Modified concentrator location problem. Show how to formulate each of the following variants of the concentrator location problem that we consider in Exercise 4.7 as a shortest path problem. Assume in each case that all the customer lie on a path.
(a) The cost of connecting each customer to a concentrator is negligible, but each concentrator can handle at most five customers.
(b) Several types of concentrators are available at each node; each type of concentrator has its own cost and its own capacity (which is the maximum number of customers it can accommodate).
(c) In the situations considered in Exercise 4.7 and in parts (a) and (b) of this exercise, no customer can be assigned to a concentrator more that 1200 meters from the concentrator (because of line degradation of transmitted signals).
Exercise 4.7
Concentrator location on a line (Balakrishnan, Magnanti, and Wong [1989]). In the telecommunication industry, telephone companies typically connect each customer directly to a switching center, which is a device that routes calls between the users in the system. Alternatively, to use fewer cables for routing the telephone calls, a company can combine the calls of several customers in a message compression device known as a concentrator and then use a single cable to route all of the calls transmitted by those users to the switching center. Constructing a concentrator at any node in the telephone network incurs a node-specific cost and assigning each customer to any concentrator incurs a "homing cost" that depends on the customer and the concentrator location. Suppose that all of the customers lie on a path and that we wish to identify the optimal location of concentrators to service these customers (assume that we must assign each customer to one of the concentrators). Suppose further that the set of customers allocated to any concentrator must be contiguous on the path (many telephone companies use this customer grouping policy). How would you find the optimal location of a single concentrator that serves any contiguous set of customers? Show how to use the solution of these single-location subproblems (one for each interval of customers) to solve the concentrator location problem on the path as a shortest path problem.