# Case Study: The American Steel Transshipment Problem

## Problem Description

THE AMERICAN STEEL PROBLEM

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.

The table below 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 as follows:

 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.

## Problem Formulation

1. IDENTIFY THE DECISION VARIABLES

The decision variables for this problem are the same as for the transportation problem, the `Flow` of goods (cases of beer in The Beer Distribution Problem, tons of steels here) through the network. In <span style=

## Computational Model

The computational model...

## Results

Implementing the usual script file with `transshipment.mod` and `steel.dat` (with the `Cost` modification) gives us the optimal `Flow` variables:

<img width=

## Conclusions

There is quite a bit of information to summarise and many ways to present it. Some suggestions include:

1. Summarise the problem as usual and list the shipments that American Steel should make (similar to the transportation problem);
2. Summarise the problem as usual and present a table of shipments that American Steel should make;
3. Draw the network formulation for the problem (being sure to specify what the labels mean). Then draw the actual solution on top of the network formulation. You could colour code flows long arcs to show if they are at their bounds.

Implementation and Ongoing Monitoring

Given that the solution is very sensitive to the supply and demand amounts, careful consideration of the accuracy of these figures is important. Making sure that the bounds on the arcs are reliable is another concern.

Ongoing monitoring of the supply, demand and bounds will help American Steel to keep making good decisions. As shown in the parametric analysis, ongoing negotiation of transportation prices along important routes will help American Steel reduce their expenditure.

## Extra for Experts

Another common network flow problem that uses almost the same model as transshipment is the maximum flow problem. The objective of this problem is to maximise flow through the network. The only changes to `transshipment.mod` are:

1. A new variable `FlowOut` is introduced at each node;
2. `FlowOut` has lower and upper bounds, often the upper bound is set to as the `Demand` at the nodes;
3. `FlowOut` replaces `Demand` in the `ConserveFlow` constraints;
4. The `ConserveFlow` constraint becomes an equality constraint as all flow must be accounted for (no storage at the nodes);
5. `Cost` of transshipment is ignored;
6. The objective is to maximise the total flow out of the network `sum {n in NODES} FlowOut[n]`;
7. The problem no longer needs to be balanced.

Maxflow.mod <span style=

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

What to hand in Hand in your management summary.

1. American Steel are planning a marketing campaign to increase demand in their four markets Albany, Tempe, Houston and Gary. However, they would like to know the maximum demand their distribution network can handle (in each market) before they proceed. Write a script file that uses `maxflow.mod` and `steel.dat` to find the maximum demand they can supply in each market.

What to hand in Hand in your script file and a management summary of your solution.

Topic attachments
I Attachment History Action Size Date Who Comment
jpg conserve_flow.jpg r1 manage 116.4 K 2008-02-15 - 09:05 TWikiAdminUser
jpg steel_cost_parametric.jpg r1 manage 20.7 K 2008-02-15 - 09:06 TWikiAdminUser
jpg steel_formulation.jpg r1 manage 30.3 K 2008-02-15 - 09:06 TWikiAdminUser
jpg steel_maxflow_solution.jpg r1 manage 39.1 K 2008-02-15 - 09:07 TWikiAdminUser
jpg steel_network.jpg r1 manage 25.0 K 2008-02-15 - 09:08 TWikiAdminUser
jpg steel_parametric.jpg r1 manage 68.0 K 2008-02-15 - 09:08 TWikiAdminUser
jpg steel_rhs.jpg r1 manage 91.1 K 2008-02-15 - 09:09 TWikiAdminUser
jpg steel_sensitivity.jpg r1 manage 91.0 K 2008-02-15 - 09:11 TWikiAdminUser
jpg steel_solution.jpg r1 manage 37.6 K 2008-02-15 - 09:12 TWikiAdminUser
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