For info : [email protected] Goalxie/DefiSante/PolyDefiSante2019.pdf · 2019-09-10 · 9% for 2016) 5 Context and trends 6 Context and trends ... • By DRG (diagnostic relatedgroups)

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Healthcare Operation managementProfessor Xiaolan Xie

Schedule1. 13/09 Friday 13:30-16:45

2. 23/09 Monday 09:00-12h:15

3. 30/09 Monday 13:30-16:45

4. 07/10 Monday 13:30-16:45

5. 04/11 Monday 13:30-16:45

Lecture room: 158 (Session 1) + CIS Building (others)

http://www.emse.fr/~xie/DefiSanteFor info : [email protected]

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Chapter 1. Healthcare delivery and its operation

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Goal

present quantitative techniques from the perspective of health care organisations’ delivery of care, rather than their traditional manufacturing applications.

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Context and trends

Rising health expenditures (17.2% in USA, 11.0% in France, OECD35 9% for 2016)

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Context and trends

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Context and trends

• Increasing demand due to demographic change and agingpopulation.

• Shift from offer-driven to patient-centered health care with more active role of patients in health care and better informed patients

• Growing concern of health care safety and quality -> Need of traceability of health care delivery

• Arrivals of new ICT technologies (IA, telemedicine, delivery robots, RFID, HIS, PDA, e-prescription, POS, EDI, online appointment, eVisit, …)

• Hospitals are bigger and bigger and more complex(CHU-StE 2000 beds – 7259 employes, Ruijin Hospital –2000 beds + 12000 outpatients/day, Huaxi hospital – 4000+ beds, Zhengzuo hospital 10000+ beds

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French situations

• Mostly public hospitals

• Culture change on going. Words (clients, productivity, efficiency, service level, competition) better understood.

• Government responses:

• New healthcare financing: T2A, new governance, referee physician, payment by pathway, GHT (groupement hospitalier territorial)

• New health Info. Syst: carte vitale, personal health record

• Diversification of healthcare organizations to meet the diversity of healthcare demand (CHU, mid-size hospitals, clinics, home care, …)

• Better regional regulation by ARS

• Hospital responses :

• merging, reorganization,

• lean health care,

• Integrated Hospital Information Systems everywhere8

French situations

• Many obstacles to change.

• Healthcare costs hardly known,

• Increasing bigger hospitals with people used to work in isolated isles

• Lack of system thinking and spaghetti-like organization

• poor management skills and incentives of health professionals

• A labor-intensive industry facing quality human resourcesshortage

• about 10% of jobs in France and 40% in hospitals,

• feminisation and aging health professionals,

• working time regulation,

• Increasing importance of working conditions, …

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Context and trends

The bad side

• long waiting list,

• long overtime,

• poor resourceutilization,

• ...

Wild demand fluctuation

Extra-beds at ED, 2013.07 Outpatient queue, 6h AM,11/15/2011

Poor demand-supply match Poor quality of services

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Perspectives and evolution of French hospitals

• Montaigne report (2004),

l’hôpital de demain sera polymorphe et il n’y aura pas un modèle unique d’hôpital mais une variété d’établissements, recouvrant des organisations diversifiées, assurant des missions variées en fonction du contexte dans lequel ils se situent:

• Network organization - des établissements parties prenantes de réseaux, en liaison étroite avec la médecine de ville (GHT),

• Hospitals without wall des hôpitaux sans murs ou quasiment, pour gérer l’hospitalisation à domicile (HAD, SIAD, …),

• Highly specialized hospitals - des établissements organisés autour d’un plateau technique très spécialisé (bio, image, …),

• Others for hotelling & long term care - d’autres centrés sur l’hébergement et la dispensation de soins à des personnes âgées dépendantes (EPAHD, …),

• Others dedicated to emergency cares - des établissements privilégiant l’urgence et les soins de premiers recours.

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Perspectives and evolution of French hospitals

Six important trends:

• Health expenditure regulation and healthcare cost transparency

• Increasing productivity by optimization of healthcare deliveryorganization and management.

• Developing better relations with upstream and downstream parties in the healthcare value chain

• Adapting the healthcare offers

• Transforming healthcare delivery and hospital organization by new medical, technological and scientific progresses (new healthcaremodes: HAD, SAD, ...)

• Increasing regional control of healthcare offers.

But also, informatization of hospitals and the importance of ICT as drivers for healthcare delivery improvement.

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Distinctive characteristics of health care services

•Patient participation : interaction between the health care organisation and patient throughout the delivery of care

•Simultaneous production and consumption : product cannot be inspected and challenge for quality control

•Perisable capacity : operating rooms, physicians, ...

•Intangible nature of health care outputs : cannot be tested or handled before deciding on it.

•High levels of judgement and heterogeneous nature of health care: However, standardization (diagnosis and treatment process, T2A) is in process.

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A Four-level model of health care system

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A healthcare delivery system: function

Healthcare services or modes

• Medical cares

•Elective surgery

•Emergency surgery

•Day surgery

•Surgery at home

•Hospitalisation at home

•Rehabilitation

•...

Healthcare delivery system

PatientsCuredpatients

evironnementregulation, insurance,

competition

resources

The French GHT reform makes the functional design crucial

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A healthcare delivery system: human resources

A hospital is a lot of highly skilled human resources :

Surgeons

Anaesthetists

Nurses - Infirmier Anesthésiste Diplômé d’Etat (IADE)

Nurses - Infirmier de Bloc Opératoire Diplômé d’Etat (IBODE)

Nurses - Infirmier diplômé d’état (IDE)

Caregivers - Aides soignants

Stretchers - Brancardiers

Hospital attendants - Agents de Service Hospitalier (ASH)

Also: radiologists, biologists, technicians, secretaries, ...

CHU-St Etienne = 7000+ employees16

A healthcare delivery system: material resources

• Expensive technical facilities (Plateaux Techniques Medicaux):

• Operating theatres (operating rooms, induction rooms, recovery rooms)

• ICU, NICU (Intensive Care Units)

• Imaging equipment (MRI, CT scan, X-rays, …)

• Biology laboratories

• Pharmacies

• Sterilization facilities

• Hospitalization beds

• Consultation rooms

• ...

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A healthcare delivery system: material resources

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A healthcare delivery system: organisation

• Wards

• Medical services or units

• Specialties (Medecine, Surgery, Obstetrics)

• Clusters of competencies

• Hospitals

• Healthcare networks

• Logistic units

• Technical centers

• Administration

Usually with a funtional organisation.

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French health authorities

•Haute autorité de la santé (HAS)• Evaluation : évaluer l'utilité médicale de l'ensemble des actes, prestations et

produits de santé pris en charge par l'assurance maladie, • Certification : mettre en oeuvre la certification des établissements de santé • Best practice : promouvoir les bonnes pratiques et le bon usage des soins

auprès des professionnels de la santé et du grand public.

•Direction de l’Hospitalisation et de l’Organisation des Soins (DHOS)• National Healthcare organization (MOH) - Organisation de l’offre de soins à

la fois en ville et en établissement. • Décliner les priorités de santé publique en les traduisant en priorités pour le

secteur hospitalier.

•Agences Régionales de Santé (ARS)• Regional organisation of health : assurer un pilotage unifié de la santé en

région, de mieux répondre aux besoins de la population et d’accroître l’efficacité du système

• Agit dans le cadre d’un Projet Régional de Santé (PRS).20

French healthcare performance improvement agency

MeaH: Mission Nationale d’Expertise et d’Audits Hospitaliers

www.anap.fr (devenu ANAP)

• Objectifs :

– Faire émerger une meilleure organisation des activités hospitalières en conciliant (Best practice of hospital organisation):

• Service rendu au patient (patient service)

• Efficience économique (economic efficiency)

• Conditions de travail satisfaisantes pour le personnel (workingcondition)

• Exemples de chantier menés par la MEAH• Gestion des organisation des blocs opératoires (Operating theatre), • Circuit du médicament (drug distribution chain), • Radiothérapie, • Restauration, • Temps médical

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Six Quality Aims for the 21st Century Health Care System

• Safe—avoiding injuries to patients from the care that is intended to help them.

• Effective—providing services based on scientific knowledge to all whocould benefit and refraining from providing services to those not likely to benefit (avoiding underuse and overuse, respectively).

• Patient-centered—providing care that is respectful of and responsive to individual patient preferences, needs, and values and ensuring thatpatient values guide all clinical decisions.

• Timely—reducing waits and sometimes harmful delays for both thosewho receive and those who give care.

• Efficient—avoiding waste, including waste of equipment, supplies, ideas, and energy.

• Equitable—providing care that does not vary in quality because of personal characteristics such as gender, ethnicity, geographic location, and socioeconomic status.

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Six key healthcare performance dimensions OMS Europe- projet PATH

Clinical effectivenes Qualité technique, organisation et pratiques basées sur la preuve, gain en santé, résultat (individuel et global)

Patient centeredness Réactivité envers les patients: orientation du patient (rapidité de prise en charge, accès aux moyens d’aide sociale, qualité de l’accueil), satisfaction du patient

Efficiency Ressources, financière (syst. Financiers, continuité, gaspillage de ressources), taux d’encadrement, expérience (dignité, confidentialité, autonomie, communication)

safety Patient et soignants, environnement, structure, utilisation des technologies nécessaires à l’éfficience clinique

staff orientation Santé, bien être, satisfaction, développement (taux de renouvellement, emploi, absentéisme)

responsive governance Orientation vers la communauté (réponse au besoins et demandes), accessiblité, continuité des soins, promotion de la santé, équité, capacité d’adaptation à l’évolution de la demande de la population

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Health Operation Management

General OM decision:

The planning and control of the processes that transform inputs intooutputs

Example

(Individual doctor/patient consultation. Input = a patient with a request for healthcare, output = patient diagnosed or cared or cured. Resource to be managed: their time, diagnositic or therapeutic services needed

Extensions:

Individual doctor -> individual provider (a hospital dept, a hospital, a network of hospitals, …)

Scale and scope of the resources to be planned increase and the complexity of OM

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A meta-model of healthcare delivery system

INPUTS

PATIENT DEMAND(perceived need)

Other hospi & providers• Number• Specialty• Teaching• reputation

PURCHASERS(finances)

SUPPLIERS

TRANSFORMATIONPROCESSES

CLINICAL PROCESSES• Treatment modality• Treatment protocol• Provider-patient

encounters

MANAGEMENT PROCESSES• Infrastructure• Structure• Provider-patient

encounters

AN

CILA

RY

P

RO

CE

SS

ES

OUTPUTS

HEALTH STATUS

CLIENT PERCEPTION

USE OF RESOURCES

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Health OM can be defined as the analysis, design, planning

and control of the steps necessary to provide a service for a

client.

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Decision Layers

Strategic decisions:

Decisions having long term impacts on the hospital

Horizon: year or multiple years

Made by top management

On hospital-wide long-term vision, types of services and directions on material and human resources investments (hospital strategic plan, contrat d’objectif et de moyens, organisation de l’établissement, ...)

Major decisions:

Services: catchment areas, target groups/markets, specialities, case-mix

Investment: new hospital construction, new specialties, expansion

Partnership: shared resources, outsourcing, collaboration

Organisaton: units merging, mutualisation, polyvalence, working time regulation

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Decision Layers

Operational decisions:

Short term decisions

Horizon: day, week or month

Made by each operational unit.

Ensure the smooth execution of the all activities

Examples:

Surgery planning/scheduling, Admission/discharging control, nurse scheduling, inventory control, supplier relation management, …

Reactive controls: emergency add-on, surgery cancelation, …

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Decision Layers

Tactical decisions:

• Addresses the organization of the operations / execution of the health care delivery process ( i.e. the “what , where, how, when and who” ) .

• Similar to operational planning but on a longer planning horizon

Horizon: trimestre or year

Ensure the right match between available resources and activitiesof the strategic plan

Without profound changes of the structure and organisation

Examples:

• Surgery time allocation to specialties, block scheduling

• Bed allocations

• Tentering, supplier selection

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Decision Layers

Usual confusions between

• tactical & operational layers

• tactical & strategical layers

• advance operational sched & real-time execution

Decision frameworkVissers, Bertrand, De Vries (2001). A framework for production control in health care organizations.

STRATEGIC PLANNING(range of services, long-term resource requirement, shared resources, annual patient volumes, service & efficiency levels)

patient flows resources2-5 years

PATIENT VOLUME PLANNING AND CONTROL(case-mix, rough-cut resource capacity allocation)

patient flows resources1-2 years

RESOURCE PLANNING AND CONTROL(time-phased resource allocation (specialist time, # of patients per period)

patient flows resources3 months - 1 year

PATIENT GROUP PLANNING AND CONTROL(service requirement and planning guidelines per patient group))

patient flows resourcesWeeks – 3 months

PATIENT PLANNING AND CONTROL(scheduling individual patients)

patient flows resourcesDays - weeks

restrictions restrictions

Demand-supply specialty

Demand-supply season

Demand-supply peak hour

Demand-supply match

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Decision framework

Key resources

• Operating rooms

• Beds

• Diagnostic equipments

• Specialist time

• ICU

Patient groups

• By specialty (orthopaedic patients, general surgery, trauma, oncology, internal medicine, diabetics)

• By ages (services for older people, …)

• By DRG (diagnostic related groups)

• By health resources groups

• …

STRATEGIC PLANNINGpatient flowsSpecialties & production range

Patient groups as business units

ResourcesCollaboration & outsourcing

Shared resources2-5 years

PATIENT VOLUME PLANNING & CONTROL

patient flowsVolume contracts (insurance)# patients per patient group

Service levels

ResourcesRough-cut capacity checkTarget occupancy levels1-2 years

RESOURCE PLANNING & CONTROL

3 months - 1 year

PATIENT GROUPPLANNING & CONTROL

Weeks – 3 months

PATIENTPLANNING & CONTROL

Days - weeks

Restrictions on types of patients Restrictions on types of resources

patient flowsExpected patient # per group

Capacity requirement per group

ResourcesAllocation of leading resources

Batching rules for shared resources

patient flowsProject patient # per period

ResourcesAvailability of specialist capacity

patient flowsScheduling of patients for visits,

admission, exam

ResourcesAllocation of capacity to individual

patients

Restrictions on total patient vol Restrictions on amount of resources

Restrictions on detailed patient vol Restrictions on resource availability

Restrictions on timing of patient flows Restrictions on timing of resources

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A Wider Decision framework

Hans, Van Houdenhoven, Hulshof, A Framework for Health Care Planning and Control, 2012

Four areas by Four hierarchy levels

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Medical planning

• health care’s version of “technological planning”

• decision made by clinicians, medical protocols, treatments, diagnoses,

• Triage, RTD new treatments,

• The more complex and unpredictable the health care processes, the more autonomy is required for clinicians.

• Ex: acute care planned by clinicians, standard elective care planned centrally

• Ex: Benefits of early imaging test by triage nurse (prohibited by the current French law)

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Resource capacity planning

• Resource dimensioning, planning, scheduling, monitoring, and control

• On renewable resources : staff + equipment and facilities (e.g. MRI s, physical therapy equipment , bed linen, sterile instruments, operating theatres, rehabilitation rooms)

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Materials planning or Logistics

• Management of all consumable resources/materials, such as suture materials, prostheses, blood, bandages, food, etc

• Materials planning = the acquisition, storage, distribution and retrieval

• Typically encompasses functions like warehouse design, inventory management and purchasing.

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Financial planning

• Management of costs and revenues to organizational strategical objectives.

• Financial planning = investment planning, contracting (with e.g. health care insurers) , budget and cost allocation, accounting, cost price calculation, and billing.

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Hans, Van Houdenhoven, Hulshof, A Framework for Health Care Planning and Control, 2012

Application to a general hospital

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Major operation management issues

Demand forecast (care types – geography – time)

Facility location & layout

Planning / scheduling

Capacity planning

Supply chain and inventory management

Quality control

Project management

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Major operation management issues

Modeling

Tools: enterprise modelling, SADT, IDEF, Petri nets,

Performance assessment and diagnostic

Tools: simulation, queueing theory, Markov chains, statistics, Excel, ...

Design or re-engineering

Tools: simulation, optimisation, ...

Planning and control

Tools: planning, scheduling, optimisation, linear programing, heuristics, statistics,...

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Chaptre 2. Facility Layout1. Introduction2. Product layout3. Process layout4. Fixed position layout5. Facility layout supplement

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Introduction

Facility layout design is necessary when building a new facility or renovating an existing one in order to improve process flow and minimize waste space.

Facility layout largely depends on the shape and size of the building.

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Facility layout is important

The overall layout of a facility will last for long time and only minor changes are possible after the building or renovation.

Layout has enormous effect on daily operations.

Layout dictate the distance a patient or staff member travels

Layout influences the interations and communications of the staff members.

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Goals of Facility layout

Functionality:

• Placing heavily interacted departments together

• Placing apart departments that should not be close

• Ensuring space and form requirements

• Facilitating communication

Cost savings:

• Reduction of travel times

• Reduce overall space requirement

• Enabling for reduced staffing by placing similar job functionstogether

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Three basic types of Facility layout

Product Layout

Process Layout

Fixed position layout

Can be used to either a single department or an entire facility.

But also:

Retail store layout

Warehousing and storage layout (relation of unloading and loading areas)

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Types of manufacturing systems

• Job-Shop production

• Process layout, functionally similar machines are grouped

• Flow-shop production

•Product layout, machines are arranged along the manufacturing processes of a product

•Celluaire manufacturing systems

•Hybrid layout, similar parts and corresponding machines are grouped

• Project shop

•Product is fixed, personnel and equipment brought to it

• Continuous-flow process

•Chemical plants and flood industry

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Organisation of hospitals

• Mainly a functional organisation in care units and technique facility centers to which are associated all human/materail resources

• each patient travels from one unit to anotheraccording to her clinical pathway defining the cares and tests needed

• But also with pools of shared material/humanresources and human resources seconded to otherunits, ...

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Layout of an emergency department

Surgery

Radiology

E.R. triage room Emergency room admissions

Laboratories

E.R. beds Pharmacy Billing/exit

Patient A

Patient B

Patient A (broken leg) proceeds to ER triage, radiology, surgery, bed, pharmacy, billing. Patient B (pacemaker problem): ER triage, surgery, pharmacy, lab, bed, billing.

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Chaptre 2. Facility Layout1. Introduction2. Product layout3. Process layout4. Fixed position layout

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Product Layout

The product layout arranges equipment (departments) in the order of production process flow.

Generally used in mass production such as automobile assembly where the processes are standardized and there is little variation.

Product layout is generally less flexible and requires higher initial equipment costs.

But it minimizes the process cycle time and increases equipment utilisation.

Examples: hospital cafeteria, standardized biological tests.

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Product Layout

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Product Layout

Product layout, known as assembly line balancing problem, is generally determined by the product or service itself.

Most decisions concern

• assignment of basic operations to different workstations

• in order to balance the workloads

• such that each station has approximately the same cycle time, i.e. the time for one item to pass through that workstation (Why)

Two types of problems:

• Using a minimum number of workstations to achieve a given cycle time

• Minimizing the cycle time given the nb of workstations

3

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Product Layout

Station#3Station#1

Station#2

5

46

10

21

20a

hd

5c j

6b

15g

35f

15i

8e k

16l

An example of 3 stations and cycle time 70.

Waste time = ?, workstation cycle time = ?

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Product layout

• Since variability is inherent in health care, the product layout is rarely used in health care other than for supporting activities

• Although the health care process is similar for a patient group with similar diagnosis, the amount of time that patients spend in each process varies greatly. Line balancing is impossible.

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Process layout

The process layout, known as layout by functionality, groups different types of process (departments, equipments) together to provide the maximum flexibility.

Hospital groups together functions such as intensive care, surgery, emergency medecine, and radiology as separate departments

The flexibility allows accomodate the variability of patient flows and times while preserving high utilisation of resources

The downside of a process layout is the large travel time, and high material handling costs. A good layout will reduce this negative impact.

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Process layout

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Process layout

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Closeness-based method

Identify the desireness and undesireness of closeness by closeness rating chart.

Codes for desired closeness:

A – absolute necessary

E – very important

I – important

O – ordinary important

U – unimportant

X – undesirable.

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Closeness rating charting

A

E

AO

U

IE

X

I

E

U

I

U

X

X

1. Nurses’ station

2. Ambulance entrance

3. Patient Room Area

4. Laundry

5. Main entrance

6. Dietary Department

40*80

40*40

remaining

40*80

80*80

40*80

area in m Department

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Using a heuristic rule to design layout.

Step 1. Assign departments to available spaces according desired closeness relationships identified as absolutely necessary or undesirable, i.e. A and X, by starting with the most frequent department in either relationship.

A graph representation of A and X closeness can be built.

Step 2. Consider other departments with relationships E, I, O, U.

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Ambulance entrance

Nurses’Station

Patientroom area

Main entrance

Laundry

DietaryDepartment

A A

X

X

X

A and X closeness representation

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2 Ambulance entrance

1 Nurses’Station

3 Patientroom area

5 Main entrance

4 Laundry

6 DietaryDepartment

A A

X

X

X

A and X closeness representation

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Final layout

5. mainentrance

Amb.entrance

4. L

aund

ry6.

Die

tary

Dep

t.

1 nurses’station

3. Patient Room area

400 m

200 m

5

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Distance and cost-based method

This method tries to minimize the costs or repetitive distances traveled by patients and staff.

Data representing such traffic are represented in a from-to chartwhich represents the nb of trips or flows between departments.

Once the traffic information is identified, departments with most frequent traffics are assigned to adjacent locations.

Informations such as department space requirement, fixed locations, ... can be taken into account.

The problem is highly combinatorial when the nb of department is large and software tools such as CRAFT are necessary.

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Formally, the objective is to

Minimize total cost TC = ij Dij Wij Cij

where

W =[Wij] is the From-To traffic matrix

D =[Dij] is the distance matrix (Manhattan distance or Euclideandistance, why)

C =[Cij] is the unit traffic cost

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Example:

Consider a small hospital of 3 departments A, B, C. Three locations 1, 2, 3 of identical size are available.

Assume that a nurse can walk 100 feet in 30 seconds and earn $48 per hour.

Unit traffic cost Cij = 0.004$ / foot walk

location1

location2

location3

100 feet 100 feet

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Distance matrix (in feet)

Location

From/To 1 2 3

1 - 100 200

2 100 - 100

3 200 100 -

Condensed traffic matrix

Department

From/To A B C

A - 3300 1400

B - 200

C -

Traffic matrix (in trips)

Department

From/To A B C

A - 1000 300

B 2300 - 100

C 1100 100 -

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locationconfiguration 1 2 3 TC

1 A B C 2520$2 A C B 3280$3 B A C 2040$4 B C A 3280$5 C A B 2040$6 C B A 2520$

Dept.B

Dept.A

Dept.C

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Distance measures

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Complexity measures

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Complexity measures

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Distance and cost-based method : example 2

The Walters Company’s management wants to arrange the 6

departments of its hospital in a way that will minimise

interdepartmental material handling costs. They make an initial

assumption (to simplify the problem) that each department is

20x20 feet and that the building is 60 feet long and 40 feet wide.

The process layout procedure that they follow involves 6 steps.

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Step 1. Construct a « from-to-matrix » showing the flow of patients or personnals from department to department (From hospitalinformation systems + future demand forecast).

50 100 0 0 20

30 50 10 0

20 0 100

50 0

0

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Step 2. Determine the space requirements of each department.

Department

1

Department

2

Department

3

Department

4

Department

5

Department

6

room1 room2 room3

room4 room5 room6

Building dimensions and a possible department layout

36


Step 3. Develop an intial schematic diagram (qualitative RELationDiagram) showing the sequence of departments through whichpatients will have to move. Try to place departments of heavy flow next to one another.

1 2 3

45

6

50 30

100

20

50

10 20100

50

7

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Step 4. Determine the cost of this layout with traffic cost of 1€ between adjacent departments and 2€ between non adjacent departments.

Cost = 570 €.

1 2 3

45

6

50 30

100

20

50

10 20100

50

38


Step 5. Try to improve this layout by trial and error to establish a reasonable good arrangement.

Switch departments 1-2 as there is a heavy traffic between dept. 1-3. Cost = 480 €.

Department

2

Department

1

Department

3

Department

4

Department

5

Department

6

room1 room2 room3

room4 room5 room6

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Step 6. Prepare a detailed plan (Space Rel. Diagram) consideringspace or size requirements of each department; that is, arrange the departments to fit the shape of the building and obstables.

M1

M2

M3

M4

M5

M6

M1

M2

M3

M4

M5

M6

Rel. Diagram Space Rel. Diagram

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Quadratic Assignment Approach for identical dept.

Problem: Locate N identical departments on N locations in order to minimize total traffic cost.

QAP model (Quadratic Assignment Approach)

Decision variable : Xij = 1 if dept. i is located at location j

Minimize TC = ijkl cijkl Xik Xjl

Subject to

j Xij = i Xij =1

where cijkl = ijwijdkl

ij = unit moving cost, wij = inter-department flow,

dkl = inter-location distance

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Systematic Layout Planning (SLP)

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45


46


47


48


9

49


50


51


52


53


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The fixed-position layout consists of the fixed service positions where personnel and materials come together to perform the service.

Generally used in industry when the product is either too large or too delicate to more such as airplane assembly.

In inpatient hospital rooms (especially in an intensive care unit), the service position is the patien bed.

The operating table in an operating room is another example.

10

55


Designing a fixed position layout entails positiioning several service positions withing a given area, each of which may require an adjacent but separate support area.

Conflicts about space constraints and timing have to be resolved (suspended x-ray machine and overhead lighting)

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57

Chaptre 2. Facility Layout1. Introduction2. Product layout3. Process layout4. Fixed position layout5. Facility layout supplement

58


Designing a fixed position layout entails positiioning several service positions withing a given area, each of which may require an adjacent but separate support area.

Conflicts about space constraints and timing have to be resolved (suspended x-ray machine and overhead lighting)

Renovation of Mercy Family Health Center

Haji Maryam, Wang Lei, Wong Yuet, Darabi Houshang

University of Illinois at Chicago Mechanical and Industrial Department

842 W. Taylor St. Chicago, IL, 60607, USA

[email protected] [email protected] [email protected] [email protected]

Abstract: The design of healthcare facilities has to provide a comfortable environment for the patient as well as acceptable waiting and processing times to serve the patient. In this paper, we redesign the layout of a real world healthcare facility (Mercy Family Health Center) to explicitly reduce the back and forth flows of the patients in the existing area and consequently reduce the patient’s waiting time. We propose two different alternatives. We use mixed integer programming (MIP) as the basis of our design. Keywords: Facility Layout, forecast Method, MIP algorithm.

1. INTRODUCTION In mid-1990’s a few high-profile medical errors brought healthcare quality and patient safety to the fore. Using engineering tools for advance studies in transportation and financial service areas and the success of these studies bring the healthcare experts starting to take notice that engineering tools have proven effective in the service sector. Thus, the uses of engineering tools-which have long proven useful in the other service industries-, seem to be a good solution for some of the health systems’ problems. Some of these problems are related to the structure of the healthcare systems. Therefore, hospital and healthcare facility design must be sensitive and responsive to the marketplace changes. “Failure to anticipate or respond to the market spells disaster” by Miller, et al. (2002). The need for flexibility is intensified by the technological nature of the healthcare industry. Healthcare facilities must adapt to changing patient populations and changing patient needs. In Janet R. Carpmans’ addressed by Miller, et al. (2002), she calls a dynamic design “a socially responsible health facility design”. The design

approach is positive responses from the users on physical, intellectual, and emotional levels. One of the most important problems that some of the healthcare systems are facing to, is the unnecessary flows of the patient that exist in the healthcare facilities which are mostly related to the inefficient layout design of the facility. In this study, we address this type of problem and by using engineering techniques such as operation research tools, we obtain a solution. Mixed integer programming (MIP) is used to model the problem. The lessons learned from this project can be applied to other real world healthcare facilities. The audience of this paper could be the healthcare administrators and operational managers who have experienced similar problems in their facilities. The organization of this paper is as follows. In section 2, we describe our case study facility, its problems and the solutions that were generated to solve these problems. Two alternatives will be developed and evaluated. In section 3, we conclude the paper.

2. A CASE STUDY

2.1 Mercy Family Health Center Mercy Family Health center is an outpatient clinic serving with compassion, accountability, respect, excellence and quality services. It is located in Chicago and is the first chartered hospital in the state of Illinois. Its vision is to serve the sick and uneducated people in need with quality healthcare regardless of their ability to pay. Its mission is to foster an environment of healing, providing access and needed care with compassion and excellence to the diverse communities it serves. The clinic provides a wide array of primary care services including: Adult Medicine, Women Health Services, Pediatric and Genetic Counseling, and Specialties (allergy, rheumatology, cardiology, etc.). 25% of the residents near the south side of Chicago who need hospitals care come to Mercy Hospital for various medical needs. 2.2 Problem and Causes Mercy Family Health Center has a typical traditional layout (Figure 1). The east side of the building consists of the administrative offices, the pharmacy, and the Pediatrics department with its own receptions’ area and waiting area. The west side of the building consists of the Continuity Medical Clinics (General Practices), OB/GYN, and Specialties Practices with one waiting area and receptions’ area in the middle. This study concentrates on the west side of the building.

Figure 1: The original layout of the Facility The centrally located waiting area is the major problem. Although the clinic is divided into various departments, patients are not classified separately in the waiting area. This causes an uncomfortable situation for the patients who are waiting to be served. Access to the examination

rooms brings a lot of flow in the area which is absolutely unnecessary. Therefore, the patient is served more slowly and the waiting time for the patients is increased. Many patients who come to Mercy for general health care carry contagious disease, i.e. fever, flu. The packed waiting area can be the best place to spread these diseases. It is highly recommended by physicians that Obstetrics patients do not sit among others with contagious diseases since they might be more vulnerable to some viruses. Currently the departments that are closely related are located far away from each other. For example, the eye examination room is set diagonally on the opposite side of the Specialties practice area, which includes Ophthalmology. An eye patient is first served at Specialties area, and then walks across the waiting area to the eye exam room, again back to the Specialties area. It unnecessarily increases the process time and the flow over the area. While other specialties patients have to wait for a longer time to be served. Some of the spaces that are related to the other parts of the hospital facility are unnecessarily located in this area which again increases the flow in the whole clinic area. The administrative personnel of the health care center are aware that old structures not only merely fail to serve the patients adequately but fail in what even the most reluctant healthcare providers have come to recognize as a medical marketplace. The most obvious product in the medical marketplace is excellence in healthcare, and a facility’s reputation for excellence is a strong incentive to healthcare consumers to select that institution over another. The most important factors that influence the potential consumers are the design of the facility and the patient amenities that the design offers. 2.3 Solutions Our first step is to maximize the space in order to enhance flexibility and the capability of handling more patients. This can be reached by removing redundant departments. It also eliminates the unnecessary flows in the clinic area. The second step is to reorganize all the departments. Each department will have its own exam rooms and waiting area. The reason for breaking the waiting area into smaller units is to reduce the patients’ flows and therefore make the clinic operations more efficient. Also, if the related areas become closer, the physicians’ travel time between units

O.B. Specialty

C.M.C I C.M.C II

Waiting area

will be shorter and this will improve the service. A modern healthcare facility is no longer a warehouse for the sick. According to the location of each practice area, separate waiting areas are created for each. Since the patients should stay in the waiting area as brief as possible, the waiting area should be comfortable and cheerful. In this layout, patients are classified and directed to the corresponding waiting area. This is especially desired for the obstetrics clinic to provide patients with privacy and direct attention. Besides decreasing the flow of patients from the waiting area to the exam rooms, this creates a small private healthcare atmosphere. Comfort level increases. Small waiting areas are more accessible for clinicians. It is as if the healthcare center was several private practices combined. Patients are more likely to choose up-to-date with stat-of-the-art clinics over the traditional clinics. The new design emphasizes on creating a friendly, non-threatening, yet functional hospital environment. The new facility layout will be de-institutionalized. According to the relationship between departments and the importance of their closeness a from-to-chart for the flow of the patient (Table 1) was generated. Departments are defined by the following numbers in the chart.

1. COPO room 2. Nursing room 3. O.B. department 4. Specialty department 5. Dummy area 6. Eye exam room 7. E.R. 8. Hallway I 9. Reception area 10. Waiting area for O.B. department 11. Bathroom 12. Waiting area for Specialty department 13. Hallway II 14. C.M.C department I 15. Waiting area for C.M.C 16. C.M.C department II

Flows can be measured quantitatively in terms of the amount of the patients moved between departments. The chart most often used to record these flows is a from-to-chart (Tompkins, et al. 2003). The from-to-chart is a square matrix which lists all departments down the row and across the column following the overall flow pattern. The numbers in table 1 show the average number of the patients between departments per day. This average is calculated regarding to

available information of the number of visitors per day in one year.

Table 1: From-to-Chart 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 0 30 240 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 30 30 0 0 0 30 0 0 30 0 30 30 0 30 3 0 0 0 0 0 0 0 719 719 719 240 0 0 0 0 0 4 0 0 0 0 0 40 0 476 476 0 159 476 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 40 40 0 0 40 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1195 719 1195 476 0 0 0 0 9 0 0 0 0 0 0 0 0 0 719 5 476 759 0 759 0 10 0 0 0 0 0 0 0 0 0 0 180 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 120 180 0 180 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 759 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 759 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 75916 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2.4 Alternative I The first alternative is focused on the renovation of the existing facility in terms of existing number of patients in recent year assuming that the number of the patient will not change over years (fixed number of visits in future). Based on the number of the patients who visits each department in one working day, a value is allocated to the area of their waiting area. It is assumed that each waiting area has to be close to their related department because of their strong relationships. All the operation areas that have interaction with each other should be close enough. Due to the management restrictions, some of the current areas are fixed and cannot be moved. Considering this as assumption, to an MIP model is generated (Tompkins, et al. 2003) to redesign the layout of the clinic. The objective is to minimize the patients’ movements between departments. Layout algorithms can be classified according to their objective functions. There are two basic objectives: one aims at minimizing the sum of flows times distances while the other aims at maximizing an adjacency score. Generally speaking, the former, that is, the “distance-based” objective is more suitable when the input data is expressed as a from-to-chart. In this study, we consider the distance-based objective. Mathematically, the objective function can be written as

ijij

m

i

m

jij dcfz ∑∑

= =

=1 1

min (1)

Where m denotes the number of departments,

ijf denotes the flow from department i to

department j (expressed in average number of patients moved per day between departments),

ijd is the distance from department i to j . In the MIP model distance is measured rectilinearly between the centers of department i and j , and

ijc denotes the relative weight of a unit of

patient moved from department i to department j . We assume that in this specific

problem ijc =1 which means the weights of the flows between the departments are equal. Therefore, the objective function minimizes the total flow of the patients between departments in the facility. Since it is assumed that all the departments of this problem are rectangular, a mixed integer programming (MIP) can be used for solving the addressed problems in this paper. Treating the department dimensions as decision variables, the layout problem can be formulated as follows. The decision variables and the parameters that are used in this model are: Decision variables

i α : The x-coordinate of the center of department i

i β : The y-coordinate of the center of department i

ix ′ : The x-coordinate of the left (or west) side of department i

ix ′′ : The x-coordinate of the right (or east) side of department i

iy ′ : The y-coordinate of the bottom (or south) side of department i

iy ′′ : The y-coordinate of the top (or north) side of department i

xijz =1: If department i strictly to the east of

department j (zero otherwise). yijz =1: If department i strictly to the north of

department j (zero otherwise) +

ijα : If x-coordinate of the center of department

i is in the east of department j

− ijα : If x-coordinate of the center of department

i is in the west of department j +

ijβ : If y-coordinate of the center of department

i is in the north of department j −

ijβ : If y-coordinate of the center of department

i is in the south of department j Parameters

xB : The building length (measured along x-coordinate)

yB : The building width (measured along y-coordinate) M : A large number

liL : The lower limit on the length of department

i uiL : The upper limit on the length of department

i l

iW : The lower limit on the width of department

i u

iW : The upper limit on the width of department i The above parameter and variable definitions lead to the following model:

)( −+−+ +++=∑∑ ijijijijiji j

ijcfzMin ββαα (2)

Subject to:

)(

uiii

li LxxL ≤′−′′≤ for all i (3)

)(

uiii

li WyyW ≤′−′′≤ for all i (4)

)(

uiiiii

li PyyxxP ≤′−′′+′−′′≤ for all i (5)

xii Bxx ≤′′≤′≤0 for all i (6)

yii Byy ≤′′≤′≤0 for all i (7)

iii xx ′′+′= 5.0 5.0α for all i (8)

iii yy ′′+′= 5.0 5.0β for all i (9) −+ −=− ijijji αααα for all ji ≠ (10) −+ −=− ijijji ββββ for all ji ≠ (11)

)1( xijij zMxx −+′≤′′ for all ji ≠ (12)

)1( yijij zMyy −+′≤′′ for all ji ≠ (13)

1≥+++ yji

yij

xji

xij zzzz for all ji < (14)

0, ≥ii βα for all i (15)

0 , , , ≥′′′′′′ iiii yyxx for all i (16)

0 , , , ≥−+−+ijijijij ββαα for all ji ≠ (17)

, yij

xij zz 0/1 integer for all ji ≠ (18)

In this model, we assumed 6 out of 16 departments (Hallway I, Hallway II, Nursing room, COPO, Reception area, and E.R.) are fixed and can not be moved. The objective function includes 148 variables and the constraints are more than 1000 (exactly 1088). It is also assumed that x

ijz and y

ijz variables must

be an integer and binary number. The result for the objective function value (the optimal amount of patient flow) in this model is equal to 876,806.2. The produced layout is shown in figure 2.

E.R.

eye exam rm

Reception BathroomWaiting area for Specialty

Waiting area for O.B.

Waiting area for C.M.C

C.M.C 2C.M.C 1

Hallway

Hallway

SpecialtyO.B.Nursing rm

COPO

Figure 2: New layout result for alternative I 2.5 Alternative II The second alternative is based on the number of the patients that are going to visit the Mercy Family Health Center in the future. The number of the visitors to this center in the future can be calculated based on one of the forecasting methods. Statistical studies in the previous years shows that the number of the visitors of this healthcare clinic behaves as a linear trend. Therefore based on the linear behavior of the flow, double exponential smoothing method (Holts’ Method) (Nahmias, et al. 2005) is used as a forecast method. Table 2 shows the increase in the number of the visitors in general based on the calculated forecast (Table 2).

Table 2: Total number of forecasted future patients visits

Year Demand Forecast 2000 8372 8130 2001 7914 7946 2002 6956 7860 2003 7335 7849 2004 7951 7953 2005 8058 2006 8162 2007 8267 2008 8685

The other three tables show the increases in the number of the visitors in each main department (O.B. Department (Table 3), Specialty Department (Table 4), and C.M.C department (Table 5)). Table 3: Forecasted future Patient visits of O.B.

Department


Table 4: Forecasted future Patient visits of

Specialties Department


Table 5: Forecasted future patient visits of

C.M.C. Department


Based on the forecast numbers of future visits, new space requirements were defined for the areas of different departments and their corresponding waiting areas. All other assumptions are similar to those of Alternative I. Accordingly a new MIP model is generated and solved. The definition of the decision variables and the parameters are similar to those of the first model. The model minimizes the objective function to find the optimal solution for the total flow of the

patients moving between the depatments. Objective function for this alternative includes 144 variables and the constraints are more than 1000 (exactly 1084). The result for the objective function value in this model is equal to 725,473.5. The layout is shown in figure 3.

CO

POO.B.Waiting area for O.B.

C.M.C. 1

Wai

thin

g ar

ea fo

r Sp

ecia

lty

Nursing rm

BathroomWaiting area for C.M.C

Reception

E.R.

Hallway

Hallway

Eye exam rmSpecialtyC.M.C 2

Figure 3: New layout result for alternative II

3. CONCLUSION

In this Study, some of the existing problems in the Mercy Family Health Center are addressed (the large and disorganized waiting area, far distance between the departments that are strongly related and have strong relationships with each other, and finally utilizing the unused space). All these problems increase the flow of the patients between the departments. Therefore, process and waiting time for each patient increase. Consequently, the clinic space becomes an uncomfortable place for patients who are suffering from illness. As we know patient discontent is against the hospital beliefs and policies. To solve the problem, MIP model is used as a tool to redesign the layout of the clinic. Explicitly, we used the objective of minimizing the patient flow. In the first alternative the current flow of the visitors in the recent year is used and based on that the new layout is generated which gives the optimal solution related to the assumptions. In the second alternative, the forecast data is used and with the same assumptions as of the first alternative, another layout is generated with corresponding optimal solution for the total flow of the patient. The result of the first alternative satisfies the hospital request comparing to the budget that they have allocated to renovation of the clinic. Both alternatives explicitly minimize the total flow of the patients as well as implicitly decrease the waiting and service time. In this way the physicians are more available to give service to the patients. However the result of the first alternative satisfied the hospital request,

with a comparison of the two alternatives, we suggested them to use the second alternative instead of the first one. The reason is that the second alternative result can be used in long term and will be less as it considers the growth in the future number of patients.

4. ACKNOWLEDGEMENT We would like to thank Barbara Townsend-Vice President of Business Development, Katherine Freidl-Director of Mercy Family Health Center, and Daniel Vicencio, M.D.-Medical Director of Mercy Family Health Center, for their collaboration with this research study. We appreciate their valuable guidance and helpful suggestions.

REFERENCES Dettenkofer M., Seegers S., Antes G., Motschall E., Schumacher M., and F.D. Daschner, “Does the architecture of hospital facilities influence nosocomial infection rates? A systematic review”, Infection Control and Hospital Epidemiology, 25 (1), 21-25, Jan 2004 Douglas C.H., and M.R. Douglas, “Patient-friendly hospital environments: exploring the patients’ perspective”, Health Expectations, 7 (1), 61-73, Mar 2004. Douglas C.H., and M.R. Douglas, “Patient-centered improvements in healthcare built environments: perspectives and design indicators”, Health Expectations, 8 (3), 264-276, Sep 2005. HicK J.L., Hanfling D., Burstein J.L., DeAtley C., Barbisch D., Bogdan G.M., and S. Cantrill, “Healthcare facility and community strategies for patient care surge”, Annals of emergency Medicine, 44 (3), 253-261, Sep 2004. Mercy Hospital and medical Center, http://www.mercy-chicago.org/ Miller R.L., E.S. Swensson, “Hospital and Healthcare Facility Design”, Second Edition, 2002, W.W. Norton and Company, Inc Nahmias S., “Production and Operations Analysis”, Fifth Edition, 2005, McGraw-Hill Irwin Stevenson W.J., “Operation Management”, Eighth Edition, 2005, McGraw-Hill Irwin Tompkins J.A. White, Y.A. Bozer, and J.M.A. Tanchoco, “Facilities Planning”, Third Edition, 2003, John Wiley & Sons, Inc

1

Chapter 3Healthcare human resource

management

230 septembre 2015

Plan

Introduction

Planning/Scheduling

Resource dimensionning or staffing

330 septembre 2015

Introduction

Hospital

Authorities

Incentives for better healthcare cost control

Users

Increade healthcare demandAging population

Needs to rethink the models and organisations

Quality of care, reduced costs, improved working condition

Merging to benefit from the scale economy

An envisioned strategy

430 septembre 2015

Introduction

Plateau Médico-Technique (PMT)Center of technical facilities

EndoscopiaObstetrics

Interventional

radiology

Surgery Anaesthesia

10+ % of hospital budget

Aim of a regional research

project HRP²

Mutualisation of human resources

530 septembre 2015

Sharing human resources

Serviced’ORL

Serviced’orthopédie

Servicede chirurgie

digestive

Serviced’ORL

Unités d’hospitalisa-

tion

Salles d’intervention SSPI

Serviced’orthopédie


tion

Bloc d’orthopédie


Servicede chirurgie

digestive


tion

Bloc de chirurgie digestive

Bloc d’ORL



Chirurgie ambulatoir

e

Services de

Chirurgie

Urgences chirurgicale

s Unités d’hospitalisatio

n

Soins intensifs

Réanimation

Plateau médico-technique

Retour au domicile

• ORL• Orthopédie• Ophtalmologie

• Urologie• Obstétrique • …

Services de

médecine

• Radiologie• Gastro-entérologie• …


Chirurgie ambulatoir

e

Services de

Chirurgie

Urgences chirurgicale

s Unités d’hospitalisatio

n

Soins intensifs

Réanimation

Plateau médico-technique

Retour au domicile

• ORL• Orthopédie• Ophtalmologie

• Urologie• Obstétrique • …

Services de

médecine

• Radiologie• Gastro-entérologie• …

630 septembre 2015

Sharing human resources

RE-ORGANISATION

Monodisciplinary organization Integrated multidisciplinaryorganization

2

730 septembre 2015

Approach

Generation of

simulation model

Simulation with Infinite capacity

Characteristicsof PMT

of Processesof organisations

Data collection

Extrapolation

Workload profileFor each resource

Performance evaluation

Proposition of improvement

actions

Ajustment of simulationparameters

Dimensioning Human Resources (workforce per time

slot, working time, start time)

Simulation with Finite capacity

Modifying the model

Decision-aid for resource dimensioning and organization

830 septembre 2015

Objectives

Mutualisation of human resources

Design

Accompagner la conception de la nouvelle organisation:

• Dimensionner les ressources humaines

• Objectiver les choix d’organisation

Objectives

Control

Aider à la gestion des pools de personnel mutualisés

• Piloter la performance

• Aider à la planification des ressources

humaines

930 septembre 2015

Sommaire

Introduction

Planning/scheduling

Resource dimensionning or staffing

1030 septembre 2015

Dimensioning human resources

Workforce requirement

Time slots

Workload coverage

Shifts

Workload profile

Phase 1

Evaluate workforce requirement by the

workload profile

Phase 2

Determine a set of shifts covering the workload

profile

1130 septembre 2015

Phase 1: Deriving workload profilePhase 1


workload profile

Prepare the treatment of demand

Forecast demand arrivals

Simulate the system with infinite capacity

Modélisation des processus

Valueing the processes

Organisation des ressources

Workload profile



Generic model

Level of mutualisation

1230 septembre 2015

Phase 1: Deriving workload profile

TriageConsultation

llllll llllll

Exam

llllll

1st consulation

2nd consulation

Examples of process models

Emergency department

Birth deliveryStep 1. Birth delivery in an Operating Room by an obstetric physicianStep 2.1 Recovery in a ward for the womanStep 2.2 If type-2 patient, neonatal care for the newbornStep 2.3 If type-3 patient, NICU and then neonatal care for the newborn

3

1330 septembre 2015


Deriving human resource requirement with a determinstic model

TriageConsultation

llllll llllll

Exam

llllll

1st consulation

2nd consulation

Emergency department1. Activities for each ED patient5 min (triage nurse)15 min (ED physician)0.5 to 2h later5 min with proba. 20% (ED physician)2. Average physician workload per ED patient15 + 5*20% min = 16 min3. Determine arrival rate8h-9h : on average 1112-13 : on average 64. Determine workload profile 8h-9h : 176 min12-13 : 96 min

5 min

15 min

5 min

80%

30 min – 2h

5. Workforce requirement8h-9h : 3 physicians12-13 : 2 physicians

Issues not captured by the simple model1. Uncertainty, 2. Queueing effect

Patients arriving 8-9 are likely to waitmuch longer and even beyond 9h

1430 septembre 2015

Organisation of the resources

Vertical polyvalence

horizontal polyvalence

… … …

Patient ofOR 1

Patient ofOR 2

Patient ofOR 3

Reception

Transfert

Induction

Intervention

Duty of a personal

1530 septembre 2015

Phase 2: Shift constructionPhase 2

Determine a set of shifts covering the

workload profile

Explicit approach

Énumération des vacations

Sélection des vacations

Multiple approachs are available [Partouche, 1998]

Enumerate all shifts

Selection of shifts

Enumeration algorithm of shift patterns

Set covering model

[Dantzig, 1954]

aij {0,1}

1630 septembre 2015

Shift pattern enumeration

Determine the set of shift patterns fulfilling all labor regulation constraints:

• min and max duration of the shift

•Earlist and latest starting time of the shift

•Duration of a break

•Time window of the break

•Number of hours before and after the break

1730 septembre 2015


•Min and max duration (7-8h)

• Earlist and latest date of the start (7-11h)

•Duration of the break (1h)

•Time window of the break (11-14h)

•Number of hours before the break (2h) and after (1h)

avj 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 211 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 02 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 03 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 04 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 06 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 07 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 09 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 013 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 014 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 015 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 016 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 017 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 018 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 019 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 020 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 021 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 022 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 023 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0

1830 septembre 2015

Set covering model

P = 100% P = 80%

Integer linear programming model

Coverage contraintsCoverage contraints

Number of employees of shift i

Cost of an employee working shift i

Mean number of employees needed for period j

Min % of the workload to cover

4

1930 septembre 2015

A hybrid approach

Observations: The workload of the personals is random Covering the mean workload does not garantee the

avoidance of:• Under-capacity due to arrivals greater than average• Over-capacity due to arrivals less than average

Interest of a hybrid approach: Evaluating the real coverage by simulation Integration of two types of costs:

• Personal cost Overtime cost

Determine the right value of P

2030 septembre 2015

Principle of the hybrid approach

Model parameters

Simulation model

Definition of the workload profile

Evaluation of the shifts

Optimization model

PerformanceModification of the

workload profile

Workload coverage optimisation model

1st P-workload profile

Set of optimal shifts

Modified P-

workload profile

1

2

3

4

2130 septembre 2015

Results

250

270

290

310

330

350

60% 65% 70% 75% 80% 85% 90% 95% 100%

P (in %)

Total cost (in K€)

Iteration n°1

Iteration n°2

Iteration n°3

Cost of the solution of the optimisation model with

P = 100 %

Cost of the hybrid approach with

P = 82,5 %

Cost saving 21%

Cleaning personal

CHU de Saint Etienne

2230 septembre 2015

Sommaire

Introduction

Planning human resources


2330 septembre 2015

Planning shared resourcesHuman resources

Better HR planning

Better operations PMT

Personal satisfaction

Planning Anaesthesia nursesIADE - Infirmiers Anesthésistes

ASH

IADEASIBODE

IADE

MAR

MAR

Planning Anaesthesists MAR - Médecins Anesthésistes

Planning pharmacy personals

2430 septembre 2015

<

Problems of hospital personal planning

Staff planning

Determine working days & working time

Meet contraints

Repeat the same weekly or monthly sift pattern

Easy implementation

Rigid & weak adaptabily to changes

New planning for each period

Flexible

Time consuming

Noncyclic PlanningCyclic planning

Cost Soft constraints violated Equity

[Blöchliger, 2004]

[Valouxis, 2000]

MARIADE

5

2530 septembre 2015

Planning anaesthesia nurses IADE

Day-regular (DR)

Day-urgent ( DU)

Night-urgent (NU)

Supervision recovery (SR)

8 H – 16 H

8 H – 20 H

20 H – 8 H

9 H – 17 H

Demand coverage

Working time regulation

Work on night & weekend

Succession of activities

ContraintsShifts

Assign IADE to all day and night activities of a week

Maximise the equity among employees

Shared ressources In operating rooms In recovery rooms

Both urgent and elective surgeries Work of the day and night Polyvalent personal running on all duties

CH de Valence

2630 septembre 2015

Planning anaesthesia nurses

Criteria Meet working time regulation and personal preferences

(vacations, ...) Maximise the equity.

Penality score (pénibilité or arduousness perceived by staff):

Lun Mar Mer Jeu Ven Sam DimDay regular 1 1 1 1 1Day Urgency 1,2 1,2 1,2 1,2 1,2 1,4 1,4

Ning Urgency 1,4 1,4 1,4 1,4 1,4 1,6 1,6Recovery 1,6 1,6 1,6 1,6 1,6

IADE

Minimise the total penality score deviation

max minZ P P Minimise

2730 septembre 2015


ContraintsHard

C1: Nb IADE needed per time slot C2: working time per day less than 12h C3: weekly working time around 38h but less than 48h C4: no more than 3 nights per employee per week.

Soft C5: Saturday DU (resp. NU) implies Sunday DU (resp. NU) and no

work on Monday and Tuesday. C6: shift succession constraints to ensure at least 11h rest per day :

• NU during the week implies no working the next day • DU during the week implies NU or no working the next day (due to twice more

DU shift demand than NU)

IADE

2830 septembre 2015


VariablesXijk = 1 if nurse i assign to shift k on day j

Contraints C1: # of nurses per shift

per day

C2: daily working time less than 12h

C3: weekly working time less than 48h

C4: no more than 3 nights a week.

IADE

7

( )7 6

3w

ij NUj w

X

11

K

ijkk

X

1

N

ijk jki

X b

7

max7 6 1

max( 48 , 1, )

w K

k ijk ij w k

i

n X T R

T h R regime

2930 septembre 2015


Contraints

C5: Sat. DU (resp. NU) implies Sun. DU (resp. NU) and no work on Mon. and Tue.

C6 Shift succession

DU-NU followed by no working

DU followed by NU

(7 )( ) (7 )( ) (7 1) (7 2)1

1K

i w JU i w NU i w k i w kk

X X X X

IADE

(7 1) (7 ) 0i w k i w kX X

( ) ( ) ( 1)1

1K

ij JU ij NU i j kk

X X X

( ) 1 ( ) 0ij JU i j NUX X

3030 septembre 2015


Criterion

IADE

max min

max1 1

min1 1

min

J Kk ijk

j k i

J Kk ijk

j k i

P P

p XP

R

p XP

R

6

3130 septembre 2015

Planning anaesthesia nurses IADE

JR = Day RegularJU = Day UrgencyNU = Night UrgencySS = Supervision Recovery

3230 septembre 2015

Contraintes obligatoires

Contraintes souples

Planning anaesthesists MAR MAR

Assign MAR to activities and half-day of a week

Pre-operation: Consultation

Per-operation: Anesthesia

Post-operation: Visit

Minimum demand coverage

Daily and weekly working time regulation

No working post-duty

No isolated half day working

ContraintsActivities

Minimise the number of soft constraints violated

An integer programming model

Extension of the scope: Activitiess pre, per and post operations

Assignment by half-day Need to take into account the competencies

Competency requirment of demands

Maximum demande coverage

Continuity of post-operation visits

3330 septembre 2015

Objective MAR

# MAR assignment outside their specialty

# of isolated half working day

# of post-visit continuity violated

Deviation from the maximum demand coverage

Minimise

Weighting factors

3430 septembre 2015

Experimentation: CH de Valence

Resolution of 5 problems, over 7, 14 and 28 days

By two solvers CPLEX GLPK

5 « specialties »: 4 specialist groups 1 covering all other specialties

Demands: Pre-operation: min and max Per-operation: according to the surgery planning (rule: one MAR

for 2 operating rooms) Post-operation: fixed according to the workload profile

List of duties

MAR

Weights λ1 λ2 λ3 λ4

Problem 1 1 1 1 1

Problem 2 1 2 1 1

Problem 3 2 2 1 1

Problem 4 1 1 1 2

Problem 5 1 4 1 1

3530 septembre 2015

Example of results: Problem 1

Lun Mar Mer Jeu Ven Lun Mar Mer Jeu VenSpécialité MAR am pm am pm am pm am pm am pm Spécialité MAR am pm am pm am pm am pm am pmSpécialité 1 2 per 0 0 0 0 0 0 0 0 0 0 Spécialité 5 1 pre 0 0 0 0 1 1 0 0 0 0

post 0 1 0 1 0 1 0 1 0 1 per 0 0 0 0 0 0 1 1 0 0Chirurgie 3 per 0 1 0 0 0 0 1 1 0 0 Toutes 2 pre 1 0 0 0 1 0 1 0 1 0viscérale post 0 0 0 0 0 0 0 0 0 0 spécialités per 0 0 1 0 0 0 0 0 0 0et urologie 4 per 1 0 0 0 1 1 0 0 0 0 3 pre 1 0 1 1 1 0 0 0 1 0

post 0 0 0 0 0 0 0 0 0 0 per 0 0 0 0 0 1 0 0 0 15 per 0 0 0 0 0 0 0 0 0 0 4 pre 0 1 1 1 0 0 0 0 0 0

post 0 1 0 1 0 1 0 1 0 1 per 0 0 0 0 0 0 0 0 0 1Spécialité 2 6 per 0 0 1 0 1 0 1 0 0 0 5 pre 0 0 0 0 0 0 1 0 0 0

post 0 1 0 1 0 1 0 1 0 1 per 0 0 0 0 0 0 0 0 0 0Orthopédie et 7 per 0 0 0 1 0 1 0 0 0 0 6 pre 0 0 0 0 0 0 0 0 0 0neurochirugie post 0 0 0 0 0 0 0 0 0 0 per 1 0 0 0 0 0 0 0 1 0

8 per 0 0 0 0 0 0 0 1 0 0 7 pre 0 0 0 0 0 0 0 0 0 0post 0 0 0 0 0 0 0 0 0 0 per 1 1 1 0 1 0 0 1 1 1

Spécialité 3 10 per 1 0 0 0 0 0 0 0 0 0 8 pre 0 1 0 0 0 0 0 0 0 0post 0 1 0 1 0 1 0 1 0 1 per 1 0 0 1 1 1 1 0 1 0

ORL, 11 per 0 0 0 0 0 1 0 0 0 0 9 pre 0 0 0 0 0 0 0 1 1 1Ophtalmologie post 0 0 0 0 0 0 0 0 0 0 per 1 1 1 1 0 0 1 0 0 0et chir. Ambu 12 per 0 1 0 0 1 0 0 0 0 0 10 pre 0 0 0 0 0 0 0 0 0 0

post 0 0 0 0 0 0 0 0 0 0 per 0 0 1 0 1 0 1 0 1 0Spécialité 4 4 per 0 0 0 0 0 0 0 0 1 0 11 pre 0 0 1 0 0 0 0 0 0 0

post 0 0 0 0 0 0 0 0 0 0 per 0 1 0 1 0 0 0 0 0 1Maternité 8 per 0 0 1 0 0 0 0 0 0 1 12 pre 1 0 0 0 0 1 0 0 0 0gynécologie post 0 0 0 0 0 0 0 0 0 0 per 0 0 0 0 0 0 0 0 0 0obstétrique 11 per 0 0 0 0 0 0 0 0 1 0 13 pre 0 0 0 0 0 0 1 0 0 0et pédiatrie post 0 0 0 0 0 0 0 0 0 0 per 0 0 0 0 0 0 0 0 0 0

13 per 1 0 1 0 1 0 0 0 1 0 14 pre 0 0 0 0 0 0 0 0 0 0post 0 1 0 1 0 1 0 1 0 1 per 0 0 0 0 0 0 0 1 0 0

14 per 1 1 1 1 1 1 1 0 0 0post 0 0 0 0 0 0 0 0 0 0

15 per 0 1 1 1 0 0 1 1 1 1post 0 0 0 0 0 0 0 0 0 0

Objective = 21

MAR

3630 septembre 2015

Planning pharmacy personal

Motivated by the restructuring of CH Villefranche (2 times bigger)

Need of a decision aid tool to generate pharmacy personal planning

Personals: 21 employees (4 pharmacists, 7 pharmacy assistants for preparation)

Various duties : • gestion, appro et distribution des médicaments

(armoires informatisées ou non)• Guichet• Préparation chimio• Gestion de gaz médicaux

Objectives: robust planning, reactivity to perturbations, equity between personals (rotation on all duties)

7

3730 septembre 2015


A set of n tasks on a H days horizon Parameters of a task:

Task duration pi Frequency Ti Contraints of the days Date of execution ti (if fixed) Earliest Date ri Latest Date di Min delay between two executions

A set of m resources of different competencies Bij = 1 if resource j can execute task i Soft contraints: breaks, workload balancing Decisions :

Assign tasks to resources Planning the execution scheduling (day-date)

3830 septembre 2015


Example of data

1

Chapter 3

Workforce scheduling

- 2 -

Plan

• Introduction

• Days-off scheduling

• Shift scheduling

• Cyclic staffing probme

• Crew scheduling

- 3 -

Intoduction

• Workforce allocation & personnel scheduling deal with the arrangement of work schedules and the assignment of personnel to shits to cover the demand for resources that vary over time.

• In service environments the operations are often prolonged and irregular, and the staff requirements fluctuate over time.

• The schedules are typically subject to various constraints dictated by equipment requirements, union rules, ...

- 4 -

Days-off scheduling

An elementary personnel assignment problem.

The problem is to find the minimum number of employees to cover a 7-days-a-week operation so that the following constraints are satisfied:

1. The demand per day, nj, j = 1, ..., 7 (Sunday to Saturday), is met

2. Each employee is given k1 out of every k2 weekends off

3. Each employee works exactly 5 out of 7 days (Sunday to Saturday)

4. Each employee works no more than 6 consecutive days.

Days j 1 2 3 4 5 6 7Sun Mon Tues Wed Thurs Fri Sat

Requirement 1 0 3 3 3 3 2 k1/k2 = 1/3

- 5 -

Days-off scheduling : lower bounds

Weekend constraint

(k2 – k1)W >= k2 max(n1, n7)

Where W is the minimum size of the workforce

Total demand constraint:

5W >= nj

Maximum daily demand constraint:

W >= max (n1, ..., n7)Days j 1 2 3 4 5 6 7

Sun Mon Tues Wed Thurs Fri SatRequirement 1 0 3 3 3 3 2

2 1 7

2 1

max ,k n nW

k k

k1/k2 = 1/3

7

15j

j

W n

- 6 -

Days-off scheduling : Algorithm

W = min workforce, n = max(n1, n7)

Step1. (Schedule the weekends off)

Assign the 1st weekend off to the first W-n employees

Assign the 2nd weekend off to the second W-n employees

This process is continued cyclically.

S S M M T W T F S S M M T W T F S S M M T W T F S1 x x x2 x x3 x x

2

- 7 -


uj = W- nj, j= 2, ..., 6, uj = n – nj, j = 1, 7

Step2. (Determine the additional off-day pairs)

Construct a list of n pairs of off days, numbered 1 to n.

Choose day k such that uk = max(u1, ..., u7)

Choose day l (l k), such that ul > 0; if ul = 0 for all l k, set l = k

Add the pair (k, l) to the list and decrease uk and ul by 1.

Repeat the process n times.

Days j 1 2 3 4 5 6 7Sun Mon Tues Wed Thurs Fri Sat

uj 1 3 0 0 0 0 0

(2, 1), Sunday-Monday

(2, 2), Monday-Monday (non distinct pairs)

- 8 -


Set i = 1Step3. (Categorize emplyees in week i)Type T1 : weekend i off, no days needed during week i, weekend i+1 offType T2 : weekend i off, 1 off day needed during week i, weekend i+1 onType T3 : weekend i on, 1 off day needed during week i, weekend i+1 offType T4 : weekend i on, 2 off days needed during week i, weekend i+1 on

|T3| + |T4| = n, |T2| + |T4| = n (as n people working each weekend)Pair Each employee of T2 with one of T3

Step 4 (Assign off-day pairs in week i)Assign the n pairs of days, starting from the top off the list as follows:First assign pairs of days to the employees of T4Then, to each employee of T3 and his companion of T2, assign the one of T3 the

earliest day of the pair.Set i = i+1 and return to step 3.

- 9 -


S S M M T W T F S S M M T W T F S S M M T W T F S1 x x x2 x x3 x x

Week 1 : T2 = 1, T3 = 2, T4 = 3

Week 2 : T2 = 2, T3 = 3, T4 = 1

Week 3 : T2 = 3, T3 = 1, T4 = 2

S S M M T W T F S S M M T W T F S S M M T W T F S1 x x x x x x x2 x x x x x x3 x x x x x x

The schedule generated by the days-off scheduling algorithm is always feasible.

- 10 -

Shift scheduling

A cycle (one day, one or several weeks) is fixed.

Each work assignment pattern over a cycle has its own cost.

Problem:

m time intervals/periods in the predetermined cycle

bi personnel are required for period i

b different shift patterns, and each employee is assigned to one and only one pattern

(a1j, a2j, ..., amj) = shift pattern j with aij = 1 if period i is a work period.

cj = cost of patern j

Determine the number of employees of each pattern in order to minimise the total cost.

- 11 -

Shift scheduling

1 1 2 2

1 1 2 2

minimise ...subject to

... , 1,...,integer.

n n

i i in n i

i

c x c x c x

a x a x a x b i m

x

- 12 -

Shift scheduling

Pattern Hours of Work Total Hours Cost1 10 AM to 6PM 8 50 €2 1pm tp9pm 8 60 €3 12pm to 6pm 6 30 €4 10am to 1pm 3 15 €

Hours Staffing requirement10am-11am 311am-12am 412am-1pm 61pm - 2 pm 42 pm - 3pm 73pm-4pm 84pm-5pm 75pm-6pm 66pm-7pm 47pm-8pm 78pm-9pm 8

What if overtime is allowed?

3

- 13 -

Shift scheduling

• The integer programming formulation of the general personnel scheduling problem (with arbitrary 0-1 A matrix) is NP-hard

• The special case with each column containing a contiguous set of ones is easy and the solution of the LP-relaxation is always integer.

- 14 -

Cyclic staffing problem

The objective is to minimise the cost of

assigning people to an m-period cyclic schedule

so that

sufficient workers are present during time period i, in order to meet requirement bi,

and each person works a shift of k consecutive periods and is free the other m-k periods.

1 0 0 1 1 1 11 1 0 0 1 1 11 1 1 0 0 1 1

, min /1 1 1 1 0 0 11 1 1 1 1 0 00 1 1 1 1 1 00 0 1 1 1 1 1

A cX AX B

(5,7) cyclic staffing

Each column is a possible shift

- 15 -

Cyclic staffing problem : algorithm

Step 1. Solve the linear relaxation of the problem to obtain xi’

If (xi’) are integer, STOP

Step 2. Form two linear programs LP’ and LP’’ from the relaxation of the original problem by adding respectively the constraints:

1 2 1 2

1 2 1 2

... ' ' ... '

... ' ' ... ' ¨.

n n

n n

x x x x x x

and

x x x x x x

LP’’ has an optimal solution that is integer

If LP’ does not have a feaible solution, then the solution of LP’’ is the optimal solution

If LP’ has a feasible solution, then it has an optimal solution that is integer and the best of LP’ and LP’’ solutions is the optimal solution.

- 16 -

Cyclic staffing problem : algorithm

31 0 0 1 141 1 0 0 1

, 61 1 1 0 040 1 1 1 070 0 1 1 1

3.6, 4.8,5.5,3.7,5.243.3

A b

c

Optimal

- 17 -

CREW SCHEDULING

• Crew scheduling problems are very important in transportation especially in airline industry

• Consider a set of m jobs, or flight legs.

• A flight leg is characterized by a point of departure and a point of arrival, as well as an approximate time interval during which the flight has to take place.

• There is a set of n feasible and permissible combinations of flight legs that one crew can handle, round trips or tours.

• Each round trip j, has a cost cj.

• Crew schedule determines round trips to select in order to minimize the total cost under the constraint that each flight leg iscovered exactly once by one and only one round trip.

- 18 -

CREW SCHEDULING

• Each column in the A matrix is a round trip, and each row is a flight leg that must be covered exactly once by one round trip.

• Set partitioning problem.

1 1 2 2

1 1 2 2

minimise ...subject to

... , 1,...,integer.

n n

i i in n i

i

c x c x c x

a x a x a x b i m

x

4

- 19 -

CREW SCHEDULING

2 1

4

3 5

depot

52

32

3 4

2

2 12

Truck routing network

route 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

cj 8 10 4 4 2 14 10 8 8 10 11 12 6 6 5

1 0 0 0 0 1 1 1 1 0 0 0 0 0 0

0 1 0 0 0 1 0 0 0 1 1 1 0 0 0

0 0 1 0 0 0 1 0 0 1 0 0 1 1 0

0 0 0 1 0 0 0 1 0 0 1 0 1 0 1

0 0 0 0 1 0 0 0 1 0 0 1 0 1 1

1

Chapter 4Capacity planning of emergency

health services

Plan

• ED operations

• A simple shift scheduling model

• Introduction to Markov chains

• Queueing models and key results

• Queueing models of Emergency departments

• Hospital capacity planning by M/M/c models

• Physician Staffing for Emergency Departments with Time-Varying Demand

• Exact prediction of waiting time by uniformization

ED of Ruijin Hospital

Top 1 hospital in Shanghai

+12000 outpatient visits / day

+800 emergency patients / day

An comprehensive Emergency Department (ED)

18 physicians of 11 different specialties

110 nurses

1 EICU, 3 dedicated OR, 1 observation room, X dedicated wards

Dedicated diagnostic facilities (CT, ECG, ...), pharmacy

Almost no transfers from ED to other parts of the hospital

ED organization at Ruijin Hospitalre

ssus

cita

tion

area

Exam phar

mac

y

Consultation area

Ambu

lanc

e Pay

Tria

ge

Examinations:X ray, B-mode ultason, CT, Blood test, Urine test

11 ED specialties:• Internal medecine• General surgery• Obstetrics• Neurology• Neuro-surgery• Burn

Urology• Pneumology• Ophthalmology &am

otorhinolaryngology• Dermatology• Orthopedics

ED patient flows of Ruijin HospitalEmergency Severity Index (ESI)

• 5-level triage acuity system.

• Acuity 1 : the most serious

• Acuity 5 : the least urgent and often represents office or clinic-type patients.

Frequent registration/pays

Second visit to physicians (D) after exams

f1 f2 f3 f4

f5

5% 19% 37% 22%

17%

a = 24.29 p/h, b = 0.625 p/h

Beyond ED

2

ED capacity requirement Continuity of care 24h

a day.

At least one physician in ED at anytime.

Significant fluctuation of patient arrival during a day

0

5

10

15

20

25

30

35

1 3 5 7 9 11 13 15 17 19 21 23

num

ber

of p

atie

nts

per

hour

hour of dayArrival rates from the emergency

department in Ruijin hospital

ED capacity

Current capacity plan does not take into account fluctuated demand.

Crowding during peak hours;

Low human resource utilization during low arrival periods

Question : how to better match capacity and demand by exploiing the flexibility in human resource management (Working time, organisation, polyvalence, mutualisation, …)?

Dilemma : resource utilisation vs service level

0

1

2

3

4

5

0

5

10

15

20

25

30

Arrival rate

Physician #

Question 1: How to plan ED shifts to optimize the overall service quality (waiting time)

Question 2: How many ED physicians are needed to meet waiting time targets1: less than 20 minutes for at least 80% of patients2: less than 1h for at least 95% of patients

Typical capacity mgt issues of the chapter

23

Similar issues for maternity services

Basic question: How many beds?Dilemma: occupancy vs service level?

Typical questions:

• Assuming the target occupancy level of 75%, what is the probability of delay for lack of beds for a hospital with s = 10, 20, 40, 60, 80, 100, 150, 200 beds.

• What is the size of an obsterics unit (nb of beds) necessary to achieve a probability of delayed admission not exceeding 1% while keeping the target occupancy level of 60%, 70%, 75%, 80%, 85%?

Plan

• ED operations








1210 septembre 2019


Workforce requirement

Time slots

Workload coverage

ShiftsWorkload profile

Phase 1


workload profile

Phase 2

Determine a set of shifts covering the workload

profile

3

1310 septembre 2019

Phase 1: Deriving workload profilePhase 1


workload profile

Prepare the treatment of demand

Forecast demand arrivals

Simulate the system with infinite capacity


Valueing the processes


Workload profile



Generic model

Level of mutualisation

1410 septembre 2019


TriageConsultation

llllll llllll

Exam

llllll

1st consulation

2nd consulation

Examples of process models

Emergency department

Birth deliveryStep 1. Birth delivery in an Operating Room by an obstetric physicianStep 2.1 Recovery in a ward for the womanStep 2.2 If type-2 patient, neonatal care for the newbornStep 2.3 If type-3 patient, NICU and then neonatal care for the newborn

1510 septembre 2019


Deriving human resource requirement with a determinstic model

TriageConsultation

llllll llllll

Exam

llllll

1st consulation

2nd consulation

Emergency department1. Activities for each ED patient5 min (triage nurse)15 min (ED physician)0.5 to 2h later5 min with proba. 20% (ED physician)2. Average physician workload per ED patient15 + 5*20% min = 16 min3. Determine arrival rate8h-9h : on average 1112-13 : on average 64. Determine workload profile 8h-9h : 176 min12-13 : 96 min

5 min

15 min

5 min

80%

30 min – 2h

5. Workforce requirement8h-9h : 3 physicians12-13 : 2 physicians

Issues not captured by the simple model1. Uncertainty, 2. Queueing effect

Patients arriving 8-9 are likely to waitmuch longer and even beyond 9h

1610 septembre 2019

Organisation of the resources

Vertical polyvalence

horizontal polyvalence

… … …

Patient of

OR 1Patient of

OR 2Patient of

OR 3

Reception

Transfert

Induction

Intervention

Duty of a personal

1710 septembre 2019

Phase 2: Shift constructionPhase 2

Determine a set of shifts covering the

workload profile

Explicit approach

Énumération des vacations

Sélection des vacations

Enumerate all shifts

Selection of shifts

Enumeration algorithm of shift patterns

Set covering model

[Dantzig, 1954]

aij {0,1}

1810 septembre 2019


Determine the set of shift patterns fulfilling all labor regulation constraints:

• min and max duration of the shift

•Earlist and latest starting time of the shift

•Duration of a break

•Time window of the break

•Number of hours before and after the break

4

1910 septembre 2019


•Min and max duration (7-8h)

• Earlist and latest date of the start (7-11h)

•Duration of the break (1h)

•Time window of the break (11-14h)

•Number of hours before the break (2h) and after (1h)

aij 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 211 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 02 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 03 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 04 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 06 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 07 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 09 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 012 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 013 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 014 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 015 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 016 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 017 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 018 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 019 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 020 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 021 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 022 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 023 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0

2010 septembre 2019

Set covering model

P = 100% P = 80%

Integer linear programming model

Coverage contraintsCoverage contraints

Number of employees of shift i

Cost of an employee working shift i

Mean number of employees needed for period j

Min % of the workload to cover

2110 septembre 2019

A toy example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240,5 0,5 0,5 0,5 0,5 1 4 10 10 8 4 8 6 6 4 6 10 15 8 6 4 2 0,5 0,5

You are asked to help improving the nurse planning of an Emergency Department (ED). From the historic data, you have the following demand forecast on the number arrivals at the ED during a day.

From the statistics, 60% of the ED patients are regular patients and need only 15 minutes of the nursing care. However 40% of the ED patients are true emergency patients and require about 1h nursing care at ED before transfer to the wards.

You are asked to :

• Derive the workload profile of a typical day.• Enumerate all possible shift patterns. Shifts of 8h start either 7h-9h (20€/h),

or 15h-17h (22€/h), or 23-01h (25€/h). Shifts of 12h start either 7h-9h (21€/h) or 19h-21h (23€/h).

• Determine the optimal shifts with a coverage P = 100%, 80%, et 120%. What is the utilization ratio of the nurses

Solvable by excel solver

2210 septembre 2019

A toy example

hour arrival unit loadhourly

load1 0,5 0,55 0,2752 0,5 0,55 0,2753 0,5 0,55 0,2754 0,5 0,55 0,2755 0,5 0,55 0,2756 1 0,55 0,557 4 0,55 2,28 10 0,55 5,59 10 0,55 5,5

10 8 0,55 4,411 4 0,55 2,212 8 0,55 4,413 6 0,55 3,314 6 0,55 3,315 4 0,55 2,216 6 0,55 3,317 10 0,55 5,518 15 0,55 8,2519 8 0,55 4,420 6 0,55 3,321 4 0,55 2,222 2 0,55 1,123 0,5 0,55 0,27524 0,5 0,55 0,275

0

1

2

3

4

5

6

7

8

9

1 6 11 16 21

Workload profile

2310 septembre 2019

Observations

The workload of the personals is random

Covering the mean workload does not garantee the avoidance of:

– Under-capacity due to arrivals greater than average

– Over-capacity due to arrivals less than average

The set covering model neglects the queueing effectand is not appropriate for service level measurement

• Patients arriving in peak periods are likely to wait much longer

It neglects the patient over flow between periods

• Patients arriving in peak periods are likely to be served in later periods

Cover demand of peak periods in peak periods might lead to exagerated human resource requirement.

Plan

• ED operations








5

25

Stochastic processes

Server

Queue

N(t) : nb of customersin the queue

Customer arrival

A stochastic process {Xt, t T} : a random variable defined on the same state space E and evolving as time t goes on.

Example: the queue length N(t) at time t

26

Stochastic process

Discrete events

Continuous event

Discrete time

Continuous time

Memoryless

A CTMC is a continuous time and memoryless discrete event stochastic process.

Continuous Time Markov Chain (CTMC)

27

Homogenuous CTMC

Definition : A CTMC {X(t), t > 0} is homogeneous iff

P[X(t+s)= j X(t) = i] = P[X(t+s)= j X(t) = i] = pij(s)

Homogeneous memoryless:In reliability, we only say "a machine that does not fail at age t is as good as new"

Only homogeneous CTMC will be considered first.

28

Continuous Time Markov Chain (CTMC)

Definition : a stochastic process with discrete state space and continuous time {X(t), t > 0} is a continuous time Markov Chain (CTMC) iff

P[X(t+s)= j X(u), 0≤u≤t] = P[X(t+s)= j X(t)], t, s, j

Memoryless:In a CTMC, the past history impacts on the future evolution of the system via the current state of the system

29

Exponential distribution

T = EXP()• Probability density function (pdf) :

– fT(t) = dFT(t)/dt = e-t

• Distribution Function (cdf) : – FX(t) = P{X ≤ t} = 1 -e-t

• Mean : E[T] = 1/ • Standard deviation: T = 1/ • Coeficient of variation: cvT = / E[T] = 1

often called event rate (failure rate, repairrate, production rate, ...)

• Memoryless remaining life : P[T – t ≤ x| T ≥ t] = P[T ≤ x]

pdf

cdf

30

(Homogenuous) Poisson process

A Poisson process is a stochastic process N(t) such that• N(0) = 0• N(t) increments by +1 after a time T (called inter-arrival

time) random distributed according to an exponential distribution of parameter .

An arrival process is said Poisson if the inter-arrival times are exponentially distributed.

6

31

Key conditions for memoryless

Memoryless times

• All times (activity times, repair times, lifetimes, …) are exponentially distributed.

• X = EXP(m-1) : P(X x) = 1 – e-x/m, E[X] = m, X = m

• Memoryless remaining life : P[X – t ≤ x| X ≥ t] = P[X – t ≤ x]

Memoryless events

• All events (arrivals, machine failures, …) occur according to a POISSON process

• A POISSON event e of frequency (also called event rate)

time between occurrences of e = EXP()

• Memoryless remaining time to event at any independentobservation time = EXP()

32

A single server queue

Exponential service time at

rate

Queue


Poisson customer arrival at rate

33

Markov chain representation

s1

s2

s3

s4

12Freq. of event e12



state

34



rate

Queue



0 1 2 3

35

Steady-state distribution

Steady-state distribution = probability distribution afterinfinite time

i = probability of being in state i in steady-state

Alternative definition (under ergodicity condition)

i = percentage of time of state i over infinite time

36

Determination of the steady-state distribution

s1

s2

s3

s4

12Probability flow

13Probability flow

41Probability flow

Probability flow of event eij=12=frequency of event eij

Flow balance equation

Total flow in = Total flow out

Holds for any state or subset of states

Normalisation equationi i = 1

41 = 12 + 13

7

37


0 1 2 3

Flow balance equation

Total flow in = Total flow out

Normalisation equationi i = 1

Online derivation

38

Steady state distribution of a CTMC

Conditions for existence of steady state :

1. Strongly connected Markov chain (irreducible)

2. Finite number of states or stable system (positive recurrent)

Plan

• ED operations








40

Definition of a queueing system

Customer arrivals

Departure of impatient customers

Departure of served customers

• A queueing system can be described as follows:"customers arrive for a given service, wait if the service cannot start

immediately and leave after being served"

• The term "customer" can be men, products, machines, ...

41

Notation of Kendall

Kendall notation of queueing systems

T/X/C/K/P/Z

– T: probability distribution of inter-arrival times– X: probability distribution of service times– C: Number of servers– K: Queue capacity– P: Size of the population– Z: service discipline

In this course,– K = : unlimited queue capacity– P = : infinity population– Z = FIFO: First In First Out service

42

Notation of Kendall

T/X/C– T: probability distribution of inter-arrival times– X: probability distribution of service times– C: Number of servers

•T or X can take the following values:

– M : markovian (i.e. exponential)– G : general distribution– D : deterministic

M/M/1 = Markovian arrival & service single server queueM/M/n = Markovian arrival & service n-servers queue

8

43

Little’s Laws

For any stable system,

L = TH×W(Number = Throughput Delay)

where• L : average number of customers in the system• W : average response time• TH : average throughput rate

Queueing system

LTH TH

W44



rate

Queue


0 1 2 3

Stability condition :

45

M/M/1 queueStationary distribution:

n = n(1-), n≥0

where = / is called traffic intensity.

Ls = Number of customers in the system = ) = /()

Ws = Sojourn time in the system = ) = 1/()

Lq = queue length = 2/() Ls

Wq = average waiting time in the queue = /() Ws

= departure rate =

Server utilization ratio =

Server idle ratio = P0 = 1 -

P{n > k} = Probability of more than k customers = k+1

46

M/M/c queue – Erlang C system

N(t) is a birth and death process with• The birth rate .• The deadth rate is not constant and is equal to N(t) if N(t) C and C

if N(t) > C.Stability condition : < c.

N(t) : number of customers in the system

Exponentially distributed service tim

Poisson arrivals

47

11

00 ! ! 1

n cc

n

a a

n c

, 0n

n c ca

nc

Stationary probability distribution:a offered load c a/c traffic intensityn = an/n! 0, 0 < n c


0 1 2 c

c

C+1

c c

48

C(c,a) = Waiting probability of an incoming customer= c + c+1 + ...

wq = random waiting time of a customer (Moment generating funct)

T = Waiting time target

(T) = Service level= P(wq ≤ T)

1

0

! 1,

1

! ! 1

c

cn cc

n

a

cC c a

a a

n c

0, with probability 1 ,

, with probability ,q

C c aw

EXP c C c a

1 , c TT C c a e

Erlang C formula


9

49


50

M/M/c with impatient customers –Erlang B

• Similar to M/M/C queue except the loss of customerswhich arrive when all servers are busy.

0 1 2 c

c

51


Steady state distribution :a offered load c traffic intensityn = an/n! 0, 0 < n c

Percentage of lost customers = C

Server utilization ratio = (1 – C) /C

Insensitivity to service time distribution:n depends on the distribution of service time T onlythrough its mean, i.e. with = E[T]

1

00 !

nC

n

a

n

52


0

!,

!

c

c c nn

a cB c a

a n

1 ,a B c a

Erlag loss function or Erlang B formula= Percentage of lost customers or overflow probability

Accepted load

53

Normal approximation for staffing Erlang Loss systems

Condition: high offered load (a > 4) and high targeted service level

N(t) = number of patients : approximately normally distributed

E[N(t)] a

In M/M/∞ system, N(t) =d POISSON(a), i.e. E[N(t)] = a, Var[N(t)] = a

Square-Root-Staffing-Formula for a delay probability

c a a

1N a c a

P Delay P N t c Pa a

Where is the cdf of the standard normal distribution


54

0

!,

!

c

c nn

a cB c a

a n

Computation issues of Erlang B and C formula

1

0

! 1,

! ! 1

c

n cc

n

a

cC c a

a a

n c

, 1 / , : the reciprocalR c a B c a

1, 1 1,R c a R c a

11, 1 1 1,C c a R c a

1, ,B c a R c a 0, 1B a

0, 1C a

!!! recursion for the same offered load !!!

a

c c

Loss proba Waiting proba

10

55


arrival rate 10service rate 3

offerred load a 3,333target wait time 0,1 Erlang C KPI Erlang B

queue length

# in system wait time syst time On time Blocking accepted

c R B C Lq Ls Wq Ws proba proba load0 1 1 1 1 0

1 3,3333 1,3000 0,7692 3,3333 - 4,76 - 1,42 - 0,47 - 0,14 - 5,71 0,7692 0,7692

2 1,6667 1,7800 0,5618 2,0833 - 5,20 - 1,87 - 0,52 - 0,18 - 2,10 0,5618 1,4607

3 1,1111 2,6020 0,3843 1,2165 - 12,16 - 8,83 - 1,21 - 0,88 - 0,34 0,3843 2,0523

4 0,8333 4,1224 0,2426 0,6577 3,2886 6,6219 0,3289 0,6622 0,4615 0,2426 2,5247

5 0,6667 7,1836 0,1392 0,3267 0,6533 3,9867 0,0653 0,3987 0,8019 0,1392 2,8693

6 0,5556 13,9305 0,0718 0,1482 0,1853 3,5186 0,0185 0,3519 0,9334 0,0718 3,0940

7 0,4762 30,2540 0,0331 0,0613 0,0557 3,3890 0,0056 0,3389 0,9796 0,0331 3,2232

8 0,4167 73,6096 0,0136 0,0231 0,0165 3,3498 0,0016 0,3350 0,9943 0,0136 3,2880

Plan

• ED operations








21

Poisson arrival

The assumption of Poisson arrivals has been shown to bereasonable for unschedulinged patients (Young 1965). Example : arrivals to Obstetrics, ICU, ED

Theoretical basis : Poisson approximation of large number of independent binomial trials

Most commonly used arrival process in modelling service systems and hown empirically to a good approximation (police, fire, EMS, bank, call centers, …

Overestimation of the waiting time for scheduled patients (surgery, …)

For outpatients, bad for scheduled ones or good for walk-ins

Young, J.P. 1965. Stabilization of Inpatient Bed Occupancy through Control of Admissions. Journalof the American Hospital Association 39: 41–48

21

Exponential distribution

The performance predicted by M/M/c model is fairly insensitveto the exponential assumption provided the coefficient of variation of the service times is close to 1.

Evidence from Kingman’s approximation LoS of obsteric patients : Av = 2.9d, Cv = 1.04 (Green &

Nguyen, 01) LoS of ICU patients : Av = 18d, Cv = 1.1 – 1.6 (Green &

Nguyen, 01). Underestimation of the congestion ED consultation: Av = 24-30min, Cv = ??? (Green et al. 2006) Outpatient: Cv = 0,35 – 0,85 (Cayirli & Veral 03) Assumptions widely used in service system operations

2 2/ / / /

2G G s M M s A T

q q

C CW W

21

Wait or abandon

M/M/c model assume patients wait till been served. Good for obsteric and ICU patients that cannot be placed off

service and so often do wait. Good also for best hospitals such as Ruijin for which patients

do wait for a bed Questionable assumption for ED services which have

significant ratio of LWBS (Left Without Been Seen)

21

Servers and service time

The servers can be : physicians, nurses, beds, operating rooms, diagnositic equiment

The service time depends on the capacity planning perspective.

Physician capacity for ED patients: Service time = consultation time (in minutes) Intra-day fluctuation of arrival is important

Bed capacity of obsteric patients: Service time = LoS (in days) Intra-day fluctuation is not important but intra-week

one is

11

21

Basic queueing models

Single queue for multiple identical servers

Drawback of the ED queue : problem with the same-doctor rule

Doct 1

Doct N

Bed 1

Bed 2

Bed N

EDobsetricICU

21

Erlang-R models

G. B. Yom-Tov, A. Mandelbaum, Erlang-R: A Time-Varying Queue with Reentrant Customers, in Support of Healthcare Staffing, MSOM, 2014Drawbacks : problem with the same-doctor rule return patients usually need different consultation time

1

2

Needy (s servers)rate

1-p

pContent (delay)rate

ArrivalPoisson t

21

Queueing network models

N. Izady, D. Worthington, Setting staffing requirements for time dependent queueing networks: The caseof accident and emergency departments, EJOR, 2012Same drawbacks as the Erlang-R model.

Plan

• ED operations








21

An introductory example

A hospital is exploring the level of staffing needed for a booth in the local mall, where they would test and provide information on the diabetes. Previous experience has shown that, on average, every 6.67 minutes a new person approaches the booth. A nurse can completetesting and answering questions, on average, in twelve minutes.

Assuming s = 2, 3, 4 nurses, a hourly cost of 40€ per nurse and a customer waiting cost of 75€ per hour waiting.

Determine the following: patient arrival rate, service rate, overallsystem utilisation, nb of patients in the system (Ls), the average queue length (Lq), average time spent in the system (Ws), average waitingtime (Wq), probability of no patient, probability of waiting C(c,a), total system costs.

66


arrival rate 9

service rate 5

offerred load a 1,800

M/M/c queue KPI

Erlang formulaqueue length

# in system wait time syst time Cost

c R B C Lq Ls Wq Ws pi_00 1 1 1

1 1,8000 1,5556 0,6429 1,8000 - 4,05 - 2,25 - 0,45 - 0,25 - 0,80 - 263,75

2 0,9000 2,7284 0,3665 0,8526 7,6737 9,4737 0,8526 1,0526 0,0526 655,5263

3 0,6000 5,5473 0,1803 0,3547 0,5321 2,3321 0,0591 0,2591 0,1460 159,9088

4 0,4500 13,3274 0,0750 0,1285 0,1052 1,9052 0,0117 0,2117 0,1616 167,8873

12

22

An introductory example

performance mesure 2 nurse 3 nurses 4 nursesPatien arrival rate 9 9 9service rate 5 5 5

Overall system utilisation 90% 60% 45%L (system) 9,47 2,33 1,91Lq 7,67 0,53 0,11w (system) - in hours 1,05 0,26 0,21Wq - in hours 0,85 0,06 0,01

no patient probability (idle) 0,05% 14,60% 16,16%patient waiting proba 85,26% 35,50% 12,85%

Total system cost € per hour 655 160 168

23

Target occupancy level of obsterics units

Obsterics is generally operated independently of otherservices, so its capacity needs can be determined separately.

Good fit of a standard M/M/s queueing model Most obsterics patients are unscheduled -> Poisson

arrivals. CV of LOS is typically very close to 1 -> exponential

service time.

How many Beds for an obstetrics unit

24

Dilemma efficiency vs service quality

« Optimal » bed occupancy level = 85% (from Green, « How many hospital beds »)Basis for bed capacity decision by government & hospitalmanagementSince obsterics patients are considered emergent, the American College of Obsterics and Gynecology (ACOG) recommends thatoccupancy levels of obsterics units not exceeding 75%. Delay targetNo standard target but Schneider suggested that the proba of delay for an obstetrics bed < 1%Schneider, D. 1981. A Methodology for the Analysis of Comparability of Services and Financial Impact of Closure of Obstetrics Services. Medical Care 19: 395–409

Bed capacity of maternity services

24

Q1 : Assuming the target occupancy level of 75%, what isthe probability of delay for lack of beds for a hospital withs = 10, 20, 40, 60, 80, 100, 150, 200 beds.

Lesson : For the same

occupancy level, the probability of delaydecreases with the size of the service.

Size matters!!!


0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 50 100 150 200

Pdelay

Bed #

25

Q2 : What is the size of an obsterics unit (nb of beds) necessary to achieve a probability of delay not exceeding1% while keeping the target occupancy level of 60%, 70%, 75%, 80%, 85%?

Lesson : Achieving high occupancylevel while having small

probability of delay is onlypossible for obsterics unit of

large hospitals.

Capacity cut should be made with clear understanding of

the impact. Simple and naiveanalysis based on average

could lead to bad decisions.


26

Impact of seasonalityConsider an obsterics unit with 56 beds which experiences a significant degree of seasonality with occupancy level varying from a low of 68% in January to about 88% in July.

What is the probability of delay in January and in July?

If, as is likely, there are several days when actual arrivals exceed the month average by 10%, what is the probability of delay for these days in July?

Lesson : Capacity planning should not be based only on the yearly average. Extra bed capacity should be planned for predictable demand increase during

peak times.

Bed capacity and seasonality

13

27

Impact of clinical organisationConsider the possiblity of combining cardiac and thoracic surgery patients as thoracic patients are relatively few and require similar nursing skills as cardiac patients.The average arrival rate of cardiac patients is 1,91 bed requests per day and that of thoracic patients is 0,42. No additional information is available on the arrival pattern and we assume Poisson arrivals. The average LOS (Length Of Stay) is 7,7 days for cardiac patients and 3,8 days for thoracic patients.What is the number of beds for cardiac patients and thoracic patients in order to have average patient waiting time for a bed E(D) not exceeding 0,5, 1, 2, 3 days? What is the number of beds if all patients are treated in the same nursing unit?Delay in this case measures the time a patient coming out of surgery spends waiting in a recovery unit or ICU until a bed in the nursing unit is available. Long delays cause backups in operating rooms/emergency rooms, surgery cancellation and ambulance diversion.

Bed capacity reducing through merging

27

Lesson : Personal and equipment flexibility and service pooling can achieve higher

occupancy level and reduction of beds.

However, priority given to one patient group could significantly degrade the waiting time of other patients if all treated in the same nursing unit.

Bed capacity reducing through merging

27

Staffing Emergency Department under Service Level Constraints

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

0,5 0,5 0,5 0,5 0,5 1 4 10 10 8 4 8 6 6 4 6 10 15 8 6 4 2 0,5 0,5

You are asked to help improving the nurse planning of an Emergency Department (ED). From the historic data, you are able to obtain the following demand forecast on the number arrivals at the ED:


27

0

2

4

6

8

10

12

14

16

1 6 11 16 21

Patient arrivals


Goal 1: Planning min cost shifts to meet loss probability target (<5%, 1%) (Erlang B)

Preliminary goals :

• Derive the workload profile of a typical day.

• Enumerate all possible shift patterns. Shifts of 8h start either 7h-9h (20€/h), or 15h-17h (22€/h), or 23-01h (25€/h). Shifts of 12h start either 7h-9h (21€/h) or 19h-21h (23€/h).

Goal 2: Planning min cost shifts to meet waiting time targets (Erlang C)1: less than 20 minutes for at least 80% of patients2: less than 1h for at least 95% of patients


Pros :

• Simple analytical closed formula available.

• Quite robust to modelling errors

• Can be extended to network structures (ED, ICU, Wards, …)

• Adapted to time varying demand with SIPP (Stationary Independent Period by Period) or Lag SIPP of Green et al.

Cons: Not applicable when peak arrivals are significantly higher than

maximum possible capacity -> overflow

Pros and Cons

14

Plan

• ED operations








ED model

Doct 1

Doct pt

ED

0

10

20

30

40

50

60

1 3 5 7 9 11 13 15 17 19 21 23

Number of arrival patients

Nb of physicians changing according to the shift scheduling

ED model

Key variables

△ Length of each period

λt # of patients arrived in period

ut # of patients served in period

pt # of physicians in period

qt # of patients overflowed

Doct 1

Doct pt

ED

0

10

20

30

40

50

60

1 3 5 7 9 11 13 15 17 19 21 23


Basic assumptions

A1. We simplify the ED service process as a single-stage multi-server queuing system: patients arrive, wait in a common queue, consult with a physician and then leave the ED.

A2. Patients arrive according to a Poisson process at rate λt and are served on a first come first serve (FCFS) basis.

A3. The consultation times are exponentially distributed of rate μ.

A4. The consultation time is short with respect to the period length. If the ED is not overloaded (λt<pt), steady state is reached within the period and each patient consultation begins and completed in t.

A5. No patient leaves without being served (LWBS).

A6. The ED has sufficient capacity to clear the workload at the end of the day.

Waiting time without overloaded perio

The main problem with this approach is that the stability condition might not hold for periods of peak arrival in which the limited number of physicians is not sufficient

M/M/s2,

1cW c

1

0

1

,! ! ! 1

c k cc

ck

c c c

c c k c

Use standard steady-state waiting time of stable M/M/s queues of arrival rate and c servers:

15

Basic Waiting time approximation

Rely on two simply yet efficient ideas.

Idea 1: use different waiting time estimations for patients served in a period and overflow patients.Av. waiting time of patients served in t WM/M/s(ut /Δ, pt)

Total waiting time of overflow patients to t+1 Foverflow(qt, 𝜆t)

Why ??? Inter-arrival times of last arrivals

1

2

1

11, if

2,

1,

2

t

t

qt t

t ti t toverflow

t t

tt t t

i t

q qi q

F q

q i q otherwise


Idea 2: Rather than employing sophisticated nonstationary queuing analysis, we transform this performance evaluation problem into an optimization problem by selecting ut and qt to minimize the total patient waiting time.

11

1 1 1

1

M/M/s

,...,1 min

, \{1

,

}

0

,

, ,

, ,

T

Tt t

t t t t

T

t

t

t

overflowt t t

t

t

u u

t

t

APP W

q u

q q u t T

q

u p t T

u W Fp q

t

u

T

W t T

u q

this simple idea is surprisingly efficient because the “artificial decision” variables ut and qt

closely match the relatedsimulated performance measures


11

1 1 1

1

M/M/s

,...,min

, \{1}

0

,

, ,

, ,

,

T

Tt t

t t t t

T

t t

overflowt tt t t

t

u u

t

t

W

q u

q q u t T

q

u p t T

u W F qu p

t T

W t T

u q

Key variables

△ Length of each period

λt # of patients arrived in period

ut # of patients served in period

pt # of physicians in period

qt # of patients overflowed

Improving Waiting time approximation

Drawbacks of APP1

APP1 relies on the assumption of the stationary arrival of all patients served in each period.

While this assumption is reasonable for patients arriving in the period, it is not reasonable for overflow patients.

Overflow patients are already waiting at the beginning of the period and are served consecutively by physicians

Improving Waiting time approximation Improving Waiting time approximation

1 1

11

2

t t t t

tt

u p u pW

p

16

Improving Waiting time approximation

1 1

1 M/M/s1

, ,2

t t t toverflowt

t t t t t tt

u p u puW u u W p F q

p


11

1 1 1

1

11

1 1

1 M/M/s

,...,2 min

, \{1}

0

,

min , ,

1

2

, , ,

, ,

T

Tt t

t t t t

T

t t

t t t

t t t t

tt

overflowtt t t t t

t

u u

t

APP W

q u

q q u t T

q

u p t T

u q u t T

u p u pW

p

uu u W p F q

t T

t T

u q

Goodness of the approximations

Simulation

APP1 APP2

Waiting time DevWaiting

time Dev

Day1 73.569±0.277 93.050 20.94% 80.104 8.16%

Day2 137.842±0.371 187.587 26.52% 149.687 7.91%

Day3 84.518±0.157 98.885 14.53% 86.658 2.47%

Day4 101.240±0.269 137.112 26.16% 111.346 9.08%

Day5 73.634±0.221 92.699 20.57% 78.595 6.31%

Day6 96.436±0.326 133.269 27.64% 109.196 11.69%

Day7 86.053±0.265 111.835 23.05% 95.394 9.79%

Avg. 93.327 122.062 23.54% 101.569 8.11%

Total waiting times for the actual staffing of Ruijin Hospital

Goodness of the approximationsTotal waiting times for the actual staffing of a larger ED

Simulation

APP1 APP2

Waiting time Dev Waiting time Dev

Day1 166.933±0.420 197.974 15.68% 174.926 4.57%

Day2 152.995±0.566 175.738 12.94% 155.648 1.70%

Day3 181.325±0.281 198.916 8.84% 178.834 -1.39%

Day4 109.462±0.354 131.546 16.79% 117.607 6.93%

Day5 206.368±0.729 246.464 16.27% 209.708 1.59%

Day6 163.323±0.432 197.623 17.36% 173.120 5.66%

Day7 154.806±0.539 181.155 14.55% 161.460 4.12%Avg. 162.173 189.917 14.61% 167.329 3.08%

Goodness of the approximationsNb of patients served per period

0

5

10

15

20

25

1 3 5 7 9 11 13 15 17 19 21 23

APP1

APP2

served

patients at each period

Time0

5

10

15

20

25

30

35

40

1 3 5 7 9 11 13 15 17 19 21 23

APP1

APP2

Simulation

served patien

ts at ea

ch period

Time

Goodness of the approximationsHourly patient watiting time

0

2

4

6

8

10

12

14

16

1 3 5 7 9 11 13 15 17 19 21 23

APP1

APP2

Simulation

Hourly patient waiting time

Time0

5

10

15

20

25

30

35

40

1 3 5 7 9 11 13 15 17 19 21 23

APP1

APP2

Simulation

Hou

rly pa

tien

t waiting

tim

e

Time

17

Goodness of the approximationsNumber of overflow patients

0

5

10

15

20

25

30

1 3 5 7 9 11 13 15 17 19 21 23

APP1

APP2

Simulation

Overflow patients per hour

Time0

5

10

15

20

25

30

35

40

45

1 3 5 7 9 11 13 15 17 19 21 23

APP1

APP2

Simulation

Overflow patients per hour

Time

Staffing model

A7. Each physician has only one working shift in one day, i.e., a physician’s daily working time is continuous and cannot be interrupted.

A8. The shift length is more than LBD and less than UBD but can start at any time.

A9. The total working time of all N physicians should not exceed the ED physician time budget TW

A10 (hand-shaking). For each period t with new physicians starting their shifts, there should be at least one physician working both periods t‒1 and t.

Staffing model

Decision variables:

st: number of physicians starting their shifts in period t

et: number of physicians completing their shifts at the beginning of period t

11 : min Tt tMIP W Staffing model11 : min T

t tMIP W

1

1 1 1

12

1

1

1

1

1

1

: min

1

, \{1} 2

, 3

, 4

1, 5

0, 6

, 7

, 8

, , ,

Tt t

T

t i ii

T

tt

T

tt

t t

T

t tt

t LBD

t t ii t

t UBD

t ii t

t t t

t

MIP W

p s e p

p s e p t T

s N t T

p t T

p s t T

s e t T

p s e t T

p e t T

e s

T

p t T

W

st: nb starting at t

et: nb completing at t

pt # of physicians

N: total # of servers

LBD : shift lower bound

UBD : shift upper bound

TW : physician time budget

△ Length of a period

(t) = 1 + (t mod T )

Staffing model

The model should be combined with an evaluation method.

Feasible shifts can be easily derived.

Two mathematical models MIP1 and MIP2 are obtained by combining the staffing model with the two approximation methods APP1 and APP2.

The models can be linearized.

MIP models can only be solved exactly for small-size problems.

An VSN algorithm is proposed in the paper to solve realistic size problems.

11 : min Tt tMIP W Comparison of optimal vs actual shifts11 : min T

t tMIP W

0

10

20

30

40

50

60

1 3 5 7 9 11 13 15 17 19 21 23


0

1

2

3

4

5

6

7

8

1 3 5 7 9 11 13 15 17 19 21 23

MIP2-VNS staffingReal staffing

Nb of physicians

Actual staffing does not consider the hand shaking constraint while recognizing its necessity

18

Comparison of optimal vs actual shifts11 : min Tt tMIP W

1.6 3.0

4.9 8.4

17.6

34.7

61.3

4.0 8.2

14.0

24.0

40.3

66.7

133.6

0

20

40

60

80

100

120

140

0.8 0.9 1 1.1 1.2 1.3 1.4

VNS Staffing

Actual Staffing

Arrival rate

Mean waiting time(min.)

4.0

8.7

26.8

9.2

19.7

53.0

0

10

20

30

40

50

60

0.8 0.9 1

VNS Staffing

Actual Staffing


Arrival rate

Sensitivity wrt arrival rate

Ruijin day1 Larger H

Day4 23.4

12.0

4.9 3.2

1.7 1.1 0.7

51.5

24.8

13.8

9.7 5.0 3.5 2.3

0.0

10.0

20.0

30.0

40.0

50.0

60.0

8 9 10 11 12 13 14

VNS Staffing

Actuall Staffing


physician number

Comparison of optimal vs actual shifts11 : min Tt tMIP W

Sensitivity wrt physician number

Ruijin day1

Larger HDay4

Plan

• ED operations








Uniformization

Problem of general Markov chains:

• Each Continuous Time Markov Chain ischaracterized by the transition rates ij of all possible transitions.

• The sojourn time Ti in each state i is exponentiallydistributed with rate (i) = j≠i ij, i.e. E[Ti] = 1/(i)

• Transitions different states are unpaced and asynchronuous depending on (i).

Any continuous-time Markov chain can be converted to a discrete-time chain through a process called« uniformization ».

Uniformization by example

1 0

1 0

Uniformized Markov chain by adding fictitious events

Each state has the same event rate of

Unpaced event change at rate in state 1 and in state 0


1 0

Uniformized Markov chain

Each state has the same event rate of

L = EXP(), M = EXP(),

Time in state i : Ti = min(L, M) = EXP()

Transition proba: • p10|T1 = P[L=T1|T1] = /()• p01|T2 = P[M=T2|T2] = /()

19


1 0

Governed by two independentrandom processes:

• A single Poisson event clock at rate ()

• Independent state transition

P

EXP()

If 1, real event wp

If 0, real event wp

Poisson clock

Uniformization

In order to synchronize (uniformize) the transitions at the same pace, we choose a uniformization rate

MAX{(i)}

« Uniformized » Markov chain with

•transitions occur only at instants generated by a common a Poisson process of rate (also called standard clock)

•state-transition probabilities

pij = ij /

pii = 1 - (i)/

where the self-loop transitions correspond to fictitious events.

Uniformization

S1 S2

a

b

S1 S2

a/1-a/

b/

1-b/

CTMC

DTMC by uniformization

Step1: Determine rate of the states

(S1) = a, (S2) = b

Step 2: Select an uniformization rate

≥ max{(i)}

Step 3: Add self-loop transitions to states of CTMC.

Step 4: Derive the corresponding uniformized DTMC

S1 S2

a

b

Uniformized CTMC

-a -b

Uniformization

Rates associated to states

Uniformizatin Rate = +

Uniformized ED queue of ct servers

Uniformization rate = uniformized event clock rate

• γt =λt+ct×μ

Transition probability (state = nb of patients):

...

ct×μ

λt

μ (ct－1)×μ

0 ct... ct-1

ct×μ

ct-2

2μ μ(ct－2)×μct×μ

λt λt λt λt

, 1 , 1

min , , if 1

, if 1

1 , if

0, if 1

t t

t t

iji i i i

c i j i

j ip

p p j i

j i


State at k-th uniformization event

Where ki is the probability of i patients in the system at k-th uniformization event.

...

ct×μ

λt

μ (ct－1)×μ

0 ct... ct-1

ct×μ

ct-2


λt λt λt λt

( )q t i

1,0

1

k i kj jij

k k

p

P

Matrix form

20


State at the beginning of the next period

Where N(t) = Poi(t) is the number of uniformization events in period t.

...

ct×μ

λt

μ (ct－1)×μ

0 ct... ct-1

ct×μ

ct-2


λt λt λt λt

( )q t i

0

0

1!

1!

t

t

n

ti niN t i

n

n

tnN t i

n

et E

n

et E

n

Matrix form


Total expected waiting time in the period

Where WT is the total waiting time in period t.

...

ct×μ

λt

μ (ct－1)×μ

0 ct... ct-1

ct×μ

ct-2


λt λt λt λt

( )q t i

0 !

tn

t

n

eW t E WT N t n

n


Poisson process : given n events in period t, the event times are order statistics of n i.i.d. rv Xi = U(0, )

( )q t i

0 0

0 0 0

0 0

1

1 !

where 1 !

t

t

n

kj tk j

nnt

kj tn k j

k t kjk j

n

tk

n k

E WT N t n j cn

eW t j c

n n

B j c

eB

n n


...

ct×μ

λt

μ (ct－1)×μ

0 ct... ct-1

ct×μ

ct-2


λt λt λt λt

( )q t i

max max

0 0

N q

k t kjk j

W t B j c

Uniformized ED queue of ct servers ( )q t i

max

10 0

0

Algorithm A1: Probability distribution and waiting time evaluation

Step 1. Initialization : , ( ), 0, ( 1) 0

Step 2. For n=0 to Nmax,

Step 2.1: , ( 1) (

tt

q

n t njj

B e t W t t

W t B j c t t

1

11 1

1)!

Step 2.2: ,1 !

t

t

n

tn

n

tn n n n t

e

n

eP B B

n

max max max

0 0 0

1 ,!

1 !

t

t

nN N qt

n k t kjn k j

n

tk

n k

et W t B j c

n

eB

n n

Conclusions

The uniformization method has been combined with a meta-heuristic for weekly physician shift scheduling.

It has also been extended to evaluate the probability of meeting a waiting time target.

1

Chapter 5

Surgery planning

- 2 -

Agenda

• Introduction

• Operating theatre

• Surgery planning

• Open scheduling

• Bloc scheduling

• Daily Surgery scheduling & real-time control

- 3 -

Introduction

• Surgery interventions are organised around some expensive technical facilities (plateau médico-technique) : operating theatres (operating rooms, recovery rooms), imaging services, biology labs, sterilization facilities.

• The central part is the operating theatre where the efficiency in terms of Cost-Quality-Delay is a must.

• Mutation from a monospecialty with ad hoc organization to a multi-specialities with better organisation due to budget constraints and more strict safety regulations.

• Operating theatre consumes about 10% of hospital budget and constitutes a melting pot between different systems and different actors with different visions of its operation.

• The reforms of French health system imposes operating theatres to meet some criteria of efficient management that the actors are not prepared to.

- 4 -

Context

The operating theatre is the heart of a hospital:

Consultations of surgeons, anaesthesists, medical units, imaging facilities, biology labs, stretchers, sterilization, emergercy departments, …

A flexible and rigorous organization

A management of « patient flow »

- 5 -

Context

Patient arrivals

Waitinglists

Transfer

Leave the hospital

Surgery & Recovery

- 6 -

Context

Five-level planning framework(Vissers et al. 2001. A framework for production control in health care

organizations. Production planning & control, 12(6), 591-604)

o Strategic planning (2-5 years)

o Patient volumes planning & control (1-2 years)

o Resources planning & control (3 months - 1 year)

o Patient group planning & control (weeks - 3 months)

o Patient planning & control (days - weeks)

2

- 7 -

Context

Hospital: a profit maximizer or a cost minimizer

o Department

Surgeon group: surgeons, pathologies

Ward: a fixed number of beds, accommodating its patients for recovery

o Surgical center

Operating rooms: shared by all surgeon groups in a time-phased pattern

Operating block: the smallest time unit for which an operating room can be assigned to a

surgeon group for performing surgeries

o Patient: elective inpatients

Patient grouping: an iso-process grouping procedure

pathology, surgeon group, surgery duration, the length of stay (LOS), rewards, etc.

Patient case mix:

the number of patients of every patient group that can be treated annually

- 8 -

Context

Case mix planning problem

S: surgeon groupP: patient groupORs: operating rooms

How many beds areallocated to each ward?

How ORs are assigned tovarious surgeon groups?

How many patients fromeach patient group can be treated annually?

- 9 -

Context

Master Surgery Scheduling

- 10 -

Context


- 11 -

..h ..

17h 00

16h 00

15h 00

14h 00

13h 00

12h 00

11h 00

10h 00

9h 00Monday

Patient 7

OR 1

Patient 4

Patient 11

Patient 3

Patient 1

Patient 5

Patient 19

Patient 2

Patient 20

Patient 17

Patient 8

Patient 15

Patient 6

Patient 12

Tuesday Wednesday

OR 2OR 1 OR 2

Surgery planning and scheduling

Context

- 12 -

Plan

• Introduction



• Open scheduling

• Bloc scheduling


3

- 13 -

Operating theatre : macros processes

• Capacity Planning/Allocation consists in determining (over a semestre to 1 year):

• Number of surgery blocs per surgeon,• OR sessions of the operating rooms,• Modes of control for periods of over- or under- capacity

• Surgery Programming includes:• Determination of a preliminary surgey program,• Determination of the modes of anaesthesia,• Management of the operating rooms, equipments, materials and

ancillaries,• Assignment of personals,• Management of surgery program changes till day D-1,• Determine the « final » surgery program

• Regulation consists in managing the adjustments during the day D concerning the over-usages and the exchanges of surgery blocs, personal assignment, allocation of material resources.

- 14 -

• Registration allows to record the set of interventions performed a day D (name of the operators, durations of the interventions, natures of the incidences, … ).

• Feedback loop a deux objectifs :• réajuster la planification en fonction de l'activité constatée, des files

d'attente estimées, des capacités des unités de soins, des effectifs non médicaux disponibles, …,

• réévaluer les durées d'intervention par praticien et type d'intervention (nécessaires à la programmation)


- 15 -

Allocation programming supervision

D-7 / D-1Year NSemestre S

D

registration

D

Feedback loop

Year N+1Semestre S+1

Monthly

Mission : superviser les différentes catégories intervenants pour optimiser et sécuriser la prise en chargeFréquence : quotidienne

Mission : adaptation des processus et rappels formalisés pour le respect des processus validésFréquence : Mensuelle

Weekly

« operating theatreboard »

« Commission des utilisateurs

du bloc »Mission : adaptation des vacationsFréquence : Annuelle (voire semestriel)

« Supervision cell »

« Coordonnateur »

Missions : arbitrer les priorités médicale relatives à la prise en charge immédiate (= du jour)

« Chef du bloc »

Missions : veiller à l’application des règles, contribuer à l’évolution des règles pour améliorer l’efficience, la sécurité et les conditions de travail.


- 16 -

Operating theatre: Components

Operating theatres (mono disciplinary or shared)

ED operating theatres

Recovery rooms

Induction rooms

Endoscopia operating rooms

Obstetric operating theatre

Stretchers

Obstetric labor rooms

Interventional radiology ORs

ED

Sterilisations

Wards

Two scopes:

- 17 -

WardsOperating theatre

OR1

OR2

ORn

Recovery

ICU

Operating theatre: trajectory

- 18 -

WardOperating theatre

OR

OR

OR

ICU


Recovery

4

- 19 -

WardsOperating theatre

OR

OR

OR

Recovery

ICU


- 20 -

Operating theatre: some observations

Turn dedicated operating rooms (OR) into shared polyvalent ones improves the economic efficiency.

Induction outside the OR (in an induction room) improves the OR usage

Keep the right balance between the surgery volume target and all human and material resources.

SFAR + ASA recommendation: 1.5-2 recovery beds per OR

- 21 -

Operating theatre: performance indicators

Three types of patients concerning the surgery programming:

• Elective patients programmed at D-8 : elective surgery patients of week W are known the week W-1 (in general, before Thursday noon) to allow the validation of the surgery program of the operating roomsand the personal planning,

• Semi-urgent patients programmed at D-1: patients known the daybefore their intervention

• Patients not yet known at the beginning of D.

A good surgery program depends directly on the capacity to anticipate the demands of different operators. This anticipation requires the knowledgeof the patients to be operated at least one week before.

Health professionals agree that a ratio of 80 to 85% of patients known at D-8 is signal of efficient surgery programming.

Of course, one has to take into account emergency surgeries and the catchment area and target market of the hospital.

- 22 -

Operating Room time usage:

• Hospital OR time provision - « temps de mise à disposition » (TMD), a decision of hospital management based on the openingtimes of each room – OR sessions,

• OR timed used - « temps réel d'occupation des salles » (TROS), part of the TMD actualy used for an intervention, i.e. interval frompatient arrival in the room to end of cleaning of the room

• Conventional OR time « temps conventionnel MeaH », base OR times defined by Agency MeaH to benchmark different hospitals. For each OR :

• Conventional Day OR time of 10 hours (08h30 -18h30)• Conventional «continuity of care duty » of 14 hours (18h30 - 08h30)


- 23 -

Ratio 1 (allocation) : Hospital OR time provision / MeaH convention68 - 90% with one at 111%.

Ratio 2 (programing & regulation) : Time used / Time programmed45% - 77%, mean 62%. National objective (CTN) : 75 - 80%.

Ratio 3 (productivity) : Timed used / MeaH convention31% - 63%.


- 24 -

Surgery occupation time - TROS :• T1 patient preparation time : patient arrival to induction,• T2 induction time: induction to incision,• T3 surgery intervention time,• T4 duration of bandage,• T5 : cleaning.TUC = time needing a medical specialist (operator and anaesthestist).


5

- 25 -

For each « Day » conventional surgery:

• T1 – close to mean12mn.• T2 - 3 hospital groups: < 20mn [3], mean (20mn) [4], >30mn [1]• T3 - 3 hospital groups: < 50mn [3], mean (55mn) [4], >100mn [1]• T4 - 3 hospital groups: < 11mn [4], mean (11mn) [3], 18mn [1]• T5 - 3 hospital groups: < 10mn [1], mean (14mn) [4], >14mn [3]

• Total TROS - 3 hospital groups : 1h15 - 1h30 [3], 1h30 - 2h [4], 3h [1]

• Time with medical personal TUC (T2+T3) =66% TROS• Time with surgeon T3 = 48% TROS

This explains the general feeling of « waste time » of the operators and their wish to operate in 2 ORs simultaneously.


- 26 -

OR time provision (TMD) and overtime

Ratio of overtime is a significant capacity regulation issue.

General agreement (CTN), 2% overtime seems unavoidable

Most hospitals visited by MeaH found ways to improve their overtime ratio and hence reducing overtime cost.


- 27 -

OR time allocated to physicians - Temps de vacation (TVO)• TVO = arrival of the first patient to departure of the last patient• TVO does not include OR closing (at the end of the day) or bio cleaning

between morning and afternoon surgery sessions.• This concept is « surgeon oriented » and corresponds to time during which

his/her activities are possible

Allocated OR time usage - Temps réel d’occupation des vacations (TROV)TROV = actual TVO used + standard bio cleaning after each intervention


- 28 -

Analysis of OR staff occupancy (not including physicians)

Time in hospital of a personal : Temps de Présence d'un Professionnel (TPP)

TTP =

Time in OR (TTPS : Temps de Présence du Personnel en Salle)

+ waiting between interventions

+ working on other tasks (administrative, storing devices, checking materials, …).

Profession agreed goal : TTPS = 80% TTP


- 29 -

Human resources: • Operators (surgeons) assurant l'acte opératoire (leur nombre est un

indicateur de la complexité de la planification et de la programmation),

• anaesthesia by anaesthesists and anaesthesia nurses (IADE)• Bandage, instrumentist and operation attendance by specialised

nurses of operating rooms (IBODE),• Cleaning and patient transportation by hospital attendants and

stretchers,• Team supervision, programming regulation and secretary by lead

nurse.

Personal Efficiency or productivity : • Staff time in OR or in the hospital per 100h of TROS

• Number of staff hours during T2+T3 of TROS with respect to standard mean useful time.


- 30 -

analyse en fonction du type de programmation

analyse en fonction de la période (semaine, week-end, de chaque

jour de la semaine)

Taux d’occupation moyen sur la période observée des salles par

quart d’heureTemps Réel d’Occupation de

la Salle (= temps pris en compte dans la T2A)


6

- 31 -

Plan

• Introduction



• Open scheduling

• Bloc scheduling

• Modified Bloc scheduling

• Surgery scheduling

- 32 -

Surgery programming models

• Surgery programming is an efficient tool of operating theatre management.

• It consists in building a « provisional » planning of surgery interventions to be realised during a period based on surgery demand forecast.

• It determines the activities of the operating theatre (surgeons, anaesthesists, nurses IADE – IBODE), and also the wards, ICUs, pharmacy, radiology, etc.

- 33 -

• Surgery program built various ways: (i) in chronologic order as information arrives, (ii) periodically by allowing rescheduling of earlier decisions.

• Open scheduling is always centralized and is simple to organise.

• Cons: dysfunction such as under-usage of resources, overtimes, and surgery cancelations if it is realised in an inappropriate way.

• Open scheduling in two phases: intervention planning and scheduling.


Open scheduling model : propose for each period (week or month) a surgery program that is completely independent of previous decisions, i.e. starts from an empty sheet.

..h ..

17h 00

16h 00

15h 00

14h 00

13h 00

12h 00

11h 00

10h 00

9h 00

Lundi

Patient 7

Salle 1

Patient 4

Patient 11

Patient 3

Patient 1

Patient 5

Patient 19

Patient 2

Patient 20

Patient 17

Patient 8

Patient 15

Patient 6

Patient 12

Mardi Mercredi

Salle 2Salle 1 Salle 2

- 34 -

• Widely used especially in North America

• Efficiency strongly depends on the design of a good surgery program pattern, in order to adapt to actual activities.

• Pro: no need of centralised information.

• Head of the operating theatre can fill the pattern with its planning and does not manage the detailed scheduling.

• Caution: difficulty of building a good surgery program pattern.


..h ..

17h 00

16h 00

15h 00

14h 00

13h 00

12h 00

11h 00

10h 00

9h 00

Lundi

Salle 1

Mardi

Salle 1

Chirurgien

Dr. Dupont

Service

orthopédie

Chirurgie

générale

Groupe chirurgical

DR. Durant

Dr. Martin

Service ORL

Block scheduling model :

relies on a fixed surgery program pattern to allocate OR-blocks (time slots of an Operating Room) to surgeon groups or specialties.

Each surgeon group places its interventions within its OR-blocks.

A weekly pattern

- 35 -

Modified Block scheduling similar to block scheduling but withadaptation mechanisms.

• The concept of OR-blocks becomes elastic with the possibility of lengthening and shortening.

• It allows greater flexibility in surgery programming.

• Based on the evolution of the surgery program, the head of the operating theatre has the possiblity to adjust the lengths of OR-blocks or to allow general access to some OR-blocks.


- 36 -

Modified Bloc scheduling

Improving the Block Scheduling model by the following:

• OR-blocks of elastic length. The length can vary according to an accepted ratio as a percentage deviation from a nominal length

• Insertion of unassigned OR-blocks in the surgery program pattern

• Possiblity of the lead nurse to adjust the length of or remove some OR blocks that are poorly filled or abandonned.

Unassigned blocks allow absorbing occasional demand surges of surgeon groups SP.

Block release time:

• This practice monitors the fill rate of OR-blocks. If it is below some threshold of economic breakeven, the OR-block is adjusted or turned unassigned.

• The adjustment occurs at block release time usually 120 to 48h before the validation of the surgery program. The lead nurse then invites SP to plan new interventions to improve the efficiency.

7

- 37 -

Modified Bloc scheduling

..h ..

17h 00

16h 00

15h 00

14h 00

13h 00

12h 00

11h 00

10h 00

9h 00Lundi

Salle 1

Mardi

Salle 1

SurgeonDr. Dupont

Serviceorthopedics

General surgery

Surgeon GroupDr. DurantDr. Dupont

Service ORL

Demand of surgeons

Drs. Dupont & Durant

Demand

orthopedics

Demand

General surgery

..h ..

17h 00

16h 00

15h 00

14h 00

13h 00

12h 00

11h 00

10h 00

9h 00Lundi

Salle 1

Mardi

Salle 1

SurgeonDr. Dupont

Serviceorthopedics

General surgery

Surgeon GroupDr. DurantDr. Dupont

Unassigned

2 mois - 120 or 48 h During 120 or 48 h

- 38 -

Surgery programming rules in practice

• Fixed hours system, aims at optimizing the resource usage. Interventions are programmed if they finish before the fixed closing time, leading to potential under-utilization and intervention postponement or cancelation.

• Any workday. It focuses on the satisfaction of patients and surgeons and allows them to choose the date of intervention, leading to potential over-utilisation of the operating rooms.

• Reasonnable time system, allows higher occupation of operating rooms with reasonable surgery intervention delays. The programming is realised by the head nurse in order to minimise the waiting times of interventions.

- 39 -

Key decision phases

• Trajectory of a surgery patient : pre-operation, per-operation, post-operation.

• During the pre-operation phase, after the surgery and anaesthesia consultations, a hospitalization date is given to the patient.

• Depending on the strategy, this date can be changeable or not.

- 40 -

Key decision phases

• During the per-operation phase,

• Decisions concerning the organization of the operating theatre are to be made: (i) assignment of each invention to an OR and a time slot, (ii) organization of relevant logistics (stretchers, recovery beds, sterilized medical devises, consumables, …).

• These decisions are under the responsibility of the head nurse of the operating theatre.

• She is in charge of

• planning the surgery activities over several weeks by partitioning the interventions in order to minimise the deviationof intervention dates;

• daily scheduling of patient flows by proposing detailedschedules for each operating room, recovery beds and relevant logistics.

- 41 -

Key decision phases

• During the post-operation phase,

• Only decisions on booking resources for cares during the stay, according to the evolution of the health condition.

• In most cases, these decisions do not need to be planned.

• Among the three phases, the per-operation phase has the greatest potential of performance improvement.

• The two others are more or less hotel-like activities.

- 42 -

Plan

• Introduction



• Open scheduling

• Bloc scheduling


8

- 43 -

Open scheduling

• Open scheduling consists in proposing, for each period, a surgery planning without any pre-assignment.

• To reduce complexity of open planning, most hospitals (i) pre-assign operating rooms (OR) to surgery specialties, and (ii) forces the surgeons to realise their interventions of a day in the same OR.

• Pros: more flexible adaptable to perturbations ad it does not fix a priori the dates, hours and the number of interventions.

• Cons: (i) efficiency strongly depends to the expertise of the head nurse and her ability to negotiate with surgeons; (ii) heavy workload of the head nurse for surgery program consolidation, coordination of different specialties, and decision-making of difficult situations.

• Condition: Software tools to aid surgery programming of a large operating theatre.

- 44 -

Open scheduling

• The problem arises after the surgery and anaesthesia consultation. The nurse proposes a hospitalisiation date or take note of patient’s preferences.

• Based on patient’s preferences and the availability of the surgeon, a date of intervention is proposed later on.

• Determination of surgery intervention dates should take into acount the following constraints:

Patients is admitted the day before their intervention and a bed is needed during the whole Length of Stay (LoS)

Each patient needs a time slot in an OR for her intervention

Each intervention should be performed in a given time window

Surgeons are not always available and can perform a limited number of intervention per day.

Patients might also have availability constraint.

- 45 -

Mathematical model for open scheduling

Decision variable :

Xisj = 1 if intervention i planned in room s on day j

Xisj = 0 if not.

Criterion (minimisation of deviation from most preferred date) :

1 1 1

N T S

isj isji j s

Minimise Z C X

• Cisj = number of days in advance or after patient’s preferred date if Xisj = 1.

- 46 -

Constraints:

1) Realise all interventions

1 1

1, 1,...,T S

isjj s

X i N

2) Capacity Rcapsj of OR-day (s, j)

1

N

i isj sji

Dur X Rcap


- 47 -

3) Capacity Maxjm of Surgeon-day (m, j)

with Surgi = the set of surgeries of surgeon m.

5) Bed capacity Bcapj on day j

1

1 1 1

, 1,..., 1i

jN S

isk ji s k j LoS

X Bcap j T

1m

S

i isj jmi Surg s

Dur X Max

with LoSi = Length of Stay of patient i


- 48 -

6) Time window [DMINi, DMAXi] of intervention i

Xisj = 0, if j < DMINi or j > DMAXi

7) Integrity constraints

0,1 , , ,isjX i s j


9

- 49 -

Xisj = 0, if j < DMINi or j > DMAXi

0,1 , , ,isjX i s j

1

1 1 1

, 1,..., 1i

jN S

isj ji s k j LoS

X Bcap j T

1m

N S

i isj jmi Surg s

Dur X Max

1 1

1, 1,...,T S

isjj s

X i N

1

N

i ipj sji

Dur X Rcap

1 1 1

N T S

isj isji j s

Minimise Z C X


- 50 -

Mathematical model = Integer Linear Programming

Solvable

by commercial solvers such as CPLEX, XPRESS

or

by heuristic methods

• Bin-packing-based heuristics (First fit, best fit, worst fit, decreasing fit)

• Local optimization (pairwise echange, patient relocation, OR room rescheduling).


- 51 -

Problem : How to plan elective cases when the operating rooms capacity is shared with emergency patients

References :• M. Lamiri, X.-L. Xie, A. Dolgui and F. Grimaud (2008). "A stochastic model for

operating room planning with elective and emergency surgery demands",

European Journal of Operational Research, 185/3, 1026-1037

• Mehdi Lamiri, Xiaolan Xie and Shuguang Zhang (2008), "Column generation for

operating theatre planning with elective and emergency patients," IIE

Transactions, 40(9): 838 – 852.

• M. Lamiri, F. Grimaud, and X. Xie (2009). “Optimization methods for a stochastic

surgery planning problem,” International Journal of Production Economics,

120(2): 400-410.

Open scheduling under both elective & emergency demands

- 52 -

Plan

• Introduction



• Open scheduling

• Bloc scheduling


- 53 -

Bloc scheduling model

• Il assigns OR-blocks to surgeon groups (SP).

• A SP can be a surgeon or several surgeons of the same specialty or not which share the same OR-blocks.

• Block scheduling consists in building a weekly surgery program pattern for regular periods or for particular periods such as holiday seasons.

• Each SP places its interventions within its OR-blocks at their convenience.

• This model relies on knowledge of detailed demand forecast of each SP in order to allocate enough OR times decomposed into OR blocks of consistent length with respect to the SP’s intervention durations.

• The surgery program pattern, reminiscent of the master schedule of MRP in production control, is called Master Surgery Schedule.

- 54 -

Bloc scheduling

..h ..

17h 00

16h 00

15h 00

14h 00

13h 00

12h 00

11h 00

10h 00

9h 00Lundi

Salle 1

Mardi

Salle 1

Chirurgien

Dr. Dupont

Service

orthopédie

Chirurgie

générale

Groupe chirurgical

DR. Durant

Dr. Martin

Service ORL

10

- 55 -

Bloc scheduling

• Pros: simplicity of implementation.

• Cons:

• Extreme difficulty for building a good MSS as over-dimensionning OR-blocks leads to low productivity and under-dimensionning OR-blocks leads to tensions among surgeons and intervention delay/cancelation.

• Rigid surgery programming imposing surgeons a constant and regular activity volume at fixed dates.

- 56 -

Building a cyclic master surgery schedule

Problem:

Ensure that each surgeon (or surgeon group) obtains a specific nb of OR blocks

With surgery-dependent OR capacity requirement and stochastic LoS

In order to minimize the expected bed shortage

o Source : Jeroen Belien é Erik Demeulemeester, EJOR, 2007

- 57 -


Problem setting:

Data:

LoS distribution / surgery type (To be defined)

bi : nb of OR blocks available on day i

rs : nb of OR blocks required by surgeon s

Decision :

xis = nb of OR blocks of each surgeon s on each day i of the planning cycle (weekly or bi-weekly)

- 58 -


- 59 -


LoS (Length of Stay): the whole period in the hospital of each patient during which a bed is needed throughout.

Deterministic LoS :

Asd = set of hospitalization days if operated on day d

Stochastic LoS :

psd = probability of staying d days after surgery by surgeon s

Ex : appendicitis surgery: (2, 20%), (3, 50%), (4, 30%)

LOS Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed Thu Fri Sat Sun

3 S

3 S

3 S

10 S

10 S

hospitalization days

- 60 -


11

- 61 -


Solution

Analytical expressions of mean, variance, percentile

Linearization of the non-linear model

Repetitive MIP heuristic, Quadratic MIP, Simulated annealing

- 62 -

Extension to Case mix planning

To advance the resource efficiency

o Approach Matching the patient demand and the resource provision

Coordinating the resource allocation within the whole hospital

To maximize the overall financial contribution of given resources

o Long-term decision making Patient case mix that can be treated annually

Resource capacity allocation to each specialty

o Solution method Integer linear program modeling

Branch-and-price algorithms (Ma et Demeulemeester 2013)

- 63 -

Case mix planning

Major demand data

Demand range per patient group

Resource requirement per group-p patient

• OR capacity: durp

• Bed requirement

LoSp represented by

Apd = set of hospitalization days if operated on day d

LOS Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed Thu Fri Sat Sun

3 S

3 S

3 S

10 S

10 S

hospitalization days

- 64 -

Case mix planning

• Integer linear programming (ILP) model

o the number of patients of group p that receive surgery on day a

o the number of beds allocated to ward w

o the number of blocks assigned to surgeon group s on day a

o Cycle: one week

. .

, ,

,

, ,

,

, , , , , ,

w p d

s

ww W

p a wp P a A

s as S

p p a s ap P

p p a pa A

p a w s a

s t y B E D S

x y w W d D

z B L O C K S a A

d u r x z L E N G T H s S a A

L B x U B p P

x y z w W s S p P a A

max p pap P a A

r x

bed capacity

integer

admission range

OR blocks

:pax

:wy

:saz

Objective: to maximize the overall financial contribution

Constraint: resource capacity

- 65 -

Case mix planning

To improve the resource utilization

o Variability: e.g., the variable LOS the bed shortage phenomenon

o Approach Adjusting the resource capacity allocation

Building balanced master surgery schedules

To coordinate the capacity utilization of various resources at each specialty

o Decision-making Long-term: the bed capacity allocation

Medium-term: (seasonal) balanced master surgery schedules

o Solution method Mixed integer programming models

- 66 -


• Master Surgery Scheduling model

o the number of patients of group p that receive surgery on day a

o the number of beds allocated to ward w

o the number of blocks assigned to surgeon group s on day a

o Cycle: one week

. .

,

, ,

,

, , , , , ,

s

ww W

s as S

p p a s ap P

p a pa A

p a w s a

s t y B E D S

z B L O C K S a A

d u r x z L E N G T H s S a A

x T H R p P

x y z w W s S p P a A

min ww W

TEBS

b e d c a p a c i ty

integer

Planned case mix

OR blocks

:pax

:wy

:saz

Objective: to minimize the Total Expected Bed Shortage (TEBS)

12

- 67 -


,0

2, ,

0

2 2

,

,

(1 )

. .

,

, , ,

, , , ,

w

w

ww W

wd pa p kT d ap P a A k

wd pa p kT d a p kT d ap P a A k

wd w wd wd

wd w wd w

y BEDS

x prob

x prob prob

y w

s

W d

w W d

t

D

D

2min w wd w w w ww W d D

Building a balancedcyclic MSS:

bed occupancy

on each ward-day

OR assignments

patient admission

integer/nonnegative

, , ,

, , ,s

sark k ars S k K

p pa sark kp P r R k K

z Length ORday a A r R

dur x z Length s S a A

, ,

, , 0,1 , 0, , , , , , , .

pa pa A

pa w sark wd

x THR p P

x y z p s w a d r k

daily bed deficit

bed surplus bed demands variance

Approach: balancing the daily bed occupancy of each wardto minimize the total expected bed shortages (TEBSs)

- 68 -

Plan

• Introduction



• Open scheduling

• Bloc scheduling


- 69 -

Daily operation management of an operating theatre

1. Schedule the most critical resources: operating rooms OR (daily surgery scheduling) .

2. Schedule the working time of all relevant surgery team members

3. Schedule the logistics of patients and the working time of stretchers (brancardiers)

4. Schedule the logistics of all consumables and medical devices

- 70 -

Daily surgery scheduling

Assignment and sequencing of

surgery interventions of the day

To

ORs

- 71 -

A mathematical model

i=1

1

Variable:

1, if patient i assigned to OR m

0, if not.

Minimise Cmax

subject to:

Cmax ,

1,

0,1 .

im

N

i im

M

imm

im

X

Dur X m

X i

X

- 72 -

Bin Packing Equivalence

Daily surgery scheduling Bin Packing

Surgery interventions objects

Surgery duration Size of an object

ORs or OR-blocks Bin

OR session or block length Bin capacity

Nb of ORs to open Nb of Bins to use

Placing an intervention in an OR Placing an object in a bin

Closing time of the operating theatre Maximal bin capacity

Equivalence also with the parallel machine scheduling

13

- 73 -

Dynamic Bin Packing Rules

First Fit

• Insert the next object in the first bin (without exceeding the max bin capacity if given)

Best Fit

• Insert the next object in bin with the least remaining capacity(without exceeding the max bin capacity if given)

Worst Fit

• Insert the next object in bin with the largest remaining capacity(without exceeding the max bin capacity if given)

Objects are known dynamically at their arrival

Observations: methods easily extendable to include complicated features such as non identical ORs, patient/surgeon preferences/constraints, …

- 74 -

Static Bin Packing Rules

Step 1: Sort the objects using one of the priority rule.

• FIFO : First Come First Serve (First In First Out)

• LPT: Longest Processing Time First (good for workload balancing)

• SPT: Shortest Process Time First (good for waiting time)

• EDD: Earliest Due Date first (good for meeting due date)

Step 2: Apply one of the dynamic Bin-Packing rules

Special combinations:

First Fit Decreasing = LPT + First Fit First Fit Increasing = SPT + First Fit

Best Fit Decreasing = LPT + Best Fit Best Fit Increasing = SPT + Best Fit

Worst Fit Decreasing = LPT + Worst Fit Worst Fit Increasing = SPT + Worst Fit

- 75 -

Worst-case performances

Minimise the number of bins for a given capacity

• First Fit Decreasing, Best Fit Decreasing : B <= 11/9OPT + 1

• First Fit : B <= 17/10OPT + 2

where B = nb of bins by rules, OPT = minimal nb

Minimise the maximal bin capacity for a given number B of bins

• Best Fit Decreasing : Cmax/Cmax* <= 4/3 – 1/3B

The case of a given number B of bins can be solved exactly by a pseudo-polynomial dynamic programming algorithm.

1

Dynamic Daily Surgery Scheduling

Department of Healthcare Engineering

Centre for Health Engineering

Ecole des Mines de Saint Etienne, France

[email protected]

Centre for Healthcare Engineering

Dept. Industrial Engr. & Management

Shanghai Jiao Tong University, China

[email protected]

Xiaolan XIE

- 2 -

Field observations of surgery scheduling

- 3 -

Ruijin Hospital (since 1907 by French missionaries)

Teaching hospital of the medical school of the Shanghai Jiao Tong University

Top 1 hospital in Shanghai

+12000 outpatient visits / day

A 23-floor outpatient consultation building

- 4 -

Field observation of the operating theatre of Ruijin Hospital

An integrated operating theatre of 21 OR and a second one recently constructed

60-70 elective surgery interventions + 10 emergency surgeries / day

No integrated surgery planning but each surgery speciality is given an amount of total OR time

Each speciality decides the surgeries to perform the next day

The operating theatre (OT) is responsible for daily OR assignment and the OR program execution.

- 5 -


Special features of the Ruijin Hospital

Queue of elective patients never empty

Availability of patients to be operated in short notice

Availability of surgeons to operate each day

Large variety of surgeons : top surgeons, senior surgeons, ordinary surgeons

Strong demand to operate at the OT opening in the morning to avoid endless waiting

Strong concern of OT personal overtime

- 6 -


Issues to be addressed

Promising surgery starting times to meet surgeon's demand for reliable surgery starting

(Tell me early enough when I start my surgery)

Surgery team overtime management

(How to guarantee the on-time end of duty of surgery teams?)

Outpatient surgery appointment when servers respond to congestion

2

- 7 -

Managing surgeon appointment times

- 8 -

Why surgeon appointments not used in practice

• Not used in practice to avoid potential OR capacity loss

Research question

How to provide surgeon appointment guarantee while ensuring appropriate OR capacity usage?

Observed Daily OR utilization

• But OR capacity usage is not always high over the day

- 9 -

Related work

Static scheduling for a single OR

Surgeon appointment scheduling (AS):

Two surgeries: AS solved by a newsvendor model (Weiss, 1990)

A fixed sequence of surgeries: stochastic linear program solved by SAA

and L-shape algo to determine the allowance of each surgery, or

equivalently, the arrival time (Denton 2003).

Others: discrete appointment (Begen et al, 2011), robust appointment

(Kong et al, 2011)

Sequence scheduling: The problem is to jointly determine the position and

arrival time of each surgery (Denton 2007; Mancilla 2012).

- 10 -

Related work

Dynamic scheduling for a single OR

Arrival scheduling: The demand of surgeries is uncertain, surgeries are

processed as FCFS rule. The problem is to dynamically determine the

arrival time upon each application(Erdogan 2011).

Sequence scheduling: The demand of surgeries is also uncertain. The

problem is to jointly determine the position and arrival time of each

surgery upon each application (Erdogan 2012).

- 11 -

Our focus

Multi-OR setting

- 12 -

Our focus

Multi-OR setting

Single-OR

Multi-OR

A1 A2 A3 An

A1/A2 A3 A4 An

No OR assignment

Dynamic OR assignment

3

- 13 -

Our focus

Two inter-related problems:

• Determining surgeon arrival times by taking into

account OR capacities and random surgery

durations.

• Dynamic surgeon-to-OR assignment of during the

course of a day as surgeries progress by taking

into account planned surgeon arrival times.

- 14 -

Assumptions of our work

A1: Emergency surgeries in dedicated ORs and hence neglected.

A2: Identical ORs and surgeries assignable to any OR.

A3: At most one surgery per surgeon each day.

A4: Promised starting or appointment time informed at the end of day D-1 (Surgeon appointment scheduling or proactive problem).

A5: Surgeons not available before the promised times.

A6: Dynamic surgery-to-OR assignment during the course of the day upon the surgery completion events.

- 15 -

Dilemma of promising surgery starting time

Promise too early

Surgery 1

promised start of surgeon 2

Surgery 2

Surgery 1

promised start of surgeon 2

Surgery 2

Promise too late

surgeon waiting

OR idleOR overtime

Easy if known OR time but OR times are uncerain

- 16 -

Data

J set of surgery interventions or surgeons

N number of identical ORs

T length of OR session

pi() random duration of surgery i in scenario

bi unit time waiting cost of surgeon i

c1 unit OR idle time cost

c2 unit OR overtime cost

Similar to parallel machine scheduling but with planned job release dates and random service time.

- 17 -

Dynamic Surgery Assignment of Multiple Operating Rooms with Planned Surgeon Arrival Times

Zheng Zhang, Xiaolan Xie, Na Geng

In IEEE Trans. Automation Science and Engineering

- 18 -

Plan

Promising surgery starting times

Real time OR assignment strategies

Some numerical results

Conclusion and perspective

4

- 19 -

Decision variables

si promised surgery starting time of surgeon i

Approximation assumption: fixed assignment & sequencing

xir = 1/0 assignment of surgery i to OR r

yij = 1 if surgery i precedes j in the same OR

= 0 if not

Auxiliary scenario-based random variables

Cir() completion time of surgery i on OR r

Ir() idle time of OR r

Or() overtime of OR r

Wi() waiting time of surgeon i

- 20 -

Model for promising surgery starting times

Assign each surgery to an OR ∑r xir = 1

Relation between assignment & sequencing yij + yji ≥ xir + xjr -1

Promised start before the end of the session si ≤ T

Scenario-dependent completion time xir pi() ≤ Cir ()

Cir () ≤ M xir

Cjr () Cir () + pj() - M (1- yij) - M(2- xir - xjr )

Scenario-dependent OR idle time Cir () ≤ Ir () + iJ xir pi()

Scenario-dependent OR overtime Or () Cir () - T

Scenario-dependent surgeon waiting time rE Cir() = si + Wi() + pi()

OR idle costOR overtime

costsurgeon

waiting cost

min E{c1 ∑r Ir() + c2 ∑r Or() + ∑i biIi()}

- 21 -

Proposed solution

1. Convertion into mixed-integer linear programming model by Sample Average Approximation by using a given number of randomly generated samples

2. Heuristic for large size problem based on

a) Local search for surgery-to-OR assignment optimization

b) Surgery sequencing rule based on optimal sequencing of the two-surgery case

c) Optimal promised start time by SAA and MIP

- 22 -

Plan





- 23 -

Dynamic surgery assignment optimization

At time 0, start surgeries planned at time 0

At the completion time t* of a surgery in OR r*,

select a surgery i* to be the next surgery in OR r*

among all remaining ones J*

Surgery i* starts at time max{ t*, si* } in OR r* after the arrival of the surgeon at time si*

An Event-Based Framework

- 24 -

Dynamic surgery assignment optimization

Surgery i* is selected in order to minimize E[ TC(t*, i*, J*)]

where

E[ TC(t*, i*, J*)] is the minimal total cost similar to promised time planning model

by conditioning on all completed surgeries and ages of all on-going surgeries

by scheduling i* as the next surgery on OR r*

5

- 25 -

Two-stage stochastic programming approximation

• At k-th surgery completion event at time tk

where J\J(k-1) is the set of remaining surgeries

• The first stage cost is the OR-idle

or surgeon waiting cost induced by surgery l

• lk is the second stage cost, i.e. the total cost induced by

remaining surgeries plus OR overtimes.

\ 1mink lk

l J J klkV g

ˆlk l k l k ls t t sg

- 26 -

The second stage cost

\ 1 \minlk jlk

j J J k l

where• jlk is the expected stage cost induced by surgery j

• if surgery l is selected at event k and surgery j at event k+1

Jensen's inequality is used to speedup the OPLA rule.

One-period look-ahead (OPLA) approximation

- 27 -

The second stage cost (cont'd)

Min. cost of two dynamic assignment rules:

• Rule 1 (minimal stage cost first): Remaining surgeries assigned in the scenario-independent order of minimal expected first stage cost, i.e. the surgery in selected at event n > k minimizes the stage n cost induced by in.

• Rule 2 (FCFS): Remaining surgeries are selected in non-decreasing order of their surgeon arrival times si

Jensen's inequality and another valide inequality are used to speedup the MPLA rule.

Multi-period look-ahead (MPLA) approximation

- 28 -

Lower bound of the dynamic surgery assignment

• Based on perfect information, i.e. all surgery duration

realizations pj() are known at the beginning of the day, i.e.

randomness known at time 0+

• The lower bound problem is similar to the proactive

problem but with

o given promised surgery start times

o scenario-dependent surgery assignment xir() and

sequencing yij()

- 29 -

Dynamic surgery assignment policies

Policy Static:

No real time rescheduling

OR assignment / sequencing decisions of promised time planning model are followed

Policy FIFO:

Dynamic surgery assignment in FIFO order of surgeon arrival times

Policy I:

Dynamic surgery assignment optimization with OPLA

Policy II:

Dynamic surgery assignment optimization with MPLA

- 30 -

Plan

Background and motivation

Problem setting





6

- 31 -

Optimality gap

Observations

• Optimality gap is relatively small

• High surgery duration variation degrades the optimality gap

• High workload reduces the optimality gap

• MPLA better than OPLA

GAP = (costX- LB) / LB

(,)GAPI(%) GAPII(%)

Ave. Min. Max. Ave. Min. Max.(0.3,0.75) 7.4 0.1 14.7 6.3 0.1 12.8(0.7,0.75) 8.5 5.1 14.8 7.7 3.8 18.4(0.3,1.25) 5.6 1.3 11.2 4.1 1.0 8.3(0.7,1.25) 7.8 1.9 17.3 6.0 1.6 9.6

(80 3-OR instances)

- 32 -

Value of dynamic scheduling

OR# (,)VDS (%)

Ave. Min. Max.

3 (0.3,75) 10.6 2.6 22.9

(0.7,75) 14.8 5.5 26.9

(0.3,125) 7.4 3.9 14.1

(0.7,125) 11.1 5.7 15.5

Ave. 11.0 4.4 19.9

6 (0.3,75) 25.4 18.7 31.6

(0.7,75) 29.2 24.7 39.9

(0.3,125) 11.1 7.1 15.5

(0.7,125) 19.1 12.8 24.1

Ave. 21.2 15.8 27.8

12 (0.3,75) 33.6 30.1 37.9

(0.7,75) 36.0 28.9 42.1

(0.3,125) 18.6 17.2 20.4

(0.7,125) 26.1 23.9 30.1

Ave. 28.6 25.0 32.6

Observations

• Dynamic surgery scheduling always helps.

• The benefit is more important for larger OT.

• Dynamic surgery scheduling is able to cope efficiently with surgery uncertainties.

• VDS decreases as the workload of OT increases.

: variation parameter of surgery time: workload

VDS = (costStatic - costDyna) / costStatic

- 33 -

Value of dynamic scheduling optimization

Observations

• VOS increases as OR# increases.

• VOS increases as increases, i.e. the variance of surgery durations increases.

• VOS decreases as increases, i.e. the workload of OT increases.

OR# (,)VOS (%)

Ave. Min. Max.

3 (0.3,75) 2.8 0.0 14.4

(0.7,75) 5.4 0.0 26.5

(0.3,125) 2.3 0.0 7.0

(0.7,125) 3.1 0.0 10.2

Ave. 3.4 0.0 14.5

6 (0.3,75) 5.4 -0.1 13.6

(0.7,75) 6.0 -0.1 11.3

(0.3,125) 2.9 0.0 5.0

(0.7,125) 5.0 0.6 8.7

Ave. 4.8 0.1 9.6

12 (0.3,75) 7.0 5.8 7.8

(0.7,75) 9.3 6.1 11.8

(0.3,125) 5.0 3.4 6.8

(0.7,125) 6.4 4.7 9.2

Ave. 6.9 5.0 8.9

: variation parameter of surgery time: workload

VOS = (costFIFO - costDynaOpt) / costFIFO

- 34 -

Value of proactive decisions

Observations

• Proactive decision is very important to dynamic assignment scheduling.

• The arrival times that optimize the proactive model may not be adjustable to the dynamic assignment scheduling.

• Joint optimization of promised start times and dynamic assignment policies is an open research issue.

VOS = (costX - costX) / costX

where costX is the average cost of the strategy X but with promised start times determined with deterministic surgery duration.

(,)VPSI(%) VPSII(%)

Ave. Min. Max. Ave. Min. Max.(0.3,0.75) 7.2 -15.2 23.3 7.0 -20.9 22.6(0.7,0.75) 6.8 -11.1 20.4 6.4 -14.4 20.4(0.3,1.25) 9.8 1.1 23.1 10.0 0.9 21.6(0.7,1.25) 10.1 1.1 19.2 10.1 3.2 17.9

- 35 -

Plan





- 36 -

Optimal surgery promised starting times for a given OR assignment / sequencing?

Features of surgeries planned to start at OR opening?

Time slacks in promised times vs surgery OR time and waiting cost?

Design of efficient optimization algorithms for promised time planning and real time rescheduling?

Promising time planning under starting time reliability constraints?

Open issues

7

- 37 -

Simulation-based Optimization of Surgery Appointment Scheduling

Zheng Zhang, Xiaolan Xie

To appear in IIE Transactions

- 38 -

Outline

• BACKGROUND AND MOTIVATION

• SURGERY APPOINTMENT SCHEDULING PROBLEM

• SAMPLE PATH ANALYSIS

• STOCHASTIC APPROXIMATION

• NUMERICAL EXPERIMENTS

• CONCLUSION AND PERSPECTIVE

- 39 -

Our focus

Example :

1st released OR allocated to surgeon 3,

2nd released OR to surgeon 4, ....

Multi-OR

A1/A2 A3 r1An

FCFS assignment

r2 A4

Surgeon appointment optimization for a given sequence of

surgeries assigned to ORs on a FIFO basis.

- 40 -

Outline







- 41 -

Modeling

Parameters

n surgeries\surgeons

m ORs with regular capacity T for each OR

pi(): surgery duration with known distribution

/ /i: unit OR idling cost / overtime cost / surgeon waiting cost

Decisions

Surgeon arrival time A = [Ai] such that:

A1 = A2 = … Am = 0 ≤ Am+1 ≤ Am+2 ≤ … ≤ An

- 42 -

Modeling

Sample-path cost function

C[i](): i-th surgery completion event time.

C[i]() depends on A and and can be solved using a simple recursion.

1

1 0

( , )n m

i i ii m i m n pi m p

f A C A A C C T

Waiting cost Idling cost Overtime cost

8

- 43 -

Modeling

Expected cost function

Objective

( ) ,g A E f A

1

min ( )

0, 1, ...,

, , ..., 1

A

i

i i

g A

A i mA

A A i m n

- 44 -

Outline







- 45 -

Sample path analysis

LEMMA . The sample path cost function f(A,) is

• differentiable with probability 1 and

• Lipschitz-continuous throughout with finite Lipschitz constant

Why?

1 2 1 2 1 2( , ) ( , ) , ,f A f A K A A A A

- 46 -


Sample path cost function differentiable when

• There is no simultaneuous event occurrences (no

simultaneous departure, no simultaneous arrival-

departure),

• i.e. one event at a time (set of sample paths of proba 1

1

1 0

[i]

( , )

C ( ): i-th surgery completion event time

n m

i i ii m i m n pi m p

f A C A A C C T

- 47 -


Proof of Lipschitz differentiable

: min min max

min min min max min max

min min min max min max

th thk k k k

k k k

th th th thk k k l l k l l

k k k l k l

th th th thk k k l l k l l

k k k l k l

Lemma i a i b a b

i a i b i b a b i b a b

i b i a i a b a i b b a

If , (Lypshitz continue), then

min min max

k k

th thk k k k

k k k

x x K k

i x i x x x K

- 48 -


THEOREM 1 (unbiasednes of sample path gradient). The

objective function g(A) is continuously differentiable on , and the

gradient of g(A) exists for all A∈ with

, ,A AE f A E f A

The noisy sample-path gradient is on average correct!

Why?

9

- 49 -


- 50 -

Sample path analysis : partial derivative at interior point

\{ }

\{ }

\{ }

A:

B:

C: 1

D: 1

i

i

i

i

jj BP i

ji j BP i

jj BP i

f

A

Ai

i BP2(i) j

A.

B.

i

Ai waiting

i BP2(i) BP3(i)C.

Ai

i BP2(i) BP3(i)D.

Ai

waiting

waiting waiting

waiting waiting overtime

[i-m]

[i-m]

[i-m]

[i-m] BP4(i)

waiting

= unit OR idling cost

overtime cost

i = surgeon waiting cost

Busy Period approach

A. i does not initiate BP(i)

B. i initiates BP(i) but not the last BP of the OR

C. i initiates the last BP of the OR without overtime

D. i initiates the last BP of the OR with overtime

- 51 -

Sample path analysis : directional derivative at boundary point

Boundary point A with Ak = Ak+1 = … = Al

0

0

... , ,, lim

... , ,, lim

, if 0

, if 0

1 1 , if

1 0 , if

i

i

lk i

v jj i

ii l

u jj k

i i i

i

i i

j j

j

j m j mj

f A e e f Af A

f A e e f Af A

x W

W

C T j n mx

W x j n m

Left-hand directional derivative

Right-hand directional derivative

- 52 -

Sample path analysis : improving direction

At an interior point, i.e. Ai-1 < Ai < Ai+1

At a boundary point A with Ak = Ak+1 = … = Al

Select two surgeries i < j such that

Determine the improving direction

,f A d

, 0, , 0i jv uf A f A

,..., ,0,...,0, ,...,i i j jv v u ud f f f f

- 53 -

Outline







- 54 -

Stochastic approximation

1k k k kA A s d

where

is an improving direction according to sample-path gradient ,

= is a converging step-size

min is the orthogonal projection into the feasible set

k k

k

d f A

as

k

y

x y x

Hill-climbing with noisy sample-path gradient

10

- 55 -

Outline







- 56 -

Convergence of stochastic approximation

BAD NEWS:

The sample path cost function is not quasiconvex.

Counter-example: p() = {9, 4, 4, 1};

2 ORs, OR session T=10;

idle time cost = 1; no overtime cost; Unit waiting cost 3=1, 4=3.

Three arrival time vectors:

A1=(0, 0, 4, 7.5)

A2=(0, 0, 6, 8.5)

A = A1 + (1-)A2

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 0,2 0,4 0,6 0,8 1

f(A, )

- 57 -


By randomly perturbing p around {9, 4, 4, 1}, we implement the stochastic

approximation algorithm.

Evolution of arrival times visited by the stochastic approximation algorithm in

Example 1, when applying it over 200 sample paths.

- 58 -


Hopeful news: The sample path cost fuction f(A,) is strongly unimodal.

Properties verified experimentally:

• Unimodality of the expected cost function

• Convergence of the stochastic approximation algorithm.

- 59 -

Convergence of stochastic approximation: numerical evidence

Log normal distribution Uniform distribution

var, wkload 0.3,0.75 0.7,0.75 0.3,1.25 0.7,1.25 0.3,0.75 0.7,0.75 0.3,1.25 0.7,1.25

Initial dispersion

3-OR 5.0 4.9 6.5 7.0 5.4 4.8 6.6 6.8

6-OR 6.5 6.7 8.5 9.5 6.5 6.6 10.3 9.8

9-OR 8.0 7.4 11.2 10.5 7.9 7.7 10.5 10.5

Final dispersion

3-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Final grad

3-OR 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

6-OR 0.0 0.0 0.1 0.1 0.0 0.0 0.1 0.1

9-OR 0.0 0.2 0.1 0.3 0.0 0.2 0.2 0.3

- 60 -

Allowances of Multi-OR vs single OR settings

Optimal allowance shapedome shape in 1-OR, zigzag shape in 2-OR

2-OR vs 1-ORsmaller allowances, half total allowance, highly uneven

Increasing surgery duration

variability (o vs o)smoothing 2-OR allowances, increasing 1-OR allowance variability

Higher waiting cost (o vs o)larger allowances in both settings but rather insensitive in the 2-OR setting

11

- 61 -

Allowances vs OR#

Zigzag shape

1 large allowance followed by m-1 small allowances

Total m-OR allowance = 1/m of total-1-OR allowance

Higher OR# and higher duration variation smoother allowances

- 62 -

Allowances vs OR#

Two-parameter heuristic

Larger 1st allowance followed by constant allowances

- 63 -

Value of dynamic assignment and proactive solution

Three strategies

Strategy I : no dynamic surgery-to-OR assignment

Strategy II : same appointment times, FIFO surgery-to-OR assignment

Strategy III : same surgeon arrival sequence, FIFO surgery-to-OR assignment, simulation-based optimized appointment times

Value of dynamic assignment (VDA)percentage improvement of strategy II over strategy I

Value of proactive anticipation and dynamic assignment (VPD)percentage improvement of strategy III over strategy I - 64 -

Value of dynamic assignment and proactive solution

VDA > 0, VPD > 0 , VPD > VDA : dynamic assignment and the proactive anticipation of dynamic assignments always pay

Higher OR number : increasing VDA and VPD due to scale effect and benefit of well planned arrivals.

Higher duration variability: increasing VDA and VPD implying the importance of careful appointment planning and dynamic scheduling.

Higher waiting costs: higher VPD but smaller VDA implying the importance of appointment time optimization.

Higher workload: smaller VPD and VDA due to unimprovability of overloaded syst

Impact of case-mix: • larger VPD when surgeries are identical due to their interchangeability.

• smaller VDA when surgeries are identical due to suboptimal appointment timesValue of dynamic assignment (VDA)Value of proactive anticipation and dynamic assignment (VPD)

- 65 -

Outline







- 66 -

Summary

A more realistic model of AS which has m servers; patients are served

in a pre-determined order but are flexible to any server.

Our aim is to proactively optimize the arrival times under the FCFS

dynamic assignment strategy.

We formulate a simulation-based optimization model to smooth integer

assignments, and derivate a continuous and differentiable cost function.

The proposed stochastic approximation algorithm is able to solve

realistic-sized instances and significantly improve the initial solution.

12

- 67 -

Managing surgery team overtime

“Branch and Price for Chance Constrained Bin Packing”

Zhang, Denton, Xie

INFORMS Journal On Computing, to appear

- 68 -

Motivation

ORs: critical resources that require high utilization

Unpredictable overtime causes high nurse turnover rate

Nurses ask for ...

• Some ORs to have low overtime

• Predictable completion times

Challenges:

• Fixed number of ORs

• Uncertain service time

• High cost of overtime

- 69 -

A chance constrained OR scheduling setting

Chance constraint (r , ) of an OR r

The surgery team of the OR r completes its daily due before time T + r with probability

where

= regular OR session time (T)

r = allowable overtime

Chance constraint = End-of-duty guarantee

Examples: No overtime with proba 90% : r = 0, = 0.1

at most 1h overtime with proba 95% : r = 1, = 0.05

- 70 -

A chance constrained OR scheduling setting

An informal setting

Decisions:Surgeries-to-ORs assignment

Constraints: For each chance constrained OR:

P(OR overtime k) ≥ 1 -

Objective: Minimize the expected overtime

A version of chance constrained extensible bin-packing problem

- 71 -

A stochastic programming formulation

Decision variables:

(1a) = Minimize total expected ovetime

(1b) = Assign each surgery to an OR

(1c) = Determine the overtime

(1d) = Chance constraints

I, R set of surgeries and set of ORs

di() duration of surgery i under scenario

T regular OR session time

set of ORs of chance constraint k

xir binary var equal to 1 if surgery i is assigned to OR r

or() overtime of OR r under scenario

Defining elements:

CkR

- 72 -

Solving Stoch. Prog. formulation: Branch-and-Price

Master problem

Decision variables

p column containing surgeries to be allocated in the same OR

p binary var equal to 1 if the column p is selected

13

- 73 -

Solving Stoch. Prog. formulation

Key ideas of branch-and-price

1. Branch on constraints

• Select a pair: (i, j )

• Left side (in the same bin): yip = yjp

• Right side (in separate bins): yip + yjp 1

2. Enforcing the antisymmetry constraints due to identical ORs

- 74 -


Pricing problem

Decision variables

yip binary var equal to 1 if surgery i is in column p

ckp binary var equal to 1 if column p is type-k chance constrained

op() overtime of column p

Stochastic knapsack problem

- 75 -


Pricing problem solution acceleration

• Tight upperbound withprobabilisticcovers and probabilisticpackings (Song et al., 2014).

: 1

: 1

:

Probability cover

Probability pack

: 1

max

Complementary

,

1,

set

ki k

i A

k

k

k

ki i ki

i ii

i ii Q P i P

ii C

F A P d T

F C

F P

Q P i P F P i

UB Y E O

d Y T O

Y M P Y P

Y C C

- 76 -

Robust optimization formulation

Assumptions:

A1. Given first two-moments (mi, i) of surgery durations.

A2. Unknown probability distributions of surgery durations.

Chance constraints replaced by worst-case chance constraints:

where D is the set of all distributions matching the first two moments:

inf 1i ir kD

i I

P d x T

d

2 2 2,i i i i iD E d m E d m d

- 77 -

Robust optimization formulation: key result

Theorem: For any random variable X of mean m and standard deviation s, the worst-chance probability CP is reached by a three-point distribution such that

22 2

22

2 2

1, if

, if ,

, if ,

k

k k

k

k kk

m T

CP m T m m TT m

mm T m m T

T

Under the mild assumption CV (-1 – 1)0.5,

the robust optimization formulation can be converted into a deterministic mixed-integer-programming model.

- 78 -

Case study

• Experiments are based on real data of 21 surgical days.

• Number of ORs:

m = 3 + 3 + 3;

OR session time T = 10h;

Overtime threshold

dk {0.0; 2.0h; }.

• Number of surgeries:

dk [11; 37].

14

- 79 -

Performance of Branch-and-Price

Performance of different methods for the stochastic model

• Simple size: 500• Computation time limit: 15,000 seconds.• Probability guarantee: 1 - = 0.9.

- 80 -

Value of Robust Optimization

Worst-case probability

Experimental setting: 1- = 0.9 (stochastic), 1- = 0.9 (robust), n [21; 25]

• Extensive form of robust optimization can be solved by Cplex

• The unachieved probability of stochastic solution could be 0.16

• The average overtime of robust solution could be 2 times higher

Average overtime

90%

- 81 -

Value of Robust Optimization

Worst-case probability

Experimental setting: 1- = 0.9 (stochastic), 1- = 0.7 (robust), n [21; 25]

• More robust solution with slighter higher overtime

Average overtime

90%

- 82 -

Conclusions

• The Branch-and-price can effectively solve the real-size problem instances

• The robust optimization problem can be much easier to solve than the stochastic problem

• The robust optimization can provide more robust solution with slightly higher overtime

- 83 -

Accounting for congestion behavior in appointment scheduling

“Appointment Scheduling Problem When the Server Responds to Congestion”

Zhang, Berg, Denton, Xie

IISE, to appear

- 84 -

Evidences from the literature

• Outpatient clinic: physicians tend to speedup when they perceive congestion in the waiting area (Rising et al. 1973; Cayirli et al. 2008);

• Emergency department: triage-ordered testing and task reduction are used to reduce service time (Batt and Terwiesch 2012);

• ICU/ED: delays in receiving intensive care can result in longer lengths of stay in the ICU (Chan et al. 2015).

15

- 85 -

An outpatient procedure case

• Data for a one year period

• Samples are classified by surgeon and procedure type

• Specific records on patient waiting time, pre-procedure time, procedure time and post-procedure time.

We look at the impact of waiting time on different service times

- 86 -

A case in the context of outpatient procedures

• Negative correlation Between pre-procedure time and waiting time

• No correlation between procedure time, post-procedure time and waiting time

- 87 -

Related work

• Although there is a vast literature on appointment scheduling, none of the existing studies considered endogenous randomness.

• Congestion was incorporated in queuing models by Chan et al (2014), Vericourt and Jennings(2011), …

• However, appointment systems have a little number of customers and they need to determine arrival times.

- 88 -

Research questions

• Can the appointment scheduling problem be solved when the endogenous randomness is incorporated?

• How important is it to anticipate a congestion response from the server when scheduling appointments?

• Why is the dome shaped rule that is claimed "optimal", in practice, not widely implemented?

- 89 -

Problem setting

A1/A2

FCFS assignment

Appointment optimization

for a given sequence of customers

to a single server system with congestion response behaviour

in order to minimize the total cost related to

• Customer waiting (lower service quality)

• Service time reduction (lower quality service)

• Overtime.

- 90 -

Problem setting

A1/A2

FCFS assignment

Decision variables:

xi = customer-i allowance or interarrival time between i-1 and i

2 2

1 1

2

min

, ,

, , , ,

,

n nw s oi i i i i

i i

i i i i

i i i i

n

i n ni

E c w c Z d c o

w w Z x i

Z f w d i

o x w Z T

: waiting cost

: service reduction cost

: overtime cost

: normal service time

: actual service time

, ,

wi

si

o

i

i

i i i i

c

c

c

d

Z

Z f w d

16

- 91 -

Problem setting : congestion behavior models

A1/A2

FCFS assignment

Linear response model

- 92 -


A1/A2

FCFS assignment

Logic Regression response model

2

11 i

ii i i w

Z de

- 93 -


A1/A2

FCFS assignment

Linear response model with customer no-show

0, if no-show

1 , if show and

1 , if show and

ii i i i i

i

i i i i

Z d w w tt

d w t

- 94 -

Solution approaches

Under mild continuity condition of the server reponse model,

• Stochastic-optimization with unbiased sample path gradients

Under linear response model

• Stochastic linear Mixed Interger Programming

- 95 -

Computational results : Comparison of SimOpt and SMIP

• Identical customers

• 500 samples are used for the SMIP, and 107 samples for the SimOpt.

• Costs are evaluated based on 106 samples.

• The SimOpt is much more efficient than solving the SMIP

• Across all instances, the SimOpt solved the global optimum

- 96 -

Computational results : Solution

• Allowances increase with variability and waiting cost

• Congestion reduces allowances

• Congestion makes allowances more flat

17

- 97 -

Computational results : comparison with heuristics

• Our method always finds the best solution

• Mean-value solution may outperform the Dome solution when congestion occurs

- 98 -

Computational results : comparison with heuristics

• Our method always finds the best solution

• Mean-value solution may outperform the Dome solution when congestion occurs

- 99 -

Conclusions

• Simulation-based Optimization can efficiently solve the congestion anticipated AS problems

• Variability and waiting coefficient affect the allowance and cost, while congestion behavior helps to lower the cost and smooth the allowances

• Ignoring the congestion is costly; the dome-shaped solution may perform worse than the mean-value solution

- 100 -

General conclusions

- 101 -

What next?

Joint optimization of surgery sequence and surgeon

appointment times.

simulation-based discrete optimization + stochastic approximation

Chance constraints of surgery starts

Dynamic control of overtime allocation

Surgeon behavior

Joint scheduling of inpatient and day surgeries

1

Chapter 2Appointment scheduling

- 2 -

Plan

• Basis of Appointment Systems

• Individual-block/Variable-interval rule

• Variable-block/Fixed-interval rule

• A static appointment model with dynamic features

• Managing the Appointments with Waiting Time Targets and Random Walk-ins

- 3 -

Intoduction

• Rising healthcare expenditure and increasing demand lead a growing pressure on health service providers to improve efficiency.

• Appointment scheduling systems lie at the intersection of efficiency and timely access to health services.

• Timely access is important for realizing good medical outcomes. It is also an important determinant of patient satisfaction.

• The ability to provide timely access is determined by a variety of factors including both strategy designs (location, capacity, size) and operation management.

• The issue of capacity planning is addressed in another chapter and an example of resource allocation in multisite service systems can be found in Chao et al.

Chao, X., Liu, L. and Zheng, S. (2003) Resource allocation in multisite service systems with intersite customer flows. Management Science, 49(12), 1739–1752

- 4 -

Dimensions of appointment systems

Goal : to find an Appointment System (AS) for which a particular measure of performance is optimized in a clinical environment—an application of resource scheduling under uncertainty.

• Nature of decision-making

• Modeling clinic environments

• Mesures of performance

• Appointment system design

• Analysis methods

T. Cayiri & E. Veral, « Outpatient scheduling in health care : a review of literature , » Production and Operations Management, 12/4, 2003

- 5 -

Nature of decision making

Static vs dynamic appointment scheduling systems

Static systems :

• all decisions must be made prior to the beginning of a clinic session (day or half-day), which is the most common appointment system.

• Most of the literature concentrates on the static problem.

Dynamic case :

• the schedule of future arrivals revised continuously over the course of the day based on the current state of the system

• Applicable when patient arrivals can be regulated dynamically, which involves patients already admitted to a hospital or clinic.

??? Possible with on-line apps?- 6 -

Modelling clinic environment

Regarded as a queuing system characterized by the followings:

• Number of services (single or multi-stage)

• Number of doctors (single or multi-server)

• Number of appointments per clinic session

• Arrival process (deterministic or stochastic)

• Punctuality of patients

• Presence of no-shows

• Presence of regular & emergency walk-ins (preemtive or not)

• Presence of companions

• Service times (empirical or theoretical distribution)

• Lateness and interrruption of doctors

• Queue discipline

2

- 7 -


Number of services (single or multi-stage)

• Mostly a single queue system where patients queue for a single service (rationing: doctors = bottleneck, time to consultation is the most important criterion)

• A few simulation studies on clinic environments includingregistration, pre-examination, post-examination, x-ray, laboratory, checkout, etc

Doct 1

Doct N

Outpatient unit

- 8 -


Number of doctors (servers)

• Mostly a single server

• Doctors have their own waiting list

• Although pooling improves waiting times & utilisation, random doctor assignment is undesirable due to the desirableone-to-one doctor-patient relationship

• Sending patients to the first available doctor is howeverpracticed in some countries

Warning : Multi-server setting is however realistic for diagnostic equipment (IMR, CT, …)

- 9 -


Number of Appointments per Clinic Session

• There exists a positive relationship between waiting times and the number of appointments in a clinic session (N).

• the effect of N is mitigated by no-shows and variability of consultation times, and thus cannot be easily generalized

- 10 -


The arrival process

Unpunctuality of patients: • Empirical evidence : early more often than late. • patient earliness may also be undesirable, since it creates

excessive congestion in the waiting area

Presence of no-shows: • Empirical evidence : 5-30% (even 40%) no-shows depending

on the specialties. • Numerical observation : no-show proba is a major factor

affecting the performance and the choice of AS• Depending on different variables (such as age, socioeconomic

level, etc.)• Countermeasures: overbooking but also mechanism to

discouraging no-shows (blacklist?)

- 11 -


The arrival processUnpunctuality of patients: Presence of no-shows:

Presence of walk-ins (regular and emergency):

• Rarely in UK hospital clinics used mainly for referred patients

• But must be anticipated and planned for in some US clinicsthat are patient’s GP and respeonsible for patient’s total care, whether emergent or not.

• Common in Chinese hospitals providing on-line appointment

• Observation : walk-in varying across specialties and throughoutthe day, but not from day to day

Presence of companions• Importan for waiting room area designBalking or reneging behavior

- 12 -


Service times

• sum of all the times a patient is claiming the doctor’s attention

• Common model: homogenuous patients with iid service times independent of the arrival process (no response to congestion)

• Empirical data: unimodal and right-skewed, CV = 0.35 – 0.85

• Erlang or exponential distribution in analytical studies

• Observation 1: optimal solutions mostly dependent on mean and variance

• Observation 2: high variability of service times deteriorates bothpatient wait and doctor’s idle time

• Observation 3: shorter mean service time -> lower waiting time.

• Technologies help reducing the consultation time

3

- 13 -


- 14 -


Lateness and Interruption Level of Doctors

• Agreement among existing studies is that patient waiting times are highly sensitive to this factor.

• If the doctor does not start the clinic on time, a delay factor builds up from the start that ripples throughout the clinicsession.

• Another doctor-related factor is the interruption level (also called the “gap times”).

• Game theory may be useful in modeling patient and doctor arrivals by considering the conflicting interests of both parties.

• patients arrive arrive late knowing that they will have to wait.

• doctors may arrive late, being afraid that the first patient will be late.

- 15 -


Queueing discipline

• Common model: FIFO

• Reasonable for puctual patients

• Unrealistic in the presence of unpunctuality as doctors wouldnot keep idle waiting for the next appointment in the presence of other waiting patients

• In practice, the following priority order is used :

• Emergencies

• second consultations

• scheduled patients

• lowest priority to walk-ins seen on a FCFS basis

• Calling patients in the order of arrival is used in practice but destroy the purpose of an AS.

- 16 -

Performance mesures

Cost-based mesures

• Min E[TC] = E[W] Cp + E[I] Cd + E[O] Co

• Cp= patient waiting cost, Cd= doc idling cost, Co= overtime cost

• Common model : identical linear cost

• Klassen and Rohleder (1996) : One patient waits 40 minutes ≠ 20 patients wait 2 minutes each

• In the presence of unpunctual patients and/or walk-ins, the assumption of homogeneous waiting costs may need to be relaxed

• For regular patients, there might be a threshold over which patients’ tolerance declines steeply. Some survey results indicate that tolerance diminishes after about 30 minutes.

• UK national standard, 75% within 30 min of their appointment

- 17 -

Performance mesures

Cost-based mesures

• Min E[TC] = E[W] Cp + E[I] Cd + E[O] Co

• Cp= patient waiting cost, Cd= doc idling cost, Co= overtime cost

• In practice, Co/Cd = 1.5 (overtime cost 50% more)

• Cd/Cp = 1 to 100

• Fries & Marathe (1981):

easier to estimate the costs relative to the server, which are usually available via standard cost accounting,

but the costs of waiting involve a different type of analysis where the issues of goodwill, service, and “costs to the society” place a value on patients’ waiting time

- 18 -

Performance mesures

Time-based mesures

• True waiting time = Si – max(Ai, ai) or max(0, Si – max(Ai, ai)) Si = starting time Ai = Appointment time ai = actual arrival time

• Negative true waiting = voluntary and not due to the AS

• Flow-time = total time in the clinic.

• Flow time rarely used as patients generally do not mind the service time

• Idle time:

• Overtime

• There may be a maximal acceptable level of each. Ex: % seenwithin 30 min of their appointment

4

- 19 -

Performance mesures

Congestion mesures

• Mean & distribution of the queue length

• Mean & distribution of the number in the system

- 20 -

Performance mesures

Fairness mesures

• Fairness = uniformity of performance of an AS across patients

• Mean & variance of waiting time + queue size of patients accordingto their place in the AS

• Double penalties of patients at the end of the clinic session

waiting times tend to increase over time, whereas

Consultations time tend to decrease due to congestion response

- 21 -

Performance mesures

Other mesures

• Productivity of the doctor

• Mean doctor utilization

• Delays between requests and granted appointment

• % of urgent patients served

• % of patients receiving the requested slots

- 22 -

Appointment System Design

The AS design can be broken down into a series of decisions regarding:

1) the appointment rule,

2) the use of patient classification, if any, and

3) the adjustments made to reduce the disruptive effects of walk-ins, no-shows, and/or emergency patients

- 23 -

Appointment rules

XXXXXXXXX

Block size N

Clinical session

Single-block

X X X X X X X X X X

Individual-block/Fixed-intervalai = constantni = 1

X

X

X X X X X X X X X X

Individual-block/Fixed-interval with an initial blockai = constantn1 > 1, ni = 1

X X X X X

X X X X X

Multiple-block/Fixed-intervalni = mai = constant

Three variablesn1=initial blockni = Size of block iai = time of interval i, also called allowance

X

X

X X X X X

X X X X X

Multiple-block/Fixed-interval with an initial blockn1>ni = mai = constant

X

X X

X X X

X X X X X

X X X X X

Variable-block/Fixed-interval ni variableai = constant

X X X X X

Individual-block/ Variable-interval ni variableai = variable

- 24 -

Appointment rules

• Single-block rule : the most primitive “date only” AS ensuring doctor productivity by excessive patient waiting time.

• Individual-block/Fixed-interval rule with an initial block :

• Keep an inventory of patients to hedge against the risk of the unpunctuality or no-show of the 1st patient

• Bailey’s rule (n1 = 2, ni = 1, ai = mean service time ) -> reasonable balance between patient waiting and doctor idle (1952)

• Multiple-block/Fixed-interval rule :

• Usually (ni = m, ai = m)

• Possibly more suitable when the mean consultation time is short (Nuffield Trust (1965))

• Lack of rigorous research on the circumstances multiple block rule performs better

5

- 25 -

Appointment rules

• Individual-block/Variable-interval rule:

• The “dome” shape appointment intervals: initially increase toward the middle of the session and then decrease

• Shown to be optimal for i.i.d. service times and uniform waiting costs for all patients

- 26 -

Patient classification

• Common model : homogenuous and served in FCFA (first-call first appointment) basis

• Purposes of patient classification: sequence patients at booking, and adjust appointment intervals

• Radiology examination times depend on factors such as patient’s age, physical mobility, and type of service.

• Classification scheme: new/return, variability of service times (low/high-variance patients), and type of procedure

• Limited interest in outpatient settings in which the schedule has to be ready in advance and the arriving requests are handled dynamically

• Realistic application: the patients classified into a manageable number of groups and assigned to pre-marked slots when they call for appointments. Ex: new patients before 10:00, return patients10:00 to 12:00.

• Drawback: reduced flexibility and potential lost capacity

- 27 -

Adjustments

• Whenever relevant, no-shows, walk-ins, urgent patients, and/or emergencies need to be planned for

• No-shows cannot be entirely eliminated by administrative mechanisms

• There is a tendency of walk-in aming lower social classes and it is not fair to deny walk-in access to clinics.

• When the 2nd consultation is frequent (orthopedics), some allowance should be made for the additional demand imposed on doctors

• Suggestion from literature: the patient load (i.e., percent of available appointments filled) be adjusted based on the expected number of walk-ins and no-shows.

- 28 -

Adjustments

• Adjustment for no-shows

• None

• Overbooking extra patients to predetermined slots

• Decreasing appointment intervals proportionally

• Adjustment for walk-ins, second consultations, urgent patients, and/or emergencies

• None

• Leaving some predetermined slots open

• Increasing appointment intervals proportionally

- 29 -

Adjustments

• Adjustment for no-shows

• None

• Overbooking extra patients to predetermined slots

• Decreasing appointment intervals proportionally

• Adjustment for walk-ins, second consultations, urgent patients, and/or emergencies

• None

• Leaving some predetermined slots open

• Increasing appointment intervals proportionally

- 30 -

Analysis methodologies

• Analystical studies

• Queueing theory

• Mathematical programming

• Dynamic programming

• Nonlinear programming

• Stochastic programming

• Simulation studies

• Ho et al. (1992, 1999,1995) evaluate 50 appointment rules under various operating environments.

• No rule that performs well under all circumstances and a simple heuristic is proposed to choose an appointment rule for a clinic given , CV, N, and Cp/Cd ratio

• They find that , CV, and N affect AS performance in the order of decreasing importance.

6

- 31 -

Plan






- 32 -

Individual-block/Variable-interval rule by Lindley equation

• AS = Individual-block/Variable-interval rule

• Single server

• N scheduled patients (with a given appointment sequence)

• Served in FAFS discipline (First Appointment First Served)

• Different general distributed service times

Not considered in this basic model :

No-shows (see extension)

Unpunctuality

X X X X X


- 33 -


1

1

1 1 1

1 1

1

Sample waiting time of patient i

Idle time of doctor waiting for patient i

Overtime

interval between i and i+1, also job c ll aa ed

i i i i

N

N N ii

i

i i i i

I x W p

O W p x T

x

W W p x

servic

llowan

e time

c

e

of iip

1 1 1

i i

i i i

W I

W p x

Why?X X X X X


- 34 -


Optimal appointment scheduling

1 1

1 1 1

1

1

min

Subject to

, 2,...,

0, 0, 0, 0

N N

p i d i oi i

i i i i i

N

N N ii

i i

E C W C I C O

W I W p x i N

O u T W p x

W I O u

Can be converted into a linear programming model by sample average over K randomly generated scenario:

1 1 1

1

1min

where ,...,

K N N

p i k d i k o kk i i

T

k k N k

C W C I C OK

p p

- 35 -


Case without overtime cost, i.e. Co = 0

1 1

1 1 1

min

Subject to

, 2,...,

0, 0

N N

p i d ii i

i i i i i

i i

E C W C I

W I W p x i N

W I

1 1

1 1 1

ˆ ˆmin

Subject to

ˆ ˆ ˆ ˆ ˆ , 2,...,

ˆ ˆ0, 0

N N

p i d ii i

i i i i i

i i

E C W C I

W I W p x i N

W I

ˆ ˆ,

ˆ ˆ,

i i i i i i

i i i i

p p b a x x b a

W W a I I a

Special case:

1

1

1

1

ˆ

,

ˆ 0

i i i

N

i i ii

i

b aN

E p

x ax

Optimal allowance margin independent of mean

- 36 -


Solution methods :

• L-shape algorithm with sequence bounding in Denton & Gupta 2003

• Bender’s decomposition in Chen & Robinson 2014

• Stochastic approximation with sample path gradient (to be addressed in surgery scheduling part for a multi-server setting)

7

- 37 -


Numerical results

Homogenuous patients

Unit doctor idle cost

Patient wait cost =

No overtime cost

Transformed service time distribution- 38 -


= 0,1N = 8

• ‘Dome’ pattern allowances with more time to patients in the middle.

• The 1st allowance much lower than the others, and varies only slightly with N, for fixed.

• The final allowance is also somewhat lower.

• The intermediate allowances are all about the same.

From Robinson & Chen, 2003

- 39 -


From Robinson & Chen, 2003

• Best two-parameterpolicy within 1% of the true optimum

- 40 -


Extension to no-shows

1 1 1 1

1 1 1 1

1

1

Sample waiting time of i-th patient

Idle time of doctor waiting for patient i

Overtime

interval between i and i+1

i i i i i

i i i i i

N

N N N ii

i

i

W W p x

I x W p

O T W p x

x

p

show-up in

service

dicator

time of i

of i, becoming known at i i iBernoulli P x

Patient specific no-show probability

- 41 -



1 1

1 1 1 1

1

11

min

Subject to

, 2,...,

0, 0, 0, 0, 0

N N

p i i d i oi i

i i i i i i

N

N N i ii

i i i

E C W C I C O

W I W p x i N

O u T W p x

x W I O u

Can be converted into a linear programming model by sample average over K randomly generated scenario:

1 1 1

1 1

1min

where ,..., ; ,...,

K N N

p i i k d i k o kk i i

T

k k N k k N k

C W C I C OK

p p

- 42 -

Plan



• Variable-block/Fixed-interval rule by queueing analysis



8

- 43 -

Variable-block/Fixed-interval rule by queueing analysis

• AS = Variable-block/Fixed (or Given)-interval rule

• Single server

• N scheduled patients

• Served in FAFS discipline (First Appointment First Served)

• homogenuous patient population

• Exponential service time distribution

Not considered in this basic model :

No-shows (see extension)

Unpunctuality

Walk-ins

- 44 -


Basic result:

where

• qt = queue length at the beginning of block t

• xt = block size

• Lt = Poi(t) number of departures in block t

• = service rate

1t t t tq q x L

!

k

tP L k ek

- 45 -


Waiting time of patients arriving in t:

1 1 1

1

11 1

2

tx

t t t t t ti

W q i q x x x

Total wait experienced in a period

1

0 2

2 1

0

1 1! 1

11 1 0.5 2 1 1

2

,! !

t t t

k k n

k m

k kx

k x k

TW f q x

f n e m n k nk k

n n n n n n

x e x ek k

Why?

- 46 -


Idle time

Overtime

1

1! 1

1

!

t t

k

t t tk q x

t t t t t t

k

k x

I e k q xk k

q x q x q x

x ek

11 1

11

2 T TO q q

- 47 -


Evaluating a Variable-block/Fixed-interval rule by Markov chain

• k(t) = probability of k patients at the beginning of t

• Transition probability

• Queue distribution :

• (t+1) = (t) Pt

• State space bound

• k(t) = 0, for all k > N

1

0, if

, otherwise!

t

t

i x jijt t t

t

j i x

p P q j q ie

i x j

- 48 -


Optimal Variable-block/Fixed-interval rule

• In some well-chosen neighborhood (Kaandorp & Koole, 2007),

the cost function is multimodular in block size vector x and hence

any local optimum is global optimum.

9

- 49 -


Stochastic programming of Variable-block/Fixed-interval rule

1

1

1

11 1

2 1

min

0.5 1

11 1 0.5 2 1 1

2

T

p t d Ot

t t t t

t t t

T T

t

E C TW C T O N C O

q q x L

TW f q x

O q q

L Poi

f n n n n n n n

1 1, 1

1 0 0

0

, 1 , 10 0

0

min 0.5 1

0, 1,...,

, 1,...,

1, 1,..., 1

T N N

p it d O i T dt i i

N

it ti

N N

i t i t jt ti j

N

iti

E C f i z C C i i z C T N

iz x t T

iz x jz L t T

z t T

zit = 1 iff qt + xt = i

- 50 -


Basic result:

where

• qt = queue length at the beginning of block t

• xt() = Binomial(Xt, 1-) : Xt = block size, = no show proba

• Lt = Poi(t) number of departures in block t

1t t t tq q x L

1

!

tkt X k

t

k

t

XP x k

k

P L k ek


- 51 -


The Markov chain analysis extends directly

The multi-modularity and the optimality of local search under some well-choisen neighborhood also extend.


- 52 -


Stochastic programming of Variable-block/Fixed-interval rule withno-shows

1 1, 1

1 0 0

0 1

, 1 , 10 1 0

0

1

min 0.5 1

0, 1,...,

, 1,...,

1, 1,..., 1

1, 1,...,

T N N

p it d O i T dt i i

N N

it i iti i

N N N

i t i i t jt ti i j

N

iti

T

itt

E C f i z C C i i z C T N

iz J x t T

iz J x jz L t T

z t T

x i N

1,1 1

, 1,..., 1 (anti-symmetry)

1

T T

it i tt t

i

tx tx i N

J Bernoulli

- 53 -

Plan






- 54 -

From

Kumar Muthuraman, Mark Lawley, « A stochastic overbooking model for outpatient clinical scheduling with no-shows » IIE Transactions, 40, 820-837, 2008

A static appointment model with many dynamic features

10

- 55 -

Clinical booking model

• Single server

• Clinical session divided into I slots of length ti

• Patients call in sequentially before the start of the session

• These « call-ins » can be scheduled to one of the I slots or rejected.

• Rejection of a patient terminates the booking process.

• The appointment decision is made dynamically when call-in.

• Patients scheduled for each slot have a patient-specific no-show probability and arrive independently of other patients

• Patients are served in FIFO order

• Service times are exponentially distributed at rate

• Patients are categorized into J groups and the group attribute is known at the call-in time. Each group-j patient has a probability pjof showing up.

- 56 -

Performance measures

• r = reward of serving a patient

• ci = cost of each patient overflowed from slot i to slot i+1

- 57 -

Scheduling policy

• It is sequential in the sense that it assigns patients as they call

• It is myopic in the sense that it does not consider future arrivals when making the assignment.

• For each new call-in, it enumerates all possible assignments for the new patient and selects the assignment that maximizes the objective function generated by all scheduled patients.

• The algorithm rejects the patient and terminates when there is no way to schedule the patient without hurting the objective (Why ?)

- 58 -

Evaluation of a schedule with n call-ins

,n n n nim i ik

i m i k

f r mQ c kQ Q R

- 59 -


1

number of pattients arrving for slot

number of pattients overflowing from slot

,

maximal number of departures in slot

ni

ni

n n ni i i i

n n n ni i i

i i

i

i i

X i

Y i

Y Y X L

f E r X c Y

L i

L Poi t

Q R

- 60 -


1 1, 1

1

1

1, ''

1,

appointment of pattient

group attribute of pattient

1 , if

, if

, if

' , if , 0

, if

n n

i

n

n

n nim j i m j nn

imnim n

nim n

n n nim i i m ik n

m k

n nL i m k l ik

k l

i n

j n

Q p Q p i iQ

Q i i

R i i

R P L m k R Q i i m

f l R Q

, 0n

i i

i i m

L Poi t

11

- 61 -

Properties

1 ,i n

n

It i i

i ni i

r c e i

1min i n

nn

It i i

iii i

r c e

Unimodality

- 62 -

Numerical results

Three types of patients of no-show proba (0.9, 0.5, 0.1)Extension to include patient preference

- 63 -

Numerical results

• Unique local maximum• More appointments and

lower overflow than Round-Robin

• Similar overflow probability per patient

Total overflow

Overflow per patient

- 64 -

Plan






- 65 -

From

Xingwei Pan, Na Geng, Xiaolan Xie, Jing Wen, « Managing appointments with waiting time targets and random walk-ins”, submitted

Exercices Layout

1. Using the flow data in the table below, measure the flow dominace and estimate complexity of the facility layout problem.

From-To-Matrix A B C D E F A - 10 12 0 40 0 B 34 - 59 0 0 12 C 0 10 - 25 0 0 D 0 23 12 - 21 0 E 9 14 0 0 - 32 F 0 27 14 0 0 - Develop a corresponding Flow Graph and try to create a reasonable layout by locating ressources with high flow close to each other.

2. Using the flow data of Exercise 1, and the added information that department C and E must be separated, create a REL chart. Solve the facility location problem for this chart, assuming those departments have equal size.

3. Consider the REL chart below and assume that all departments are similar in size.

Using quantitative method based on Total Closeness Rating, create initial layout and try to improve it by pair-wise exchange.

4. The data for four-department layout problem are given below. cost 1 2 3 4 Flow 1 2 3 4 Distance 1 2 3 4

1 5 5 10 1 10 20 5 1 10 20 52 5 20 20 2 10 15 20 2 10 15 203 5 20 10 3 20 12 10 3 20 12 104 10 10 10 4 5 20 10 4 5 20 10

1. Setup the quadratic assignment problem to find the optimal layout. 2. Estimate an initial lower bound for total location cost. 3. Improve the solution by exchanging positions of the departments.

I

X

I U

I

I E

E

U

A

U

O O

A

A

MMaacchhiinnee11

MMaacchhiinnee22

MMaacchhiinnee33

MMaacchhiinnee44

MMaacchhiinnee55

MMaacchhiinnee66

4. The Snow-Bird Hospital is a small emergency-oriented facility located in a popular ski resort area. Its new administrator, Mary Lord, decides to reorganise the hospital, using the process-layout method. The current layout is shown below.

The only physical restriction is the need to keep the entrance and initial processing room in its current location. All other departments or room (each of 10 feet square) can be moved. Mary’s first step is to analyse records in order to determine the number of trips made by patients between departments in an average month. The data are shown below. The objective Ms. Lord decides is to layout the rooms so as to minimise the total distance walked by patients who enter for treatment.

1 2 3 4 5 6 7 8 100 100 1. entrance 50 20 2. exam. 1 30 30 3. Exam. 2 20 20 4. X-ray 20 10 5. Lab 30 6. Op. room 7. Rec. Room 8. Cast-setting Given the above information, redo the layout of Snow-Bird Hospital to improve its efficiency in terms of patient flow.

Entrance/ initial

processing

Exam Room 1

Exam Room 2

X-ray

Lab tests/ EKG

Operating room

Recovery room

Cast-setting room

4400’’

1100’’

1100’’

Exo Human resources 1 (Day-off scheduling). Consider the days-off scheduling problem with the following daily requirements : Day j 1 2 3 4 5 6 7 Sun Mon Tues Wed Thurs Fri Sat requirement 3 5 5 5 7 7 3 1 weekends off over 3. 2. Consider a (5, 7)-cyclic staffing problem with b = (4, 9, 8, 8, 8, 9, 4) and c = (6, 5, 6, 7, 7, 7, 7). 3. (Employee schedule) After faithfully serving the OR profession for 50 years, you decide to retire and open a restaurant. Among the hundreds of details with opening a restaurant, you need to hire and schedule employees. Based on the foot traffic of other restaurants in the area, you expect that you will need the following number of employees each day: Day j 1 2 3 4 5 6 7 Mon Tues Wed Thurs Fri Sat Sun Employees needed 4 5 5 10 12 12 2 Your employees will work four consecutive days and then have three days off. They will be paid 100€ for each day they work. In your rush to get the restaurant started, you haphazardly hire 17 employees. Five will start on Monday, five will start on Thursday and seven will start on Friday. This schedule satisfies the above work requirements, but you have no idea how optimal this is. Questions:

1) How much money would you save each week from your current schedule if you optimized your workforce?

2) How much additional money would you save or lose each week if you switched your employees to a “five days on, two days off” schedule at 80€ per day?

4. (Planning nurse shifts) You are asked to help improving the nurse planning of an Emergency Department (ED). From the historic data, you are able to obtain the following demand forecast on the number arrivals at the ED: period (h) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24arrivals 0,5 0,5 0,5 0,5 0,5 1 4 10 10 8 4 8 6 6 4 6 10 15 8 6 4 2 0,5 0,5

From the statistics, 60% of the ED patients are regular patients and need only 15 minutes of the nursing care. However 40% of the ED patients are true emergency patients and require about 1h nursing care at ED before transfer to the wards. You are asked to :

a) Derive the workload profile of a typical day. b) Enumerate all possible shift patterns. Shifts of 8h start either 7h-9h (20€/h), or 15h-17h

(22€/h), or 23-01h (25€/h). Shifts of 12h start either 7h-9h (21€/h) or 19h-21h (23€/h). c) Determine the optimal shifts with a coverage P = 100%, 80%, et 120%. What do you think of

the solutions ? How to know which one is the best ?

9 (Staffing : number of nurses) A hospital is exploring the level of staffing needed for a booth in the local mall, where they would test and provide information on the diabetes. Previous experience has shown that, on average, every 6.67 minutes a new person approaches the booth. A nurse can complete testing and answering questions, on average, in twelve minutes. Assuming s = 2, 3, 4 nurses, a hourly cost of 40€ per nurse and a customer waiting cost of 75€ per hour. DDeetteerrmmiinnee tthhee ffoolllloowwiinngg:: ppaattiieenntt aarrrriivvaall rraattee,, sseerrvviiccee rraattee,, oovveerraallll ssyysstteemm uuttiilliizzaattiioonn,, nnbb ooff ppaattiieennttss iinn tthhee ssyysstteemm ((LLss)),, tthhee aavveerraaggee qquueeuuee lleennggtthh ((LLqq)),, aavveerraaggee ttiimmee ssppeenntt iinn tthhee ssyysstteemm ((WWss)),, aavveerraaggee wwaaiittiinngg ttiimmee ((WWqq)),, pprroobbaabbiilliittyy ooff nnoo ppaattiieenntt ((PP00)),, pprroobbaabbiilliittyy ooff wwaaiittiinngg ((PPww)),, ttoottaall ssyysstteemm ccoossttss.. 10. (Dimensioning the number of beds : Target occupancy level) CCoonnssiiddeerr oobbsstteerriiccss uunniittss iinn hhoossppiittaallss.. OObbsstteerriiccss iiss ggeenneerraallllyy ooppeerraatteedd iinnddeeppeennddeennttllyy ooff ootthheerr sseerrvviicceess,, ssoo iittss ccaappaacciittyy nneeeeddss ccaann bbee ddeetteerrmmiinneedd wwiitthhoouutt rreeggaarrdd ttoo ootthheerr sseerrvviicceess.. IItt iiss aallssoo oonnee ffoorr wwhhiicchh tthhee uussee ooff aa ssttaannddaarrdd MM//MM//ss qquueeuueeiinngg mmooddeell iiss qquuiittee ggoooodd.. MMoosstt oobbsstteerriiccss ppaattiieennttss aarree uunnsscchheedduulleedd aanndd tthhee aassssuummppttiioonn ooff PPooiissssoonn aarrrriivvaallss hhaass bbeeeenn sshhoowwnn ttoo bbee aa ggoooodd oonnee iinn ssttuuddiieess ooff uunnsscchheedduulleedd hhoossppiittaall aaddmmiissssiioonnss.. IInn aaddddiittiioonn,, tthhee ccooeeffffiicciieenntt ooff vvaarriiaattiioonn ((CCVV)) ooff tthhee lleennggtthh ooff ssttaayy ((LLOOSS)),, wwhhiicchh iiss ddeeffiinneedd aass tthhee rraattiioo ooff tthhee ssttaannddaarrdd ddeevviiaattiioonn ttoo tthhee mmeeaann,, iiss ttyyppiiccaallllyy vveerryy cclloossee ttoo 11 ssaattiissffyyiinngg tthhee sseerrvviiccee ttiimmee aassssuummppttiioonn ooff tthhee MM//MM//ss mmooddeell.. Since obsterics patients are considered emergent, the American College of Obsterics and Gynecology (ACOG) recommends that occupancy levels of obsterics units not exceeding 75%. Many hospitals have obsterics units operating below this level. However, some have eliminated beds to reduce « excess » capacity and costs and 20% of NY hospitals had obsterics units that would be considered over-utilized by this standard. Assuming the target occupancy level of 75%, what is the probability of delay for lack of beds for a hospital with s = 10, 20, 40, 60, 80, 100, 150, 200 beds. 11 (Dimensioning the number of beds) Evaluation of capacity based on a delay target leads to very important conclusion. Though there is no standard delay target, it has been suggested that the probability of delay for an obsterics bed should not exceed 1%. What is the size of an obsterics unit (nb of beds) necessary to achieve a probability of delay not exceeding 1% while keeping the target occupancy level of 60%, 70%, 75%, 80%, 85%? 12. (Dimensioning the number of beds: Impact of seasonality) Consider an obsterics unit with 56 beds which experiences a significant degree of seasonality with occupancy level varying from a low of 68% in January to about 88% in July. What is the probability of delay in January and in July? If, as is likely, there are several days when actual arrivals exceed the month average by 10%, what is the probability of delay for these days in July? 13 (Dimensioning the number of beds : Impact of clinical organization) Consider the possiblity of combining cardiac and thoracic surgery patients as thoracic patients are relatively few and require similar nursing skills as cardiac patients. The average arrival rate of cardiac patients is 1,91 bed requests per day and that of thoracic patients is 0,42. No additional information is available on the arrival pattern and we assume Poisson arrivals. The average LOS (Length Of Stay) is 7,7 days for cardiac patients and 3,8 days for thoracic patients. What is the number of beds for cardiac patients and thoracic patients in order to have average patient waiting time for a bed E(D) not exceeding 0,5, 1, 2, 3 days? What is the number of beds if all patients are treated in the same nursing unit? Delay in this case measures the time a patient coming out of surgery spends waiting in a recovery unit or ICU until a bed in the nursing unit is available. Long delays cause backups in operating rooms/emergency rooms, surgery cancellation and ambulance diversion. 14. (Planning nurse shifts) You are asked to help improving the nurse planning of an Emergency Department (ED). From the historic data, you are able to obtain the following demand forecast on the number arrivals at the ED: period (h) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24arrivals 0,5 0,5 0,5 0,5 0,5 1 4 10 10 8 4 8 6 6 4 6 10 15 8 6 4 2 0,5 0,5


Assume that the ED patients arrive according to a Poisson process with time slot dependent rate and the service time of each patient is exponentially distributed with mean equal to the previous average. You are asked to :

d) Derive the workload profile of a typical day. e) Enumerate all possible shift patterns. Shifts of 8h start either 7h-9h (20€/h), or 15h-17h

(22€/h), or 23-01h (25€/h). Shifts of 12h start either 7h-9h (21€/h) or 19h-21h (23€/h). f) Determine the optimal shifts meeting the waiting time target WT with probabilty % (WT =

30 min – = 90%, WT = 1h – = 90%, WT = 30 min – = 80%). What do you think of the solutions ? How to know which one is the best ?

Operating theatre planning/scheduling 1 (surgery planning). Vous êtes chargé de planifier un bloc opératoire. Le bloc opératoire dispose de 8h par jour sur une semaine de 5 jours. Les opérations suivantes ont été inscrites:

Case OR time (in 10 min) Pre‐assigned date (ri)

1 5 3

2 22 5

3 21 5

4 21 4

5 23 3

6 22 5

7 5 2

8 15 3

9 7 1

10 9 2

11 4 5

12 9 4

13 13 2

14 16 2

15 15 1

16 3 1

17 11 4

18 16 5

19 13 2

20 17 2 On vous demande de: 1/ vérifier si la capacité du bloc opératoire est suffisante 2/ trouver un planning permettant d'opérer un nombre maximal de patients (préciser la méthode utilisée) 3/ formuler comme un problème de programmation linéaire le problème de planification afin de minimiser le coût total sous les conditions suivantes : (i) chaque jour dispose de 2h supplémentaire au coût de 300€/h; (ii) un jour de retard coûte 500€ par patient; (iii) la non planification d'un patient i coûte (5-ri+3)*500 . 4) proposer une méthode heuristique au problème 3/.

2/ (Affectation des salles) Les patients suivants sont planifiés pour être opérés lundi dans les trois salles de votre bloc opératoire.

Case OR time (in 10 min) Spécialité

1 11 3

2 6 2

3 18 3

4 12 2

5 3 1

6 13 2

7 13 1

8 5 1

9 12 1

10 9 2

11 8 2

12 5 3

13 6 1

14 24 2

15 3 3 a/ déterminer une affectation afin de minimiser la durée d'ouverture du bloc opératoire. Expliquer votre méthode; b/ proposer une formulation mathématique afin de minimiser le coût total sous les conditions suivantes : (i) l'ouverture du bloc coûte 500 euros par heure; (ii) une salle est associée à chaque spécialité; (iii) opérer un patient dans une salle qui n'est pas de la spécialité correspondante coûte C avec C = 100€, 200€, 300€, 400€. c) proposer une méthode pour résoudre le problème b/. d) proposer une formulation mathématique et une méthode garantissant à 95% de chance la fermeture d’une salle après 8h et minimisant la somme des heures supplémentaires de autres salles. Pour cette question, chaque OR time a un écart-type de % de sa moyenne avec ( = 10%, 50%) 3/ (Ordonnancement) Les patients suivants doivent être affectés et ordonnancés dans un bloc opératoire de 2 salles d'opération et 3 lits de réveil:

Case OR time (in 10 min) Recovery time (in 10 min)

1 11 14

2 6 7

3 18 23

4 12 15

5 3 3

6 13 16

7 13 16

8 5 6

9 12 15

10 9 11 Deux cas doivent être considérés: (i) chaque patient doit être transféré dans un lit de réveil à la fin de l'opération; (ii) un patient peut commencer le réveil dans une salle d'opération en attendant la libération d'un lit de réveil. a/ proposer un affectation des salles afin de minimiser l'ouverture des salles; b/ déterminer un ordonnancement des patients dans les salles d'opération et les lits de réveil c/ proposer et tester d'autres règles d'ordonnancement.

Case mix and Master Surgery Scheduling Without loss of generality, it is assumed that each department has three surgeon groups with each handling 3 pathologies in the testing instances. Moreover, for each patient group, the lower bound on the number of admitted patients per cycle is set as one, while the upper bound is generated randomly from a discrete uniform probability distribution within the interval [3, 6]. Similarly, other data parameters are also generated from discrete uniform distribution functions. Specifically, the average LOS per pathology is distributed in the interval [4, 10], the expected surgery duration is distributed in the interval [60, 240], and the mean treatment reward is distributed in the interval [12, 30]. In addition, the total number of beds and the number of operating rooms are fixed as different amounts in various instances.

min 1 3 4 60 12 480

max 1 6 10 240 30 480

DRG group Lp Up LOSp DURp rp LENGTH

1 1 3 4 236 14 480

2 1 4 5 106 13 480

3 1 4 5 122 27 480

4 1 4 9 157 15 480

5 1 4 5 223 20 480

6 1 3 6 227 12 480

7 1 4 8 69 28 480

8 1 3 9 217 29 480

9 1 6 5 220 17 480

10 1 5 7 224 26 480

11 1 4 7 152 21 480

12 1 6 9 66 21 480

13 1 6 7 130 12 480

14 1 4 5 125 26 480

15 1 6 7 167 15 480

16 1 5 4 137 19 480

17 1 6 9 193 24 480

18 1 3 6 151 20 480

19 1 3 8 112 28 480

20 1 3 5 127 19 480

21 1 4 4 125 20 480

22 1 5 10 96 16 480

23 1 4 7 157 16 480

24 1 4 10 238 18 480

25 1 5 10 100 28 480

26 1 3 6 227 28 480

27 1 6 5 203 20 480

28 1 4 5 156 13 480

29 1 6 5 127 15 480

30 1 3 7 111 16 480

31 1 3 9 96 14 480

32 1 5 4 65 17 480

33 1 3 10 129 29 480

34 1 3 8 186 14 480

35 1 3 6 119 17 480

36 1 5 8 232 23 480

37 1 6 8 122 12 480

38 1 4 5 68 22 480

39 1 4 4 116 25 480

40 1 5 4 75 22 480

41 1 6 9 217 26 480

42 1 3 8 147 18 480

43 1 5 9 98 29 480

44 1 3 5 112 22 480

45 1 6 4 82 26 480

46 1 3 10 175 21 480

47 1 3 8 117 21 480

48 1 3 7 68 12 480

49 1 4 7 119 13 480

50 1 6 5 159 12 480

51 1 4 5 60 28 480

52 1 5 7 163 25 480

53 1 3 5 216 26 480

54 1 6 8 225 19 480

55 1 6 9 114 22 480

56 1 5 7 223 12 480

57 1 3 5 101 24 480

58 1 3 7 183 14 480

59 1 6 8 157 19 480

60 1 3 10 228 22 480

61 1 6 7 60 22 480

62 1 5 8 84 24 480

63 1 6 7 109 26 480

64 1 5 6 104 25 480

65 1 5 9 81 22 480

66 1 5 10 127 21 480

67 1 5 10 196 19 480

68 1 5 6 120 22 480

69 1 3 8 208 25 480

70 1 4 4 101 22 480

71 1 3 4 184 20 480

72 1 5 6 240 30 480

73 1 6 4 214 14 480

74 1 4 6 141 18 480

75 1 5 8 130 16 480

76 1 4 5 225 20 480

77 1 4 5 129 22 480

78 1 3 6 187 29 480

79 1 3 6 87 14 480

80 1 5 4 197 29 480

81 1 4 9 219 14 480

82 1 5 7 173 20 480

83 1 3 5 77 13 480

84 1 5 8 86 24 480

85 1 6 9 145 24 480

86 1 4 4 139 23 480

87 1 3 4 122 13 480

88 1 4 8 133 20 480

89 1 5 4 209 23 480

90 1 5 6 86 13 480

91 1 6 9 205 17 480

92 1 6 5 168 20 480

93 1 6 6 141 25 480

94 1 6 9 149 22 480

95 1 4 5 110 21 480

96 1 4 10 128 16 480

97 1 3 8 90 18 480

98 1 4 6 167 20 480

99 1 5 5 78 20 480

100 1 6 10 67 26 480

101 1 5 6 100 13 480

102 1 3 6 218 18 480

103 1 4 7 81 28 480

104 1 6 4 95 20 480

105 1 3 7 90 25 480

106 1 3 4 165 21 480

107 1 5 6 68 20 480

108 1 3 6 199 19 480

109 1 3 9 93 21 480

110 1 3 9 154 22 480

111 1 3 9 220 18 480

112 1 5 5 174 14 480

113 1 4 5 90 30 480

114 1 6 8 233 21 480

115 1 5 6 163 27 480

116 1 6 5 60 19 480

117 1 6 6 220 29 480

118 1 6 10 163 13 480

119 1 3 9 186 29 480

120 1 6 7 190 19 480

121 1 5 5 137 23 480

122 1 4 8 174 22 480

123 1 3 8 74 15 480

124 1 4 7 92 17 480

125 1 5 6 101 13 480

126 1 4 10 110 27 480

127 1 3 4 199 22 480

128 1 4 10 123 19 480

129 1 5 5 100 14 480

130 1 3 4 131 23 480

131 1 3 4 220 22 480

132 1 6 7 95 21 480

133 1 5 10 213 27 480

134 1 6 7 213 14 480

135 1 4 6 221 14 480

136 1 3 10 234 15 480

137 1 4 5 231 24 480

138 1 5 5 205 14 480

139 1 5 10 173 22 480

140 1 6 6 171 19 480

141 1 6 10 79 19 480

142 1 4 10 163 24 480

143 1 5 7 97 13 480

144 1 6 9 160 29 480

145 1 4 5 206 21 480

146 1 3 6 124 21 480

147 1 3 7 103 18 480

148 1 4 4 219 29 480

149 1 4 10 190 26 480

150 1 3 7 126 23 480

151 1 3 8 147 28 480

152 1 4 6 146 15 480

153 1 5 8 238 28 480

154 1 6 7 188 25 480

155 1 5 4 161 21 480

156 1 4 4 194 19 480

157 1 3 7 61 27 480

158 1 6 9 150 17 480

159 1 4 5 98 21 480

160 1 4 8 165 12 480

161 1 3 7 148 20 480

162 1 5 5 93 26 480

163 1 4 8 184 20 480

164 1 3 8 154 12 480

165 1 3 6 236 18 480

166 1 4 5 201 18 480

167 1 3 8 168 19 480

168 1 4 10 239 30 480

169 1 3 9 86 25 480

170 1 5 10 133 15 480

171 1 6 4 131 21 480

172 1 5 9 123 24 480

173 1 6 9 133 16 480

174 1 5 7 109 18 480

175 1 4 9 193 26 480

176 1 3 7 92 14 480

177 1 5 9 125 12 480

178 1 4 7 67 30 480

179 1 4 4 80 13 480

180 1 6 4 192 14 480

181 1 4 8 176 21 480

182 1 3 4 137 19 480

183 1 4 7 229 25 480

184 1 6 8 91 16 480

185 1 5 4 77 27 480

186 1 5 10 220 21 480

187 1 5 9 170 29 480

188 1 5 7 77 25 480

189 1 6 10 208 29 480

190 1 6 7 67 14 480

191 1 6 9 140 14 480

192 1 4 6 63 28 480

193 1 4 5 170 19 480

194 1 4 8 191 13 480

195 1 5 10 176 28 480

196 1 4 4 71 18 480

197 1 5 7 183 12 480

198 1 4 5 101 13 480

199 1 6 10 237 18 480

200 1 3 8 66 19 480

201 1 3 10 221 26 480

202 1 6 5 168 16 480

203 1 6 7 102 15 480

204 1 3 9 74 22 480

205 1 6 9 226 23 480

206 1 4 6 178 18 480

207 1 5 7 190 27 480

208 1 4 4 71 26 480

209 1 6 9 80 23 480

210 1 6 10 136 17 480

211 1 4 9 128 23 480

212 1 6 6 143 27 480

213 1 5 7 70 25 480

214 1 5 7 171 20 480

215 1 4 9 223 23 480

216 1 3 8 83 30 480

217 1 4 10 189 27 480

218 1 4 6 160 13 480

219 1 4 4 146 23 480

220 1 4 6 189 29 480

221 1 4 9 163 29 480

222 1 5 4 219 24 480

223 1 5 7 142 19 480

224 1 4 10 170 14 480

225 1 5 6 177 29 480

Documents

For info : [email protected] Goalxie/DefiSante/PolyDefiSante2019.pdf · 2019-09-10 · 9% for 2016) 5 Context and trends 6 Context and trends ... • By DRG (diagnostic relatedgroups)