Difference: SetsInAMPL (7 vs. 8)

Revision 82008-03-11 - MichaelOSullivan

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META TOPICPARENT name="AMPLSyntax"
<-- Ready to Review - done - Lauren-->

Sets in AMPL

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  • Using a Table You give a table using the first index set for the rows and the second index set for the columns, then you place a + where an element exists and a - where there is no element. This is good for dense sets.
    
    
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    set ARCS: Cincinnati ‘Kansas City’ Chicago Albany Houston Tempe Gary :=
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    set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary :=
     Youngstown + + + + - - - Pittsburgh + + + - - - + Cincinnati - - - + + - -
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    ‘Kansas City’ - - - - + + -
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    'Kansas City' - - - - + + -
     Chicago - - - - - + + ;
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      You can create sets where the elements are ordered using the ordered keyword during definition.
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    Up to here - Mike
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    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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    Deleted:
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    set MONTHS ordered; 
    AMPL will puts the elements in this set in the order they appear in the data file (or {\tt 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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    Set Example

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    Consider the following AMPL statement from the The American Steel Planning Problem. We use ord in the creation of TIME_ARCS:

    # 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.

    >
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    Consider the following AMPL statement from the The American Steel Planning Problem. We use ord in the creation of TIME_ARCS:

    # 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.

     

    Set Operations

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    There are many operations we can perform on sets (see Set Expressions). We have seen that cross creates all pairs of two sets, so TIME_NODES cross TIME_NODES creates a set of all pairs of TIME_NODES.

    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)}); 
    or you could loop over {\tt MONTHS} and {\tt ARCS} simultaneously
    set TRANSPORT_ARCS := union {t in MONTHS, (m, n) in ARCS} {(m, t, n, t)}; 

    >
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    There are many operations we can perform on sets (see Set Expressions). We have seen that cross creates all pairs of two sets, so TIME_NODES cross TIME_NODES creates a set of all pairs of TIME_NODES.

    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)});
    
    or you could loop over MONTHS and ARCS simultaneously:
    set TRANSPORT_ARCS := union {t in MONTHS, (m, n) in ARCS} {(m, t, n, t)};
    
     

    Set Membership and Subsets

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    We have seen how loop over a set using the in keyword. This keyword also provides a logical check if an element is in a set, e.g., (m, n) in ARCS is true if the pair (m, n) is in the set ARCS and false otherwise. We may restrict a set to be a subset of an existing set by using the keyword {\tt within}, e. g.,
    set ARCS within NODES cross NODES; 
    means each arc is created between two nodes.

    >
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    We have seen how loop over a set using the in keyword. This keyword also provides a logical check if an element is in a set, e.g., (m, n) in ARCS is true if the pair (m, n) is in the set ARCS and false otherwise. We may restrict a set to be a subset of an existing set by using the keyword {\tt within}, e. g.,
    set ARCS within NODES cross NODES;
    
    means each arc is created between two nodes.
     

    Ordered Set Operators

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    The final condition on TIME_ARCS
        ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; 
    creates the "storage" arcs. We could use next(t, MONTHS) = u or t = prev(u, MONTHS) except for a problem when t is June or u is April, respectively. When you use prev or next you must be careful of the first and last members of the set respectively. However, you can use first or last to check if you are using these elements.

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    The final condition on TIME_ARCS
        ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )};
    
    creates the "storage" arcs. We could use next(t, MONTHS) = u or t = prev(u, MONTHS) except for a problem when t is June or u is April, respectively. When you use prev or next you must be careful of the first and last members of the set respectively. However, you can use first or last to check if you are using these elements.

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    Restricted Sets

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    When using display or printf statements we saw that we could restrict the members of a set being printed, e.g.,
    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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    >
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    When using display or printf statements we saw that we could restrict the members of a set being printed, e.g.,
    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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    Multi-dimensional Sets

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    We have already seen different ways of declaring 2-dimensional sets. We have now encountered higher dimensional sets, e.g., TIME_ARCS has 4 dimensions. We generated TIME_ARCS automatically, but we could have specified it using a data file.
    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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    We have already seen different ways of declaring 2-dimensional sets. We have now encountered higher dimensional sets, e.g., TIME_ARCS has 4 dimensions. We generated TIME_ARCS automatically, but we could have specified it using a data file.

    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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    Efficient Generation of Sets

    When creating models, generating sets by looping over all possibilities and removing those that don't fit some conditions (e.g., like the TIME_ARCS statement) can take a long time if there are many possibilities. This is time that could be spent solving the model! If possible you should try to create sets by building them up from smaller building block, rather than by creating an enormous set and pruning it. For example, instead of using this statement
    # 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 
    to create TIME_ARCS you could use these statement
    set TRANSPORT_ARCS := union {t in MONTHS, (m, n) in ARCS} {(m, t, n, t)}; set STORAGE_ARCS := union {t in MONTHS, (m, n) in ARCS : t <> last(MONTHS)}   {(m, t, n, next(t, MONTHS)}; set TIME_ARCS within TIME_NODES cross TIME_NODES := TRANSPORT_ARCS union STORAGE_ARCS; 
    Rather than looping over all possibilities and only keeping those that are appropriate, the new statement only loops over smaller sets that can be used to build up the TIME_NODES set efficiently.

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    -- MichaelOSullivan - 27 Feb 2008

     
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