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Algorithm development for solving the emergencyvehicle location problem with stochastic

travel times and unequal vehicle utilizations

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Authors Paz Avila, Luis Albert, 1964-

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Algorithm development for solving the emergency vehicle location problem with stochastic travel times and unequal vehicle utilizations

Paz Avila, Luis Alberto, M.S.

The University of Arizona, 1988

U M I 300 N. Zeeb Rd. Ann Arbor, MI 48106

ALGORITHM DEVELOPMENT FOR SOLVING

THE EMERGENCY VEHICLE LOCATION PROBLEM

WITH STOCHASTIC TRAVEL TIMES AND

UNEQUAL VEHICLE UTILIZATIONS

by

Luis Alberto Paz Avila

A Thesis Submitted to the Faculty of the

COMMITTE ON SYSTEMS AND INDUSTRIAL ENGINEERING

In Partial Fulfillment of the Requirements

for the Degree of

MASTER OF SCIENCE

WITH A MAJOR IN INDUSTRIAL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 8 8

2

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for extended quotations from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

i ^ j n h i ' DATE Jeffrey B. Goldberg

Assistant Professor of Systems and Industrial Engineering

3

ACKNOWLEDGEMENTS

For his assistance and guidance during my college career and specially during the period of my thesis work, I wish to express my deep gratitude to my advisor, Dr. Jeffrey B. Goldberg. His invaluable enthusiasm, motivation, and support were key factors in the successful preparation of this thesis.

I would also like to thank Dr. Ferenc Szidarovszky for his contribution and advice in the solution of the system of nonlinear simultaneous equations defining the vehicle utilizations.

Last but not least, I would like to dedicate this thesis to my wife, Amarillys, for her love, support, and infinite patience during my years in college, and to my mother, Negda, whose love and confidence on me have been great motivators for the successful completion of my academic studies.

4

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS . 5

LIST OF TABLES 6

ABSTRACT 7

CHAPTER 1: INTRODUCTION 8

CHAPTER 2: LITERATURE REVIEW 11

CHAPTER 3: MODEL DESCRIPTION 17

Formulation 17

CHAPTER 4: HEURISTIC DEVELOPMENT 21

Criteria 21

Category 1 21

Category 2 24

Category 3 27

Category 4 30

Discussion of Common Steps 32

CHAPTER 5: GENERATION OF PROBLEM SETS 35

Tucson Emergency Medical System 35

Procedure for Generating the Test Problems 36

CHAPTER 6: COMPUTATIONAL TESTING 45

Overall Aggregate Results 46

CHAPTER 7: CONCLUSIONS AND FUTURE RESEARCH 63

APPENDIX A 65

REFERENCES 69

5

LIST OF FIGURES

Figure Page

1 Flow Chart of T&B Heuristic 2 2

2 Flow Chart of VNZ2 Heuristics 25

3 Flow Chart of VNZ1 Heuristics 2 8

4 Flow Chart of SINGLE Heuristics 31

5 Map #1 40

6 Map #2 41

7 Map #3 4 2

8 Map #4 4 3

9 Map #5 4 4

6

LIST OF TABLES

Table Page

1 a Number of Pairwise Interchange Evaluations 47

1 b Number of Pairwise Interchange Evaluations 48

1 c Number of Pairwise Interchange Evaluations 49

2 a Total Expected Coverage 51

2 b Total Expected Coverage 52

2 c Total Expected Coverage 53

3 Number of "Best" (Tolerance = 1 call) 56

4 Number of "Best" (Tolerance = 2 calls) 57



7 a Deviation from "Best" 60

7 b Deviation from "Best" 61

7 c Deviation from "Best" 62

7

ABSTRACT

This thesis deals with the problem of locating emergency vehi

cles in an urban area. An optimization model is formulated that ex

tends previous work by allowing stochastic travel times, unequal ve

hicle utilizations, and backup service. The heart of the model is a

procedure similar to the Hypercube approximation model. Ten pair-

wise interchange heuristics are developed and tested on 240 test

problems. Demand and service time components of the test data

have been generated using characteristics of the Tucson Emergency

Medical System. Geographical components of the test data have been

generated using actual city shapes as models. It is believed that

these test problems are more indicative of actual emergency vehicle

location problems than those previously presented in the literature.

8

CHAPTER 1

INTRODUCTION

The emergency vehicle location problem consists of determining

the location of vehicle bases of an emergency service system so as to

best achieve some level or combination of levels of service. In the

location problem , the location of each vehicle may be fixed or

mobile. For example, in ambulance applications it is usual to have

each ambulance positioned at a fire station house, a hospital, or any

other "fixed" location. On the other hand, for police patrol, the

"location" of each unit is mobile, corresponding to the areas that the

unit patrols in its sector.

Typical urban emergency service systems are police, fire, and

ambulance service systems. Common to these systems is the parti

tioning of the geographical region being served into districts of ser

vice. The "districting" problem can be formulated as follows: given

a region with a certain spatial distribution of demands for service

and given N response units or vehicles that are spatially distributed

throughout the region, "how should the region be partitioned into ar

eas of primary responsability so as to optimize some measure of

service". Obviously, the vehicle location problem and the districting

problem are closely related: in order to "optimally" locate the units

one assumes a given partitioning of the service region and in order to

"optimally" partition the service region one assumes a given set of

vehicle locations. Given the vehicle base locations, the districts are

formed based on closest distance.

Among the urban emergency service systems that have re

ceived increased attention in resource allocation studies are Emer

9

gency Medical Service (EMS) systems. An EMS system consists of a

fleet of ambulances that provides public paramedical service to the

population of a given geographical area. In most cases, the geo

graphical area is a city and its suburbs, divided into zones. Data col

lection is at the zone level. A zone comprises a subset of the total

system demand and is bounded by streets, free ways, railroads,

rivers or any type of landmark. Demand for service is generated by

the population via telephone communication (911 service). The

system usually has a Central Communications division that responds

to a particular call by dispatching an ambulance that will provide the

necessary medical assistance. Ambulance dispatching is made by

selecting the idle ambulance, i.e, an ambulance available for service,

which is nearest to the zone where the call originated. If there are

no idle ambulances, the call waits in queue for service (infinite line

capacity system) or, more likely, a private ambulance must be

called (loss or zero line capacity system). In most cases, calls are

served in a first -come-first-served basis although other medically

accepted criteria may be used.

Regardless of the dispatching criterion employed, a critical

factor in the performance of EMS systems is response time. Re

sponse time is defined to be the time between the time of emergency

call notification and the time the ambulance arrives on the scene.

Total call service time for an ambulance includes dispatching time,

travel time to the call location, service time at the scene, possible

transportation to a hospital, possible time at the hospital, and return

travel time to base (i.e, fire station house). Dispatching time is gen

erally constant and small relative to travel time. The service time,

not including travel time to scene, is dependent on zone location.

This is due to the fact that some zones are further from hospitals

than others and that different emergencies occur in different zones.

The purpose of this thesis is to develop and test the computa

tional effectiveness of ten pairwise interchange heuristics for solving

1 0

an emergency vehicle location model presented by Goldberg et al

[1988], In the past, this model has been used purely in a "system

evaluation" or single run mode. Due to the discrete nature of the

base location decision in the model, it was determined that pairwise

interchange is an appropiate technique. Note that the model is

broadly applicable and has been validated in Tucson, Arizona.

Among its features are its ability to handle stochastic travel times

and unequal vehicle utilizations.

The thesis is organized into 7 chapters. Chapter 2 presents a

literature review of the relevant research related to the work re

ported here. Chapter 3 introduces the model developed by Goldberg

et al [1988] and compares it with other studies. Chapter 4 discusses

the development of the ten heuristics and their detailed algorithms.

Chapter 5 discusses the generation of 240 test problems. Actual city

shapes are used as geographic models and the characteristics of data

from the Tucson EMS system are used for generating demand and

service time data. Chapter 6 presents the computational results of

the testing of the ten heuristics on the 240 test problems. Conclu

sions and future research directions are discussed in Chapter 7.

1 1

CHAPTER 2

LITERATURE REVIEW

This chapter presents a survey of the relevant work done in the

past in vehicle location problems. Past work in vehicle location has

centered on three approaches: queueing, mathematical programming,

and simulation. Each one will be discussed separately. In most of

the approaches presented, solution methods were tested using

problem sets whose characteristics do not resemble those of actual

systems. In some cases, problem sizes were too small to guarantee a

true measure of algorithmic performance for "real-world" problems.

The most widely diseminated queueing approaches for locating

multiple facilities are the hypercube model (denoted Hypercube),

Larson [1974], and the hypercube approximation (denoted A-Hyper-

cube), Larson [1975]. Both models are descriptive in that they yield

system operating characteristics that can be used to compute a host

of objectives. Strengths include the ability to evaluate cooperation

between vehicles, dispatch vehicles from mobile locations, and eval

uate a wide variety of output measures. Weaknesses include expo

nential assumptions for service time distribution and computational

intractability for problems with many vehicles. A-Hypercube reme

dies the computational problems by showing that the vehicle utiliza

tions can be approximated by solving a system of nonlinear equa

tions whose size depends on the number of vehicles. Larson [1974]

used a test problem with nine units and 18 equal size atoms or zones

equally spaced along a linear region. Calls were distributed uni

formly over the linear region. The method was also tested on prob

lems with a total of 12 units, registering marked increases in com

putational effort.

1 2

Hypercube requires that the service time is solely dependent on

the vehicle, not the call location. To incorporate location service time

characteristics, it is necessary to develop a method that estimates the

mean service time for each vehicle based on the calls served. Jarvis

[1975] developed the mean service calibration method to estimate

mean service time for each vehicle. The general process is:

1. Initialize the current estimate of mean service time of vehicle j to the average service time for the entire area.

2. Using the current estimates for mean service time, evaluate Hypercube to obtain the probability that each vehicle serves each zone.

3. Using the probability estimates for each vehicle serving each zone, derive a new estimate of the average service time for each vehicle.

4. If the new estimate of average service time is close to the current estimate the algorithm stops since an equilibrium point has been reached. If convergence has not been achieved, replace the current estimate of mean service time with the new estimate and return to step 1.

Jarvis [1975] and Halpern [1977] showed that the service time mean

is sufficient to obtain reasonable estimates of system performance.

The major shortcoming of this approximation method is that Hyper

cube must be evaluated in each iteration.

Optimization models using Hypercube or A-Hypercube as a

function evaluation subroutine include Jarvis [1975]'s location-

allocation method, Berman and Larson [1982]'s method for the

congested median problem, Benveniste [1985]'s location-allocation

method, and Berman, Larson and Parkan [1987]'s method for the

stochastic queue p-median problem. Goldberg et al. [1988]'s model

1 3

with stochastic travel times and unequal vehicle utilizations is the

model formulation for which heuristic procedures are developed in

this thesis. All of these models and their companion solution

methods are attempts at locating cooperating service facilities on a

network. Berman and Larson [1982] presented a simple 5-node

network with three facilities to be located as a computational

example. Berman, Larson, and Parkan [1987] used a three-server

network with 10 nodes for computational testing. Each node behaved

as a Poisson generator of service requests.

Mathematical programming approaches fall into two categories:

set covering and median. These are discussed briefly since the work

presented in this study does not build off these directly. Also, these

approaches generally have restrictive assumptions that make model

validation difficult.

Set covering approaches have been discussed in Toregas et al .

[1971], Church and ReVelle [1974], Daskin [1983], and Saydam and

McKnew [1987]. Toregas et al. formulated a 0-1 integer program to

decide on the minimum number of bases to use so that every zone

was covered. That is for each zone, the expected travel time from at

least one open base was within the specified time limit. Computa

tional experience was gained by testing on a 30-node 90-arc net

work. Church and ReVelle withdrew the coverage requirement while

adding a demand factor. Their model was to locate bases so that the

maximum number of calls could be reached within the specified

limit. Both of these papers assume that a vehicle that is based within

the limit can always cover the call. They do not take into considera

tion vehicle utilization. If a vehicle is busy, it may not be able to

reach a call in time, eventhough it is within the specified limit.

Daskin's paper rectifies this shortcoming by formulating an

objective that represents the expected number of covered calls based

on the number of vehicles that cover a particular zone. Daskin's

1 4

objective is based on ideas similar to Larson's system of nonlinear

equations in A-Hypercube. The formulation of Daskin's model is:

n M maximize X Z 0 - P) P'"1 hj Yjj

j=li=l

M n subject to I Yij - I aij Xj < 0

i=l i=l

n I X j < M

i=l

Xj = 0, 1,2, .... , M for all i

Yjj = 0, 1 for all i,j

where M = the maximum number of facilities to be located, Xj = the number of facilities located at node i,

_ j 1 if node j is covered by at least i facilities iJ 10 if node j is covered by less than i facilities

hj = the demand generated at node j.

Saydam and McKnew [1987] transformed Daskin's model to a sepa

rable convex program and then solved the model to optimality using

branch and bound.

One weakness of Daskin's approach is that it assumes that all

vehicles have identical utilizations. Also, Daskin (and all other set

covering approaches) assume that travel times are deterministic.

This assumption leads to properties such as opening more bases

within the critical response time of a zone is better than opening

fewer bases. This is not true in general. The location of a base is

critical, not simply whether or not the base covers a certain zone.

1 5

Besides the queueing models that incorporate median type

objectives, other approaches that have been considered include

Hakimi [1964] (deterministic p-median), Church and Weaver [1983]

(stochastic median), Church and Weaver [1985] (vector assignment

p-median), and Pirkful and Schilling [1988] (capacitated p-

median).

Francis and White [1974] discuss a multifacility planar location

model where there are prespecified interaction between certain fa-

cility-customer combinations. These interactions are part of the

problem formulation and do not depend upon the distance between

the customers and the facilities to be located. Such problems occur in

manufacturing and warehousing. In contrast, the model to be de

veloped here assumes service assignment is location dependent.

Closest assignment of facilities to demand is an example of location

dependent assignment.

Pirkful and Schilling [1980] generated 180 test problems from

a 50x100 grid in a fashion somewhat similar to the method em

ployed in this study. However, the data they generated do not pos

sess characteristics of "real-world" systems. Church and Weaver

[1983] used Mirchandani [1980]'s 10-node 3-state stochastic network

to test their subgradient procedure. They also generated a three

state network from Toregas [1971 ]'s network. Church and Weaver

[1985] presented a more thorough computational testing by using

three different data sets of moderate size: Toregas' 30-node 90-arc

network (3-15 facilities), the 49-node Talala India set found in

Hillsman [1980], and the 55-demand point Swain[1971] data set (2-

25 facilities). In general, the strengths of these models is that opti

mal solution procedures have been developed that can accomodate

practical problems. Weaknesses include requiring assumptions such

as non-cooperation between vehicles, the probability of each system

state is known and the fraction of call served by the closest, second

closest, etc., must be known for each zone.

1 6

Simulation has been used to evaluate system performance in

numerous papers including Savas [1969], Berlin and Liebman [1972],

and Goldberg et al. [1988]. Simulation models can be formulated with

a great deal of detail and hence can be validated. Also they provide

a wealth of output measures.

1 7

CHAPTER 3

MODEL FORMULATION

In this chapter a new model that relaxes the assumptions of

service time independence on call location, and equal vehicle utiliza

tion is presented. The model objective is to maximize the expected

number of calls reached within a critical response time. Note that

this specific time limit does not effect the structure of the model. In

this objective the assumption of deterministic travel times is relaxed.

The model moves away from "coverage" type objectives to objectives

that more precisely measure the probability of successful emergency

service. From a practical point of view, the model is broadly applica

ble and has been extensively validated in Tucson, Arizona (Goldberg

et al[ 1988]). Limitations on applicability due to remaining assump

tions and data availability are also discussed in the chapter.

3.1 Model Formulation.

Assume that we are given an area that is broken into I zones

(indexed by i). In the area there are J potential base locations

(indexed by j). If the set of open bases is known, then for each zone

we can determine a preference ordering of the open bases. This

preference ordering represents the scheduling order for servicing

calls in the zone. Denote k as the kth position on the preference list

of open bases for any zone. The following additional notation is re

quired:

1. Decision Variables - Xj = 1 if base j is open, 0 else,

1 8

- Xjjjc= 1 if base j is the kth closest open base to zone i, 0

else, - pj is the utilization of a vehicle located at base j,

2. Data - dj is the expected number of calls for zone i,

- tjj is the mean travel time from base j to zone i,

- Pjj is the probability that a vehicle at base j can reach a

call at zone i within the critical response time,

- f is the total number of bases to open in the area, - S j is the mean service time per call in zone i excluding

travel time from a base to i,

- TT is the total time available for a vehicle in the study,

and

- M is a large integer number.

The following assumptions are required in the model. First, it is assumed that the probability that a vehicle is busy is pj and is un

affected by the state of the system. This is the "independence as

sumption" used in Daskin's model. Second, we do not consider the stochastic nature of travel times in determining pj , only the mean

travel is used. Third, the model assumes that there is a strict or

dering of the bases preferred for each zone. If two bases are equally

distant from a zone, then one base will be preferred over the other

for dispatches when both are idle. This assumption requires that

zone size and construction be carefully considered. Fourth, the model

assumes that all calls are answered by a vehicle originating from its

base, not enroute back to the base. This assumption is largely true

when utilizations are small and simplifies the amount of travel time

required. However, the variance of the predicted travel time will

increase since the true origination point of the vehicle is unknown

for some calls. The assumptions are discussed further when dis

cussing the applicability of the model. The mathematical model is

(denoted Problem P):

1 9

maximize ^ d i i

I-XKPip Xijk (l-Pj) <n.XX i r g P r > ] k j g=l r

(IP)

I Xijk = 1 for each ( i , j ) pair (2P) k

subject to ^ Xjjk < 1 for each (i,j) pair (3P) k

I X j < f (4P) j

I-IXijk := MXj (5P) i k

X r I k X i j k < £ k X i r k

k k

for each base pair (j,r) where j is closer than r to i (6P)

k-1

Z - l d i ( t i j + S i ) ( 1 - P j ) ( £ X i j k n - IX i r g P r ) }

Pj = 8-Ll f0r j (7P)

Xj e [0, 1] , Xijk e [0, 1] , 0 < pj < 1 (8P)

The objective function represents the expected number of calls

reached within the critical response time for each zone. The term P j j Xjjk (1- pj) represents the probability that the vehicle at base j is

idle, j is the kth closest open base from i and can get to i in time.

The next term represents the probability that the first through (k-

2 0

l)st closest bases to i all are busy. So, for each i, the probability that

each base j successfully services calls is computed. This total proba

bility is multiplied by the demand to get the expected number of

calls reached successfully. Note that for k = 1, the product is defined

to be 1.

Constraint 2P states that one open base must take a particular

closeness rank while constraint 3P states that only one rank can be

assigned to each open base for any particular zone. Constraint 4P

limits the number of facilities to open and constraint 5P is the logical

relationship between opening a base and allowing it to take closeness

ranking. Constraint 6P requires that if base j is closer than base r to

zone i, then j must be given the lower closeness rank when both are

open. Note that if only j is open, then it can be given any rank relative to r, hence the need for the Xr indicator on the left hand side of

the constraint. Finally, Constraint 7P represents the nonlinear uti

lization equations for each open base. The service and travel time

per call are multiplied by the probability that j is assigned the call. Note that if j is not open, then pj will be set to 0 since Xjjk = 0 for all

(i,k). Constraint 7P implicitly assumes that queueing is rare since the

possibility of having all vehicles busy is not considered. Constraint

7P is similar to the utilization equations used in Larson [1975] for the

0-queue case. However, the dependence factor has been deleted and

factors for unequal travel and service times have been added.

2 1

CHAPTER 4

HEURISTIC DEVELOPMENT

This chapter discusses the overall structure of the ten heuristic

procedures used for the computational study. The heuristics are

presented by categories according to the number of pairwise inter

changes that are evaluated in every major iteration. In three of the

four specific categories, the heuristics differ from each other in the

criterion used to select the open base that will potentially leave the

incumbent set of open vehicle bases. The major steps that are com

mon to all heuristics are individually discussed at the end of the

chapter.

4.1 Criteria.

The following criteria will be used in the appropiate heuristics

to select from the current set of open bases, the base(s) that will be

evaluated for possible interchanges:

1. Success Total: number of emergency calls reached

successfully by a vehicle.

2. Success Ratio: percentage of the total allocated number of

emergency calls reached successfully by a vehicle.

3. Utilization: total time in service divided by the total time

available for a vehicle.

4 . 2 C a t e g o r y 1 : F u l l P a i r w i s e I n t e r c h a n g e ( T & B ) .

This category includes one heuristic developed by Teits and

Bart [1971]. This heuristic, labeled T&B in this study, evaluates all

Figure 1. Flow Chart of T&B Heuristic.

generate initial solution

setup current lists of open and closed bases

by iteratively evaluating all possible pairwise interchanges between the

current list of open bases and the current list of closed bases, find the

greatest objective value improvement

yes no update current solution and objective value

STOP

was an improving pairwise interchange found?

2 3

possible pairwise interchanges between the current set of open bases

and the current set of closed bases. The heuristic is very similar to

the Steepest-Descent Pairwise-Interchange procedure used for find

ing solutions to the Quadratic Assignment Location Problem [Francis

and White, 1974] in that all possible single pairwise interchanges are

considered. Figure 1 shows an idea-oriented flow chart of the

heuristic. A more detailed description of the steps of the heuristic are

as follows:

Step 0: Find an initial set of F locations using the initial solution

generator. Denote this set as the current solution. Set

the current objective value equal to the objective

function value generated by the initial solution ( using

steps 3,4, and 5 below). Set the improvement flag to 1

and go to step 1.

Step 1: If the improvement flag equals 0, stop, the incumbent

solution is the heuristic solution. Else (improvement

on last iteration), using the incumbent solution , set up

two lists. List O represents the open bases while list C

represents the closed bases. Set the improvement flag

equal to 0. Set the counters of the lists, a and b, equal to

1. Go to step 2.

Step 2: Create the candidate solution. Start with the open list O ,

delete 0(a) and add C(b). Go to step 3.

Step 3: Rank order the candidate solution for each zone according

to the travel time expected values tjj. This ranking represents the Xjjk values. Go to step 4.

Step 4: Using the rank order, solve the nonlinear equations (7P) for the pj values. Go to step 5.

2 4

Step 5: Using the pj values and the rank order evaluate the

objective function for the candidate solution. If the

objective is larger than the current objective value,

replace the current solution with the candidate solution,

update the current objective and set the improvement

flag to 1. Add one to b. If b is greater than J - F then

add one to a and set b equal to 1. If a is greater than F

then go to step 1, else go to step 2.

A total of F(J-F) pairwise interchanges are considered in every

major iteration of the T&B heuristic (a major iteration starts every

time stepl is executed). Thus for large problems, the amount of

computational effort required may be large, depending on the initial

solution, the number of major iterations, and the effort required to

evaluate a particular set of open bases.

4.3 Category 2: Two open, all closed pairwise interchanges

(VNZ2)

This category includes heuristics where two bases out of the

current set of open bases are selected for interchange. For each one

individually all possible pairwise interchanges with the set of closed

bases are evaluated. Three heuristics are contained in this category,

each one differing from the others in the criterion used to select the

two open bases for the pairwise interchanges. The general structure

of these heuristics is similar to the first phase of the improvement

procedure devised by Vollmann, Nugent , and Zartler[1968] for

finding a least-cost assignment to the Quadratic Assignment Location

Problem. We denote these heuristics by the term VNZ2.

Figure 2 shows an idea-oriented flow chart of the procedure.

The more detailed steps of the general procedure are as follows:

2 5

Figure 2. Flow Chart of VNZ2 Heuristics.

yes no

is s = m2? *1 set s = m2 yes no no

yes

no yes ISTOP

is improvement fla

was an improving pairwise interchange found?

generate initial solution Setup current list of closed bases

set improvement flag to 1, update current solution and objective value Setup current list of closed bases, is s = m2?

for the particular selection criterion being used, compute the values associated with the current solution. Let ml and m2 be the open bases with lowest and second lowest criterion values. Set improvement flag to 0. Set s= ml


current open base s and the bases in current list of closed bases, find the

pairwise interchange with greatest objective value improvement

2 6






and go to step 1.

Step 1: Compute for each of the ambulances in the current open

set, the values for the particular selection criterion used.

Let ml and ml be the base numbers associated with

lowest and second lowest CRITERION values computed

above, respectively. Set improvement flag i to 0, i = 1,2.

Set s to ml. Set i to 1. Set b to 1. Set up two lists. List

0 represents the open bases while list C represents the

closed bases. Go to step 2.


delete O(s) and add C(b). Go to step 3.

Step 3: Rank order the candidate solution for each zone according to the travel time expected values tjj. This ranking

represents the X values. Go to step 4.




objective is larger than the current objective value, set

the temporary solution to the candidate solution, update

the current objective value and set the improvement flag

1 to 1 . Add one to b. If b is greater than J - F ,

then go to step 6. Else go to step 2.

2 7

Step 6: If improvement flag i is 1, then replace current solution

with temporary solution. Go to step 7.

Step 7: If s = ml, then set s to m2, set b to 1, create two new

lists O and C representing the open and closed bases

respectively, and go to step 2. Else go to step 8.

Step 8: If improvement flag i is 0, i = 1,2, then STOP.

Otherwise, go to step 1.

The three heuristics in the category can be obtained by simply

using criterion 1,2, and 3, respectively, in step 1 of the general pro

cedure. In every major iteration, 2(J-F) pairwise interchanges are

considered (every time step 1 is executed, a major iteration starts).

So this method requires considerably less work than the T&B heuris

tic.

4.4 Category 3: One open, all closed pairwise interchange

(VNZ1)

The general structure of the heuristics in this category is similar

to the structure employed in category 2, except that only the open

base with lowest criterion value is selected for pairwise interchange

evaluations. There are also three heuristics in this category, each one

employing one of the three criteria introduced in section 4.1. Figure

3 shows the idea-oriented flow chart of the general procedure. The

procedure, denoted VNZ1, is:






and go to step

2 8

Figure 3. Flow Chart of VNZl Heuristics.

no yes

STOP


Setup current list of closed bases

was an improving pairwise interchange found? update

current solution and objective value

for the particular selection criterion being used, compute the values associated with the

current solution. Let ml be the open base with lowest criterion value.


current open base ml and the bases in current list of closed bases, find the

pairwise interchange with greatest objective value improvement

2 9

Step 1: Compute for each of the vehicles in the current open


Let s be the base number associated with lowest

CRITERION value computed above. Set improvement flag

to 0. Set b to 1. Set up two lists. List O represents the

open bases while list C represents the closed bases. Go to

step 2.


delete s from O, add C(b). Go to step 3.

Step 3: Rank order the candidate solution for each zone according to the travel time expected values tjj. This ranking

represents the Xjjk values. Go to step 4.




objective is larger than the current objective value, set

the temporary solution to the candidate solution, update

the current objective value, and set the improvement flag

to 1. Go to step 6.

Step 6: Add one to b. If b is greater than J - F , then go to step

7. Else go to step 2.

Step 7: If improvement flag is 1, then replace current solution

with temporary solution, and go to step 1. Else STOP.

A major iteration for this heuristic starts every time step 1 is

executed. In every major iteration (J - F) pairwise interchanges

3 0

are evaluated. So the work required is approximately half that of

VNZ2.

4.5 Category 4 : Single Interchange Heuristics (SINGLE).

This category includes heuristics that evaluate only one pair-

wise interchange between the set of open and the set of closed am

bulances. A total of three heuristics, which differ only in the crite

rion used to select the open base, are contained in this category.

Figure 4 shows the idea-oriented flow chart for this category. The

general heuristic structure is as follows:






and go to step 1.

Step 1: Compute for each of the ambulances in the current open


Let s be the base number associated with lowest

CRITERION value computed above. Set up two lists. List O

represents the open bases while list C represents the

closed bases. Go to step 2.

Step 2: From list C, find the vehicle base which is nearest to open

base s. Let b be the base number of such base. Go to

step 3.

Step 3: Create the candidate solution. Delete s from O, add b. Go

to step 4.

3 1

Figure 4. Flow Chart of SINGLE Heuristics.

no yes

ISTOP


Setup current list of closed bases

Does the pairwise interchange improve the objective value? update

current solution and objective value

for the particular selection criterion being used, compute the values associated with the

current solution. Let ml be the open base with lowest criterion value.

find the base in the current list of closed bases which is closest to ml. Label this closed base s. Evaluate the pairwise interchange between the

current open base ml and the current closed base s.

3 2

Rank order the candidate solution for each zone according to the travel time expected values ty. This ranking

represents the values. Go to step 5.

Using the rank order, solve the nonlinear equations (7P) for the pj values. Go to step 6.

Using the pj values and the rank order evaluate the


objective is larger than the current objective value,

replace the current solution with the candidate solution,

update the current objective value and go to step 1.

Otherwise, STOP.

A major iteration of this heuristic is comprised by steps 1

through 7. As mentioned above, only one pairwise interchange is

evaluated in every iteration. Among all the categories discussed,

this category involves the least number of pairwise interchanges.

The three heuristics in this category are obtained by altering the se

lection criterion as in categories 2 and 3. We denote this category as

SINGLE.

4.6 Discussion of Common Steps.

There are four main steps which are common to all ten heuris

tics. A more detailed explanation of each one will be given next.

A. Initial Solution Generation: an initial solution for all

heuristics is obtained by ignoring vehicle utilizations

(assume vehicles are available all the time) and

ambulance rankings for zones. Thus, the problem

becomes one of opening the set of vehicle bases whose

expected coverage is largest. The steps of the

procedure are as follows:

Step 4:

Step 5:

Step 6:

3 3

Step 1: Set counter N to 1. Go to step 2.

Step 2: Set up two lists. List O represents the set of open

bases and list C represents the list of closed bases.

Let O = { } and C = { k: k = }. Go to step 3.

Step 3: For every base in list C compute the expected

coverage

I

ECij = £ pij ' di > for j = 1, .. ,f and tjj < T i=l

where T is the critical response time.

Go to step 4.

Step 4: From list C, select the base with highest expected

coverage. Label this base as s. go to step 5

Step 5: Delete s from C and add it to O. Add one to N. Go

to step 6

Step 6: If N is less than or equal to f, then update the

expected coverage for every base in list C as follows:

ECjj = ECjj - Pjs • dj > f°r every j such that tjj and

t i j < T ,

and go to step 4.

Else, STOP. List O is the initial set of open bases.

B. Determining the base ranking for every zone: the

process of setting up the base rankings requires selecting

the open bases from a list of the potential bases, and then

sorting the travel time values. The selection and sort can

be done simultaneously using O (IJ2) steps. The I term

appears since the process must be done for each zone

while the J2 term appears since the list of bases must be

scanned J times for sorting and selection.

3 4

C Solving the nonlinear equations (7P): the system

of nonlinear simultaneous equations is solved by using

the Seidal iteration technique. At every iteration,

updated values of utilizations previously estimated are

used to compute the remaining utilizations. Appendix A

contains a detailed description of the technique along

with some convergence theorems.

D. Evaluating the objective function: this step can be

done in order O (If) steps. However, a step can require

up to f-1 multiplications to evaluate the probability that

the vehicles from more preferred bases are busy. In any

case, the complexity is bounded by o (If).

3 5

CHAPTER 5

GENERATION OF PROBLEM SETS

The generation of test problems comprises a great part of the

work done in this study. 240 test problems with geographical and

data components closely related to characteristics of actual EMS

systems were generated. In light of the "real-world" characteristics

of the data, it is strongly believed that the test problems generated

in this study are more indicative of actual emergency vehicle location

problems than those previously presented in the literature. It is

hoped that they can be used as a test bed for future research in this

area.

5.1 Tucson Emergency Medical System: an overview.

Goldberg et al. [1988] validated the analytical model presented

in chapter 3 using data from the Tucson EMS system. The city of

Tucson has a population of approximately 365,000 people spread

over more than 140 square miles. The Tucson Fire Department

Paramedic Service handles more than 20,000 requests yearly using 7

vehicles. The data used by Goldberg [1988] was collected from Jan

uary 1, 1986 to June 30, 1986. The city was divided into 405 de

mand zones with a data set of 9,350 requests for emergency ser

vice. For service calls, the following data were collected:

- zone where the call originated,

- ambulance that serviced the call,

- time the call entered the system,

- time the assigned ambulance arrived on scene, and

- time the assigned ambulance again became idle.

3 6

For the Tucson data set , it was assumed that all demand for a

zone occurs at the zone population center. Service time for an

emergency call included service time at the scene, possibly transport

to the hospital, and possible time at the hospital. Travel time to the

accident scene averaged 20% of the total service time (including

travel time). A strong assumption used in the model was that calls

are serviced by a vehicle starting from its base. This assumption ne

glects the fact that vehicles become available for service after leav

ing the hospital or completing an on-scene emergency service. The

low vehicle utilizations of approximately 0.15 that characterizes the

Tucson data seems to justify the assumption. The critical response

time adopted in the Tucson EMS system is 8 minutes.

5.2 Procedure for Generating the Test Problems.

The first step in the generation of the 240 test problems was

the selection of the geographical components of the areas. Five

different areas were generated from a 25x25 grid. By assigning ap-

propiate probability weights to the columns and rows of the grid,

unit square zones were randomly generated so as to give each of the

five areas a desired shape. Figures 5, 6, 7, 8, and 9, at the end of

the chapter, show the different maps. Each map contains 300 zones;

map 5 (because of the "river" crossing the area diagonally) had to

be generated by directly shaping the grid area with l's. For each

map, 20 potential vehicle bases for vehicles were generated by

assigning a probability weight of 1/300 to each zone and then

randomly choosing 20 zones out of the total of 300. Assumptions

made at this step were that every potential vehicle base must reside

in a zone and that no zone can contain more than one potential base.

Uniformity of vehicle bases is appropiate since federal fire service

regulations state that all zones in an urban area must be reachable

within a specific time limit.

3 7

Service time and demand data were generated by constructing a

histogram of the Tucson data used by Goldberg et al. [1988]. In

terval widths for the histograms were varied appropiately in order

to eliminate "gaps" in the range of values realized by the data. To

generate the demand and service time data for every zone, an inter

val was randomly generated from the respective cummulative his

togram and then a value within the interval was uniformly picked.

This combined strategy of eliminating gaps in the data and generat

ing uniformly within a selected interval avoids neglecting any value

in the ranges characterizing the Tucson data.

The rectilinear distance between the centers of mass of the

zones and the potential bases was used to generate the travel times.

Given the distance Dij between a zone and a base, a preliminary

travel time was generated at random from a uniform distribution

with mean Dij and a width value of 2. Between a base and a zone

sharing the same geographical location, the travel time was gener

ated from an uniform distribution with end points 0.25 and 0.75

time units. The preliminary travel times to a given zone were finally

adjusted in such a way as to make the mean travel time to the zone

equal to a percentage of the service time for the zone. This percent

age was generated from an uniform distribution with endpoints 18%

and 22%. This adjustment was made to try to "calibrate" the travel

times so as to provide the generated maps with similar statistical

characteristics between travel and service times as observed in the

Tucson data.

Another important step was the determinations of an ap-

propiate critical response time and the probabilities Pij. Since the

model moves away from the notion of "all or nothing coverage", the

deviation between a base-to-zone travel time and the critical re

sponse time is used to assign a value to the probability, Pij, that a

vehicle at base j can reach a call at zone i within the critical response

time. Naturally, the smaller the travel time between a base j and a

3 8

zone i, the greater Pij should be. Following this reasoning, a

"normal distribution standarization" technique, Goldberg et al. [1988],

was used to determine the critical response time and then the Pij's

values. A standard normal random variable was defined as

Zij ~ T tj;

where T is the critical response time and o is the standard

deviation of travel times (taken over the entire map).

The critical response time, T, was setup so that a vehicle located at

the closest base to each zone can reach the zone with a probability of

0.90. The following equation described the technique,

T = min { a • Z0.i0 + tij } for all i,j with Z0.io = normal

quantile of 0.90.

The P jj's values were determined using P { Z < Z,j }.

Four sets of travel, service, and demand data were generated

for every map. Every run consisted of four parameters namely,

1. A map .

2. A travel-service-demand data set for the map.

3. Number of open bases.

4. Average reference utilization.

The different parameter values used were

. 5 different maps.

. 4 travel-service-demand sets for each map .

. 4 number of open bases : 4, 7, 10, and 13.

. 3 reference utilizations : 0.15, 0.30, and 0.40.

3 9

By combining the different parameter values , the 240 different runs

were generated (5x4x4x3 = 240).

The reference utilization value was used to determine the total

time, TT, available for each vehicle. Given the number of ambu

lances to open and the reference utilization value, the following

equation was used to estimate a value for the total time :

I

X S j + T ; r™, i = l

where, Sj is the mean service time at zone i,

Tj is the mean travel time to zone i,

pa is the reference utilization value, and

f is the total number of bases to open.

Equation (*), which is closely related to equation 7P in the model

formulation, assumes that the total average workload is divided

equally among the open bases. This method was used to determine

if average utilization had an effect on heuristic performance. Al

though each vehicle utilization was not equal to the reference uti

lization, the average vehicle utilization was expected to be approxi

mately the reference value.

Figure 5. Mnp #1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 X X X 20 X X X X X I X X X 2 X X X X X 9 X X X X 18 X X 3 X X X X X X X X X 16 4 X X X X X X X X X X X 7 X X 5 X 11 X X X X X X X X X X 6 X X X X 15 X X X X X X 7 X X X X X X X X X X 8 X X X X X X X X 17 X 9 13 X X X X X X X X 4 X X X X 10 X X X X 12 X X X X X X X X X X 11 X X X X X X X X X 8 X X X 12 X 10 X X X X X X X X X X 13 X X X X X X X X X X X X X X 14 X X X X X X X X X X X X X X 15 X X X X 2 X X X

16 X X X X X 19 X 6 X X X 17 X X X X X X X X X X X

18 X X X X X X X X X X 14 X X 19 X X X X X X X X X 5 X X

20 X X X X X X X X X X X X X

21 X X X X X X X X X X X X X X X

22 X X X 3 X X X X X X X X

23 X X X X X X X X X X

24 X X X X X X X X X

25 X X X X X X X X X X X

Figure 6. Map #2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 X X X X X X X X X X X X X X 2 X X X X X X X X X X X X X X 3 X X 4 X X X X X X X X X X X X 4 X X 13 X X X X X X X X 5 X X X X X X X X X X X X X X X 6 X X X X X X X X X X X X X 7 16 18 X X X X 2 X X X

8 X X X X X X 12 X X X X 9 X X X X X X X 8 X X X X X 10 X X X X X X X X X X X

11 X 5 X X 14 X X X X X

12 X X X X X X X X X 17 X

13 X X X 1 X X X X X X X X

14 X X X X X X X X X 15 X X X X X 10 X X X X

16 6 X X X X X X X X X X X 17 X X X X X X X X X X X 18 X X X X 11 X X X 15 X X X X X

19 X X X X X X 3 X X X X X X 20 X X X X X X 20 X X X X 21 X X X X X X X X X X X X

22 X X 7 X X X 9 X X X 23 X X X X X X X X X X X

24 X X X X X X X X X X X

25 X X X X X X 19 X X X X X X X X X

Figure 7. Map #3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 X X X X X X X X 2 X X X X X X 18 3 X X X X X X X X X X 4 X X X X X X X 5 X X X X X X X X X 6 X X X X X X X X X 7 X X X X X X X X X 13 X X

8 11 X X 9 X X X X X X 9 X X X X X X X X X X X X X X X X X X X X X 10 X X X X X X X X X X X X X X X 17 X X X X 11 20 X X X X X X X X X X X X X X X X X X X X X 12 X X X X X X X X X X X X X X X X X X X X X 5 13 X X X X X 15 X X X X X 8 X X X

14 X X X X 7 12 X X X X X X X X X X X X X X X X 15 X X X X X X X X X X X X X X X X X X

16 X X X X X X X X X X X X X X X X 17 X X X X X X X

18 X X X X X

19 X X X X 3 X X X X X

20 X X X X X X X

21 X X X X 19 X

22 X X X X X 4 1 X 16 23 X 10 X X X X

24 X X X X X X 2 X X

25 14 X X X X X X X X X X 6 X

Figure 8. Map #4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 X 5 X X X X X X 2 X X X X X X X X 3 X 13 X 11 X X X X X 4 X X X X X X X 5 X X X X X X X X 14 6 X X X X X X X X X X X X X 7 X X X X 4 X X X X X X 8 X X X X X X X X X X X 9 X X X X X X X X X X X X X X X X X 3 X X 10 X X X X X X X X X X X X X X X X X X X X X 11 X X X X 10 X X X X X 6 X X X X X X X X X X 12 X X X X X X 16 X 1 X X X X X X 7 X X 13 X X X X X X X X X X X X X X X X 14 X X X X X 2 X X X X X X X X X X 18 X 15 15 9 X X X X 12 X X X X X X X X X X X X X X 16 X X X X X X X X X X X X X 20 X X 17 X X X X X X X X 18 X X X X X X 19 X X X X X X X X X X 20 X X X X X X 21 19 X X X X X 22 X 17 X X X X X X 23 X X X X X 8 X 24 X X X X X X X X X 25 X X X X X X X X X X X X X

(j-i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Figure 9. Map #5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

X X X X X X X X X X X X X 4 X X X X X X X X X 1 X X X X X X X X 17 X X X

X X 14 X X X X X X X X X X 19 X X X X X X X X X X X X X X X X X X X X X X

6 X X X X X 18 X X X X X X X X X X X X X X X X X X X

X X X X X X X X X X X X X X X 3 X X 15 X X X X X X X X X X X

X 12 16 X X X X X 10 X X X X X X X X X 2 X 7 X X X X

X X X X X X X 9 X X X X X X X X X X X X X X X X X

X X X X X X X X X X X

X X X X X X X X X X X X X X X X X X X X X

X X X X X X X X X

X X X X X X X X X X

X X X X X X X X X X X X X X

X X X X X X 11 X X X X X

X X X X X X X X X X X X X X X

X X X X 5 X X X X 13 X X X X X X X X X X 8 X X 20 X X X

X X

4 5

CHAPTER 6

COMPUTATIONAL TESTING

This chapter discusses the computational results derived from

testing each of the ten heuristics on the 240 problem sets. All the

heuristics were programmed in PL/1 using AT&T 7300 PC's. As

mentioned in the previous chapter, it was assumed that only one

vehicle can be located at any open base. In addition, only teams of

four bases were assigned to each zone because of the low objective

value contribution provided by the fifth and subsequent backup

bases. Evaluating a set of open bases took approximately 20 seconds.

For every run, the following measures were recorded:

. Initial solution objective value.

. Final solution objective value.

. Total demand.

. Percentage total coverage.

. Number of pairwise interchanges evaluated.

. Zone with lowest coverage and coverage value.

. Sequence of intermediate solutions with vehicle utilizations.

Analyzing such a large amount of data yields many options for

aggregating the results into measures that allow clear performance

comparisons. Consequently, computational results are presented on

overall aggregate measures. The aggregated performance measures

were the number of pairwise interchanges evaluated, the total per

cent coverage, the number of times each heuristic obtained the

highest total percent coverage, and the deviation from the highest

total percent coverage. The results are presented as to show the ef

fect of every combination of the number of open bases and the ref

4 6

erence utilization value on the performance of the heuristics. The

following heuristic labeling will be used to facilitate tabulation of re

sults:

. T&B : as before.

. VNZ2SUCT : VNZ2 heuristic with success total as criterion.

. VNZ2RAT : VNZ2 heuristic with success ratio as criterion.

. VNZ2UTIL : VNZ2 heuristic with utilization as criterion.

. VNZ1SUCT : VNZ1 heuristic with success total as criterion.

. VNZ1RAT : VNZ1 heuristic with success total as criterion.

. VNZ1UTIL : VNZ1 heuristic with utilization as criterion.

. SINGSUCT : SINGLE heuristic with success total as criterion.

. SINGRAT : SINGLE heuristic with success ratio as criterion.

. SINGUTIL : SINGLE heuristic with utilization as criterion.

6.1 Overall Aggregate Results.

Overall aggregate results were obtained by averaging and ag

gregating the performance data for each heuristic and every pair of

settings of the number of open bases and reference utilization over

all the maps and travel-service-demand data sets. So for each num

b e r o f o p e n b a s e s - r e f e r e n c e u t i l i z a t i o n - h e u r i s t i c t r i p l e t , 2 0 r u n s

were made.

Tables la, lb, and lc give values for the number of pairwise

interchanges evaluated. For each heuristic, results are presented as

a function of number of open bases and reference utilization pairs.

As expected by the structures of the heuristics, T&B had the worst

performance for this measure with its minimum value being greater

than any of the maximum values for the other heuristics. VNZ2,

VNZ1, and SINGLE heuristics rank in the given order with SINGLE

heuristics having the lowest min, average, and max values. In gen

eral, the data exhibits an decreasing trend in the average amount of

computational effort required to reach a final solution as the number

Table la. Number of Pairwise Intcrclinngc Evaluations.

SINGSUCT SINCRAT SINGUTIL Open m m . in i ii a v e in a x in i ii a v c in ax mi n a v e m a x

4 1 2.0 4 1 2.6 4 4 0.30 1 3.2 5 1 3.2 6 1 1.9 4 4 0.40 1 3.6 6 1 3.4 5 1 2.0 5 7 0.15 2 3.6 6 1 2.7 5 2 3.5 6 7 0.30 1 3.8 6 1 3.5 7 1 1.5 3 7 0.40 1 3.7 8 1 3.5 6 1 1.1 3 10 0.15 3 4.2 6 1 2.2 6 2 4.2 6 10 0.30 1 3.9 8 1 3.0 6 1 2.0 5 10 0.40 1 3.4 8 1 3.1 6 1 1.3 3 13 0.15 2 4.1 7 1 2.6 7 2 3.8 6 13 0.30 1 4.0 7 1 3.5 6 I 1.7 4 13 0.40 1 3.0 8 1 3.9 8 1 I.I 3

Table lb. Number of Pairwise Interchange Evaluations.

VNZISUCT VNZIRAT VNZIUTIL Open Utili . m i n a v e max m i n a v e m a x m i n a v e m a x

4 0.15 48 59.2 80 32 46.4 96 48 59.2 80 4 0.30 48 70.4 1 12 48 69,6 96 16 37.6 64 4 0.40 48 76.0 128 48 80.0 128 16 37.6 48 7 0.15 39 65.7 91 13 48.8 91 39 72.8 117 7 0.30 26 74.1 130 13 59.8 91 26 39.0 78 7 0.40 39 71.5 117 26 62.4 91 13 33.8 52 10 0.15 30 54.0 90 10 35.5 70 30 62.0 90 10 0.30 30 57.0 100 30 53.0 80 10 34.0 60 10 0.40 40 62.0 110 30 61.0 90 10 21.0 30 13 0.15 14 39.6 63 7 21.7 35 21 41.7 56 13 0.30 7 37.1 70 7 33.3 49 7 17.5 35 13 0.40 7 37.5 70 21 38.8 63 7 10.1 21

Tabic lc. Number of Pairwise Interchange Evaluations.

jx ...... VNZ2SUCT VNZ2H AT VNZ2UTIL T&B open Ullli . mm ave man m i n ave ma, min > T I max 1.11 n ave

64 99.2 160 64 112.0 128 64 86.4 160 64 118.4 192 32 88.0 130 52 83.2 130 52 101.4 182 78 98.8 130 52 84.5 208 78 104.0 156 52 72.8 140 40 72.0 160 60 97.0 180 60 99.0 160 20 66.0 220 60 101.0 160 20 53.0 84 28 48.3 70 42 60.9 140 28 58.8 84 14 40.6 98 42 69.3 98 14 35.7

m a x

0.15 64 97.6 0.30 64 97.6 0.40 64 115.2 0.15 52 100.1 0.30 78 124.8 0.40 78 131.3

10 0.15 60 92.0 10 0.30 40 103.0

160 192 278.4 384~ 128 192 262.4 384 160 192 256.0 320 130 455 618.8 819 130 546 664.3 819 104 546 678.0 819 140 600 845.0 1000

in X'.X ,BU ou yyu 160 20 660 ' 20 700 905 0 1100 10 0.40 80 125.0 220 60 101.0 IfiO 9n sin on 13 0.15 42 58.8 13 0.30 28 71.4 13 0.40 28 65.1

80 600 870.0 1100 84 455 632.5 819 70 455 668.9 819 84 546 696.2 910

5 0

of open bases increases (reference utilization value being fixed). This

trend is due to the fact that the number of pairwise interchanges

considered at each iteration is proportional to the number of closed

bases. Thus, as we open more bases, the number of closed bases de

creases and consequently the number of pairwise interchange evalu

ations is smaller. Another justification can be given by the fact that

since there are more resources (vehicles) in the system, a higher

coverage is provided from the initial solution making the gap be

tween the initial solution coverage and the final solution smaller.

Higher initial solutions yield faster convergence in each heuristic.

SINGLE heuristics did not show a marked trend because in this type

of heuristics the number of pairwise interchanges considered does

not depend on the number of closed bases, but is always one. T&B

displayed a concave shape for the number of pairwise interchanges

as a function of the number of open bases (for every reference uti

lization value).

Similarly, as the reference utilization became larger (for a fixed

number of open bases) the number of pairwise interchanges re

quired decreased. This trend can be explained by a similar argument

to the one given for the number of open bases:. In this case, we do

not have more vehicles, but have increased the work capacity of the

available ones, thus reaching a convergence level faster. Note that

the rate of decrease is fairly constant and small relative to the rate

exhibited by the data as a function of the number of open bases. The

fact that the initial solution coverage was the same for runs that only

differed in the reference utilization setting accounts, in part, for this

smoother trend.

Regarding the quality of the final total coverages obtained by

each heuristic, we can notice in tables 2a, 2b, and 2c the high num

ber of averages in the 90% coverage level. These tables show the to

tal expected coverage for each heuristic as a function of number of

open bases and reference utilization pairs. Total expected coverage

Table 2a. Total

SINGSUCT O p e n U t i l i . m i n a v e m a x

4 0.30 .861 .912 .952 4 0.40 .823 .890 .944 7 0.15 .916 .948 .981 7 0.30 .792 .924 .967 7 0.40 .819 .903 .947 10 0.15 .917 .950 .982 10 0.30 .877 .920 .976 10 0.40 .816 .901 .965 13 0.15 .918 .951 .983 13 0.30 .851 .918 .976 13 0.40 .802 .897 .968

Expected Coverage.

SINGRAT m i n a v e m a x

SINGUTIL m i n a v e m a x

.933 .970 .899 .936 .973 .852 .913 .948 .642 .844 .948 .716 .887 .944 .630 .801 .923 .914 .946 .981 .916 .948 .981 .902 .934 .974 .645 .820 .945 .885 .919 .963 .619 .789 .924 .916 .948 .982 .917 .950 .982 .907 .940 .975 .740 .863 .936 .877 .925 .964 .731 .833 .916 .917 .950 .983 .918 .951 .983 .909 .943 .977 .767 .877 .945 ..892 .930 .966 .758 .858 .934

Table 2b. Total I

O p e n U t i l i . VNZISUCT

m i n a v e m a x

.944 .980 4 0.30 .883 .928 .971 4 0.40 .883 .928 .971 7 0.15 .830 .912 .959 7 0.30 .891 .930 .972 7 0.40 .871 .922 .962 10 0.15 .918 .951 .984 10 0.30 .858 .927 .977 10 0.40 .831 .911 .958 13 0.15 .919 .952 .984 13 0.30 .810 .918 .977 13 0.40 .792 .905 .969

I

Expected Coverage.

VNZIRAT VNZIUTIL m i n a v e m a x m i n a v e m a x

.910 .941 .976 .909 .942 .980

.897 .932 .970 .689 .815 .902

.897 .932 .970 .689 .815 .902

.881 .919 .957 .677 .783 .852

.905 .941 .977 .641 .795 .934

.895 .931 .967 .571 .733 .898

.917 .950 .983 .918 .950 .984

.909 .945 .979 .701 .816 .896

.900 .936 .970 .636 .745 .865

.918 .951 .983 .919 .951 .984

.913 .945 .978 .754 .827 .905

.901 .936 .968 .709 .790 .898

Table 2c. Total Expectcd Coverage.

O p e n U t i l i . VNZ2SUCT

m i n a v e m a x VNZ2RAT

m i n a v e m a x VNZ2UTIL

m i n a v e m a x ni i n T&B

a v e m a x

.979 .910 .944 .980 .910 .944 .979 .911 .945 .980 4 0.30 .897 .931 .971 .897 .933 .971 .689 .840 .935 .897 .933 .972 4 0.40 .875 .918 .957 .881 .919 .959 .681 .806 .927 .882 .919 .959 7 0.15 .918 .950 .983 .915 .949 .983 .918 .950 .983 .918 .950 .983 7 0.30 .909 .939 .973 .906 .943 .978 .679 .836 .933 .912 .945 .979 7 0.40 .900 .930 .964 .902 .932 .967 .650 .765 .875 .904 .936 .970 10 0.15 .919 .951 .984 .917 .951 .984 .918 .951 .984 .919 .952 .985 10 0.30 .840 .931 .977 .911 .946 .979 .572 .805 .924 .915 .948 .980 10 0.40 .892 .926 .964 .900 .937 .971 .564 .758 .896 .907 .940 .971 13 0.15 .919 .952 .984 .918 .951 .984 .919 .952 .984 .919 .952 .984 13 0.30 .823 .925 .980 .913 .946 .979 .746 .828 .908 .915 .948 .980 13 0.40 .742 .911 .958 .905 .938 .969 .674 .780 .884 .908 .940 .972

5 4

is simply defined as the percentage of the system total call demand

that is reached successfully within the critical response time. The fol

lowing observations can be derived from the aggregate results:

. Performance of the heuristics decreased as the reference utilization

value increased.

. As the number of open bases increased (reference utilization fixed),

total coverage improved at a small, fairly constant rate. At 0.15

utilization value, the number of open bases seemed to have no

effect on total coverage. At 0.15 all heuristics achieved, on the

average, very similar coverage values in the middle 90% level.

. In general, T&B, VNZ2RAT, and VNZ1RAT heuristics seem to be the

more consistent registering smaller average ranges between

minimum and maximum values of total coverage. Heuristics using

utilization as criterion performed the worst in average total

coverage as well as min-max ranges.

Tables 3, 4, 5 , and 6 give the number of times each heuristic

achieved a coverage value "close enough" to the best coverage pro

vided. Different call tolerances are used to define the closeness mea

sure. For a one call tolerance, T&B outperformed the other heuristics

with VNZ2SUCT ranking second and so on. The SINGLE heuristics

ranked at the bottom. As the call tolerance is raised, the gap be

tween T&B and the other heuristics grew smaller with no major dif

ference in ranking except for VNZ1SUCT. VNZ1SUCT performed bet

ter than VNZ2RAT and VNZ2UTIL. In general, success total proved to

be the best criterion. The utilization criterion performed well only at

the 0,15 reference utilization level.

The more interesting results are the average deviations from

highest total expected coverage achieved for each number of open

bases, reference utilization, and map combination. Tables 7a, 7b,

5 5

and 7c summarize these results. For each number of open bases and

reference utilization pair, results are aggregated over all maps. At a

reference utilization level of 0.15, all heuristics performed very

close to T&B with average deviations ranging from less than one call

to approximately 80 calls. In general, deviations become greater as

the reference utilization increases. VNZ2RAT was more consistent on

the average than VNZ2SUCT. Analytically, one can state that the

recorded deviations indicate that no significance difference exists

among the top heuristics in total coverage achieved. Thus, prompt

ing for the sacrifice of "few" calls for the sake of less computational

effort. However, attention must be given to the nature of the service

provided by emergency systems. A few missed calls in actual sys

tem operation may cause irrevocable damage to members of the

population being served.

Table 3. Number of "Best" (Tolerance = 1 call).

SING SING SING VNZl VNZl VNZl VNZ2 VNZ2 VNZ2 Open Ulili. SUCT RAT UTIL SUCT RAT UTIL SUCT RAT UTIL T&B

4 0.15 0 "o 4 0.30 0 0 4 0.40 0 0 7 0.15 0 0 7 0.30 0 0 7 0.40 0 0 10 O.IS 0 0 10 0.30 0 0 10 0.40 0 0 13 0.1S 1 0 13 0.30 0 0 13 0.40 0 0

Total i 0

4 7 8 6 9 19 0 5 6 0 5 8 0 20 0 6 6 0 8 10 1 20 0 4 1 7 9 4 11 20 0 3 0 0 3 0 0 20 0 1 0 0 3 0 0 20 0 2 0 3 6 1 8 20 0 1 0 0 0 0 0 20 0 1 0 0 2 1 0 20 0 9 0 9 12 0 14 19 0 1 0 0 2 0 0 20 0 0 0 0 0 0 0 20

0 40 17 26 58 30 43 238

U\ o\

4 4 4 7 7 7 10 10 10 13 13 13

20 20 20 20 20 20 20 20 20 19 20 20

Table 4. Number of "Best" (ToIerancc = 2 calls).

SING SING SING VNZ1 VNZ1 VNZ1 VNZ2 VNZ2 VNZ2 Utili. S U C T RAT UTIL SUCT RAT UTIL SUCT RAT UTIL

0.15 0 0 0 9 7 9 12 10 13 0.30 0 0 0 7 7 0 8 9 2 0.40 0 0 0 6 8 0 12 12 1 0.15 1 0 0 9 2 10 13 5 15 0.30 0 0 0 3 0 0 3 0 0 0.40 0 0 0 1 0 0 4 1 0 0.15 0 0 0 7 0 9 14 3 12 0.30 0 0 0 2 0 0 1 0 0 0.40 0 0 0 1 0 0 3 2 0 0.15 2 0 2 15 0 14 18 1 18 0.30 0 0 0 1 0 0 3 0 0 0.40 0 0 0 0 0 0 1 0 0

0 61 24 42 92 43 6 1

Table 5. Number of "Best" (Tolerance = 5 calls).

SING SING SING VNZl VNZl VNZl VNZ2 VNZ2 VNZ2 Open Utili. S U C T RAT UTIL SUCT RAT UTIL SUCT RAT UTIL T&B

11 7 11 13 12 14 20 4 0.30 0 0 0 12 1 1 0 13 13 2 20 4 0.40 0 0 0 10 12 0 13 15 1 20 7 0.15 1 0 0 15 6 18 18 8 19 20 7 0.30 0 0 0 3 1 0 5 2 0 20 7 10

0.40 0 0 0 1 0 0 7 1 0 20 7 10 0.1S 2 1 2 16 3 15 17 7 19 20 10 0.30 0 0 0 2 0 0 3 3 0 20 10 0.40 0 0 0 1 0 0 4 2 0 20 13 0.15 5 0 5 18 2 17 19 8 20 20 13 0.30 0 0 0 1 1 0 4 1 0 20 >3 0.40 0 0 0 0 1 0 1 2 0 20

Total 8 1 7 90 44 61 1 17 74 75 240

KJ\ 00

Table 6. Number of "Best" (Tolerance = 10 calls).

SING SING Ope n U lili. SUCT RAT

4 0.30 ' 1 1 4 0.40 0 0 7 0.15 1 0 7 0.30 1 0 7 0.40 0 0 10 0.15 4 2 10 0.30 1 0 10 0.40 0 0 13 0.15 11 4 13 0.30 1 0 13 0.40 1 0

Total 21 7

SING VNZl VNZl VNZl VNZ2 VNZ2 VNZ2 UTIL SUCT RAT UTIL SUCT RAT UTIL T&B

13 15 14 15 20 0 12 14 0 15 17 2 20 0 13 15 0 15 18 1 20 1 19 8 19 20 1 1 20 20 0 5 3 0 9 5 0 20 0 4 1 0 9 2 0 20 5 19 7 19 20 11 20 20 0 2 2 0 5 7 0 20 0 2 1 0 5 4 0 20 12 19 12 19 20 15 20 20

! 0 2 2 0 5 5 0 20 0 0 1 0 1 4 0 20

18 111 74 70 139 75 78 240

Ul

Table 7a. Deviation From "Best".

SINGSUCT SINGRAT SINGUTIL Open U t i l i . m i n a v e m a x m i n a v e m a x m i n a v e m a x

.001 .009 .025 .001 .011 .029 .001 .008 .025 4 0.30 .001 .021 .066 .001 .021 .045 .021 .089 .272 4 0.40 .004 .029 .078 .004 .033 .173 .019 .119 .266 7 0.15 .000 .003 .006 .001 .005 .016 .001 .003 .006 7 0.30 .001 .021 .120 .005 .011 .034 .024 .125 .281 7 0.40 .003 .033 .107 .008 .017 .050 .035 .147 .326 10 0.15 .000 .002 .004 .000 .004 .008 .000 .002 .004 10 0.30 .001 .027 .074 .002 .008 .019 .023 .085 .196 10 0.40 .004 .039 .113 .003 .014 .035 .029 .107 .197 13 0.15 .000 .001 .002 .001 .002 .005 .000 .001 .003 13 0.30 .001 .030 .067 .001 .005 .015 .017 .071 .170 13 0.40 .001 .043 .127 .002 .010 .024 .018 .082 .171

Table 7b. Deviation from "Best".

VNZISUCT VNZIRAT VNZIUTIL Open Utili . m i n ave max min ave max min ave max

4 0.15 .000 .001 .004 .000 .004 .013 .000 .002 .029 4 0.30 .000 .006 .032 .000 .001 .012 .033 .119 .250 4 0.40 .000 .007 .059 .000 .001 .004 .085 .137 .246 7 0.15 .000 .000 .001 .000 .002 .008 .000 .002 .026 7 0.30 .000 .016 .080 .000 .004 .012 .020 .150 .288 7 0.40 .000 .014 .049 .001 .006 .018 .073 .203 .392 10 0.15 .000 .000 .002 .000 .002 .003 .000 .002 .026 10 0.30 .000 .021 .067 .001 .003 .005 .045 .131 .268 10 0.40 .000 .029 .076 .001 .004 .007 .066 .194 .306 13 0.15. .000 .000 .001 .000 .001 .005 .000 .001 .026 13 0.30 .000 .030 .127 .001 .003 .009 .035 .121 .190 13 0.40 .002 .035 .149 .000 .004 .009 .036 .151 .232

Table 7c. Deviation from "Best".

VNZ2SUCT VNZ2RAT Open Utili . min ave max min ave n

VNZ2UTIL T&B m i n a v e m a x m i n a v e m a x

.000 .001 .004 .000 .000 .000

.000 .093 .250 .000 .000 .000

.000 .114 .242 .000 .000 .000

.000 .000 .001 .000 .000 .000

.021 .109 .224 .000 .000 .000

.070 .172 .271 .000 .000 .000

.000 .000 .001 .000 .000 .000

.017 .143 .364 .000 .000 .000

.049 .181 .365 .000 .000 .000

.000 .000 .000 .000 .000 .000

.031 .120 .198 .000 .000 .000

.041 .160 .234 .000 .000 .000

0.15 .000 .001 .004 "1)00 .001 ~ ~005" 0.30 .000 .003 .022 .000 .001 .004 0.40 .000 .002 .011 .000 .000 .001 0.15 .000 .000 .001 .000 .001 .005 0.30 .000 .006 .016 .000 .003 .008 0.40 .000 .007 .032 .000 .004 .014

10 0.15 .000 .000 .001 .000 .001 .003 10 0.30 .000 .017 .074 .000 .002 .007 10 0.40 .000 .014 .048 .000 .003 .007 13 0.15 .000 .000 .001 .000 .001 .002 13 0.30 .000 .023 .114 .001 .002 .005 13 0.40 .000 .029 .149 .000 .002 .007

6 3

CHAPTER 7

CONCLUSIONS AND FUTURE RESEARCH

In this thesis, a vehicle location model for emergency service

systems has been presented. The strengths of the model are its abil

ity to consider location dependent service times, stochastic travel

times, unequal vehicle utilizations, and backup service. The model is

sufficiently general to allow it to be adapted to other problem set

tings and decision criteria. In addition, while the model was devel

oped within the context of emergency services, it is amenable to

other public or private sector applications concerned with service

delay penalties and demand requiring complementary services.

Effective solution procedures were developed using pairwise

interchange techniques to explote the discrete nature of the location

decisions. A number of 240 large test problems with characteristics

resembling those of the Tucson Emergency Medical System were

generated. In light of the "real-world" characteristics of the data, it

is believed that these test problems are more indicative of actual

emergency vehicle location problems than those previously pre

sented in the literature. These test problems can also be used as a

test bed for future research in this area.

In addition, results of extensive computational experiments

testing ten heuristic solution procedures were presented. These ex

periments indicate that most of the heuristics performed well in

terms of approximation to optimality regardless of the problem

structure and that there is a strong correlation between number of

pairwise interchanges evaluated and the consistency of approxima

tion performance.

6 4

From this experience, future work considerations can include

relaxing the 0-queue assumption to allow for infinite capacity, al

lowing dispatching from locations other than open bases (dispatches

on route), developing other dispatching priorities besides "closest

first", and using a combined multiobjective format to also optimize

other performance measures like total expected coverage

6 5

APPENDIX A

The material to be presented next can be found in Goldberg and

Szidarovszky [1988].

Let the system of equations (7P) be rewritten as

ft = R(ft) , (1)

with ft = ( pi pj) , and for each j e {1,2, . . . ,J} ,

Rj(ft) = (1 - Pj) ' 1 I k( i j ) - l

7X X A ij • XX P r( i , i=l m=l

m) (2)

where k(i,j) gives the index k of the only nonzero Xjjk with fixed i

and j. If there is no nonzero value for Xjjk, then simply take Ajj = 0

for those (i,j). It is also known that for all fixed i and m there is

exactly one nonzero Xjrm- Let this index r be denoted by r(i.m). This

notation was used in Rj(ft). Note that r(i,m)/=j.

Theorem: If TT is large enough, then equations (1) have a unique

solution in [0,1].

Proof: Let J(£) = (Jjf(ft)) denote the Jacobian of mapping R. If f = j then

1 Jjf(ft) -

f I

I A , j i=l

II Pr(i ,m) m=l

and if f £ j, then

66

Jjf(£L) — (1 Pj) • • / I

X A jj i=l

Vfifr( i .m)

k(LL)-i A

,m)

CiJ)- 1

11 pr(i , i m=l

7

where the summation is performed for those values of i such that

r(i,m) = f with some m < k(i,j) - 1. Hence if g.£ [0,1]J, then for all j

and f,

I I | Jjf(&) J ij

i=l

Consequently, the iteration method converges if

111(a)11< xx max £ Ajj • ||1|| < 1, j i=l

(3)

where X = (1) having l's at each element position (a matrix of l's).

We have to guarantee in evaluation that R, maps [0,1 ]J to itself. By

using relation (2) this is true if

_1_ TT I Ay

i=l < 1 , for all j

(4)

Observe finally, that (3) and (4) hold if

I TT> max Ajj • max {l.||l||}.

j i=l

Remark 1: If the row norm of the Jacobian is below unity, then it is

known (Szidarovszky and Yakowitz [1978]) that the Seidal-type

67

iterations,

n ( k + 1 ) - R f n ( k ) r , ( k ) n ( k ) n ( k ) > > Pi ~ R1 VPl >P2 'Q-l'Pj /

r t < k + 1 ) _ P r « ( k + 1 > « ( k > P2 = R2 VPl ,P2 .0-1.Pj )

r t( k + 1 ) - P r ^ ( k + 1 ^ ( k + 1 > r t < k + 1 > « < k h

pj = rj vpi ,p2 »o-i »Pj ;

has a faster convergence that the fixed-point iterations,

P r ' ) -K(^y

Remark 2: Another acceleration of convergence is the following. Observe that the equation for pj has the form,

p j = ( 1 - p j ) - Z j ( n )

where Zj does not depend on Pj. Hence from this equation we

conclude that

Zj(e) , j 1 + Zj (p.) J

Therefore, the following iteration scheme can be aggregated

pSk+1) = S1 (<L(k))

68

( k + l ) P r u = s 2 ( ^ k ) )

Pj(k+1) = Sj (p.(k))

or its Seidal-type variant. We can show that the use of this trick

decreases the elements of the Jacobian. Note that

Z • [ 1 + Z j ] - Z j • z 6Si(ft) J J

5S, (1 + Zj)2 t

z = d + Z j ) 2 < Z j

Furthermore,

5Sj(p.)

8Sj = 0

69

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7 1

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