Simulation-based Optimization

In order to minimize the cost of hiring nurses while meeting service level requirements, a simulation-based optimization model is used. This model uses the simulation to test a wide variety of shift plans that have nurses working different shifts to cover the demand of patients for

the system. The formulation of the model is a mathematical program. The model aims to minimize cost from three sources as seen in equation (1). First, the cost associated with each nurse that is assigned to a shift is summed over all shifts and each week. A shift is either 8 hours in which case the nurse works 5 days a week, or 10 hours in which case the nurse will work 4 days a week. The shifts are predetermined starting and ending times spread out over the 24 hours of the day ensuring that at least one shift is covering each hour. The number of nurses working each shift is controlled by a different variable for each shift in the simulation. The second cost is the total cost of overtime worked by all nurses in the model for each week. This is calculated by constraint (4) which says that overtime is accumulated if a nurse ends her shift past her scheduled end time. This is controlled by a random variable, 𝑦̃𝑖𝜔, described in constraint (6), which is an output from the simulation. The third term assigns a penalty cost to the total lateness, which is the time between all procedures’ scheduled start time and their actual start time as described in constraint (5). This third term ensures that the service level that is required for the system is met and also comes from a random variable, 𝑡̃𝑚𝜔, described in constraint (7), another output from the simulation. The other constraints in the model describe the range in which the model can search for an optimal solution. Constraint (2) restricts the model to a maximum number of nurses for each shift and each week. This maximum value can vary depending on the shift and requirements of the system. Constraint (3) restricts the total assignable nurses for each week over all shifts. The system can be assigned a minimum number of nurses required to ensure a high service level and also can be assigned a maximum number of nurses that are available during this current time horizon. The model uses output from the simulation model, 𝑡̃𝑚𝜔 and 𝑦̃_𝑖𝜔, which are functions of the patient arrival rate, surgery length, number of nurses working each shift, emergency patient arrival, and block schedule associated with the simulation.

Indices: Sets:

i index for nurses i ϵ N N set of nurses;

j index for shifts; j ϵ J J set of all shifts; p period of the day (30 min); p ϵ P P set of periods;

d index of days; d ϵ D D set of days;

ω week of shift plan; ω ϵ Ω Ω set of all simulation weeks; m index of procedures; m ϵ M M set of all procedures;

Parameters:

𝑐𝑗𝜔 cost of a nurse working shift j during week ω

𝑢 cost per hour of nurse overtime

𝑣 penalty cost per hour of surgery lateness

𝐵_𝑗𝜔 maximum number of nurses assignable to shift j in week ω 𝑁_𝑀𝐼𝑁 minimum total number of nurses available

𝑁_𝑀𝐴𝑋 maximum total number of nurses available

𝜑_𝑖𝜔 hours of time nurse i is scheduled to work in week ω 𝜇𝑚𝜔 scheduled start time of procedure m during week ω

Simulation Results:

𝑡_𝜔 total hours of surgery lateness per week ω 𝑦_𝜔 hours of nurse overtime per week ω

𝑦̃_𝑖𝜔 hours of time nurse i works in week ω, random variable

𝑡̃_𝑚𝜔 actual start time of procedure m during week ω, random variable

Decision Variables:

𝑥𝑗𝜔 number nurses required to work shift j in week ω

Minimize: ∑ ∑ (𝑐_𝑗𝜔 ∗ 𝑥_𝑗𝜔) 𝜔∈𝛺 𝑗∈𝐽 + ∑ (𝑢 ∗ 𝑦_𝜔) 𝜔∈𝛺 + ∑ (𝑣 ∗ 𝑡_𝜔) 𝜔∈𝛺 (1) Subject to: 𝑥𝑗𝜔 ≤ 𝐵𝑗𝜔 ∀ 𝑗 ∈ 𝐽, 𝜔 ∈ 𝛺 (2) 𝑁_𝑀𝐼𝑁≤ ∑ 𝑥_𝑗𝜔 𝑗∈𝐽 ≤ 𝑁_𝑀𝐴𝑋 ∀ 𝜔 ∈ 𝛺 (3) 𝑦_𝜔 = ∑ max{ 𝑦̃_𝑖𝜔− 𝜑_𝑖𝜔, 0} 𝑖∈𝑁 ∀ 𝜔 ∈ 𝛺 (4) 𝑡𝜔 = ∑ max { 𝑡̃𝑚𝜔− 𝜇𝑚𝜔, 0} 𝑚∈𝑀 ∀ 𝜔 ∈ 𝛺 (5) 𝑦̃𝑖𝜔 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 ∀ 𝑖 ∈ 𝑁, 𝜔 ∈ 𝛺 (6)

Using the cost minimization objective, the model generates several different shift plan scenarios, trying to find the optimal number of nurses for each shift. A scenario consists of a different number of nurses required for each of the shifts. Each scenario is run for a minimum of 5 replications in the simulation to ensure a consistent result. The replications of the models give different results due to the stochastic variables of the problem, patient arrival and surgery length. After completion of the optimization, the data is analyzed further to determine which scenario works best; it may be the case that several scenarios have the same or similar costs. Once the best scenario is chosen, the number of nurses working each shift is given as an input to the Nurse Assignment phase and the assignment model.

Input: OR Scheduling Info. Nurse Requirements Shift Alternatives OR Simulation Model Simulation-based Optimization Experimants Input: Objective Function Constraints Performance Metrics Alternative Solution Output/Input: Nurse Scheduling Plan Nurse Shift Assignment Model Input: Nurse Information Assignment History Input: Objective Function Constraints Output: Nurse Schedule

Figure 3: Nurse Planning Phase portion of Figure 1

Seen in Figure 3, the simulation-based optimization takes the objective function, constraints, and performance metrics of patient lateness and nurse overtime costs from the simulation as an input. The optimization model generates scenarios, each with a different shift plan or number of nurses working each shift, and applies them to the simulation. Each scenario’s output is given from the simulation to the optimization model so that it can be compared against

Nurse Planning Phase

other scenarios in terms of total cost. The model then decides which of the scenarios has the minimum total cost and uses that as the final shift plan or Nurse Scheduling Plan. This shift plan is used as an input in the Nurse Assignment model in the second phase.