Analyzing the Improvement Potential of Workforce Scheduling with Focus on the Planning Process and Caregiver Continuity

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IN

DEGREE PROJECT MATHEMATICS,

SECOND CYCLE, 30 CREDITS ,

STOCKHOLM SWEDEN 2019

Analyzing the Improvement

Potential of Workforce Scheduling

with Focus on the Planning

Process and Caregiver Continuity

A Case Study of a Swedish Home Care Planning

System

LIDA WANG

ENKHZUL UYANGA

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Analyzing the Improvement

Potential of Workforce Scheduling

with Focus on the Planning

Process and Caregiver Continuity

A Case Study of a Swedish Home Care Planning

System

LIDA WANG

ENKHZUL UYANGA

Degree Projects in Systems Engineering (30 ECTS credits)

Degree Programme in Industrial Engineering and Management (120 credits) KTH Royal Institute of Technology year 2019

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TRITA-SCI-GRU 2019:328 MAT-E 2019:80

Royal Institute of Technology

School of Engineering Sciences

KTH SCI

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Abstract

Swedish home care industry has been facing both external and internal problems, such as ageing population, varying quality and unsatisfactory continuity. Accordingly, workforce scheduling system, as one of the most common and useful software within home care planning nowadays, is in need of constant improvement and upgrading. This master’s thesis aimed to explore and analyze improvement potential of an established workforce scheduling system for an IT-company. The thesis was divided into two phases, of which a pre-study in Phase I tried to understand the planning process for planners and identify the perceived problems and shortcomings of the current system from a planner’s perspec-tive. Based on the analysis from the pre-study, the caregiver continuity was chosen as the research area for Phase II.

The current system was re-implemented and was modelled as an optimization problem. Furthermore, the system mainly consisted of two key parts, mixed integer linear program-ming (MILP) and heuristics. Different approaches in terms of modifications in both MILP and heuristics were applied to the re-implemented system. The performance of the mod-ifications was measured by multiple evaluation indicators. The test results showed that there was a potential to improve caregiver continuity with 1.2% to almost 13% depending on the modification type. The modifications were lastly suggested for further examina-tion regarding their practical appropriateness by applying them to the current running algorithm.

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Sammanfattning

Analys av Förbättringspotential inom Schemaläggning med Fokus på

Planeringsprocess och Personalkontinuitet:

En Fallstudie av ett Planeringssystem inom den Svenska Hemtjänsten

Den svenska hemtjänsten möter både yttre och inre problem såsom åldrande befolkning, varierande kvalitet och bristande kontinuitet. Schemaläggningssystemet som är en av de vanligaste och an-vändbaraste programvarorna inom hemtjänsten behöver därmed en ständig förbättring och upp-gradering som bemöter de existerande utmaningarna. Detta examensarbete hade som syfte att utforska och analysera förbättringspotentialen av ett etablerat schemaläggningssystem för ett IT-företag. Arbetet var indelat i två faser, varav förstudien i Fas I försökte förstå planerarnas planer-ingsprocesser och identifiera upplevda problem och brister i det nuvarande systemet utifrån ett planerares perspektiv. Baserat på analysen från förstudien, personalkontinuitet valdes som ett forskningsområde för Fas II.

Nuvarande systemet implementerades om och det modellerades som ett optimeringsproblem. Sys-temet bestod huvudsakligen av två nyckeldelar, blandat heltalslinjärprogrammering (MILP) och heuristik. Olika metoder i form av modifieringar i både MILP och heuristik tillämpades på det omimplementerade systemet. Modifieringarnas prestanda mättes sedan med flera utvärder-ingsindikatorer. Testresultaten visade att, beroende på vilken modifiering det gäller, fanns det en potential att förbättra personalkontinuiteten med 1,2% till nästan 13%. Det föreslogs slutligen att modifieringarnas praktiska lämplighet behövs undersökas ytterligare genom att applicera det på det nuvarande systemet som är i drift.

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Acknowledgements

The past eight months has been an unforgettable journey for us, as we grew both personally and academically. It was an extremely valuable experience where we had to cultivate and realize each others’ ideas but a period we would not like to go through again. We would like to express our utmost gratitude to our industry supervisor Eva and the rest of the team for their practical advise-ment and for providing us with all resources needed during the study. Our sincere thanks also go to our academic supervisor Per Enqvist for dealing with our emotional ups and downs and guiding us with his technical consultations and warm encouragements. Furthermore, we would like to thank our families and friends for always supporting and encouraging us. Our special thanks go to Eduard from Mathworks Support for helping us overcome technical difficulties encountered on Matlab. Finally, we would like to thank each other for being patient and trying our best.

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List of Figures

1.1 Forecast on population share aged over 65 based on 2017 data. . . 1

2.1 Planners’ satisfactions with the workforce scheduling system in general. . . 7

2.2 Current manual planning on average. . . 8

2.3 The main reasons for not using the optimization motor. . . 8

2.4 Capacity utilization rate for units. . . 9

2.5 Current caregiver continuity. . . 10

3.1 An illustration of a Branch-and-Bound tree. . . 15

3.2 The flowchart of the system. . . 23

3.3 Geographical positions for caretakers and the planning organization. . . 27

3.4 Results for modification 1. . . 29

3.9 Results for the combination of modifications 3 and 4. . . 34

3.10 Results for the combination of modifications 3 and 5. . . 35

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List of Tables

2.1 Home care terminology . . . 6

2.2 Ranking on the visit categories based on their planning difficulties. . . 9

2.3 Ranking on important aspects of home care planning. . . 10

3.1 Common heuristic methods. . . 17

3.2 The data structure of caretaker information. . . 19

3.3 The data structure of caregiver information. . . 20

3.4 Corresponding equations for system modifications to MILP. Note: The red sym-bols indicated changes made for each equation. . . 25

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1

Introduction

1.1 Background

The Swedish population is characterized by an increasing share of elderly. At the end of 2018, the population of Sweden was estimated to be 10.2 million people, of which 20 percent have passed the standard retirement age of 65 [1]. According to the projection of Statistics Sweden, in 2028 there will be 15 percent more people aged 65 and older compared with today. The number of people aged 80 and older is expected to increase the most, with around 50 percent until 2028. A forecast of the increased share of the population aged over 65 made by Statistics Sweden in 2017 is presented in figure 1.1.

Figure 1.1: Forecast on population share aged over 65 based on 2017 data.

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1. Introduction

services.

The decentralization of responsibility and more freedom of choices led to reduction of elderly hospitalization which has given rise to more home care services. Home care service is differ-ent from home nursing, as the latter provide professional medical services by a nurse at home. Home care services are provided by home care workers to the elderly living at home and include household-service tasks (cleaning, shopping, post- and bank-related errands etc.), assisting guid-ance to different locations and temporary relief for relatives by home care worker temporarily tak-ing over the responsibility to look after the elderly. Personal care such as bathtak-ing, getttak-ing dressed, moving around can also be included, as well as basic medical tasks such as insulin injections and treatment of wounds. The number of home care services can vary from once a month to multiple visits per day. On average, a home care user receives around seven hours of help per week [3]. The visits and the caregiver schedules are mainly created with help of workforce scheduling systems. These systems replaced the manual "white board" scheduling process in hope of reducing cost and increasing time efficiency.

Reports and documents have shown that there are large differences in quality of home care ser-vices, which can vary from very good to unacceptable [4][5][6]. There are many different factors which affect the quality of home care services, such as caregiver continuity, time duration of visits, competence of the caregivers and working methods [7].

Issues regarding caregiver continuity within home care have received more and more attention in recent years. Caregiver continuity is a measurement on how many different caregivers a home care user usually meet during a specified time period, for example two weeks. Media news and debate articles describe that the caregiver continuity within home care sector is low [8][9]. An article re-ports that elderly in Essunga municipality received home care service from 24 different workers on average during a two-week period [10]. The low caregiver continuity leads to stress and anxiety, not only for the elderly but also for the home care workers. According to Magdalena Elmersjö, as-sociate professor in social work at Södertörn university, maintaining a good relationship between the elderly and the workers is an important factor for the quality of home care services.

Another broadly debated topic is the schedules of the caregivers. One can easily find articles based on different personal stories of home care workers and they all address same issues. The daily work schedules are pre-planned minute-by-minute and leaves no time for the worker to take a break or handle unexpected situations with the home care user [11][12]. Many blame the inhumane workforce scheduling and work reporting systems for sacrificing the elderly and the caregivers for efficiency. The stressful schedules affect both the working conditions and the quality of service provided by the home care [13]. From the perspectives of the municipalities, a problem in terms of balancing between operation costs and service quality is always present. In conclusion, the Swedish home care industry has various challenges ahead where it needs to find a solution which prioritizes the human in the process of minimizing costs.

1.2 Purpose

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1. Introduction

students was to apply mathematical optimization and systems theory methods to the problem. The company’s interest was to fulfill the improvement needs of the customers, in this case the planners’. Due to confidentiality, any direct information about the IT-company, the system or any study participant could not be revealed.

1.3 Outline

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2

Phase I: Pre-study

2.1 Problem Description

It has passed more than ten years since the workforce scheduling system was first developed and then implemented in the company of this study. With the purpose of digitalizing home care and simplifying daily activities for the elderly, the system has been actively used and praised by many private as well as public home care organizations. The workforce scheduling system is mainly built on an optimization algorithm-based motor and generates optimized work schedules, in terms of minimized travel time, for caregivers. But beyond that, the system also takes some common factors, for instance, caregivers’ competences, caretakers’ requirements and time windows, into consideration. Some other parameters, such as caregiver continuity, which has been mentioned earlier in Section 1.1, has visibly grown in importance for the home care industry in general. In order to meet customers’ expectations and retain its competitive advantage in the market, the sys-tem needs thereby constant improvement and upgrading.

There has been some feedback received from their customers during the last few years, which showed that many home care planners still preferred to do scheduling and planning manually. In other words, the optimization motor was not used as effectively as the system initially was de-signed for. As the utilization rate of the optimization motor seemed to not fulfill the expectations, it could indicate some potential problems within the current system. For example, it could be due to unsatisfactory results from the optimization motor, i.e. not optimal enough, or it could also be the result from other system deficiencies. Since the company had not carried out any evaluation focusing on planners’ perspectives before, it was hard to find out planners’ opinions about the overall system and the main reason for them choosing manual planning instead of using the opti-mization motor. Hence, a comprehensive evaluation focusing on home care planners’ perspectives for the current workforce scheduling system turned out to be necessary for driving this thesis study forward.

2.2 Purpose

The purpose of the pre-study was to in a systematic and methodical way: • Understand the home care planning process of the planners.

• Identify the perceived problems and shortcomings of the system from the planners’ per-spectives.

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2. Phase I: Pre-study

2.3 Method

The pre-study was carried out in two steps. The first step was to conduct personal interviews with two full-time planners. The interviews were expected to provide the authors an overview about the general home care planning process of the planners, including daily work routines and planners’ reflections regarding their own experiences when using the workforce scheduling system. The information obtained from the interviews enabled the authors to formulate a survey, which was the main method used in step two of the pre-study.

The survey was formulated and created by using Google Forms. It consisted of 28 questions. The questions covered five different parts, including general information about the planner and his/her unit, the planner’s planning process and his/her usages of optimization motor as well as other functionalities in the system. Furthermore, the planners were asked to give, optionally, some suggestions for improving the current system. The data collected from survey responses was automatically exported to Excel and thereafter analyzed by using different built-in features, such as charts and tables.

2.4 Terminology

There were various terms that were frequently used in the home care industry. In order to make the understanding easier, the terms used throughout Chapter 2 were summarized and explained in table 2.1.

Table 2.1: Home care terminology Term Explanation

Time window The time period between an earliest and a latest arrival time that a caretaker should be visited by a caregiver.

Preferred caregiver Every caretaker has one preferred caregiver whom the care-taker would rather be helped by.

Day continuity The number of different caregivers a caretaker meets per day.

Caregiver continuity The number of different caregivers a caretaker meets during a certain period, e.g. two weeks.

Time continuity Recurring visits of a caretaker are expected to take place at the same time every time.

Visit time The time that is actually spent on visiting caretakers, i.e. duration of visits.

Back-office time The time that is usually spent on things outside of visits but still related to home care planning, for instance, reporting, attending meetings, workshops and education seminars. Capacity utilization rate Defined as the ratio of (visit time+ back-office time + travel

time)/ (total working time) for every caregiver. It is a mea-surement for productivity.

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2.5 Result

In total, there were 76 survey answers from 19 different public planning organizations received during the pre-decided period between 2019-03-12 and 2019-03-22. The results showed that most of the planning organizations had 1-2 schedule planners. The majority of the planners have been actively using the workforce scheduling system for more than one year. On average, 158 visits were needed to be planned by a planner during a workday. Furthermore, the visits were planned five days in advance and three days at a time. Other important and useful results obtained from the survey answers that could be illustrated in the form of figures and tables were presented as below. Figure 2.1 showed the satisfactions among participated planners with the current workforce schedul-ing system. The satisfaction levels were scaled from 1 to 4. Level 1 stood for least planner satis-faction, i.e. the planner was not satisfied with the current system at all, while level 4 indicated the maximum satisfaction. Based on the figure, it showed that the planners were overall satisfied with the current system, with approximately 85.6% given by level 3 to 4.

Figure 2.1: Planners’ satisfactions with the workforce scheduling system in general.

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Figure 2.2: Current manual planning on average.

Figure 2.3: The main reasons for not using the optimization motor.

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Figure 2.4: Capacity utilization rate for units.

Table 2.2 summarized the data on different types of visits in accordance with their planning diffi-culties. One could observe that dual staffing was ranked at the first place in terms of its planning difficulty. It was reasonable since it could be hard to find two caregivers with same availability and required skills to serve a certain caretaker. Attendant was ranked as second among the different types of visits according to the planners. One possible explanation could be its uncertainty and unforeseeable nature. For instance, it might be difficult to get to know and decide the necessity in advance when a caretaker needs someone to accompany him or her to hospital or other places.

Table 2.2: Ranking on the visit categories based on their planning difficulties. No. Visit Type

1 Dual staffing

2 Attendant, e.g. accompany caretaker to hospital 3 Shower

4 Washing 5 Co-habitation 6 Cleaning 7 Shopping

8 Take caretaker for a walk

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Table 2.3: Ranking on important aspects of home care planning. No. Aspects

1 Caregiver continuity

2 The preferred caregiver should, as often as possible, execute some of their caretakers’ visits

3 The preferred caregiver should, as often as possible, execute all of their caretakers’ visits

4 Day continuity 5 Time continuity 6 Short travel time

7 Fair workload distribution between caregivers 8 High capacity utilization rate

9 Fixed recurring routes 10 Others

Figure 2.5: Current caregiver continuity.

2.6 Discussion

Since the time period of this survey was limited, it excluded potentially some additional answers from other planners. Nevertheless, the total number of responses was sufficient to make basic calculations and analyses in order to fulfill the purpose of the pre-study. Since most of the partic-ipated planners were well-experienced with the current workforce scheduling system and all the responses received were quite complete, the collected data was thus considered to be representa-tive and reliable.

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the fact that planners, as a bridge between caregivers and caretakers, often have better understand-ing of the needs from both sides. Thus, planners were more likely to plan manually since they could take more humane aspects into consideration, rather than only letting the system generate the schedules. Another possible explanation could be that the planners had insufficient knowledge about how an algorithm-based optimization actually works, which in turn led to that they could not really trust the obtained results. Anyhow, considering that every planner had to handle a fairly large number of visits every day and the fact that manual planning could be very time-consuming, there was a need to further examine the system performance and find out ways to reduce the pro-portion of manual planning and increase planning efficiency.

The results showed that caregiver continuity was considered to be the most important aspect of home care, which coincided with the growing tendency in the home care industry. That is to say, from both the planners’ and the caretakers’ perspectives, maintaining a good caregiver continuity is significantly important. However, the result of current average caregiver continuity seemed not to meet the expectations. One could also observe from table 2.3 that short travel time, which is the main optimizing objective, was unexpectedly ranked sixth according to the planners. Instead of jumping to conclusions that short travel time is not as important as continuity, a more reason-able explanation might implicate that the current system has worked well so far with regard to minimizing travel time. Thus, there could be more attention paid to the aspects that still lack of performance, such as caregiver continuity. Hence, it turned out to be both interesting and valuable to investigate ways to improve caregiver continuity in the next step of this thesis study.

2.7 Conclusion from the Pre-study

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3

Phase II: Caregiver Continuity

3.1 Problem Description

Being determined in Phase I of the study, caregiver continuity was the aspect which was the most important for caregivers and caretakers, chosen by the planners. The other continuities, time and day continuity, were listed highly on the list as well, see table 2.3. In fact, the number two and three in the same table, i.e. preferred caregiver should execute some or most of their caretakers’ visits, were connected to continuity as well.

A detailed algorithm study by the thesis students showed that the current workforce schedule optimizer consisted of two key parts: mixed integer linear programming (MILP) and heuristics. The objective of the current algorithm was to get as many visits as possible planned while min-imizing the total travel time. The system aspect of actively improving caregiver continuity was however limited to clustering the caretakers and caregivers based on geographical and skill-based clusters and applying caregiver prioritization based on the caretakers’ preferences.

Documentations on the concept of preferred caregiver were found for several municipalities [14][15]. The aim of appointing a preferred caregiver for every caretaker is to create secure service and im-prove care continuity. The city of Stockholm had reported, however, that the caregivers’ need of effective planning risked the preferred caregiver continuity. The same situation has been informed by the interviewees who did not think that the system supported the concept of preferred caregiver. Another problem was related to how one measured the different continuities. The National Board of Health and Welfare has defined caregiver continuity as average number of different people from the home care organization a caretaker meets during a time period of fourteen days [16]. The system of focus in this study, had adapted this method of measuring caregiver continuity. The articles [8][9] mentioned in the Introduction had referred to caregiver continuity by this definition as well. Furthermore, there were no standardized way of measuring time and day continuity. A problem with this measurement was that it did not take the number of visits each caretaker had into account. In other words, the current standardized way of measuring caregiver continuity did not for example distinguish between a caretaker with total of ten visits during a two weeks period and another with forty visits but both had a caregiver continuity of seven.

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3. Phase II: Caregiver Continuity

3.2 Purpose

The purpose of Phase II was to study how different approaches of changes in the MILP and the heuristics of the re-implemented system would affect the caregiver, time and day continuity. The aim of the determined approaches were to improve caregiver continuity.

3.3 Theory

3.3.1 Mixed Integer Linear Programming

An integer programming problem (IP) is a linear programming problem (LP) in which some or all of the variables are required to be non-negative integers [17]. An IP in which all variables are required to be integers is called a pure integer programming problem, while a mixed integer linear programming problem (MILP) is a special case of IP where some of the variables are required to be integers. A general mathematical formulation of a MILP can be expressed in the following way [18]: minimize x,z c T" x z # (3.1) subject to A" x z # ≤ b (3.2) Aeq " x z # = beq (3.3) LB ≤ A" x z # ≤ U B (3.4) x ∈ R (3.5) z ∈ Z. (3.6)

An IP is known as a NP-complete problem, which indicates a class of problems where no poly-nomial bounded algorithm exists [19]. In other words, there is no efficient solution algorithm for such a "non-deterministic polynomial time" problem and it requires large amount of computing power to solve when a problem instances gets large. Some well-known NP-complete problems are, for example, knapsack problem and travelling salesman problem. MILP is thus also classified as NP-complete because it is more general than IP. Due to its NP-complete nature, it turns out to be difficult to solve a general MILP. The concept of LP relaxation of an integer programming problem plays a key role for solving IPs. LP relaxation works by generating an LP for any IP by omitting all integer or 0-1 (binary) constraints on variables. If the solution obtained from the LP relaxation of an IP are integers, then the optimal solution to the LP relaxation is also the optimal solution to the IP.

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basic operations. Branching stands for dividing collection of sets of solutions into subsets while bounding consists of establishing bounds on the value of the objective function over the subsets of solutions [20]. Figure 3.1 illustrates an example of a B&B tree. As the figure shows, the original problem P0is branched to two subproblems P1and P2by partitioning the set of feasible solutions

of x1into two smaller disjoint subsets. The optimal solution of this example is yielded by further

branching on x2 and solving the subproblem P3. Compared to pure IPs, B&B for MILPs only

branches on those required to be integers when branching on a fractional variable. Furthermore, construction of a good initial feasible solution is one of key issues in the solution of large combina-torial optimization problems by B&B and some heuristics may be used as preprossessing methods [21].

Figure 3.1: An illustration of a Branch-and-Bound tree.

One of the common modeling tricks of MILP is to introduce a sufficiently large positive value M together with a binary indicator, which is also called the big-M method. The method tries to control whether a linear constraint is active or not depending on other parts of the model or at the price of paying a penalty in the objective function [22]. For instance, considering a linear inequality:

αT

x ≤ x0.

Then, the reformulation of this constraint with the big-M method is in the form of: αT

x ≤ x0+ Mt,

where t ∈ {0, 1} and M is a large-enough value that guarantees that the constraint is inactive if t= 1.

3.3.2 Vehicle Routing Problem

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3. Phase II: Caregiver Continuity Minimize Z= N X i=1 N X j=1 V X k=1 ci, jxi, j,k Subject to N X i=1 V X k=1 xi, j,k= 1 ∀ j= 1, 2, ..., N − 1 N X j=1 V X k=1 xi, j,k= 1 ∀i= 1, 2, ..., N − 1 N X i=1 xi,h,k− N X j=1 xh, j,k= 0 ∀k= 1, 2, ..., V, h = 1, 2, ..., N N X i=1 Qi N X j=1 xi, j,k≤ Pk ∀k= 1, 2, ..., V N X i=1 N X j=1 ci, jxi, j,k≤ Tk ∀k= 1, 2, ..., V N−1 X j=1 xN, j,k ≤ 1 ∀k= 1, 2, ..., V N−1 X i=1 xi,N,k≤ 1 ∀k= 1, 2, ..., V xi, j,k = {0, 1} ∀i, j, k yi− yj+ Nxi, j,k ≤ N − 1 ∀1 ≤ i , j ≤ N − 1, 1 ≤ k ≤ V, where V= Number of vehicles, Pk= Capacity of vehicle k,

Tk = Maximum cost allowed for a route of vehicle k,

Qi= Demand at node i, QN = 0,

xi, j,k= 1 if pair i, j is in the route of vehicle k, 0 otherwise.

The high interest of the international research community has led to different variants of the VRP. Some of the most studied versions of the VRP are, for example, capacitated vehicle routing prob-lem (CVRP), vehicle routing probprob-lem with pickup and delivery (VRPPD) and vehicle routing problem with time windows (VRPTW). Other variants of the VRP which are also considered to be important are, for example, VRP with backhauls (VRPB), heterogeneous or mixed fleet VRP (HFVRP), periodic VRP (PVRP) and split delivery VRP (SDVRP).

3.3.3 Heuristics

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impractical.

A good heuristic is supposed to fulfill the following properties:

• A solution can be obtained with reasonable computational effort. • The solution should be near optimal (with high probability).

• The likelihood for obtaining a bad solution (far from optimal) should be low.

Since there are many heuristics that are different in nature, it is hard to provide a complete classifi-cation. In addition, many heuristics are designed to solve a specific problem such that it becomes inapplicable to other similar problems. Table 3.1 explains briefly some common heuristic methods.

Table 3.1: Common heuristic methods.

Heuristic method Explanation

Constructive heuristics A constructive heuristic constructs possible solutions to a prob-lem. In other words, single or multiple solutions can be gener-ated in constructive heuristics and the best one from the multiple solution is chosen as the final solution [27]. These methods have been widely used in classic combinatorial optimization.

Improvement heuristics An improvement heuristic is an iterative improvement of process and it starts with a initial solution to the problem. This type of algorithm tries to improve the solutions by modifying that se-quence. A common technique used in improvement heuristics is local search, which means that each step of the process carries out a movement from one solution to another solution with a better value. The algorithm terminates when there is no other accessible solution that improves the current one.

Meta-heuristics A meta-heuristic is defined as an iterative generation process which guides a subordinate heuristic by combining intelligently different concepts for exploring (global search) and exploiting (local search) the search space [28]. Some well-known exam-ples of meta-heuristics are: simulated annealing, tabu search and genetic algorithm etc. [29].

3.4 Method

3.4.1 Assumptions

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implementation of the system in Matlab:

• The schedules created from the re-implemented system did not take pauses and breaks, such as lunch or dinner breaks, into consideration.

• The schedules created from the re-implemented system did not take back-office time, such as seminars, administration and meetings, into consideration.

• Dual staffing option for visits were not considered.

• The travel time, calculated by using haversine formula in R [30] and chosen speeds for different transport ways, were used as the actual travel time between different geographical positions. The haversine formula determines the shortest distance between two points on a sphere given their longitude and latitude. The travel time was measured in whole minutes. • It was assumed that no late changes to the schedule, such as notification of illness of

care-givers and hospitalization of caretakers, occurred.

• It was assumed that there were no manual planning of visits. The results analysed were directly obtained from the optimization motor. Hence, the unserved visits were remained unserved without any further action.

• There were no differentiation between the caregivers based on their form of employment, for example, fill-in workers and permanent employees were seen as caregivers.

3.4.2 Data

The test data was provided by the IT-company. The data used in this study consisted of an anonymized data from a Swedish home care organization which had 43 caregivers and 118 care-takers during the specific time period. The time period consisted of 29 days between 2018.12.04 - 2019.01.21. Weekends and holidays, according to the Swedish calendar, were excluded. The information contained in the data could be divided into two parts: caretaker information and care-giver information. Furthermore, the caretaker information had visit as its unit while the carecare-giver information had work shift as its unit. The structure of the data could be seen in table 3.2 and table 3.3 respectively.

3.4.3 Measurement of Continuity

3.4.3.1 Caregiver Continuity

The current standardized way of measuring caregiver continuity is to count how many different caregivers a caretaker meets during a time period of fourteen days, according to the National Board of Health and Welfare (NBHW). Furthermore, caregivers who execute visits related to food dis-tribution and security alarms are excluded from the count. In this report, the caregiver continuity has been measured as:

Number of different caregivers a caretaker meets during a time period of fourteen days divided by the total number of visits the caretaker had during the same time period. The unserved visits from the system were excluded from the count.

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Table 3.2: The data structure of caretaker information. Caretaker information Explanation

Visit ID Identification number which was unique for the visit. Caretaker ID Identification number which connected the Visit ID

with the belonging caretaker.

Recurrence ID Identification number which connected same type of visits from the same caretaker.

Earliest Arrival The earliest time of arrival accepted for the specific visit, given in the form YYYY-MM-DD HH:MM:SS. Latest Arrival The latest time of arrival accepted for the specific visit, given in the form YYYY-MM-DD HH:MM:SS. Duration The estimated duration of the specific visit, given in

number of minutes.

Position Execution place of the visit given by geographical co-ordinates.

Skills Skills desired and required by the caretaker for the caregiver to be able to execute this visit.

Skills Must Have Indication on which of the skills were a must-have for the caregiver to be able to execute the visit.

Area The geographical area, in numbered format, which the visit belonged to.

caretaker test data used for this study did not contain quantitative information about the visit cate-gory which made exclusion of certain visit categories impossible.

The value of caregiver continuity could, based on this new definition, range between one and near zero. This could be explained more explicitly by an example. If a caretaker had in total 20 visits during the last fourteen days and the corresponding caregiver continuity was 0.6, it implied that the caretaker met 12 different caregivers during the specified period. Thus, when calculating the average caregiver continuity for the test period, caretakers who had only one visit during the fourteen days period were excluded in order to avoid values of ones which were caused by single visits.

3.4.3.2 Day Continuity

A standardized way of defining and measuring day continuity was not found. In this report, the day continuity has been measured as:

Number of different caregivers a caretaker meets during a day divided by the total num-ber of visits the caretaker had during the same day. The unserved visits from the system were excluded from the count.

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Table 3.3: The data structure of caregiver information. Caregiver information Explanation

Shift ID Identification number which was unique for the shift. Caregiver ID Identification number which connected the Shift ID

with the belonging caregiver.

Start time Starting time of the shift in the form YYYY-MM-DD HH:MM:SS.

End time Ending time of the shift in the form YYYY-MM-DD HH:MM:SS.

Start Position The start position of the shift given by geographical coordinates.

End Position The end position of the shift given by geographical coordinates.

Transport way The chosen transport way of the caregiver which the shift belonged to. The transport way could either be walk, car or bicycle.

Skills The skills the caregiver possessed for the specific shift. The skills were bounded to a shift, not the caregiver, because skills were counted as a perishable thing which could be learned and forgotten. Typical skills were different medicine delegations and differ-ent languages. Different characteristics of the care-giver, such as gender and allergies, were seen as skills as well. The skills were numbered and grouped for easy handling.

Areas The geographical areas, in numbered format, which the shift could serve.

3.4.3.3 Time Continuity

A standardized way of defining and measuring time continuity was not found. In this report, the time continuity was measured as:

The standard deviation, given in minutes, of the start times of recurring visits during the entire test period. The unserved visits from the system were excluded from the count.

3.4.4 Implementation

3.4.4.1 Mathematical Formulation

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Sets and indices

P= {1, ..., n}, which was the set of caretakers.

N= {0, ..., n}, which was the set of caretakers and the planning organization. Note that the location of the planning organization was presented as {0}.

K= {1, ..., k}, which was the set of caregivers.

i, j ∈ N, which corresponded to indices of caretakers and the planning organization. Parameters

xi, j,k: A binary optimization variable which indicated whether if caregiver k traveled from care-taker i to carecare-taker j.

Ci, j,k: The cost of travel time from caretaker i to caretaker j by caregiver k. Ti, j,k: The travel time from caretaker i to caretaker j by caregiver k. Ai,k: The arrival time of caregiver k at caretaker i.

Ei: The earliest arrival time at caretaker i.

Li: The latest arrival time at caretaker i.

Vi: The visit time or duration at caretaker i.

Pi,k: The execution bonus for caregiver k carrying out a visit at caretaker i. Bi,k: The matching bonus for caregiver k carrying out a visit at caretaker i. M: A sufficiently large positive number.

The modified and simplified optimization model

Minimize X k∈K X i∈N X j∈P Ci, j,kxi, j,k− X k∈K X i∈N X j∈N Pi,kxi, j,k− X k∈K X i∈N X j∈N Bi,kxi, j,k (3.7) The objective function consisted of three parts, as could be seen in equation (3.7). The first part represented the total travel cost, while the second and the third parts represented the total execution bonus and the total matching bonus for carrying out visits, respectively. Thus, for caretakers who had not been visited or had been visited by less matching caregivers, it turned out to be penalty costs for them instead.

xi, j,k =        1 0 ∀k ∈ K, i, j ∈ N, i , j (3.8) The binary variable xi, j,k, as could be seen in equation (3.8), was equal to 1 if caregiver k traveled from caretaker i to caretaker j, otherwise 0.

X

j∈P

x0, j,k≤ 1 ∀k ∈ K (3.9)

Constraint (3.9) stated that each caregiver started his/her working shift from the planning organi-zation. Furthermore, there could be maximum one caretaker j which gets served by caregiver k coming directly from the planning organization.

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Constraint (3.10) ensured that a caregiver must leave the caretaker after completing a certain visit at the caretaker.

Aj,k ≥ Ai,k+ Vi+ Ti, j,k+ M ∗ (xi, j,k− 1) ∀k ∈ K, i, j ∈ P, i , j (3.11)

Constraint (3.11) guaranteed an operative schedule. More explicitly, it demonstrated that a care-giver could arrive earliest at a caretaker after completing the previous visit and travelling from the previous place to this caretaker. Note that the constant M was used in the purpose of linearization, as known as the big-M method.

Ejxi, j,k≤ Aj,k≤ Ljxi, j,k ∀k ∈ K, i, j ∈ P, i , j (3.12)

Constraint (3.12) was the time window constraint, since every visit must be carried out between its earliest arrival time and latest arrival time.

The complete mathematical formulation of the MILP could be summarized as below:

Minimize X k∈K X i∈N X j∈P Ci, j,kxi, j,k−X k∈K X i∈N X j∈N Pi,kxi, j,k−X k∈K X i∈N X j∈N Bi,kxi, j,k Subject to X j∈P x_{0, j,k}≤ 1 ∀k ∈ K X j∈P xi, j,k= X j∈P xj,i,k ∀k ∈ K, i ∈ P, i , j Aj,k≥ Ai,k+ Vi+ Ti, j,k+ M ∗ (xi, j,k− 1) ∀k ∈ K, i, j ∈ P, i , j Ejxi, j,k≤ Aj,k ≤ Ljxi, j,k ∀k ∈ K, i, j ∈ P, i , j xi, j,k=        1 0 ∀k ∈ K, i, j ∈ N, i , j

The modified and simplified heuristic algorithm For caretaker i who has not been served from MILP:

1. Double-check the appropriateness for caregivers visiting the caretaker. 2. Try to allocate the caretaker into a caregiver k’s current schedule, without changing the order of existing caretakers.

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Figure 3.2: The flowchart of the system.

3.4.4.2 Algorithm Modifications

There were in total five algorithm modifications done in order to the improve average caregiver continuity. The technical descriptions of the modifications can be found below.

1. Prioritizing the "caretaker-based" preferred caregiver

Technical description of the modification in MILP:For every caretaker, a preferred caregiver was appointed. This was done by matching area, skills, date and times for all combinations of caregivers and caretakers. The preferred caregiver was set and unchanged for the test period. For each visit, if the visit was executed by the preferred caregiver, a bonus in form of negative weighting in the objective function was given. This was done by varying weightings in the matching bonus parameter. Several different weightings were tested. See equation (3.13) in table 3.4 for the mathematical formulation of the modification.

Technical description of the modification in heuristics: For each visit which was unserved from the MILP, the ones with preferred caregiver were prioritized. Among the unserved visits with and without preferred caregiver, the visits with worst caregiver continuity so far was prioritized. For each visit, the algorithm tried to appoint the visit for the shift which belonged to the preferred caregiver.

2. Prioritizing the"recurrent-visits-based" preferred caregiver

Technical description of the modification in MILP: For every recurrence ID, a preferred caregiver was appointed. This was done by matching area, skills, date and time for all com-binations of recurrence IDs and caregivers. The preferred caregiver was set and unchanged for the test period. For each visit, if the visit was executed by the preferred caregiver, a bonus in form of negative weighting in the objective function was given. This was done by varying weightings in the matching bonus parameter. Several different weightings were tested. In contrast to modification (1), a caretaker could have several preferred caregivers based on visit type. See equation (3.14) in table 3.4 for the mathematical formulation of the modification.

Technical description of the modification in heuristics: For each visit which was unserved from the MILP, the ones with preferred caregiver per recurrence ID were prioritized. Among the unserved visits with and without preferred caregiver, the visits with worst caregiver con-tinuity so far was prioritized. For each visit, the algorithm tried to appoint the visit for the shift which belonged the preferred caregiver.

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from the MILP, the ones with worst caregiver continuity were prioritized. For each visit, the algorithm tried to appoint the visit for the shift which belonged to the caregiver who had visited the caretaker the most during the past fourteen days.

4. Adding a new parameter in the MILP objective function which indicated the total number of different caregivers all the caretakers met during the past two weeks Technical description of the modification in MILP:In the objective function a new parame-ter indicating the total number of different caregivers all the caretakers met during the past fourteen days, a scalar, was added. The parameter was tested with several different weight-ings. For the first fourteen days, an accumulated value for the parameter was used. For examples, for day two and ten, the new parameter was calculated for days 1-2 and 1-10 respectively. See equation (3.16) in table 3.4 for the mathematical formulation of the mod-ification. The parameter was penalized linearly.

5. Adding a new step-wise weighted parameter in the MILP objective function which in-dicates the total number of different caregivers all the caretakers met during the past two weeks

Technical description of the modification in MILP: As in modification (4) but the param-eter was penalized step-wise. In other words, the worse the caregiver continuity (by the definition of NBHW), the larger the penalty got. See equation (3.20) in table 3.4 for the mathematical formulation of the modification.

Modifications (1), (2) and (3) implied changes in both MILP and heuristics of the algorithm while modifications (4) and (5) made changes in MILP only. The concept of focusing on caregiver continuity would not have been held if the modification was done to MILP-part only. Hence com-binations of modifications (4) and (5) with modification (3) will be tested as well.

Modifications (1) and (2) were done due to the fact that planners did not think that the system of focus supported the concept of preferred caregivers. Several runs of the re-implemented sys-tem, without any modification, showed that not many of the caretakers got the preferred caregivers for their visits. First of all, this was due to that many preferred caregivers did not fulfill the skill or geographical requirements of the certain visit. The system would, of-course, never go against the constraints which matches the skills and allowed areas exactly. Secondly, a few caregivers who had more skills and who could work across broader geographical areas were chosen by a lot more caretakers as their preferred caregiver. This resulted in a few caregivers who had to serve too many caretakers and eventually ended up serving one or few visits of many different caretakers. Hence, these modifications tried to reappoint the preferred caregivers for each caretaker. In addition, by appointing preferred caregiver per recurrence ID, one could avoid overusing a caregiver. The cur-rent algorithm did not exclusively prioritize preferred caregivers, thus by these modifications, the preferred caregivers were prioritized in every step of the algorithm.

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changing one.

Modifications (4) and (5) tried to affect the objective of the MILP. The current algorithm tried to serve as many visits as possible while minimizing the total travel time. By these modifications, the algorithm tried to serve as many as visits as possible by exclusively minimizing the caregiver continuity while taking the travel time into consideration.

Table 3.4: Corresponding equations for system modifications to MILP. Note: The red symbols indicated changes made for each equation.

Modification Modified equation 1 X k∈K X i∈N X j∈P Ci, j,kxi, j,k−X k∈K X i∈N X j∈N Pi,kxi, j,k−X k∈K X i∈N X j∈N B1i,kxi, j,k (3.13) where B1i,kwas a scalar and represented the matching bonus.

2 X k∈K X i∈N X j∈P Ci, j,kxi, j,k−X k∈K X i∈N X j∈N Pi,kxi, j,k−X k∈K X i∈N X j∈N B2i,kxi, j,k (3.14) where B2i,kwas a scalar and represented the matching bonus.

3 X k∈K X i∈N X j∈P Ci, j,kxi, j,k−X k∈K X i∈N X j∈N Pi,kxi, j,k−X k∈K X i∈N X j∈N B3i,kxi, j,k (3.15) where B3i,kwas a scalar and represented the matching bonus.

4 X k∈K X i∈N X j∈P Ci, j,kxi, j,k−X k∈K X i∈N X j∈N Pi,kxi, j,k−X k∈K X i∈N X j∈N Bi,kxi, j,k+ZX i∈N X k∈K Ai,k (3.16) where Z was a scalar and represented the penalty cost for caregiver continuity. Ai,k

was a binary variable, indicating 1 if the accumulated number of visits executed at caretaker k by caregiver i during a given time period, Si,k, was greater than 0, other-wise 0. More clearly,

Ai,k =        1 if Si,k> 0 0 otherwise (3.17) Thus,P

k∈KAi,k indicated the number of different caregivers a caretaker k meets

dur-ing a given time period andP

k∈K

P

i∈NAi,k indicated the total number of different

caregivers all the caretakers meet during the time period. Furthermore, in order to make the modification effectual, two new constraints were introduced:

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3. Phase II: Caregiver Continuity 5 (3.7)+Z1X i∈N X k∈K A1i,k+ Z2 X i∈N X k∈K A2i,k+ Z3 X i∈N X k∈K A3i,k+ Z4 X i∈N X k∈K A4i,k (3.20) where Z1, Z2, Z3, Z4 were scalars and represented the different penalty costs depend-ing on the size of caregiver continuity. A1i,k, A2i,k, A3i,k, A4i,k were binary variables, indicating 1 if the accumulated number of visits executed at caretaker k by caregiver iduring a given time period, Si,k, was in a specified range, otherwise 0. More clearly,

A1i,k =        1 if Si,k > 0 0 otherwise (3.21) A2i,k =        1 if Si,k > 5 0 otherwise (3.22) A3i,k =        1 if Si,k> 10 0 otherwise (3.23) A4i,k =        1 if Si,k> 15 0 otherwise (3.24) Furthermore, eight corresponding new constraints were introduced for linearization.

1 − M(1 − A1i,k) ≤ Si,k (3.25) Si,k− MA1i,k ≤ 0 (3.26) 6 − M(1 − A2i,k) ≤ Si,k (3.27) Si,k− MA2i,k ≤ 5 (3.28) 11 − M(1 − A3i,k) ≤ Si,k (3.29) Si,k− MA3i,k ≤ 10 (3.30) 16 − M(1 − A4i,k) ≤ Si,k (3.31) Si,k− MA4i,k ≤ 15 (3.32)

3.4.4.3 Evaluations

To evaluate a modification, first the modification was applied to the re-implemented algorithm and secondly, the whole algorithm was run for the test data. After each run, the evaluation was done by measuring following indicators:

• Running time measured in seconds • Optimal value

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• Average share of served caretakers • Average share of served minutes • Caregiver Continuity

– Average, standard deviation, minimum and maximum of caregiver continuity – Average caregiver continuity (according to the definition of NBHW)

• Day Continuity

– Average, standard deviation, minimum and maximum of day continuity – Average day continuity (according to the definition of NBHW)

• Time Continuity

– Average, standard deviation, minimum and maximum time continuity

Lastly, comparisons were made between the corresponding indicators of the non-modified algo-rithm and each modification.

3.5 Result

The geographical positions of the test data were illustrated in figure 3.3. The black circles in the figure indicated the positions for all the involved caretakers, while the red filled circle indicated the position of the planning organization. As shown in figure 3.3, the planning organization was located approximately at the centre of the given geographical positions. The location was quite reasonable, since caregivers were used to start as well as end their working shifts from the planning organization.

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Table 3.5 showed the results, including the evaluation indicators outlined in Section 3.4.4.3 for the unmodified algorithm. The results were primarily used as a reference for comparing with results obtained by each modification and combinations.

Table 3.5: Results showing indicators for the unmodified algorithm. Evaluation indicator Result

Running time (s) 2641 Average optimal value -22602971 Average travel time (min) 503

Total travel time (min) 14595 Average share of served caretakers (total) 89.08% Average share of served minuets (total) 39.95% Average caregiver continuity 0.3316 Average caregiver continuity (according to the

NBHW definition)

11.8925

Average day continuity 0.6489 Average day continuity (according to the

NBHW definition)

2.5677

Average time continuity (min) 16.8733

As one could see in table 3.5, the average caregiver continuity for the unmodified algorithm was 0.3316, which meant that if a caretaker had in total 100 visits during a time period, he/she had to meet around 33 different caregivers. The corresponding average caregiver continuity based on the NBHW definition was 11.8925. Moreover, as the table showed, the average day and time continu-ity for the unmodified algorithm were 0.6489 and 16.8733, respectively.

The most important results of each modification were summarized and presented in figure 3.4 to figure 3.10. In order to obtain the best results for each modification, multiple tests were performed by assigning different weighting parameters to the objective function. The figures contained two types of results for each modification, namely a continuity-related result and a non-continuity re-lated result. The continuity-rere-lated results included key evaluation indicators average caregiver continuity, average day continuity and average time continuity, while the non-continuity related results showed results for average share of served caretakers, average share of served minutes and total travel time. The complete results for all the tests could be seen in Appendix A and Appendix B.

For all the figures containing continuity-related results, the red line indicated average caregiver continuity for each test, while the orange and green lines indicated average day respective time continuity. Thus, the figures were designed in two different y-axes. The left y-axis referred to av-erage caregiver continuity and day continuity, ranging from lower bound 0 to upper bound 1. The right y-axis referred to average time continuity and had therefore minutes as the unit. Furthermore, the rectangular bars described percentage proportions of the most dominant parts of the objective function, which had the biggest impacts on the obtained objective value. The proportions were resulted from different weighting parameters assigned to the objective function.

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the dashed dark green lines and dark yellow lines showed the average share of served caretakers respective average share of served minutes from MILP. The total average share of served caretak-ers and served minutes, that is to say the results obtained from both MILP and heuristics, were displayed as the areas below the solid light green respective light yellow lines. Thus, the figures were also equipped with two different y-axes due to the different measurement units. The left y-axis was showed in percentage, ranging from 0% to 100%, while the right y-axis had minutes as the unit.

Figure 3.4 illustrated the results for modification 1. As the figure showed, the best average care-giver continuity was given by test 3, which led to the value at 0.3139. The value was reached when the objective value was almost dominated by the matching bonus.

Figure 3.4: Results for modification 1.

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value at 17.6150 minutes. Furthermore, the average served caretakers, the average served minutes and the total travel time for test 3 were not as good as other tests. The best results of these eval-uation indicators were generated by test 1, i.e. when the objective value was made up by a larger execution bonus and a smaller matching bonus. Especially, the total travel time for test 1 was much less than other tests. In addition, modification 1 could serve at most 89.08% of caretakers and 40.22% of visit time.

The continuity-related and non-continuity related results for modification 2 were showed in figure 3.5. For this modification, the best average caregiver continuity was obtained by test 2, which had the value at 0.3275. The value was also reached when the objective value was almost dominated by the matching bonus.

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indica-3. Phase II: Caregiver Continuity

tors, such as total travel time and average day continuity turned out to be better. Based on the test results, modification 2 could serve at most 88.70% of caretakers and 41.69% of visit time. Figure 3.6 showed the results for modification 3. The best average caregiver continuity for this modification was given by test 3 and had the value at 0.2945. The value was obtained when the matching bonus had apparently a much larger proportion than the execution bonus.

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The results for modification 4 were illustrated in figure 3.7. Since a new decision variable, Ai,k, was applied to the modified optimization problem, the dominant parts of the optimal objective value showed to be changed.

According to the test results, it turned out to be mainly dominated by the new-added continuity penalty and the execution bonus. As the proportion of the continuity penalty became larger, the average caregiver continuity tended to be lower. However, when the proportion of the continuity penalty was almost close to hundred percent, the value showed to increase again. As shown in figure 3.7, the best average caregiver continuity for this modification was obtained by test 3 at 0.3032. Moreover, the modification could serve at most 88.82% of caretakers and 41.43% of visit time. The overall total travel time of modification 4 were remarkably lower than the results pre-sented so far from modifications 1 to 3.

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was given by test 1 with the value at 0.3132. The modification could in total serve at most 88.71% of caretakers and 44.59% of visit time.

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modifications 3 & 4 had overall less total travel time than modifications 3 & 5. Moreover, all the tests of modifications 3 & 4 have reached the level of average caregiver continuity that were lower than 0.30.

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Figure 3.10: Results for the combination of modifications 3 and 5.

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Figure 3.11: Comparison between the best test results for each modification and the unmodified algorithm.

3.6 Discussion

As highlighted in Section 3.5, the average caregiver continuity has been improved with 1.2% to almost 13%. The biggest improvement was carried out by the combination of modifications 3 and 4. The combination not only utilized the historical data, but also took accumulated caregiver con-tinuity into account. Thus, it has tried with combined approaches at a maximum degree to improve caregiver continuity for the current system. The obtained day continuity and time continuity were slightly larger than the unmodified results, which implied an increase of 3% respective an increase of 2%. Notably, the overall day continuity and time continuity for every modification did not differ so much from each other. For day continuity, one possible explanation could be that most of the caretakers had basically only a few visits per day. The optimizing effects showed on day continuity were hence not practically noticeable. For time continuity, the results were not visible either because all the visits were already restricted by time window constraints. Therefore, since the obtained results by modifications had evident improvements on caregiver continuity and at the same time did not worsen day continuity and time continuity a lot, it could be regarded as valuable and applicable for the current system. However, as reported in Results, the total travel time varied a lot between different modifications and most of the modifications had negative impacts in terms of increasing total travel time compared to the unmodified algorithm. Considering the optimizing purpose of the original objective function was to minimize travel time while maximizing number of planned visits, it turned out to be one of the drawbacks of these modifications. But, what could not be ignored was that, due to the different types of modifications, the algorithm was more or less forced to shift its optimization emphases. The increase of travel time seemed to be a result obtained naturally by re-balancing between new contradictory objectives.

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However, what could also be observed was that, despite of which modification it concerned, the total average share of served caretakers and average share of served minutes (from both MILP and heuristics) always ended up at the levels of around 88% and 40%. Compared with the results only obtained from MILP, heuristics seemed to have at least doubled these values. On the one hand, the results underlined how efficient and important heuristics were. On the other hand, it could also be interpreted as the upper bound of how well the algorithm could perform. The considered upper bound was however dependent on the chosen test data. That is to say results might be totally different if another test data from another planning organization was used.

Observing the best test results for each modification, see figure 3.11, the concept of preferred caregiver, i.e. modifications 1 and 2, resulted in worst caregiver continuity. An explanation to this is the fact that the algorithm did not care much about the appointed caregiver if the preferred one could not take the visit. The result was also highly dependent on how well the caretakers and caregivers were matched. Different preferred caregivers would probably give different results even with the same weightings in the objective function. The practical limitation of a preferred caregiver not having enough time to perform all its caretakers’ visits was definitely a drawback of this concept compared to the other modifications. Even though modification 2 was a try to evade this issue the caregiver continuity for modification 2 was still slightly higher than modification 1. One have to examine further to check whether if this slightly higher value is actually good in practice. Say that a caretaker has in total 12 visits of 6 different visit types (2 visits per visit type) during a time period of two weeks. If the preferred caregiver per recurrence ID managed to per-form the visits it would mean at most 6 different caregivers the caretaker has to meet. By concept of preferred caregiver per caretaker, it could, for example, mean the preferred caregiver performs 8 of the visits and the rest are done by other random caregivers. The caretaker would have met 5 different caregivers in this case. Even though the latter gives better caregiver continuity value, in the long run it would probably feel better for the caretaker to meet the same 6 caregivers for the different visit types rather than always meeting 4 random caregivers every two weeks beside his/her preferred caregiver. This type of analysis could have been done by comparing the caregiver continuity with the rate of visits getting performed by its preferred caregivers. However, this type of analysis was not included in this study.

One could say that modification 3 had the highest impact on improving the caregiver continuity. This statement was supported by the fact that modification 3 gave the best result if combinations of modifications are excluded. Furthermore, the best result presented was a combination of the modifications 3 and 4. An explanation to why usage of historical data had such a high impact could be that it did not have practical limitations as in modifications 1 and 2. The fact that the caregivers are chosen based on historical data of fourteen days can result in a drawback in real life. Considering only fourteen days back in time does not guarantee that the caretaker will be meeting the same group of caregivers during a fourteen days period after a month than now. Unlike preferred caregivers in modifications 1 and 2, modification 3 does not take recurring caregivers into account which could be a disadvantage in real life for caretakers who surely have a memory which last longer than two weeks.

All the tests for the modifications were carried out on the same test data. In order to strengthen the results one could have used several test data in order to avoid test-data related errors and biases. Due to the assumptions presented in Section 3.4.1, one cannot surely know how much the modifi-cations will affect the caregiver continuity in real life planning and execution.

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the solver limit, some test days got infeasible solutions from the MILP solver. In this case, the heuristic part of the algorithm created the entire schedule. Hence, using a different solver than Matlab could have given slightly different results.

Finally, the decision on which modifications to test was based on the structure of the original system, its optimization algorithm and the results from Phase I. The authors did not find any relevant literature about staff continuity that suited the algorithm in question.

3.7 Conclusion

The purpose of Phase II was to study how different approaches of changes in the MILP and the heuristics of the re-implemented system would affect the caregiver, time and day continuity. In total, five different algorithm modifications and its combinations were tested on a predetermined test data. The aim of the modifications was to improve the caregiver continuity. Modifications in-cluded prioritizing of preferred caregiver per caretaker and recurrence ID, prioritizing of the care-giver with most execution based on historical data and reformulations of the objective function of the MILP to minimize caregiver continuity. The modifications were tested on a re-implemented system on Matlab.

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4

Discussion

The insights gained from Phase I and the results from Phase II were specific to the workforce scheduling system in focus. One cannot make conclusions based on results from Phase I about the general work process of every planner in the Swedish home care industry. Similarly, the modifi-cations tested in Phase II cannot be retested on other workforce scheduling systems and achieve the same results. The methods used in this study could, however, be applied to any workforce scheduling system and planners to examine their planning process and analyse the improvement potentials of the system.

Due to the structure and characteristics of the workforce scheduling system in focus, existing liter-ature about improving staff continuity, such as periodic workforce planning, became irrelevant to study. Hence, the modifications tested were not directly based on relevant literature which could seem made-up. In such case, one should put in mind that the general purpose of this study was to work with the system in focus.

From Phase I, the caregiver continuity was chosen as the system aspect to studied further in Phase II. Other aspects that were ranked highly on the importance list were for example day and time continuity. These aspects were monitored throughout the modifications done in Phase II but did not get more focus than that. Based on the general purpose of the thesis, these aspects could have also be chosen for Phase II. In purpose of improving the system, further research could be to improve these aspects and monitor the caregiver continuity instead. By doing this, one could get better understanding of how these different continuities relate to each other.

In Phase II, combinations of different modifications were suggested to improve the average care-giver continuity generated by the workforce scheduling system. However, one cannot surely know whether if the almost 13% increase of caregiver continuity will be seen as an improvement from the perspective of the planners. Specially since the total travel time increased by almost 7% for the best solution, this could be seen as a major drawback of the suggested modification combination. One cannot know whether if the planners are willing to sacrifice the travel time efficiency against caregiver continuity as these have contrasting relationship.

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5

Conclusion

The purpose of the thesis was to analyze the improvement potential of an established workforce scheduling system for an IT-company. In order to fulfill the general purpose, the thesis was di-vided into two phases. Firstly, a pre-study focusing on planners’ perspectives were done. By understanding the home care planning process for the planners with help of interviews and analy-sis of surveys, some shortcomings of the current system were thereafter identified. For example, the results showed that many planners still preferred working manually. Most importantly, based on the analysis of the pre-study, caregiver continuity was chosen as the focused research area for Phase II.

In Phase II, the current system was first re-implemented in Matlab so that it could be freely mod-ified by the authors. After that, different approaches of changes were applied to the MILP and heuristics, which were the two key parts of the optimization motor. The modifications were de-veloped partly based on the analysis from the pre-study and partly based on the existing potential of the MILP and heuristics. Multiple evaluation indicators were measured for each modification, including newly defined caregiver continuity, day continuity and time continuity.

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Analyzing the Improvement Potential of Workforce Scheduling with Focus on the Planning Process and Caregiver Continuity

Analyzing the Improvement

Potential of Workforce Scheduling

with Focus on the Planning

Process and Caregiver Continuity

A Case Study of a Swedish Home Care Planning

System

LIDA WANG

ENKHZUL UYANGA

Analyzing the Improvement

Potential of Workforce Scheduling

with Focus on the Planning

Process and Caregiver Continuity

A Case Study of a Swedish Home Care Planning

System

LIDA WANG

ENKHZUL UYANGA

Abstract

Sammanfattning

Analys av Förbättringspotential inom Schemaläggning med Fokus på

Planeringsprocess och Personalkontinuitet:

En Fallstudie av ett Planeringssystem inom den Svenska Hemtjänsten

Acknowledgements

Contents

List of Figures

List of Tables

1

Introduction

1.1

Background

1.2

Purpose

1.3

Outline

2

Phase I: Pre-study

2.1

Problem Description

2.2

Purpose

2.3

Method

2.4

Terminology

2.5

Result

2.6

Discussion

2.7

Conclusion from the Pre-study

3

Phase II: Caregiver Continuity

3.1

Problem Description

3.2

Purpose

3.3

Theory

3.3.1

Mixed Integer Linear Programming

3.3.2

Vehicle Routing Problem

3.3.3

Heuristics

3.4

Method

3.4.1

Assumptions

3.4.2

Data

3.4.3

Measurement of Continuity

3.4.4

Implementation

3.5

Result

3.6

Discussion

3.7

Conclusion