Solving 0–1 semidefinite programs for distributionally robust allocation of surgery blocks

Abstract

We allocate surgery blocks to operating rooms (ORs) under random surgery durations. Given unknown distribution of the duration of each block, we investigate distributionally robust (DR) variants of two types of stochastic programming models using a moment-based ambiguous set. We minimize the total cost of opening ORs and allocating surgery blocks, while constraining OR overtime via chance constraints and via an expected penalty cost in the objective function, respectively in the two types of models. Following conic duality, we build equivalent 0–1 semidefinite programming (SDP) reformulations of the DR models and solve them using cutting-plane algorithms. For the DR chance-constrained model, we also derive a 0–1 second-order conic programming approximation to obtain less conservative solutions. We compare different models and solution methods by testing randomly generated instances. Our results show that the DR chance-constrained model better controls average and worst-case OR overtime, as compared to the stochastic programming and DR expected-penalty-based models. Our cutting-plane algorithms also outperform standard optimization solvers and efficiently solve 0–1 SDP formulations.

Appendices

Appendix A: Proof of Theorem 1

Proof

Define dual variables $r_i$ for (3b), $\begin{bmatrix} H_i&p_i\\ p_i^{\mathsf {T}}&q_i\end{bmatrix}\in {\mathbb {S}}_+^{(|J|+1)\times (|J|+1)}$ for (3c), and $G_i\in {\mathbb {S}}_+^{|J|\times |J|}$ for (3d). The conic dual problem of (3) is

$$\begin{aligned}&\max _{G_i,p_i,r_i,H_i,q_i} \gamma _2\Sigma _i^0\cdot G_i + 1 - r_i + \Sigma _i^0\cdot H_i+\gamma _1 q_i \end{aligned}$$

(29a)

$$\begin{aligned} \text{ s.t. }&(s_i - \mu _i^0)^{\mathsf {T}}(-G_i)(s_i - \mu _i^0) + 2(p_i)^{\mathsf {T}}(s_i-\mu _i^0)\nonumber \\&\quad +\,r_i\le {\mathbb {I}}_{\left\{ s_i : \sum _{j\in J(i)}s_{ij} {\hat{y}}_{ij}\le T_i {\hat{z}}_i \right\} }(s_i),\ \forall s_i\in {\mathbb {R}}^{|J|}. \end{aligned}$$

(29b)

As strong duality holds for the primal and dual problems [12], satisfying the DR chance constraints (1e) with the ambiguity set ${\mathcal {D}}_i^M$, is equivalent to having solutions with objective value $\ge 1-\alpha _i$, $i\in I$. After applying Lemma 1 in Jiang and Guan [10], the dual problem (29), where its objective function is written as a constraint such that it is no less than $1-\alpha _i$, is equivalent to SDP(${\hat{y}}_i, {\hat{z}}_i$), after replacing the semi-infinite constraints (29b) with finite number of SDP constraints. The proof completes. $\square$

Appendix B: Cutting-plane algorithm for optimizing 0–1 SDP reformulations