1. Traveling Salesman Problem
and the
Open Traveling Salesman Problem

2. Traveling Salesman Problem
Objective: Given a list of cities and the distances between each pair of cities, find the shortest possible route that visits each city exactly once and returns to the origin city.
Symmetric TSP with 4 cities

3. Traveling Salesman Problem
Tucker’s Traveling Salesman Problem IP Formulation
TSP can be formulated as an integer linear program.
Label the cities with the numbers 0, ..., n and define:
For i = 0, ..., n, let
be an artificial variable, and finally take
to be the distance from city i to city j. Then TSP can be written as the following integer linear programming problem:

4. Open Traveling Salesman Problem
Objective: Same as traveling salesman problem but do not need to return to origin city .

5. Open TSP Application Valley Industrial Products: Fort Valley
Continuous plastic extrusion system for making 8’x11’ plastic sheets.
Produce in batches by color. Can produce over 80 colors.
Time/cost to change from one color to another. May make 6-8 color changes in a single day.
Produce 5 days a week, 24 hours per day. Always end in a “Clean” state (clear plastic) at the end of the week.
Objective: Minimize the cost/time of changeovers.

6. Example cost of change-overs
Open TSP Application
Valley Industrial Products: Fort Valley
 From/to c1 c2 c3 c4 c5 c6 c7 c8
CLEAN 1 2
1.5
2.1
0.9
7.5 3
1.2
1.6
3.2
1.4
1.7
1.8
3.1 4
1.21
1.3
2.5
2.9
1.34
2.6
1.9
2.2
2.3
5.1
3.5
1.1
2.4
1.91
0.8
0.95
2.76
Example cost of change-overs

7. Open Traveling Salesman Problem
Objective function
Min 𝑖=1 𝑛 𝑗=1 𝑛 𝐶𝑖𝑗∗𝑋𝑖𝑗 (1)
Subject to
   
𝑖=1 𝑛−1 𝑋𝑖𝑗≤ j=1,2,3,…,n-1 i≠j (2)
𝑖=1 𝑛−1 𝑋𝑖𝑛= (3)
𝑗=1 𝑛 𝑋𝑖𝑗= i=1,2,3,…, n-1 i≠j (4)
. 𝑢𝑖−𝑢𝑗+1≤(𝑛−1)(1−𝑋𝑖𝑗) i=1,2,3…,n-1 j=1,2,3,...n i≠j (5)
Xij E { 0,1} i,j= 1,2,3....n i≠j (6)
The objective function (1) models the total cost of change the colors that we need to produce in determine period of time. Constrains (2) ensure that at least one node will not have an entrance (3) ensure that the sequence finishes its route exactly once in the stage clean (node n). (4) ensure that from all nodes (except n) have one exit (5) avoid sub tours.

8. Open Traveling Salesman Problem
CONSTRAINTS
<= 1
NODE 1 ENTRÉE
NODE 3 ENTRÉE
NODE 5 ENTRÉE
NODE 7 ENTRÉE =
NODE CLEAN ENTRÉE
NODE 1 EXIT
NODE 3 EXIT
NODE 5 EXIT
NODE 7 EXIT
subtours -1 4
from 1 to 3
from 1 to5 -2
from 1 to 7 -3
from1 to c 3
from 3 to 1 2
from 3 to 5
from 3 to7
from 3 to c
from 5 to 1
from 5 to3
from 5 to7
from 5 to C
from 7 to 1
from 7 to 3
from 7 to 5
from 7 to c
inter c1 c2 c3 c4 c5 c6 c7 c8
CLEAN 1 2
1.5
2.1
0.9
7.5 3
1.2
1.6
3.2
1.4
1.7
1.8
3.1 4
1.21
1.3
2.5
2.9
1.34
2.6
1.9
2.2
2.3
5.1
3.5
1.1
2.4
1.91
0.8
0.95
2.76
VARIABLES
X1C
X3C
X5C
X7C u1
X17
X31
X51
X71 u3
X15
X35
X53
X73 u5
X 13
X 37
X57
X75 u7 uc
OBJECTIVE
6.45