A Chaotic Non-Dominated Sorting Genetic Algorithm for the Multi-Objective

Applied Soft Computing 13 (2013) 2790–2802

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Applied Soft Computing

j ournal homepage: www.elsevier.com/locate/asoc

A chaotic non-dominated sorting genetic algorithm for the multi-objective

automatic test task scheduling problem

∗

Hui Lu , Ruiyao Niu, Jing Liu, Zheng Zhu

School of Electronic and Information Engineering, Beihang University, Beijing 100191, China

a r t i c l e i n f o a b s t r a c t

Article history: Solving a task scheduling problem is a key challenge for automatic test technology to improve through-

Received 30 December 2011

put, reduce test time, and operate the necessary instruments at their maximum capacity. Therefore, this

Received in revised form 14 August 2012

paper attempts to solve the automatic test task scheduling problem (TTSP) with the objectives of mini-

Accepted 3 October 2012

mizing the maximal test completion time (makespan) and the mean workload of the instruments. In this

Available online 15 October 2012

paper, the formal formulation and the constraints of the TTSP are established to describe this problem.

Then, a new encoding method called the integrated encoding scheme (IES) is proposed. This encoding

Keywords:

scheme is able to transform a combinatorial optimization problem into a continuous optimization prob-

Automatic test task scheduling

lem, thus improving the encoding efﬁciency and reducing the complexity of the genetic manipulations.

Multi-objective optimization

Makespan More importantly, because the TTSP has many local optima, a chaotic non-dominated sorting genetic

NSGA-II algorithm (CNSGA) is presented to avoid becoming trapped in local optima and to obtain high quality

Chaotic operator solutions. This approach introduces a chaotic initial population, a crossover operator, and a mutation

Hybrid approach operator into the non-dominated sorting genetic algorithm II (NSGA-II) to enhance the local searching

ability. Both the logistic map and the cat map are used to design the chaotic operators, and their per-

formances are compared. To identify a good approach for hybridizing NSGA-II and chaos, and indicate the

effectiveness of IES, several experiments are performed based on the following: (1) a small-scale TTSP

and a large-scale TTSP in real-world applications and (2) a TTSP used in other research. Computational

simulations and comparisons show that CNSGA improves the local searching ability and is suitable for

solving the TTSP.

1. Introduction optimized algorithm has not yet been fully elucidated [2]. These

reasons motivated us to perform this study.

Automatic test technology offers much more efﬁciency, repeat- Because of the similarities among many of the scheduling prob-

ability, and reliability than manual operations in the modern lems, the ﬂexible job-shop scheduling problem (FJSP) is compared

manufacturing processes and is widely used in many applications. to the automatic test task scheduling problem (TTSP), and a brief

For example, in mobile phone terminal manufacturing, automatic description of our issue is provided. Both the FJSP and the TTSP

test technology is used to measure production performance; in must assign each operation or task to a machine or an instrument

spacecraft, missile, and other aerospace systems, it represents one from a set of capable machines or instruments and then sequence

of the important technologies for ensuring the safety of these sys- the assigned operations or tasks on each machine or instrument

tems. One key challenge in automatic test technology is the test [3]. However, the two problems contain some differences. First, in

task scheduling problem. This scheduling problem addresses allo- the FJSP, each operation can be performed on only one of the m

cating tasks to the available instruments to improve throughput, machines at a time, whereas in the TTSP, a task can be tested on

reduce test time, and optimize resource allocation [1]. There- more than one instrument, which requires a much more complex

fore, an effective and efﬁcient dispatching scheme is important allocation of the instruments to avoid resource conﬂicts. Second, in

and highly valued. However, although there have been several the FJSP, the operations for each job must be performed in a pre-

research projects to date within this ﬁeld with an important appli- determined order, and the priority is linear. In other words, every

cation value in modern manufacturing environments, a concrete operation has only one former operation and one latter operation.

However, the precedence relationships in the TTSP resemble a net-

work. In other words, one task could have one or more former or

latter tasks, and thus, it is more difﬁcult to obtain feasible solu-

∗ tions. As mentioned above, the TTSP is more difﬁcult to solve than

Corresponding author.

E-mail addresses: [email protected], [email protected] (H. Lu). the FJSP.

H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802 2791

The FJSP is NP-hard [3], and many similar problems, such replace the traditional velocity-displacement model. Zhang et al.

as TTSP, have this property. Thus, it is difﬁcult or impossi- [14] enabled each particle to exchange information by performing

ble to solve this type of problem with deterministic methods. a crossover operation, which did not generate real values. These

Many meta-heuristics methods and evolutionary algorithms have two approaches require adjustments or special designs and are not

been proposed. In this research, we present an algorithm, the always convenient. Similarly, chaotic technology cannot be easily

chaotic non-dominated sorting genetic algorithm (CNSGA), which applied to improve the performances of algorithms using discrete

hybridizes chaotic sequences based on the logistic map and the cat encodings because the chaotic variables are continuous variables.

map with the non-dominated sorting genetic algorithm II (NSGA- Thus, chaotic technology is suitable for the continuous search-

II). The NSGA-II method was presented by Deb in 2002 [4] and ing space, and therefore, the limitations of discrete encoding have

has been successfully used in solving shop scheduling problems stunted the development of the continuous algorithms and chaotic

[5], reactive power dispatch problems [6], and many other appli- technology in combinatorial optimization problems.

cations. This approach is effective and able to maintain a better Xia et al. [15] presented a matrix-encoding scheme for the

spread of solutions. In addition, the TTSP has many local optima, and TTSP. The column vectors of the matrix chromosome represent

chaotic operators are used to avoid becoming trapped in the local the instruments, and the row vectors express the tasks. For every

optima. Chaos theory focuses on the behavior of dynamical systems instrument, if one task uses it, that task number is entered in the

that are highly sensitive to initial conditions. Chaotic sequences row vector of that instrument. This encoding scheme will generate

have some special characteristics, such as the ergodic property and unfeasible solutions during the genetic manipulations, and if a task

the stochastic property [7]. Because of these properties, optimiza- can be tested on more than one set of instruments, it will become

tion algorithms based on chaos can escape from the local optima, invalid.

depending on their chaotic motion [8]. Thus, some researchers used As mentioned above, the commonly used TSL (OLC), the two-

optimization algorithms combined with chaos to achieve better vector representation, and the matrix-encoding are not always

results. For example, Lü et al. [9] introduced a chaotic mutation acceptable. The integrated encoding scheme (IES), inspired by ran-

into the GA to train an artiﬁcial neural network, and Guo et al. [10] dom key (RK) encoding [16], is proposed to address deﬁciencies of

used both chaos optimization of the initial population and excel- certain encoding schemes. The IES approach improves the encoding

lent individual chaotic disturbance to escape from the local optima. efﬁciency by using only one chromosome to merge the information

Although approaches that combine evolutionary algorithms and regarding both the instrument assignment and the task scheduling

chaos are applied in various ﬁelds, a few researchers have sought decisions. Additionally, the complexity of the genetic manipula-

to solve the multi-objective optimization problems using chaotic tions is reduced, and it is able to satisfy the resource constraints

operators. Thus, we contributed a substantial effort to ﬁnd sat- and solve the TTSP with multiple alternative test schemes. More

isfactory approaches for combining multi-objective optimization importantly, IES transforms a combinatorial optimization problem

algorithms with the chaos principles. into a continuous optimization problem, making the applications of

In this paper, a chaotic initial population, a crossover operator, algorithms more convenient without the consideration of whether

and a mutation operator are introduced into the NSGA-II to enhance the problem is combinatorial or continuous. This approach repre-

the local searching ability. To determine the distinct performances sents a “fast-track” between a combinatorial optimization problem

of different chaotic maps, both the logistic map and the cat map are and a continuous optimization algorithm, which makes the search

used as number generators in place of random number generators algorithms such as PSO suitable for combinatorial problems and

in the initial population, crossover operator, and mutation operator. chaos technology applicable for enhancement of the algorithm.

Thus, there are six variants of CNSGA. The remainder of this paper is organized as follows. Section 2

Additionally, the chromosome-encoding scheme is an impor- gives a summary of some proposed papers applied in scheduling

tant aspect that should be carefully considered. Both the FJSP and problems as well as adopted algorithms. Section 3 brieﬂy intro-

the TTSP are a combination of machine/instrument assignment duces certain concepts that are used for describing multi-objective

and operation/task-scheduling decisions [11]. Thus, a chromosome optimization problems. The formulation, constraints, and opti-

should include these two types of information. The FJSP has two mized objectives of the problem under study are presented in

main encoding schemes: the task sequencing list (TSL) (or the oper- Section 4. Then, in Section 5, the proposed CNSGA is described in

ations list coding (OLC)) [12] and the two-vector representation detail for solving the TTSP, including the chaotic maps, the prob-

[11]. The TSL represents the schedule in a list of operations, and lem encoding scheme, and the implementation of the CNSGA. The

each operation holds one cell. Every cell contains the operation experimental results and performance comparisons are presented

number, the job number, and the machine number. This encoding and discussed in Section 6. Finally, Section 7 contains the conclu-

scheme is understandable, but it makes the corresponding genetic sions and further research suggestions.

manipulations difﬁcult to perform. The two-vector representa-

tion is composed of a chromosome with two parts: the machine

assignment vector and the operation sequence vector. This encod- 2. Literature review

ing scheme can always represent a feasible operation sequence.

However, it still contains some disadvantages. For example, it is After reviewing the vast majority of literature on scheduling

not very efﬁcient, and decoding is inconvenient. Moreover, the problems, it was determined that only a few papers have dis-

crossover operators and the mutation operators of these two vec- cussed task scheduling in automatic test systems. Therefore, we

tors are usually different and should be considered separately; this prefer to discuss the methods used in the FJSP to provide refer-

constraint increases the complexity of the genetic manipulations. ences for the relatively new application area of the TTSP because

More importantly, the encoding scheme uses discrete encoding, of their similarities. In the literature, many meta-heuristics meth-

and thus, algorithms such as PSO cannot use it directly because ods and evolutionary algorithms have been proposed. To improve

some bits of a particle position may appear as real values, whereas some of the traditional strategies, Pezzella et al. [17] presented

the discrete encoding requires integers. Two main approaches are a new genetic algorithm (GA) for the FJSP, which integrated dif-

applied to address this type of incompatibility. One approach is ferent strategies for generating the initial population, selecting

to revise the updated particles. For example, Xia and Wu [13] the individuals for reproduction, and reproducing new individ-

rounded off the real optimum value to its nearest integer number. uals. Wang et al. [18] proposed an effective bi-population based

The other approach is to design a new particle updating method to estimation of the distribution algorithm (BEDA) to solve the FJSP

2792 H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802

with the objective of minimizing the makespan. In addition, the algorithm (CICA) that considered seven chaotic maps (logistic

inﬂuence of the parameters was considered, and the Taguchi map, tent map, iterative chaotic map with inﬁnite collapses (ICMIC

method was implemented to design the experiments. Fattahi map), sinusoidal map, circle map, gauss map and sinus map) and

et al. [19] discussed the FJSP with overlapping in operations and three chaotic methods (CICA-1, CICA-2 and CICA-3). Although their

used a simulated annealing (SA) algorithm to solve large problem analysis appears complete, the paper pruned some of the possibili-

instances. Zhang et al. [14] combined a particle swarm optimiza- ties. At the beginning, this approach used three variants of the CICA

tion (PSO) algorithm and a tabu search (TS) algorithm to solve and all of the different chaotic maps for the Griewank function. The

the multi-objective FJSP. This hybrid algorithm employed a PSO to results showed that the performance of the CICA-3 was better than

assign operations to machines and to schedule operations on each that of the CICA-1 and CICA-2. The CICA has shown a somewhat

machine. Then, TS was applied to perform a local search for the better performance when logistic and sinusoidal maps were used

scheduling sub-problem originating from each obtained solution. for generating chaotic signals. Then, only the CICA-3 and the

Li et al. [20] proposed an effective hybrid tabu search algorithm sinusoidal map were applied for four functions (Griewank, Ackley,

(HTSA) to solve the multi-objective FJSP. In addition, two adaptive Rosenborck, and Rastring functions), ignoring the CICA-1, CICA-2

neighborhood search rules were presented, three insert and swap and other chaotic maps. Therefore, the analysis is also imperfect

neighborhood structures were developed, and an efﬁcient left-shift because 84 (=7 × 3 × 4) experiments should be performed to satisfy

function was designed. logical preciseness. In addition, Tavazoei and Haeri [28] discovered

These algorithms can be categorized into hierarchical that the performances of the algorithms were different if different

approaches and integrated approaches [21]. The hierarchical chaotic maps were applied for different test problems. Therefore,

approaches decompose the problem into two sub-problems: the the results from different ﬁelds may not be compared directly, and

assignment of operations and the sequencing of operations. Fattahi complete experiments should be performed for the problems in

et al. [19] presented an assignment algorithm and a scheduling special ﬁelds before choosing the most suitable chaotic algorithm.

algorithm to ﬁnd good solutions for the assignment problem and In this research, we present an algorithm that hybridizes chaotic

the scheduling problem, respectively. Li et al. [20] proposed a operators and the NSGA-II to enhance the local searching ability. It

hybrid algorithm using a TS algorithm to produce neighboring can be categorized as an integrated approach. As mentioned above,

solutions in the machine assignment module and a variable all six variants of the CNSGA are considered to satisfy logical pre-

neighborhood search algorithm to perform a local search in the ciseness.

operation scheduling component. The hierarchical approaches

can reduce the problem’s complexity. The integrated approaches

3. Multi-objective optimization problem

are more difﬁcult to solve because the solution space expands

exponentially with the problem size, and therefore, the perfor-

In this paper, a multi-objective optimization algorithm based

mance of only one algorithm could be unsatisfactory. To improve

on NSGA-II and chaotic theory is proposed for the TTSP. Before

the efﬁciency of the algorithms, many approaches have been

explaining the details of this approach, an introduction to the multi-

suggested. Some approaches have been devoted to modifying the

objective optimization problem (MOP) is necessary [29].

original algorithm. For example, Tsutsui and Miki [22] replaced the

A multi-objective optimization problem can be described, with-

conventional crossover and mutation operators in the GA. Other

out loss of generality, as follows [30]:

researchers have proposed a variety of hybrid algorithms, which

combined the advantages of two different methods. For example, minimise F(x) = (f1(x), f2(x), . . . , fM (x))

(1)

Gao et al. [23] used a variable neighborhood descent to improve

subject to x ∈ ˝

the individuals in the GA. Xia and Wu [13] hybridized PSO and SA

to develop an easily implemented and effective hybrid approach. ˝ ˝

where is the decision space, the function F is a mapping from to

Based on the hybridization of fuzzy logic and evolutionary algo- M M

R , and R is the objective space with M objective functions. Very

rithms, Kacem et al. [24] proposed a Pareto approach. Xia et al.

often, the M objectives contradict each other, and there may exist no

[15] proposed a genetic simulated annealing algorithm (GASA),

x that can minimize all of the objectives simultaneously. Therefore,

which combined a GA and SA to solve the TTSP. It considered

a set of compromise solutions called Pareto optimal solutions can

both the parallel efﬁciency and the speedup ratio as objectives

be obtained.

and then integrated them as one ﬁtness function for optimization.

The following basic concepts adopted in this paper are often

However, this did not account for the instrument set options.

used in MOP [31].

In addition, because of the No Free Lunch Theorem (NFLT) [25],

problem-speciﬁc knowledge has been incorporated during the Deﬁnition 1 (Pareto dominance). If two feasible decision vectors

optimization process. For example, Gao et al. [11] developed a new x1 and x2 satisfy the following two conditions, then x1 dominates

genetic algorithm that was hybridized with bottleneck shifting to x2 (denoted by x1 ≺ x2).

provide guidance for the GA search process.

Although the methods detailed above can solve the TTSP, they

(1) x1 is no worse than x2 in all of the objectives:

are sometimes easily trapped in the local optima. Therefore, chaotic

∀a = , , . . . , M f x ≤ f x

operators are used to avoid becoming trapped in the local optima. 1 2 a( 1) a( 2) (2)

Many research studies use the chaos principles to improve the per-

(2) x1 is strictly better than x2 in at least one objective:

formances of evolutionary algorithms, and experimental studies

conﬁrm the beneﬁts of such a combination. Unfortunately, these

∃a = 1, 2, . . . , M fa(x1) < fa(x2) (3)

works contain some imperfections. dos Santos Coelho and Alotto

[26] considered the chaotic crossover operator using the Zaslavskii

x1 is called a Pareto optimal solution if it cannot be dominated by

map to solve multi-objective optimization problems. However,

any other solution.

two essential aspects should be discussed in depth: (1) the various

chaotic maps and (2) the different chaotic methods. Until now, Deﬁnition 2 (Pareto optimal set). For a given MOP, the Pareto opti-

few studies have been concerned with the comprehensive analysis mal set (PS) is deﬁned as follows:

of chaotic technology in evolutionary algorithms. Talatahari et al.

PS ∗ ∗

= {x ∈ ˝|¬∃

x ∈ ˝, x ≺ x}

[27] proposed a novel chaotic improved imperialist competitive (4)

H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802 2793

Table 1

Deﬁnition 3 (Pareto front). For a given MOP and the PS, the Pareto

A TTSP with four tasks and four instruments.

front (PF) is deﬁned as follows:

k k k k

w P w P

T Wj j j T Wj j j

PF = {f (x)|x ∈ PS} (5) 1 1

t w w 1 1 r1 r2 5 t3 3 r4 2 w2 t w1

1 r2 r4 3 4 4 r1 r3 4

4. Problem formulation of the TTSP 1 2

t w w 2 2 r1 4 4 r2 r4 3 2 3

w w

2 r3 1 4 r2 r3 7

4.1. The problem description

The TTSP aims to organize the execution of n tasks on m instru-

T = {t } ments. In this problem, there are a set of tasks j j=1 and a

m i i i

R = {r } P S C

set of instruments i i=1 [15]. The notiﬁcations j , j , and j

present the test time, the test start time, and the test completion

time of task tj tested on ri, respectively. Thereupon, an inequality

Ci i i

≥ S + P

j j j is obtained. For the TTSP, one task must be tested on one

or more instruments. In other words, some instruments collaborate

on one test task. A variable j is deﬁned to express whether the task

tj occupies the instrument ri. The deﬁnition is as follows:

Fig. 1. Precedence constraints between the tasks presented by a DAG.

1 if tj occupies ri Oi = j (6)

0 others

(4) The setup time of the instruments and the move time between

i i i

C = S + P

the tasks are negligible. In other words, j j j .

In typical cases, some instruments could have redundancies.

i k

P = P

For example, there could be two instruments with the same basic (5) Assume j j , to simplify the problem.

functionality but different accuracies. If one task is tested on the

instrument with lower accuracy, it can also be tested on the

4.3. Fitness function

instrument with higher accuracy. Thus, task tj could have several k

W k j

= {w }

alternative schemes to complete the test. j is used to There are two objectives to be minimized: the maximal test

j k=1

denote the alternative schemes of task tj, where kj is the number completion time (makespan) f1(x) and the mean workload of the

K = {k } instruments f (x). of schemes of tj. j j=1 is the set containing the numbers of 2

schemes that correspond to every task. Each j is a subset of R

k u ujk (1) The maximal test completion time f1(x).

and can be represented as w = {r } , and u is the number of

j jk u= jk k i

C =

1 The notiﬁcation j max j is the test completion time of k k k

w w ri ∈ w

= R instruments for j . Obviously, j . The notiﬁcation j

tj for w . Thus, the maximal test completion time of all of the

1 ≤ k ≤ kj j

tasks can be deﬁned as follows:

1 ≤ j ≤ n

Pk = i k

P w

max is used to express the test time of tj for . Ck

j j j f x =

k 1( ) max j (9) r ∈ w

i j ≤ k ≤ k

1 j

1 ≤ j ≤ n

4.2. Constraint relationships

(2) The mean workload of the instruments f2(x).

The TTSP has two types of constraints: the restriction on

First, a new notiﬁcation Q is introduced to denote the parallel

resources and the precedence constraint between the tasks.

steps. The initial value of Q is 1. Assign the instruments for all

The restriction on resources can be expressed as follows: Xkk∗

= of the tasks if jj∗ 1, Q = Q + 1. Therefore, the mean workload

k k∗

w ∩ w =/ ∅ 1 if j j∗ of the instruments can be deﬁned as follows: Xkk∗ = (7) jj∗ n m 0 others 1 i i

f x = P O

2( ) Q j j (10)

The precedence constraint between the tasks can be represented j=1 i=1

as follows:

⎧

⎨⎪ 0 if tj and tj∗ have equal priorities

Y =

jj∗ +d if tj needs to be tested before tj∗ with at least d unit time, i.e. tj tj∗ (8)

⎩⎪

−d if tj∗ tj with at least d unit time

∈

where d R . In this paper, d equals the test time of the task with

high priority. A TTSP with four tasks and four instruments is illustrated in

In addition, the hypotheses considered in this paper are the Table 1. If the precedence constraints between the tasks are 1 t2,

following [15]: t1 t3, t1 t4, t2 t4, t3 t4, then these constraints can be pre-

sented by a directed acyclic graph (DAG) G(V,E), as shown in Fig. 1,

(1) At a given time, an instrument can execute only one task. where V is the set of vertices that represent the test tasks, and E

(2) Pre-emption is not allowed. In other words, each task must be is the set of edges that represent the precedence constraint rela-

completed without interruption once it starts. tionships between the tasks. An arrow points from the task with a

(3) All of the tasks and instruments are available at time 0. higher priority to the task with a lower priority.

2794 H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802

Table 2

5. Hybrid chaotic non-dominated sorting genetic algorithm

(CNSGA) Nomenclature for six variants of the CNSGA.

The logistic map The cat map

5.1. Chaotic sequence

Initial population CNSGA-L1 CNSGA-C1

Crossover operator CNSGA-L2 CNSGA-C2

Chaos theory studies the behavior of dynamic systems. Although Mutation operator CNSGA-L3 CNSGA-C3

these systems are deterministic, small changes in the initial condi-

tions will result in completely different future behaviors. Generally

(4) Mix these two populations and prioritize all of the individuals

speaking, chaotic systems have special characteristics, such as the

that are ranked according to their non-domination level based

ergodic property and stochastic property [7]. Because of these prop-

on the objective function values through the non-dominated

erties, applying chaos theory in an optimization algorithm becomes

sorting approach.

an interesting alternative for escape from local optima [8].

(5) Choose exactly N population members and reject the individ-

A chaotic sequence can be generated by a chaotic map, which is

uals that have a lower rank.

an evolution function that exhibits some sort of chaotic behavior.

Maps that are parameterized by a discrete time usually take the

As this process continues, the new population moves gradually

form of iterated functions. There are many chaotic maps [32], and

toward the PS.

each of them has speciﬁc features. Researching the performances of

In this paper, the main framework of NSGA-II is not altered,

the different chaotic maps combined with optimization algorithms

but some details are modiﬁed via chaos. The chaotic maps are

produces interesting results.

embedded as number generators inside CNSGA in the place of

In this paper, we use both the logistic map and Arnold’s cat map

random number generators (in the initial population, crossover

to compare and analyze their behaviors and to ﬁnd an appropriate

operator, and mutation operator). Therefore, chaotic variables are

approach for combining NSGA-II with chaos.

used instead of random variables, which cause the heuristic to map

to unique regions because the chaotic map iterates to new regions.

5.1.1. The logistic map

Due to the two chaotic maps and three number generators, there

The logistic map is a well-known one-dimensional chaotic map

are six variants of the CNSGA. The nomenclature of these variants

popularized by the biologist Robert May in 1976 [33]. Mathemati-

is shown in Table 2.

cally, the logistic map can be deﬁned as follows [34]:

t+1 = t (1 − t ), t ∈ (0, 1), t = 0, 1, 2, . . . (11)

5.2.1. Problem encoding

where t is the value of the chaotic variable at the tth iteration, and Using an appropriate representation in a chromosome is very

is the so-called bifurcation parameter of the system ( ∈ [0,4]) important for solving search problems. Both the FJSP and the

[35]. In this paper, the parameter = 4 and the initial value 0 is TTSP are a combination of machine/instrument assignment and

generated randomly between 0 and 1, with 0 ∈/ {0.25, 0.5, 0.75}. operation/task scheduling decisions. Thus, a chromosome should

include these two types of information. Fig. 2 illustrates the com-

5.1.2. Arnold’s cat map monly used encoding schemes for the FJSP and the TTSP. The

Vladimir Arnold demonstrated the effects of this map using an machine/instrument and the operation/task are expressed as a and

image of a cat, hence the name [36]. Unlike the logistic map, the cat t, respectively.

map is a two-dimensional chaotic map that has been used in image The TSL represents the schedule in a list of operations, and each

encryption [37]. The transformation of an image using the cat map operation holds one cell [12]. Every cell contains the operation

will produce a new image that apparently randomizes the original number, the job number, and the machine number. For example,

image. However, after a ﬁnite number of iterations, the original the cell (2, 1, 4) means that operation 2 of job 1 is performed on

image reappears. machine 4. This encoding scheme is directly perceived, but it con-

Arnold’s cat map can be expressed as follows: tains too much information in each cell, which causes complex

genetic manipulations and many unfeasible solutions.

p = p

t+1 ( t + qt )mod n

t The two-vector representation cuts a chromosome into two

= 0, 1, 2, . . . (12)

parts, the machine assignment vector and the operation sequence

qt+1 = (pt + 2qt )mod n

vector [11], which can avoid unfeasible solutions. However,

The initial values p0 and q0 are generated randomly between 0 because the two vectors are divided and each of them contains

and 1.

5.2. Implementation of CNSGA for the TTSP

NSGA-II is an improved version of NSGA. It has been suggested

that NSGA-II overcomes some of the shortcomings of NSGA, such

as the high computational complexity of non-dominated sorting,

the lack of elitism, and the need to specify the sharing parameter.

The computational complexity of NSGA-II is O(MN ), where M is

the number of objectives and N is the population size [4]. The main

procedure of NSGA-II can be described as follows:

(1) Initialize a population and begin the iteration.

(2) Select a parent population by tournament selection depend-

ing on the non-dominated rank and the crowded-comparison

operator.

(3) Generate a new population from the parent population through

crossover and mutation operations.

Fig. 2. Selected commonly used encoding schemes.

H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802 2795

Table 3 Table 4

Example of the integrated encoding scheme. Calculation of the objective values.

Decision variables xi 0.8147 0.9058 0.1270 0.6324 The task sequence is stored in an array s.

parrallel r is an array to record the instruments that can work in parallel.

Task sequence tj 3 4 1 2

c time is an array to record the completion time of the tasks on their occupied k 1121

k instruments.

w { } { } { } { }

j r4 r1,r3 r2,r4 r1

p time is an array to record the work time of the instruments.

j 2 4 3 4

01 parallel r = w

S[1]

02 start = 1

03 Q = 1

04 for i = 2 to n //for every task

different types of information, the crossover operators and the k

w ∩ parallel

05 if r = ∅

S[i]

mutation operators of these two vectors are usually different and k

parallel r = parallel r ∪ w

S[i]

should be considered separately. Thus, the genetic manipulations 07 else

are inconvenient. 08 Q++

The matrix-encoding scheme was proposed by Xia et al. to 09 for j=start to i − 1

k k

10 c time[w ] = C

solve the TTSP [15]. The column vectors of the matrix chromo- S[j] S[j]

k k k

11 p time[w ] = p time[w ] + P

some present the instruments, while the row vectors express the S[j] S[j] S[j]

12 end for

tasks. For every instrument, if one task uses it, that task num- k

13 parallel r = w

S[j]

ber is entered in the row vector of the instrument. Obviously, this

14 start=i

encoding scheme could not solve the TTSP with multiple alternative

15 end if

schemes because it contains no information about the instruments 16 end for

(with no information about a). Additionally, it will generate unfea- 17 Q++

18 for j=start to i

sible solutions during the genetic manipulations. k k

19 c time[w ] = C

S[j] S[j]

As mentioned above, a more suitable encoding approach for the k k k

20 p time[w ] = p time[w ] + P

S[j] S[j] S[j]

TTSP is necessary. For this reason, a new encoding scheme, called

21 end for

the integrated encoding scheme (IES), is proposed and is inspired by

22 f1(x) = max(c time)

f x = sum p time

random key (RK) encoding. This encoding scheme can use only one 23 2( ) Q ( )

chromosome to contain the information about both the processing

sequence of the tasks and the occupancy of the instruments for each

task. Thus, the encoding efﬁciency is improved and the subsequent

(2) It makes full use of decision variables by using only one chro-

genetic manipulations need not be performed separately. Addition-

mosome to merge the information about both the instrument

ally, this encoding scheme transforms a combinatorial optimization

assignment and the task scheduling decisions and improves the

problem into a continuous optimization problem, which can make

encoding efﬁciency.

the search algorithm suitable for the TTSP and make it possible to

(3) It reduces the complexity of the genetic manipulations.

use chaos technology to enhance the algorithm. We are now ready

(4) It does not generate unfeasible solutions that do not satisfy the

to describe the IES.

resource constraints.

Assume that the TTSP to be encoded is as shown in Table 1.

(5) It is able to solve the TTSP with multiple alternative schemes to

The main concept of the IES is to use relationships between the

complete the test.

decision variables to express the sequence of tasks and the values

(6) It is able to transform a combinatorial optimization problem

of the variables to represent the occupancy of the instruments for

into a continuous optimization problem, which makes search

each task. This concept is illustrated in Table 3.

algorithms suitable for the TTSP and chaos technology applica-

The entries in the ﬁrst row of Table 3 are the decision vari-

ble for enhancement of the algorithm.

ables, which range between 0 and 1. They are sorted in ascending

order, and the rank of every variable denotes a test task index in

5.2.2. Population initialization (CNSGA-L1 and CNSGA-C1)

sequence. Thus, the second row (or the task sequence) is obtained.

The initial population, comprised of N individuals, should be

More speciﬁcally, the smallest variable is 0.1270, which represents

generated when the iteration number is zero. One of the N indi-

t1. The second smallest variable is 0.6324, which represents t2, and 1 2 i n

viduals can be denoted by xs = {xs , xs , . . . , xs, . . . , xs }, s = 1, 2, . . .,

so on. On the other hand, the instrument assignment can also be

. . .

N, i = 1, 2, , n. In previous research, xs is randomly initialized or

obtained from the decision variables. If we want to know which i

k xs = rand() (NSGA-II). In this paper, the logistic map initialization

instruments will be occupied by the task ti, j should be ascer-

and the cat map initialization are used. For the logistic map ini-

tained. In other words, we should know the value of k, which can be i i

tialization, we generate t using Eq. (11); in other words, xs = t

calculated by the decision variable corresponding to tj, as follows:

(CNSGA-L1). Similarly, for the cat map initialization, we use the cat

map to replace the logistic map. Note that the cat map is a two-

k = [xij × 10]mod kj + 1 (13)

dimensional chaotic map according to Eq. (12) and that only the

i i

x = p

where xij represents the decision variable that corresponds to tj, and ﬁrst dimension is applied; in other words, s t (CNSGA-C1).

kj is the number of schemes of tj. Square brackets are used to obtain

the nearest integer, which is not more than xij × 10. For example, for

5.2.3. Evaluation

the task t4, the corresponding decision variable is 0.9058, and the

The ﬁtness function is used to measure the solutions. In this

number of schemes is k4 = 3. Then, the value of k can be calculated

paper, the makespan and the mean workload of the instruments

as follows according to Eq. (13): k = [0.9058 × 10]mod k4 + 1 =

are the objectives for optimization. Individuals are evaluated and

9mod3 + 1 = 1. Thus, w = {r1, r3} is applied. Obviously, n decision

4 sorted based on these objectives through the non-dominated

variables should be used to encode n tasks. In other words, the

sorting approach and then are allotted a rank equal to their non-

decision space has n dimensions.

domination levels. Because of the complexity of the instrument

Certain advantages of the IES can be summarized as follows:

assignment, the procedure is described in detail in Table 4.

Assume that the obtained solution is shown in Table 3, and

(1) It never generates duplication of a certain task.

hence, S = {3, 4, 1, 2}. The calculation procedure of the objective

2796 H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802

where u is a random number that ranges from 0 to 1. That is to say,

u = rand() (NSGA-II). c is the distribution index for the crossover

operator.

For CNSGA, u is created through the chaotic operator instead of

through random generation. In other words, if the logistic map is

used according to Eq. (11), u = t (CNSGA-L2). Otherwise, if the cat

map is used according to Eq. (12), u = pt (CNSGA-C2).

5.2.5. Mutation (CNSGA-L3 and CNSGA-C3)

For NSGA-II, if a randomly generated number rm is less than the

mutation probability pm, the algorithm will apply the polynomial

mutation [4]. For a solution xs, the polynomial mutation is described

as follows:

∗ u l

xs = xs + (xs − xs) × ıs (17)

u l

where xs and xs are the upper and lower bounds of xs, respectively.

1/(m+1)

(2us) − 1, if us < 0.5 ıs = (18)

1/(m+1)

1 − (2 × (1 − us)) , others

where us is a random number that ranges from 0 to 1. That is to say,

us = rand() (NSGA-II). m is the distribution index for the mutation operator.

u j

Similar to the crossover scheme, for CNSGA, we have s = t

(CNSGA-L3) when using the logistic map according to Eq. (11), and

u j

s = pt (CNSGA-C3) when using the cat map according to Eq. (12).

6. Experiments and results

Two types of experiments are performed to study the effect of

CNSGA. Experiment 1 is based on a real-world instance taken from a

missile system to show the effectiveness of the CNSGA in solving the

TTSP. Experiment 2 aims to solve a TTSP proposed by Xia et al. [15]

Fig. 3. Gantt chart of a schedule. and compares the results. For the TTSP, restrictions on the resources

are always considered, while precedence constraints between the

values for this solution is shown below, and the Gantt chart is tasks are considered only in Experiment 2. The parameter settings

shown in Fig. 3. for all experiments are shown in Table 5, where n is the number of

decision variables. Experiment 1 was run 10 times, whereas Exper-

iment 2 was run 20 times to match the process detailed in the

When Q = 1, parallel r = {r4, r1, r3}, c time = {4, 0, 4, 2}, and

{ } literature. In the tables of the experiments, the uppercase char-

p time = 4, 0, 4, 2 .

acters N, L, and C represent the normal NSGA-II, the CNSGA using

When Q = 2, parallel r = {r2, r4, r1}, c time = {8, 5, 4, 5}, and

the logistic chaotic operator and the CNSGA using the cat chaotic

p time = {8, 3, 4, 5}.

operator, respectively. In all of the cases, the best performances are

denoted in bold.

Thus, f1(x) = 8 and f2(x) = (1/2) × (8 + 3 + 4 + 5) = 10.

The source codes of NSGA-II can be downloaded from the web-

site of the Kanpur Genetic Algorithms Laboratory [38]. CNSGA is

5.2.4. Crossover (CNSGA-L2 and CNSGA-C2)

implemented in C with Microsoft Visual Studio 2010. All of the

The classical NSGA-II randomly generates a number rc to deter-

experiments are simulated on personal computers running Win-

mine whether the crossover operation should be applied. If rc is

dows 7 with an Intel (R) Core (TM) 2, Q9550, 2.83 GHz, and 2.00 GB

less than the crossover probability pc, the algorithm will perform

RAM.

the simulated binary crossover (SBX) operator [4]. According to

1 i n

x = {x , . . . , x , . . . , x } x = SBX, two child individuals c1 c1 c1 c1 and c2

1 i n

, . . . , x , . . . , x } x = 6.1. Performance metrics c2 c2 c2 can be generated by a pair of parents p1

{x1 , . . ., i n 1 i n

x , . . . , x } x = {x , . . . , x , . . . , x }

p1 p1 p1 and p2 p2 p2 p2 , as follows:

For multi-objective optimization, both the convergence to the

xi = 1 − ˇ xi i Pareto-optimal set and the maintenance of diversity in solutions

+ + ˇ x

c1 [(1 ) p1 (1 ) p2] (14)

2 of the Pareto-optimal set should be considered [4]. Thus, several

metrics referring to these two aspects are adopted in this paper to

xi 1 i i

c = [(1 + ˇ)xp + (1 − ˇ)xp ] (15)

2 2 1 2 compare the performances of CNSGA and NSGA-II. These metrics

ˇ have different applicable scopes. The convergence metric and the

is generated in the following manner:

diversity metric can be applied only if the true PF is known, while

1/(c +1)

u the others do not need to satisfy this condition.

(2 ) , if u ≤ 0.5 ˇ = (16)

− u 1/(c +1)

(1/2(1 )) , others

(1) Convergence metric [4]:

H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802 2797

Table 5

Parameter settings for all of the experiments.

Population size (N) Maximal generation (gen) Crossover probability (pc) Mutation probability (pm) Distribution index (c, m)

Exp.1 t1–t6 10 20 0.9 1/n (20, 20)

Exp.1 t1–t20 70 120 0.9 1/n (20, 20)

Exp.2 80 70 0.9 1/n (20, 20)

Table 6

This metric is used to measure the extent of convergence

A real-world TTSP taken from a missile system.

to a known set of Pareto-optimal solutions. Its deﬁnition is as

k k k k

w P W w P follows: T Wj j j T j j j N t w1 t w1 1 1 r1,r4 2 11 11 r2,r3 6 1 2 2 = w r ,r 5 w r ,r 8 di (19) 1 3 5 11 2 5 N w3 r ,r 3 w3 r ,r 8 i= 1 6 8 11 6 7 1 t w1 t w1 2 2 r2 3 12 12 r2 11 w2 2

r4 4 w r5 13

where d is the minimum Euclidean distance of every solution 2 12 i 3 3

w w

2 r6 3 12 r6 13

that is obtained by an algorithm from the solutions in PF. The w4 t w1

2 r7 4 13 13 r2 4

t 1 2

smaller the value of this metric, the better the convergence w w 3 3 r3 5 13 r6 5

w2 3 w toward PF. 3 r5 5 13 r8 4 t 1 1

4 w r4 22 t14 w r3 7

(2) Diversity metric [4]: 4 14 2 2

w w

4 r8 20 14 r5 8

This metric measures the extent of spread achieved among 1 3

t w w

5 5 r7 23 14 r7 8

the obtained solutions. It is deﬁned as follows: t 1 1 w w 6 6 r1,r4 7 t15 15 r8 2 2 1 N−1 w r3,r7 8 t16 w r2 9 ¯ 6 16

d + d + |d − d| f l i w3 2 i= w = 1 6 r6,r8 8 16 r5 7 (20) t w1 3

d + d ¯ 7 r1,r2 2 w r8 6

+ N − d

f l ( 1) 7 16 w2 t w1 7 r1,r7 2 17 17 r1,r2 10 3 2

w r3,r8 3 w r5,r7 12

where di is the distance between the neighboring solutions, and 7 17 1 3

t w r ,r 5 w r ,r 11

d¯ 8 8 1 3 17 5 8

is the mean of d . The parameters d and d are the Euclidean

i f l w2 1 w

8 r1,r5 4 t18 18 r6 15

distances between the extreme solutions and the boundary w3 t w1 8 r4,r7 2 19 19 r2 8

1 2

solutions of the obtained non-dominated set. The metric t w w 9 9 r1,r4 11 19 r4 7

w2 3

takes a higher value when the distributions of the solutions are 9 r3,r4 13 19 r5 7 w3 4 r ,r 12 w r 6 worse. 9 7 8 19 8 t w1 t w1

10 10 r1 9 20 20 r3 4

(3) Coverage metric C [39]: w2 2 w

10 r4 10 20 r6 4

Assume that A and B are two sets of non-dominated solutions. w3 3 w

10 r8 10 20 r8 5

C(A, B) is the ratio of the solutions in B that are dominated by at

least one solution in A. Hence,

∈ B| is small, it could be calculated using the method of exhaustion. The

|{x ∃y ∈ A

: dominates x}|

C A, B =

)

( |B| (21) exhaustive result is shown in Fig. 4, which represents the objective

space. Although the objective space contains hundreds of points,

C(A, B) = 1 means that all of the solutions in B are dominated

the decision space has 103,680 solutions. This result indicates that

by solutions in A, while C(A, B) = 0 means that no solution in

many local optima exist for the TTSP. The other sub-experiment

B is dominated by a solution in A. Generally speaking, C(A,

instance is a large-scale TTSP, which contains all of the 20 tasks

B) 1 − C(B, A).

t1–t20 to be tested. The scale of this TTSP is much larger than the

(4) Spacing metric S [40]:

small-scale TTSP, and the best solutions are thus quite difﬁcult to

This metric is proposed to measure the distribution of the obtain.

obtained solutions. It is deﬁned as follows: N S = 1 ¯ 2

(di − d) (22)

N − 1 i=1

M i j

d =

|f x − f x | i, j = , . . . , N i =/ j where i minj a=1 a( ) a( ) , 1 and . The

d¯

parameter is the mean of di. A smaller value for S means that

the solutions are more uniformly spread.

(5) CPU time:

CPU time is commonly used to compare the efﬁciency of algo-

rithms because it is always desirable to obtain high quality

solutions within an acceptable CPU time.

6.2. Experiment 1: a real-world TTSP

Experiment 1 shows the effectiveness of CNSGA in solving the

TTSP. This experimental instance is based on a real-world TTSP

illustrated in Table 6. We divide Experiment 1 into two sub-

experiment instances.

One sub-experiment instance is a small-scale TTSP for which

only the tasks t1–t6 should be tested. Because the scale of this TTSP Fig. 4. Exhaustive result for a small-scale TTSP.

2798 H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802

Table 7

Comparisons of NSGA-II, CNSGA L1 and CNSGA C1 for a small-scale TTSP.

TTSP C CPU time

t1–t6

N L C N L C (L,N) (N,L) (C,N) (N,C) N L C

Mean 1.8294 1.3536 1.2845 1.4984 1.4985 1.4588 0.0400 0.0300 0.0900 0.0200 0.0293 0.0297 0.0288

Var 0.2889 0.5846 0.4642 0.0095 0.0058 0.0090 0.0160 0.0090 0.0232 0.0040 6.6778e−6 2.6456e−5 7.9556e−6

Max 2.2720 2.5220 2.1264 1.6125 1.6125 1.6125 0.4000 0.3000 0.4000 0.2000 0.0360 0.0440 0.0360

Min 0.3000 4.0000e−7 2.6667e−7 1.3411 1.4380 1.3145 0.0000 0.0000 0.0000 0.0000 0.0270 0.0260 0.0270

Table 8

Comparisons of NSGA-II, CNSGA L2 and CNSGA C2 for a small-scale TTSP.

TTSP C CPU time

t1–t6

N L C N L C (L,N) (N,L) (C,N) (N,C) N L C

Mean 1.8294 1.3508 1.1781 1.4984 1.5545 1.5076 0.0600 0.0000 0.0500 0.0000 0.0293 0.0293 0.0299

Var 0.2889 0.8025 0.6676 0.0095 0.0052 0.0050 0.0182 0.0000 0.0117 0.0000 6.6778e−6 1.6456e−5 3.3433e−5

Max 2.2720 2.1264 2.3207 1.6125 1.6125 1.6125 0.4000 0.0000 0.3000 0.0000 0.0360 0.0390 0.0460

Min 0.3000 2.6667e−7 2.6667e−7 1.3411 1.4380 1.4387 0.0000 0.0000 0.0000 0.0000 0.0270 0.0260 0.0270

Table 9

Comparisons of NSGA-II, CNSGA L3 and CNSGA C3 for a small-scale TTSP.

TTSP C CPU time

t1–t6

N L C N L C (L,N) (N,L) (C,N) (N,C) N L C

Mean 1.8294 1.0014 1.3341 1.4984 1.5061 1.4849 0.0700 0.0200 0.0500 0.0200 0.0293 0.0282 0.0280

Var 0.2889 0.6936 0.4258 0.0095 0.0112 0.0080 0.0223 0.0040 0.0117 0.0040 6.6778e−6 1.5111e−6 2.0000e−6

Max 2.2720 1.9176 2.0220 1.6125 1.6125 1.6125 0.4000 0.2000 0.3000 0.2000 0.0360 0.0310 0.0310

− −

Min 0.3000 3.3333e 7 3.3333e 7 1.3411 1.2636 1.3411 0.0000 0.0000 0.0000 0.0000 0.0270 0.0270 0.0260

Table 10

Comparisons of NSGA-II, CNSGA L1 and CNSGA C1 for a large-scale TTSP.

TTSP S C CPU time

t1–t20

N L C (L,N) (N,L) (C,N) (N,C) N L C

Mean 3.2588 3.0712 4.4070 0.2000 0.3257 0.4029 0.1200 4.5185 4.5142 4.5356

Var 3.8007 2.6643 4.4793 0.0502 0.0755 0.0606 0.0156 1.7625e−4 4.0176e−4 8.9764e−4

Max 7.5878 6.0341 8.6745 0.5429 0.8000 0.8714 0.3000 4.5470 4.5590 4.6120

Min 0.8298 0.9705 1.7982 0.0000 0.0000 0.0286 0.0000 4.5010 4.4930 4.5000

The NSGA-II and the six variants of CNSGA are compared for the and the coverage metric C. In terms of the diversity metric ,

two sub-experiment instances using several performance metrics. most variants of CNSGA have slightly greater values. However,

First, the small-scale TTSP is considered. The true PF of this TTSP there are only four points on the PF, which may cause inaccura-

is calculated using the method of exhaustion, and the values for cies in the measurement. The CPU times for all of the cases are

the true PF are [(23, 70/3); (28, 19.5); (31, 18); (36, 14.6)]. The not that different because the main frameworks of the CNSGA

results obtained from the NSGA-II and the six variants of the and the NSGA-II are almost the same, and the chaotic opera-

CNSGA for the small-scale TTSP are shown in Tables 7–9. Because tors are not time-consuming. The boxplots for the above results

the true PF is known, the convergence metric , the diversity are shown in Fig. 5. It can be helpful to observe the perform-

metric , the coverage metric C, and the CPU time are used ances of the NSGA-II and the CNSGA for the small-scale TTSP

to measure the performances of the algorithms. For all of the in terms of the convergence metric and the diversity metric

cases, the CNSGA performs better with the convergence metric .

Table 11

Comparisons of NSGA-II, CNSGA L2 and CNSGA C2 for a large-scale TTSP.

TTSP S C CPU time

t1–t20

N L C (L,N) (N,L) (C,N) (N,C) N L C

Mean 3.2588 3.2316 1.8468 0.3629 0.3129 0.3786 0.1943 4.5185 4.5155 4.5353

Var 3.8007 3.3652 0.7463 0.0661 0.0565 0.0769 0.0709 1.7625e−4 5.0225e−4 2.4001e−4

Max 7.5878 6.2953 3.2470 0.6857 0.6143 0.8000 0.8286 4.5470 4.5600 4.5640

Min 0.8298 0.4465 0.1886 0.0000 0.0000 0.0000 0.0000 4.5010 4.4900 4.5150

H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802 2799

Fig. 5. Boxplots of and for small-scale TTSP using NSGA-II and six variants of CNSGA.

Second, the large-scale TTSP is solved. The results obtained from not be calculated, and the spacing metric S, the coverage metric

NSGA-II and the six variants of CNSGA for the large-scale TTSP are C and the CPU time are applied instead. With the spacing met-

shown in Tables 10–12 and Fig. 6. The true PF is unknown. There- ric S, the results show that CNSGA C2, CNSGA L3 and CNSGA C3

fore, the convergence metric and the diversity metric could perform better, while there are small differences between the

Table 12

Comparisons of NSGA-II, CNSGA L3 and CNSGA C3 for a large-scale TTSP.

TTSP S C CPU time

t1–t20

N L C (L,N) (N,L) (C,N) (N,C) N L C

Mean 3.2588 2.4888 2.4782 0.4314 0.2000 0.3986 0.2429 4.5185 4.5659 4.5324

Var 3.8007 1.5197 1.2180 0.0767 0.0472 0.0627 0.0725 1.7625e−4 0.0020 2.8784e−4

Max 7.5878 5.0589 4.7477 0.7286 0.6000 0.8714 0.7000 4.5470 4.6570 4.5660

Min 0.8298 1.1002 1.2160 0.0000 0.0000 0.0000 0.0000 4.5010 4.5190 4.5180

2800 H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802

Fig. 6. Comparisons of NSGA-II and six variants of CNSGA for a large-scale TTSP.

results of other variants. In terms of the coverage metric C, only is the makespan, and the second objective is used to handle the

CNSGA L1 performs worse than NSGA-II. Furthermore, CNSGA C2 constraints [41].

performs better than CNSGA L2, and CNSGA L3 performs better According to the experiments above, the cat population initial-

than CNSGA C3. ization and the logistic mutation operator are used in the CNSGA.

The comparisons of the GA, GASA and CNSGA are shown in Table 13,

6.3. Experiment 2: a TTSP for two units under test (UUTs)

Table 13

This experiment comes from the literature [15] and aims to solve

Comparisons of GA, GASA and CNSGA for a TTSP with two UUTs.

the TTSP with two UUTs. Each UUT contains 15 tasks and occupies 8

instruments. Because the UUTs are independent of each other [15], Makespan CPU time Percentage of

optimal solution

the TTSP can be equivalent to a one-UUT TTSP that contains 30 tasks

and occupies 8 instruments. Every task has only one scheme used to GA 78 17.7 s 0.8

GASA 78 8.1 s 0.94

complete it. Note that the TTSP solved by GASA contains the prece-

CNSGA 68 8.7 s 0.85

dence constraints between the tasks. For CNSGA, the ﬁrst objective

H. Lu et al. / Applied Soft Computing 13 (2013) 2790–2802 2801

the encoding efﬁciency. Furthermore, because the TTSP is a com-

binatorial optimization problem and has many local optima, the

proposed algorithm CNSGA introduces the chaotic initial popu-

lation, the chaotic crossover operator, and the chaotic mutation

operator into NSGA-II, using both the logistic map and the cat

map, to enrich the local search behavior. Because of the ergodicity

and the pseudo-randomness of chaos, the local searching ability of

CNSGA is better than that of the normal NSGA-II. This scenario has

been conﬁrmed by computational results obtained from two types

of experiments. According to the different capabilities of the logis-

tic and the cat chaotic operators, the CNSGA approach using the cat

population initialization, the cat or logistic crossover operator, and

the logistic mutation operator performs well and is very suitable for

solving the TTSP. Future work includes study of the performances of

additional chaotic maps and discovery of the reasons for their spe-

cial properties. In addition, other multi-objective algorithms can be

combined with chaotic operators to achieve better effects.

Acknowledgments

Fig. 7. Gantt chart obtained by CNSGA for the two UUTs TTSP. We would like to thank the anonymous reviewers for their

helpful comments in improving our manuscript. This research is

supported by the National Natural Science Foundation of China

Table 14

Statistical results of the CNSGA in 20 runs. under Grant No. 61101153.

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