# A Simheuristic Algorithm for Solving the Stochastic Omnichannel Vehicle Routing Problem with Pick-up and Delivery

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Details on the Stochastic OM-VRP

## 4. Methodology

#### 4.1. Savings-Based Heuristic

- In the first stage, a dummy solution is created, where each route in this dummy solution serves one node $i\in N$, which can be either a consumer or a retail store. Routes depart from the depot, travel to the node, and then return to the depot.
- The second stage of LH, named as CWS${}_{1}$, is presented in the Figure 3, where the dummy routes from stage 1 are merged using a maximum-savings criterion [36]. Each box from Figure 3 is numbered to aid the following explanations.Initially, a savings list (SL) is constructed (step 1). This list considers all possible pairs of nodes–i.e., edges–from the problem. For each edge $\{i,j\}$, the corresponding savings value is calculated as ${s}_{ij}={t}_{0i}+{t}_{j0}-{t}_{ij}$, where ${t}_{ij}$ represents the deterministic travel time between nodes i and j. That is, candidate solutions are generated using ‘ideal’ traffic conditions for the edge-traversal times.Initially, all edges from SL are eligible. The list is sorted in descending order of the savings value (step 2), and the edge with the highest saving is selected (step 3). At this stage, the selection of edges is restricted to guarantee the assignment of a retail center to each consumer in the problem (step 5). In other words: a route containing one single customer can only be merged with a route containing a retail center, i.e., the selection is restricted to eligible edges $\{i,j\}$, where node j is a consumer in a dummy route and i is a node in a route with a retailer that can supply consumer j. According to Martins et al. [9], these attempts at only merging routes containing at least one retail center, which can supply a single consumer, are made first in order to avoid infeasible solutions–i.e., solutions in which some customers are not assigned to any retailer. This approach of addressing solution feasibility first is based on the observation that the availability of feasibility restoring merges will only decrease as the algorithm progresses.Based on CWS route-merging conditions (step 6), the two corresponding routes, i and j, of an edge $\{i,j\}$ (obtained in steps $4.1$ and $4.2$) can be merged only if: (i) nodes i and j are exterior in their respective routes (a node is exterior if it is adjacent to the depot); (ii) i and j belong to different routes; (iii) the maximum tour length is not violated; and (iv) the vehicle capacity is not violated.The selected edge is deleted from SL (step 9) only if: (a) the corresponding merge is performed (step 7); or (b) at least one of the CWS constraints ((i)–(iv)) are violated (in step 6). Otherwise, the edge becomes temporarily ineligible (step 10), but it is not removed from the list since subsequent merges might restore eligibility. This can occur when a different retail center is merged into a route, increasing the available processed inventory for subsequent consumers. For example, when selecting an edge $(i,j)$, the evolving route of i may have insufficient processed inventory for customer j at this time. However, if in subsequent iterations route i is merged with another route containing retailers, the evolving route of i may then be able to serve customer j.When a merge is successfully performed (step 7), the entire SL becomes eligible (step 8), since a new inventory scenario is generated.At the end of this stage, all the consumers are supplied by the retail centers, guaranteeing a feasible final solution. Notice that this is achieved without solving separate assignment and routing problems, as done in Abdulkader et al. [1].
- Finally, the third stage (CWS${}_{2}$) tries to improve the solution generated in the previous step. To do that, the algorithm cycles through the SL list, which includes the remaining saving edges that were discarded in the previous step, with the aim of identifying more beneficial merges. Unlike the procedure used in the CWS${}_{1}$ stage, in this phase all the customers are already assigned to a retail center, so step 5 of Figure 3 is not required. Hence, all edges are eligible. The process attempts all the available merging possibilities which may improve the solution. Each time a new edge is selected from the SL list, it is removed from the list, whether the corresponding merge is performed or not–due to it violating any of the constraints (i)–(iv). In each new iteration, the highest saving edge is selected to restart the merge process. This process is repeated until SL is empty. At the end of the procedure, a feasible solution is generated, without the necessity of repair operations.

#### 4.2. Introducing a Local Search

#### 4.3. Extending to a Biased-Randomized Algorithm

Algorithm 1: brSelection. |

Data: savings list $SL$, parameter $\beta \in \left(\right)open="["\; close="]">0,1$ |

1$l\leftarrow $ getNumberOfEligibleEdgesFromList ($SL$); |

2 Randomly select position $x\in \{1,\dots ,l\}$ according to distribution Geom ($\beta $); |

3$e\leftarrow $ selectTheXthEligibleEdgeFromList ($x,SL$); |

4returne; |

_{BRLH}).

Algorithm 2: BRLH. |

Data: set of nodes V, geometric distribution parameter $\beta $ |

1$sol\leftarrow $ createDummySolution (V); |

2$sol\leftarrow $ CWS${}_{1}$ ($sol$, $\beta $); |

3$sol\leftarrow $ CWS${}_{2}$ ($sol$, $\beta $); |

4$sol\leftarrow $ localSearch ($sol$); |

5return$sol$ |

#### 4.4. Extending to a Simheuristic Approach

_{BRLH}). Initially, a solution is generated by our BRLH in line 1 (Algorithm 2), by employing the greedy approach (i.e., $\beta \approx 1$). A short simulation is then performed on this initial solution (line 2), in order to estimate its average stochastic cost. This initial solution is set as the best-found stochastic solution cost (line 4). While the termination criterion is not met (line 6), different solutions are generated by BRLH (line 7). The deterministic cost of the initial solution is considered for guiding the search. Therefore, a solution is accepted for being submitted to the simulation module (line 10) only if its deterministic cost is smaller than the best-found deterministic solution cost plus $m\%$ of its value (line 9). This solution filtering approach reduces the amount of time spent on testing unpromising solutions in computationally expensive simulations. Moreover, by allowing the acceptance of moderately worse solutions, controlled by the parameter m, a better exploration of the solution space can be achieved [63]. At this stage, ${q}_{short}$ Monte Carlo simulation runs are used to test the accepted solution. Each simulation run replaces the deterministic travel times of a solution with randomly sampled ones–according to the assumed probability distribution. From this complete simulation process, the average stochastic cost of each solution is computed. Every time a new best stochastic cost is found (line 15), this solution is introduced into a pool of ‘elite’ solutions E (line 17). This process is repeated while the termination criterion is not met. On this reduced set of solutions, ${q}_{long}$ Monte Carlo simulation runs are performed (line 23) in order to generate more accurate results for solutions in stochastic environments. During the simulation process we also obtain an estimate of the reliability rate of a solution [64]. This estimate is computed as the rate at which all routes show completion times lower than the maximum allowed travel time. At the end, the set of elite solutions is sorted in descending order of their expected cost (line 25), and the best-found stochastic solution is provided to the manager.

Algorithm 3: Sim_{BRLH}. |

## 5. Results and Discussion

_{BRLH}–in which $\beta $ is (uniformly) randomly selected in the interval $[0.45,0.75]$–with the solutions obtained by the two-phase heuristic (AH) and multi-ant colony metaheuristic (MAC) proposed by Abdulkader et al. [1]. Their methodologies were performed on four 2.1 GHz processors with 16-cores each and a total of 256 GB RAM. That is, we initially focus on comparing each algorithm in terms of deterministic travel cost. Table 3, Table 4 and Table 5 present the results obtained for tight, relaxed, and abundant inventory scenarios, respectively. For each problem instance (I), we present results for: the cost of the best-found solution obtained by the different methodologies; the average cost of our MS

_{BRLH}; the CPU time (in seconds) required by each methodology; and their percentage gaps. The best results returned by the solution methodologies are highlighted in bold. Figure 4 and Figure 5 present how both the gap and the cost of the solutions, i.e., the objective function (OF) value, behave according to the employed solution approach and inventory scenario, respectively.

_{BRLH}algorithm is able to improve previous results (from LH) by 9–12%, on average (column gap (2)–(1)). When comparing the MS

_{BRLH}with the AH heuristic (column gap (3)–(2)), our approach is able to reduce solution costs by up to 26% in short computational times (about 18 s on average). On the other hand, when comparing our results with those generated by the MAC approach (column gap (4)–(2)), our solutions are between 9% and 21% worse, on average. Particularly, in the abundant inventory scenario, MS

_{BRLH}’s results are only 9% worse than MAC. Notice, however, that the processing time required by MAC is substantially larger in all inventory scenarios. By analyzing Figure 4, it is evident that MS

_{BRLH}performs better in the abundant inventory scenario, being able to find one better solution and several others with a maximum gap of 8%. Moreover, we can observe a variability of around 10% in the gap between MS

_{BRLH}and MAC on average. This variability is reduced to around 8% for the relaxed and abundant scenarios. These results demonstrate the robustness of our solution approach for the deterministic case. When analyzing Figure 5, which presents the overall performance of each solution approach for each inventory scenario, we can observe that our multi-start strategy is more efficient than both LH and AH heuristics, by generating solutions with a lower cost. To complement these box-plots, an ANOVA test was run for each inventory scenario. The p-values associated with the tight, relaxed, and abundant inventory scenarios were, respectively, of $0.001$, $0.000$, and $0.000$. Also, the Fisher’s LSD test suggests significant differences in all tree scenarios between MAC and AH, between MS-BRLH and AH, as well as between MAC and LH. However, cost differences between MAC and MS-BRLH were not significant in any scenario, despite the fact that MAC employed a noticeably higher amount of computing time than MS-BRLH.

_{BRLH}with the best-known solutions, in terms of gap. As introduced, these instances require approximately 15 s of processing time, given their magnitude.

_{BRLH}) against the solutions generated for the stochastic scenario (Sim

_{BRLH}). Since the only difference between the deterministic and stochastic scenarios is that stochastic delays are added to edge traversal times, we can consider the deterministic cost of the best solutions generated by MS

_{BRLH}as a lower bound (LB) of the stochastic travel times of the best Sim

_{BRLH}solution. Moreover, since MS

_{BRLH}does not account for stochastic travel times, we can consider the stochastic travel time of the best MS

_{BRLH}solution as an upper bound (UB) for the stochastic travel time of the best Sim

_{BRLH}solutions. Table 6 provides both LBs and UBs values and the best-found stochastic travel times obtained by our Sim

_{BRLH}. The solutions reported in the Sim

_{BRLH}column are the best-found stochastic travel times.

_{BRLH}solution costs are between the LB and the UB, as expected. For 24 problem instances, the solution returned by our simheuristic is better than the best deterministic solution when it is tested in the stochastic scenario (the $UB$ column). From this, we can assert that our Sim

_{BRLH}is able to generate competitive results for the stochastic scenario. The reliability value is calculated by simulation for each solution and represents the probability that all routes are completed within maximum tour duration. For visualizing this trade-off between the deterministic cost of the solutions and their reliability rate, which are conflicting objectives, a Pareto frontier of non-dominated is presented. Accordingly, Figure 7a–c present the non-dominated solutions for three different instances ($b17$, $b37$, and $b57$), each one belonging to a different inventory scenario. A solution is non-dominated if, no other solution has a greater reliability and a lower or equal travel cost, or if no other solution has a lower travel cost and a greater or equal reliability level. The $b17$ solution was randomly chosen, while the $b37$ and $b57$ solutions are for the same problem but set in the two other inventory scenarios. The square orange dot represents the best deterministic solution found by our MS

_{BRLH}, while the remaining ones, round and blue, represent different solutions with a higher reliability rate, but with higher operating costs.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The optimal solution routes for Table 1.

**Figure 4.**Gap between our best-found solutions (from MS

_{BRLH}) and the MAC’s results, for each inventory scenario.

**Figure 5.**Comparison of the solutions cost (OF value) from each solving approach, for each inventory scenario.

**Figure 7.**Set of non-dominated solutions of problem instances $b17$ (

**a**), $b37$ (

**b**) and $b57$ (

**c**). (

**a**) Non-dominated solutions for problem instance $b17$ (tight inventory). (

**b**) Non-dominated solutions for problem instance $b37$ (relaxed inventory). (

**c**) Non-dominated solutions for problem instance $b57$ (abundant inventory).

Node | X | Y | ${\mathit{OP}}_{\mathit{i}}$ | ${\mathit{OD}}_{1}$ | ${\mathit{OD}}_{2}$ | ${\mathit{OD}}_{3}$ | Inventory | ||
---|---|---|---|---|---|---|---|---|---|

${\mathit{P}}_{\mathbf{1}}$ | ${\mathit{P}}_{\mathbf{2}}$ | ${\mathit{P}}_{\mathbf{3}}$ | |||||||

0 | 48 | 23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 91 | 20 | 44 | 0 | 0 | 0 | 1 | 0 | 2 |

2 | 25 | 9 | 36 | 0 | 0 | 0 | 1 | 0 | 1 |

3 | 56 | 68 | 49 | 0 | 0 | 0 | 1 | 2 | 0 |

4 | 71 | 1 | 43 | 0 | 0 | 0 | 1 | 2 | 1 |

5 | 26 | 94 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

6 | 30 | 15 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

7 | 4 | 51 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

8 | 35 | 78 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

9 | 79 | 72 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

10 | 16 | 33 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

11 | 61 | 6 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

12 | 78 | 89 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

13 | 77 | 61 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

$\mathit{\beta}$ | m | $\mathit{\mu}$ | $\mathit{\sigma}$ | ${\mathit{q}}_{\mathit{short}}$ | ${\mathit{q}}_{\mathit{long}}$ | ${\mathit{time}}_{\mathit{max}}$ |
---|---|---|---|---|---|---|

$[0.45,0.75]$ | $20\%$ | 0 | $\{1.55,1.9,2.5\}$ | 100 | 1000 | $(r+c)\times 0.342$ |

**Table 3.**Comparison of the results obtained by our methodologies (LH and MS

_{BRLH}) with those obtained by Abdulkader et al. [1]’s methods (AH and MAC) in the tight inventory scenario.

I | $\left|\mathit{R}\right|$ | $\left|\mathit{C}\right|$ | 1 | 2 | 3 | 4 | Avg. Cost | Time (sec.) | Gap | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

LH | MS_{BRLH} | AH | MAC | (2) | (1) | (2) | (3) | (4) | (1)–(2) | (3)–(2) | (4)–(2) | |||

b1 | 10 | 25 | 1277.5 | 1110.9 | 1631.6 | 1002.5 | 1119.6 | 0 | 7 | 0 | 7 | −13% | −32% | 11% |

b2 | 10 | 50 | 1641.1 | 1378.2 | 2057.5 | 1192.0 | 1392.0 | 0 | 8 | 0 | 47 | −16% | −33% | 16% |

b3 | 10 | 75 | 2663.7 | 2437.2 | 3006.2 | 1815.4 | 2450.7 | 0 | 25 | 0 | 79 | −9% | −19% | 34% |

b4 | 10 | 100 | 2415.1 | 1930.3 | 2830.2 | 1529.0 | 1980.7 | 0 | 15 | 0 | 286 | −20% | −32% | 26% |

b5 | 10 | 150 | 2678.2 | 2395.3 | 3478.7 | 1905.2 | 2408.9 | 0 | 38 | 0 | 576 | −11% | −31% | 26% |

b6 | 15 | 25 | 1540.5 | 1389.4 | 1774.4 | 1313.7 | 1400.6 | 0 | 4 | 0 | 7 | −10% | −22% | 6% |

b7 | 15 | 50 | 2059.0 | 1769.3 | 2461.8 | 1522.3 | 1803.7 | 0 | 0 | 0 | 44 | −14% | −28% | 16% |

b8 | 15 | 75 | 3105.3 | 2620.5 | 3545.1 | 2101.8 | 2630.3 | 0 | 6 | 0 | 131 | −16% | −26% | 25% |

b9 | 15 | 100 | 3121.5 | 2836.9 | 3529.0 | 2329.5 | 2860.5 | 0 | 1 | 0 | 209 | −9% | −20% | 22% |

b10 | 15 | 150 | 4292.4 | 3787.2 | 4916.8 | 3012.2 | 3797.4 | 0 | 45 | 0 | 430 | −12% | −23% | 26% |

b11 | 20 | 25 | 2035.2 | 1817.1 | 2432.6 | 1611.3 | 1838.4 | 0 | 10 | 0 | 11 | −11% | −25% | 13% |

b12 | 20 | 50 | 2335.4 | 2109.1 | 2695.3 | 1800.9 | 2112.8 | 0 | 1 | 0 | 50 | −10% | −22% | 17% |

b13 | 20 | 75 | 3212.7 | 2765.1 | 3936.7 | 2406.0 | 2796.2 | 0 | 12 | 0 | 127 | −14% | −30% | 15% |

b14 | 20 | 100 | 3025.2 | 2842.7 | 3826.1 | 2483.8 | 2881.4 | 0 | 20 | 0 | 327 | −6% | −26% | 14% |

b15 | 20 | 150 | 3934.3 | 3308.0 | 4496.1 | 2679.2 | 3332.1 | 0 | 50 | 0 | 708 | −16% | −26% | 23% |

b16 | 25 | 25 | 2019.4 | 1847.2 | 2254.9 | 1669.6 | 1858.0 | 0 | 8 | 0 | 13 | −9% | −18% | 11% |

b17 | 25 | 50 | 2665.9 | 2434.8 | 3020.8 | 1965.6 | 2442.8 | 0 | 17 | 0 | 46 | −9% | −19% | 24% |

b18 | 25 | 75 | 3207.6 | 2853.4 | 3963.5 | 2449.8 | 2885.8 | 0 | 18 | 0 | 136 | −11% | −28% | 16% |

b19 | 25 | 100 | 4064.0 | 3551.0 | 4933.9 | 2788.5 | 3588.7 | 0 | 42 | 0 | 257 | −13% | −28% | 27% |

b20 | 25 | 150 | 3782.7 | 3512.8 | 4721.3 | 2890.3 | 3525.5 | 0 | 17 | 0 | 712 | −7% | −26% | 22% |

Average | 0 | 17 | 0 | 210 | −12% | −26% | 19% |

**Table 4.**Comparison of the results obtained by our methodologies (LH and MS

_{BRLH}) with those obtained by Abdulkader et al. [1]’s methods (AH and MAC) in the relaxed inventory scenario.

I | $\left|\mathit{R}\right|$ | $\left|\mathit{C}\right|$ | 1 | 2 | 3 | 4 | Avg. Cost | Time (sec.) | Gap | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

LH | MS_{BRLH} | AH | MAC | (2) | (1) | (2) | (3) | (4) | (1)–(2) | (3)–(2) | (4)–(2) | |||

b21 | 10 | 25 | 1233.0 | 1030.0 | 1571.6 | 879.2 | 1048.9 | 0 | 5 | 0 | 10 | −16% | −34% | 17% |

b22 | 10 | 50 | 1490.8 | 1275.7 | 1920.6 | 1083.7 | 1293.9 | 0 | 4 | 0 | 85 | −14% | −34% | 18% |

b23 | 10 | 75 | 2468.0 | 2021.9 | 2699.2 | 1591.5 | 2049.5 | 0 | 18 | 0 | 167 | −18% | −25% | 27% |

b24 | 10 | 100 | 1885.0 | 1632.2 | 2305.1 | 1437.7 | 1645.3 | 0 | 7 | 0 | 528 | −13% | −29% | 14% |

b25 | 10 | 150 | 1998.6 | 1980.4 | 2700.4 | 1520.5 | 1981.8 | 0 | 39 | 0 | 1836 | −1% | −27% | 30% |

b26 | 15 | 25 | 1591.4 | 1268.0 | 1665.2 | 1180.8 | 1308.9 | 0 | 4 | 0 | 11 | −20% | −24% | 7% |

b27 | 15 | 50 | 1940.7 | 1652.1 | 2320.7 | 1329.3 | 1660.9 | 0 | 9 | 0 | 73 | −15% | −29% | 24% |

b28 | 15 | 75 | 2436.3 | 2101.7 | 3016.5 | 1692.4 | 2160.4 | 0 | 3 | 0 | 279 | −14% | −30% | 24% |

b29 | 15 | 100 | 2648.3 | 2395.4 | 3302.4 | 2016.4 | 2412.5 | 0 | 25 | 0 | 567 | −10% | −27% | 19% |

b30 | 15 | 150 | 3373.2 | 2819.0 | 3919.0 | 2399.6 | 2847.6 | 0 | 18 | 0 | 1407 | −16% | −28% | 17% |

b31 | 20 | 25 | 1835.5 | 1679.1 | 1993.5 | 1495.8 | 1682.7 | 0 | 11 | 0 | 16 | −9% | −16% | 12% |

b32 | 20 | 50 | 2320.5 | 1960.7 | 2713.0 | 1656.9 | 1965.7 | 0 | 6 | 0 | 76 | −16% | −28% | 18% |

b33 | 20 | 75 | 2404.6 | 2267.2 | 3393.3 | 1799.6 | 2269.8 | 0 | 28 | 0 | 262 | −6% | −33% | 26% |

b34 | 20 | 100 | 2751.9 | 2469.3 | 3127.5 | 2018.5 | 2490.6 | 0 | 34 | 0 | 740 | −10% | −21% | 22% |

b35 | 20 | 150 | 3157.4 | 2818.0 | 3742.2 | 2291.0 | 2824.5 | 0 | 32 | 0 | 2141 | −11% | −25% | 23% |

b36 | 25 | 25 | 1844.0 | 1683.7 | 2032.1 | 1550.0 | 1700.9 | 0 | 13 | 0 | 15 | −9% | −17% | 9% |

b37 | 25 | 50 | 2663.9 | 2322.3 | 3130.5 | 1939.5 | 2357.5 | 0 | 22 | 0 | 73 | −13% | −26% | 20% |

b38 | 25 | 75 | 2790.7 | 2559.7 | 3433.2 | 2088.6 | 2569.7 | 0 | 30 | 0 | 283 | −8% | −25% | 23% |

b39 | 25 | 100 | 3352.9 | 3038.1 | 3824.5 | 2244.1 | 3053.6 | 0 | 27 | 0 | 656 | −9% | −21% | 35% |

b40 | 25 | 150 | 2971.1 | 2830.6 | 3447.9 | 2229.4 | 2837.7 | 0 | 37 | 0 | 2077 | −5% | −18% | 27% |

Average | 0 | 19 | 0 | 565 | −12% | −26% | 21% |

**Table 5.**Comparison of the results obtained by our methodologies (LH and MS

_{BRLH}) with those obtained by Abdulkader et al. [1]’s methods (AH and MAC) in the abundant inventory scenario.

I | $\left|\mathit{R}\right|$ | $\left|\mathit{C}\right|$ | 1 | 2 | 3 | 4 | Avg. Cost | Time (sec.) | Gap | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

LH | MS_{BRLH} | AH | MAC | (2) | (1) | (2) | (3) | (4) | (1)–(2) | (3)–(2) | (4)–(2) | |||

b41 | 10 | 25 | 805.4 | 760.5 | 897.6 | 711.3 | 760.5 | 0 | 1 | 0 | 16 | −6% | −15% | 7% |

b42 | 10 | 50 | 1014.2 | 870.4 | 1287.8 | 875.2 | 871.2 | 0 | 8 | 0 | 143 | −14% | −32% | −1% |

b43 | 10 | 75 | 1463.5 | 1259.6 | 1531.1 | 1132.1 | 1266.4 | 0 | 9 | 0 | 358 | −14% | −18% | 11% |

b44 | 10 | 100 | 1379.4 | 1284.9 | 1636.5 | 1224.1 | 1294.1 | 0 | 30 | 0 | 978 | −7% | −21% | 5% |

b45 | 10 | 150 | 1499.3 | 1364.3 | 1551.8 | 1273.9 | 1385.3 | 0 | 43 | 0 | 2085 | −9% | −12% | 7% |

b46 | 15 | 25 | 1137.5 | 1024.3 | 1264.3 | 996.9 | 1028.8 | 0 | 6 | 0 | 22 | −10% | −19% | 3% |

b47 | 15 | 50 | 1247.6 | 1135.1 | 1488.1 | 1080.3 | 1141.0 | 0 | 17 | 0 | 159 | −9% | −24% | 5% |

b48 | 15 | 75 | 1595.3 | 1355.2 | 1815.2 | 1252.4 | 1361.2 | 0 | 17 | 0 | 559 | −15% | −25% | 8% |

b49 | 15 | 100 | 2021.3 | 1777.8 | 2242.4 | 1594.0 | 1798.9 | 0 | 24 | 0 | 1167 | −12% | −21% | 12% |

b50 | 15 | 150 | 2059.6 | 1869.2 | 2459.5 | 1691.4 | 1873.6 | 0 | 26 | 0 | 4126 | −9% | −24% | 11% |

b51 | 20 | 25 | 1507.2 | 1414.7 | 1660.9 | 1302.9 | 1418.2 | 0 | 11 | 0 | 33 | −6% | −15% | 9% |

b52 | 20 | 50 | 1464.7 | 1366.8 | 1740.7 | 1301.0 | 1368.4 | 0 | 7 | 0 | 156 | −7% | −21% | 5% |

b53 | 20 | 75 | 1797.7 | 1591.5 | 2096.8 | 1421.8 | 1599.7 | 0 | 18 | 0 | 605 | −11% | −24% | 12% |

b54 | 20 | 100 | 2066.2 | 1881.8 | 2226.4 | 1640.6 | 1883.9 | 0 | 23 | 0 | 1370 | −9% | −15% | 15% |

b55 | 20 | 150 | 2214.0 | 2025.1 | 2518.2 | 1763.3 | 2030.0 | 0 | 52 | 0 | 5321 | −9% | −20% | 15% |

b56 | 25 | 25 | 1423.2 | 1368.1 | 1550.7 | 1311.6 | 1372.3 | 0 | 7 | 0 | 36 | −4% | −12% | 4% |

b57 | 25 | 50 | 1670.8 | 1559.7 | 1835.4 | 1468.1 | 1570.2 | 0 | 11 | 0 | 203 | −7% | −15% | 6% |

b58 | 25 | 75 | 2047.5 | 1845.3 | 2276.9 | 1654.9 | 1847.5 | 0 | 7 | 0 | 791 | −10% | −19% | 12% |

b59 | 25 | 100 | 1856.4 | 1797.8 | 2061.9 | 1575.7 | 1801.0 | 0 | 15 | 0 | 1262 | −3% | −13% | 14% |

b60 | 25 | 150 | 1968.1 | 1837.3 | 2347.8 | 1653.3 | 1849.1 | 0 | 60 | 0 | 4549 | −7% | −22% | 11% |

Average | 0 | 19 | 0 | 1197 | −9% | −19% | 9% |

**Table 6.**Analysis of the results obtained by our Sim

_{BRLH}on scenarios of tight, relaxed and abundant inventory.

I | Tight Inventory | I | Relaxed Inventory | I | Abundant Inventory | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

LB | Sim_{BRLH} | UB | LB | Sim_{BRLH} | UB | LB | Sim_{BRLH} | UB | |||

b1 | 1110.9 | 2164.5 | 2167.0 | b21 | 1030.0 | 2096.3 | 2154.6 | b41 | 760.5 | 1731.0 | 1858.7 |

b2 | 1378.2 | 2084.7 | 2084.7 | b22 | 1275.7 | 1996.6 | 1996.6 | b42 | 870.4 | 1574.1 | 1636.8 |

b3 | 2437.2 | 3465.3 | 3465.3 | b23 | 2021.9 | 3043.9 | 3043.9 | b43 | 1259.6 | 2239.4 | 2239.4 |

b4 | 1930.3 | 3184.3 | 3184.3 | b24 | 1632.2 | 2891.1 | 2989.5 | b44 | 1284.9 | 2553.4 | 2553.4 |

b5 | 2395.3 | 3790.2 | 3790.2 | b25 | 1980.4 | 3363.6 | 3363.6 | b45 | 1364.3 | 2753.9 | 2753.9 |

b6 | 1389.4 | 2628.3 | 2775.6 | b26 | 1268.0 | 2487.9 | 2487.9 | b46 | 1024.3 | 2269.2 | 2269.2 |

b7 | 1769.3 | 2544.9 | 2544.9 | b27 | 1652.1 | 2446.5 | 2446.5 | b47 | 1135.1 | 1919.3 | 1944.4 |

b8 | 2620.5 | 3715.6 | 3715.6 | b28 | 2101.7 | 3178.6 | 3178.6 | b48 | 1355.2 | 2420.6 | 2442.4 |

b9 | 2836.9 | 4209.9 | 4209.9 | b29 | 2395.4 | 3725.6 | 3725.6 | b49 | 1777.8 | 3138.7 | 3146.9 |

b10 | 3787.2 | 5258.7 | 5258.7 | b30 | 2819.0 | 4269.7 | 4269.7 | b50 | 1869.2 | 3308.8 | 3312.6 |

b11 | 1817.1 | 3261.5 | 3401.5 | b31 | 1679.1 | 3099.3 | 3099.3 | b51 | 1414.7 | 2814.0 | 3095.4 |

b12 | 2109.1 | 2978.8 | 2983.5 | b32 | 1960.7 | 2819.4 | 2832.9 | b52 | 1366.8 | 2233.4 | 2250.7 |

b13 | 2765.1 | 3917.1 | 3917.1 | b33 | 2267.2 | 3412.4 | 3412.4 | b53 | 1591.5 | 2732.9 | 2732.9 |

b14 | 2842.7 | 4312.3 | 4312.3 | b34 | 2469.3 | 3891.5 | 3914.8 | b54 | 1881.8 | 3289.1 | 3304.1 |

b15 | 3308.0 | 4813.7 | 4813.7 | b35 | 2818.0 | 4311.4 | 4311.4 | b55 | 2025.1 | 3510.5 | 3510.5 |

b16 | 1847.2 | 3460.4 | 3591.2 | b36 | 1683.7 | 3267.0 | 3273.6 | b56 | 1368.1 | 2944.6 | 3180.0 |

b17 | 2434.8 | 3377.2 | 3393.7 | b37 | 2322.3 | 3266.1 | 3268.6 | b57 | 1559.7 | 2497.4 | 2497.4 |

b18 | 2853.4 | 4082.6 | 4082.6 | b38 | 2559.7 | 3782.0 | 3788.0 | b58 | 1845.3 | 3065.6 | 3065.6 |

b19 | 3551.0 | 5039.2 | 5039.2 | b39 | 3038.1 | 4548.9 | 4548.9 | b59 | 1797.8 | 3287.3 | 3287.3 |

b20 | 3512.8 | 5088.2 | 5088.2 | b40 | 2830.6 | 4384.5 | 4385.9 | b60 | 1837.3 | 3380.0 | 3380.0 |

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## Share and Cite

**MDPI and ACS Style**

Martins, L.d.C.; Bayliss, C.; Copado-Méndez, P.J.; Panadero, J.; Juan, A.A.
A Simheuristic Algorithm for Solving the Stochastic Omnichannel Vehicle Routing Problem with Pick-up and Delivery. *Algorithms* **2020**, *13*, 237.
https://doi.org/10.3390/a13090237

**AMA Style**

Martins LdC, Bayliss C, Copado-Méndez PJ, Panadero J, Juan AA.
A Simheuristic Algorithm for Solving the Stochastic Omnichannel Vehicle Routing Problem with Pick-up and Delivery. *Algorithms*. 2020; 13(9):237.
https://doi.org/10.3390/a13090237

**Chicago/Turabian Style**

Martins, Leandro do C., Christopher Bayliss, Pedro J. Copado-Méndez, Javier Panadero, and Angel A. Juan.
2020. "A Simheuristic Algorithm for Solving the Stochastic Omnichannel Vehicle Routing Problem with Pick-up and Delivery" *Algorithms* 13, no. 9: 237.
https://doi.org/10.3390/a13090237