A two-stage robust optimization approach for the master surgical schedule problem under uncertainty considering downstream resources

Abstract

This paper addresses a planning decision for operating rooms (ORs) that aim at supporting hospital management. Focusing on elective patients, we determined the master surgical schedule (MSS) on a one-week time horizon. We assigned the specialties to available sessions and allocated surgeries to them while taking into consideration the priorities of the outpatients in the ambulatory surgical discipline. Surgeries were selected from the waiting lists according to their priorities. The proposed approach considered operating theater (OT) restrictions, patients’ priorities and accounted for the availability of both intensive care unit (ICU) beds and post-surgery beds. Since the management decisions of hospitals are usually made in an uncertain environment, our approach considered the uncertainty of surgery duration and availability of ICU bed. Two robust optimization approaches that kept the model computationally tractable are described and applied to deal with uncertainty. Computational results based on a medium-sized French hospital archives have been presented to compare the robust models to the deterministic counterpart and to demonstrate the price of robustness.

Appendix: Overview of the robust optimization approach by [18]

We consider the linear programming problem:

$$ \max \sum\limits_{j=1}^{n} c_{j}x_{j} $$

(57)

subject to:

$$ \sum\limits_{j=1}^{n} a_{ij}x_{j}\leq b_{i} \qquad \forall i $$

(58)

$$ x_{j} \geq 0 \qquad \forall j $$

(59)

The variable x_j, the cost c_j, indexes and the dual variables defined in this section are independent.

We consider $J^{\prime }_{i}$ the set of coefficient a_ij that are subject to uncertainty. The coefficients a_ij belongs to a symmetric range $[\overline {a}_{ij}-\widehat {a}_{ij},\overline {a}_{ij}+\widehat {a}_{ij}]$, where $\overline {a}_{ij}$ is the nominal value and $\widehat {a}_{ij}\geq 0$ its deviation. A random variable of deviation can be introduced as $\tau _{ij}=(a_{ij}-\overline {a}_{ij})/\widehat {a}_{ij}$, τ_ij takes values in [− 1, 1]. The decisions variables are assumed to be nonnegative. Hence, the worst case will be achieved at the right-hand side of the range $[\overline {a}_{ij}-\widehat {a}_{ij},\overline {a}_{ij}+\widehat {a}_{ij}]$. Therefore, the random variable τ is assumed to be positive (0 ≤ τ_i ≤ 1). The confidence range $[\overline {a}_{ij}-\widehat {a}_{ij},\overline {a}_{ij}+\widehat {a}_{ij}]$ does not have to be symmetric. For every i a parameter ∇_i is introduced. This parameter is the sum of the total deviations of the nominal values of the uncertain coefficients in the same constraint i. This parameter is called “the budget of robustness”; its role is to protect the constraint against uncertainty and to adjust the robustness of the method against the degree of conservatism. This parameter is not necessarily an integer but takes values in $[0,card(J^{\prime }_{i})]$; if the parameter is zero no deviation is allowed on the coefficients of the i^th constraint and this case is the deterministic case, whereas, if ∇_i is equal to $card(J^{\prime }_{i})$ all the parameters are likely to deviate and this case is the one considered in [93].

We consider the following nonlinear formulation of constraint Eq. 58:

$$ \sum\limits_{j=1}^{n} \overline{a}_{ij} x_{ij}+\beta_{i}(x^{*},\nabla_{i}) \leq b_{i} \qquad \forall i $$

(60)

where:

$$ \begin{array}{@{}rcl@{}} \beta_{i}(x^{*},\nabla_{i})=\underset{\{S^{\prime}_{i}\cup \{t_{i}\}|S^{\prime}_{i}\subseteq J^{\prime}_{i}, card(S^{\prime}_{i})=\left\lfloor\nabla_{i}\right\rfloor, t_{i}\in J^{\prime}_{i}\setminus S^{\prime}_{i}\}}{\max} \left\{{\sum\limits_{j\in S^{\prime}_{i}}\widehat{a}_{ij} x_{j}^{*}+(\nabla_{i}-\left\lfloor\nabla_{i}\right\rfloor)\widehat{a}_{it_{i}} x_{j}^{*}}\right\} \end{array} $$

(61)

The formulation Eq. 61 is the protection function of constraint i. The protection function protects the left-side of the inequality to be lower than b_i for different values of a_ij; the protection function is defined for every constraint with uncertain coefficient. It is an optimization problem in itself; it depends on the protection level ∇_i; when ∇_i = 0, then β_i = 0 and the problem is deterministic [79].

In the following formulation, the protection function is transformed into an optimization problem; the formulation Eq. 61 can be replaced by the following linear problem:

$$ \max \sum\limits_{j \in J^{\prime}_{i} }\widehat{a}_{ij}x_{j}^{*}w_{ij} $$

(62)

subject to:

$$ \sum\limits_{j \in J^{\prime}_{i}} w_{ij}\leq \nabla_{i} $$

(63)

$$ 0 \leq w_{ij} \leq 1 \qquad \forall j\in J^{\prime}_{i} $$

(64)

$x_{j}^{*}$ is not variable in the problem, w_ij is the new variable corresponding to τ_ij described below, to prove the equivalence of problems Eq. 61 and Eqs. 62-64, the optimal solution value of problem Eqs. 62-64 consists of $\left \lfloor \nabla _{i}\right \rfloor$ variables at 1 and one variable at $\nabla _{i}-\left \lfloor \nabla _{i}\right \rfloor$, it is equivalent to selecting a subset : ${\{S^{\prime }_{i}\cup \{t_{i}\}|S^{\prime }_{i}\subseteq J^{\prime }_{i}, t_{i}\in J^{\prime }_{i}\setminus S^{\prime }_{i}, card(S^{\prime }_{i})=\left \lfloor \nabla _{i}\right \rfloor }\}$ with the corresponding objective value ${\sum }_{j\in S^{\prime }_{i}}\widehat {a}_{ij} x_{j}^{*}+(\nabla _{i}-\left \lfloor \nabla _{i}\right \rfloor )$ $\widehat {a}_{it_{i}} {x}_{j}^{*}$ [18].

The model Eqs. 62-64 is nonlinear when x is considered as variable, nonetheless, by using its duality, it can be linearly expressed. By strong duality, since problem Eq. 62-64 is feasible and bounded for all ∇_i, then the duality is also feasible and bounded:

$$ \min \nabla_{i}z_{i}+\sum\limits_{j \in J^{\prime}_{i}} p_{ij} $$

(65)

subject to:

$$ z_{i}+p_{ij}\geq \widehat {a}_{ij}x_{j}^{*} \qquad \forall j \in J^{\prime}_{i} $$

(66)

$$ p_{ij} \geq 0 \qquad \forall j\in J^{\prime}_{i} $$

(67)

$$ z_{i} \geq 0 $$

(68)

Finally, we replace the protection function with the dual model Eqs. 65-68, then, the robust model is as follows:

$$ \max \sum\limits_{j=1}^{n}c_{j} x_{j} $$

(69)

subject to:

$$ \sum\limits_{j=1}^{n} \overline{a}_{ij} x_{j}+ z_{i}\nabla_{i}+\sum\limits_{j \in J^{\prime}_{i}} p_{ij} \leq b_{i} \qquad \forall i $$

(70)

$$ z_{i}+p_{ij} \geq \widehat{a}_{ij}x_{j} \qquad \forall i,j\in J^{\prime}_{i} $$

(71)

$$ x_{j} \geq 0 \qquad \forall j $$

(72)

$$ p_{ij} \geq 0 \qquad \forall i,j\in J^{\prime}_{i} $$

(73)

$$ z_{i} \geq 0 \qquad \forall i $$

(74)