Difference: SetsInAMPL (5 vs. 6)

Revision 62008-03-09 - LaurenJackson

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META TOPICPARENT name="AMPLSyntax"
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Sets in AMPL

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  1. Description
  2. Declaring a Set
  3. Set Expressions
  4. Defining a Set
  5. Ordered Sets
  6. Set Example
  7. Restricted Sets
  8. Multi-Dimensional Sets
  9. Effcient Generation of Sets
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  1. Description
  2. Declaring a Set
  3. Set Expressions
  4. Defining a Set
  5. Ordered Sets
  6. Set Example
  7. Restricted Sets
  8. Multi-Dimensional Sets
  9. Efficient Generation of Sets
 

Description

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Declaring a Set

Changed:
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<		Sets are declared using the set keyword followed by a label, possibly some attributes and either a set literal or set expression. The most common attribute is set by the within keyword. This specifies that the set will only contain elements from the following set definition: 
set ARCS within NODES cross NODES; # Elements of ARCS must have both elements in NODES
If you need a multi-dimensional set, but don't have the 1-dimensional sets to construct it yet you can use the dimen keyword: 
set ROUTES dimen 2;
There are some other set attributes, but we will not use them.
>
>		Sets are declared using the set keyword followed by a label, possibly some attributes and either a set literal or set expression. The most common attribute is set by the within keyword. This specifies that the set will only contain elements from the following set definition: 
set ARCS within NODES cross NODES; # Elements of ARCS must have both elements in NODES
If you need a multi-dimensional set, but don't have the 1-dimensional sets to construct it yet you can use the dimen keyword: 
set ROUTES dimen 2;
There are some other set attributes, but we will not use them.
  Set literals can be defined as a list of elements:
Changed:
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<
{'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'}
or a sequence of numbers: 
param start;
param end > start;
param step;

set NUMBERS := start .. end by step;
If the by step is missing, the step is assumed to be 1 
set NUMBERS := 1..5; # NUMBERS = {1, 2, 3, 4, 5}
Note Automatic set generation can only be done in the [[AMPLProcess#model][model environment], in the data environment you must define the set explicitly: 
set NUMBERS := 1 2 3 4 5; # NUMBERS = {1, 2, 3, 4 5}
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>
{'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'}
or a sequence of numbers: 
param start; param end > start; param step;  set NUMBERS := start .. end by step;
If the by step is missing, the step is assumed to be 1 
set NUMBERS := 1..5; # NUMBERS = {1, 2, 3, 4, 5}
Note Automatic set generation can only be done in the [[AMPLProcess#model][model environment] should this be a link? - Lauren , in the data environment you must define the set explicitly: 
set NUMBERS := 1 2 3 4 5; # NUMBERS = {1, 2, 3, 4 5}
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Set Expressions

Set expressions take the following form (which can be extended to higher dimensional sets):
  1. Define the elements that make up the set, e.g., s in SUPPLY_NODES, before the colon : operator;
Changed:
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  1. Use a logical expression (after the :) to indicate if an element (or pair of elements, or “tuple” of elements) should be included in the set.

Generic Set Expression 

{ <e> in <S>, [<f> in <T>, <g> in <U>, ...] :
  <logical expression involving e [f, g, ...]>}
Set expressions may also involve one or more set operators:
  1. A union B gives the set of elements in either A or B;
  2. A inter B gives the set of elements in both A and B;
  3. A diff B gives the set of elements in A that are not in B;
  4. A symdiff B gives the set of elements in either A or B but not both;
  5. A cross B gives the two-dimensional set of all pairs a $\in$ A, b $\in$ B.
    This can also be defined by {a in A, b in B}.

You will see examples of set expressions throughout the rest of this page.

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  1. Use a logical expression (after the :) to indicate if an element (or pair of elements, or “tuple” of elements) should be included in the set.
 
Added:
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>		Generic Set Expression Fix underlining - Lauren 
{  in , [ in ,  in , ...] :   } Set expressions may also involve one or more _set operators_: 
  1.  =A union B= gives the set of elements in either =A= or =B=; 
  2.  =A inter B= gives the set of elements in both =A= and =B=; 
  3.  =A diff B= gives the set of elements in =A= that are not in =B=; 
  4.  =A symdiff B= gives the set of elements in either =A= or =B= but not both; 
  5.  =A cross B= gives the two-dimensional set of all pairs =a= $\in$ =A=, =b= $\in$ =B=. This can also be defined by ={a in A, b in B}=. 
You will see examples of set expressions throughout the rest of this page. 
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Defining a Set

Changed:
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<		 Sets are usually defined in a data file: 
set NODES := Youngstown Pittsburgh Cincinnati ‘Kansas City’ Chicago
             Albany Houston Tempe Gary ;
although they may be defined during declaration using either an explicit set literal or using a set expression: 
set KIND := {'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'};

set COMPONENT := {C in CLASSES : (kind[C] = 'HOST'  ) or
				 (kind[C] = 'DEVICE') or
				 (kind[C] = 'HUB'   ) or
				 (kind[C] = 'SWITCH')};

set FABRIC := NODE union LINK;
and sets may also be defined dynamically: 
set SEARCH within VERTICES;

let SEARCH := {v in VERTICES: (v, w) in EDGES};
>
>

Sets are usually defined in a data file: 

set NODES := Youngstown Pittsburgh Cincinnati ‘Kansas City’ Chicago              Albany Houston Tempe Gary ;
although they may be defined during declaration using either an explicit set literal or using a set expression: 
set KIND := {'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'};  set COMPONENT := {C in CLASSES : (kind[C] = 'HOST'  ) or              (kind[C] = 'DEVICE') or              (kind[C] = 'HUB'   ) or              (kind[C] = 'SWITCH')};  set FABRIC := NODE union LINK;
and sets may also be defined dynamically: 
set SEARCH within VERTICES;  let SEARCH := {v in VERTICES: (v, w) in EDGES};

 

Defining 2-Dimensional Sets

Changed:
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<		There are three different ways to define 2-dimensional sets. The "best" way to use depends on the set.
  1. Using a List You simply list the elements in the set. This is good for sparse sets.
    model;
set ARCS within NODES cros NODES;

data;

set ARCS := (Youngstown, Albany), (Youngstown, Cincinnati), ... ;
  2. 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.
    set ARCS:   Cincinnati ‘Kansas City’ Chicago Albany Houston Tempe Gary :=
Youngstown         +          +         +      +       -      -    -
Pittsburgh         +          +         +      -       -      -    +
Cincinnati         -          -         -      +       +      -    -
‘Kansas City’      -          -         -      -       +      +    -
Chicago            -          -         -      -       -      +    + ;
  3. Using an Array You define a list of column indices for each row index. This is a good for sets with a few elements for each row.
    set ARCS :=
(Youngstown, *) Cincinnati ‘Kansas City’ Chicago Albany
(Pittsburgh, *) Cincinnati ‘Kansas City’ Chicago Gary
(Cincinnati, *) Albany Houston ...

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>
>		There are three different ways to define 2-dimensional sets. The "best" way to use depends on the set.
  1. Using a List You simply list the elements in the set. This is good for sparse sets.
    model; set ARCS within NODES cros NODES; data; set ARCS := (Youngstown, Albany), (Youngstown, Cincinnati), ... ;
  2. 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.
    set ARCS: Cincinnati ‘Kansas City’ Chicago Albany Houston Tempe Gary := Youngstown + + + + - - - Pittsburgh + + + - - - + Cincinnati - - - + + - - ‘Kansas City’ - - - - + + - Chicago - - - - - + + ;
  3. Using an Array You define a list of column indices for each row index. This is a good for sets with a few elements for each row.
    set ARCS := (Youngstown, *) Cincinnati ‘Kansas City’ Chicago Albany (Pittsburgh, *) Cincinnati ‘Kansas City’ Chicago Gary (Cincinnati, *) Albany Houston ...

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Ordered Sets

Changed:
<
<		 You can create sets where the elements are ordered using the ordered keyword during definition. 
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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>

You can create sets where the elements are ordered using the ordered keyword during definition. 

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

Changed:
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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

Changed:
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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)};
>
>		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)};

 

Set Membership and Subsets

Changed:
<
<		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.
>
>		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

Changed:
<
<		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

Changed:
<
<		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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>
>		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

Changed:
<
<		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

Changed:
<
<		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

>
>		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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