parts, or 2) same part and
different machines.
The PPS problem herein is to determine a process plan and
schedule, which tells, decision maker how, when and in which
sequence to allocate these operations of parts to suitable
manufacturing resources effectively. Moreover, this process plan
and schedule should satisfy the multiply objectives concurrently
while maintaining the feasibility of its. This problem can be
defined as follows.

Operation sequencing: determine the operation sequences for
all the parts so that the precedence relationships among all
the operations are not violated.

Operation selection: determine the resources selection
according to the feature geometry and available machining
resources.

Part scheduling: determine how and when to allocate the
manufacturing resources to the parts.
t TC : tool change time . It is needed when two adjacent
operations belong to 1) different parts, or 2) same part and
different machines, or 3) same part, same machine and different
tools.
2.2 Mathematical Formulation
processing time for an operation consists of the preparation time
and the machining time for the operation.
(2)
t mijP = t mijPRE  t mijM
As the two objectives of PPS problems, minimizing
t SC : set-up change time. It is needed when two adjacent
operations belong to 1) different parts, or 2) same part and
different machines, or 3) same part, same machine and different
TADs.
PRE
t mij : preparation time of operation oij by machine m . The
preparation time for an operation consists of machine change time,
tool change time and set-up time for the operation.
PRE
(1)
t mij
= t MC  t TC  t SC
P
: processing time of operation oij by machine m . The
t mij
The 2012 International Symposium on Semiconductor Manufacturing Intelligence
u m : workload of machine m .
um =  i =1  ji=1 t mijM xmijM
I
J
(3)
u : average workload of machine.
u=
1
M

M
(4)
u
m =1 m
C
t mij
: completion time of oij by machine m .
Mathematical Model:
The mathematical model for minimization of makespan and
variation of workload can be stated as follows:
C
(5)
min t = maxtmij
M
m ,i , j
min wP =
s.t.  t
C
mkh
1
t
M
P
mkh

t
M
(6)
(um  u ) 2
m =1
C
mij
x
M M
mij mkh ijkh
x
y
 0,   i, j  ,  k , h  , m
(7)
rijh yihij = 0, (i, j ), h
(8)
yijij = 0, (i, j )
(9)

(10)
M
M
m =1 mij
x
= 1, (i, j )
M
xmij
= 0,   i, j   Am , m
(11)
yijkh  0,1 ,   i, j  ,  k , h 
(12)
x  0,1 , m, (i, j )
(13)
C
tmij
 0, m, (i, j )
(14)
M
mij
eval ( si )=q ( si ) 1 p ( si )1 , i=1,2, ..., popSize
Equation (5) describes the one objective of minimization of
makespan. Minimization of variation of workload is defined as Eq.
(6). Equation (7) imposes that any machine cannot be selected for
one operation until the predecessor is completed. The precedence
constraint is defined as Eq. (8). Equation (9) ensures the feasible
operation sequence. The feasible resource selection are defined as
Eq. (10) and (11). Equations (12), (13) and (14) impose
nonnegative condition.
3. Hybrid Sampling Strategy-based Evolutionary
Algorithm (HSS-EA)
3.1 Pareto dominating and dominated relationship based fitness function (PDDR-FF)
The fitness assignment strategy plays an important role in
MOEAs. The GPSI-FF makes the best use of Pareto dominance
relationship to evaluate individuals. The GPSI-FF of an individual
si is calculated by the following function:
eval ( si )= p ( si )q ( si )c , i=1,2, ..., popSize
(15)
where p(si) is the number of individuals, which can be dominated
by the individual si, q(si)is the number of individuals which can
dominate the individual si, and c is population size. The bigger
value is better.
(a) GPSI-FF
mainly lie in the number of individuals which are dominated by
this individual, p(si), because c is a constant and q(si) is 0.
Moreover, the individuals locating around the central region of the
Pareto front with bigger domination area can dominate individuals
more than edge region. It causes that the higher GPSI-FF values
can tend the solutions toward the central of the Pareto front.
However, the difference between nondominated and
dominated individuals can be decreased by GPSI-FF values (as
shown in Fig. 2a). In Fig. 2a, individual A has bigger fitness value
than B and C, while B>C. In fact, nondominated individuals C
should be better than B because B is a dominated individual. This
unclear difference between nondominated and dominated
individuals causes that more dominated individuals could be kept
while nondominated ones would be removed in the evolving
process. It can reduce the convergence performance of algorithm,
unless an enough large size of its archive is set to store a sufficient
number of individuals.
For covering the disadvantage of GPSI-FF, a new PDDR-FF
-based fitness function is proposed to evaluate the individuals. The
PDDR-FF of an individual si is calculated by the following
function:
The smaller value is better.
According to the Eq. (16), PDDR-FF can set the obvious
difference values between the nondominated and dominated
individuals. If the individual belongs to nondominated one, its
fitness value will not exceed one. The fitness value of individual
which is dominated by other will exceed one. Furthermore, ones
with different numbers of dominating are also given the different
fitness values even though they are all nondominated individuals.
It is obvious that the nondominated individuals locating around
the central region of Pareto font with bigger domination area will
have smaller values (near to 0) than the edge points (near to 1).
From Fig. 2b, individual C is better than B while C is worse
than B in Fig. 2a. Moreover, nondominated individual A has
smaller value than C because the p(A) > p(C).
Moreover, the total time complexity of the PDDR-FF is
O(mN2).
3.2 The Proposed Algorithm
In selection phase of EA, the PDDR-FF based sampling
strategy has the advantage with the tendency converging toward
the center area of the Pareto front, but drawback to the edge
region. It causes bad distribution performance. The sampling
strategy of VEGA prefers the edge rather than certer regions of
Pareto front that it causes VEGA cannot achieve better
distribution performance. So it is natural, reasonable and possible
to combine these two methods to improve the overall performance
and reduce the computation time of the algorithm. Figure 3 shows
the description of HSS-EA.
(b)PDDR-FF
Figure 2: The comparisons of GPSI-FF and PDDR-FF
In GPSI-FF, the fitness values of nondominated individuals
(16)
Figure 3: The description of HSS-EA
Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem
The strong convergence capability of VEGA and PDDR-FF
ensures that the HSS-EA has the ability to converge to the true
Pareto front both in central and edge regions. The preferences for
the edge area of the Pareto front in VEGA and the central area of
the Pareto front in PDDR-FF guarantee that the HSS-EA
distributes along the Pareto front evenly. Moreover, less
computing time makes that HSS-EA has higher efficiency than
other approaches.
Since the multiobjective PPS problem belongs to the
classical combinatorial optimization and NP-hard problems,
higher efficacy and efficiency make that HSS-EA seems well
suited to solve such kind of problems. In addition, some
problem-dependent crossover, mutation and local search methods
are used to improve the performance of the algorithm.
3.3 Main Framework of HSS-EA
The evolving process of one generation of HSS-EA is shown
in Fig. 4. A(t) represents the archive at generation t and P(t)
represents the population at generation t. The solution procedure
of one generation includes 4 phases.
The smallest |A(t)| individuals in A’(t) are copied to form A(t+1).
This archive updating mechanism likes a elitist sampling strategy
to keep the better individuals with better PDDR-FF values.
3.4 Genetic Coding
The chromosome consists of multiple vectors, i.e. operation
sequence vector (v1), machine vector (v2), tool vector (v3) and
TAD vector (v4). The detailed initialization process of a
chromosome includes three steps.
Step 1: Deciding operation sequence
Operations are randomly selected from eligible operations to
generate feasible operation sequences. After one operation being
selected, it will be removed from the precedence graph and the
eligible operations will be updated. In the same manner, the
feasible operation sequence can be easily generated.
Step 2: Assigning resources
Machine and tool are selected randomly for each operation
from available machine and tool candidates. TAD is also selected
randomly from all TAD candidates for each operation. The
generated chromosome is shown in Fig. 5.
1
node ID j :
operation
o21
sequence v1(j):
machine v2(j): m4
2
3
4
5
6
7
8
9
10
11
12
13
14
o42
o41
o43
o23
o44
o31
o22
o11
o13
o33
o32
o12
o14
m3
m3
m3
m4
m2
m2
m2
m4
m3
m2
m4
m3
m3
tool v3(j):
t1
t10
t6
t3
t11
t1
t6
t12
t8
t8
t6
t6
t7
t2
TAD v4(j):
-z
-y
-y
-z
+y
-x
+z
-z
+z
+x
-x
-z
-z
-z
Figure 5: A chromosome of the example
Step 3: Generating schedule
After generating the chromosome, the schedule can be
generated. When generating it, an operation can be started
whenever its predecessor has been finished and the machine to
process it is available.
3.5 Genetic Operators
Figure 4: The evolving process in one generation
Phase 1: Selection (sampling strategy of VEGA)
In this phase, the sampling strategy of VEGA is used to select
the better individuals into part of mating pool. For two objectives
PPS problem, individuals are selected with replacement according
to objective 1 into sub population 1 while ignoring objective 2
until the size of the sub population 1 (half of population size) is
reached. In the same manner, individuals are selected for objective
2 into sub population 2 without considering objective 1 until sub
population 2 is full.
Phase 2: Generation of mating pool (hybrid sampling)
In this study, the sub populations and A(t) are combined to
form a mating pool. In the mating pool, sub-pop-1 stores the good
individuals for one objective, and sub-population 2 holds the good
individuals for the other objective. The archive saves the
individuals with good PDDR-FF values. Since there are two
objectives in this PPS problem, the sizes of those two sub
populations and archive are set as the same as half the population
size. Therefore, in the mating pool, one-third of the individuals
serve one objective, one-third the other objective, and the left
one-third both the two objectives. The archive mechanism tries to
cover the selection bias of VEGA. These three parts of the mating
pool make the solutions converge to the Pareto front evenly.
Phase 3: Reproduction and local search
Some problem-dependent crossover and mutation operators
are used to reproduce new chromosomes. Moreover, a local search
mechanism is proposed to improve the quality of individuals after
the reproduction process.
Phase 4: Archive maintenance (PDDR-FF based Elitist
sampling strategy)
The individuals of A(t) and P(t) are combined to form a
temporary archive A’(t). Thereafter, the PDDR-FF values of all
individuals in A’(t) are calculated and sorted in a ascending order.
3.5.1 Crossover operator
In this study, one-cut point order-based crossover is adopted.
This crossover is mainly applied according to operation sequence
vector. The locus of machine vector, tool vector and TAD vector
are changed with same alleles of operation sequence vector
Notation pc is defined as crossover probability.
The algorithm is described as follows.
Step 1: According to the length of a parent, a cut point is
generated randomly. Then, each parent is divided
into two parts, a left side and a right side.
Step 2: The left side of offspring 1 comes from the left side
of parent 1. The operator constructs the right side of
offspring 1 according to the order of operations in
parent 2. According to this order, the operator
constructs the right side of offspring 1 with
operations of parent 2, whose operation IDs are the
same as operations of the right side in parent 1.
Step 3: Offspring 2 is created in the same manner.
3.5.2 Mutation operator
Three resource mutation operators proposed by Zhang et al.
[8] are used. Notation pm are defined as mutation probability.
Machine mutation changes the machine to perform an
operation so as to reduce the machine change times. The
algorithm is described as follows.
Step 1: A node ID is chosen randomly, and the
corresponding machine Mold of this node ID is
marked as the old machine.
Step 2: A new machine Mnew is chosen randomly from the
alternative machine set of this node ID to replace the
old machine.
Step 3: Mark all the other machines in the machine vector
whose old machine is Mold. If there exists an
The 2012 International Symposium on Semiconductor Manufacturing Intelligence
alternative machine Mnew in any one of v2, Mold is
replaced with Mnew.
Tool mutation uses the similar manner to operate on tool
vector after the machine mutation.
TAD mutation also uses the similar mechanism to operate
on TAD vector after both the machine mutation and the tool
mutation.
3.5.3 Selection operator
In this study, a binary tournament selection (BTS) with
replacement is used. Two individuals are chosen randomly from
the population and the better (smaller) one is copied into the
mating pool.
3.6 Local Search
The local search is proposed to reduce the changeovers (the
machine change times, tool change times and TAD change times).
It is applied for every chromosome after the reproduction. The
pseudocode of local search is listed in Fig. 6.
HSS-EA, GPSI-FFGA, NSGA-II and SPEA2 are run 30
times to compared the results with each other. It should be noted
that the same selection, crossover, mutation, local search, and
different fitness functions and archive mechanisms for four
methods are used. Moreover, the parameters of all four methods
are the same, except for the size of archive. The archive sizes of
HSS-EA and GPSI-FFGA are set to be half the population size, 50,
while of NSGA-II and SPEA2 are set to be the same as the
population size, 100. GPSI-FFGA uses the same framework as
HSS-EA, except the different fitness function.
4.1 Performance Measures
Let Sj be a solution set for each method (j=1, 2, 3, 4). PF* is
a known reference Pareto solutions. Because no researches have
published their reference Pareto solutions for this problem,
therefore, PF*in this study comes from combining all of the
obtained Pareto set with 30 runs by 4 methods. In this study, the
following three performance measures are considered.
Coverage C(S1,S2) is the percent of the individuals in S2
which are weakly dominated by S1 [21]. The larger C(S1,S2) means
that S1 outperforms S2 in convergence.
Generational distance GD(Sj) finds an average minimum
distance of the solutions of Sj from PF* [22]. The smaller GD of
Sj means better Sj in approaching PF*.
Spacing SP(Sj) is the standard deviation of the closest
distances of individuals by Sj [23]. Smaller SP means better
distribution performance.
The C, GD are used to verify convergence performance while
SP is used to check the distribution performance.
4.2 Discussion of the Results
Figure 7 shows the 50% attainment surface by using HSS-EA,
GPSI-FFGA, NSGA-II and SPEA2 with 30 runs. From Fig. 7, it is
clear that HSS-EA is better than GPSI-FFGA, NSGA-II and
SPEA2 around both central and edge region of the Pareto front.
Figure 7: 50% attainment surface by HSS-EA, GPSI-FFGA, NSGA-II and
SPEA2
Figure 6: Pseudocode of local search
4 Experiments and Discussion
Because it is very difficult to get real world problem data on
PPS, 4 parts (each part has 20, 16, 14 and 7 operations,
respectively) problem suggested by Li and Mcmahon [2] and Li et
al. [10] are used.
All the simulation are performed on Core 2 Quad processor
(3.0 GHz clock), and the program is written in C# language. The
adopted parameters are listed as follows: population size, 100;
maximum generation, 2000; archive size, 50; crossover
probability, 0.70; mutation probability, 0.60.
Figure 8 shows the numerical comparison of the
box-and-whisker plots for C, GD and SP and the CPU times by
HSS-EA, GPSI-FFGA, NSGA-II and SPEA2. From Fig. 8a,b,c, it
is easy to see that the HSS-EA is slightly better than GPSI-FFGA,
and is obviously better than NSGA-II and SPEA2 on C measure.
The GD measure as shown in Fig. 8d also indicates that HSS-EA
can get smaller value than NSGA-II and SPEA2 while almost the
same as GPSI-FFGA. The distribution performances, SP, as
shown in Fig. 8e indicates that HSS-EA is obviously better than
GPSI-FFGA, while better than NSGA-II and SPEA2. Without
special mechanism to preserve the diversity evenly, HSS-EA can
also achieve satisfactory distribution performance. From the
comparisons and analysis as shown in Fig. 8f, it is clear that
HSS-EA is much faster than NSGA-II and SPEA2.
Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem
In general, the convergence and distribution performance of
HSS-EA is better than GPSI-FFGA while the efficiency is
equivalent, and the convergence and distribution performance is
also better than NSGA-II and SPEA2 and the efficiency is
obviously better than them.
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Figure 8: C, GD, SP and CPU times by HSS-EA, GPSI-FFGA, NSGA-II,
and SPEA2
Such better convergence and distribution performances
should mainly attribute to the hybrid sampling strategy of VEGA's
preference for the edge region of the Pareto front and PDDR-FF's
tendency converging toward the center area of the Pareto front.
They preserve better performances both in efficacy and efficiency.
Especially, these two mechanisms can also keep diversity evenly
without special distribution mechanisms like NSGA-II and
SPEA2.
[10]
[11]
[12]
5 Conclusions
[13]
In this study, a hybrid sampling strategy-based evolutionary
algorithm (HSS-EA) approach was proposed to solve the
multiobjective PPS problem while considering minimization of
makespan and minimization of variation of machine workload.
For covering the disadvantage of GPSI-FF, a new Pareto
dominating and dominated relationship-based fitness function
(PDDR-FF )was proposed to evaluate the individuals. The
proposed method tactfully united the advantages of VEGA and
PDDR-FF to solve multiobjective PPS problem. The sampling
strategy of VEGA has a preference for the edge region of the
Pareto front and the PDDR-FF-based sampling strategy has the
tendency converging toward the center area of the Pareto front.
The proposed method could preserve both the convergence and
distribution performances while reducing the computation time.
The binary tournament selection, problem-dependent order-based
crossover, resource-based mutation, local search could improve
the abilities of exploration and exploitation.
Complete numerical comparisons indicted that HSS-EA was
better than GPSI-FFGA in efficacy while the efficiency was
equivalent, and both convergence and distribution performance
was also better than NSGA-II and SPEA2 and the efficiency was
obviously better than NSGA-II and SPEA2.
Acknowledgement
This work is partly supported by the Dongseo Frontier Project
Research Fund of Dongseo University in 2011 and the KOFST
project in 2010-2011 at Hanyang University.
[14]
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