Traveling Salesman Problem and the Open Traveling Salesman

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’x 11’ 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.

Open TSP Application Valley Industrial Products: Fort Valley From/t o c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 CLEAN c 1 0 1 2 1. 5 2. 1 0. 9 7. 5 3 1. 2 c 2 1. 5 0 2 1. 6 3. 2 1. 4 3 1. 7 2. 1 c 3 2 1. 4 0 1. 8 2 3. 1 2 4 1. 21 c 4 1. 2 3 2 0 1. 3 2. 5 2. 9 3 1. 34 c 5 2. 1 1. 4 2. 6 0 1. 9 2. 2 2. 3 5. 1 c 6 1. 5 1. 7 1. 9 2 3. 1 0 3. 5 4 1. 1 c 7 2. 4 2. 1 1. 91 1. 7 3. 2 2. 1 0 0. 8 0. 95 c 8 3. 2 3. 1 1. 5 1. 9 2. 4 2. 1 3. 5 0 2. 76 CLEAN 0 0 0 0 0 Example cost of change-overs

Open Traveling Salesman Problem

Open Traveling Salesman Problem CONSTRAI NTS c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 CLEAN VARIABLE S OBJECTIVE c 2 c 3 0 1. 5 2 1. 2 2. 5 1. 5 2. 4 3. 2 0 X 1 C X 17 X 15 X 13 1 0 1. 4 3 2. 1 1. 7 2. 1 3. 1 0 0 0 1 0 6. 45 c 4 c 5 2 2 0 2 1. 4 1. 91 1. 5 0 X 3 C X 31 X 35 X 37 1. 5 1. 6 1. 8 0 2. 6 2 1. 7 1. 9 0 0 1 c 6 c 7 2. 1 3. 2 2 1. 3 0 3. 1 3. 2 2. 4 0 X 5 C X 51 X 53 X 57 0. 9 1. 4 3. 1 2. 5 1. 9 0 2. 1 0 0 0 1 0 c 8 CLEAN 3 1. 2 1. 7 2. 1 4 1. 21 3 1. 34 2. 3 5. 1 4 1. 1 0. 8 0. 95 0 2. 76 0 0 7. 5 3 2 2. 9 2. 2 3. 5 0 X 7 C X 71 X 73 X 75 0<= 1<= 1 0 0 0 u 1 u 3 u 5 u 7 uc 1= 1= 1= subtours 0 2 1 3 4 -1<= 0<= -2<= -3<= 2<= 0<= -1<= -2<= 4 2 3 0 1 1 NODE 1 ENTRÉE NODE 3 ENTRÉE NODE 5 ENTRÉE NODE 7 ENTRÉE NODE CLEAN 1 ENTRÉE 1 NODE 1 EXIT 1 NODE 3 EXIT 1 NODE 5 EXIT 1 NODE 7 EXIT from 1 to 43 0 from 1 to 5 4 from 1 to 7 4 from 1 to c 4 from 3 to 1 4 from 3 to 5 0 from 3 to 7 4 from 3 to c 4 from 5 to 1 0 from 5 to 3 4 from 5 to 7 4 from 5 to C 4 from 7 to 1 4 from 7 to 3 4 from 7 to 5 0 from 7 to c inter