Traveling Salesman Problem and the Open Traveling Salesman Problem

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

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:

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

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.

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

Open Traveling Salesman Problem Objective function   Min 𝑖=1 𝑛 𝑗=1 𝑛 𝐶𝑖𝑗∗𝑋𝑖𝑗 (1) Subject to     𝑖=1 𝑛−1 𝑋𝑖𝑗≤1 j=1,2,3,…,n-1 i≠j (2) 𝑖=1 𝑛−1 𝑋𝑖𝑛=1 (3) 𝑗=1 𝑛 𝑋𝑖𝑗=1 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.

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