operating room allocation using mixed integer mip report

8/8/2019 Operating Room Allocation Using Mixed Integer MIP Report

1/26

OPERATING ROOM ALLOCATION USING MIXED

INTEGER PROGRAMMING(MIP)

Seminar Report

Submitted in partial fulfillment of the requirements for the award of the

degree of

Master of Technologyin

Industrial Engineering and Management

by

KAILAS SREE CHANDRAN (Roll No.: M100447ME)

Department of Mechanical Engineering

NATIONAL INSTITUTE OF TECHNOLOGY CALICUT

November 2010


2/26

CERTIFICATE

This is to certify that the report entitled OPERATING ROOM

ALLOCATION USING MIXED INTEGER PROGRAMMING(MIP) is a

bonafide record of the Seminar presented by KAILAS SREE CHANDRAN

(Roll No.: M100447ME), in partial fulfilment of the requirements for the award

of the degree of Master of Technology in Industrial Engineering and

Technology from National Institute of Technology Calicut.

Dr. R. Sridharan

Faculty-in-Charge

(MEA692- Seminar)

Dept. of Mechanical Engineering

Dr. S. Jayaraj

Professor & Head

Dept. of Mechanical Engineering

Place : NIT Calicut

Date :


3/26

ACKNOWLEDGEMENT

I am deeply indebted to my guide Dr. R. Sridharan, Professor, Department of Mechanical

Engineering, for his invaluable guidance, consistent encouragement and suggestions

throughout the course of the work.

I wish to express my sincere thanks to Dr. S. Jayaraj, Professor and Head, Department of

Mechanical Engineering, for providing the necessary facilities to carry out this work.

Last but not the least, I extend hearty thanks to all our teachers whose constant support

and encouragement helped me to complete this seminar in time.

KAILAS SREE CHANDRAN


4/26

ABSTRACT

In most hospitals, the waiting period for surgeries are high and efforts

made in the efficient utilization of resources can prove to be productive in long

term. The aim of this seminar is to present a model which can optimally allocate

the Operating Rooms of the hospital and to minimize the total length of stay of

patient. The model is a mixed integer programming (MIP) methodology which

determines a weekly operating room(OR) allocation template that minimizes

patients cost measured as their length of stay. A number of patient type priority

(e.g., emergency over inpatient) and clinical constraints (e.g., maximum number

of hours allocated to each specialty, surgeon and staff availability) are included in

the formulation. The optimal solution from the analytical model is entered into asimulation model that captures some of the randomness of the processes (e.g.,

surgery time, demand and arrival time). The simulation model outputs the average

length of stay for each specialty and the room utilization. On a case example of a

Los Angeles County Hospital, how the reduction in hospital length of stay

pertaining to surgery is shown. A proposal for a new constraint which takes into

account the Post-Operative Care(POC) capacity is also explained.


5/26

CONTENTS

List of Symbols ii

List of Tables iii

1. Introduction 11.1. Introduction 11.2. Problem Background 11.3. Objective of the Model 31.4. Outline of the Report 3

2. Literature Review 42.1. Literature Review 4

3. Modeling 53.1. Problem Identification 53.2. Problem Formulation 63.3. Notations 63.4. Assumptions 73.5. Decision Variables 83.6.

Objective Function 9

3.7. Constraints 103.8. Simulation Modelling 123.9. Proposal: Post-Operative Care Beds Capacity Constraint 13

4. Case study 144.1. Description 144.2. Solving the Model and Results 14

5. Conclusion and Scope for Future Works 175.1. Conclusion 175.2. Scope for Future Works 17

References 18


6/26

ii

LIST OF SYMBOLS

: Penalty rate for unmet demand.

: Penalty rate for undersupply of OR hours to a specialty.

h: Total amount of idle time of all non-emergency ORs.

i: Index for room type.

j: Index for medical specialty.

k, l: Indices for days.

s: Amount of staffed hours per day.

D: Set of days.

I: Set of room types.

J: Set of medical specialties.

Subscripts

kl: The number of days delayed if a surgery is postponed from day kto day l.

ai: number of operating rooms of type i.

bjk: The amount of idle time of the OR allocated to specialtyj on day k.

cjk: The maximum number of operating rooms that specialtyj can utilize on day k.

ejk: Emergency patients surgery demand for specialtyj on day k, (hours).

ojk: Non-emergency patients surgery demand for specialtyj on day k, (hours).

pj: Oversupply of OR hours to specialtyj.

qj: Undersupply of OR hours to specialtyj.

ujk: Specialtyjs unmet non-emergency demand on day k.

xijk: The number of operating rooms of type i allocated to specialtyj on day k.

yjk: Amount of the emergency ORs staffed hours allocated to specialtyj on day k.

zjkl: Specialtyjs non-emergency demand postponed from day kto day l.


7/26

iii

LIST OF TABLES

4.1 OR capacity allocated to each specialty per week 15

4.2 Summary of simulation results for different s 16

4.3 Performance(simulation results) summary with the original demand 16


8/26

1

CHAPTER 1

INTRODUCTION

1.1INTRODUCTIONSurgery is an important activity in most hospitals and clinics since it is

estimated to generate around two thirds of hospital revenues. On the other hand, it

accounts for approximately 40% of hospital resource costs, including the costs of

personnel (surgeons, anaesthetists, nurses, etc.) and facilities (operating rooms,

intensive care beds, etc.). Surgery takes place in a context of challenging trends

such as heavy expenditure on health care, increasing rates in health care costs, and

rising surgery demand due to aging populations and technological advances thathave broadened the scope of surgical interventions. In this context, hospital

management is subject to ever mounting pressures to control surgical costs while

ensuring quality of care for surgical patients in a timely manner. A successful cost

containment strategy must integrate decision-making at all levels: strategic,

tactical, and operational. At the operational level, one of the main problems is

surgical case scheduling.

In many hospitals, a large percentage of patients undergo some type of

surgery when admitted or during their stay in a hospital. To reduce a patients

length of stay and also for the sake of the patients health, it is ideal that a surgery

is performed as soon as it is requested or judged to be needed. However, it is

common in a hospital that patients have to wait in Bed for their surgery for a

couple of days or sometimes a longer time (especially in public hospitals).

In a Hospital environment, significant amount of time and resource is

allocated for the working of operating theatres. This is made more complicated by

the demand being uncertain. If optimum allocation strategies are devised, then

worthy savings in time and there by resources can be achieved.

1.2PROBLEM BACKGROUNDIn many hospitals, surgery planning and scheduling is carried out as

follows or in a similar fashion. At the beginning of each week or each month,

the surgery planning office or a similar unit of the hospital builds an Operating


9/26

2

Room(OR) allocation template, or so-called weekly "Block Time Schedule",

which allocates blocks of OR capacity to emergency surgery and various

specialties non-emergency surgery. Oftentimes, each block of OR capacity is one

day of staffed hours of an operating room.

Before each working day, the doctors determine which inpatients in their

specialty will have surgery performed on the following day. When making these

decisions, they usually first accommodate outpatient surgeries scheduled for the

next day because these have been previously scheduled many days in advance.

Also, they take into account the number of blocks/rooms allocated to their

specialty on that day, and the order and degree of urgency of all the active

inpatients' surgery requests.

During a working day, surgeons try to finish as many scheduled surgeries

as possible, in a predetermined order. In addition, emergency surgery demand

arises almost every day, and surgeons try to operate upon these emergency

patients as soon as possible because of their critical condition. Usually, they are

sent into the emergency OR immediately, as long as it is available. If the

emergency OR is busy when needed, the emergency patient is operated in one of

the non-emergency rooms allotted to the particular specialty in which the patient

or the needed surgery belongs. As a result, some scheduled inpatient and

outpatient surgeries may have to be postponed to a later date or rescheduled.

Also, surgeries scheduled for the afternoon may not be performed because they

are too much behind schedule, and they cannot be completed within the staffed

hours if started.

Though there may be some real-time adjustments to the Block Time

Schedule (i.e., one specialtys surgery is performed in an OR allocated to another

specialty or occasionally some non-emergency surgery in the emergency OR)

during the working days, surgeons do follow it as much as possible, because each

specialty may require special medical equipment or prior preparations in the OR,

and any change to the schedule may cause confusion in the daily work and take an

extra amount of setup or switching time. Therefore, the quality of the OR

allocation template is crucial to any operational performance measure pertaining

to surgery, including the inpatients' in-hospital cost or length of stay.


10/26

3

1.3OBJECTIVE OF THE MODELPublic hospitals are non-profit organizations, and their prime operational

objective is to provide medical services to their patients at a reasonable cost. In

such an environment, a significant amount of time and resources are invested in

operating theatres. If optimum allocation strategies are devised for the same,

worthy savings can be achieved. For the study, a hospital with a certain number of

operating theatres and specialties has been considered. With the given conditions,

the allocation strategy could be bettered so that the time the patient stays at

hospital is minimized. The objective of the model is to optimally allocate the

Operating Rooms(OR) of the hospital to different medical specialties based on

surgery demand and to minimize the total length of stay of patient (Zhang B,

2009).

1.4OUTLINE OF THE REPORTThe report starts with the literature review of various surgery planning journals

and relate them to this work. Then the model is explained with its assumptions,

objective function and its constraints. Modelling of simulation is also explained

and a proposal for a new constraint in the model, i.e. Post-Operative care bedscapacity constraint is explained. Then a case study is conducted and results are

presented which shows the template from the model is performing better than the

original allocation template. Finally scope for future works is explained.


11/26

4

CHAPTER 2

LITERATURE REVIEW

2.1 LITERATURE REVIEW

The main approaches that have been adopted for surgery planning are

mathematical programming (Ogulata 2003, Blake 2002, Ozkarahan 2000), and

simulation (Dexter 1999). Mathematical programming (especially, integer or

mixed integer programming) models have shown to be useful in capacity planning

or resource allocation in many systems, including in the healthcare setting; while

valid simulation models are useful in estimating the actual performance of a

planning solution beforehand. Here the methodology consists of bothapproaches.

As for the objective, much research has aimed at maximizing OR

utilization, due to its high operational cost (Dexter and Traub, 2002, Ozkarahan

2000). However, at hospitals with fixed or nearly fixed annual budgets, allocating

OR time based on utilization can adversely affect the hospital financially, and

suggested considering not only OR time but also the resulting use of hospital beds.

In line with this idea, there have recently been some studies on the impact of

surgery schedules on the use of the other resources in hospitals. One

distinguishing characteristic of this work is the focus on minimizing inpatients

length of stay waiting for surgery, resulting from the block time schedule.

More and more attention has been paid to managing uncertainty in surgery

planning and scheduling and improving the punctuality of the schedule realized.

Giving incentives to hospital workers to improve their on-time performance is

crucial to reducing delays in surgery or other health services, and the punctuality

of the first service of the day has a significant impact on subsequent

services(Lapierre 1999). The actual surgery demand tends to be higher than

recorded or forecasted, while the supply of OR resources often suffers such

uncertainties as staffing or equipment shortages, which has an effect equivalent to

the former.


12/26

5

CHAPTER 3

MODELING

3.1 PROBLEM IDENTIFICATION

The public health care sector is severely strained with the ever increasing

population .The hospitals are unable to cope up with the increasing demands of

the teeming millions as the resources and facilities available for satisfying the

demand are limited .This has resulted in long serpentine queues which cripples the

health sector reforms and initiatives. Operating rooms are considered among the

most costly hospital facilities and it often becomes a bottleneck in the hospital.

The efficient management of the operating theatres will therefore result in animproved performance of the hospital as a whole as it is highly interrelated with

the other facilities.

An unnecessarily long length of stay(LOS) for inpatients is one of the

common issues in the hospitals. A high LOS is due to inefficient scheduling

procedures used in surgical and ancillary services, since most inpatients require

one or both of these during their stay at the hospital. In most hospitals, a large

percentage of inpatients undergo some type of surgery during their stay in a

hospital. To reduce inpatient length of stay and also from the patients health point

of view, it is ideal that a surgery is performed as soon as it is requested. However,

in reality, inpatients may need to wait in bed for their surgery for a day or two or

even longer (especially in public hospitals). Outpatients also may not always

undergo their surgery on schedule, although this is not directly counted in the

hospitals expenditures. Thus, it is of great interest to hospital administrators to

reduce inpatients' LOS. There are many possible reasons for delays. For example,

a patient who is considered to need surgery immediately might be given

preference over another patient who has been waiting for a week, and has a

scheduled surgery. Or, a hospital may face a situation where a large number of

emergency patients needing surgery use up most of the available Operating Room

capacity(Zhang B, 2009). As a result, the delayed inpatients stay in the hospital

longer, incurring higher non-reimbursable costs for the hospital and the delayed

outpatients may remain in a long outpatient queue (often in the form of a waiting


13/26

6

list) waiting for surgery and also continue to compete for operating room

capacity with inpatients, affecting LOS indirectly. Thus the unnecessarily long

length of stay in the hospital due to the inefficient scheduling procedure is the

problem which needs immediate attention .This can be solved by developing an

efficient scheduling procedure using the operations research technique.

3.2 PROBLEM FORMULATION

The mixed integer linear programming technique is used to solve the

problem of operating room scheduling. The model determines an allocation

template and each specialtys weekly OR time (uniquely determined by the

template) based on the objective of minimizing inpatients length of stay in the

hospital. Based on the study conducted on the operating theatre systems the

various constraints are also identified. The mixed integer programming method is

adopted as some of the decision variables are constrained to take integer

values(Zhang B, 2009).

The decision variables identified to be analysed in the system includes

Number of Operating Rooms of a particular type allocated to a specialty onany day.

Amount of Emergency ORs staffed hours allocated to a specialty on any day. A Specialtys postponed non-emergency demand from a day to another. A Specialtys unmet non-emergency demand on any day. Amount of idle time of the OR allocated to a specialty on a day. Total amount of idle time of all non-emergency ORs. Oversupply of OR hours to a specialty relative to its desired level. Undersupply of OR hours to a specialty relative to its desired level.3.3 NOTATIONS

I: set of room types.

J: set of medical specialties.

D: set of days.

i: index for room type. A room can be considered a different type due to its

location or special medical equipment.


14/26

7

j: index for medical specialty.

k, l: indices for days.

s: amount of staffed hours per day.

ai: number of operating rooms of type i.

ejk: emergency patients surgery demand for specialty j on day k, measured in

hours.

ojk: non-emergency patients (including inpatients and outpatients) surgery

demand for specialtyj on day k, measured in hours.

cjk: the maximum number of operating rooms that specialtyj can utilize on day k,

determined by the number of surgeons and the amount of equipment or any other

necessary medical resources that each specialty has.

kl: the number of days delayed if a surgery is postponed from day kto day l.

: the equivalent number of days delayed if some surgery demand is not met in

the model (or the penalty rate for unmet demand).

: the penalty rate for undersupply of OR hours to a specialty, relative to a desired

level determined by the percentage of total non-emergency surgery demand for

each specialty. Inclusion of this penalty term in the objective function serves the

purpose of smoothing the OR capacity. The value should be much smaller than

.

3.4 ASSUMPTIONS

1. The number of working days can be defined by the end user subject to themaximum of a 7 workdays.

2.

Staffed hours can also be given by the user. Overtime work is notmodelled.

3. The demand pattern for a week is known beforehand. The model does notaccount for the uncertainty in the demand.

4. Emergency capacity for each day is user defined.5. Only weekdays' surgery demand is considered in the model. However,

patients' stay in the hospital on Saturdays and Sundays does incur cost just

like on weekdays. Therefore,


15/26

8

kl = {7 7 (3.1)

Note that ifk= l or k> l, day l represents a weekday following the weekfor day k.

6. The surgery demand is measured by the amount of OR hours. Forexample, if specialty j, on average, has 2 emergency patients who need

surgery on Wednesday and the average length of this specialty's

emergency surgery is 1.6 hours, then the surgery demand of specialty j's

emergency patients on Wednesday or ej3is 3.2 hours.

7.

Inpatients' in-hospital cost is incurred by the delay in meeting surgerydemand. Because surgery demand is measured in OR hours, inpatients' in-

hospital cost or length of stay is measured by "OR hours days". It is

obtained by multiplying the postponed demand volume (i.e. the amount of

OR hours postponed) by the number of days between the day that amount

of demand arises and the day it is met.

8. All emergency surgery demand must be met on the day it arises. Non-emergency or inpatients' and outpatients' surgery demand can be delayed.

9. If some non-emergency patients' surgery demand cannot be met on therequested day, it can be met on the remaining days of the current week, on

any day in the following week, or become unmet (equivalent to being met

days late).

10.Each specialty performs their non-emergency surgeries only in the non-emergency OR(s) allocated to them.

11.Each specialty can perform their emergency surgeries either in theemergency OR or in the non-emergency OR(s) allocated to them.

12.Specialtyj is at most allocated cjkORs on day k.13.Post-Operative Care capacity(in the proposal).

3.5 DECISION VARIABLES

xijk: the number of operating rooms of type i allocated to specialtyj on day k. The

entire set ofxijks determines the allocation template.


16/26

9

yjk: the amount of the emergency ORs staffed hours allocated to specialty j on

day k.

zjkl: specialtyjs non-emergency demand postponed from day kto day l.

ujk: specialtyjs unmet non-emergency demand on day k.

bjk: the amount of idle time of the OR allocated to specialtyj on day k.

h: the total amount of idle time of all non-emergency ORs.

pj: oversupply of OR hours to specialtyj, relative to its desired level.

qj: undersupply of OR hours to specialtyj, relative to its desired level.

3.6 OBJECTIVE FUNCTION

The objective function aims at minimising the patients length of stay in

the hospital. In order to represent the patients length of stay in the hospital three

penalty terms are identified.The objective function of the formulation consists of

three cost or penalty terms. The first two represent the inpatients length of stay

caused by the delay in meeting surgery demand within one cycle (or for up to 7

days) and by unmet demand, respectively. The third term represents the total

penalty caused by the undersupply of OR hours to each specialty, relative to its

desired level determined by the percentage of total non-emergency surgery

demand for each specialty. This penalty term is less dominant, considering our

practical objective of minimizing inpatients length of stay waiting for their

surgery; yet inclusion of this term is useful in determining, among solutions

yielding the same or similar sum of the first two terms or total length of stay, the

one that leads to the most reasonable allocation of the non-emergency OR idle

time (in the sense that the larger non-emergency surgery demand a specialty has,

the more non-emergency OR idle time it tends to occupy) and thus would perform

the best when subject to actual demand and time uncertainty(Zhang B, 2009).

Minimize

(

)

(32)


17/26

10

3.7 CONSTRAINTS

The number of operating rooms available for each of the speciality for a

particular day (both emergency and non-emergency operating rooms) represents

the main resources constraint. The amount of available OR hours and the total

idle time are the other resource constraint identified. Also the model should make

sure that the emergency demand for a day should be met whereas the non-

emergency demand can be postponed or delayed indefinitely(Zhang B, 2009).

i (33)

jk jk (34)

(jk jk ) jk jk jk (35)

(36)

(37)

(38)

(39) jk jk (310)

ijk jk jkl jk jk 0 (311)

ijkInteger,

(312)


18/26

11

All the decision variables can take only positive values. The decision

variable representing the number of operating rooms can take only positive

integral values.

Constraint-1, Eq. (3.3) guarantees that all the operating rooms areallocated to some specialty each day.

Constraint-2, Eq. (3.4) ensures that on any day each specialty has at leastthe OR capacity to meet the non-emergency demand decided to be

postponed to that day.

Constraint-3, Eq. (3.5) states that specialty j's non-emergency surgerydemand on day kmust be met either on that day, some remaining day in

the current week, some day in the next week, or unmet (that is, met days

late).

Constraint-4, Eq. (3.6) defines h as the sum of idle hours of all the non-emergency ORs over one week.

Constraint-5, Eq. (3.7) defines pj and qj, respectively, as the oversupplyand undersupply of non-emergency OR (idle) time to specialty j, relative

to a desired level determined by the percentage of total non-emergency

surgery demand for specialty j. More specifically, given the weekly total

of non-emergency OR idle hours, it is desired that each specialty occupies

the amount proportional to its share of the total non-emergency surgery

demand; pj and qj represent the difference between the actual allocation

and the desired level for specialtyj.

Constraint-6, Eq. (3.8) guarantees that at most s hours of emergencydemand is met in the emergency OR each day.

Constraint-7, Eq. (3.9) ensures that specialtyj is at most allocated cjk ORson day k.

Constraint-8, Eq. (3.10) guarantees that the daily emergency OR capacityallocated to each specialty does not exceed their emergency demand.

Constraint-9, Eq. (3.11) is the non-negativity constraint on all the decisionvariables.

Constraint-10, Eq. (3.12) defines eachxijk variable to be an integer.


19/26

12

3.8 SIMULATION MODELING

In reality, OR capacity and surgery demand are stochastic and/or

dynamic. Also, surgery demand is discrete in nature or measured by the number

of surgeries, instead of by hours as assumed in our MIP model. The degree in

which the randomness and discreteness of the variables impacts the optimality of

the template determined by the analytical model depends on the specific data or

the problem instance under consideration. Therefore, after obtaining the template

from the MIP model, a simulation model is used to assess the quality of the

template generated by the MIP. Also, since the MIP model strives to enhance the

robustness of the template by smoothing the OR capacity, the smoothing constant

or specifically the value be determined by testing in the simulation modelthe templates resulting from different values(Zhang B, 2009).The following features are included in the surgery simulation model.

Each specialty has two queues waiting for surgery: inpatients andoutpatients. There is a single queue of all emergency patients who need

surgery. All ORs are modelled as servers in the queuing system and the

number of non-emergency ORs available to each specialty changes

throughout the week as determined by the template.

All types of patient arrivals are modelled as renewal processes. Each specialty's inpatient, outpatient, and emergency patients surgery

lengths are random variables fit from historical data. Pre-surgery set-up

time and post-surgery OR cleaning time, if significant, is also added to the

model (Spangler 2004).

Emergency surgeries are performed in the emergency OR immediately aslong as it is available. If not, they are performed in an available non-

emergency OR allocated to that specialty. If no non-emergency OR of

the needed specialty is currently available, emergency patients wait

until the emergency OR or one of the respective specialtys non-

emergency OR becomes available. Each specialtys non-emergency

surgeries are only performed in the non-emergency OR(s) allocated to

them.


20/26

13

Emergency patients have the highest priority to be served, then outpatients,and lastly inpatients.

In the case of inpatient surgeries, the simulation model has an end -of-shift protocol. If there are 90 minutes or less remaining for the end of theshift, then a search is done through each specialtys inpatient queue

(assuming that all of them have been prepped) to determine which patient

has a surgery time less than or equal to the remaining time before the

shift ends. If such a patient can be found, then he/she goes into surgery. If

not, the room remains unoccupied till the end of the shift.

3.9 PROPOSAL: POST-OPERATIVE CARE BEDS CAPACITY

CONSTRAINT

A new constraint taking into consideration the number of beds available in

post-operative care has been introduced.

(313)Constraint-11, Eq. (3.13) states that the sum of the total non-emergency

demand and the demand that has been postponed to day kfrom l minus the non-

emergency demand that has been postponed from day kfor a specialtyj must be

less than or equal to the number of hours available on post-operative care.


21/26

14

CHAPTER 4

CASESTUDY

4.1 DESCRIPTION

The Los Angeles County (LAC) General Hospital is used as a case study

to demonstrate the modelling approaches(excluding the proposed constraint).

LAC General Hospital is a large urban health centre serving a largely poor

population. It is also the trauma centre for central Los Angeles, with the

busiest emergency department, measured in admissions. Approximately 85% of

the patients admitted to beds in the hospital enter through the emergency

department.

4.2 SOLVING THE MODEL AND RESULTS

At the time of this analysis, Los Angeles County General Hospital was

using 19 operating rooms with 16 specialties and 1 emergency OR. The capacity

data and January 2005s demand data was fed into the MIP model and used

CPLEX 9.0 with default settings to solve the problem on a 3.2 GHz CPU with

2GB RAM.

Four different values (0, 0.5, 0.75, 1) of the smoothing constant (), andthe solver gave an optimal solution or a close-to-optimal solution in less than 2

hours of CPU time for all four scenarios. The weekly OR capacity allocations

for the actual template followed in January 2005 and for the four templates

determined by the MIP model (with different values) are shown in Table 4.1.Template i: Actual Allocation

Template ii: Determined by MIP Model = 0Template iii: Determined by MIP Model = 0.5Template iv: Determined by MIP Model = 0.75Template v: Determined by MIP Model = 1

In order to evaluate the different templates, a simulation analysis was

performed based on the features discussed in Section 3.8. AweSim! Version 3.0

(Pritsker and OReilly, 1999) was used as the simulation software. In order to


22/26

15

develop the surgery simulation model, the operating room process of LAC

General Hospital was closely observed over a number of days. Furthermore, four

months of data from the hospitals information system was requested. The data

included admit date, discharge date, surgery start date and time, surgery end date

and time. Based on the data analysis, the emergency patient and inpatient

demands for each specialty were assumed to follow a stationary Poisson

Process. That is, the inter-arrival times of requests were modelled as exponential

random variables with mean equal to the inverse of the average daily demand. For

outpatients the daily demand for each specialty does depend on the day of week.

Thus, the average demand for each day was computed for each specialty type. In

this case, the arrival process for outpatients was modelled as a non-stationary

Poisson Process with the arrival rate changing in each day. Furthermore, the

surgery times were assumed to be lognormal random variables with a

constant 30 minute cleaning time after surgery. Prior studies (Spangler, 2004),

and the data analysis for LAC General Hospital shows that the lognormal

distribution is a reasonable model(Zhang B, 2009).

Table 4.1: OR capacity allocated to each specialty per week

Unit: OR hour Templatei Templateii Templateiii Templateiv Templatev

Emergency 40 40 40 40 40

Burns 32 32 32 32 24

Cardiac 48 40 40 40 40

Colorectal 24 40 32 24 24

Foregut 16 16 16 16 16

HNS 88 80 88 88 88

Neuro 40 40 40 40 40

Ortho 192 160 168 176 184

Trauma 8 24 24 24 24

Tumor 24 24 24 24 24Urology 40 40 40 40 40

GSNTE 16 40 40 40 40

Plastics 24 24 24 24 24

Hepatobiliary 24 24 24 24 24

Ophthalmology 80 72 72 72 72

OMFS 32 32 32 32 32

Vascular 24 32 24 24 24


23/26

16

The simulation results based on running the model for 100 weeks with a

warm-up period of 10 weeks are shown in Table 4.2. Template v yields the

shortest inpatients length of stay waiting for surgery and also the smallest

standard deviation of non-emergency OR utilization. Notice that the function of

inpatients average wait with respect to does not necessarily have a convexstructure due to many complicating factors.

Table 4.2: Summary of simulation results for different sTemplate ii iii iv v 0 0.5 0.75 1

Inpatients Average Wait (day) 1.64 1.81 1.90 1.54

Standard Deviation of Non-Emergency OR Utilizations 14.56 10.63 10.39 9.80

Table 4.3 provides summary comparison of output statistics between the actual

template (i) and the best template generated by the model (v). The inpatients average

length of stay waiting for their surgery reduces from 1.86 days in the scenario of

using the actual hospital template to 1.54 days when using the models template. Also,

the emergency patients average wait reduces by nearly 18% and all the other

performance measures stay relatively equivalent.

Table 4.3: Performance(simulation results) summary with the original demand

Template i v

Emergency Patients Average Wait(day) 0.62 0.51

Emergency Operating Room Utilization (%) 48.35 47.28

Inpatients Average Wait(day) 1.86 1.54

Outpatients Average Wait(day) 0.34 0.33

Average Non-Emergency Operating Room Utilization (%) 63.39 65.91


24/26

17

CHAPTER 5

CONCLUSION AND SCOPE FOR FUTURE WORKS

5.1 CONCLUSION

The researches done in medical fields are focused on developing

advanced technologies in equipment arena and in developing cures for diseases.

Scheduling and optimal allocation of available resources is an area which deserves

a look in especially considering the paucity of resources seen nowadays.

A mixed integer programming model was developed to determine optimal

operating room allocation to each specialty. A simulation analysis was used to

assess the performance of the operating room template. The methodology was

illustrated on a case example of Los Angeles County General Hospital, and the

analysis showed that the average patients waiting time for surgery could be

reduced with an efficient allocation of operating room time.

5.2 SCOPE FOR FUTURE WORKS

The templates generated by the optimization model could perform poorly

in practice when there are high variances associated with surgery length and

volatile patient arrival patterns since the optimization model does not account for

uncertainty in the problem parameters. Therefore, future research can focus on

incorporating uncertainty into the analytical model. Also, since the problem sizes

were relatively small the MIP could be solved to optimality or near-optimality in

the scenarios performed in this study. However, for larger problem sizes

specialized algorithms or heuristics may be necessary in order to solve the model.

Also the model could be expanded to take into account staff shifts at the hospital.

Further, the feasibility of the proposed Post-Operative Care constraint

can be determined by solving the model using CPLEX and the performance can

be analyzed by simulating the new model.


25/26

18

REFERENCES

[1]. Aida J., Atidel B. and Hadj A., 2006. Operating Rooms Scheduling, Int.J.Production Economics,99, 52-62.

[2]. Blake, J. T., Dexter F., and Donald D., 2002, Operating RoomManagers Use of Integer Programming for Assigning Block Time to

Surgical Groups: A Case Study,Anesthesia and Analgesia, 94, 143-148.

[3]. Cardoen B. and Erik D., 2009. Optimizing A Multiple Objective SurgicalCase Sequencing Problem,Int. J. Production Economics, 119, 354-366.

[4]. Dexter, F., Macario A., Traub R. D., and Lubarsky D. A., 1999, AnOperating Room Scheduling Strategy to Maximize the Use of Operating

Room Block Time: Computer Simulation of Patient Scheduling and

Survey of Patients Preferences for Surgical Waiting Time, Anesthesia

and Analgesia, 89, 7-20.

[5]. Dinh N.P. and Andreas K., 2008. Surgical Case Scheduling as GeneralizedJob Shop Scheduling Problem, European Journal of operation Research,185, 1011-1025.

[6]. Fei H., Meskens N., 2009, A planning and scheduling problem for anoperating theatre using an open scheduling strategy, Computers &

Industrial Engineering, 78, 144-54.

[7]. Guinet A. and Chaabane S., 2003, Operating Theatre Planning, Int.J.Production Economics,85, 6981.

[8]. Kourie, D. G., 1975, A Length of Stay Index to Monitor Efficiency ofService to General Surgery In-Patients, Operations Research Quarterly,

26, 63-69.

[9]. Lapierre, S. D., and McCaskey S., 1999, Improving On-Time Performancein Health Care Organizations: A Case Study, Health Care Management

Science, 2, 27-34.


26/26

19

[10]. Marshall, A., Vasilakis C., 2005, Length of Stay-Based Patient FlowModels: Recent Developments and Future Directions, Health Care

Management Science, 8, 213-220.

[11]. Ogulata, S. N., and Erol R., 2003, A Hierarchical Multiple CriteriaMathematical Programming Approach for Scheduling General

Surgery Operations in Large Hospitals, Journal of Medical Systems,

27, 259-270.

[12]. Ozkarahan, I., 2000, Allocation of Surgeries to Operating Rooms byGoal Programing, Journal ofMedical Systems, 24, 339-378.

[13]. Spangler W. E., Strum D. P., and J. H. May, 2004, Estimating ProcedureTimes for Surgeries by Determining Location Parameters for the

Lognormal Model,Health Care Management Science, 72, 97-104.

[14]. Vincent A. and Xiaolan X., 2009, Operating theatre scheduling withpatient recovery in both operating rooms and recovery beds., Computers &

Industrial Engineering, 45, 112-124.

[15]. Zhang B., Murali P. K., Dessouky M. and Belson D., 2009. A MixedInteger Programming Approach For Allocating Operating Room Capacity,

Journal of the Operational Research Society, 60, 663-673.

operating room allocation using mixed integer mip report

Documents