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

33


Delivery ProblemDelivery Problem

TroublesTroublesDiversity of ProblemsDiversity of Problems

NP-hardnessNP-hardness

44


Vehicle Routing ProblemVehicle Routing Problem

Standard FormulationStandard Formulation

Vqi

Vehicle Capacity

Demand

Vqi

i

55



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

66



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

77

Split Delivery ProblemSplit Delivery Problem

Vqi


Several Routes for Some VerticesSeveral Routes for Some Vertices

Split even ifSplit even if Vqi

88



ExampleExample

2 3 42 3

1010

10

8

8 4

1

Depot

Destinations

Demand

Distance

99



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



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

1111


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


,,...,3,2,1

niqq i

K

k

ki

,,...2,1,,...,3,2,1

Kknixqqn

j

kiji

ki

1212



Split Delivery ProblemSplit Delivery Problemvs.vs.

Vqi SDPSDP

Each solution of VRP is solution of SDPEach solution of VRP is solution of SDP

split delivery problem jan pelikán , jan fábry , václav kořenář university of economics...

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