TSP

1
TSP VRP IP Models

• Topics
• TSP VRP Problem
• History, IP Formulation Branch and Bound
• Extensions
• Heuristic Algorithms
• Meta-Heuristic
• How to Build Heuristics

2
Traveling Salesman Problem
TSP The TSP involves finding the minimum
traveling cost for visiting a fixed set of
customers. VRP The vehicle must visit each
customer exactly once and return to its point of
origin also called depot. The objective function
is the total cost of the tour.
3
Traveling Salesman Problem
The total number of solutions is (n-1)! /2 if the
distances are symmetric. For example, if there
are 50 customers to visit, the total number of
solutions is 49!/23.04x1062.
4
Traveling Salesman Problem
Solution
If the depot is located at node 1, then the
optimal tour is 1-5-2-3-4-1 with total cost equal
to 11.
5
How many solutions exist?
A solution can be represented by the sequence of
customers being served
This problem has 5 x 4 x 3 x 2 x 1 120 possible
solutions.
In general, for n customers being served by one
vehicle, n! solutions exist.
40! 8.16 x 1047
For 40 customers
A computer that can evaluate one million
solutions per second evaluates
3.15 x 1013
solutions per year.
More than 1034 years of computing!
10,000,000,000,000,000,000,000,000,000,000,000
years.
6
TSP History

• Dantzig, Fulkerson, Johnson (1954) solved TSP
  with 49 cities in U.S.

7
TSP History

• Groetschel (1977) found the optimal tour of 120
  cities from what was then West Germany

8
TSP History

• Padberg and Rinaldi (1987) found optimal tour of
  532 ATT switch locations in U.S.

9
TSP History

• Applegate et al. (1998) found optimal tour of the
  13,509 cities in the U.S. with populations
  greater than 500

10
Applications of TSP

• Printed circuit board (PCB) manufacturing
• the scheduling of a machine to drill holes in a
  circuit board or other object
• 1990, problems in VLSI fabrication were asked to
  solve 1.2 million city TSP problem

11
Applications of TSP

• Many more applications of TSP comes from
  Logistics, transportation
• Scheduling of buses to pick the school children
• Pickup and Ride (Ryder, Mayfloweretc)
• ..
• Air force Logistics -- UAV route planner
• Yet TSP VRP provide a perfect test ground for
  the development of novel ideas
• Simulated Annealing, GA, Tabu Search
• Langragean relaxation, Branch and Cut, Column
  Generation.

12
Solving the TSP

• Heuristics Quickly find a good (but not
  necessarily optimal) solution
• Nearest-neighbor heuristic
• Cheapest-insertion heuristic
• Lin-Kernigan and many others
• Branch-and-bound Guarantees optimality but may
  require large amounts of computer time
• A special BB approach Branch and But, exists
  to solve the TSP

13
Traveling Salesman Problem
The total number of solutions is (n-1)! /2 if the
distances are symmetric. For example, if there
are 50 customers to visit, the total number of
solutions is 49!/23.04x1062.
14
Traveling Salesman Problem
Solution
If the depot is located at node 1, then the
optimal tour is 1-5-2-3-4-1 with total cost equal
to 11.
15
Traveling Salesman Problem
Mathematical Programming
Inputs n number of customers including the
depot cij cost of traveling from customer i to
j Decision variables
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Traveling Salesman Problem
(1) (2)
Constraints (1) and (2) ensure that each customer
is visited exactly once.
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Traveling Salesman Problem
minimize 2x123x132x143x153x234x24...5x45 s
.t. x12x13x14x15 1 for node 1
x21x31x41x51 1 for node 1 ... ...
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Traveling Salesman Problem
Is this solution feasible to our formulation? We
need additional constraints, so-called subtour
elimination constraints.
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Traveling Salesman Problem
20
Traveling Salesman Problem
Subtour elimination constraints for the example
2
3
4
1
5
21
To See How It Works

• Cutting Plane Approach
• http//www.tsp.gatech.edu/cpapp/wi29.html
• The Algorithm
• 1) Solve the LP,
• 2) Find Violated Constraints
• 3) Add the constraint to LP
• 4) Go to 1), Resolve the Problem.

22
Vehicle Routing Problem
The vehicle routing problem involves finding a
set of trips, one for each vehicle, to deliver
known quantities of goods to a set of
customers. The objective is to minimize the
travel costs of all trips combined. The most
basic Vehicle Routing Problem (VRP) is the
single-depot capacitated VRP.
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IP Formulation

• VRP Capacitated TSP
• Variable to keep track of load
• Variations 1 Time Window
• Variable to keep track of time
• Variations 2 Pickup and Delivery
• Variable to keep track of sequence
• Other Variations
• Multi-Depot, Order Splitting, Drivers lunch,

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Extensions in the VRP Literature

• Location Routing LRP
• Fleet Size and Mix FSMVRP
• VRP With Time Windows VRPTW
• General Pickup and Delivery GPDP
• Dial-A-Ride DARP
• Periodic VRP PVRP
• Inventory Routing IRP
• Dynamic VRP DVRP
• Capacitated Arc Routing Problem CARP

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Further Topics

• TSP is Notoriously Hard to Solve Why
• The LP solution is not that meaningful
• The LP bound is not tight, .
• Integer Programming Winter, 2004.
• If you want to know why certain IP are harder to
  solver, certain are earier to solver
• Better solution techniques --

26
WebSites

• TSPBIB WebSite
• http//www.densis.fee.unicamp.br/moscato/TSPBIB_h
  ome.html
• TSP at Gatech
• http//www.tsp.gatech.edu/
• More to be added.

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Heuristic and Meta-Heuristic

• Heuristics Quickly find a good (but not
  necessarily optimal) solution
• Greedy Construction Heuristic
• Nearest-neighbor heuristic
• Cheapest-insertion heuristic
• Local Search (Improvement) Heuristic
• 2-Opt, 3-opt, K-Opt, Lin-Kernigan
• Meta-Heuristic
• GRASP, SA,GA, TABU Search

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Illustration
Goal min f(x)
f(x)
x
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Construction Heuristic for TSP

• Greedy Construction Heuristics
• A greedy sequential algorithm constructs a
  solution by a series of steps that are Greedy
• At each step, we make a decision that looks like
  a good thing to do now (Myopic), and
  sequential- we make a sequence of decisions,
  never reversing /revising an earlier decision.
• In general, it does not guarantee optimality.

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Construction Heuristic for TSP 1 Nearest Neighbor
Step 1. Start with any node as the beginning of
the path. Step 2. Find the node closest to the
last node added to the path. Step 3. Repeat Step
2 until all nodes are contained in the path.
Then, join the first and last nodes.
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Nearest Neighborhood Start at Node 1
Start with node 1.
Nodes added Tour 1 1 2 1-2 5 1-2-5 3 1-2-5-3
4 1-2-5-3-4 1-2-5-3-4-1
2
3
2
1
2
4
1
2
4
5
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Random Search

• Random Search
• At each step, randomly select a city to the tour
• Finish when a tour is formed.
• Advantages
• Quick, in a limited amount of time, many
  solutions can
• Sampling of the solution space
• Disadvantages

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Solution Quality

• Knapsack problem
• Backpack 8 units of space, 4 items to pick
• Item Value in terms of dollars 2,5,7,9
• Item Cost in terms of space units 1,3,5,7
• Construction Heuristic
• Pick the Most Valuable Item
• Pick the Most Valuable Per Unit

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Solution Quality

• Solution Quality
• For Heuristic 1 (1,4) , Value 11
• For Heuristic 2 (1,4), Value 11.
• Optimal Solution (2,3), Value 12
• None of them gives the Optimal solution
• This is true for any heuristic
• Theoretically, for a NP-Hard problem, there is no
  polynomial algorithm

35
Heuristic Search
Goal min f(x)
f(x)
x
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Neighborhood Search/Local Search

• The procedure
• 1) Begin with a feasible solution x
• 2) For the current solution x, we define a
  neighborhood of the solution, which consists of
  solutions close to x in some sense
• 3) Then, we check whether some solutions in the
  neighborhood of x is better than x
• 4) If there exist x which is better than x, set
  x x as the current solution, go to 1)
• 5) Otherwise, stop, x is a local optimum
• It is in essence an improvement heuristic

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Representation of a TSP Solution

• A Solution can also be represented as a
  permutation of the nodes starting at node 1
  1-3-4-2-6-5

3
1
4
5
2
6
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Neighborhood Structures

• Neighborhood Structure
• A commonly used neighborhood structure is k-opt,
  where k edges are removed from the current cycle
  and k new edges are added
• The number of k-opt neighbors of a Hamiltonian
  cycle on n nodes is ?(nk)
• Beyond k 3, the search becomes prohibitive in
  practice because of large running times
• This allows for very limited changes, but even
  2-opt has been found to produce solutions within
  10 of optimal for benchmark problems

39
Traveling Salesman Problem
Two Exchange Heuristics (2-opt)
Starting with any tour, we consider the effect of
removing any two arcs in the tour and replacing
them with the unique set of two arcs that form a
different tour. If we find a tour with a lower
cost than the current tour, the we use it as the
new tour. When no possible exchange can produce
a tour that is better than the current tour, we
stop.
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2 Opt Illustration
Suppose the initial tour is given as 1-2-3-4-5
with cost 15.
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2 Opt Illustration

• Procedure
• Current Sol X (1,2,3,4,5) ? 15
• Swap (2,3) (1,3,2,4,5) ? 18
• Swap (2,4) (1,4,3,2,5) ?
• Swap (2,5) (1,5,3,4,2) ?
• Swap (3,4) (1,2,4,3,5) ?
• Swap (3,5) (1,2,5,4,3) ?
• Swap (4,5) (1,2,3,5,4) ?
• The best of them is, which will be used as the
  current solution X and the
  process continues

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A good neighborhood for the knapsack problem ?

• Knapsack problem
• Backpack 8 units of space, 4 items to pick
• Item Value in terms of dollars 2,5,7,9
• Item Cost in terms of space units 1,3,5,7
• Solution Representation
• 1 Item is picked, 0 Otherwise
• (1, 0, 0, 1)
• What is a good neighborhood?

43
Heuristic Search
Goal min f(x)
f(x)
x
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How to Overcome Local Optimal

• What is we get stuck in a bad local optimum?
• We would like to somehow get out of the local
  optimum and move towards a better solution
• Meta-heuristic A master strategy that guides and
  modifies other heuristics to product a solution
  beyond local optimal.
• GRASP
• Simulated Annealing
• Genetic Algorithm
• Tabu Search
• Metaheuristictian People who study
  meta-heuristic

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Theoretic Studies

• Neighborhood search has been used successfully in
  solving real life problems
• Vehicle routing, VLSI circuit design, network
  design, partitioning problems,
• However, there are very few theoretical result
  about neighborhood search (mostly negative)