The Transportation Problem in AMPL

AMPL Formulation

The formulation of the transportation problem is AMPL is a straightforward translation of the mathematical programme for the transportation problem.

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


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:


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

Balanced transportation models are preferred as there is no confusion about the relational operators for the supply and demand constraints. We can use script file to balance any transportation problem automatically:


model transportation.mod;

param costFromDummy {DEMAND_NODES} default 0;
param costToDummy   {SUPPLY_NODES} default 0;

param difference;

# Add the problem date file here
# e.g., data brewery.dat;

let difference := (sum {s in SUPPLY_NODES} Supply[s])
                - (sum {d in DEMAND_NODES} Demand[d]);
if difference > 0 then
  let DEMAND_NODES := DEMAND_NODES union {'Dummy'};
  let Demand['Dummy'] := difference;
  let {s in SUPPLY_NODES} Cost[s, 'Dummy'] := costToDummy[s];
else if difference < 0 then
  let SUPPLY_NODES := SUPPLY_NODES union {'Dummy'};
  let Supply['Dummy'] := - difference;
  let {d in DEMAND_NODES} Cost['Dummy', d] := costFromDummy[d];
}; # else the problem is balanced

# Make sure the problem is balanced
check : sum {s in SUPPLY_NODES} Supply[s] = sum {d in DEMAND_NODES} Demand[d];

option solver cplex;


display Flow;
Note the check statement to ensure that the balancing has been done properly before solving. Also, note that costToDummy and costFromDummy allow for the definition of costs on any flow from/to a dummy node in the data file.

Both the AMPL model file transportation.mod and script file are attached.

-- MichaelOSullivan - 02 Apr 2008

Topic attachments
I Attachment History Action Size Date WhoSorted ascending Comment
Unknown file formatmod transportation.mod r1 manage 1.2 K 2008-04-02 - 10:41 MichaelOSullivan  
Unknown file formatrun r1 manage 0.9 K 2008-04-02 - 10:42 MichaelOSullivan  

This topic: OpsRes > WebHome > AMPLGuide > TransportationProblemInAMPL
Topic revision: r4 - 2008-04-02 - MichaelOSullivan
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