# Case Study: The American Steel Transshipment Problem

## Problem Description

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 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

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

Table 2 Monthly Demands

 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

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 respectively. 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).

## Computational Model

### Computational Model in AMPL  Computational Model in AMPL

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;
```

### Computational Model in PuLP/Dippy  Computational Model in PuLP/Dippy

We can define the PuLP/Dippy model using functions in `transshipment_funcy.py`

## Results

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.

## Conclusions

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

Table 3 Optimal Shipment Plan

 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

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

What to hand in Hand in your management summary.

Topic revision: r24 - 2014-05-22 - MichaelOSullivan

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