4.1. Basic Model
Following a thorough examination of operational processes at 27 hospitals in Hong Kong and Mainland China, the most crucial factors were identified and distilled to represent each hospital’s situation accurately and streamline the complex scenario. The model is structured as follows: when considering K types of healthcare services in an ambulatory care center or K departments, the planned time frame consists of a single session with consecutive time slots divided into equal intervals based on center requirements. The appointment-scheduling challenge revolves around two critical decisions:
We introduce the following variables and indices to develop the model.
Indexes:
i: Index of starting time points i, where i = 0, 1, …, T – 1;
j: Index of ending time point j, where j= 1, 2, …, T;
T: Total points within a scheduling system in the context of healthcare operations;
k: Index of healthcare department k, where k = 1, 2, …, K;
K: Total number of numbers of distinct healthcare departments;
: Capacity of health care department ;
: Revenue from an appointment generated from time i to j within department k;
: Number of accepted appointments from time i to j within department k;
: The number of uncertain demand of appointments from time i to j within department k;
: The probability of showing up for the accepted appointment from time i to j within department k;
: Number of overbooked appointments at time within department k.
Initially, at time 0, it is assumed that there are no appointments, while all scheduled appointments are expected to be completed by the end time T. To manage overbooking effectively when starting with no appointments and needing to complete all by time , we should initially schedule more appointments than the number of available slots, based on anticipated no-shows and cancellations. This requires the use of historical data to estimate the right amount of overbooking and adjusting schedules dynamically to handle any deviations. Clear patient communication and contingency plans for managing unexpected delays are crucial to ensuring that care is delivered efficiently and on time.
In practice, the hospital operates from 9 a.m. to 6 p.m., denoted as time 0 to T in this study. Another assumption is that each accepted appointment spans at least one time slot, such as from time 1 to time 2. The network can be represented by nodes that signify the beginning and end times of each appointment, as depicted in
Figure 1.
For a particular time slot t and t ϵ {1, 2, …, T − 1} in our planning horizon, the formula below indicates the utilization rate of department k, k ϵ {1, 2, …, K} at time t, which embraces the existence of no-shows:
For all
, the capacity of department that has been utilized is represented in the first term. The capacity which was occupied previously and is vacant again is shown in the second term. Finally, for the latest accepted appointments which will continue, at least one should be represented by the third term. Due to the problem of no-shows after the appointments are made in healthcare, the overbooking policy is introduced to manage the appointment scheduling. For each specialty department k with constraints on their capacities, decision makers in hospitals may choose to invite more patients and accept more appointments than the available appointment session to mitigate the effect of no-shows. Therefore, the constraints will be modified as follows:
The revenue produced by fulfilling the consultation sessions is shown below:
The equations above seem to outline a linear programming problem (LPP). However, the parameters and are uncertain within the initial time frame. Since complete elimination of uncertainty is challenging in this complex scenario, our approach involves acknowledging and integrating it into our model. To address uncertain parameters, we employ linear interval programming based on interval analysis. This method is designed to manage no-shows for appointments by incorporating overbooking.
4.3. The Proposed Approach
With intervals involved given in Alefeld and Herzberger [
43], the linear programming problem (ILP) will be converted:
To maximize the return under uncertainty constraints, which are represented as interval numbers, the objective of the interval linear programming is changed from the ordinary one. Interval linear programming (ILP) is a mathematical optimization technique that addresses decision-making under uncertainty, where parameters are described by intervals rather than precise values. This approach is particularly useful in situations where data are uncertain, imprecise, or subject to variability. The objective is to maximize or minimize a linear combination of decision variables , where each variable is multiplied by an interval coefficient .
In simple terms, the expected value of any uncertain variable typically lies at the midpoint of its interval. When presenting uncertain returns within an interval, the lower bound represents the pessimistic return, while the upper bound signifies the optimistic return. The ILP discussed earlier can be likened to a hybrid model combining elements of expected value modeling with uncertainty and a pessimistic decision-making approach. The objective function to maximize can be regarded as a variant version of max–min problem. The constraint requirements specify the fact that the feasible solutions to ILP indicate the cost factors in the worst-case scenario. The average costs are less than or equal to the maximum possible value and the expected average value of the resources with uncertainties, respectively.
Definition 2 (Lai et al. [
42])
. Let be defined as a feasible solution to a non-inferior solution to ILP if, and only if, no other feasible solution is present such that Based on the Definition 2, the non-inferior solution to ILP can be derived in the following bi-objective programming:
To tackle multiple-objective decision making (MODM) programming, a linear combination of objective functions is suggested by Chankong and Haimes [
44]. Hence, the objective function of BIL can be rewritten as follows:
Eventually, the ILP solution can be generated from the parametric linear programming (PLP) model as follows:
From (12),
indicates the risk appetite for decision makers. The aforementioned equations have some strength for calculation. The non-inferior solution is comparatively easy to be calculated by the suggested interval linear programming (λ) corresponding to various values of
. The distinguished characteristic of the suggested ILP (λ) is that the model has already been in linear programming form; hence, it is able to be solved efficiently via simple different linear model packages [
45], while
can be determined.