Research ArticleStochastic Machine Scheduling to Minimize Waiting TimeRelated Objectives with Emergency Jobs
Lianmin Zhang1 Lei Guan2 and Ke Zhou3
1 School of Management and Engineering Nanjing University Nanjing 210093 China2 School of Management and Economics Beijing Institute of Technology Beijing 100081 China3 Systems Engineering and Engineering Management The Chinese University of Hong Kong Hong Kong
Correspondence should be addressed to Lei Guan guanleibiteducn
Received 14 March 2014 Accepted 24 April 2014 Published 11 May 2014
Academic Editor Xiaolin Xu
Copyright copy 2014 Lianmin Zhang et alThis 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
We consider a new scheduling model where emergency jobs appear during the processing of current jobs and must be processedimmediately after the present job is completed All jobs have randomprocessing times and should be completed on a singlemachineThe most common case of the model is the surgery scheduling problem where some elective surgeries are to be arranged in anoperation roomwhen emergency cases are coming during the operating procedure of the elective surgeries Two objective functionsare proposed to display this practice in machine scheduling problem One is the weighted sum of the waiting times and the otheris the weighted discounted cost function of the waiting times We address some optimal policies to minimize these objectives
1 Introduction
Machine scheduling problems have attracted researchers fordecades since they play an important role in various appli-cations from the areas of operations research managementscience and computer science There are lots of works ofthe literature published on these problems However mostof them study the deterministic case where all the infor-mation about jobs and machines is completed without anyuncertainty for example job processing times are assumedto be exactly known in advance But it is really hard to knowthe exact values of these parameters in practical situationsand thus such an assumption is hardly justifiable Insteadone sometimes can only roughly estimate the values ofthese parameters or their probability distributions As AlbertEinstein said ldquoAs far as the laws of mathematics refer toreality they are not certain as far as they are certain they donot refer to realityrdquo
In addition themajority ofmachine scheduling problemsstudied in the literature assume that all jobs are preparedfor processing before scheduling In reality however it is acommon phenomenon that new jobs may come randomlyfrom time to time The necessity to account for random
coming of jobs is a key drive of developing stochasticapproaches to scheduling problems
In this paper we study the single machine stochasticscheduling problem with emergency jobs In our study wefocus on optimal policies to schedule current jobs with someobjectives on a single machine when emergency jobs appearduring the processing of current jobs and must be processedimmediately after the present job is completed All processingtimes of jobs are not known The most common case ofthe model is the surgery scheduling problem where someelective surgeries are to be arranged in an operation roomwhen emergency cases are coming during the operatingprocedure of the elective surgeries Those emergency casesmust be operated as soon as possible However since thepresent surgery occupies the operation room they shouldwait to operate until the current surgery is completed
As indicated by Rothkopf and Smith [1] there are twobasic classes of delay costs considered in scheduling prob-lems The first class includes linear costs which considerno discount of the value of money over time whereas theother class involves exponential functions to represent thediscounts as a function of time In our research two objectivefunctions are expected to minimize One is the weighted sum
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 837910 5 pageshttpdxdoiorg1011552014837910
2 Discrete Dynamics in Nature and Society
of jobsrsquo waiting time and the other is the weighted discountedfunction of jobsrsquo waiting times Our first objective belongs tothe first class while our second objective function belongs tothe second class We will give optimal policies to solve thoseproblems
The remainder of this paper is organized as follows InSection 2 the related works of the literature will be reviewedNotations and some definitions of the model are provided inSection 3 And specially Section 31 investigates the optimalpolicy to schedule jobs with the objective to minimize thetotal weighted waiting time of jobs The optimal policy forscheduling jobs to minimize the discount cost of the waitingtime of jobs is included in Section 32 Concluding remarksand further research are presented in Section 4
2 Literature Review
There has been an enormous amount of work on singlemachine scheduling We do not intend to do a completereview of results in the area and restrict our attention toworksof the literature directly related to the matter of this paper
The first is scheduling jobs on machines minimizing thetotal weighted completion time of jobs It has attracted agreat deal of attention partly because of its importance asa fundamental problem in scheduling and also because ofnew applications for instance in manufacturing compileroptimization and parallel computing Researchers have stud-ied it extensively in different environments such as allowingdifferent release times of jobs multimachine schedulingand online scheduling We here just review the schedulingproblem on a single machine and all jobs have the samerelease times
Perhaps the most famous scheduling policy with thisobjective in the deterministic case is Smithrsquos rule also knownas the WSPT (weighted shortest processing time) rule Itschedules jobs in the order of nonincreasing ratio 119908119895119901119895where 119901119895 denotes the deterministic processing time of job 119895
and 119908119895 is the weight assigned to job 119895 and it is shown to beoptimal for scheduling jobs on a single machine by Smith[2] Extending this rule to stochastic environment firstlyRothkopf [4 5] develops the WSEPT (weighted shortestexpected processing time) rule which schedules jobs in theorder of nonincreasing ratio 119908119895E[119875119895] Rothkopf [4 5]proves that the WSEPT rule is optimal for single machinescheduling with identical release dates The techniqueadopted is to reduce the problem to the deterministic case bytaking the expectations of the processing times This policyhowever fails in the multimachine case with general process-ing times even if the weights are identical that is119908119894 equiv 119908 Formore details about this research we refer readers to see [3]
Another stream related to our research is to minimizetotal weight discounted cost function such as Rothkopf[4 5] Recently Cai et al [6] consider the problem ofscheduling a set of jobs on a singlemachine subject to randombreakdownsThey study the preemptive-repeat model whichaddresses the situation where if a machine breaks downduring the processing of a job the work done on the job priorto the breakdown is lost and the job will have to be startedfrom the beginning again when the machine resumes its
work They obtain the optimal policy for a class of problemsto minimize the expected discounted cost from completingall jobs For more results in the area see the survey [3]
What differentiates our paper from the above works ofliterature is that we consider a new scheduling problemwherethe emergency jobs are constantly coming and should beplaced as soon as possible To the best of our knowledge thereis no previous research studying this problem Therefore weopen up a new research direction for the scheduling problemIn addition different from previous works of the literatureconsidering the completion time of jobs we focus on thewaiting time of jobs It is more reasonable in our schedulingenvironment
3 Models and Optimal Policies
We here consider the stochastic scheduling problem withemergency jobs Specially what we want to study is how toschedule current jobs on a single machine when emergencyjobs appear during the processing of jobs and must beprocessed immediately after the present job is completedWithout loss of generality in the following we assume thatthe emergent stream of jobs has a Poisson process with rate 120582The processing time 119885119895 for the emergency job 119895 is arbitrarilydistributed with the mean 120583
Let 119869 = 1 2 119899 be a set of jobs to be processednonpreemptively on a single machine In other words if a jobstarts to process other jobs must wait to be processed untilthe present job is completed Assume that the processing time119875119895 of current job 119895 is a random variable We denote119882119895 as thewaiting time of job 119895 and let 120574119895 be the unit waiting cost of job119895 In the following we will propose two objectives and studythe optimal polices to minimize those functions
31 Objective E[ sum 120574119894119882119894] In this subsection we will firstfocus on the objective function to minimize total weightedwaiting cost written as E[sum 120574119894119882119894] for short We here willaddress that the well-known policy in stochastic singlemachine schedulingWSEPT is also an optimal policy for ourmodel
Theorem 1 WSEPT that is sequencing jobs in nonincreasingorder of 120574119895E[119875119895] is an optimal nonpreemptive policy tominimize total weighted waiting cost
Proof Consider the schedule 1 2 119899 denoted asΠ For thefirst job 1 its waiting time is 0 Based on the fact that new jobsrsquocoming follows the Poisson process it is easy to know thatthe expected number of emergency jobs before job 2 starts toprocess is (120582(1 minus 120582))E[1198751] Therefore the expected waitingtime of job 2 is (1 + 120582120583(1 minus 120582))E[1198751] since each new job hasmean processing time 120583 Let119862119895 be the completion time of job119895 Similarly we can obtain that
E [119882119894] = E [119862119894minus1] +120582120583E [119875119894minus1]
1 minus 120582
E [119862119894minus1] = E [119882119894minus1] + E [119875119894minus1]
E [1198750] = 0
(1)
Discrete Dynamics in Nature and Society 3
Thus we have
E[
119899
sum119894=1
120574119894119882119894] =
119899
sum119894=1
120574119894E [119882119894]
=
119899
sum119894=1
120574119894 (E [119862119894minus1] +120582120583E [119875119894minus1]
1 minus 120582)
=
119899
sum119894=1
120574119894E [119882119894minus1] +
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus1]
=
119899
sum119894=1
120574119894E [119882119894minus2] +
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus2]
+
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus1]
= (120582120583 minus 120582 + 1
1 minus 120582)
119899
sum119894=1
120574119894
119894minus1
sum119896=1
E [119875119896]
(2)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895E[119875119895] that is1205741E[1198751] ge 1205742E[1198752] ge sdot sdot sdot ge 120574119899E[119875119899] If this sequencedenoted by Π for brevity is not an optimal policy theremust exist job 119895 in some optimal policy such that job119895 + 1 is processed before job 119895 Without loss of generalitylet 1 2 119895 minus 1 119895 + 1 119895 119895 + 2 119899 denoted by Π
1015840 besuch optimal policy and we hence have Φ(Π) gt Φ(Π
1015840)
Comparing these two policies and observing that the waitingcost of jobs 1 2 119895 minus 1 119895 + 2 119899 is not changed weobtain
Φ (Π) minus Φ (Π1015840)
= (120582120583 minus 120582 + 1
1 minus 120582)
times (
119895+1
sum119894=119895
120574119894
119894minus1
sum119896=1
E [119875119896] minus 120574119895+1
119895minus1
sum119896=1
E [119875119896]
minus120574119895
119895minus1
sum119896=1
E [119875119896] minus 120574119895E [119875119895+1])
= (120582120583 minus 120582 + 1
1 minus 120582) (120574119895+1E [119875119895] minus 120574119895119864 [E119895+1]) lt 0
(3)
Therefore Φ(Π) lt Φ(Π1015840) which leads to a contradiction
With the adjacent pairwise interchange we can know thatWSEPT is an optimal policy This completes the proof
Remarks (1) In fact the optimal policy for the objectivewith minimizing total expected waiting cost is equal to that
with minimizing total expected completion timeThis can beshown according to the following equation
E [sum120574119894119862119894] = E [sum120574119894 (119882119894 + 119875119894)]
= E [sum120574119894 (119882119894)] + sum120574119894E [119875119894]
(4)
It is well known that WSEPT optimally minimizes the totalexpected completion time for singlemachine problem whichcoincides with the result we obtained But here we consider anew schedulingmodel Hence this theorem extends the well-known optimal policy to a new scheduling environment
(2) The theorem shows that it is simple to construct thepolicy to minimize the total weighted waiting time of jobsThe only information for the optimal policy is the weightand the expected processing time of jobs while the exactdistribution of processing times of jobs is needed so as toobtain optimal policy with simple structure in most works ofthe literature in stochastic scheduling
32 Objectivesum120574119894E[1minusexp(minus120572119882119894)] In this section we studythe objective ofweighted discounted cost of thewaiting timesthat is sum120574119894E[1 minus exp(minus120572119882119894)] Here 120572 is the discount factorBefore giving the optimal policy for ourmodel wewill releasetwo lemmas
Lemma 2 Considerinfin
sum119909=0
120582119909
119909= 119890120582 (5)
Proof Let 119909 be a random variable and be followed withPoisson distribution We then have
infin
sum119909=0
119890minus120582
sdot120582119909
119909= 1 (6)
By multiplying 119890120582 to both sides of the equality we have
suminfin119909=0(120582119909119909) = 119890
120582
The other lemma which will be used is stated as follows
Lemma 3 For any 119886 if 119909 is followed with Poisson distributionthen E[119890119886119909] = 119890
120582(119890119886minus1)
Proof Since 119909 is followed with Poisson distribution weobtain
E [119890119886119909] =
infin
sum119909=0
119890119886119909
sdot120582119909119890minus120582
119909= 119890minus120582infin
sum119909=0
120582119909119890119886119909
119909
= 119890minus120582infin
sum119909=0
(120582119890119886)119909
119909= 119890minus120582
119890119890119886120582
= 119890120582(119890119886minus1)
(7)
We now can prove our second main theorem as follows
4 Discrete Dynamics in Nature and Society
Theorem 4 Sequencing jobs with the nonincreasing order of
120574119895
1 minus 119891119895(8)
are an optimal policy to minimize the objective sum120574119894E[1 minus
exp(minus120572119882119894)] where
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] (9)
Proof Assume that 1 2 119899 is a feasible schedule Let 119884119895 bethe random variable that denotes the total time of processingjob 119895 and processing emergence jobs arriving during theprocessing of job 119895 And let 119873119895 denote the random variableas the number of emergence jobs during the processing ofjob 119895 Observing that the emergence jobs arrive according tothe Poisson distribution with rate 120582 we know that randomvariable (119873119895 | 119875119895) has a Poisson distribution with mean(120582(1 minus 120582))119875119895 Thus
Based on the fact that all 119884119896119896=119899119896=1 are independent random
variables we thus can know that
E [119890minus120572119882119895] = E [119890
minussum119895minus1
119896=1120572119884119896] =
119895minus1
prod119896=1
119891119896 (11)
Therefore the objective function under sequence 11989511198952 119895119899 results is
119899
sum119894=1
119903119895119894
(1 minus
119894minus1
prod119896=1
119891119896) (12)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895(1 minus 119891119895) that is 1205741(1 minus
1198911) ge 1205742(1minus1198912) ge sdot sdot sdot ge 120574119899(1minus119891119899) If this sequence denotedby Π is not an optimal policy there must exist 119895 in someoptimal policy such that job 119895 + 1 is processed before job 119895Without loss of generality let 1 2 119895minus1 119895+1 119895 119895+2 119899denoted by Π
1015840 be such optimal policy and we hence haveΦ(Π) gt Φ(Π
1015840) Comparing these two policies and observing
that the waiting cost of jobs 1 2 119895 minus 1 119895 + 2 119899 is notchanged we obtain
Φ (Π) minus Φ (Π1015840)
=
119895+1
sum119894=119895
119903119894(1 minus
119894minus1
prod119896=1
119891119896) minus 119903119895+1(1 minus
119895minus1
prod119896=1
119891119896)
minus 119903119895(1 minus 119891119895+1
119895minus1
prod119896=1
119891119896)
= minus119903119895 (1 minus 119891119895+1)
119895minus1
prod119896=1
119891119896 + 119903119895+1 (1 minus 119891119895)
119895minus1
prod119896=1
119891119896
= (119903119895+1 (1 minus 119891119895) minus 119903119895 (1 minus 119891119895+1))
119895minus1
prod119896=1
119891119896 lt 0
(13)
Therefore we have Φ(Π) lt Φ(Π1015840) which leads to a contra-
dictionWith the adjacent pairwise interchange we can knowthat nonincreasing order of 120574119895(1 minus 119891119895) is an optimal policyThis completes the proof
RemarkThe structure of the optimal policy is a little complexcomparing to the optimal policy in the last section In addi-tion to weight and processing time of jobs this optimal policyis also related to the parameters 120582 1198851 120572 It is reasonablesince the objective function is really complex It is howeveralso easy to calculate or estimate those values with the helpof computer Besides if the processing time of current andemergent jobs is followed with some special distribution theoptimal policy will have very simple structure One exampleis stated as follows
Example Assume that 119875119894 sim exp(]119895) and 119885119894 sim exp(120583) We caneasily calculate that
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] = E [119890(minus120572minus120582+120582(120583(120583+120572)))119875
119895]
= E [119890(minus120572minus(120582120572(120583+120572)))119875
It is easy to verify that the optimal policy sequences jobs innonincreasing order of 120574119895(]119895 + 120572 + (120582120572(120583 + 120572)))
4 Conclusions and Future Research
In this paper we consider a new stochastic scheduling modelbased on the emergency jobsWe give optimal policies for twoclasses of objectives one is the total waiting time of jobs andthe other is the weighted sum of an exponential function ofthe waiting times
One can consider other objectives in scheduling areasuch as the weighted number of late jobs Specially we canstudy the environment that each job 119895 has a due date 119863119895which follows the exponential distribution We believe that itis reasonable to study this case Taking the surgery scheduling
Discrete Dynamics in Nature and Society 5
as an example again we can consider the due date for eachelective surgery One reason is that each elective surgeryshould have a due date to wait for operating Otherwise thepatient will die or will leave to search for another hospital forhelpThe other reason is that in some sense this date can alsobe as the commitment to the patients And sometimes thiscommitment is consistent with patientsrsquo conditionThereforeit can be interpreted as a random variable In additionDenton et al [7] consider the overtime to themodel and studyhow case sequencing affects patient waiting time operatingroom idling time (surgeonwaiting time) and operation roomovertimeTherefore maybe we can study the objective whichincludes both the weighted number of late jobs and overtime
E[
[
119870
sum119896=1
119900119896max
0 sum119894isin119878119896
119875119894 minus Δ 119896
+ sum119895
120574119895(119882119895 minus 119863119895)+]
]
(15)
where 119900119896 is the unit overtime cost Δ 119896 is the width of the 119896thregular time and 119878119896 is the subset of cases scheduled in the 119896thday
Another direction we can study is to take the waitingcost of emergency jobs into consideration The reason isthat emergency surgery which cannot be treated in time willresult in substantial losses It reflects that the unit waitingtime cost of emergency jobs is very large in the model Itthus is necessary to operate these cases as soon as possibleClearly sometimes it is nonsense if the new coming jobsfollow the Poisson process since its influence on the time lineis the same Thus in this case maybe we can take anotherdistribution for the emergency jobsrsquo coming Sometimes itis reasonable since in practice emergency jobsrsquo appearance isrelated to the timing For example Summala andMikkola [8]show that the fatal accidents appear more frequently during3amndash5am and 2pm-3pm which is very close to emergencysurgeries
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
We thank the Guest Editor and the three anonymous refereesfor their helpful comments and suggestions The secondauthor is supported by the fundamental research funds ofBeijing Institute of Technology (no 20122142014)
References
[1] M H Rothkopf and S A Smith ldquoThere are no undiscoveredpriority index sequencing rules for minimizing total delaycostsrdquo Operations Research vol 32 no 2 pp 451ndash456 1984
[2] W E Smith ldquoVarious optimizers for single-stage productionrdquoNaval Research Logistics Quarterly vol 3 pp 59ndash66 1956
[3] X Q Cai X Y Wu L Zhang and X Zhou Schedulingwith Stochastic Approaches Sequencing and Scheduling with
Inaccurate Data Nova NewYork 2014 edited byUSotskov andF Werner
[4] M H Rothkopf ldquoScheduling independent tasks on parallelprocessorsrdquoManagement Science vol 12 pp 437ndash447 1966
[5] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 pp 707ndash713 1966
[6] X Q Cai X Sun and X Zhou ldquoStochastic scheduling subjectto machine breakdowns the preemptive-repeat model withdiscounted reward and other criteriardquo Naval Research Logisticsvol 51 no 6 pp 800ndash817 2004
[7] B Denton J Viapiano and A Vogl ldquoOptimization of surgerysequencing and scheduling decisions under uncertaintyrdquoHealthCare Management Science vol 10 no 1 pp 13ndash24 2007
[8] H Summala and T Mikkola ldquoFatal accidents among car andtruck drivers effects of fatigue age and alcohol consumptionrdquoHuman Factors vol 36 no 2 pp 315ndash326 1994
of jobsrsquo waiting time and the other is the weighted discountedfunction of jobsrsquo waiting times Our first objective belongs tothe first class while our second objective function belongs tothe second class We will give optimal policies to solve thoseproblems
The remainder of this paper is organized as follows InSection 2 the related works of the literature will be reviewedNotations and some definitions of the model are provided inSection 3 And specially Section 31 investigates the optimalpolicy to schedule jobs with the objective to minimize thetotal weighted waiting time of jobs The optimal policy forscheduling jobs to minimize the discount cost of the waitingtime of jobs is included in Section 32 Concluding remarksand further research are presented in Section 4
2 Literature Review
There has been an enormous amount of work on singlemachine scheduling We do not intend to do a completereview of results in the area and restrict our attention toworksof the literature directly related to the matter of this paper
The first is scheduling jobs on machines minimizing thetotal weighted completion time of jobs It has attracted agreat deal of attention partly because of its importance asa fundamental problem in scheduling and also because ofnew applications for instance in manufacturing compileroptimization and parallel computing Researchers have stud-ied it extensively in different environments such as allowingdifferent release times of jobs multimachine schedulingand online scheduling We here just review the schedulingproblem on a single machine and all jobs have the samerelease times
Perhaps the most famous scheduling policy with thisobjective in the deterministic case is Smithrsquos rule also knownas the WSPT (weighted shortest processing time) rule Itschedules jobs in the order of nonincreasing ratio 119908119895119901119895where 119901119895 denotes the deterministic processing time of job 119895
and 119908119895 is the weight assigned to job 119895 and it is shown to beoptimal for scheduling jobs on a single machine by Smith[2] Extending this rule to stochastic environment firstlyRothkopf [4 5] develops the WSEPT (weighted shortestexpected processing time) rule which schedules jobs in theorder of nonincreasing ratio 119908119895E[119875119895] Rothkopf [4 5]proves that the WSEPT rule is optimal for single machinescheduling with identical release dates The techniqueadopted is to reduce the problem to the deterministic case bytaking the expectations of the processing times This policyhowever fails in the multimachine case with general process-ing times even if the weights are identical that is119908119894 equiv 119908 Formore details about this research we refer readers to see [3]
Another stream related to our research is to minimizetotal weight discounted cost function such as Rothkopf[4 5] Recently Cai et al [6] consider the problem ofscheduling a set of jobs on a singlemachine subject to randombreakdownsThey study the preemptive-repeat model whichaddresses the situation where if a machine breaks downduring the processing of a job the work done on the job priorto the breakdown is lost and the job will have to be startedfrom the beginning again when the machine resumes its
work They obtain the optimal policy for a class of problemsto minimize the expected discounted cost from completingall jobs For more results in the area see the survey [3]
What differentiates our paper from the above works ofliterature is that we consider a new scheduling problemwherethe emergency jobs are constantly coming and should beplaced as soon as possible To the best of our knowledge thereis no previous research studying this problem Therefore weopen up a new research direction for the scheduling problemIn addition different from previous works of the literatureconsidering the completion time of jobs we focus on thewaiting time of jobs It is more reasonable in our schedulingenvironment
3 Models and Optimal Policies
We here consider the stochastic scheduling problem withemergency jobs Specially what we want to study is how toschedule current jobs on a single machine when emergencyjobs appear during the processing of jobs and must beprocessed immediately after the present job is completedWithout loss of generality in the following we assume thatthe emergent stream of jobs has a Poisson process with rate 120582The processing time 119885119895 for the emergency job 119895 is arbitrarilydistributed with the mean 120583
Let 119869 = 1 2 119899 be a set of jobs to be processednonpreemptively on a single machine In other words if a jobstarts to process other jobs must wait to be processed untilthe present job is completed Assume that the processing time119875119895 of current job 119895 is a random variable We denote119882119895 as thewaiting time of job 119895 and let 120574119895 be the unit waiting cost of job119895 In the following we will propose two objectives and studythe optimal polices to minimize those functions
31 Objective E[ sum 120574119894119882119894] In this subsection we will firstfocus on the objective function to minimize total weightedwaiting cost written as E[sum 120574119894119882119894] for short We here willaddress that the well-known policy in stochastic singlemachine schedulingWSEPT is also an optimal policy for ourmodel
Theorem 1 WSEPT that is sequencing jobs in nonincreasingorder of 120574119895E[119875119895] is an optimal nonpreemptive policy tominimize total weighted waiting cost
Proof Consider the schedule 1 2 119899 denoted asΠ For thefirst job 1 its waiting time is 0 Based on the fact that new jobsrsquocoming follows the Poisson process it is easy to know thatthe expected number of emergency jobs before job 2 starts toprocess is (120582(1 minus 120582))E[1198751] Therefore the expected waitingtime of job 2 is (1 + 120582120583(1 minus 120582))E[1198751] since each new job hasmean processing time 120583 Let119862119895 be the completion time of job119895 Similarly we can obtain that
E [119882119894] = E [119862119894minus1] +120582120583E [119875119894minus1]
1 minus 120582
E [119862119894minus1] = E [119882119894minus1] + E [119875119894minus1]
E [1198750] = 0
(1)
Discrete Dynamics in Nature and Society 3
Thus we have
E[
119899
sum119894=1
120574119894119882119894] =
119899
sum119894=1
120574119894E [119882119894]
=
119899
sum119894=1
120574119894 (E [119862119894minus1] +120582120583E [119875119894minus1]
1 minus 120582)
=
119899
sum119894=1
120574119894E [119882119894minus1] +
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus1]
=
119899
sum119894=1
120574119894E [119882119894minus2] +
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus2]
+
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus1]
= (120582120583 minus 120582 + 1
1 minus 120582)
119899
sum119894=1
120574119894
119894minus1
sum119896=1
E [119875119896]
(2)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895E[119875119895] that is1205741E[1198751] ge 1205742E[1198752] ge sdot sdot sdot ge 120574119899E[119875119899] If this sequencedenoted by Π for brevity is not an optimal policy theremust exist job 119895 in some optimal policy such that job119895 + 1 is processed before job 119895 Without loss of generalitylet 1 2 119895 minus 1 119895 + 1 119895 119895 + 2 119899 denoted by Π
1015840 besuch optimal policy and we hence have Φ(Π) gt Φ(Π
1015840)
Comparing these two policies and observing that the waitingcost of jobs 1 2 119895 minus 1 119895 + 2 119899 is not changed weobtain
Φ (Π) minus Φ (Π1015840)
= (120582120583 minus 120582 + 1
1 minus 120582)
times (
119895+1
sum119894=119895
120574119894
119894minus1
sum119896=1
E [119875119896] minus 120574119895+1
119895minus1
sum119896=1
E [119875119896]
minus120574119895
119895minus1
sum119896=1
E [119875119896] minus 120574119895E [119875119895+1])
= (120582120583 minus 120582 + 1
1 minus 120582) (120574119895+1E [119875119895] minus 120574119895119864 [E119895+1]) lt 0
(3)
Therefore Φ(Π) lt Φ(Π1015840) which leads to a contradiction
With the adjacent pairwise interchange we can know thatWSEPT is an optimal policy This completes the proof
Remarks (1) In fact the optimal policy for the objectivewith minimizing total expected waiting cost is equal to that
with minimizing total expected completion timeThis can beshown according to the following equation
E [sum120574119894119862119894] = E [sum120574119894 (119882119894 + 119875119894)]
= E [sum120574119894 (119882119894)] + sum120574119894E [119875119894]
(4)
It is well known that WSEPT optimally minimizes the totalexpected completion time for singlemachine problem whichcoincides with the result we obtained But here we consider anew schedulingmodel Hence this theorem extends the well-known optimal policy to a new scheduling environment
(2) The theorem shows that it is simple to construct thepolicy to minimize the total weighted waiting time of jobsThe only information for the optimal policy is the weightand the expected processing time of jobs while the exactdistribution of processing times of jobs is needed so as toobtain optimal policy with simple structure in most works ofthe literature in stochastic scheduling
32 Objectivesum120574119894E[1minusexp(minus120572119882119894)] In this section we studythe objective ofweighted discounted cost of thewaiting timesthat is sum120574119894E[1 minus exp(minus120572119882119894)] Here 120572 is the discount factorBefore giving the optimal policy for ourmodel wewill releasetwo lemmas
Lemma 2 Considerinfin
sum119909=0
120582119909
119909= 119890120582 (5)
Proof Let 119909 be a random variable and be followed withPoisson distribution We then have
infin
sum119909=0
119890minus120582
sdot120582119909
119909= 1 (6)
By multiplying 119890120582 to both sides of the equality we have
suminfin119909=0(120582119909119909) = 119890
120582
The other lemma which will be used is stated as follows
Lemma 3 For any 119886 if 119909 is followed with Poisson distributionthen E[119890119886119909] = 119890
120582(119890119886minus1)
Proof Since 119909 is followed with Poisson distribution weobtain
E [119890119886119909] =
infin
sum119909=0
119890119886119909
sdot120582119909119890minus120582
119909= 119890minus120582infin
sum119909=0
120582119909119890119886119909
119909
= 119890minus120582infin
sum119909=0
(120582119890119886)119909
119909= 119890minus120582
119890119890119886120582
= 119890120582(119890119886minus1)
(7)
We now can prove our second main theorem as follows
4 Discrete Dynamics in Nature and Society
Theorem 4 Sequencing jobs with the nonincreasing order of
120574119895
1 minus 119891119895(8)
are an optimal policy to minimize the objective sum120574119894E[1 minus
exp(minus120572119882119894)] where
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] (9)
Proof Assume that 1 2 119899 is a feasible schedule Let 119884119895 bethe random variable that denotes the total time of processingjob 119895 and processing emergence jobs arriving during theprocessing of job 119895 And let 119873119895 denote the random variableas the number of emergence jobs during the processing ofjob 119895 Observing that the emergence jobs arrive according tothe Poisson distribution with rate 120582 we know that randomvariable (119873119895 | 119875119895) has a Poisson distribution with mean(120582(1 minus 120582))119875119895 Thus
Based on the fact that all 119884119896119896=119899119896=1 are independent random
variables we thus can know that
E [119890minus120572119882119895] = E [119890
minussum119895minus1
119896=1120572119884119896] =
119895minus1
prod119896=1
119891119896 (11)
Therefore the objective function under sequence 11989511198952 119895119899 results is
119899
sum119894=1
119903119895119894
(1 minus
119894minus1
prod119896=1
119891119896) (12)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895(1 minus 119891119895) that is 1205741(1 minus
1198911) ge 1205742(1minus1198912) ge sdot sdot sdot ge 120574119899(1minus119891119899) If this sequence denotedby Π is not an optimal policy there must exist 119895 in someoptimal policy such that job 119895 + 1 is processed before job 119895Without loss of generality let 1 2 119895minus1 119895+1 119895 119895+2 119899denoted by Π
1015840 be such optimal policy and we hence haveΦ(Π) gt Φ(Π
1015840) Comparing these two policies and observing
that the waiting cost of jobs 1 2 119895 minus 1 119895 + 2 119899 is notchanged we obtain
Φ (Π) minus Φ (Π1015840)
=
119895+1
sum119894=119895
119903119894(1 minus
119894minus1
prod119896=1
119891119896) minus 119903119895+1(1 minus
119895minus1
prod119896=1
119891119896)
minus 119903119895(1 minus 119891119895+1
119895minus1
prod119896=1
119891119896)
= minus119903119895 (1 minus 119891119895+1)
119895minus1
prod119896=1
119891119896 + 119903119895+1 (1 minus 119891119895)
119895minus1
prod119896=1
119891119896
= (119903119895+1 (1 minus 119891119895) minus 119903119895 (1 minus 119891119895+1))
119895minus1
prod119896=1
119891119896 lt 0
(13)
Therefore we have Φ(Π) lt Φ(Π1015840) which leads to a contra-
dictionWith the adjacent pairwise interchange we can knowthat nonincreasing order of 120574119895(1 minus 119891119895) is an optimal policyThis completes the proof
RemarkThe structure of the optimal policy is a little complexcomparing to the optimal policy in the last section In addi-tion to weight and processing time of jobs this optimal policyis also related to the parameters 120582 1198851 120572 It is reasonablesince the objective function is really complex It is howeveralso easy to calculate or estimate those values with the helpof computer Besides if the processing time of current andemergent jobs is followed with some special distribution theoptimal policy will have very simple structure One exampleis stated as follows
Example Assume that 119875119894 sim exp(]119895) and 119885119894 sim exp(120583) We caneasily calculate that
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] = E [119890(minus120572minus120582+120582(120583(120583+120572)))119875
119895]
= E [119890(minus120572minus(120582120572(120583+120572)))119875
It is easy to verify that the optimal policy sequences jobs innonincreasing order of 120574119895(]119895 + 120572 + (120582120572(120583 + 120572)))
4 Conclusions and Future Research
In this paper we consider a new stochastic scheduling modelbased on the emergency jobsWe give optimal policies for twoclasses of objectives one is the total waiting time of jobs andthe other is the weighted sum of an exponential function ofthe waiting times
One can consider other objectives in scheduling areasuch as the weighted number of late jobs Specially we canstudy the environment that each job 119895 has a due date 119863119895which follows the exponential distribution We believe that itis reasonable to study this case Taking the surgery scheduling
Discrete Dynamics in Nature and Society 5
as an example again we can consider the due date for eachelective surgery One reason is that each elective surgeryshould have a due date to wait for operating Otherwise thepatient will die or will leave to search for another hospital forhelpThe other reason is that in some sense this date can alsobe as the commitment to the patients And sometimes thiscommitment is consistent with patientsrsquo conditionThereforeit can be interpreted as a random variable In additionDenton et al [7] consider the overtime to themodel and studyhow case sequencing affects patient waiting time operatingroom idling time (surgeonwaiting time) and operation roomovertimeTherefore maybe we can study the objective whichincludes both the weighted number of late jobs and overtime
E[
[
119870
sum119896=1
119900119896max
0 sum119894isin119878119896
119875119894 minus Δ 119896
+ sum119895
120574119895(119882119895 minus 119863119895)+]
]
(15)
where 119900119896 is the unit overtime cost Δ 119896 is the width of the 119896thregular time and 119878119896 is the subset of cases scheduled in the 119896thday
Another direction we can study is to take the waitingcost of emergency jobs into consideration The reason isthat emergency surgery which cannot be treated in time willresult in substantial losses It reflects that the unit waitingtime cost of emergency jobs is very large in the model Itthus is necessary to operate these cases as soon as possibleClearly sometimes it is nonsense if the new coming jobsfollow the Poisson process since its influence on the time lineis the same Thus in this case maybe we can take anotherdistribution for the emergency jobsrsquo coming Sometimes itis reasonable since in practice emergency jobsrsquo appearance isrelated to the timing For example Summala andMikkola [8]show that the fatal accidents appear more frequently during3amndash5am and 2pm-3pm which is very close to emergencysurgeries
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
We thank the Guest Editor and the three anonymous refereesfor their helpful comments and suggestions The secondauthor is supported by the fundamental research funds ofBeijing Institute of Technology (no 20122142014)
References
[1] M H Rothkopf and S A Smith ldquoThere are no undiscoveredpriority index sequencing rules for minimizing total delaycostsrdquo Operations Research vol 32 no 2 pp 451ndash456 1984
[2] W E Smith ldquoVarious optimizers for single-stage productionrdquoNaval Research Logistics Quarterly vol 3 pp 59ndash66 1956
[3] X Q Cai X Y Wu L Zhang and X Zhou Schedulingwith Stochastic Approaches Sequencing and Scheduling with
Inaccurate Data Nova NewYork 2014 edited byUSotskov andF Werner
[4] M H Rothkopf ldquoScheduling independent tasks on parallelprocessorsrdquoManagement Science vol 12 pp 437ndash447 1966
[5] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 pp 707ndash713 1966
[6] X Q Cai X Sun and X Zhou ldquoStochastic scheduling subjectto machine breakdowns the preemptive-repeat model withdiscounted reward and other criteriardquo Naval Research Logisticsvol 51 no 6 pp 800ndash817 2004
[7] B Denton J Viapiano and A Vogl ldquoOptimization of surgerysequencing and scheduling decisions under uncertaintyrdquoHealthCare Management Science vol 10 no 1 pp 13ndash24 2007
[8] H Summala and T Mikkola ldquoFatal accidents among car andtruck drivers effects of fatigue age and alcohol consumptionrdquoHuman Factors vol 36 no 2 pp 315ndash326 1994
120574119894 (E [119862119894minus1] +120582120583E [119875119894minus1]
1 minus 120582)
=
119899
sum119894=1
120574119894E [119882119894minus1] +
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus1]
=
119899
sum119894=1
120574119894E [119882119894minus2] +
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus2]
+
119899
sum119894=1
120574119894 (120582120583 minus 120582 + 1
1 minus 120582)E [119875119894minus1]
= (120582120583 minus 120582 + 1
1 minus 120582)
119899
sum119894=1
120574119894
119894minus1
sum119896=1
E [119875119896]
(2)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895E[119875119895] that is1205741E[1198751] ge 1205742E[1198752] ge sdot sdot sdot ge 120574119899E[119875119899] If this sequencedenoted by Π for brevity is not an optimal policy theremust exist job 119895 in some optimal policy such that job119895 + 1 is processed before job 119895 Without loss of generalitylet 1 2 119895 minus 1 119895 + 1 119895 119895 + 2 119899 denoted by Π
1015840 besuch optimal policy and we hence have Φ(Π) gt Φ(Π
1015840)
Comparing these two policies and observing that the waitingcost of jobs 1 2 119895 minus 1 119895 + 2 119899 is not changed weobtain
Φ (Π) minus Φ (Π1015840)
= (120582120583 minus 120582 + 1
1 minus 120582)
times (
119895+1
sum119894=119895
120574119894
119894minus1
sum119896=1
E [119875119896] minus 120574119895+1
119895minus1
sum119896=1
E [119875119896]
minus120574119895
119895minus1
sum119896=1
E [119875119896] minus 120574119895E [119875119895+1])
= (120582120583 minus 120582 + 1
1 minus 120582) (120574119895+1E [119875119895] minus 120574119895119864 [E119895+1]) lt 0
(3)
Therefore Φ(Π) lt Φ(Π1015840) which leads to a contradiction
With the adjacent pairwise interchange we can know thatWSEPT is an optimal policy This completes the proof
Remarks (1) In fact the optimal policy for the objectivewith minimizing total expected waiting cost is equal to that
with minimizing total expected completion timeThis can beshown according to the following equation
E [sum120574119894119862119894] = E [sum120574119894 (119882119894 + 119875119894)]
= E [sum120574119894 (119882119894)] + sum120574119894E [119875119894]
(4)
It is well known that WSEPT optimally minimizes the totalexpected completion time for singlemachine problem whichcoincides with the result we obtained But here we consider anew schedulingmodel Hence this theorem extends the well-known optimal policy to a new scheduling environment
(2) The theorem shows that it is simple to construct thepolicy to minimize the total weighted waiting time of jobsThe only information for the optimal policy is the weightand the expected processing time of jobs while the exactdistribution of processing times of jobs is needed so as toobtain optimal policy with simple structure in most works ofthe literature in stochastic scheduling
32 Objectivesum120574119894E[1minusexp(minus120572119882119894)] In this section we studythe objective ofweighted discounted cost of thewaiting timesthat is sum120574119894E[1 minus exp(minus120572119882119894)] Here 120572 is the discount factorBefore giving the optimal policy for ourmodel wewill releasetwo lemmas
Lemma 2 Considerinfin
sum119909=0
120582119909
119909= 119890120582 (5)
Proof Let 119909 be a random variable and be followed withPoisson distribution We then have
infin
sum119909=0
119890minus120582
sdot120582119909
119909= 1 (6)
By multiplying 119890120582 to both sides of the equality we have
suminfin119909=0(120582119909119909) = 119890
120582
The other lemma which will be used is stated as follows
Lemma 3 For any 119886 if 119909 is followed with Poisson distributionthen E[119890119886119909] = 119890
120582(119890119886minus1)
Proof Since 119909 is followed with Poisson distribution weobtain
E [119890119886119909] =
infin
sum119909=0
119890119886119909
sdot120582119909119890minus120582
119909= 119890minus120582infin
sum119909=0
120582119909119890119886119909
119909
= 119890minus120582infin
sum119909=0
(120582119890119886)119909
119909= 119890minus120582
119890119890119886120582
= 119890120582(119890119886minus1)
(7)
We now can prove our second main theorem as follows
4 Discrete Dynamics in Nature and Society
Theorem 4 Sequencing jobs with the nonincreasing order of
120574119895
1 minus 119891119895(8)
are an optimal policy to minimize the objective sum120574119894E[1 minus
exp(minus120572119882119894)] where
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] (9)
Proof Assume that 1 2 119899 is a feasible schedule Let 119884119895 bethe random variable that denotes the total time of processingjob 119895 and processing emergence jobs arriving during theprocessing of job 119895 And let 119873119895 denote the random variableas the number of emergence jobs during the processing ofjob 119895 Observing that the emergence jobs arrive according tothe Poisson distribution with rate 120582 we know that randomvariable (119873119895 | 119875119895) has a Poisson distribution with mean(120582(1 minus 120582))119875119895 Thus
Based on the fact that all 119884119896119896=119899119896=1 are independent random
variables we thus can know that
E [119890minus120572119882119895] = E [119890
minussum119895minus1
119896=1120572119884119896] =
119895minus1
prod119896=1
119891119896 (11)
Therefore the objective function under sequence 11989511198952 119895119899 results is
119899
sum119894=1
119903119895119894
(1 minus
119894minus1
prod119896=1
119891119896) (12)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895(1 minus 119891119895) that is 1205741(1 minus
1198911) ge 1205742(1minus1198912) ge sdot sdot sdot ge 120574119899(1minus119891119899) If this sequence denotedby Π is not an optimal policy there must exist 119895 in someoptimal policy such that job 119895 + 1 is processed before job 119895Without loss of generality let 1 2 119895minus1 119895+1 119895 119895+2 119899denoted by Π
1015840 be such optimal policy and we hence haveΦ(Π) gt Φ(Π
1015840) Comparing these two policies and observing
that the waiting cost of jobs 1 2 119895 minus 1 119895 + 2 119899 is notchanged we obtain
Φ (Π) minus Φ (Π1015840)
=
119895+1
sum119894=119895
119903119894(1 minus
119894minus1
prod119896=1
119891119896) minus 119903119895+1(1 minus
119895minus1
prod119896=1
119891119896)
minus 119903119895(1 minus 119891119895+1
119895minus1
prod119896=1
119891119896)
= minus119903119895 (1 minus 119891119895+1)
119895minus1
prod119896=1
119891119896 + 119903119895+1 (1 minus 119891119895)
119895minus1
prod119896=1
119891119896
= (119903119895+1 (1 minus 119891119895) minus 119903119895 (1 minus 119891119895+1))
119895minus1
prod119896=1
119891119896 lt 0
(13)
Therefore we have Φ(Π) lt Φ(Π1015840) which leads to a contra-
dictionWith the adjacent pairwise interchange we can knowthat nonincreasing order of 120574119895(1 minus 119891119895) is an optimal policyThis completes the proof
RemarkThe structure of the optimal policy is a little complexcomparing to the optimal policy in the last section In addi-tion to weight and processing time of jobs this optimal policyis also related to the parameters 120582 1198851 120572 It is reasonablesince the objective function is really complex It is howeveralso easy to calculate or estimate those values with the helpof computer Besides if the processing time of current andemergent jobs is followed with some special distribution theoptimal policy will have very simple structure One exampleis stated as follows
Example Assume that 119875119894 sim exp(]119895) and 119885119894 sim exp(120583) We caneasily calculate that
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] = E [119890(minus120572minus120582+120582(120583(120583+120572)))119875
119895]
= E [119890(minus120572minus(120582120572(120583+120572)))119875
It is easy to verify that the optimal policy sequences jobs innonincreasing order of 120574119895(]119895 + 120572 + (120582120572(120583 + 120572)))
4 Conclusions and Future Research
In this paper we consider a new stochastic scheduling modelbased on the emergency jobsWe give optimal policies for twoclasses of objectives one is the total waiting time of jobs andthe other is the weighted sum of an exponential function ofthe waiting times
One can consider other objectives in scheduling areasuch as the weighted number of late jobs Specially we canstudy the environment that each job 119895 has a due date 119863119895which follows the exponential distribution We believe that itis reasonable to study this case Taking the surgery scheduling
Discrete Dynamics in Nature and Society 5
as an example again we can consider the due date for eachelective surgery One reason is that each elective surgeryshould have a due date to wait for operating Otherwise thepatient will die or will leave to search for another hospital forhelpThe other reason is that in some sense this date can alsobe as the commitment to the patients And sometimes thiscommitment is consistent with patientsrsquo conditionThereforeit can be interpreted as a random variable In additionDenton et al [7] consider the overtime to themodel and studyhow case sequencing affects patient waiting time operatingroom idling time (surgeonwaiting time) and operation roomovertimeTherefore maybe we can study the objective whichincludes both the weighted number of late jobs and overtime
E[
[
119870
sum119896=1
119900119896max
0 sum119894isin119878119896
119875119894 minus Δ 119896
+ sum119895
120574119895(119882119895 minus 119863119895)+]
]
(15)
where 119900119896 is the unit overtime cost Δ 119896 is the width of the 119896thregular time and 119878119896 is the subset of cases scheduled in the 119896thday
Another direction we can study is to take the waitingcost of emergency jobs into consideration The reason isthat emergency surgery which cannot be treated in time willresult in substantial losses It reflects that the unit waitingtime cost of emergency jobs is very large in the model Itthus is necessary to operate these cases as soon as possibleClearly sometimes it is nonsense if the new coming jobsfollow the Poisson process since its influence on the time lineis the same Thus in this case maybe we can take anotherdistribution for the emergency jobsrsquo coming Sometimes itis reasonable since in practice emergency jobsrsquo appearance isrelated to the timing For example Summala andMikkola [8]show that the fatal accidents appear more frequently during3amndash5am and 2pm-3pm which is very close to emergencysurgeries
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
We thank the Guest Editor and the three anonymous refereesfor their helpful comments and suggestions The secondauthor is supported by the fundamental research funds ofBeijing Institute of Technology (no 20122142014)
References
[1] M H Rothkopf and S A Smith ldquoThere are no undiscoveredpriority index sequencing rules for minimizing total delaycostsrdquo Operations Research vol 32 no 2 pp 451ndash456 1984
[2] W E Smith ldquoVarious optimizers for single-stage productionrdquoNaval Research Logistics Quarterly vol 3 pp 59ndash66 1956
[3] X Q Cai X Y Wu L Zhang and X Zhou Schedulingwith Stochastic Approaches Sequencing and Scheduling with
Inaccurate Data Nova NewYork 2014 edited byUSotskov andF Werner
[4] M H Rothkopf ldquoScheduling independent tasks on parallelprocessorsrdquoManagement Science vol 12 pp 437ndash447 1966
[5] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 pp 707ndash713 1966
[6] X Q Cai X Sun and X Zhou ldquoStochastic scheduling subjectto machine breakdowns the preemptive-repeat model withdiscounted reward and other criteriardquo Naval Research Logisticsvol 51 no 6 pp 800ndash817 2004
[7] B Denton J Viapiano and A Vogl ldquoOptimization of surgerysequencing and scheduling decisions under uncertaintyrdquoHealthCare Management Science vol 10 no 1 pp 13ndash24 2007
[8] H Summala and T Mikkola ldquoFatal accidents among car andtruck drivers effects of fatigue age and alcohol consumptionrdquoHuman Factors vol 36 no 2 pp 315ndash326 1994
Theorem 4 Sequencing jobs with the nonincreasing order of
120574119895
1 minus 119891119895(8)
are an optimal policy to minimize the objective sum120574119894E[1 minus
exp(minus120572119882119894)] where
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] (9)
Proof Assume that 1 2 119899 is a feasible schedule Let 119884119895 bethe random variable that denotes the total time of processingjob 119895 and processing emergence jobs arriving during theprocessing of job 119895 And let 119873119895 denote the random variableas the number of emergence jobs during the processing ofjob 119895 Observing that the emergence jobs arrive according tothe Poisson distribution with rate 120582 we know that randomvariable (119873119895 | 119875119895) has a Poisson distribution with mean(120582(1 minus 120582))119875119895 Thus
Based on the fact that all 119884119896119896=119899119896=1 are independent random
variables we thus can know that
E [119890minus120572119882119895] = E [119890
minussum119895minus1
119896=1120572119884119896] =
119895minus1
prod119896=1
119891119896 (11)
Therefore the objective function under sequence 11989511198952 119895119899 results is
119899
sum119894=1
119903119895119894
(1 minus
119894minus1
prod119896=1
119891119896) (12)
Without loss of generality we assume that jobs areordered in nonincreasing order of 120574119895(1 minus 119891119895) that is 1205741(1 minus
1198911) ge 1205742(1minus1198912) ge sdot sdot sdot ge 120574119899(1minus119891119899) If this sequence denotedby Π is not an optimal policy there must exist 119895 in someoptimal policy such that job 119895 + 1 is processed before job 119895Without loss of generality let 1 2 119895minus1 119895+1 119895 119895+2 119899denoted by Π
1015840 be such optimal policy and we hence haveΦ(Π) gt Φ(Π
1015840) Comparing these two policies and observing
that the waiting cost of jobs 1 2 119895 minus 1 119895 + 2 119899 is notchanged we obtain
Φ (Π) minus Φ (Π1015840)
=
119895+1
sum119894=119895
119903119894(1 minus
119894minus1
prod119896=1
119891119896) minus 119903119895+1(1 minus
119895minus1
prod119896=1
119891119896)
minus 119903119895(1 minus 119891119895+1
119895minus1
prod119896=1
119891119896)
= minus119903119895 (1 minus 119891119895+1)
119895minus1
prod119896=1
119891119896 + 119903119895+1 (1 minus 119891119895)
119895minus1
prod119896=1
119891119896
= (119903119895+1 (1 minus 119891119895) minus 119903119895 (1 minus 119891119895+1))
119895minus1
prod119896=1
119891119896 lt 0
(13)
Therefore we have Φ(Π) lt Φ(Π1015840) which leads to a contra-
dictionWith the adjacent pairwise interchange we can knowthat nonincreasing order of 120574119895(1 minus 119891119895) is an optimal policyThis completes the proof
RemarkThe structure of the optimal policy is a little complexcomparing to the optimal policy in the last section In addi-tion to weight and processing time of jobs this optimal policyis also related to the parameters 120582 1198851 120572 It is reasonablesince the objective function is really complex It is howeveralso easy to calculate or estimate those values with the helpof computer Besides if the processing time of current andemergent jobs is followed with some special distribution theoptimal policy will have very simple structure One exampleis stated as follows
Example Assume that 119875119894 sim exp(]119895) and 119885119894 sim exp(120583) We caneasily calculate that
119891119895 = E [119890(minus120572minus120582+120582119864[119890minus1205721198851 ])119875
119895] = E [119890(minus120572minus120582+120582(120583(120583+120572)))119875
119895]
= E [119890(minus120572minus(120582120572(120583+120572)))119875
It is easy to verify that the optimal policy sequences jobs innonincreasing order of 120574119895(]119895 + 120572 + (120582120572(120583 + 120572)))
4 Conclusions and Future Research
In this paper we consider a new stochastic scheduling modelbased on the emergency jobsWe give optimal policies for twoclasses of objectives one is the total waiting time of jobs andthe other is the weighted sum of an exponential function ofthe waiting times
One can consider other objectives in scheduling areasuch as the weighted number of late jobs Specially we canstudy the environment that each job 119895 has a due date 119863119895which follows the exponential distribution We believe that itis reasonable to study this case Taking the surgery scheduling
Discrete Dynamics in Nature and Society 5
as an example again we can consider the due date for eachelective surgery One reason is that each elective surgeryshould have a due date to wait for operating Otherwise thepatient will die or will leave to search for another hospital forhelpThe other reason is that in some sense this date can alsobe as the commitment to the patients And sometimes thiscommitment is consistent with patientsrsquo conditionThereforeit can be interpreted as a random variable In additionDenton et al [7] consider the overtime to themodel and studyhow case sequencing affects patient waiting time operatingroom idling time (surgeonwaiting time) and operation roomovertimeTherefore maybe we can study the objective whichincludes both the weighted number of late jobs and overtime
E[
[
119870
sum119896=1
119900119896max
0 sum119894isin119878119896
119875119894 minus Δ 119896
+ sum119895
120574119895(119882119895 minus 119863119895)+]
]
(15)
where 119900119896 is the unit overtime cost Δ 119896 is the width of the 119896thregular time and 119878119896 is the subset of cases scheduled in the 119896thday
Another direction we can study is to take the waitingcost of emergency jobs into consideration The reason isthat emergency surgery which cannot be treated in time willresult in substantial losses It reflects that the unit waitingtime cost of emergency jobs is very large in the model Itthus is necessary to operate these cases as soon as possibleClearly sometimes it is nonsense if the new coming jobsfollow the Poisson process since its influence on the time lineis the same Thus in this case maybe we can take anotherdistribution for the emergency jobsrsquo coming Sometimes itis reasonable since in practice emergency jobsrsquo appearance isrelated to the timing For example Summala andMikkola [8]show that the fatal accidents appear more frequently during3amndash5am and 2pm-3pm which is very close to emergencysurgeries
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
We thank the Guest Editor and the three anonymous refereesfor their helpful comments and suggestions The secondauthor is supported by the fundamental research funds ofBeijing Institute of Technology (no 20122142014)
References
[1] M H Rothkopf and S A Smith ldquoThere are no undiscoveredpriority index sequencing rules for minimizing total delaycostsrdquo Operations Research vol 32 no 2 pp 451ndash456 1984
[2] W E Smith ldquoVarious optimizers for single-stage productionrdquoNaval Research Logistics Quarterly vol 3 pp 59ndash66 1956
[3] X Q Cai X Y Wu L Zhang and X Zhou Schedulingwith Stochastic Approaches Sequencing and Scheduling with
Inaccurate Data Nova NewYork 2014 edited byUSotskov andF Werner
[4] M H Rothkopf ldquoScheduling independent tasks on parallelprocessorsrdquoManagement Science vol 12 pp 437ndash447 1966
[5] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 pp 707ndash713 1966
[6] X Q Cai X Sun and X Zhou ldquoStochastic scheduling subjectto machine breakdowns the preemptive-repeat model withdiscounted reward and other criteriardquo Naval Research Logisticsvol 51 no 6 pp 800ndash817 2004
[7] B Denton J Viapiano and A Vogl ldquoOptimization of surgerysequencing and scheduling decisions under uncertaintyrdquoHealthCare Management Science vol 10 no 1 pp 13ndash24 2007
[8] H Summala and T Mikkola ldquoFatal accidents among car andtruck drivers effects of fatigue age and alcohol consumptionrdquoHuman Factors vol 36 no 2 pp 315ndash326 1994
as an example again we can consider the due date for eachelective surgery One reason is that each elective surgeryshould have a due date to wait for operating Otherwise thepatient will die or will leave to search for another hospital forhelpThe other reason is that in some sense this date can alsobe as the commitment to the patients And sometimes thiscommitment is consistent with patientsrsquo conditionThereforeit can be interpreted as a random variable In additionDenton et al [7] consider the overtime to themodel and studyhow case sequencing affects patient waiting time operatingroom idling time (surgeonwaiting time) and operation roomovertimeTherefore maybe we can study the objective whichincludes both the weighted number of late jobs and overtime
E[
[
119870
sum119896=1
119900119896max
0 sum119894isin119878119896
119875119894 minus Δ 119896
+ sum119895
120574119895(119882119895 minus 119863119895)+]
]
(15)
where 119900119896 is the unit overtime cost Δ 119896 is the width of the 119896thregular time and 119878119896 is the subset of cases scheduled in the 119896thday
Another direction we can study is to take the waitingcost of emergency jobs into consideration The reason isthat emergency surgery which cannot be treated in time willresult in substantial losses It reflects that the unit waitingtime cost of emergency jobs is very large in the model Itthus is necessary to operate these cases as soon as possibleClearly sometimes it is nonsense if the new coming jobsfollow the Poisson process since its influence on the time lineis the same Thus in this case maybe we can take anotherdistribution for the emergency jobsrsquo coming Sometimes itis reasonable since in practice emergency jobsrsquo appearance isrelated to the timing For example Summala andMikkola [8]show that the fatal accidents appear more frequently during3amndash5am and 2pm-3pm which is very close to emergencysurgeries
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
We thank the Guest Editor and the three anonymous refereesfor their helpful comments and suggestions The secondauthor is supported by the fundamental research funds ofBeijing Institute of Technology (no 20122142014)
References
[1] M H Rothkopf and S A Smith ldquoThere are no undiscoveredpriority index sequencing rules for minimizing total delaycostsrdquo Operations Research vol 32 no 2 pp 451ndash456 1984
[2] W E Smith ldquoVarious optimizers for single-stage productionrdquoNaval Research Logistics Quarterly vol 3 pp 59ndash66 1956
[3] X Q Cai X Y Wu L Zhang and X Zhou Schedulingwith Stochastic Approaches Sequencing and Scheduling with
Inaccurate Data Nova NewYork 2014 edited byUSotskov andF Werner
[4] M H Rothkopf ldquoScheduling independent tasks on parallelprocessorsrdquoManagement Science vol 12 pp 437ndash447 1966
[5] M H Rothkopf ldquoScheduling with random service timesrdquoManagement Science vol 12 pp 707ndash713 1966
[6] X Q Cai X Sun and X Zhou ldquoStochastic scheduling subjectto machine breakdowns the preemptive-repeat model withdiscounted reward and other criteriardquo Naval Research Logisticsvol 51 no 6 pp 800ndash817 2004
[7] B Denton J Viapiano and A Vogl ldquoOptimization of surgerysequencing and scheduling decisions under uncertaintyrdquoHealthCare Management Science vol 10 no 1 pp 13ndash24 2007
[8] H Summala and T Mikkola ldquoFatal accidents among car andtruck drivers effects of fatigue age and alcohol consumptionrdquoHuman Factors vol 36 no 2 pp 315ndash326 1994
