P.2008.09 - Optimization model to minimize the makespan in a Hospital's Gastroenterology Service.pdf

7/27/2019 P.2008.09 - Optimization model to minimize the makespan in a Hospital's Gastroenterology Service.pdf

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Los cuadernos de PYLO

- Logstica Hospitalaria -

Optimization model to minimize the makespan in a

hospitals gastroenterology service

Los textos publicados en la serie de los informes de investigacin de la Universidad de los Andes slo

comprometen la responsabilidad de sus autores

C. Serrano, A.M. Jimenez, C.A.

Amaya, N.Velasco.

Universidad de los Andes

PYLO

H 2008 9

Diciembre 2008


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Optimization model to minimize the makespan in a hospitalsgastroenterology service

C. Serrano, A.M. Jimenez, C. A. Amaya, PhD. y N. Velasco, PhD.

Departamento de Ingeniera Industrial, Universidad de los Andes, Bogot, Colombia.

Abstract: This work presents an application case in the gastroenterology service (GS) of a hospital. Itspurpose is to minimize the delays in the transportation of inpatients from their hospital room to the GS, byscheduling patient appointments. The service is represented as a flexible job shop of three stations. The firstone assigns beds to outpatients. In the second one, patients are examined by the specialists and, the last one isthe recovery room. The second station, identified as the bottleneck of the system, is made up of three identicalrooms. Each one is equipped with three dedicated machines. Inside each room, three processes must becarried out. Each process consumes different kinds of resources and these are shared between rooms.

Although several processes can operate simultaneously, their execution is limited by resource capacityconstraints. Furthermore, one resource cannot be used for two procedures at the same time. The objective is tominimize the makespan. A computer tool for solving scheduling problems with heterogeneous parallelmachines was developed based on several heuristics. At the end, 94% of the results obtained using the toolwere better solutions compared with historical data.

Keywords: Hospital logistics, Gastroenterology service, Flexible job shop, Heuristics, Parallel machines.

1. INTRODUCTION

The problem arises in the gastroenterology service(GS) of a private hospital in Bogota, Colombia.Most of service delays are associated with thetransportation of inpatients from the differenthospital rooms to the GS, and the other way round.In the actual situation, a scheduling of the servicesrooms is not established, even if the demand ofinpatients is known at least one day before andoutpatients appointments are booked through a GScall-centre. In consequence, there is no schedule forthe inpatients transportation. Such transportation

requires a setup time and an adequate coordinationbetween the medical personnel and support serviceinvolved in this activity. An improper coordinationbetween the previous actors consumes more timeand occupies more resources than the ones needed

The GS offers three (3) different types of exams:colonoscopy, endoscopy and gastric ball removal.There also exists a combined procedure which

consists of a colonoscopy and an endoscopy, both in

the same appointment. On the other hand, theservice attends two types of demand. An unknowndemand corresponding to patients who come fromthe emergency room (ER), and a known butvariable demand which is divided between twogroups; inpatients and outpatients. Within thiscircumstance, as soon as the gastroenterologistbecomes idle he makes the requirement for aninpatient. As this event is not planned, thetransportation of inpatients can last long dependingon the availability of the nurse who prepares the

patient and supports in charge of the transportationof inpatients in the hospital. During this time, otherpatients can arrive to the service and thegastroenterologist will need to examine them. Whenthe required inpatient arrives reaches the service,there might be no rooms or resources available withwhich he could be examined. As a result, he mustwait.


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The objective of this study is to decrease thedelays presented in the service in order to examinepatients in a more efficient way. It is important tokeep in mind resource constraints related topersonnel and equipment,and also the demands coming from inpatients,outpatients, and the ER. The aggregated value ofthis project is the development of an easy-to-usecomputer tool designed for the medical personnel.The purpose of this tool is to support the GSscheduling and the transportation of inpatients.With its use, the expectation is to increase theservice level given to patients represented in lower(or none) waiting times and to increase resourcesavailability as a consequence of a more efficient useof them.

The remainder of this article is organized asfollows: First, the characterization of the GS isexposed. Then, the methodology used to model the

system is presented and a brief review of theliterature relating to parallel machine problems withallocation and resource constraints is provided.Afterwards, a proposed mathematical representationis shown, followed by a description of severalheuristics for computing GS schedules. Thecomputer tool and the analysis of numericalexperiments based on historical data are presented.Finally, the findings are summarized and somerecommendations are given for the practicalimplementation of the tool.

2. CHARACTERIZATION OF THE GS2.1.The ServiceThe service is composed by three (3) identical

rooms, each one equipped with three differentmachines dedicated to each type of basic exam. Inthe actual system configuration, there is only onegastroenterologist and one anaesthesiologistavailable to examine patients. This means that,although there are 3 rooms, only one patient can beseen at a time. In the same way, the cleaningpersonnel dedicated to wash and clean the machinesafter every exam performed can process just onemachine at a time. In addition of the describedrooms, there is a recovery room which has capacityfor six (6) patients or six hospital beds at the same

time. Inpatients occupy neither a hospital bed norrecovery room space as they come in their ownhospital bed and must be carried to their roomsimmediately after the exam is performed. Onlyoutpatients occupy hospital beds in the GS.

2.2.Patients FlowIn the transportation of a patient to the service,

five (5) actors take part: the GSs secretary (ES)who calls the floors assistant (FA) to make therequirement for the inpatient to the service; thepatients nurse who prepares the patient, and thesupport service (SS) who is in charge of thepatients transportation to the GS. The same appliesto return the inpatient to its hospital room.

The hospital has four (4) floors, an intensivecare unit (ICU) and an ER from which thetransportation of inpatients is also performed. Eachfloor is provided with a FA who is in charge of

receiving all the information concerning inpatientsthat belong to the same floor. In the same way, eachfloor is provided with one SS who is in charge ofthree activities: transportation of patients,transportation of laboratory samples, andtransportation of supplies. The SS carries a radiocommunicator in order to keep in touch with theFA.

When an outpatient arrives to the GS, the nextsteps are followed. First, the payment of the

examination procedure is done. Then, the patientmust change clothes and occupy an availablehospital bed. Afterwards, the patient is sedated orinjected with anaesthesia, as required, by theanaesthesiologist inside the room. This process isfollowed by the exam done by thegastroenterologist. Finally, the patient goes to therecovery room. Figure 1 illustrates this flow. Eachtime a machine is used it must be washed by thecleaning personnel. After the cleaning is done, themachine will be available for another procedure in

the corresponding room.More over when an inpatient arrives to the GS,

the same activities are performed without theprocesses of assigning a hospital bed, changingclothes and the recovery.


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Figure 1. Patients flow in the service.

2.3. DemandSeveral observations must be done with respect

to the demand of gastroenterology exams inside thehospital. When concerning inpatients, 29.73%comes from the ER; 28.38% comes from the secondfloor; 21.62% comes from the fourth floor and the

remaining comes from the first and the third floors.It is important to highlight the fact that an importantnumber of inpatients come from the ER. This showsthat, as this demand is random, the service must beprepared in order to examine the unscheduledattendance of patients from the ER (free capacity).In addition, the exam with the greatest inpatientsdemand is the endoscopy (55.41%). The remainingpercentage of patients demands either acolonoscopy or an endoscopy-colonoscopyprocedure in the same proportion.

Concerning outpatients, the most frequentlydemanded exam is the endoscopy (53.36%),followed by colonoscopy (21.08%), endoscopy-colonoscopy (24.44%), and gastric ball removal(1.12%). Generally speaking, the demand ofoutpatients is higher on Tuesdays, Wednesdays andThursdays and minimum on Mondays.

2.4.Process and Delay TimesA preliminary analysis was done to determine

the factors which can affect the stay time of patientsinside the GS such as the patients age and the typeof exam required. The first conclusion was thatthere is no relation between the patients age andoutpatients stay time in the service; as well as nocorrelation exists between the gastroenterologistwho performs the exam and the patients stay time,concluding that the stay time is notgastroenterologist dependant. In addition, patients

stay time in the service is independent from theanaesthesiologist, the cleaning personnel, the day ofthe week and the time at which the exam isperformed.

Average times were found for the processing

times to quantitatively characterize the system.These times include: patients change of clothes,setup of patients inside the room (sedation,anaesthesia), setup of machines (special cleaning),setup of the gastroenterologist (wearing medicalgloves, medical mask, goggles and medical suit),setup of the room (cleaning at the end of the day),the processing time of the exam and the patientsrecuperation time. Table 1 report these times.Deterministic times were assumed.

Processing Times in the Gastroenterology Service

CommentsAverage Times

Minimum - Maximum(mins)

Setup time of the patientoutside the room

3 - 7

Setup time of the patientinside the room

Gastric BallRemoval

60

Anotherexam

10 - 15

Setup time of hospitalbeds

3 - 7

Setup time of machines 10 - 15Setup time ofgastroenterologist

3 - 5

Setup time of the room Per room 20Cleaning time of the GS 30 - 60

Examination Time

Endoscopy 3 - 5Colonoscopy 710Endoscopy-

Colonoscopy1015

Gastric BallRemoval

10

Recuperation Time 60Table 1. Processing times in the GS.


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Due to the lack of exam scheduling in theservice, there exist delay times. Inpatients and themedical personnel in the service have to toleratethese delays, caused by the uncoordinated flow ofinformation, to transport inpatients into the GS andout from it. Graphs 1 and 2 show the averagewaiting times for inpatients to arrive and to releasethe GS, represented as delays in the system.

Graph 1. Average waiting time until SS picks up thepatient

Graph 2. Average waiting time for the patient to arrive tothe service.

It is important to highlight the fact that, whenthe waiting times are greater than 15 minutes, it ismore likely to have an outpatient arriving, and sothe requirement of medical personnel in the GS toexamine it. In consequence, the service will beblocked when the inpatient arrives as no resourceswill be available. In the same way, delays in pickingup inpatients block the service area reducing thecapacity of the recovery room as inpatients mustwait inside the rooms until SS arrives.

3. MODELLING THE PROBLEMThe actual situation can be modelled as a

flexible job shop with three (3) stations. Station 1being the hospital bed assignment for outpatients,Station 2 is composed by the three (3) examinationrooms and Station 3 corresponds to the recoveryroom. This configuration is shown in Figure 2.

3.1.Stations 1 and 3In Station 1 the hospital beds assignment is done

as well as the change of bed sheets. This processlasts in average 5 minutes. In Station 3 takes placethe outpatient recuperation which lasts for about 60minutes. These times are independent from thepatient and the resources used. These two stationsshare resources such as the hospital beds and the

nurse. This happens because a hospital bed isassigned to a patient in Station 1 and is freed onlyuntil the patient finishes with his recuperation inStation 3. In this way, both stations are composedby six (6) hospital beds which can be occupied inparallel by outpatients.

Figure 2. GS divided into stations.

If the system is seen in steady state, Station 3dominates Station 1 because these stations shareresources and the processing time in Station 3 isgreater. A patient/hospital bed is releasedtheoretically every 11 minutes (cycle time) underthis scenario.

3.2.Station 2This is the bottleneck station. The processes

performed in this station are: sedation oranaesthesia injection; the corresponding exam andmachine cleaning. Sedation or anaesthesia injectionprocess is performed exclusively by theanaesthesiologist, inside the room. Similarly, thegastroenterologist performs the exam and thecleaning personnel wash the used machines.Calculating cycle time in this station is not as

simple as in Stations 1 and 3 because it depends onthe sequence of the performed exams. However,considering various scenarios, the cycle time, wouldresult in 28.32 minutes in average with the actualresources configuration. This cycle time is greaterthan the cycle time found for stations 1 and 3 whichconfirms that Station 2 as the bottleneck station.


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The analysis for this station reveals somecharacteristics of the system: the machine cleaningprocess can be done simultaneously with otherprocesses (sedation or anaesthesia injection, exam,doctor setup) only when the machine being cleanedup is not the same one required in the examperformed. Considering Station 2 as the bottleneckstation, it is necessary to find the way of schedulingthe exams in each room keeping in mind theavailability of the required resources. The scheduleneeded must be one that minimizes the time inwhich the last patient leaves the service (makespan).In this way, a minimal waiting time for the patientsand greater resource utilization can be obtained. Byscheduling the GSs processes, it is expected thatthe actual delays in the transportation of inpatientswill be eliminated as the time for each patient toarrive to the GS will be known. So, it will bepossible to program the SS for patient transportationto be done on time.

4. LITERATURE REVIEWTo solve the non-identical parallel machines

problem or the unrelated parallel machines problem,a lot of research has been done. Using the standardnotation, this problem can be written as max||CR with processing times ijp in machine i , for job j

(See Lawler et al [5]). According to Garey andJohnson [11] this is an NP-hard problem in the

strong sense. So, a lot of effort has been dedicatedin order to design approximating algorithms. Themost important contributions are the ones done byHorowitz and Sahni [4], Ibarra and Kim [12], Deand Morton [13], Davis and Jaffe [3], Potts [2],Lenstra et al. [7] and by Hariri and Potts [1].

If GS rooms are regarded as parallel andidentical machines, the Longest Processing Timefirst (LPT) heuristic gives a schedule near theoptimal to minimize the makespan [10]. In the caseof the GS, the problem has an additional resource

constraint and so, the given solution using thisheuristic might not be necessarily the optimal, but atleast can produce feasible solutions.

The Least Flexible Job first rule (LFJ) [8] givesan optimal schedule for the problem

max|,1| CMpPm jj . This is a problem with

mparallel machines with processing times all equalto 1 and dedicated machines for each job. To obtain

optimality, it must be ensured that the machine sets

jM in which each job j might be processed, are

nested. This is, that each set of jobs jM must meet

at least one of the following conditions for jobs j and k:

1) kj MM 2) kj MM 3) jk MM 4) kj MM The imposed constraints in this heuristic, in

order to obtain an optimal schedule, are not valid inthe GS problem because the pi are not equal to 1and also, because the subsets of jobs Mj are notnested. Nonetheless, this heuristic is adapted for theGS because the availability of the resourcesrequired will be taken into consideration as thesubset of rooms which can be used by a specificpatient.

Although the GS problem is similar to a max||CR problem, there is an important constraint not yetpresented: machines cannot be used at the sametime because there is only one gastroenterologist inthe actual configuration of the GS. Various patientscannot be sedated or injected with anaesthesiabecause there is only one anaesthesiologist. Theseare resource constraints.

Not much research has been done to approachthis type of problems.

The problem cannot be analyzed as a singlemachine problem because the time to perform anexam, depending on the exam performed, can beshort in relation with other processes in the system.Nonetheless, an important research was done byKellerer et al. [6] where resources were supposed tobe consumed in a simultaneous way during theprocess. In the GS problem, resources such as thegastroenterologist, the anaesthesiologist and thecleaning personnel are used but not in a

simultaneous way but in a sequential way.According to Kellerer et al. [6] this problem can

be described as a max|| CresPDm problem inwhich PDm means there are mparallel dedicatedmachines, 1 is the total amount of resources inthe system, 1 represents the units available of

resource )1( and 1j the units of


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resource that job j consumes at any momentduring the process. If different jobs require the sameresource at the same time, they can be processedsimultaneously only if their total consumption ofthe resource does not exceed the quantity . It is

assumed that there is only one machine that canprocess job type i. The objective to be evaluated isalso the makespan minimization and no pre-emption

of any job is allowed. In the case of the GSproblem, there are three machines available toprocess job type i even if these machines areavailable in different rooms.

It might not be easy to obtain an optimalschedule for the GS problem but a solution near tothe optimal can be found by generating anapproximating algorithm which can be used todevelop an easy-to-use computer tool for themedical personnel, in order to obtain a betterappointment schedule. For this application case, thementioned heuristics (LPT and LFJ) and Kellerer etal. procedures were adapted to generate anapproximating algorithm to find asserted schedules.Four heuristics were proposed with the aim ofcomparing their results and determining which onegives the minimum makespan.

5. MATHEMATICAL DESCRIPTION OF THEPROBLEM

In this section, the model formulation for a

restricted version of the problem is introduced. Forthis model two assumptions were made. The firstone state that a room cannot be used until the lastmachine used at this room finishes its cleaningprocess. Relaxing this assumption would result inmachines being cleaned (i.e. colonoscopy machine)and used (i.e. endoscopy machine) in the same roomat the same time. Note that under this scenario anavailable colonoscopy machine from one particularGS room cannot be used to perform a colonoscopy-endoscopy exam if the endoscopy machine is being

cleaned. The same applies the other way round.Generally speaking, relaxing the above

constraint makes the problem more complexallowing a machine to be cleaned (outside the room)at the same time a patient is being sedated, injectedwith anaesthesia or examined inside the room. Thiscould happen only if the required machine for theexam is not the same one that is being cleaned.Summarizing, both scenarios (restricted and relaxed

versions of the model) execute the three processesin its natural sequential way: (1) sedation, (2) examand (3) machine cleaning, with the difference thatthe restricted version does not admit simultaneousprocesses intra-room. However, the two versionsallow simultaneous activities inter-room (machinecleaning at the same time an exam is taking place, ifeach machine belongs to a different room).

In the second assumption for the restrictedversion, it is supposed that the recovery room hasinfinite capacity, so the availability of hospital bedsis not considered in the mathematical formulation.As mentioned above, in the actual systemconfiguration there is only one anaesthesiologist,one gastroenterologist and one cleaning person. Inaddition, Station 2 is recognized as the bottleneckstation of the system. Thus it is reasonable to workwith the restricted version of the problem with theaim of computing optimal solutions and compare

them with the results given by heuristics.Although the mathematical model for the

restricted version is developed in subsection 5.2, thecomplete constraints of the problem are enunciatedfor future research.

5.1.Whole System Constraints(1) Only one patient can be examined in each

room at a time.(2) Patients are examined only once, exactly at

one room.(3) Patients cannot be examined if the requiredresource is not available, keeping in mindthat:a. The gastroenterologist examines one

patient at a timeb. The anaesthesiologist examines one

patient at a timec. A machine will only be used if it is

available (clean).d. The cleaning personnel processes one

machine at a time.(4) Processes precedence must be respected(sedation, exam, machine cleaning, andrecovery).

(5) Only one machine is needed to examine apatient, except in the endoscopy-colonoscopy appointment.

(6) Sedation or anaesthesia injection and examprocesses must be performed inside a room.


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The cleaning process is performed outsidethe room.

(7) Machine cleaning and patient sedation orexamination can be done at the same time,only if the required machine for the exam isnot the same one being cleaned and does notbelong to the same room as the one beingused to examine the patient.

5.2.Restricted Version of the SystemLet Adenote the set of patients, indexed by i,

and B denote the set of rooms, indexed by j. Cdenotes the set of processes done in the GS.{1=anaesthesia, 2=examination, 3=machinecleaning, 4=recuperation}, indexed by k. tProci,k isthe duration of process k for patient i;Preci,j is aparameter that equals 1 if process iprecedes processj and 0, if not. Consider the following decisionvariables:

xi,j,k: Process kstarting time in roomjfor patient iyi,j : 1 if patient iis examined in roomj. 0, if notzi,j : 1, if patient iprecedes in time patientj. 0, if not

The makespan objective is similar to min( max (xi,j,k) ) CkBjAi ,, . Constraints for the restrictedversion of the problem are enunciated as follows:

5.2.1.1. Examine only one patient per room)]1()1()1[(*)Pr( ,,,3,3,,1,, jbjabaajajb yyzMoctxx

CkBjAbaba 1,3,},,|{

5.2.1.2. All patients are examined only once,in only one room

Bj

jiy 1, Ai

5.2.1.3. Patient cannot be examined if therequired resource is not available, respectingprecedence and room occupancy.

)]1()1()1[(*)Pr( ,,,1,1,,1,, dbcabaacadb yyzMoctxx

)]1()1()1[(*)Pr( ,,,2,2,,2,, dbcabaacadb yyzMoctxx)]1()1()1[(*)Pr( ,,,3,3,,3,, dbcabaacadb yyzMoctxx

CkBdcdcAbaba 3,2,1},,|{},,|{

Note that patient ais sequenced before patient bin different rooms cand d.

5.2.1.4. Respect the precedence of processes

)1(*)Pr( ,,,,,, jikikjilji yMoctxx

1Pr|,,, ,lkecClkBjAi

5.2.1.5. Respect patients precedence1,, ijji zz },|{ Ajiji

5.2.1.6. The nature of variables0,, kjix CkBjAi ,,

}1,0{,ji

y BjAi ,

}1,0{,jiz Aji,

5.3.Model ResultsResults for the optimization model were

obtained with Xpress-MP Professional release2007. Although the systems complexity was

reduced due to mentioned assumptions, the softwareis able to compute optimal results in a short time foreight (8) patients or less. For nine (9) or morepatients, the software takes more than half an hourto process the model without giving optimal results.Because in daily operation the GS does not examineless than nine (9) patients a day, the results obtainedby the software cannot be used as optimal.Nevertheless, various historical scenarios wereprocessed for half an hour in the software obtainingresults near to the optimal schedule. From these notoptimal results, it can be observed that they areconsistent and generate schedules that are coherentwith resource constraints and with the type ofexams. Furthermore, 76.5% of the results presenteda shorter makespan than the observed in historicaldata. Thus, the model is a good approximation tothe problem. However, this model is arepresentation of a different system from the realsituation and a better mathematical approach has tobe found in order to determine a way of schedulingthe examination appointments with the realconstraints.

6. HEURISTICS6.1.

Heuristic 1: LPT (Longest Processing Timefirst)

This heuristic consists in ordering the jobs bytheir total processing times from longer to shorter.The processing times in this case, corresponds tothe total expected time of the patient inside the GS:the sedation or injection of anaesthesia time, plusthe exam duration and the time spent at the recovery


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room. In this way, the patients (outpatients andinpatients) will be ordered from first to last in thefollowing way: gastric ball removal, endoscopy-colonoscopy, colonoscopy, endoscopy withsedation, endoscopy without sedation. The steps tobe taken in order to generate schedules are:

Step 1: Choose a patient with the longest totalaverage processing time in the GS as specified

before and who has not yet been scheduled.Step 2: Assign this patient to an available room ifthe required machine(s) are also available. Beginthe process in this room when anaesthesiologistand gastroenterologist are freed from the lastpatient and ready to examine the patient. Updateclock and availability time for used machine(s).Step 3: Return to Step 1 until all patients areexamined.

6.2.Heuristic 2: RDM (Resource DependentMachines)

Based on Kellerer et al.s [6] work, the GSsproblem is similar to the max|311|3 CresPD problem.For this problem, the authors do not present anapproximating algorithm noting its complexity.Nevertheless, it was possible to adapt the schemedeveloped for the problem max|11| CresPDm . Thesteps needed to generate this heuristic are:

Step 1: Create an artificial machine representingthe joint process endoscopy-colonoscopy. Set

time equal to zero.Step 2: Find an available machine in the currenttime.Step 3: Determine if the room corresponding tothe machine in Step 2 is available. (Yes: go toStep 4; No: return to Step 2).Step 4: Find an unscheduled patient who requiresthe selected machine.Step 5: Assign the patient to a room and wait forthe gastroenterologist and anaesthesiologist toexecute procedures. Update clock andavailability time for used machine(s).Step 6: Return to Step 2, until all patients arescheduled.

6.3.Heuristic 3: LFJ (Least Flexible Job First)Assuming that all patients are ready at time zero

(initial time), as well as resources are available;jobflexibility is calculated each time counting the

amount of available rooms to examine a patient.Room availability implies that the requiredmachines are also available at time t. The sequencerule gives priority to patients with the least numberof rooms available (least job flexibility) to enter theservice.Flexibility is computed each time a machineis freed as resource availability varies in time. Thesteps to implement this heuristic are the following:

Step 1: For each unscheduled patient, find itsflexibility.Step 2: Choose the patient with the minimumflexibility.Step 3: Let this patient occupy any of itsavailable rooms. Wait for the gastroenterologistand the anaesthesiologist. Examine the patient.Update clock and availability time for usedmachine(s).Step 4: Return to Step 1, until all patients areexamined.

6.4.Heuristic 4: LPT + LFJHeuristic 4 is just a hybrid between heuristic 1

(LPT) and heuristic 3 (LFJ). In this way, patientswill be ordered by the LPT rule and then heuristic 3(LFJ) will be imposed. The steps to generate thisheuristic are:

Step 1: Create a list of patients ordered by theirtotal processing time in the GS (as specified inheuristic 1).

Step 2: In this order, compute the amount ofrequired machines that are available in idlerooms, at that time, for each unscheduled patient.Step 3: If no machines are available for anypatient, advance the clock to the first resourcerelease. Else, continue with Step 4.Step 4: Choose a patient who has the leastamount of rooms available to perform an exam.Step 5: Let this patient occupy any of itsavailable rooms. Wait for the gastroenterologistand the anaesthesiologist. Examine the patient.

Update clock and availability time for usedmachine(s).Step 6: Return to Step 2, until all patients arescheduled.

7. COMPUTER TOOL DESCRIPTIONThe computer tool was designed in Microsoft

Excels Visual Basic as it is the only software


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resource that the personnel use in the GS. Patientsinformation was saved in a data table. Input dataincluded the type of appointment (Ambulatory, FirstFloor, ER, etc.), patient full name, type of exam,sedation or injection of anaesthesia requirementsand patient age. Figure 3 illustrates thecorresponding user interface. On the other hand,system parameters such as process times andresource quantities can be modified, as well as olddata can be erased, using the respective yellowbuttons shown in Figure 3. The blue button is usedto run heuristics and obtain schedules.

7.1.Changing Systems ParametersWhen the Parameter Configuration button is

clicked, the tool will show three (3) windows whichallow the user to modify the following parameters:processing times for each type of exam, thepatients setup time (sedation or anaesthesia), the

gastroenterologist setup time and cleaning time forthe machines. This button also allows the user tochange the number of hospital beds available andthe number of rooms in the GS, as well as medicalpersonnel in charge. Figure 4 shows an example ofone of these interface windows.

Figure 3. Data table of the patients information

Figure 4. Example of an interface window of the tool

7.2.Running the Heuristics

The button Generate Schedule allows the user tochoose one of the four proposed heuristics. Then itdisplays a second data table with the resultingexamination schedule. This data table contains, inaddition, some patient information: (1) timedifference between scheduled appointments, (2)scheduled appointments starting time (first one at7:00 am), (4) the assigned room (5) the estimatedtime for a patient to leave the GS, and (6) the totaltime the patient stays in the service. Other three (3)columns are available for the user in order togenerate future statistics to improve systemmodelling. These three columns correspond to: thereal patient arrival time, the real time in whichpatient entered the service and the time the patientleaves the service.

8.NUMERICAL RESULTSThe computer tool takes less than half a second

to compute and print results. This fact supports itspractical use, in addition with the following results.

8.1.Heuristics RankingAfter 34 evaluated instances using historical

data from May and June 2008, comparative resultsfrom heuristics are shown in Graph 3.

Graph 3. Heuristics Ranking

For the considered instances the best heuristic tosolve the GS problem and to minimize themakespan was LPT, followed by the hybridheuristic LPT + LFJ. The worst heuristic was foundto be the LFJ one.

Following the obtained schedules with LPTheuristics, inpatients and patients coming from theER will always be examined after outpatients in aparticular day because they (inpatients) occupy theleast number of resources and so, their totalprocessing time in the service is the lowest. SS ofeach hospital floor must be available and preparedto transport patients from their hospital room to the


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GS and the other way round at these scheduledtimes. It will not interfere with ER patients as theymust have a setup time to prepare themselves forthe exam which can be done earlier.

Comparing LPT results with the relaxed versionof the mathematical model, 76.5% of the resultsobtained by the heuristic were better in terms ofmakespan. The difference between the makespan

given by the relaxed system and the one given bythe heuristic LPT is in average 14,26 minutes. Thisresult reveals the fact that the schedule given by theheuristic is better than the one obtained with themathematical model. Nevertheless, both are quiteclose and coherent.

8.2.Systems ImprovementsAn average finishing time for the last patient to

release the system cannot be determined becausethe amount of patients examined varies from day to

day. However it is possible to compare theoreticalfinishing times given by the LPT heuristic with dayto day historical data. Graph 4 illustrates theseresults. The conclusion is that in 94% of the cases,the LPT heuristic found a better solution. Inaverage, the improvement in the makespan was onehour, an important improvement to consider incases of emergency arrivals and extraordinary nonprogrammed patients. In the same way, nurses andthe cleaning personnel will have enough time toclean up the service in order to have it ready for the

next day without having to leave later at night,avoiding overtime costs for the GS administration.

Graph 4. Difference between Historical and Theoretical

finishing times of the last patient.

The stay time of patients in the GS alsoimproves due to GS programming. A comparisonwas done between the average historical andheuristic results, as Table 2 shows. Differences varyfrom 5 to 22 minutes. A difference of 5 minutesmight not be very important; that means that in theactual situation, patients who require the endoscopy

or colonoscopy exams have low waiting times.Nevertheless, decreasing the stay time by 18 to 22minutes for the other procedures is a very importanttime reduction. It is important to underline the factthat the difference between these two times can alsobe interpreted as more efficient resource utilizationderiving in a better service level.

DIFFERENCE BETWEEN HISTORICALAND THEORETICAL STAY TIME

Endo ColoG. Ballremoval

Endo-colo

Historical 01:34 01:48 02:37 01:40

Theoretical 01:25 01:30 02:15 01:35

Difference 00:09 00:18 00:22 00:05Table 2.Stay time improvement

However, the most important result is thatmedical personnel will be able to examine thepatient immediately after the patient arrives to the

GS (assuming that the patient arrives at the exactappointment time). So, the patient will no longerhave to sit down waiting for his turn (if it is anoutpatient). To accomplish this result, werecommend to advice the patient to arrive at least 15minutes before the examination appointment forpayment and paper processes before the exam.

Historical data reveals that waiting times in thewaiting room can be up to one and a half hours. Ifthe theoretical schedule is performed in the realsituation, the waiting time will get near cero for all

patients and no more delays will be presented in thetransportation of inpatients.

9. CONCLUSIONThis work shows that, by scheduling the service,

it is possible to improve the service level (measuredin waiting times and stay time in the system), andthe utilization of resources. Nevertheless, the use ofthe computer tool will introduce changes in theGSs organizational culture, specifically in activities

related with the appointments assignation. The maindifference is that once a patient requests anappointment, a confirmation process will berequired for implementing the computer tool. Thismeans that the time of an appointment will not beknown in the same moment the patient asks for one.Only until the complete list of patients who will bescheduled to a specific day is known, the computer


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tool will be used and so the time of theappointments will be revealed. Considering thisfact, we propose the following steps to scheduleexamination appointments (for outpatients).

Step 1: Indicate patients to have the whole dayavailable or at least the whole morning (as it isrequired due to anaesthesia procedure). Up to 20outpatients can be scheduled for a day.

Step 2: Explain patients what are the medicalconditions and requirements to perform eachexam (depending on the type of exam andanaesthesia).Step 3: Advice patients that the exact time of theappointment will be confirmed at most one daybefore the appointment. As a general rule, wesuggest that appointments for a specific day mustnot be scheduled until two days before that day.Step 4: Ask patients to arrive 15 minutes beforethe real appointment time for payment and paper

work. Patients must also be warned about thefact that arriving earlier would not alter thesequence in which patients are attended. Butarriving late will make them lose theirappointment.

If the last steps are too hard to implement, timeintervals can be used in order to give the patient anapproximate time for the appointment, dependingon the required exam. The time intervals to be usedare shown in Table 3. The only difference is that the

patient will have an idea of the possible time hisappointment will be scheduled. Either way, it willnot be until two days before the appointment thatthe exact time will be revealed. The length of thetime intervals shown in Table 3 considers thedifference in the demand for each exam type. Notethat for this scenario only LPT can be used and thatfor inpatients it will not be necessary to use Table 3as inpatient cases are submitted directly by thegastroenterologist. Inpatient schedule must be usedto schedule SSs activities for programmingtransportation at right times.

TIME INTERVALS BETWEEN

EXAMINATION APPOINTMENTS

Exam Time interval

Gastric ball removal 07:00 09:00Endoscopy-colonoscopy 07:00 10:00Colonoscopy 09:00 11:00Endoscopy 10:00 13:00

Table 3. Time intervals for the appointments schedule

If none of these alternatives is accepted toschedule examination appointment dates in the GS,either by the medical personnel or because theperception that patients will reject them, somepolicies may be used to schedule the appointmentsand obtain a schedule similar to the one generatedby the LPT heuristic. The policies are:

Schedule gastric ball removals as earlier aspossible. No other appointments should bescheduled for the next 60 minutes (consideringactual system configuration).Schedule exams in the LPT suggested order andkeep Table 3 in mind.Time between appointments for endoscopies andcolonoscopies (not to the same patient) must be15 minutes.When two different types of procedures arescheduled one after the other, the time between

these appointments must oscillate between 20and 30 minutes (except for gastric ballremovals).If various endoscopy-colonoscopies arescheduled one after the other, time between theseappointments must be at least 18 minutes.If the patient needs an appointment for anendoscopy without sedation, it must bescheduled 30 minutes after the previousscheduled patient. It is preferable if the previouspatient required an endoscopy with sedation.

Although the schedule that could be obtainedusing these recommendations will not be asefficient as the obtained by the computer tool, thesepolicies will improve the service level and resourcesutilization.

10. ACKNOWLEDGEMENTSThe author would like to thank Fair Isaac

Corporation for providing the Xpress-MP softwarelicenses under the Academic Partner Programsubscribed with the Universidad de los Andes; alsoDr. Fernando Sierra and his team for their supportand help during the development of this project.The author also thanks ngela Jimnez, NubiaVelasco and Ciro Amaya for their support and helpduring the development of this project.


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[8] M. Pinedo, Scheduling: Theory, algorithmsand systems, Prentice Hall, Second Edition(2002) 103-104.

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