1)Formulate the problem mathematically. Be careful to dene your data, variables, constraints, and objective function.

2) Single sourcing happens when a customer only wants their demand to come from a single warehouse (point of origin). Reformulate the problem to accommodate for single sourcing.

3) Due to internal regulation, no warehouse is allowed to work with more than 3 customers. Is the solution to part (2) still optimal? If not, resolve the problem adding the new constraint, and obtain the new optimal solution.

In class, we discussed about the facility location problem and provided one of its formulations. Opening a facility is associated with a fixed cost that needs to be paid before the facility can be used. On top of that, different facilities have different capacities; that is, they can only provide so many items to the customers. Assume that a company is deciding where to open up their new warehouses from a set of 6 potential sites. Their goal is to satisfy all 10 of their customers in this location. The cost of shipping one item from a warehouse to a customer is shown in Table 1. Each facility in order to be "opened" comes with a fixed cost (the cost to open the facility) and a capacity (the amount of products that can be shipped from that facility): these are shown in Table 2. Last, the demands of the customers are known to be 25, 15, 20, 10, 15, 15, 25, 20, 30, and 20, respectively. Customer 1 3 6 5 7 10 12 7 5 8 10 co con Warehouses 2 3 4 5 5 8 12 10 7 10 12 8 3 8 10 12 6 7 10 9 8 5 8 6 6 4 7 5 10 6 6 8 6 9 4 4 10 7 5 4 12 8 3 4 07 vo AOOOOO 6 11 10 10 5 5 4 3 7 6 6 8 oct n E Table 1: The shipping costs from each facility (warehouses) to each customer. 50 CON Warehouse Fixed Cost Capacity 1750000 1800000 3000000 120 2250000 100 1500000 60 1000000 50 . Table 2: The fixed cost and capacity of each potential site. In class, we discussed about the facility location problem and provided one of its formulations. Opening a facility is associated with a fixed cost that needs to be paid before the facility can be used. On top of that, different facilities have different capacities; that is, they can only provide so many items to the customers. Assume that a company is deciding where to open up their new warehouses from a set of 6 potential sites. Their goal is to satisfy all 10 of their customers in this location. The cost of shipping one item from a warehouse to a customer is shown in Table 1. Each facility in order to be "opened" comes with a fixed cost (the cost to open the facility) and a capacity (the amount of products that can be shipped from that facility): these are shown in Table 2. Last, the demands of the customers are known to be 25, 15, 20, 10, 15, 15, 25, 20, 30, and 20, respectively. Customer 1 3 6 5 7 10 12 7 5 8 10 co con Warehouses 2 3 4 5 5 8 12 10 7 10 12 8 3 8 10 12 6 7 10 9 8 5 8 6 6 4 7 5 10 6 6 8 6 9 4 4 10 7 5 4 12 8 3 4 07 vo AOOOOO 6 11 10 10 5 5 4 3 7 6 6 8 oct n E Table 1: The shipping costs from each facility (warehouses) to each customer. 50 CON Warehouse Fixed Cost Capacity 1750000 1800000 3000000 120 2250000 100 1500000 60 1000000 50 . Table 2: The fixed cost and capacity of each potential site