Adaptive layout for operating theatre in hospitals: different mathematical models for optimal layouts

Abstract

The adaptive layout for operating theatre (ALOT) problem in hospitals seeks to determine the ‘most efficient’ layout placement of a set of health-care operating facilities, corridors and elevators in a designated area subject to a set of constraints on professional standards. Such standards include regulations on: hygiene, safety and security of stakeholders (doctors, medical staff, patients and visitors); movements of technologies; and specifications of operating rooms (functions, orientations, space sizes, and desired closeness). Existing ALOT layouts are mostly generated from designs based on experiential judgments of experts. Due to the lack of scientific rigor and huge impact of layout design on the efficiency and effectiveness of an operating theater, the paper proposes mixed integer linear programming models to find optimal layouts under three different design variants: ALOT with multiple sections; ALOT with multiple rows and ALOT with multiple floors. Each variant has different demands for personnel, patients, and technologies over a planning horizon. Operating facilities can exchange functions at rearrangement costs from one period to another to meet the changing demands. The general objective consists of two sub-objectives: the first sub-objective is to minimize the total sum of the rearrangement and travel costs whereas the second sub-objective is to maximize the total sum of desired closeness among facilities. Computational experiences are presented on a set of quasi-real data instances for a hospital in France. They demonstrate the effectiveness of the formulations in providing optimal layouts for realistic-sized instances. Conclusion and future research directions are presented.

Appendices

Appendix 1: Constraints linearization

Constraint (7) was linearized using the following Eqs. (63) and (64):

$$ x^{t}_{c} - x^{t}_{o} \ge \left( {\frac{{\delta_{c} + \delta_{o} }}{2}} \right)\quad \forall c,o = {\text{n}} + 1, \ldots ,{\text{n}} + \# {\text{c}}; \forall t \in {\mathcal{P}} $$

(63)

$$ x^{t}_{o} - x^{t}_{c} \ge \left( {\frac{{\delta_{c} + \delta_{o} }}{2}} \right)\quad \forall c,o = {\text{n}} + 1, \ldots ,{\text{n}} + \# {\text{c}}; \forall t \in {\mathcal{P}} $$

(64)

To linearize Eqs. (15) and (16), Eqs. (65)–(69) were used with the additional binary variables $\theta_{ij}^{t}$ and $\eta_{ij}^{t}$ to decide whether a facility ‘i’ is strictly to the right of a facility ‘j’ or it is strictly above facility ‘j’ at period ‘t’, respectively. Similarly, we conduct linearization of constraints (17) and (18), (30) and (31); and (32) and (33) as follows.

$$ x^{t}_{i} - x^{t}_{j} + X_{max} \cdot \left( {1 - \theta_{ij}^{t} } \right) \ge \frac{{l^{t}_{i} + l^{t}_{j} }}{2}\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(65)

$$ y^{t}_{i} - y^{t}_{j} + Y_{max} \cdot \left( {1 - \eta_{ij}^{t} } \right) \ge \frac{{d^{t}_{i} + d^{t}_{j} }}{2} \quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(66)

$$ \theta_{ij}^{t} + \theta_{ji}^{t} + \eta_{ij}^{t} + \eta_{ji}^{t} \ge 1\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(67)

$$ \theta_{ij}^{t} + \theta_{ji}^{t} \le 1\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(68)

$$ \eta_{ij}^{t} + \eta_{ji}^{t} \le 1\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(69)

To linearize Eq. (21), the computational parameters $X^{t}_{ij}$ and $Y^{t}_{ij}$ are temporally defined by

$$ \begin{aligned} X^{t}_{ij} & = \left| {x^{t}_{i} - x^{t}_{j} } \right|\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} \\ Y^{t}_{ij} & = \left| {y^{t}_{i} - y^{t}_{c} } \right| + \left| {y^{t}_{c} - y^{t}_{j} } \right|\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} \\ \end{aligned} $$

They are used inside Eqs. (70)–(75) to perform the linearization of the two parts of constraint (21).

$$ Y^{t}_{ij} \ge \left( {y^{t}_{i} - y^{t}_{c} } \right) + \left( {y^{t}_{c} - y^{t}_{j} } \right)\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(70)

$$ Y^{t}_{ij} \ge \left( {y^{t}_{i} - y^{t}_{c} } \right) + \left( {y^{t}_{j} - y^{t}_{c} } \right)\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(71)

$$ Y^{t}_{ij} \ge \left( {y^{t}_{c} - y^{t}_{i} } \right) + \left( {y^{t}_{c} - y^{t}_{j} } \right)\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(72)

$$ Y^{t}_{ij} \ge \left( {y^{t}_{c} - y^{t}_{i} } \right) + \left( {y^{t}_{j} - y^{t}_{c} } \right)\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(73)

$$ X^{t}_{ij} \ge \left( {x^{t}_{i} - x^{t}_{j} } \right)\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(74)

$$ X^{t}_{ij} \ge \left( {x^{t}_{j} - x^{t}_{i} } \right)\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall t \in {\mathcal{P}} $$

(75)

To linearize Eq. (46), we conduct the same linearization process as follows.

$$ Z^{t}_{ij} \le H_{f} \mathop \sum \limits_{f = 1}^{\# f} f \cdot \left( {V^{t}_{if} - V^{t}_{jf} } \right)\quad \forall i,j = 1, \ldots {\text{n}}; \forall f \in {\mathcal{F}};\forall t \in {\mathcal{P}} $$

(76)

$$ Z^{t}_{ij} \le H_{f} \mathop \sum \limits_{f = 1}^{\# f} f \cdot \left( {V^{t}_{jf} - V^{t}_{if} } \right)\quad \forall i,j = 1, \ldots {\text{n}}; \forall f \in {\mathcal{F}};\forall t \in {\mathcal{P}} $$

(77)

Equations (23) and (50) involve the product of two variables, to linearize them the three following equations will be introduced to linearize Eq. (50) similarly can be done for Eq. (23):

$$ A^{t}_{ijc} \ge W^{t}_{ic} + W^{t}_{jc} - 1\quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(76)

$$ A^{t}_{ijc} \le W^{t}_{ic} \quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(77)

$$ A^{t}_{ijc} \le W^{t}_{jc} \quad \forall i,{\text{j}} = 1, \ldots ,{\text{n}};\forall c = n + 1, \ldots ,n + \# c;\forall t \in {\mathcal{P}} $$

(78)

Appendix 2: Example of traveling frequency for a small operating theater

The table below gives an example of the number of travel frequencies between each pair of facilities for each type of actors (k: 1 = doctor; 2 = medical staff; 3 = other staff; and 4 = patient). For instance, doctor (k = 1) performed at total of 60 operations in OR1 and 45 operations in OR2 leading total of 105 operations. The associated travels by other actors in support of 105 operations are provided, e.g., 105 patients (k = 4) are coming from outside the hospital, outdoor (0). Other data are reported in the table as collected from the information system of the hospital based on the type of surgical operations.

Label	Facility	Actor	Facility
		k	0	1	2	3	4	5	6	7
0	Outdoor	1		0	0	0	105	0	0	0
		2		210	0	0	0	0	0	105
		3		210	0	0	0	0	0	0
		4		105	0	0	0	0	0	0
1	Induction	1	0	0	0	0	0	0	0	0
		2	0	0	120	90	0	0	105	0
		3	105	0	60	45	0	0	0	0
		4	0	0	60	45	0	0	0	0
2	OR1	1	0	0	0	0	60	0	0	0
		2	0	0	0	0	0	60	60	0
		3	0	0	0	0	0	60	0	0
		4	0	0	0	0	0	60	0	0
3	OR2	1	0	0	0	0	45	0	0	0
		2	0	0	0	0	0	45	45	0
		3	0	0	0	0	0	45	0	0
		4	0	0	0	0	0	45	0	0
4	Scrub	1	0	0	60	45	0	0	0	0
		2	105	0	0	0	0	0	0	0
		3	0	0	0	0	0	0	0	0
		4	0	0	0	0	0	0	0	0
5	PACU	1	0	0	0	0	0	0	0	0
		2	105	0	0	0	0	0	0	0
		3	0	0	0	0	0	0	0	0
		4	105	0	0	0	0	0	0	0
6	Decontamination	1	0	0	0	0	0	0	0	0
		2	210	0	0	0	0	0	0	210
		3	0	0	0	0	0	0	0	0
		4	0	0	0	0	0	0	0	0
7	Sterile arsenal	1	0	0	0	0	0	0	0	0
		2	0	105	60	45	0	0	0	0
		3	0	0	0	0	0	0	0	0
		4	0	0	0	0	0	0	0	0

Appendix 3: Illustrations of final adaptive layout designs for the different variants

See Figs. 5, 6 and 7.

Fig. 5

ALOT with multi-section solution for S16-3 over a three-period planning horizon

Fig. 6

ALOT with multi-floor solution for F14-3 over a three-period planning horizon

Fig. 7

ALOT multi-row solution for R24-3 over a three-period planning horizon