HHMM Inc., a clothing retail firm that operates in the USA is planning to expand its operations in Europe by opening warehouses. The firm has identified m cities across Europe where its major clients are, and it has defined n potential locations where to locate the new warehouses. Each city has a demand ai to be satisfied. The department of logistics has defined a transportation cost matrix between each city and each potential location, where tij is the unit transport cost between city i and potential location j.

The objective of the firm is to find the minimum number of warehouses needed and where to locate them, in order to obtain total coverage (delivery of the product) of its clients within T hours. Define parameters, variables, constraints and objective, and formulate the problem.

After solving the model, the board of directors is not satisfied with the results obtained, since the cost of opening new warehouses in Europe may vary across countries and cities. Therefore, it is decided to modify the objective used. The new objective is to minimize opening costs fj in order to cover all the clients within T hours. Formulate the new objective.

Still not satisfied with the results obtained, the board decides to abandon the objective of total covering within T, and it is decided to obtain the number and location of the new warehouses by minimizing total costs (fixed costs + transportation costs). Redefine variables, constraints and objective to formulate the problem.