A Two-Stage Method of Dimensioning and Scheduling Service Providers under Patient Demand Uncertainty

Table 1

A brief classification of the models reviewed in the literature.

Author	Year	Service provider dimensioning	Shift scheduling	Objective	Uncertainty	Solution approach
Klinz et al. [17]	2006	—	✓	Minimize the total number of work shifts and the general unhappiness of all nurses	—	Heuristic
Topaloglu and Selim [18]	2010	—	✓	Minimize deviations from nurse preferences and hospital regulations	Fuzzy	Exact
Landa-silva and Le [19]	2008	—	✓	Minimize deviations from nurses’ satisfaction and work regulations	—	Meta-heuristic
Ohki [20]	2012	—	✓	Minimize the penalty function to evaluate shift schedules	—	Meta-heuristic
El Adoly et al. [33]	2011	—	✓	Maximize the quality of objectives concerning the importance of constraints	—	Meta-heuristic
Maenhout and Vanhoucke [8]	2013	✓	✓	Minimize the penalty associated with different types of nurses	—	Exact
Santos et al. [22]	2016	—	✓	Minimize the penalty of assignment	—	Heuristic
Ingels and Maenhout [23]	2015	—	✓	Minimize the allocation penalty and change the nurse schedule	—	Exact and simulation
Dohn and Mason [24]	2013	—	✓	Minimize penalties from under-and over-coverage and minimize the total cost of all roster lines	—	Column generation
						Branch and price
Bagheri et al. [25]	2016	—	✓	Minimize the normal and overtime hours of nurses	Stochastic	Sample average
						Approximation
Punnakitikashem et al. [26]	2013	—	✓	Minimize the excess workload on nurses and the cost of staffing	Stochastic	Benders and Lagrangian
Chen et al. [10]	2016	✓	✓	First stage: Minimize the number of nurses. Second stage: Minimize the penalty of the soft constraints of nurses’ preferences	—	Exact
Ang et al. [27]	2018	—	✓	Minimize the maximum and average deviations from target nurse-patient ratios	—	Exact
Hamid et al. [28]	2020	—	✓	Minimize the sum of incompatibility among nurses and the total cost of staffing and maximize the satisfaction of nurses with their assigned shifts	—	Meta-heuristic
Pham and Dao [29]	2021	—	✓	Minimize the total cost of assigning nurses to different shifts (morning, evening, night, and day-off)	—	Hybrid metaheuristic
Hassani and Behnamian [30]	2021	—	✓	Minimizing the total cost of allocating shifts to nurses, reserve nurses required, overtime and underemployed costs of a particular type of shift, cost of mismatching the nurse preferences with the roster	Robust scenario-based optimization	Meta-heuristic
Kheiri et al. [31]	2021	—	✓	Minimizing violation of eight soft constraints	—	Hyper-heuristic with statistical Markov model
This study	2022	✓	✓	First stage: minimize the number of service providers. Second stage: minimizes regular work hours, overtime hours, and the cost of idle hours	Stochastic	First stage: exact
						Second stage: improved sample average approximation