OPERATING ROOM ALLOCATION USING

MIXED INTEGER LINEAR PROGRAMMING (MILP)

PROJECT REPORT

Submitted by

SREENATH G V

SASTHA KRISHNAN V

SARAH MATHEW

KAILAS SREE CHANDRAN

To

The University of Kerala

In partial fulfilment of the requirements for the award of the degree

Of

Bachelor of Technology in Mechanical Stream - Industrial Engineering

DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING, THIRUVANANTHAPURAM

April 2010

DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING

THIRUVANANTHAPURAM

CERTIFICATE

This is to certify that the Project Report entitled “OPERATING ROOM

ALLOCATION USING MIXED INTEGER LINEAR PROGRAMMING

(MILP)” submitted by “Sreenath G V, Sastha Krishnan V, Sarah Mathew and

Kailas Sree Chandran” to the University of Kerala in partial fulfilment of the

requirements for the award of the Degree of Bachelor of Technology in Mechanical

Stream - Industrial Engineering, is a bonafide record of the work carried out by them

under my/our guidance and supervision. The contents of this work in full or in parts,

have not been submitted to any other institute or university for the award of any

degree or diploma.

SRI. K. SUNIL KUMAR PROF. E. ABDUL RASHEED

Lecturer Head of The Dept Of Mechanical Engineering

Dept. of Mechanical Engg; College of Engineering

(GUIDE) Thiruvananthapuram

ACKNOWLEDGEMENT

It is with great pleasure, we place on record our deep sense of gratitude and

indebtedness to our guide Sri. K. SUNIL KUMAR, Lecturer, Department of

Mechanical Engineering, College of Engineering, Thiruvananthapuram, for helping us

to carryout this work successfully.

We take this opportunity to express our most sincere thankfulness to Prof. E.

ABDUL RASHEED, Head of the Department, Department of Mechanical

Engineering, College of Engineering, Thiruvananthapuram, for his valuable

directions.

We are also thankful to Prof. P. VINCENT, Former Head of the Department,

Smt. MINI R.S., Lecturer and Staff Advisor, Department of Mechanical Engineering,

College of Engineering, Thiruvananthapuram for their direct and indirect helps for the

success of this project.

Last but not least, we are grateful to our parents, friends and classmates for

their help and co-operation for the successful completion of this project.

SREENATH G V

SASTHA KRISHNAN V

SARAH MATHEW

KAILAS SREE CHANDRAN

ABSTRACT

Public 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.

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. Our aim was

to optimally allocate the Operating Rooms of the hospital and to minimize the total

length of stay of patient. For that, a methodology for allocating operating room

capacity to specialties has been developed. It consists of a finite-horizon mixed

integer programming (MIP) model which determines a weekly operating room (OR)

allocation template that minimizes inpatients' cost measured as their length of stay. For the purpose, a hospital with a certain number of operating theatres and specialties

has been considered.

A basic model and two variants of it have been developed. A number of clinical

constraints (e.g. maximum number of hours allocated to each specialty, surgeon and

staff availability) are included in the formulation of the basic model. The first variant

of the model brings in the emergency situations that could arise for each specialty at

the hospital. The second variant considers the post-operative care for the patient at the

hospital. The model output gives optimized allocation of operating rooms to each

speciality for each day.

On a case example of a Pushpagiri Institute of Medical Sciences and Research Centre,

Thiruvalla, we show how the hospital length of stay pertaining to surgery can be

reduced.

CONTENTS

1. CHAPTER 1: Introduction

1.1 Introduction 01

1.2 Motivation for the Project 02

2. CHAPTER 2: Study

2.1 Study of Operating Theatres 03

2.2 Study of MATLAB 04

2.3 Study of IBM ILOG CPLEX 17

2.4 Study of Operating Theatre Scheduling 32

2.5 Study of MILP 35

3. CHAPTER 3: Modeling

3.1 Problem Identification 38

3.2 Problem Formulation 39

3.3 Objective Function 40

3.4 Constraints 41

3.5 Notation 41

3.6 Assumptions 43

3.7 Decision Variables 44

3.8 Models 45

4. CHAPTER 4: Case Study

4.1 Data Collection 51

4.2 Data analysis and Interpretation 51

4.3 Solving the Model 52

4.4 Results 54

5. CHAPTER 5: Conclusion and Future Works 56

APPENDICES

Appendix 1: Collected Data 57

Appendix 2: MATLAB code 59

Appendix 3: Screenshots of MATLAB 70

REFERENCE 71

LIST OF FIGURES

Figure 2.1 The desktop of MATLAB 07

Figure 4.1 Operating Rooms allocation for Speciality 1 55

Figure 4.2 Operating Rooms allocation for Speciality 2 55

Figure AP 3.1 MATLAB Programming Window 70

Figure AP 3.2 MATLAB Model running window 70

LIST OF TABLES

Table 2.1 Concert Technology Libraries 20

Table 2.2 Summary of Optimizers 22

Table: 4.1 Percentage of Surgery in each department 52

Table 4.2 Non-Emergency demand 53

Table 4.3 Emergency demand 53

Table 4.4 Results 54

Table AP 1.1 Average daily Out Patient Department attendance 57

Table AP 1.2 Number of surgeries for the year 2009 58

1

CHAPTER 1

1.1 INTRODUCTION

Surgery 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, anesthetists, 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 that have 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 (SCS)[1].

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 patient‘s length of stay and

also for The sake of the patient‘s 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[2].

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

2

uncertain. If optimum allocation strategies are devised, then worthy savings in time

and there by resources can be achieved. In the present project, an MILP model has

been formulated with the objective of reducing the length of stay of patient in the

hospital for treatment.

While trying to maximize the utilization of the OR, managers are confronted with the

complexity of the case mix and high costs for overtime of the OR staff. Typically, in

hospitals with a complex case mix with a high variability in the duration of surgical

cases, utilization rates are significantly lower than 100%. For example, 100%

utilization for a cancer medical centre is utopian, since a considerable number of the

surgical cases cannot be completed within the planned duration because of the

unpredictability of the disease. Typically in a cancer centre the duration of surgical

cases is unpredictable and they can last much longer than expected. This causes

frequent overtime in operating rooms of a cancer centre[3].

1.2 MOTIVATION FOR THE PROJECT

Public hospitals are non-profit organizations, and their prime operational objective is

to provide medical services to their patients at a reasonable cost. With better

utilization of available resources, treatment can be provided to more people. This is

significant in the sense that the government is constrained on resources and more

patients needs to be treated with the available resources.

Our work focuses on reducing the delay described above, or, in particular, minimizing

Patients' length of stay waiting for their surgery by developing a methodology for

allocating OR capacity to different medical specialties.

3

CHAPTER 2

2.1 STUDY OF OPERATION THEATRES

In many hospitals, patients are categorized into two types: emergency patients, non-

emergency patients, in their patient database and also when the hospital staff analyze

and discuss their operational processes (e.g., surgery planning and scheduling).

Another classification scheme of patients is by medical specialty, like Burns, Cardiac,

or Trauma patients. Some specialties have both types of patients undergoing surgery,

but others only have one or two[6].

Operating rooms can be considered of two types: emergency and non-emergency. A

hospital often has only a few emergency ORs and all the others as non-emergency

ORs. The emergency ORs are solely allocated to the emergency patients who need

surgery, and usually all specialties‘ surgeries can be performed in that room. The non-

emergency ORs are often allotted to different specialties. Though a non-emergency

OR assigned to a specialty is intended for its non-emergency surgeries (i.e., inpatient

and outpatient surgeries), the room can also accommodate for emergency surgeries of

that particular specialty. Indeed, many medical situations require that the emergency

patients be given a higher priority in accessing the non-emergency ORs than the non-

emergency patients[1].

Before each working day (usually one day in advance), 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

4

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[8].

2.2 STUDY OF MATLAB

2.2.1 Introduction

MATLAB is a powerful computing system for handling the calculations involved in

scientific and engineering problems. The name MATLAB stands for

MATrixLABoratory, because the system was designed to make matrix computations

particularly easy. The MATLAB software package is used for computation in

engineering, science, and applied mathematics. It offers a powerful programming

language, excellent graphics, and a large standard library. The focus in MATLAB is

on computation, not mathematics: symbolic expressions and manipulations are not

possible, except through the optional Symbolic Toolbox. All variables must have

5

values, and all results are numerical and potentially inexact, thanks to the rounding

errors inherent in computer arithmetic. The emphasis on numerics is typical for most

work in scientific computation. Compared to other numerically oriented languages,

such as C++ and Fortran, MATLAB is usually found to be much easier to use.

However, its execution speed can be slower. This gap is not always as dramatic as

popular lore has it, and it can often be narrowed or closed with good MATLAB

programming. Moreover, one can link other types of code into MATLAB, or vice

versa, and MATLAB has some optional support for parallel computing. Still,

MATLAB is usually not the tool of choice for high-performance computing.

Whatever you think of these or other limitations of MATLAB, they have not held

back its popularity: a recent search for ―matlab‖ on the books section of Amazon.com

turned up 8,543results! Rapid code development and interaction with data often trump

execution speed, and the integrated graphics and expert routines that come with

MATLAB can be decisively helpful. Even for speed hungry users, MATLAB can be a

valuable environment in which to explore and fine-tune algorithms before creating

production code in another environment. Successful computing languages and

environments reflect a distinctive set of values. In MATLAB, those values include an

emphasis on experimentation and interaction with data and algorithms; syntax that is

compact, friendly, and interactive (rather than tightly constrained and verbose); a

kitchen-sink mentality for providing functionality; and a predilection for vectors,

matrices, and arrays[5].

2.2.2 History

MATLAB was created in the late 1970s by Cleve Moler, then chairman of the

computer science department at the University of New Mexico.He designed it to give

6

his students access to LINPACK and EISPACK without having to learn Fortran. It

soon spread to other universities and found a strong audience within the applied

mathematics community. Jack Little, an engineer, was exposed to it during a visit

Moler made to Stanford University in 1983. Recognizing its commercial potential, he

joined with Moler and Steve Bangert. They rewrote MATLAB in C and founded The

Math Works in 1984 to continue its development. These rewritten libraries were

known as JACKPAC. In 2000, MATLAB was rewritten to use a newer set of

libraries for matrix manipulation, LAPACK[16].

MATLAB was first adopted by control design engineers, Little's specialty, but

quickly spread to many other domains. It is now also used in education, in particular

the teaching of linear algebra and numerical analysis, and is popular amongst

scientists involved with image processing.

2.2.3 A Quick Tour

When you start MATLAB, you get a multi paneled desktop, as seen in Figure 2.1.

The layout and behavior of the desktop and its components are highly customizable.

The component that is the heart of MATLAB is called the Command Window,

located in the middle by default. Here you can give MATLAB commands typed at the

prompt, shown as >>. Unlike Fortran and other compiled computer languages,

MATLAB is an interpreted environment- you give a command, and MATLAB tries to

execute it right away, then awaits another. At the left you can see the Current

Directory window. In general MATLAB is aware only of files in the current directory

(folder) and on the list known as its path, which can be customized. Commands for

working with the directory and path include cd, what, which, add path, and edit path

7

(or you can choose Set path ... from the File menu).You can add files to a directory

on the path and thereby add commands to MATLAB[16].

Figure 2.1: The desktop of MATLAB

At the top right is the Workspace window. The Work space shows you what variable

names are currently defined and some information about their contents. At start-up it

is, naturally, empty. This represents another break from compiled environments:

variables created in the work space persist for you to examine and modify, even after

code execution stops. Below the Work space window is the Command History

window. As you enter commands, they are recorded here. This record persists across

different MATLAB sessions, and commands or blocks of commands can be copied

from here or saved to files. Thus the Command History is very useful if you realize

belatedly that you need to save some or all of what you have been doing interactively.

As you explore MATLAB, you may encounter some toolboxes. These are

8

individually packaged sets of capabilities that provide in-depth expertise on particular

subject areas. There is no need to issue a command to load them- once installed, they

appear on the path and are available automatically.

2.2.4 Graphical Versus Command-Line Usage

Originally, MATLAB was entirely a command-line environment, and it retains a bias

in that direction. But it is possible to access a great deal of the functionality from

graphical widgets such as menus, buttons, and soon. These interfaces are especially

useful to beginners, because they layout the available choices clearly. In particular,

take time to right-click (or Control-click on a Mac®) on various objects to see what

you might be able to do with them. As a rule, graphical interfaces can be more natural

for certain types of inter-active work, such as annotating a single graph, whereas

typed commands remain better for complex, precise, repeated, or reproducible tasks.

Much of the time, you can choose which ever mode of operation suits you. For

instance, you can write a function that customizes any figure‘s appearance, but you

can also save aspects of a current figure‘s style as a template just by pointing and

clicking. Moreover, you can create your own graphical interfaces and even distribute

them with your code as a package for non-MATLAB users. In the end, an advanced

MATLAB user should be able to exploit both modes of operation[5].

2.2.5 Help

MATLAB is huge. Nobody can tell you everything that you personally will need to

know; nor could you remember it all anyway. It is essential that you become familiar

with the online help. There are two levels of help:

9

• If you need quick help on the syntax of a command, type help in the

commandwindow. For example, help plot shows directly in the Command Window all

the ways in which you can use the plot command. Typing help by itself gives you a

list of categories that themselves yield lists of commands.

• Typing doc followed by a command name brings up more extensive help in a

separate window. For example, doc plot is better formatted and more informative than

help plot. In the left panel one sees a hierarchical, browsable display of all the online

documentation. Typing doc alone or selecting Help from the drop-down menus brings

up the window at a root homepage.

Starting in MATLAB7.7, the Command Window includes a function browser whose

icon is , sitting next to the prompt. Clicking on it yields a searchable, hierarchical

listing of available functions, with help available on one more click. Merely pausing

after typing the name of a function and its opening parenthesis brings up a context-

sensitive box with syntactic suggestions, as well. The Getting Started with MATLAB

manual is a good place to get a more leisurely and thorough introduction than the one

to follow here. Depending on your installation, the documentation may be available in

PDF form for printing and offline reading, or you can find it on the Web at:

www.mathworks.com/access/helpdesk/help/pdf_doc/matlab/getstart.pdf

Complementary to the online help is MATLAB Central, found on the Web at

www.mathworks.com/matlabcentral. It includes a discussion forum and the File

Exchange, which consists of code contributions by MATLAB users and friends. It‘s a

good place to check when you suspect that you may be trying to reinvent the wheel.

10

2.2.6 Saving And Loading Work

If you enter save myfile, all defined variables will be saved to a file called myfile.mat

in the current directory. This file format is particular to MATLAB. You can also

select a subset of variables to be saved by typing their names after the file name. If

you later enter load myfile, the saved variables are returned to the workspace,

overwriting any presently defined values assigned to the same names. If you highlight

commands in the Command History window, right-click, and select Create M-File

you can save the typed commands to a text file. This can be very helpful for recreating

what you have done. Another way to save both input and output is to use the diary

command with a file name. This causes all subsequent commands and results to be

recorded in a text file on disk. MATLAB is also capable of loading and saving other

common file formats, such as formatted text files, spread sheet files, and common

graphics and video files. To load such a file, it‘s often easiest to double-click it in the

Current Directory window, and follow the resulting prompts[5].

MATLAB, the application, is built around the MATLAB language. The simplest way

to execute MATLAB code is to type it in at the prompt, >> , in the Command

Window, one of the elements of the MATLAB Desktop. In this way, MATLAB can

be used as an interactive mathematical shell. Sequences of commands can be saved in

a text file, typically using the MATLAB Editor, as a script or encapsulated into a

function, extending the commands available.

2.2.7 Variables

Variables are defined with the assignment operator, =. MATLAB is a weakly

dynamically typed programming language. It is a weakly typed language because

11

types are implicitly converted. It is a dynamically typed language because variables

can be assigned without declaring their type, except if they are to be treated as

symbolic objects, and that their type can change. Values can come from constants,

from computation involving values of other variables, or from the output of a

function.

MATLAB has several functions for rounding fractional values to integers:

round(X): round to nearest integer, trailing 5 rounds to the nearest integer

away from zero.

fix(X): round to nearest integer toward zero (truncate).

floor(X): round to the nearest integer toward minus infinity (round to the

nearest integer less than or equal to X).

ceil(X): round to the nearest integer toward positive infinity (round to the

nearest integer greater than or equal to X).

Vectors/Matrices

MATLAB is a "Matrix Laboratory", and as such it provides many convenient ways

for creating vectors, matrices, and multi-dimensional arrays. In the MATLAB

vernacular, a vector refers to a one dimensional (1×N or N×1) matrix, commonly

referred to as an array in other programming languages. A matrix generally refers to a

2-dimensional array, i.e. an m×n array where m and n are greater than or equal to 1.

Arrays with more than two dimensions are referred to as multidimensional arrays.

12

MATLAB provides a simple way to define simple arrays using the syntax:

init:increment:terminator. The increment value can actually be left out of this syntax

(along with one of the colons), to use a default value of 1. Indexing is one-based,

which is the usual convention for matrices in mathematics, although not for some

programming languages.

Matrices can be defined by separating the elements of a row with blank space or

comma and using a semicolon to terminate each row. The list of elements should be

surrounded by square brackets: []. Parentheses: () are used to access elements and

sub-arrays (they are also used to denote a function argument list).

A square identity matrix of size n can be generated using the function eye, and

matrices of any size with zeros or ones can be generated with the functions zeros and

ones, respectively.

Most MATLAB functions can accept matrices and will apply themselves to each

element. For example, mod(2*J,n) will multiply every element in "J" by 2, and then

reduce each element modulo "n". MATLAB does include standard "for" and "while"

loops, but using MATLAB's vectorized notation often produces code that is easier to

read and faster to execute.

Unlike many other languages, where the semicolon is used to terminate commands, in

MATLAB the semicolon serves to suppress the output of the line that it concludes (it

serves a similar purpose in Mathematica.)

Graphics

Function plot can be used to produce a graph from two vectors x and y.

13

Three-dimensional graphics can be produced using the functions surf, plot3 or mesh.

Structures

MATLAB supports structure data types. Since all variables in MATLAB are arrays, a

more adequate name is "structure array", where each element of the array has the

same field names. In addition, MATLAB supports dynamic field names (field look-

ups by name, field manipulations etc). Unfortunately, MATLAB JIT does not support

MATLAB structures, therefore just a simple bundling of various variables into a

structure will come at a cost.

2.2.8 Function Handles

MATLAB supports elements of lambda-calculus by introducing function handles, a

references to functions, which are implemented either in .m files or anonymous/nested

functions.

2.2.9 Classes

MATLAB supports classes, however the syntax and calling conventions are

significantly different than in other languagues, because MATLAB does not have

reference data types. For example, a call to a method object.method(); cannot

normally alter any variables of object variable. To create an impression that the

method alters the state of variable, MATLAB toolboxes use evalin() command, which

has its own restrictions.

Object-Oriented Programming

MATLAB's support for object-oriented programming includes classes, inheritance,

virtual dispatch, packages, pass-by-value semantics, and pass-by-reference semantics.

14

2.2.10 Interactions With Other Languages

MATLAB can call functions and subroutines written in the C programming language

or Fortran. A wrapper function is created allowing MATLAB data types to be passed

and returned. The dynamically loadable object files created by compiling such

functions are termed "MEX-files" (for MATLAB executable).

Libraries written in Java, ActiveX or .NET can be directly called from MATLAB and

many MATLAB libraries (for example XML or SQL support) are implemented as

wrappers around Java or ActiveX libraries. Calling MATLAB from Java is more

complicated, but can be done with MATLAB extension, which is sold separately by

MathWorks.

As alternatives to the MuPAD based Symbolic Math Toolbox available from

MathWorks, MATLAB can be connected to Maple or Mathematica.

2.2.11 Limitations

MATLAB is a proprietary product of The MathWorks, so users are subject to vendor

lock-in. Although MATLAB Builder can deploy MATLAB functions as library files

which can be used with .NET or Java application building environment, future

development will still be tied to the MATLAB language.

MATLAB, like Fortran, Visual Basic and Ada, uses parentheses, e.g. y = f(x), for

both indexing into an array and calling a function. Although this syntax can facilitate

a switch between a procedure and a lookup table, both of which correspond to

mathematical functions, a careful reading of the code may be required to establish the

intent.

15

Mathematical matrix functions generally accept an optional argument to specify a

direction, while others, like plot, do not, and so require additional checks. There are

other cases where MATLAB's interpretation of code may not be consistently what the

user intended (e.g. how spaces are handled inside brackets as separators where it

makes sense but not where it doesn't, or backslash escape sequences which are

interpreted by some functions like fprintf but not directly by the language parser

because it wouldn't be convenient for Windows directories). What might be

considered as a convenience for commands typed interactively where the user can

check that MATLAB does what the user wants may be less supportive of the need to

construct reusable code[20].

Array indexing is one-based which is the common convention for matrices in

mathematics, but does not accommodate any indexing convention of sequences that

have zero or negative indices. For instance, in MATLAB the DFT (or FFT) is defined

with the DC component at index 1 instead of index 0, which is not consistent with the

standard definition of the DFT in any literature. This one-based indexing convention

is hard coded into MATLAB, making it difficult for a user to define their own zero-

based or negative-indexed arrays to concisely model an idea having non-positive

indices. A workaround can be constructed to create an ancillary frequency labeling

array with element values equal to the index less 1 that would be used instead of the

index, or the MATLAB programmer can remember to subtract 1 from the index

obtained from functions that return an index such as find(), min(), max().

MATLAB built-in data types are always passed by value. Therefore, all input

parameters to a function are usually copied (later MATLAB releases introduced lazy

16

copy where if an input parameters are not being altered, a copy is not being made,

however this has certain restrictions). Instances of user-defined classes are also copied

and passed by value as the default, however user-defined classes can be made to

exhibit reference behavior by inheriting from the abstract handle class. Variables of

such classes store a reference to the instance. An alternative to using references is

global variables, however MATLAB JIT does not support globals, as well as

structures, therefore the code performance will degrade[4].

The MATLAB editor does not have code completion, references searches, or

refactoring. Since MATLAB is weakly typed, these tools would be hard, if not

impossible, to provide in future releases since it is unknown which methods of an

object can be called without knowing its type. Without code completion, there is a

tendency to use short cryptic names for variables, function and methods, making code

hard to read. Without code completion or reference searches it can take a long time to

text search for what functions and methods can be called in a given context. Without

refactoring, it is time consuming and error prone to change variable, method, or

function names, or to modify an API[4].

Productivity of a development team working on a large software project in MATLAB

will likely be several times slower and result in a higher defect rate than development

in a language and IDE with type checking, code completion, reference search,

refactoring tools, and unit testing support.

It is very difficult to employ agile programming practices, such as Extreme

Programming, in MATLAB due to the lack of refactoring tools.

17

MATLAB lacks any native support for GPU acceleration such as OpenCL,

Microsoft's DirectCompute, or nVidia's CUDA.

2.2.12 Alternatives

MATLAB has a number of competitors.

There are free open source alternatives to MATLAB, in particular GNU Octave,

FreeMat, and Scilab which are intended to be mostly compatible with the MATLAB

language (but not the MATLAB desktop environment). Among other languages that

treat arrays as basic entities (array programming languages) are APL and J, Fortran 95

and 2003, as well as the statistical language S (the main implementations of S are S-

PLUS and the popular open source language R).

There is one web / cloud based alternative - Monkey Analytics - which provides a

modern web interface on top of GNU Octave or Python (including matplotlib,

NumPy, SciPy and more).

There are also several libraries to add similar functionality to existing languages, such

as Perl Data Language for Perl and SciPy together with NumPy and Matplotlib for

Python.

2.3 STUDY OF IBM ILOG CPLEX

2.3.1 Introduction

IBM® ILOG® CPLEX® is a tool for solving linear optimization problems,

commonly referred to as Linear Programming (LP) problems, of the form:

Maximize (or Minimize) c 1 x 1 + c 2 x 2 +...+ c n x n

18

subject to a 11 x 1 + a 12 x 2 +...+ a 1n x n ~ b 1

a 21 x 1 + a 22 x 2 +...+ a 2n x n ~ b 2

...

a m1 x 1 + a m2 x 2 +...+ a mn x n ~ b m

with these bounds l 1 ≤ x 1

u1≥x1

...

l n ≤ x n

un≥xn

where ~ can be ≤, ≥, or =, and the upper bounds u i and lower bounds l i may be

positive infinity, negative infinity, or any real number.

The elements of data you provide as input for this LP are:

Objective function coefficients c 1 , c 2 , ... , c n

Constraint coefficients a 11 , a 21 , ... , a n1

...

a m1 , a m2 , ... , a mn

Right hand sides b 1 , b 2 , ... , b m

Upper and lower bounds u 1 , u 2 , ... , u n and l 1 , l 2 , ... , l n

19

The optimal solution that CPLEX® computes and returns is:

Variables x 1 , x 2 , ... , x n

CPLEX® also can solve several extensions to LP:

Network Flow problems, a special case of LP that CPLEX® can solve much faster by

exploiting the problem structure. In Quadratic Programming (QP) problems, the LP

objective function is expanded to include quadratic terms. Quadratically Constrained

Programming (QCP) problems include quadratic terms among the constraints. In fact,

CPLEX® can solve Second Order Cone Programming (SOCP) problems[7].

Mixed Integer Programming (MIP) problems, where any or all of the LP, QP, or QCP

variables are further restricted to take integer values in the optimal solution and where

MIP itself is extended to include constructs like Special Ordered Sets (SOS) and semi-

continuous variables.

2.3.2 CPLEX Components

CPLEX® comes in various forms to meet a wide range of users' needs:

The CPLEX Interactive Optimizer is an executable program that can read a

problem interactively or from files in certain standard formats, solve the

problem, and deliver the solution interactively or into text files. The program

consists of the file cplex.exe on Windows platforms or cplex on UNIX

platforms.

Concert Technology is a set of C++, Java, and .NET class libraries offering an API

that includes modeling facilities to allow the programmer to embed CPLEX®

20

optimizers in C++, Java, or .NET applications. Concert Technology libraries. lists the

files that contain the libraries.[11]

Table 2.1: Concert Technology Libraries

Microsoft Windows UNIX

C++

ilocplex.lib

concert.lib

libilocplex.a libconcert.a

Java cplex.jar cplex.jar

.NET

ILOG.CPLEX.dll

ILOG.Concert.dll

The Concert Technology libraries make use of the Callable Library .

The CPLEX Callable Library is a C library that allows the programmer to

embed CPLEX® optimizers in applications written in C, Visual Basic,

FORTRAN, or any other language that can call C functions. The library is

provided in files cplexXXX.lib and cplexXXX.dll on Windows platforms, and

in libcplex.a, libcplex.so, and libcplex.sl on UNIX platforms.

The Python API for CPLEX® a full-featured Python application programming

interface supporting all aspects of CPLEX® optimization.

The CPLEX connector for The MathWorks MATLAB enables a user to define

optimization problems and solve them within MATLAB using either the

MATLAB Toolbox or a CPLEX® class in the MATLAB language.

21

2.3.3 Compatible Platforms

CPLEX® is available on Windows, UNIX, and other platforms. The programming

interface works the same way and provides the same facilities on all platforms.

2.3.4 Optimizer Options

Default settings will result in a call to an optimizer that is appropriate to the class of

problem you are solving. However you may wish to choose a different optimizer for

special purposes. An LP or QP problem can be solved using any of the following

CPLEX® optimizers: Dual Simplex, Primal Simplex, Barrier, and perhaps also the

Network Optimizer (if the problem contains an extractable network substructure).

Pure network models are all solved by the Network Optimizer. QCP models,

including the special case of SOCP models, are all solved by the Barrier optimizer.

MIP models are all solved by the Mixed Integer Optimizer, which in turn may invoke

any of the LP or QP optimizers in the course of its computation. Optimizers

summarizes these possible choices[19].

The choice of optimizer or other parameter settings may have a very large effect on

the solution speed of your particular class of problem. The CPLEX User's Manual

describes the optimizers, provides suggestions for maximizing performance, and notes

the features and algorithmic parameters unique to each optimizer.

2.3.5 File Formats

BAS files are text files governed by Mathematical Programming System (MPS)

conventions (that is, they are not binary) for saving a problem basis. They are

documented in BAS file format: advanced basis.

22

Table 2.2 Summary of Optimizers

LP Network QP QCP MIP

Dual Optimizer yes yes

Primal Optimizer yes yes

Barrier Optimizer yes yes yes

Mixed Integer

Optimizer

yes

Network Optimizer Note 1 yes Note 1

Note 1: The problem must contain an extractable network substructure.

BZ2 is not a file format specific to CPLEX® . Rather, this file extension indicates that

a file (possibly in one of the formats that CPLEX® reads) has been compressed by

BZIP2. On most platforms, CPLEX® can automatically uncompress such a file and

then read data from the file in one of the formats briefly described here

CLP is the format CPLEX® uses to represent the conflicting constraints and bounds

(a subset of an infeasible model) that were found by the conflict refiner.

CSV files contain comma-separated values. Concert Technology offers facilities in

CPLEX® for reading and writing such files. See the CPLEX C++ API Reference

Manual for details, especially the classes IloCsvReader, IloCsvLine, and

IloCsvReader::Iterator.

23

DPE is the format CPLEX® uses to write a problem in a binary SAV file after the

objective function of the problem has been perturbed.

DUA format, governed by MPS conventions, writes the dual formulation of a problem

currently in memory so that the MPS file can later be read back in and the dual

formulation can then be optimized explicitly. This file format is largely obsolete now

since you can use the command set presolve dual in the Interactive Optimizer to tell

CPLEX® to solve the dual formulation of an LP automatically. (You no longer have

to tell CPLEX® to write the dual formulation to a DUA file and then tell CPLEX® to

read the file back in and solve it.)

EMB is the format CPLEX® uses to save an embedded network it extracts from a

problem. EMB files are written in MPS format.

FLT is the format CPLEX® uses to save filters for solution pools.

GZ is not a file format specific to CPLEX® . Rather, this file extension indicates that

a file (possibly in one of the formats that CPLEX® reads) has been compressed by

gzip , the GNU zip program. On most platforms, CPLEX® can automatically

uncompress a gzipped file and then read data from a file in one of the formats briefly

described here.

LP (Linear Programming)is a CPLEX® -specific file formatted for entering

problems in an algebraic, row-oriented form. In other words, LP format allows you to

enter problems in terms of their constraints. When you enter problems interactively in

the Interactive Optimizer, you are implicitly using LP format. CPLEX® also reads

24

files in LP format. The section LP file format: matrix models describes the

conventions and use of this format.

MIN format for representing minimum-cost network-flow problems was introduced

by DIMACS in 1991.

MPS is an industry-standard, ASCII-text file format for mathematical programming

problems. This file format is documented in MPS file format: industry standard.

Besides the industry conventions, CPLEX® also supports extensions to this format

for CPLEX® -specific cases, such as names of more than eight characters, blank

space as delimiters between columns, etc. The extensions are documented in Special

records in MPS files: CPLEX extensions.

MST is an XML format available with the CPLEX® MIP optimizer. It is a text

format CPLEX® uses to enter a starting solution for a MIP. MST file format: MIP

starts documents this file format.

NET is a CPLEX® -specific ASCII format for network-flow problems. It supports

named nodes and arcs. NET file format: network flow models offers a fuller

description of this file format.

ORD is a format available with the CPLEX® MIP optimizer. It is used to enter and to

save priority orders for branching. It may contain branching instructions for individual

variables. ORD file format: priorities and branching orders documents this file format.

PPE is the format CPLEX® uses to write a problem in a binary SAV file after the

righthand side has been perturbed.

25

PRE is the format CPLEX® uses to write a presolved, reduced problem formulation

to a binary SAV file. Since a presolved problem has been reduced, it does not

correspond to the original problem.

PRM is the format CPLEX® uses to read and write nondefault values of parameters

in a file. PRM file format: parameter settings documents the format and conventions

for reading and writing such files through the Callable Library.

REW is a format to write a problem in MPS format with disguised row and column

names. This format is simply an MPS file format with all variable (column) and

constraint (row) names converted to generic names. Variables are relabeled x1

through xn , and rows are renamed c1 through cm . This format may be useful, for

example, for problems that you consider highly proprietary.

RLP is the LP format using generic names in the Interactive Optimizer.

SAVis a CPLEX® -specific binary format for reading and writing problems and their

associated basis information. CPLEX® includes the basis in a SAV file only if the

problem currently in memory has been optimized and a basis exists. This format

offers the advantage of being numerically accurate (to the same degree as your

platform) in contrast to text file formats that may lose numerical accuracy. It also has

the additional benefit of being efficient with respect to read and write time. However,

since a SAV file is binary, you cannot read nor edit it with your favorite text editor.

SOL files are XML formatted files that contain solution information; they may also

provide an advanced start for an optimization.

26

XML as a file format is available to C++ users of Concert Technology to serialize

models and solutions (that is, instances of IloModel and IloSolution). XML file

format: serialized models and solutions explains more about this serialization API.

2.3.6 Using IBM ILOG CPLEX with MATLAB

IBM ILOG CPLEX for MATLAB® can also solve several extensions to LP:

Quadratic Programming (QP) problems, where the LP objective function is

expanded to include quadratic terms.

Quadratically Constrained Programming (QCP) problems that include

quadratic terms among the constraints. In fact, CPLEX can solve Second

Order Cone Programming (SOCP) problems.

Mixed Integer Programming (MIP) problems, where any or all of the LP, QP,

or QCP variables are further restricted to take integer values in the optimal

solution and where MIP itself is extended to include constructs like Special

Ordered Sets (SOS), semi-continuous variables, and indicator variables.

Least Squares (LSQ) problems, where the objective is to minimize a norm.

The problem can be constrained linearly or quadratically, and the variables

may be restricted to take integer values in the solution.

The standard MATLAB vector and matrix format is used for the elements of data that

you need to provide. For example, the IBM ILOG CPLEX for MATLAB Toolbox

function ‗cplexlp‘ solves the problem specified by

min f*x

st. Aineq*x <= bineq

27

Aeq*x = beq

lb <= x <= ub

where f, bineq, beq, lb, and ub are MATLAB vectors, and Aineq and Aeq are

MATLAB matrices.

The vector x returned by the function call

x = cplexlp(f,Aineq,beq,Aeq,beq,lb,ub)

contains the optimal solution to the specified linear programming problem.

Provided in IBM ILOG CPLEX for MATLAB is both a toolbox of functions and a

class API. The toolbox contains functions for solving optimization problems, where

the input matrices are provided to the function and results returned. With the class

API, objects can be created, and those objects carry a state[19].

The benefits of using the Cplex class API include the ability to:

build up a model by manipulating a Cplex object.

use computation methods such as Cplex.solve() and Cplex.refineConflict()

that modify the object so results can be queried as needed.

perform restarts after manipulation.

attach an output parser, a GUI with stop buttons, and other controls.

2.3.7 Overview of Toolbox

The toolbox provides the functions cplexlp, cplexqp and cplexbilp to solve linear

programming problems (LP), quadratic programming problems (QP) and binary

integer programming problems (BILP)[7].

28

The toolbox provides functions that support the solution of the basic problem types

handled by CPLEX® are:

cplexlp for linear programming problems (LP),

cplexqp for quadratic programming problems (QP) and

cplexbilp for binary integer programming problems (BILP).

Functions that support the solution of additional problem types handled by CPLEX

are provided. These functions are:

cplexqcp for quadratically constrained programming problems (QCP),

cplexmilp for mixed integer linear programming problems (MIP),

cplexmiqp for mixed integer quadratic programming problems (MIQP) and

cplexmiqcp for mixed integer quadtratically constrained mixed integer

programming problems (MIQCP).

The solution of least square problems is supported through the functions:

cplexlsqlin for linearly constrained least squares problems,

cplexlsqmilp for linearly constrained mixed integer least squares problems,

cplexlsqmiqcp for quadratically constrained mixed integer least squares

problems,

cplexlsqnonneglin for nonnegative least squares problems,

cplexlsqnonnegmilp for nonnegative mixed integer least squares problems

and

29

cplexlsqnonnegmiqcp for nonnegative quadratically constrained mixed

integer least squares problems.

Options, also called parameters, can be set to control the solution of problems. The

toolbox provides two types of options input. One type is the CPLEX parameters, and

the other type is the MATLAB® toolbox options. You can use either or both of these

types of options. If you use both, the CPLEX parameters will override the MATLAB

options.

You can set the options with the toolbox function cplexoptimset and access the

options with cplexoptimget.

The advantage of the toolbox design is that you can reuse your code where you had

used MATLAB Optimization Toolbox functions to solve linear programming,

quadratic programming, binary integer programming, linearly constrained least

squares, and nonnegative least squares problems.

2.3.8 CPLEX Class API

While the IBM® ILOG® CPLEX® for MATLAB® Toolbox functions provide the

ability to solve a multitude of mathematical programming problems, the toolbox

design does not support restart. To enable users to use decomposition algorithms, the

Cplex Class API is also provided in IBM ILOG CPLEX for MATLAB.

The Cplex class stores the model and provides methods for the solution, analysis,

manipulation and reading/writing of the model file. All of the data associated with the

problem is stored in the properties of a Cplex object. These class properties are

standard MATLAB data structures and can be manipulated directly within MATLAB.

30

However, modifying the problem using methods provided in the Cplex class enforces

consistency, such as ensuring that vectors are of the proper length[11].

The documentation of the Cplex class provides an introduction of properties and

methods of the Cplex class in more detail.

The properties of the Cplex class include:

Cplex.Model stores the data of the model

Cplex.Solution stores the solution of the model

Cplex.Param stores the parameters (options) of the model

Cplex.Start stores the start of the LP model

Cplex.MipStart stores the start of the MIP model

Cplex.InfoCallback pointer to an informational callback

Cplex.Conflict stores the conflict information of a conflicted model

Cplex.DisplayFunc pointer to a function which provides control of display of

output

The following informative methods are provided:

Cplex.getVersion returns the CPLEX version

Cplex.getProbType returns the problem type of the model

The following methods are provided for reading from and writing to files:

Cplex.readModel

31

Cplex.writeModel

Cplex.readBasis

Cplex.writeBasis

Cplex.readMipStart

Cplex.writeMipStart

Cplex.writeParam

Cplex.writeConflict

The following methods are provided to solve and analyze the model, solution and

mipstart:

Cplex.solve

Cplex.populate

Cplex.feasOpt

Cplex.refineMipStartConflict

Cplex.terminate

The following methods are provided to solve, set and query parameters:

Cplex.tuneParam

Cplex.setDefault

Cplex.getChgParam

32

2.4 STUDY OF OPERATING THEATRE SCHEDULING

2.4.1 Introduction

During the last decades, cost pressures on hospitals have increased dramatically. This

emphasis on cost containment has forced hospital executives to run their organizations

in a more business-like manner. The constant challenge is to provide high-quality

service at ever reduced costs. In order to achieve this purpose, inefficient use of

resources should be identified and actions should be taken to eliminate these sources

of waste. Operations research techniques are increasingly being used to assist in this

complicated task[22].

2.4.2 Block scheduling in hospitals

There are many ways to set up schedule in a hospital. A strict block schedule will

block all or most available time, which usually creates a lot of unused and, therefore,

non-productive operating room time. When using this method, the blocks may not be

released at all or are not released until the day before the surgery date. In another

method, the open schedule, cases are scheduled on a first come–first served basis.

Such schedules can frustrate surgeons, as they may not be able to schedule their cases

back-to-back if another surgeon schedules a case behind them. Therefore, surgeons

who can anticipate their schedules far in advance, such as ophthalmologists or plastic

surgeons, can live with this method far better than OB/GYN or general surgeons, who

often schedule within one to three days of surgery and cannot get a time that works

for them[18].

Modified block scheduling is the most widely used method and affords most centers

maximum flexibility. In this system, a physician, practice, or specialty may be granted

33

a block of time, which can range from two hours to an entire day, in which to use the

operating room. The center board of managers—usually delegated to the medical

executive committee— reviews block use and manage the blocks to maximize use of

operating room time. A center should strive to achieve 70% utilization of all operating

room time. Eighty-five percent is probably the maximum anyone can hope to achieve

in any surgery setting.

In a modified block scheduling format, the center reserves some of the available

surgery time for surgeons who can fill the block as required by your center‘s policies.

For example, you may require that a physician fill 60%–70% or more per quarter to

keep the block, depending on the maturity of the medical centre[17].

A new medical center will probably set the requirements a little lower, but it must

diligently review block time use and eventually push to achieve a higher percentage.

Unblocked times are called open blocks, which physicians use when they cannot fill a

block alone. OB/GYN surgeons or general surgeons, depending upon their practice,

may fit into this category. Blocktime should be released by the office as early as

possible, if the surgeon will not be using a block due to vacation, seminar, or another

reason. The medical centre releases the blocks according to its policies, usually 72

hours to one week in advance if the surgeon has not filled the block. Whatever part of

the assigned block is released becomes open time. The surgery scheduler can fill this

new open time with any cases that were added to the schedule close to the date of

surgery, as cases from OB/GYNs and general surgeons often are. This method allows

the surgery scheduler to accommodate the needs of more surgeons, maximize

operating room use, and make the staff more productive.

34

Most medical centers include a statement in their scheduling policies that prohibits

―ghost scheduling‖ or ―phantom scheduling,‖ which is when the office schedules

nonexistent cases to hold block time to prevent another physician from taking it[15].

2.4.3 Multi-Mode Blocking Job Shop (MMBJS)

In the MMBJS, n jobs are performed on m resources. Each job consists of a sequence

of operations. The execution of an operation requires a set of resources. Such a set of

resources is called a mode. The modes of any two consecutive operations of any job

have no common resources. There might be more than one mode available for each

operation, justifying the term multi-mode. Once a mode is chosen for an operation, its

processing entails a processing time during which the resources of the mode are

occupied simultaneously, and possibly a setup time and a cleanup time. All these

times can depend on the mode.

Resources are not always available all the time. Therefore, to each mode is attached a

time interval during which the mode is available. There is no buffer between

resources, so that if a job has finished an operation and the job‘s next operation cannot

be started, the resources of the finished operation remain blocked until the next

operation is started. Because of this blocking, one needs to distinguish between the

completion time of an operation, which equals the sum of its starting time and

processing time, and its departure time (the time the job leaves its current processing

stage to enter its next stage or leave the system). Scheduling with blocking is of

emergent research interest for its applicability in various shop environments, e.g. in

job shop or flow shop[13].

35

2.5 STUDY OF MIXED INTEGER LINEAR PROGRAMMING

(MILP)

MIXED-integer linear programming (MILP) methods have attracted attention because

of their modeling capability and because powerful solvers are available commercially.

The use of MILP for modeling and control problems have been used for cooperative

reconnaissance ,air traffic management, coordinated robot motion unmanned aerial

vehicle (UAV) path planning, and cooperative control[21].

We have seen that the max flow problem is a special case of the min cost flow

problem which is again a special case of linear programming. Thus, an LP solver

should be sufficient to tackle any of the problems we have seen so far. However, if we

use a general LP solver based on, say, the simplex method to solve max flow

instances or max flow instances rather than an implementation of the Ford-Fulkerson

method or Klein's algorithm, it seems that we may lose an important property of the

latter algorithms, namely the integrality property: If the Ford-Fulkerson algorithm or

Klein's algorithm are given integer-valued inputs, the solutions returned are also

guaranteed to be integer-valued. We saw previously that this integrality property is

often very important when modeling combinatorial optimization instances as max

flow or min cost flow instances[21].

A good example is the reduction of maximum cardinality matching to max flow:

There, we constructed a reduction r converting bipartite graphs G into max flow

instances r(G) with the property that any matching of size v in G leads to a flow of

value v in r(G) and any flow with an integer valued flow along every edge of value v

in r(G) leads to a matching of size v in G. This reduction enables us to use any max

36

flow Algorithm with the integrality property, such as Ford-Fulkerson, to find

maximum cardinality matchings. Suppose we now instead try to use the simplex

algorithm to find a maximum matching in a bipartite graph by expressing the

appropriate max flow instance as a linear program. The value of the solution found

will give us the cardinality of a maximum matching as desired, but it is not a priori

clear that we can use the solution to extract the matching itself.

Indeed, this only seems to be the case if we can prove an integrality theorem for the

simplex algorithm. Fortunately, it turns out that we can prove such an integrality

theorem for a special class of linear programs that include the linear programs that

express max flow and min cost flow instances. This is captured in the following

loosely formulated (in the sense that we are not being specific about which LP

formulations we have in mind) fact[19].

Fact 1 Max flow instances with integer-valued capacities and min cost flow instances

with integer-valued capacities and balances can be expressed as linear programs in

such a way that every basic feasible solution to the resulting linear program is integer-

valued. In particular, the optimal basic solution returned by the simplex algorithm is

integer-valued.

Of course, we might choose to solve our linear program by methods other than the

simplex algorithm. In particular, we might prefer an interior point algorithm. If we do,

we may conceivably have a solver that is guaranteed to return optimal solutions but

not necessarily basic optimal solutions. If this is the case, we can easily build a \shell"

around the solver that converts any non-basic optimal solution found to a basic

optimal solution and Fact 1 again applies. Fact 1 takes care of the integrality property

37

for the special class of linear programs corresponding to flow instances. However,

being able to impose integrality constraints on the solutions we seek is relevant for

many other optimization tasks that can be expressed using linear constraints and for

these, Fact 1 may not apply. Therefore, we shall look at a generalization of linear

programming that allows us to explicitly express such integrality constraints:

A Mixed Integer Linear Program (MILP) is an optimization instance of the following

kind: Find a point x 2 Rn maximizing (or minimzing) a linear form hc; xi so that a

given set of linear inequalities and/or equations and a given set of integrality

constraints xi 2 Z are satisfied. A point x satisfying the constraints is called a feasible

solution to the program[6].

Thus, the only addition to the LP formalism we make is the ability to require certain

components of the solution to be integer valued. We shall see, however, that this

seemingly modest added expressiveness in general makes the task of finding the

optimal solution painfully harder. In particular, all algorithms for solving MILPs have

exponential worst case complexity and unlike the case of the simplex algorithm which

is also exponential but only on artificially constructed instances, the exponential

running time of any of the MILP solvers we know (and shall describe later in this

course) is typical behavior which we have to live with[6].

Powerful software packages such as CPLEX can solve MILPs efficiently for problems

in which the number of binary variables is of reasonable size. However, a major

disadvantage of MILP is its computational complexity. Because MILP is NP-hard

,computational requirements can grow significantly as the number of binary variables

needed to model the problem increases.

38

CHAPTER 3

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 (OR) 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 an improved performance of the

hospital as a whole as it is highly interrelated with the other facilities[6].

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 patient‟s 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 hospital‟s 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

39

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. 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 list) waiting for

surgery and also continue to “compete” for operating room capacity with inpatients,

affecting LOS indirectly[3]. 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

specialty‟s 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[6].

Here we have developed a basic model without considering the emergency and has

improved this model to consider the case of emergency demand also. The following

represents the general description of the various aspects identified and represented in

the models(with improvements).

40

The decision variables identified to be analysed in the system includes

• Number of Operating Rooms of a particular type allocated to a specialty on any

day.

• Amount of Emergency OR‟s staffed hours allocated to a specialty on any day.

• A Specialty‟s postponed non-emergency demand from a day to another.

• A Specialty‟s 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 OR‟s.

• 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 OBJECTIVE FUNCTION

The objective function aims at minimising the patient‟s length of stay in the hospital.

In order to represent the patient‟s 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

41

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[6].

3.4 CONSTRAINTS

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

representing the number of operating rooms can take only positive integral values.

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. In case of the final model, we have brought in

the constraint representing the number of post operative beds capacity as it has been

identified as one of the limited facilities. The various constraints are elaborated along

with each of the model.

3.5 NOTATION

I: set of room types.

J: set of medical specialties.

D: set of days.

42

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

or special medical equipment.

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 specialty j on day k, measured in hours.

cjk: the maximum number of operating rooms that specialty j 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.

ρkl: the number of days delayed if a surgery is postponed from day k to day l.

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

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

β: 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 β value should be much smaller than

θ.

43

3.6 ASSUMPTIONS

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

maximum of a 7 workdays.

2. Staffed hours can also be given by the user. Overtime work is not modelled.

3. The demand pattern for a week is known before hand. The model does not

account for the uncertainty in the demand.

4. 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,

Note that if k = l or k > l, day l represents a weekday following the week for

day k.

5. Emergency capacity for each day is user defined.

6. The surgery demand is measured by the amount of OR hours. For example, 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 ej3 is 3.2 hours.

7. Inpatients' in-hospital cost is incurred by the delay in meeting surgery demand.

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

44

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 the

requested 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 θ 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 the emergency

OR or in the non-emergency OR(s) allocated to them.

12. Specialty j is at most allocated cjk ORs on day k.

13. Post Operative Care capacity

3.7 DECISION VARIABLES

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

entire set of xijk‟s determines the allocation template.

yjk: the amount of the emergency OR‟s staffed hours allocated to specialty j on day k.

zjkl: specialty j‟s non-emergency demand postponed from day k to day l.

ujk: specialty j‟s unmet non-emergency demand on day k.

45

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

h: the total amount of idle time of all non-emergency OR‟s.

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

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

3.8 MODELS

3.8.1 Basic Model

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 (or

equivalent to being postponed θ days, by our notation), 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[6].

46

Min ∑ ∑ (ρkl ∑ zjkl) + θ ∑ ∑ ujk + β∑qj

kϵD lϵD jϵJ jϵJ kϵD jϵJ

Subjected To

∑ xijk = ai For all i,k (1)

jϵJ

s ∑ xijk ≥ ∑ zilk For all j,k (2)

iϵI iϵI

s ∑ xijk – bjk + ujk = Ojk For all j,k (3)

iϵI

h = ∑ ∑ bjk (4)

jϵJ kϵD

h∑ ojk

kϵD

∑ ∑ ojk For all j (5)

jϵJ kϵD

∑ xijk ≤ cjk For all j,k (6)

iϵI

h, pj, qj, xijk, bjk, yjk, zjkl, ujk ≥ 0 For all I, j, k (7)

Xijk is an integer. (8)

Constraint (1) guarantees that all the operating rooms are allocated to some

specialty each day.

47

Constraint (2) ensures that on any day each specialty has at least the OR

capacity to meet the non-emergency demand decided to be postponed to that

day.

Constraint (3) states that specialty j's non-emergency surgery demand on day k

must be met either on that day, some remaining day in the current week, some

day in the next week, or unmet (that is, met θ days late).

Constraint (4) defines h as the sum of idle hours of all the non-emergency ORs

over one week.

Constraint (5) defines pj and qj, respectively, as the oversupply and

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

specialty j.

Constraint (6) ensures that specialty j is at most allocated cjk ORs on day k.

Constraint (7) is the non-negativity constraint on all the decision variables.

Constraint (8) defines each xijk variable to be an integer.

3.8.2 Model 1 (including the emergency case)

Min ∑ ∑ (ρkl ∑ zjkl) + θ ∑ ∑ ujk + β∑qj

kϵD lϵD jϵJ jϵJ kϵD jϵJ

48

Subjected To

∑ xijk = ai For all i,k (1)

jϵJ

s ∑ xijk ≥ ejk – yjk + ∑ zilk For all j, (2)

iϵI iϵI

s∑xijk – (ejk – yjk+ ∑zjlk) – bjk + ∑zjkl + ujk= Ojk iϵI lϵD lϵD

For all j,k (3)

h = ∑ ∑ bjk (4)

jϵJ kϵD

h∑ ojk

kϵD

∑ ∑ ojk For all j (5)

jϵJ kϵD

∑yjk ≤ s For all k (6)

jϵJ

∑ xijk ≤ cjk For all j,k (7)

iϵI

yjk ≤ ejk For all j,k (8)

h, pj, qj, xijk, bjk, yjk, zjkl, ujk ≥ 0 For all i, j, k (9)

Xijk is an integer. (10)

49

Constraint (1) guarantees that all the operating rooms are allocated to some

specialty each day.

Constraint (2) ensures that on any day each specialty has at least the OR

capacity to meet

the sum of its emergency demand on that day and non-emergency demand

decided to be postponed to that day.

Constraint (3) states that specialty j's non-emergency surgery demand on day k

must be met either on that day, some remaining day in the current week, some

day in the next week, or unmet (that is, met θ days late).

Constraint (4) defines h as the sum of idle hours of all the non-emergency ORs

over one week.

Constraint (5) defines pj and qj, respectively, as the oversupply and

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

specialty j.

Constraint (6) guarantees that at most s hours of emergency demand is met in

the emergency OR each day.

Constraint (7) ensures that specialty j is at most allocated cjk ORs on day k.

50

Constraint (8) guarantees that the daily emergency OR capacity allocated to

each specialty does not exceed their emergency demand.

Constraint (9) is the nonnegativity constraint on all the decision variables.

Constraint (10) defines each xijk variable to be an integer.

3.8.3 Model 2 (introducing the post operative care beds capacity constraint)

Min ∑ ∑ (ρkl ∑ zjkl) + θ ∑ ∑ ujk + β∑qj

kϵD lϵD jϵJ jϵJ kϵD jϵJ

Subjected To

ojk + ∑ zijk – ujk ≤ POC hours For all j,k (11)

iϵI

The constraints (1)-(10) remain unaltered for this variant of the basic model.

However a new constraint taking into consideration the number of beds available in

post operative care has been introduced. The constraint (11) states that the sum of the

total non-emergency demand and the demand that has been postponed to day k from l

minus the non emergency demand that has been postponed from day k for a specialty j

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

51

CHAPTER 4

4. CASESTUDY

A case study has conducted to demonstrate the modeling approaches.

4.1 DATA COLLECTION

The data for the model was collected from Pushpagiri Institute of Medical Sciences

and Research Centre, Thiruvalla.

The collected data contained basic statistical information about the hospital such as

the number of operating theatres, number of staffed hours of the theatre, etc.,

information about the average daily out patient attendance in various departments of

the hospital and the total number of surgeries performed in 2008.

The data collected has been summarized in the appendix.

4.2 DATA ANALYSIS AND INTERPRETATION

The hospital has separate intensive care units and post-operative rooms. However, for

convenience, both of these can be clubbed together and termed as Post-Operative Care

(POC).

As part of the analysis, it is necessary to find the number of people each day who are

sent in for surgery. In order to find this approximate percentage 2 and 3 have been

used. This method of calculation is better explained with the help of an example.

Taking the case of the General Surgery Department;

Total number of surgeries performed in the year : 2326 + 2781 = 5107

Total average daily OPD attendance = 111

52

Total daily OPD attendance for a year = 111 * 365 = 40515

Percentage of patients going in for surgery = 5107 / 40515 * 100 = 12.6

Hence it can be seen that approximately 13% of the total number of patients coming

in per day at the general surgery department would undergo surgical treatment.

This analysis was carried out for all the four departments and the results have been

tabulated below:

Table 4.1 Percentage of Surgery in each department

Department Approx. Percentage for Surgery

GS including paediatric

surgery 13

Orthopaedics 5

Oto-Rhino-Laryngology 2

Ophthalmology 1

4.3 SOLVING THE MODEL

Based on the data collected a set of sample data was drawn for input to the model.

This input data considers the General Surgery and Orthopedics department of the

hospital and is as given below:

Number of operating theatre types : 2

Number of staffed hours : 8 hours

Delay for unmet demand : 1 day

Number of operating days : 5

Number of Specialties : 2

Penalty rate for undersupply : 0.1

Maximum number of operating rooms for Specialty 1 : 3

Maximum number of operating rooms for Specialty 2 : 3

53

Number of operating rooms of type 1 : 3

Number of operating rooms of type 2 : 2

The input data for non-emergency demand for the operating day is tabulated below:

Table 4.2 Non-Emergency demand

Day Specialty 1 Specialty 2

Monday 20 10

Tuesday 18 11

Wednesday 22 9

Thursday 19 12

Friday 21 13

The input data for the emergency demand for the operating days is tabulated below:

Table 4.3 Emergency demand

Day Specialty 1 Specialty 2

Monday 3 2

Tuesday 6 3

Wednesday 2 3

Thursday 3 1

Friday 2 2

The MILP model was solved on computer with the following configuration:

Intel Core 2 Duo 2.0 GHz CPU

3GB RAM

250GB Hard Disk

The softwares used to code and optimize the model were:

1. MATLAB Version R2008b for coding.

2. CPLEX 9.0 with default settings for optimization.

54

4.4 RESULTS

The results of the MATLAB model give the minimized length of stay of the patient in

the hospital and also gives the optimal allocation of operating rooms to each specialty

for every operating day.

Optimized Length of stay is 208 hours.

Table 4.4 Results

DAY SPECIALITY 1 SPECIALITY 2

Type 1 Type 2 Type 1 Type 3

Monday 2 2 1 0

Tuesday 1 0 3 1

Wednesday 3 1 0 1

Thursday 0 1 2 2

Friday 2 2 1 1

Two graphs are plotted based on the allocation template for Speciality 1 and

Speciality 2. From the graphs, we can find that for specialty 1,two operating rooms

are each allocated for each of the two types of operating rooms available on Monday.

For Tuesday and Thursday we find that no operating rooms are scheduled for one of

the types which are due to lack of demand on those days. The graphs for the second

specialty also can be interpreted in the same manner.

55

Figure 4.1 Operating Rooms allocation for Speciality 1

Figure 4.2 Operating Rooms allocation for Speciality 2

Monday Tuesday Wednesday Thursday Friday

SPECIALITY 1 Type 1 2 1 3 0 2

SPECIALITY 1 Type 2 2 0 1 1 2

0

0.5

1

1.5

2

2.5

3

3.5O

R R

oo

ms

Allocation for Speciality 1

Monday Tuesday Wednesday Thursday Friday

SPECIALITY 2 Type 1 1 3 0 2 1

SPECIALITY 2 Type 3 0 1 1 2 1

0

0.5

1

1.5

2

2.5

3

3.5

OR

Ro

om

s

Allocation for Speciality 2

56

CHAPTER 5

5. CONCLUSION AND FUTURE WORKS

Usually we find that 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. The methodology was illustrated on a case example

of Pushpagiri Institute of Medical Sciences and Research Centre, Thiruvalla, and the

analysis showed that the average inpatient waiting time for surgery could be reduced

with an efficient allocation of operating rooms.

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.

Some of the areas where there is considerable scope for improvement is in

incorporating other commonly seen constraints like Staff shifts, taking a detailed view

of the various equipments needed in all the specialties.

57

APPENDICES

APPENDIX 1: Data Collected from Pushpagiri Institute of Medical

Sciences and Research Centre, Thiruvalla.

Basic Information:

No. of Operating Theatres: 10

No. of Post-Operative Rooms: 4

No. of Pre-Anesthesia Rooms: 1 hall

Capacity of Each POR: 7 with required equipment

No. of beds in the Surgical ICU: 12

No. of beds in the Septic ICU: 4

No. of beds in the Burns ICU: 4

No. of Operations performed at a time in one OT: 1

No. of surgeons on GS: 25; divided into 4 units

Time for elective surgeries in a day: 8:30 AM - 3:30 PM

Emergency slots: 1 theatre Round the clock

Duration Range for an operation: Variable; 1-3 hours, up to 14 for transplants

Shifts of Doctors & nurses: 4 shifts; 8:30 AM - 3:30 PM; 1 - 9 PM; 3:30 - 11:30 PM;

11:30 PM - 7:30 AM

Table AP 1.1 Average daily Out Patient Department attendance

GS Ortho OR

Laryn Ophthalmology

Old 84 61 37 11

New 27 17 11 2

Total 111 78 48 13

58

Table AP 1.2 Number of surgeries for the year 2009

Department Major Minor

GS including Pediatric

Surgery 2326 2781

Orthopaedics 732 747

Oto-Rhino-Laryngology 219 157

Ophthalmology 30 20

59

APPENDIX 2: MATLAB code for MILP Model 2

function projcpy()

try

%DEFINE DATA

no_of_types_of_theatres = input('Enter the number of operating theatres types:');

no_of_staffed_hours = input('Enter the number of staffed hours:');

delay_for_unmet_demand = input('Enter the delay for unmet demand:');

no_of_operating_days = input('Enter the number of operating days:');

no_of_specialities = input('Enter the number of specialities:');

penalty_rate = input('Enter the penalty rate:');

no_of_beds_in_POC = input('Enter the number of bed in post-operative care:');

total_hours_in_POC = no_of_beds_in_POC*168;

%For the multiplication factor delay

for k = 1:no_of_operating_days,

for l = 1:no_of_operating_days,

if k==l,

delay(k,l) = 7,

end

if k<l,

delay(k,l) = l-k,

end

if k>l,

delay(k,l) = 7-k+l,

end

end

end

%For the total non emergency demand (o)

for j= 1:no_of_specialities

for k = 1:no_of_operating_days

display(sprintf('enter the total non emergency demand for the day %d for specialty

%g',k,j));

o(j,k) = input(':');

end

end

%For the maximum number operating rooms (c)

for j=1:no_of_specialities

for k=1:no_of_operating_days

display(sprintf('enter the maximum no. of operating rooms for the day %d for

specialty %g',k,j));

c(j,k) = input(':');

end

60

end

%For the number of operating rooms of type i (f)

for i = 1:no_of_types_of_theatres

display(sprintf('enter the number of operating rooms of type %d',i));

f(i) = input(':');

end

%For the emergency demand (em)

for j=1:no_of_specialities

for k=1:no_of_operating_days

display(sprintf('enter the emergency demand for the day %d for specialty

%g',k,j));

em(j,k) = input(':');

end

end

%Build Model

cplex = Cplex('proj');

cplex.Model.sense = 'minimize';

%OBJECTIVE FUNCTION

%For the demand that has been postponed from the previous week

a = 1;

for j = 1:no_of_specialities,

for k = 1:no_of_operating_days,

for l = 1:no_of_operating_days,

obj(a) = delay(k,l);

ctype(a) = char('C');

a = a+1;

end

end

end

%For the unmet non emergency demand

for j = 1:no_of_specialities,

for k = 1:no_of_operating_days,

obj(a) = delay_for_unmet_demand;

ctype(a) = char('C');

a = a+1;

end

end

%For the undersupply

for j = 1:no_of_specialities,

61

obj(a) = penalty_rate;

ctype(a) = char('C');

a = a+1;

end

%For the oversupply

for j = 1:no_of_specialities,

obj(a) = 0;

ctype(a) = char('C');

a = a+1;

end

%For the amount of idle time

for j = 1:no_of_specialities,

for k = 1:no_of_operating_days,

obj(a) = 0;

ctype(a) = char('C');

a = a+1;

end

end

%For the number of operating rooms engaged in a day

for i = 1:no_of_types_of_theatres,

for j = 1:no_of_specialities,

for k = 1:no_of_operating_days,

obj(a) = 0;

ctype(a) = char('I');

a = a+1;

end

end

end

%For the total idle time

obj(a) = 0;

ctype(a) = char('C');

a=a+1;

%For the emergency ORs staffed hours

for j = 1:no_of_specialities,

for k = 1:no_of_operating_days,

obj(a) = 0;

ctype(a) = char('C');

a = a+1;

end

end

a=a-1;

62

%Constraints formulation

rows

=((no_of_types_of_theatres*no_of_operating_days)+5*(no_of_specialities*no_of_op

erating_days)+1+no_of_operating_days);

A = zeros(rows,a);

%Number of operating rooms constraint

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days));

for i = 1:no_of_types_of_theatres

for k = 1:no_of_operating_days

for j = 1:no_of_specialities

A(((i-1)*no_of_operating_days)+k,d+((k-1)*no_of_specialities)+((i-

1)*no_of_operating_days*no_of_specialities)+j)= 1;

end

end

end

%Amount of idle time constraint

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities)));

e=((no_of_types_of_theatres*no_of_operating_days)+(2*(no_of_specialities*no_of_

operating_days)));

for j= 1:no_of_specialities

for k= 1:no_of_operating_days

A(e+1,d+k+((no_of_operating_days)*(j-1)))=1;

end

end

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days)+(no_of_types_of_theatres*no_of_specialities*no_of_operating_days));

A(e+1,d+1)=-1;

%Max number of operating rooms for a speciality constraint

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days));

e=((no_of_types_of_theatres*no_of_operating_days)+(2*(no_of_specialities*no_of_

operating_days)+1+no_of_operating_days)+no_of_specialities);

for i= 1:no_of_types_of_theatres

for j = 1:no_of_specialities

for k = 1:no_of_operating_days

A(e+((j-1)*no_of_operating_days)+k,d+((j-1)*no_of_operating_days)+((i-

1)*no_of_specialities*no_of_operating_days)+k)=1;

63

end

end

end

%OR capacity constraint for each speciality

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days));

for j=1:no_of_specialities

for k=1:no_of_operating_days

for i=1:no_of_types_of_theatres

A(((no_of_types_of_theatres*no_of_operating_days)+(j-

1)*no_of_operating_days)+k,d+((j-

1)*no_of_operating_days*no_of_types_of_theatres)+...

((k-1)*no_of_types_of_theatres)+i)=no_of_staffed_hours;

end

for l=1:no_of_operating_days

A(((no_of_types_of_theatres*no_of_operating_days)+(j-

1)*no_of_operating_days)+k,((j-

1)*no_of_operating_days*no_of_types_of_theatres)+...

((k-1)*no_of_types_of_theatres)+l)=-1;

end

end

end

%Total non-emergency demand for a specialty constraint

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days));

e=((no_of_types_of_theatres*no_of_operating_days)+(no_of_specialities*no_of_oper

ating_days));

for j=1:no_of_specialities

for k=1:no_of_operating_days

for i=1:no_of_types_of_theatres

A(e+(j-1)*no_of_operating_days+k,d+((j-

1)*no_of_operating_days*no_of_types_of_theatres)+((k-

1)*no_of_types_of_theatres)+i)= no_of_staffed_hours;

end

end

end

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities)));

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+((j-1)*no_of_operating_days)+k,d+((j-1)*no_of_operating_days)+k)=-1;

end

64

end

d=((no_of_operating_days*no_of_operating_days*no_of_specialities));

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k) = 1;

end

end

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days)+(no_of_types_of_theatres*no_of_specialities*no_of_operating_days)+

1);

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k) = 1;

end

end

%For the emergency hours equal staffed hours

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days)+(no_of_types_of_theatres*no_of_specialities*no_of_operating_days)+

1);

e=((no_of_types_of_theatres*no_of_operating_days)+(2*(no_of_specialities*no_of_

operating_days)+1)+no_of_specialities);

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k) = 1;

end

end

%For the emergency OR hours allocated

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_op

erating_days)+(no_of_types_of_theatres*no_of_specialities*no_of_operating_days)+

1);

e=((no_of_types_of_theatres*no_of_operating_days)+(3*(no_of_specialities*no_of_

operating_days)+1+no_of_operating_days)+no_of_specialities);

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k) = 1;

end

end

65

%For the POC constraint

e=((no_of_types_of_theatres*no_of_operating_days)+(4*(no_of_specialities*no_of_

operating_days)+1+no_of_operating_days)+no_of_specialities);

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k) = 1;

end

end

d=((no_of_operating_days*no_of_operating_days*no_of_specialities));

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k) = -1;

end

end

%Under supply constraint

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_specia

lities*no_of_operating_days)+(2*(no_of_specialities)));

e=(no_of_types_of_theatres*no_of_operating_days)+2*(no_of_specialities*no_of_op

erating_days)+1;

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k)=1;

end

end

d=(no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_special

ities*no_of_operating_days);

e=(no_of_types_of_theatres*no_of_operating_days)+2*(no_of_specialities*no_of_op

erating_days)+1;

for j=no_of_specialities

A(e+j,d+1)=1;

end

d=(no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_special

ities*no_of_operating_days)+no_of_specialities;

e=(no_of_types_of_theatres*no_of_operating_days)+2*(no_of_specialities*no_of_op

erating_days)+1;

for j=no_of_specialities

A(e+j,d+1)=1;

end

sum=0;

for j=1:no_of_specialities

for k=1:no_of_operating_days

66

sum= sum+o(j,k);

end

end

for j=1:no_of_specialities

g(j)=0;

for k=1:no_of_operating_days

g(j)=g(j)+o(j,k) ;

end

end

d=((no_of_operating_days*no_of_operating_days*no_of_specialities)+2*(no_of_spe

cialities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_types_of_theatres*

no_of_operating_days*no_of_specialities));

for j=1:no_of_specialities

for k=1:no_of_operating_days

A(e+(j-1)*no_of_operating_days+k,d+((j-1)*no_of_operating_days)+k)=-

(g(j)/sum);

end

end

%Constraint bounndary

rs=(no_of_types_of_theatres*no_of_operating_days)+(5*(no_of_specialities*no_of_o

perating_days))+1+no_of_operating_days+no_of_specialities;

left_hand=zeros(rs,1);

right_hand=zeros(rs,1);

%Boundary for operating room constraint

for i=1:no_of_types_of_theatres

for k=1:no_of_operating_days

right_hand(((i-1)*no_of_operating_days)+k,1)= f(i);

left_hand(((i-1)*no_of_operating_days)+k,1)= f(i);

end

end

%Boundary for amount of idle time constraint

row=(no_of_types_of_theatres*no_of_operating_days)+2*(no_of_specialities*no_of_

operating_days);

right_hand(row+1,1)=0;

left_hand(row+1,1)=0;

%Boundary for max number of operating room for a speciality constraint

rows =

((no_of_types_of_theatres*no_of_operating_days)+(2*(no_of_specialities*no_of_ope

rating_days)+1+no_of_operating_days)+no_of_specialities);

67

for j=1:no_of_specialities

for k=1:no_of_operating_days

right_hand(((rows+((j-1)*no_of_operating_days)+k)),1)=c(j,k);

left_hand(((rows+((j-1)*no_of_operating_days)+k)),1)=0;

end

end

%Boundary for OR capacity

for j=1:no_of_specialities

for k=1:no_of_operating_days

right_hand((no_of_types_of_theatres*no_of_operating_days)+((j-

1)*no_of_operating_days)+k,1)=0;

left_hand((no_of_types_of_theatres*no_of_operating_days)+((j-

1)*no_of_operating_days)+k,1)=0;

end

end

%Boundary for non-emergency demand

rows=((no_of_types_of_theatres*no_of_operating_days)+(no_of_specialities*no_of_

operating_days));

for j=1:no_of_specialities,

for k=1:no_of_operating_days

left_hand(((rows+((j-1)*no_of_operating_days)+k)),1)= o(j,k)+em(j,k);

right_hand((rows+(((j-1)*no_of_operating_days)+k)),1)= o(j,k)+em(j,k);

end

end

%Boundary for emergency OR hours and staffed hours

rows=((no_of_types_of_theatres*no_of_operating_days)+(2*(no_of_specialities*no_

of_operating_days)+1)+no_of_specialities);

for j=1:no_of_specialities,

for k=1:no_of_operating_days

left_hand(((rows+((j-1)*no_of_operating_days)+k)),1)= 0;

right_hand((rows+(((j-1)*no_of_operating_days)+k)),1)= no_of_staffed_hours;

end

end

%Boundary for allocated OR hours constraint

rows=((no_of_types_of_theatres*no_of_operating_days)+(3*(no_of_specialities*no_

of_operating_days)+1+no_of_operating_days)+no_of_specialities);

for j=1:no_of_specialities,

for k=1:no_of_operating_days

left_hand(((rows+((j-1)*no_of_operating_days)+k)),1)= 0;

right_hand((rows+(((j-1)*no_of_operating_days)+k)),1)= em(j,k);

end

end

68

%Boundary for POC

rows=((no_of_types_of_theatres*no_of_operating_days)+(4*(no_of_specialities*no_

of_operating_days)+1+no_of_operating_days)+no_of_specialities);

for j=1:no_of_specialities,

for k=1:no_of_operating_days

left_hand(((rows+((j-1)*no_of_operating_days)+k)),1)= 0;

right_hand((rows+(((j-1)*no_of_operating_days)+k)),1)= total_hours_in_POC-

o(j,k);

end

end

%Boundary for under supply constraint

rows=(no_of_types_of_theatres*no_of_operating_days)+2*(no_of_specialities*no_of

_operating_days)+1;

for j=1:no_of_specialities

for k=1:no_of_operating_days

left_hand(((rows+((j-1)*no_of_operating_days)+k)),1)= 0;

right_hand((rows+(((j-1)*no_of_operating_days)+k)),1)=0;

end

end

%Specifying the upper and lower boundary for the variables(optional)

lb=zeros(length(obj),1);

ub=ones(length(obj),1)*Inf;

% Solving using Cplex

cplex.Model.obj = obj';

cplex.Model.ctype = ctype;

cplex.Model.A = A;

cplex.Model.lb = lb;

cplex.Model.ub = ub;

cplex.Model.lhs = left_hand;

cplex.Model.rhs = right_hand;

cplex.solve();

cplex.writeModel('projcpy.lp');

%Display

display(cplex.Solution.status);

solf=cplex.Solution.x;

fprintf ('\nMinimum Time: %f \n', cplex.Solution.objval);

f=((no_of_operating_days*no_of_operating_days*no_of_specialities)+(no_of_special

ities*no_of_operating_days)+(2*(no_of_specialities))+(no_of_specialities*no_of_ope

rating_days))+1;

69

%Solution matrix

for i = 1:no_of_types_of_theatres,

for j = 1:no_of_specialities,

for k = 1:no_of_operating_days,

B(i,j,k)=solf(f);

f=f+1;

end

end

end

display(B);

catch m

disp (m.message);

end

end

70

APPENDIX 3: Screenshots of MATLAB

Figure AP 3.1 MATLAB Programming Window

Figure AP 3.2 MATLAB Model running window

71

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