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December  2017, 7(4): 417-433. doi: 10.3934/naco.2017026

A robust multi-trip 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

Received  October 2016 Revised  August 2017 Published  October 2017

Fund Project: This paper was prepared at the occasion of The 12th International Conference on Industrial Engineering (ICIE 2016), Tehran, Iran, January 25-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Assoc. Prof. A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

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 multi-trip 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.

Citation: Erfan Babaee Tirkolaee, Alireza Goli, Mani Bakhsi, Iraj Mahdavi. A robust multi-trip vehicle routing problem of perishable products with intermediate depots and time windows. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 417-433. doi: 10.3934/naco.2017026
References:
[1]

B. Adenso-DíazM. González and E. Garcia, A hierarchical approach to managing dairy routing, Interfaces, 28 (1998), 21-31.   Google Scholar

[2]

A. AgraM. ChristiansenR. FigueiredoL. M. HvattumM. Poss and C. Requejo, The robust vehicle routing problem with time windows, Computers & Operations Research, 40 (2013), 856-866.  doi: 10.1016/j.cor.2012.10.002.  Google Scholar

[3]

M. AlinaghianH. Amanipour and E. B. Tirkolaee, Enhancement of inventory management approaches in vehicle routing-cross docking problems, Journal of Supply Chain Management Systems, 3 (2014), 27-34.   Google Scholar

[4]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. doi: 10.1515/9781400831050.  Google Scholar

[5]

D. Bertsimas and M. Sim, Robust discrete optimization and network flows, Mathematical programming, 98 (2003), 49-71.  doi: 10.1007/s10107-003-0396-4.  Google Scholar

[6]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[7]

I. M. ChaoB. L. Golden and E. Wasil, A new heuristic for the multi-depot vehicle routing problem that improves upon best-known solutions, American Journal of Mathematical and Management Sciences, 13 (1993), 371-406.   Google Scholar

[8]

B. CrevierJ. F. Cordeau and G. Laporte, The multi-depot vehicle routing problem with inter-depot routes, European Journal of Operational Research, 172 (2007), 756-773.  doi: 10.1016/j.ejor.2005.08.015.  Google Scholar

[9]

G. B. Dantzig and J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959), 80-91.  doi: 10.1287/mnsc.6.1.80.  Google Scholar

[10]

B. Fleischmann, The vehicle routing problem with multiple use of the vehicles, Working paper, 1990. Google Scholar

[11]

F. P. GoksalI. Karaoglan and F. Altiparmak, A hybrid discrete particle swarm optimization for vehicle routing problem with simultaneous pickup and delivery, Computers & Industrial Engineering, 65 (2013), 39-53.   Google Scholar

[12]

C. E. GounarisW. Wiesemann and C. A. Floudas, The robust capacitated vehicle routing problem under demand uncertainty, Operations Research, 61 (2013), 677-693.  doi: 10.1287/opre.1120.1136.  Google Scholar

[13]

K. GovindanA. JafarianR. Khodaverdi and K. Devika, Two-echelon multiple-vehicle location-routing problem with time windows for optimization of sustainable supply chain network of perishable food, International Journal of Production Economics, 152 (2014), 9-28.   Google Scholar

[14]

E. Hadjiconstantinou and R. Baldacci, A multi-depot period vehicle routing problem arising in the utilities sector, Journal of the Operational Research Society, 49 (1998), 1239-1248.   Google Scholar

[15]

W. HoG. T. HoP. Ji and H. C. Lau, A hybrid genetic algorithm for the multi-depot vehicle routing problem, Engineering Applications of Artificial Intelligence, 21 (2008), 548-557.   Google Scholar

[16]

C. I. HsuS. F. Hung and H. C. Li, Vehicle routing problem with time-windows for perishable food delivery, Journal of Food Engineering, 80 (2007), 465-475.   Google Scholar

[17]

H. S. Hwang, A food distribution model for famine relief, Computers & Industrial Engineering, 37 (1999), 335-338.   Google Scholar

[18]

I. KaraG. Laporte and T. Bektas, A note on the lifted Miller-Tucker-Zemlin subtour elimination constraints for the capacitated vehicle routing problem, European Journal of Operational Research, 158 (2004), 793-795.  doi: 10.1016/S0377-2217(03)00377-1.  Google Scholar

[19]

Z. LiR. Ding and 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), 10567-10603.   Google Scholar

[20]

S. H. MirmohammadiE. B. TirkolaeeA. Goli and S. Dehnavi-Arani, The periodic green vehicle routing problem with considering of time-dependent urban traffic and time windows, International Journal of Optimization in Civil Engineering, 7 (2017), 143-156.   Google Scholar

[21]

J. R. Montoya-TorresJ. L. FrancoS. N. IsazaH. F. Jiménez and N. Herazo-Padilla, A literature review on the vehicle routing problem with multiple depots, Computers & Industrial Engineering, 79 (2015), 115-129.   Google Scholar

[22]

A. Olivera and O. Viera, Adaptive memory programming for the vehicle routing problem with multiple trips, Computers & Operations Research, 34 (2007), 28-47.  doi: 10.1016/j.cor.2005.02.044.  Google Scholar

[23]

A. Osvald and L. Z. Stirn, A vehicle routing algorithm for the distribution of fresh vegetables and similar perishable food, Journal of Food Engineering, 85 (2008), 285-295.   Google Scholar

[24]

N. PrindezisC. T. Kiranoudis and D. Marinos-Kouris, A business-to-business fleet management service provider for central food market enterprises, Journal of Food Engineering, 60 (2003), 203-210.   Google Scholar

[25]

L. Sun and B. Wang, Robust optimisation approach for vehicle routing problems with uncertainty, Mathematical Problems in Engineering, 2015 (2015), 1-8.  doi: 10.1155/2015/901583.  Google Scholar

[26]

L. TansiniM. E. Urquhart and O. Viera, Comparing assignment algorithms for the multi-depot VRP, Reportes Técnicos, (2001), 01-08.   Google Scholar

[27]

C. D. Tarantilis and C. T. Kiranoudis, A meta-heuristic algorithm for the efficient distribution of perishable foods, Journal of food Engineering, 50 (2001), 1-9.   Google Scholar

[28]

E. B. Tirkolaee and A. Goli, Supply Chain Management Decisions: Location, Routing and Inventory Models and Optimization Methods, LAP Lambert Academic Publishing, 2016. Google Scholar

[29]

P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Second Edition, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973594.  Google Scholar

[30]

P. M. Verderame and C. A. Floudas, Multisite planning under demand and transportation time uncertainty: Robust optimization and conditional value-at-risk frameworks, Industrial & Engineering Chemistry Research, 50 (2010), 4959-4982.   Google Scholar

[31]

T. VidalT. G. CrainicM. GendreauN. Lahrichi and W. Rei, A hybrid genetic algorithm for multi depot and periodic vehicle routing problems, Operations Research, 60 (2012), 611-624.  doi: 10.1287/opre.1120.1048.  Google Scholar

[32]

T. YaoS. Reddy Mandala and B. Do Chung, Evacuation transportation planning under uncertainty: a robust optimization approach, Networks and Spatial Economics, 9 (2009), 171-189.  doi: 10.1007/s11067-009-9103-1.  Google Scholar

show all references

References:
[1]

B. Adenso-DíazM. González and E. Garcia, A hierarchical approach to managing dairy routing, Interfaces, 28 (1998), 21-31.   Google Scholar

[2]

A. AgraM. ChristiansenR. FigueiredoL. M. HvattumM. Poss and C. Requejo, The robust vehicle routing problem with time windows, Computers & Operations Research, 40 (2013), 856-866.  doi: 10.1016/j.cor.2012.10.002.  Google Scholar

[3]

M. AlinaghianH. Amanipour and E. B. Tirkolaee, Enhancement of inventory management approaches in vehicle routing-cross docking problems, Journal of Supply Chain Management Systems, 3 (2014), 27-34.   Google Scholar

[4]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. doi: 10.1515/9781400831050.  Google Scholar

[5]

D. Bertsimas and M. Sim, Robust discrete optimization and network flows, Mathematical programming, 98 (2003), 49-71.  doi: 10.1007/s10107-003-0396-4.  Google Scholar

[6]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[7]

I. M. ChaoB. L. Golden and E. Wasil, A new heuristic for the multi-depot vehicle routing problem that improves upon best-known solutions, American Journal of Mathematical and Management Sciences, 13 (1993), 371-406.   Google Scholar

[8]

B. CrevierJ. F. Cordeau and G. Laporte, The multi-depot vehicle routing problem with inter-depot routes, European Journal of Operational Research, 172 (2007), 756-773.  doi: 10.1016/j.ejor.2005.08.015.  Google Scholar

[9]

G. B. Dantzig and J. H. Ramser, The truck dispatching problem, Management Science, 6 (1959), 80-91.  doi: 10.1287/mnsc.6.1.80.  Google Scholar

[10]

B. Fleischmann, The vehicle routing problem with multiple use of the vehicles, Working paper, 1990. Google Scholar

[11]

F. P. GoksalI. Karaoglan and F. Altiparmak, A hybrid discrete particle swarm optimization for vehicle routing problem with simultaneous pickup and delivery, Computers & Industrial Engineering, 65 (2013), 39-53.   Google Scholar

[12]

C. E. GounarisW. Wiesemann and C. A. Floudas, The robust capacitated vehicle routing problem under demand uncertainty, Operations Research, 61 (2013), 677-693.  doi: 10.1287/opre.1120.1136.  Google Scholar

[13]

K. GovindanA. JafarianR. Khodaverdi and K. Devika, Two-echelon multiple-vehicle location-routing problem with time windows for optimization of sustainable supply chain network of perishable food, International Journal of Production Economics, 152 (2014), 9-28.   Google Scholar

[14]

E. Hadjiconstantinou and R. Baldacci, A multi-depot period vehicle routing problem arising in the utilities sector, Journal of the Operational Research Society, 49 (1998), 1239-1248.   Google Scholar

[15]

W. HoG. T. HoP. Ji and H. C. Lau, A hybrid genetic algorithm for the multi-depot vehicle routing problem, Engineering Applications of Artificial Intelligence, 21 (2008), 548-557.   Google Scholar

[16]

C. I. HsuS. F. Hung and H. C. Li, Vehicle routing problem with time-windows for perishable food delivery, Journal of Food Engineering, 80 (2007), 465-475.   Google Scholar

[17]

H. S. Hwang, A food distribution model for famine relief, Computers & Industrial Engineering, 37 (1999), 335-338.   Google Scholar

[18]

I. KaraG. Laporte and T. Bektas, A note on the lifted Miller-Tucker-Zemlin subtour elimination constraints for the capacitated vehicle routing problem, European Journal of Operational Research, 158 (2004), 793-795.  doi: 10.1016/S0377-2217(03)00377-1.  Google Scholar

[19]

Z. LiR. Ding and 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), 10567-10603.   Google Scholar

[20]

S. H. MirmohammadiE. B. TirkolaeeA. Goli and S. Dehnavi-Arani, The periodic green vehicle routing problem with considering of time-dependent urban traffic and time windows, International Journal of Optimization in Civil Engineering, 7 (2017), 143-156.   Google Scholar

[21]

J. R. Montoya-TorresJ. L. FrancoS. N. IsazaH. F. Jiménez and N. Herazo-Padilla, A literature review on the vehicle routing problem with multiple depots, Computers & Industrial Engineering, 79 (2015), 115-129.   Google Scholar

[22]

A. Olivera and O. Viera, Adaptive memory programming for the vehicle routing problem with multiple trips, Computers & Operations Research, 34 (2007), 28-47.  doi: 10.1016/j.cor.2005.02.044.  Google Scholar

[23]

A. Osvald and L. Z. Stirn, A vehicle routing algorithm for the distribution of fresh vegetables and similar perishable food, Journal of Food Engineering, 85 (2008), 285-295.   Google Scholar

[24]

N. PrindezisC. T. Kiranoudis and D. Marinos-Kouris, A business-to-business fleet management service provider for central food market enterprises, Journal of Food Engineering, 60 (2003), 203-210.   Google Scholar

[25]

L. Sun and B. Wang, Robust optimisation approach for vehicle routing problems with uncertainty, Mathematical Problems in Engineering, 2015 (2015), 1-8.  doi: 10.1155/2015/901583.  Google Scholar

[26]

L. TansiniM. E. Urquhart and O. Viera, Comparing assignment algorithms for the multi-depot VRP, Reportes Técnicos, (2001), 01-08.   Google Scholar

[27]

C. D. Tarantilis and C. T. Kiranoudis, A meta-heuristic algorithm for the efficient distribution of perishable foods, Journal of food Engineering, 50 (2001), 1-9.   Google Scholar

[28]

E. B. Tirkolaee and A. Goli, Supply Chain Management Decisions: Location, Routing and Inventory Models and Optimization Methods, LAP Lambert Academic Publishing, 2016. Google Scholar

[29]

P. Toth and D. Vigo, Vehicle Routing: Problems, Methods, and Applications, Second Edition, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973594.  Google Scholar

[30]

P. M. Verderame and C. A. Floudas, Multisite planning under demand and transportation time uncertainty: Robust optimization and conditional value-at-risk frameworks, Industrial & Engineering Chemistry Research, 50 (2010), 4959-4982.   Google Scholar

[31]

T. VidalT. G. CrainicM. GendreauN. Lahrichi and W. Rei, A hybrid genetic algorithm for multi depot and periodic vehicle routing problems, Operations Research, 60 (2012), 611-624.  doi: 10.1287/opre.1120.1048.  Google Scholar

[32]

T. YaoS. Reddy Mandala and B. Do Chung, Evacuation transportation planning under uncertainty: a robust optimization approach, Networks and Spatial Economics, 9 (2009), 171-189.  doi: 10.1007/s11067-009-9103-1.  Google Scholar

Figure 1.  The scheme of the supply chain of the problem
Figure 2.  CPU time comparison of the BBO and CPLEX
Figure 3.  Sensitive analysis for $T_{max}$ parameter
Table 1.  Indices and sets
$NC=\{1, 2, \dots, nc\}$Set of customers
$ND=\{1, 2, \dots, \ nk\}$Set of intermediate depots
$NT=\{1, 2, \dots, nt\}$Set of customers and intermediate depots
$NV=\{1, 2, \dots, nv\}$Set of vehicles
$NP=\{1, 2, \dots, np\}$Set of trips
$i, \ j, \ k, \ l$Indices of network nodes
$v$Index of vehicles
$p$Index of trips
$S$An optional subset of customers
$M$A Large number
$NC=\{1, 2, \dots, nc\}$Set of customers
$ND=\{1, 2, \dots, \ nk\}$Set of intermediate depots
$NT=\{1, 2, \dots, nt\}$Set of customers and intermediate depots
$NV=\{1, 2, \dots, nv\}$Set of vehicles
$NP=\{1, 2, \dots, np\}$Set of trips
$i, \ j, \ k, \ l$Indices of network nodes
$v$Index of vehicles
$p$Index of trips
$S$An optional subset of customers
$M$A Large number
Table 2.  Parameters of problems
$t_{ij}$Traveling time from customer $i$ to customer $j$
${QG}_k$Quantity of product at intermediate depot $k$
${VC}_{v\ }$Capacity of vehicle $k$
${DE}_j$Forecasted demand of customer $j$
$\widetilde{{DE}_j}$Non-deterministic demand of customer $j$ that belongs to the interval ${ [{DE}_j-\widehat{{DE}_j}, {DE}_j+\widehat{{DE}_j}]}$
$d_{ij}$Distance between customer $i$ and customer $j$
${DI}_V$Maximum distance range of vehicle $v$
$Pe$Penalty cost of earliness in servicing customers
$Pl$Penalty cost of tardiness in servicing customers
${CV}_v$Usage cost of vehicle $v$
$e_{i}$Lower bound of initial interval to service customer $i$
$l_{i}$Upper bound of initial interval to service customer $i$
${ee}_{i}$Lower bound of secondary interval to service customer $i$
${ll}_{i}$Upper bound of secondary interval to service customer $i$
$T_{\max}$Maximum time of vehicle usage
$ul$Unit loading time of vehicles in nodes with demand
$uu$Unit unloading time of vehicles in unloading platform
$t_{ij}$Traveling time from customer $i$ to customer $j$
${QG}_k$Quantity of product at intermediate depot $k$
${VC}_{v\ }$Capacity of vehicle $k$
${DE}_j$Forecasted demand of customer $j$
$\widetilde{{DE}_j}$Non-deterministic demand of customer $j$ that belongs to the interval ${ [{DE}_j-\widehat{{DE}_j}, {DE}_j+\widehat{{DE}_j}]}$
$d_{ij}$Distance between customer $i$ and customer $j$
${DI}_V$Maximum distance range of vehicle $v$
$Pe$Penalty cost of earliness in servicing customers
$Pl$Penalty cost of tardiness in servicing customers
${CV}_v$Usage cost of vehicle $v$
$e_{i}$Lower bound of initial interval to service customer $i$
$l_{i}$Upper bound of initial interval to service customer $i$
${ee}_{i}$Lower bound of secondary interval to service customer $i$
${ll}_{i}$Upper bound of secondary interval to service customer $i$
$T_{\max}$Maximum time of vehicle usage
$ul$Unit loading time of vehicles in nodes with demand
$uu$Unit unloading time of vehicles in unloading platform
Table 3.  Non-decision variables
${{tt}_i}^p$Presence time of vehicles at intermediate depot $i$ or customer node $i$ in trip $p$
${tt}_{i}$Presence time of vehicles at intermediate depot $i$ or customer $i$
${{dd}_i}^p$Distance traveled by vehicle in customer $i$ in trip $p$
${{sd}_i}^p$Distance traveled by vehicle in customer $i$ in trip $p$ or initial distance traveled by vehicle in trip $p$ and intermediate depot $i$
${{ddd}_j}^v$Distance traveled by each vehicle when arriving at intermediate depot $i$
${LT}^p_v$Total loading time of vehicle $v$ in trip $p$
${UT}^p_v$Total unloading time of vehicle $v$ in trip $p$
$\alpha$Factor for converting total traveled distance to total transportation cost
${{tt}_i}^p$Presence time of vehicles at intermediate depot $i$ or customer node $i$ in trip $p$
${tt}_{i}$Presence time of vehicles at intermediate depot $i$ or customer $i$
${{dd}_i}^p$Distance traveled by vehicle in customer $i$ in trip $p$
${{sd}_i}^p$Distance traveled by vehicle in customer $i$ in trip $p$ or initial distance traveled by vehicle in trip $p$ and intermediate depot $i$
${{ddd}_j}^v$Distance traveled by each vehicle when arriving at intermediate depot $i$
${LT}^p_v$Total loading time of vehicle $v$ in trip $p$
${UT}^p_v$Total unloading time of vehicle $v$ in trip $p$
$\alpha$Factor for converting total traveled distance to total transportation cost
Table 4.  Decision variables
$x_{ij}^{vp}$Equals to 1, if vehicle $v$ traverses node $i$ to node $j$ in trip $p$; 0, otherwise.
$y_{j vk}$Equals to 1, if the demand of node $j$ is satisfied by intermediate depot $k$ and vehicle $v$; 0, otherwise.
$F_{vk}$Equals to 1, if vehicle $v$ is used in intermediate depot $k$; 0, otherwise.
${YE}_i$Earliness in order to service customer $i$
${YL}_i$Tardiness in order to service customer $i$
$x_{ij}^{vp}$Equals to 1, if vehicle $v$ traverses node $i$ to node $j$ in trip $p$; 0, otherwise.
$y_{j vk}$Equals to 1, if the demand of node $j$ is satisfied by intermediate depot $k$ and vehicle $v$; 0, otherwise.
$F_{vk}$Equals to 1, if vehicle $v$ is used in intermediate depot $k$; 0, otherwise.
${YE}_i$Earliness in order to service customer $i$
${YL}_i$Tardiness in order to service customer $i$
Table 5.  Solution details of the instance problem
Vehicle 1 in intermediate depot 1Intermediate 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 2Intermediate depot 2 -customer 5 -customer 6 -customer 7 -Intermediate depot 2
Vehicle 1 in intermediate depot 1Intermediate 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 2Intermediate depot 2 -customer 5 -customer 6 -customer 7 -Intermediate depot 2
Table 6.  Solution details of the instance problem
Problem $n$DepotCustomersTypes of Vehicles
P15322
P27342
P39453
P410463
P513584
P614594
P7155105
P8206115
P9308227
P1040103010
Problem $n$DepotCustomersTypes of Vehicles
P15322
P27342
P39453
P410463
P513584
P614594
P7155105
P8206115
P9308227
P1040103010
Table 7.  Calculation of robust parameters
$\Gamma _{k} $ $\Gamma _{vp} $ $\rho$
$\left\lceil C^{\prime} /K^{\prime} \right\rceil $ $\left\lceil V^{\prime} /P^{\prime} \right\rceil $(0.4, 0.8, 1)
$\Gamma _{k} $ $\Gamma _{vp} $ $\rho$
$\left\lceil C^{\prime} /K^{\prime} \right\rceil $ $\left\lceil V^{\prime} /P^{\prime} \right\rceil $(0.4, 0.8, 1)
Table 8.  Calculation of robust parameters
Problem$\rho$Objective valueCPU Time (Sec)
DPRPDPRP
10.46032.26853.8590.22.25
$(\Gamma _{k} =1)$0.88118.738
$(\Gamma _{vp} =1)$18444.959
20.412046.213928.066.799.66
$(\Gamma _{k} =2)$0.817417.6
$(\Gamma _{vp} =1)$119273.68
30.424074.628076.63198.73308.06
$(\Gamma _{k} =2)$0.833606.15
$(\Gamma _{vp} =1)$136593.37
40.412055.313938.58902.31102.64
$(\Gamma _{k} =2)$0.815974.48
$(\Gamma _{vp} =1)$117108.88
50.412071.413883.441449.011649.68
$(\Gamma _{k} =2)$0.816050.13
$(\Gamma _{vp} =2)$117182.43
60.412079.814016.311853.142309.01
$(\Gamma _{k} =2)$0.816143.44
$(\Gamma _{vp} =2)$117272.91
70.412761.115026.22520.082590.18
$(\Gamma _{k} =2)$0.818273.9
$(\Gamma _{vp} =2)$119473.44
80.417952.6121548.523801.84106.63
$(\Gamma _{k} =2)$0.826857.1
$(\Gamma _{vp} =2)$127496.22
90.423721.429184.445682.126612.09
$(\Gamma _{k} =3)$0.835858.69
$(\Gamma _{vp} =3)$137241.65
100.430816.936681.3610841.1512394.88
$(\Gamma _{k} =4)$0.845953.11
$(\Gamma _{vp} =4)$151462.99
Problem$\rho$Objective valueCPU Time (Sec)
DPRPDPRP
10.46032.26853.8590.22.25
$(\Gamma _{k} =1)$0.88118.738
$(\Gamma _{vp} =1)$18444.959
20.412046.213928.066.799.66
$(\Gamma _{k} =2)$0.817417.6
$(\Gamma _{vp} =1)$119273.68
30.424074.628076.63198.73308.06
$(\Gamma _{k} =2)$0.833606.15
$(\Gamma _{vp} =1)$136593.37
40.412055.313938.58902.31102.64
$(\Gamma _{k} =2)$0.815974.48
$(\Gamma _{vp} =1)$117108.88
50.412071.413883.441449.011649.68
$(\Gamma _{k} =2)$0.816050.13
$(\Gamma _{vp} =2)$117182.43
60.412079.814016.311853.142309.01
$(\Gamma _{k} =2)$0.816143.44
$(\Gamma _{vp} =2)$117272.91
70.412761.115026.22520.082590.18
$(\Gamma _{k} =2)$0.818273.9
$(\Gamma _{vp} =2)$119473.44
80.417952.6121548.523801.84106.63
$(\Gamma _{k} =2)$0.826857.1
$(\Gamma _{vp} =2)$127496.22
90.423721.429184.445682.126612.09
$(\Gamma _{k} =3)$0.835858.69
$(\Gamma _{vp} =3)$137241.65
100.430816.936681.3610841.1512394.88
$(\Gamma _{k} =4)$0.845953.11
$(\Gamma _{vp} =4)$151462.99
Table 9.  Sensitivity analysis of $T_{max}$ parameter
P3Objective value for different $T_{max}$ values
360480550600
DP24461.3924074.623517.3523517.35
RP ($\rho$=0.4)28363.8528076.6328076.6328076.63
RP ($\rho$=0.8)34017.1533606.1533109.9933109.99
RP ($\rho$=1)37076.5536593.3733857.533857.5
AVE30979.73530587.687529640.367529640.3675
P3Objective value for different $T_{max}$ values
360480550600
DP24461.3924074.623517.3523517.35
RP ($\rho$=0.4)28363.8528076.6328076.6328076.63
RP ($\rho$=0.8)34017.1533606.1533109.9933109.99
RP ($\rho$=1)37076.5536593.3733857.533857.5
AVE30979.73530587.687529640.367529640.3675
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