split delivery problem jan pelikán , jan fábry , václav kořenář university of economics...
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Split DeliverySplit DeliveryProblemProblem
Jan PelikánJan Pelikán, , Jan FábryJan Fábry, Václav Kořenář, Václav KořenářUniversity of Economics PragueUniversity of Economics Prague
______________________________________________________________________________________________________________________________________________________MME 2005, Hradec KrálovéMME 2005, Hradec Králové
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Delivery ProblemDelivery Problem
Standard Problem of ORStandard Problem of OR
Fuel & Time Consumption Fuel & Time Consumption
SavingsSavings
Number of Used VehiclesNumber of Used Vehicles
* * * * * * * * * * * * * * * * * * * * * *
Using Optimum RepeatedlyUsing Optimum Repeatedly Any Improving SolutionAny Improving Solution
Impact on the EnvironmentImpact on the Environment}}
WHAT
HOW
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Delivery ProblemDelivery Problem
TroublesTroublesDiversity of ProblemsDiversity of Problems
NP-hardnessNP-hardness
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Vehicle Routing ProblemVehicle Routing Problem
Standard FormulationStandard Formulation
Vqi
Vehicle Capacity
Demand
Vqi
i
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Vehicle Routing ProblemVehicle Routing Problem
Miler-Tucker-ZemlinMiler-Tucker-Zemlin
n
i
n
jijij xc
1 1
minimizeminimize
subject tosubject to,,...,3,2,1
1njx
n
iij
,,...,3,2,11
n
jij nix
,,,...,3,2,,...,2,1,)1( jinjniuxVqu jijji
.,...,2,1,,01 niVuu i
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Vehicle Routing ProblemVehicle Routing Problem
Vehicles with Different CapacitiesVehicles with Different Capacitiesminimizeminimize
subject tosubject to
K
k
n
i
n
j
kijij xc
1 1 1
,,...2,1,,...,3,2,1 1
Kknjxxn
i
n
i
kji
kij
,,...2,1,12
1 Kkxn
j
kj
,,...3,2,11 1
nixK
k
n
j
kij
,,...,2,1,,,...,3,2,,...,2,1,)1( KkjinjniuxVqu kj
kijkj
ki
.,...,2,1,,...,3,2,,01 KkniVuu kki
k
Each vertex lies exactly on 1 route
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Split Delivery ProblemSplit Delivery Problem
Vqi
________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Several Routes for Some VerticesSeveral Routes for Some Vertices
Split even ifSplit even if Vqi
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Split Delivery ProblemSplit Delivery Problem
ExampleExample
2 3 42 3
1010
10
8
8 4
1
Depot
Destinations
Demand
Distance
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Split Delivery ProblemSplit Delivery Problem
Example – w/o SplitExample – w/o Split
2 3 42 3
1010
10
8
8 4
1
10V
3 Routes: (1-2-1); (1-3-1); (1-4-1)3 Routes: (1-2-1); (1-3-1); (1-4-1)Total Distance = 60Total Distance = 60
1010
________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Split Delivery ProblemSplit Delivery Problem
Example – SplitExample – Split
2 3 42 3
1010
10
8
8 4
1
10V
2 Routes: (1-2-3-1); (1-3-4-1)2 Routes: (1-2-3-1); (1-3-4-1)Total Distance = 45Total Distance = 45
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
minimizeminimize
subject tosubject to
K
k
n
i
n
j
kijij xc
1 1 1
,,...2,1,,...,3,2,1 1
Kknjxxn
i
n
i
kji
kij
,,...2,1,12
1 Kkxn
j
kj
,,...,2,1,,,...,3,2,,...,2,1,)1( KkjinjniuxVqu kj
kijkj
ki
.,...,2,1,,...,3,2,,01 KkniVuu kki
k
Split Delivery ProblemSplit Delivery Problem
,,...,3,2,1
niqq i
K
k
ki
,,...2,1,,...,3,2,1
Kknixqqn
j
kiji
ki
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________________________________________________________________________________________________________________________________________________________Split Delivery ProblemSplit Delivery Problem MME 2005, Hradec Králové MME 2005, Hradec Králové
Vehicle Routing ProblemVehicle Routing Problem
Split Delivery ProblemSplit Delivery Problemvs.vs.
Vqi SDPSDP
Each solution of VRP is solution of SDPEach solution of VRP is solution of SDP
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