# Effective Optimisation of the Patient Circuits of an Oncology Day Hospital: Mathematical Programming Models and Case Study

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## Abstract

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## 1. Introduction

## 2. Motivation: The Problem of the ODH in Santiago de Compostela (Spain)

## 3. Literature Review

**Mathematical tools for decision support in health management**. Ref. [4] provides a review of recent optimisation studies that present decision-support tools for the design and planning of outpatient appointment systems. In ref. [5], as an aid for healthcare managers with the COVID-19 patient prioritisation and scheduling problem, a tool was developed based on artificial intelligence, using the neural networks method, and operations research, using a mathematical fuzzy interval model. The results of this study indicated that the combination of both models provides an effective evaluation under conditions of scarce initial information to select a suitable list. The proposed approach achieves one goal: to minimise mortality rates under the constraints of available resources in each hospital. The main objective of [6] is the efficient and balanced use of equipment and resources in hospital operating theatres. In this context, data sets from one hospital were used through the methods of goal programming and constraint programming. The main objective of [7] is the design and application of a binary scheduling model to support the decision making process, especially with regard to manpower scheduling in organisations with stochastic demand. The results were applied to the personnel allocation process in the ambulance service station in Subotica (Serbia).

**Optimizing the operation of oncology day centers or primary care clinics.**There are several works addressing the tasks of optimising oncology day centres or primary care clinics appointment scheduling, with mathematical, artificial intelligence and simulation scheduling techniques being the most commonly used. Below we describe some of the works that address problems that are quite similar to the Santiago de Compostela ODH. In this work we focus on daily optimization of patient circuits, which could be extended to a weekly planning. Naturally, this task is integrated into a more general problem such as the schedule corresponding to the instant of initiation of treatment plans, which has been addressed by different authors, who have highlighted its relevance. Incidents such as a patient failing to keep appointments and the availability of resources clearly impact on the problem at hand. Ref. [8], in their work to quantify the association between cancer treatment delay and mortality, concludes that cancer treatment delay is a problem in healthcare systems worldwide. It is now possible to quantify the impact of delay on mortality in order to prioritise and model. Even a four-week delay in cancer treatment—surgical indications, systemic treatment and radiotherapy—is associated with increased mortality in seven types of cancer. Policies focused on minimising delays in cancer treatment initiation could improve population survival outcomes. Ref. [9], following a systematic review, concludes that a comprehensive strategic approach, including realignment of resources, operational efficiency and process improvement, holds the most promise for improving the efficiency and effectiveness of outpatient and ambulatory services, thereby reducing waiting times and improving health outcomes. These three broad areas identified are complementary and offer a comprehensive approach to policy improvement in these areas. In research by [10], they perform a baseline measurement of lung cancer patients’ waiting times for systemic therapy across the UK. The authors understand that the continued introduction of new therapies will have a significant effect on service demands and recommend that health service managers model the likely impact on resource needs, and suggest the use of the so-called C-PORT tool developed by the UK Department of Health. A study conducted by [11] at a single radiotherapy cancer treatment centre shows that the majority of outpatient consultations (80%) were seen within 20 min of their scheduled time. The reported delays were due to clinic workflow and the coordination of multiple appointments throughout the day. Findings from such studies, the authors believe, can help formulate strategies to improve efficiency and patient satisfaction. The problems identified by [12] in the course of an audit of patient waiting time and physician consultation time in a primary care clinic were addressed by the aforementioned paper which has formulated strategies to improve waiting and consultation time, including increasing staffing, implementing an algorithm for a staggered appointment system for patient follow-up and improving the queuing system for walk-in patients attending the clinic. The results shown in [13] provide some insights into waiting time, which is a barrier to healthcare delivery in mainland China. The authors show that adopting improvements in outpatient management software, following detailed analysis of patient data, patient surveys and patient interviews, can be an effective way to deal with long waiting time.

**Applications of deterministic mathematical programming to the optimization of oncology patient circuits.**Ref. [14] presents the situation of a clinical centre to which patients come to receive chemotherapy sessions with a certain periodicity, forming regular cycles of treatment sessions. Two objectives are pursued. On the one hand, to reduce as much as possible the delays in the patients’ cycles, given that excessive delays in the time between chemotherapy sessions greatly diminish their effectiveness. On the other hand, to reduce the hospital costs associated with the hours of work that are carried out. Ref. [15] addresses the problem of minimising patient waiting times during a working day in a centre offering oncology appointments and chemotherapy treatments in a context where it is already known how many and what type of patients need to be seen. Initially, a bi-objective optimisation problem is posed, trying to achieve a balanced workload throughout the working day both in terms of the use of the chemotherapy chairs and in terms of the consultations with the oncologists. Once the process has been completed, and a new method of appointment scheduling has been obtained, the simulation will be used to corroborate that the desired level of balanced workload can be achieved, as well as to compare it with the existing method and verify that the new method is more efficient. In [16], in the context of a centre to which patients come for chemotherapy, the problem of minimising delays in patient treatments and the total working time of the centre is addressed. Specifically, the aim is to find the optimal starting day of treatment for each patient so that their chemotherapy cycle is completed as soon as possible, as well as for the centre’s professionals to achieve this goal with as few working hours as possible. However, the problem is also posed by taking the number of resources as a decision variable to find out what would be the appropriate number of professionals and chairs to achieve optimal performance (if the hospital were in a position to devote a larger budget to increase such resources). The modelling in the paper by [17] studies a way to proceed when, for a given working day, patients in the centre are assigned appointments with their oncologist and chemotherapy treatments. Furthermore, it is assumed that the appointments with each patient’s oncologist are scheduled in advance, and the problem faced is to determine the schedules of each patient’s chemotherapy treatment sessions. Thus, this work is close to the problem at hand, although the constraints and the objective itself are not identical to those of ODH. However, it is noteworthy that the work has the great advantage that the proposed model can be solved exactly and very quickly, for the planning of a working day with a number of patients around 100, which is a good property inherited by the models introduced in our paper. In [18], treatments are scheduled for new patients and nursing needs are taken into account, taking into account last minute cancellations and multiple objectives. Ref. [19] uses a model similar to those considered in vehicle routing problems to balance resource use and minimise waiting times. They make simplifications such as not setting a limit on the number of nurses, although they plan a set of days. They only accurately solve problems with about 30 patients and generally use a two-stage heuristic involving a constructive algorithm and a local search. They perform a sensitivity analysis and use a second model that adjusts appointments robustly to longer than expected treatments.

**Applications of stochastic mathematical programming to the optimization of oncology patient circuits.**Ref. [20] addressed the problem of adjusting outpatient time and appointments along with the optimal number of physicians, for an outpatient appointment system in an individual block/fixed interval class, by using an adaptive penalty genetic algorithm. The length of service time for medical consultation, the time required for laboratory tests, and the time deviation from the appointment time are modelled by random variables. No-show patients are also included in the system. Using the adaptive penalty scheme, optimisation constraints are handled automatically and numerically. The solution methodology is easily applicable to other appointment systems. Ref. [21] presents a two-stage stochastic integer schedule to design patient appointment schedules under uncertainty of treatment times. The goal is to minimise a trade-off between the expected waiting times of patients and the expected total time to treat them. It is shown that solving this optimisation problem requires prohibitive computational time, so a heuristic algorithm is developed to find approximate solutions. Ref. [22] proposes such a model that is solved by averaging sample scenarios, which is an approach that can result in affordable computational times. This approach has served as a source of inspiration for the second model introduced in this paper. The work is motivated by a multidisciplinary oncology clinic that communicates the diagnosis and explains the treatment plan to its patients. In addition, regular patients are also seen by the clinicians. Therefore, all clinicians involved need a work plan in which various types of patients can be scheduled. These work plans are designed to optimise the waiting time of the patients and the waiting time of the clinicians. In [23], patient waiting times, chemotherapy chair downtime and nurse overtime are minimised using a stochastic programming algorithm. The work [24] includes stochastic programming aimed at minimising waiting times and the implementation of the methods with OpenSolver.

**Optimization of oncology patient circuits through simulation techniques.**A discrete event simulation model to explore appointment scheduling in a general hospital outpatient chemotherapy department has been developed by [25]. They consider different statistical distributions of the times involved in the problem. They identified an efficient schedule that kept bed utilisation at a tolerable level, restricting excess waiting time in a clinical setting. The authors suggest a scheduling method based on infusion time for the outpatient chemotherapy department. The study [26] proposes a feasible solution that increases resource utilisation without affecting patient service. The proposed simulation model shows how, with a better balance in the appointment system, the clinic could increase the number of patients seen by 18% while maintaining the same total patient time in the system. In other words, the model establishes a new schedule for infusion chairs that would allow more patients to enter the system and maintain the workload of nurses and pharmacists. The model distributes patients to available slots without exceeding the capacity of human resources. By means of simulation, waiting times, allocation of wards to specialists, multiple clinics and the use of resident and senior doctors are simultaneously considered in [27]. The convenience of computerised data collection is highlighted. Ref. [28] analyses the effects of possible improvements to the circuit by simulation. Consultation with the oncologist and pharmacy work are highlighted as key steps. They recommend citation for all stages and improved information flow between them.

**Other methods applied to optimize the management of oncology patient circuits**. Research by [29] addresses an oncology patient admission and allocation problem. The probability of patient appointment cancellation, as an important indicator of possible future gaps, is considered by the author in making admission and patient allocation decisions. The aim of walk-in admission by general outpatient clinics is to reduce the negative impact of patient absences and cancellations, and to improve the utilisation and accessibility of the clinic. This study presents a learning-based outpatient management (LGOM) system that focuses specifically on the management of patient admission and allocation during the day. The author develops a Markov decision process model to capture the patient admission decision process in the general outpatient clinic. Admission and patient allocation decisions are made from the perspective of maximising the long-term benefit of the system. LGOM is trained with simulation data and then applied to the real situation and the policy is updated to reflect the actual information. Ref. [30] applied a decision tree analysis to predictors that were significantly correlated with patient attendance behaviour to assess the probability of no-shows. The authors then developed a dynamic appointment scheduling procedure using different over-demand strategies for different numbers of appointments. A computer simulation was used to evaluate the effectiveness of the dynamic procedure against two other methods of randomly and uniformly assigning appointments. The dynamic scheduling procedure resulted in increased scheduling efficiency through overbooking, but with less than 5% risk of appointment conflicts (i.e., two patients presenting at the same time). In [31,32], discrete event simulation is incorporated into a kaizen approach aimed at reducing waiting times and better distributing the workload of hospital staff. The inclusion of this approach in a lean process has enabled staff to participate in a team-based problem-solving approach.

## 4. Contribution of the Paper

- The first contribution was to measure the times involved in the circuit for all patients visiting the centre during a normal working week. Despite the difficulty of this task, the collaboration of each and every one of the CHUS professionals involved has been very valuable and has not only served to feed the models developed in the work, but has also shed light on the true reality of the times of the different stages of the circuit. This is of great use to the aforementioned professionals and in making different decisions aimed at improving the service.
- Then, the model proposed in [17] was selected as it represents a problem very close to that of ODH and solves it very efficiently, and its objective and restrictions were adapted to fully represent our problem. The data collected have been used for the definition of the model parameters. A data-driven procedure is used to determine the ready times at which each patient can receive cancer treatment. While providing promising initial results, this model is deterministic and is not able to capture the underlying stochasticity in the mix of patients.
- Thus, a second model is proposed, which generalises the previous one and, following the [22] approach, it is based on the data driven construction of different scenarios obtained from cluster analysis methods, that take into account the stochasticity of the times of the different stages of the process and the consequent variety in the patient mix. This second model achieves the objective of not only scheduling treatment appointments but also rescheduling appointments with oncologists from the original ones. In addition, an improvement in patient waiting times is achieved by comparing the results provided by the model with the available data. All models are programmed with the AMPL [33] language and solved with the Gurobi ( https://www.gurobi.com/, last accessed 20 Decemer 2021) solver in a fast way (seconds). The PC used for this work was a Lenovo Intel(R) Core(TM) i7-1065G7, 8 GB RAM, Windows 10 64-bit operating system.
- Finally, the results obtained in both models feed a third and last model created to assign the nurses in charge of providing the treatment to the patients.

## 5. Materials and Methods

#### 5.1. Deterministic Model for Treatment Appointments

#### 5.1.1. Parameters

- P: it establishes the number of patients with a chemotherapic appointment.
- K: it establishes the number of chemotherapic chairs available.
- T: it establishes the number of time slots used to split the 14 daily working hours.
- ${l}_{p}$ with p∈$\{1,...,P\}$: chemotherapic treatments’ duration of each patient.
- ${r}_{p}$ with p∈$\{1,...,P\}$: it establishes, for each patient, the first time slot where his chemotherapic treatment could be scheduled.
- $Ndis{p}_{t}$ with t∈$\{1,...,T\}$: number of available nurses inside the chemotherapic room for each time slot.

#### 5.1.2. Variables

- ${x}_{p,t}$ with p $\in \{1,...,P\}$ and t $\in \{1,...,T\}$: binary variable that is equal to 1 if the p patient begins his treatment in the time slot t.
- ${C}_{max}$: integer variable that establishes which time slot will be the first one with no treatment associated.
- ${\lambda}_{1}$ and ${\lambda}_{2}$: positive real variables. We will use them in order to ponder both parts of the bi-objective function.

#### 5.1.3. Mathematical Model

- Important Differences between the Models of the ODH and Those Presented in Previous Related Investigations:

#### 5.2. Stochastic Model of Oncologist and Treatment Appointments

#### 5.2.1. Parameters

#### 5.2.2. Variables

- $y[p,t]$ with p $\in \{1,...,P\}$ and t $\in \{1,...,T\}$: binary variable, for scenario 1, that is equal to 1 if the p patient begins his treatment in the time slot t.
- $f[p,t]$ with p $\in \{1,...,P\}$ and t $\in \{1,...,T\}$: binary variable, for scenario 2, that is equal to 1 if the p patient begins his treatment in the time slot t.
- $x[p,t]$ with p $\in \{1,...,P\}$ and t $\in \{1,...,T\}$: binary variable, for scenario 3, that is equal to 1 if the p patient begins his treatment in the time slot t.
- $z[p,t]$ with p $\in \{1,...,P\}$ and t $\in \{1,...,T\}$: binary variable, for scenario 4, that is equal to 1 if the p patient begins his treatment in the time slot t.
- $c[p,t]$ with p $\in \{1,...,P\}$ and t $\in \{1,...,T\}$: (first stage) binary variable that is equal to 1 if the p patient is scheduled for the oncologist in the time slot t.

#### 5.2.3. Objective Function of the Mathematical Model

- $\sum _{t=1}^{T}$$(t\xb7y[p,t\left]\right)$ represents the period in which he starts his treatment,
- $\sum _{t=1}^{T}$$(t\xb7c[p,t\left]\right)$ represents the period in which he is scheduled for the oncologist.

- $({\displaystyle \sum _{t=1}^{T}}$$(t\xb7y[p,t])-({\displaystyle \sum _{t=1}^{T}}$$(t\xb7c[p,t\left]\right))-2.6)$ is the waiting time of patient p from the theoretical exit of the appointment with the oncologist to the start of treatment, and
- $\sum _{p=1}^{P}$$({\displaystyle \sum _{t=1}^{T}}$$(t\xb7y[p,t])-({\displaystyle \sum _{t=1}^{T}}$$(t\xb7c[p,t\left]\right))-2.6)$ is the total waiting time from the theoretical exits of the appointment with the oncologist to the start of treatments, in scenario 1.

#### 5.3. Model of Nurse-to-Patient Allocation

#### 5.3.1. Parameters

- P: it establishes the number of patients with a chemotherapic appointment.
- T: it establishes the number of time slots used to split the 14 working hours.
- $Ndis{p}_{t}$ wth t∈$\{1,...,T\}$: number of available nurses inside the chemotherapic room for each time slot.
- ${y}_{p,t}$ with p∈$\{1,...,P\}$ and t∈$\{1,...,T\}$: binary parameter that is equal to 1 if patient p has his appointment scheduled for time slot t.

#### 5.3.2. Variables

- ${e}_{i,p,t}$ with p ∈$\{1,...,P\}$, t $\in \{1,...,T\}$ and i $\in \{1,...,Ndis{p}_{t}\}$: binary variable that is equal to 1 if patient p begins his appointment during time slot t assisted by nurse i.

#### 5.3.3. Mathematical Model

## 6. Results of the Case Study

#### 6.1. Summary of the Collected Data

#### 6.2. Results of the Deterministic Model

#### 6.3. Results of the Stochastic Model

## 7. Discussion, Conclusions and Framework for a Further Research Agenda

#### Framework for a Further Research Agenda

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AMPL | A Mathematical Programming Language |

CHUS | Hospital Clínico Universitario of Santiago de Compostela |

ODH | Onco-haematological Day Hospital |

IDIS | Healt Research Institute of Santiago de Compostela |

LGOM | Learning-based outpatient management |

ONCOMET | Translational Medical Oncology Group |

OpenSolver | Excel add-in for solving optimization models |

SCHA | Santiago de Compostela Health Area |

SERGAS | Galician Health Service |

## Appendix A. The Two-Stage Stochastic Model in Extensive Form

- param $P>0;$ #Number of patients assigned to the working day
- param $l\left\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\right\}$; #Duration of patients’ chemotherapy treatments
- param $T>0$; #Number of time intervals that constitute the working day
- param $Ndisp\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$; #Number of available nurses during each time interval
- param $q\ge 1$ integer; #Minutes consumed by delay of appointments, their duration and preparation of
- chemotherapy substances (scenario 1)
- param $u\ge 1$ integer; #Idem scenario 2
- param $v\ge 1$ integer; #Idem scenario 3
- param $s\ge 1$ integer; #Idem scenario 4
- param $K>0$; #Number of available chemotherapy chairs
- param $N>0$; #Number of nurses working during the day
- param M; #Number of treatments that a nurse can simultaneously supervise
- set E1; #Cancer type I patients
- set E2; #Cancer type II patients
- set E3; #Cancer type III patients

- var c$\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P,t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ binary; #Medical check-ups schedule
- var y$\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P,t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ binary; #Chemotherapy treatments schedule at scenario 1
- var f$\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P,t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ binary; #Idem at scenario 2
- var x$\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P,t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ binary; #Idem at scenario 3
- var z$\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P,t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ binary; #Idem at scenario 4

- 0.287*sum $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$((sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*y[p,t]))-(sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*c[p,t]))- 2.6)
- +0.272*sum $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$((sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*f[p,t]))-(sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*c[p,t]))-4.2)
- +0.364*sum $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$((sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*x[p,t]))-(sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*c[p,t]))-2.2)
- +0.078*sum $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$((sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*z[p,t]))-(sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$(t*c[p,t]))-1.8);

- subject to restriction0 $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..15\}$ c[p,t]=0; #Impossible to program medical check-ups before 09:20
- subject to restriction0b $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}61..T\}$ c[p,t]=0; #Impossible to program medical check-ups after 13:00
- subject to restriction0c $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}16..60\}$ c[p,t]=1; #Each patient has his medical check-up programmed
- subject to restriction1 $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ y[p,t]=1; #At scenario 1, each patient begins his treatment
- subject to restriction1b $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ f[p,t]=1; #At scenario 2, idem
- subject to restriction1c $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ x[p,t]=1; #At scenario 3, idem
- subject to restriction1d $\{p\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..P\}$:
- sum$\{t\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}1..T\}$ z[p,t]=1; #At scenario 4, idem
- subject to restriction2 {p in 1..P}:
- (sum{t in 1..T}(t*y[p,t]))-(sum{t in 1..T}(t*c[p,t]))-q-1≥0;
- #Each chemotherapy treatment begins after substances are ready and after the medical check-up is finished (scenario 1)
- subject to restriction2b {p in 1..P}:
- (sum{t in 1..T}(t*f[p,t]))-(sum{t in 1..T}(t*c[p,t]))-u-1≥0; #At scenario 2, idem
- subject to restriction2c {p in 1..P}:
- (sum{t in 1..T}(t*x[p,t]))-(sum{t in 1..T}(t*c[p,t]))-v-1≥0; #At scenario 3, idem
- subject to restriction2d {p in 1..P}:
- (sum{t in 1..T}(t*z[p,t]))-(sum{t in 1..T}(t*c[p,t]))-s-1≥0; #At scenario 4, idem
- subject to restriction3 {t in 1..T}:
- sum{p in 1..P, a in max(1,t-l[p]+1)..t} y[p,a]≤K; #At scenario 1, there aren’t more ongoing treatments than chairs
- subject to restriction3b {t in 1..T}:
- sum{p in 1..P, a in max(1,t-l[p]+1)..t} f[p,a]≤K; #At scenario 2, idem
- subject to restriction3c {t in 1..T}:
- sum{p in 1..P, a in max(1,t-l[p]+1)..t} x[p,a]≤K; #At scenario 3, idem
- subject to restriction3d {t in 1..T}:
- sum{p in 1..P, a in max(1,t-l[p]+1)..t} z[p,a]≤K; #At scenario 4, idem
- subject to restriction4 {t in 1..T}:
- sum{p in 1..P}((1-1/M)*y[p,t])≤Ndisp[t]; #At scenario 1, nurses can begin one treatment for time interval
- subject to restriction4b {t in 1..T}:
- sum{p in 1..P}((1-1/M)*f[p,t])≤Ndisp[t]; #At scenario 2, idem
- subject to restriction4c {t in 1..T}:
- sum{p in 1..P}((1-1/M)*x[p,t])≤Ndisp[t]; #At scenario 3, idem
- subject to restriction4d {t in 1..T}:
- sum{p in 1..P}((1-1/M)*z[p,t])≤Ndisp[t]; #At scenario 4, idem
- subject to restriction5 {t in 1..T}:
- sum{p in 1..P}((1/M)*(sum{a in max(1,t-l[p]+1)..t}y[p,a]))≤Ndisp[t];#At each time interval, it is not possible to have more than 16 ongoing treatments for each available nurse (scenario 1)
- subject to restriction5b {t in 1..T}:
- sum{p in 1..P}((1/M)*(sum{a in max(1,t-l[p]+1)..t}f[p,a]))≤Ndisp[t]; #At scenario 2, idem
- subject to restriction5c {t in 1..T}:
- sum{p in 1..P}((1/M)*(sum{a in max(1,t-l[p]+1)..t}x[p,a]))≤Ndisp[t]; #At scenario 3, idem
- subject to restriction5d {t in 1..T}:
- sum{p in 1..P}((1/M)*(sum{a in max(1,t-l[p]+1)..t}z[p,a]))≤Ndisp[t]; #At scenario 4, idem
- subject to restriction6 {t in 16..60}:
- sum{p in E1, a in t..t+2} c[p,t]≤3; #Cancer type I medical specialists can only handle 1 patient per 15 min
- subject to restriction6b {t in 16..60}:
- sum{p in E2, a in t..t+2} c[p,t]≤3; #Cancer type II idem
- subject to restriction6c {t in 16..60}:
- sum{p in E3, a in t..t+2} c[p,t]≤3; #Cancer type III idem
- #Following constraints ensure that every 15 min the number of starting treatments do not exceed nurses working capacity:
- subject to restriction7 {t in 1..24}:
- sum{p in 1..P, a in t..t+2} y[p,a]≤5;
- subject to restriction7b {t in 1..24}:
- sum{p in 1..P, a in t..t+2} f[p,a]≤5;
- subject to restriction7c {t in 1..24}:
- sum{p in 1..P, a in t..t+2} x[p,a]≤5;
- subject to restriction7d {t in 1..24}:
- sum{p in 1..P, a in t..t+2} z[p,a]≤5;
- subject to restriction8 {t in 25..84}:
- sum{p in 1..P, a in t..t+2} y[p,a]≤6;
- subject to restriction8b {t in 25..84}:
- sum{p in 1..P, a in t..t+2} f[p,a]≤6;
- subject to restriction8c {t in 25..84}:
- sum{p in 1..P, a in t..t+2} x[p,a]≤6;
- subject to restriction8d {t in 25..84}:
- sum{p in 1..P, a in t..t+2} z[p,a]≤6;
- subject to restriction9 {t in 85..108}:
- sum{p in 1..P, a in t..t+2} y[p,a]≤3;
- subject to restriction9b {t in 85..108}:
- sum{p in 1..P, a in t..t+2} f[p,a]≤3;
- subject to restriction9c {t in 85..108}:
- sum{p in 1..P, a in t..t+2} x[p,a]≤3;
- subject to restriction9d {t in 85..108}:
- sum{p in 1..P, a in t..t+2} z[p,a]≤3;
- subject to restriction10 {t in 109..165}:
- sum{p in 1..P, a in t..t+2} y[p,a]≤2;
- subject to restriction10b {t in 109..165}:
- sum{p in 1..P, a in t..t+2} f[p,a]≤2;
- subject to restriction10c {t in 109..165}:
- sum{p in 1..P, a in t..t+2} x[p,a]≤2;
- subject to restriction10d {t in 109..165}:
- sum{p in 1..P, a in t..t+2} z[p,a]≤2;

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**Figure 2.**Subsequent stages in the circuit of oncology patients after the theoretical instant of the appointment with the oncologist and preceding the moment when their treatment begins.

**Figure 3.**Illustration of the construction of 3 scenarios for patients by means of the k-means algorithm making use of 3 times involved in waiting times.

**Figure 4.**Boxplot and histogram of the oncological check-ups’ delays. Axis X: delay time in minutes. Axis Y: number of patients.

**Figure 5.**Boxplot and histogram of the oncological check-ups’ durations. Axis X: check-ups’ durations in minutes. Axis Y: number of patients.

**Figure 6.**Boxplot and histogram of the durations of substance preparation. Axis X: durations of substance preparation in minutes. Axis Y: number of patients.

**Figure 7.**Boxplot and histogram of the durations of the treatments. Axis X: durations of the treatments in minutes. Axis Y: number of patients.

**Figure 8.**Comparison of the number of oncology treatments in progress actually being performed and those proposed by the deterministic optimization model (120 min of margin) at each point in time on Monday, 11 January 2021. Axis X: time of the day between 8:00 am and 10:00 pm. Axis Y: number of treatments in progress.

**Figure 9.**Time elapsed from the opening of the centre to the start of treatment (blue colour) and treatment duration (red colour) on Monday 11 January 2021: actual data (top) and data proposed by the deterministic model with time margins of 120 min (bottom). Patients have been ordered from down to up from the longest to the shortest treatments. X-axis: time of day between 8:00 and 22:00 h. Y-axis: patient number.

**Figure 10.**Actual oncologist appointment time and stochastic model proposed appointment time for patients on Monday 11 January 2021. X-axis: time of day between 8:00 and 13:00 h. Y-axis: patient number.

$\mathit{Scenario}$ | $\mathit{Scenario}$ | $\mathit{Scenario}$ | $\mathit{Scenario}$ | |
---|---|---|---|---|

1 | 2 | 3 | 4 | |

Average length of oncology appointments | 2.6 | 4.2 | 2.2 | 1.8 |

% of patients in each scenario | 28.7 | 27.2 | 36.4 | 7.7 |

**Table 2.**Total waiting minutes over the different ready times considered provided by the deterministic model.

$\mathit{At}\phantom{\rule{4pt}{0ex}}\mathit{Present}$ | $\mathit{Model}\phantom{\rule{4pt}{0ex}}1$ | $\mathit{Model}\phantom{\rule{4pt}{0ex}}2$ | $\mathit{Model}\phantom{\rule{4pt}{0ex}}3$ | $\mathit{Model}\phantom{\rule{4pt}{0ex}}4$ | $\mathit{Patients}$ | |
---|---|---|---|---|---|---|

Monday | 7339 | 9255 | 7900 | 6250 | 4600 | 56 |

Tuesday | 10,382 | 13,375 | 11,675 | 9275 | 7115 | 72 |

Wednesday | 7351 | 9790 | 8190 | 6360 | 4480 | 61 |

Thursday | 5906 | 7790 | 6360 | 4860 | 3360 | 50 |

Friday | 5810 | 8075 | 6615 | 5085 | 3555 | 51 |

Total | 36,788 | 48,285 | 40,740 | 31,830 | 23,110 | 290 |

**Table 3.**Total waiting minutes obtained with the deterministic approach, using Model 1 for the ready times, with different durations of the considered time intervals.

Duration of the Time Intervals: | 5 min | 10 min | 15 min |
---|---|---|---|

Monday | 9255 | 9770 | 10,815 |

Tuesday | 13,375 | 14,120 | 15,420 |

Wednesday | 9790 | 10,320 | 10,890 |

Thursday | 7790 | 8180 | 8250 |

Friday | 8075 | 8470 | 9285 |

Total | 48,285 | 50,860 | 54,660 |

**Table 4.**Treatment duration, actual start time and start time proposed by the deterministic model with 120 min of margin, expressed in number of 5-min periods, for the 56 patients on Monday 11 January 2021.

$\mathit{Patient}$ $\mathit{Number}$ | $\mathit{Treatment}$ $\mathit{Duration}$ | $\mathit{Real}$ $\mathit{Time}$ | $\mathit{Proposed}$ $\mathit{Time}$ | $\mathit{Patient}$ $\mathit{Number}$ | $\mathit{Treatment}$ $\mathit{Duration}$ | $\mathit{Real}$ $\mathit{Time}$ | $\mathit{Proposed}$ $\mathit{Time}$ |
---|---|---|---|---|---|---|---|

1 | 29 | 72 | 61 | 29 | 54 | 60 | 50 |

2 | 30 | 65 | 46 | 30 | 44 | 46 | 47 |

3 | 19 | 55 | 49 | 31 | 82 | 65 | 61 |

4 | 23 | 77 | 81 | 32 | 27 | 59 | 57 |

5 | 25 | 56 | 41 | 33 | 18 | 54 | 45 |

6 | 27 | 50 | 57 | 34 | 24 | 66 | 50 |

7 | 37 | 50 | 53 | 35 | 21 | 67 | 45 |

8 | 24 | 46 | 57 | 36 | 45 | 74 | 53 |

9 | 22 | 57 | 57 | 37 | 27 | 84 | 61 |

10 | 30 | 60 | 49 | 38 | 13 | 57 | 41 |

11 | 65 | 43 | 76 | 39 | 39 | 45 | 44 |

12 | 59 | 49 | 72 | 40 | 46 | 42 | 50 |

13 | 82 | 50 | 52 | 41 | 28 | 80 | 49 |

14 | 69 | 51 | 74 | 42 | 33 | 91 | 81 |

15 | 12 | 66 | 53 | 43 | 12 | 80 | 65 |

16 | 9 | 80 | 69 | 44 | 40 | 41 | 44 |

17 | 12 | 53 | 57 | 45 | 49 | 39 | 47 |

18 | 12 | 65 | 43 | 46 | 44 | 60 | 66 |

19 | 40 | 55 | 41 | 47 | 30 | 90 | 67 |

20 | 8 | 82 | 69 | 48 | 84 | 42 | 52 |

21 | 21 | 53 | 42 | 49 | 39 | 81 | 85 |

22 | 35 | 63 | 55 | 50 | 36 | 93 | 69 |

23 | 18 | 65 | 61 | 51 | 30 | 84 | 58 |

24 | 18 | 63 | 65 | 52 | 22 | 68 | 65 |

25 | 46 | 38 | 44 | 53 | 19 | 72 | 53 |

26 | 43 | 38 | 39 | 54 | 26 | 70 | 41 |

27 | 45 | 52 | 47 | 55 | 24 | 84 | 61 |

28 | 8 | 80 | 65 | 56 | 61 | 56 | 73 |

$\mathit{Scenario}\phantom{\rule{4pt}{0ex}}1$ | $\mathit{Scenario}\phantom{\rule{4pt}{0ex}}2$ | $\mathit{Scenario}\phantom{\rule{4pt}{0ex}}3$ | $\mathit{Scenario}\phantom{\rule{4pt}{0ex}}4$ | |
---|---|---|---|---|

Check-ups’ delays | 49 | 14 | 15 | 41 |

Oncological check-ups | 13 | 21 | 11 | 9 |

Substance preparation | 63 | 42 | 90 | 147 |

Treatments | 125 | 77 | 116 | 197 |

Patients | 59 | 56 | 75 | 16 |

% Patients | 28.6% | 27.2% | 36.4% | 7.8% |

$\mathit{At}\phantom{\rule{4pt}{0ex}}\mathit{Present}$ | $\mathit{Stochastic}\phantom{\rule{4pt}{0ex}}\mathit{Model}$ | $\mathit{Improvement}\phantom{\rule{4pt}{0ex}}(\%)$ | $\mathit{Patients}$ | |
---|---|---|---|---|

Monday | 7339 | 6029 | 17.74 | 56 |

Tuesday | 10,382 | 7544 | 27.3 | 72 |

Wednesday | 7351 | 6392 | 13.0 | 61 |

Thursday | 5906 | 5239 | 11.3 | 50 |

Friday | 5810 | 5344 | 8.0 | 51 |

Total | 36,788 | 30,548 | 17.0 | 290 |

**Table 7.**Mean value of waiting minutes on Monday, 11 January 2021, waiting time in the randomized experiment realizations, probabilities of each scenario, actual waiting time and computation time (s).

$\mathit{At}$ $\mathit{Present}$ | $\mathit{Scenario}$ 1 | $\mathit{Scenario}$ 2 | $\mathit{Scenario}$ 3 | $\mathit{Scenario}$ 4 | $\mathit{Average}$ $\mathit{Result}$ | $\mathit{Computing}$ $\mathit{Time}$ | |
---|---|---|---|---|---|---|---|

7339 | 6550 | 3585 | 6385 | 10,975 | 6029 | 215 | |

Probabilities | 28.6 | 27.2 | 36.4 | 7.8 |

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**MDPI and ACS Style**

González-Maestro, A.; Brozos-Vázquez, E.; Casas-Méndez, B.; López-López, R.; López-Rodríguez, R.; Reyes-Santias, F.
Effective Optimisation of the Patient Circuits of an Oncology Day Hospital: Mathematical Programming Models and Case Study. *Mathematics* **2022**, *10*, 62.
https://doi.org/10.3390/math10010062

**AMA Style**

González-Maestro A, Brozos-Vázquez E, Casas-Méndez B, López-López R, López-Rodríguez R, Reyes-Santias F.
Effective Optimisation of the Patient Circuits of an Oncology Day Hospital: Mathematical Programming Models and Case Study. *Mathematics*. 2022; 10(1):62.
https://doi.org/10.3390/math10010062

**Chicago/Turabian Style**

González-Maestro, Adrián, Elena Brozos-Vázquez, Balbina Casas-Méndez, Rafael López-López, Rosa López-Rodríguez, and Francisco Reyes-Santias.
2022. "Effective Optimisation of the Patient Circuits of an Oncology Day Hospital: Mathematical Programming Models and Case Study" *Mathematics* 10, no. 1: 62.
https://doi.org/10.3390/math10010062