Planning of Treatment at Rehabilitation Clinics Using a Two Stage Mixed-Integer Programming Approach

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Planning of Treatment at Rehabilitation Clinics

Using a Two Stage Mixed-Integer Programming

Approach

Department of Mathematics, Linköping University Tobias König

LiTh-MAT-EX–2021/01–SE

Credits: 30 hp Level: A

Supervisor: Björn Morén,

Department of Mathematics, Linköping University Examiner: Torbjörn Larsson,

Department of Mathematics, Linköping University Linköping: February 2021

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Abstract

This thesis presents a method for planning patient intake and assignment of treat-ment personnel at rehabilitative care clinics. The rehabilitation process requires patients to undergo a series of treatments spanning several weeks, requiring ther-apists of di↵erent disciplines. We have developed a two stage mixed-integer pro-gramming model which plans when each admitted patient will receive treatment and assigns therapists. In addition, the model provides support to decide when to admit new patients and when to hire additional sta↵ in order to maximise the clinic’s patient throughput. Numerical results based on a real rehabilitation clinic are presented and discussed.

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Acknowledgments

My supervisor, Björn Morén, has provided invaluable help and encouragement. Had it not been for his continuous support I would not have been able to finish this thesis. For that I am very grateful.

I would also like to thank Torbjörn Larsson, my examiner, for giving me the idea and the opportunity to write this thesis.

Finally, I would like to thank my friend and opponent Viktor Wingquist for his helpful comments.

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1

Introduction

1.1 Objective of This Thesis

This thesis aims to develop a framework to assist the administrators at rehabili-tation clinics, where rehabilirehabili-tation is organised for treatment programmes span-ning several weeks and consisting of both group activities and individual treat-ment, with the following tasks:

• Help creating schedules that give each member of the sta↵ an even work-load each week, not exceeding the predetermined time allocated to each specific member for treatment of patients.

• Determine how many new patients to admit each week to the clinic. • Maximise the clinic patient throughput.

This removes some of the scheduling burden by having an optimisation algorithm assigning which members of the sta↵ should meet a certain patient in a specific week for which activity. It also estimates the number of patients to admit for assessment each week based on historical patient type distribution data. What is left to do for the administrators is to schedule the meeting between patient and sta↵ member to a specific day and time during the planned week.

1.2 Methodology

Problems similar to the one considered in this thesis have been studied (see Chap-ter 3), although the combination of group loyalty – requiring patients to meet the same therapists at multiple occations – and planning in advance when to take in new patients for assessments, which increases complexity, is more uncommon.

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2 1 Introduction

We have studied the processes of rehabilitation clinics and a mixed-integer programming (MIP) model describing these processes with appropriate simplifi-cations has been developed. This model proved to be too complex to solve with commercial solvers in a reasonable time and a two stage model was developed to reduce the number of variables. This two stage model has then been solved with the commercial solver Gurobi with programme and sta↵ data from an ac-tual clinic.

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1.3 Structure of Thesis 3

1.3 Structure of Thesis

This thesis is structured into the following chapters:

Problem Formulation Describes the problem addressed and provides a general background.

Literature Overview Discusses relevant literature. It is divided into two parts: healthcare specific planning and ing literature, and general planning and schedul-ing literature.

Assumptions and Simplifications Presents and motivates simplifications and as-sumptions made in the models.

MIP Model Presents the mixed-integer programming (MIP) model of the problem.

Results Presents results from solving the MIP model. Discussion and Future Research Here observations about the results and

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2

Problem Formulation

This chapter describes the problem addressed and provides a general background.

2.1 Rehabilitation Clinics

When a patient is considered to be in need of rehabilitation of some kind, he or she can be referred to a rehabilitation clinic (treating the condition he or she might su↵er from) for assessment. Rehabilitation clinics generally treat more than one condition. For example, one rehabilitation clinic might treat both pa-tients su↵ering from chronic pain, and fatigue syndrome.

A clinic can choose how many patients to admit for assessment. Each admit-ted patient will either be diagnosed as needing one of the treatments the clinic provides or it is found that the patient is not in need of any care the clinic pro-vides. If a patient is diagnosed as needing rehabilitation available at the clinic, the clinic can not turn the patient away. It is therefore important to not admit too many patients.

Patients are placed in a group with other patients who need the same or sim-ilar rehabilitation and go through a treatment programme in order to recover. Figure 2.1 illustrates the flow of patients through a clinic.

Figure 2.1: High level rehabilitation process for one patient

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6 2 Problem Formulation

2.2 Treatment Programmes

Typically a treatment programme will span many months and consists of both individual treatment and group treatment. The group treatments requiregroup loyalty; a patient is part of the same group for every group treatment.

The activities in a programme must be performed in a certain order (prece-dence), and activities may have to be separated by a certain number of weeks (time separation). For example, a programme may require all patients to first receive individual treatment from a psychologist before a group therapy activity can start. See Figure 2.2, where elongated boxes represent repeating visits.

In addition togroup loyalty many activities require sta↵ loyalty, meaning that

for each patient, certain activities have to be performed by the same member of the treatment sta↵. For example, the psychologist doing the individual treatment and group treatment should be the same person.

Figure 2.2: Illustration of an admitted patient’s treatment programme path

In addition to precedence and time separation within the programme, no patient should have to wait more than a certain amount of time after assessment before beginning treatment.

2.3 Planning Challenges

Treatment scheduling is complicated and therefore time consuming. There are many possible ways to compose a schedule and it might keep one administrator working almost full time creating viable schedules for a clinic.

Another challenge is to decide how many patients should be accepted for as-sessment each week since a clinic’s capacity to admit another patient is generally not only limited by the available time the current week, but also on the clinics fu-ture schedule. For example, the limiting factor on how many new patients can be assessed a given week might not be limited by the work load on disciplines doing assessments, but rather the work load on some other discipline many weeks into the future.

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2.3 Planning Challenges 7

All sta↵ are not working full time with patient treatment. Some also perform other tasks such as administration or supplementary training. If present week scheduling does not take future scheduling into account this can lead to uneven work load. Many of the treatment appointments require sta↵ loyalty for the pa-tients. This, in combination with the structure of the treatment programmes, activities assigned for a member of the sta↵ one week can greatly a↵ect the activ-ities he or she has to perform other weeks. For example, in Figure 2.2 if there is a sta↵ loyalty requirement between the first group activity and the (individual) activity preceding it, and one sta↵ member is assigned the first group activity for two groups the same week; he or she will then also have to perform up to 2n indi-vidual activities two weeks prior to the group activity given that there are at most

n patients in each group. With the current method of scheduling it happens that

for some weeks and members of the sta↵ the time required for treating patients overshoots the pre-allocated treatment time.

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3

Literature Overview

Planning and scheduling is as old as the field of mathematical programming it-self. In healthcare, the nurse rostering problem – to assign nurses to shifts – has been widely studied.

In more recent years, optimisation techniques have been applied to planning and scheduling in a variety of healthcare specific areas such as operating room scheduling, cancer clinic scheduling, emergency room planning and rehabilita-tion clinic scheduling.

The relevant literature can be divided into two parts: healthcare specific plan-ning and scheduling, and general planplan-ning and scheduling.

3.1 Healthcare Specific Planning and Scheduling

The literature relevant for this thesis has been gathered by searches on keywords and phrases such as outpatient care pathway scheduling, multi appointment health-care/rehabilitation scheduling, care pathway rehabilitation scheduling and some

vari-ations thereof.

In addition, three review articles have been used: Multi-disciplinary planning in health care: a review by Leeftlink et al 2018 [4], Literature review on multi-appointment scheduling problems in hospitals by Marynissen et al 2018 [5], and Outpatient appointment systems in healthcare: A review of optimization studies by

Ahmadi-Javid et al 2016 [1].

Ahmadi-Javid et al have studied articles on outpatient care – a patient whose treatment does not require an overnight stay in a hospital or clinic – scheduling between 2003 and 2016, and note that there was a drastic increase of publications in the years 2012 to 2016, with 73% of the papers reviewed being published in this period. The review article by Marynissen et al found no paper on the subject of multi-appointment scheduling in hospitals published before 1995, and

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10 3 Literature Overview

of the 56 papers included in the review they found an increasing trend of papers published between 1995 and 2017.

In the literature reviews both Marynissen et al and Leeftlink et al found that heuristic methods were more common than exact alorithms. Of the total of 45 pa-pers whose solution method was classified as either exact or heuristic by Marynis-sen et al, 12 used exact algorithms. Leeftlink et al have a finer classification, and out of a total of 33 articles on the sub-topico↵ line resource to patient assignment,

13 used exact algorithms.

Two articles and a master’s thesis have provided particularly relevant informa-tion: Decision support for rehabilitation hospital scheduling by Schimmelpfeng et al, Integral multidisciplinary rehabilitation treatment planning by [8] Braaksma et al [2], and a master thesis by Ronneberg and Odegaard [7].

Schimmelpfeng et al [8] have made a model for rehabilitation scheduling with a combination of in- and outpatients with the objective of maximising the amount of treatments over a planning horizon, where the treatment for specific patients is a pre-determined or individually prescribed care pathway consisting of repeti-tions of rehabilitative procedures with only intra-day precedence. They employ two ways of solving the problem. One "monolithic" model attempting to solve the entire scheduling in one run, and one "hierarchical" decomposition to reduce computational time. Treatments are prescribed to patients and placed on a spe-cific day during the planning horizon, then activities are placed in time slots during that day and filled with patients.

There are three major di↵erences between the problem formulations of Schim-melpfeng et al and that of this thesis: [8] has no patient intake estimation, only intra-day precedence, and group loyalty is exogenously enforced. In this thesis the original plan was to use a hierarchical approach to plan weekly and solve the intra-week problem was inspired by [8], but our model only solves the weekly planning problem.

Braaksma et al [2] have studied a similar problem to [8] where the model only considers scheduling of outpatients. This paper deals with scheduling patients after they have been given a treatment plan, and schedules every patient indepen-dently based future free resources. They seek to make an optimal schedule based on a variety of factors primarily based on patient satisfaction, such as short wait-ing times, weekly recurrwait-ing treatment times, and number of omitted treatments. A major di↵erence between the scheduling methodology in [2] and that of this thesis is that the former schedule patients iteratively one by one as they arrive to the clinic.

The master’s thesis by Ronneberg and Odegaard [7] studies scheduling and plan-ning of psychiatric diagnostic pathways where treatment pathway and appoint-ment duration are uncertain. Ronneberg and Odegaard seek to schedule all in-coming patients while minimizing time limits and take uncertainties, such as patients not showing up, and extended diagnostic pathways into account. In

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ad-3.2 Comparison 11

dition they allow rescheduling of already scheduled patients.

A di↵erence between the solution methods in all three of these papers and that of this thesis is that all of them incorporate some forms of relaxation of constraints, such as number of treatments in [2] or time dependencies in [7].

3.2 Comparison

Below similarities and di↵erences between the reviewed literature and this thesis are summarised.

3.2.1 Scheduling Methodology and Re-Planning

In this thesis future plans are based on the expected outcomes of assessments with the objective of placing assessments and treatments so that the throughput of a clinic is maximised. If a patient is admitted, he or she is placed in one of the planned treatment programmes and this programme is locked in the sense that it will not be re-planned. If outcomes from assessments di↵er from the expected number of patients, all planned activities that have not yet been assigned a pa-tient is re-planned in order to make an optimal plan based on the new state of admitted patients.

Estimating how many patients to admit for assessment is not done by any of the reviewed literature, instead they focus on scheduling patients when they arrive to the clinic. How re-planning is handled varies between the reviewed literature; all employ a model where the schedule is updated daily or weekly. Braaksma et al schedule patients one by one as they arrive to the clinic and once a patient has been planned he or she is not re-planned. Ronneberg and Odegaard also schedule patients as they arrive to the clinic, but allow for rescheduling for some activities if it improves the schedule.

3.2.2 Staff and Group Loyalty

Ronneberg and Odegaard [7] handle sta↵ loyalty by penalising the use of more than one therapist per patient. Braaksma et al require complete sta↵ loyalty; ev-ery activity requiring a specific discipline needs to be performed by the same therapist. In Schimmelpfeng et al, sta↵ loyalty is disregarded altogether unless a therapist is manually assigned to a set of activities. In this thesis, sta↵ loyalty between activities are enforced by constraints, similarly to Braaksmas et al ap-proach. However, all activities requiring therapists of a certain discipline need not necessarily be performed by the same therapist.

Schimmelpfeng et al are the only ones to consider group activities. Group loy-alty is enforced similarly to how sta↵ loyloy-alty is: manually enforced. In this thesis there is complete group loyalty, and the groups are automatically generated.

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12 3 Literature Overview

3.3 General Planning and Scheduling Literature

Literature not specifically concerning healthcare, but relevant to this thesis, con-sists of a book: Complex Scheduling by Bruker and Knust [3] and an article: The Project Portfolio Selection and Scheduling Problem: Mathematical Model and Algo-rithms by Naderi et al [6].

Complex Scheduling by Bruker and Knust [3] describes several aspects of

ing and bases its disposition in "The resource constrained planning and schedul-ing problem" (a very general problem). They discuss both modellschedul-ing and solvschedul-ing aspects for a variety of scheduling scenarios.

Naderi [6] discusses "the project portfolio planning and selection problem". To select, within some scheduling horizon under resource constraints, the most prof-itable projects out of a portfolio of projects each consisting of a set of operations. This is similar to the objective of this thesis by viewing a patient, or group of pa-tients, as projects made up of the activities in a rehabilitation programme. Naderi evaluates di↵erent heuristics to solve this problem.

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4

Assumptions and Simplifications

A MIP model of a clinic’s activity is developed in Chapter 5. The model plans activities weekly. This chapter introduces simplifications and assumptions to the rehabilitation process and explains how the model will be used.

4.1 Planning Process

The automatic planning is performed weekly with the previously established activities, outcome of the assessments, available treatment personnel and their available time assigned to treatment as input to plan next week’s activities, see Figure 4.1.

Figure 4.1: Planning process

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14 4 Assumptions and Simplifications

The output from the mixed-integer model (MIP) can be divided into two distinct sets: a set of active activities and a set of planned activities. The active activities are activities in a programme that have been assigned a patient, while a planned activity is part of a programme that has not yet been assigned a patient.

The active activities can not be replanned. The planned activities can be re-planned as they do not yet have a patient assigned. Replanning will occur in cases where the outcome of the assessment programmes di↵er from the expected outcome.

4.2 Simplifications

A number of simplifications have been introduced to make the modelling easier and reduce the total number of activities to schedule.

4.2.1 Assessment Programme

The assessment programme typically consists of a sequence of meetings between the patient being assessed and therapists of di↵erent disciplines, and discussions between the involved therapists. It is reasonable to assume that all of the activ-ities of the assessment programme are performed within the same week. There-fore the assessment programme can be modelled as a single activity when doing weekly planning. For example for a rehabilitation clinic having provided data, the assessment programme consists of the activities listed in Table 4.1

Table 4.1: Example of assessment programme

Number Activity Present Time

1 Individual meeting Patient, Physician 45 minutes 2 Individual meeting Patient, Psychologist 45 minutes 3 Individual meeting Patient, Physiotherapist 45 minutes 4 Assessment_Conference Physician, Psychologist,_{Physiotherapist} 30 minutes

where activities 1, 2 and 3 have to be performed before activity 4. Under the assumption that all the activities in the assessment programme are performed within the same week, or even the same day, the entire programme is modelled as a single multidisciplinary activity consuming one hour and 15 minutes for all involved therapists, see Figure 4.2.

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4.3 Assumptions 15

Figure 4.2: The assessment programme is modelled as a single group activity The grouping of assessments decreases the number of variables in the model, speeding up solution times and simplifying modelling.

4.2.2 Treatment Programmes

Similarly to the assessment programme, some of the activities in the treatment programmes are grouped together in order to reduce the number of variables in the model and simplify modelling. Activities that can be assumed to happen the same week are grouped.

Some activities are completely excluded from the treatment programme, specif-ically activities that should happen many weeks after the last activity of the bulk of the programme. Including these activities would force the scheduling horizon to be too long and they can be manually placed in a week where treatment sta↵ workload is low.

4.3 Assumptions

4.3.1 Patient Intake Estimation

It is assumed that patients arrive to a clinic at a rate which the clinic decides. Which programme a patient should be part of is decided by the assessment pro-gramme and not known a-priori. While the outcome of an assessment propro-gramme is unknown, the average distribution of outcomes can be estimated from histori-cal data. Here it is assumed that the probability that a patient after going through assessment is placed into programme of type »p» is constant throughout the plan-ning horizon. After assessment, each patient is placed in a programme specific queue awaiting treatment.

4.3.2 Treatment Rooms

It is assumed that there are enough treatment rooms for all planned activities. In the model the constraints concerning treatment sta↵ and treatment rooms would

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16 4 Assumptions and Simplifications

be very similar resource constraints and here it is assumed that the treatment room constraints would be dominated by treatment sta↵ constraints and they are therefore left out.

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5

Mixed-Integer Programming Model

Here the mixed-integer programming model describing the problem is given. First a complete model will be presented. However, with large sta↵ and long planning horizon this problem is too large to be solved on a standard computer. In order to solve it, a two stage solution approach has been developed. Stage one places activities in time, treating a clinic as having only one sta↵ member of each discipline with treatment time for each equal to the aggregated time per disci-pline. Stage two assigns specific sta↵ members to the activities from stage one. In other words: Stage one decides when an activity happens, and stage two who performs it.

5.1 Notation Common to Both Models

5.1.1 Indices, Sets and Parameters

Every patient is identified with the three indices (p, i, n) where »p» is the type of programme, »i» the group the patient is in and »n» is a number the patient is assigned within its group.

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18 5 Mixed-Integer Programming Model Indices Index Description t Time (weeks) j Activity k Sta↵ member p Programme type i Group number n Patient number d Discipline Sets Set Description

O Set of all sta↵ in the clinic Od Set of all sta↵ of discipline d

O_jp Set of all sta↵ that can perform activity »j» in pro-gramme »p»

¯

O_jp Set of sta↵ of only one arbitrary discipline needed for activity »j» in programme »p»

OH External sta↵ D Set of all disciplines

Djp Set of disciplines needed to perform activity »j» in

pro-gramme »p»

AI

p Individual activities in programme »p»

AG_p Group activities in programme »p»

P_jpI Individual activities preceding activity »j» in pro-gramme »p»

P_jpG Group activities preceding activity »j» in programme »p»

D_pI Pairs of activities that must lie separated within some interval in time

M_px Set of pairs of activities who have to be performed by the same person. x denotes if the activity pairs are: I Both single patient activites, G Both group activites,

M One activity of each.

AI_pm Set of activities performed by »m» persons from the

sta↵

P_tB Set of all assessment programmes available at time »t»

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5.1 Notation Common to Both Models 19 Parameters

Parameter Description

µ_p Probability that an assessed patient is put in pro-gramme »p»

n_pi Number of possible groups of type »p»

n_pn Maximum number of patients in each group

w_k Cost of using an external therapist of dicipline k

v_jp Number of consecutive weeks activity »j» of pro-gramme »p» is performed

T_kt Time allocated for treatment week »t» for sta↵ mem-ber »k»

T_jkpI Duration of activity »j» in the individual part of pro-gramme »p» if performed by sta↵ member »k»

T_jkpG Duration of activity »j» in the group part of

pro-gramme »p» if performed by sta↵ member »k»

TB Duration of the assessment programme

ck Cost of using sta↵ member k 2 OH in an assessment

programme.

D_jminˆjp Minimal time separation between »j» and » ˆj» in

pro-gramme »p»

D_jmaxˆjp Maximal time separation between »j» and » ˆj» in

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20 5 Mixed-Integer Programming Model

5.2 Complete Model

Here the complete model is described.

5.2.1 Variables

The variables x and ¯x are decision variables for treatment activities, B is the de-cision variable for assessments, and y and g are auxiliary variables defined by x and ¯x.

Variables

B_tk _{2 N} Number of assessments performed by sta↵ member »k» in week t x_jktpin = 8 > > > < > > > :

1 if activity »j» for patient »i,n» in the individual part of programme »p» occurs at time »t» with sta↵ »k» 0 otherwise ¯x_jktpi = 8 > > > < > > > :

1 if activity »j» for group »i» in the group part of pro-gramme »p» occurs at time »t» with sta↵ »k»

0 otherwise y_tpin = 8 > > > < > > > :

1 if the individual part of programme »p» starts for patient »i,n» at time »t»

0 otherwise G_tpi = 8 > > > < > > > :

1 if the group therapy part of programme »p» starts for group »i» at time »t»

0 otherwise

Îtp Number of patients in queue for rehabilitation in

programme »p» at time »t»

Itp Integer variable for the number of patients in queue

for rehabilitation in programme »p» at time »t»

In this chapter, when referencing an index, 8 means for all existing indices. For example x_jktpin _ Pt

ˆt=1yˆtpin8j, t, p, i, n should be interpreted as 8t 2 [1, T ], for all programs p at the clinic, 8j 2 AI

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5.2 Complete Model 21

Auxiliary Variables

The entire model could be written with only the variables x, ¯x and B, but to make it more readable the variables y, G and I are defined.

Both g and y is the first activity of the individual and group therapy part of the programmes, respectively.

ytpin = X k2 ¯O1,p x1ktpin 8t, p, i, n (5.1) g_tpi = X k2 ¯O1,p ¯x_1ktpi _{8t, p, i} (5.2)

5.2.2 Constraints and Objective Function

Objective Function

The objective is to maximise the throughput of patients at the clinic during the planning period minus the cost, measured relative to the value of rehabilitating one patient, of using external sta↵ for performing assessments.

max N_patients= X p2PR T X t=1 npi X i=1 npn X n=1 y_tpin T X t=1 X k2OH cB_kB_tk (5.3) Patient Queue

The queue of patients waiting for rehabilitation in programme »p» week »t», in-cluding patients that have gone through assessment at week »t» Itp is given by

Îtp = Ît 1,p+ µpNB X k2OL Btk npi X i=1 npn X n=1 yt 1,pin 8t, p (5.4) Itp  Îtp (5.5) I_tp Î_tp "_p (5.6) The value of I0,p is given by the outcomes of the assessments, and is the amount of real patients in queue when the scheduling is made.

The expected queue of patients to each programme is the number of patients assessed as needing rehabilitation in programme »p» minus the amount of pa-tients receiving their first treatment that week plus the remaining queue from the week before.

Constraint (5.4) is the expected queue to each programme. (5.5) - (5.6) define the queue in integer values by rounding the value of ˆI_tpdown. " is the largest dec-imal value ˆItp can attain. Constraints (5.5) - (5.6) assume that after the dµp1e:th

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value of "_p the number of assessments that needs to be performed before the queue reaches one can be changed.

I_{t w}_B_,p _ t X ˆt=t wB np,i X i=1 np,n X n=1 y_ˆtpin _{8t, p} (5.7)

Constraint (5.7) forces programmes to start at a high enough rate in order that no patient has to be in the queue for more than three weeks before their first appointment. This model does not assign patients an index, but rather the lowest index should be paired with the patient who has been waiting the longest. This constraint could be removed by adding weights in the objective to punish patients waiting more than w_Bweeks.

Precedence Constraints

These constraints describe the programme: X k2 ¯Ojp xjktpin  1 |PI jp| + |PjpI M| t X ˆt=1 ✓ X ˆj2PI jp X ˆk2 ¯O_ˆjp x_{ˆjˆk ˆtpin}+ X ˆj2PI M jp X ˆk2 ¯O_ˆjp ¯x_{ˆjˆk ˆtpi}◆ 8tpin (5.8) X k2 ¯Ojp ¯x_jktpi 1 |PG jp| + |PjpGM| T X ˆt=t ✓ X ˆj2PG jp X ˆk2 ¯Oˆjp x_{ˆjˆk ˆtpi}+ X ˆj2PGM jp X ˆk2 ¯Oˆjp ¯x_{ˆjˆk ˆtpi}◆ _{8tpin (5.9)}

constraints (5.8) and (5.9) assure that all activities in the precedence set for activ-ity »j» is performed before, or at the same time, as »j» itself. The two constraints are similar with (5.8) concerns precedence for individual activities while (5.9) concerns group activities. (5.9) is on a di↵erent form than the more intuitive for-mulation of (5.8). Constraints can not be formulated with group variables being less than individual variables if groups should be allowed to not be completely filled. X k2 ¯Ojp xjktpin t+D_{j ˆjp}max X ˆt=t+Dmin j ˆjp X k2 ¯O_ˆjp x_{ˆjk ˆtpin} 8(j, ˆj) 2 DI p, 8t, p, i, n (5.10) ¯xjktpi  t+D_{j ˆjp}max X ˆt=t+Dmin j ˆjp X k2 ¯O_ˆjp ¯x_{ˆjk ˆtpi} 8(j, ˆj) 2 DG p, 8t, p, i (5.11) X k2 ¯Ojp x_jktpin _ t+Dmax_{j ˆjp} X ˆt=t+Dmin j ˆjp X k2 ¯O_ˆjp ¯x_{ˆjk ˆtpi} _{8(j, ˆj) 2 D}I M p , 8t, p, i, n (5.12)

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5.2 Complete Model 23

Constraints (5.10) - (5.12) assure activities that should be separated in time in-deed are separated.

Staff Constraints

These constraints consider the sta↵ working hours, and allocation.

X p2PR npi X i=1 ✓ X j2AI p npn X n=1 T_jkpI t X ˆt=t vjp+1 x_{jk ˆtpin}+ X j2AG p T_jkpG t X ˆt=t vjp+1 ¯x_{jk ˆtpi}◆+ TBB_tk _{ T}_k,t _{8t, k} (5.13)

Constraint (5.13) makes sure the sta↵ works less or equal hours with treating and assessing patients than specified.

T X t=1 x_jktpin= T X t=1 x_ˆjktpin _{8(j, ˆj) 2 M}_pI, _{8k, p, i, n} (5.14) T X t=1 x_jktpin_ T X t=1 ¯x_ˆjktpi _{8(j, ˆj) 2 M}M p , 8k, p, i, n (5.15) T X t=1 ¯x_jktpi = T X t=1 ¯x_ˆjktpi 8(j, ˆj) 2 MG p, 8k, p, i (5.16)

Some actvivities must be performed by the exact same member of the sta↵. (5.14) - (5.16) assures this by linking these activities together by forcing the decision variables to have the same index k.

Some General Constraints

X ˆtt

y_ˆtpin X

ˆtt

y_ˆtpi,n+1 _{8p, i , n = 1, . . . , n}_p,n 1, t (5.17)

This constraint orders the variables so that the groups are filled in ascending or-der.

The following constraints ensure that each activity in the programmes are per-formed at most one time and that all the activities are perper-formed.

T X t=1 X k2Od x_jktpin = T X t=1 y_tpin _{8j 2 A}I_p, _{8d 2 D, 8p, i, n} (5.18) T X t=1 X k2Od ¯x_jktpi = T X t=1 g_tpi _{8j 2 A}G_p, _{8d 2 D, 8p, i} (5.19)

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24 5 Mixed-Integer Programming Model T X t=1 y_tpin _ T X t=1 g_tpi _{8p, i, n} (5.20) T X t=1 g_tpi _{ 1 8p, i} (5.21)

Constraints (5.18) - (5.21) force all activities in programme »p» to be performed for all patients »n» in group »i» if one activity is performed.

5.3 Two Stage Model

In the following two sections the two stage model is presented. Stage one places activities in time during the planning horizon and stage two assigns sta↵ mem-bers to the activities.

In order to di↵erentiate between the decision variables in the complete model, the variables in stage one and two have (1) or (2) superscipted for stage one and stage two respectively.

The constraints in both stage one and stage two are very similar to the con-straints in the complete model. It is described after the list of concon-straints in each stage how the constraints in the respective stages corresponds to the constraints in the complete model.

5.4 Stage One

The stage one model is essentially the complete model but instead of considering individual sta↵ members, disciplines are considered. Constraints (5.14) - (5.16) are removed, the constrains for sta↵ working hours are reformulated so that no discipline works more than the combined time for all sta↵ members of one disci-pline. The other di↵erence between the complete model and the stage one model is that in stage one the objective function contains penalties which aim spread out activities for patients in the same group to di↵erent weeks are added to the objective function.

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5.4 Stage One 25

5.4.1 Variables

Variables

Bt 2 N Number of assessments week t

x(1)_jtpin = 8 > > > < > > > :

1 if activity »j» for patient »i,n» in the individual part of programme »p» occurs at time »t»

0 otherwise ¯x(1)_jtpi = 8 > > > < > > > :

1 if activity »j» for group »i» in the group part of pro-gramme »p» occurs at time »t»

0 otherwise y_tpin(1) = 8 > > > < > > > :

1 if the individual part of programme »p» starts for patient »i,n» at time »t»

0 otherwise g_tpi(1) = 8 > > > < > > > :

1 if the group therapy part of programme »p» starts for group »i» at time »t»

0 otherwise

Itp Integer variable for the number of patients in queue

vt,p,i Continuous variable to penalise high work-load

The penalty variable v_t,p,i is a weight added to the objective function to disfavour accumulation of activities within each group requiring the same discipline.

5.4.2 Indices, Sets and Parameters

The indices, sets and parameters described in Section 5.1 are the same in both stages one and two of the two stage model. However, there are two additional parameters in the stage one model listed below

Parameters

Parameter Description

Tv _{Number of hours each discipline can treat patients}

within one group before further time causes the weights to become active

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5.4.3 Auxiliary Variables

y_tpin(1) = x(1)_1tpin _{8t, p, i, n} (5.22)

g_tpin(1) = ¯x(1)_1tpi _{8t, p, i} (5.23) These variables are equivalent to those in the complete model.

5.4.4 Objective Function

max Npatients = X p2PR T X t=1 npi X i=1 npn X n=1 ytpin w T X t=1 X p2PR npi X i=1 vtpi (5.24) The term wPT t=1 P p2PRP npi

i=1vtpi in the objective function is not present in the

objective function in the complete stage or stage two.

5.4.5 Constraints

Îtp = Ît 1,p+ µpBt npi X i=1 npn X n=1 y(1)_{t 1,pin} 8t, p (5.25) I_tp  Îtp (5.26) I_tp Î_tp "_p (5.27) It wB,p  t X ˆt=t 3 npi X i=1 npn X n=1 y(1)_ˆtpin 8t, p (5.28) x(1)_jtpin _ 1 |PI jp| + |PjpI M| t X ˆt=1 ✓ X ˆj2PI jp x(1)_ˆjˆtpin+ X ˆj2PI M jp ¯x(1)_ˆjˆtpi◆ _8tpin (5.29) ¯x(1)_jtpi 1 |PG jp| + |PjpGM| T X ˆt=t ✓ X ˆj2PG jp x(1)_ˆjˆtpi+ X ˆj2PGM jp ¯x(1)_ˆjˆtpi◆ _8tpin (5.30) x(1)_jtpin  t+D_{j ˆjp}max X ˆt=t+Dmin j ˆjp x(1)_ˆjˆtpin 8(j ˆj) 2 DI p, 8t, p, i, n (5.31) ¯x(1)_jtpi _ t+D_{j ˆjp}max X ˆt=t+Dmin j ˆjp ¯x(1)_ˆjˆtpi _{8(j ˆj) 2 D}G p, 8t, p, i (5.32)

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5.4 Stage One 27 x(1)_jtpin  t+D_{j ˆjp}max X ˆt=t+Dmin j ˆjp ¯x(1)_ˆjˆtpi 8(j ˆj) 2 DI M p , 8t, p, i, n (5.33) X p2P npi X i=1 ✓ X j2AI p npn X n=1 T_jdpI X ˆt=t vjp+1 x(1)_{jk ˆtpin}+ X j2AG p T_jdpG t X ˆt=t vjp+1 ¯x(1)_{jk ˆtpi}◆+ X i2PB t T_dBB_t _ X k2Od T_kt _{8t, d} (5.34) X j2AI p npn X n=1 x(1)_jtpin+ X j2AG p ¯x(1)_jtpi _{ T}v_{+ v} tpi (5.35) X t y_tpin(1) X t y_tpi,n+1(1) _{8p, i , n = 1, . . . , n}_pn 1 (5.36) T X t=1 xjtpin= T X t=1 ytpin 8j 2 AIp, 8d 2 D, 8p, i, n (5.37) T X t=1 ¯x_jtpi = T X t=1 gtpi 8j 2 AGp, 8d 2 D, 8p, i (5.38) T X t=1 y_tpin _ T X t=1 g_tpi _{8p, i, n} (5.39) T X t=1 g_tpi _{ 1 8p, i} (5.40)

Constraints (5.25) - (5.33) are equivalent to (5.4) - (5.12) in the complete model. (5.34) is similar to (5.13), but instead of one constraint per sta↵ member, there is one per discipline. In this stage there is no equivalent to (5.14) - (5.16) as this stage operates on an aggregated level without individual sta↵ members. (5.35) introduces weights for each group and week to favour a solution where activities are spread and do not accumulate to certain weeks. Constrains (5.36) - (5.40) are equivalent to (5.17) - (5.21) in the complete model.

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5.5 Stage Two

In this stage, variables are created from the solution of stage one. Variables x_jktpin(2) and ¯x(2)_jktpionly exist if x_jtpin(1) = 1 and ¯x(1)_jtpi = 1 from stage one. The variables in this stage still have a time index t. This index is not needed, as stage one has decided when all activities should happen, but is kept for readability. The second stage assigns sta↵ members to activities placed in time in stage one by maximising the number of patients whose activities planned in stage one can be assigned sta↵.

Sets Unique to Stage Two

Sets

Set Description

Gp Groups in programme »p»

Ppi Patients in group »i» of programme »p»

J_tpinI All individual activities in programme »p» at time »t»

J_tpiG All group activities in programme »p» at time »t»

The above mentioned sets are created from the stage one model:

Gp = n i2 {1, . . . , npi}| T X t=1 g_tpi(1) = 1o (5.41) Ppi = n {n 2 {1, . . . , npn}| T X t=1 y_tpin(1) = 1o (5.42) J_tpinI =nj2 AI p|x(1)jtpin = 1 o (5.43) J_tpiG =nj2 AG p| ¯x(1)jtpi = 1 o (5.44) (5.45)

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5.5 Stage Two 29

Variables

The superscript (2) indicates that the variables are part of the second stage model. Variables

B_t,k _{2 N} Number of assessments week t performed by sta↵ member »k» x(2)_jktpin = 8 > > > < > > > :

1 if activity »j» for patient »i,n» in the individual part of programme »p» is assigned sta↵ member »k» 0 otherwise ¯x(2)_jktpi = 8 > > > < > > > :

1 if activity »j» for group »i» in the group part of pro-gramme »p» is assigned sta↵ member »k»

0 otherwise y_tpin = 8 > > > < > > > :

1 if the individual part of programme »p» starts for patient »i,n» 0 otherwise g_tpi(2) = 8 > > > < > > > :

1 if the group therapy part of programme »p» starts for group »i»

0 otherwise

Itp Integer variable for the number of patients in queue

Auxiliary Variables ytpin = X k2 ¯O1,p x_1ktpin(2) 8p, i, n (5.46) g_tpi(2) = X k2 ¯O1,p ¯x(2)_1ktpi _{8p, i} (5.47)

5.5.1 Objective Function

The objective function in this model is similar to the objective function in the complete model. In this stage the problem can be considered maximising the number of patients planned from stage one that is assigned treatment sta↵, minus the cost of using external sta↵ to perform assessments.

max N_patients = X p2PR X i2Gp X n2Ppi X k2 ¯O1,p y_tpin T X t=1 X k2OH cB_kB_tk (5.48)

5.5.2 Constraints

ˆI_tp = ˆI_{t 1,p}+ µ_pN_B X k2OL X i2PB t B_tik npi X i=1 npn X n=1 y_{t 1,pin} 8t, p (5.49)

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30 5 Mixed-Integer Programming Model Itp  ˆItp (5.50) Itp ˆItp ✏p (5.51) I_{t 3,p} _ t X ˆt=t 3 np,i X i=1 np,n X n=1 y_ˆtpin _{8t, p} (5.52) X p2P npi X i=1 ✓ X j2AI p npn X n=1 T_jkpI X ˆt=t vjp+1 x(2)_{jk ˆtpin}+ X j2AG p T_jkpG t X ˆt=t vjp+1 ¯x(2)_{jk ˆtpi}◆+ TBB_tk _{ T}_k,t _{8t, k} (5.53) T X t=1 x(2)_jktpin= T X t=1 x(2)_ˆjktpin _{8(j, ˆj) 2 M}_pI, _{8k, p, i, n} (5.54) T X t=1 x(2)_jktpin  T X t=1 ¯x(2)_ˆjktpi _{8(j, ˆj) 2 M}M p , 8k, p, i, n (5.55) T X t=1 ¯x(2)_jktpi = T X t=1 ¯x(2)_ˆjktpi _{8(j, ˆj) 2 M}G p, 8k, p, i (5.56) X ˆtt y_tpin X ˆtt y_tpi,n+1 8p, i , n = 1, . . . , np,n 1 (5.57) X t X k x(2)_jktpin=X t y_tpin 8j 2 AI p8j, p, i, n (5.58) X t X k2OP x(2)_jktpin=X t y_tpin _{8j 2 A}I_p,3_{[ A}I_p,3_{8j, p, i, n} (5.59) T X t=1 y_t,p,i,n _ T X t=1 g_t,p,i(2) (5.60) T X t=1 X k2Ojp ¯x(2)_j,k,t,p,i = T X t=1 g_t,p,i(2) 8j 2 AG p, p, i (5.61)

Constraints (5.49) - (5.52) correspond to constraints (5.4) - (5.7) in the complete model. The precedence constraints present in the complete model and stage one are omitted from this stage, since it has already been decided by stage one. Con-straints (5.53) - (5.60) correspond to conCon-straints (5.13) - (5.21) in the complete model.

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5.6 Time Dependencies Between Activities 31

5.6 Time Dependencies Between Activities

All possible variables do not need to exist. For example if activity »j» in pro-gramme »p» is required to happen at least five weeks before » ˆj», and activity » ˆj» can happen at the end of the planning horizon, there is no reason to include

x_jktpin for t 2 [T 4, T ] as all these variables will always be 0.

By restricting one activity in each programme to an interval in time, variables for other activities need only be declared for »t» in a interval in relation to the restricted activities’ placement in time.

To calculate which indices »t» need to be defined for activities »j» the pro-grammes are represented as graphs with activities as vertices, and precedence relations as edges with Dmin

j ˆjp or Djmaxˆjp as weights. By solving the shortest

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6

Results

This chapter presents the results from solving the MIP model with Gurobi (a commercial MIP solver) with the assumptions made in Chapter 4. Specifically, stage one of the two stage model is solved over a period of 71 weeks and the full two stage model over a period of 30 weeks with sta↵ and programme data from a real clinic. Exact descriptions for the treatment programmes and sta↵ composition is not presented in this thesis as the clinic who provided it does not want it available to the public.

6.1 Hardware and Software

The two stage optimisation problem described in chapter 5 has been solved with Gurobi 8.1 with Table Python as a modelling language on an Intel i7-2820QM 2.3 GHz processor with 8 GB of RAM. Gurobi uses branch and bound together with cuts and heuristic methods.

The two stage MIP model solves the problem of deciding what the clinic’s activities each week should be in order to maximise patient throughput. In the following section the models have been applied to an empty plan with no previ-ously planned activities.

6.2 Data

Plans generated by the models are large and unsuitable for detailed analyses. In Figures 6.1 - 6.3 the results are therefore presented on an aggregate level with sta↵ utilisation per discipline instead of per sta↵ member. For each discipline there are two curves, one dashed representing the total available time each week, and one solid representing the output from the respective stage.

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34 6 Results

The data for the computations is based on real data supplied by a rehabili-tation clinic. The clinic wishes to be unnamed and therefore the names of the activities and the sta↵ members have been anonymised.

This clinic has two programmes: exhaustion and chronical pain. Below is a summary of the relevant data about the clinic and its treatment programmes.

Table 6.1: Clinic data

Planning horizon 30 weeks

Physicians 6

Psychologists 13

Physiotherapists 12

Occupational Therapists 2 Number of programmes 2

Table 6.2: Programme data

Treatment programme exhaustion chronical pain

Group activities 7 11

Individual activities 9 7

µ 0.085 0.56

wB 3 3

6.3 Longer Planning Horizon

Stage one is a less complicated model, and it is generally solved relatively quickly and allows for a longer planning horizon. For example, the entire duration for which real data exists (71 weeks, Figure 6.1) can be solved in stage one in only a few minutes. The stage two model is more computationally demanding. There-fore the planning horizon needs to be shorter when solving it.

Figure 6.1 visualises the sta↵ utilisation per discipline when solving the stage one model without the penalty term that spreads out activities over the weeks. Without the weights, stage one is a simplification and relaxation of the complete model with the objective function representing patient throughput, and whose optimal objective value is an upper bound for the complete model.

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6.4 Shorter Planning Horizon 35

Figure 6.1: Sta↵ utilisation over 71 weeks

6.4 Shorter Planning Horizon

The solution times for stage two are significantly longer than for stage one, and it is not solved to optimality. In order to solve the second stage model to a rea-sonable optimality gap, a shorter planning horizon must be used. The planning horizon 30 weeks has been chosen as it is twice the average duration of the the treatment programmes. In Table 6.3 the model statistics are presented.

Table 6.3: Model statistics

Stage one Stage two Rows (the number of

variables) 32178 87841

Columns (the number

of constraints) 47861 51853

Solution time [seconds] 213 28000

Gap [%] 1.06 11.9

Optimal value 379 293

6.4.1 Staff Utilisation

Figures 6.2 and 6.3 visualise the sta↵ utilisation from stages one and two. The solid lines are the the sta↵ utilisations for the output from stage two, which is the final plan. The dashed lines are the sta↵ utilisations from stage one, which are given for comparison.

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36 6 Results

Figure 6.2: Sta↵ utilisation with planning horizon 30 weeks, stage one

Figure 6.3: Sta↵ utilisation with planning horizon 30 weeks, stage two

The sta↵ utilisation in stage two, see Figure 6.3, is slightly more uneven than in stage one, as shown in Figure 6.2. In both figures boundary e↵ects can be noticed. For example the occupational therapists are completely unplanned during the first seven weeks as neither of the two programmes has activities requiring occu-pational therapists during the first weeks of treatment. Towards the end of the scheduling horizon, utilisation of all disciplines declines rapidly, since no new

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treatments are started.

In Figure 6.4 the sta↵ utilisation between the stage one and stage two models are compared as a ratio between the stage two and one models solutions. Stage two assigns treatment sta↵ to activities in the treatment programmes planned in stage one, so the planned time spent on activities in the treatment programmes from stage two can not exceed that from stage one. However, as the assessments in stage two are not decided by stage one, there are some weeks when the ratio exceeds 100% because stage two has placed assessments in di↵erent weeks than stage one.

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38 6 Results

6.4.2 Patient Throughput

Figures 6.5 and 6.6 show the number of assessments performed each week and the number of patients starting treatment each week, respectively. The clinic does not start any new treatments of patients after week 14. The exhaustion pro-gramme takes at least 16 weeks to complete, so there can be no new propro-grammes of this type started after week 14. The chronic pain programme takes 15 to 16 weeks to complete.

Figure 6.5: Assessments performed per week

Note that there are assessments taking place after the start of the last patients’ treatmentst. This is an e↵ect of how the patient queue constraints (5.4), (5.27) and (5.51) are constructed. The reason (5.4) is not violated at week 15 is that the queue remains empty as long as not enough assessments have been performed to increase the integer queue variable by one.

Figure 6.7 shows the total number of patients in an active programme each week. Boundary e↵ects are further discussed in Chapter 7.

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Figure 6.6: Treatments started

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40 6 Results

6.4.3 Example Patient Plan

In Table 6.4 a patient starting treatment week 3 in the programme for chronic pain. This patient is in group number 32 and has been assigned Psychologist 3, Physician 3, Occupational therapist 1 and Phystotherapist 6.

Table 6.4: One patient’s plan

Week Activity Week Activity

3 Start activity 13 Group Activity AB

Multidisciplinary activity B, 2

4 No activities 14 Group Activity AB

Multidisciplinary activity B, 3 5 Group Activity AB

Multidiscipline activity B 1 15 No activities

6 Group Activity AB 16 Psychologist activity 2 Physiotherapist activity 3 Group activity C

Occupational therapist activ-ity 2

7 Group Activity AB 17 Group activity C

8 Group Activity AB

Lecture 1 18 Group activity C

9 Group Activity AB Physiotherapist activity 2 Multidisciplinary activity A, 1

19 Group Activity C

Multidiscipline activity A, 2

10 Group Activity AB 20 Group activity C

11 Group Activity AB 21 Group Activity C

Multidiscipline activity B, 4 12 Group Activity AB

Occupational therapist activ-ity 1

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7

Discussion and Future Research

In the first section of this chapter particular observations about the results are presented. In the second section suggestions for potential further developments are given.

7.1 Discussion

7.1.1 Boundary Effects

The results of the 30 week planning horizon describe the startup and shutdown phases of a clinics activity as well as a relatively short steady state region around the middle of the period. When planning rehabilitation care which is performed in programmes, there will be boundary e↵ects towards the end of the scheduling horizon if it is required that all patients starting treatment should finish their programmes within the period.

Similar to the boundary e↵ects towards the end of the planning horizon, there is a startup phase when the clinic starts from no previous activities. This startup phase is very clear in Figure 6.7 where the number of patients actively receiving care can be seen to steadily rise.

There are boundary e↵ects at the start and the end of the planning horizon as shown in Figure 6.6 describing the number of patients receiving care at the clinic each week. The boundary e↵ects at the end of the planning horizon is due to the fact all patients have to finish treatment within the planning horizon. The startup phase, when the number of patients steadily rise, is due to the fact that the planning starts from an empty plan.

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42 7 Discussion and Future Research

7.1.2 Practical Usage

Because of uncertainties in assessment outcome and solution times, it is for prac-tical purposes neither necessary nor viable to simulate the clinic’s activity for a duration as long 71 weeks. The plan is meant to be generated for a shorter period (such as 30 weeks) and then updated weekly to account for assessment outcomes. When replanning in practise, the outcomes of the assessments from the previous week and the activities that have been assigned patients should be input to the model in order to update the future plan.

When sufficiently many replans after the initial startup period have been performed, the plan should be steady-state-like without displaying the startup boundary e↵ects mentioned above. The boundary e↵ects towards the planning horizon is always pushed one week further away, and not present in part of the plan that is actually implemented.

The output from the second stage can then be used to make the intra week schedule for the upcoming weeks.

7.1.3 Balancing Disciplines

As can be seen from the first 20 weeks in Figure 6.2, the limiting resource in the solution from stage one is the psychologist time. In the stage two solution, see Figure 6.3, the psychologist time remains high for most of the weeks. However, in this stage constraints for sta↵ loyalty are added, and the model has been solved to the requirement that there may only be one sta↵ member per discipline for each group, so even if the optimality gap in stage two would reach 0%, the sta↵ utilisation might not be as good as in stage one.

It can also be noted in the results of the 71 week plan that the available psy-chologist time rapidly increases after week 20, which results in better utilisation of all disciplines during the following weeks.

Stage one could be used as a tool for making strategic decisions about the sta↵ composition.

7.1.4 Assessments

There are more assessments being performed than patients starting rehabilita-tion. This is because not all patients that are assessed need, or are suitable for, treatment. In the model one assessment generates a fraction, µ based on histori-cal data, of a patient which is added to a (non integer) queue, ˆI. The total number of patients estimated to need care at the clinic is µ times the number of assess-ments performed. The total number of treatassess-ments started will be slightly higher than the total number of patients estimated to need care at the clinic because the non integer queue is rounded upwards every week. The choice of rounding up-ward is for safety. If too many assessments generate patients needing care there is a risk that the planning problem becomes unfeasible when replaning the up-coming week.

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7.2 Future Research 43

7.2 Future Research

Among the most important limitations of the mathematical modelling are the assumptions made about the patient queue. The variable B_tk is in this thesis modelled as a deterministic variable. In reality the outcome of the assessments is stochastic and an investigation of a stochastic programming approach to plan-ning problem could improve the plans. However, stochastic programming is com-putationally expensive.

The second stage model is computationally expensive, and Gurobi struggles to find optimality gaps smaller than in the order of 10% even with long compu-tation times. Shimmelpfeng [8] mention that the possibility to quickly replan is highly valued by the treatment personel. Therefore investigation of heuristic methods for the second stage model should be interesting.

It could be valuable to investigate allowing members of the sta↵ to overshoot the time allocated for treatment slightly. Especially clinics with total sta↵ loyalty. It could be the case that allowing one sta↵ member to work only a little more than the allocated time for treatment allow one more patient to admitted to the clinic. For example in the computations performed in Chapter 6, Psychologist 1 has 28 hours allocated for treatment. However, Psychologist 1 is in no week planned to spend more than 27.75 hour on treatment. By allowing Psychologist 1 to perform one additional 45 minute treatment activity pushing the planned time up to 28.5 hours it might be possible to admit one more patient.

Allowing overtime could be done by penalising it in the objective function, or investigating how increasing the right hand side of the constraints which assure the sta↵ do not work too much will a↵ect the solutions.

Finally, modelling and solving the intra-week problem of assigning all activi-ties to time slots within the weeks is a natural next step.

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[2] A Braaksma, N Kortbeek, G.F Post, and F Nollet. Integral multidisciplinary rehabilitation treatment planning. Operations Research for Health Care, 3: 145–159, 2014.

[3] P. Brucker and S. Knust. Complex Scheduling. Springer, 2006.

[4] Gréanne Leeftink, Ingeborg Bikker, I.M.H. Vliegen, and R.J. Boucherie. Multi-disciplinary planning in health care: A review. Health Systems, pages 1–24, 2018.

[5] J. Marynissen and E. Demeulemeester. Literature review on multi-appointment scheduling problems in hospitals. European Journal of Oper-ational Research, 272:407–419, 2019.

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Planning of Treatment at Rehabilitation Clinics Using a Two Stage Mixed-Integer Programming Approach