Solving the Green Open Vehicle Routing Problem Using a MembraneInspired Hybrid Algorithm
Abstract
:1. Introduction
 (1)
 A novel threelevel nested membrane structure is designed with respective algorithms. To be specific, the skin membrane acts as the first level, where a genetic algorithm is mainly exploited to search for solutions to the routing problems. Six adjacent inner membranes act as the second level, where different tabu search algorithms are exploited to find tentative solutions. The elementary membrane in each level2 membrane acts as the third level, where neighbourhood search operations are exploited to facilitate adjusting the search direction of the corresponding level2 membrane.
 (2)
 Communication channels between the level2 membranes and their inner membranes are designed to exchange solutions to favourably find better solutions. Communication channels also exist between the skin membrane and the level2 membranes, where the crossover operator in the genetic operators is leveraged to retain satisfactory gene segments. In addition, the tabu search with different attractors is adopted to help the genetic algorithm escape from the local optimum. The convergence curve cliffs after each communication justify the effectiveness of the communication channels.
 (3)
 Experiments are carried out on largescale realworld problem instances, i.e., a Beijing 100nodes set and a Jingdong 1000nodes instance. The results demonstrate that our method significantly outperformed the hybrid tabu search [17], tabu search, and genetic algorithm, respectively. In particular, the computation time observed when comparing the performance on the Jingdong 1000nodes instances and the Beijing 100nodes instances further demonstrates the superiority of our algorithm in solving largescale problems.
2. Related Work
2.1. Algorithms for the OVRP
2.2. Membrane Algorithms
3. Problem Formulation
4. The Proposed Method
Algorithm 1 Pseudocode of MIHA 
Require: maximum number of iterations ${I}_{max}$, iteration number before MCA ${I}_{MCA}$ 
Ensure:${x}_{best}$. 

4.1. Initialisation
Algorithm 2 Pseudocode of initialisation 
Require: Operators: Random, NNH, mNNH, ${I}_{1}$, EDF, SWTF 
Ensure: Implement the initialisation of MIHA 

 (1)
 Random heuristic: It randomly chooses routes that satisfy constraints (5)–(17).
 (2)
 Nearest neighbourhood heuristic (NNH): It generates a set of routes according to the distance from the current node. The nearest customer to the depot is chosen as the start node ${x}_{0}$ of the first route. Then it chooses an unassigned customer ${x}_{1}$, who is nearest to ${x}_{0}$. It repeats the same procedure until no feasible candidate nodes for the current route can be found. This also means that the current route is completed. Then, it allocates a new vehicle for the next route and constructs the route in a similar way until all customer nodes are assigned.
 (3)
 Modified nearest neighbourhood heuristic (mNNH): It generates a set of routes according to both the demand of the next customer and the distance from the current node. During the shipping process, a vehicle can offload ${q}_{i}$ payload after servicing customer i. The unit distance payload of customer j on arc $(i,j)$ is defined as $\overline{\Delta {f}_{ij}}={q}_{j}/{d}_{ij}$. The mNNH creates a number of routes sequentially by considering the $\overline{\Delta {f}_{ij}}$ as an objective. First, it chooses the customer c that satisfies $c=argma{x}_{c\in {N}_{ucs}}\left\{\right\overline{\Delta {f}_{0c}}\left\right\}$, where ${N}_{ucs}$ represents the unassigned customer set, which includes the initial current node of the first route. Next, the feasible customer $nc$ that satisfies $nc=argma{x}_{nc\in {N}_{ucs}}\left\{\right\overline{\Delta {f}_{c,nc}}\left\right\}$ is selected as the next node c and added to the current route. Then, it repeats choosing the next node and appending it to the current route. If no more feasible nodes can be added, the current route is completed, and another route will be constructed in a similar way until all customers have been assigned.
 (4)
 Insert ${I}_{1}$ heuristic: As first proposed by Solomon [47], the customer ${u}^{*}$ is chosen based on the Equations (19) and (20) and then inserted to the route according to the insert ${I}_{1}$ heuristics. Moreover, the feasible and desired position of the selected ${u}^{*}$ in the route is decided by Equations (21) and (22) as follows, where ${y}_{{j}_{u}}$ is the new time for service to begin at customer j, given that u is on the route. The main idea is to use several criteria to insert a new customer into the current partial route at every iteration.$${c}_{2}({i}^{*},{u}^{*},{j}^{*})=maximum\left[{c}_{2}(i,u,j)\right],$$$${c}_{2}(i,u,j)=\lambda {d}_{0u}{c}_{1}(i,u,j),\phantom{\rule{4pt}{0ex}}\lambda \ge 0$$$${c}_{1}({i}^{*},u,{j}^{*})=min\left[{c}_{1}(i,u,j)\right],$$$${c}_{1}(i,u,j)={\alpha}_{1}({d}_{iu}+{d}_{uj}\mu {d}_{ij})+{\alpha}_{2}({y}_{{j}_{u}}{y}_{j}),\phantom{\rule{4pt}{0ex}}\mu ,{\alpha}_{1},{\alpha}_{2}\ge 0,\phantom{\rule{4pt}{0ex}}{\alpha}_{1}+{\alpha}_{2}=1.$$
 (5)
 Earliest deadline first heuristic: It selects the customer with the earliest (or tightest) deadline for service at each step.
 (6)
 Shortest waiting time first heuristic: It selects the customer with the shortest waiting time.
4.2. GA in Skin Membrane
4.2.1. Crossover Operator
4.2.2. Mutation Operator
 (1)
 Random: This operation randomly removes a customer node from a given route and inserts it into another feasible position of the origin route.
 (2)
 Splitlongest: This operation searches for the route with the highest total cost and breaks the route into two parts at a random point.
 (3)
 Mergeshortest: This operation searches for the two routes of the chromosome with the smallest total cost and appends one to the other.
4.3. Tabu Search in Level2 Membranes
Algorithm 3 Level2 Membrane Search Algorithm 

 (1)
 Random operator: It randomly exchanges the position of two nodes of a given solution, provided that no constraint is violated.
 (2)
 Highcostnode operator: It removes a highcost customer node defined as ${u}^{*}=argma{x}_{u\in N}\{{d}_{iu}+{d}_{uj}\}$, where i is the preceding customer and j is the succeeding customer, and inserts the node into another position.
 (3)
 Longwaittime operator: It relocates the customer with a long wait time node defined as ${u}^{*}=argma{x}_{u\in N}\{{a}_{u}{e}_{u}\}$, where ${e}_{u}$ is the arrival time of customer u.
4.4. Neighbourhood Search in Level3 Membranes
Algorithm 4 Level3 Membrane Search Algorithm 

4.5. Communications between the Level2 Membrane and Skin Membrane
Algorithm 5 Membrane Communication Algorithm 

4.6. Speed Optimisation
Algorithm 6 Speed and Departuretime Optimisation Algorithm (SDTOA) 

5. Computational Results
5.1. Parameter Analysis
5.1.1. ${p}_{level3}$
5.1.2. ${I}_{MCA}$
5.2. Effectiveness of Search in Level3 Membranes
5.3. Effectiveness of Tabu Search
5.4. Effectiveness of GA in Skin Membrane
5.5. Effectiveness of the Membrane Structure
5.6. Computational Result of LargerScale Problems
5.7. Comparison with Other Algorithms
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Notations
G  Complete directed graph 
N  Node set 
${N}_{0}$  Customer set 
A  Arc set 
Q  Vehicle capacity 
w  The weight of a vehicle 
${q}_{i}$  The demand of customer i 
${s}_{i}$  The time cost of the route ending with i 
${y}_{i}$  The actual starting time for serving node i 
${d}_{ij}$  The length of arc $(i,j)$ 
${f}_{ij}$  The amount of freight flow on arc $(i,j)$ 
${x}_{ij}$  Binary flag variable 1 
${z}_{ij}^{r}$  Binary flag variable 2 
${v}_{ij}^{r}$  The travel speed of vehicle r on arc $(i,j)$ 
$[{a}_{i},{b}_{i}]$  The time window of customer i 
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Notation  Description  Typical Value 

$\eta $  Diesel engine efficiency  0.45 
${C}_{r}$  Rolling resistance coefficient  0.01 
$\rho $  Density of air (kg/m${}^{3}$)  1.2041 
$\psi $  Conversion factor (g/s to litre/s)  737 
$\kappa $  Heating value of the typical diesel fuel (kilojoule/g)  44 
g  Constant of gravitation (m/s${}^{2}$)  9.81 
${n}_{tf}$  Vehicle drive train efficiency  0.45 
$\xi $  Fueltoair mass ratio  1 
$\tau $  Acceleration (m/s${}^{2}$)  0 
$\theta $  Angle of the road  0 
${v}^{u}$  Highest speed (m/s)  27.8 (or 100 km/h) 
${v}^{l}$  Lowest speed (m/s)  5.5 (or 20 km/h) 
${f}_{c}$  Cost of fuel and CO${}_{2}$ emissions (GBP/litre)  1.4 
${f}_{d}$  Cost of driver wage (GBP/s)  0.0022 
Q  Vehicle capacity (kg)  4000 
w  Vehicle curb weight (kg)  3500 
f  Vehicle fixed cost (GBP/day)  0 
V  Engine displacement (litre)  4.5 
k  Engine friction factor (kilojoule/rev/litre)  0.25 
${N}_{e}$  Engine speed (rev/s)  38.34 
A  Area of frontal surface (m${}^{2}$)  7.0 
${C}_{d}$  Aerodynamics drag coefficient  0.6 
Notation  B  Typical Values 

${I}_{MCA}$  Iterations between MCAs  150 
${I}_{max}$  Maximum iteration number  500 
${p}_{level3}$  Probability of level3 search  0.8 
$Ar$  Archive size  100 
$Ns$  Neighbourhood size  100 
L  Tabulist size  30 
P  Population size  100 
${p}_{mutation}$  Rate of mutation  0.8 
${p}_{crossover}$  Rate of crossover  0.2 
$({\alpha}_{1},{\alpha}_{2},\mu ,\lambda )$  ${I}_{1}$ parameters  $(0.5,0.5,1,1)$ 
${\mathit{p}}_{\mathit{l}\mathit{e}\mathit{v}\mathit{e}\mathit{l}3}$  Best Solution  Mean Solution  Worst Solution  SD  ET 

0  10,875.6553  10,928.4416  10,957.5933  22.3405  98.4386 
0.1  10,882.5664  10,935.8356  10,978.9560  25.6220  99.7357 
0.2  10,897.0627  10,922.4866  10,946.9486  14.2856  106.7122 
0.3  10,894.6855  10,935.1849  10,981.7598  22.6840  114.1316 
0.4  10,893.8999  10,927.5347  10,960.1311  20.8581  132.3853 
0.5  10,870.5658  10,927.1250  10,965.2899  28.0809  139.0837 
0.6  10,893.7139  10,922.8795  10,946.2359  19.7116  153.5391 
0.7  10,859.4377  10,918.7540  10,956.2807  23.5941  158.2609 
0.8  10,849.7328  10,907.9905  10,959.0818  28.7783  172.7458 
0.9  10,885.8673  10,922.2359  10,945.8091  16.2176  172.7154 
1  10,864.7334  10,914.4491  10,939.8273  21.2864  179.3093 
${\mathit{I}}_{\mathit{MCA}}$  Best Solution  Mean Solution  Worst Solution  SD  ET 

15  10,887.9339  10,915.9258  10,933.9978  12.4846  180.9297 
25  10,887.3386  10,907.1512  10,946.9533  17.9590  179.3292 
50  10,864.7334  10,914.4491  10,939.8273  21.2864  179.3093 
75  10,887.7475  10,914.0151  10,947.0650  18.9695  175.5044 
100  10,899.9133  10,917.2956  10,930.3250  9.1312  174.3383 
125  10,881.1171  10,913.0205  10,956.5405  19.0231  180.1102 
150  10,847.0901  10,900.6532  10,947.3101  26.0596  150.6687 
175  10,866.3854  10,908.5777  10,944.5708  23.6946  158.3528 
200  10,878.1705  10,911.7143  10,945.2829  18.0546  184.6337 
Instance  Best Solution  Mean Solution  Worst Solution  SD  ET  

MIHA  BJ60_01  10,864.7334  10,914.4491  10,939.8273  21.2864  179.3093 
BJ60_02  10,217.4691  10,310.1949  10,395.6818  54.3404  167.1625  
BJ60_03  11,321.6848  11,423.8391  11,489.1644  53.3224  168.3861  
BJ60_04  11,800.7563  11,847.7284  11,915.0534  31.7880  179.6456  
BJ60_05  10,890.8377  10,909.8830  10,947.0117  14.5797  190.0507  
BJ60_06  11,875.3018  11,916.5380  11,942.8835  21.5401  157.5884  
BJ60_07  12,528.3080  12,634.5020  12,685.8953  54.0499  138.2126  
BJ60_08  11,783.3490  11,867.4716  11,932.7904  45.1478  162.5045  
BJ60_09  11,573.5410  11,705.3953  11,908.5312  107.6724  173.2149  
BJ60_10  12,939.9900  13,196.1938  13,269.0754  90.2917  161.0214  
Average  11,579.5971  11,672.6195  11,742.5914  49.4019  167.7096  
MIHA${}_{level3}$  BJ60_01  10,876.9472  10,929.2492  10,959.7533  23.8597  99.9533 
BJ60_02  10,354.6481  10,439.4020  10,529.0454  47.3483  98.3095  
BJ60_03  11,417.6826  11,510.4497  11,651.4713  77.5482  99.3993  
BJ60_04  11,810.6944  11,850.3646  11,947.2880  36.0053  104.0135  
BJ60_05  10,900.3834  10,930.1086  10,996.6298  29.6276  115.6167  
BJ60_06  11,884.6734  11,934.8017  11,994.1760  37.2390  100.8201  
BJ60_07  12,570.4124  12,737.4635  12,916.3772  116.3777  88.5942  
BJ60_08  11,816.6496  11,865.4873  11,937.1039  42.1579  94.7336  
BJ60_09  11,652.5596  11,744.5783  11,860.8162  74.2931  92.3574  
BJ60_10  13168.2194  13,240.3696  13,322.1267  43.8999  85.6219  
Average  11,645.2870  11,718.2275  11,811.4788  52.8356  97.9418 
Instance  Best Solution  Mean Solution  Worst Solution  SD  ET  

MIHA  BJ60_01  10,864.7334  10,914.4491  10,939.8273  21.2864  179.3093 
BJ60_02  10,217.4691  10,310.1949  10,395.6818  54.3404  167.1625  
BJ60_03  11,321.6848  11,423.8391  11,489.1644  53.3224  168.3861  
BJ60_04  11,800.7563  11,847.7284  11,915.0534  31.7880  179.6456  
BJ60_05  10,890.8377  10,909.8830  10,947.0117  14.5797  190.0507  
BJ60_06  11,875.3018  11,916.5380  11,942.8835  21.5401  157.5884  
BJ60_07  12,528.3080  12,634.5020  12,685.8953  54.0499  138.2126  
BJ60_08  11,783.3490  11,867.4716  11,932.7904  45.1478  162.5045  
BJ60_09  11,573.5410  11,705.3953  11,908.5312  107.6724  173.2149  
BJ60_10  12,939.9900  13,196.1938  13,269.0754  90.2917  161.0214  
Average  11,579.5971  11,672.6195  11,742.5914  49.4019  167.7096  
MIHA${}_{TS}$  BJ60_01  10,887.9165  10,928.7764  10,958.6152  22.1538  179.1811 
BJ60_02  10,350.4335  10,433.3471  10,569.4033  66.3863  165.9635  
BJ60_03  11,410.8506  11,546.6879  11,693.6700  83.6334  183.0199  
BJ60_04  11,808.0504  11,876.0743  12,002.3412  60.3121  190.7105  
BJ60_05  10,893.5351  10,908.2957  10,941.6042  15.4682  213.1253  
BJ60_06  11,930.4763  11,993.7405  12,040.6174  29.2162  162.8483  
BJ60_07  12,790.4131  12,945.8023  13,093.2836  95.8863  150.9155  
BJ60_08  11,840.7762  11,980.0637  12,070.6884  64.6119  170.2074  
BJ60_09  11,700.6390  12,040.8509  12,195.6716  140.5182  169.8237  
BJ60_10  13,352.3191  13,452.7316  13,528.2388  54.9095  152.6279  
Average  11,696.5410  11,810.6370  11,909.4134  63.3096  173.8423 
Instance  Best Solution  Mean Solution  Worst Solution  SD  ET  

MIHA  BJ60_01  10,864.7334  10,914.4491  10,939.8273  21.2864  179.3093 
BJ60_02  10,217.4691  10,310.1949  10,395.6818  54.3404  167.1625  
BJ60_03  11,321.6848  11,423.8391  11,489.1644  53.3224  168.3861  
BJ60_04  11,800.7563  11,847.7284  11,915.0534  31.7880  179.6456  
BJ60_05  10,890.8377  10,909.8830  10,947.0117  14.5797  190.0507  
BJ60_06  11,875.3018  11,916.5380  11,942.8835  21.5401  157.5884  
BJ60_07  12,528.3080  12,634.5020  12,685.8953  54.0499  138.2126  
BJ60_08  11,783.3490  11,867.4716  11,932.7904  45.1478  162.5045  
BJ60_09  11,573.5410  11,705.3953  11,908.5312  107.6724  173.2149  
BJ60_10  12,939.9900  13,196.1938  13,269.0754  90.2917  161.0214  
Average  11,579.5971  11,672.6195  11,742.5914  49.4019  167.7096  
MIHA${}_{GA}$  BJ60_01  10,889.6733  10,920.5130  10,947.4318  20.3041  162.6233 
BJ60_02  10,267.1195  10,333.7142  10,415.6234  49.4848  149.6058  
BJ60_03  11,349.3097  11,473.1526  11,595.0291  77.7908  149.0490  
BJ60_04  11,804.8311  11,843.4759  11,937.9748  39.7545  150.1856  
BJ60_05  10,898.5533  10,917.0069  10,942.3023  14.3160  168.1142  
BJ60_06  11,882.7327  11,939.8357  11,973.6663  30.1972  170.9655  
BJ60_07  12,534.6436  12,658.8725  12,849.5941  85.0550  129.9516  
BJ60_08  11,806.8383  11,876.4740  11,934.8815  39.6358  138.7723  
BJ60_09  11,630.3241  11,687.7664  11,850.7900  63.9084  140.8432  
BJ60_10  13,120.3543  13,262.1483  13,355.0285  75.4520  128.9989  
Average  11,618.4380  11,691.2960  11,780.2322  49.5899  148.9109 
Instance  Best Solution  Mean Solution  Worst Solution  SD  ET  

MIHA  BJ60_01  10,864.7334  10,914.4491  10,939.8273  21.2864  179.3093 
BJ60_02  10,217.4691  10,310.1949  10,395.6818  54.3404  167.1625  
BJ60_03  11,321.6848  11,423.8391  11,489.1644  53.3224  168.3861  
BJ60_04  11,800.7563  11,847.7284  11,915.0534  31.7880  179.6456  
BJ60_05  10,890.8377  10,909.8830  10,947.0117  14.5797  190.0507  
BJ60_06  11,875.3018  11,916.5380  11,942.8835  21.5401  157.5884  
BJ60_07  12,528.3080  12,634.5020  12,685.8953  54.0499  138.2126  
BJ60_08  11,783.3490  11,867.4716  11,932.7904  45.1478  162.5045  
BJ60_09  11,573.5410  11,705.3953  11,908.5312  107.6724  173.2149  
BJ60_10  12,939.9900  13,196.1938  13,269.0754  90.2917  161.0214  
Average  11,579.5971  11,672.6195  11,742.5914  49.4019  167.7096  
MIHA${}_{MS}$  BJ60_01  11,328.7061  11,450.7661  11,645.5626  102.4337  16.3996 
BJ60_02  10,706.7220  10,825.8266  10,952.4319  77.1375  16.0414  
BJ60_03  12,036.9001  12,135.7632  12,194.5568  54.3717  15.1175  
BJ60_04  12,367.3172  12,561.4857  12,854.7533  147.9147  17.2651  
BJ60_05  11,232.0918  11,294.0261  11,371.0893  52.7390  16.8575  
BJ60_06  12,288.4131  12,446.6655  12,589.8602  89.2907  16.7956  
BJ60_07  13,243.3081  13,393.8310  13,479.3281  84.1279  13.8604  
BJ60_08  12,172.4397  12,363.8495  12,507.0350  110.1176  13.1230  
BJ60_09  12,293.5838  12,422.8764  12,574.3870  75.6798  14.9571  
BJ60_10  13,356.4703  13,585.7384  13,783.8378  130.1142  13.2258  
Average  12,102.5952  12,248.0829  12,395.2842  92.3927  15.3643 
Instance  Best Solution  Mean Solution  Worst Solution  SD  ET 

BJ100_01  19,067.0291  19,198.1247  19,286.2464  77.2545  177.5238 
BJ100_02  20,054.6213  20,212.3116  20,335.2703  79.4795  177.5122 
BJ100_03  19,611.4194  19,739.1057  19,807.4854  69.5798  187.4094 
BJ100_04  18,185.2382  18,288.2393  18,365.8098  66.8545  183.9430 
BJ100_05  18,450.2424  18,650.0880  18,721.6565  78.3719  177.8560 
BJ100_06  17,185.9916  17,292.5207  17,429.9383  100.3186  191.1868 
BJ100_07  18,018.7958  18,130.9018  18,231.9109  58.4036  203.7468 
BJ100_08  18,077.4404  18,245.1333  18,386.8610  102.6939  190.7556 
BJ100_09  17,317.3479  17,414.3330  17,513.4761  57.2721  181.7808 
BJ100_10  20,888.4245  20,996.4199  21,191.1938  87.1091  165.4292 
Average  18,685.6551  18,816.7178  18,926.9849  77.7338  183.7144 
Algorithm  Best Solution  Mean Solution  Worst Solution  SD  ET 

MIHA  140,577.0284  141,085.6778  141,823.7589  365.4070  2331.9363 
Instance  BJ60  BJ100  JD1000 

ET  150.6687  183.7144  2331.9363 
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Niu, Y.; Yang, Z.; Wen, R.; Xiao, J.; Zhang, S. Solving the Green Open Vehicle Routing Problem Using a MembraneInspired Hybrid Algorithm. Sustainability 2022, 14, 8661. https://doi.org/10.3390/su14148661
Niu Y, Yang Z, Wen R, Xiao J, Zhang S. Solving the Green Open Vehicle Routing Problem Using a MembraneInspired Hybrid Algorithm. Sustainability. 2022; 14(14):8661. https://doi.org/10.3390/su14148661
Chicago/Turabian StyleNiu, Yunyun, Zehua Yang, Rong Wen, Jianhua Xiao, and Shuai Zhang. 2022. "Solving the Green Open Vehicle Routing Problem Using a MembraneInspired Hybrid Algorithm" Sustainability 14, no. 14: 8661. https://doi.org/10.3390/su14148661