Group Title: systems analysis of optimal manpower utilization in health maintenance organizations
Title: A Systems analysis of optimal manpower utilization in health maintenance organizations
Full Citation
Permanent Link:
 Material Information
Title: A Systems analysis of optimal manpower utilization in health maintenance organizations
Physical Description: xx, 276 leaves. : illus. ; 28 cm.
Language: English
Creator: Schneider, Donald Paul
Publication Date: 1973
Copyright Date: 1973
Subject: Medical care -- Mathematical models   ( lcsh )
Health facilities -- Personnel management   ( lcsh )
Industrial and Systems Engineering thesis Ph. D
Dissertations, Academic -- Industrial and Systems Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 266-274.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098190
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000585246
oclc - 14204169
notis - ADB3879


This item has the following downloads:

PDF ( 10 MBs ) ( PDF )

Full Text



Donald Paul Schneider




I wish to acknowledge the considerable contribution to this work

by Dr. Kerry E. Kilpatrick who acted as the Chairman of my Supervisory

Committee. His direction and insight contributed to both the form and

content of this dissertation.

I also thank the remainder of my committee, Dr. H. Donald Ratliff,

Dr. Thom J. Hodgson, Dr. Frank R. Sloan and Richard C. Reynolds, M.D.,

for their efforts which contributed to the completion of this work. I

would also like to thank Dr. Stephen D. Roberts for his guidance and

support in several stages of this research; Dr. Barney L. Capehart for

his guidance in the early stages of my graduate studies; Dr. Donald R.

Miller for his aid and helpful suggestions, and Dr. M. E. Thomas for

his support throughout my stay at the University of Florida.

I would also like to thank the members of the health care community

who provided valuable advice, criticism, and the facilities so that the

research could be developed, tested and validated. These include Dr.

Douglas Fenderson, Director of the Office of Special Programs, Bureau

of Health Manpower Education, National Institutes of Health; Mr. John

Braun, Chief, Physician's Assistant Staff, Office of Special Programs,

Bureau of Health Manpower Education, National Institutes of Health;

Dr. L. F. Krystynak of the University of Florida, Mr. Richard Bohn,

President of the Metropolitan Health Council of Indianapolis, Indiana;

Dale R. Benson, M.D., director of the Southeast Health Center of Indi-

anapolis; Mr. Tim Payne, administrator of the Southeast Health Center;

Mr. Dave Norton, consultant to the Florida Health Care Plan, Inc.; Mr.

Richard Freeman, Special Assistant to the Deputy Secretary of Health,

Education and Welfare; Mr. Dave Whelan, President of Health Management

Group, Inc.; and Dr. Judith Liebman of the University of Illinois.

Finally, I thank my wife Ester for her encouragement and patience

throughout the course of this work.

This dissertation was supported in part by the Health Systems Re-

search Division of the University of Florida.


ACKNOWLEDGMENTS . . . . . . . . .

LIST OF TABLES . . . . . . . . .

LIST OF FIGURES . . . . . . . . .

KEY TO SYMBOLS . . . . . . . . . .

ABSTRACT . . . . . . . . . . .


1 INTRODUCTION . . . . . . . .

Background . . . . . . . . .

Scope and Purpose . . . . . . .

Research Objectives . . . . . . .

Organization of Dissertation . . . . .


Health Maintenance Organizations . . . .

General Terminology . . . . . .
Legislative Background . . . . .
Research Viewpoint . . . . . .

Allied Health Manpower . . . . .

Physician Extenders . . . . . .
Physician Extenders in Primary Care . .
Physician Extender Utilization: The Team
Delegation of Medical Services . . .

Motivation for Mathematical Modeling . . .

Previous Studies . . . . . .

Systems Analysis as
Process . . .
Planning the Benefit
Proposed BMO Design







. .

. . .

an Aid to the HMO Design
. . . . . . . . 27
s To Be Provided by an HMO 28
Process . . . . ... 30



Preliminary Models . . . . . . . . ... 35

Introduction . . . . . . . . ... 35
System Schematic for Model Development . . .. .36
Tradeoff Decisions To Be Incorporated in the
Models . . . . . . . . ... . 38
Preliminary Model Development . . . . .. 41
Development of the Medical Care Aspects of the
Model . . . . . . . . . . . 44
Further Development of Financial Aspects of the
Model . . . . . . . .. . . . 51
Additional Model Refinements . . . . .. 58
A Comparison of Models Ml, M2, and M3 ...... 64

Development of Planning Models . . . . . ... .66

Development of the Overall Planning Model .... .66
Minimum Cost Model for Fixed Services . . ... .69
A Special Case of the Minimum Cost Model ... .72
Subscriber Maximization Model . . . . .. 75
Minimal Use of Professional Manpower ...... .77
Concluding Remarks ... . . . . . . 79


Introduction . . . . . . . . .. . . 81

An Overview of Health Care Data Collection . . 81
Purpose of the Chapter . . . . . ... .81

Medical Classification Systems . . . . . ... .83

The Criteria for a Medical Classification System 83
A Review of Existing MCS . . . . . ... .84
A Formulation of a New MCS . . . . ... .90

Time Requirements, Delegation and Utilization Data . 93

Time Requirements . . . . . . . ... 93
Delegation Guidelines . . . . . . ... .99
Utilization of Services . . . . . ... .108

Medical Cost Data . . . . . . . .... 113

Introduction . . . . . . . . . 113
Cost Calculations . . . . . . . ... 115
Concluding Remarks . . . . . . ... .120



Introduction . . . . . . . . . . . 122

Solution Process for Linear Programming and Mixed
Integer Programming . . . . . . . ... 122

Validation Process . . . . . . . . . 124

Problems Involved in the Validation Process .... .124
Formulation of a Validation Process . . . ... .125
Steps Taken To Validate the Models . . . ... .126
Implications of the Model Assumptions . . ... .131

Nominal Solutions . . . . . . . .... . .134

Comparison to the Prepaid Group Practice . . .. .134
Presentation of Nominal Solutions . . . ... .136
Comparison of Nominal Solutions to an Existing
Prepaid Plan . . . . . . . . ... 145

Case Examples . . . . . . . . ... . . 147

Use of the Models in the Design of an HMO in
Indianapolis, Indiana . . . . . . ... 147
Evaluation of the Indianapolis HIO Case Study . . 163
Case Study of the Design of an HMO in Daytona
Beach, Florida . . . . . . . ... 166
Evaluation of Daytona Beach HMO Study . . ... 173

Concluding Remarks . . . . . . . .... . .175

QUESTIONS . . . . . . . . ... ..... 176

Introduction . . . . . . . . ... . .176

Sensitivity Analysis . . . . . . . . ... 177

Facility Costs . . . . . . . . ... 177
Physician Extender Salary . . . . . . .. 179
Indirect Supervision Guidelines . . . . .. 182

Parametric Analysis . . . . . . . . ... 190

PE Utilization As a Function of Subscriber Levels
and the Integer Restrictions . . . . .. 190
Optimal Delegation With a Scarcity of PE's .... .195
Maximum Subscribers Per Physician Under a Scarcity
of PE's . . . . . . . . ... . .201


Remarks . . . . . . . . . . . . 202


Introduction . . . . . . . . .. . . 206

Results and Conclusions . . . . . . . .. 206

Development of the Models and Data Base . . .. .206
Results From the Models . . . . . ... .209
Results of the Validation Process . . . ... .213
An Overview of the Results and Conclusions . 215

Areas fcr Further Research . . . . . . ... .216




VARIABLES . . . . . . . . . . . 254


REFERENCES . . ... . . . . . . . . . 266

BIOGRAPHICAL SKETCH . . . . . . . . ... . .275




1 Example of Personnel and Time Requirements ...... 47

2 Relative Sizes of Models Ml, M2 and M3 . . . ... 65

3 Sample Output for Analysis of Utilization, Time
Requirements and Delegation . . . . .... 95

4 MD and RN Time Requirements for Medical Services . .. 100

5 MD, PE and RN Time Requirements for Medical Services .102

6 Maximum Percent of Visits Under Indirect Supervision
for Four Delegation Assumptions . . . . .. 109

7 Comparison of Demographic Characteristics for the
Kaiser Sample and the PGP ............ 111

8 Percent Utilization for Adult Medicine, Pediatrics
and OB/GYN in the KCBDCS . . . . . . 112

9 1972 PGP Budget. . . . . . . . .... .115

10 Nonoffice Visit Expenses for PGP . . . . .... 117

11 Nonhospital Departmental Costs for Three Primary Care
Departments in PGP . . . . . . . .... 120

12 Cost and Manpower Comparison of PGP and the Overall
Planning Model . . . . . . . ..... .. 135

13 Comparison of OWM Nominal Solutions to PGP . . ... 139

14 Delegation Analysis for Nominal Continuous Variable
Solution . . . . . . . . ... ..... 141

15 Nominal Solutions for MC Model . . . . .... 143

16 Nominal Solutions for the SM Model . . . . .. 144

17 Parametric Results for Indianapolis HMO Design .... .151


18 Variable Expenditures at Indianapolis HMO . . ... .156

19 Maximum Subscriber Levels for Indianapolis HMO As a
Function of Staffing Pattern . . . . . ... 161

20 Projected Demographic Characteristics for FHCP .... .168

21 Projected Utilization Rates Per Thousand Members for
FHCP . . . . . . . . . . . . 168

22 Continuous Optimal Solution for FHCP . . . ... 170

23 Integer Staffing Levels for FIICP . . . . ... .170

24 Parametric Results for Various Integer Staffing Levels
for FHCP . . . . . . . . ... .. .. . 172

25 Suggested Dynamic Hiring Plan for Adult Medicine at
FHCP -. . . . . . . . . . . . 173

26 Manpower Utilization Response to Increases in Facility
Cost . . . . . . . . .. . . . 178

27 PE Use As a Function of Salary Increment From the
Nominal . . . . . . . . .. . . 180

28 An Analysis of Six Different Indirect Supervision
Guidelines . . . . . . . . .. . . 184

29 Maximum Shifts From the Nominal in MD and PE Utilization
Caused By Changes in the Indirect Supervision Guide-
lines . . . . . . . . ... . . 186

30 PE Utilization As a Function of Basic MD Supervision
Time . . . . . . . . ... .. . .. 189

31 Results of Integer Manpower Restrictions For 8000-30000
Subscriber Levels . . . . . . . ... .193

32 Optimal Delegation and MD Requirements Under a
Scarcity of PE's . . . . . . . . ... 197

33 Marginal MD and Cost Savings As Additional PE's
Are Used . . . . . . . . ... .... . 199

34 Maximum Subscriber Sizes Per MD Under a Scarcity
of PE's . . . . . . . . .. .. . 203



A-I Principle Sections of the International Classification
of Diseases . . . . . . . . . ... . 222

A-2 California Relative Value Classification for Office
and Home Visits . . . . . . . . ... 223

A-3 Major Categories of a Medical Service Classification
System for General Practice . . . . . ... .224

A-4 Patterns of Medical Care for the Primary Care
Specialties . . . . . . . . ... .. . 225

A-5 Diagnostic Categories for which GEOMET Specifications
of Care Were Prepared . . . . . . ... 226

A-6 Sample Listing of Elements of Care for Pediatrics . 228

A-7 Sample Specifications of Care . . . . . .. 229

A-8 Assignment of Proxy Specifications of Care ...... .231

A-9 Clinical Subgroups in the Kaiser Clinical Behavior
Disease Classification System . . .... .. . .233

A-10 Ancillary Task Listing for General Medical Practice 234

B-1 Physicians' Willingness To Delegate Activities to a
Trained OB/GYN Assistant . . . . . .... 235

B-2 Summary of Tasks for a Physician Assistant ...... .236

B-3 Survey by Physicianh Specialty for Possible Duties of
Physician's Assistant . . . . . . ... .238

B-4 Physician Response for Delegation of Tasks to Physician
Assistants . . . . . . . .... ..... 240

B-5 Feasible Delegation of Elements of Care . . ... .251

C-l Names Assigned to Personnel Classes . . . ... .254

C-2 Names Assigned to the Medical Classification System . 255

C-3 Principal Input Data Coefficients for the Mathematical
Models . . . . . . . . ... .. .. . 258

D-l Age-Sex Breakdown for Potential Enrollees in SHC . . 261


D-2 Projected Visits Per Year for SHC as a Function of
Subscriber Level . . . . . . . . . 263

D-3 Fixed Expenditures at SHC . . . . . . ... .265





1 Present Decision Process for Determining Benefits . 29

2 Proposed Decision Process for Determining Benefits . 29

3 Three-Stage Hierarchy in the Planning of HMO's .... .31

4 Present HHO Design Dynamics . . . . . . .. 32

5 Proposed 1MO Design Dynamics . . . . . . .. 33

6 A Schematic Representation of an HMO Structure .... .36

7 Medical Care Structure Showing Relation Between Re-
sources and Requirements . . . . . . ... .37

8 Tri-Level Classification System . . . . .... 91

9 Flow Diagram of the TLCS Computer Program . . ... 94

10 Subscribers Per MD As a Function of PE/MD Ratio . .. 204



i refers to type i personnel (including teams)
j refers to service j
k refers to health care teams
m refers to department m

aijm composite coefficient (see equation (3.56))

AD administrative costs (dollars per year)

AD administrative costs allocated to department m (dollars per year)

ADD composite coefficient (see equation (3.66))

B* upper limit on yearly budget for operating and salary expenses
for medical departments included in the optimization (dollars per

b.. number of type j services a type i personnel can perform per year
for a given level of technology (medical services per man-year)

b* number of type j services a type iel* personnel can perform per
Year for a given level of technology (medical services per man-year)

c.. labor cost for one type i to perform one service j (dollars per
1 medical service)

CLT average cost per laboratory test (dollars per test)

CXR average cost per x-ray service (dollars per service)

d. demand for service J (dj = Sd') (medical services per year)

d'. demand for type j service per subscriber per year (medical services
per subscriber per year)

e' amortized cost for a type i personnel due to technological cost
(dollars per year)

e. initial cost for a type i personnel due to technological cost
S (dollars)

EXT external sources of revenue (dollars per year)


f. level of independence for type i personnel to perform service j

FTE full time equivalent professionals in department m (man-years)

g amortized construction cost (dollars per unit area per year)

g initial construction cost (dollars per unit area)

h. composite coefficient (see equation (3.61))

h* composite coefficient (see equation (3.62))

I set of all medical personnel including teams

I* set of professional manpower

I* ancillary manpower

Ip set of physician extender personnel including teams led by physician

I set of personnel excluding teams

It set of health care teams

Im set of all personnel in department m

I set of personnel whose productivity is enhanced by additional

J set of services considered for provision or to be provided

J subset of J for which V = 1

Jm set of services considered for provision or to be provided in de-
partment m

J set of services class i*el* personnel supervise

j* supervisory services

K arbitrary, large constant

M set of all medical departments under consideration

MAN management and planning cost per FTE professional (dollars per man-
year per year)


MAX.. maximum percent of service j that can be carried out by per-
sonnel i in the indirect supervision mode

MX. maximum fraction of type j services provided by nontypical means

N. number of type i personnel employed (man-year)

NLT number of lab tests ordered in department m per service provided
(tests per medical service)

NXR number of x-rays ordered in department m per service provided
(x-rays per medical service)

n, number of type j services demanded and not provided by ordinary
means (medical services per year)

o maintenance and utility costs per unit of area (dollars per unit
area per year)

OM other medical costs not included in the optimization (dollars per

P yearly profit rate or rate at which capital fund accumulates

P amortization rate for initial capitalization

P. maximum number of type i personnel available (man-years)

PER personnel cost per FTE employee (dollars per man-year)

PV administrative cost per patient visit (dollars per patient visit)

qkij man-years of personnel type i per man-year of team k providing
service j (akij is defined to be one for the team leader)

R average yearly fee paid by subscribers for nonhospital. services
(dollars per year)

r. patient fee for service j (dollars per medical service)

S number of subscribers

s. yearly salary of type i personnel (includes overhead salary such
as retirement, vacation, insurance, etc.) (dollars per year)

SUi maximum fraction of time type i personnel will engage in super-
1 vision of ancillary personnel
l if N.>O
T. f\ iI'
i 0 if N. = 0

t'm amortized cost of equipment in department m per FTE professional
(dollars per man-year per year)

t initial cost of equipment in department m per FTE professional
(dollars per man-year)

u. per unit cost of providing type j service by extraordinary means
(dollars per medical service)

p. department m provided
Vm 0 department m not provided

w. space required per type i person (units of area per one man-

x number of type i personnel assigned to service j (man-years)

x*. number of supervisory level personnel assigned to the supervisory
service j* (man-years)

x! fractional part of personnel i group which are idle (man-years)

Y* maximum initial capitalization (dollars)

Y total floor space available (units of area)

Y floor space to be constructed for department m (Y = m w.N.)
S (units of area per department)

Pjm composite coefficient (see equation (3.58))

0'j composite coefficient (see equation (3.57))
*jm composite coefficient (see equation (3.59))

yj composite coefficient (see equation (3.64))

Yjm composite coefficient (see equation (3.63))

ym composite coefficient (see equation (3.65))

Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy



Donald Paul Schneider

August, 1973

Chairman: Dr. Kerry E. Kilpatrick
Major Department: Industrial and Systems Engineering

Mixed integer programming and linear programming models are pro-

posed as aids to decision makers in the design and evaluation of health

manpower requirements in the office care setting of Health Maintenance

Organizations (HMO). Although special emphasis is given to the poten-

tial role of physician's assistants, nurse practitioners and nuise mid-

wives in HMO's,the staffing requirements for physicians, registered

nurses and licensed practical nurses are also investigated.

Four basic mathematical models are.developed to analyze in detail

the design relationships between resources and requirements in HMO's.

The models are used to examine the interaction between effective man-

power utilization, technology utilization, facility requirements and

local inputs such as available capital and existing health care facil-

ities. Another principal feature of the models is that they reflect

the total cost for outpatient medical care services delivered at the

HMO. The objectives used in the models pertain to either minimum cost

or the minimum feasible use of physicians through the substitution of

allied health personnel.


A new three-level hierarchial medical classification system is de-

veloped which relates to the following manpower planning considerations:

training and delegation, morbidity statistics, and manpower utilization.

The new medical classification system was used to define and collect

data regarding time requirements, delegation possibilities and patient

utilization. In addition medical cost data were collected at a major

prepaid group practice. Data relating to both direct and indirect costs

are presented to fulfill the data requirements of the models.

The models are validated through a seven-step process that in-

cludes: comparisons to two existing prepaid group practices; use of the

models in rhe design of two HMO's; face validity; sensitivity analysis;

parametric analysis; examination of internal validity; and an exami-

nation of data validity. Two detailed case studies are presented which

demonstrate the flexibility and usefulness of the models in actual HMO

planning. One case study was principally concerned with staffing re-

quirements, benefits to be offered and the capitation rate while the

other was chiefly concerned with the possible use of physician's assis-

tants and a dynamic hiring plan as the HMO subscriber size increased.

An extensive variety of results are presented relating to the use

of allied health personnel in the primary care specialties in HMO's.

An analysis of the potential cost and manpower effect extensive use of

allied health personnel would have on large HMO's shows a 4 to 10 percent

cost reduction and a 25 to 50 percent reduction in physician require-

ments depending on the medical specialty. An optimal delegation

analysis shows that routine examinations, well child care, chronic ill-

nesses, and diseases with a high emotional content are most economically

and medically suited for delegation to physician's assistant level personnel

sonnel. Results are also presented to show that one physician's assis-

tant level person can replace about .6 of a physician in an HMO and

that the optimal ratio of physician's assistant level personnel to phy-

sicians is 1.55, 2.10, and .47 in adult medicine, pediatrics, and ob-

stetrics/gynecology, respectively.

In addition, sensitivity analyses of physician's assistant utili-

zation as a function of facility cost, salary, and delegation guide-

lines are presented. Parametric analyses of physician's assistant

utilization as a function of the integer manpower restrictions and the

size of the HMO are also presented.




The last decade has witnessed a growing national recognition that

the U.S. health care system has failed to meet the expectations of the

general populace (1). The plight of the rural poor and inner city poor

has been especially noticeable. The overall national shortage of phy-

sicians and trained health personnel plus the disinclination of the

professionals to settle in rural and impacted urban areas is creating

a doubly critical problem (1). In addition an increasing proportion

of physicians are not engaged in patient care at all but are engaged

in research, administration, teaching, government service or consul-

tation activities (1). Of those involved in patient care, the ratio

of family practitioners (general practitioners, internists, and pedi-

atricians) to the population fell by 33 percent from 1950 to 1965 (2).

The demand for health care is being stimulated by two additional

factors: a changed attitude toward health and health care and greater

financial support for health care. It has only been in the last two

decades when health insurance, Medicare, Medicaid, and many other

public and private financing programs became available and translated

medical needs into medical demand (1). Health care expectations have

risen to the point where many people believe health care is a right

rather than a privilege. In 1966 the American government in P.L. 89-749

assumed a commitment "to assure comprehensive health services of high

quality for every person."

Compounding these problems is the fact that too much of the phy-

sician's time is ineffectively utilized in routine and semi-clerical

tasks. Much of the work physicians and nurses do can be characterized

by routine tasks such as reading electrocardiograms, following a pa-

tient's vital signs manually, and administering and interpreting chem-

ical, biochemical and physical tests (3). In addition, large portions

of time and attention are devoted to the well or the so called "worried

well" (4).

Finally, costs of health care are rising rapidly. From 1965 to

1970 physician fees rose by an annual rate of from 5.4 percent to 9.2

percent (1). The average daily service charge for hospitals rose 279

percent from 1960 to 1970 (5). These cost increases are caused by

specific factors such as rising labor costs; the increasing employment

of more highly skilled personnel; the changing status and higher pay

of the house staff of hospitals; the rise in the cost of construction

and supplies; the increase in number and sophistication of diagnostic

tests and therapeutic procedures; the changing mix of the patient

population with a trend toward more serious illness; the persistence of

too many economically inefficient small units; and the rising costs of

administrative overhead (1).

Out of consideration for these signs and symptoms of failure of

the current health care system is the growing recognition that new

health care systems must be encouraged and developed. Thus in January,

1971, President Nixon called for a new health care system in America (6).

The central feature of his plan is the Health Maintenance Organization

(HMO) which is similar in concept to neighborhood health centers but

fundamentally embodies the principals of prepaid group practice (PPGP).

Scope and Purpose

This dissertation reviews the existing descriptive and conceptual

literature on HMO's and PPGP's and utilizes this information to model

mathematically HMO's. A systems analysis approach utilizing mixed in-

teger programming and linear programming models is proposed which pro-

vides an aid to decision makers in the design and evaluation of health

manpower requirements in the outpatient segment of the HMO.

To model the manpower needs a medical classification system was

developed to facilitate the development of the relationship between

the system requirements in terms of typical patient visits and system

resources in terms of trained health manpower. The models and medical

classification system are used to evaluate the potential role of a

physician extender (PE) in the HMO setting. There is a large number

of different types of PE's involved in health care and a few examples

include physician assistants, pediatric nurse practitioners, and nurse

midwives. The models are developed to examine the interaction between

effective manpower utilization, technology utilization, facility re-

quirements and local inputs such as available capital and existing

health care facilities. Another principle feature of the models is

that they reflect the total cost for outpatient medical care services

delivered at the HMO.

The analytical models introduced in this dissertation can be used

to evaluate staffing requirements for any type of medical service pro-


vided in the outpatient setting by an HMO. However, due to data limita-

tions the scope of the solutions presented are narrowed to the primary

specialties; adult medicine, pediatrics and obstetrics/gynecology

(OB/GYN). The manpower types considered are thus the physicians and

physician extenders commonly used in the above specialties as well as

registered nurses (RN) and licensed practical nurses (LPN). A principal

byproduct of a systems analysis and mathematical model is a systematic

framework in which data can be collected and analyzed. In the health care

setting such a framework is particularly needed (7). Thus within the

systems viewpoint developed in the models an extensive amount of data is

presented relating to the above specialties, manpower, and the cost of

medical care.

The overall result of this research is the presentation of a sys-

tems methodology which will aid in the evaluation and design of HMO's.

In audition, through use of the models and the data collected, definite

design guidelines for the utilization of medical manpower in HMO's are

established. Since the analytical models presented are very flexible,

they allow the comparison of an existing HMO to an optimal design which

incorporates many of the local inputs such as utilization rates, patterns

of medical care, salary structure and other items that are relevant to

the particular HMO being evaluated. The study compares two existing

prepaid group practices to the design given by the analytical models

and the models also are used to aid in the design of two HMO's. One of

the HMO's was in the process of converting from a neighborhood health

center to an MO and the other HMO was in its preoperational planning

stage. Through the comparison of results to existing HMO's and through

the use of the models in the design of emerging HMO's, the viability

and validity of the design approach was determined.

Overall, the goal is to carry out a research effort in which systems

analysis and operations research techniques are used and discussed, but

end with results and conclusions which can be utilized by persons not

necessarily acquainted with these techniques.

Research Objectives

This section briefly summarizes the specific research objectives

that are carried out in this dissertation. The objectives can be divided

into three principal areas: development of mathematical models to aid

in the design of IZO's; collecting the data required to solve the models;

and using the models to examine various policy questions and to aid in

the design of two emerging lMZO's.

Four mathematical models are developed to analyze in detail the

design relationships between resources and requirements in HMO's. These

four models solve the following basic types of problems:

(a) preoperational planning for which minimum

cost solutions are sought for the services

to be provided, manpower needed, delegation

policy, facilities needed and particular

technology innovations;

(b) preoperational planning for which the

optimal staffing, delegation policy and

facility is sought to minimize the capi-

tation rate;

(c) manpower planning for which the optimal

allied personnel policy is sought to

maximize the IMO subscribers per phy-

sician; and

(d) manpower planning for which the optimal

allied personnel policy is sought to

minimize the number of physicians re-


These four models are mixed integer programs (or linear programs if the

integer restrictions are dropped) and are developed in such a manner that

solutions are obtained at a relatively low cost.

The mathematical models were used to provide a framework in which

data were collected. As a first step in the data collection effort, a new

medical classification system is presented which relates to the following

manpower planning considerations: training and delegation, morbidity

statistics, and manpower utilization. In addition to previously pub-

lished data, an extensive amount of original data collected at a major

prepaid group practice is presented. In summary,data are presented for

the following areas:

(a) manpower time requirements;

(b) delegation possibilities to PE's;

(c) medical utilization by diagnosis;

(d) direct medical costs; and

(e) indirect medical costs.

The third major research objective involves the use of the models

and the data to derive solutions. As a first step in the validation

process the results of the models are compared to two major prepaid

group practices. In addition the models are used to derive solutions

for the following problems:

(a) an analysis of the potential cost and

manpower effect extensive PE utilization

would have on a major prepaid practice;

(b) an analysis of optimal delegation guide-

lines for PE's;

(c) a case study for the optimal manpower re-

quirements and resulting cost for a

neighborhood health center planning to

convert to an HMO in Indianapolis, Indiana;

(d) a case study for the possible utilization

of PE's in an emerging HMO in Daytona Beach,


(e) a sensitivity analysis of PE utilization as

a function of facility cost, PE salary, and

PE delegation guidelines;

(f) a parametric analysis of PE utilization as

a function of the integer manpower restrictions

and size of the HMO; and

(g) a parametric analysis of the maximum subscribers

per physician in an HMO as a function of PE

utilization and HMO size.

Organization of the Dissertation

In Chapter 2 the two main components of the medical system under

study are presented. The concepts and properties of HMO's are discussed

and a more detailed look at health manpower is provided. Special emphasis

is given to a discussion of PE's. In addition a motivation for using

mathematical models to aid in the design of the HMO is presented. This

motivation includes the framework for HMO design that is carried out by

the models.

In Chapter 3 a system schematic and tradeoffs to be incorporated

are presented. Preliminary models are developed to show the basic

structural relationships in the HMO system. The preliminary models are

then enriched in a step-by-step fashion to the point where a detailed

model is presented. The detailed model provides the basis for the de-

velopment of four additional models which are aimed at specific HMO and

manpower planning questions.

In Chapter 4 a new medical classification system is presented. This

system is then used to present data relating to manpower time require-

ments, delegation possibilities and patient utilization. In addition,

both direct and indirect costs for prepaid medical care are presented.

In Chapter 5 the solution method for the models is briefly dis-

cussed and a detailed examination of the validation process is presented.

This includes a suggested sequence of steps which can be used to validate

a prescriptive mathematical model. Solutions from the models are pre-

sented and compared to two major prepaid group practices and an analysis

of the potential effect of PE utilization is explored. In addition two

case studies are presented which give additional data and demonstrate

some of the types of HMO analysis that can be carried out with the models.

In Chapter 6 sensitivity analyses and parametric analyses are pre-

sented. The sensitivity analyses include an examination of facility

costs, PE salary, and delegation and supervision guidelines. The para-

metric analyses include an examination of the integer manpower restric-

tions, HMO size, a scarcity of PE's and the maximum subscribers per


physician. In addition the potential national manpower implications are

briefly discussed.

A summary of the results and conclusions of this research and sug-

gestions for further research are presented in Chapter 7.



Health Maintenance Organizations

General Terminology

In many ways the Neighborhood Health Center (NHC) is a forerunner of

the R10 without the HMO financing system. NHC's employ nurses and para-

medical aids in expanded involvements, provide local employment for

neighborhoods, invite consumer participation in the decision processes,

affiliate with a hospital for referral and inpatient care, and provide

outreach services to the community for health education and family health

counseling (8). The NHC has been a very viable concept and is widely

accepted and used by the residents in its vicinity. However, although

the NHC's serve their communities well, they are in reality a separate

health care system and may be supplanted by HMO's in the near future.

Another concept central to HMO's is the relation between and defi-

nition of group practice and prepaid group practice. Group practice has

been described as "the application of medical services by three or more

full-time physicians formally organized to provide medical care, consul-

tation, diagnosis, and/or treatment through the joint use of equipment

and personnel, and with the income from medical practice distributed in

accordance with methods previously determined by the group." (9, p. 598)

Groups may be single disciplinary or multi-disciplinary in nature and

exist in the form of partnerships or as corporations.

Prepaid group practice (PPGP) is a version of group practice in

which the patient population prepays for health care at a specified

yearly or monthly rate. PPGP attempts to mutualize through capitation

the cost of comprehensive medical care for the population at risk and

removes the fee-for-service barrier to care. The large PPGP's include

teams of full-time physicians representing all of primary care and most

of the minor specialties so that comprehensive services can be provided

through an integrated single system (9). Services are not only available

at a hospital, but in neighborhood ambulatory care units which provide

primary care. Generally PPGP's have a scale of operation which permits

the extensive use of ancillary personnel, and the capitation method of

payment, it has been suggested, tends to provide an incentive for inte-

gration of physician extender health teams and to maintain the health of

the patient in an effort to deliver health care economically.

The claimed advantages of prepaid group practice are numerous and

include: provision of a comprehensive range of outpatient services;

continuity of health care in one setting; pooling of resources to make

possible the most efficient use of manpower, money, medical technology

and equipment; quick and efficient use of consultants; and an emphasis

on preventive medicine (10). Other authors point out additional ad-

vantages such as increased productivity and a better division of labor

(11) and peer review and better doctor-patient relationships (12). In

addition, capitation has been shown to be effective in decreasing sur-

gical rates for such procedures as tonsillectomies, adenoidectomies and

hysterectomies (13). Many of the above advantages have been observed in

the Kaiser-Permanente system which is the largest prepaid group practice

in the United States. As an example, the subscribers of Kaiser-Permanente

reported savings of 30-40 percent per family for medical care (14). Also

with the low surgical rates and the emphasis on ambulatory care, the

number of short term general hospital beds in the Kaiser system is about

1.6 per thousand members compared to the national rate of 4.1 per thous-

and population (13).

Legislative Background

As a result of a continuing dialog concerning the alternatives to

the present health care system, health care has again entered the national

political arena. In August, 1970, Senator Kennedy (D-Mass.) introduced

the Health Security Act (15). This bill called for sweeping changes in

the national health system revolving about a National Health Insurance

which would virtually replace private health insurance. This bill also

emphasized moving the medical care system toward organized programs of

health services, with special emphasis on teams of professional, tech-

nical and support personnel, and sought to move the health system toward


Partially because of Senator Kennedy's bill, health care came to

the forefront of national politics and in President Nixon's 1971 State of

the Union address (6), he set forth broad proposals for improving Amer-

ica's health care. These proposals included: a national health in-

surance program; increasing the number of doctors and other health per-

sonnel; making greater use of medical assistants to slow the rise in

costs; and new programs to encourage better preventive medicine. He

later followed this up with a paper to Congress detailing his health care

proposals (16). In this report, the President was highly complimentary

with regard to PPGP and made it a cornerstone of his health care pro-

gram under the name Health Maintenance Organization. The main objec-

tive of his HMO would be to foster cost consciousness in the group

practice setting. The cost savings would arise due to economics of

scale and the use of ancillary medical personnel where possible. His

plan called for a national commitment to help HMO's get started, along

with a restructuring of private health insurance to make HMO coverage

optional. It should be noted that the Administration's HMO plans are

broad enough to include individually practicing physicians and community

health facilities, bound together by contractual and professional agree-

ments and serving the enrolled population side by side with the fee-for -

service practice (17). This type of plan is exemplified by the San

Joaquin Medical Care Foundation. The President also asked for the repeal

of laws in 22 states which either limit group practice of medicine or the

use of physician's assistants. He also called for a greater number of

people in the allied health areas to help use existing medical manpower

more effectively and the Secretary of Health, Education, and Welfare was

directed to focus research in the field of health care services on new

techniques for improving the productivity of our medical system. In

addition to the health care legislation summarized above, there have been

several other major health care bills under consideration by Congress

(18-22) and further background material is given in reference (23).

Research Viewpoint

The HMO concept has been proposed as a potential cure for a number

of problems present in the American health care system. Among the most

frequently cited of these problems are rising costs, the episodic rather

than preventive nature of health care delivery, and maldistribution of

services which has resulted in inadequate access to care in inner city

and rural areas. Numerous viewpoints on the most desirable structure of

an HMO have baen expressed by spokesmen of the Nixon Administration (17,

24, 25), the American Public Health Association (26), the American Medi-

cal Association (27) and others. These structures differ in many impor-

tant details but are sufficiently similar to generalize for the purposes

of this research.

It is assumed that the precise concept of an HIO (or whatever term

may subsequently replace it) will be in continual flux. For present

purposes the following structural elements are taken as a minimum. An

HMO is an entity which

(a) serves an enrolled population who contract

with the delivery system for provision of a

range of health services;

(b) is managed in a manner to insure legal,

fiscal and professional accountability;


(c) provides prenegotiated comprehensive health

services to all subscribers directly through

its own staff and supporting resources or

through other health delivery entities for a

fixed payment paid on a periodic basis with-

out regard to the frequency,extent, or kind

of service actually provided during the


The basic argument for the workability of HMOs is that with a fixed

annual capitation fee it is in the best interests of both the HMO and

the subscribers to maintain each subscriber in a high degree of health

to minimize the utilization of costly services such as inpatient hos-

pitalization. Other advantages offered by HMO's are expected to be:

(a) a continuity of care through a variety

of specialists in one location;

(b) lower cost to the patient (28-30);

(c) less physician involvement in clerical

and managerial details (31);

(d) regular working hours for physicians

(31); and

(e) continuing education and peer review of

physicians (31).

Allied Health Manpower

Physician Extenders

Although there are hundreds of kinds of manpower utilized in the

delivery of health services, the patient receiving medical care is in

most frequent contact with either a physician or a nurse. Recently a

new role has emerged which is intended to supplement the physician by

relieving him of routine duties not requiring his extended training. Al-

though these persons act under the supervision of a physician,they are

not usually involved in the type of direct patient nursing care associated

with the traditional nurses' role.

As of March,1971, at least 125 programs were in operation or in

advanced planning to train persons for this new role (32). The personnel

trained in these programs are known by a plethora of names; a partial

list follows:

(a) physician associate,

(b) physician assistant,

(c) family health practitioner,

(d) pediatric nurse associate,

(e) public health nurse practitioner,

(f) family nurse practitioner,

(g) opthalmic nurse,

(h) nurse midwife,

(i) nurse anaesthetist, and

(j) family nurse clinician.

Although some of these programs are relatively well-established, most are

quite new. To avoid the multitude of names, in this research the above

personnel categories will generally be referred to as physician extenders


Physician Extenders in Primary Care

This dissertation deals principally with the delivery of primary

care in adult medicine, OB/GYN and pediatrics. Physician extender types

of importance to these areas are discussed in this section.

There are two general types of physician extender programs evolving

in the United States. One type is typically called a physician assistant

(PA) program and is principally aimed at persons with at least a high

school education and perhaps some college education and also with prior

health care experience such as a medical corpsman. The other type seeks

to expand the role of the nurse and is usually denoted by the term nurse

plus other modifiers such as practitioner, clinician, or midwife. These

programs generally accept RN's and give them further specialized training.

There are many different types of physician assistant programs in

existence. However a general definition of a TYPE A assistant was pro-

posed by the American Association of Medical Colleges' Task Force on

Physician's Assistants Programs.

Type A, within this definition of an assistant
to the physician, is capable of approaching the
patient, collecting historical and physical data,
organizing the data, and presenting it in such a
way that the physician can visualize the medical
problem and determine the next appropriate diag-
nostic or therapeutic step. He is also capable
of assisting the physician by performing diag-
nostic and therapeutic procedures and coordinating
the role of other more technical assistants. It
is recognized that he functions under the general
supervision and responsibility of the physician,
though he might, under special circumstances and
under defined rules, operate away from the immed-
iate surveillance of the physician. To properly
perform at this level, the assistant must possess
enough knowledge of medicine to permit a degree
of interpretation of findings and a degree of in-
dependent action within these defined rules and
circumstances. (33, p. 102)

The first PA training program was at Duke University (34) and Estes

(35) pointed out that the tasks physicians perform can be divided into

those requiring the complex judgement their education prepared them for

and those that require technical skills that can be learned by repi-

tition. It is these technical skills that Duke trained the PA's to

perform,with the additional goal to prepare the PA to "do anything which

the doctor can program him to do." (36, p. 33) The use of TYPE A or

generalist PA's has also been defined and analyzed in the University of

Washington's MEDEX program (37). Ex-corpsmen in this program follow a

three-month academic program followed by a twelve-month preceptorship

that includes three days of formal education per month. The Washington

MEDEX program has been used as a model to set up MEDEX programs in many

other universities (34). The use of ex-corpsmen as input to the PA

programs has to be considered as temporary due to the reduced future

supply of corpsmen; thus if PA's are to become a standard part of the

health care system, people without previous medical experience will have

to be trained or an increased number of nurses could be admitted to PA

programs (34).

An important example of the nurse expander type program is the

pediatric nurse practitioner (PNP). The first formal training program

was established by Silver and Ford (38) in Denver and since then the

American Nurses Association and the American Academy of Pediatrics have

issued a joint statement defining this concept and established guidelines

for programs of continuing education (39). Twenty-four training programs

for PNP's were listed in July 1971 (40). At the University of Colorado,

the PNP is a graduate nurse with a baccalaureate degree who has received

four months of intensive theory and practice in pediatrics at the Uni-

versity of Colorado Medical Center (41). During their training, emphasis

is placed on patient interviewing techniques, performing a complete phy-

sical, various aspects of parent-child relationships, child development

and counseling techniques (41). They also learn to assist in both the

management of healthy children and those with a variety of acute and

chronic disorders (42).

Another major category of the expanded role of the nurse is the

nurse midwife (NMW). According to the American College of Nurse Mid-

wifery definition, the NMW is "a registered nurse who, by virtue of her

added skill gained through an organized program of study and clinical

experience, recognized by the American College of Nurse-Midwifery, has

extended the limits of her practice into the area of management of care

of mothers and babies through the maternity cycle so long as progress

meets criteria accepted as normal." (43, p. 354)

Although there are many other types of training programs for many

medical specialties, the PA, PNP and NMW programs were specifically men-

tioned due to their involvement in the primary care areas: PA's are

usually trained to assist general practitioners or physicians in internal

medicine; PNP's are trained to assist pediatricians; and NMW's are

trained to assist physicians in 0B/GYN. These are the three primary care

areas focused on in this peper. Since many of the programs produce health

care providers with different titles but very similar capabilities, this

will be simplified by referring to all as PE's and recalling the three

primary care prototypes described above.

Physician Extender Utilization: The Team Concept

Utilization of auxiliary personnel in primary medical care delivery

is usually done in the context of a "health care team." At present,

tradition, licensing, registration, and practice act restrictions dictate

that the auxiliary personnel be supervised by an MD (44-51). Although it

is likely that this configuration will continue for some time to come,

some research (52) has been directed toward family health teams for HMO's

that are totally comprised of allied health personnel. Granting more

autonomy to allied health professionals raises questions of professional

acceptance, legal accountability, and patient acceptance. In recogni-

tion of these issues the AMA (27) has called for more research into the

questions of which type of assistant to develop, the tasks they can

assume, their acceptance by patients and physicians, and their impact on

costs and productivity.

This dissertation will explicitly consider three health care team


(a) a team comprised of a physician, an RN, and

possibly an LPN;

(b) a team comprised of a physician extender,

an RN, and possibly an LPN; Pnd,

(c) a team comprised of a physician, a phy-

sician extender, and an RN.

A PE in the second type of team noted above would be acting under in-

direct supervision by an MD. Under that configuration the MD would

typically confer briefly with the PE regarding the patient and the PE's

findings, or the MD may briefly see the patient and read over the PE's

findings. In the third type of team, the PE would be acting unaer direct

supervision. This would not necessarily imply that the MD is in the room

at all times with the PE. However, it does imply that the MD has direct

control over the particular patient visit. The PE would typically take

the patient history and perform a preliminary workup or physical exam.

This portion of the visit is time consuming and does not require the

highly developed skills of the MD. The MD would then take these find-

ings and carry on from that point with the patient. It's possible the

PE would later be used to carry out tests, a treatment regimen, or to

explain a home treatment regimen to the patient.

Delegation of Medical Services

The question of which functions can be carried out by a physician

extender is at the core of the problem of PE utilization. For the pur-

poses of this study, a broad definition of the PE types is not enough to

fully analyze potential PE utilization. A more detailed answer of what

allied health personnel are prepared to do and what professional per-

sonnel are prepared to delegate to them is required.

Although the use of PE's is relatively new in the United States,

there has been considerable use of similar level personnel in other

countries. As far back as the 1700's "feldshers" were functioning in

Russia (44). In urban areas the feldsher works as an assistant to the

physician but in the rural areas the feldsher frequently performs in a

primary care role (53). An even more striking example of the utilization

of allied health personnel exists in China. China has several levels of

assistants for the physicians. One level of assistant, with a limited

formal training of about three months and on the job training, provides

certain elements of primary care in rural areas and another level is

typically comprised of housewives with ten or more days training who

work on a neighborhood level (54). With regard to health care teams,

a professor of orthopedics in China noted:

In the countryside--and I am sure this will
cause eyebrows to be raised--we made no distinction
between nurses and doctors....In fact, doctors and
nurses were in essence doing the same job to the
best of their abilities, and their abilities de-
pended as much on their adaptability and sense of
responsibility as on the type and duration of
training which they had undergone.(55, p. 192)

In Cuba, the nurses routinely give intravenous injections and make house

calls (56). In many other countries (most but not all are developing

nations) paramedical aids are an integral part of the health care system


However, within the United States the organized use of personnel on

the PE level is relatively new. As a consequence there has been some

uncertainty as to their possible function and the specific training pro-

grams needed. Since PE's will perform in a subsidiary capacity, several

surveys or studies to elicit the opinions of physicians have been per-

formed (33, 61-71). In a survey of 3,425 internists the American Society

of Internal Medicine found that internists believed many elements of their

practice could be delegated to an allied health worker (61). The Amer-

ican Academy of Pediatrics surveyed 5,799 pediatricians and found that

over 70 percent favored delegation of recording the patient history and

counseling on child care, feeding and development (62). About 25 percent

favored delegation of well child and sick child examinations. It has also

been reported that 50-65 percent of the physicians in Wisconsin indicate

a need for a physician assistant (63). Patient acceptance has been good

(64) and over 50 percent found the care provided by a pediatrician and

a pediatric nurse practitioner better than that received from a phy-

sician alone (65).

The major problem involved in utilizing the studies noted above is

their lack of a common medical classification system upon which dele-

gation can be analyzed. The studies used tasks, functions, services,

procedures, typical patient visits and sometimes a combination of these.

However, the modeling effort in this dissertation requires a single

consistent classification system in which delegation can be analyzed.

There are many classification systems for medical patients but unfor-

tunately few are useful for analysis of manpower and delegation decisions.

The most widely known is the International Classification of Diseases,

Abstracted (ICDA) but it is of more use in hospital admissions where

diagnosis is a feasible classification index. The California Relative

Value Units (CRVU) are also used to some extent, but it is too vague in

the area of patient office visits. There are several compilations of

tasks performed in the health care system (71, 72), but tasks do not lend

themselves well to viable patterns of delegation and analysis.

Instead of the above methods of medical classification the concept

of a medical service or typical patient visit will be used in this study.

This concept allows the model to concentrate only on viable patterns of

delegation, simplify the computational details and increases the in-

tuitive appeal of the results (73). Previous medical service listings

have been reported for general practice (73) and for the primary care

specialties (74). Since the former is limited to general practice and

the latter is too general, a new medical classification is presented in

Chapter 4 which enables the analysis of the relationship between train-

ing, delegation and manpower utilization to be fully explored.

Motivation for Mathematical Modeling

Previous Studies

Formal PE programs were started in the United States only a few

years ago. As a result there have been few studies which have analyzed

the use of PE's. Likewise the HMO proposal was first introduced in 1970

and also suffers from a lack of analysis. In both areas, discussions

regarding the concept behind paramedical personnel and HMO's are rela-

tively plentiful and,while these studies have guided the overall goal

of this paper, they do not provide a concrete basis on which to start

an analysis. However, there have been several exploratory studies per-

formed which are fairly directly related to the goals of this paper.

The research most useful to review was performed by Shuman (75,

76), Golladay, Smith and Miller (73), Reinhardt (77), and Gnldstein and

Horowitz (78). This collection of studies is of particular interest

since they represent the spectrum of techniques which have been used to

analyze health manpower usage. Shuman and Golladay et al. arrived at

totally different normative mathematical models to analyze manpower util-

ization, while Reinhardt used a descriptive model in the form of pro-

duction functions to analyze the increased efficiency resulting from the

use of paramedical aides. On the other hand, Goldstein and Horowitz took

a personnel management approach to increasing manpower efficiency in a

hospital. Each of these will be discussed in greater detail in the

following paragraphs.

Shuman's work dealt mainly with regional health manpower planning

and the substitution between various personnel classes. Both papers by

Shuman are of interest but since the aspects essential to this study are

contained in (76) only the latter work will be reviewed. Shuman con-

siders three ways by which productivity may be increased: introduction

of technology, transfer of tasks to less skilled personnel, and organi-

zational changes. A key point he makes is that the manpower problem is

part of a larger problem: the determination of an efficient means for

the delivery of health services. Only then can it be determined whether

shortages actually exist for MD's, nurses or allied health personnel.

This rather straight forward point has been largely neglected in most

manpower studies and casts doubt on many of their conclusions. In other

words the emphasis should be on the shortages of health services not on

a manpower shortage. Starting with this insight into the problems in the

health care system, Shuman then formulated an analytic model to attempt

to determine the proper mix of manpower to provide the services. Shu-

man's work was one of the first attempts to model analytically the

problem and consequently there are many problems left unanswered. The

principle shortcomings revolve around the size of the model as it is

structured and the lack of data and manpower analysis contained in the

study. Several aspects of Shuman's models are discussed in Chapter 3.

Golladay, Smith and Miller (73) investigated the optimal role for

paraprofessionals in the health care system and their potential impact on

the productivity of the physician. The study developed an analytical

model of primary care practice which enabled them to explore the impli-

cations of delegation for physician productivity, per patient costs,

and demands for all categories of medical workers. The most unique

feature of the study involved the data collection for task analysis in

medical practice. The analytic model was derived to answer these ques-


1) what is the optimal staffing pattern for a

practice and how is it related to the size

of the patient population;

2) how many hours of patient contact time would

be required to satisfy the medical demands

of a specified patient population; and,

3) which activities would be delegated in an

efficiently run practice?

Their model is described in Chapter 3. Golladay, Smith and Miller con-

cluded that

1) use of one PA may increase the productivity

of a physician by 74 percent; and

2) from a monetary standpoint, MD's would

profit from using PA's.

In contrast to the normative models proposed by Shuman and Gelladay et

al., Reinhardt derived a descriptive model utilizing a production function

approach. This type of approach is the most common quantitative technique

used to analyze manpower utilization and is frequently used by econo-

mists. Reinhardt set out to answer two questions:

1) to what extent is it possible to raise the

output per physician hour through use of

paramedical personnel; and

2) can physicians in group practice use aux-

iliary personnel more efficiently than

physicians in solo practice?

His conclusion was that MD's should employ 3-4 aides rather than the 1-2

aides they now employ and that this would increase medical production by

20 percent. He also concluded group practice resulted in 11 percent to

16 percent higher production per MD than solo practice. Note these fig-

ures are conservative since they represent the use of allied personnel

in 1965 and 1967 and not an optimal or even near-optimal task delegation.

His work does however firmly lead to the conclusion that allied personnel

will lead in practice to increased medical care production.

Goldstein and Horowitz used personnel management techniques in an

attempt at better utilization of medical manpower. Their study involved

establishing what tasks were performed in the hospital, what training

level was required to perform each of the tasks and what level personnel

were presently performing the task. The two major goals of the study


(1) study and analyze the hiring-in requirements

and the duties and functions of paramedical

personnel in a single hospital; and,

(2) to recommend changes to restructure occu-

pations and to improve the utilization of

manpower in that hospital.

Their study did not attempt any optimization but it did result in concrete

recommendations for better utilization of manpower. It is also an ap-

proach that is intuitively appealing to many people including those who

have an inherent distrust of mathematics and model building. Its prin-

cipal drawback is that it relies upon studying a well established insti-

tution such as a hospital in order to arrive at results. It also limited

itself to reshuffling tasks paramedical personnel perform, and not sub-

stituting paramedical personnel for MD's.

Systems Analysis as an Aid to the HMO Design Process

The approach taken by Shuman, Golladay et al., and Reinhardt and the

approach taken in this dissertation can be termed systems analysis. Sys-

tems analysis has been described by Kershaw and McKean as the "comparison

of alternative means of carrying out some function, when those means are

rather complicated and comprise a number of interrelated elements."

(79, p. 1) The essential feature of a systems analysis is thus the rec-

ognition of the interrelatedness of parts. The system should be defined

so as to encompass all parts that are related in a significant fashion


An HMO is a system that consists of staff, facilities and materials

and is organized to deliver medical care. However, the output of the HMO

is difficult to quantify. It can be argued, for example, that the out-

put of a medical system should be measured in terms of the improvement

or stabilization of the health of the patients. In practice however,

this concept is extremely difficult to apply and in this dissertation

the output is measured in terms of medical services provided. The math-

ematical models will allow an analysis of the interrelated factors to be

performed and will allow the examination of the substitution of various

alternative resource combinations. In addition to manpower, the models

should also permit the analysis of the effects of automation and tech-

nology upon output. An approach to incorporating technology into the

models is discussed in the next chapter. Although extremely important,

questions on the quality of care provided and the effects of alternative

system configurations on the general health of the population covered

are considered to be beyond the scope of this dissertation.

Planning the Benefits To Be Provided by an HMO

There are two basic approaches to examining the choice of services.

The first approach (see Figure 1) is the traditional method which is most

commonly proposed. This technique assumes that legislative groups in

consultation with various health personnel will decide what services an

IIO should provide. The principal weakness is that it is a decision made

without regard to the other variables in the planning process. Note that

Figure 1 denotes a system with no feedback loops.

Begin Local Services Staff Facilities Finish
Planning Inputs to be Required Required Planning

Figure 1: Present Decision Process
for Determining Benefits

The second approach recognizes the problems attendant to legislating

the variety of services an HMO must offer. This approach leads to the

best array of services tailored to the local situation (see Figure 2).

For example, it may be too expensive in a rural area to offer specialty

care to HMO subscribers, but at the same time an HMO might possibly pro-

vide many other services at a lower price and comparable quality than can

be found in the traditional mode of health service. The models are de-

signed to formulate a set of services based on the available capital and

on the level of fees the patients are willing to pay. The services of-

fered naturally depend very heavily upon the population base since a

large HMO, through economics of scale, may be able to offer many spec-

ialized services in an economical manner, whereas a small HMO may under-

utilize the manpower and facilities for that service, thereby making it

more expensive than the traditional system.

Services -- Staff

Begin Local Inputs tFs Finish
Planning Facilities Planning

Figure 2: Proposed Decision Process
for Determining Benefits

The models are designed such that they can be useful in the pro-

posed decision process depicted by Figure 2. The individual medical

services used in the models are detailed enough to distinguish between

time requirements and training requirements. Sets of these services can

be considered for the various medical specialties and the models are de-

signed so that the specialties can be included or excluded from the HMO

depending on economic factors. The models also allow the decision maker

to fix some or all sets of services as necessary for provision in the HMO

due to political or other conditions.

Proposed HMO Design Process

This research assumes a three-level hierarchy in the design of an

HMO (see Figure 3). The first level describes the financing, social

characteristic, incentives, generalities and overall objectives of HMO's.

This is the level that is most frequently discussed in the literature.

The second level deals with HMO's on a more specific basis. This level

of decision making would be concerned with the size of subscriber group,

levels of technology, general personnel guidelines, capitation rate, re-

lations with hospitals, etc. The third level examines NMO's on a micro-

scopic level. The decision on this level would be concerned with the

type and number of various personnel classes, specific technology utili-

zation, task allocation, patient flow, etc.

This three-level hierarchy completely specifies the design of an

HMO. Because the models developed in this dissertation can best be used

to analyze levels two and three they are presented as a systematic

technique to decision making on these levels.

Level 1 Financing
Social Characteristics
Overall Objectives

Size of Relations
Level 2 s Services Personnel Capitation with
Level 2 MO's Guidelines Rate t

Specific Number of Specific Assignment
Level 3 type of each type Technology of Services
Personnel of Personnel Utilization

Figure 3 Three-Stage Hierarchy in the Planning of IMO's

Figure 3 showed the relationship which exists between the levels of

decision making for planning an HMO. Figures 4 and 5 present the dynam-

ics of the decision process. Figure 4 presents an approximation of the

dynamics involved in the present HMO design concept while Figure 5 pre-

sents the dynamic relationships which are modeled in this paper. Note

that Figure 5 portrays the strong interaction which exists between sub-

systems of the HMO while Figure 4 shows a weaker interaction which occurs

in the present design process.

As was noted previously, the research effort is primarily aimed at

effective manpower utilization. However, as Figure 5 shows, the manpower

utilization is strongly interactive with the other subsystems of the HMO.

JC -




These other subsystems will be examined only as far as is needed to

perceive their input to the mathematical models. The optimal design, or

even design, of these other subsystems is beyond the scope of this re-


In the next chapter, models are developed which will permit the

evaluation of alternative designs of 1MO's. To examine the design of

HO1's, three measures of effectiveness are utilized and developed by the

mathematical models:

1) minimizing cost to the subscriber;

2) using the least number of professional

manpower to serve a given set of sub-

scribers (and maintain quality); and,

3) using a given set of professionals such

that the number of services they provide

are maximized.

Using these objectives, the models can help optimally design certain

aspects of the HMO. Thus, they can be used to present design guidelines

and enable one to perform sensitivity analysis on the inputs.



Preliminary Models


In this chapter several models are proposed to answer some of the

questions which arise in the overall design process for HMO's. These

models are designed to represent a useful formulation for part of the

HMO system and to be solvable in a practical sense. Each of the models

involves the assignment of personnel to services and the staffing levels

required. Beyond this, the models can be dichotomized as:

(a) maximize subscribers by optimal allocation and mix

of professional manpower and facility resources,

and ancillary manpower, or

(b) minimize resources used to provide a fixed set of

services or requirements.

Initially, a very simple model is derived. A step by step motiva-

tion and refinement is then carried out; culminating in a more complex,

but more flexible and refined model. The complex model is then used to

derive four other principle models. Throughout the development new

notation will be introduced in the text; additionally, a complete listing

is given in the Key To Symbols.

System Schematic for Model Development

A recurring HMO theme, as expressed in the literature noted in the

first two chapters, is the cost-effectiveness through utilization of

group practice and the capitation incentives. To examine the cost-

effectiveness of HMO's one must examine the costs in the IMO structure.

A schematic of the structure of an HMO is given in Figure 6.





Figure 6 A Schematic Representation of the O10 Structure

The inpatient care and dental care portions of an HMO are major

problems in themselves and can be effectively isolated from the analysis

of office based medical care. This does not imply there are no relation-

ships between inpatient and ambulatory care, but only that the relation-

ships can be handled external to this analysis. For example, an increased

use of ambulatory care usually occurs in prepaid practice due to the

decreased use of inpatient facilities. However since utilization of

ambulatory care is an input to the models, this increase can be noted

and used to determine the input to the models which would then reflect

the ambulatory care manpower and costs resulting from the increased use

of ambulatory care facilities. Also, within the O10 structure there are

additional functions other than the medical care function. There are

support personnel, such as clerical, administrative, laboratory, x-ray,

and other ancillary aides, and management personnel for planning and

direction and these functions are reflected in the models. The models

focus on medical manpower, but to look at overall cost-effectiveness the

models should reflect the costs of non-medical care functions for two

principal reasons:

(a) to examine the effect the costs of these non-medical

functions have on the medical manpower; and

(b) to examine the full costs for medical services and

lIMO's as a whole (excluding inpatient and dental).

Within the medical care structure there are two main factions-those

who subscribe to the plan for medical services and those who work for the

plan and provide medical services. This is shown in Figure 7.

REQUIREMENTS: Subscribers serce
for services

Medical Capability to
RSOURC: Personnel provide services

Figure 7 Medical Care Structure Showing Relation
Between Resources and Requirements

The upper loop must be expressed in some taxonomy of health care

delivery while the lower loop must be expressed in manpower types and

capabilities. In addition, to be combined in the last block, the man-

power capabilities must be expressable in terms of the taxonomy of health

care delivery. This is an extremely difficult concept to put into prac-

tice. In previous related studies Golladay, Smith and Miller (73)

approached this empirically for a small sample, Pondy (81) took a related

but more superficial approach and Shuman (76) did not attempt to put the

concept into practice.

Tradeoff Decisions To Be Incorporated in the Models

Chapter 2 presented a detailed view of IMO's and health manpower

and resulted in a systems viewpoint of the interactions between the two.

Figures 6 and 7 further helped to sharpen the focus to the point where

mathematical models could be developed. At this point the particular

decision making areas to be considered in the models will be explicitly

stated. When normative mathematical models make decisions, they make

them by trading off resources and requirements against each other in such

a way that the objective function reaches its extremum. The constraints

act to complicate the decision making to the point where mathematical

programming techniques must usually be used to solve the problem.

As a result of the systems development it has been determined that

several factors are important and should be considered by the mathemati-

cal models in their decision making. The major tradeoffs that are incor-

porated in the minimum cost models are summarized below.

1. The models tradeoff delegating a service to the lowest

cost person (who can perform the service) versus the

additional cost of supervision by the MD.

2. The models tradeoff delegating a service to the lowest

cost person (who can perform the service) versus the

supervision time required from the MD. Since supervision

time is usually a scarce resource this implies also

selecting those services to be delegated as well as to whom

to delegate the service,

3. The models tradeoff the cost of providing a service by an

MD and nurse versus the cost of providing the service by

an MD, PE and nurse versus providing the service by a PE

and nurse. In the second case, MD time decreases but MD

plus PE time is greater than in the first case. In the

third case, MD time for the service becomes zero, but

additional MD time is required for the indirect supervi-

sion of the PE.

4. The models tradeoff the cost of higher cost personnel

versus the integer restrictions on manpower levels. Thus

for example, if delegating a service that requires .2 man

years would require that one more PE would be hired with

the subsequent .8 man years of idleness; it may be better

to delegate the service to another, perhaps higher salaried


5. The models tradeoff the total cost of a service in the HMO

versus what it can be purchased for outside the IN0. Note

that the total cost for a service in an HMO is interrelated

with all other services offered. This tradeoff can be

done on a service by service level or on a medical

specialty level.

6. The models tradeoff the cost for additional technology

versus the higher productivity or lower salary levels


Only the major factors have been included in the tradeoffs summarized

above. There are other subsidiary factors such as capital requirements,

capability limitations, and scarcity of manpower that are potentially

involved in most of the above tradeoffs.

This chapter also includes models concerned with the minimal use of

MD's or for maximizing subscribers per MD for which the six tradeoffs

listed above are still relevant, but the key is minimal use of MD time.

In some cases the two give the same result, but there are many exceptions.

In addition this latter type of model includes two additional factors:

(a) as subscriber levels increase this means that more

patient visits are eligible for PE's to handle; and

(b) a budgetary limit is imposed, thus there may be ways

to add additional subscribers but they would be too

costly (in the models this is taken as the revenue =

expenses point not the marginal revenue = marginal

expenses point).

The above tradeoffs are explicitly designed into the models. There

may be other types of tradeoffs made depending on the exact use of the

models or the data used for solving the models, but these additional

tradeoffs are expressable in terms of the above tradeoffs.

Preliminary Model Development

Figures 6 and 7 represent two different but related structures in

the HMO. Figure 6 diagrams what could be viewed as a cost structure

while Figure 7 diagrams a medical care structure. On the most elemental

level Figure 7 could be modeled by viewing the subscriber demands as the

requirements vector and the medical personnel as the resources. Two

objectives present themselves for this model:

(a) minimize the medical manpower needs for a given

number of subscribers; or

(b) maximize the subscribers for a given number of

medical personnel.

The first objective takes the requirements as fixed and the resources to

be variable while the second objective is the converse of the first.

Use the definitions:

I set of medical personnel of type i,

I* set of MD's,

I* set of non-MD's,

J set of medical services of type j,

xij manpower type i performing service j (man-years),

d. demand for service j (medical services per year),

b.. rate at which manpower type i produce service j (medical

services per man-year),

N. number of manpower type i employed (man-years),

s. salary for type i personnel (dollars per year),

i iel

S= e
J jeJ

With these definitions we can meet the medical care requirements with

the constraint

Z b..x.. > d. jeJ. (3.1)
i 1i 13 3

We can also define the number of manpower type i employed as

E x..-N. < 0 V icl (3.2)
S131 1
j i

and the first objective becomes

min Z N. (3.3)

or if constraint (3.2) is not used we have

min S x.. (3.4)

The model given by (3.1) (3.4) will be called M1 and is a linear

programming model which essentially characterizes the models used by

Golladay, Smith and Miller (73) and by Pondy (81). To arrive at the

Golladay, Smith and Miller model the converse of the b.. coefficient is

used. The b.. coefficient in (3.1) converts resources to activities

while Golladay, Smith and Miller use a coefficient in constraint (3.2)

to convert activities to resources. This allows them to define health

care teams or technologies in a straight forward manner but has the

undesirable property of forcing the inputs from each member of the team

to occur in fixed ratios-to change the ratios a new team must be defined.

In addition Golladay, Smith and Miller take the MD input as fixed and

seek to minimize the salary of ancillary personnel as their objective.

This places weighting factors in equation (3.4) and the summation is

taken over those i for non-MD's and constraint (3.2) is eliminated for

non-MD's. The resultant model (where k is the team or technology) is:

min E E siaikx (3.5)
iEl* k

subject to

Saikxk i N V iEI* (3.6)

E xk = n V jEJ (3.7)

The Pondy model is very similar. Both can be efficiently solved as a

minimum cost network flow problem with nonnegative gains (80).

Thus both the Golladay, Smith and Miller model and the Pondy model

follow directly from the elementary constraints on resources and require-

ments. This structure does not allow a full exploration of the relation-

ship expressed in Figure 7. It also ignores the cost structure in which

the medical system operates. It can be noted that both models were

primarily focused on the fee-for-service setting.

The model resulting from the second objective easily follows if the

following definitions are made. Let

S number of subscribers,

d' services of type j demanded per subscriber.
Here the resources are fixed and the requirements are variable. Thus

the model is:

max S (3.8)

subject to

x.. Ni ieI (3.9)

Sb..xij d! S < 0 jeJ (3.10)
i 13 3j J -

This model arises from the implication that the present shortage of

medical manpower is an overriding concern in the health care setting

and thus seeks to provide medical care to the most people given the man-

power available. No corresponding model appears in the literature. The

weaknesses are the same as were noted above and in addition this model

ignores the economics of medical care entirely.

Development of the Medical Care Aspects of the Model

Several general weaknesses of the Ml model have been pointed out.

At this point the model will be developed further along the scope of

Figure 7. This further development will lead to a model which exhibits

greater flexibility and depth in its portrayal of prepaid group practice.

The concept of minimizing medical manpower cost will be used.

Medical care is provided either by individuals or by teams. Within

a health care team the leader would typically be an MDi however,teams of

PE's and nurses are also possible. In the former case, direct supervi-

sion by MD's is implicit within the team. However, in the latter case

indirect supervision by physicians would normally be required. The

Golladay, Smith and Miller model (73) and Pondy model (81) explicitly

consider teams but do not consider the latter problem. On the other hand

a model proposed by Shuman (76) considers indirect supervision but not

teams. With the following definitions, model Ml can be revised to pro-

vide the needed generality.


I set of individual health care personnel,
It set of health care teams,
I set of manpower including teams,

qkij man-years of manpower type i per man-year of team k
providing service j,

k kIt '
-* t
I= I I v I

The particular introduction and definition of qkij is the key to the

generalization of the model. Using that definition (3.1) and (3.4) are

unaffected. The only change occurs in (3.2) which becomes

ij + s kijxkj Ni 0 V iel (3.11)
j kj

In addition, a method to view indirect supervision must be introduced,

not only for PE-nurse teams but also for PE's or nurses in an individual

capacity. This concept, as used by Shuman (76), provided for fixed

supervision levels regardless of the service being performed or the

health care personnel involved. The constraint can be modified to

reflect these considerations by defining:

f.. level of independence for type i personnel to perform

service j; fj = 0 implies no indirect supervision required

while f.. = 1 implies full indirect supervision,
Ji* set of services class i el personnel supervise,

xi*j* j* is the supervisory service provided by personnel i

The supervisory constraint involves the quality of health care in that

it constrains the number of personnel a professional staff member may

supervise and the level of independence exercised by the paramedical

assistants. The general supervisory constraint is given by

S E f..bx.. Sj bi j bi*j*xi*j* (3.12)
jeJi* iCI* 1

It is generally accepted that MD's do not want to be burdened by an

excessive amount of administrative work or supervision of ancillary

personnel. The amount of supervision can easily be limited by making

x i** an upper bounded variable which would not make the model any more
difficult to solve. Thus, if SU. is defined as the maximum fraction of
time type i personnel will engage in indirect supervision of ancillary

personnel, the upper bound is given by

xi*j, < SUi'N. V i*cI* (3.13)
ij* -- i 1

Thus far the model is specified by (3.1), (3.4), and (3.11) (3.13)

and a further tightening up on the definitions of resources and require-

ments is desirable.

The resource considered thus far in the model is the available

health manpower. Since an HMO has as a principle objective the delivery

of effective health care in a cost-conscious manner, the resource should

in many cases be evaluated in monetary terms. Also, the resources used

here represent health care employers; thus in many cases the employees

will be full time employees only. This constraint is usually beneficial

to both the employee and the employer. Full time employees can be

represented by integer variables in the model. Thus, let Ni be defined

as an integer variable and define

x! fractional part of personnel group i which are slack or

idle (man-years),

c.. labor cost for type i personnel or team to perform one type

j service (dollars per medical service).

The result is

c.. = s./b.. V
i1j i ij

cij = I/bijnCls S n inj

iIs ,

9 iCI

Also let

b = units of time worked by team leader/year
ij units of team leader time/medical service

An example will help to clarify the meaning of these terms and

their relation to qkij. Suppose for service 1 the time and personnel

requirements are as listed in Table 1.

Table 1 Example of Personnel and Time Requirements




M.D. and R.N.

time to perform service 1 (min.)

10 (M.D.) and 5 (R.N.)

Also suppose an M.D. works 1440 hours per year in office visits and the

R.N. works 2000 hours per year. Then if the M.D. is designated the team


b3 = 1440/10/60 = 8640 services/year, and

q311 = 1, and for the RN,

q321 = 5/60 8640/2000 = .360

If the MD's salary is $30,000 and the RN's salary is $10,000 then the

direct labor cost per unit of service is given by


c31 = (1/8640) ((30,000) (1) + (10,000) (.360)) = 3.89

Returning to the development of the model, recall the objective is

to minimize the labor cost for all services provided which is given by

min s S (b. ic. ) x. (3.15)
j i 1( 1J 13

Incorporating the integer manpower constraint for N., constraint (3.11)


E x.. + E r qkij + x! N. = 0 V ieI (3.16)
Sj kj 1

and the objective becomes

min ES (b.i.c.) x.i + S s.x! (3.17)
j i ij 13 i I I

Note that the objective can be written as

min E s.N. (3.18)
icis L I

but for clarity it will be written as the two terms in (3.17).

It is also possible to generalize and refine the requirements con-

straint which is given by constraint (3.1). Physically, it is impossible

for an HMO to deliver more services than are requested by the subscribers.

It is however possible to deliver fewer services than are demanded but

this would not be consistent with the concept of prepaid practice. The

remaining possibility is that some health services be provided by

extraordinary means. Thus a new variable n. can be defined as

n. number of type j services demanded per year and not
provided by ordinary means.
provided by ordinary means.

This can be incorporated in constraint (3.1) to give

E b..x.. + n. = d. V jeJ, j#j* (3.19)
i i

The n. variable has several possible interpretations within an HMO

structure. In a multiclinic HMO it may be desirable to schedule some

patients into a more heavily staffed clinic during peak periods.

Another interpretation is that the HMO may contract out to other health

care providers for their overload patients. A third possibility is that

the n. represents those services performed on an overtime basis due to

understaffing. A fourth possibility is that the n. can be used as a

planning variable to determine if the HMO should provide the service or

if it can be provided cheaper to the subscribers outside the HMO. This

alternative will be developed later in the chapter.

Several restrictions must be placed on the n. variable for reason-

able solutions to arise from the model. To develop these restrictions

define the coefficients:

u. per unit cost for provision of type j services by

extraordinary means, and

MX. maximum fraction of services provided by extraordinary


Constraint (3.19) shows the requirements can be provided by n. which has

no cost in the objective function. To rectify this, (3.17) can be modi-

fied to give


min E (b..c..) x.. + E s.xx + E u.nn .
i ij i j

In addition from a standpoint of convenience to the patient and paper-

work problems for the 4MO it is desirable to limit the services to be

provided by extraordinary means. This can easily be included in the

model using an upper bound constraint

n. < MX. d. V jeJ (3.21)

MX. can be set to zero if all patients are to be seen under normal care

for service j or set to non-zero if it is feasible to use one of the

extraordinary means listed above. Typically MX. would range from 0.0-0.1.

The modified model, to be denoted M2, is summarized below:

min S S (b..c..) x, + s.x! + E u.n. (3.22)
i i i j

subject to

S s fij..b.i.ij b i*eI* (3.23)
jeJi. iC* J

E xij + S Z q kijkj + x! N. = 0 I ieI (3.24)
j kj k

E b.jx + nj = d. j j j* (3.25)
i 13 iJ 3 3

x.j.j < SU N. i*el* (3.26)

n. < MX.- d. f jeJ (3.27)

This model allows for a much fuller and more general representation of

the system represented by Figure 7 than model Ml. If Ni is not

restricted to integer levels, M1 and M2 are approximately of the same

order of difficulty for solution purposes. If N. is restricted to

integer levels the result is a mixed integer program which is more diffi-

cult to solve than the linear program required for model Ml.

Further Development of Financial Aspects of the Model

Thus far the financial aspects of the HMO have not been fully

investigated. Health manpower costs have been considered, but they

represent less than half of the total cost in an HMO. This fact in

itself provides some motivation to include the remaining costs in the

HO0 model. In addition, an objective of the model is to determine the

cost of medical service. This determination will involve the cost of

particular medical services, the costs by specialty or department, and

the cost for the entire clinic or HMO. To further develop the model

along the concepts expressed in Figure 6 it is necessary to examine

manpower overhead costs, facility overhead costs, ancillary costs, the

fee structure and the technology cost.

An HMO will in most cases be organized in departments along medical

specialty lines. Thus it may have an adult medicine, pediatrics, OB/GYN

and other specialty departments. This departmentalization and ready

access for patients to any department leads to little overlap in the

functions carried out in the various departments. Model M2 did not

reflect any departmental structure, but this will now be corrected.


V zero-one decision variable; if V = 1 department m is offered
m m
and if V = 0 department m is not offered;

M set of all medical departments under consideration;

Im set of personnel in department m;

J set of services considered for provision;

let J = (1,2,...j1* l,jl*'. mJ* ...j*) where j* is the
1 1 r-i m m
supervisory service provided by set I* personnel to supervise

set I* personnel in department m;

J' subset of J for which V = 1;
Jm set of services considered for provision or to be provided in

department m.

J = (U Jm)U( V j*)
m m

The V variable can thus be used as a decision variable to decide whether
it is economically desirable to include a department in a clinic or in

an entire HMO or whether these services can best be obtained elsewhere.

If the decision is to not include the department in one clinic this would

be provided at the central YZ0 facility. If an entire HMO decided not

to maintain a given department, they could contract out to another health

care provider or remove that service from their list of benefits. If the

departments are fixed, then the V can be set to zero or one, which in

turn makes the model easier to solve. The V variable must be introduced

in (3.22), (3.25), and (3.27) which respectively become

min E Z (b.ijci) xij + S s.x!
J3ij i 1

+ E (1 V ) S d.u. + u.n (3.28)
m m J j J

E bi x. + n. d.V = 0 V jeJ, j#j* (3.29)
jij i m

. < MX. d.
3-- 1 J

V jeJm, meM (3.30)

Standard accounting practices lead to the dichotomy between revenue

producing cost centers and non-revenue producing cost centers. Since

services are offered on a prepaid basis no medical department is strictly

a revenue producing cost center. However, there is still a clear dis-

tinction between the direct services performed by the medical departments

and the indirect services performed by the administrative, clerical and

managerial personnel. The position of ancillary services such as labora-

tory, x-ray, physical therapyetc.,is not as clear. In the development

of the cost model there will be three main overhead cost categories:

manpower overhead costs, facility overhead costs, and ancillary costs.

The costs for these functions will be reflected in the model but an

optimization of these functions will not be carried out.

There are three basic variables used to allocate overhead costs to

the revenue cost centers: departmental services performed, departmental

budget, or number of departmental employees. For the purpose of defining

the model the manpower overhead costs are broken down and allocated in

three basic parts:

(a) medical records, membership, appointments, etc. (allocated

on basis of patient visits to department);

(b) management, planning, legal, etc. (allocated on basis

of FTE professionals in a department); and

(c) personnel, administrators of employee benefits, etc.

(allocated on the basis of FTE employees in a department).

With the above guidelines, define:

AD manpower overhead or administrative cost for department m

(dollars per year),

PV manpower overhead cost per patient visit (dollars per

patient visit),

MAN management and planning cost per FTE professional (dollars

per man-year),

PER personnel cost per FTE employee (dollars per man-year),

b* number of type j services type iCI* personnel can perform

per year for a given level of technology.

The FTE professionals needed to provide medical services for a depart-

ment is given by

FTE = (V n )/bt (3.31)
m jem m J j

which can then be used to define the manpower overhead costs for the m


AD = PV E Z b..x..
m jeJm ieIm 1

+MAN E (d.V n.)/b* + PER N. (3.32)
jeJ m ilm

This cost can now be included in the objective function to give

min E (bijci.) x.i + E s.x! + E ujn.
ij 113 13 i 11 3j

+ E (1 Vm) m d.u. + AD (3.33)
m jeJ m

For this study, general office equipment and fixtures are included under

plant operations and maintenance or under construction or rental cost and

the equipment cost category has been reserved for medical equipment.

Since facility and equipment are long lived assets, they should not be

treated as an operating cost, but rather should be treated as a

capitalization cost and be amortized. To continue with the model

development define the following terms:

g initial cost per unit area for construction (dollars per

unit area),

g' amortized cost per unit area for construction (dollars per

unit area per year),

oc maintenance and utility costs (dollars per unit area per year),

P amortization rate for initial capitalization,

t cost of equipment in department m per FTE professional

(dollars per man-year),

t' amortized cost of equipment in department m per FTE profes-
sional (dollars per man-year per year),

w. space required per type i person (units of area per


Y total floor space available,

Y floor space to be constructed for department m

(Y = E wiNi),
m icIm i i
Y* maximum initial capitalization (dollars).

If an HMO is in a planning stage, there may be an upper limit on

capital available for constructing an HMO or if it is already in opera-

tion there may be a limit on floor space available. In the latter case

this would imply

E Y < Y (3.34)
m --

while in the former case


E Y =Y .

A constraint similar to (3.35) will be used; recognizing that (3.34)

could easily be incorporated at a later point. For an IMO in the plan-

ning stage with a maximum amount of capital available the following

constraint arises:

g E w.N. + E t FTE < Y* (3.36)
i. ii m a -
i m

The capitalization costs must also be amortized and appear in the objec-

tive function which will be discussed shortly. However, the plant opera-

tions and maintenance cost have not yet been developed. This cost is

o 2 w.l. .
c. 1 1

Now all three of the facility overhead costs can be included in (3.33)

to give

min 2E (b.c..) xi + E s x' + Z u.n.
i j 13 3 1J i J J

+ E (1 V ) E d.u. + E AD + o E w.N.
m mjem J J m m i I

+ g' i w.N. + Z t' FTE (3.37)
Sii m m
i m

The overhead costs have all been included except the ancillary

costs. In this model this cost category will be used for the laboratory

and x-ray departments. These departments could be included in the regu-

lar medical department formulation with no change in model M2 or sub-

sequent models. However, to examine these two departments in the same

detail as the primary care departments was outside the scope of the study.

Thus they have been included as a separate category. If enough informa-

tion and data were available, this part of the formulation could be

dropped and laboratory and x-ray manpower utilization and costs could be

examined in the same manner as the primary care departments. To proceed,

define the following coefficients:

CLT average cost per laboratory test (dollars per test),

CXR average cost per x-ray service (dollars per service),

NLT number of laboratory tests ordered in department m per service

provided (tests per medical services),

NXRm number of x-ray services ordered in department m per service

provided (x-ray services per medical service).

The units between CLT and NLT and CXR and NXR should be consistent.
m- m
For example, if NXRm is given in series per medical service, then CXR

should be in dollars per series. From the above definition, the labora-

tory and x-ray costs are thus:

E (CLT NLT + CXR NXR ) Z E b..x.. (3.38)
m m m .6m is6m ij j (
m jej iel

Now (3.38) can be included in (3.37):

min E (b..c. ) x.. + s.x' + u.n.
i j i j

+ S (1 V ) S d.u. + E AD + o E w.N
m mjm m m c i

+ g' E w.N. + Z t'FTE
1 1 m m
i m

+ (CLT NLT + CXR NXR ) SE b x, (3.39)
mm jeJ islm 1

At this point, the objective function (3.39) is much more detailed

and versatile than the objective function for model M2 which is given by

(3.22). The function given by (3.39) is certainly more complex than that

expressed previously but the difficulty of solution has remained on the

same level. In fact if the V decision variables are set to either 0 or
1 (in (3.22) they are implied to be 1) then the present model and M2

are almost equal in difficulty of solution. The real change has occurred

in the data requirements. Information that was left out of model M2 is

needed in the present model. If the full cost for an HMO is not desired,

but rather, only the medical manpower costs are sought, then the extra

coefficients in the present model can be set to zero. It is felt,

however, that the extra effort required to find the additional coeffi-

cients results in a more meaningful and richer solution.

Additional Model Refinements

The model development could stop at this point but there are several

additional features which can easily be added and will further refine

the model without sacrificing solution capability. The first feature

presented below involves further refinement of the medical care structure

and the other three further refine the cost structure. In the previous

discussion involving indirect supervision of physician extenders and

nurses, several constraints arose and are given by (3.12) and (3.13).

These constraints arise mainly from the MD's viewpoint of the time

requirements for him to supervise lower level personnel. However,

another viewpoint can arise from the patient care aspect of medical care.

The physician extender is not qualified to diagnose and treat every

patient that comes to the HMO for medical care under the indirect super-

vision mode. In fact, for many health services, the physician extender

will only be used in a team role under the direct supervision of a physi-

cian. The details of this limitation will be fully explored in the next

chapter. For the moment it suffices to note that the above limitation

results in upper bounded variables. Let MAX.. be the maximum percent of

service j that can be carried out by personnel i in the indirect super-

vision role. This coefficient will be used for those indices ieI' which

refer to physician extenders. The upper bound is thus:

xij. MAXij dj V jeJ, icP (3.40)

Note that it is not necessary to include the V variable here since if

V = 0 constraint (3.29) and the objective function will force x.i = 0
m ij
in an optimal solution. This point is important since it means that

(3.40) is an upper bounded variable which adds very little to the solu-

tion difficulty.

The other three model refinements involve the cost structure and

the first to be discussed is the fee structure. Even though HMO's will

operate on a capitation basis a small fee for service may be collected.

This fee should be small so that sick patients are not discouraged from

seeking health care thereby subverting one of the prime reasons for pre-

paid medical care. This fee is sometimes charged to discourage the use

of a physician's time by patients who are basically not ill. The co-

payment is also used as a marketing device to decrease the fixed capita-

tion rate and thus make the plan more attractive when compared to other

insurance plans which may have a lower fixed cost but less comprehensive

coverage. The fee is frequently $1 or $2 per office visit and if the

fee for service j is denoted r., then the income from fees is


Sj CJ bijxij
j -

A further refinement can be made to the cost of equipment. There

are certain pieces of equipment that change the fundamental nature of

the delivery of a health care service. Examples of this are multiphasic

testing units for use in comprehensive physical exams and SMA-12 blood

chemical analyzers for use in laboratories. Both of these examples also

have the property of being extremely expensive. Since both large cost

and alternative health care delivery patterns could have a major effect

on the solution, this type of equipment will be separated from the

remainder of the medical equipment used. Technology cost was used by

Shuman (76) by introducing a fourth subscript onto all the problem

variables. This is not only cumbersome but also adds considerably to

the number of variables. In addition several levels of technology for

an entire O10 cannot be analyzed in a practical sense. The data not

only do not exist; it would be difficult and very costly to design an

experiment under which the data could be collected. Instead it is pos-

sible to evaluate the effect of specific technological advances on the

productivity of specific personnel providing specific services. Define

the terms:

e. initial cost for a type i person due to technology cost


e! amortized value for e (dollars per year),
1 i
I' set of personnel whose productivity is enhanced by additional


T. one if Ni > 0 and 0 if N. = 0 for iTe'

Then the technology cost is given by

S e.T.
iel' 1 1

and should be added to the capitalization constraint to give

g E w.N. + t FTE + S e.T < Y* (3.42)
Si m m m i-
i m iel'

In addition, to ensure the zero-one variable T. takes on its proper

value, the following constraint is necessary

N. K T. < 0, where K >> N. (3.43)
1 1 --

Several points should be made before proceeding. First note that the

set I' can include those personnel who do not need the additional tech-

nology cost to be effective but rather technology just revises their pro-

ductivity coefficient. It can also include those who would not be hired

if the technology is not employed such as technicians hired to run a

multiphasic testing unit. The second point is that the above term is

not central to the solution of the model and can be dropped if tech-

nology changes are not being considered in the decision process.

The last addition to the model involves an operating constraint for

HMO's. HMO's provide medical care on a capitation basis and must operate

within a budget defined by the capitation income plus any external

sources of revenue. Thus the yearly operating expenditures must be less

than the budgeted amount to cover cost of operation. The relation

between the budgeted amount and the capitation is developed first and

then the budgetary constraint is developed. The budget is developed for

a non-profit institution. For a profit making institution appropriate

changes can easily be made. Define the coefficients:

B* -upper limit on yearly budget for operating and salary expenses

for medical departments included in the optimization (dollars

per year),

OM other medical costs; this would include the costs of operating

departments which are not included in the optimization,

inpatient care, out of area professional services, additional

cost of premium plans, etc. (dollars),

P rate at which capital fund accumulates as a fraction of gross

income from subscriber capitation fees,

R capitation rate (dollars per subscriber per year),

XT external sources of revenue such as planning grants or

endowments (dollars per year).

ince the initial capitalization, Y* is being amortized at the rate

P the result is


Now s

B* = (1 P) S R P Y* OM + EXT


Now. a budgetary constraint can be written for which B* is fixed which

implies the capitation rate is fixed, or B* can be defined as a variable

which in turn will define the capitation rate. The former alternative

will be used here recognizing it is trivial to study a variable capita-

tion rate if it is later desired. The terms given in (3.41), (3.44) and

the yearly expenses from the objective function (3.39) can be combined to

give the budget constraint:

S E (b..c..) x + sx + o E w.N. E r (S b x.)
j i 1 c i j I ij ij

+ S (CLT NLT + CXR NXR ) ES b..x..
Sm m jeJ ielm 13 1

+ (PV S S b x.i + MAN (d. V n )/b
m jejm iclI jeJm j m j

+ PER E N.) < B* (3.45)

The entire model will be called M3 and can now be summarized here. The

objective was given by (3.39) and must be revised to reflect the income

fee and amortized technology cost. In addition AD and FTE will be
m m
replaced by their definitions given by (3.32) and (3.31) respectively.

The result is the model M3:

min g' w.N. + SE (b..c ) xi. + S s.x' + E (1 V ) S d.u.
i1 j 1i i m m jeJm j

+ S u.n. + S t' ( E (dVm n.)/bt) + o Z w.N. E r. (E b. x..)
j J m jcJ c i J i

+ e eT T. + E (CLT NLT + CXR NXR ) S E S b .x..
iel' m m jem m 13 13

+ S (PV E m b..x. + MAN S (d.V n.)/b + PER Z N.)
m jem iIm 13 jem jm n ieI


subject to

Sxij + Z E qkijkj + x' N. = 0
j1 kj

V iC ,

V jeJ, j#j* ,

Z b..x.. + n. d.V = 0
i13 3 m

S E f ib x b x <0 V i*dl* ,
J iJi* iCl* ij j

SE (b..c ) x + s + o S w.N. E r. ( b..x..)
j iJ i 1 1 ci j 1 i i 1 11

+ S (CLT NLT + CXR NXR ) 5 Z b ix.
m m m jeJm islm 13 13
m jejm icim

+ (PV S b..x.. +MAN S (d.V n.)/b*
m jeJm icm 13 jej m J j

+ PER E Ni) < B*

g E w.N + E t E (d.V n.)/bt + e.T. < Y*
i1 m jcJm m i i < Y*






Ti K N. 0 ieI' (3.52)

xi*j* SUi Ni V i*e* (3.53)

n. < MX. d. V jeJ, (3.54)


Thus the problem is to find the xij, N., T. and V to minimize (3.46)

subject to (3.47) (3.55). Note chat the optimal basis will always be

such that constraint (3.49) is tight. This is because the xi*j. in

(3.49) will always have a positive cost; thus if a slack variable from

(3.49) appeared in the final basis, the objective function could

obviously be reduced by reducing the slack variable to zero, thereby

reducing the x ,.* and thus its contribution to the objective function.

This means that (3.493) could be used to limit xij, from the fcrmula-

tion and consequently eliminate (3.49). However, this would change

(3.53) to a constraint rather than an upper bound and thus would accom-

plish little in addition to the notational problems which would arise.

Thus (3.49) will be retained in the formulation.

A Comparison of Models Ml, M2,and M3

The formulation has been developed in three stages through models

M1, M2 and M3. It is instructive at this point to compare the features

and sizes of the three models. The features of the three models are:

M1 very simple model of the basic resource and requirements


M2 generalized and detailed refinement of the medical care

structure presented in Ml; and

M3 generalized and detailed description of the medical cost

structure in addition to the medical care refinements of

model M2.

Let the cardinality of the sets I, Is, I', and J and M be demoted by 0(I),

0(IS), 0(I'), 0(J), and 0(M). Also assume that the matrix of xij is

about 20 percent dense; about 50 percent of the personnel classes are

required to be integer; and that O(I') is about a fourth the 0(IS).

Then, if for example, there were 10 personnel classes, 15 teams, 60

services and 3 departments the size of the models would be as follows

(strictly speaking Ml would have to be developed slightly to allow


Table 2 Relative Sizes of Models Ml, M2 and M3

model Continuous Integer Zero-One Cntraints Upper
Variables Variables Variables Bounds

Ml 300 0 0 70 0
M2 370 5 0 73 63
M3 370 5 5 75 123

In solving linear programs, the number of constraints is Lhe major

factor in the time and difficulty of the solution. Upper bounds can be

added to a model with very little extra solution time and they do not re-

quire the addition of constraints. Also, the number of problem variables

only slightly affects the solution speed. Thus in going from a very

simplistic model to a much more detailed model the refinements were prin-

cipally made in such a way that the difficulty of the model changed very

slightly. Note also that in interpreting solutions to linear programs

the principal quantities of interest are the number of variables and

number of constraints. Here again very little additional difficulty is

encountered in the interpretation of the results.

The exception to the above paragraph is the integer and zero-one

variables. Solving a mixed integer linear program is more difficult

than solving a linear program. However, in this case the problem is

well within the solution range of mixed integer programming techniques.

Additionally, the zero-one variables could be set at an assumed value

as was implicit in Ml and M2 and the integer restriction on all Ni could

be dropped which would convert M3 into a linear program of about the

same difficulty of Ml or M2, but M3 would remain a much more fertile

model for decision making. The major caveat would then be the addi-

tional data requirements, but for an HMO with an adequate accounting

system the extra data would essentially be available. Thus model M3 will

be used for further development and analysis in the remainder of the


Development of Planning Models

Development of the Overall Planning Model

The M3 model will be algebraically simplified to define the Overall

Planning Model (OPM). OPM can be used in a preliminary planning stage

of an IMO or with some simplifications it can be used to refine operation

of an MO0. The model is formulated to solve the following basic problem:

given a fixed capitation rate and a projected subscriber

base find

(1) manpower to be hired,

(2) delegation of services,

(3) facilities required,

(4) particular technological innovations to be

utilized (if any).

The M3 model has many terms and coefficients that can be combined. Thus

the model can be simplified by defining:

a i = (c, rj + CLT NLT + CXR NXR + V) b.. (3.56)
ijm i J m m 1

OT = (MAN + t') d./b4 d.u. (3.57)
Sm j j

P =MAN d./b (3.58)

0* = t d/b (3.59)
jm m j 3

h! = w1 (g' + Oc) + PER (3.60)

h. = o w. + PER (3.61)

h = gw (3.62)

ym = j (t' + MAN)/b4 (3.63)

y. = MAN/bt (3.64)

m = t /b (3.65)

ADD = E d.u.. (3.66)

Substitute (3.56) (3.66) into (3.46) (3.55) and the resulting formu-

lation is the OPM (note the constant term, ADD, is being carried with

the formulation):

min S E E a.. x. + E S V + S h;N
m jjm iem ijm j1 m jeJm j m icI i

+ E E y. n. + E s.x' + S e!T. + ADD
m jejm jm J iI s i 1 iEI' 1 i

subject to

x + + qkjx + x' Ni = 0 V iIs
j kj

S f ib ijxi bi x = 0 V i*Il* ,
jxJi* ie i v *l*

xij < SU.N. V iI* ,
-~" 11

E bijxij+ n. d V = 0
i 1313 3

n. < MX. d.
3j 3 3

V jeJ meM ,

V jeJm, V meM ,

xij MAX.. d V jej, iIp ,

ZS Z a. x. + Z h.N. +S S V
m jcm ilm ijm ies 1 1 m ejJm 3mm

+ E s s + Yjj < B* ,
ici sixi j -

S hN. + E e.T. + Z E V + E E y n < Y*,
iel 1s iCI m j mm m jeJm m j -

N. K T. < 0
1 1 -

V il' .











Additionally N. is integer, V and T. are zero-one, and all x i, x' and
1 m I ij 1
n. > 0.
The OPM will be used extensively in the remaining analysis. It is

the basic model from which other models will be derived and it will also

be used extensively to arrive at computational results. One of the prin-

cipal difficulties for solution of the ORI is the mixed integer feature

of the model. However, there are many instances where the model could

be linearized. There are many HMO's where the decision to utilize a

certain technology level and provide certain medical departments may be

made for political, legislative, or intangible reasons. In this case

the zero-one variables would be removed from the model. There are also

many HMO's that effectively do not reflect an integer manpower constraint

and it will be shown in later results that assuming continuous manpower

is a reasonable assumption in many cases. Thus often the integer restric-

tions can be removed entirely from the model. However, they are in the

formulation if needed and this in turn allows the effects of removing

these restrictions to be analyzed.

Minimum Cost Model for Fixed Services

It was noted that the OPM could be made computationally easier if

the zero-one variables V which determined which departments would be
provided are eliminated. In the OPM departments that are economically

infeasible to offer would be reflected by zero values for V and for the

corresponding slack variables, n., which would appear in the basis

(making it degenerate). The basis size could be made smaller and many

of the zero-one variables eliminated if it is decided a priori which

services are to be offered. Aside from making the model more amenable,

fixing the services to be provided is essentially the approach now being

proposed in the legislative programs. Thus another question which

arises is:

given a fixed set of medical departments and a fixed

subscriber base, find the optimal staffing, delegation,

and facility to minimize the subscriber fee.

For this formulation define the set:

M' = m : V = 11, (3.77)

J' = j : jeJ meM'} j* (3.78)

and also let

E= E
m meM'

2 = E
j jeJ'

Recall also from (3.44) that B* is a function of R and Y*. Note that in

this model and in the OPM one could treat Y* as a variable to find the

optimal initial capital outlay. From a best solution standpoint if con-

straint (3.75) is active then it implies that the limit on original

capital is actually acting to increase the yearly cost of operating an

M0O. This will never happen if Y* is treated as a decision variable.

Note also that if all demands are not met the Z u.n. appears in the
yearly budget constraint (3.86). These services are paid for by the HMO

and provided through another source. Thus the Minimum Cost (MC) model

is given by (3.79) (3.88) and involves finding the x., x, n, N. T.

and R (also possibly Y*) to

min R (3.79)

subject to

Sx.ij + E qkijxij + x! Ni = 0 V isl* (3.80)
S kj 3 1 1

S E f .bijxij b xij* = 0 V i*cI* (3.81)
jeJi* iel* ii ij* i

ij* SUi Ni ieI* (3.82)

E b.ix.j + n. = d. jcJ', j#j* (3.83)

n. MX. d. t jEJ', j j* (3.84)

x..< MAX. d. F jej, iel (3.85)

EZ Z Z a.. x.. + E s.x5 + Z hN. + S (y. + u.) n.
m jcJ ilm njm iI s i I ie s i i

(1 P) S R < P Y* OM S 0Z (3.86)
c nm jm
m jeJm

E htNi + E e.T. + 2 yZ n < Y* S m Z (3.87).
iIs 1 1 jeJm j m jej j

N. K T. < 0 V iel' (3.88)
1 1 -

Additionally, N. is integer, T. is zero-one and xi., x!, n. and R> 0.

Making the same assumptions as were previously made regarding the density

of the x.. matrix this problem has the following characteristics:

(a) .2 0(I) 0(J') + 0(I) + O(J' + +0(M') continuous variables,

(b) .5 0(Is) integer variables,

(c) .2 0(Is) zero-one variables,

(d) 1.5 0(Is) + 0(J') + 2 constraints, and

(e) 2 0(J') + 0(1*) upper bounds.

Thus with 10 personnel classes, 15 teams, 60 services to be provided,

and 3 departments this problem would have about 370 continuous variables,

5 integer variables, 2 zero-one variables, 75 constraints, and 123 upper

bounds. This is easier to solve than the OPM but the MC model is also a

mixed integer program.

In conclusion, this model is useful if the services to be provided

are dictated by political, local or legislative reasons. It can also

be used to formulate a reasonable set of services if the objective is to

minimize the capitation rate for a fixed set of services rather than

minimize the subscribers total medical bill as the OPM does, whereas

the MC will give the staffing, delegation and facilities required for

minimizing the cost to the subscribers for a given set of services.

A Special Case of the Minimum Cost Model

The MC model can be further simplified by using it in an operational

planning mode rather than the preliminary planning stage since it is

assumed the professional personnel and facility are already fixed. The

planning question at the operational stage can be stated as:

given a fixed set of services, professional staff and

facilities, what is the optimal subscriber size, delegation

and ancillary staff to minimize the subscriber fee.

For this case T. and some N. are fixed and the integer restrictions on
i 1
the remaining N. are dropped; thus the model is no longer a mixed integer
program. Constraint (3.80) becomes

E x + Z qkijxij = N. V iCI* (3.89)
j k j k

and the initial capital constraint (3.87) is replaced by a constraint on

the amount of floor space available for medical services,

S w.N. < Y (3.90)
i 1 1

and the T. definition constraint can be dropped from the formulation. A
problem arises with constraint (3.86) which now has a nonlinear term
problem arises with constraint (3.86) which now has a nonlinear term

S R since both S and R are now assumed variable. However, this non-

linearity can be eliminated by a double transformation. First define

Q = S R (3.91)

and then (3.79) becomes

min Q/S (3.92)

and (3.86) is again a linear constraint with Q replacing S R. The new

model thus consists of (3.80) (3.86), (3.90) and (3.92). This model

is a fractional linear program since for a physically reasonable solu-

tion to the model S > 0. A slight modification of the simplex is made

in the Charnes-Cooper Algorithm (82) to solve this problem. However,

the model can also be easily transformed to a linear program by the


r = 1/S (3.93)

and then define

S= rxii Q' = rQ ,

x' = rx S' = rS ,
1 i

n'. = rn N' = rN.. (3.94)
1 3 1 1

Now (3.93) and (3.94) are substituted into the linear fractional model

to give

min Q' (3.95)

subject to

E x!. + E E q .jx + x"' N.r = 0 V ieI* (3.96)
13 kj 13 1 1

S x'. + ES q .x: + x'' N! = 0 V ie* (3.97)
j 1 k j k 1 1

Z f b x' b x = 0 V i*cl* (3.98)
Ji* icI* ij i ij i*j* 1*j*

x,, SU'. N.r < 0 V ile* (3.99)
i*j I 1 -

S b.jx j + ni = 0 V jcJ', j#j* (3.100)
i j

n' MX. d S' < 0 V jeJ', jIj* (3.101)

x MAX.. d' S' < 0 V jej, iei (3.102)

C S S a,. x. + E s.x' + E h.N!
m jejm iclm ijm + iI s is i i

+ E h.N.r + E (y. + u.) n'. (1 P) Q'
iei* j

+ r (P Y* + OM + jm) < 0 (3.103)
m jeJ

E w.N: rY < 0 (3.104)
*1 1

S' = 1 (3.105)

Thus (3.95) (3.105) is a linear program and can be solved in the usual

manner. In practice it may be preferable to use the Charnes-Cooper

Algorithm on the linear fractional program rather than redefining all

the variables and requirements vector as was done above. Whichever model

is used, the result is a useful operational planning tool which is

especially useful if an HMO wants to change from the traditional MD-nurse

medical practice to the three level MD-PE-nurse medical practice. The

model defined by (3.95) (3.105) is a little lengthier to solve since

the upper bounded variables were replaced by generalized upper bounding

constraints. Although generalized upper bounding algorithms have been

formulated (e.g. Geoffrian (83)) they are not as fast as upper bounding

nor as available in packaged linear programming routines. However, the

generalized upper bounding constraints can be treated as standard con-

straints with the resulting larger basis.

Subscriber Maximization Model

One of the reasons HIO's are being proposed is because they poten-

tially will offer care to more subscribers per physician than the tradi-

tional form of practice thus providing better utilization of this scarce

manpower resource. Thus a valid objective would be to maximize the

number of subscribers an HMO can serve for a fixed professional staff

and subscriber fee. Again for this problem, the original capital Y*,

can be assumed to be an input or a variable. The problem can thus be

stated as:

given a fixed professional staff and subscriber fee,

find the optimal delegation, facility, and hiring

policy for allied personnel to maximize the number

of people who can be served.

This model is derived through modification of the OPM and involves find-

ing the x.., N., T. and S (possibly also Y*) to

max S (3.106)

subject to

Sx.ij + sE q kij + x! = N. V iel* (3.107)
j kj

E x.i + E qkijx + xt N. = 0
kj k j 1J 1 1

V iei* (3.108)

i i b x b x = 0 V i*1* (3.109)
Ji* j ij i i*j*

x ij* < SUi. Ni iel* (3.110)

S b..x. + n. d'S = 0 V jej', j#j* (3.111)
1J 1j 3 J

n. MX. d'S < 0 V jej', jij* (3.112)

xij MAX.i djS < 0 jeJ', j#j* (3.113)

E E S a. x. + s.x + E h.N. + Z (y + u.) n.
Sjjm im im ij 13 s i* 1 j 3

(1 P) S R+ SE B. S + P Y* < OM E h.N. (3.114)
m j6Jm jm c 1 1

Sh* N. + E e.T. + E S y n, + E S
ie 1 il' m jeJm n mj m jeJm jm

< Y* S hN.
i* 1 i (3.115)

N. K T. < 0 V iel' (3.116)
1 1 -

Note, for this model and 0* must be redefined in terms of d' rather

than d.. Constraint (3.107) refers to the professional level personnel

which are at a fixed level, whereas (3.108) refers to the ancillary per-

sonnel and N. is an integer variable. Note the term S u.n. in (3.114)
1 j J 3
and the use of n. as a slack variable in (3.111). This is somewhat of a

paradox since the purpose of the model is to maximize the number of sub-

scribers served by the HMO, but the above use of n. allows services to

be bought outside of the HMO. However, the n. should not be used as a

surplus variable in (3.111) and have the corresponding term in (3.114)

eliminated. To do this might unreasonably restrict the number of

subscribers. An example will clarify this point. Suppose only one

general surgeon is available and he can provide services for 10,000

people. However, the remainder of the staff can provide services for

their specialties for 30,000 people. The use of n. as a surplus vari-

able would lead to an answer of 10,000 subscribers, but using it as a

slack variable would lead to a solution of 30,000 subscribers with

two-thirds of the members being referred to a surgeon outside of the

HMO and paid for by the HMO. This latter solution is certainly more


Minimal Use of Professional Manpower

In an attempt to minimize the use of manpower for which a shortage

exists, an alternative approach to the SM model can be taken. It is

plausible that in some cases it will be known what subscriber base is

desired for geographical or political reasons. Thus the problem can

be summarized as:

given a fixed subscriber base, a fixed fee, and a fixed

set of services ,find the optimal staffing, delegation

and facility to minimize the professional level per-

sonnel used.

This minimization will naturally need to account for the quality of

service provided. In this case, it would be a major consideration and

the indirect supervision constraints will play an important part in

determining the plausibility of the final solution. This Minimal Pro-

fessional Manpower Model (MPIM) is given below and involves finding the

x..' x" n., N., Ti and Y* to

min Z Ni (3.117)

subject to:

S x.. + q + x' N. = 0 V iel (3.118)
j1 kj k1iij 1 1

S E fi..b ix bi xi = 0 1i*I* (3.119)
j 1Ji* 13 ii ij i*j* i*j*
jSJi* ieI*

Xi*j, < SU.i N. V iel* (3.120)

Sb.x.ij + n. = d. V jeJ', j#j* (3.121)

n. < MX. d. V jeJ', j#j* (3.122)

x. < MAX.. d. V jeJ, ielp (3.123)

Z Sm a.. x.. + h.N. + s.xI
m jc i m 13 1 is 1

+ E y n. + P Y* < (1 P) S R OM E jm (3.124)
j j c -m jeJm

Sh* N. + E e.T. + E S Y* n. Y* < S E (3.125)
ieis i 1 ieI' m jeim 3m m jem jm

N. K T. < 0 V iel'. (3.126)
1 i -

Additionally N. is integer, T. is zero-one and x.., x!, n., and
1 1 13 1 J
Y* > 0. The size of this problem, using the assumptions as for OPM, is:

(a) .2 0(I) 0(J') continuous variables,

(b) .5 0(I) integer variables,

(c) .2 0(I) zero-one variables,

(d) 1.5 0(I) + 0(J) + 2 constraints,

(e) 2 O(J') + 0(1*) upper bounds.

This formulation can also be used if the facility is fixed by making the

following changes:

(a) assureY* doesn't include paramedical equipment costs, Ti;

(b) in (3.124) add P E e!T. to the left hand side; and
c. 1 1i
(c) replace constraint (s.125) with E w.. < Y .
i -

Concluding Remarks

This chapter has presented a wide array of problem formulations

ranging in complexity from 1L to M3 and ranging in purpose from OPM to

MPMT. The internal validity of these models was partially developed

along with the development of the model in going from 1M to M3. Further

validation and solutions to the models will be covered in the remaining

chapters. The development of the complex model was carried out in such

a manner that, excluding the integer restrictions, 13 is of the same

order of difficulty to solve as the simplest model. The major difficulty

is the additional data requirements imposed by model M3; however, the

data needed to solve the mode are presented in the next chapter.

In summary, the models presented are mixed integer programs. One

of the models exhibited a quadratiz term but this was removed by a trans-

formation that resulted in a linear program. The models presented are

general and flexible enough to answer many HMO planning questions. A

partial list would include:

(a) assignment of personnel to medical services;

(b) selection of medical departments to offer;

(c) selection of needed facilities;

(d) optimal subscriber levels;

(e) optimal hiring policy;

(f) maximize subscribers per health professional;

(g) minimize the number of health professionals utilized;

(h) find the minimum cost structures for different base


(i) examine cost and manpower effects of P.E. utilization;

(j) examine P.E. utilization as.a function of subscriber

levels, P.E. salary, and supervisory levels;

(k) examine effect of integer manpower restrictions;

(1) examine imputed cost for medical services;

(m) examine different levels of responsibility for P.E.'s;

(n) examine which services are best delegated;

(o) examine usage of P.E.'s under a scarcity of P.E.'s; and

(p) examine which P.E. activities save MD's the most time.

These questions and others will be analyzed and at least partially

answered in the remaining chapters,

University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs