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An investigation of the most important factors for sustainable product development using evidential reasoning
A robust multitrip vehicle routing problem of perishable products with intermediate depots and time windows
1.  Mazandaran University of Science and Technology, Department of Industrial Engineering, Babol, Iran 
2.  Department of Industrial Engineering, Yazd University, Yazd, Iran 
3.  Department of Industrial and System Engineering, Isfahan University of Technology, Isfahan, Iran 
4.  Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran 
Distribution of products within the supply chain with the highest quality is one of the most important competitive activities in industries with perishable products. Companies should pay much attention to the distribution during the design of their optimal supply chain. In this paper, a robust multitrip vehicle routing problem with intermediate depots and time windows is formulated to deals with the uncertainty nature of demand parameter. A mixed integer linear programming model is presented to minimize total traveled distance, vehicles usage costs, earliness and tardiness penalty costs of services, and determine optimal routes for vehicles so that all customers' demands are covered. A number of random instances in different sizes (small, medium, and large) are generated and solved by CPLEX solver of GAMS to evaluate the robustness of the model and prove the model validation. Finally, a sensitivity analysis is applied to study the impact of the maximum available time for vehicles on the objective function value.
References:
[1] 
B. AdensoDíaz, M. González, E. Garcia, A hierarchical approach to managing dairy routing, Interfaces, 28 (1998), 2131. 
[2] 
A. Agra, M. Christiansen, R. Figueiredo, L. M. Hvattum, M. Poss, C. Requejo, The robust vehicle routing problem with time windows, Computers & Operations Research, 40 (2013), 856866. doi: 10.1016/j.cor.2012.10.002. 
[3] 
M. Alinaghian, H. Amanipour, E. B. Tirkolaee, Enhancement of inventory management approaches in vehicle routingcross docking problems, Journal of Supply Chain Management Systems, 3 (2014), 2734. 
[4] 
A. BenTal, L. El Ghaoui and A. Nemirovski, Robust Optimization Princeton University Press, 2009. 
[5] 
D. Bertsimas, M. Sim, Robust discrete optimization and network flows, Mathematical programming, 98 (2003), 4971. doi: 10.1007/s1010700303964. 
[6] 
D. Bertsimas, M. Sim, The price of robustness, Operations Research, 52 (2004), 3553. doi: 10.1287/opre.1030.0065. 
[7] 
I. M. Chao, B. L. Golden, E. Wasil, A new heuristic for the multidepot vehicle routing problem that improves upon bestknown solutions, American Journal of Mathematical and Management Sciences, 13 (1993), 371406. 
[8] 
B. Crevier, J. F. Cordeau, G. Laporte, The multidepot vehicle routing problem with interdepot routes, European Journal of Operational Research, 172 (2007), 756773. doi: 10.1016/j.ejor.2005.08.015. 
[9] 
G. B. Dantzig, J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959), 8091. doi: 10.1287/mnsc.6.1.80. 
[10] 
B. Fleischmann, The vehicle routing problem with multiple use of the vehicles, Working paper, 1990. 
[11] 
F. P. Goksal, I. Karaoglan, F. Altiparmak, A hybrid discrete particle swarm optimization for vehicle routing problem with simultaneous pickup and delivery, Computers & Industrial Engineering, 65 (2013), 3953. 
[12] 
C. E. Gounaris, W. Wiesemann, C. A. Floudas, The robust capacitated vehicle routing problem under demand uncertainty, Operations Research, 61 (2013), 677693. doi: 10.1287/opre.1120.1136. 
[13] 
K. Govindan, A. Jafarian, R. Khodaverdi, K. Devika, Twoechelon multiplevehicle locationrouting problem with time windows for optimization of sustainable supply chain network of perishable food, International Journal of Production Economics, 152 (2014), 928. 
[14] 
E. Hadjiconstantinou, R. Baldacci, A multidepot period vehicle routing problem arising in the utilities sector, Journal of the Operational Research Society, 49 (1998), 12391248. 
[15] 
W. Ho, G. T. Ho, P. Ji, H. C. Lau, A hybrid genetic algorithm for the multidepot vehicle routing problem, Engineering Applications of Artificial Intelligence, 21 (2008), 548557. 
[16] 
C. I. Hsu, S. F. Hung, H. C. Li, Vehicle routing problem with timewindows for perishable food delivery, Journal of Food Engineering, 80 (2007), 465475. 
[17] 
H. S. Hwang, A food distribution model for famine relief, Computers & Industrial Engineering, 37 (1999), 335338. 
[18] 
I. Kara, G. Laporte, T. Bektas, A note on the lifted MillerTuckerZemlin subtour elimination constraints for the capacitated vehicle routing problem, European Journal of Operational Research, 158 (2004), 793795. doi: 10.1016/S03772217(03)003771. 
[19] 
Z. Li, R. Ding, C. A. Floudas, A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization, Industrial & Engineering Chemistry Research, 50 (2011), 1056710603. 
[20] 
S. H. Mirmohammadi, E. B. Tirkolaee, A. Goli, S. DehnaviArani, The periodic green vehicle routing problem with considering of timedependent urban traffic and time windows, International Journal of Optimization in Civil Engineering, 7 (2017), 143156. 
[21] 
J. R. MontoyaTorres, J. L. Franco, S. N. Isaza, H. F. Jiménez, N. HerazoPadilla, A literature review on the vehicle routing problem with multiple depots, Computers & Industrial Engineering, 79 (2015), 115129. 
[22] 
A. Olivera, O. Viera, Adaptive memory programming for the vehicle routing problem with multiple trips, Computers & Operations Research, 34 (2007), 2847. doi: 10.1016/j.cor.2005.02.044. 
[23] 
A. Osvald, L. Z. Stirn, A vehicle routing algorithm for the distribution of fresh vegetables and similar perishable food, Journal of Food Engineering, 85 (2008), 285295. 
[24] 
N. Prindezis, C. T. Kiranoudis, D. MarinosKouris, A businesstobusiness fleet management service provider for central food market enterprises, Journal of Food Engineering, 60 (2003), 203210. 
[25] 
L. Sun, B. Wang, Robust optimisation approach for vehicle routing problems with uncertainty, Mathematical Problems in Engineering, 2015 (2015), 18. doi: 10.1155/2015/901583. 
[26] 
L. Tansini, M. E. Urquhart, O. Viera, Comparing assignment algorithms for the multidepot VRP, Reportes Técnicos, (2001), 0108. 
[27] 
C. D. Tarantilis, C. T. Kiranoudis, A metaheuristic algorithm for the efficient distribution of perishable foods, Journal of food Engineering, 50 (2001), 19. 
[28] 
E. B. Tirkolaee and A. Goli, Supply Chain Management Decisions: Location, Routing and Inventory Models and Optimization Methods LAP Lambert Academic Publishing, 2016. 
[29] 
P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Second Edition, Society for Industrial and Applied Mathematics, 2014. 
[30] 
P. M. Verderame, C. A. Floudas, Multisite planning under demand and transportation time uncertainty: Robust optimization and conditional valueatrisk frameworks, Industrial & Engineering Chemistry Research, 50 (2010), 49594982. 
[31] 
T. Vidal, T. G. Crainic, M. Gendreau, N. Lahrichi, W. Rei, A hybrid genetic algorithm for multi depot and periodic vehicle routing problems, Operations Research, 60 (2012), 611624. doi: 10.1287/opre.1120.1048. 
[32] 
T. Yao, S. Reddy Mandala, B. Do Chung, Evacuation transportation planning under uncertainty: a robust optimization approach, Networks and Spatial Economics, 9 (2009), 171189. doi: 10.1007/s1106700991031. 
show all references
References:
[1] 
B. AdensoDíaz, M. González, E. Garcia, A hierarchical approach to managing dairy routing, Interfaces, 28 (1998), 2131. 
[2] 
A. Agra, M. Christiansen, R. Figueiredo, L. M. Hvattum, M. Poss, C. Requejo, The robust vehicle routing problem with time windows, Computers & Operations Research, 40 (2013), 856866. doi: 10.1016/j.cor.2012.10.002. 
[3] 
M. Alinaghian, H. Amanipour, E. B. Tirkolaee, Enhancement of inventory management approaches in vehicle routingcross docking problems, Journal of Supply Chain Management Systems, 3 (2014), 2734. 
[4] 
A. BenTal, L. El Ghaoui and A. Nemirovski, Robust Optimization Princeton University Press, 2009. 
[5] 
D. Bertsimas, M. Sim, Robust discrete optimization and network flows, Mathematical programming, 98 (2003), 4971. doi: 10.1007/s1010700303964. 
[6] 
D. Bertsimas, M. Sim, The price of robustness, Operations Research, 52 (2004), 3553. doi: 10.1287/opre.1030.0065. 
[7] 
I. M. Chao, B. L. Golden, E. Wasil, A new heuristic for the multidepot vehicle routing problem that improves upon bestknown solutions, American Journal of Mathematical and Management Sciences, 13 (1993), 371406. 
[8] 
B. Crevier, J. F. Cordeau, G. Laporte, The multidepot vehicle routing problem with interdepot routes, European Journal of Operational Research, 172 (2007), 756773. doi: 10.1016/j.ejor.2005.08.015. 
[9] 
G. B. Dantzig, J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959), 8091. doi: 10.1287/mnsc.6.1.80. 
[10] 
B. Fleischmann, The vehicle routing problem with multiple use of the vehicles, Working paper, 1990. 
[11] 
F. P. Goksal, I. Karaoglan, F. Altiparmak, A hybrid discrete particle swarm optimization for vehicle routing problem with simultaneous pickup and delivery, Computers & Industrial Engineering, 65 (2013), 3953. 
[12] 
C. E. Gounaris, W. Wiesemann, C. A. Floudas, The robust capacitated vehicle routing problem under demand uncertainty, Operations Research, 61 (2013), 677693. doi: 10.1287/opre.1120.1136. 
[13] 
K. Govindan, A. Jafarian, R. Khodaverdi, K. Devika, Twoechelon multiplevehicle locationrouting problem with time windows for optimization of sustainable supply chain network of perishable food, International Journal of Production Economics, 152 (2014), 928. 
[14] 
E. Hadjiconstantinou, R. Baldacci, A multidepot period vehicle routing problem arising in the utilities sector, Journal of the Operational Research Society, 49 (1998), 12391248. 
[15] 
W. Ho, G. T. Ho, P. Ji, H. C. Lau, A hybrid genetic algorithm for the multidepot vehicle routing problem, Engineering Applications of Artificial Intelligence, 21 (2008), 548557. 
[16] 
C. I. Hsu, S. F. Hung, H. C. Li, Vehicle routing problem with timewindows for perishable food delivery, Journal of Food Engineering, 80 (2007), 465475. 
[17] 
H. S. Hwang, A food distribution model for famine relief, Computers & Industrial Engineering, 37 (1999), 335338. 
[18] 
I. Kara, G. Laporte, T. Bektas, A note on the lifted MillerTuckerZemlin subtour elimination constraints for the capacitated vehicle routing problem, European Journal of Operational Research, 158 (2004), 793795. doi: 10.1016/S03772217(03)003771. 
[19] 
Z. Li, R. Ding, C. A. Floudas, A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization, Industrial & Engineering Chemistry Research, 50 (2011), 1056710603. 
[20] 
S. H. Mirmohammadi, E. B. Tirkolaee, A. Goli, S. DehnaviArani, The periodic green vehicle routing problem with considering of timedependent urban traffic and time windows, International Journal of Optimization in Civil Engineering, 7 (2017), 143156. 
[21] 
J. R. MontoyaTorres, J. L. Franco, S. N. Isaza, H. F. Jiménez, N. HerazoPadilla, A literature review on the vehicle routing problem with multiple depots, Computers & Industrial Engineering, 79 (2015), 115129. 
[22] 
A. Olivera, O. Viera, Adaptive memory programming for the vehicle routing problem with multiple trips, Computers & Operations Research, 34 (2007), 2847. doi: 10.1016/j.cor.2005.02.044. 
[23] 
A. Osvald, L. Z. Stirn, A vehicle routing algorithm for the distribution of fresh vegetables and similar perishable food, Journal of Food Engineering, 85 (2008), 285295. 
[24] 
N. Prindezis, C. T. Kiranoudis, D. MarinosKouris, A businesstobusiness fleet management service provider for central food market enterprises, Journal of Food Engineering, 60 (2003), 203210. 
[25] 
L. Sun, B. Wang, Robust optimisation approach for vehicle routing problems with uncertainty, Mathematical Problems in Engineering, 2015 (2015), 18. doi: 10.1155/2015/901583. 
[26] 
L. Tansini, M. E. Urquhart, O. Viera, Comparing assignment algorithms for the multidepot VRP, Reportes Técnicos, (2001), 0108. 
[27] 
C. D. Tarantilis, C. T. Kiranoudis, A metaheuristic algorithm for the efficient distribution of perishable foods, Journal of food Engineering, 50 (2001), 19. 
[28] 
E. B. Tirkolaee and A. Goli, Supply Chain Management Decisions: Location, Routing and Inventory Models and Optimization Methods LAP Lambert Academic Publishing, 2016. 
[29] 
P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Second Edition, Society for Industrial and Applied Mathematics, 2014. 
[30] 
P. M. Verderame, C. A. Floudas, Multisite planning under demand and transportation time uncertainty: Robust optimization and conditional valueatrisk frameworks, Industrial & Engineering Chemistry Research, 50 (2010), 49594982. 
[31] 
T. Vidal, T. G. Crainic, M. Gendreau, N. Lahrichi, W. Rei, A hybrid genetic algorithm for multi depot and periodic vehicle routing problems, Operations Research, 60 (2012), 611624. doi: 10.1287/opre.1120.1048. 
[32] 
T. Yao, S. Reddy Mandala, B. Do Chung, Evacuation transportation planning under uncertainty: a robust optimization approach, Networks and Spatial Economics, 9 (2009), 171189. doi: 10.1007/s1106700991031. 
 Set of customers 
 Set of intermediate depots 
 Set of customers and intermediate depots 
 Set of vehicles 
 Set of trips 
 Indices of network nodes 
 Index of vehicles 
 Index of trips 
 An optional subset of customers 
 A Large number 
 Set of customers 
 Set of intermediate depots 
 Set of customers and intermediate depots 
 Set of vehicles 
 Set of trips 
 Indices of network nodes 
 Index of vehicles 
 Index of trips 
 An optional subset of customers 
 A Large number 
 Traveling time from customer 
 Quantity of product at intermediate depot 
 Capacity of vehicle 
 Forecasted demand of customer 
 Nondeterministic demand of customer 
 Distance between customer 
 Maximum distance range of vehicle 
 Penalty cost of earliness in servicing customers 
 Penalty cost of tardiness in servicing customers 
 Usage cost of vehicle 
 Lower bound of initial interval to service customer 
 Upper bound of initial interval to service customer 
 Lower bound of secondary interval to service customer 
 Upper bound of secondary interval to service customer 
 Maximum time of vehicle usage 
 Unit loading time of vehicles in nodes with demand 
 Unit unloading time of vehicles in unloading platform 
 Traveling time from customer 
 Quantity of product at intermediate depot 
 Capacity of vehicle 
 Forecasted demand of customer 
 Nondeterministic demand of customer 
 Distance between customer 
 Maximum distance range of vehicle 
 Penalty cost of earliness in servicing customers 
 Penalty cost of tardiness in servicing customers 
 Usage cost of vehicle 
 Lower bound of initial interval to service customer 
 Upper bound of initial interval to service customer 
 Lower bound of secondary interval to service customer 
 Upper bound of secondary interval to service customer 
 Maximum time of vehicle usage 
 Unit loading time of vehicles in nodes with demand 
 Unit unloading time of vehicles in unloading platform 
 Presence time of vehicles at intermediate depot 
 Presence time of vehicles at intermediate depot 
 Distance traveled by vehicle in customer 
 Distance traveled by vehicle in customer 
 Distance traveled by each vehicle when arriving at intermediate depot 
 Total loading time of vehicle 
 Total unloading time of vehicle 
 Factor for converting total traveled distance to total transportation cost 
 Presence time of vehicles at intermediate depot 
 Presence time of vehicles at intermediate depot 
 Distance traveled by vehicle in customer 
 Distance traveled by vehicle in customer 
 Distance traveled by each vehicle when arriving at intermediate depot 
 Total loading time of vehicle 
 Total unloading time of vehicle 
 Factor for converting total traveled distance to total transportation cost 
 Equals to 1, if vehicle 
 Equals to 1, if the demand of node 
 Equals to 1, if vehicle 
 Earliness in order to service customer 
 Tardiness in order to service customer 
 Equals to 1, if vehicle 
 Equals to 1, if the demand of node 
 Equals to 1, if vehicle 
 Earliness in order to service customer 
 Tardiness in order to service customer 
Vehicle 1 in intermediate depot 1  Intermediate depot 1 customer 1 customer 2 Intermediate depot 2 customer 3 customer 4 Intermediate depot 1 customer 10 customer 9 customer 8 Intermediate depot 1 
Vehicle 2 in intermediate depot 2  Intermediate depot 2 customer 5 customer 6 customer 7 Intermediate depot 2 
Vehicle 1 in intermediate depot 1  Intermediate depot 1 customer 1 customer 2 Intermediate depot 2 customer 3 customer 4 Intermediate depot 1 customer 10 customer 9 customer 8 Intermediate depot 1 
Vehicle 2 in intermediate depot 2  Intermediate depot 2 customer 5 customer 6 customer 7 Intermediate depot 2 
Problem   Depot  Customers  Types of Vehicles 
P1  5  3  2  2 
P2  7  3  4  2 
P3  9  4  5  3 
P4  10  4  6  3 
P5  13  5  8  4 
P6  14  5  9  4 
P7  15  5  10  5 
P8  20  6  11  5 
P9  30  8  22  7 
P10  40  10  30  10 
Problem   Depot  Customers  Types of Vehicles 
P1  5  3  2  2 
P2  7  3  4  2 
P3  9  4  5  3 
P4  10  4  6  3 
P5  13  5  8  4 
P6  14  5  9  4 
P7  15  5  10  5 
P8  20  6  11  5 
P9  30  8  22  7 
P10  40  10  30  10 
  
  (0.4, 0.8, 1) 
  
  (0.4, 0.8, 1) 
Problem  Objective value  CPU Time (Sec)  
DP  RP  DP  RP  
1  0.4  6032.2  6853.859  0.2  2.25 
 0.8  8118.738  
 1  8444.959  
2  0.4  12046.2  13928.06  6.79  9.66 
 0.8  17417.6  
 1  19273.68  
3  0.4  24074.6  28076.63  198.73  308.06 
 0.8  33606.15  
 1  36593.37  
4  0.4  12055.3  13938.58  902.3  1102.64 
 0.8  15974.48  
 1  17108.88  
5  0.4  12071.4  13883.44  1449.01  1649.68 
 0.8  16050.13  
 1  17182.43  
6  0.4  12079.8  14016.31  1853.14  2309.01 
 0.8  16143.44  
 1  17272.91  
7  0.4  12761.1  15026.2  2520.08  2590.18 
 0.8  18273.9  
 1  19473.44  
8  0.4  17952.61  21548.52  3801.8  4106.63 
 0.8  26857.1  
 1  27496.22  
9  0.4  23721.4  29184.44  5682.12  6612.09 
 0.8  35858.69  
 1  37241.65  
10  0.4  30816.9  36681.36  10841.15  12394.88 
 0.8  45953.11  
 1  51462.99 
Problem  Objective value  CPU Time (Sec)  
DP  RP  DP  RP  
1  0.4  6032.2  6853.859  0.2  2.25 
 0.8  8118.738  
 1  8444.959  
2  0.4  12046.2  13928.06  6.79  9.66 
 0.8  17417.6  
 1  19273.68  
3  0.4  24074.6  28076.63  198.73  308.06 
 0.8  33606.15  
 1  36593.37  
4  0.4  12055.3  13938.58  902.3  1102.64 
 0.8  15974.48  
 1  17108.88  
5  0.4  12071.4  13883.44  1449.01  1649.68 
 0.8  16050.13  
 1  17182.43  
6  0.4  12079.8  14016.31  1853.14  2309.01 
 0.8  16143.44  
 1  17272.91  
7  0.4  12761.1  15026.2  2520.08  2590.18 
 0.8  18273.9  
 1  19473.44  
8  0.4  17952.61  21548.52  3801.8  4106.63 
 0.8  26857.1  
 1  27496.22  
9  0.4  23721.4  29184.44  5682.12  6612.09 
 0.8  35858.69  
 1  37241.65  
10  0.4  30816.9  36681.36  10841.15  12394.88 
 0.8  45953.11  
 1  51462.99 
P3  Objective value for different  
360  480  550  600  
DP  24461.39  24074.6  23517.35  23517.35 
RP (  28363.85  28076.63  28076.63  28076.63 
RP (  34017.15  33606.15  33109.99  33109.99 
RP (  37076.55  36593.37  33857.5  33857.5 
AVE  30979.735  30587.6875  29640.3675  29640.3675 
P3  Objective value for different  
360  480  550  600  
DP  24461.39  24074.6  23517.35  23517.35 
RP (  28363.85  28076.63  28076.63  28076.63 
RP (  34017.15  33606.15  33109.99  33109.99 
RP (  37076.55  36593.37  33857.5  33857.5 
AVE  30979.735  30587.6875  29640.3675  29640.3675 
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