Minimize the costs of shipping goods from factories to warehouses and customers, and warehouses to customers, while not exceeding the supply available from each factory or the capacity of each warehouse, and meeting the demand from each customer. Cost of shipping ($ per product)         Destinations     Warehouse 1 Warehouse 2 Warehouse 3 Warehouse 4   Factory 1 $0.50 $0.50 $1.00 $0.20   Factory 2 $1.50 $0.30 $0.50 $0.20         Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Factory 1 $1.75 $2.50 $1.50 $2.00 $1.50 Factory 2 $2.00 $2.50 $2.50 $1.50 $1.00       Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Warehouse 1 $1.50 $1.50 $0.50 $1.50 $3.00 Warehouse 2 $1.00 $0.50 $0.50 $1.00 $0.50 Warehouse 3 $1.00 $1.50 $2.00 $2.00 $0.50 Warehouse 4 $2.50 $1.50 $0.20 $1.50 $0.50 Number of products shipped             Warehouse 1 Warehouse 2 Warehouse 3 Warehouse 4 Total   Factory 1 0 20,000 0 15,000 35,000   Factory 2 45,000 0 11,000 0 56,000   Total 45,000 20,000 11,000 15,000   Capacity 45,000 20,000 30,000 15,000         Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Total   Factory 1 10,000 0 0 15,000 0 25,000 Factory Factory 2 0 0 0 0 0 0 Capacity   Total products shipped out of factory 1 60,000 60,000   Total products shipped out of factory 2 56,000 60,000       Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Total   Warehouse 1 0 23,000 0 17,000 5,000 45,000   Warehouse 2 20,000 0 0 0 0 20,000   Warehouse 3 0 0 0 0 11,000 11,000   Warehouse 4 0 0 15,000 0 0 15,000   Total 30,000 23,000 15,000 32,000 16,000   Demands 30,000 23,000 15,000 32,000 16,000                     Total cost of shipping $237,000           Problem               A company has 2 factories, 4 warehouses and 5 customers. It wants to minimize the cost of shipping its product from the factories to the warehouses, the factories to the customers, and the warehouses to the customers. The number of products received by a warehouse from the factory should be the same as the number of products leaving the warehouse to the customers. How should the company distribute the products?                 Solution               1) The variables are the number of products to ship from the factories to the warehouses, the factories to the customers, and the warehouses to the customers. These are defined in worksheet Transport2 as   Factory_to_warehouse, Factory_to_customer, Warehouse_customer.       2) The logical constraints are all defined via the Assume Non-Negative option:       Factory_to_warehouse >= 0             Factory_to_customer >= 0             Warehouse_customer >= 0           The other constraints are               Total_from_factory <= Factory_capacity           Total_to_customer >= Demand           Total_to_warehouse <= Warehouse_capacity         Total_to_warehouse = Total_from_warehouse       3) The objective is to minimize cost, given by Total_cost.                         Remarks               Please note that the last constraint must be an '=' , because otherwise products would start piling up at the warehouse. It would be possible to make this a multi-period model where storage at the warehouses would be possible and even desired, if transportation prices would fluctuate during the different time periods. In   worksheet Transport3 we will look at a multi-product situation.