Transportation Problem 2 (2-stage-transport)

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.

Transportation Problem 2 (2-stage-transport)

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