## 4 A General Solving Methodology
Finally, the resulting BR algorithms are embedded into a multi-start (MS) framework in order to generate many alternative solutions. Then, the best-found solution is updated and returned at the end of this procedure. ```
Data: set of nodes \(V\), parameter \(\beta\in[0,1]\) \(sol\leftarrow\) createDummySolution(\(V\)) \(SL\leftarrow\) createSavingsList(\(sol\)) whilethere are edges in \(SL\)do \(\leftarrow\) selectEdgeFromList(\(\beta,SL\)) \(i\leftarrow\) getOrigin(\(e\)) \(j\leftarrow\) getEnd(\(e\)) \(iRoute\leftarrow\) getEvolvingRouteOfNode(\(i\)) \(jRoute\leftarrow\) getEvolvingRouteOfNode(\(j\)) ifall route-merging conditions are satisfiedthen \(sol\leftarrow\) mergeRoutesUsingEdge(\(e,iRoute,jRoute,sol\))  end if \(SL\leftarrow\) deleteEdgeFromList(\(e,SL\)) end return\(sol\)
```

**Algorithm 1**Biased-Randomized Algorithm. ```
Data: set of nodes \(V\), parameter \(\beta\in[0,1]\) \(sol\leftarrow\) createDummySolution(\(V\)) \(SL\leftarrow\) createSavingsList(\(sol\)) whilethere are edges in \(SL\)do \(\leftarrow\) selectEdgeFromList(\(\beta,SL\)) \(i\leftarrow\) getOrigin(\(e\)) \(j\leftarrow\) getEnd(\(e\)) \(iRoute\leftarrow\) getEvolvingRouteOfNode(\(i\)) \(jRoute\leftarrow\) getEvolvingRouteOfNode(\(j\)) ifall route-merging conditions are satisfiedthen \(sol\leftarrow\) mergeRoutesUsingEdge(\(e,iRoute,jRoute,sol\))  end if \(SL\leftarrow\) deleteEdgeFromList(\(e,SL\)) end return\(sol\)
```

**Algorithm 2**Biased-Randomized Algorithm. ```
Data: set of nodes \(V\), parameter \(\beta\in[0,1]\) \(sol\leftarrow\) createDummySolution(\(V\)) \(SL\leftarrow\) createSavingsList(\(sol\)) \(\leftarrow\) sort(\(SL\)) whilethere are edges in \(SL\)do \(\leftarrow\) selectEdgeFromList(\(\beta,SL\)) \(i\leftarrow\) getOrigin(\(e\)) \(j\leftarrow\) getEnd(\(e\)) \(iRoute\leftarrow\) getEvolvingRouteOf  
3. The last stage refers to the incorporation of both simulation and fuzzy components into the BR-MS framework, so that promising solutions are processed to estimate their expected costs (Algorithm 2). For the VRP and TOP, the uncertain variables represent the customer demands and the travel times, respectively. * For stochastic variables, a new value is assigned to each random element based on its probability distribution. For stochastic variables, the MCS is used to estimate them. * For fuzzy variables, the new value of each element is based on its fuzzy function. Accordingly, fuzzy variables are estimated through the FIS. This procedure is explained more thoroughly in Section 5. Once the deterministic version of each problem is solved, their respective solutions are submitted to a exploratory stage, in which only a low number of simulation (\(q_{short}\)) runs are performed to avoid jeopardizing the time of the metaheuristic component [79]. These'short' simulation runs are applied only to solutions that meet an acceptance criterion (line 8). Altering these stochastic and fuzzy values involves a re-evaluation of both the objective function and the constraints, so that the expected cost/reward of each promising solution can be computed. These short simulation runs allow multiple elite solutions to be found (line 11). In this way, once the BR-MS main loop is completed, a larger number of simulation (\(q_{long}\)) runs are executed for each elite solution (line 17). Consequently, the algorithm is able to obtain more accurate values of the output variables. Finally, a reduced set of best-found solutions is returned. From this set, managers can assess not only the expected costs/rewards but also the risk or reliability values associated with each solution, as described in [PERSON] et al. [6].