propose in teams a mathematical programming model to optimize it. Clearly state the sets, parameters, decision variables, objective function and constraints.

Lets assume a major grocery firm desires to re-design its distribution system bringing various food products to its convenience stores in the city, according to City Logistics principles. The firm distributes 10 products. Products are identified by the letter p, with p = 1, 2, , 10.

For this strategic study, the firm has divided the city into 200 zones, identified by the letter z = 1, 2, , 200. Consider that there is a daily demand for each zone z, of standard boxes of product p to be delivered at the beginning of each day. To satisfy this demand, the firm makes use of 3 large warehouses, located in the vicinity but not close to the city, where the products acquired around the country and imported from abroad are collected, inspected, sorted and packaged prior to expedition to the Distribution Centres located within the city. Each warehouse, w = 1, 2, 3, has a daily storage capacity of total standard boxes and also a daily storage capacity per product p. In addition, consider that there are associated storage costs (per standard box) of product p at every warehouse w.

The firm owns and use 9 distribution centers (DC), identified by the letter d = 1, 2, , 9. The current study aims to identify the distribution centers among these 9 that will be opened. For each distribution centre d, the firm estimates that it will have to pay a cost to open it. Once opened, each distribution centre d could handle (reception, storage, delivery preparation, etc.) a particular total number of standard boxes and also a specific number of standard boxes per product p. Additionally, consider that there is a unit handling cost at every distribution centre d, irrespective of the type of product p.

Consider that there are unit transportation costs in the system (per standard box) from warehouse w to DC d and from DC d to city zone z, all of these costs are also irrespective of the type of product p.

It has been decided to build an optimization model to help determine the new design of the distribution system. In particular, the model must help decide

-Which distributions centers d to open;

-The quantities of each product p to store in each warehouse w;

-The flows from warehouses w to DCs d and to customer zones z from each product p;

The model must aim to minimize the total cost of the system.