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> >  < Ready to Review  done  Lauren>  
Sets in AMPL  
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Description  
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Declaring a Set  
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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 NODESIf you need a multidimensional set, but don't have the 1dimensional 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 NODESIf you need a multidimensional set, but don't have the 1dimensional 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:  
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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}  
> >  {'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 ExpressionsSet expressions take the following form (which can be extended to higher dimensional sets):
 
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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:
You will see examples of set expressions throughout the rest of this page.  
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> >  Generic Set Expression Fix underlining  Lauren
{ in , [ in , in , ...] : } Set expressions may also involve one or more _set operators_:  
Defining a Set  
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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 2Dimensional Sets  
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< <  There are three different ways to define 2dimensional sets. The "best" way to use depends on the set.
 
> >  There are three different ways to define 2dimensional sets. The "best" way to use depends on the set.
 
Ordered Sets  
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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  
> >  You can create sets where the elements are ordered using the 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
 
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 timestaged 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 storageThere are many concepts within this one statement, let's look at them one at a time.  
> >  Consider the following AMPL statement from the The American Steel Planning Problem. We use # The set of timestaged 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 storageThere 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 timestaged 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 timestaged 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  
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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.  
> >  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.
 
> >  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.
 
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 nodesWe 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 nonsupply nodes (since nonsupply 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] = ...  
> >  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 nodesWe 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 nonsupply nodes (since nonsupply 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] = ...
 
Multidimensional Sets  
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< <  We have already seen different ways of declaring 2dimensional 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’ ... ... ;  
> >  We have already seen different ways of declaring 2dimensional 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’ ... ... ;
 
Efficient Generation of Sets  
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< <  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 timestaged 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 storageto 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.
 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 timestaged 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 storageto 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.  MichaelOSullivan  27 Feb 2008  
