U.S. patent application number 16/442947 was filed with the patent office on 2020-10-22 for network design and optimization.
The applicant listed for this patent is Anthem, Inc.. Invention is credited to Ariel Bayewitz, Yue Geng, KuoLing Huang, Yan Jiang, Changhyeok Lee, Adarsh Ramesh, Shawn Wang.
Application Number | 20200334775 16/442947 |
Document ID | / |
Family ID | 1000004348302 |
Filed Date | 2020-10-22 |
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United States Patent
Application |
20200334775 |
Kind Code |
A1 |
Huang; KuoLing ; et
al. |
October 22, 2020 |
NETWORK DESIGN AND OPTIMIZATION
Abstract
A computer-implemented method for selecting one or more
providers having one or more specialties for a network available to
members while satisfying one or more constraints, the method
comprising: at a first computing device: receiving, from a second
computing device separate and distinct from the first computer, a
network provider request to select an optimized network of
providers, the request including: a desired objective including at
least one of provider cost minimization, provider average quality
maximization and provider total volume maximization, one or more
geographical designations, computing an optimized provider network
of one or more selected providers including: applying a model that
produces a non-optimized provider network of one or more providers
having one or more specialties that satisfies the desired objective
for the one or more geographical designations of the members
without consideration of the availability in the geographical
designation of one or more providers having one or more
specialties, and applying a linear decomposition to the
pre-optimized provider network to generate an optimized provider
network; providing the optimized provider network to the second
computing device.
Inventors: |
Huang; KuoLing; (Chicago,
IL) ; Geng; Yue; (Chicago, IL) ; Jiang;
Yan; (Wilmette, IL) ; Ramesh; Adarsh;
(Schaumburg, IL) ; Lee; Changhyeok; (Santa Clara,
CA) ; Bayewitz; Ariel; (Teaneck, NJ) ; Wang;
Shawn; (Kildeer, IL) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Anthem, Inc. |
Indianapolis |
IN |
US |
|
|
Family ID: |
1000004348302 |
Appl. No.: |
16/442947 |
Filed: |
June 17, 2019 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
62835286 |
Apr 17, 2019 |
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06Q 50/22 20130101;
G06F 17/12 20130101 |
International
Class: |
G06Q 50/22 20060101
G06Q050/22; G06F 17/12 20060101 G06F017/12 |
Claims
1. A computer-implemented method for selecting one or more
providers having one or more specialties for a network available to
members while satisfying one or more constraints, the method
comprising: at a first computing device: receiving, from a second
computing device separate and distinct from the first computer, a
network provider request to select an optimized network of
providers, the request including: a desired objective including at
least one of provider cost minimization, provider average quality
maximization and provider total volume maximization, one or more
geographical designations, computing an optimized provider network
of one or more selected providers including: applying a model that
produces a non-optimized provider network of one or more providers
having one or more specialties that satisfies the desired objective
for the one or more geographical designations of the members
without consideration of the availability in the geographical
designation of one or more providers having one or more
specialties, and applying a linear decomposition to the
pre-optimized provider network to generate an optimized provider
network; providing the optimized provider network to the second
computing device.
2. The method of claim 1, wherein without consideration of the
availability in the geographical designation of one or more
providers having one or more specialties includes forgoing
considering whether a ratio of total members in a particular
geographical designation have access to a particular specialty is
greater than a member coverage ratio threshold.
3. The method of claim 2, wherein without consideration of the
availability in the geographical designation of one or more
providers having one or more specialties includes forgoing
considering whether a single member in each of the geographical
designations will have access to each of the particular
specialties;
4. The method of claim 3, wherein without consideration of the
availability in the geographical designation of one or more
providers having one or more specialties includes for each
particular geographical designation, each particular provider
and/or each particular provider specialty, forgoing considering
whether all members in the particular geographical designation have
access to the particular provider specialty offered by the
particular provider;
5. The method of claim 1, wherein the average cost of the selected
providers is less than a predetermined provider cost threshold.
6. The method of claim 1, wherein the average volume of the
patients seen by the selected providers is greater than a
predetermined provider volume threshold.
7. The method of claim 1, wherein the average quality of the
selected providers is greater than a predetermined provider quality
threshold.
8. The method of claim 1, wherein the linear decomposition is
Benders decomposition.
9. The method of claim 1, wherein the model that produces a
non-optimized provider network is min c .about. T y .about. + s
.di-elect cons. S Q ~ s ( y ~ ) , s . t . y ~ .di-elect cons. ,
where ##EQU00025## Q ~ s ( y .about. ) : = max 0 T w . s , s . t .
w . s .di-elect cons. , 0 .ltoreq. w z s .ltoreq. 1 , .A-inverted.
z , ##EQU00025.2## wherein: {tilde over (c)} is a vector of
objective coefficients, {tilde over (y)} is a vector of binary
variables used to provide a non-optimized provider network, y
represents a feasible set for {tilde over (y)}, S represents a set
of specialties, {tilde over (Q)}.sub.s({tilde over (y)}) is a
recourse subproblem that represents the coverage ratio requirement
for each specialty, w.sub.zs is a vector of decision variables that
determines if members located at one of the geographical
designations have an access to specialty s considered in network, W
represents the feasible set for {tilde over (w)}, z is the one or
more geographical designations.
10. A system comprising: one or more memory units each operable to
store at least one program; and at least one processor
communicatively coupled to the one or more memory units, in which
the at least one program, when executed by the at least one
processor, causes the at least one processor to perform the steps
of claims 1-9.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit of U.S. Provisional
Patent Application No. 62/835,286 filed on Apr. 17, 2019 entitled
"NETWORK DESIGN AND OPTIMIZATION", which is incorporated by
reference in its entirety.
BACKGROUND
[0002] This application may provide details of a Network Design and
Optimization (NDO) project. The purpose of an NDO may be to create
an online solution platform that may help business clients
construct an effective healthcare provider network solution. The
NDO may be required to meet specific business requirements such as
product scope, specialty required, minimum retained volume, network
coverage ratio and cost/quality baseline. More importantly, it may
be cost-effective and the solution time may be limited to 10
minutes. To address these challenging business requirements, a
fractional mixed-integer linear optimization model, accompanying
with several computational enhancements that speedup the solution
time without compromising the solution quality as a bid may be
presented herein. Computational results on 396 problems drawn from
different product scopes using real life data may show that the
proposed solution needs only 35 seconds of computations on average
to find the optimal solution. The solution described herein may
improve the cost-saving target by 16% on average, when compared
with a current baseline solution.
[0003] Depending on the product scope and other business
requirements, the created optimization model may involve 100,000
binary decision variables and model constraints, which undoubtedly
complicates the model complexity. In our computational experience,
leading commercial optimization software packages such as Gurobi
may require an hour of computations to obtain the optimized network
solution. Unfortunately, such excessive solution time must be
prohibited, as the purpose of NDO is to provide a network solution
in a real-time manner (typically within a 10-minute time limit). In
lieu of this obstacle, additional computational techniques that
successfully speed up the solution process without compromising the
solution quality as a bid may be presented herein.
[0004] The invention is to create an online solution platform for
business users to create a customized healthcare provider network
solution that satisfies the business requirements such as product
scope, specialty required, minimum retained volume, network
coverage ratio and cost/quality baseline. At the core of the
solution is an optimization model that is enhanced with our
in-house computational innovations to speed up the solution
process. Computational results on a large number of problems drawn
from different product scopes using real life data clearly
demonstrate the efficiency and effectiveness of our solution
approach.
SUMMARY
[0005] In one embodiment there is a computer-implemented method for
selecting one or more providers having one or more specialties for
a network available to members while satisfying one or more
constraints, the method comprising: at a first computing device:
receiving, from a second computing device separate and distinct
from the first computer, a network provider request to select an
optimized network of providers, the request including: a desired
objective including at least one of provider cost minimization,
provider average quality maximization and provider total volume
maximization, one or more geographical designations, computing an
optimized provider network of one or more selected providers
including: applying a model that produces a non-optimized provider
network of one or more providers having one or more specialties
that satisfies the desired objective for the one or more
geographical designations of the members without consideration of
the availability in the geographical designation of one or more
providers having one or more specialties, and applying a linear
decomposition to the pre-optimized provider network to generate an
optimized provider network; providing the optimized provider
network to the second computing device.
[0006] In one embodiment, wherein without consideration of the
availability in the geographical designation of one or more
providers having one or more specialties includes forgoing
considering whether a ratio of total members in a particular
geographical designation have access to a particular specialty is
greater than a member coverage ratio threshold.
[0007] In one embodiment, wherein without consideration of the
availability in the geographical designation of one or more
providers having one or more specialties includes forgoing
considering whether a single member in each of the geographical
designations will have access to each of the particular
specialties.
[0008] In one embodiment, wherein without consideration of the
availability in the geographical designation of one or more
providers having one or more specialties includes for each
particular geographical designation, each particular provider
and/or each particular provider specialty, forgoing considering
whether all members in the particular geographical designation have
access to the particular provider specialty offered by the
particular provider;
[0009] In one embodiment, the average cost of the selected
providers is less than a predetermined provider cost threshold.
[0010] In one embodiment, the average volume of the selected
providers is greater than a predetermined provider volume
threshold.
[0011] In one embodiment, the average quality of the selected
providers is greater than a predetermined provider quality
threshold.
[0012] In one embodiment, the linear decomposition is bender's
decomposition.
[0013] In one embodiment, one or more memory units each operable to
store at least one program; and at least one processor
communicatively coupled to the one or more memory units, in which
the at least one program, when executed by the at least one
processor, causes the at least one processor to perform the steps
specified above.
BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS
[0014] The foregoing summary, as well as the following detailed
description of aspects of the disclosed invention, will be better
understood when read in conjunction with the appended drawings of
an exemplary aspect. It should be understood, however, that the
invention is not limited to the precise arrangements and
instrumentalities shown. In the drawings:
[0015] FIG. 1 illustrates a block diagram that illustrates an
example system optimizing network design according to at least one
aspect of the present invention;
[0016] FIG. 2 is an exemplary illustration of a business objective
and optimization according to at least one aspect of the present
invention;
[0017] FIG. 3 is an exemplary illustration of an algorithm
according to at least one aspect of the present invention;
[0018] FIG. 4 is an exemplary illustration of a graph of
performance profiles comparing solution time performances of the
basic model and the enhanced model according to at least one aspect
of the present invention;
[0019] FIG. 5A is an exemplary illustration of network
optimization, according to at least one aspect of the present
invention;
[0020] FIG. 5B is an exemplary illustration of network
optimization, according to at least one aspect of the present
invention; and
[0021] FIG. 6 is an exemplary illustration of a flow chart for
recommending one or more providers to a user according to at least
one aspect of the present invention.
DETAILED DESCRIPTION
[0022] In some aspects, methods and systems described herein
provide the technical details for a Network Design and Optimization
(NDO) project. The purpose of NDO may be to provide an online
solution platform that may help business clients create a cost and
quality effective healthcare provider network in a real time
manner. Included in this network are a set of providers that not
only meet the business requirements such as product scope,
specialty required, minimum retained volume, network coverage
ratio, cost/quality baseline; more importantly, they have to be
cost-effective. As such, an optimization model as described herein
was developed to, among other things, solve this specific provider
selection problem.
[0023] The disclosure described herein may describe: [0024] 1. An
optimization model in Section 1.1 for NDO. [0025] 2. How this model
may improve the cost-saving by 16% on average as shown in Section
2.1. [0026] 3. It may be shown in Section 1.4 that the proposed
optimization model may have a special dual block-angular structure,
for which the integrality requirements and some constraints
associated with each block may be eliminated safely. Benders
decomposition may be applied for solving this dual block-angular
structure problem. In one embodiment, it may be shown that each sub
problem can be solved in closed-form to generate the Benders cuts.
This may further simplify the model complexity and thus potentially
improve the solution time. These enhancements may speed up the
solution time by almost 14 folds.
[0027] Solution time and solution quality may be equally important
since, as described herein, one of the goals of the system may be
to deliver an effective provider network solution efficiently.
[0028] FIG. 1 shows a block diagram that illustrates a system 100
for improving personalized provider search according to at least
one aspect of the present invention. While some example features
are illustrated, various other features have not been illustrated
for the sake of brevity and so as not to obscure pertinent aspects
of the example aspects disclosed herein. To that end, in at least
one aspect, the system 100 may include one or more computers or
servers, non-transitory memory operable to store one or more
computer programs and one or more processors to implement the one
or more computer programs. For example, the system 100, shown in
FIG. 1, may include client device 110, server device 120 and
network 130.
[0029] Client device 110 may be a computing device for receiving
inputs from a user (e.g., a member), requesting data from server
device 120 via network 130 and/or displaying data from service
device 120 at the request of a user. Examples of a client device
110 may include a smart phone, tablet or a personal computer, among
others.
[0030] Server device 120 may be any computing device, including one
or more software modules (e.g., a scoring module) for receiving
and/or responding to requests for data from client device 110.
Examples of data may include web page data, hypertext markup
language (HTML), text, video, audio as a free form speech
describing symptoms and conditions, picture, software, executable,
interpretable, byte-code, and binary files. In some aspects, the
server device 120 may be a plurality of computing devices that
process the request from the client device 110. The server device
120 may be configured to process requests from other computing
devices in parallel with the request from the client device
110.
[0031] In one aspect, server device 120 is a web server that hosts
a website. Client device 110 may be configured to request provider
recommendations from server device 120 based on a hypertext
transfer protocol (HTTP). Server device 120 may respond to a
provider recommendation request by sending provider recommendation
data (e.g., an ordered list of providers) to client device 110. In
one aspect, provider recommendation data may include web page data
included on an HTML web page. While the server device 120 may be
configured for HTTP/HTML requests and responses, as described in
the exemplary aspect above, the system 100 is not limited to the
use of HTML or HTTP, and that aspects of the present invention can
be used with any computer communication language or network
protocol suitable for the purposes of the described communications
between client device 110 and server device 120.
[0032] Client device 110 may include communication infrastructure
111, processor 112, memory 113, user interface 114 and
communication interface 115. Server device 120 may include
communication infrastructure 121, processor 122, memory 123 and
communication interface 125.
[0033] Processor 112 or processor 122 may be any type of processor,
including but not limited to a special purpose digital signal
processor. Processor 112 is connected to a communication
infrastructure 111 (for example, a bus or network). Processor 112
is connected to a communication infrastructure 121 (for example, a
bus or network). Various software implementations are described in
terms of this exemplary computer system.
[0034] Memory 113 or memory 123 may include one or more of random
access memory (RAM), a hard disk drive and a removable storage
drive, such as a floppy disk drive, a magnetic tape drive, or an
optical disk drive, etc. The removable storage drive may read from
and/or writes to a removable storage unit. The removable storage
unit can be a floppy disk, a magnetic tape, an optical disk, etc.,
which is read by and written to a removable storage drive. Memory
113 and/or memory 123 may include a computer usable storage medium
having stored therein computer software programs and/or data to
perform any of the computing functions of client device 110 and/or
server 120. Computer software programs (also called computer
control logic), when executed, enable client device 110 and/or
server 120 to implement aspects of the present invention as
discussed herein. Accordingly, such computer software programs
represent controllers of client device 110 and/or server 120.
Memory 123 may include one or more data stores that store data such
as web page data, software files or any other types of data files.
Server device 120 may retrieve the data from memory 123 before
transmitting to client device 110 via network 130. Memory 123 may
include member characteristics, provider characteristics,
member-provider interaction characteristics, feature bias
weightings, member/provider bias weightings, and learnt weightings,
among other described herein.
[0035] User interface 114 may be produced by a program that
controls a display (not shown) of client device 110. User interface
114 may include one or more peripheral user interface components,
such as a keyboard or a mouse. The user may use the peripheral user
interface components to interact with client device 110. User
interface 114 may receive user inputs, such as mouse inputs or
keyboard inputs from the mouse or keyboard user interface
components. User interface 114 may display data, such as web pages,
on the display of client device 110 using a web browser. While the
user interface 114 may be configured for displaying data using a
web browser, as described in the exemplary aspect above, user
interface 114 is not limited to displaying data using a web
browser, and that aspects of the present invention may contemplate
using other display devices or software suitable for the purposes
of the displaying the data.
[0036] Communication interface 115 and/or communication interface
125 allow data to be transferred between client device 110 and
server device 120 via network 130. Examples of communication
interface 115 or communication interface 125 may include a modem, a
network interface (such as an Ethernet card), a communication port,
a Personal Computer Memory Card International Association (PCMCIA)
slot and card, etc. Data transferred via communication interface
115 or communication interface 125 are in the form of signals,
which may be electronic, electromagnetic, optical, or other signals
capable of being transmitted or received by communication
interface.
[0037] Network 130 connects client device 110 and server device 120
by carrying signals. Network 130 may be implemented using wire or
cable, fiber optics, a phone line, a wireless link, a cellular
phone link, a radio frequency link, or any other suitable
communication channel. For instance, network 130 may be implemented
using a combination of channels. Network 130 may be implemented as
an intranet and/or an internet.
[0038] Referring to FIG. 2, there is shown the business objective
200 in accordance with an exemplary embodiment of the present
invention.
[0039] A method of implementing the optimization model, as
described herein, may be:
[0040] Step 1. User inputs requirements through solution platform
webpage.
[0041] Step 2. Upon completion, user submits the problem to backend
computation engine.
[0042] Step 3. Backend computation engine performs the computations
and returns the results.
[0043] Step 4. If a valid network solution is available, solution
platform reports the network solution performance, as well as
sensitivity analysis around the current solution.
[0044] Step 5. If a valid network solution is not available,
solution platform reports the root-cause that results in the model
infeasibility.
[0045] Step 6. Users fix the input based on the infeasibility
report and resubmit the problem.
[0046] Further, as described herein, an optimization engine has
been developed that systematically and automatically creates a cost
and quality effective healthcare provider network in a real time
manner. Included into this network may be a set of providers that
not only meet the business requirements such as product scope,
specialty required, minimum retained volume, network coverage
ratio, cost/quality baseline; more importantly, they have to be
cost-effective.
[0047] Our model equipped with our in house enhancements allows
optimization software package to explore astronomical number of
possibilities (trillion number of solutions is very common in
optimization) and find out the best one in minutes, if not seconds.
Without optimization, such network solutions must be created
manually and the solution optimality cannot be guaranteed.
[0048] Our online platform solution provides a very clean
interactive user interface, which helps business users input the
model parameters and create the network solution without the need
of any technical background. Furthermore, it provides additional
diagnostics that help users to construct valid network solutions
easily, as well as other post-analysis tools for users to profile
the details of the constructed solution clearly.
[0049] Healthcare insurance companies might likely use this
invention when they try to construct an optimized healthcare
provider network.
[0050] I. Network Design and Optimization
[0051] Details of the proposed optimization model for NDO are
presented herein. The basic model formulation will be described. It
is then shown that the model has an equivalent but simplified
format with improved model complexity. Finally, Benders
decomposition on this enhanced model is applied, and further shown
that each decomposed subproblem may be solved in closed-form.
1.1 Notation Set
[0052] S: A set of specialties.
[0053] S.sub.g.OR right.S: A set of specialties with geo-access
requirements.
[0054] G: A set of provider groups.
[0055] G.sub.c.OR right.G: A set of provider groups with scorable
cost index.
[0056] G.sub.q.OR right.G: A set of provider groups with scorable
quality index.
[0057] Z: A set of member zip codes.
Sets Related to Manual Override
[0058] G.rho.: A set of providers with PCP.
[0059] G.sub.S: A set of providers with multi-specialty.
[0060] G.sub.I: A set of providers that must be included into the
network. Users are allowed to have G.rho..OR right.G.sub.1,
G.sub.S.OR right.G.sub.I, or both.
[0061] G.epsilon.: A set of providers that must be excluded into
the network.
[0062] Gx:=G.sub.I.orgate.G.epsilon.. Without loss of generality,
we assume G.sub.I.andgate.G.epsilon.={O}. Data
[0063] v.sub.g: Overall episode volume for provider group
g.epsilon.G.sub.c. Note that v.sub.g.gtoreq.1 for all g.di-elect
cons.G.sub.c.
[0064] v.sub.g,s: Episode volume for provider group g.di-elect
cons.G.sub.c at specialty level s.di-elect cons.S. If provider g
does not provide specialty s, then let v.sub.g,s:=0. Note that
.gamma..sub.s.di-elect cons..sub.sv.sub.g,s.ltoreq.v.sub.g (the
equality happens only if v.sub.g,s is sufficiently large).
Moreover, v.sub.g,s.gtoreq.1 for all g.di-elect cons.G.sub.c and
s.di-elect cons.S.
[0065] c.sub.g, c.sub.g, c.sub.g: Lower bound, average, and upper
bound value of the cost index for provider group g.di-elect
cons.G.sub.c.
[0066] q.sub.g, q.sub.g, q.sub.g: Lower bound, average, and upper
bound value of the quality index for provider group g.di-elect
cons.G.sub.q.
[0067] m.sub.z: Number of members in zip code z.di-elect
cons.Z.
[0068] a(g, s): The zip code of provider g with specialty s.
[0069] d(z, a(g, s)): Distance between a member zip code z and a
provider-specialty's zip code a(g, s).
User-Defined Parameters
[0070] c.sup.b: Maximum cost for a provider group to be considered
in network.
[0071] q.sup.b: Minimum quality allowed for a provider group to be
considered in network.
[0072] c*: Maximum average cost requirement for the entire
network.
[0073] q*: Minimum average quality requirement the entire
network.
[0074] d*.sub.s: Maximum distance that defines the member
accessibility (in close proximity) to specialty s.
[0075] m*.sub.s: Minimum percentage of members who have access to
specialty s in close proximity (coverage ratio).
[0076] v*: Minimum percentage of the volume requirement.
[0077] v*.sub.s: Minimum percentage of the volume requirement at
specialty level s.
Indicator
[0078] b.sub.zgs: Let b.sub.zgs:=1 if a(g, s) exists and d(z, a(g,
s)).ltoreq.d*.sub.s, or 0 otherwise.
Decision Variables
[0079] y.sub.g.di-elect cons.: Given provider g.di-elect cons.G,
let y.sub.g:=1 if g is considered in network, or 0 otherwise.
[0080] w.sub.zs.di-elect cons.: Given member zip code z.di-elect
cons.Z and a specialty s.di-elect cons.S.sub.g with a geo-access
requirement, let w.sub.zs:=1 if members located at zip code z have
an access to specialty s considered in network, or 0 otherwise.
Objective Functions
[0081] Depending on the user's preference, three different
objective functions may be used for optimization.
[0082] Average Cost Minimization:
min g .di-elect cons. G c v g c g y g g .di-elect cons. G c v g y g
. ( 1 ) ##EQU00001##
[0083] Average Quality Maximization:
max g .di-elect cons. G q v g q g y g g .di-elect cons. G q v g y g
. ( 2 ) ##EQU00002##
[0084] Total Volume Maximization:
max g .di-elect cons. G q = v g y g . ( 3 ) ##EQU00003##
[0085] Constraints
[0086] Each provider in G.sub.I (resp. G.sub..epsilon.) must be
included in (resp. excluded from) network.
y.sub.g=1,.A-inverted..sub.g.di-elect cons.G.sub.I. (4)
y.sub.g=0,.A-inverted..sub.g.di-elect cons.G.sub..epsilon.. (4)
[0087] Each provider group may satisfy the maximum cost index
constraint to be considered in network. Depending on the user's
preference, the cost index may be chosen from its lower bound
value, average value, or upper bound value.
c.sub.gy.sub.g.ltoreq.c.sup.b,.A-inverted.g.di-elect
cons.G.sub.c\G.sub.x. (6)
[0088] Each provider group may satisfy the minimum quality index
constraint to be considered in network. Depending on the user's
preference, the quality index may be chosen from its lower bound
value, average value, or upper bound value.
q.sub.gy.sub.g.ltoreq.q.sup.b,.A-inverted.g.di-elect
cons.G.sub.g\G.sub.x. (7)
[0089] The network may satisfy the minimum overall volume ratio
requirement.
g .di-elect cons. G c v g y g .gtoreq. v * g .di-elect cons. G c v
g . ( 8 ) ##EQU00004##
[0090] The network may satisfy the minimum volume ratio requirement
at specialty level.
g .di-elect cons. G c v g , s y g .gtoreq. v s * g .di-elect cons.
G c v g , s , .A-inverted. s .di-elect cons. S . ( 9 )
##EQU00005##
[0091] The network may satisfy the maximum average cost
requirement.
g .di-elect cons. G c v g c g y g .ltoreq. c * g .di-elect cons. G
c v g c g . ( 10 ) ##EQU00006##
[0092] The network may satisfy the minimum average cost
requirement.
g .di-elect cons. G c v g q g y g .gtoreq. c * g .di-elect cons. G
c v g q g . ( 11 ) ##EQU00007##
[0093] Each specialty may satisfy minimum coverage ratio.
z .di-elect cons. Z m z w zs .gtoreq. m s * z .di-elect cons. Z m z
, .A-inverted. s .di-elect cons. S g . ( 12 ) ##EQU00008##
[0094] Determine if members in each zip code may have coverage for
each specialty.
g .di-elect cons. G b zgs y g .gtoreq. w zs , .A-inverted. z
.di-elect cons. Z , s .di-elect cons. S g . ( 13 ) b zgs y g
.ltoreq. b zgs w zs , .A-inverted. z .di-elect cons. Z , s
.di-elect cons. S g , g .di-elect cons. G . ( 14 ) ##EQU00009##
[0095] 1.2 Linearization of the Objective Function
[0096] The average cost objective function (1) and the average
quality objective function (2) are not linear, but they both may be
be linearized with additional variable substitution and big-M
reformulation. In the following the details for linearizing (1) are
presented. (2) can be linearized in a similar manner.
[0097] Let x.sup.0:=1/(.SIGMA..sub.g.di-elect
cons.G.sub.ev.sub.gy.sub.g) and x.sub.g: y.sub.gx.sub.0,
.A-inverted.g.di-elect cons.Gc. Moreover, let
v.sub.min:=min.sub.g.di-elect cons.G.sub.cv.sub.g) Note that
v.sub.min.gtoreq.1. Now, objective function (1) can be rewritten
as
min g .di-elect cons. G c c g v g x g g .di-elect cons. G c v g , (
15 ) ##EQU00010##
with the following additional constraints:
[0098] Make sure x.sub.0 is well-defined.
g .di-elect cons. G c .upsilon. g y g .gtoreq. 1. ( 16 )
##EQU00011##
[0099] Variable substitution.
g .di-elect cons. G c .upsilon. g x g = g .di-elect cons. G c
.upsilon. g . ( 17 ) ##EQU00012##
[0100] Bound on z.
0 .ltoreq. z .ltoreq. g .di-elect cons. G c .upsilon. g y g
.upsilon. min . ( 18 ) ##EQU00013##
[0101] Bound on xg.
0 .ltoreq. x g .ltoreq. g .di-elect cons. G c .upsilon. g y g
.upsilon. g y g . ( 19 ) ##EQU00014##
[0102] Big-M formulation for x.sub.g.
z - g .di-elect cons. G c .upsilon. g .upsilon. min ( 1 - y g )
.ltoreq. x g .ltoreq. z , .A-inverted. g .di-elect cons. G c . ( 20
) ##EQU00015##
[0103] 1.3 Basic Model Formulation
[0104] Let {tilde over (y)}:=(x; y; z). The optimization model can
be rewritten as
min c ~ T y ~ + s .di-elect cons. S Q s ( y ~ ) , s . t . y ~
.di-elect cons. , ( 21 ) ##EQU00016##
where
y:={{tilde over (y)}:constraints(4)-(11),(16)-(20),y.sub.g.di-elect
cons..A-inverted.g.di-elect cons.G,}
and
Q.sub.s({tilde over (y)}):=max 0.sup.Tw.sub. S
s.t. constraights (12),(13),(14),
w.sub.xs.di-elect cons.,z.di-elect cons.Z,s.di-elect cons.S.sub.g.
(22)
[0105] Here {tilde over (c)} may be a column vector with proper
dimension that stores the objective coefficients from (15), and
define w.sub. s:={w.sub.zs:.A-inverted.Z.di-elect cons.z} is
defined. Observe that this optimization model has a dual
block-angular structure [4], where each block may be associated
with a specific specialty s, and is linked to others with
constraints (13) and (14). Each block may involve all binary
variables; moreover, its dimensionality may depend on the number of
zip codes and providers in constraint (14). This unavoidably may
increase the model complexity as the combination of zip codes and
providers may be very large in practice.
[0106] To reduce the model complexity, it is shown that, for each
block the integrality of binary variable w.sub.zs may be relaxed
and the constraint (14) may be removed safely.
[0107] 1.4 Enhanced Model Formulation
[0108] Now, let
min c ~ T y ~ + s .di-elect cons. S Q ~ s ( y ~ ) , s . t . y
.di-elect cons. , where ( 23 ) Q ~ s ( y ~ ) := max 0 T w zs s . t
. constraints ( 12 ) , ( 13 ) , 0 .ltoreq. w zs .ltoreq. 1. ( 24 )
##EQU00017##
[0109] Lemma 2.1. For any
{tilde over (y)}.di-elect cons.Y, if Q.sub.s({tilde over (y)}) for
all s.di-elect cons.S are feasible, then so are {tilde over
(Q)}.sub.s({tilde over (y)})
Proof. This is true since
Q.sub.s({tilde over (y)}).OR right.{tilde over (Q)}.sub.s({tilde
over (y)}) for all s.di-elect cons.S
[0110] Lemma 2.2. For any
[0111] {tilde over (y)}.di-elect cons., for any s.di-elect cons.S
if Q.sub.s({tilde over (y)}) is not feasible, then either is {tilde
over (Q)}.sub.s({tilde over (y)}) Proof. Given {tilde over
(y)}.di-elect cons., it is assumed Q.sub.s({tilde over (y)}) has no
feasible solution, whereas {tilde over (Q)}.sub.s({tilde over (y)})
has. Hence, there exists a fractional solution w.sub.zs* such that
w.sub.zs*.di-elect cons. for some .di-elect cons.. However, .left
brkt-top.w.sub.zs*.right brkt-bot..A-inverted.z.di-elect cons. also
satisfies constraints (12), (13), (14), and thus is a feasible
solution to Q.sub.s({tilde over (y)}), contradicting to the
assumption. Thus the result.
[0112] Let {tilde over (y)}* and w.sub.zs*.A-inverted.s.di-elect
cons.S, z.di-elect cons.Z bean optimal solution of (23). The
following result show may that the problem (21) and problem (23)
equivalent.
[0113] Theorem 2.1. Problem (21) and problem (23) have the same
optimal objective value. Moreover, if
w.sub.zs*.A-inverted.s.di-elect cons.S, z.di-elect cons.Z is an
integral solution, then y* and w.sub.zs* also satisfy (21).
otherwise, a corresponding integral solution can be obtained by
rounding any fractional component of w.sub.zs* up to 1.
[0114] Proof. The first part of Theorem 2.1 is a direct result of
Lemma 2.1 and Lemma 2.2, since Q.sub.s({tilde over (y)}) and {tilde
over (Q)}.sub.s({tilde over (y)}) are both feasibility problems.
For the second part, any .left brkt-top.w.sub.zs*.right
brkt-bot..A-inverted.z.di-elect cons. also satisfies constraints
(12), (13), (14), and thus is a feasible solution to Q.sub.s({tilde
over (y)}). This completes the proof
[0115] Theorem 2.1 allows solving a relaxation problem with
significantly fewer constraints and binary variables in each block,
and thus potentially improves the model complexity and solution
times.
[0116] 1.5 Solving the Enhanced Model with Benders
Decomposition
[0117] In this section it is shown that, problem (23) can be
efficiently solved with an integer L-shaped method (or Benders
decomposition). Since each recourse subproblem involves only
continuous variables, the recourse function {tilde over
(Q)}.sub.s({tilde over (y)}) is a piecewise linear convex function,
and can be approximated with standard Benders' cuts. All
subproblems {tilde over (Q)}.sub.s({tilde over (y)}) have a null
objective function, and hence, only feasibility cut is required for
the recourse function approximation. The generation of the
feasibility cuts requires the dual unbounded ray (certificate of
the primal infeasibility) of {tilde over (Q)}.sub.s({tilde over
(y)}), and this process can be expensive when the dimensionality of
{tilde over (Q)}.sub.s({tilde over (y)}) is large. To speed up the
solution, it is demonstrated in the following that, such unbounded
ray can be obtained in closed-form.
[0118] Given {tilde over (y)}.di-elect cons.y, rewrite {tilde over
(Q)}.sub.s({tilde over (y)}) as:
max 0 T w ( 25 ) s . t . z .di-elect cons. Z m z w zs .gtoreq. m s
* z .di-elect cons. Z m s , ( 26 ) g .di-elect cons. G b zgs y g
.gtoreq. w zs , .A-inverted. z .di-elect cons. Z , ( 27 ) 0
.ltoreq. w zs .ltoreq. 1. ( 28 ) ##EQU00018##
[0119] The dual formulation of {tilde over (Q)}.sub.s({tilde over
(y)}) is
min - m s * .lamda. + z .di-elect cons. Z ( g .di-elect cons. G b
zgs y g ) .pi. zs + z .rho. zs ( 29 ) s . t . = m z .lamda. + .pi.
zs + .rho. zs .gtoreq. 0 , .A-inverted. z .di-elect cons. Z , ( 30
) .lamda. * := 1 , .rho. zs * := 0 , .pi. zs * := { m z , if g
.di-elect cons. G b zgs y g = 0 , 0 , otherwise , ( 31 )
##EQU00019##
where .lamda., .pi..sub. s, .rho..sub. s, are Lagrangian
multipliers corresponding to constraints (26), (27), and (28),
respectively.
[0120] Now it is assumed that {tilde over (Q)}.sub.s({tilde over
(y)}) is not feasible. If .SIGMA..sub.g.di-elect
cons.Gb.sub.zgsy.sub.g.gtoreq.1 for all z.di-elect cons.S, then it
is claimed our network optimization problem has no feasible
solution, as coverage constraint (26) can never be satisfied. On
the other hand, if .SIGMA..sub.g.di-elect cons.Gb.sub.zgsy.sub.g=0
for some z*.di-elect cons.Z, then an unbounded ray may be obtained
with the following:
.lamda. * := 1 , .rho. zs * := 0 , .pi. zs * := { m z , if g
.di-elect cons. G b zgs y g = 0 , 0 , otherwise , ( 32 )
##EQU00020##
Observing that for any p>0, p(.lamda.*, .pi..*s, .rho..*s)
always satisfies constraints (30) and (31) and the objective
function value decreases as p increases, hence (.lamda.*, .pi..*s,
.pi..*s) is a valid unbounded ray.
[0121] Theorem 2.2. Given y.di-elect cons., assuming that {tilde
over (Q)}.sub.s({tilde over (y)}) is not feasible for some
s.di-elect cons.S. If .SIGMA..sub.g.di-elect
cons.Gb.sub.zgsy.sub.g.gtoreq.1 for all z.di-elect cons.Z, then
optimization problem (21) has no feasible solution. Otherwise, (32)
gives a dual unbounded ray for {tilde over (Q)}.sub.s({tilde over
(y)}), and it can be calculated with O(|G||Z|) operations. As such,
a valid feasibility cut is given by
g .di-elect cons. G z .di-elect cons. Z .pi. zs * b zgs y g
.gtoreq. 0. ( 33 ) ##EQU00021##
[0122] Referring to FIG. 3, there is shown a modification of the
integer L-shaped method for solving model (23). In this algorithm,
a branch-and-cut search tree may be developed over the integer
variables in the master problem, and the Benders' cut may be
generated during the search.
[0123] 2. Computational Study
[0124] In this section computational results are presented. All
these enhancements may be implemented in an optimization engine
referred to as ndo-engine-disc, which is developed in Python 2.7.
All computations are performed on 24-physical-core
(48-logical-core) Intel Xeon E5-2670 2.30 GHz CPU with 128 GB RAM
running Linux Red Hat 6.9; however, no parallelization was used in
this computational study.
[0125] Our problem test set consists of 396 problems drawn from
different product scopes. All problem data were collected from
claim data, and user-defined parameters were selected carefully to
align with business goals. Cost minimization (1) was chosen as the
objective function for all these problems. Constraints (4)-(9),
(12)-(14), and (16)-(20) are imposed into the model. PCP
requirement and multi-specialty requirements are not imposed. Other
parameters are summarized in Table 1.
[0126] GUROBI [1] Version 6.5 may be used for solving the
underlying mixed-integer linear programs with additional customized
parameter settings listed in Table 2. In our computational runs a
time limit of an hour may be set for each problem in each run.
TABLE-US-00001 TABLE 1 Problem Profile Requirement Possible Values
Selection Criteria State CO, CT, GA, IN, ME, Randomly selected NH,
NV, NY, VA Product Commercial NA LOB PPO, HMO Randomly selected
Specialty 23 most commonly All selected used specialties* 1*:
Ophthalmology, General Surgery, Orthopedic Surgery, Cardiology,
Dermatology, Neurology, Gastroenterology, Otolaryngology, Urology,
Chiropractic, Endocrinology, Family Practice, General Practice,
Internal Medicine, Interventional Cardiology, Licensed Clinical
Social Wrkr, Nephrology, Neurosurgery, Pediatric Medicine, Physical
Medicine and Rehab, Podiatry, Pulmonary Disease.
TABLE-US-00002 TABLE 2 Customized parameter settings for GUROBI
Parameter Description Value MIPGapAbs Absolute MIP optimality gap
1.E-3 MIPGap Relative MIP optimality gap 1.E-3 BarHomogeneous IPM
method 1 (HSD-IPM)* PrePasses Presolve level 10 Method Root method
1 (Simplex) or 2 (IPM)** 1*: Homogeneous and self-dual
interior-point method (IPM). 2**: Simplex method is used if the
number of variables is less than 10000; otherwise, we used IPM.
[0127] To provide a summary of the results, we use performance
profiles [5]. Consider a set A of n.sub.a algorithms, a set P of
n.sub.p problems, and a performance measure m.sub.a,p, e.g.,
computation time. Compare the performance on problem p by algorithm
a with the best performance by any algorithm on this problem using
the following performance ratio
r p , a = m p , a m p * , ##EQU00022##
where m.sub.p*:=min {m.sub.p,a|a.di-elect cons.A}. Obtain an
overall assessment of the performance of the algorithm by defining
the following value
.rho. a ( .tau. ) = 1 n p cardinality { p .di-elect cons. P | r p ,
a .ltoreq. .tau. } . ##EQU00023##
[0128] This may represent the probability for algorithm a that the
performance ratio r.sub.p,a is within a factor .tau. of the best
possible ratio. The function p.sub.a( ) represents the distribution
function for the performance ratio.
[0129] 2.1 Performance of the Network Solution
[0130] Present and discuss computational results on the test
problems. Benchmark the effectiveness of the proposed optimization
model by comparing its optimal objective value with that from a
line model, which maximizes the total volume (3) and is subjected
to the simple variable bounded constraints (4)-(7). As such, an
optimal solution can be obtained without explicit optimization but
simply filtering out irrelevant or constraint-violated providers.
Note this optimal solution also satisfy constraints (8)-(9) and
(12)-(14). More importantly, emphasize that this baseline solution
is used by business users nowadays to create a network
solution.
[0131] Table 3 may summarize the computational results on 396
problems. Column "State" gives the product scopes associated with a
particular state. Columns "Baseline value" and "Cost-minimized
value" respectively provide the overall objective values of the
baseline model and the proposed optimization model. Column
"Improvement" shows the overall improvements in percentage. "AVG"
in the last row of the table provides the arithmetic means.
[0132] Observe that the proposed average cost of the
cost-minimization model consistently outperforms that of the
baseline model across nine different states. Specifically, the
improvement ranges from 10.19% (VA problems) to 25.34% (NY
problems), with an overall improvement of 16.08%. This is a very
significant cost-saving in practice and it clearly demonstrates the
effectiveness of our proposed optimization model. Finally, note
that NY problems in general are much larger than others, and thus
the cost-savings are even more significant.
TABLE-US-00003 TABLE 3 Computational comparisons of cost-minimized
value and baseline value Baseline Cost-minimized Improvement State
value value (%) CO 1.1068 0.9627 14.97 CT 1.0868 0.9436 15.18 GA
1.1044 0.9412 17.34 IN 1.1064 0.9473 16.82 ME 1.0987 0.9772 12.43
NH 1.1210 0.9774 14.69 NV 1.1052 0.9387 17.74 NY 1.1129 0.8879
25.34 VA 1.0740 0.9747 10.19 AVG 1.1018 0.9501 16.08
[0133] 2.2 Performance of the Enhanced Model
[0134] The solution times for solving basic optimization model (21)
with that for the enhanced optimization model (23) are compared.
Table (4) summarizes the computational results on 396 problems.
Column "State" gives the product associated with a particular
state. Columns "Basic model` and "Enhanced model` respectively
provide the overall solution times in seconds required for solving
the basic optimization model and the enhanced optimization model.
Column "Improvement" shows the overall improvements in percentage.
"AVG" in the last row of the table provides the arithmetic
means.
[0135] Observe that the enhanced optimization model improves the
solution times dramatically, with the improvement ranges from 300%
to almost 4,000%. The overall improvement is 1,388%. Recall that
the purpose of NDO is to provide an online solution platform that
allows users to create a network solution within a 10 minutes of
time-limit. The basic model failed to satisfy these requirements on
solving most IN, NY, and CT problems. In comparison, the enhanced
model successfully solves all problems within the time-limit. In
average, the basic model requires 7 minutes of computations whereas
the enhanced model needs only 35 seconds. This clearly demonstrates
the effectiveness of the enhanced model.
[0136] Referring now to Table 4, there is shown the performance
profiles comparing solution time performances of the basic model
and the enhanced model.
TABLE-US-00004 TABLE 4 Computational comparisons of basic
optimization model and enhanced optimization model Basic Enhanced
Improvement State model model (%) CO 200.70 7.98 2414.75 CT 591.15
14.74 3911.08 GA 303.35 24.72 1126.97 IN 724.18 95.20 660.69 ME
24.08 2.62 818.55 NH 21.99 2.68 720.24 NV 10.65 2.67 299.60 NY
1767.11 151.01 1070.22 VA 220.62 19.66 1022.34 AVG 429.32 35.70
1338.27
Concluding Remarks
[0137] In this application, an optimization model for the Network
Design and Optimization project is described. The model solution
improves the cost-saving target by 16% on average, when compared
with a baseline solution used nowadays. Moreover, additional
computational enhancements that significantly speedup the solution
time by almost 14 folds on average are developed. Computational
results on a large number of problems drawn from different product
scopes using real life data clearly demonstrate the efficiency and
effectiveness of the proposed solution.
EXAMPLES
[0138] Goal: Given N providers, each with its own cost and volume,
the goal is to select K providers among them such that the average
cost of the selected providers (called network) is minimized. The
business requirements must be satisfied.
[0139] Model data (see page 4, Data Section)
[0140] c.sub.g (g=1, 2, . . . , N): cost of provider g.
[0141] v.sub.g (g=1, 2, . . . , N): volume of provider g.
[0142] Decision variables (see page 5, Decision variables
Section)
[0143] y.sub.g (g=1, 2, . . . , N): y.sub.g=1 if provider g is
selected, 0 o/w.
[0144] Average cost (see page 6, eq (1))
[0145] .SIGMA.c.sub.gv.sub.gy.sub.g/.SIGMA.v.sub.gy.sub.g: Total
cost weighted by volume/Total volume
[0146] Volume Constraint (see page 7, eq (9))
[0147] .SIGMA.v.sub.gy.sub.g.gtoreq.v*.SIGMA.v.sub.g: Provided
volume>=Required volume, where v* is given in %.
Example 1
[0148] Given three candidate providers (N=3).
Our goal is to select some (or all) of them such that the average
cost is minimized, the minimum volume (v*=4/6) can be
satisfied.
TABLE-US-00005 Provider 1 2 3 Cost 1 3 3 Volume 2 1 3 Weighted cost
1 * 2 = 2 3 * 1 = 3 3 * 3 = 9
[0149] All possible solutions for example 1
TABLE-US-00006 Selected Providers 1 2 3 1, 2 1, 3 2, 3 1, 2, 3 None
Total provided 2 1 3 3 5 4 6 0 volume Total provided 2/6 1/6 3/6
3/6 5/6 4/6 6/6 0 volume/Total volume Total weighted 2 3 9 5 11 12
14 0 cost Average cost 1 3 3 5/3 11/5 3 14/6 NA
[0150] Recall our minimum volume requirement v*=4/6. From row 2,
only the following three networks can provide sufficient volume:
(1,3), (2,3), (1,2,3)
From row 4, it is shown that network (1,3) gives the best average
cost (11/5). A network problem with 3 providers and 1 volume
constraint may be successfully solved in this way.
Model in Example 1
[0151] Model
[0152] Objective:
.SIGMA.c.sub.gv.sub.gy.sub.g/.SIGMA.v.sub.gy.sub.g(2y.sub.1+3y.sub.2+9y.s-
ub.3)/(2y.sub.1+1y.sub.2+3y.sub.3)
[0153] Volume requirement:
.SIGMA.v.sub.gy.sub.g.gtoreq.v*.SIGMA.v.sub.g(2y.sub.1+3y.sub.2+9y.sub.3)-
.gtoreq.(4/6)(2+1+3)
[0154] Solution Verification
[0155] Suppose providers 1 and 3 are selected, then, y_1=y_3=1, and
y_2=0.
[0156] The objective value (average cost)=(2+9)/(2+3)=11/5
The provided volume=(2+3)=5.gtoreq.(4/6)(2+1+3)=4.
[0157] For network with 3 providers to pick, the total
possibilities are 2{circumflex over ( )}3=8. What about for a
network with 1,000 providers? The total possibilities are
2.sup.1000=(2.sup.10).sup.100.about.10000.sup.100
[0158] Model 2: Making Model 1 More Practical
[0159] Model Data
[0160] m.sub.z (z=1, 2, . . . , M): Total members located at zone z
(see page 5, Data Section).
[0161] S: Possible specialties (see page 3, Data Section).
[0162] Decision variables (see page 5, Decision variables
Section)
[0163] w.sub.zs (z=1, 2, . . . , s=1, 2, . . . ): w.sub.zs=1 if
members at zone z has access to specialty s, or 0 o/w.
[0164] Coverage ratio requirement (see page 8, eq (15))
[0165] .SIGMA.m.sub.zw.sub.zs.gtoreq.m.sub.s*.SIGMA.m.sub.z: For
specialty s, total covered members>=Total required covered
members, where m.sub.s* is given in %.
[0166] Impose some relationships between y.sub.g and w.sub.zs.
[0167] Indicator (see page 5, Indicator Section).
[0168] b.sub.zgs=1 means a provider g can provide specialty s for
members located at z.
[0169] b.sub.zgs are given.
[0170] Linking y.sub.g and w.sub.zs (see page 8, eq (16) and
(17))
[0171] .SIGMA.b.sub.zgsy.sub.g.gtoreq.w.sub.zs: A member at zone z
will not have access to specialty s, unless at least a provider g
being able to provide specialty s for members at zone z is
selected.
[0172] b.sub.zgsy.sub.g.ltoreq.b.sub.zgsw.sub.zs: If provider g is
picked, then all covered members at z must have access to specialty
provided s. Note: This is done through indicator b.sub.zgs.
[0173] Given three candidate providers (g=1, 2, 3) as usual.
[0174] Only one specialty (s=1). Three zone (z=a, b, c).
TABLE-US-00007 Zone a b c Members (m.sub.z) 1 2 3 Provider 1:
b.sub.z,g =1,s =1 1 0 0 Provider 2: b.sub.z,g =2,s =1 1 1 0
Provider 3: b.sub.z,g =3,s =1 1 0 1
[0175] Provider 1 can provide service for zone a. Provider 2 can
provide service for member at zone a and b. Provider 3 can provide
service for members at zone a and c.
[0176] Objective and volume, as defined in example 1. Our goal is
to select some (or all) of them such that average cost is
minimized, the minimum volume (v*=4/6) can be satisfied. The
minimum covered ratio (m*=5/6) can be satisfied. Consider the
following three networks: (1,3), (2,3), (1,2,3)
TABLE-US-00008 Selected Providers 1, 3 2, 3 1, 2, 3 Covered zone a,
c a, b, c a, b, c Total covered members 4 6 6 Total covered
members/ 4/6 6/6 6/6 Total members Average cost 11/5 3 14/6
[0177] Objective:
.SIGMA.c.sub.gv.sub.gy.sub.g/.SIGMA.v.sub.gy.sub.g
(2y.sub.1+3y.sub.2+9y.sub.3)/(2y.sub.1+1y.sub.2+3y.sub.3)
[0178] Volume: .SIGMA.v.sub.gy.sub.g.gtoreq.v*.SIGMA.v.sub.g
(2y.sub.1+3y.sub.2+9y.sub.3).gtoreq.(4/6) (2+1+3)
[0179] Coverage:
.SIGMA.m.sub.zw.sub.zs.gtoreq.m.sub.s*.SIGMA.m.sub.z
(1w.sub.as+2w.sub.bs+3w.sub.cs).gtoreq.(5/6) (1+2+3)
[0180] Linking: .SIGMA.b.sub.zgsy.sub.g.gtoreq.w.sub.zs
[0181] For z=a, (y.sub.1+y.sub.2+y.sub.3).gtoreq.w.sub.a,1. For
z=b, (y.sub.2).gtoreq.w.sub.b,1. For z=c,
(y.sub.3).gtoreq.w.sub.c,1.
[0182] Linking: b.sub.zgsy.sub.g.ltoreq.b.sub.zgsw.sub.zs:
For z=a, g=1,y.sub.1.ltoreq.w.sub.a,1. For z=b, g=1,
y.sub.1.ltoreq.w.sub.b,1=1. For z=c, g=1,
y.sub.1.ltoreq.w.sub.c,1=1. And so forth.
[0183] Solution Verification
Suppose providers 1, 2, 3 are selected, then,
y.sub.1=y.sub.2=y.sub.3=1. Suppose members in every zone can be
covered, then w.sub.a1=w.sub.b1=w.sub.c1=1. The objective value
(average cost)=(2+3+9)/(2+1+3)=14/6 The provided
volume=(2+3+1)=6.gtoreq.(4/6) (2+1+3)=4. The provided
coverage=(1+2+3).gtoreq.(5/6) (1+2+3)
[0184] Linking: .SIGMA.b.sub.zgsy.sub.g.gtoreq.w.sub.zs
For z=a, g=1, 1.ltoreq.w.sub.a,1=1. For z=b, g=1,
0.ltoreq.w.sub.b,1=1. For z=c, g=1, 1.ltoreq.w.sub.c,1=1. And so
forth.
[0185] Linking: b.sub.zgsy.sub.g.ltoreq.b.sub.zgsw.sub.zs:
For z=a, g=1, y.sub.1=1.ltoreq.w.sub.a,1=1. For z=b, g=1,
y.sub.1=1.ltoreq.w.sub.b,1=1. For z=c, g=1, y.sub.1=1 w.sub.c,1=1.
And so forth.
Complexity of Example 2
[0186] 3 providers to pick (y.sub.g), and 3 associated coverage
variable (w.sub.zs) are available. The total possibilities are
2.sup.(3+3)=64.
A practical model may involve 1000+ providers, 100+ zones, and 75+
specialties.
[0187] Optimization Algorithm
Advanced optimization algorithm has internal mechanisms that
prevents from exploring the whole solution space (full
enumeration), for example, if y.sub.3=0, then
wa1=w.sub.b1=w.sub.c1=0. (called constraint propagation). Its
complexity in general depends on number of variables and
constraints, and the variable type (continuous/discrete; in our
case, they are all discrete).
[0188] Discrete variable screws up the complexity, and makes
problem strongly NP hard.
[0189] Our model is very difficult to solve due to large number of
discrete variables and constrains.
[0190] What if, the integrality requirements are relaxed and some
constraints are removed, without compromising the solution quality
as a bid.
[0191] Specifically, the integrality requirements of w.sub.zs, and
linking constraint: b.sub.zgsy.sub.g.ltoreq.b.sub.zgsw.sub.zs (for
all z, g, s) can be removed safely.
[0192] The worst case complexity can be cut down from O(2.sup.N+ZS)
to O(2.sup.N), where Z is the number of zones, N is the number of
providers, and S is the number of specialties, and usually
ZS>>N.
[0193] Moreover, at most O(ZNS) constraints can be dropped
safely.
[0194] Specific Model Structure
[0195] Referring now to FIG. 5A, there is shown an exemplary
embodiment of the present invention. If the problem is written down
manually, the nonzero elements will only arise in only certain
blocks.
[0196] Referring now to FIG. 5B, there is shown an exemplary
embodiment of the present invention. A decomposition algorithm
called Benders decomposition can benefit from this structure.
Conceptually, it may only consider the upper-left block of
constraints, as shown in FIG. 5B, and ignore the rest first. The
algorithm generates the valid constraints that can represent the
ignored structure on the fly. Generations of this valid
constraints, however, are not trivial. In most cases, it may
require w.sub.zs to be continuous.
[0197] The integrality requirement of w.sub.zs can be relaxed.
[0198] In one embodiment, it has been shown that the problem
structure can be significantly simplified without compromising the
solution quality as a bid.
[0199] In one embodiment, it has been shown that the simplified
model structure can be solved with an efficient decomposition
algorithm.
[0200] Computational experiments on a large number of cases
demonstrate that, our proposed method speeds up the solution time
14 folds. In particular, several cases failed to solve originally
can be now solved with this new approach.
[0201] This solution platform can also perform sensitivity analysis
(what-if analysis) that further provides solution analytics to
profile the robustness of the constructed network solution. For
example, if 5% of cost-saving is sacrificed, how much network
coverage can further be improved?
[0202] The solution platform can leverage providers with incomplete
input data to improve the model feasibility. Specifically, if all
providers with complete input data are not sufficient to construct
a valid network solution, the solution platform may further include
providers with incomplete data to explore a feasible network
solution.
[0203] If a valid network solution still cannot be constructed, the
solution platform will report the reason that results in the model
infeasibility, with hope that such information helps users
modifying their input easily.
[0204] Referring to FIG. 6, there is shown a flow chart in
accordance with an exemplary embodiment of the present invention.
In one embodiment, there is a computer-implemented method for
selecting one or more providers (e.g., providers 202 shown in FIG.
2) having one or more specialties (e.g., Ophthalmology, General
Surgery, Orthopedic Surgery, Cardiology, Dermatology, Cardiology,
etc.) for a network available to members while satisfying one or
more constraints (e.g., constraints, (6)-(14) defined above).
[0205] In one embodiment, the method comprises, at a first
computing device: receiving 602, from a second computing device
separate and distinct from the first computer, a network provider
request to select an optimized network of providers (e.g.,
providers 204 in FIG. 2). In one embodiment, the request includes:
a desired objective including at least one of provider cost
minimization (e.g., average cost minimization equation, (1)),
provider average quality maximization (e.g., average quality
maximization equation, (2)) and provider total volume maximization
(e.g., provider total volume maximization (3)), one or more
geographical designations (e.g., d(z, a(g, s)): Distance between a
member zip code z and a provider-specialty's zip code a(g, s)).
[0206] In one embodiment, the method comprises, at a first
computing device: computing 604 an optimized provider network
(e.g., narrow network 204) of one or more selected providers.
Computing 604 an optimized provider network may include: applying
606 a model that produces a non-optimized provider network (that
satisfies enhanced model formulation equation 23) of one or more
providers having one or more specialties that satisfies the desired
objective for the one or more geographical designations of the
members without consideration of the availability in the
geographical designation of one or more providers having one or
more specialties.
[0207] In one embodiment, the method comprises, at a first
computing device: applying 608 a linear decomposition (e.g.,
equations 25-28 and/or equations 29-31) to the pre-optimized
provider network to generate an optimized provider network.
[0208] In one embodiment, the method comprises, at a first
computing device: providing 610 the optimized provider network to
the second computing device.
[0209] In one embodiment, without consideration of the availability
in the geographical designation (e.g., d(z, a(g, s)): distance
between a member zip code z and a provider-specialty's zip code
a(g, s)) of one or more providers (e.g., providers 202 shown in
FIG. 2), having one or more specialties includes forgoing
considering whether a ratio of total members in a particular
geographical designation having access to a particular specialty
(e.g., cardiology) is greater than a member coverage ratio (e.g.,
as specified in constraint equation (12)) threshold.
[0210] In one embodiment, without consideration of the availability
in the geographical designation (e.g., d(z, a(g, s)): Distance
between a member zip code z and a provider-specialty's zip code
a(g, s)) of one or more providers having one or more specialties
includes forgoing considering whether a single member in each of
the geographical designations will have access to each of the
particular specialties.
[0211] In one embodiment, without consideration of the availability
in the geographical designation (e.g., d(z, a(g, s)): Distance
between a member zip code z and a provider-specialty's zip code
a(g, s)) of one or more providers having one or more specialties
includes for each particular geographical designation, each
particular provider and/or each particular provider specialty,
forgoing considering whether all members in the particular
geographical designation have access to the particular provider
specialty offered by the particular provider (e.g., as specified in
constraint equation (13)).
[0212] In one embodiment, of the average cost (e.g., average cost
equation (1)) of the selected providers (e.g., providers 202 in
FIG. 2) is less than a predetermined provider cost threshold (e.g.,
as specified in constraint equation (14)).
[0213] In one embodiment, the average volume of the patients seen
by the selected providers is greater than a predetermined provider
volume threshold.
[0214] The average quality (e.g., average quality maximization,
equation (2)) of the selected providers is greater than a
predetermined provider quality threshold.
[0215] In one embodiment, the linear decomposition (e.g.,
linearization of the objective function (15)) is Benders
decomposition (e.g., Benders decomposition further explained in
equations (25)-(28)).
[0216] In one embodiment, the model that produces a non-optimized
provider network is
min c ~ T y ~ + s .di-elect cons. S Q ~ s ( y ~ ) , s . t . y ~
.di-elect cons. , where ##EQU00024## Q ~ s ( y ~ ) : = max 0 T w .
s , s . t . w . s .di-elect cons. , 0 .ltoreq. w z s .ltoreq. 1 ,
.A-inverted. z , ##EQU00024.2##
wherein: {tilde over (c)} is a vector of the objective
coefficients, {tilde over (y)} is a vector of the binary variables
that provide the network solution, e.g., a provider is included or
not, y represents the feasible set for {tilde over (y)}, including
business requirements such as product scope, specialty required,
minimum retained volume, and cost/quality baseline, S represents
the set of specialties, {tilde over (Q)}.sub.s({tilde over (y)}) is
a recourse subproblem that represents the coverage ratio
requirement for each specialty s.di-elect cons.S, w.sub.zs is a
vector of decision variables that determines if members located at
z have an access to specialty s considered in network, W represents
the feasible set for {tilde over (w)}. Specifically, w addresses
minimum coverage ratio from the business requirement, and
determines if members in each zip code may have coverage for each
specialty. In some embodiments, w.sub.zs was originally formulated
as a binary decision variable, but w.sub.zs's integrality
requirement can be relaxed safely without comprising the solution
quality. In some embodiments, w.sub.zs is rounded to an integer
number in the end of solution process to ensure the integrality and
feasibility of the network solution. In some embodiments, such
rounded w.sub.zs solution has been shown to be the optimality
solution to the original network problem, i.e., without integrality
relaxation.
[0217] In a numerical experiment, relaxing the integrality
requirement of w.sub.zs can speed up the solution time by almost 14
folds on average.
[0218] In at least one embodiment, there is included one or more
computers having one or more processors and memory (e.g., one or
more nonvolatile storage devices). In some embodiments, memory or
computer readable storage medium of memory stores programs, modules
and data structures, or a subset thereof for a processor to control
and run the various systems and methods disclosed herein. In one
embodiment, a non-transitory computer readable storage medium
having stored thereon computer-executable instructions which, when
executed by a processor, perform one or more of the methods
disclosed herein.
[0219] It will be appreciated by those skilled in the art that
changes could be made to the exemplary embodiments shown and
described above without departing from the broad inventive concept
thereof. It is understood, therefore, that this invention is not
limited to the exemplary embodiments shown and described, but it is
intended to cover modifications within the spirit and scope of the
present invention as defined by the claims. For example, specific
features of the exemplary embodiments may or may not be part of the
claimed invention, different components as opposed to those
specifically mentioned may perform at least some of the features
described herein, and features of the disclosed embodiments may be
combined. As used herein, the terms "about" and "approximately" may
refer to + or -10% of the value referenced. For example, "about 9"
is understood to encompass 8.2 and 9.9.
[0220] It is to be understood that at least some of the figures and
descriptions of the invention have been simplified to focus on
elements that are relevant for a clear understanding of the
invention, while eliminating, for purposes of clarity, other
elements that those of ordinary skill in the art will appreciate
may also comprise a portion of the invention. However, because such
elements are well known in the art, and because they do not
necessarily facilitate a better understanding of the invention, a
description of such elements is not provided herein.
[0221] It will be understood that, although the terms "first,"
"second," etc. are sometimes used herein to describe various
elements, these elements should not be limited by these terms.
These terms are only used to distinguish one element from another.
For example, a first element could be termed a second element, and,
similarly, a second element could be termed a first element,
without changing the meaning of the description, so long as all
occurrences of the "first element" are renamed consistently and all
occurrences of the second element are renamed consistently. The
first element and the second element are both elements, but they
are not the same element.
[0222] As used herein, the term "if" may be, optionally, construed
to mean "upon" or "in response to determining" or "in response to
detecting" or "in accordance with a determination that," depending
on the context. Similarly, the phrase "if it is determined" or "if
[a stated condition or event] is detected" is, optionally,
construed to mean "upon determining" or "in response to
determining" or "upon detecting [the stated condition or event]" or
"in response to detecting [the stated condition or event]" or "in
accordance with a determination that [a stated condition or event]
is detected," depending on the context.
[0223] The terminology used herein is for the purpose of describing
particular implementations only and is not intended to be limiting
of the claims. As used in the description of the implementations
and the appended claims, the singular forms "a", "an" and "the" are
intended to include the plural forms as well, unless the context
clearly indicates otherwise. It will also be understood that the
term "and/or" as used herein refers to and encompasses any and all
possible combinations of one or more of the associated listed
items. It will be further understood that the terms "comprises"
and/or "comprising," when used in this specification, specify the
presence of stated features, integers, operations, elements, and/or
components, but do not preclude the presence or addition of one or
more other features, integers, operations, elements, components,
and/or groups thereof.
[0224] As used herein, the term "if" may be construed to mean
"when" or "upon" or "in response to determining" or "in accordance
with a determination" or "in response to detecting," that a stated
condition precedent is true, depending on the context. Similarly,
the phrase "if it is determined (that a stated condition precedent
is true)" or "if (a stated condition precedent is true)" or "when
(a stated condition precedent is true)" may be construed to mean
"upon determining" or "in response to determining" or "in
accordance with a determination" or "upon detecting" or "in
response to detecting" that the stated condition precedent is true,
depending on the context.
[0225] Further, to the extent that the method does not rely on the
particular order of steps set forth herein, the particular order of
the steps should not be construed as limitation on the claims. The
claims directed to the method of the present invention should not
be limited to the performance of their steps in the order written,
and one skilled in the art can readily appreciate that the steps
may be varied and still remain within the spirit and scope of the
present invention.
* * * * *