A Variable Neighborhood Descent Matheuristic for the Drone Routing Problem with Beehives Sharing
Abstract
:1. Introduction
 We formulate the drone delivery with shared BHs as a locationrouting mathematical model where the nonlinear nature of energy consumption rates in drone battery and the uncertainty of flight duration are taken into account.
 We develop an efficient matheuristic approach integrating the VND with a cut generation procedure that provides promising results outperforming the Gurobi offtheshelf solver.
2. Literature Review
3. Drone Routing Problem with Shared Beehives
4. The VNDBased Matheuristic
 (1)
 The current solution p is not an energyfeasible solution and the battery capacity for some of the drone tours is violated (we refer to the tours exceeding the battery charge as the “infeasible tours”). In this case, corresponding to each infeasible tour ${\mathcal{P}}_{\pi},\phantom{\rule{0.166667em}{0ex}}\pi \in \{1,\cdots ,{N}_{d}\}$ in p, a valid nogood cut (38) is generated and added to the master problem.$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \mathcal{NG}\left(\pi \right):\phantom{\rule{0.166667em}{0ex}}\sum _{\begin{array}{c}(f,\phantom{\rule{0.166667em}{0ex}}j)\in {\mathcal{P}}_{\pi}\\ f\in D\end{array}}\sum _{\begin{array}{c}r\in L\end{array}}(1{w}_{fj}^{r})+\sum _{\begin{array}{c}(i,\phantom{\rule{0.166667em}{0ex}}j)\in {\mathcal{P}}_{\pi}\\ i,\phantom{\rule{0.166667em}{0ex}}j\in C\end{array}}\sum _{\begin{array}{c}r\in L\end{array}}(1{y}_{ij}^{r})+\sum _{\begin{array}{c}(i,\phantom{\rule{0.166667em}{0ex}}f)\in {\mathcal{P}}_{\pi}\\ f\in D\end{array}}(1{w}_{if}^{1})\ge rhs\hfill \end{array}$$Note that $rhs$ is a prespecified integer input parameter. From now on, we refer to (38) as the cut type I. Any cut in the form of (38) is, in fact, a valid cut that ensures corresponding to each infeasible tour ${\mathcal{P}}_{\pi}$, at least $rhs$ customers/BHs are removed. Obviously, the higher the value of $rhs$ is, the more emphasis on seeking diverse solutions is imposed. Once the cuts are generated, we can specify the structure of neighborhood as follows:Let ${\left\{{\mathcal{P}}_{\pi}\right\}}_{\pi =1}^{s}$ be the set of infeasible tours in p, the neighborhood ${\mathcal{N}}_{rhs}\left(p\right)\phantom{\rule{0.166667em}{0ex}}({\mathcal{N}}_{rhs}\subseteq X)$ is defined as ${\mathcal{N}}_{rhs}\left(p\right)=\left\{\overline{p}\right\overline{p}\in {\bigcap}_{\pi =1}^{s}\mathcal{NG}\left(\pi \right)\}$ where X denotes the master problem solution space. We should note that the parameter $rhs$ is interpreted as the size of neighborhood ${\mathcal{N}}_{rhs}$.
 (2)
 The current solution p is an energyfeasible solution, not necessarily optimal. Therefore, we cut off the current solution from the search space introducing a single nogood cut cut type II removing the current solution from the search space.$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \mathcal{NG}\left(1\right)\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\sum _{(i,j)\in S}\sum _{r\in L}(1{y}_{ij}^{r})+\sum _{(i,j)\notin S}\sum _{r\in L}{y}_{ij}^{r}\ge rhs\hfill \end{array}$$
Algorithm 1:Generate cut $\left(p\right)$ procedure 

Algorithm 2:Pseudocode of build route $\left(p\right)$ procedure 

 (i)
 The new solution ${p}^{\prime}$ is energyfeasible, therefore, we replace the current solution p with ${p}^{\prime}$ and update the incumbent, if necessary (Lines 13–17).
 (ii)
 The new solution ${p}^{\prime}$ is not energyfeasible but the amount of violation is within an acceptable prespecified threshold $\alpha $. In such case, we replace the current solution p with ${p}^{\prime}$ (Line 19). This can be regarded as a diversification mechanism to allow a controlled exploration of infeasible promising solutions.
 (iii)
 None of the two previous cases holds. Hence, we move to the next neighborhood (Line 21).
Algorithm 3:VNDM 

5. Computational Experiments
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Sets  
C  Set of customers 
D  Set of potential BHs 
${D}^{\prime}$  Set of two auxiliary BHs $({D}^{\prime}=\{0,\phantom{\rule{0.166667em}{0ex}}m+1\})$ 
$L=\{1,\cdots ,N\}$  Set of positions, indexed by r 
Parameters  
${N}_{d}$  Number of drones 
${d}_{i}$  Demand of customer i 
Q  Drone payload capacity 
k  Energy consumptionrelated parameter $k=\sqrt{\frac{{g}^{3}}{2\rho \xi h}}$ 
W  Drone frame mass 
M  Drone battery mass 
g  Force applied by gravity per unit mass 
$\rho $  Fluid density of air 
$\xi $  Area of spinning blade disc 
h  Number of rotors 
$\Gamma $  Total mass of an unloaded drone $(W+M)$ 
${N}_{BH}^{f}$  Maximum number of drones that BH f can host 
${N}_{s}$  Maximum number of BHs to use 
${t}_{ij}$  Travel time along arc $(i,j)$ 
B  Drone battery charge 
${M}_{fj}^{\prime},\phantom{\rule{0.166667em}{0ex}}{M}_{ij}^{\prime \prime}$  BigM numbers 
Decision variables  
${x}_{i}^{r}$  Binary variable that takes 1 if customer i is visited in position r; otherwise, 0 
${y}_{ij}^{r}$  Binary variable that takes 1 if customer j is visited right after customer i in position r and there are exactly $r1$ 
customers left to be visited after; otherwise, 0  
${w}_{fj}^{r}$  Binary variable that takes 1 if customer j is visited right after BH f and there are exactly $r1,\phantom{\rule{0.166667em}{0ex}}(r\ne 1)$ 
customers left to be visited after; otherwise, 0  
${v}_{fj}$  Total load carried by the drone traveling over arc $(f,\phantom{\rule{0.166667em}{0ex}}j)$ upon departure from BH f 
${u}_{ij}$  Total load carried by the drone traveling over arc $(i,\phantom{\rule{0.166667em}{0ex}}j)$ after serving customer i 
${z}_{f}$  Binary variable takes 1 if BH f is used; otherwise, 0 
${e}_{i}$  The accumulated energy consumption upon arrival at node $i\in C\cup D$ 
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VNDM  AOA  

Instance  $\left\mathbf{C}\right$  $\left\mathbf{D}\right$  ${\mathbf{N}}_{\mathbf{d}}$  ${\mathbf{N}}_{\mathbf{BH}}^{\mathbf{f}}$  Obj  CPU  Obj  CPU  LB  $\mathbf{Gap}(\%)$ 
A1101  10  5  2  2  2.02  17.89  2.02  7.09  0.00  
A1102  1.80  1.01  1.80  2.02  0.00  
A1103  1.84  43.51  1.84  4.11  0.00  
A1104  1.57  2.32  1.57  2.77  0.00  
A1105  1.96  0.55  1.96  2.75  0.00  
A1151  15  5  3  2  2.09  1.87  2.09  33.38  0.00  
A1152  2.52  2.65  2.52  48.49  0.00  
A1153  2.00  3.00  2.00  37.94  0.00  
A1154  2.24  1.49  2.24  42.86  0.00  
A1155  2.35  4.03  2.33  47.55  0.85  
A1201  20  5  4  3  2.60  13.10  2.60  336.83  0.00  
A1202  2.67  6.74  2.67  442.88  0.00  
A1203  2.68  8.33  2.68  496.25  0.00  
A1204  2.38  6.58    TL  2.38  0.00  
A1205  2.72  9.35    TL  2.61  4.04  
A1251  25  5  7  5  4.42  11.24    TL  4.01  9.28 
A1252  4.21  64.47    TL  4.20  0.24  
A1253  4.25  TL    TL  3.60  15.29  
A1254  3.91  15.11    TL  3.89  0.51  
A1255  4.21  4.98    TL  3.99  5.23  
A1301  30  5  8  6  4.31  44.59    TL  4.20  2.55 
A1302  3.56  28.57    TL  3.51  1.40  
A1303  4.89  236.27    TL  4.80  1.84  
A1304  4.48  2.74    TL  4.39  2.01  
A1305  3.66  50.79    TL  3.60  1.64  
A1351  35  5  8  6  4.61  166.97    TL  4.55  1.30 
A1352  5.86  384.00    TL  5.59  4.61  
A1353  4.95  66.00    TL  4.87  1.62  
A1354  4.91  52.98    TL  4.85  1.22  
A1355  5.57  39.88    TL  5.51  1.08  
A1401  40  5  8  6  7.14  235.75    TL  6.89  3.50 
A1402  5.65  181.48    TL  5.61  0.71  
A1403  7.01  TL    TL  6.82  2.71  
A1404  6.66  TL    TL  6.20  6.91  
A1405  6.18  220.52    TL  5.95  3.72  
Average  97.96  2.06 
VNDM  AOA  

Instance  $\left\mathbf{C}\right$  $\left\mathbf{D}\right$  ${\mathbf{N}}_{\mathbf{d}}$  ${\mathbf{N}}_{\mathbf{BH}}^{\mathbf{f}}$  Obj  CPU  Obj  CPU  LB  $\mathbf{Gap}(\%)$ 
A1101  1.64  35.67  1.64  2.85  0.00  
A1102  10  5  2  2  1.56  7.30  1.56  2.68  0.00  
A1103  1.40  3.02  1.40  1.87  0.00  
A1104  1.13  2.26  1.13  1.96  0.00  
A1105  1.86  3.31  1.86  2.81  0.00  
A1151  2.05  2.09  2.05  35.10  0.00  
A1152  15  5  3  2  2.25  3.24  2.25  30.17  0.00  
A1153  2.13  5.30  2.13  32.12  0.00  
A1154  2.35  3.94  2.35  41.78  0.00  
A1155  1.87  3.31  1.87  51.74  0.00  
A1201  2.59  5.07  2.59  305.30  0.00  
A1202  20  5  4  3  2.56  8.78  2.56  440.93  0.00  
A1203  2.68  10.75  2.68  425.31  0.00  
A1204  2.49  6.50    TL  2.19  12.16  
A1205  2.83  9.38    TL  2.31  18.49  
A1251  3.41  1.71    TL  3.00  12.05  
A1252  25  5  7  5  4.10  21.79    TL  3.95  3.69 
A1253  3.62  7.27    TL  3.01  16.97  
A1254  3.47  3.25    TL  3.20  7.73  
A1255  3.63  27.93    TL  3.23  11.10  
A1301  3.55  2.46    TL  3.29  7.11  
A1302  30  5  8  6  3.49  2.20    TL  3.22  7.66 
A1303  3.92  3.64    TL  3.74  4.57  
A1304  3.65  2.26    TL  3.20  12.37  
A1305  4.03  2.45    TL  3.27  18.91  
A1351  4.13  155.01    TL  4.01  2.76  
A1352  35  5  8  6  5.15  107.19    TL  5.12  0.54 
A1353  4.38  80.72    TL  4.33  1.01  
A1354  4.95  66.00    TL  3.46  30.14  
A1355  4.78  30.97    TL  3.70  22.61  
A1401  4.87  107.53    TL  4.78  1.87  
A1402  40  5  8  6  5.22  160.56    TL  5.17  0.98 
A1403  4.72  112.01    TL  4.71  0.28  
A1404  5.08  153.48    TL  4.95  2.68  
A1405  4.96  86.51    TL  4.87  1.95  
Average  35.57  5.65 
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Bruni, M.E.; Khodaparasti, S. A Variable Neighborhood Descent Matheuristic for the Drone Routing Problem with Beehives Sharing. Sustainability 2022, 14, 9978. https://doi.org/10.3390/su14169978
Bruni ME, Khodaparasti S. A Variable Neighborhood Descent Matheuristic for the Drone Routing Problem with Beehives Sharing. Sustainability. 2022; 14(16):9978. https://doi.org/10.3390/su14169978
Chicago/Turabian StyleBruni, Maria Elena, and Sara Khodaparasti. 2022. "A Variable Neighborhood Descent Matheuristic for the Drone Routing Problem with Beehives Sharing" Sustainability 14, no. 16: 9978. https://doi.org/10.3390/su14169978