Developing a Monte Carlo Simulation-Based Decision Support Tool for Patient Sequencing Optimisation in Health Services: A Methodological Study

Abstract

This study presents the design, development and validation of a Monte Carlo simulation-based decision support tool for the patient sequencing problem in health services. The tool was designed on the basis of the analytic profile of a systematic dataset of 84 articles retrieved from the Web of Science database and coded on 13 dimensions. Built on an event-driven queue architecture, the model integrates a multi-server structure, stochastic service times with lognormal and gamma distributions, patient-level no-show and cancellation probabilities, walk-in arrivals, right-skewed timing deviation and a threshold-based cost function. The tool evaluates six sequencing rules in parallel and presents performance through a Pareto comparison. Matured through four iterative development cycles with expert critique, it is positioned as an experimental simulation platform that provides health managers with evidence at the level of policy comparison.

Keywords: patient sequencing, appointment scheduling, Monte Carlo simulation, decision support system, health management, stochastic optimisation, event-driven queue

1. Introduction

The patient sequencing problem in health services is a core operational decision area that covers determining in which order and when patients will be served. Behind its apparent simplicity lies deep mathematical and managerial complexity: service times are not known exactly in advance, patients may not attend their appointments, emergency cases may disrupt the planned flow, and multiple performance criteria (waiting time, cost, capacity utilisation, patient satisfaction) must be balanced simultaneously (Denton & Gupta, 2003; Zacharias & Pinedo, 2014).

The operations research literature offers a rich methodological body for this problem. Integer programming (Jiang et al., 2017), stochastic optimisation (Mak et al., 2015), distributionally robust optimisation (Bertsimas et al., 2019), queueing theory (Hassin & Mendel, 2008) and heuristic algorithms (Zacharias & Pinedo, 2014) are the main approaches. These sophisticated models, however, are not adequately transferred to health management practice because of two critical gaps.

The first is a lack of empirical validation: most studies use analytic or model-based data, and validation with real hospital data remains very limited. The second is an accessibility gap: the mathematical models developed are not turned into tools that health managers can use directly.

The aim of this study is to develop an interactive, Monte Carlo simulation-based decision support tool for patient sequencing and to document systematically its methodological foundations, iterative development and limitations. Its original contribution is threefold: (a) the 13-dimension profile of a systematic 84-article dataset directly informed the tool’s module structure; (b) the engine, built on an event-driven queue architecture, enables policy comparison under uncertainty; and (c) a design research process matured through four iterative cycles of expert critique was followed.

2. Conceptual Framework

2.1. The Sequencing Problem in Health Services

Sequencing determines the order in which patients are served, whereas scheduling covers planning exactly when each patient will be served. The two decision levels interact: when the sequence changes, the optimal schedule also changes (Denton & Gupta, 2003). The main factors that make the problem difficult in health services are service time variability, no-show uncertainty, emergency arrivals and multidimensional performance criteria.

Begen and Queyranne (2011) formulated appointment scheduling mathematically under discrete random service times. Kong et al. (2013) optimised arrival sequencing in stochastic service delivery systems using copositive cones. Mak et al. (2015) showed that effective schedules can be built even when distributional information is limited. These studies show the mathematical maturity of the field, but the clinical applicability of their solutions has largely not been tested.

2.2. Decision Support Systems and Simulation

Monte Carlo simulation produces outcome distributions by repeatedly realising uncertain parameters through random sampling. By presenting the probability distribution of outcomes rather than a single optimal solution, it offers a more informative decision frame for health managers: a manager can assess risk by seeing not only the average cost but also the cost in the worst 10% of scenarios (P90) (Cayirli & Veral, 2003).

2.3. Research Gap

Three main gaps emerge from the literature: (a) most existing models perform single-objective optimisation, (b) the rate of uncertainty modelling is low, and (c) the models developed are not turned into interactive tools. This study aims to develop an integrated decision support tool that addresses all three gaps together.

3. Methodology

3.1. Research Design

The study adopts the design science research paradigm proposed by Hevner et al. (2004), which requires designing, developing, evaluating and iteratively improving an innovative artefact for a practical problem. Four iterative development cycles were followed, with systematic corrections in each cycle in response to expert critique.

3.2. Literature Search and Coding

To guide the design, 232 articles addressing the sequencing problem in health services in an operations management context were retrieved from Web of Science and coded on 13 dimensions: methodology, performance metric, decision variable, sector, uncertainty, patient behaviour, time horizon, data type, multi-objectivity, capacity structure, solution quality, technology integration and theoretical framework. The 84 articles with sequencing as a decision variable were filtered, and the tool’s module structure was derived from the analytic profile of this subset.

3.3. Design of the Simulation Model

The model is built on an event-driven queue architecture that merges scheduled patients, walk-in patients and overbooked patients into a single chronological queue by arrival time. In each Monte Carlo replication four stochastic processes are realised simultaneously: (a) non-attendance is drawn from a Bernoulli distribution and classified as 30% cancellation and 70% no-show while preserving the total rate; (b) service times are sampled from a lognormal or gamma distribution; (c) timing deviation is modelled with a right-skewed distribution; and (d) walk-in patients are generated by a Poisson process.

The cost function is threshold-based: waiting cost has three tiers (c_w×1 between 0 and θ₁, c_w×1.5 between θ₁ and θ₂, c_w×3 above θ₂), and overtime cost has two tiers (c_o×1 between 0 and θ₃, c_o×2 above θ₃). All thresholds are user-adjustable parameters.

3.4. Iterative Development Process

The first version was a deterministic single-server prototype. The second added Monte Carlo simulation, the lognormal distribution, threshold costs and Pareto comparison. The third added the multi-server structure, walk-ins, cancellations and dynamic overbooking; expert review, however, identified critical problems such as a non-chronological walk-in queue, overlap between cancellation and no-show probabilities, the absence of direction in the hybrid score and main-thread blocking. The fourth version corrected all of them.

4. Architecture of the Tool

The tool consists of six core modules: data entry and scenario definition, a sequencing rules engine, a stochastic modelling engine, a threshold cost function, multi-metric performance evaluation, and dynamic overbooking and scenario analysis. At system level, the number of servers (s), working time (T), slot length (d), cost coefficients, thresholds, distribution type, walk-in rate and number of Monte Carlo replications are defined. At patient level, expected service time, variance, non-attendance probability and patient type are entered.

Six sequencing rules are evaluated simultaneously: FCFS (first come, first served), SPT (shortest processing time first), LPT (longest processing time first), SVF (smallest variance first), NSR (lowest no-show risk first) and HYB (a user-defined weighted hybrid score). In the HYB rule, the direction (minimise or maximise) can be selected separately for each dimension.

For each rule, five metrics are computed from the Monte Carlo distribution: mean waiting time, P90 waiting time, overtime, cost and waiting fairness. Results are presented as a Pareto comparison with min-max normalisation. The source code and interactive demo link are given in Appendix 1.

5. Application and Validation

5.1. Example Scenario

To illustrate the tool, an outpatient clinic scenario with 12 patients was defined. Expected service times range from 15 to 45 minutes, service time variances from 9 to 144 min² and non-attendance probabilities from 5% to 30%. The patient mix consists of four new referrals, four chronic follow-ups and four control visits. Clinic parameters were: total working time T = 480 minutes (8 hours), slot length d = 30 minutes, one server (s = 1), waiting cost c_w = 1, idle cost c_i = 2, overtime cost c_o = 3, and thresholds θ₁ = 15, θ₂ = 30, θ₃ = 15 minutes. The lognormal distribution was used for service times, with 1,000 Monte Carlo replications per rule.

5.2. Comparison of Sequencing Rules

Table 1. Monte Carlo performance comparison of sequencing rules (n = 12, s = 1, 1,000 replications)

Note. The sixth rule, the user-defined hybrid rule (HYB), is not shown because its results depend on the weights the user chooses.

The results reveal clear performance profiles. The SVF rule produced the lowest cost and the fairest waiting distribution, with a mean cost of 204 units and waiting fairness (standard deviation) of 6.0 minutes. SPT performed very close to SVF with a mean waiting time of 3.7 minutes. FCFS delivered intermediate performance, while LPT performed worst with a mean wait of 13.2 minutes and a cost of 322 units. Although NSR sequences by no-show risk, in this scenario it produced waiting times close to, but slightly higher than, SPT.

These findings confirm that no rule is best on all metrics at once. SVF is superior on cost and fairness, while SPT is marginally better on mean waiting time. This Pareto structure shows that health managers may prefer different rules depending on which metric they prioritise.

5.3. Effect of the Multi-Server Structure

When the number of servers was increased to s = 2, waiting times fell dramatically: mean waiting time dropped to 0.1–0.2 minutes for all rules. This improvement was offset by higher idle cost, however; total cost rose to 720–766 units in the two-server scenario. Adding a server thus reduces waiting almost to zero while increasing total cost roughly 3.5-fold, making the capacity-cost trade-off concrete.

5.4. Dynamic Overbooking Analysis

Table 2. Effect of the overbooking level on system performance (SPT rule, s = 1, 1,000 MC replications)

Adding one or two overbooked patients increases cost by only 3–11% while the number of attending patients rises from 10.2 to 12.1. From +3 onwards, overtime begins to rise markedly (7.4 minutes) and reaches 49.4 minutes at +5. For this scenario, +2 overbooked patients is therefore the optimal balance point: throughput increases while cost stays acceptable.

5.5. Sensitivity to the Number of Patients

Table 3. Sensitivity analysis by number of patients (SPT rule, s = 1, 1,000 MC replications)

Cost increases linearly from n = 6 to n = 14 (97 → 252) but jumps to 428 units at n = 16. The reason is that at n = 16 total expected service time (16 × 30 = 480 minutes) reaches the clinic’s working time (T = 480 minutes); from this point, stochastic service time variability starts to trigger overtime (mean 26.6 minutes). P90 cost reaches 732 units, 1.7 times the mean. As capacity limits are approached the risk profile deteriorates dramatically, so health managers should consider not only average performance but also tail performance in capacity planning.

5.6. Model Validation

Structural validation was carried out at three levels. First, in deterministic mode (with stochastic features switched off), it was confirmed that the SPT rule minimises mean flow time, consistent with Smith’s rule. Second, when the no-show rate and variance were set to zero, Monte Carlo results coincided with deterministic results. Third, as the number of servers increased, waiting time decreased monotonically and idle time increased, the basic prediction of queueing theory. Expert evaluations across the four iterative cycles made it possible to detect and correct structural problems such as the walk-in chronological queue error and the probability overlap.

6. Discussion

The tool connects with the existing literature in several ways. First, it extends the basic cost function of Denton and Gupta (2003) to a threshold structure, reflecting the non-linear nature of waiting cost in health services. Second, it meets the limited distributional information requirement of Mak et al. (2015) by working only with mean and variance parameters. Third, it carries Zacharias and Pinedo’s (2014) analysis of the interaction between no-shows and overbooking into a dynamic, simulation-based platform.

The problems that emerged during iterative development (a non-chronological walk-in queue, probability overlap, a missing direction distinction) show that simulation-based decision support tools require systematic validation. The walk-in queue error in particular revealed that calculation logic contrary to queueing theory can produce visually plausible outputs, making the difference between formal and structural correctness concrete.

For health management practice, the tool’s main contribution is evidence at the level of policy comparison. An outpatient clinic manager can enter local parameters and compare the effects of different sequencing rules, not only on averages but together with risk measures such as P90. The Pareto presentation shows that no rule is best on every dimension and guides the manager towards a conscious trade-off decision.

7. Conclusion and Recommendations

This study presented the design, development and maturation over four iterative cycles of a Monte Carlo simulation-based decision support tool for patient sequencing. The tool integrates an event-driven queue architecture, a multi-server structure, stochastic service times, patient-level no-show and cancellation modelling, walk-in integration, a threshold cost function and Pareto-based performance evaluation. Future work should validate the tool empirically with real hospital data, incorporate multi-room resource structures and staff heterogeneity, add a preemptive priority mechanism and improve computational performance through Web Worker integration.

8. Limitations

The study has five main limitations. First, the model has no preemptive priority mechanism and supports only non-preemptive queues. Second, multi-room resource structures and staff heterogeneity are not modelled. Third, cost coefficients are user-defined rather than empirically calibrated. Fourth, dynamic rescheduling after a cancellation or no-show is not implemented. Fifth, the Monte Carlo computation runs on the main thread, so interface performance may decline in very large simulations. Because of these limitations, the tool should be positioned not as a final solution but as an experimental simulation tool serving policy comparison.

Appendix 1: Source Code and Interactive Demo

The source code and interactive demo of the tool: https://claude.ai/public/artifacts/e495dabf-8dce-439e-85d0-73ab77d9c661

How to cite

This article is the English version of the technical report: Kurutkan, M. N. (2026). Sağlık hizmetlerinde hasta sıralama optimizasyonu için Monte Carlo simülasyon tabanlı karar destek aracı geliştirme: Metodolojik bir çalışma (Technical report). Düzce University, Department of Health Management. https://www.researchgate.net/publication/403513569

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