The Transportation Problem in AMPL

AMPL Formulation

The formulation of the transportation problem is AMPL is a straighforward translation of the matehmatical programme for the transportation problem.

The sets ${\cal S}$ and ${\cal D}$ are declared as SUPPLY_NODES and DEMAND_NODES respectively:

set SUPPLY_NODES;
set DEMAND_NODES;

The supply $s_i, i \in {\cal S}$ and demand $d_j, j \in {\cal D}$ are declared as integer parameters:

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

The cost $c_{ij}$ is declared over the SUPPLY_NODES and DEMAND_NODES:

param Cost {SUPPLY_NODES, DEMAND_NODES};

Now, the mathematical programme follows directly:

var Flow {SUPPLY_NODES, DEMAND_NODES} >= 0, integer;

minimize TotalCost:
  sum {i in SUPPLY_NODES, j in DEMAND_NODES} Cost[i, j] * Flow[i, j];

subject to UseSupply {i in SUPPLY_NODES}:
  sum {j in DEMAND_NODES} Flow[i, j] = Supply[i];

subject to MeetDemand {j in DEMAND_NODES}:
  sum {i in SUPPLY_NODES} Flow[i, j] = Demand[j];
Note that we assume the transportation is balanced.

Adding Bounds

In the main discussion of transportation problems, we saw that adding bounds to the flow variables allowed us to easily either bound the transportation of good from a supply node to a demand node or remove an arc from the problem altogether.

We can add bounds to our AMPL formulation by declaring 2 new parameters with defaults:

param Lower {SUPPLY_NODES, DEMAND_NODES} integer default 0;
param Upper {SUPPLY_NODES, DEMAND_NODES} integer default Infinity;
and adding them to the Flow variable declaration:
var Flow {i in SUPPLY_NODES, j in DEMAND_NODES}
  >= Lower[i, j], <= Upper[i, j], integer;

Balancing Transportation Problems

-- MichaelOSullivan - 02 Apr 2008

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Topic revision: r3 - 2008-04-02 - MichaelOSullivan
 
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