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< Ready to Review  done  Lauren> Sets in AMPL  
Line: 221 to 221  
 MichaelOSullivan  27 Feb 2008  
Changed:  
< < 
 
> > 

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< Ready to Review  done  Lauren> Sets in AMPL  
Line: 36 to 36  
Generic Set Expression  
Changed:  
< <  {  
> >  { <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:  
Line: 54 to 54  
Sets are usually defined in a data file:  
Changed:  
< <  set NODES := Youngstown Pittsburgh Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary ;  
> >  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:  
Changed:  
< <  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;  
> >  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:  
Changed:  
< <  set SEARCH within VERTICES; let SEARCH := {v in VERTICES: (v, w) in EDGES};  
> >  set SEARCH within VERTICES; let SEARCH := {v in VERTICES: (v, w) in EDGES};  
Defining 2Dimensional SetsThere are three different ways to define 2dimensional sets. The "best" way to use depends on the set.  
Changed:  
< < 
 
> > 
set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary := Youngstown + + + +    Pittsburgh + + +    + Cincinnati    + +   'Kansas City'     + +  Chicago      + + ;
set ARCS := (Youngstown, *) Cincinnati ‘Kansas City’ Chicago Albany (Pittsburgh, *) Cincinnati ‘Kansas City’ Chicago Gary (Cincinnati, *) Albany Houston ...  
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Line: 82 to 117  
AMPL will put 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:  
Changed:  
< <  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  
> >  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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Line: 90 to 132  
Consider the following AMPL statement from the The American Steel Planning Problem. We use ord in the creation of TIME_ARCS :  
Changed:  
< <  # 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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage 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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage arcs  
There are many concepts within this one statement, let's look at them one at a time.  
Line: 99 to 147  
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:  
Changed:  
< <  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 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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< Ready to Review  done  Lauren> Sets in AMPL  
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Return to top
Declaring a Set  
Changed:  
< <  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 multidimensional set, but don't have the 1dimensional sets to construct it yet you can use the set ROUTES dimen 2;There are some other set attributes, but we will not use them here.  
> >  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 multidimensional set, but don't have the 1dimensional sets to construct it yet you can use the set ROUTES dimen 2;There are some other set attributes, but we will not use them here.  
Set literals can be defined as a list of elements:  
Changed:  
< <  {'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 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 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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Line: 59 to 36  
Generic Set Expression  
Changed:  
< <  { <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:  
Line: 80 to 54  
Sets are usually defined in a data file:  
Changed:  
< <  set NODES := Youngstown Pittsburgh Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary ;  
> >  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:  
Changed:  
< <  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;  
> >  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:  
Changed:  
< <  set SEARCH within VERTICES; let SEARCH := {v in VERTICES: (v, w) in EDGES};  
> >  set SEARCH within VERTICES; let SEARCH := {v in VERTICES: (v, w) in EDGES};  
Defining 2Dimensional SetsThere are three different ways to define 2dimensional sets. The "best" way to use depends on the set.  
Changed:  
< < 
 
> > 
 
Return to top  
Line: 148 to 78  
You can create sets where the elements are ordered using the ordered keyword during definition.  
Changed:  
< <  set MONTHS ordered;  
> >  set MONTHS ordered;  
AMPL will put 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:  
Changed:  
< <  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  
> >  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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Line: 169 to 90  
Consider the following AMPL statement from the The American Steel Planning Problem. We use ord in the creation of TIME_ARCS :  
Changed:  
< <  # 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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage 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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage arcs  
There are many concepts within this one statement, let's look at them one at a time.  
Line: 184 to 99  
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:  
Changed:  
< <  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 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
We have seen how to loop over a set using the  
Changed:  
< <  set ARCS within NODES cross NODES;means each arc is created between two nodes.  
> >  set ARCS within NODES cross NODES;means each arc is created between two nodes.  
Ordered Set Operators
The final condition on  
Changed:  
< <  ( (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.  
> >  ( (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.  
Return to top
Restricted SetsWhen using display or printf statements we saw that we could restrict the members of a set being printed, e.g.,  
Changed:  
< <  display {(m, n) in ARCS : Supply[m] > 0}; # Display all arcs from supply nodes  
> >  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 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).  
Changed:  
< <  var UnderProduction {(n, t) in TIME_NODES : Supply[n, t] > 0} >= 0, integer;  
> >  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.  
Changed:  
< <  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] = ...  
> >  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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Line: 238 to 132  
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.  
Changed:  
< <  set TIME_ARCS within TIME_NODES cross TIME_NODES;  
> >  set TIME_ARCS within TIME_NODES cross TIME_NODES;  
List  
Changed:  
< <  set TIME_ARCS := (Youngstown, April, Albany, April) (Youngstown, April, Youngstown, May) ... ;  
> >  set TIME_ARCS := (Youngstown, April, Albany, April) (Youngstown, April, Youngstown, May) ... ;  
Table  
Changed:  
< <  set TIME_ARCS : = (*, May, *, May) Cincinnati 'Kansas City' Albany ... := Youngstown + + + ... Pittsburgh + +  ... ... ;  
> >  set TIME_ARCS : = (*, May, *, May) Cincinnati 'Kansas City' Albany ... := Youngstown + + + ... Pittsburgh + +  ... ... ;  
Array  
Changed:  
< <  set TIME_ARCS := (*, May, *, May) (Youngstown, Cincinnati) ... ... ;  
> >  set TIME_ARCS := (*, May, *, May) (Youngstown, Cincinnati) ... ... ;  
or  
Changed:  
< <  set TIME_ARCS := (Youngstown, May, *, May) Cincinnati 'Kansas City' ... ... ;  
> >  set TIME_ARCS := (Youngstown, May, *, May) Cincinnati 'Kansas City' ... ... ;  
Return to top
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  
Changed:  
< <  # 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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage 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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage arcs  
to create TIME_ARCS you could use these statements  
Changed:  
< <  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;  
> >  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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< Ready to Review  done  Lauren> Sets in AMPL  
Line: 152 to 152  
set MONTHS ordered;  
Changed:  
< <  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:  
> >  AMPL will put 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  
Line: 194 to 194  
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.,  
> >  We have seen how to 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 within , e. g.,  
set ARCS within NODES cross NODES; 
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< Ready to Review  done  Lauren> Sets in AMPL  
Line: 20 to 20  
Return to top
Declaring a Set  
Changed:  
< <  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 NODES
If you need a multidimensional set, but don't have the 1dimensional sets to construct it yet you can use the set ROUTES dimen 2;There are some other set attributes, but we will not use them here.  
Set literals can be defined as a list of elements:  
Changed:  
< <  {'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 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 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}  
Return to top  
Line: 145 to 171  
# The set of timestaged arcs  
Changed:  
< <  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  
> >  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 transportation arcs ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage arcs  
There are many concepts within this one statement, let's look at them one at a time. 
Line: 1 to 1  

< Ready to Review  done  Lauren> Sets in AMPL  
Line: 246 to 246  
Return to top
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 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 storage
to create
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  MichaelOSullivan  27 Feb 2008  

Line: 1 to 1  

< Ready to Review  done  Lauren> Sets in AMPL  
Line: 96 to 96  
+ where an element exists and a  where there is no element. This is good for dense sets.
 
Changed:  
< <  set ARCS: Cincinnati ‘Kansas City’ Chicago Albany Houston Tempe Gary :=  
> >  set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary :=  
Youngstown + + + +    Pittsburgh + + +    + Cincinnati    + +    
Changed:  
< <  ‘Kansas City’     + +   
> >  'Kansas City'     + +   
Chicago      + + ;  
Line: 121 to 121  
You can create sets where the elements are ordered using the ordered keyword during definition.  
Changed:  
< <  Up to here  Mike  
> >  set MONTHS ordered;
AMPL will puts the elements in this set in the order they appear in the data file (or
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  
Deleted:  
< <  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  
Changed:  
< <  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.
 
> > 
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 storage There are many concepts within this one statement, let's look at them one at a time.  
Set Operations  
Changed:  
< <  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 MONTHS and 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.
 
> > 
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  
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 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 nodes
We can do this when creating sets, e. g.,
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.,
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  
Changed:  
< <  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 SetsWhen creating models, generating sets by looping over all possibilities and removing those that don't fit some conditions (e.g., like theTIME_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 
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< Ready to Review  done  Lauren> Sets in AMPL  
Line: 23 to 23  
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:  
Changed:  
< <  {'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}  
> >  {'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 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}  
Return to top  
Line: 32 to 32  
 
Changed:  
< <  Generic Set Expression Fix underlining  Lauren
{ in , [ in , in , ...] : } Set expressions may also involve one or more _set operators_:  
> >  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.  
Defining a Set  
Changed:  
< <  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  
Changed:  
< <  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  
Changed:  
< <  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
 
> > 
You can create sets where the elements are ordered using the ordered keyword during definition.
Up to here  Mike
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 ExampleConsider 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 
Line: 1 to 1  

 
Changed:  
< <  < Ready to Review >  
> >  < Ready to Review  done  Lauren>  
Sets in AMPL  
Changed:  
< <  
> >  
Description  
Line: 20 to 20  
Return to top
Declaring a Set  
Changed:  
< <  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:  
Changed:  
< <  {'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}  
Return to top
Set ExpressionsSet expressions take the following form (which can be extended to higher dimensional sets):
 
Changed:  
< < 
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.  
> > 
 
Added:  
> >  Generic Set Expression Fix underlining  Lauren
{ in , [ in , in , ...] : } Set expressions may also involve one or more _set operators_:  
Defining a Set  
Changed:  
< < 
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  
Changed:  
< <  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  
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  
> >  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  
Changed:  
< < 
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  
Changed:  
< <  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  
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.
 
> >  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  
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 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  
Changed:  
< <  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  
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 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  

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