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 the transshipment problem. We will build the file transshipment.mod.

The set is declared as NODES:

set NODES;

The net demand is declared as an integer parameter (note there is no >= 0):

param NetDemand {NODES} integer, default 0;

Note that the set NODES and the parameter NetDemand can be easily created from SUPPLY_NODES, DEMAND_NODES, Supply, etc, e.g.,:

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

The set ARCS is defined between pairs of nodes and costs and bounds are also defined:

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;

Now, the mathematical programme follows directly:

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];

-- MichaelOSullivan - 06 Oct 2014