# Sets¶

## Declaration¶

Sets can be declared using instances of the Set and RangeSet classes or by assigning set expressions. The simplest set declaration creates a set and postpones creation of its members:

model.A = pyo.Set()


The Set class takes optional arguments such as:

• dimen = Dimension of the members of the set
• doc = String describing the set
• filter = A Boolean function used during construction to indicate if a potential new member should be assigned to the set
• initialize = An iterable containing the initial members of the Set, or function that returns an iterable of the initial members the set.
• ordered = A Boolean indicator that the set is ordered; the default is True
• validate = A Boolean function that validates new member data
• within = Set used for validation; it is a super-set of the set being declared.

In general, Pyomo attempts to infer the “dimensionality” of Set components (that is, the number of apparent indices) when they are constructed. However, there are situations where Pyomo either cannot detect a dimensionality (e.g., a Set that was not initialized with any members), or you the user may want to assert the dimensionality of the set. This can be accomplished through the dimen keyword. For example, to create a set whose members will be tuples with two items, one could write:

model.B = pyo.Set(dimen=2)


To create a set of all the numbers in set model.A doubled, one could use

def DoubleA_init(model):
return (i*2 for i in model.A)
model.C = pyo.Set(initialize=DoubleA_init)


As an aside we note that as always in Python, there are lot of ways to accomplish the same thing. Also, note that this will generate an error if model.A contains elements for which multiplication times two is not defined.

The initialize option can accept any Python iterable, including a set, list, or tuple. This data may be returned from a function or specified directly as in

model.D = pyo.Set(initialize=['red', 'green', 'blue'])


The initialize option can also specify either a generator or a function to specify the Set members. In the case of a generator, all data yielded by the generator will become the initial set members:

def X_init(m):
for i in range(10):
yield 2*i+1
model.X = pyo.Set(initialize=X_init)


For initialization functions, Pyomo supports two signatures. In the first, the function returns an iterable (set, list, or tuple) containing the data with which to initialize the Set:

def Y_init(m):
return [2*i+1 for i in range(10)]
model.Y = pyo.Set(initialize=Y_init)


In the second signature, the function is called for each element, passing the element number in as an extra argument. This is repeated until the function returns the special value Set.End:

def Z_init(model, i):
if i > 10:
return pyo.Set.End
return 2*i+1
model.Z = pyo.Set(initialize=Z_init)


Note that the element number starts with 1 and not 0:

>>> model.X.pprint()
X : Size=1, Index=None, Ordered=Insertion
Key  : Dimen : Domain : Size : Members
None :     1 :    Any :   10 : {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
>>> model.Y.pprint()
Y : Size=1, Index=None, Ordered=Insertion
Key  : Dimen : Domain : Size : Members
None :     1 :    Any :   10 : {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
>>> model.Z.pprint()
Z : Size=1, Index=None, Ordered=Insertion
Key  : Dimen : Domain : Size : Members
None :     1 :    Any :   10 : {3, 5, 7, 9, 11, 13, 15, 17, 19, 21}


Additional information about iterators for set initialization is in the [PyomoBookII] book.

Note

For Abstract models, data specified in an input file or through the data argument to AbstractModel.create_instance() will override the data specified by the initialize options.

If sets are given as arguments to Set without keywords, they are interpreted as indexes for an array of sets. For example, to create an array of sets that is indexed by the members of the set model.A, use:

model.E = pyo.Set(model.A)


Arguments can be combined. For example, to create an array of sets, indexed by set model.A where each set contains three dimensional members, use:

model.F = pyo.Set(model.A, dimen=3)


The initialize option can be used to create a set that contains a sequence of numbers, but the RangeSet class provides a concise mechanism for simple sequences. This class takes as its arguments a start value, a final value, and a step size. If the RangeSet has only a single argument, then that value defines the final value in the sequence; the first value and step size default to one. If two values given, they are the first and last value in the sequence and the step size defaults to one. For example, the following declaration creates a set with the numbers 1.5, 5 and 8.5:

model.G = pyo.RangeSet(1.5, 10, 3.5)


## Operations¶

Sets may also be created by storing the result of set operations using other Pyomo sets. Pyomo supports set operations including union, intersection, difference, and symmetric difference:

model.I = model.A | model.D # union
model.J = model.A & model.D # intersection
model.K = model.A - model.D # difference
model.L = model.A ^ model.D # exclusive-or


For example, the cross-product operator is the asterisk (*). To define a new set M that is the cross product of sets B and C, one could use

model.M = model.B * model.C


This creates a virtual set that holds references to the original sets, so any updates to the original sets (B and C) will be reflected in the new set (M). In contrast, you can also create a concrete set, which directly stores the values of the cross product at the time of creation and will not reflect subsequent changes in the original sets with:

model.M_concrete = pyo.Set(initialize=model.B * model.C)


Finally, you can indicate that the members of a set are restricted to be in the cross product of two other sets, one can use the within keyword:

model.N = pyo.Set(within=model.B * model.C)


## Predefined Virtual Sets¶

For use in specifying domains for sets, parameters and variables, Pyomo provides the following pre-defined virtual sets:

• Any = all possible values
• Reals = floating point values
• PositiveReals = strictly positive floating point values
• NonPositiveReals = non-positive floating point values
• NegativeReals = strictly negative floating point values
• NonNegativeReals = non-negative floating point values
• PercentFraction = floating point values in the interval [0,1]
• UnitInterval = alias for PercentFraction
• Integers = integer values
• PositiveIntegers = positive integer values
• NonPositiveIntegers = non-positive integer values
• NegativeIntegers = negative integer values
• NonNegativeIntegers = non-negative integer values
• Boolean = Boolean values, which can be represented as False/True, 0/1, ’False’/’True’ and ’F’/’T’
• Binary = the integers {0, 1}

For example, if the set model.O is declared to be within the virtual set NegativeIntegers then an attempt to add anything other than a negative integer will result in an error. Here is the declaration:

model.O = pyo.Set(within=pyo.NegativeIntegers)


## Sparse Index Sets¶

Sets provide indexes for parameters, variables and other sets. Index set issues are important for modelers in part because of efficiency considerations, but primarily because the right choice of index sets can result in very natural formulations that are conducive to understanding and maintenance. Pyomo leverages Python to provide a rich collection of options for index set creation and use.

The choice of how to represent indexes often depends on the application and the nature of the instance data that are expected. To illustrate some of the options and issues, we will consider problems involving networks. In many network applications, it is useful to declare a set of nodes, such as

model.Nodes = pyo.Set()


and then a set of arcs can be created with reference to the nodes.

Consider the following simple version of minimum cost flow problem:

\begin{array}{lll} \mbox{minimize} & \sum_{a \in \mathcal{A}} c_{a}x_{a} \\ \mbox{subject to:} & S_{n} + \sum_{(i,n) \in \mathcal{A}}x_{(i,n)} & \\ & -D_{n} - \sum_{(n,j) \in \mathcal{A}}x_{(n,j)} & n \in \mathcal{N} \\ & x_{a} \geq 0, & a \in \mathcal{A} \end{array}

where

• Set: Nodes $$\equiv \mathcal{N}$$
• Set: Arcs $$\equiv \mathcal{A} \subseteq \mathcal{N} \times \mathcal{N}$$
• Var: Flow on arc $$(i,j)$$ $$\equiv x_{i,j},\; (i,j) \in \mathcal{A}$$
• Param: Flow Cost on arc $$(i,j)$$ $$\equiv c_{i,j},\; (i,j) \in \mathcal{A}$$
• Param: Demand at node latexmath:i $$\equiv D_{i},\; i \in \mathcal{N}$$
• Param: Supply at node latexmath:i $$\equiv S_{i},\; i \in \mathcal{N}$$

In the simplest case, the arcs can just be the cross product of the nodes, which is accomplished by the definition

model.Arcs = model.Nodes*model.Nodes


that creates a set with two dimensional members. For applications where all nodes are always connected to all other nodes this may suffice. However, issues can arise when the network is not fully dense. For example, the burden of avoiding flow on arcs that do not exist falls on the data file where high-enough costs must be provided for those arcs. Such a scheme is not very elegant or robust.

For many network flow applications, it might be better to declare the arcs using

model.Arcs = pyo.Set(dimen=2)


or

model.Arcs = pyo.Set(within=model.Nodes*model.Nodes)


where the difference is that the first version will provide error checking as data is assigned to the set elements. This would enable specification of a sparse network in a natural way. But this results in a need to change the FlowBalance constraint because as it was written in the simple example, it sums over the entire set of nodes for each node. One way to remedy this is to sum only over the members of the set model.arcs as in

def FlowBalance_rule(m, node):
return m.Supply[node] \
+ sum(m.Flow[i, node] for i in m.Nodes if (i,node) in m.Arcs) \
- m.Demand[node] \
- sum(m.Flow[node, j] for j in m.Nodes if (j,node) in m.Arcs) \
== 0


This will be OK unless the number of nodes becomes very large for a sparse network, then the time to generate this constraint might become an issue (admittely, only for very large networks, but such networks do exist).

Another method, which comes in handy in many network applications, is to have a set for each node that contain the nodes at the other end of arcs going to the node at hand and another set giving the nodes on out-going arcs. If these sets are called model.NodesIn and model.NodesOut respectively, then the flow balance rule can be re-written as

def FlowBalance_rule(m, node):
return m.Supply[node] \
+ sum(m.Flow[i, node] for i in m.NodesIn[node]) \
- m.Demand[node] \
- sum(m.Flow[node, j] for j in m.NodesOut[node]) \
== 0


The data for NodesIn and NodesOut could be added to the input file, and this may be the most efficient option.

For all but the largest networks, rather than reading Arcs, NodesIn and NodesOut from a data file, it might be more elegant to read only Arcs from a data file and declare model.NodesIn with an initialize option specifying the creation as follows:

def NodesIn_init(m, node):
for i, j in m.Arcs:
if j == node:
yield i
model.NodesIn = pyo.Set(model.Nodes, initialize=NodesIn_init)


with a similar definition for model.NodesOut. This code creates a list of sets for NodesIn, one set of nodes for each node. The full model is:

import pyomo.environ as pyo

model = pyo.AbstractModel()

model.Nodes = pyo.Set()
model.Arcs = pyo.Set(dimen=2)

def NodesOut_init(m, node):
for i, j in m.Arcs:
if i == node:
yield j
model.NodesOut = pyo.Set(model.Nodes, initialize=NodesOut_init)

def NodesIn_init(m, node):
for i, j in m.Arcs:
if j == node:
yield i
model.NodesIn = pyo.Set(model.Nodes, initialize=NodesIn_init)

model.Flow = pyo.Var(model.Arcs, domain=pyo.NonNegativeReals)
model.FlowCost = pyo.Param(model.Arcs)

model.Demand = pyo.Param(model.Nodes)
model.Supply = pyo.Param(model.Nodes)

def Obj_rule(m):
return pyo.summation(m.FlowCost, m.Flow)
model.Obj = pyo.Objective(rule=Obj_rule, sense=pyo.minimize)

def FlowBalance_rule(m, node):
return m.Supply[node] \
+ sum(m.Flow[i, node] for i in m.NodesIn[node]) \
- m.Demand[node] \
- sum(m.Flow[node, j] for j in m.NodesOut[node]) \
== 0
model.FlowBalance = pyo.Constraint(model.Nodes, rule=FlowBalance_rule)


for this model, a toy data file (in AMPL “.dat” format) would be:

set Nodes := CityA CityB CityC ;

set Arcs :=
CityA CityB
CityA CityC
CityC CityB
;

param : FlowCost :=
CityA CityB 1.4
CityA CityC 2.7
CityC CityB 1.6
;

param Demand :=
CityA 0
CityB 1
CityC 1
;

param Supply :=
CityA 2
CityB 0
CityC 0
;


This can also be done somewhat more efficiently, and perhaps more clearly, using a BuildAction (for more information, see BuildAction and BuildCheck):

model.NodesOut = pyo.Set(model.Nodes, within=model.Nodes)
model.NodesIn = pyo.Set(model.Nodes, within=model.Nodes)

def Populate_In_and_Out(model):
# loop over the arcs and record the end points
for i, j in model.Arcs:

model.In_n_Out = pyo.BuildAction(rule=Populate_In_and_Out)


### Sparse Index Sets Example¶

One may want to have a constraint that holds

$\forall \; i \in I, k \in K, v \in V_k$

There are many ways to accomplish this, but one good way is to create a set of tuples composed of all model.k, model.V[k] pairs. This can be done as follows:

def kv_init(m):
return ((k,v) for k in m.K for v in m.V[k])
model.KV = pyo.Set(dimen=2, initialize=kv_init)


We can now create the constraint $$x_{i,k,v} \leq a_{i,k}y_i \;\forall\; i \in I, k \in K, v \in V_k$$ with:

model.a = pyo.Param(model.I, model.K, default=1)

model.y = pyo.Var(model.I)
model.x = pyo.Var(model.I, model.KV)

def c1_rule(m, i, k, v):
return m.x[i,k,v] <= m.a[i,k]*m.y[i]
model.c1 = pyo.Constraint(model.I, model.KV, rule=c1_rule)