(r7) TransshipmentProblemInAMPL < OpsRes

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---+ The Transshipment Problem in AMPL

The formulation of the transshipment problem in AMPL we present here is a straightforward translation of the alternative mathematical programme for [[TransshipmentProblem][the transshipment problem]]. We will build the file [[%ATTACHURL%/transshipment.mod][<tt>transshipment.mod</tt>]].

The set %${\cal N}$% is declared as =NODES=:
<pre>
set NODES;
</pre>

The net demand %$\hat{d}_j, j \in {\cal N}$% is declared as an *integer* parameter (note there is no <tt>>= 0</tt>):
<pre>
param NetDemand {NODES} integer, default 0;
</pre>
Setting the default =NetDemand= to be 0 means that no values need to be entered for transshipment nodes.

Note that the set =NODES= and the parameter =NetDemand= can be easily created from =SUPPLY_NODES=, =DEMAND_NODES=, =Supply=, etc, e.g.,:
<pre>
set SUPPLY_NODES;
set DEMAND_NODES;
set TRANSSHIPMENT_NODES;

set NODES := SUPPLY_NODES union
             TRANSSHIPMENT_NODES union
             DEMAND_NODES;

param Supply {SUPPLY_NODES} >= 0, integer;
param Demand {DEMAND_NODES} >= 0, integer;

param NetDemand {n in NODES} integer
  := if n in SUPPLY_NODES then -Supply[n]
     else if n in DEMAND_NODES then Demand[n]; # else 0 by default
</pre>
Note that no default value is needed for =NetDemand= as it is explicitly defined for all nodes.

The set =ARCS= is defined between pairs of nodes and costs and bounds are also defined:
<pre>
set ARCS within NODES cross NODES;

param Cost {ARCS};

param Lower {ARCS} >= 0, integer, default 0;
param Upper {(i, j) in ARCS} >= Lower[i, j], integer, default Infinity;
</pre>

Now, the mathematical programme follows directly:
<pre>
var Flow {(i, j) in ARCS} >= Lower[i, j], <= Upper[i, j], integer;

minimize TotalCost:
  sum {(i, j) in ARCS} Cost[i, j] * Flow[i, j];

subject to ConserveFlow {j in NODES}:
  sum {(i, j) in ARCS} Flow[i, j] - sum {(j, k) in ARCS} Flow[j, k] >= NetDemand[j];
</pre>

-- Main.MichaelOSullivan - 06 Oct 2014