# Question

The Audiofile Company produces boomboxes. However, management has decided to subcontract out the production of the speakers needed for the boomboxes. Three vendors are available to supply the speakers. Their price for each shipment of 1,000 speakers is shown below.

In addition, each vendor would charge a shipping cost. Each shipment would go to one of the company’s two warehouses. Each vendor has its own formula for calculating this shipping cost based on the mileage to the warehouse. These formulas and the mileage data are shown below.

Whenever one of the company’s two factories needs a shipment of speakers to assemble into the boomboxes, the company hires a trucker to bring the shipment in from one of the warehouses. The cost per shipment is given next, along with the number of shipments needed per month at each factory.

Each vendor is able to supply as many as 10 shipments per month. However, because of shipping limitations, each vendor is able to send a maximum of only 6 shipments per month to each warehouse. Similarly, each warehouse is able to send a maximum of only 6 shipments per month to each factory.

Management now wants to develop a plan for each month regarding how many shipments (if any) to order from each vendor, how many of those shipments should go to each warehouse, and then how many shipments each warehouse should send to each factory. The objective is to minimize the sum of the purchase costs (including the shipping charge) and the shipping costs from the warehouses to the factories.

(a) Draw a network that depicts the company’s supply network. Identify the supply nodes, transshipment nodes, and demand nodes in this network.

(b) Formulate this problem as a minimum cost flow problem by inserting all the necessary data into this network. Also include a dummy demand node that receives (at zero cost) all the unused supply capacity at the vendors.

(c) Formulate and solve a spreadsheet model for this problem.

In addition, each vendor would charge a shipping cost. Each shipment would go to one of the company’s two warehouses. Each vendor has its own formula for calculating this shipping cost based on the mileage to the warehouse. These formulas and the mileage data are shown below.

Whenever one of the company’s two factories needs a shipment of speakers to assemble into the boomboxes, the company hires a trucker to bring the shipment in from one of the warehouses. The cost per shipment is given next, along with the number of shipments needed per month at each factory.

Each vendor is able to supply as many as 10 shipments per month. However, because of shipping limitations, each vendor is able to send a maximum of only 6 shipments per month to each warehouse. Similarly, each warehouse is able to send a maximum of only 6 shipments per month to each factory.

Management now wants to develop a plan for each month regarding how many shipments (if any) to order from each vendor, how many of those shipments should go to each warehouse, and then how many shipments each warehouse should send to each factory. The objective is to minimize the sum of the purchase costs (including the shipping charge) and the shipping costs from the warehouses to the factories.

(a) Draw a network that depicts the company’s supply network. Identify the supply nodes, transshipment nodes, and demand nodes in this network.

(b) Formulate this problem as a minimum cost flow problem by inserting all the necessary data into this network. Also include a dummy demand node that receives (at zero cost) all the unused supply capacity at the vendors.

(c) Formulate and solve a spreadsheet model for this problem.

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