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the analyst in answering typical managerial what-if
questions, while navigating the cube. We distinguish
between two types of what-if questions:
• Questions related to a system of drill-down equa-
tions. For example, “How is the profit in the year
2010 affected when the profit for a certain product
is changed with one percent in the first quarter in
The Netherlands, c.p.?”
• Questions related to a system of business model
equations. For example, “How is the profit in the
year 2010 for a certain product affected when its
unit price is changed with one additional unit in
the sales model, c.p.?”
This paper is structured as follows. In Subsection
1.1 we discuss related work. In Section 2 we discuss
sensitivity analysis in systems that consist of purely
drill-down equations. In Section 3 we elaborate on
sensitivity analysis in systems that consist of purely
business model equations and mixed systems of equa-
tions. Finally, in Section 4 we draw some conclu-
sions.
1.1 Related Work
The variables, parameter values, and assumptions
of any business or economic model are subject to
change. Sensitivity analysis, generally defined, is
the investigation of these potential changes and their
impacts on conclusions to be drawn from the model
(e.g. (Baird, 1990)). There are many possible appli-
cations of sensitivity analysis, described here within
the categories of decision support, communication,
increased understanding or quantification of the sys-
tem, and model development (Pannell, 1997). There
is a very large literature on procedures and techniques
for sensitivity analysis (Clemson et al., 1995). Two
general classes of techniques for sensitivity analy-
sis are the implicit function theorem (Currier, 2000;
Heckman, 2000) and monotone comparative statics
(Milgrom and Shannon, 1994). These are methods
for characterizing whether an increase in a parame-
ter causes the dependent variable to increase or de-
crease. Historically the implicit function theorem was
used for this purpose and the implicit function theo-
rem not only tells you whether the dependent vari-
able increases or decreases but also the magnitude of
change. In contrast, monotone comparative statics
tells you only “up” or “down”, i.e., it gives an ordi-
nal rather than cardinal answer. In our research, we
focused solely on quantitative what-if analysis within
the multi-dimensional database.
To the best of our knowledge, (Balmin et al.,
2000) and (Lakshmanan et al., 2007) are the only pub-
lished research works that address sensitivity analysis
in OLAP databases in a significant way. In (Balmin
et al., 2000), the authors have developed the SESAME
system for the processing of hypothetical queries. For
this system query algebra operators are proposed that
are suitable for spreadsheet-style what-if computa-
tions. In the system hypothetical queries are mod-
eled as a list of hypothetical modifications on the data
in the fact table. A shortcoming of their approach is
that it lacks a good mathematical underpinning, to de-
cide whether a certain change is allowed in the model
or not, as opposed to our approach. In (Lakshmanan
et al., 2007), a different perspective is taken on what-
if analysis. They focus on what-if analysis related to
changes in dimensions and their hierarchical struc-
ture. However, our focus is on data-driven what-if
scenarios, as opposed to structural ones.
In many OLAP software products, sensitivity
analysis is not possible at the moment. If one wants
to do sensitivity analysis in these products one has
to copy the data to a reporting environment, for ex-
ample MS Excel, to compute manually the impact of
changes in certain cells of the data cube. An excep-
tion is the software product Clickview (Cliqview Cor-
poration, 2017), where a fixed change in a base vari-
able can be induced in a system of additive drill-down
measures, to determine its impact on non-base vari-
ables. The difference with our approach is that we can
induce variable changes in systems of additive and av-
erage drill-down measures and under certain condi-
tions in non-linear systems of business equations. For
this purpose we have designed a prototype application
for sensitivity analysis in MS Excel with Pivot tables,
with additional features implemented in Visual Basic.
2 SENSITIVITY ANALYSIS IN A
SYSTEM OF DRILL-DOWN
EQUATIONS
In this section we investigate the influence of a change
in a measure value of a cell in any OLAP cube, on a
higher level value of the same measure in the aggrega-
tion lattice. Or in formal notation, what is the effect
of changing y(c
0
) to y(c
0
) + δ on a dependent vari-
able y(c) in the upset of c
0
. To solve this consider the
lattice L
0
with top cube C
p
= [i
1
,i
2
,...,i
n
] and base
cube C
q
= [ j
1
, j
2
,..., j
n
]. Notice that L
0
is a sublat-
tice of L and L
0
= {↓ c} ∩ {↑ c
0
}. The values of the
measure y in the cube C
q
are denoted by x(c
0
i
), and
are called the base variables where i = 1,2,...,|C
q
|,
and the values of the measure y in {↑ C
q
} are de-
noted by y(c), and are called the non-base variables.
We distinguish between the original values of a mea-
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