set MONTHS ordered;

AMPL will puts the elements in this set in the order they appear in the data file (or let statement). AMPL also understands the following operations for ordered sets:

ord(e, ORD_SET)   # The position of e in ORD_SET
first(ORD_SET)    # The first element in ORD_SET
last(ORD_SET)     # The last element in ORD_SET
prev(e, ORD_SET)  # The element before e in ORD_SET
next(e, ORD_SET)  # The element after e in ORD_SET
member(i, ORDSET) # The element at position i in ORD_SET

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# The set of time-staged arcs
set TIME_ARCS within TIME_NODES cross TIME_NODES :=   { (m, t) in TIME_NODES, (n, u) in TIME_NODES :
     ( ( (m, n) in ARCS) and (t = u) ) or # The arcs used for transportation
     ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The arcs used for storage

There are many concepts within this one statement, let's look at them one at a time.

Some set operations may be looped over indexing sets. For example, to generate all the transportation arcs in the time-staged network you could use the following statement:

set TRANSPORT_ARCS := union {t in MONTHS} (union {(m, n) ARCS} {(m, t, n, t)});

set TRANSPORT_ARCS := union {t in MONTHS, (m, n) in ARCS} {(m, t, n, t)};

set ARCS within NODES cross NODES;

    ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )};

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display {(m, n) in ARCS : Supply[m] > 0}; # Display all arcs from supply nodes

We can do this when creating sets, e. g., TIME_NODES, or when using a set as an index for variables, parameters or constraints. For example, rather than setting the upper bound of UnderProduction to be 0 for all non-supply nodes (since non-supply nodes don't produce anything) we could only create this variable for the supply nodes (in fact this may be preferable since there will be less variables).

var UnderProduction {(n, t) in TIME_NODES : Supply[n, t] > 0} >= 0, integer;

We could make sure this variable is only added to the constraints for the supply nodes (e.g., ConserveFlow constraints) by using a conditional expression.

subject to ConserveFlow {(n, t) in TIME_NODES}:
   sum {(m, s) in TIME_NODES: (m, s, n, t) in TIME_ARCS}
     Shipment[m, s, n, t] + Supply[n, t] -
     if Supply[n, t] > 0 then UnderProduction[n, t] = ... 

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set TIME_ARCS within TIME_NODES cross TIME_NODES;

List

set TIME_ARCS := (Youngstown, April, Albany, April)
                  (Youngstown, April, Youngstown, May) ... ;

Table

set TIME_ARCS : =
(*, May, *, May)  Cincinnati 'Kansas City' Albany ... :=
Youngstown         +          +            +      ...
Pittsburgh         +          +            -      ...
... ;

Array

set TIME_ARCS := (*, May, *, May) (Youngstown, Cincinnati) ... ... ;

or

set TIME_ARCS := (Youngstown, May, *, May) Cincinnati 'Kansas City' ... ... ;

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