Reallocation of unoccupied beds among requesting wards

Abstract

Disruptions can cause demand fluctuation, thus overcrowding at hospital wards, and can make the waiting list of patients longer. Bed management at hospitals is one of the solutions to deal with overcrowding. In addition, Resilience Engineering (RE) is an approach that can help organizations to bounce back to their desired performance state after a disruption. In this paper, the concept of RE has been used to improve the bed management of hospitals during and after disruption. More precisely, bed sharing among hospital wards has been introduced as a collaboration strategy and its impact on the length of patients’ waiting list as the major performance index is investigated. Relationship priority between different wards, patients’ gender, patients’ length of stay and the number of rooms in every ward are the major factors considered in our modeling. A mixed integer linear programming optimization model with the objective of minimizing the patients’ waiting time in a hospital has been proposed for the real-world problems. The main contribution of the present paper is proposing a resiliency-based modeling of bed management in hospitals. Due to the complexity and making the proposed model applicable to the real world problems, a simulated annealing algorithm is used to solve the model and a new procedure for creating initial solution is presented. The results show that applying resilience strategy has a considerable impact on improving the hospital’s performance index.

Appendices

Appendix 1

Respective constraints are replaced with the following:

$$ dq_{sijt} = p_{sit} \left( {1 - mP_{{si\left( {j - 1} \right)t}} - mP_{sijt} + a1_{sijt} } \right) + a3_{sijt} - a4_{sijt} $$

(42)

Suppose that x is a continuous variable with lower bound L and upper bound U, and d is a binary variable. In order to linearize $y = x \times d$, we can use the following mixed integer programming formulation:

$$ Ld \le y \le Ud $$

(43)

$$ L\left( {1 - d} \right) \le x - y \le U\left( {1 - d} \right) $$

(44)

Suppose that $d_{1} , d_{2}\, and \,d_{3}$ are binary variables. In order to linearize $d_{3} = d_{1} d_{2}$, we can use the following mixed integer programming formulation:

$$ d_{3} \le d_{1} $$

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$$ d_{3} \le d_{2} $$

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$$ d_{3} \ge d_{1} + d_{2} - 1 $$

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Thus, the mentioned linearization techniques are used to linearize the proposed model as follows. Constraints (48)–(62) are added to the formulation.

$$ a1_{sijt} = mP_{{si\left( {j - 1} \right)t}} mP_{sijt} \quad \forall s,i,j,t $$

(48)

$$ a1_{sijt} \le mP_{{si\left( {j - 1} \right)t}} \quad \forall s,i,j,t $$

(49)

$$ a1_{sijt} \le mP_{sijt} \quad \forall s,i,j,t $$

(50)

$$ a1_{sijt} \ge mP_{{si\left( {j - 1} \right)t}} + mP_{sijt} - 1\quad \forall s,i,j,t $$

(51)

$$ a2_{sijt} = mP_{{si\left( {j - 1} \right)t}} \left( {1 - mP_{sijt} } \right)\quad \forall s,i,j,t $$

(52)

$$ a2_{sijt} \le mP_{{si\left( {j - 1} \right)t}} \quad \forall s,i,j,t $$

(53)

$$ a2_{sijt} \le \left( {1 - mP_{sijt} } \right)\quad \forall s,i,j,t $$

(54)

$$ a2_{sijt} \ge mP_{{si\left( {j - 1} \right)t}} - mP_{sijt} \quad \forall s,i,j,t $$

(55)

$$ a3_{sijt} = a2_{sijt} \mathop \sum \limits_{k = 1}^{j} dq_{{sik\left( {t - 1} \right)}} \quad \forall s,i,j,t $$

(56)

$$ a3_{sijt} \le {\text{M}}a2_{sijt} \quad \forall s,i,j,t $$

(57)

$$ a3_{sijt} \ge \mathop \sum \limits_{k = 1}^{j} dq_{{sik\left( {t - 1} \right)}} + M\left( {a2_{sijt} - 1} \right)\quad \forall s,i,j,t $$

(58)

$$ a3_{sijt} \le \mathop \sum \limits_{k = 1}^{j} dq_{{sik\left( {t - 1} \right)}} \quad \forall s,i,j,t $$

(59)

$$ a4_{sijt} = a2_{sijt} xt_{sit} \quad \forall s,i,j,t $$

(60)

$$ a4_{sijt} \le {\text{M}}a2_{sijt} \quad \forall s,i,j,t $$

(61)

$$ a4_{sijt} \ge xt_{sit} + M\left( {a2_{sijt} - 1} \right)\quad \forall s,i,j,t $$

(62)

$$ a4_{sijt} \le xt_{sit} \quad \forall s,i,j,t $$

(63)

Appendix 2

We generated 81 test problem by random parameters which are stated in below table for each test problem and compared the objective function and the time for Cplex and SA. As can be seen, the objective function obtained by two mentioned methods are the same for the whole test problems. All of the test problems can be categorized as small problem. The runtime for Cplex is less than the time required by SA to reach the optimal solution.

See Table 3.

Table 3 Comparison between the proposed algorithm and Cplex

Appendix 3

In order to investigate the effect of randomness on the performance of the algorithm, we have run each test problems five time and investigate the objective function and required time for each problem. As can be seen, the objective functions for each type of problems are the same and the proposed algorithm reached these values in similar times for each type of problems.

See Table 4.

Table 4 Run times and objective function values for the case study under two scenarios for 5 trials

Appendix 4

See Table 5.

Table 5 Comparing runtime and objective function between simulated annealing and Cplex