Parameter Value Range Necessity
Patient Characteristics
\(\text{Mean}\) \((0,∞)\) Optional
\(\text{SCV}\) \([0.1,1.5]\) Required
\(q\) \([0,\frac{3-2\cdot\text{SCV}}{5})\) Optional
\(w\) \([0,1]\) Optional
Schedule Characteristics
\(n\) \([2,35]\) 2 out of 3
\(ω\) \([0.05,0.99]\) 2 out of 3
\(T\) \((n\cdot\text{Mean},∞)\) 2 out of 3
\(Δ\) \([0,∞)\) Optional
Objective Function
\[ \min_{t_1,\ldots, t_n} \omega\sum_{i=1}^n \mathbb{E}I_i^{k_1} + (1-\omega)\sum_{i=1}^n \mathbb{E}W_i^{k_2} \]
Background Information
Parameter Details
\(\text{Mean}\)
The mean service time.
\(\text{SCV}\)
The variance divided by the squared mean.
\(q\)
The no-show probability.
\(w\)
The walk-in probability.
\(n\)
The number of patients that need to be scheduled.
\(ω\)
The weight factor in the objective function.
\(T\)
The targeted expected session end time.
\(Δ\)
The arrival times will be rounded to multiples of \(Δ\). If entered then the actual expected makespan will differ due to rounding.
2 out of 3 Basic Functionality
\(n\) & \(ω\)
The optimal schedule will be computed for these parameters, which also gives the corresponding expected makespan.
\(n\) & \(T\)
The optimal schedule will given such that exactly \(n\) patients are served in expectation within \(T\) giving the implied value of \(ω\).
\(ω\) & \(T\)
The optimal schedule will be given such that for the value of \(ω\) a maximum number of patients \(n\) are served in expectation within \(T\).
The webtool solves the minimization problem (\(t_1, \ldots, t_n\) are the patients' arrival epochs) \[ \min_{t_1,\ldots, t_n} \omega\, \sum_{i=1}^n \mathbb E[I_i^{k_1}] + (1-\omega)\, \sum_{i=1}^n \mathbb E[W_i^{k_2}]. \] \(I_i\) is the idle time prior to the \(i\)-th patient's arrival and \(W_i\) the \(i\)-th patient's waiting time.