Data Conversion of TimeDependent Identifiers
Timedependent data
When you are working with timedependent data, it is usually not sufficient to provide and work with a single fixedtime scale. The following examples serve as an illustration.
Demand data is available in a database on a daybyday basis, but is needed in a mathematical program for each horizon period.
Production quantities are computed per horizon period, but are needed on a daybyday basis.
For all of the above data weekly or monthly overviews are also required.
The procedures Aggregate
and Disaggregate
With the procedures Aggregate
and Disaggregate
you can instruct
AIMMS to perform an aggregation or disaggregation step from one time
scale to another. Both procedures perform the aggregation or
disaggregation of a single identifier in one time scale to another
identifier in a second time scale, given a timetable linking both time
scales and a predefined aggregation type. The syntax is as follows.
Aggregate
(timeslotdata, perioddata, timetable, type[, locus])Disaggregate
(perioddata, timeslotdata, timetable, type[, locus])
Time slot and period data
The identifiers (or identifier slices) passed to the Aggregate
and
Disaggregate
procedures holding the timedependent data must be of
equal dimension. All domain sets in the index domains must coincide,
except for the time domains. These must be consistent with the domain
and range of the specified timetable.
Different conversions
As was mentioned in Introduction, timedependent data can be interpreted as taking place during a period or at a given moment in the period. Calendar data, which takes place during a period, needs to be converted into a periodbased representation by allocating the data values in proportion to the overlap between time slots and horizon periods. On the other hand, calendar data which takes place at a given moment, needs to be converted to a periodbased representation by linearly interpolating the original data values.
Aggregation types
The possible values for the type argument of the Aggregate
and
Disaggregate
procedures are the elements of the predefined set
AggregationTypes
given by:
summation
,average
,maximum
,minimum
, andinterpolation
.
Reverse conversion
All of the above predefined conversion rules are characterized by the following property.
The disaggregation of period data into time slot data, followed by immediate aggregation, will reproduce identical values of the period data.
Aggregation followed by disaggregation does not have this property. Fortunately, as the horizon rolls along, disaggregation followed by aggregation is the essential conversion.
The summation
rule
The conversion rule summation
is the most commonly used
aggregation/disaggregation rule for quantities that take place during
a period. It is appropriate for such typical quantities as production
and arrivals. Data values from a number of consecutive time slots in the
calendar are summed together to form a single value for a multiunit
period in the horizon. The reverse conversion takes place by dividing
the single value equally between the consecutive time slots.
The average
, maximum
, and minimum
rules
The conversion rules average
, maximum
, and minimum
are less
frequently used aggregation/disaggregation rules for quantities that
take place during a period. These rules are appropriate for such
typical quantities as temperature or capacity. Aggregation of data from
a number of consecutive time slots to a single period in the horizon
takes place by considering the average or the maximum or minimum value
over all time slots contained in the period. The reverse conversion
consists of assigning the single value to each time slot contained in
the period.
Illustration of aggregation
this table demonstrates the aggregation and disaggregation taking place for each conversion rule. The conversion operates on a single period consisting of 3 time slots in the calendar.
Conversion rule 
Calendar to horizon 
Horizon to calendar 


3 
1 
2 
3 


6 
1 
1 
1 


2 
3 
3 
3 


3 
3 
3 
3 


1 
3 
3 
3 
Interpolation
The interpolation
rule should be used for all quantities that take
place at a given moment in a period. For the interpolation
rule
you have to specify one additional argument in the Aggregate
and
Disaggregate
procedures, the locus. The locus of the
interpolation
defines at which moment in a periodas a value between
0 and 1the quantity at hand is to be measured. Thus, a locus of 0
means that the quantity is measured at the beginning of every period, a
locus of 1 means that the quantity is measured at the end of every
period, while a locus of 0.5 means that the quantity is measured
midway through the period.
Interpolation for disaggregation
When disaggregating data from periods to time slots, AIMMS interpolates linearly between the respective loci of two subsequent periods. For the outermost periods, AIMMS assigns the last available interpolated value.
Interpolation for aggregation
AIMMS applies a simple rule for the seemingly awkward interpolation of
data from unitlength time slots to variablelength horizon periods. It
will simply take the value associated with the time slot in which the
locus
is contained, and assign it to the period. This simple rule
works well for loci of 0 and 1, which are the most common values.
Illustration of interpolation
this table demonstrates aggregation and disaggregation of a horizon of 3 periods, each consisting of 3 time slots, for loci of 0, 1, and 0.5. The underlined values are the values determined by the reverse conversion.
Locus 
Horizon data 


0 
3 
9 

0 
\(\underline{0}\) 
1 
2 
\(\underline{3}\) 
5 
7 
\(\underline{9}\) 
9 
9 
1 
0 
0 
\(\underline{0}\) 
1 
2 
\(\underline{3}\) 
5 
7 
\(\underline{9}\) 
0.5 
0 
\(\underline{0}\) 
1 
2 
\(\underline{3}\) 
5 
7 
\(\underline{9}\) 
9 
Example
Consider the calendar DailyCalendar
, the horizon ModelPeriods
and the timetable TimeTable
declared in Calendars,
Horizons and Creating Timetables, along with the
identifiers
DailyDemand(d)
,Demand(h)
,DailyStock(d)
, andStock(h)
.
The aggregation of DailyDemand
to Demand
can then be
accomplished by the statement
Aggregate( DailyDemand, Demand, TimeTable, 'summation' );
Assuming that the Stock
is computed at the end of each period, the
disaggregation (by interpolation) to daily values is accomplished by the
statement
Disaggregate( Stock, DailyStock, TimeTable, 'interpolation', locus: 1 );
Userdefined conversions
If your particular aggregation/disaggregation scheme is not covered by
the predefined aggregation types available in AIMMS, it is usually not
too difficult to implement a custom aggregation scheme yourself in
AIMMS. For instance, the aggregation by summation from DailyDemand
to Demand
can be implemented as
Demand(h) := sum( d in TimeTable(h), DailyDemand(d) );
while the associated disaggregation rule becomes the statement
DailyDemand(d) := sum( h  d in TimeTable(h), Demand(h)/Card(TimeTable(per)) );