## 4 Proposed Solution
### Specific Heuristic Algorithm
As observed before, \(T_{b_{i}}\) is following a normal distribution with mean  
\[\mu_{T_{b_{i}}}=\mu_{Dt}+2\cdot\mu_{d}(s_{k}^{1})+\mu_{d}(s_{j}^{1})+2\cdot \mu_{Ct}\]  
and standard deviation  
\[\sigma_{T_{b_{i}}}=\sqrt{\left(\sigma_{Dt}\right)^{2}+2\cdot\left(\sigma_{d}(s _{k}^{1})\right)^{2}+\left(\sigma_{d}(s_{j}^{1})\right)^{2}+2\cdot\left(\sigma _{Ct}\right)^{2}}.\]  
Since \(\{\overline{S_{k}},\overline{S_{k}},\overline{S_{j}}\}\) is a possible scheduling type, the chance constraint (3) is satisfied. In order to check this equation, based on normal distribution property [34], the following inequality \[P[T_{b_{i}}\leq l(b_{i})]\geq Cl\]  
is equivalent to  
\[\frac{l(b_{i})-\mu_{T_{b_{i}}}}{\sigma_{T_{b_{i}}}}\geq V_{Cl},\]  
where \(V_{Cl}\) is the value corresponding to the normal variable \((x\sim N(0,1))\) with a cumulative distribution function \(Cl\), i.e., \(P[x\leq V_{Cl}]=Cl\) and this value is tabulated. Part (ii): Scheduling of the time blocks, consisting of the following four steps executed sequentially for each time block \(b_{i}\) to schedule. _Step 3: For each possible scheduling type of time block \(b_{i}\), obtain the sets of real scheduling_. Given a possible scheduling type, a real scheduling is composed by a set of patients with the types of surgeries equal to the possible scheduling types. For example, considering the scheduling type \(\{\overline{S_{k}},\overline{S_{k}},\overline{S_{j}}\}\), a real scheduling is composed from two patients \(w^{\prime}\) and \(w^{\prime\prime}\) on which a surgery \(s_{k}\in\overline{S_{k}}\) should be performed and one patient \(w^{\prime\prime\prime}\) on which a surgery \(s_{j}\in\overline{S_{j}}\) should be performed. The first real scheduling corresponding to \(\{\overline{S_{k}},\overline{S_{k}},\overline{S_{j}}\}\) is chosen by taking the patients with a lower number of preference order (corresponding to the higher preference). However, all other combinations of patients satisfying the chosen scheduling type having no patients with preference order greater than the maximum preference order of the patient in the first real scheduling will be considered. Let us assume, for example, the list of patients given in Table 1 with \(\mathcal{W}=\{w_{1},w_{2},w_{3},w_{4},w_{5},w_{6}\}\) such that patients \(w_{1}\), \(w_{3}\), and \(w_{5}\) have assigned the surgery \(s_{1}\), the patient \(w_{2}\) has assigned the surgery \(s_{2}\), the patient \(w_{4}\) the surgery \(s_{3}\), and the patient \(w_{6}\) the surgery \(s_{4}\). Assuming the set of surgeries \(\mathcal{S}=\{\overline{S_{1}},\overline{S_{2}},\overline{S_{3}}\}\) with \(\overline{S_{1}}=\{s_{1},s_{2}\}\), \(\overline{S_{2}}=\{s_{3}\}\) and \(\overline{S_{3}}=\{s_{4}\}\) and two possible scheduling types \(\{\overline{S_{1}},\overline{S_{1}},\overline{S_{2}}\}\) and \(\{\overline{S_{1}},\overline{S_{2}},\overline{S_{3}}\}\), the first real scheduling considered for \(\{\overline{S_{1}},\overline{S_{1}},\overline{S_{2}}\}\) is \(\{w_{1},w_{2},w_{4}\}\). The maximum order of patients in this real scheduling is 4 (corresponding to \(w_{4}\)), hence the following real scheduling will be considered as well: \(\{w_{1},w_{3},w_{4}\}\) and \(\{w_{2},w_{3},w_{4}\}\). For the scheduling type \(\{\overline{S_{1}},\overline{S_{2}},\overline{S_{3}}\}\), the first real scheduling is \(\{w_{1},w_{4},w_{6}\}\). Since the maximum preference order of patients is 6, the following real scheduling will be considered as well: \(\{w_{2},w_{4},w_{6}\}\), \(\{w_{3},w_{4},w_{6}\}\) and \(\{w_{4},w_{5},w_{6}\}\). All of this real scheduling is shown in Table 2.