# The m-traveling-salesman problem is a variant of the traveling-salesman problem in which m…

The m-traveling-salesman problem is a variant of the traveling-salesman problem in which m salesmen stationed at a home base, city 1, are to make tours to visit other cities, 2, 3, . . . , n. Each salesman must be routed through some, but not all, of the cities and return to his (or her) home base; each city must be visited by one salesman. To avoid redundancy, the sales coordinator requires that no city (other than the home base) be visited by more than one salesman or that any salesman visit any city more than once.

Suppose that it cost cij dollars for any salesman to travel from city i to city j.

a) Formulate an integer program to determine the minimum-cost routing plan. Be sure that your formulation does not permit subtour solutions for any salesman.

b) The m-traveling-salesman problem can be reformulated as a single-salesman problem as follows: We replace the home base (city 1) by m fictitious cities denoted 1’ , 2’ , . . . , m’. We link each of these fictitious cities to each other with an arc with high cost  and we connect each fictitious city i’ to every city j at cost ci’j = cij . Figure illustrates this construction for m = 3.

Suppose that we solve this reformulation as a single-traveling-salesman problem, but identify each portion of the tour that travels from one of the fictitious cities i’ through the original network and back to another fictitious city j’ as a tour for one of the m salesmen. Show that these tours solve the m-traveling-salesman problem. [Hint. Can the solution to the single-salesman problem ever use any of the arcs with a high-cost M? How many times must the single salesman leave from and return to one of the fictitious cities?]

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