Transportation Logistics
An introduction to VRP
Transportation Logistics
Part V: An introduction toVRP
c© R.F. Hartl, S.N. Parragh 1 / 120
Transportation Logistics
An introduction to VRP
A classification
Example 1
A furniture retailer is planning tosupply a given number ofcustomers with the furniture theyordered, using the companyowned vehicle fleet. Each vehicleis of limited capacity. Whichcustomers should be put onwhich vehicle’s tour (starting andending at the warehouse) and inwhich order, such that the totaldistance traveled is minimized?
Example 2
On a given winter day, the roadmaintenance authority has tomake sure that all major roadsare plowed. How shall theavailable snow plowing vehiclesby employed and which roadsegment shall be plowed bywhich vehicle such that the totaldistance traveled is minimal?
Source: Domschke (1997) Logistik: Rundreisen und Touren, Chapter 5.
c© R.F. Hartl, S.N. Parragh 2 / 120
Transportation Logistics
An introduction to VRP
A classification
Example 1
A furniture retailer is planning tosupply a given number ofcustomers with the furniture theyordered, using the companyowned vehicle fleet. Each vehicleis of limited capacity. Whichcustomers should be put onwhich vehicle’s tour (starting andending at the warehouse) and inwhich order, such that the totaldistance traveled is minimized?
→ node routing problem
Example 2
On a given winter day, the roadmaintenance authority has tomake sure that all major roadsare plowed. How shall theavailable snow plowing vehiclesby employed and which roadsegment shall be plowed bywhich vehicle such that the totaldistance traveled is minimal?
Source: Domschke (1997) Logistik: Rundreisen und Touren, Chapter 5.
c© R.F. Hartl, S.N. Parragh 3 / 120
Transportation Logistics
An introduction to VRP
A classification
Example 1
A furniture retailer is planning tosupply a given number ofcustomers with the furniture theyordered, using the companyowned vehicle fleet. Each vehicleis of limited capacity. Whichcustomers should be put onwhich vehicle’s tour (starting andending at the warehouse) and inwhich order, such that the totaldistance traveled is minimized?
→ node routing problem
Example 2
On a given winter day, the roadmaintenance authority has tomake sure that all major roadsare plowed. How shall theavailable snow plowing vehiclesby employed and which roadsegment shall be plowed bywhich vehicle such that the totaldistance traveled is minimal?
→ arc-routing problem
Source: Domschke (1997) Logistik: Rundreisen und Touren, Chapter 5.
c© R.F. Hartl, S.N. Parragh 4 / 120
Transportation Logistics
An introduction to VRP
A classification
Example 1
A furniture retailer is planning tosupply a given number ofcustomers with the furniture theyordered, using the companyowned vehicle fleet. Each vehicleis of limited capacity. Whichcustomers should be put onwhich vehicle’s tour (starting andending at the warehouse) and inwhich order, such that the totaldistance traveled is minimized?
→ node routing problem
Example 2
On a given winter day, the roadmaintenance authority has tomake sure that all major roadsare plowed. How shall theavailable snow plowing vehiclesby employed and which roadsegment shall be plowed bywhich vehicle such that the totaldistance traveled is minimal?
→ arc-routing problem
Source: Domschke (1997) Logistik: Rundreisen und Touren, Chapter 5.
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Transportation Logistics
An introduction to VRP
A classification
The ’basic problem’: capacitated VRP (CVRP)
constraints: capacity of the vehicles
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Transportation Logistics
An introduction to VRP
A classification
Classical extensions
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Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
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Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
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Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
time windows (TW)
c© R.F. Hartl, S.N. Parragh 10 / 120
Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
time windows (TW)
soft time windows (STW)
c© R.F. Hartl, S.N. Parragh 11 / 120
Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
time windows (TW)
soft time windows (STW)
multiple time windows (MTW)
c© R.F. Hartl, S.N. Parragh 12 / 120
Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
time windows (TW)
soft time windows (STW)
multiple time windows (MTW)
split deliveries (SD)
c© R.F. Hartl, S.N. Parragh 13 / 120
Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
time windows (TW)
soft time windows (STW)
multiple time windows (MTW)
split deliveries (SD)
backhauls (B)
c© R.F. Hartl, S.N. Parragh 14 / 120
Transportation Logistics
An introduction to VRP
A classification
Classical extensions
multiple depots (MD)
heterogeneous fleet (HF)
time windows (TW)
soft time windows (STW)
multiple time windows (MTW)
split deliveries (SD)
backhauls (B)
pickup and delivery (PD)
c© R.F. Hartl, S.N. Parragh 15 / 120
Transportation Logistics
An introduction to VRP
A classification
Classical extensions
VRP
MDVRP VRPB VRPTW VRPPD
MDVRPB MDVRPTW VRPBTW VRPPDTW
MDVRPBTW
c© R.F. Hartl, S.N. Parragh 16 / 120
Transportation Logistics
An introduction to VRP
CRVP
The capacitated vehicle routing problem
Notation
xij =
{
1, if arc (ij) is part of the solution,
0, otherwise.
cij = the costs to traverse arc (i, j)
di = demand of customer i
C = vehicle capacity
A...set of arcs,V ...set of verticesn...number of customers0...depotS...subest of customers
r(S) = ⌈∑
i∈S di/C⌉...the min. number of vehicles needed to serve the nodes in S.
(CVRP is formulated on a complete directed graph)
c© R.F. Hartl, S.N. Parragh 17 / 120
Transportation Logistics
An introduction to VRP
CRVP
The capacitated vehicle routing problem
CVRP 1 - capacity cut constraints (CCCs)
∑
i∈V
∑
j∈V
cijxij → min (1)
∑
i∈V \{j}
xij = 1 ∀j ∈ V \ {0}, (2)
∑
j∈V \{i}
xij = 1 ∀i ∈ V \ {0}, (3)
∑
i∈S
∑
j /∈S
xij ≥ r(S) ∀S ⊆ V \ {0}, S 6= Ø, (4)
xij ∈ {0, 1} ∀i, j ∈ V. (5)
set of CCC: cardinality grows exponentially with n
c© R.F. Hartl, S.N. Parragh 18 / 120
Transportation Logistics
An introduction to VRP
CRVP
The capacitated vehicle routing problem
CVRP 2 - generalized subtour elmination constraints (GSECs)
∑
i∈V
∑
j∈V
cijxij → min (6)
∑
i∈V \{j}
xij = 1 ∀j ∈ V \ {0}, (7)
∑
j∈V \{i}
xij = 1 ∀i ∈ V \ {0}, (8)
∑
i∈S
∑
j∈S
xij ≤ |S| − r(S) ∀S ⊆ V \ {0}, S 6= Ø, (9)
xij ∈ {0, 1} ∀i, j ∈ V. (10)
set of GSECs: cardinality grows exponentially with n
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Transportation Logistics
An introduction to VRP
CRVP
The capacitated vehicle routing problem
CVRP 3 - adapted Miller-Tucker-Zemlin (MTZ) subtourelimination constraints
∑
i∈V
∑
j∈V
cijxij → min (11)
∑
i∈V \{j}
xij = 1 ∀j ∈ V \ {0}, (12)
∑
j∈V \{i}
xij = 1 ∀i ∈ V \ {0}, (13)
ui − uj + Cxij ≤ C − dj ∀i, j ∈ V \ {0}, i 6= j, such that di + dj ≤ C (14)
xij ∈ {0, 1} ∀i, j ∈ V, (15)
di ≤ ui ≤ C ∀i ∈ V \ {0} (16)
ui ∀i ∈ V \ {0} gives the load of the vehicle after visiting cust. iset of MTZ subtour elim: polynomial cardinality
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Transportation Logistics
An introduction to VRP
CRVP
Given vehicle fleet m
∑
j∈V
x0j ≤ m (17)
∑
i∈V
xi0 ≤ m (18)
∑
j∈V
x0j = m (19)
∑
i∈V
xi0 = m (20)
c© R.F. Hartl, S.N. Parragh 21 / 120
Transportation Logistics
An introduction to VRP
Constructive heuristics
Constructive heuristics
Clarke and Wright savings algorithm
Sequential and parallel insertion heuristics
Cluster first route second heuristics
Petal algorithms
Route first cluster second heuristics
c© R.F. Hartl, S.N. Parragh 22 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Clarke and Wright savings algorithm
⋆ Merging (0, . . . , i, 0) and (0, j, . . . , 0) into(0, . . . , i, j, . . . , 0) results in a distance saving ofsij = ci0 + c0j − cij .
c© R.F. Hartl, S.N. Parragh 23 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Clarke and Wright savings algorithm
⋆ Merging (0, . . . , i, 0) and (0, j, . . . , 0) into(0, . . . , i, j, . . . , 0) results in a distance saving ofsij = ci0 + c0j − cij .
Initialization Computation of savings: for each i and j(i 6= j) compute the savings sij = ci0 + c0j − cij; create asingle customer route for each customer i: (0, i, 0); putsavings into a list in decreasing order.
c© R.F. Hartl, S.N. Parragh 24 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Clarke and Wright savings algorithm
⋆ Merging (0, . . . , i, 0) and (0, j, . . . , 0) into(0, . . . , i, j, . . . , 0) results in a distance saving ofsij = ci0 + c0j − cij .
Initialization Computation of savings: for each i and j(i 6= j) compute the savings sij = ci0 + c0j − cij; create asingle customer route for each customer i: (0, i, 0); putsavings into a list in decreasing order.
Step 1 Best feasible merge: start with the first entry of thesavings list. For each saving sij, determine if there exist tworoutes, one with arc or edge (0, j) and one with arc (i, 0),that can be merged in a feasible way. If yes combine the two:delete (0, j) and (i, 0), introduce (i, j).
c© R.F. Hartl, S.N. Parragh 25 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example - Data
0
1
2
3
4
5
6
7
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 10
0 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
di 0 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 26 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 27 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 28 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 29 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 30 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 31 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 32 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 33 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 34 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 35 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 36 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 37 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 38 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 39 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 40 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 41 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 42 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 43 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 44 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 45 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 46 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 47 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 48 / 120
Transportation Logistics
An introduction to VRP
Savings algorithm
Savings - Example
0
1
2
3
4
5
6
7
8
9
10
sij 1 2 3 4 5 6 7 8 9 10
1 x 5.8 5.3 4.9 2.7 0.0 0.3 0.6 1.7 3.12 5.8 x 6.5 2.7 1.2 0.1 0.0 1.4 2.9 4.53 5.3 6.5 x 2.0 0.6 0.7 0.1 2.6 5.4 7.54 4.9 2.7 2.0 x 5.8 1.4 1.7 0.0 0.1 0.65 2.7 1.2 0.6 5.8 x 2.6 2.7 0.4 0.1 0.06 0.0 0.1 0.7 1.4 2.6 x 4.1 4.0 4.3 2.17 0.3 0.0 0.1 1.7 2.7 4.1 x 1.8 1.4 0.68 0.6 1.4 2.6 0.0 0.4 4.0 1.8 x 6.0 4.49 1.7 2.9 5.4 0.1 0.1 4.3 1.4 6.0 x 8.610 3.1 4.5 7.5 0.6 0.0 2.1 0.6 4.4 8.6 x
di 2 1 1 2 1 1 2 1 1 2
C = 4
c© R.F. Hartl, S.N. Parragh 49 / 120
Transportation Logistics
An introduction to VRP
Sequential and parallel insertion
A sequential insertion algorithm
Step 1 If all vertices belong to a route → STOP. Otherwise,initialize a new tour with an unrouted vertex k. New currenttour (0, k, 0). Choose k according to a certain criterion, e.g.random, closests to 0, farthest from 0 ...
Step 2 Do cheapest feasible insertion into the current touruntil no additional vertex can be feasibly inserted. In thiscase, go to step 1.
c© R.F. Hartl, S.N. Parragh 50 / 120
Transportation Logistics
An introduction to VRP
Sequential and parallel insertion
A sequential insertion algorithm
Step 1 If all vertices belong to a route → STOP. Otherwise,initialize a new tour with an unrouted vertex k. New currenttour (0, k, 0). Choose k according to a certain criterion, e.g.random, closests to 0, farthest from 0 ...
Step 2 Do cheapest feasible insertion into the current touruntil no additional vertex can be feasibly inserted. In thiscase, go to step 1.
⋆ One route is filled after the other.
c© R.F. Hartl, S.N. Parragh 51 / 120
Transportation Logistics
An introduction to VRP
Sequential and parallel insertion
A parallel insertion algorithm
Step 1 Initialize the minimum number of routes ⌈∑
i di/C⌉with seed customers (chosen randomly, closest to 0, farthestfrom 0 ...)
Step 2 Start with customer 1. Sequentially, insert onecustomer after the other at its best insertion position. If acustomer cannot be inserted in a feasible way in any of theroutes, add an additional route.
c© R.F. Hartl, S.N. Parragh 52 / 120
Transportation Logistics
An introduction to VRP
Sequential and parallel insertion
A parallel insertion algorithm
Step 1 Initialize the minimum number of routes ⌈∑
i di/C⌉with seed customers (chosen randomly, closest to 0, farthestfrom 0 ...)
Step 2 Start with customer 1. Sequentially, insert onecustomer after the other at its best insertion position. If acustomer cannot be inserted in a feasible way in any of theroutes, add an additional route.
⋆ Several routes are considered and filled simultaneously.
c© R.F. Hartl, S.N. Parragh 53 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Cluster first route second heuristics
Sweep algorithm
Fisher and Jaikumar Algorithm
c© R.F. Hartl, S.N. Parragh 54 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm
⋆ Only applicable to planar instances (polar coordinates necessary:angle and ray length).
c© R.F. Hartl, S.N. Parragh 55 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm
⋆ Only applicable to planar instances (polar coordinates necessary:angle and ray length).Clusters are formed by rotating a ray centered at the depot:
Clustering Start with the unassigned vertex with the smallestanlge. Assign vertices to vehicle k as long as the capacity (orthe route length) is not exceeded. If there are still unroutedvertices, initialize a new vehicle.
Routing Solve a TSP for each cluster (exact or heuristicmethods, e.g. nearest neighbor).
c© R.F. Hartl, S.N. Parragh 56 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
.
.
.
.
c© R.F. Hartl, S.N. Parragh 57 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, ....
c© R.F. Hartl, S.N. Parragh 58 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, ....
c© R.F. Hartl, S.N. Parragh 59 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
1
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 ....
c© R.F. Hartl, S.N. Parragh 60 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, ...
c© R.F. Hartl, S.N. Parragh 61 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
5
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, 5 ...
c© R.F. Hartl, S.N. Parragh 62 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
5
7
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, 5 .7, ..
c© R.F. Hartl, S.N. Parragh 63 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
5
7
6
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, 5 .7, 6, ..
c© R.F. Hartl, S.N. Parragh 64 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
5
7
6
8
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, 5 .7, 6, 8 ..
c© R.F. Hartl, S.N. Parragh 65 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
5
7
6
8
9
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, 5 .7, 6, 8 .9, .
c© R.F. Hartl, S.N. Parragh 66 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Clustering
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
3
2
14
5
7
6
8
9
10
i x y di angle0 0 0 0 0.001 2 4 2 63.42 3 2 1 33.73 5 1 1 11.34 -2 4 2 116.65 -3 2 1 146.36 -3 -4 1 233.17 -2 -1 2 206.68 1 -3 1 288.49 4 -5 1 308.710 5 -2 2 338.2
C = 4
3, 2, 1 .4, 5 .7, 6, 8 .9, 10 .
c© R.F. Hartl, S.N. Parragh 67 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 10
0 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
.
.
.
.
c© R.F. Hartl, S.N. Parragh 68 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0 ....
c© R.F. Hartl, S.N. Parragh 69 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2 ....
c© R.F. Hartl, S.N. Parragh 70 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1 ....
c© R.F. Hartl, S.N. Parragh 71 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3 ....
c© R.F. Hartl, S.N. Parragh 72 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 ....
c© R.F. Hartl, S.N. Parragh 73 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0 ...
c© R.F. Hartl, S.N. Parragh 74 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5 ...
c© R.F. Hartl, S.N. Parragh 75 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4 ...
c© R.F. Hartl, S.N. Parragh 76 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 ...
c© R.F. Hartl, S.N. Parragh 77 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0 ..
c© R.F. Hartl, S.N. Parragh 78 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7 ..
c© R.F. Hartl, S.N. Parragh 79 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6 ..
c© R.F. Hartl, S.N. Parragh 80 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
0
3
2
14
5
7
6
8
9
10
cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6-8 ..
c© R.F. Hartl, S.N. Parragh 81 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
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cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6-8-0 ..
c© R.F. Hartl, S.N. Parragh 82 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
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-3
-2
-1
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cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6-8-0 .0 .
c© R.F. Hartl, S.N. Parragh 83 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
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cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6-8-0 .0-10 .
c© R.F. Hartl, S.N. Parragh 84 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
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cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6-8-0 .0-10-9 .
c© R.F. Hartl, S.N. Parragh 85 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Sweep algorithm - Example
Routing (Nearest neighbor)
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5
-4
-3
-2
-1
0
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cij 0 1 2 3 4 5 6 7 8 9 100 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
0-2-1-3-0 .0-5-4-0 .0-7-6-8-0 .0-10-9-0 .
c© R.F. Hartl, S.N. Parragh 86 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm
⋆ Not directly applicable to VRP with distance constraint.⋆ Assumption: size of the vehicle fleet is given.
Clustering
Step 1 Seed selection: choose seed vertices jk ∈ V toinitialize each cluster k.Step 2 Allocation of customers to seeds: Compute the costaik of allocating customer i to cluster k:aik = min{c0i + cijk + cjk0, c0jk + cjki + ci0} − (c0jk + cjk0).Step 3 Generalized assignment: Solve a GAP with aik, weightsdi (demand) and vehicle capacity C to obtain the clusters.
Routing Solve a TSP for each cluster.
c© R.F. Hartl, S.N. Parragh 87 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm
The generalized assignment problem (GAP)
xik =
{
1, if i assigned to cluster k,
0, otherwise.(21)
min
∑
i
∑
k
aikxik (22)
∑
k
xik = 1 ∀i (23)
∑
i
dixik ≤ C ∀k (24)
xik ∈ {0, 1} (25)
⋆ The GAP is more difficult than the AP.c© R.F. Hartl, S.N. Parragh 88 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Clustering
0
1
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4 vehiclesseed vertices: 2,5,6,10.aik 2 5 6 101 3.1 6.3 8.9 5.82 0.0 6.0 7.1 2.73 3.7 9.6 9.5 2.74 6.3 3.1 7.5 8.35 6.0 0.0 4.6 7.26 9.9 7.4 0.0 7.97 4.5 1.8 0.4 3.98 4.9 6.0 2.3 1.99 9.9 12.7 8.5 4.210 6.3 10.7 8.6 0.0
c© R.F. Hartl, S.N. Parragh 89 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Clustering
0
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4 vehiclesseed vertices: 2,5,6,10compute aik values.aik 2 5 6 101 3.1 6.3 8.9 5.82 0.0 6.0 7.1 2.73 3.7 9.6 9.5 2.74 6.3 3.1 7.5 8.35 6.0 0.0 4.6 7.26 9.9 7.4 0.0 7.97 4.5 1.8 0.4 3.98 4.9 6.0 2.3 1.99 9.9 12.7 8.5 4.210 6.3 10.7 8.6 0.0
c© R.F. Hartl, S.N. Parragh 90 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Clustering
0
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3
4
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4 vehiclesseed vertices: 2,5,6,10compute aik values.aik 2 5 6 101 3.1 6.3 8.9 5.82 0.0 6.0 7.1 2.73 3.7 9.6 9.5 2.74 6.3 3.1 7.5 8.35 6.0 0.0 4.6 7.26 9.9 7.4 0.0 7.97 4.5 1.8 0.4 3.98 4.9 6.0 2.3 1.99 9.9 12.7 8.5 4.210 6.3 10.7 8.6 0.0
c© R.F. Hartl, S.N. Parragh 91 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Clustering
0
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3
4
5
6
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solve the GAP(e.g. with Excel solver)
c© R.F. Hartl, S.N. Parragh 92 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Clustering
0
1
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3
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5
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solve the GAP(e.g. with Excel solver)xik 2 5 6 101 1 0 0 02 1 0 0 03 0 0 0 14 0 1 0 05 0 1 0 06 0 0 1 07 0 0 1 08 0 0 1 09 0 0 0 110 0 0 0 1
⋆ The GAP can also be solved heuristically (e.g. greedy).
c© R.F. Hartl, S.N. Parragh 93 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Routing (e.g. nearest neighbor)
0
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cij 0 1 2 3 4 5 6 7 8 9 10
0 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
c© R.F. Hartl, S.N. Parragh 94 / 120
Transportation Logistics
An introduction to VRP
Cluster first route second heuristics
Fisher and Jaikumar Algorithm - Example
Routing (e.g. nearest neighbor)
0
1
2
3
4
5
6
7
8
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cij 0 1 2 3 4 5 6 7 8 9 10
0 0.0 4.5 3.6 5.1 4.5 3.6 5.0 2.2 3.2 6.4 5.41 4.5 0.0 2.2 4.2 4.0 5.4 9.4 6.4 7.1 9.2 6.72 3.6 2.2 0.0 2.2 5.4 6.0 8.5 5.8 5.4 7.1 4.53 5.1 4.2 2.2 0.0 7.6 8.1 9.4 7.3 5.7 6.1 3.04 4.5 4.0 5.4 7.6 0.0 2.2 8.1 5.0 7.6 10.8 9.25 3.6 5.4 6.0 8.1 2.2 0.0 6.0 3.2 6.4 9.9 8.96 5.0 9.4 8.5 9.4 8.1 6.0 0.0 3.2 4.1 7.1 8.27 2.2 6.4 5.8 7.3 5.0 3.2 3.2 0.0 3.6 7.2 7.18 3.2 7.1 5.4 5.7 7.6 6.4 4.1 3.6 0.0 3.6 4.19 6.4 9.2 7.1 6.1 10.8 9.9 7.1 7.2 3.6 0.0 3.210 5.4 6.7 4.5 3.0 9.2 8.9 8.2 7.1 4.1 3.2 0.0
c© R.F. Hartl, S.N. Parragh 95 / 120
Transportation Logistics
An introduction to VRP
Petal algorithms
Petal algorithms
1 Route generationGenerate a bundle of routes (one route is called a petal),using constructive methods (e.g. sweep (starting from eachvertex once), randomized sequential insertion, etc.)
2 Route selectionSelect the best combination of routes by solving a setpartitioning problem.
c© R.F. Hartl, S.N. Parragh 96 / 120
Transportation Logistics
An introduction to VRP
Petal algorithms
Petal algorithms
The set partitioning problem
xr =
{
1, if petal/route r selected,
0, otherwise.
air = parameter (0/1) indicating whether customer i is on route r
cr = costs of petal r
min∑
r
crxr (26)
∑
r
airxr = 1 ∀i (27)
xr ∈ {0, 1} ∀r (28)
c© R.F. Hartl, S.N. Parragh 97 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics
RoutingUse a TSP solution method to generate one giant tourthrough all the customers.
PartitioningInsert the depot in such a way that feasible VRP tours areobtained (respecting the capacity and/or the distanceconstraint). Exact and heuristic methods possible.
c© R.F. Hartl, S.N. Parragh 98 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics
RoutingUse a TSP solution method to generate one giant tourthrough all the customers.
PartitioningInsert the depot in such a way that feasible VRP tours areobtained (respecting the capacity and/or the distanceconstraint). Exact and heuristic methods possible.
⋆ Only applicable if the there is no constraint on the number ofvehicles.
c© R.F. Hartl, S.N. Parragh 99 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Assume the following TSP tour through all vertices:
0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 0
0 1 2 3 4 5 6 7 8 9 10
di 0 2 1 1 2 1 1 2 1 1 2
c© R.F. Hartl, S.N. Parragh 100 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Assume the following TSP tour through all vertices:
0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 0
0 1 2 3 4 5 6 7 8 9 10
di 0 2 1 1 2 1 1 2 1 1 2
Option 1 (heuristic)Insert the depot whenever necessary:
c© R.F. Hartl, S.N. Parragh 101 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Assume the following TSP tour through all vertices:
0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 0
0 1 2 3 4 5 6 7 8 9 10
di 0 2 1 1 2 1 1 2 1 1 2
Option 1 (heuristic)Insert the depot whenever necessary:0 - 1 - 2 - 3 - 0 - 4 - 5 - 6 - 0 - 7 - 8 - 9 - 0 - 10 - 0
c© R.F. Hartl, S.N. Parragh 102 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Assume the following TSP tour through all vertices:
0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 0
0 1 2 3 4 5 6 7 8 9 10
di 0 2 1 1 2 1 1 2 1 1 2
Option 1 (heuristic)Insert the depot whenever necessary:0 - 1 - 2 - 3 - 0 - 4 - 5 - 6 - 0 - 7 - 8 - 9 - 0 - 10 - 0
Option 2 (exact)Find the best partitioning (shortest path algorithms!):
c© R.F. Hartl, S.N. Parragh 103 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
01
2
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3.6
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di1 22 13 14 25 16 17 28 19 110 2
C = 4
c© R.F. Hartl, S.N. Parragh 104 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
01
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di1 22 13 14 25 16 17 28 19 110 2
C = 4
1 2 3 4 5 6 7 8 9 10 0
9 7.2 10.2 9 7.2 10 4.4 6.4 12.8 10.8
1-2:10.3
1-2-3:14.0
2-3:10.9
2-3-4:17.9
3-4:17.2
3-4-5:18.6
4-5:10.3
4-5-6:17.7
5-6:14.6
5-6-7:15.0
6-7:10.4
6-7-8:14.9
7-8:9.0
7-8-9:15.9
8-9:13.2
8-9-10:15.3
9-10:15
arc weights from node i to node k: c0i + cj0 + lij , with lij costs of going from i to j in TSP tour and j directpredecessor of k in TSP tour.
c© R.F. Hartl, S.N. Parragh 105 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited
c© R.F. Hartl, S.N. Parragh 106 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.0
c© R.F. Hartl, S.N. Parragh 107 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.0
c© R.F. Hartl, S.N. Parragh 108 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.3
c© R.F. Hartl, S.N. Parragh 109 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.0
c© R.F. Hartl, S.N. Parragh 110 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.0
c© R.F. Hartl, S.N. Parragh 111 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.3
c© R.F. Hartl, S.N. Parragh 112 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.36 31.7;4 34.7;6 39.2;6 -;1 -;1 7; 31.7
c© R.F. Hartl, S.N. Parragh 113 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.36 31.7;4 34.7;6 39.2;6 -;1 -;1 7; 31.77 34.7;6 39.2;6 47.6;7 -;1 8; 34.7
c© R.F. Hartl, S.N. Parragh 114 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.36 31.7;4 34.7;6 39.2;6 -;1 -;1 7; 31.77 34.7;6 39.2;6 47.6;7 -;1 8; 34.78 39.2;6 47.6;7 50;8 9; 39.2
c© R.F. Hartl, S.N. Parragh 115 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.36 31.7;4 34.7;6 39.2;6 -;1 -;1 7; 31.77 34.7;6 39.2;6 47.6;7 -;1 8; 34.78 39.2;6 47.6;7 50;8 9; 39.29 47.6;7 50;8 10; 47.6
c© R.F. Hartl, S.N. Parragh 116 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.36 31.7;4 34.7;6 39.2;6 -;1 -;1 7; 31.77 34.7;6 39.2;6 47.6;7 -;1 8; 34.78 39.2;6 47.6;7 50;8 9; 39.29 47.6;7 50;8 10; 47.610 50;8 0; 50
c© R.F. Hartl, S.N. Parragh 117 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
Find the shortest path in the auxiliary graph (Dijkstra!)
D[1], D[2], D[3], D[4], D[5], D[6], D[7], D[8] D[9] D[10] D[0]
n V[1] V[2] V[3] V[4] V[5] V[6] V[7] V[8] V[9] V[10] V[0] visited0 0;1 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 1; 0.01 9;1 10.3;1 14;1 -;1 -;1 -;1 -;1 -;1 -;1 -;1 2; 9.02 10.3;1 14;1 26.9;2 -;1 -;1 -;1 -;1 -;1 -;1 3; 10.33 14;1 26.9;2 28.9;3 -;1 -;1 -;1 -;1 -;1 4; 14.04 23;4 24.3;4 31.7;4 -;1 -;1 -;1 -;1 5; 23.05 24.3;4 31.7;4 38;5 -;1 -;1 -;1 6; 24.36 31.7;4 34.7;6 39.2;6 -;1 -;1 7; 31.77 34.7;6 39.2;6 47.6;7 -;1 8; 34.78 39.2;6 47.6;7 50;8 9; 39.29 47.6;7 50;8 10; 47.610 50;8 0; 50
The shortest path:
1 - 4 - 6 - 8 - 0
c© R.F. Hartl, S.N. Parragh 118 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
Route first cluster second heuristics - Example
01
2
3
4
5 6
7
8
9
10
3.6
5.1
4.53.6 5
2.2
3.2
6.4
4.5
2.2
2.2
7.6
2.26
3.2
3.6
3.6
3.2
5.4
di1 22 13 14 25 16 17 28 19 110 2
C = 4
1 2 3 4 5 6 7 8 9 10 0
9 7.2 10.2 9 7.2 10 4.4 6.4 12.8 10.8
1-2:10.3
1-2-3:14.0
2-3:10.9
2-3-4:17.9
3-4:17.2
3-4-5:18.6
4-5:10.3
4-5-6:17.7
5-6:14.6
5-6-7:15.0
6-7:10.4
6-7-8:14.9
7-8:9.0
7-8-9:15.9
8-9:13.2
8-9-10:15.3
9-10:15
c© R.F. Hartl, S.N. Parragh 119 / 120
Transportation Logistics
An introduction to VRP
Route first cluster second heuristics
References
Paolo Toth, and Daniele Vigo (2002) The Vehicle RoutingProblem, SIAM. (Chapters 1 and 5)
W. Domschke (1997) ’Logistik: Rundreisen und Touren’Oldenbourg.
c© R.F. Hartl, S.N. Parragh 120 / 120