Scheduling patient appointment in an infusion center: a mixed integer robust optimization approach

Abstract

Infusion centers are experiencing greater demand, resulting in long patient wait times. The duration of chemotherapy treatment sessions often varies, and this uncertainty also contributes to longer patient wait times and to staff overtime, if not managed properly. The impact of such long wait times can be significant for cancer patients due to their physical and emotional vulnerability. In this paper, a mixed integer programming infusion appointment scheduling (IAS) mathematical model is developed based on patient appointment data, obtained from a cancer center of an academic hospital in Central Virginia. This model minimizes the weighted sum of the total wait times of patients, the makespan and the number of beds used through the planning horizon. A mixed integer programming robust slack allocation (RSA) mathematical model is designed to find the optimal patient appointment schedules, considering the fact that infusion time of patients may take longer than expected. Since the models can only handle a small number of patients, a robust scheduling heuristic (RSH) is developed based on the adaptive large neighborhood search (ALNS) to find patient appointments of real size infusion centers. Computational experiments based on real data show the effectiveness of the scheduling models compared to the original scheduling system of the infusion center. Also, both robust approaches (RSA and RSH) are able to find more reliable schedules than their deterministic counterparts when infusion time of patients takes longer than the scheduled infusion time.

Appendices

Appendix A: The RSA model

In this section, we define the RSA model for a general case, to obtain the robust solution of an uncertain optimization problem. At the end of the section, it is explained how to use the RSA structure to elicit the robust solutions of the uncertain IAS model. Sets, parameters and variables of the RSA model are as follows:

Sets:

• O Set of all constraints

• P Set of all variables

• $U=\{ k:L_{k}^{*} \geq 0\}$ Set of uncertain constraints

Parameters:

• C The objective cost

• A The constraint matrix

• B The left hand side matrix

• M A big number

• ρ The average number of the uncertain constraints that would need slacks

• ϕ The minimum slack level of uncertain constraints

• ψ The maximum slack level of uncertain constraints

• Z^∗ The deterministic optimal cost

• δ The percentage of cost increase, compared to the deterministic optimal cost

Variables:

• w_l A positive variable

• $w_{l}^{*}$ The optimal solution of the deterministic model

• s_k A positive variable that shows the slack level of constraint k

• [–] σ_k A binary variable that takes 1, if s_k > 0; otherwise 0

• θ_k A positive variable that shows the total difference between the slack level and the maximum CV of constraint k

• $L_{k}^{*}$ The maximum CV of the uncertain constraint k

The basic idea of the RSA model, as a mathematical model for dealing with such uncertainty, is borrowed from the LR approach [12], where at first the maximum CV of each uncertain constraint is calculated, a slack variable is added to each uncertain constraint, then the difference between the slack variables and the maximum CV of the uncertain constraints are minimized. Consider the general optimization model below:

$$ \min \sum\limits_{l \in P}\ c_{l} w_{l} $$

(20)

$$ \sum\limits_{l \in P}\ a_{kl} w_{l} \leq b_{k}, \forall k \in O \ $$

(21)

$$ w_{l} \geq 0, \forall l \in P \ $$

(22)

Where c_l, a_kl, and b_k are the cost, the right-hand side, and the left-hand side parameters, respectively, and w_l is a positive variable. Constrain matrix A is an uncertain parameter that can take values like $\Tilde {a}_{kl} \in [a_{kl},a_{kl}+\hat {a}_{kl}]$, which might be different than its nominal level (a_kl). $w_{l}^{*}$ is the optimal solution of the general deterministic model (20) -(22). $L_{k}^{*}$ can be defined as the maximum CV of constraint k considering $w_{l}^{*}$ and a realization as the upper level of the uncertain parameter ($a_{kl}+\hat {a}_{kl}$) [12]:

$$ L_{k}^{*} = \sum\limits_{l}\ (a_{kl}+\hat{a}_{kl}) w_{l}^{*} - b_{k} , \forall k \in O \ $$

(23)

Set U is defined as the set of constraints with $L_{k}^{*} \geq 0$:

$$ U=\{ k:L_{k}^{*} \geq0\} \ $$

(24)

By adding slacks to constraints ∈ U, they would be protected from becoming infeasible if the value of the uncertain parameter changes. The higher the slack, the less possible infeasibilities. On the other hand, by adding slacks, the problem will be solved with higher costs than the optimal solution. The optimal level of slacks should be found to have a good balance between uncertainty protection and the additional objective costs.

In the RSA, a slack allocation problem is solved by minimizing the difference between the slack level of uncertain constraints with their $L_{k}^{*}$, to guarantee lower CV. The number of constraints that need slacks in average (parameter ρ), and the minimum (parameter ϕ) and maximum (parameter ψ) slack level for each uncertain constraint can be estimated based on the data. It is important to consider a minimum level of protection for each uncertain constraint to avoid high CVs, and the average number of constraints with slacks and the maximum slack level is helpful in controlling the costs. Parameter δ, which determines the percentage of cost increase, compared to the deterministic optimal cost, should be chosen based on the preference of the people that are planning the schedule.

After finding $L_{k}^{*}$ of each uncertain constraint of the model, estimating ρ, ϕ and ψ parameters and deciding about δ, the general RSA can be formulated as follows:

$$ \min \sum\limits_{k \in U}\ \theta_{k} $$

(25)

$$ \sum\limits_{l}\ a_{kl} w_{l} +s_{k}= b_{k}, \forall k \in U \ $$

(26)

$$ \sum\limits_{k}\ \sigma_{k} \leq \rho \ $$

(27)

$$ \ s_{k} \geq \phi \sigma_{k}, \forall k \in U \ $$

(28)

$$ \ s_{k} \leq \psi \sigma_{k}, \forall k \in U \ $$

(29)

$$ \ s_{k} + \theta_{k} \geq L_{k}^{*} - M(1-\sigma_{k}), \forall k \in U \ $$

(30)

$$ \sum\limits_{l \in P}\ c_{l}w_{l} \leq (1+\delta)Z^{*} \ $$

(31)

$$ w_{l} \geq 0, \forall l \in P \ $$

(32)

$$ \sigma_{k} \in{\{0,1\}}, \theta_{k} \geq 0, s_{k} \geq 0, \forall k \in U\ $$

(33)

The objective of the above model (25) is to minimize the total difference between the slack level and the maximum CV of each uncertain constraint, which is calculated by Eq. 30. The slack variables are added to the model through Eqs. 26. Equation 27 limits the number of active slack variables of the model to ρ. Equations 28 and 29 limit the slack level to [ϕ, ψ], if s_k of the uncertain constraint is not 0. Equation (31) limits the total cost increase of the problem compared to the deterministic optimal solution, and Eqs. 32 and 33 are the variable definition.

Appendix B: The CW initialization & the ALNS algorithms

The initial schedule is built using the “CW initialization procedure” algorithm, shown here. The insertion-based initial solution algorithm first places a number of patients into the schedule randomly, then finds the insertion cost (objective function value) of all the remaining patients to the feasible appointments, and finally inserts patients into the schedule with the least cost.

The structure of the ALNS with the simulated annealing (SA) is shown in the “ALNS with SA” algorithm. The SA is used as the local search framework of the ALNS in this paper. In the ALNS with SA, X_best, X_current and X_new are the best solution (schedules) found, the solution at the start of the iteration and the solution at the end of the iteration respectively. Also, LP shows the list of patients removed by the selected removal operator ($r_{k}^{*}$) that will be inserted again later using the selected insertion operator ($i_{t}^{*}$) in each iteration, and x_partial shows the partial solution (schedule). The cost of the solution is measured through a cost function denoted by C(X). X_new is the temporary solution found at the end of the iteration that would be accepted if C(X_new) < C(X_current). Even if C(X_new) > C(X_current), X_new is accepted with the probability of $e^{-(C(X_{new} )-C(X_{current} ))/T}$ to prevent stocking in a local optimum, where T is the initial temperature defined within SA. After each iteration, T is gradually decreased by multiplying by the cooling rate, h. The initial probability of using removals and insertion operators is equal at the start and is calculated using the following:

$$ {P_{k}^{r}}=1/ \| R \|, \forall k \in R, \text{and}, {P_{t}^{i}}=1/ \| I \|, \forall t \in I $$

(34)

Where ∥R∥ and ∥I∥ are the number of removal and insertion operators; then a score is assigned to each operator based on its performance through the roulette wheel procedure. The operators are selected dynamically after a certain number of iterations [9].

The ALNS parameter setting is mostly done according to Demir et al. [9] as shown in Table 8.

Table 8 The ALNS parameters

In Table 8, N_i shows the total number of iterations and T and h are used in the SA as a part of the solution acceptance criteria. N_w, r_p, and σ₁, σ₂ and σ₃ are used to update the scores of removal and insertion operators in the roulette wheel procedure, and $\underline {s}$ and s̄ are the minimum and maximum number of removable patients by removal operators in the iterations.

In this paper, we apply six removal and four insertion operators, shown in Table 9. Some of these operators are extracted from the ALNS literature and revised according to our scheduling framework, and the rest are produced heuristically.

Table 9 The removal and insertion operators