research article a single-machine two-agent scheduling...
TRANSCRIPT
Research ArticleA Single-Machine Two-Agent SchedulingProblem by a Branch-and-Bound and ThreeSimulated Annealing Algorithms
Shangchia Liu1 Wen-Hsiang Wu2 Chao-Chung Kang3
Win-Chin Lin4 and Zhenmin Cheng5
1 Business Administration Department Fu Jen Catholic University New Taipei City 24205 Taiwan2Department of Healthcare Management Yuanpei University Hsinchu 30015 Taiwan3Department of Business Administration and Graduate Institute of Management Providence UniversityShalu Taichung 43301 Taiwan
4Department of Statistics Feng Chia University Taichung 40724 Taiwan5 Business College Beijing Union University Beijing 100101 China
Correspondence should be addressed to Shangchia Liu 056298mailfjuedutw
Received 30 June 2014 Accepted 11 August 2014
Academic Editor Chin-Chia Wu
Copyright copy 2015 Shangchia Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In the field of distributed decisionmaking different agents share a common processing resource and each agent wants tominimizea cost function depending on its jobs onlyThese issues arise in different application contexts including real-time systems integratedservice networks industrial districts and telecommunication systems Motivated by its importance on practical applications weconsider two-agent scheduling on a single machine where the objective is to minimize the total completion time of the jobs of thefirst agent with the restriction that an upper bound is allowed the total completion time of the jobs for the second agent For solvingthe proposed problem a branch-and-bound and three simulated annealing algorithms are developed for the optimal solutionrespectively In addition the extensive computational experiments are also conducted to test the performance of the algorithms
1 Introduction
Multiagent simulation is a branch of artificial intelligence thatoffers a promising approach to dealing with multistakeholdermanagement systems such as common pool resources Itprovides a framework which allows analysis of stakeholdersrsquo(or agentsrsquo) interactions and decision making For examplein forest plantation comanagement Purnomo and Guizol [1]further pointed out ldquoComanagement of forest resources is aprocess of governance that enables all relevant stakeholders toparticipate in the decision-making processes Illegal loggingand forest degradation are currently increasing and loggingbans are ineffective in reducing forest degradation At thesame time interest in forest plantations and concern aboutpoverty problems of neighboring people whose livelihoodsdepend on forest services and products continue to increaserapidly Governments have identified the development of
small forest plantations as an opportunity to provide woodsupplies to forest industries and to reduce poverty Howeverthe development of small plantations is very slow due toan imbalance of power and suspicion between communitiesand large companiesrdquo In blood cell population dynamicsBessonov et al [2] gave some possible explanations of themechanism for recovery of the systemunder important bloodloss or blood diseases such as anemia Agnetis et al [3]also indicated that multiple agents compete on the usageof a common processing resource in different applicationenvironments and different methodological fields such asartificial intelligence decision theory operations researchand so forth
Scheduling with multiple agents has received growingattention in recently years Agnetis et al [4] and Baker andSmith [5] were among the pioneers to introduce the concept
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 681854 8 pageshttpdxdoiorg1011552015681854
2 Discrete Dynamics in Nature and Society
of multiagent into scheduling problems Yuan et al [6]discussed two dynamic programming recursions in Bakerand Smith [5] and proposed a polynomial-time algorithm forthe same problem Cheng et al [7] addressed the feasibilitymodel of multiagent scheduling on a single machine whereeach agentrsquos objective function is to minimize the totalweighted number of tardy jobs Ng et al [8] proposed a two-agent scheduling problem on a single machine where theobjective is to minimize the total completion time of thefirst agent with the restriction that the number of tardy jobsof the second agent cannot exceed a given number Agnetiset al [3] discussed the complexity some single-machinescheduling problems with several agents Cheng et al [9]studied multiagent scheduling on a single machine wherethe objective functions of the agents are of the max-formAgnetis et al [10] used a branch-and-bound method to solveseveral two-agent scheduling problems on a single machineLee et al [11] considered a multiagent scheduling problem ona single machine in which each agent is responsible for hisown set of jobs and wishes to minimize the total weightedcompletion time of his own set of jobs Recently Leung etal [12] generalized the single machine problems proposed byAgnetis et al [4] to the environment with multiple identicalmachines in parallel Yin et al [13] considered several two-agent scheduling problems with assignable due dates on asingle machine where the goal was to assign a due date froma given set of due dates and a position in the sequence toeach job so that the weighted sum of the objectives of bothagents is minimized For different combinations of the objec-tives which include the maximum lateness total (weighted)tardiness and total (weighted) number of tardy jobs theyprovided the complexity results and solve the correspondingproblems if possible Yin et al [14] investigated a schedulingenvironment with two agents and a linear nonincreasingdeterioration with the objective of scheduling the jobs suchthat the combined schedule performs well with respect to themeasures of both agents Three different objective functionswere considered for one agent including the maximumearliness cost total earliness cost and total weighted earlinesscost while keeping the maximum earliness cost of the otheragent below or at a fixed level 119880 They proposed the opti-mal (nondominated) properties and present the complexityresults for the problems Yin et al [15] addressed a two-agentscheduling problem on a single machine where the objectiveis to minimize the total weighted earliness cost of all jobswhile keeping the earliness cost of one agent below or at afixed level 119876 A mixed-integer programming (MIP) modelwas first formulated to find the optimal solution which isuseful for small-size problem instances and then a branch-and-bound algorithm incorporating with several dominanceproperties a lower bound and a simulated annealing heuristicalgorithm were provided to derive the optimal solution formedium- to large-size problem instances For more recentworks with two-agent issues readers can refer to Nong et al[16] Wan et al [17] Luo et al [18] Yin et al [19] and soforth In addition for more multiple-agent works with time-dependent readers can refer to Liu and Tang [20] Liu et al[21] Cheng et al [22 23] Wu et al [24] Li and Hsu [25] Yinet al [26ndash28] Wu [29] and Li et al [30] and so forth
For the importance of multiple agents competing on theusage of a common processing resource in different appli-cation environments and different methodological fields weconsidered two-agent scheduling on a single machine Theobjective is to minimize the total completion time of the jobsof the first agent with the restriction that an upper bound isallowed the total completion time of the jobs for the secondagent which has been shown binary NP-hard by Agnetis etal [4] The problem is described as follows There are 119899 jobswhich belong to one of the agents AG
0or AG
1 For each job
119895 there is a normal processing time 119901119895and an agent code 119868
119895
where 119868119895= 0 if 119869
119895isin AG
0or 119868119895= 1 if 119869
119895isin AG
1 All the jobs
are available at time zero Under a schedule 119878 let 119862119895(119878) be the
completion time of job 119895 The objective of this paper is to findan optimal schedule to minimizesum119899
119895=1119862119895(119878)(1minus119868
119895) subject to
sum119899
119895=1119862119895(119878)119868119895lt 119880 where 119880 is a control upper bound
The remainder of this paper is organized as followsIn Section 2 we present some dominance properties anddevelop a lower bound to speed up the search for the optimalsolution followed by discussions of a branch-and-bound andthree simulated annealing (SA) algorithms We present theresults of extensive computational experiments to assess theperformance of all of the proposed algorithms under differentexperimental conditions in Section 3 We conclude the paperand suggest topics for further research in the last section
2 Dominance Properties
We will develop some adjacent dominance rules by using thepairwise interchange method Assume that 119878
1and 1198782denote
two given job schedules in which the difference between 1198781
and 1198782is a pairwise interchange of two adjacent jobs 119894 and 119895
That is 1198781= (120590 119894 119895 120590
1015840) and 119878
2= (120590 119895 119894 120590
1015840) where 120590 and
1205901015840 each denote a partial sequence In addition let 119905 be the
starting time of the last job in 120590
Property 1 If jobs 119894 119895 isin AG0 and 119901
119894lt 119901119895 then 119878
1dominates
1198782
Property 2 If jobs 119894 119895 isin AG1sum119894isin120590capAG
1
119862119894+2119905+2119901
119894+119901119895lt 119880
and 119901119894lt 119901119895 then 119878
1dominates 119878
2
Next we give a proposition to determine the feasibility ofthe partial schedule Let (120587 120587119888) be a sequence of jobs where120587 is the scheduled part with 119896 jobs and 120590119888 is the unscheduledpart with (119899minus119896) jobs Among the unscheduled jobs let 119901
(1)=
min119869119895isin120587119888capAG1
119901119895 Moreover let 119862
[119896]be the completion times
of the last job in120587 Also let1205871015840 and12058710158401015840 denote the unscheduledjobs in AG
0and AG
1arranged in the smallest processing
times (SPT) order respectively
Property 3 If there is a job 119894 isin AG1such that sum
119894isin120587capAG1
119862119894+
119862[119896]+ 119901(1)gt 119880 then (120587 120587119888) is not a feasible sequence
Property 4 If all unscheduled jobs are belonging toAG0 then
(120587 1205871015840) dominates (120587 120587119888)
Property 5 If all unscheduled jobs are belonging toAG1 then
(120587 120587119888) can be determined by (120587 12058710158401015840) That is (120587 12058710158401015840) either isfeasible solution or can be deleted
Discrete Dynamics in Nature and Society 3
21 Lower Bound Assume that PS is a partial schedule inwhich the order of the first 119896 jobs is determined and letUS be the unscheduled part with (119899 minus 119896) jobs Amongthe unscheduled jobs there are 119899
0jobs from agent AG
0
and 1198991jobs from agent AG
1 Moreover let 119862
[119896]denote the
completion times of the 119896th job in PS The completion timefor the (119896 + 119895)th job is
119862[119896+119895]
ge 119862[119896]+
119895
sum
119894=1
119901(119896+119894)
for 1 le 119895 le 1198990 (1)
Then a lower bound can be obtained as follows119899
sum
119895=1
119862119895(119878) (1 minus 119868
119895) ge
119896
sum
119895=1
119862119895(119878) (1 minus 119868
119895) +
1198990
sum
119895=1
119862(119895)(119878) (2)
where 119862(119895)= 119862[119896]+ sum119895
119894=1119901(119896+119894)
22 Simulated Annealing Algorithms Simulated annealinghas become one of the most popular metaheuristic methodsto solve combinatorial optimization problems since it wasproposed by Kirkpatrick et al [31] For example Kim etal [32] applied the method in scheduling of raw-materialunloading from ships at a steelworks Sun et al [33] used thetechnique for allocating product lots to customer orders insemiconductor manufacturing supply chains Moreover themethod has the advantage of avoiding getting trapped in alocal optimum because of its hill climbing moves which aregoverned by a control parameter Thus we will use SA toderive near-optimal solutions for our problem The steps ofSA algorithms are summarized as follows
221 Initial Sequence Three initial sequences for the threeSA algorithms are adopted for our problem For the firstinitial sequence in SA
1 the sequence of jobs of AG
1is
arranged according to the smallest processing times (SPT)order followed by arranging the sequence of jobs of AG
0in
the shortest processing time (SPT) order In order to get agood initial one two more initial sequences are consideredFor the second initial sequence in SA
2 the pairwise inter-
change is used to the initial sequence produce by SA1 For the
third initial sequence in SA3 the NEHmethod [34] is applied
to the initial sequence produce by SA1
222 Neighborhood Generation The pairwise interchange(PI) neighborhood generation method is adopted in thealgorithms
223 Acceptance Probability When a new feasible sequenceis generated it is accepted if its objective value is smallerthan that of the original sequence otherwise it is acceptedwith a probability that decreases as the process evolves Theprobability of acceptance is generated from an exponentialdistribution
119875 (accept) = exp (minus120572 times ΔTC) (3)
where 120572 is a control parameter and ΔTC is the change in theobjective value In addition we use the method suggested by
Ben-Arieh and Maimon [35] to change 120572 in the 119896th iterationas follows
120572 =119896
120575 (4)
where 120575 is an experimental constant After preliminary trials120575 = 6 is used in our experiments
If the total completion time of jobs from agent AG0
increases as a result of a random pairwise interchange thenew sequence is accepted when 119875(accept) gt 119903 where 119903 israndomly sampled from the uniform distribution 119880(0 1)
224 Stopping Rule All three proposed SAs are stopped after100119899 iterations where 119899 is the number of jobs
3 Computational Experiments
The extensive computational experiments were conducted totest the performances of the branch-and-bound algorithmand the three simulated annealing algorithms All the algo-rithms were coded in Fortran using Compaq Visual Fortranversion 66 and performed the experiments on a personalcomputer powered by an Intel(R) Core(TM)2 Quad CPU266GHz with 4GB RAM operating under Windows XPThe job processing times were generated from a uniformdistribution over the integers 1ndash100 For the control upperbound 119880 we arrange the jobs of AG
1by the smallest
processing times (SPT) order and compute out the totalcompletion times of the jobs of AG
1 (recorded as TC
1)Then
following the sequence of the jobs of AG1 we arrange the
sequence of jobs of AG0in the shortest processing time (SPT)
order and compute out the total completion times of jobs ofAG1and AG
0 (recoded as TC
1+2) We let 119880 = (1 minus 120572)TC
1+
120572TC1+2
where 0 lt 120572 lt 1 Moreover120572was taken as the valuesof 025 05 and 075 while the proportion of the jobs of agentAG1at pro = 025 05 and 075 in the testsFor the branch-and-bound algorithm the average and
standard deviation of the numbers of nodes and the executiontimes (in seconds) were recorded For the SA heuristicsthe mean and standard deviation percentage errors wererecorded The percentage error of a solution obtained froma heuristic algorithm was given by
119881 minus 119881lowast
119881lowast (5)
where 119881 and 119881lowast are denoted as the total completion time
of the heuristic and the optimal solution respectively Thecomputational times of the heuristic algorithms were notrecorded because they all were fast in generating solutions
The computational experiments were divided into twoparts In the first part of the experiments three numberof jobs were tested at 119899 = 8 12 and 16 As a result27 experimental situations were examined 50 instanceswere randomly generated for each case The results weresummarized in Table 1 that includes the CPU time (mean andstandard deviation) and the number of nodes for the branch-and-bound algorithm
4 Discrete Dynamics in Nature and Society
Table1Perfo
rmance
oftheb
ranch-and-bo
undandheuristicalgorithm
s(119899=8sim
16)
119899Pro
120572
Branch-and
-bou
ndalgorithm
SA1
SA2
SA3
SA4
Valid
samples
ize
CPUtim
eNum
bero
fnod
esErrorp
ercentages
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
8
025
025
50000
001
526
318
028
146
028
146
028
146
028
146
050
50000
001
748
588
129
412
093
311
086
309
024
169
075
50001
001
1383
923
342
729
283
573
227
480
062
246
050
025
50001
001
2229
1371
001
006
001
006
000
000
000
000
050
50001
001
2500
1412
050
178
014
056
020
086
004
020
075
50001
001
3183
1139
092
203
127
226
148
276
026
102
075
025
50001
001
1091
217
000
000
000
000
000
000
000
000
050
50001
001
1091
217
000
000
000
000
000
000
000
000
075
50001
001
1089
196
036
118
011
048
049
133
003
024
Average
001
001
1538
709
075
199
062
152
062
159
016
079
12
025
025
50078
086
97087
113419
029
093
061
230
030
138
005
020
050
5013
619
6175858
261636
186
441
399
781
145
386
077
195
075
50392
524
535992
751273
478
725
440
682
530
843
212
426
050
025
501983
2275
2278911
2794764
002
007
004
020
001
005
000
000
050
502791
3386
3405293
4448908
064
124
053
116
032
073
017
045
075
503538
2151
4692766
3096131
207
216
237
243
237
274
069
092
075
025
50630
218
7940
033044
61000
000
000
000
000
000
000
000
050
50630
218
7944
39304555
000
000
000
000
000
000
000
000
075
50660
182
862132
262277
044
066
074
122
055
119
015
033
Average
1204
1026
1515165
1370825
112
186
141
244
114
204
044
090
16
025
025
4884682
169136
62738963
129480137
036
092
039
106
035
116
018
077
050
46143263
241156
105554536
181805362
199
346
249
428
188
438
095
229
075
331446
76262913
106507557
197947458
272
314
779
1200
430
664
183
228
050
025
228044
31605859
429572906
333869433
000
000
000
000
000
000
000
000
050
20726799
569796
387550386
314387045
010
029
000
000
011
051
000
000
075
3749301
577630
411927486
331822988
093
107
691
617
665
411
093
107
075
025
13950254
305315
625205281
214788228
000
000
000
000
000
000
000
000
050
13947874
3044
52625205281
214788228
000
000
000
000
000
000
000
000
075
7996598
365310
652692683
258677186
041
073
024
063
103
106
000
000
Average
6164
31377952
378550564
241951785
072
107
198
268
159
198
043
071
Discrete Dynamics in Nature and Society 5
0
2
4
6
025 050 075
Aver
age n
umbe
r of n
odes
pro = 025
pro = 050
pro = 075
120572
times10
6
Figure 1 Performance of the branch-and-bound algorithms (119899 = 12)
For the performance of the branch-and-bound algorithmit can be observed from Table 1 that the number of nodesand the mean of the CPU time increase when 119899 becomesbigger The difficult situations occur at 119899 = 16 Especiallythe instances with a bigger value of 120572 (120572 = 075) are difficultto solve than those with a smaller one (120572 = 025 050)Moreover the instances with a bigger value of pro (pro = 05075) are difficult to solve than those with a smaller one (pro= 025) The most difficult case is located at pro = 05 075and 120572 = 075 as the number of instances which can be solvedout within less 10minus9 nodes were declined to 10 or below Inaddition as shown in Figure 1 and Table 1 the instances withpro = 05 took more nodes or CPU time than others and thetrend became clear as the value of 120572 got bigger
For the performance of the proposed SAs it can beseen in Table 1 that the means of the percentage error ofSA1(between 00 and 478) are lower than those of SA
2
(between 00 and 779) and those of SA3(between 00
and 665) Moreover the trend of the standard deviationsof the percentage error keeps the similar pattern As shownin Table 1 the standard deviations of the percentage errorof SA
1 SA2 and SA
3were between 00 and 729 00
and 120 and 00 and 843 respectively It released thatthe performance of SA
1was slightly better than the other
two Furthermore Figure 2 and Table 1 indicated that themeans of the percentage error of SA
1 SA2 and SA
3are
slightly affected by the parameter 120572 For example the meansof the percentage errors of SA
1 SA2 and SA
3were lager
than 2 at 120572 = 025 Specifically the behavior became clearat pro = 025 and 05 It also can be observed that there isno absolutely dominance relationship among three proposedSAs Due to get a good quality solution we further combinedthree proposed SAs into one (recorded as SA
4and SA
4=
minSA119894 119894 = 1 2 3) The means of the percentage error of
SA4were located between 00 and 212 Meanwhile its
standard deviations of the percentage error were between
000
100
200
300
Mea
n er
ror (
)
SA1SA2
SA3SA4
025 050 075120572
Figure 2 Performance of the SA algorithms (119899 = 12)
00 and 426 Moreover the impacts of pro and 120572 werenot present for proposed three SAs
In the second part of the experiments the performancesof the proposed SA heuristics were further tested for largenumbers of jobs Three different numbers of jobs wereconsidered at 119899 = 20 40 and 60 The proportion of thejobs of agent AG
1at pro = 025 05 and 075 in the tests
Moreover 120572 was taken as the values of 025 05 and 075 Asa result 27 experimental situations were tested 50 instanceswere randomly generated for each situation The relativedeviance percentage with respect to the best known solutionwas reported for each instance The mean execution timeandmean relative deviance percentage were also recorded foreach SA heuristicThe relative deviation percentage RDPwasgiven by
SA119894minus SAlowast
SAlowast (6)
where SA119894is the value of the objective function generated by
SA119894and SAlowast = minSA
119894 119894 = 1 2 3 is the smallest value of
the objective function obtained from the three SA heuristicsThe results were summarized in Table 2
As shown in Figure 3 and Table 2 it can be seen that theRPDmeans of SA
1 SA2 and SA
3are getting slightly bigger as
the value of 120572 increases In general the RPDmeans of SA3are
lower than those of SA1and SA
2 Furthermore all of the RDP
means of SA1 SA2 and SA
3were less than 2 Figure 2 also
indicated that there is no absolutely dominance relationshipamong three proposed SAs
4 Conclusions
Thispaper explored the single-machine two-agent schedulingproblem where the objective is to minimize the total com-pletion time of the jobs belonging to the first agent withthe restriction that total completion time of jobs from thesecond agent has an upper bound Due to the fact that theproblem under study is binary NP hard a branch-and-bound
6 Discrete Dynamics in Nature and Society
Table 2 RPD of heuristic algorithms (119899 = 20sim60)
119899 Pro 120572
SA1 SA2 SA3CPU time RPD CPU time RPD CPU time RPD
Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std
20
02525 00019 00051 045 144 00019 00051 041 119 00016 00047 051 17550 00019 00051 139 267 00019 00051 187 551 00022 00055 089 18875 00019 00051 286 481 00019 00051 496 680 00019 00051 201 366
0525 00019 00051 002 007 00019 00051 000 001 00016 00047 001 00650 00019 00051 051 081 00019 00051 039 067 00022 00055 051 07875 00019 00051 105 152 00022 00055 119 208 00022 00055 130 203
07525 00019 00051 000 000 00013 00043 000 000 00019 00051 000 00050 00019 00051 000 000 00016 00047 000 000 00019 00051 000 00075 00019 00051 050 058 00019 00051 058 075 00019 00051 034 055
Average 00019 00051 075 132 00018 00050 104 189 00019 00052 062 119
40
02525 00066 00078 010 030 00072 00079 016 052 00066 00078 000 00050 00072 00079 092 151 00078 00079 122 209 00072 00079 039 14775 00078 00079 366 443 00084 00079 391 584 00081 00079 282 404
0525 00066 00078 002 006 00066 00078 000 000 00066 00078 000 00050 00075 00079 039 043 00078 00079 031 057 00078 00079 046 07975 00084 00079 081 081 00091 00078 103 169 00091 00078 071 112
07525 00066 00078 000 000 00063 00077 000 000 00063 00077 000 00050 00063 00077 000 000 00063 00077 000 000 00063 00077 000 00075 00072 00079 032 045 00078 00079 025 036 00081 00079 025 031
Average 00071 00078 069 089 00075 00078 076 123 00073 00078 051 086
60
02525 00159 00022 007 020 00178 00055 009 023 00147 00037 002 01150 00175 00051 148 272 00197 00069 120 188 00163 00044 028 08875 00197 00069 238 251 00219 00077 374 510 00206 00074 258 410
0525 00147 00037 001 003 00147 00037 000 000 00144 00043 000 00050 00175 00060 039 057 00184 00061 027 035 00181 00066 023 04275 00209 00075 066 103 00231 00079 062 073 00222 00078 066 090
07525 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00050 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00075 00175 00051 021 027 00197 00069 022 028 00194 00067 021 025
Average 00169 00051 058 082 00182 00060 068 095 00171 00056 044 074
000
100
200
Aver
age o
f RPD
()
SA1SA2
SA3
025 050 075120572
Figure 3 Performance of the SA algorithms (119899 = 60)
algorithm incorporating several dominance properties anda lower bound was proposed for the optimal solutionThree simulated annealing algorithms were also proposedfor near-optimal solutions The computational results showthat with the help of the proposed heuristic initial solutionsthe branch-and-bound algorithm performs well in terms ofnumber of nodes and execution time when the number ofjobs is fewer than or equal to 16Moreover the computationalexperiments also show that the proposed combined SA
4
performs well since its mean error percentage was less than212 for all the tested cases Further research lies in devisingefficient and effective methods to solve the problem withsignificantly larger numbers of jobs
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Discrete Dynamics in Nature and Society
of multiagent into scheduling problems Yuan et al [6]discussed two dynamic programming recursions in Bakerand Smith [5] and proposed a polynomial-time algorithm forthe same problem Cheng et al [7] addressed the feasibilitymodel of multiagent scheduling on a single machine whereeach agentrsquos objective function is to minimize the totalweighted number of tardy jobs Ng et al [8] proposed a two-agent scheduling problem on a single machine where theobjective is to minimize the total completion time of thefirst agent with the restriction that the number of tardy jobsof the second agent cannot exceed a given number Agnetiset al [3] discussed the complexity some single-machinescheduling problems with several agents Cheng et al [9]studied multiagent scheduling on a single machine wherethe objective functions of the agents are of the max-formAgnetis et al [10] used a branch-and-bound method to solveseveral two-agent scheduling problems on a single machineLee et al [11] considered a multiagent scheduling problem ona single machine in which each agent is responsible for hisown set of jobs and wishes to minimize the total weightedcompletion time of his own set of jobs Recently Leung etal [12] generalized the single machine problems proposed byAgnetis et al [4] to the environment with multiple identicalmachines in parallel Yin et al [13] considered several two-agent scheduling problems with assignable due dates on asingle machine where the goal was to assign a due date froma given set of due dates and a position in the sequence toeach job so that the weighted sum of the objectives of bothagents is minimized For different combinations of the objec-tives which include the maximum lateness total (weighted)tardiness and total (weighted) number of tardy jobs theyprovided the complexity results and solve the correspondingproblems if possible Yin et al [14] investigated a schedulingenvironment with two agents and a linear nonincreasingdeterioration with the objective of scheduling the jobs suchthat the combined schedule performs well with respect to themeasures of both agents Three different objective functionswere considered for one agent including the maximumearliness cost total earliness cost and total weighted earlinesscost while keeping the maximum earliness cost of the otheragent below or at a fixed level 119880 They proposed the opti-mal (nondominated) properties and present the complexityresults for the problems Yin et al [15] addressed a two-agentscheduling problem on a single machine where the objectiveis to minimize the total weighted earliness cost of all jobswhile keeping the earliness cost of one agent below or at afixed level 119876 A mixed-integer programming (MIP) modelwas first formulated to find the optimal solution which isuseful for small-size problem instances and then a branch-and-bound algorithm incorporating with several dominanceproperties a lower bound and a simulated annealing heuristicalgorithm were provided to derive the optimal solution formedium- to large-size problem instances For more recentworks with two-agent issues readers can refer to Nong et al[16] Wan et al [17] Luo et al [18] Yin et al [19] and soforth In addition for more multiple-agent works with time-dependent readers can refer to Liu and Tang [20] Liu et al[21] Cheng et al [22 23] Wu et al [24] Li and Hsu [25] Yinet al [26ndash28] Wu [29] and Li et al [30] and so forth
For the importance of multiple agents competing on theusage of a common processing resource in different appli-cation environments and different methodological fields weconsidered two-agent scheduling on a single machine Theobjective is to minimize the total completion time of the jobsof the first agent with the restriction that an upper bound isallowed the total completion time of the jobs for the secondagent which has been shown binary NP-hard by Agnetis etal [4] The problem is described as follows There are 119899 jobswhich belong to one of the agents AG
0or AG
1 For each job
119895 there is a normal processing time 119901119895and an agent code 119868
119895
where 119868119895= 0 if 119869
119895isin AG
0or 119868119895= 1 if 119869
119895isin AG
1 All the jobs
are available at time zero Under a schedule 119878 let 119862119895(119878) be the
completion time of job 119895 The objective of this paper is to findan optimal schedule to minimizesum119899
119895=1119862119895(119878)(1minus119868
119895) subject to
sum119899
119895=1119862119895(119878)119868119895lt 119880 where 119880 is a control upper bound
The remainder of this paper is organized as followsIn Section 2 we present some dominance properties anddevelop a lower bound to speed up the search for the optimalsolution followed by discussions of a branch-and-bound andthree simulated annealing (SA) algorithms We present theresults of extensive computational experiments to assess theperformance of all of the proposed algorithms under differentexperimental conditions in Section 3 We conclude the paperand suggest topics for further research in the last section
2 Dominance Properties
We will develop some adjacent dominance rules by using thepairwise interchange method Assume that 119878
1and 1198782denote
two given job schedules in which the difference between 1198781
and 1198782is a pairwise interchange of two adjacent jobs 119894 and 119895
That is 1198781= (120590 119894 119895 120590
1015840) and 119878
2= (120590 119895 119894 120590
1015840) where 120590 and
1205901015840 each denote a partial sequence In addition let 119905 be the
starting time of the last job in 120590
Property 1 If jobs 119894 119895 isin AG0 and 119901
119894lt 119901119895 then 119878
1dominates
1198782
Property 2 If jobs 119894 119895 isin AG1sum119894isin120590capAG
1
119862119894+2119905+2119901
119894+119901119895lt 119880
and 119901119894lt 119901119895 then 119878
1dominates 119878
2
Next we give a proposition to determine the feasibility ofthe partial schedule Let (120587 120587119888) be a sequence of jobs where120587 is the scheduled part with 119896 jobs and 120590119888 is the unscheduledpart with (119899minus119896) jobs Among the unscheduled jobs let 119901
(1)=
min119869119895isin120587119888capAG1
119901119895 Moreover let 119862
[119896]be the completion times
of the last job in120587 Also let1205871015840 and12058710158401015840 denote the unscheduledjobs in AG
0and AG
1arranged in the smallest processing
times (SPT) order respectively
Property 3 If there is a job 119894 isin AG1such that sum
119894isin120587capAG1
119862119894+
119862[119896]+ 119901(1)gt 119880 then (120587 120587119888) is not a feasible sequence
Property 4 If all unscheduled jobs are belonging toAG0 then
(120587 1205871015840) dominates (120587 120587119888)
Property 5 If all unscheduled jobs are belonging toAG1 then
(120587 120587119888) can be determined by (120587 12058710158401015840) That is (120587 12058710158401015840) either isfeasible solution or can be deleted
Discrete Dynamics in Nature and Society 3
21 Lower Bound Assume that PS is a partial schedule inwhich the order of the first 119896 jobs is determined and letUS be the unscheduled part with (119899 minus 119896) jobs Amongthe unscheduled jobs there are 119899
0jobs from agent AG
0
and 1198991jobs from agent AG
1 Moreover let 119862
[119896]denote the
completion times of the 119896th job in PS The completion timefor the (119896 + 119895)th job is
119862[119896+119895]
ge 119862[119896]+
119895
sum
119894=1
119901(119896+119894)
for 1 le 119895 le 1198990 (1)
Then a lower bound can be obtained as follows119899
sum
119895=1
119862119895(119878) (1 minus 119868
119895) ge
119896
sum
119895=1
119862119895(119878) (1 minus 119868
119895) +
1198990
sum
119895=1
119862(119895)(119878) (2)
where 119862(119895)= 119862[119896]+ sum119895
119894=1119901(119896+119894)
22 Simulated Annealing Algorithms Simulated annealinghas become one of the most popular metaheuristic methodsto solve combinatorial optimization problems since it wasproposed by Kirkpatrick et al [31] For example Kim etal [32] applied the method in scheduling of raw-materialunloading from ships at a steelworks Sun et al [33] used thetechnique for allocating product lots to customer orders insemiconductor manufacturing supply chains Moreover themethod has the advantage of avoiding getting trapped in alocal optimum because of its hill climbing moves which aregoverned by a control parameter Thus we will use SA toderive near-optimal solutions for our problem The steps ofSA algorithms are summarized as follows
221 Initial Sequence Three initial sequences for the threeSA algorithms are adopted for our problem For the firstinitial sequence in SA
1 the sequence of jobs of AG
1is
arranged according to the smallest processing times (SPT)order followed by arranging the sequence of jobs of AG
0in
the shortest processing time (SPT) order In order to get agood initial one two more initial sequences are consideredFor the second initial sequence in SA
2 the pairwise inter-
change is used to the initial sequence produce by SA1 For the
third initial sequence in SA3 the NEHmethod [34] is applied
to the initial sequence produce by SA1
222 Neighborhood Generation The pairwise interchange(PI) neighborhood generation method is adopted in thealgorithms
223 Acceptance Probability When a new feasible sequenceis generated it is accepted if its objective value is smallerthan that of the original sequence otherwise it is acceptedwith a probability that decreases as the process evolves Theprobability of acceptance is generated from an exponentialdistribution
119875 (accept) = exp (minus120572 times ΔTC) (3)
where 120572 is a control parameter and ΔTC is the change in theobjective value In addition we use the method suggested by
Ben-Arieh and Maimon [35] to change 120572 in the 119896th iterationas follows
120572 =119896
120575 (4)
where 120575 is an experimental constant After preliminary trials120575 = 6 is used in our experiments
If the total completion time of jobs from agent AG0
increases as a result of a random pairwise interchange thenew sequence is accepted when 119875(accept) gt 119903 where 119903 israndomly sampled from the uniform distribution 119880(0 1)
224 Stopping Rule All three proposed SAs are stopped after100119899 iterations where 119899 is the number of jobs
3 Computational Experiments
The extensive computational experiments were conducted totest the performances of the branch-and-bound algorithmand the three simulated annealing algorithms All the algo-rithms were coded in Fortran using Compaq Visual Fortranversion 66 and performed the experiments on a personalcomputer powered by an Intel(R) Core(TM)2 Quad CPU266GHz with 4GB RAM operating under Windows XPThe job processing times were generated from a uniformdistribution over the integers 1ndash100 For the control upperbound 119880 we arrange the jobs of AG
1by the smallest
processing times (SPT) order and compute out the totalcompletion times of the jobs of AG
1 (recorded as TC
1)Then
following the sequence of the jobs of AG1 we arrange the
sequence of jobs of AG0in the shortest processing time (SPT)
order and compute out the total completion times of jobs ofAG1and AG
0 (recoded as TC
1+2) We let 119880 = (1 minus 120572)TC
1+
120572TC1+2
where 0 lt 120572 lt 1 Moreover120572was taken as the valuesof 025 05 and 075 while the proportion of the jobs of agentAG1at pro = 025 05 and 075 in the testsFor the branch-and-bound algorithm the average and
standard deviation of the numbers of nodes and the executiontimes (in seconds) were recorded For the SA heuristicsthe mean and standard deviation percentage errors wererecorded The percentage error of a solution obtained froma heuristic algorithm was given by
119881 minus 119881lowast
119881lowast (5)
where 119881 and 119881lowast are denoted as the total completion time
of the heuristic and the optimal solution respectively Thecomputational times of the heuristic algorithms were notrecorded because they all were fast in generating solutions
The computational experiments were divided into twoparts In the first part of the experiments three numberof jobs were tested at 119899 = 8 12 and 16 As a result27 experimental situations were examined 50 instanceswere randomly generated for each case The results weresummarized in Table 1 that includes the CPU time (mean andstandard deviation) and the number of nodes for the branch-and-bound algorithm
4 Discrete Dynamics in Nature and Society
Table1Perfo
rmance
oftheb
ranch-and-bo
undandheuristicalgorithm
s(119899=8sim
16)
119899Pro
120572
Branch-and
-bou
ndalgorithm
SA1
SA2
SA3
SA4
Valid
samples
ize
CPUtim
eNum
bero
fnod
esErrorp
ercentages
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
8
025
025
50000
001
526
318
028
146
028
146
028
146
028
146
050
50000
001
748
588
129
412
093
311
086
309
024
169
075
50001
001
1383
923
342
729
283
573
227
480
062
246
050
025
50001
001
2229
1371
001
006
001
006
000
000
000
000
050
50001
001
2500
1412
050
178
014
056
020
086
004
020
075
50001
001
3183
1139
092
203
127
226
148
276
026
102
075
025
50001
001
1091
217
000
000
000
000
000
000
000
000
050
50001
001
1091
217
000
000
000
000
000
000
000
000
075
50001
001
1089
196
036
118
011
048
049
133
003
024
Average
001
001
1538
709
075
199
062
152
062
159
016
079
12
025
025
50078
086
97087
113419
029
093
061
230
030
138
005
020
050
5013
619
6175858
261636
186
441
399
781
145
386
077
195
075
50392
524
535992
751273
478
725
440
682
530
843
212
426
050
025
501983
2275
2278911
2794764
002
007
004
020
001
005
000
000
050
502791
3386
3405293
4448908
064
124
053
116
032
073
017
045
075
503538
2151
4692766
3096131
207
216
237
243
237
274
069
092
075
025
50630
218
7940
033044
61000
000
000
000
000
000
000
000
050
50630
218
7944
39304555
000
000
000
000
000
000
000
000
075
50660
182
862132
262277
044
066
074
122
055
119
015
033
Average
1204
1026
1515165
1370825
112
186
141
244
114
204
044
090
16
025
025
4884682
169136
62738963
129480137
036
092
039
106
035
116
018
077
050
46143263
241156
105554536
181805362
199
346
249
428
188
438
095
229
075
331446
76262913
106507557
197947458
272
314
779
1200
430
664
183
228
050
025
228044
31605859
429572906
333869433
000
000
000
000
000
000
000
000
050
20726799
569796
387550386
314387045
010
029
000
000
011
051
000
000
075
3749301
577630
411927486
331822988
093
107
691
617
665
411
093
107
075
025
13950254
305315
625205281
214788228
000
000
000
000
000
000
000
000
050
13947874
3044
52625205281
214788228
000
000
000
000
000
000
000
000
075
7996598
365310
652692683
258677186
041
073
024
063
103
106
000
000
Average
6164
31377952
378550564
241951785
072
107
198
268
159
198
043
071
Discrete Dynamics in Nature and Society 5
0
2
4
6
025 050 075
Aver
age n
umbe
r of n
odes
pro = 025
pro = 050
pro = 075
120572
times10
6
Figure 1 Performance of the branch-and-bound algorithms (119899 = 12)
For the performance of the branch-and-bound algorithmit can be observed from Table 1 that the number of nodesand the mean of the CPU time increase when 119899 becomesbigger The difficult situations occur at 119899 = 16 Especiallythe instances with a bigger value of 120572 (120572 = 075) are difficultto solve than those with a smaller one (120572 = 025 050)Moreover the instances with a bigger value of pro (pro = 05075) are difficult to solve than those with a smaller one (pro= 025) The most difficult case is located at pro = 05 075and 120572 = 075 as the number of instances which can be solvedout within less 10minus9 nodes were declined to 10 or below Inaddition as shown in Figure 1 and Table 1 the instances withpro = 05 took more nodes or CPU time than others and thetrend became clear as the value of 120572 got bigger
For the performance of the proposed SAs it can beseen in Table 1 that the means of the percentage error ofSA1(between 00 and 478) are lower than those of SA
2
(between 00 and 779) and those of SA3(between 00
and 665) Moreover the trend of the standard deviationsof the percentage error keeps the similar pattern As shownin Table 1 the standard deviations of the percentage errorof SA
1 SA2 and SA
3were between 00 and 729 00
and 120 and 00 and 843 respectively It released thatthe performance of SA
1was slightly better than the other
two Furthermore Figure 2 and Table 1 indicated that themeans of the percentage error of SA
1 SA2 and SA
3are
slightly affected by the parameter 120572 For example the meansof the percentage errors of SA
1 SA2 and SA
3were lager
than 2 at 120572 = 025 Specifically the behavior became clearat pro = 025 and 05 It also can be observed that there isno absolutely dominance relationship among three proposedSAs Due to get a good quality solution we further combinedthree proposed SAs into one (recorded as SA
4and SA
4=
minSA119894 119894 = 1 2 3) The means of the percentage error of
SA4were located between 00 and 212 Meanwhile its
standard deviations of the percentage error were between
000
100
200
300
Mea
n er
ror (
)
SA1SA2
SA3SA4
025 050 075120572
Figure 2 Performance of the SA algorithms (119899 = 12)
00 and 426 Moreover the impacts of pro and 120572 werenot present for proposed three SAs
In the second part of the experiments the performancesof the proposed SA heuristics were further tested for largenumbers of jobs Three different numbers of jobs wereconsidered at 119899 = 20 40 and 60 The proportion of thejobs of agent AG
1at pro = 025 05 and 075 in the tests
Moreover 120572 was taken as the values of 025 05 and 075 Asa result 27 experimental situations were tested 50 instanceswere randomly generated for each situation The relativedeviance percentage with respect to the best known solutionwas reported for each instance The mean execution timeandmean relative deviance percentage were also recorded foreach SA heuristicThe relative deviation percentage RDPwasgiven by
SA119894minus SAlowast
SAlowast (6)
where SA119894is the value of the objective function generated by
SA119894and SAlowast = minSA
119894 119894 = 1 2 3 is the smallest value of
the objective function obtained from the three SA heuristicsThe results were summarized in Table 2
As shown in Figure 3 and Table 2 it can be seen that theRPDmeans of SA
1 SA2 and SA
3are getting slightly bigger as
the value of 120572 increases In general the RPDmeans of SA3are
lower than those of SA1and SA
2 Furthermore all of the RDP
means of SA1 SA2 and SA
3were less than 2 Figure 2 also
indicated that there is no absolutely dominance relationshipamong three proposed SAs
4 Conclusions
Thispaper explored the single-machine two-agent schedulingproblem where the objective is to minimize the total com-pletion time of the jobs belonging to the first agent withthe restriction that total completion time of jobs from thesecond agent has an upper bound Due to the fact that theproblem under study is binary NP hard a branch-and-bound
6 Discrete Dynamics in Nature and Society
Table 2 RPD of heuristic algorithms (119899 = 20sim60)
119899 Pro 120572
SA1 SA2 SA3CPU time RPD CPU time RPD CPU time RPD
Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std
20
02525 00019 00051 045 144 00019 00051 041 119 00016 00047 051 17550 00019 00051 139 267 00019 00051 187 551 00022 00055 089 18875 00019 00051 286 481 00019 00051 496 680 00019 00051 201 366
0525 00019 00051 002 007 00019 00051 000 001 00016 00047 001 00650 00019 00051 051 081 00019 00051 039 067 00022 00055 051 07875 00019 00051 105 152 00022 00055 119 208 00022 00055 130 203
07525 00019 00051 000 000 00013 00043 000 000 00019 00051 000 00050 00019 00051 000 000 00016 00047 000 000 00019 00051 000 00075 00019 00051 050 058 00019 00051 058 075 00019 00051 034 055
Average 00019 00051 075 132 00018 00050 104 189 00019 00052 062 119
40
02525 00066 00078 010 030 00072 00079 016 052 00066 00078 000 00050 00072 00079 092 151 00078 00079 122 209 00072 00079 039 14775 00078 00079 366 443 00084 00079 391 584 00081 00079 282 404
0525 00066 00078 002 006 00066 00078 000 000 00066 00078 000 00050 00075 00079 039 043 00078 00079 031 057 00078 00079 046 07975 00084 00079 081 081 00091 00078 103 169 00091 00078 071 112
07525 00066 00078 000 000 00063 00077 000 000 00063 00077 000 00050 00063 00077 000 000 00063 00077 000 000 00063 00077 000 00075 00072 00079 032 045 00078 00079 025 036 00081 00079 025 031
Average 00071 00078 069 089 00075 00078 076 123 00073 00078 051 086
60
02525 00159 00022 007 020 00178 00055 009 023 00147 00037 002 01150 00175 00051 148 272 00197 00069 120 188 00163 00044 028 08875 00197 00069 238 251 00219 00077 374 510 00206 00074 258 410
0525 00147 00037 001 003 00147 00037 000 000 00144 00043 000 00050 00175 00060 039 057 00184 00061 027 035 00181 00066 023 04275 00209 00075 066 103 00231 00079 062 073 00222 00078 066 090
07525 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00050 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00075 00175 00051 021 027 00197 00069 022 028 00194 00067 021 025
Average 00169 00051 058 082 00182 00060 068 095 00171 00056 044 074
000
100
200
Aver
age o
f RPD
()
SA1SA2
SA3
025 050 075120572
Figure 3 Performance of the SA algorithms (119899 = 60)
algorithm incorporating several dominance properties anda lower bound was proposed for the optimal solutionThree simulated annealing algorithms were also proposedfor near-optimal solutions The computational results showthat with the help of the proposed heuristic initial solutionsthe branch-and-bound algorithm performs well in terms ofnumber of nodes and execution time when the number ofjobs is fewer than or equal to 16Moreover the computationalexperiments also show that the proposed combined SA
4
performs well since its mean error percentage was less than212 for all the tested cases Further research lies in devisingefficient and effective methods to solve the problem withsignificantly larger numbers of jobs
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
21 Lower Bound Assume that PS is a partial schedule inwhich the order of the first 119896 jobs is determined and letUS be the unscheduled part with (119899 minus 119896) jobs Amongthe unscheduled jobs there are 119899
0jobs from agent AG
0
and 1198991jobs from agent AG
1 Moreover let 119862
[119896]denote the
completion times of the 119896th job in PS The completion timefor the (119896 + 119895)th job is
119862[119896+119895]
ge 119862[119896]+
119895
sum
119894=1
119901(119896+119894)
for 1 le 119895 le 1198990 (1)
Then a lower bound can be obtained as follows119899
sum
119895=1
119862119895(119878) (1 minus 119868
119895) ge
119896
sum
119895=1
119862119895(119878) (1 minus 119868
119895) +
1198990
sum
119895=1
119862(119895)(119878) (2)
where 119862(119895)= 119862[119896]+ sum119895
119894=1119901(119896+119894)
22 Simulated Annealing Algorithms Simulated annealinghas become one of the most popular metaheuristic methodsto solve combinatorial optimization problems since it wasproposed by Kirkpatrick et al [31] For example Kim etal [32] applied the method in scheduling of raw-materialunloading from ships at a steelworks Sun et al [33] used thetechnique for allocating product lots to customer orders insemiconductor manufacturing supply chains Moreover themethod has the advantage of avoiding getting trapped in alocal optimum because of its hill climbing moves which aregoverned by a control parameter Thus we will use SA toderive near-optimal solutions for our problem The steps ofSA algorithms are summarized as follows
221 Initial Sequence Three initial sequences for the threeSA algorithms are adopted for our problem For the firstinitial sequence in SA
1 the sequence of jobs of AG
1is
arranged according to the smallest processing times (SPT)order followed by arranging the sequence of jobs of AG
0in
the shortest processing time (SPT) order In order to get agood initial one two more initial sequences are consideredFor the second initial sequence in SA
2 the pairwise inter-
change is used to the initial sequence produce by SA1 For the
third initial sequence in SA3 the NEHmethod [34] is applied
to the initial sequence produce by SA1
222 Neighborhood Generation The pairwise interchange(PI) neighborhood generation method is adopted in thealgorithms
223 Acceptance Probability When a new feasible sequenceis generated it is accepted if its objective value is smallerthan that of the original sequence otherwise it is acceptedwith a probability that decreases as the process evolves Theprobability of acceptance is generated from an exponentialdistribution
119875 (accept) = exp (minus120572 times ΔTC) (3)
where 120572 is a control parameter and ΔTC is the change in theobjective value In addition we use the method suggested by
Ben-Arieh and Maimon [35] to change 120572 in the 119896th iterationas follows
120572 =119896
120575 (4)
where 120575 is an experimental constant After preliminary trials120575 = 6 is used in our experiments
If the total completion time of jobs from agent AG0
increases as a result of a random pairwise interchange thenew sequence is accepted when 119875(accept) gt 119903 where 119903 israndomly sampled from the uniform distribution 119880(0 1)
224 Stopping Rule All three proposed SAs are stopped after100119899 iterations where 119899 is the number of jobs
3 Computational Experiments
The extensive computational experiments were conducted totest the performances of the branch-and-bound algorithmand the three simulated annealing algorithms All the algo-rithms were coded in Fortran using Compaq Visual Fortranversion 66 and performed the experiments on a personalcomputer powered by an Intel(R) Core(TM)2 Quad CPU266GHz with 4GB RAM operating under Windows XPThe job processing times were generated from a uniformdistribution over the integers 1ndash100 For the control upperbound 119880 we arrange the jobs of AG
1by the smallest
processing times (SPT) order and compute out the totalcompletion times of the jobs of AG
1 (recorded as TC
1)Then
following the sequence of the jobs of AG1 we arrange the
sequence of jobs of AG0in the shortest processing time (SPT)
order and compute out the total completion times of jobs ofAG1and AG
0 (recoded as TC
1+2) We let 119880 = (1 minus 120572)TC
1+
120572TC1+2
where 0 lt 120572 lt 1 Moreover120572was taken as the valuesof 025 05 and 075 while the proportion of the jobs of agentAG1at pro = 025 05 and 075 in the testsFor the branch-and-bound algorithm the average and
standard deviation of the numbers of nodes and the executiontimes (in seconds) were recorded For the SA heuristicsthe mean and standard deviation percentage errors wererecorded The percentage error of a solution obtained froma heuristic algorithm was given by
119881 minus 119881lowast
119881lowast (5)
where 119881 and 119881lowast are denoted as the total completion time
of the heuristic and the optimal solution respectively Thecomputational times of the heuristic algorithms were notrecorded because they all were fast in generating solutions
The computational experiments were divided into twoparts In the first part of the experiments three numberof jobs were tested at 119899 = 8 12 and 16 As a result27 experimental situations were examined 50 instanceswere randomly generated for each case The results weresummarized in Table 1 that includes the CPU time (mean andstandard deviation) and the number of nodes for the branch-and-bound algorithm
4 Discrete Dynamics in Nature and Society
Table1Perfo
rmance
oftheb
ranch-and-bo
undandheuristicalgorithm
s(119899=8sim
16)
119899Pro
120572
Branch-and
-bou
ndalgorithm
SA1
SA2
SA3
SA4
Valid
samples
ize
CPUtim
eNum
bero
fnod
esErrorp
ercentages
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
8
025
025
50000
001
526
318
028
146
028
146
028
146
028
146
050
50000
001
748
588
129
412
093
311
086
309
024
169
075
50001
001
1383
923
342
729
283
573
227
480
062
246
050
025
50001
001
2229
1371
001
006
001
006
000
000
000
000
050
50001
001
2500
1412
050
178
014
056
020
086
004
020
075
50001
001
3183
1139
092
203
127
226
148
276
026
102
075
025
50001
001
1091
217
000
000
000
000
000
000
000
000
050
50001
001
1091
217
000
000
000
000
000
000
000
000
075
50001
001
1089
196
036
118
011
048
049
133
003
024
Average
001
001
1538
709
075
199
062
152
062
159
016
079
12
025
025
50078
086
97087
113419
029
093
061
230
030
138
005
020
050
5013
619
6175858
261636
186
441
399
781
145
386
077
195
075
50392
524
535992
751273
478
725
440
682
530
843
212
426
050
025
501983
2275
2278911
2794764
002
007
004
020
001
005
000
000
050
502791
3386
3405293
4448908
064
124
053
116
032
073
017
045
075
503538
2151
4692766
3096131
207
216
237
243
237
274
069
092
075
025
50630
218
7940
033044
61000
000
000
000
000
000
000
000
050
50630
218
7944
39304555
000
000
000
000
000
000
000
000
075
50660
182
862132
262277
044
066
074
122
055
119
015
033
Average
1204
1026
1515165
1370825
112
186
141
244
114
204
044
090
16
025
025
4884682
169136
62738963
129480137
036
092
039
106
035
116
018
077
050
46143263
241156
105554536
181805362
199
346
249
428
188
438
095
229
075
331446
76262913
106507557
197947458
272
314
779
1200
430
664
183
228
050
025
228044
31605859
429572906
333869433
000
000
000
000
000
000
000
000
050
20726799
569796
387550386
314387045
010
029
000
000
011
051
000
000
075
3749301
577630
411927486
331822988
093
107
691
617
665
411
093
107
075
025
13950254
305315
625205281
214788228
000
000
000
000
000
000
000
000
050
13947874
3044
52625205281
214788228
000
000
000
000
000
000
000
000
075
7996598
365310
652692683
258677186
041
073
024
063
103
106
000
000
Average
6164
31377952
378550564
241951785
072
107
198
268
159
198
043
071
Discrete Dynamics in Nature and Society 5
0
2
4
6
025 050 075
Aver
age n
umbe
r of n
odes
pro = 025
pro = 050
pro = 075
120572
times10
6
Figure 1 Performance of the branch-and-bound algorithms (119899 = 12)
For the performance of the branch-and-bound algorithmit can be observed from Table 1 that the number of nodesand the mean of the CPU time increase when 119899 becomesbigger The difficult situations occur at 119899 = 16 Especiallythe instances with a bigger value of 120572 (120572 = 075) are difficultto solve than those with a smaller one (120572 = 025 050)Moreover the instances with a bigger value of pro (pro = 05075) are difficult to solve than those with a smaller one (pro= 025) The most difficult case is located at pro = 05 075and 120572 = 075 as the number of instances which can be solvedout within less 10minus9 nodes were declined to 10 or below Inaddition as shown in Figure 1 and Table 1 the instances withpro = 05 took more nodes or CPU time than others and thetrend became clear as the value of 120572 got bigger
For the performance of the proposed SAs it can beseen in Table 1 that the means of the percentage error ofSA1(between 00 and 478) are lower than those of SA
2
(between 00 and 779) and those of SA3(between 00
and 665) Moreover the trend of the standard deviationsof the percentage error keeps the similar pattern As shownin Table 1 the standard deviations of the percentage errorof SA
1 SA2 and SA
3were between 00 and 729 00
and 120 and 00 and 843 respectively It released thatthe performance of SA
1was slightly better than the other
two Furthermore Figure 2 and Table 1 indicated that themeans of the percentage error of SA
1 SA2 and SA
3are
slightly affected by the parameter 120572 For example the meansof the percentage errors of SA
1 SA2 and SA
3were lager
than 2 at 120572 = 025 Specifically the behavior became clearat pro = 025 and 05 It also can be observed that there isno absolutely dominance relationship among three proposedSAs Due to get a good quality solution we further combinedthree proposed SAs into one (recorded as SA
4and SA
4=
minSA119894 119894 = 1 2 3) The means of the percentage error of
SA4were located between 00 and 212 Meanwhile its
standard deviations of the percentage error were between
000
100
200
300
Mea
n er
ror (
)
SA1SA2
SA3SA4
025 050 075120572
Figure 2 Performance of the SA algorithms (119899 = 12)
00 and 426 Moreover the impacts of pro and 120572 werenot present for proposed three SAs
In the second part of the experiments the performancesof the proposed SA heuristics were further tested for largenumbers of jobs Three different numbers of jobs wereconsidered at 119899 = 20 40 and 60 The proportion of thejobs of agent AG
1at pro = 025 05 and 075 in the tests
Moreover 120572 was taken as the values of 025 05 and 075 Asa result 27 experimental situations were tested 50 instanceswere randomly generated for each situation The relativedeviance percentage with respect to the best known solutionwas reported for each instance The mean execution timeandmean relative deviance percentage were also recorded foreach SA heuristicThe relative deviation percentage RDPwasgiven by
SA119894minus SAlowast
SAlowast (6)
where SA119894is the value of the objective function generated by
SA119894and SAlowast = minSA
119894 119894 = 1 2 3 is the smallest value of
the objective function obtained from the three SA heuristicsThe results were summarized in Table 2
As shown in Figure 3 and Table 2 it can be seen that theRPDmeans of SA
1 SA2 and SA
3are getting slightly bigger as
the value of 120572 increases In general the RPDmeans of SA3are
lower than those of SA1and SA
2 Furthermore all of the RDP
means of SA1 SA2 and SA
3were less than 2 Figure 2 also
indicated that there is no absolutely dominance relationshipamong three proposed SAs
4 Conclusions
Thispaper explored the single-machine two-agent schedulingproblem where the objective is to minimize the total com-pletion time of the jobs belonging to the first agent withthe restriction that total completion time of jobs from thesecond agent has an upper bound Due to the fact that theproblem under study is binary NP hard a branch-and-bound
6 Discrete Dynamics in Nature and Society
Table 2 RPD of heuristic algorithms (119899 = 20sim60)
119899 Pro 120572
SA1 SA2 SA3CPU time RPD CPU time RPD CPU time RPD
Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std
20
02525 00019 00051 045 144 00019 00051 041 119 00016 00047 051 17550 00019 00051 139 267 00019 00051 187 551 00022 00055 089 18875 00019 00051 286 481 00019 00051 496 680 00019 00051 201 366
0525 00019 00051 002 007 00019 00051 000 001 00016 00047 001 00650 00019 00051 051 081 00019 00051 039 067 00022 00055 051 07875 00019 00051 105 152 00022 00055 119 208 00022 00055 130 203
07525 00019 00051 000 000 00013 00043 000 000 00019 00051 000 00050 00019 00051 000 000 00016 00047 000 000 00019 00051 000 00075 00019 00051 050 058 00019 00051 058 075 00019 00051 034 055
Average 00019 00051 075 132 00018 00050 104 189 00019 00052 062 119
40
02525 00066 00078 010 030 00072 00079 016 052 00066 00078 000 00050 00072 00079 092 151 00078 00079 122 209 00072 00079 039 14775 00078 00079 366 443 00084 00079 391 584 00081 00079 282 404
0525 00066 00078 002 006 00066 00078 000 000 00066 00078 000 00050 00075 00079 039 043 00078 00079 031 057 00078 00079 046 07975 00084 00079 081 081 00091 00078 103 169 00091 00078 071 112
07525 00066 00078 000 000 00063 00077 000 000 00063 00077 000 00050 00063 00077 000 000 00063 00077 000 000 00063 00077 000 00075 00072 00079 032 045 00078 00079 025 036 00081 00079 025 031
Average 00071 00078 069 089 00075 00078 076 123 00073 00078 051 086
60
02525 00159 00022 007 020 00178 00055 009 023 00147 00037 002 01150 00175 00051 148 272 00197 00069 120 188 00163 00044 028 08875 00197 00069 238 251 00219 00077 374 510 00206 00074 258 410
0525 00147 00037 001 003 00147 00037 000 000 00144 00043 000 00050 00175 00060 039 057 00184 00061 027 035 00181 00066 023 04275 00209 00075 066 103 00231 00079 062 073 00222 00078 066 090
07525 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00050 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00075 00175 00051 021 027 00197 00069 022 028 00194 00067 021 025
Average 00169 00051 058 082 00182 00060 068 095 00171 00056 044 074
000
100
200
Aver
age o
f RPD
()
SA1SA2
SA3
025 050 075120572
Figure 3 Performance of the SA algorithms (119899 = 60)
algorithm incorporating several dominance properties anda lower bound was proposed for the optimal solutionThree simulated annealing algorithms were also proposedfor near-optimal solutions The computational results showthat with the help of the proposed heuristic initial solutionsthe branch-and-bound algorithm performs well in terms ofnumber of nodes and execution time when the number ofjobs is fewer than or equal to 16Moreover the computationalexperiments also show that the proposed combined SA
4
performs well since its mean error percentage was less than212 for all the tested cases Further research lies in devisingefficient and effective methods to solve the problem withsignificantly larger numbers of jobs
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Discrete Dynamics in Nature and Society
Table1Perfo
rmance
oftheb
ranch-and-bo
undandheuristicalgorithm
s(119899=8sim
16)
119899Pro
120572
Branch-and
-bou
ndalgorithm
SA1
SA2
SA3
SA4
Valid
samples
ize
CPUtim
eNum
bero
fnod
esErrorp
ercentages
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
8
025
025
50000
001
526
318
028
146
028
146
028
146
028
146
050
50000
001
748
588
129
412
093
311
086
309
024
169
075
50001
001
1383
923
342
729
283
573
227
480
062
246
050
025
50001
001
2229
1371
001
006
001
006
000
000
000
000
050
50001
001
2500
1412
050
178
014
056
020
086
004
020
075
50001
001
3183
1139
092
203
127
226
148
276
026
102
075
025
50001
001
1091
217
000
000
000
000
000
000
000
000
050
50001
001
1091
217
000
000
000
000
000
000
000
000
075
50001
001
1089
196
036
118
011
048
049
133
003
024
Average
001
001
1538
709
075
199
062
152
062
159
016
079
12
025
025
50078
086
97087
113419
029
093
061
230
030
138
005
020
050
5013
619
6175858
261636
186
441
399
781
145
386
077
195
075
50392
524
535992
751273
478
725
440
682
530
843
212
426
050
025
501983
2275
2278911
2794764
002
007
004
020
001
005
000
000
050
502791
3386
3405293
4448908
064
124
053
116
032
073
017
045
075
503538
2151
4692766
3096131
207
216
237
243
237
274
069
092
075
025
50630
218
7940
033044
61000
000
000
000
000
000
000
000
050
50630
218
7944
39304555
000
000
000
000
000
000
000
000
075
50660
182
862132
262277
044
066
074
122
055
119
015
033
Average
1204
1026
1515165
1370825
112
186
141
244
114
204
044
090
16
025
025
4884682
169136
62738963
129480137
036
092
039
106
035
116
018
077
050
46143263
241156
105554536
181805362
199
346
249
428
188
438
095
229
075
331446
76262913
106507557
197947458
272
314
779
1200
430
664
183
228
050
025
228044
31605859
429572906
333869433
000
000
000
000
000
000
000
000
050
20726799
569796
387550386
314387045
010
029
000
000
011
051
000
000
075
3749301
577630
411927486
331822988
093
107
691
617
665
411
093
107
075
025
13950254
305315
625205281
214788228
000
000
000
000
000
000
000
000
050
13947874
3044
52625205281
214788228
000
000
000
000
000
000
000
000
075
7996598
365310
652692683
258677186
041
073
024
063
103
106
000
000
Average
6164
31377952
378550564
241951785
072
107
198
268
159
198
043
071
Discrete Dynamics in Nature and Society 5
0
2
4
6
025 050 075
Aver
age n
umbe
r of n
odes
pro = 025
pro = 050
pro = 075
120572
times10
6
Figure 1 Performance of the branch-and-bound algorithms (119899 = 12)
For the performance of the branch-and-bound algorithmit can be observed from Table 1 that the number of nodesand the mean of the CPU time increase when 119899 becomesbigger The difficult situations occur at 119899 = 16 Especiallythe instances with a bigger value of 120572 (120572 = 075) are difficultto solve than those with a smaller one (120572 = 025 050)Moreover the instances with a bigger value of pro (pro = 05075) are difficult to solve than those with a smaller one (pro= 025) The most difficult case is located at pro = 05 075and 120572 = 075 as the number of instances which can be solvedout within less 10minus9 nodes were declined to 10 or below Inaddition as shown in Figure 1 and Table 1 the instances withpro = 05 took more nodes or CPU time than others and thetrend became clear as the value of 120572 got bigger
For the performance of the proposed SAs it can beseen in Table 1 that the means of the percentage error ofSA1(between 00 and 478) are lower than those of SA
2
(between 00 and 779) and those of SA3(between 00
and 665) Moreover the trend of the standard deviationsof the percentage error keeps the similar pattern As shownin Table 1 the standard deviations of the percentage errorof SA
1 SA2 and SA
3were between 00 and 729 00
and 120 and 00 and 843 respectively It released thatthe performance of SA
1was slightly better than the other
two Furthermore Figure 2 and Table 1 indicated that themeans of the percentage error of SA
1 SA2 and SA
3are
slightly affected by the parameter 120572 For example the meansof the percentage errors of SA
1 SA2 and SA
3were lager
than 2 at 120572 = 025 Specifically the behavior became clearat pro = 025 and 05 It also can be observed that there isno absolutely dominance relationship among three proposedSAs Due to get a good quality solution we further combinedthree proposed SAs into one (recorded as SA
4and SA
4=
minSA119894 119894 = 1 2 3) The means of the percentage error of
SA4were located between 00 and 212 Meanwhile its
standard deviations of the percentage error were between
000
100
200
300
Mea
n er
ror (
)
SA1SA2
SA3SA4
025 050 075120572
Figure 2 Performance of the SA algorithms (119899 = 12)
00 and 426 Moreover the impacts of pro and 120572 werenot present for proposed three SAs
In the second part of the experiments the performancesof the proposed SA heuristics were further tested for largenumbers of jobs Three different numbers of jobs wereconsidered at 119899 = 20 40 and 60 The proportion of thejobs of agent AG
1at pro = 025 05 and 075 in the tests
Moreover 120572 was taken as the values of 025 05 and 075 Asa result 27 experimental situations were tested 50 instanceswere randomly generated for each situation The relativedeviance percentage with respect to the best known solutionwas reported for each instance The mean execution timeandmean relative deviance percentage were also recorded foreach SA heuristicThe relative deviation percentage RDPwasgiven by
SA119894minus SAlowast
SAlowast (6)
where SA119894is the value of the objective function generated by
SA119894and SAlowast = minSA
119894 119894 = 1 2 3 is the smallest value of
the objective function obtained from the three SA heuristicsThe results were summarized in Table 2
As shown in Figure 3 and Table 2 it can be seen that theRPDmeans of SA
1 SA2 and SA
3are getting slightly bigger as
the value of 120572 increases In general the RPDmeans of SA3are
lower than those of SA1and SA
2 Furthermore all of the RDP
means of SA1 SA2 and SA
3were less than 2 Figure 2 also
indicated that there is no absolutely dominance relationshipamong three proposed SAs
4 Conclusions
Thispaper explored the single-machine two-agent schedulingproblem where the objective is to minimize the total com-pletion time of the jobs belonging to the first agent withthe restriction that total completion time of jobs from thesecond agent has an upper bound Due to the fact that theproblem under study is binary NP hard a branch-and-bound
6 Discrete Dynamics in Nature and Society
Table 2 RPD of heuristic algorithms (119899 = 20sim60)
119899 Pro 120572
SA1 SA2 SA3CPU time RPD CPU time RPD CPU time RPD
Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std
20
02525 00019 00051 045 144 00019 00051 041 119 00016 00047 051 17550 00019 00051 139 267 00019 00051 187 551 00022 00055 089 18875 00019 00051 286 481 00019 00051 496 680 00019 00051 201 366
0525 00019 00051 002 007 00019 00051 000 001 00016 00047 001 00650 00019 00051 051 081 00019 00051 039 067 00022 00055 051 07875 00019 00051 105 152 00022 00055 119 208 00022 00055 130 203
07525 00019 00051 000 000 00013 00043 000 000 00019 00051 000 00050 00019 00051 000 000 00016 00047 000 000 00019 00051 000 00075 00019 00051 050 058 00019 00051 058 075 00019 00051 034 055
Average 00019 00051 075 132 00018 00050 104 189 00019 00052 062 119
40
02525 00066 00078 010 030 00072 00079 016 052 00066 00078 000 00050 00072 00079 092 151 00078 00079 122 209 00072 00079 039 14775 00078 00079 366 443 00084 00079 391 584 00081 00079 282 404
0525 00066 00078 002 006 00066 00078 000 000 00066 00078 000 00050 00075 00079 039 043 00078 00079 031 057 00078 00079 046 07975 00084 00079 081 081 00091 00078 103 169 00091 00078 071 112
07525 00066 00078 000 000 00063 00077 000 000 00063 00077 000 00050 00063 00077 000 000 00063 00077 000 000 00063 00077 000 00075 00072 00079 032 045 00078 00079 025 036 00081 00079 025 031
Average 00071 00078 069 089 00075 00078 076 123 00073 00078 051 086
60
02525 00159 00022 007 020 00178 00055 009 023 00147 00037 002 01150 00175 00051 148 272 00197 00069 120 188 00163 00044 028 08875 00197 00069 238 251 00219 00077 374 510 00206 00074 258 410
0525 00147 00037 001 003 00147 00037 000 000 00144 00043 000 00050 00175 00060 039 057 00184 00061 027 035 00181 00066 023 04275 00209 00075 066 103 00231 00079 062 073 00222 00078 066 090
07525 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00050 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00075 00175 00051 021 027 00197 00069 022 028 00194 00067 021 025
Average 00169 00051 058 082 00182 00060 068 095 00171 00056 044 074
000
100
200
Aver
age o
f RPD
()
SA1SA2
SA3
025 050 075120572
Figure 3 Performance of the SA algorithms (119899 = 60)
algorithm incorporating several dominance properties anda lower bound was proposed for the optimal solutionThree simulated annealing algorithms were also proposedfor near-optimal solutions The computational results showthat with the help of the proposed heuristic initial solutionsthe branch-and-bound algorithm performs well in terms ofnumber of nodes and execution time when the number ofjobs is fewer than or equal to 16Moreover the computationalexperiments also show that the proposed combined SA
4
performs well since its mean error percentage was less than212 for all the tested cases Further research lies in devisingefficient and effective methods to solve the problem withsignificantly larger numbers of jobs
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
0
2
4
6
025 050 075
Aver
age n
umbe
r of n
odes
pro = 025
pro = 050
pro = 075
120572
times10
6
Figure 1 Performance of the branch-and-bound algorithms (119899 = 12)
For the performance of the branch-and-bound algorithmit can be observed from Table 1 that the number of nodesand the mean of the CPU time increase when 119899 becomesbigger The difficult situations occur at 119899 = 16 Especiallythe instances with a bigger value of 120572 (120572 = 075) are difficultto solve than those with a smaller one (120572 = 025 050)Moreover the instances with a bigger value of pro (pro = 05075) are difficult to solve than those with a smaller one (pro= 025) The most difficult case is located at pro = 05 075and 120572 = 075 as the number of instances which can be solvedout within less 10minus9 nodes were declined to 10 or below Inaddition as shown in Figure 1 and Table 1 the instances withpro = 05 took more nodes or CPU time than others and thetrend became clear as the value of 120572 got bigger
For the performance of the proposed SAs it can beseen in Table 1 that the means of the percentage error ofSA1(between 00 and 478) are lower than those of SA
2
(between 00 and 779) and those of SA3(between 00
and 665) Moreover the trend of the standard deviationsof the percentage error keeps the similar pattern As shownin Table 1 the standard deviations of the percentage errorof SA
1 SA2 and SA
3were between 00 and 729 00
and 120 and 00 and 843 respectively It released thatthe performance of SA
1was slightly better than the other
two Furthermore Figure 2 and Table 1 indicated that themeans of the percentage error of SA
1 SA2 and SA
3are
slightly affected by the parameter 120572 For example the meansof the percentage errors of SA
1 SA2 and SA
3were lager
than 2 at 120572 = 025 Specifically the behavior became clearat pro = 025 and 05 It also can be observed that there isno absolutely dominance relationship among three proposedSAs Due to get a good quality solution we further combinedthree proposed SAs into one (recorded as SA
4and SA
4=
minSA119894 119894 = 1 2 3) The means of the percentage error of
SA4were located between 00 and 212 Meanwhile its
standard deviations of the percentage error were between
000
100
200
300
Mea
n er
ror (
)
SA1SA2
SA3SA4
025 050 075120572
Figure 2 Performance of the SA algorithms (119899 = 12)
00 and 426 Moreover the impacts of pro and 120572 werenot present for proposed three SAs
In the second part of the experiments the performancesof the proposed SA heuristics were further tested for largenumbers of jobs Three different numbers of jobs wereconsidered at 119899 = 20 40 and 60 The proportion of thejobs of agent AG
1at pro = 025 05 and 075 in the tests
Moreover 120572 was taken as the values of 025 05 and 075 Asa result 27 experimental situations were tested 50 instanceswere randomly generated for each situation The relativedeviance percentage with respect to the best known solutionwas reported for each instance The mean execution timeandmean relative deviance percentage were also recorded foreach SA heuristicThe relative deviation percentage RDPwasgiven by
SA119894minus SAlowast
SAlowast (6)
where SA119894is the value of the objective function generated by
SA119894and SAlowast = minSA
119894 119894 = 1 2 3 is the smallest value of
the objective function obtained from the three SA heuristicsThe results were summarized in Table 2
As shown in Figure 3 and Table 2 it can be seen that theRPDmeans of SA
1 SA2 and SA
3are getting slightly bigger as
the value of 120572 increases In general the RPDmeans of SA3are
lower than those of SA1and SA
2 Furthermore all of the RDP
means of SA1 SA2 and SA
3were less than 2 Figure 2 also
indicated that there is no absolutely dominance relationshipamong three proposed SAs
4 Conclusions
Thispaper explored the single-machine two-agent schedulingproblem where the objective is to minimize the total com-pletion time of the jobs belonging to the first agent withthe restriction that total completion time of jobs from thesecond agent has an upper bound Due to the fact that theproblem under study is binary NP hard a branch-and-bound
6 Discrete Dynamics in Nature and Society
Table 2 RPD of heuristic algorithms (119899 = 20sim60)
119899 Pro 120572
SA1 SA2 SA3CPU time RPD CPU time RPD CPU time RPD
Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std
20
02525 00019 00051 045 144 00019 00051 041 119 00016 00047 051 17550 00019 00051 139 267 00019 00051 187 551 00022 00055 089 18875 00019 00051 286 481 00019 00051 496 680 00019 00051 201 366
0525 00019 00051 002 007 00019 00051 000 001 00016 00047 001 00650 00019 00051 051 081 00019 00051 039 067 00022 00055 051 07875 00019 00051 105 152 00022 00055 119 208 00022 00055 130 203
07525 00019 00051 000 000 00013 00043 000 000 00019 00051 000 00050 00019 00051 000 000 00016 00047 000 000 00019 00051 000 00075 00019 00051 050 058 00019 00051 058 075 00019 00051 034 055
Average 00019 00051 075 132 00018 00050 104 189 00019 00052 062 119
40
02525 00066 00078 010 030 00072 00079 016 052 00066 00078 000 00050 00072 00079 092 151 00078 00079 122 209 00072 00079 039 14775 00078 00079 366 443 00084 00079 391 584 00081 00079 282 404
0525 00066 00078 002 006 00066 00078 000 000 00066 00078 000 00050 00075 00079 039 043 00078 00079 031 057 00078 00079 046 07975 00084 00079 081 081 00091 00078 103 169 00091 00078 071 112
07525 00066 00078 000 000 00063 00077 000 000 00063 00077 000 00050 00063 00077 000 000 00063 00077 000 000 00063 00077 000 00075 00072 00079 032 045 00078 00079 025 036 00081 00079 025 031
Average 00071 00078 069 089 00075 00078 076 123 00073 00078 051 086
60
02525 00159 00022 007 020 00178 00055 009 023 00147 00037 002 01150 00175 00051 148 272 00197 00069 120 188 00163 00044 028 08875 00197 00069 238 251 00219 00077 374 510 00206 00074 258 410
0525 00147 00037 001 003 00147 00037 000 000 00144 00043 000 00050 00175 00060 039 057 00184 00061 027 035 00181 00066 023 04275 00209 00075 066 103 00231 00079 062 073 00222 00078 066 090
07525 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00050 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00075 00175 00051 021 027 00197 00069 022 028 00194 00067 021 025
Average 00169 00051 058 082 00182 00060 068 095 00171 00056 044 074
000
100
200
Aver
age o
f RPD
()
SA1SA2
SA3
025 050 075120572
Figure 3 Performance of the SA algorithms (119899 = 60)
algorithm incorporating several dominance properties anda lower bound was proposed for the optimal solutionThree simulated annealing algorithms were also proposedfor near-optimal solutions The computational results showthat with the help of the proposed heuristic initial solutionsthe branch-and-bound algorithm performs well in terms ofnumber of nodes and execution time when the number ofjobs is fewer than or equal to 16Moreover the computationalexperiments also show that the proposed combined SA
4
performs well since its mean error percentage was less than212 for all the tested cases Further research lies in devisingefficient and effective methods to solve the problem withsignificantly larger numbers of jobs
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Discrete Dynamics in Nature and Society
Table 2 RPD of heuristic algorithms (119899 = 20sim60)
119899 Pro 120572
SA1 SA2 SA3CPU time RPD CPU time RPD CPU time RPD
Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std
20
02525 00019 00051 045 144 00019 00051 041 119 00016 00047 051 17550 00019 00051 139 267 00019 00051 187 551 00022 00055 089 18875 00019 00051 286 481 00019 00051 496 680 00019 00051 201 366
0525 00019 00051 002 007 00019 00051 000 001 00016 00047 001 00650 00019 00051 051 081 00019 00051 039 067 00022 00055 051 07875 00019 00051 105 152 00022 00055 119 208 00022 00055 130 203
07525 00019 00051 000 000 00013 00043 000 000 00019 00051 000 00050 00019 00051 000 000 00016 00047 000 000 00019 00051 000 00075 00019 00051 050 058 00019 00051 058 075 00019 00051 034 055
Average 00019 00051 075 132 00018 00050 104 189 00019 00052 062 119
40
02525 00066 00078 010 030 00072 00079 016 052 00066 00078 000 00050 00072 00079 092 151 00078 00079 122 209 00072 00079 039 14775 00078 00079 366 443 00084 00079 391 584 00081 00079 282 404
0525 00066 00078 002 006 00066 00078 000 000 00066 00078 000 00050 00075 00079 039 043 00078 00079 031 057 00078 00079 046 07975 00084 00079 081 081 00091 00078 103 169 00091 00078 071 112
07525 00066 00078 000 000 00063 00077 000 000 00063 00077 000 00050 00063 00077 000 000 00063 00077 000 000 00063 00077 000 00075 00072 00079 032 045 00078 00079 025 036 00081 00079 025 031
Average 00071 00078 069 089 00075 00078 076 123 00073 00078 051 086
60
02525 00159 00022 007 020 00178 00055 009 023 00147 00037 002 01150 00175 00051 148 272 00197 00069 120 188 00163 00044 028 08875 00197 00069 238 251 00219 00077 374 510 00206 00074 258 410
0525 00147 00037 001 003 00147 00037 000 000 00144 00043 000 00050 00175 00060 039 057 00184 00061 027 035 00181 00066 023 04275 00209 00075 066 103 00231 00079 062 073 00222 00078 066 090
07525 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00050 00141 00047 000 000 00141 00047 000 000 00141 00047 000 00075 00175 00051 021 027 00197 00069 022 028 00194 00067 021 025
Average 00169 00051 058 082 00182 00060 068 095 00171 00056 044 074
000
100
200
Aver
age o
f RPD
()
SA1SA2
SA3
025 050 075120572
Figure 3 Performance of the SA algorithms (119899 = 60)
algorithm incorporating several dominance properties anda lower bound was proposed for the optimal solutionThree simulated annealing algorithms were also proposedfor near-optimal solutions The computational results showthat with the help of the proposed heuristic initial solutionsthe branch-and-bound algorithm performs well in terms ofnumber of nodes and execution time when the number ofjobs is fewer than or equal to 16Moreover the computationalexperiments also show that the proposed combined SA
4
performs well since its mean error percentage was less than212 for all the tested cases Further research lies in devisingefficient and effective methods to solve the problem withsignificantly larger numbers of jobs
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 7
Acknowledgment
This research was supported in part by the foundation ofBeijingMunicipal Education on Science andTechnology PlanProject (KM 20121417015)
References
[1] H Purnomo and P Guizol ldquoSimulating forest plantation co-management with a multi-agent systemrdquo Mathematical andComputer Modelling vol 44 no 5-6 pp 535ndash552 2006
[2] N Bessonov I Demin L Pujo-Menjouet and V Volpert ldquoAmulti-agent model describing self-renewal of differentiationeffects on the blood cell populationrdquo Mathematical and Com-puter Modelling vol 49 no 11-12 pp 2116ndash2127 2009
[3] A Agnetis D Pacciarelli and A Pacifici ldquoMulti-agent singlemachine schedulingrdquo Annals of Operations Research vol 150no 1 pp 3ndash15 2007
[4] A Agnetis P B Mirchandani D Pacciarelli and A PacificildquoScheduling problems with two competing agentsrdquo OperationsResearch vol 52 no 2 pp 229ndash242 2004
[5] K R Baker and J C Smith ldquoA multiple-criterion model formachine schedulingrdquo Journal of Scheduling vol 6 no 1 pp 7ndash16 2003
[6] J J Yuan W P Shang and Q Feng ldquoA note on the schedulingwith two families of jobsrdquo Journal of Scheduling vol 8 no 6 pp537ndash542 2005
[7] T C Cheng C T Ng and J J Yuan ldquoMulti-agent scheduling ona single machine to minimize total weighted number of tardyjobsrdquo Theoretical Computer Science vol 362 no 1ndash3 pp 273ndash281 2006
[8] C T Ng T C Cheng and J J Yuan ldquoA note on the complexityof the problem of two-agent scheduling on a single machinerdquoJournal of Combinatorial Optimization vol 12 no 4 pp 387ndash394 2006
[9] T C Cheng C T Ng and J J Yuan ldquoMulti-agent schedulingon a single machine with max-form criteriardquo European Journalof Operational Research vol 188 no 2 pp 603ndash609 2008
[10] A Agnetis G de Pascale and D Pacciarelli ldquoA Lagrangianapproach to single-machine scheduling problems with twocompeting agentsrdquo Journal of Scheduling vol 12 no 4 pp 401ndash415 2009
[11] K Lee B C Choi J Y Leung and M L Pinedo ldquoApproxima-tion algorithms for multi-agent scheduling to minimize totalweighted completion timerdquo Information Processing Letters vol109 no 16 pp 913ndash917 2009
[12] J Y Leung M Pinedo and G Wan ldquoCompetitive two-agentscheduling and its applicationsrdquo Operations Research vol 58no 2 pp 458ndash469 2010
[13] Y Yin S-R Cheng T C Cheng C-C Wu and W-H WuldquoTwo-agent single-machine scheduling with assignable duedatesrdquo Applied Mathematics and Computation vol 219 no 4pp 1674ndash1685 2012
[14] Y Yin S-R Cheng and C-C Wu ldquoScheduling problemswith two agents and a linear non-increasing deterioration tominimize earliness penaltiesrdquo Information Sciences vol 189 pp282ndash292 2012
[15] Y Yin C-C Wu W-H Wu C-J Hsu and W-H Wu ldquoAbranch-and-bound procedure for a single-machine earliness
scheduling problem with two agentsrdquo Applied Soft ComputingJournal vol 13 no 2 pp 1042ndash1054 2013
[16] Q Q Nong T C Cheng and C T Ng ldquoTwo-agent schedulingto minimize the total costrdquo European Journal of OperationalResearch vol 215 no 1 pp 39ndash44 2011
[17] G Wan S R Vakati J Y Leung and M Pinedo ldquoSchedulingtwo agents with controllable processing timesrdquo European Jour-nal of Operational Research vol 205 no 3 pp 528ndash539 2010
[18] W Luo L Chen and G Zhang ldquoApproximation schemes fortwo-machine flow shop scheduling with two agentsrdquo Journal ofCombinatorial Optimization vol 24 no 3 pp 229ndash239 2012
[19] Y Yin S-R Cheng T C E Cheng W-H Wu and C-CWu ldquoTwo-agent single-machine scheduling with release timesand deadlinesrdquo International Journal of Shipping and TransportLogistics vol 5 no 1 pp 75ndash94 2013
[20] P Liu and L Tang ldquoTwo-agent scheduling with linear deterio-rating jobs on a single machinerdquo in Computing and Combina-torics Proceedings of the 14th Annual International ConferenceCOCOON 2008 Dalian China June 27ndash29 2008 vol 5092of Lecture Notes in Computer Science pp 642ndash650 SpringerBerlin Germany 2008
[21] P Liu N Yi and X Zhou ldquoTwo-agent single-machine schedul-ing problems under increasing linear deteriorationrdquo AppliedMathematical Modelling vol 35 no 5 pp 2290ndash2296 2011
[22] T C E Cheng W-H Wu S-R Cheng and C-C Wu ldquoTwo-agent scheduling with position-based deteriorating jobs andlearning effectsrdquo Applied Mathematics and Computation vol217 no 21 pp 8804ndash8824 2011
[23] T C Cheng Y H Chung S C Liao and W C Lee ldquoTwo-agent singe-machine scheduling with release times to minimizethe total weighted completion timerdquo Computers amp OperationsResearch vol 40 no 1 pp 353ndash361 2013
[24] C-C Wu S-K Huang and W-C Lee ldquoTwo-agent schedulingwith learning considerationrdquo Computers and Industrial Engi-neering vol 61 no 4 pp 1324ndash1335 2011
[25] D-C Li and P-H Hsu ldquoSolving a two-agent single-machinescheduling problem considering learning effectrdquo Computersand Operations Research vol 39 no 7 pp 1644ndash1651 2012
[26] Y Yin M Liu J Hao and M Zhou ldquoSingle-machine schedul-ing with job-position-dependent learning and time-dependentdeteriorationrdquo IEEE Transactions on Systems Man and Cyber-netics Part A Systems and Humans vol 42 no 1 pp 192ndash2002012
[27] Y Yin T C E Cheng and C C Wu ldquoScheduling with time-dependent processing timesrdquo Mathematical Problems in Engi-neering vol 2014 Article ID 201421 2 pages 2014
[28] Y Yin W-H Wu T C E Cheng and C C Wu ldquoSingle-machine scheduling with time-dependent and position-dependent deteriorating jobsrdquo International Journal ofComputer Integrated Manufacturing 2014
[29] W-H Wu ldquoSolving a two-agent single-machine learningscheduling problemrdquo International Journal of Computer Inte-grated Manufacturing vol 27 no 1 pp 20ndash35 2014
[30] D-C Li P-H Hsu and C-C Chang ldquoA genetic algorithm-based approach for single-machine scheduling with learningeffect and release timerdquoMathematical Problems in Engineeringvol 2014 Article ID 249493 12 pages 2014
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Discrete Dynamics in Nature and Society
[31] S Kirkpatrick J Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983
[32] B-I Kim S Y Chang J Chang et al ldquoScheduling of raw-material unloading from ships at a steelworksrdquo ProductionPlanning and Control vol 22 no 4 pp 389ndash402 2011
[33] Y Sun J W Fowler and D L Shunk ldquoPolicies for allocatingproduct lots to customer orders in semiconductor manufactur-ing supply chainsrdquo Production Planning and Control vol 22 no1 pp 69ndash80 2011
[34] M Nawaz E E Enscore Jr and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983
[35] D Ben-Arieh and O Maimon ldquoAnnealing method for PCBassembly scheduling on two sequentialmachinesrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 6 pp361ndash367 1992
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of