Conditional Expressions, Assignments and Equations#

Introduction#

This chapter deals with the way in which conditional assignments, expressions and equations are made in GAMS. The index operations already described are very powerful, but it is necessary to allow for exceptions of one sort or another. For example, heavy trucks may not be able to use a particular route because of a weak bridge, or some sectors in an economy may not produce exportable products. Exceptions such as these may easily be modeled with a logical condition combined with the where operator, a very powerful feature of GAMSPy introduced in this chapter.

The ‘where’ Statement#

The where operator is a very powerful feature in GAMSPy. The general syntax for a conditional expression is:

term.where[logical_condition]

Here, term can be a number, a (indexed) symbol, and also a complex expression. The where operator may be read as under the condition that the following logical_condition evaluates to TRUE.

Consider the following simple condition, where u[i], s[i] and v[i] are parameters with index i:

u[i].where[~ s[i]] = v[i]

Note that the term is the parameter u[i] and the logical condition is the expression ~ s[i]. If the condition is not satisfied, no assignment is made. To make it clear, this conditional assignment may be read as: ‘given that s[i] is not existing, u[i] equals v[i]’.

Logical conditions may take various forms, they are introduced in the next section. Conditional expressions may be used in the context of assignments, indexed operations and equations. These topics are covered in later sections of this chapter.

Note

Logical conditions used with the where operator cannot include variables. However, variable attributes are allowed.

Note

where operator is equivalent to the dollar condition $ in GAMS.

Logical Conditions#

Logical conditions are special expressions that evaluate to a value of either TRUE or FALSE. Logical conditions may be numerical expressions and numerical relations and they may refer to set membership. In the following subsections this is shown in the context of simple conditional assignments with the where operator on the left-hand side.

Numerical Expressions#

Numerical expressions may serve as logical conditions: a result of zero is treated as the logical value FALSE and a non-zero result is treated as the logical value TRUE. The following example illustrates this point.

from gamspy import Container, Set, Parameter, Number
m = Container()

i = Set(m, name="i", records=["i1","i2","i3","i4","i5"])
s = Parameter(m, "s", domain = i,
              records = [["i1", 3],
                         ["i2", 5],
                         ["i3", 6]])
u = Parameter(m, "v", domain = i)

u[i].where[2*s[i]-6] = 7

In [1]: u.records
Out[1]:
     i      value
0   i2        7.0
1   i3        7.0

Here the numerical expression is the logical condition. The numerical expression is zero if a[i] equals 3, and non-zero otherwise. Hence the logical value of the expression is FALSE for a[i] = 3 and TRUE for all other values of a[i]. The assignment is only made if the numerical expression evaluates to TRUE, otherwise no assignment is made.

Note

Values of the extended range arithmetic such as float("inf") are also allowed in logical conditions. If the result of a numerical expression used as a logical condition takes any of these values, the logical value is TRUE.
Observe that gamspy.math() functions are also allowed in logical conditions. If they evaluate to zero, the logical condition is FALSE, otherwise it is TRUE.

Numerical Relational Operators#

Numerical relational operators compare two numerical expressions and return a logical value. Consider the following examples.

from gamspy import Container, Set, Parameter, Number
m = Container()

i = Set(m, name="i", records=["i1","i2","i3","i4","i5"])
s = Parameter(m, "s", domain = i,
              records = [["i1", 3],
                         ["i2", 5],
                         ["i3", 6]])
u = Parameter(m, "v", domain = i)

u[i].where[s[i] >= 5] = u[i] + 10

In [1]: u.records
Out[1]:
     i       value
0   i2        11.0
1   i4        10.0

The assignment u[i].where[s[i] >= 5] = u[i] + 10 depends on whether s[i] is greater or equal to 5. If this is the case, an assignment is made, otherwise not.

Bitwise Operators#

Bitwise operators can be used to combine two or more logical conditions to build complex logical expressions. For example, if several expressions are required to be TRUE simultaneously, they may be connected with the python bitwise operator &. For all available bitwise operators in python read more here. Another way to construct complex logical conditions is by nesting them. For details, see subsection Nested Conditions below.

The following somewhat artificial examples serve as illustration.

from gamspy import Container, Set, Parameter
m = Container()

i = Set(m, name="i", records=["i1","i2","i3","i4","i5"])

s = Parameter(m, "s", domain = i,
              records = [["i1", 3],
                         ["i2", 5],
                         ["i4", 8]])

t = Parameter(m, "t", domain = i,
              records = [["i1", 13],
                         ["i2", 13],
                         ["i3", 13],
                         ["i4", 13]])

u = Parameter(m, "u", domain = i,
              records = [["i2", 1]])

v = Parameter(m, "v", domain = i,
              records = [["i1", 7],
                         ["i3", 2]])

u[i].where[~ s[i]] = v[i]
u[i].where[s[i] & u[i] & t[i]] = s[i]
u[i].where[s[i] | v[i] | t[i]] = 4

Note that there are three conditional assignments for the parameter u. In the first assignment the logical condition is ~ s[i]. This condition holds for all entries of s that are not specified and therefore zero by default: s['i3'] and s['i5']. Hence u['i3'] and u['i5'] are assigned the values of v['i3'] and v['i5'] respectively. The value of v['i3'] is 2 and the value of v['i5'] is zero by default. After the first assignment we have u['i2']=1 and u['i3']=2, all other values of u are zero. Note that the logical condition failed for u['i2'] and therefore its value remained unchanged.

In [1]: u.records
Out[1]:
     i      value
0   i2        1.0
1   i3        2.0

The logical condition in the second assignment is TRUE for those labels of the set i that have non-zero entries in the parameters s, u and t simultaneously. This condition holds only for i2. Therefore u['i2']=s['i2']=5 and all other values of u remain unchanged, resulting in non-zero values only for u['i2'] and u['i3'].

In [2]: u.records
Out[2]:
     i      value
0   i2        5.0
1   i3        2.0

The logical condition in the last assignment evaluates to TRUE for all labels of the set i that have at least one non-zero entry in the parameters s, v and t. This holds for all labels except for i5. Therefore u['i5'] stays zero and all other values of u are changed to 4.

     i      value
 i1        4.0
 i2        4.0
 i3        4.0
 i4        4.0

These examples demonstrate the power of the where operator combined with bitwise operators. Even more complex logical conditions are possible; see subsection Mixed Logical Conditions below for details.

Set Membership and Set Functions#

Apart from numerical and relational expressions, set membership and functions referencing set elements may be used as a logical condition. Consider the following example as illustration for set membership as logical condition.

from gamspy import Container, Set, Parameter
m = Container()

i = Set(m, name="i", records=["i1","i2","i3","i4","i5"])
j = Set(m, name="j", records=["i1","i2","i3"], domain = i)

s = Parameter(m, "s", domain = i,
              records = [["i1", 3],
                         ["i2", 5],
                         ["i3", 11],
                         ["i4", 8],
                         ["i5", 1]])

t = Parameter(m, "t", domain = i)

t[i].where[j[i]] = s[i] + 3

In [1]: t.records
Out[1]:
     i      value
0   i1        6.0
1   i2        8.0
2   i3       14.0

Note that the set j is a subset of the set i and that the parameter t is declared but not defined. The conditional expression t[i].where[j[i]] in the last line restricts the assignment to the members of the subset j since only they satisfy the condition j[i]. The values for t['i4'] and t['i5'] remain unchanged. In this case this means that they are zero (by default). Note that there is an alternative formulation for this type of conditional assignment; for details see subsection Filtering Sets in Assignments below.

Note

Only the membership of subsets and dynamic sets may be used as logical conditions.

The use of set membership as a logical condition is an extremely powerful feature of GAMSPy, see section Conditional Equations below for more examples.

Logical conditions may contain the method sameas or set operators that return particular values depending on the position of elements in sets, the size of sets or the comparison of set elements to each other or text strings. In the following example we have two sets of cities and we want to know how many of them feature in both sets.

from gamspy import Container, Set, Parameter, Sum, Domain
m = Container()

i = Set(m, name="i", records=["Beijing","Calcutta","Mumbai","Sydney","Johannesburg","Cairo "])
j = Set(m, name="j", records=["Rome","Paris","Boston","Cairo","Munich","Calcutta","Barcelona "])

b = Parameter(m, "b")

b[...] = Sum(Domain(i,j).where[i.sameAs(j)],1)

In the assignment statement we Sum over both sets and we use sameAs to restrict the domain of the indexed operation to those label combinations (i,j) where sameAs evaluates to TRUE. Thus only identical elements are counted.

The operators ord and card are frequently used to single out the first or last element of an ordered set. For example, we may want to fix a variable for the first and last elements of a set:

from gamspy import Container, Set, Variable, Ord, Card
m = Container()

i = Set(m, name="i", records=["Beijing","Calcutta","Mumbai","Sydney","Johannesburg","Cairo"])
j = Set(m, name="j", records=["Rome","Paris","Boston","Cairo","Munich","Calcutta","Barcelona"])

x = Variable(m, "x", domain=[i])

x.fx[i].where[Ord(i) == 1]       = 3
x.fx[i].where[Ord(i) == Card(i)] = 7

In the first assignment the variable x is fixed for the first element of the set i and in the second assignment x is fixed for the final element of i.

Note

As an alternative to the formulation above, one could also use the set attributes first and last to get the same result:

x.fx[i].where[i.first] = 3
x.fx[i].where[i.last]  = 7

Mixed Logical Conditions#

The building blocks introduced in the subsections above may be combined to generate more complex logical conditions. These may contain standard arithmetic operations, numerical relational operations and logical/bitwise operations. All operations, their symbols and their order of precedence are given below. Note that 1 denotes the highest order of precedence and 7 denotes the lowest order of precedence. As usual, the default order of precedence holds only in the absence of parentheses and operators (symbols) on the same level are evaluated from left to right.

Type of Operation	Operation	Operator	Order of precedence
Standard arithmetic operation	Exponentiation, Floor division	**, //	1
Standard arithmetic operation	Multiplication, Division	*, /	2
Standard arithmetic operation	Unary operators: Plus, Minus	+, -	3
Standard arithmetic operation	Binary operators: Addition, Subtraction	+, -	3
Numerical Relational operation	All	<, <=, ==, !=, >=, >	4
Logical operation	Negation	~	5
Logical operation	Logical Conjunction	&	6
Logical operation	All other logical operations	\|, ^, <<, >>	7

Note

We recommend to use parentheses rather than relying on the order of precedence of operators. Parentheses prevent errors and make the intention clear.

Consider the following example:

x - 5*y & z - 5
(x - (5*y)) & (z-5)

These two complex logical conditions are equivalent. However, the parentheses make the second expression easier to understand.

Some simple examples of complex logical conditions, their numerical values and their logical values are given below.

Logical Condition	Numerical Value	Logical Value
(1 < 2) + (3 < 4)	2	TRUE
(2 < 1) & (3 < 4)	0	FALSE
(4*5 - 3) + (10/8)	18.25	TRUE
(4*5 - 3) \| (10 - 8)	1	TRUE
(4 & 5) + (2*3 <= 6)	2	TRUE
(4 & 0) + (2*3 < 6)	0	FALSE

Nested Conditions#

An alternative way to model complex logical conditions is by nesting them. The syntax is:

term.where[logical_condition1.where[logical_condition2.where[...]]]

Note that in nested conditions all succeeding expressions after the where operator must be enclosed in parentheses. The nested expression is equivalent to the following conditional expression that uses the logical operator & instead of the nesting:

term.where[logical_condition1 & logical_condition2 & ...]

Consider the following example.:

from gamspy import Container, Set, Parameter
m = Container()

i = Set(m, name="i", records=["i1","i2","i3","i4","i5"])
j = Set(m, name="j", records=["i1","i2","i3"], domain = i)
k = Set(m, name="k", records=["i1","i2"], domain = i)

u = Parameter(m, "u", domain = i)

v = Parameter(m, "v", domain = i,
              records = [["i1", 7],
                         ["i3", 2]])

u[i].where[j[i].where[k[i]]] = v[i]

In [1]: u.records
Out[1]:
     i      value
0   i1        7.0

Note

We recommend to use the logical & operator instead of nesting conditions, because this formulation is easier to read.

Conditional Assignments#

A conditional assignment is an assignment statement with a where condition on the left-hand side or on the right-hand side. Most examples until now were conditional assignments with the where operator on the left.

Warning

he effect of the where condition is significantly different depending on which side of the assignment it is located.

The next two subsections describe the use of the where condition on each side of the assignment. Note that in many cases it may be possible to use either of the two forms of the where condition to describe an assignment. We recommend to choose the clearer formulation.

Note that if the logical condition in an assignment statement refers to set membership, then under certain conditions the restriction may be expressed without the use of the where operator. For details, see section Filtering Sets in Assignments below.

where[] on the Left#

If the where condition is on the left-hand side of an assignment, an assignment is made only in case the logical condition is satisfied. If the logical condition is not satisfied then no assignment is made and the previous content of the parameter on the left will remain unchanged. In case the parameter on the left-hand side of the assignment has not previously been initialized or assigned any values, zeros will be used for any label for which the assignment was suppressed.

Consider the following example. Note that the parameter sig has been previously defined in the model.

rho[i].where[sig[i] <> 0] = (1/sig[i]) - 1

In this assignment rho[i] is calculated and the where condition on the left protects against dividing by zero. If any of the values associated with the parameter sig turns out to be zero, no assignment is made and the previous values of rho[i] remain. As it happens, rho[i] was not previously initialized, and therefore all the labels for which sig[i] is zero will result in a value of zero.

Now recall the convention that non-zero implies TRUE and zero implies FALSE. The assignment above could therefore be written as:

rho[i].where[sig[i]]  =  (1/sig[i]) - 1

In the following examples i is a set and s and t are parameters.

s[i].where[t[i]] = t[i]
s[i].where[(t[i]-1) > 0] = t[i]**0.5

Note that the first assignment is suppressed if the value of the parameter t equals zero. The second assignment is suppressed for values of the parameter t that are smaller or equal to 1.

where[] on the Right#

If the where condition is on the right-hand side of an assignment statement, an assignment will always be made. In case the logical condition is not satisfied the value of zero is assigned. Example:

u[i].where[s[i] >= 5] = 7

Now we move the where condition to the right-hand side:

u[i] = Number(7).where[s[i] >= 5]

This is equivalent to:

if (s[i] >= 5)   then (u[i] = 7),    else (u[i] = 0)

Note that an if-then-else type of construct is implied, but the else operation is predefined and never made explicit. The else could be made explicit with the following formulation:

u[i] = Number(7).where[s[i] >= 5] + Number(0).where[s[i] < 5]

The use of this feature is more apparent in instances when an else condition needs to be made explicit. Consider the next example. The set i is the set of plants, and we are calculating mur[i], the cost of transporting imported raw materials. In some cases a barge trip must be followed by a road trip because the plant is not alongside the river and we must combine the separate costs. The assignment is:

mur[i] = (1.0 + 0.0030 * ied[i,'barge']).where[ied[i,'barge']]
       + (0.5 + 0.0144 * ied[i,'road' ]).where[ied[i,'road' ]]

This means that if the entry in the distance parameter ied is not zero, then the cost of shipping using that link is added to the total cost. If there is no distance entry, there is no contribution to the cost, presumably because that mode is not used.

Consider another example for a conditional assignment with the where operator on the right:

b = Sum(i, t[i]).where[a > 0] + 4

Here a and b are scalars, i is a set and t is a parameter. If the scalar a is positive, the scalar b is assigned the sum of all values of the parameter t plus 4. If a is zero or negative, b becomes just 4. Note that the sum is only computed if the condition holds, this potentially makes the program faster.

Conditional Indexed Operations#

We have seen how exceptions in assignments are modeled with where conditions. where conditions are also used in indexed operations, where they control the domain of operations. This is conceptually similar to the conditional assignment with the where on the left.

Consider the following example adapted from a gas trade model for interrelated gas markets. Here the set i contains supply regions and the parameter supc models supply capacities. The scalar tsupc is computed with the following statement:

tsupc  =  Sum(i.where[supc[i] != float("inf")], supc[i])

This assignment restricts the Sum to the finite values of the parameter supc.

In indexed operations the logical condition is often a set. This set is called the conditional set and assignments are made only for labels that are elements of the conditional set. This concept plays an important role in dynamic sets.

Multi-dimensional sets are introduced in section Multi-Dimensional Sets. In the example used there a two-dimensional set is used to define the mapping between countries and ports. Another typical example for a multi-dimensional set is a set-to-set mapping that defines the relationship between states and regions. This is useful for aggregating data from the state to the regional level. Consider the following example:

from gamspy import Container, Set, Parameter, Sum
import pandas as pd

m = Container()

r = Set(m, name = "r", description = "regions")
s = Set(m, name = "s", description = "states")

c = pd.Series(
    index=pd.MultiIndex.from_tuples([("north", "vermont"),
                                    ("north", "maine"),
                                    ("south", "florida"),
                                    ("south", "texas")])
)

corr = Set(m, name = "corr",
        domain = [r,s],
        uels_on_axes=True,
        domain_forwarding = True,
        records = c)

y = Parameter(m, "y", domain = r, description = "income for each region")
income = Parameter(m, "income", domain = s, description = "income for each state",
                  records = [["florida", 4.5],
                            ["vermont", 4.2],
                            ["texas", 6.4],
                            ["maine", 4.1]])

The set corr links the states to their respective regions, the parameter income is the income of each state. The parameter y is computed with the following assignment statement:

y[r] = Sum(s.where[corr[r,s]], income[s])

The conditional set corr[r,s] restricts the domain of the summation: for each region r the summation over the set s is restricted to the label combinations (r,s) that are elements of the set corr[r,s]. Conceptually, this is analogous to the Boolean value TRUE or the arithmetic value non-zero. The effect is that only the contributions of vermont and maine are included in the total for north, and south is the sum of the incomes from only texas and florida.

In [1]: y.records
Out[1]:
        r   value
0   north     8.3
1   south    10.9

Note that the summation above can also be written as:

y[r] = Sum(s,income[s].where[corr[r,s]])

In this formulation the parameter income is controlled by the conditional set corr instead of the index s. Note that both formulations yield the same result, but the second alternative is more difficult to read.

Note that if the logical condition in the context of indexed operations refers to set membership, then under certain conditions the restriction may be expressed without the use of the where operator. For details, see section Filtering Controlling Indices in Indexed Operations below.

Conditional Equations#

The where operator is also used for exception handling in equations. The next two subsections discuss the two main uses of where operators in the context of equations: in the body of an equation and over the domain of definition.

Dollar Operators within the Algebra of Equations#

A where operator in the algebraic formulation of an equation is analogous to the where on the right of assignments, as presented in section where[] on the Right. Assuming that “the right” means the right of the '=' then the analogy is even closer. As in the context of assignments, an if-else operation is implied. It is used to exclude parts of the definition from some of the generated constraints.

from gamspy import Container, Set, Variable, Equation, Sum
m = Container()

i =  Set(m,
         name = "i",
         description = "sectors",
         records = ["light-ind","food+agr","heavy-ind","services"])
t =  Set(m,
         name = "t",
         domain = i,
         description = "tradables",
         records = ["light-ind","food+agr","heavy-ind"])

x = Variable(m,"x",domain = i, description = "quantity of output")
y = Variable(m,"y",domain = i, description = "final consumption")
e = Variable(m,"e",domain = i, description = "quantity of exports")
n = Variable(m,"n",domain = i, description = "quantity of imports")

mb = Equation(m, "mb", domain = i, description = "material balance")

mb[i] = x[i] >= y[i] + (e[i] - n[i]).where[t[i]]

Note that in the equation definition in the last line, the term (e[i] - m[i]) on the right-hand side of the equation is added only for those elements of the set i that also belong to the subset t[i], so that the element services is excluded.

Further, conditional indexed operations may also feature in expressions in equation definitions. In the following example, note that the set i contains the supply regions, the set j contains the demand regions, and the two-dimensional set ij is the set of feasible links; the variable x denotes the shipment of natural gas and the variable s denotes the regional supply.

sb[i] = Sum(j.where[ij[i,j]), x[i,j])  <=  s[i]

Similar to the assignment example seen before, the conditional set ij[i,j] restricts the domain of the summation: for each supply region i the summation over the demand regions j is restricted to the label combinations (i,j) that are elements of the set of feasible links ij[i,j].

Control over the Domain of Definition#

In case constraints should only be included in the model if particular conditions are met, a where condition in the domain of definition of an equation may be used to model this restriction. Such a where condition is analogous to the where[] control on the left of assignments. Assuming that “the left” means the left of the '=' then the analogy is even closer.

Note

The where control over the domain of definition of equations restricts the number of constraints generated to less than the number implied by the domain of the defining sets.

Consider the following example:

gple[w,wp,te].where[ple[w,wp]] = yw[w,te] - yw[wp,te] <= dpack

Here w, wp and te are sets, ple is a two-dimensional parameter, yw is a variable and dpack is a scalar. Note that the where condition restricts the first two indices of the domain of the equation to those label combinations that have non-zero entries in the two-dimensional parameter ple.

Sometimes the desired restriction of an equation may be achieved either way: through a condition in the algebra or a condition in the domain of definition. Compare the following two lines, where eq1 and eq2 are equations, i and j are sets, b is a scalar, s is a parameter and x is a two-dimensional variable.

eq1[i].where[b] = Sum(j, x[i,j])          >= -s[i]
eq2[i]          = Sum(j, x[i,j]).where[b] >= -s[i].where[b]

In the first line the where condition is in the domain of definition, in the second line the where conditions are in the algebraic formulation of the equation. If b is non-zero, the generated equations eq1 and eq2 will be identical. However, if b is 0, no equation eq1 will be generated, but for each i we will see a trivial equation eq2 of the form 0 >= 0.

Note that if the logical condition in the domain of definition of an equation refers to set membership, then under certain conditions the restriction may be expressed without the use of the where operator. For details, see section Filtering the Domain of Definition below.

Filtering Sets#

If the logical condition refers to set membership, the restriction modeled with a where condition may sometimes be achieved without the where operator. Consider the following statement, where i and j[i] are sets, and u and s are parameters:

u[i].where[j[i]] = s[i]

Note that the assignment is made only for those elements of the set i that are also elements of the subset j. This conditional assignment may be rewritten in a shorter way:

u[j] = s[j]

In this statement the assignment has been filtered through the condition without the where operator by using the subset j as the domain for the parameters u and s. This formulation is cleaner and easier to understand. It is particularly useful in the context of multi-dimensional sets (tuples), and it may be used in assignments, indexed operations and the domain of definition of equations.

Filtering Sets in Assignments#

Suppose we want to compute the transportation cost between local collection sites and regional transportation hubs for a fictional parcel delivery service. We define sets for the collection sites and transportation hubs and a two-dimensional set where the collection sites are matched with their respective hubs:

from gamspy import Container, Set, Parameter, Variable, Equation, Sum
import pandas as pd
m = Container()

i =  Set(m, "i", description = "local collection sites")
j =  Set(m, "j", description = "regional transportation hubs")

c = pd.Series(
    index=pd.MultiIndex.from_tuples([("boston", "newyork"),
                                    ("miami", "atlanta"),
                                    ("houston", "atlanta"),
                                    ("chicago", "detroit"),
                                    ("phoenix", "losangeles")])
)

r =  Set(m, "r", domain = [i,j],
         uels_on_axes=True,
         domain_forwarding = True,
         description = "regional transportation hub for each local collection site",
         records = c)

dist = pd.DataFrame(
    [("miami", "newyork", 1327),
     ("miami", "detroit", 1387),
     ("miami", "losangeles", 2737),
     ("miami", "atlanta", 665),
     ("boston", "newyork", 216),
     ("boston", "detroit", 699),
     ("boston", "losangeles", 3052),
     ("boston", "atlanta", 1068),
     ("chicago", "newyork", 843),
     ("chicago", "detroit", 275),
     ("chicago", "losangeles", 2095),
     ("chicago", "atlanta", 695),
     ("houston", "newyork", 1636),
     ("houston", "detroit", 1337),
     ("houston", "losangeles", 1553),
     ("houston", "atlanta", 814),
     ("phoenix", "newyork", 2459),
     ("phoenix", "detroit", 1977),
     ("phoenix", "losangeles", 398),
     ("phoenix", "atlanta", 1810)],
    columns=["i", "j", "distance in miles"],
)

distance = Parameter(m, "distance", domain = [i,j],
                     description = "distance in miles",
                    records = dist)

shipcost = Parameter(m, "shipcost", domain = [i,j],
                     description = "cost of transporting parcels from a local collection site to a regional hub per unit")

factor = 0.009

shipcost[i,j].where[r[i,j]] = factor*distance[i,j]

In [1]: shipcost.records
Out[1]:
          i          j      value
0    boston    newyork      1.944
1     miami    atlanta      5.985
2   houston    atlanta      7.326
3   chicago    detroit      2.475
4   phoenix losangeles      3.582

The distance between collection sites and transportation hubs is given in the parameter distance. The last line is a conditional assignment for the parameter shipcost. This assignment is only made if the label combination (i,j) is an element of the set r. Note that in each instance the indices i and j appear together. Thus the assignment may be simply written as:

shipcost[r] = factor*distance[r]

Note that the assignment is explicitly restricted to the members of the set r; the where operator is not necessary. Observe that if the indices i or j appear separately in any assignment, the above simplification cannot be made. For example, consider the case where the shipping cost depends not only on the factor and the distance between collection sites and regional hubs, but also on the congestion at the regional hub. We introduce a new parameter congestfac that models the congestion at each regional hub and is indexed only over the set j:

congestfac = Parameter(m, "congestfac", domain = j,
                       description = "congestion factor",
                       records = [["newyork", 1.5],
                                 ["detroit", 0.7],
                                 ["losangeles", 1.2],
                                 ["atlanta", 0.9]])

The new cost of shipment is computed as follows:

shipcost[i,j].where[r[i,j]] = factor * congestfac[j] * distance[i,j]

Note that this conditional assignment cannot be reformulated as:

shipcost[r] = factor * congestfac[j] * distance[r]

In the representation above the index j appears on the right-hand side, but not on the left-hand side. GAMSPy will flag this assignment as an error. However, the following representation will work:

shipcost[r[i,j]] = factor * congestfac[j] * distance[r]

In this formulation the set r is explicitly denoted as a tuple of the sets i and j. The set j may then appear on the right-hand side.

Filtering Controlling Indices in Indexed Operations#

Similarly, the controlling indices in indexed operations may be filtered through the conditional set without the use of the where operator. We continue with the shipping cost example from the last subsection. The total cost of shipment is obtained through the equation that follows. We also include the variable definitions for clarity.

shipped = Variable(m, "shipped", domain = [i,j])
totcost = Variable(m, "totcost")
costequ = Equation(m, "costequ")

costequ = totcost == Sum(Domain(i,j).where[r[i,j]], shipcost[i,j]*shipped[i,j])

Here the variable shipped is the number of parcels shipped from the local collection site i to the regional transportation hub j, and the variable totcost is the total cost of all shipments. Note that she summation in the equation is restricted to the label combinations that are elements of the set r. Alternatively, the equation above may be written as:

costequ = totcost == Sum(r, shipcost[r]*shipped[r])

In this formulation the summation is performed explicitly only over the elements of the set r, no where condition is necessary. However, if the expression in the equation included a term dependent only on index j, then we would have to reformulate differently. Suppose the equation included also the congestion factor congestfac that is indexed only over j:

costequ = totcost == Sum(Domain(i,j).where[r[i,j]], factor*congestfac[j]*distance[i,j]*shipped[i,j])

In this case the equation needs to be simplified in the following way:

costequ = totcost == Sum(r[i,j], factor*congestfac[j]*distance[r]*shipped[r])

Like before, the domain of the indexed operation Sum is the set r. But this time the domain of r has to be named explicitly, so that the parameter congestfac which is indexed only over the set j is permitted in the scope of the indexed operation. Note that this reasoning is analogous to the reasoning for filtering sets in assignments in the subsection above.

Filtering the Domain of Definition#

The rules for filtering sets that we have introduced in subsections Filtering Sets in Assignments and Filtering Controlling Indices in Indexed Operations also apply in the context of equation domains. We continue with the parcel transport example introduced above and add a binary variable bin, the parameter bigM and the equation connect to the model. Recall that shipped[i,j] is a variable and r[i,j] is a set.

bigM = Parameter(m, "bigM",domain = [i,j])
bin = Variable(m, "bin", domain = [i,j], type = "binary")

connect = Equation(m, "connect", domain = [i,j])

connect[i,j].where[r[i,j]] = shipped[i,j] <= bigM[i,j]*bin[i,j]

The where condition restricts the domain of definition of the equation connect to those label combinations of the sets i and j that are elements of the set r. The equation relates the continuous variable shipped[i,j] to the binary variable bin[i,j]. Note that each domain in the quation is the index pair (i,j). So the equation may be simplified as follows:

connect[r] = shipped[r] <= bigM[r]*bin[r]

In this formulation the domain of the equation is explicitly restricted to the members of the set r, without the use of a where condition. Note that if the right-hand side of the equation contained any term that was indexed over i or j separately, then the domain of definition of the equation would have to be simplified as:

connect[r[i,j]]

The reasoning is the same as in the case of assignments and indexed operations.