math models of or: traveling salesman problemeaton.math.rpi.edu/faculty/mitchell/courses/matp... ·...

Math Models of OR:Traveling Salesman Problem

John E. Mitchell

Department of Mathematical SciencesRPI, Troy, NY 12180 USA

November 2018

Mitchell Traveling Salesman Problem 1 / 19

Outline

1 Examples

2 Proving Optimality

3 Subtours

4 Reoptimizing after adding subtour elimination constraints


Examples

Outline

1 Examples


3 Subtours



Examples

The Traveling Salesman Problem

TSP WebpageBill Cook, Waterloo

Robert BoschOberlin College

Find shortest route that visits each city exactly once.

Largest problem solved to date has more than 85,000 cities.

Applications: VLSI, vehicle routing (UPS, school buses,...)Mitchell Traveling Salesman Problem 4 / 19

Proving Optimality

Outline

1 Examples


3 Subtours



Proving Optimality

How do we know we are optimal?

We’ve found a good tour somehow.How do we know that we can’t do better?Look at relaxations.Venn diagram:

set of tours

relaxationof set oftours


Proving Optimality

How do we know we are optimal?We’ve found a good tour somehow.How do we know that we can’t do better?Look at relaxations.Venn diagram:

optimaltour

optimalrelaxation

Optimal relaxation gives lower bound on length of optimal tour


Proving Optimality

Constructing a Relaxation


Proving Optimality

Constructing a Relaxation

Use exactly two edges incident to each vertex.


Proving Optimality

Optimization Formulation

Use binary variables:

xe =

⇢1 if edge e is in tour0 otherwise

Let E denote the set of edges and V the set of vertices.Let ce be the cost of edge e.

Let �(v) denote the edges incident to vertex v .

minxP

e2E cexe

subject toP

e2�(v) xe = 2 for all v 2 V

and xe = 0 or 1 for all e 2 E .


Subtours

Outline

1 Examples


3 Subtours



Subtours

May have subtours

r

s

t

u

v

w


Ledge, foreach vertex

0 A

Subtours

May have subtours

r

s

t

u

v

w

Optimal tour


Subtours

May have subtours

r

s

t

u

v

wSolution to relaxation:use 2 edges at each vertex


§""""""

Subtours

May have subtours

r

s

t

u

v

wUse subtour elimination constraintto tighten relaxation:

xrs + xst + xrt 2


{ r, t ,u} :Xr t t x ru t xc .us 2 .

a

Subtours

Refinements

Can add subtour elimination constraints for any subset U ✓ V ,with 3 |U| |V |/2.Too many subtour elimination constraints to use them all, so addthem selectively as needed.(How many possible constraints do we have?)Can also relax the restriction that x be binary: use 0 xe 1.Then have linear program, which can be solved efficiently.Extensions of this approach can solve instances of the TSP withtens of thousands of cities (and hundreds of millions of edges).


Subtours

Refinements



Seca, K s l U l - l .

Subtours

Refinements



Reoptimizing after adding subtour elimination constraints

Outline

1 Examples


3 Subtours




Using dual simplex to reoptimize

r

s

t

u

v

w1

1

1

1

1

1

5

5

5

edge lengths ce shown

Use two edges incident to each vertex. Eg: xrs + xrt + xru = 2.

Have 0 xij 1 for each edge (i , j). Only impose upper boundexplicitly in simplex tableau for the long edges.



Initial tableau

xrs xrt xst xuv xuw xvw xru xsv xtw sru ssv stw0 1 1 1 1 1 1 5 5 5 0 0 02 1 1 0 0 0 0 1 0 0 0 0 02 1 0 1 0 0 0 0 1 0 0 0 02 0 1 1 0 0 0 0 0 1 0 0 02 0 0 0 1 1 0 1 0 0 0 0 02 0 0 0 1 0 1 0 1 0 0 0 02 0 0 0 0 1 1 0 0 1 0 0 01 0 0 0 0 0 0 1 0 0 1 0 01 0 0 0 0 0 0 0 1 0 0 1 01 0 0 0 0 0 0 0 0 1 0 0 1


L - x r s t x r e +quos t e a l

-

C m(= Xew + Sew


Optimal tableau for relaxation

xrs xrt xst xuv xuw xvw xru xsv xtw sru ssv stw�6 0 0 0 0 0 0 4 4 4 0 0 0

1 1 0 0 0 0 0 12

12 �1

2 0 0 01 0 1 0 0 0 0 1

2 �12

12 0 0 0

1 0 0 1 0 0 0 �12

12

12 0 0 0

1 0 0 0 1 0 0 12

12 �1

2 0 0 01 0 0 0 0 1 0 1

2 �12

12 0 0 0

1 0 0 0 0 0 1 �12

12

12 0 0 0

1 0 0 0 0 0 0 1 0 0 1 0 01 0 0 0 0 0 0 0 1 0 0 1 01 0 0 0 0 0 0 0 0 1 0 0 1

r

s

t

u

v

w


9 basic variables:column of identitymatr ix

i n\ See x r u = x s v

e x e - =O


Add subtour elimination contraint

xrs xrt xst xuv xuw xvw xru xsv xtw sru ssv stw xrst�6 0 0 0 0 0 0 4 4 4 0 0 0 0

1 1 0 0 0 0 0 12

12 �1

2 0 0 0 01 0 1 0 0 0 0 1

2 �12

12 0 0 0 0

1 0 0 1 0 0 0 �12

12

12 0 0 0 0

1 0 0 0 1 0 0 12

12 �1

2 0 0 0 01 0 0 0 0 1 0 1

2 �12

12 0 0 0 0

1 0 0 0 0 0 1 �12

12

12 0 0 0 0

1 0 0 0 0 0 0 1 0 0 1 0 0 01 0 0 0 0 0 0 0 1 0 0 1 0 01 0 0 0 0 0 0 0 0 1 0 0 1 02 1 1 1 0 0 0 0 0 0 0 0 0 1

xrs + xrt + xst 2, slack variable xrst



Pivot to dual canonical form


1 1 0 0 0 0 0 12

12 �1

2 0 0 0 01 0 1 0 0 0 0 1

2 �12

12 0 0 0 0

1 0 0 1 0 0 0 �12

12

12 0 0 0 0

1 0 0 0 1 0 0 12

12 �1

2 0 0 0 01 0 0 0 0 1 0 1

2 �12

12 0 0 0 0

1 0 0 0 0 0 1 �12

12

12 0 0 0 0

1 0 0 0 0 0 0 1 0 0 1 0 0 01 0 0 0 0 0 0 0 1 0 0 1 0 01 0 0 0 0 0 0 0 0 1 0 0 1 0

�1 0 0 0 0 0 0 � 12� �1

2 �12 0 0 0 1

� � � � � � 8 8 8 � � � �

Optimize using dual simplex.Mitchell Traveling Salesman Problem 17 / 19


Result of first dual simplex pivot


0 1 0 0 0 0 0 0 0 �1 0 0 0 10 0 1 0 0 0 0 0 �1 0 0 0 0 12 0 0 1 0 0 0 0 1 1 0 0 0 �10 0 0 0 1 0 0 0 0 �1 0 0 0 10 0 0 0 0 1 0 0 �1 0 0 0 0 12 0 0 0 0 0 1 0 1 1 0 0 0 �1

�1 0 0 0 0 0 0 0 � 1� �1 1 0 0 21 0 0 0 0 0 0 0 1 0 0 1 0 01 0 0 0 0 0 0 0 0 1 0 0 1 02 0 0 0 0 0 0 1 1 1 0 0 0 �2

� � � � � � � 0 0 � � � �

Optimize using dual simplex.Mitchell Traveling Salesman Problem 18 / 19


Result of second dual simplex pivot


0 1 0 0 0 0 0 0 0 �1 0 0 0 11 0 1 0 0 0 0 0 0 1 �1 0 0 �11 0 0 1 0 0 0 0 0 0 1 0 0 10 0 0 0 1 0 0 0 0 �1 0 0 0 11 0 0 0 0 1 0 0 0 1 �1 0 0 �11 0 0 0 0 0 1 0 0 0 1 0 0 11 0 0 0 0 0 0 0 1 1 �1 0 0 �20 0 0 0 0 0 0 0 0 �1 0 1 0 21 0 0 0 0 0 0 0 0 1 0 0 1 01 0 0 0 0 0 0 1 0 0 1 0 0 0

r

s

t

u

v

wOptimal in relaxation

Feasible in TSPOptimal in TSP


math models of or: traveling salesman problemeaton.math.rpi.edu/faculty/mitchell/courses/matp... ·...

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