Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol69.Issue1.2001:

Int. J. Production Economics 69 (2001) 23}37

Coherence analysis methods for production systems by
performance aggregation
Yves Ducq*, Bruno Vallespir, Guy Doumeingts
Laboratoire d'Automatique et de Productique, GRAI Group, University Bordeaux 1, 351 Cours de la Libe& ration, 33405 Talence Cedex, France
Received 2 April 1998; accepted 2 May 2000

Abstract
This paper aims at describing methods for the coherence analysis in production systems. This coherence between the
various objectives of the decision system is analysed in order to ensure that the achievement of local performances allows
the achievement of the global ones, assigned to the whole enterprise. The coherence analysis methods are usable for
quantitative objectives, whatever the kinds of performance which is concerned (cost, quality, lead time, #exibility, etc.).
The coherence analysis is based on the aggregation and comparison of expected performances (objectives) between the
various decision levels. Various methods are developed in the frame of the concepts of the GRAI Model. ( 2001
Elsevier Science B.V. All rights reserved.
Keywords: Performance; GRAI Model; Coherence; Objectives; Aggregation

1. Introduction
The improvement of Enterprise Performance in
order to get more bene"ts is a continuous expectation of the enterprise Sta! Management.

Nevertheless, even if these objectives are always
the same for more than 20 years, the means to
achieve them have considerably evolved, due to the
enterprise environment and competition on the one
side and the evolution of production and production management techniques on the other. One of
the main changes is, for the decision makers, to
control their production system more and more
precisely in order to improve continuously its per-

* Corresponding author. Tel.: #33-556842407; fax: #33556846644.
E-mail address: ducq@lap.u-bordeaux.fr (Y. Ducq).

formances, mainly in terms of cost, lead time and
product quality, and also in terms of reactivity and
#exibility [1}5].
This performance, which was yesterday monocriterion, mainly based on cost, is today multicriteria [6,7].
In order to perform this accurate control, the
decision makers must have a coherent objective
system. It means that each decisional level must
contribute to the achievement of global performance of the enterprise de"ned by the top management.

The objective of the research works presented in
this paper is to contribute to a method, based on
the GRAI Model [18], which allows to analyse and
to correct the coherence of an objective system in
order to ensure that the local (i.e. operational)
objectives contribute to the achievement of global
(i.e. strategic) objectives, it means that the local

0925-5273/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 5 4 - 2

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Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

performance achievement contributes to the
achievement of global performance.
Thus, after the slight presentation of the GRAI
Model concepts, the de"nition of the concept of
objective will be given. Then, a typology is presented for the decomposition of &father' objective in

&son' objectives. In a third time, the various
methods to analyse and to correct the coherence
will be presented, each one being adapted to each
kind of decomposition.

2. State of the art of coherence analysis methods
There does not exist much work concerning
coherence analysis methods in the domain of production systems but some methods exists in other
domains. We are going to draw the state of the art
and conclusions on these various methods.
Some domains really deal with this problem.
It is particularly the case of hierarchical planning
methods [9] or arti"cial intelligent domain [10]
with COVADIS or methods like AGLAE [11] or
SACCO [12]. These methods are iterative and use
complex algorithms. The main problem of these
methods is that they require very accurate quantitative data which are not always available in
an enterprise even if performance indicators are
implemented.
In the domain of production system, [13] proposes to analyse the coherence in four steps: validation, static coherence analysis, syntactic coherence

analysis and semantic coherence analysis.
Other domains are concerned with the coherence
problem but they do not solve this problem at all. It
is particularly the case of data base management
systems, work#ow and groupware software systems. Finally, in the new approaches of production
management systems and particularly multi-agent
systems [14], fractal systems [15] or bionic
and holonic systems, the coherence problem is
considered but not solved completely [16].
Therefore, all the methods allow to detect essential concepts which must be taken into account
when analysing the coherence and particularly the
production systems.
The "rst concept is the necessity to have both
a top-down and a bottom-up approach. The "rst

one is to analyse the decomposition of global objectives, the second one to analyse the contribution of
local objectives.
The second concept is that it is necessary to have
the clearest view of the physical (controlled) system
before analysing the coherence and that the coherence analysis must be based on the decomposition

and on the performance of the physical (controlled)
activities.
The third notion that must stay in mind is the
analysis of the relationships between the decisional
activities decomposition and the physical activities
decomposition. Indeed, the coherence of a production system depends on these relationships.
The last notion coming from the state of the art is
the requirement to have a reference model or a reference base when we want to compare heterogeneous elements.
In order to answer to these "ve notions, we
decide to perform our works in the frame of the
GRAI Model concepts.

3. The GRAI Model concepts
The coherence we propose to study is analysed
between objectives of decision centres in the frame
of the GRAI Model [8].
The decisional system is very complex and the
GRAI Model proposes to decompose it in order to
facilitate its modelling.
Two decomposition axes are de"ned (Fig. 1).

A vertical axis linked to the nature of decisions:
The "rst criterion is a temporal one linked with the
classical decomposition by level of decision: strategic, tactical, operational (Fig. 1):
Strategic: the decisions which allow to de"ne the
goals to achieve on one considered horizon,
Tactical: the decisions which allow to set up the
resources and the products on a middle term, and
Operational: decisions which allow to execute the
product transformation activities by the resources
on a short-term horizon.
We assign to each level a temporal characteristic:
f Horizon: The interval of time over which the
decision extends (i.e. remains valid).

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

25

Fig. 1. The global GRAI Model for the enterprise.

f Period: The interval of time after which we reconsider the set of decisions. In such a structure, the
horizon is sliding.
A horizontal axis linked to the nature of decisions:
The second criteria of decomposition is the functional activities decomposition criteria.
For the enterprise, six functions are taken into
account: to manage sales, to manage design, to
manage engineering, to manage manufacturing, to
manage assembling, and to manage delivery (Fig. 1).
There can exist other functions according to the
type of enterprise and the development of reference
model will bring new answer to this question. The
interest of this model is to facilitate the integration
between decisional levels and between functions.
The matrix structure allows to co-ordinate the
functional view (vertical) and the process view
(horizontal) by decisional level (Fig. 1).
Moreover, there is a strong relationship between
the physical system decomposition and the decisional levels, each level controlling a part more or
less aggregated of the physical system.
A decision centre is conceptually de"ned as the

cross between a function and a decision level.
There the two kinds of links between two decision centres: information link (simple arrow) and
decision (hierarchical) link represented by a &deci-

sion frame' (double arrow). Through the decision
frame, a decision centre transmits to another decision
centre the objectives, the decision variables, the
criteria and the constraints that this last decision
centre has to take into account in its decision.
This "rst model for the enterprise can be decomposed more accurately by taking the basic principles of the GRAI Model into account to manage
the synchronisation of activities.
At each decisional level, the performance objective imposes the need to synchronise in time the
product and resource availability to perform the
activity with the highest level of performance.
Thus, there are three basic types of functional
activities: The product management activities, the
planning activities and the resource management
activities (Fig. 2).
In the Fig. 2, the product management activities
are linked with the purpose of the function, which

is: to transform raw materials and components into
"nal products according to the objectives,
constraints, and criteria (optimisation of some
features). We "nd in product management, the
classical functions: to buy, to purchase, to store.
The resource management activities are linked
with the means which transform the material #ow:
human and technical.

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Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

Fig. 2. The global GRAI Model for the control of each function.

The planning activities realise the synchronisation between the two previous activities: a good
management allows the short-term level (operational level) to synchronise the available means and
the products to be transformed.
To these three basic functions, we propose to add
the decisional activities linked to the quality management (linked to products), and to the maintenance management (linked to resources).

We get the grid represented in Fig. 2. This basic
decomposition can be applied to each function of
the global grid of Fig. 1.
By combining Figs. 1 and 2, a general grid of
6 functions]5 columns"30 columns could be
built. In practice, one focuses on a main function
which will depend on the studied domain, the
others being represented by only one column.

4. De5nition of objectives
Based on literature survey [17}19], the following
syntactic and semantic de"nition is chosen [20]:
An objective expresses the intention of going
from an existing performance status to the ex-

Fig. 3. Di!erence between objective and performance indicator
concepts.

pected performance status for the physical system controlled by a decision centre. This objective must be expressed with a verb explaining the
expected trend (i.e. to increase, to decrease, to

maintain2) associated to a considered performance domain (i.e. cost, quality, lead time, #exibility2).
So, each objective is associated to a decision centre
which controls a speci"c activity to make it achieve
a speci"c level of performance.
The Fig. 3 shows the di!erence between an objective and a performance indicator, placed on the
performance axis.

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

5. Graph of a priori decomposition
5.1. Building of the graph
As it is explained above, in the GRAI Model, and
particularly in the GRAI grid, the objectives are
decomposed between decision centres through the
decision frames.
But the decision frames are relationships between decision centres and not directly between
objectives. It means that if one decision centre,
containing two objectives, is related with a decision
frame to another decision centre, containing also
two objectives, it is not possible to know a priori
the respective relationships between objectives.
That is why we propose, "rst, to use a graph of
decomposition to represent the a priori relationships between objectives (Fig. 4). These relationships are elaborated according "rst to the decision
frames and than to the experience of the decision
makers.
Then, the coherence analysis between the objectives aims to verify the relevancy of each decomposition relationship between the objectives of the
graph.
For instance, in Fig. 4, there is a decision frame
between the decision centres (to manage products
} Level 10) MP10 and MP20. Then, the decision
makers determined there is a priori a relationship

27

between O
and O
and no relationship
MP10
MP20v1
between O
and O
. The further analysis
MP10
MP20v2
will be able to con"rm this situation.
5.2. First analysis of the graph of decomposition
The goal of the "rst analysis of the graph is to
detect if there exist objectives which are not decomposed or which are not the decomposition of
another.
In Fig. 4, the objective O
is not decomMR10v2
posed in another objective at the same or at the
upper level of decision. It means that this objective
will not be achieved by acting at the operational
level.
In order to correct this incoherence, two solutions are proposed:
f to add an objective at the upper decision level
(MR20),
f to remove the objective O
. This solution is
MR10v2
dangerous because this objective is the decomposition of O
and its removal can be fatal for
PL10
the achievement of this last objective.

6. Typology of decomposition between objectives
Based on the GRAI Model concepts and on the
de"nition of objectives, the decomposition between
father (i.e. superior) objective and son (i.e. inferior)
objectives can have di!erent forms. The kind of
decomposition between the objectives has a great
in#uence in the way the coherence is analysed.
Starting from the previous de"nition, two classes
of decomposition are de"ned: homogeneous and
heterogeneous decomposition. These two decompositions are explained next.
6.1. Homogeneous decomposition

Fig. 4. The graph of decomposition.

The "rst kind of decomposition concerns one
father objective and one son objective expressed
in the same domain of performance (i.e. expressed
on the same performance axis). This performance
domain must be either the cost, the lead time or
the product quality rate (number of correct
products/total number of products). This three

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Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

Grid). This objective is decomposed in only one son
quantitative objective &To decrease inventory cost
by 5%' at the same level (no. 20) for the function &to
manage products'. So, the performance domain is
the cost for both objectives. The decomposition is
then homogeneous.
6.2. Heterogeneous decomposition
Fig. 5. An example of homogeneous decomposition.

performance domains are called &reference performance domains'. We propose only three reference
domains but others can be de"ned as in [21] or
[22], depending on the enterprise strategy. In fact,
these performance reference domains are the performance domains of the enterprise global objectives.
The main reason for choosing the performance
domains of the global objectives as reference
performance domains is that these objectives
are the most important for the enterprise and all the
lower objectives and actions must be achieved
taking these global objectives into account.
Anyway, the reference performance domains of
the enterprise, even if they are di!erent than the
cost, lead time and quality rate, must be able to be
quanti"ed precisely in order to achieve them.
An example of homogeneous decomposition is
presented in Fig. 5.
In Fig. 5, the qualitative objective at the upper
level (father objective), &to decrease production cost'
is the "rst objective of the decision centre PL20 (i.e.
the decision centre at the cross of the function &to
plan' and the decisional level no. 20 of the GRAI

The heterogeneous decomposition includes two
kinds of decomposition between father and son
objectives:
f either the two objectives are expressed in di!erent performance domains (Fig. 6a),
f or they are expressed in the same performance
domain but this domain is not a performance
reference domain (i.e. cost, lead time, or quality
rate) (Fig. 6b).
In Fig. 6a, the performance domains of father objective &to decrease production cost' is then the cost.
The performance domain of the son objective &to
de"ne several suppliers for speci"c parts' is the
number of suppliers. So, the two performance
domains are di!erent.
In Fig. 6b, the performance domain of father
objective &to adapt the capacity to cover 80% of
load' is the resource utilisation rate. The performance domain of the son objective &to increase the
utilisation rate of human and technical resources
until 86%' is also the utilisation rate. So, they have
the same performance domain but the latter is
di!erent from the three performance reference
domains (i.e. cost, lead time, or quality rate).

Fig. 6. Two cases of heterogeneous decomposition of objectives.

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

29

Then, for each kind of decomposition, we will
present a speci"c method to analyse the coherence
of quantitative objectives.

7. Relationships between the decompositions of
physical and decision systems
As it is de"ned previously, an objective is de"ned
for a decision centre and represents the performance to achieve by the activity controlled by this
centre.
In the GRAI Model, each decision level (10, 20,
30,2), controls a part more or less aggregated of
the physical system. It means that the level 10
controls the physical system in its globality
although the level 20 controls a part less
aggregated of this system (for instance, a speci"c
shop).
Thus, if we consider a father objective, at the level
20, and a son objective at the level 30, there must
exist a decomposition between the activity &mother'
controlled by the level 20 and the activity &daughter' controlled by the level 30 as represented in
Fig. 7.
This conclusion shows that the coherence analysis between objectives depends strongly on the
decomposition of activities they control.
In Fig. 7, the decisional level 20 controls the
activity 20-2 of the physical system. The objectives
of the various decision centres of the level 20 (as for
instance O
) are assigned to this activity. The
PL20v2
performance of the activity 20-2 is measured with
PI
(but there can exist several PIs). In the same
20v2
way, the level 30 controls the detailed activities 30-1
and 30-2. So, the objectives O
and
PL30v1
O
can be assigned to one or both of these
PL30v2
detailed activities. The performance of detailed
activity 30-1 is measured by PI
and that of the
30v1
activity 30-2 by PI
.
30v2
This "gure shows that the coherence between the
objective O
(i.e. the performance to achieve
PL20v2
by the activity 20-2) and the objectives O
(i.e.
PL30v1
the performance to achieve by the activity 30-1) and
O
(i.e. the performance to achieve by
PL30v2
the activity 30-2) depends on the decomposition of
the detailed activities in the global activity. In the
case of Fig. 7, the decomposition is sequential.

Fig. 7. Relationship between the decompositions of physical
and decision systems.

Therefore, it is necessary to ensure that all
considered objectives are assigned to the activities
during the exploitation phase of the manufacturing
system and not during the design phase of this
system

8. Typology of decomposition of physical system
activities
8.1. Typology of decomposition of one activity in two
detailed activities
In fact, the coherence analysis consists "rst
in the aggregation of the performances related
to son objectives, and assigned to detailed
activities, and in comparing the result of
this aggregation to the performance related
to the father objective and assigned to the global
activity.
As mentioned above, the coherence depends on
the decomposition of physical system activities.
But this aggregation depends also on the
performance domain which is considered. In the
following, we will consider "rst only the reference
performance domains, the other domains being
considered especially for the heterogeneous
decomposition.

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Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

Fig. 8. Sequential decomposition.

Fig. 10. AND decomposition.

Fig. 9. OR decomposition.
Fig. 11. Generalised sequential decomposition.

So, three kinds of decomposition between activities are de"ned:
The sequential decomposition (Fig. 8):
In the decomposition of Fig. 8, the product can
change activity only if the previous one is "nished.
Each detailed activity has the following performances:
f for A : C , LT , Qr ,
1 1
1
1
f for A : C , LT , Qr .
2 2
2
2
The OR decomposition (Fig. 9):
In the decomposition of Fig. 9, the product can
be processed either by the activity A or A , the
1
2
result of the transformation being the same, but the
activities have di!erent performances in terms of
cost, lead time and quality rate.
Each detailed activity has the following performances:
f for A : C , LT , Qr ,
1 1
1
1
f for A : C , LT , Qr .
2 2
2
2
The AND decomposition (Fig. 10):
By the decomposition of Fig. 10, the product can
be processed simultaneously by the activities
A and A , the result of the transformation being
1
2
considered after the two processes.
Each detailed activity has the following performances:
f for A : C , LT , Qr ,
1 1
1
1
f for A : C , LT , Qr .
2 2
2
2

8.2. Generalisation of decomposition of one activity
in several detailed activities
The previous typology is interesting when one
global activity is decomposed in two detailed activities. In a lot of cases, one activity is decomposed in
several sequential ones for instance. Nevertheless,
the previous decomposition could be used several
times but in order to save time in the calculus of the
aggregation, we propose to generalise the previous
approach for one activity decomposed in several
activities. The typology is kept and three cases are
then considered.
The generalised sequential decomposition (Fig. 11):
In the decomposition of Fig. 11, the product can
change activity only if the previous one is "nished.
Each detailed activity has the following performances:
f for A : C , LT , Qr ,
1 1
1
1
f for A : C , LT , Qr ,
2 2
2
2
f for A : C , LT , Qr .
p p
p
p
The generalised OR decomposition (Fig. 12):
By the decomposition of Fig. 12, the product can
be processed either by activity A or A or A , the
1
2
p
result of the transformation being the same, but the
activities have di!erent performances in terms of
cost, lead time and quality rate.

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

31

Fig. 12. Generalised OR decomposition.

Fig. 14. Formula to aggregate the detailed performances.

ture and to elaborate the links between the physical
and the decision systems, that is, the links between
the various more or less aggregated activities and
the various decision centres of the GRAI grid.
Fig. 13. Generalised AND decomposition.

Each detailed activity has the following performances:
f for A : C , LT , Qr ,
1 1
1
1
f for A : C , LT , Qr ,
2 2
2
2
f for A : C , LT , Qr .
p p
p
p
The generalised AND decomposition (Fig. 13):
By the decomposition of Fig. 13, the product can
be processed simultaneously with the activities
A , A and A , the result of the transformation
1 2
p
being considered after the various processes.
Each detailed activity has the following performances:
f for A : C , LT , Qr ,
1 1
1
1
f for A : C , LT , Qr ,
2 2
2
2
f for A : C , LT , Qr .
p p
p
p
8.3. Combination of the various decompositions for
the representation of the physical system
Based on the previous decomposition forms, it is
possible to decompose any physical system struc-

9. Coherence analysis for homogeneous
decomposition of objectives
After having elaborated the decomposition
typology for the physical activities, it is necessary
now to de"ne the aggregation formula in order to
calculate the global performance of the global
activity by knowing the detailed performances of
the detailed activities.
These aggregation formulas depend, of course,
on the decomposition but also on the reference
performance domain.
Fig. 14 presents the result of aggregated performance for the mother activity according to the
decomposition form and the reference performance
domain.
It is necessary to mention that the formula given
in Fig. 14 represents the most pessimistic case.
Thus, whatever is the decomposition of the physical activities and then the combination of the previous forms, it is always possible to calculate the
possible performance of the mother global activity
knowing the expected performance for the detailed
activities.

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Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

It is then possible to compare this calculated
global possible performance with the expected
global performance (objective) and then to analyse
the coherence between father and son objectives.
When the reference performance domains (i.e.
those of the global objectives) are di!erent, it is
necessary to establish the same formula for each
new domain.
Nevertheless, the absolute coherence (no di!erence between possible and expected global performance) is not frequent.
That is why it is necessary to de"ne an accepted
di!erence between the two results over which the
objectives are not coherent.
The di!erence is equal to 10% of the performance of the father objective. When the di!erence
will be less than 10%, the father and son objectives
will be considered as coherent.

10. Coherence analysis for heterogeneous
decomposition of objectives
This part aims at contributing to a method to
analyse the coherence of objectives which are
expressed in the same performance domain the
later being di!erent from a reference performance
domain or between objectives which are expressed
in di!erent performance domains. Indeed, the performance domain of objectives can be completely
di!erent according to the decision level. Particularly, at the short-term level, they can be di!erent
from the reference performance domains. First, this
di!erence is due to the perception of the physical
system of the decision makers. Because each decision level has a di!erent view of the physical
system, the decision makers have di!erent criteria
to evaluate the performance of the activities they
control. Moreover, for the same decision level, this
di!erence exists for decision makers of di!erent
functions because they do not control the same
entities (resources, or products, or both).
10.1. Basis of reference domains
10.1.1. Multi-criteria analysis concepts
The basic concept of the multi-criteria analysis is,
when comparing two di!erent things, to express

them on the same basis of criteria and to compare
each expression on this basis.
In the state of the art, there exists a lot of mathematical methods for mono or multi-criteria analysis (constrain optimisation, simplex, etc.).
Nevertheless, for the qualitative problems, these
methods are not adapted because they require too
much information [23].
That is why it was decided to express each objective on the basis of several criteria in order to
compare them. These criteria are in fact performance domains in which each objective will be
expressed.
10.1.2. Choice of the performance domains of the
basis
The choice of the performance domains of the
basis will condition further results of the coherence
analysis. That is why this choice must be relevant.
For the same reasons as previously, these performance domains will be the reference performance
domains used for the coherence analysis in the
homogeneous decomposition (i.e. cost, lead time
and quality rate or other domains of the global
objectives of the enterprise).
10.1.3. Performance vector
Each objective represents a performance to
achieve. So, it can be expressed on a performance
axis as explained previously.
Then, each performance P (e.g. a resource utilis1
ation rate) and then each objective can be expressed
with a vector with coordinates n , n , and n , on
11 12
13
the basis of the three reference performance
domains as presented below:

CD

n
11
[C LT Qr].
P " n
12
1
n
13
10.2. Normalisation coezcient
This part aims at de"ning how to calculate the
various coe$cients n of the previous vector.
ij
These coe$cients represent the relationship between the considered performance P and each of
1
the reference performance.

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

When the performance P will evolve in the
1
sense of the objective, then the various reference
performance domains will also evolve (if n O0).
ij
For instance, the improvement of the utilisation
rate for a resource makes the cost and the lead time
to evolve in relation to this resource.
It is then possible to conclude that the coe$cients n represent the ratio between variation of
ij
the considered performance P and the variation of
1
the reference performance.
It implies that if both performances evolve in
a di!erent direction, then the coe$cient will be
negative.
One can write then
C !C
c
*C
*/*5*!- " .
" &*/!n "
11 *P
P
!P
P
1 &*/!1 */*5*!1
1
It is possible then to explain how to calculate the
various coe$cients.
Indeed, if a decision maker knows that a variation of x% of the performance P makes the
1
reference performance C of y%, to evolve then it is
possible to calculate the coe$cients without knowing the absolute expected performances.
It is possible to demonstrate easily that
yC(0)
.
n "
11 xP (0)
1
In the same way, it is then possible to calculate all
the coe$cients.
For instance, considering a decision centre with
three objectives expressed in three di!erent performances P , P , P , the following matrix, called
1 2 3
&normalisation matrix &or' matrix of coe$cients' is
obtained:

CD C
c

n

DC D

n
n
p
12
13
1
lt " n
n
n
p .
21
22
23
2
qr
n
n
n
p
31
32
33
3
Based on this matrix, one obtains for the domain c@
the relationships
11

33

Fig. 15. Decomposition of one global activity in two detailed
activities.

Fig. 15 shows a global activity A decomposed
into two detailed activities A and A .
1
2
The objectives O
and O
in charge of
MR20v1
MR20v2
the decision centre of the function &to manage resources' at the decision level 20, are assigned to the
controlled activity A.
The objectives O
and O
in charge of
MR30v1
MR30v2
the decision centre of the function &to manage resources' at the decision level 30, are assigned to the
controlled activity A .
1
The objective O
and O
in charge of
MR30v1
MR30v3
the decision centre of the function &to manage resources' at the decision level 30, are assigned to the
controlled activity A .
2
The normalisation matrix for the activity A is
presented in (Fig. 16).
The objectives of the detailed activities are
f O
: to increase the utilisation rate of human
MR30v1
and technical production resources to 88%,
f O
: to keep 10% of the capacity for load,
MR30v2
f O
: to improve the quality control reaching
MR30v3
a quality rate of 90% for activity A .
2
The normalisation matrices for the activities
A and A are presented in Fig. 17.
1
2
Now, it is necessary to de"ne the procedure to
aggregate the detailed performances in order to be
able to compare them with the global ones.

c"+n p .
ij i

10.4. Matrix of contribution %

10.3. Example of normalisation matrix

The previous normalisation matrix shows the
relationships between three objectives of a decision
centre and the reference performances, these relationships being valid inside one decision centre.

An example of controlled activities decomposition is presented in Fig. 15.

34

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

Fig. 16. Normalisation matrix for activity A.

Fig. 17. Normalisation matrices for activities A and A .
1
2

This paragraph shows the contribution of a son
decision centre in the achievement of the father
decision centre objectives.
By considering two decision centres CD1 and
CD2 related by a decision frame, it is shown that
there exists a decomposition between the objectives
of CD1 and CD2 (Fig. 18).
The decision centre CD1 has a performance variation vector:

CD

p
11
p " p
.
1
12
p
13
The decision centre CD2 has a performance variation vector:

CD

p
21
.
p " p
22
2
p
23

Fig. 18. Decision frame and objective decomposition between
two decision centres.

The decision centre CD1 contributes to the improvement of the reference performances through
the matrix N .
1
The decision centre CD2 contributes to the
improvement of the reference performances
through the matrix N .
2
It is important to mention that we keep the
principle that CD2 is not always the only one
decision centre related to CD1 and that it contributes in a certain quantity to the improvement of the

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

35

performance, it means the achievement of the
objectives of CD1.
This partial contribution is formalised below.
Then,

CD CD
CD CD
c
p
1
11
lt "N p
1 12
1
qr
p
1
13
and

c
p
2
21
.
lt "N p
1 22
2
qr
p
2
23
The contribution of a variation of the reference
performances of CD2 &p ' to the variation of
3%&2
reference performance of CD1 &p ' is represented
3%&1
by the contribution matrix % with
2,1
p "% p #C,
3%&1
2,1 3%&2
where C is the contribution to the reference performance of the other decision centres related to CD1.
One can also write

CDC

DC D

c
z
z
z
1
11
12
13
lt " z
z
z
1
21
22
23
qr
z
z
z
1
31
32
33
hggiggj
%

c
2
lt #C.
2
qr
2

2,1

The coe$cients z depend on the decomposition of
ij
the physical activities controlled by CD1 and CD2.
Actually, the coe$cients z are not quanti"ed
ij
values but operators, it means sum & or products
%. These operators can also be Min or Max
operators.
10.5. Operators of the matrix of contribution %
The concepts for the operators of the contribution matrix are similar to those of the homogeneous decomposition.
Indeed, this decomposition depends, as in the
previous case on
f the control links, and then the assignment of
each objective to a part of the physical system,

Fig. 19. Aggregation formula of normalisation coe$cients.

f the kind of decomposition of physical activities.
One could think "rst that the aggregation formula
is the same as previously but one distinction is
important.
Previously, for the homogeneous decomposition,
the formulas are absolute values of performances
expressed in particular units.
In the current case, the coe$cients of normalisation are without unit because they are rates of
variations. That is why the formulas for the homogeneous decomposition are not valid for the contribution matrix %.
By elaborating, for each kind of decomposition
of physical activities, the formula for the performance aggregation, we obtain the formula of
Fig. 19.
The operators in the "gure above are then placed
in the contribution matrix, depending on the decomposition form between the activities controlled
by CD1 and CD2 and on the reference performance.
10.6. Example of performance aggregation and
comparison
Considering the example of Section 10.3, the
contribution matrix concerns sequential decomposition. The result of the aggregation is then presented in Fig. 20.

36

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

Fig. 20. Result of the aggregation in the case of Section 10.3.

As for the homogeneous decomposition, the
absolute coherence does not exist, and we de"ne
an accepted di!erence between the two results
over which the objectives are not coherent.
The di!erence is equal to 10% of the normalisation coe$cients of father objective. When the di!erence will be less than 10%, the father and son
objectives will be de"ned as coherent.
The result of the comparison is presented in
Fig. 21.
In this case, the di!erence being less than 10% of
the father objectives, the son objectives are then
coherent with the father ones.

11. Conclusion
This paper presents a method to analyse the
coherence in the decomposition of quantitative
objectives.
This method is based on the decomposition of
physical activities on which are assigned the various objectives.

Fig. 21. Comparison between the aggregated expected performances of detailed activities (son objectives) with those expected
for the global activity.

This method is valid if the decomposition is
made with homogeneous performance as for heterogeneous performances. For these two kinds of
decompositions, formulas are de"ned to aggregate
the performance related to the son objectives and
criteria of di!erence (10%) are accepted to compare
the result of this aggregation to the performance
related to the father objective.

Y. Ducq et al. / Int. J. Production Economics 69 (2001) 23}37

An example is presented in order to illustrate the
application of the method in the case of heterogeneous decomposition.
This example shows that the coherence analysis
requires many pieces of information concerning the
quantitative expected performances of each controlled activities. This information is often di$cult
to obtain but their de"nition and their collection is
essential in order to de"ne clearly the evolution of
the system.
When the decision makers cannot give precise
quantitative objectives, it is necessary to analyse
also the coherence. So, the next step of our research
work is to elaborate a method for the coherence
analysis of qualitative objectives.

[9]
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[11]
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[13]

[14]

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