{ <e> in <S>, [<f> in <T>, <g> in <U>, …] :   <logical expression involving e [f, g, …]>} 

set NODES := Youngstown Pittsburgh Cincinnati 'Kansas City' Chicago
             Albany Houston Tempe Gary ;

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

1. Using a List You simply list the elements in the set. This is good for sparse sets.

      model;
      set ARCS within NODES cross 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       -           -            -       -      -       +     +
 ;

1. 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 ...

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

# 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 transportation arcs
	( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage arcs

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

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

set ARCS within NODES cross NODES; # 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; 

{'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'} 

param start; param end > start; param step; set NUMBERS := start .. end by step; 

set NUMBERS := 1..5; # NUMBERS = {1, 2, 3, 4, 5} 

set NUMBERS := 1 2 3 4 5; # NUMBERS = {1, 2, 3, 4 5} 

{  in , [ in ,  in , …] :   } 

set NODES := Youngstown Pittsburgh Cincinnati 'Kansas City' Chicago              Albany Houston Tempe Gary ; 

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

1. Using a List You simply list the elements in the set. This is good for sparse sets.

   model; set ARCS within NODES cross 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 ... 

set MONTHS ordered; 

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 

# 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 transportation arcs      ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The storage arcs 

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

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

set ARCS within NODES cross NODES; 

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

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

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

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] = ...  

set TIME_ARCS within TIME_NODES cross TIME_NODES; 

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

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

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

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

# 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 transportation arcs      ( (m = n) and (ord(t, MONTHS) + 1 = ord(u, MONTHS)) )}; # The 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; 

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;

{'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'}

param start;
param end > start;
param step;
set NUMBERS := start .. end by step;

set NUMBERS := 1..5; # NUMBERS = {1, 2, 3, 4, 5}

set NUMBERS := 1 2 3 4 5; # NUMBERS = {1, 2, 3, 4 5}

# 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 statements

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

set MONTHS ordered;

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

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

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

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

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

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

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

set ARCS within NODES cross NODES;

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

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

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

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

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

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

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set TIME_ARCS within TIME_NODES cross TIME_NODES;

List

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

Table

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

Array

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

or

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

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{'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'} 

param start; param end > start; param step;  set NUMBERS := start .. end by step; 

set NUMBERS := 1..5; # NUMBERS = {1, 2, 3, 4, 5} 

set NUMBERS := 1 2 3 4 5; # NUMBERS = {1, 2, 3, 4 5} 

{ <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 A, b 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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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};

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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Up to here - Mike

set MONTHS ordered; 

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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<-- Ready to Review - done - Lauren-->

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

set ARCS within NODES cross NODES; # Elements of ARCS must have both elements in NODES 

set ROUTES dimen 2; 

{'HOST', 'DEVICE', 'SWITCH', 'HUB', 'LINK', 'SUPERLINK'} 

param start; param end > start; param step;  set NUMBERS := start .. end by step; 

set NUMBERS := 1..5; # NUMBERS = {1, 2, 3, 4, 5} 

set NUMBERS := 1 2 3 4 5; # NUMBERS = {1, 2, 3, 4 5} 

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.

{  in , [ in ,  in , ...] :   } Set expressions may also involve one or more _set operators_: 
 =A union B= gives the set of elements in either =A= or =B=; 
 =A inter B= gives the set of elements in both =A= and =B=; 
 =A diff B= gives the set of elements in =A= that are not in =B=; 
 =A symdiff B= gives the set of elements in either =A= or =B= but not both; 
 =A cross B= gives the two-dimensional set of all pairs =a=  =A=, =b=  =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. 
[[#top][Return to top]] 
 

Sets are usually defined in a data file:

set NODES := Youngstown Pittsburgh Cincinnati ‘Kansas City’ Chicago              Albany Houston Tempe Gary ; 

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

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

set MONTHS ordered; 

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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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 

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

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

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

set ARCS within NODES cross NODES; 

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

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

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

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

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set TIME_ARCS within TIME_NODES cross TIME_NODES; 

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

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

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

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

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

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; 

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-- MichaelOSullivan - 27 Feb 2008