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. | ||||||||