Retail Distribution

1
Retail Distribution

• Inside a City or small Region

2
The Traveling Salesman Problem

• Given a set of cities and distances among
  them, find a closed tour, not visiting any city
  more than once, and passing through all cities

3
Exemple
4
Heuristic

• An aproximate procedure
• Does not garantie optimality
• Is not bad
• Easy to perform

5
Triangle Heuristic

• Start with any partial solution, e.g the convex
  envelop of the cities
• Por each remaining point, find the triangle with
  a vertex on it and two other points in the
  partial solution, whose angle at the point is
  larger
• Insert the new point between the oposite existing
  points
• Use the new solution as the new partial solution
• Keep going

6
Initial Tour
7
Initial Tour
8
Insertion
9
Solution
10
Can you improve it?
11
Vehicle routing
12
Euclidean Distances

• Y calculables por una simple fórmula

13
Examples
Straight line p 2
Manhattan p 1
14
Ejemplos
Intermedias p 1.5
Superiores p 3
15
Problem Data
16
Angle Heuristic

• Assume that each vehicle has a capacity of 8
  customers plus euclidean 2-distance

17
Angle Heuristic
18
Angle Heuristic
19
Angle Heuristic
20
Angle Heuristic
21
Angle Heuristic
22
Angle Heuristic
23
Angle Heuristic
24
Angle Heuristic
25
Angle Heuristic
26
Angle Heuristic
27
Angle Heuristic
28
Improvement Algorithm
29
Initial
30
Improvement?
?
31
Improvement?
d(0,a) d(b,0) - d(a,b)
a
?
b
32
Procedimiento (más usado)

• Compute all improvements
• Order in decreasing order
• Keep adding
• Fullfilling constraints

33
Optimization?

• Too complicated!
• However use aproximations
• The most usual one
• Assume a route for each point
• The route passes through the point, Shall we use
  it?
• Is it better to assign the point to other routes?
• Its an assignment problem
• Let yi 1 if you use route i 0 otherwise
• Let xji 1 if node j is assigned to the route
  containing i

34
VILP
Savins?
d(0,j) -d(0,i)d(i,j)
Cost of opening route at i fixed d(i,0)d(0,i)
?
j
i
35
Formulation
36
Resumen

• All construct routes
• No closed tours
• Need TSP
• But disturbs the solution
• Heuristics
• Available software with many variations