Personnel and Patient Scheduling in the High Demanded Hospital Services: A Case Study in the Physiotherapy Service

Abstract

High demand but limited staffs within some services of a hospital require a proper scheduling of staff and patients. In this study, a hierarchical mathematical model is proposed to generate weekly staff scheduling. Due to computational difficulty of this scheduling problem, the entire model is broken down into manageable three hierarchical stages: (1) selection of patients, (2) assignment of patients to the staff, (3) scheduling of patients throughout a day. The developed models were tested on the data collected in College of Medicine Research Hospital at Cukurova University using GAMS and MPL optimization packages. From the results of the case study, the presented hierarchical model provided a schedule that ensures to maximize the number of selected patients, to balance the workload of physiotherapist, and to minimize waiting time of patients in their treatment day.

Appendices

Appendix 1

Table 5 Patient candidate list in physiotherapy and rehabilitation service of C.U. research hospital

Appendix 2

• Stage I: Patient Acceptance Planning

  $${\text{Max}}\,\sum\limits_{i = 1}^N {p_i x_i } $$

  (1)

  Subject to:

  $$\sum\limits_{i = 1}^N {x_i t_i \, \leqslant T} $$

  (2)

• Stage II: Assignment to Physiotherapists

  $${\text{Min }}W_{\text{1}} d_{\text{1}}^{\text{ + }} + W_2 d_2^ + + W_3 d_3^ + + W_4 d_4^ + + W_5 d_4^ - $$

  (3)

  Subject to:

  $$d_1^ + = \sum\limits_{j = 1}^S {\left| {G_j - \mathop G\limits^ - } \right|} \quad i{\text{ = 1,}}...S{\text{;}}$$

  (4)
  $$G_{j} = {\sum\limits_{i = 1}^n {t_{i} } }Y_{{ij}} \quad \begin{array}{*{20}c} {j{\text{ = 1,}}...n{\text{;}}} \\ {j{\text{ = 1,}}...S{\text{;}}} \\ \end{array} $$

  (5)
  $$G_j = \sum\limits_{i = 1}^n {G_j } /S\,\,\,\,\,\,\,\,\,\,j = 1, \ldots S$$

  (6)
  $$d^{ + }_{{k + 1}} = {\sum\limits_{j = 1}^S {{\left| {N_{{jk}} - {\mathop {N_{k} }\limits^ - }} \right|}} }\,\,\,\,\,\,\begin{array}{*{20}c} {j{\text{ = 1,}}...S{\text{;}}} \\ {k{\text{ = 1,2;}}} \\ \end{array} $$

  (7)
  $${\mathop {N_{{jk}} }\limits^ - } = \,{\sum\limits_{^{{{\mathop {j = 1}\limits_{tc_{i} = k} }}} } {Y_{{ij}} } }\,\,\,\,\begin{array}{*{20}c} {j{\text{ = 1,}}...S{\text{;}}} \\ {k{\text{ = 1,2;}}} \\ \end{array} $$

  (8)
  $${\mathop {N_{k} }\limits^ - } = {\sum\limits_{j = 1}^S {N_{{jk}} } }/S\,\,\,\,\,\,\,\,\,\begin{array}{*{20}c} {i{\text{ = 1,}}...n{\text{;}}} \\ {k{\text{ = 1,2;}}} \\ \end{array} \,$$

  (9)
  $$d^{ + }_{4} - d^{ - }_{4} = {\sum\limits_{i = 1}^n {{\sum\limits_{j = 1}^S {Y_{{ij}} } }} }t_{i} - hS\quad \begin{array}{*{20}c} {j{\text{ = }}\,\,{\text{1,}}...S{\text{;}}} \\ {i{\text{ = }}\,\,{\text{1,}}...n{\text{;}}} \\ \end{array} $$

  (10)
  $${\sum\limits_{j = 1}^S {Y_{{ij}} } } \leqslant 1\,\,\,\,\,\,\,\,\,j{\text{ = 1,}}...S{\text{;}}$$

  (11)
  $$G_{j} \leqslant h\quad j{\text{ = 1,}}...S{\text{;}}$$

  (12)

• Stage III: Scheduling Patients

  $${\text{Min }}d_5^ + $$

  (13)

  Subject To:

  $$\sum\limits_{i = 1}^n {Z_{il} = 1} \,\,\,\,\,\,\,\,\,\,l = 1, \ldots m$$

  (14)
  $$TI_l = \sum\limits_{i = 1}^n {t_i Z_{il} } \,\,\,\,\,\,\,\,\,\,l = 1, \ldots m$$

  (15)
  $$d_5^ + = \sum\limits_{l = 1}^m {\left| {TI_l - \overline {TI} } \right|} $$

  (16)
  $$\overline {Tl} = \sum\limits_{l = 1}^m {{{TI_l } \mathord{\left/ {\vphantom {{TI_l } m}} \right. \kern-\nulldelimiterspace} m}} $$

  (17)