Download pdf - Unrelated parallel-machine scheduling with position-dependent deteriorating jobs and resource-dependent processing time

Optim LettDOI 10.1007/s11590-012-0594-1

ORIGINAL PAPER

Unrelated parallel-machine schedulingwith position-dependent deteriorating jobsand resource-dependent processing time

Chou-Jung Hsu · Dar-Li Yang

Received: 18 January 2012 / Accepted: 16 November 2012© Springer-Verlag Berlin Heidelberg 2012

Abstract This paper is to analyze unrelated parallel-machine scheduling resourceallocation problems with position-dependent deteriorating jobs. Two general resourceconsumption functions, the linear and convex resource, are investigated. The objec-tives are to minimize the cost function that includes the weights of total load, totalcompletion time, total absolute deviation of completion time, and total resource cost.Moreover, we try to minimize the cost function that includes the weights of total load,total waiting time, total absolute deviation of waiting time, and total resource cost.Although each job processing time can be compressed through incurring an additionalcost, we show that the problems are polynomial time solvable when the number ofmachines is fixed.

Keywords Scheduling · Parallel-machine · Deterioration effects ·Assignment problem · Resource allocation

1 Introduction

There are various situations in which the processing time of jobs may be subjectto change due to deterioration and/or learning phenomena or additional resourcesconsumption. Machine scheduling problems with deterioration and/or learning effectshave been extensively studied in the last two decades in various machine settings and

C.-J. Hsu (B)Department of Industrial Engineering and Management, Nan Kai,University of Technology, Nan-Tou 542, Taiwane-mail: [email protected]

D.-L. YangDepartment of Information Management, National Formosa University,Yun-Lin 632, Taiwan

123

C.-J. Hsu , D.-L. Yang

performance measures. The deterioration job scheduling problem was initiated byBrowne and Yechiali [5] and Gupta and Gupta [11]. They presented an optimal solutionfor expected makespan minimization problem of single-machine scheduling under lin-ear deteriorating conditions. Since then, numbers of plentiful research related to thedeteriorating jobs have been conducted in scheduling under different machine environ-ments. Browne and Yechiali [4], Kunnathur and Gupta [14], Mosheiov [19], Bachmanand Janiak [2], and Low et al. [18] pointed out several real-life situations in whichdeteriorating jobs might occur. These include shops with deteriorating machines, delayof maintenance or cleaning, national defense, fire fighting, hospital emergency wards,steel rolling mills, and metal forming process where any delay in processing a job ispenalized by incurring additional time for accomplishing the job. For details on thisstream of research, reader may refer to the comprehensive surveys by Alidaee andWomer [1], Cheng et al. [7], and a book by Gawiejnowicz [8]. More recent papersthat have considered scheduling with deteriorating jobs include Gawiejnowicz andKononov [9], Koulamas et al. [15], Ng et al. [20], Wei and Wang [35], Lee et al. [16],Wang et al. [28], Yang and Wang [39], Lu et al. [17], Wang and Wang [29,30], Wanget al. [31], Wang et al. [27], Wei et al. [36], Yang et al. [38], and Zhao and Tang [42].

The manufacturing process can always be influenced by the resources. Vickson [26]was one of the pioneers to study a shop scheduling problem with controllable process-ing time. He argued that the objective of minimizing the total flow time and the totalprocessing cost incurred due to the job processing time compression. Van Wassenhoveand Baker [34] explored the single machine scheduling problems in which the objec-tive function is to minimize the maximum completion penalty. After 1982, problemswith fixed processing time dependent on resources have been widely investigated inthe last three decades in various machine settings and performance measures. Fordetails on this stream of research, reader may refer to the updated survey by Shabtayand Sreiner [24].

It is natural to study problems combining scheduling deteriorating jobs and resourceallocation cause of the characteristic of deterioration effects. However, to the bestof our knowledge, there exist only a few papers dealing with the resource alloca-tion and deteriorating jobs simultaneously. Zhao and Tang [41] found single-machinescheduling with deteriorating jobs where the release time of the jobs depends on theamounts of resource allocation. They presented two optimal algorithms for the prob-lems to minimize the sum of earliness penalties subject to no tardy jobs, to minimizethe resource consumption with makespan constraints and to minimize the makespanwith the total resource consumption constraints, respectively. Yang et al. [40] inves-tigated single-machine scheduling with deterioration and learning effects under thegroup resource consumption constraints. Zhu et al. [43] explored two single-machinescheduling problems with proportional linear deterioration of job processing time withresource constraint and proved that they can be solved in polynomial time algorithms.Wang et al. [27] studied single-machine scheduling with learning effect and resource-dependent processing time. The objectives were to minimize the cost functions, includ-ing makespan, total completion time, total absolute differences in completion timeand total resource cost. Moreover, the cost functions included makespan, total waitingtime, total absolute differences in waiting time and total resource cost. They presentedpolynomial time algorithms for linear and convex resource consumption functions.

123

Unrelated parallel-machine scheduling with position-dependent

Zhu et al. [44] explored the single-machine group scheduling problems with learningeffect and resource allocation. They showed that the problems for minimizing theweighted sum of makespan and total resource cost were polynomially solvable. Moreimportantly, the problems for minimizing the weighted sum of total completion timeand total resource cost have polynomial solutions under certain conditions.

This study is motivated by Bachman and Janiak [2] and many earlier studies relatedto position-dependent learning and/or deterioration effects. In Bachman and Janiak [2],they gave an example regarding the steel production in the process of preheating ingotsby gas to prepare them for hot rolling on the blooming mill. Before the ingots could behot rolled, they had to achieve the required temperature. However, the preheating timeof the ingots depended on their starting temperature. For example, the longer ingotswaited for the start of the preheating process, the lower the temperature was. Therefore,the longer the preheating process lasted. The preheating time could be shortened bythe increase of the gas flow intensity. For instance, the more gas was consumed, theshorter the preheating process lasted. Thus, the ingot preheating time depended on thestarting moment of the preheating process and the amount of gas consumed.

However, in the realistic production settings, there is rarely a single-machine envi-ronment. In many branches of industry and logistics, there arise problems of order-ing jobs on machines [12,21–23,25,32,33,37]. Therefore, this research investigatesunrelated parallel-machine scheduling with position-dependent deteriorating jobs andresource-dependent processing time. Two general resource consumption functions areexplored.

2 Problem formulation

The problem this paper aims to examine can be formally described as follows. Thereare n independent jobs J = {J1, J2, . . ., Jn} to be processed on m unrelated parallelmachines (M j , j = 1, 2, . . ., m) (i.e., processing job Ji on machine M j requiresdifferent time units). Let n j denote the number of jobs assigned to machine M j andP(n, m) = (n1, n2, . . ., nm) denote a job-allocation vector, where

∑mj=1 n j = n. We

assume, as in most practical situations, that m < n. The jobs are non-preemptiveand they are all available for processing at time zero. Each machine can handle atmost one job at a time and cannot stand idle until the last job assigned to it hasfinished processing. Each job can be processed on any one of the m unrelated parallelmachines. Let pi j denote the processing time for job Ji (i = 1, 2, . . ., n) if processedby machine M j ( j = 1, 2, . . ., m). Let pi jr denote the actual processing time forjob Ji when scheduled in position r of machine M j . In this research, we consider thefollowing models:

A linear resource consumption function:

pi jr = pi j rai j − bi j ui j , r = 1, 2, . . ., n j , 0 ≤ ui j ≤ ui j < pi j/bi j (1)

where ai j ≥ 0, bi j , r , ui j , and ui j are the deteriorating index of Ji , the positivecompression rate of Ji , the position of Ji , the amount of resource allocated to of Ji ,

123


and the upper bound on the amount of resource that can be allocated to Ji if it isscheduled on machine M j , respectively.

A convex resource consumption function:

pi jr = (pi j r

ai j /ui j)k

, ui j > 0, (2)

where k is a positive constant.For convenience, C j

max, TC j , TW j , TADC j , and TADW j denote makespan, thetotal completion time, the total waiting time, the total absolute deviation of job com-pletion time, and the total absolute deviation of job waiting time on machine M j ,respectively. Then, the total load, the total completion time, the total waiting time,the total absolute deviation of job completion time, and the total absolute devia-tion of job waiting time on all machines are

∑mj=1 C j

max,∑m

j=1 TC j ,∑m

j=1 TW j ,∑m

j=1 TADC j , and∑m

j=1 TADW j , respectively.Let [i j] denote the i th processing job on machine M j . The objective is to determine

the resource allocation u∗ = (u∗

1, u∗2, . . ., u∗

n

)and the job sequence π∗ ∈ � which

minimize the corresponding value of the following cost functions:

f (π, u) = α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j], (3)

g(π, u) = α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j], (4)

where � is the set of all permutations with respect to the job-allocation vector, weightsαi ≥ 0, i = 1, 2, . . ., 4, are given constants, v[i j] is the per time unit cost associatedwith the resource allocation that is used to compress the processing time of i th job onmachine M j . Then, using the three-field notation introduced by Graham et al. [10],the corresponding scheduling problems are denoted by

Rm

∣∣∣∣∣∣pi jr = pi j r

ai j − bi j ui j

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j],

Rm


ai j − bi j ui j

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j],

123


Rm

∣∣∣∣∣∣pi jr = (

pi j rai j /ui j

)k

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j], and

Rm

∣∣∣∣∣∣pi jr = (

pi j rai j /ui j

)k

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j],

respectively.

3 Optimal solutions

In this section, we prove that the proposed problems can be solved in polynomial time.Note that C j

max = ∑n ji=1 p[i j], TC j = ∑n j

i=1

(n j −i +1

)p[i j], TW j = ∑n j

i=1(n j −i

)p[i j], TADC j =

∑n ji=1 (i − 1)

(n j −i +1

)p[i j] (Kanet [13]), and TADW j =

∑n ji=1 i

(n j −i

)p[i j] (Bagchi [3]).

3.1 Case of the linear resource consumption function

For the actual processing time is a linear resource consumption function, Eq. (3) canbe calculated as follows:

f (π, u) = α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

= α1

m∑

j=1

n j∑

i=1

p[i j] + α2

m∑

j=1

n j∑

i=1

(n j − i + 1

)p[i j]

+α3

m∑

j=1

n j∑

i=1

(i − 1)(n j − i + 1

)p[i j] + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

[α1 + α2

(n j − i + 1

) + α3(i − 1)(n j − i + 1

)]p[i j]

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

123


=m∑

j=1

n j∑

i=1

w[i j] p[i j] + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j](

p[i j]ia[i j] − b[i j]u[i j]) + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j] p[i j]ia[i j] +m∑

j=1

n j∑

i=1

(α4v[i j] − w[i j]b[i j]

)u[i j] (5)

where w[i j] = α1 + α2(n j − i + 1

) + α3(i − 1)(n j − i + 1

). Taking the deriva-

tive by u[i j] to Eq. (5), we see that d f (π,u)du[i j] = α4v[i j] − w[i j]b[i j] for j = 1, 2, . . .,

m and i = 1, 2, . . ., n j . Then, the sign of d f (π,u)du[i j] must be the opposite of the sign

of α4v[i j] − w[i j]b[i j] in order to minimize f . Therefore, if α4v[i j] − w[i j]b[i j] < 0,we will allocate the maximal feasible amount of resource to the i th job on machineM j ; if α4v[i j] − w[i j]b[i j] ≥ 0, we should not allocate any resource to the i th job onmachine M j . These imply that the following Lemma holds.

Lemma 1 The optimal resource allocation of scheduling problem of

Rm


ai j − bi j ui j

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

can be expressed as follows:

u∗[i j] ={

u[i j], if α4v[i j] − w[i j]b[i j] < 0,

0, if α4v[i j] − w[i j]b[i j] ≥ 0,

where u∗[i j], 1 ≤ j ≤ m and 1 ≤ i ≤ n j , represents the optimal resource allocationof the i-th job on machine M j .

Define the value of ci jr by

ci jr ={

wr j pi j rai j , if α4vi j − wr j bi j ≥ 0,

wr j pi j rai j + (α4vi j − wr j bi j

)ui j , if α4vi j − wr j bi j < 0,

it represents the minimum possible cost resulting from assigning job i in position ron machine M j . Furthermore, let xi jr be a 0/1 variable such that xi jr = 1 if Ji isscheduled in position r on machine M j , and xi jr = 0 otherwise. Therefore, if thenumber (n j ) of jobs on machine j is known, the addressed scheduling problem canbe formulated as the following assignment problem and solved in O(n3); see, forexample, Brucker [6].

123


Minimizen∑

i=1

m∑

j=1

n j∑

r=1

ci jr xi jr ,

Subject tom∑

j=1

n j∑

r=1

xi jr = 1, i = 1, 2, . . ., n,

n∑

i=1

xi jr = 1, j = 1, 2, . . ., m, r = 1, 2, . . ., n j ,

xi jr = 0 or 1, i = 1, 2, . . ., n, j = 1, 2, . . ., m, r = 1, 2, . . ., n j .

The constraints make sure that each job is scheduled exactly once and each positionon each machine is taken by one job.

Next, the question is how many job-allocation vectors exist. Note that n j maybe 0, 1, 2, . . ., n for j = 1, 2, . . ., m. So if we get the numbers of jobs on the firstm − 1 machines, the number of jobs processed on the last machine is then determineduniquely due to

∑mj=1 n j = n. Therefore, the upper bound on the number of job-

allocation vectors is (n + 1)m−1. Based on the above analysis, we have the followingresult.

Theorem 1 The following scheduling problem can be solved in O(nm+2

)time:

Rm


ai j − bi j ui j

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j] .

Similarly, Eq. (4) can be expressed as follows:

g(π, u) = α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

= α1

m∑

j=1

n j∑

i=1

p[i j] + α2

m∑

j=1

n j∑

i=1

(n j − i

)p[i j] + α3

m∑

j=1

n j∑

i=1

i(n j − i

)p[i j]

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j] p[i j] + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j](

p[i j]ia[i j] − b[i j]u[i j]) + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

123


=m∑

j=1

n j∑

i=1

w[i j] p[i j]ia[i j] +m∑

j=1

n j∑

i=1

(α4v[i j] − w[i j]b[i j]

)u[i j]

where w[i j] = α1 + α2(n j − i

) + α3i(n j − i

). Applying the above analysis in a

similar manner to the model (3), if α4v[i j]−w[i j]b[i j] < 0, we will allocate the maximalfeasible amount of resource to the i th job on machine M j ; if α4v[i j] − w[i j]b[i j] ≥ 0,we should not allocate any resource to the i th job on machine M j . These imply thatthe following Lemma holds.


Rm


ai j − bi j ui j

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]


u∗[i j] ={

u[i j], if α4v[i j] − w[i j]b[i j] < 0,

0, if α4v[i j] − w[i j]b[i j] ≥ 0,

where u∗[i j], 1 ≤ j ≤ m and 1 ≤ i ≤ n j , represents the optimal resource allocationof the ith job on machine M j .

Define the value of ci jr by

ci jr ={

wr j pi j rai j , if α4vi j − wr j bi j ≥ 0,

wr j pi j rai j + (α4vi j − wr j bi j

)ui j , if α4vi j − wr j bi j < 0,

it represents the minimum possible cost resulting from assigning job i in position ron machine M j . Furthermore, if the number (n j ) of jobs on machine j is known, theoptimal sequence is obtained via the following assignment problem.

Minimizen∑

i=1

m∑

j=1

n j∑

r=1

ci jr xi jr ,

Subject tom∑

j=1

n j∑

r=1

xi jr = 1, i = 1, 2, . . ., n,

n∑

i=1xi jr = 1, j = 1, 2, . . ., m, r = 1, 2, . . ., n j

xi jr = 0 or 1, i = 1, 2, . . ., n, j = 1, 2, . . ., m, r = 1, 2, . . ., n j .

Taking advantage of the analysis in the previous problem, the following Theoremalso holds.

123



)time:

Rm


ai j − bi j ui j

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

3.2 Case of the convex resource consumption function

In the same way as previous sub-section, Eq. (3) can be calculated as follows:

f (π, u) = α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j] p[i j] + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j](

p[i j]ia[i j]/u[i j])k + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j] (6)

where w[i j] = α1+α2(n j − i + 1

)+α3(i−1)(n j − i + 1

). Taking the first derivative

to Eq. (6) with respect to u[i j], j = 1, 2, . . ., m, i = 1, 2, . . ., n j , equating it to zeroand solving it for u∗[i j], we obtain

u∗[i j] = (kw[i j]/α4v[i j]

) 1k+1 × (

p[i j]ia[i j])k

k+1 . (7)

Since the objective is a convex function, Eq. (7) provides necessary and sufficientcondition for optimality. Thus, we conclude that the following Lemma holds.


Rm

∣∣∣∣∣∣pi jr = (

pi j rai j /ui j

)k

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]



) 1k+1 × (

p[i j]ia[i j])k

k+1 , j = 1, 2, . . ., m, i = 1, 2, . . ., n j ,

where w[i j] = α1 + α2(n j − i + 1

) + α3(i − 1)(n j − i + 1

).

123


By substituting Eq. (7) into Eq. (6), we obtain a unified expression for the costfunction for the objective under an optimal resource allocation and as a function ofthe job sequence:

f (π, u∗(π)) =m∑

j=1

n j∑

i=1

A × [α1 + α2

(n j − i + 1

) + α3(i − 1)(n j − i + 1

)] 1k+1

× (p[i j]ia[i j]v[i j]

) kk+1 ,

where A =(

k−kk+1 + k

1k+1

)× α

kk+14 .

Similar to the analysis in the previous sub-section, we define the value of di jr by

di jr = A × [α1 + α2

(n j − r + 1

) + α3(r − 1)(n j − r + 1

)] 1k+1 × (

pi j rai j vi j

) kk+1 ,

it represents the minimum possible cost resulting from assigning job i in position ron machine M j . Therefore, if the number (n j ) of jobs on machine j is known, theoptimal sequence is obtained via the following assignment problem.

Minimizen∑

i=1

m∑

j=1

n j∑

r=1

di jr xi jr ,

Subject tom∑

j=1

n j∑

r=1

xi jr = 1, i = 1, 2, . . ., n,

n∑

i=1

xi jr = 1, j = 1, 2, . . ., m, r = 1, 2, . . ., n j

xi jr = 0 or 1, i = 1, 2, . . ., n, j = 1, 2, . . ., m, r = 1, 2, . . ., n j .

Taking advantage of the analysis in the previous sub-section, we conclude that thefollowing Theorem holds.


)time:

Rm

∣∣∣∣∣∣pi jr = (

pi j rai j /ui j

)k

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TC j + α3

m∑

j=1

TADC j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j] .

123


Similarly, Eq. (4) can be expressed as follows:

g(π, u) = α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j] p[i j] + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]

=m∑

j=1

n j∑

i=1

w[i j](

p[i j]ia[i j]/u[i j])k + α4

m∑

j=1

n j∑

i=1

v[i j]u[i j],


) + α3i(n j − i

). Taking advantage of the analysis in

the previous problem, one can immediately conclude that the following Lemma 4 andTheorem 4 also hold.


Rm

∣∣∣∣∣∣pi jr = (

pi j rai j /ui j

)k

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j]



) 1k+1 × (

p[i j]ia[i j])k

k+1 , j = 1, 2, . . ., m, i = 1, 2, . . ., n j ,


) + α3i(n j − i

).


)time:

Rm

∣∣∣∣∣∣pi jr = (

pi j rai j /ui j

)k

∣∣∣∣∣∣α1

m∑

j=1

C jmax + α2

m∑

j=1

TW j + α3

m∑

j=1

TADW j

+α4

m∑

j=1

n j∑

i=1

v[i j]u[i j] .

4 Conclusions

This research is aimed to investigate the unrelated parallel-machine scheduling dete-riorating jobs with resource-dependent processing time. Two general resource con-sumption functions are explored. In this study, the objectives are to minimize the cost

123


functions that include the weights of total load, total completion time, total absolutedeviation of completion time, and total resource cost. In addition, we try to mini-mize the cost functions that include the weights of total load, total waiting time, totalabsolute deviation of waiting time, and total resource cost. If the number of machinesis fixed, we showed that the proposed problems were polynomial time solvable.

Acknowledgments We are grateful to two anonymous reviewers for their helpful comments on an earlierversion of this article. This research is supported in part by the National Science Council of the Republic ofChina, under Grant No. NSC 101-2221-E-252-005. Professor Yang is currently a Fulbright visiting scholarto Oregon State University and he is supported in part by the National Science Council of Republic ofChina, under grant number NSC-101-2918-I-150-002.

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