Question 1 You assist your client to select a location x = (x1 , x2) for a service facility that will serve K = 50 customers by providing a single (identical) commodity to each customer. The facility can be located anywhere within the unit square 0 x1 1, 0 x2 1 (which can model e.g., a 10 km by 10 km rectangular region scaled to the unit square). The customers are modeled as points pk = (pk1 , pk2) for k = 1, ... , K located within the unit square. Each customers yearly demand is assumed to be a known value, and we assume that all demands are satisfied. However, customers are assumed to be of different relative weight, proportionate to the size of their yearly demand for the commodity. This aspect is expressed by assigning weights wk for k = 1, ... , K to the customers. To illustrate the problem, please see the figure below that shows the unit square (blue), a possible (but not optimized) location for the facility (black dot), and the locations of the weighted customers (red dots of radius wk for k = 1, ... , K). Assume that the distance between the facility location x and customer pk is expressed by the so-called Manhattan (l1-norm) distance function d(x, pk) = |x1 - pk1| + | x2 - pk2|. Formulate a decision model that optimizes the location of the facility. The quality of a location is expressed by the weighted sum of the Manhattan distances between the facility and the customers

. Question 2 Determine the convexity / nonconvexity properties of your facility location problem. Based on the discussion presented in the lectures, state whether this facility location problem is expected to be easy or hard to solve.

Question 3 Assume now that the regions |x1 x2| > 0.3, x1 + x2 > 1.5, x12 x2 + 0.4 < 0, and x12 + 3 x22 < 0.5 must be excluded from consideration for the possible location of the facility. Compared to the answer to 1.2, state whether this facility location problem is expected to be easier or harder to solve.

Question 4 Suggest an initial solution that is likely to be a good guess of the solution to this facility location problem. Briefly explain your choice.