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.

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 then 5000 tons/month due the shortage of rail cars.

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 |

The current monthly demand at American Steel's four field warehouses is shown in Table 2.

Field Warehouses |
Monthly Demand |

Albany, N.Y. | 3000 |

Houston | 7000 |

Tempe | 4000 |

Gary | 6000 |

The Youngstown and Pittsburgh mills can produce up to 10,000 tons and 15,000 tons of steel per month, respectively. The management wants to know the least cost monthly shipment plan.

The American Steel Problem can be solved as a transshipment problem. The supply at the supply nodes is the maximum production at the steel mills, i.e., 10,000 and 15,000 for Youngstown and Pittsburgh repsectively. The demand at demand nodes in given by the demand at the field warehouses and the other nodes are transshipment nodes. The costs and bounds on flow through the network are also given. The most compact formulation for this problem is a network formulation (see The Transshipment Problem for details).

We can use the AMPL model file `transshipment.mod`

(see The Transshipment Problem in AMPL for details) to solve the American Steel Transshipment Problem. We need a data file to describe the nodes, arcs, costs and bounds. The node set is simply a list of the node names:

set NODES := Youngstown Pittsburgh Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary ;

Note that `Kansas City`

must be enclosed in `'`

and `'`

because of the whitespace in the name.

The arc set is 2-dimensional and can be defined in a number of different ways:

# List of arcs set ARCS := (Youngstown, Albany), (Youngstown, Cincinnati), ... , (Chicago, Gary) ;

# Table of arcs set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary := Youngstown + + + + - - - Pittsburgh + + + - - - + . . .

# Array of arcs set ARCS := (Youngstown, *) Cincinnati 'Kansas City' Chicago Albany (Pittsburgh, *) Cincinnati 'Kansas City' Chicago Gary . . . (Chicago, *) Tempe Gary ;

Since the node set is small and the arc set is "dense", i.e., many of the node pairs are represented in the arc set, we choose a table to define `ARCS`:

set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary := Youngstown + + + + - - - Pittsburgh + + + - - - + Cincinnati - - - + + - - 'Kansas City' - - - - + + - Chicago - - - - - + + ;

The `NetDemand`

can be defined easily from the supply and demand. Since the default is 0 we don't need to define `NetDemand`

for the transshipment nodes:

param NetDemand := Youngstown -10000 Pittsburgh -15000 Albany 3000 Houston 7000 Tempe 4000 Gary 6000 ;

We can use lists, tables or arrays to define the parameters for the American Steel Transhippment Problem,
but in this case we will use a list and define multiple parameters at once. This allows us to "cut-and-paste" the parameters from the problem description. Note the use of
`.`

for parameters where the defaults will suffice (e.g., `Lower`

for `Youngstown`

and `Albany`

):

param: Cost Lower Upper:= Youngstown Albany 500 . 1000 Youngstown Cincinnati 350 . 3000 Youngstown 'Kansas City' 450 1000 5000 Youngstown Chicago 375 . 5000 . . . Chicago Gary 120 . 4000 ;

Note that the cost is in $/1000 ton, so we must divide the cost by 1000 in our script file before solving:

reset; model transshipment.mod; data steel.dat; let {(m, n) in ARCS} Cost[m, n] := Cost[m, n] / 1000; option solver cplex; solve; display Flow;

Using `transshipment.mod`

, and the data and script files defined in Computational Model we can solve the American Steel Transshipment Problem to get the optimal `Flow`

variables:

If the total supply is greater than the total demand, the transshipment problem will solve, but flow may be left in the network (in this case at the Pittsburgh node). In `transshipment.mod`

we `check`

that `sum {n in NODES} NetDemand[n] <= 0` to ensure a problem is feasible before solving.

If total supply is less than demand (hence the problem is infeasible) we can add a *dummy* supply node (see with arcs to all the demand nodes. The optimal solution will show the "best" nodes to send the extra supply to.

In order to minimise the monthly shipment costs, American Steel should follow the shipment plan shown in Table 3.

From/To | Cincinnati |
Kansas City |
Chicago |
Albany |
Houston |
Tempe |
Gary |
---|---|---|---|---|---|---|---|

Youngstown |
3000 | 3000 | 3000 | 1000 | |||

Pittsburgh |
2000 | 3000 | 3000 | 2000 | |||

Cincinnati |
2000 | 3000 | |||||

Kansas City |
4000 | 2000 | |||||

Chicago |
2000 | 4000 |

As with many network problems, it can be illuminating to display the solution graphically as shown in Figure 1.

**Figure 1** Optimal Shipment Plan

- Solve the American Steel Transshipment Problem. Write a management summary of your solution.
**What to hand in**Hand in your management summary.

Topic revision: r21 - 2008-04-29 - MichaelOSullivan

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