Lecture 02_Travelling Salesman Problem Flashcards
Evalation of a transportation network
- distances [linear, real distance]
- travel times [depending on daytime]
- costs [e.g.toll]
- capacities [e.g. pipeline, truck size]
Travelling Salesman Problem
What is the optimal route to connect all locations in one tour?
- In which order should locations be visited?
- find the shortest path
- find one tour containing all locations
TSP
Examples of Applications
1. Delivery of goods
* Sequence of delivery or pick-up
* Sub-problem of vehicle routing
2. Order picking in warehouses
3. Vehicle deployment problems
* Avoidance of idle times
4. Production control
* Sequence dependent set-up costs Cutting stock problems
Sub-Tour Elimination
TSP
Optimization Problem
z(i)−z(j)+n* x(ij) <= n−1
for i,j=2,…,n; i cannot be j
Case 1
x(ij) = 0
z(i) + 1<= z(j) + n
[ie. location is not visited]
- redundant constraint, as if it were not there
Case 2
x(ij) = 1
z(i) + 1 <= z(j)
the count of customers reached need to be smaller than the customers left
n = number of customers
Heuristics for the TSP
3 methods
I. Construction Procedures
1. Nearest neighbour
2. Succesive Insertion
II. Improvement Procedures
- exchange operators e.g. 2-Opt
Nearest Neighbour Procedure
Heuristic for TSP
Construction Procedure
- start at an aribtrary node
- go from current node to closest node which hasn’t been visited
Successive Insertion
Heuristics for TSP
Construction Procedure
- insert any node, which hasn’t been visited, in the best possible way into the so far constructed tour
2-Opt
Heuristics for TSP
Improvement Procedure
- remove 2 edges and replace these with 2 different edges
- iterate until no further improvement possible
Note
- symmetric vs. asymmetric matrix
- 2-Opt is different from City Swap
Variants of 2-Opt
Variant 1
- largest improvement (steepest descent)
- check all exchange options for current solution and choose best option
- continue until no further improvement possible
Variant 2
- next improvement (decent procedure)
- realize every improvement immediately and choose improved solution as a starting point for further exchanges
- continue until no further improvement possible
Vehicle Routing Problem
VRP
What if locations can be visited in different tours (e.g. different vehicles)
- assign locations to tours and then determine optimal order (route)
VRP
Depot
Tour
Route
Standard problem
Depot
- start and end of all delivery trips
Tour
- set of all customers, who are supplied by one vehicle in one tour
Route
- sequence of customers of one tour
Standard Problem
- one depot
- single planning period
- no time windows
Discuss the differences in applying the 2-Opt heuristic in a symmetric and asymmetric TSP?
Symmetric:
- easier, faster to solve
-> enough to calculate distances of the nodes you switch
Asymmetric:
- you also have to calculate the distance within the switched edges (difference distance between 2-4 and 4-2)
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