stakeholders
Jurn van Kampen S3214931
Master’s thesis
MSc Supply Chain Management
Faculty of Economics and Business University of Groningen University of Groningen
Supervisors
dr. I. Bakir (First assessor, University of Groningen) dr. E. Ursavas (Second assessor, University of Groningen)
Abstract
Facing the growing pressure for higher efficiency, Home Care (HC) providing organizations try to optimize their schedule. HC scheduling is a very complex process involving several different stakeholders’, often contradictory, interests. This research aims at developing a model that satisfies the different interest of the stakeholders. The results of this study show that our proposed model increases all different performance indicators for all the stakeholders. Furthermore, this study contributes to the theory by developing a novel model that satisfies the needs of the different stakeholders.
Keywords: Home Care scheduling, workforce scheduling, continuity of care, optimization, integer
Table of content
Abstract ... 1 1. Introduction ... 3 2. Related Literature ... 5 2.1. Workforce Scheduling ... 5 2.2. Workforce Scheduling in HC ... 6 3. Problem description ... 9 4. Model design ... 10 5. Numerical studies ... 14 5.1. Parameters selection ... 145.2. Testing historical data ... 18
1. Introduction
According to studies, 37% of the European population will be over the age of 60 by 2050 (Euronews, 2015). Unless the productivity of the healthcare provided is improved, the care of the aging population will require a third of the labour force to be employed in healthcare. As this will be difficult to achieve, the need to increase the capacity of the current system as well as the efficiency of the healthcare provided is of great importance (Nordlander, Lamorgese, Viet, Nguyen, & Montemanni, 2016). Given that people generally prefer to live at home for as long as possible, a fact that proved to increase the quality of life, makes that the demand for Home Care (HC) is on the rise (Di Gaspero & Urli, 2014). The increasing demand for the provided HC, leads to an increasing demand for cost efficiency (Mosquera, Smet, & Vanden Berghe, 2018). A major efficiency gain can be made if HC scheduling became more efficient. However, the HC scheduling process is a complex issue, with many factors that need to be taken into consideration, such as the worktime constraints of the employees, patient preferences, caregiver preferences, qualification requirements, and more importantly; continuity of care (clients’ preference to have the same caregiver) (Cabana & Jee, 2004; Wirnitzer, Heckmann, Meyer, & Nickel, 2016).
Despite the need to be more cost effective in HC, Wirnitzer et al. (2016) state that HC scheduling is mostly done manually. There are different stakeholders of the schedule, such as caregivers, clients and the company; and they have all different interests. Clients have a preferred day of the week and a preferred caregiver to receive care from (Bender, Kalcsics, Nickel, & Pouls, 2018; Braekers, Hartl, Parragh, & Tricoire, 2016; Bredström & Rönnqvist, 2008; Hiermann et al., 2015). Caregivers have clients they prefer to serve and they prefer to work a certain amount of hours (Lanzarone & Matta, 2014; Mutingi, 2013; Yalçindaǧ, Cappanera, Grazia Scutellà, Şahin, & Matta, 2016). The company aims at making the schedule more efficient by reducing the travel times of their employees. Many of these needs are currently not incorporated in the available scheduling tools, resulting in increased risk of low service quality or incompliance (Cappanera & Scutellà, 2015). For example, continuity of care is often missing while it is considered as one of the most important service quality determinants (Bachouch, Guinet, & Hajri-gabouj, 2011). Furthermore, most clients prefer to have a stable schedule, as they are over 65 years old or they have additional disabilities. In addition, most current available scheduling tools can be seen as incomplete, given that their outcomes lead to low satisfaction by either the patient, caregiver or HC provider. The facts presented above result in scheduling tools that are not suitable for use in HC (Shi, Boudouh, & Grunder, 2017). These facts lead to an increasing pressure to enhance the perceived quality of the schedule while being more cost efficient. This leads to the following research question:
How can the quality of the HC schedule be improved for all stakeholders by means of an integer programming model, and what is the interaction among the different objectives of the stakeholders?
German medium size HC provider. The requirements for the scheduling tool were given by the project partner and the final scheduling tool is tested on data provided by the project partner. The scheduling tool and the underlying optimization models can be adapted to be used by other HC providers. Although HC scheduling is still done manually in most cases, research in the field of HC scheduling is increasing in the last years (Fikar & Hirsch, 2017). The existing literature could be divided in two categories: Models that focus on long-term scheduling (weekly or longer) (Cappanera & Scutellà, 2013; Wirnitzer et al., 2016) and the majority; models focusing on short-term scheduling with a time horizon of one day (Bachouch et al., 2011; Fikar & Hirsch, 2017; Rasmussen, Justesen, Dohn, & Larsen, 2012). Nevertheless, focusing on short-term scheduling has many drawbacks. Short-term scheduling reduces the flexibility of the schedules, since some patients need to be visited several times a week and several constraints are related to a horizon of a week, for example the hours per week an employee can work. Besides discrepancies in the time horizons, there are differences as well in the way of solving the model, some studies use heuristics while other use exact methods (Defraeye & Van Nieuwenhuyse, 2016; Di Mascolo, Espinouse, & Hajri, 2017; Emiliano, Telhada, & Carvalho, 2017). Model objectives do vary in the literature, with some aiming at reducing travel time or distance and others focusing on patient satisfaction or to decrease overtime (Di Mascolo et al., 2017; Fikar & Hirsch, 2017). Studies only focusing on increasing the schedule efficiency can indeed achieve that, but the risk of the schedule being unsatisfactory to clients remains unaddressed. While, on the other hand, studies that focus only on increasing the service level can lead to a very inefficient schedule. Additionally, the literature shows that there are many different constraints, for example fairness among the employees, preferences, soft or hard time constraints, breaks, and continuity of care (Defraeye & Van Nieuwenhuyse, 2016; Di Mascolo et al., 2017; Fikar & Hirsch, 2017; Maya Duque, Castro, Sörensen, & Goos, 2015).
2. Related Literature
The related theory for HC scheduling is discussed in this chapter. Firstly, the general workforce scheduling theory is presented. Secondly, workforce scheduling in HC is discussed. The model developed in this study is partly derived of the theory discussed here and the problem description of the company, discussed in the next section.
2.1.
Workforce Scheduling
Workforce scheduling involves the scheduling of employees to jobs or shifts in order to cover the demand for a certain time period. Defraeye & Van Nieuwenhuyse (2016) found that this research field is growing extensively. Nevertheless, most models lack life implications. Based on the lack of real-life implications, Brucker, Qu and Burke (2011) concluded that new models have to be able to deal with different circumstances. The work of Garaix et al. (2018) can be seen as very relevant for our research. They made a model for workforce scheduling that could satisfy several of our objectives. Only their model is about assigning employees to tasks and in our case the task is the client, and the clients have preferences as well. Furthermore, Pinedo (2014) describes in his well-known book the day-off scheduling, crew scheduling, or shift scheduling problem. These problems appear to be more applicable in general and are hardly applicable for HC scheduling. The aim of most models is to minimize labour costs, while less attention was paid to the service level (Castillo, Joro, & Li, 2009). Nowadays there is an increasing attention to the service level (Othman, Gouw, & Bhuiyan, 2012). Workforce scheduling is especially relevant for the service industry. In the service industry, demand prediction or forecasting can be very challenging. In the last years the attention for finding solutions by means of optimization models has increased (Brucker, Qu, & Burke, 2011). It can be concluded, based on the above, that in the past the emphasis was on creating a schedule that would reduce costs, instead of increasing the service level. In addition, the attention to the satisfaction of the clients and employees was lacking.
There are various constraints for the scheduling, such as labour and minimum capacity constraints. All these lead to very complex scheduling problems (Pinedo, 2014). A special case in the service industry is Health Care scheduling. Scheduling problems in Health Care can be characterized as very difficult. There is an increasing pressure to decrease the budget while certain quality standards should be met (Brucker et al., 2011). Pinedo (2014) described several scheduling problems in Health Care, such as the assignment of personnel to operating rooms, emergency rooms, and surgery scheduling, although most of these problems are significantly different that HC scheduling. Another related and often mentioned scheduling problem is the nurse rostering problem. Drake (2013) found in his research that managers should better understand the impact of schedules on the performance of the organization, which highlights the importance of scheduling for companies’ performance. The nurse rostering problem can be viewed as a nurse-shift or nurse-task scheduling problem (Santos, Toffolo, Gomes, & Ribas, 2016). The nurse scheduling problem has several constraints that can be applicable for the HC scheduling problem; nurses’ workload, nurses’ skill level, and preferences (Cheang, Lim, & Rodrigues, 2003). Many heuristics are developed and used to solve the nurse scheduling problem in a simplified way (Legrain, Bouarab, & Lahrichi, 2015). Many of the above-mentioned problems are the same for HC. There are several constraints related to the nurses: the hours per day and hours per week they can work for instance. Additionally, the demand is unstable, and not all nurses being able to do every task. On the other hand, there are also differences related to HC scheduling, the travel distances should be taken into consideration, and most importantly the clients have a more direct role in the quality of the schedule since they have more preferences that should be met. The next section discusses related theory about HC scheduling.
2.2.
Workforce Scheduling in HC
The academic attention to HC is increasing (Fikar & Hirsch, 2017; Wirnitzer et al., 2016). Moreover, there is a trend to find solutions via optimization models (Burke, De Causmaecker, Berghe, & Van Landeghem, 2004). Most HC scheduling models differ on the time horizon, solution method, objectives, and constraints (Di Mascolo et al., 2017; Fikar & Hirsch, 2017; Maya Duque et al., 2015). Therefore, the different time horizons are first discussed, followed by the different methods used in the literature. Afterwards, the different objectives are addressed. Lastly, the different constraints are discussed.
The time horizon differs per study; some have a time horizon of one day, while others of a week, and some studies have a time horizon of more than one week (Di Mascolo et al., 2017; Fikar & Hirsch, 2017). Scheduling problems can be analysed on a strategic, tactical, or operational level. On a strategic level, the focus is on decisions like in which regions we want to operate (Emiliano et al., 2017). Scheduling is positioned on tactical or operational level. Scheduling is tactical when the time horizon is weekly, while it is more operational when it is short time scheduling (normally one day) (Nordlander et al., 2016). A weekly scheduling horizon offers more options, since some clients can be served on more days, and employees are mostly available on more days of the week. Nevertheless, most studies focus on a short-term scheduling (Yalçindaǧ et al., 2016). Since several constraints are weekly (how often clients need care per week, how many hours caregivers can work per week) we opt for a model that creates a weekly schedule.
The type of solution method differs among literature. Some studies use exact solution methods instead of heuristics (Di Mascolo et al., 2017; Fikar & Hirsch, 2017; Maya Duque et al., 2015). Maya Duque et al. (2015) found that several studies use a hybrid form, combining exact approaches and heuristics. The exact approaches are mostly translated in some extension of a Vehicle Routing Problem (VRP) (Cissé et al., 2017; Frifita, Masmoudi, & Euchi, 2017; Lin, Hung, Liu, & Tsai, 2018; Riazi et al., 2014). Most problems are converted to Vehicle Routing Problem with time windows, or a Travelling Salesman Problem (TSP). The heuristics vary a lot, with the ones used the most being the Tabu Search heuristic and Local Search. In this study the problem is solved in an exact way by means of an integer programming model, which incorporates some simplifying assumptions, to have an acceptable computation time.
Table 2.1: Overview of how often objectives are used in the investigates studies, adapted from Di Mascolo et al. (2017)
Furthermore, Table 2.1 shows that there are few studies that incorporate the preferences of the patients regarding the caregivers. The low amount of studies that incorporate the preferences is evidence of why the current scheduling tools are not seen as sufficient. Besides the focus on minimization of travel distances, some focus on overtime and the waiting times as well (Hiermann et al., 2015; Mutingi, 2013; Trautsamwieser & Hirsch, 2011). This focus could be more beneficial for cases with a high scarcity of manpower, but very often this does not constitute the main objective of a HC provider. Besides cost minimization, the objective of continuity of care is not broadly applied (Carello & Lanzarone, 2014; Kazemi, 2016; Lanzarone & Matta, 2014). Another (relatively new objective) is the fairness of workload, whereby the amount of workload is compared among the caregivers and, if needed, adjusted to create a more balanced workload spreading (Cappanera & Scutellà, 2013, 2015; Hertz & Lahrichi, 2009).
Currently, there is an increasing focus on service quality. Models that only focus on creating an efficient schedule are not seen as sufficient anymore. Therefore, more researchers are incorporating the preferences of the clients and the caregivers (Bertels & Fahle, 2006; Bredström & Rönnqvist, 2008; Trautsamwieser & Hirsch, 2011). Besides the preferences, continuity of care is very important (Cabana & Jee, 2004). The research of Cabana and Jee (2004) concluded that clients benefit from having the same caregiver for a longer period. This leads to the need of incorporating multiple objectives such as decreasing travel distances while increasing service quality. Maya Duque et al. (2015) developed a progressive model, based on two phases that focus on the travel distance minimizer and increasing service quality, by means of continuity of care and clients’ preferences. Nevertheless, their model did not aim for making a fair workload balance. Another study that is very relevant for this case, is the research of Guven-Kocak et al. (2017), which focuses on continuity of care as well, but does not take the preference of the day and the workload balance into consideration. This study extends this work by incorporating availability constraints and creating a fair workload among the caregivers. For our research the following objectives are chosen; minimize workload difference among the caregivers, maximize the day preferences of the clients, maximize the caregiver preferences of the clients, maximize the clients’ preferences of the caregiver, continuity of care, and lastly minimize the distances.
Objective
Distance total travelling cost / distance/ time 80%
working time/overtime 23%
number of routes 5%
number of caregivers 7%
material production and delivery costs 4% cost of adding a new patient to the planning 4%
rejection costs 5%
holding costs (a vera genumber of pa tients i n s ys tem) 4%
Cost s
Caregivers
Patients
preferences (ca regi ver, gender, vi s i t time) 6%
respect of a reference caregiver 4% number of different caregivers for a patient 8%
number of patients served 4%
number of uncovered visits 7%
waiting time 19%
workload difference 35%
violating of the geographic coherence 4%
Satisf actio
n
The minimizing of the distances is secondary since our project partner emphasized the importance of satisfying the needs of the clients and caregivers first.
Beside the different objectives, different studies have different constraints. Table 2.2 presents the most used constraints (Di Mascolo et al., 2017). A constraint that is widely used is the time windows, where clients have a time and day when they are available and when they are not available (Bertels & Fahle, 2006; Bredström & Rönnqvist, 2008; Mutingi, 2013; Rasmussen et al., 2012; Trautsamwieser & Hirsch, 2011). Furthermore, another constraint that is often used, is the skill requirements, sometimes called qualification; the skills of the caregiver should match with the requirements of the client (Bertels & Fahle, 2006; Bredström & Rönnqvist, 2008; Mutingi, 2013; Rasmussen et al., 2012; Trautsamwieser & Hirsch, 2011). A more obvious, and not always incorporated constraint, is the limited working time of the caregiver. There are several legal obligations about the amount of hours a caregiver can work (Cappanera & Scutellà, 2013; Mutingi, 2013; Rasmussen et al., 2012). In addition, a constraint that has to deal with contractual obligations is the constraint about the breaks, as many companies are obliged to give employees a break after a certain time. This constraint is surprisingly hardly incorporated (Maya Duque et al., 2015). Furthermore, the preferences of the caregiver are rarely incorporated, even though this could lead to low employee satisfaction. In our study the day availability and the maximum hours a caregiver can work per week and per day are incorporated. For the clients and caregivers, the day availability is incorporated. Furthermore, the skill availability of the caregiver is taken into consideration and the maximum number of visits per day per caregiver as well.
Table 2.2: Overview how often constraints are used in the investigated studies, adapted from Di Mascolo et al. (2017)
Starting at the first / finishing at the last patient’s home 4%
Starting / finishing at the HC structure 39%
Starting / finishing at home 24%
Vehicle capacity 2%
Caregiver preferences 4%
Qualification 63%
Travel time depending on public transport 2%
Constraints related to the lifetime of some conveyed products (blood, chemotherapy) 4%
Multiplicity of employment contracts (full-time and part-time) 13%
Days off 9%
Breaks 26%
Availability days 7%
Availability time windows 50%
Maximum working time for the week/day 74%
Limited number of caregivers per patient 23%
Always same caregiver for a given patient 19%
Patients’ dislikes 7% Patients’ preferences 23% Synchronization 21% Disjunction 2% Precedence 9% Preference days 5%
Patterns (same time slots visits every week) 9%
Soft time windows in hard time windows 5%
Soft time windows 19%
Hard time windows 58%
Starting hour 9%
Constraints concerning the patient Assignment
constraints
Temporal constraints
3. Problem description
This section explains shortly the problem of our project partner and the context of the problem. Our project partner is a German HC provider. Our project partner provides HC in different regions in Germany. Even though the provider operates in different regions in Germany, data of only one specific region is used for this research, as all regions have a comparable situation. The selected region has 175 clients and 18 caregivers. This region can be reconsidered as representative of other regions, and other HC organizations, since the circumstances are more or less the same. There are several clients, several caregivers, the clients have preferences about the caregivers and a specific day, the caregivers do so as well. There are several limitations; the number of hours a caregiver can work, day availability for the clients and caregivers, skill requirement, and the travelled distances should be minimized.
Currently, the scheduling is done manually. Due to time constraints the company focuses now on a feasible schedule where all clients receive care and, if possible, by their standard caregiver on their favourite day. The preferences of the client, caregiver and distances are not taken into consideration. The current available scheduling tools do not lead to a satisfactory schedule, which is quite common in the industry (Fikar & Hirsch, 2017), as there are too many factors that are not taken into consideration. The proposed scheduling tool has a time horizon of one week. The time horizon of one week is most appropriate, since there are several weekly constraints. Clients and the caregivers can be available on several days of the week, which increases the flexibility of the schedule. Our project partner aims for a scheduling tool that primary increases the quality of the schedule for the clients and caregivers and secondary decreases the traveling times. The clients assess the quality of the schedule in terms of compliance with their preferences. They have a preferred caregiver, preferred day of the week, and a preferred time. Besides the clients, the caregivers assess the quality of the schedule as well, in terms of compliance with their preferences. They have preferred clients and they prefer a fair workload balance among all caregivers.
4. Model design
This section explains the mathematical model that we developed. The mathematical model aims firstly to schedule the caregivers to the clients in such a way that the quality of the schedule for the clients and caregivers is as high as possible, while secondly reducing the travelled distances. The quality is determined based on the preferences of the clients, the continuity of care, the preferences of the caregiver, and the fairness of the workload balance among the caregivers. Furthermore, several constraints should be taken into consideration such as the maximal hours a caregiver can work per day, the availability of the clients, and skill requirements. The travelled distances are calculated via Bing maps driving distances, the fastest way mode was used for calculating the distances. An automatic tool was used to calculate all these distances. Several simplifying assumptions are made for the model. Firstly, all clients need to have just one caregiver per visit. There are no situations where a client needs multiple caregivers per visit. Secondly, there are no constraints related to the amount of time between visits. Thirdly, for calculating the distances the assumption has been made that caregivers go home after every visit; routing is not included. This assumption has been made since several clients do not have a time window but require to receive care at an exact time. Lastly, the total number of hours of care needed per week should be equal or lower than the maximum of the total number of hours caregivers can work per week. Furthermore, some clients require a standard caregiver, and some clients do not have a standard caregiver i.e.: They do not mind having a different caregiver. Some preferences of clients and caregivers were generated at random due to lack of information.
Sets
D
set of days of the weekI
set of clientsJ
set of caregivers Decision variable𝑥
. 𝑖𝑗 𝑑1, 𝑖𝑓 𝑐𝑙𝑖𝑒𝑛𝑡 𝑖 𝑖𝑠 𝑠𝑒𝑟𝑣𝑒𝑑 𝑏𝑦 𝑐𝑎𝑟𝑒𝑔𝑖𝑣𝑒𝑟 𝑗 𝑜𝑛 𝑑𝑎𝑦 𝑑
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
ParametersADCjd Maximal number of hours that caregiver j can work on a day d. It is 0 if caregiver j is not available on day d. Otherwise the maximum is given by the labour union law. ADKid Availability of client i on day d, if available 1 otherwise 0
AW1j Maximal number of hours that caregiver j can work per week
CAWj Availability of caregiver j in the chosen week, if available 1 otherwise 0 Dij Distance between caregiver j and client i
HCi Number of hours client i needs care per visit L Number of hours of work in the selected week MHi Number of visits client i needs in the chosen week
NAAi If client i is not available in the afternoon, if not available 1 otherwise 0 NAMi If client i is not available in the morning, if not available 1 otherwise 0
PC1ij Preferred caregivers j for client i, most clients have several preferred caregivers PC2ji Preferred client i for caregiver j, most caregivers have several preferred clients PC3ij The standard caregiver j for client i, most clients have one standard caregiver, they
are used to receive care from this caregiver. This is related to the continuity of care PDid Preferred day d of the week client i prefers to receive care, might be multiple days Sij If caregiver j has the right skills to serve client i, if yes 1 otherwise 0
UF Difference in number of hours that is perceived as unfair (e.g. 8) W1 Weight assigned to objective 1, clients preferences for caregivers W2 Weight assigned to objective 2, clients preferences for days W3 Weight assigned to objective 3, caregivers preferences for clients W4 Weight assigned to objective 4, continuity of care
W5 Weight assigned to objective 5, caregivers preferences for clients
W5 ∙ ∑. i ∈ I ∑. j ∈ J ∑ Dij ∙
𝑥
.ijd d ∈ D (e)∑
.
i ∈ I𝑥
.ijd≤ 4
∀ d ∈ D, j ∈ J
(10)
∑
.
j ∈ J𝑥
.ijd≤ 1
∀ d ∈ D, i ∈ I
(11)
𝑥
. ij d ∈ { 0,1} ∀ i ∈ I, d ∈ D, j ∈ J (12)5. Numerical studies
Firstly, the influence of the parameter selection on the performance indicators is analysed. This was done by means of a sensitivity analysis for the different parameters. This analysis led to a chosen set of parameters. Secondly, with these chosen parameters the model was tested on historical data. The model was coded in PuLP 1.6.0. The model was solved with the open source solver COIN-OR. The computations were performed on a PC running on Windows 10, 2,1 GHz, and 8 GB ram. The computation time was on average less than 2 minutes.
5.1.
Parameters selection
There are several parameters that can be chosen to add weight to certain objectives. The model consists of different objectives with different weights, and the workload balance is done by means of a hard constraint. The parameter of this hard constraint can be chosen before running the model. All these parameters lead to a specific objective or constraint, for example parameter W1 is the weight of objective preferences of the clients. Table 5.1 shows the performance indicators and their description. In this section the interaction between the parameters is discussed and analysed by means of a sensitivity analysis. A random week is chosen to compare the interaction among the parameters values and to come up with a proposed set of parameters. This proposed set can be used for running the model and make to compare the results of the schedule of the model with the schedule of the company without the model.
Parameter Directly related performance indicator
Description of the performance indicator
W1 Preferences clients The sum of clients that receive care of their preferred caregivers per
week
W2 Preferred day of the week The sum of clients that receive care on their preferred day per week
W3 Preferences caregivers The sum of caregivers that give care to their preferred clients per week
W4 Continuity of care The sum of clients that receive care of their “standard” caregivers per
week
W5 Total travelled distance The sum of the distance travelled by the caregivers per week
UF Workload balance The sum of the absolute value of the relative difference of the number
of hours a caregiver wants to work and the number of hours a caregiver will work per week
Table 5.1: Overview of the parameters
changing, while afterwards there is hardly any change. Furthermore, it can be seen that the total travelled distances are increasing together with the preferences of the clients. This could mean there is trade-off between these factors. In addition, the day preferences and preferred caregivers seem to be unrelated to the changed parameter while the workload balance is reduced when the parameter increases.
Figure 5.1: Overview of the influence of changing the weight of preferences clients’ parameter
Secondly, the parameter related to the objective about the preferences of the caregivers is analysed. Figure 5.2 depicts the results. The day preferences are not influenced by the changing parameter. The continuity of care and the distances are both negatively influenced, while distances increase if the parameter increases and the continuity of care decreases. This is in contrast to the first case where the continuity of care was not influenced significantly, while in this case the indicator is more sensitive.
Figure 5.2: Overview of the influences of changing weight preferences caregiver parameter
4400 4600 4800 5000 5200 5400 5600 50 70 90 110 130 150 170 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 K M t rav e lle d Su m o f o b jec tiv e
Weight of preferences clients
Preferences clients Preferences caregivers Workload difference
Day preferences clients Continuity of care Distances
4400 4600 4800 5000 5200 5400 5600 60 80 100 120 140 160 180 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 K M t rav e lle d Su m o f o b jec tiv e s
Weight preferences caregivers
Preferences caregivers Preferences clients Workload difference
In addition, the parameter of the workload constraints is analysed. This parameter determines the maximum hours of difference that is allowed between the desired number of hours a caregiver wants to work per week and the actual amount they work. Figure 5.3 shows the results. The first notable observation is the significant decrease in distances in the begin. This can be due to the fact that if the parameter is very low there is not enough freedom for the model to assign caregivers to clients who live closes by since the hard constraints about the workload balance would not be met. Another point to observe is that after the parameter reaches the value 8 the changes are minimum. Furthermore, it can be seen that the preferences of the caregivers, continuity of care and the distances are, slightly, positively affected by the increase of the parameter. This effect, though, is almost diminished if the parameter reaches the value 10. Again, in this case the day preferences are hardly influenced by the parameter.
Figure 5.3: Overview of the influence of increasing the parameter of the maximum workload differences
Figure 5.4 shows that the changing parameter values for the objective day preferences only slightly influences the other performance indicators. The effect is on first sight a bit unexpected. The expectations were that if the parameter would have higher values the other performance indicators would decrease. This can be explained due to the fact that the day preferences of the clients are distributed in a way that it is not constraining other objectives i.e.: There is enough freedom to satisfy both objectives. 4400 4600 4800 5000 5200 5400 5600 20 40 60 80 100 120 140 160 180 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 D istan ce s tr av e lle d Su m o b jec tiv e
Maximum hours of difference for the workload constraints
Workload difference Preferences clients Preferences caregivers
Day preferences clients Continuity of care Distances
4400 4600 4800 5000 5200 5400 5600 0 20 40 60 80 100 120 140 160 0 2,5 5 10 15 20 D istan ce tr av e lle d Su m o b ject iv e
Weight of day preferences
Figure 5.4: Overview of changing the day preferences parameter
Figure 5.5 depicts the influence of the parameter related to the objective of the distances. The parameter has a negative value since the objective should be to decrease the distances. If the parameter value decreases the distances decrease as well, just like the preferences of the clients, caregivers and the continuity of care. Although this effect is stronger with the smaller values of this parameter, it continues till larger values of this parameter. All this shows a clear trade-off between these performance indicators. Furthermore, the day preferences are hardly influenced while the workload differences do not show a stable relation.
Figure 5.5: Overview of decreasing the parameter of the objective of the distances
Lastly, the parameter related to the continuity of care is presented. Figure 5.6 shows the results. Here it can be seen as well, that if the parameters value of the continuity of care increase, the distances do as well, as do the clients’ preferences. The preferences of the caregivers and the day preferences of the clients are hardly influenced by the changes. Furthermore, the workload difference is slightly influenced by the changing parameter.
Figure 5.6: Overview of increasing the parameter of the continuity of care
This all brings us to the following conclusions of the sensitivity analyses of the parameters. Firstly, the day preferences are hardly influenced by the changes of the other parameters. Furthermore, there is a strong interaction between the distances and the other objectives such as continuity of care,
2500 3000 3500 4000 4500 5000 5500 6000 0 20 40 60 80 100 120 140 160 180 0 -2 -4 -6 -8 10 -12 -14 -16 -18 -20 -22 -24 -26 -28 -30 Dis ta n ce tra ve lle d Su m o b jec tiv e Weight distances
Distances Preferences clients Preferences caregivers
Workload difference Day preferences clients Continuity of care
3500 4000 4500 5000 5500 6000 50 70 90 110 130 150 170 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 KM t ra ve lle d Ou tp u t o b jec tiv e
Parameter continuity of care
Continuity of care Preferences clients Preferences caregivers
preferences caregivers, and clients. This indicates a trade-off, an efficient schedule in terms of low travel distances or satisfying the preferences of the clients and caregivers. The same trade-off holds for a fair workload; the more unfair the schedule is in terms of workload distribution, the lower the travelled distances.
Table 5.2 depicts the chosen parameters used for the comparison to the company’s data. The last column shows how the results are compared to optimal results. The optimal results are the best results possible for this case, that are calculated by only incorporating that objective. The column compared to the optimal is the percentage of the optimal or the absolute difference for the workload difference. The chosen parameters are based on the sensitivity analysis and the importance of certain performance indicators for our project partner. Our project partner emphasized the importance of the continuity of care and preferred day of the clients, that why, for example, the distances have a relative low value.
Parameter value Performance indicator Result Optimal Compared to optimal
10 Preferences clients 158 170 93%
10 Preference caregivers 111 153 73%
5 Workload difference 59 0 -59
5 Day preferences clients 147 148 99%
-1 Distances 5526 2428 44%
40 Continuity of care 147 153 96%
Table 5.2: Chosen parameters values for the model
5.2.
Testing historical data
After the parameters are chosen, the model is tested on historical data from our project partner. Eight weeks are used to test the model and compare the results of the model to the results of the historical data. Table 5.3 shows the characteristics of the chosen weeks. The number of clients fluctuated quite a lot per week. This might have been due to holidays, hospitals stay or other reasons, all of which be considered as normal in the HC sector. The performance indicators, which were discussed above, are compared to the historical data and the schedules the model produces.
Week 1 2 3 4 5 6 7 8
Number of clients 101 121 116 95 94 95 92 84
Number of caregivers 13 15 14 14 15 14 12 12
Number of hours care 290,5 318,5 326,5 290,5 278,5 295 268 260,5 Number of hours caregivers want
to work 335 380 355 334 330 360 315 315
Table 5.3: Characteristics of tested data
Graph 5.7: Overview of the performance improvements 0% 10% 20% 30% 40% 50% 60% 1 2 3 4 5 6 7 8 Im p ro ve m e n t as % o f h isto ri cal d ata Week
Performance improvements
Preferences clients Preference caregivers Workload difference
6. Conclusion
In this research we investigated a HC scheduling problem. An integer programming model was made with different objectives to satisfy all different needs of the stakeholders, namely the clients, the caregivers and the company. The results of the sensitivity analysis showed that changes in most parameters influence other performance objectives as well. The changes of the parameter for the preferred day of the clients do not influence the other performance indicators, while there is a trade-off between the distances travelled and the continuity of care. In addition, the changes of the parameter value for the workload balance do strongly influence the performance indicator of the travelled distances. On the other hand, the changes of the parameter for the workload balance only slightly influence the performance indicator of the continuity of care. Furthermore, the model was tested for eight weeks and the model achieved excellent performance. All the performance indicators were improved for all weeks. In addition, the model is solved on average in less than two minutes. Moreover, the results show that the value for several different stakeholders can be improved compared to the current scheduling practices of our project partner.
6.1.
Limitations
Firstly, this research could not incorporate exponential functions due to limitations of the solver used. This limited the model, for example an exponential penalty for workload differences would significantly enhance the quality of that objective. In addition, some assumptions which made the research possible can be seen as not perfectly realistic. The assumption that every caregiver travel back home after every visit is not always the case. Although this can be seen as a limitation, it can be assumed that if clients are scheduled to caregivers that live closer by the travel distances would be reduced. Furthermore, there are several operational constraints that limits the usability of a TSP. In our case the caregivers wanted to receive care at an exact time, they did not have a time window.
6.2.
Further research
For future research, we recommend for cases where a time window applies an extension of the model with a TSP per day per caregiver. The TSP could also incorporate time windows and time preferences of the clients and caregivers. This can be seen as a second phase, as the first phase would be the model developed in this research and the second phase would be the route per day per caregiver. To maintain two phases, the complexity of the model would remain low, resulting in a model that can be solved without requiring too many resources in time, software and hardware. This extension could only be possible for cases where clients have a time window. Another possible extension of the model would be incorporating an update of the model based on the last solution. More specifically, it happens quite often that a client or caregiver is unavailable at the last moment, resulting in a new schedule needing to be made for that or the coming days. The new schedule, in that case should be based on the past schedule of that week to minimize changes. For further research it is also recommended that the same data set is used, now there are several studies with all different data sets what makes it difficult to compare the different models.
6.3.
Scientific contribution
stakeholders cannot be seen as sufficient anymore. Furthermore, the model incorporates several constrains that leads to a unique model that is more applicable as well.
