American Steel, an Ohio-based steel manufacturing company, produces steel at its two steel mills located at Youngstown and Pittsburgh. The company distributes finished steel to its retail customers through the distribution network of regional and field warehouses shown below:

The network represents shipment of finished steel from American Steel's two steel mills located at Youngstown (node 1) and Pittsburgh (node 2) to their field warehouses at Albany, Houston, Tempe, and Gary (nodes 6, 7, 8 and 9) through three regional warehouses located at Cincinnati, Kansas City, and Chicago (nodes 3, 4 and 5). Also, some field warehouses can be directly supplied from the steel mills.

American Steel are planning their production levels, inventory levels and shipment plans for the next quarter (3 months). Table 1 presents the minimum and maximum flow amounts of steel that may be shipped between different cities along with the cost per 1000 ton/month of shipping the steel. For example, the shipment from Youngstown to Kansas City is contracted out to a railroad company with a minimal shipping clause of 1000 tons/month. However, the railroad cannot ship more than 5000 tons/month due the shortage of rail cars.

Table 1 Arc Costs and Limits | From node | To node | Cost | Minimum | Maximum | | Youngstown | Albany | 500 | - | 1000 | | Youngstown | Cincinnati | 350 | - | 3000 | | Youngstown | Kansas City | 450 | 1000 | 5000 | | Youngstown | Chicago | 375 | - | 5000 | | Pittsburgh | Cincinnati | 350 | - | 2000 | | Pittsburgh | Kansas City | 450 | 2000 | 3000 | | Pittsburgh | Chicago | 400 | - | 4000 | | Pittsburgh | Gary | 450 | - | 2000 | | Cincinnati | Albany | 350 | 1000 | 5000 | | Cincinnati | Houston | 550 | - | 6000 | | Kansas City | Houston | 375 | - | 4000 | | Kansas City | Tempe | 650 | - | 4000 | | Chicago | Tempe | 600 | - | 2000 | | Chicago | Gary | 120 | - | 4000 |

One added consideration is union action in Kansas City during June that will increase the transportation costs out of Kansas City by 5%.

The monthly demand for the next quarter at American Steel's four field warehouses is shown in Table 2.

Table 2 Monthly Demands

| | Demand | | Field Warehouses | April | May | June | | Albany, N.Y. | 3000 | 2000 | 4000 | | Houston | 7000 | 5000 | 6000 | | Tempe | 4000 | 1000 | 3000 | | Gary | 5000 | 6000 | 7000 |

The Youngstown and Pittsburgh mills can produce up to 10,000 tons and 15,000 tons of steel per month, respectively. However, the Youngstown mill is undergoing maintenance in May and can only produce 5000 ton of steel.

Finally, there is a holding cost of $25/1000 ton at the mills, $10/1000 ton at the regional warehouses and $25/1000 ton at the field warehouses.

The management want to know the best production levels, inventory levels and shipments to minimise their operating cost during the quarter.

1. Solve the American Steel Planning Problem. Write a management summary of your solution.

   Hint You can use transshipment.mod but with 2-dimensional nodes incorporating location and time, e.g., (Youngstown, April).

   What to hand in Your model, data and script files. Your management summary.