Minimize the costs of producing 3 different goods, and shipping them 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 to make products Product 1 Product 2 Product 3 Factory 1 \$4 \$5 \$3 Factory 2 \$2 \$8 \$6 Product 1 Product 2 Product 3 Cost Factory 1 0 0 0 \$0 Factory 2 0 0 0 \$0 Total Cost \$0 Cost of shipping (\$ per product) Destinations Warehouse 1 Warehouse 2 Warehouse 3 Warehouse 4 Factory 1 Product 1 \$0.50 \$0.50 \$1.00 \$0.20 Product 2 \$1.00 \$0.75 \$1.25 \$1.25 Product 3 \$0.75 \$1.25 \$1.00 \$0.80 Factory 2 Product 1 \$1.50 \$0.30 \$0.50 \$0.20 Product 2 \$1.25 \$0.80 \$1.00 \$0.75 Product 3 \$1.40 \$0.90 \$0.95 \$1.10 Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Factory 1 Product 1 \$2.75 \$3.50 \$2.50 \$3.00 \$2.50 Product 2 \$2.50 \$3.00 \$2.00 \$2.75 \$2.60 Product 3 \$2.90 \$3.00 \$2.25 \$2.80 \$2.35 Factory 2 Product 1 \$3.00 \$3.50 \$3.50 \$2.50 \$2.00 Product 2 \$2.25 \$2.95 \$2.20 \$2.50 \$2.10 Product 3 \$2.45 \$2.75 \$2.35 \$2.85 \$2.45 Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Warehouse 1 Product 1 \$1.50 \$0.80 \$0.50 \$1.50 \$3.00 Product 2 \$1.00 \$0.90 \$1.20 \$1.30 \$2.10 Product 3 \$1.25 \$0.70 \$1.10 \$0.80 \$1.60 Warehouse 2 Product 1 \$1.00 \$0.50 \$0.50 \$1.00 \$0.50 Product 2 \$1.25 \$1.00 \$1.00 \$0.90 \$1.50 Product 3 \$1.10 \$1.10 \$0.90 \$1.40 \$1.75 Warehouse 3 Product 1 \$1.00 \$1.50 \$2.00 \$2.00 \$0.50 Product 2 \$0.90 \$1.35 \$1.45 \$1.80 \$1.00 Product 3 \$1.25 \$1.20 \$1.75 \$1.70 \$0.85 Warehouse 4 Product 1 \$2.50 \$1.50 \$0.60 \$1.50 \$0.50 Product 2 \$1.75 \$1.30 \$0.70 \$1.25 \$1.10 Product 3 \$1.50 \$1.10 \$1.50 \$1.10 \$0.90 Number of products shipped Warehouse 1 Warehouse 2 Warehouse 3 Warehouse 4 Total Factory 1 Product 1 0 0 0 0 0 Product 2 0 0 0 0 0 Product 3 0 0 0 0 0 Factory 2 Product 1 0 0 0 0 0 Product 2 0 0 0 0 0 Product 3 0 0 0 0 0 Total Product 1 0 0 0 0 Product 2 0 0 0 0 Product 3 0 0 0 0 Capacity Product 1 35,000 20,000 30,000 15,000 Product 2 30,000 25,000 15,000 24,000 Product 3 20,000 20,000 25,000 20,000 Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Total Factory 1 Product 1 0 0 0 0 0 0 Product 2 0 0 0 0 0 0 Product 3 0 0 0 0 0 0 Factory 2 Product 1 0 0 0 0 0 0 Product 2 0 0 0 0 0 0 Product 3 0 0 0 0 0 0 Capacity Total products shipped out of factory 1 Product 1 0 0 Product 2 0 0 Product 3 0 0 Total products shipped out of factory 2 Product 1 0 0 Product 2 0 0 Product 3 0 0 Customer 1 Customer 2 Customer 3 Customer 4 Customer 5 Total Warehouse 1 Product 1 0 0 0 0 0 0 Product 2 0 0 0 0 0 0 Product 3 0 0 0 0 0 0 Warehouse 2 Product 1 0 0 0 0 0 0 Product 2 0 0 0 0 0 0 Product 3 0 0 0 0 0 0 Warehouse 3 Product 1 0 0 0 0 0 0 Product 2 0 0 0 0 0 0 Product 3 0 0 0 0 0 0 Warehouse 4 Product 1 0 0 0 0 0 0 Product 2 0 0 0 0 0 0 Product 3 0 0 0 0 0 0 Total Product 1 0 0 0 0 0 Product 2 0 0 0 0 0 Product 3 0 0 0 0 0 Demands Product 1 30,000 23,000 15,000 32,000 16,000 Product 2 20,000 15,000 22,000 12,000 18,000 Product 3 25,000 22,000 16,000 20,000 25,000 Total cost of shipping \$0 Total cost of production \$0 Total Cost \$0 Problem A company wants to minimize the cost of shipping three different products from factories to warehouses and customers and from warehouses to customers. The production of each product at each plant depends on the distribution. How many products should each factory produce and how should the products be distributed in order to minimize total cost while meeting demand? Solution Notice that this is an extension of the transportation model as seen in the Transport3 worksheet. This time the factories do not produce a fixed amount. The amounts produced are now variables. 1) The variables are the number of products to make in the factories, the number of products to ship from factories to warehouses, factories to customers, and warehouses to customers. In worksheet Prodtran these are given the names Products_made, Factory_to_warehouse, Factory_to_customer, and Warehouse_to_customer. 2) The logical constraints are all defined via the Assume Non-Negative option: Products_made >= 0 Factory_to_warehouse >= 0 Factory_to_customer >= 0 Warehouse_to_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. This is defined in the worksheet as Total_cost. Remarks This is one of the more complex models in this series of examples. If the number of products, factories and warehouses becomes large, the number of variables in a model like this one becomes very large. Also bear in mind the degree of coordination between business units that may be needed in order to implement the optimal solution. For these reasons, some users prefer to split problems like this one into a set of smaller, simpler models.