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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 The arc set is 2dimensional 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 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 "cutandpaste" the parameters from the problem description. Note the use of
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; </twistyPlugin> </twistyPlugin twikiMakeVisibleInline>
We can define the PuLP/Dippy model using functions in transshipment_funcy.py
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Using
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 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.  
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In order to minimise the monthly shipment costs, American Steel should follow the shipment plan shown in Table 3.
As with many network problems, it can be illuminating to display the solution graphically as shown in Figure 1. Figure 1 Optimal Shipment Plan
