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Topic #4b: Short-Haul Freight Transportation Traveling Salesman Problem, TSP Vehicle Routing Problem, VRP (Node Routing Problem) – Capacity and Length Constraints (7.4) – Time Window Constraint (7.5) – VRP Heuristics

Brief introduction to other short-haul problems

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Traveling Salesman Problem (TSP) Min.

∑∑ c

ij

xij

∑i xij = 1 ∀j ∑ xij = 1 ∀i i

j ≠i

j

xij = 0 or 1 s.t.

feasible

xij=1 if the tour passes from i to j,=0 otherwise. cij≡ the distance (cost/time) from i to j. (p. 252)

2


Illustration of Sub-tour Connectivity Constraint

Sub-tour Elimination Constraint If only some of these constraints are considered, it can generate a lower bound. How strong is it? (p. 252)

3


Illustration of Vehicle Routing Problem (VRP)

4


MIP of Capacitated VRP

s.t.

5


MIP of Capacitated VRP (CVRP)

xvij = 1 if the route of the vth vehicle passes from i to j,=0 otherwise. cij = the distance (cost/time) from i to j. di = the load at i. Tv = the size of vehicle v. 6


Set Partitioning Formulation of VRPs

What are the advantages and disadvantages?

(p. 267)

7


Example of VRP Heuristics Savings Heuristic Identify the distance matrix Identify the savings matrix Assign customers to vehicles or routes Sequence customers within routes

(Chopra and Meindl, p. 437 - 444)

8


Vehicle Capacity = 200 units

20 15

3/43 1/48

4/92

10 2/36

5

5/57 6/16

0 0 9/57 -5

8/30

10

15 7/56 20 10/47

-10 -15

5

12/55

25

11/91

13/38

-20 (Chopra and Meindl, p. 437 - 444)

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Distance Matrix DC DC 1 2 3 4 5 6 7 8 9 10 11 12 13

0 12 8 17 15 15 20 17 8 6 16 21 11 15

1 0 9 8 9 17 23 22 17 18 23 28 22 27

2

0 10 8 9 15 13 9 12 14 18 14 20

(Chopra and Meindl, p. 437 - 444)

3

0 4 14 20 20 19 22 22 26 24 30

4

0 11 16 16 16 20 19 22 21 28

5

0 6 5 11 17 9 11 14 22

6

0 4 14 20 8 7 16 23

7

0 10 16 4 6 12 20

8

0 6 8 13 5 12

9

0 14 19 7 9

10

0 5 9 16

11

0 13 20

12

13

0 8

0

10


Illustration of Saving

(p. 272)

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Saving Matrix DC DC 1 2 3 4 5 6 7 8 9 10 11 12 13

1 0 11 21 18 10 9 7 3 0 5 5 1 0

2

0 15 15 14 13 12 7 2 10 11 5 3

3

0 28 18 17 14 6 1 11 12 4 2

4

0 19 19 16 7 1 12 14 5 2

5

0 29 27 12 4 22 25 12 8

6

0 33 14 6 28 34 15 12

7

0 15 7 29 32 16 12

8

0 8 16 16 14 11

9

0 8 8 10 12

10

0 32 18 15

11

0 19 16

12

0 18

13

0

S(x,y)= Dist(DC,x) + Dist(DC,y) - Dist(x,y) (Chopra and Meindl, p. 437 - 444)

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Grouping Result

20 15

(1,3,4) =183

3/43

1/48

4/92

10 2/36

5

(2,9) =93

0 0 9/57 -5

5

(6,7,8,11) =193

6/16 8/30

10

15 7/56 20 10/47

-10 -15

5/57

12/55

25

11/91

(5,10,13,13) =197

13/38

-20 (Chopra and Meindl, p. 437 - 444)

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Sequence customers within routes It is actually a TSP. With (5,10,12,13) as an example: – Farther insert: » » » »

1st: 5(30), 10(32), 12(22), 13(30) — (DC,10,DC) 2nd: 5(40), 12(36), 13(46) — (DC,10,13,DC) 3nd: 5(55), 12(48) — (DC,5,10,13,DC) …, final result: (DC, 5,10,12,13, DC): 56

– Nearest insert: » Starting from 12(22), final result: (DC, 5,10,12,13, DC): 56

– Nearest neighbor » Starting from 12(22), final result: (DC, 12,10,5,13, DC): 66

– Sweep: final result: (DC, 5,10,12,13, DC): 56 (Chopra and Meindl, p. 437 - 444)

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Example for Route Improvement 20 Consider the route (DC, 12, 10, 5, 13, DC)

3/43

15

1/48

4/92

10

2/36

5

5/57

0 -5

0

9/57

-10 -15

5

8/3010 12/55

15

6/16

7/56 20

10/47

25

11/91

13/38

-20 (Chopra and Meindl, p. 437 - 444)

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Route Improvement Procedure

2-OPT: Break (DC, 12, 10, 5, 13, DC) as (12, 10, 5) and (13, DC)

3-OPT: break (DC, 12, 10, 5, 13, DC) as (DC), (5,10) and (12, 13) with 8 combinations for re-connection

The possibility of Inter-route improvement. (Chopra and Meindl, p. 437 - 444)

16


Result of Saving Matrix Method

Total = 175

(Chopra and Meindl, p. 437 - 444)

17


Another Version of Saving Heuristic

(p. 271)

18


Another Version of Saving Heuristic

(p. 271)

19


Various Features of VRP Researches

20


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Other Short-Haul Transportation Problems Arc Routing Problem (7.6) – Garbage Collection, Mail Delivery, Maintenance etc.

Real-Time Vehicle Routing and Dispatching (7.7) – Dial-a-ride system etc.

Integrated Location and Routing (7.8) Vendor-Managed Inventory Routing (7.9)

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