Optimistic decision making using an

International Journal of Artificial Intelligence & Applications (IJAIA) Vol. 6, No. 2, March 2015
DOI : 10.5121/ijaia.2015.6201 1
OPTIMISTIC DECISION MAKING USING AN
APPROXIMATE GRAPHICAL MODEL
BOUTOUHAMI Khaoula and KHELLAF Faiza
RIIMA, University of Sciences and Technology Houari Boumediene., Algeria
ABSTRACT
Min-based qualitative possibilistic networks are one of the effective tools for a compact representation of
decision problems under uncertainty. The exact approaches for computing decision based on possibilistic
networks are limited by the size of the possibility distributions. Generally, these approaches are based on
possibilistic propagation algorithms. An important step in the computation of the decision is the
transformation of the DAG (Direct Acyclic Graph) into a secondary structure, known as the junction trees
(JT). This transformation is known to be costly and represents a difficult problem. We propose in this paper
a new approximate approach for the computation of decision under uncertainty within possibilistic
networks. The computing of the optimal optimistic decision no longer goes through the junction tree
construction step. Instead, it is performed by calculating the degree of normalization in the moral graph
resulting from the merging of the possibilistic network codifying knowledge of the agent and that codifying
its preferences.
KEYWORDS
Possibilistic decision theory, Min-based possibilistic networks, Moral graph, Data fusion, optimistic
criteria.
1. INTRODUCTION
More and more, problems of decision making under uncertainty [1] [5]take an important place in
Artificial Intelligence (AI) applications. Several decision making tools [2][20]have been
developed to assist decision makers in their tasks: simulation techniques, dynamic
programming[21], logical decision models[7] and graphical decision models (e.g [4] [12][13]
[19]).
Several non-classical theories of uncertainty have been proposed in order to deal with uncertain
and imprecise data such as evidence theory [17], Spohn's ordinal conditional functions and
possibility theory [24] issued from fuzzy sets theory [23].
This paper focuses on graphical decision models which provide efficient decision tools by
allowing a compact representation of decision problems under uncertainty [19]. A decision
problem is a choice between a list of possible alternatives taking into account the knowledge of
an agent (knowledge is sometimes tainted with uncertainties) as well as his/her preferences. The
results of his/her decision are expressed by a set of utilities.
The qualitative possibilistic decision model allows a progressive expression of preferences as
well as knowledge of the decision-maker. This model offers two qualitative criteria of utilities for

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the approach of decision under uncertainty: the pessimistic decision criterion and the optimistic
decision criterion. Interest in the issue of the calculation of qualitative decision continues to grow
and many approaches and models have been proposed [11][16].
In addition to the calculation of decision, the aim of this method is to improve other methods and
overcome their limits as regards the presentation form, the calculation time as well as the ease of
understanding. In our work we focus on graphical decision models that provide effective tools for
decision problems under uncertainty using a compact representation. Several evaluation methods
have been proposed to select the optimal decision. Among these methods, there is an exact
approach based on possibilistic networks spread. This approach requires a transformation of an
original graph into a secondary structure called the junction tree [15] which is used then in
various calculations. In this work, our goal is to propose a new approximate approach to compute
the optimal optimistic decision. Our approach is based on the moral graph associated with the
result of merging the networks representing the agent's beliefs and preferences. This approach has
a polynomial complexity [2]. Indeed, it avoids the transformation of the initial graph into a
junction tree which is known to be an intractable problem (NP-hard). Using the approximate
approach provides very close answers to the exact marginal distributions [2].
The reminder of the paper is organized as follows. The next section briefly recalls the
fundamental concepts of possibility theory and min-based possibilistic networks. The main results
of merging min-based possibilistic networks are also briefly presented in this section. Section 3
describes the new approach and its use in calculating the optimal optimistic decision and section 4
concludes the paper.
2. BACKGROUND
2.1.Basic concepts of possibility theory
Probability theory is the fundamental uncertainty theory used in classical decision theory. Despite
its fame, probability theory presents some limits since it cannot model qualitative uncertainty and
total ignorance is represented by equiprobability which formalizes randomness rather than
ignorance. In order to avoid limits of probability theory, non-classical uncertainty theories have
been developed. Possibility theory was initially proposed by Zadeh [24] and was developed by
Dubois and Prade [8]. This theory offers a suitable framework to handle uncertainty since it
allows the representation of qualitative uncertainty.
This section gives a brief refresher on possibility theory which is issued from fuzzy sets theory
[24], [18] and represents a main approach in dealing with uncertainty. Let ࣰ = ሼAଵ, Aଶ, … , A୬ሽbe
a set of variables. We denote by D୅ = ሼaଵ, … . , a୬ሽthedomain associated with the variable
Aa௜denotes any instance ofA. The universe of discourse is denoted by =××஺೔ ∈ೇ
‫ܦ‬஺, which is
the Cartesian product of all variable domains inࣰ. Each element ω ∈ is calledan interpretation
which represents a possible state of the world. It is denotedby ω = (aଵ, … . , a୬) orω =
(ܽଵ⋀ܽ, … ⋀ܽ௡). Whereሼa୧ |1 ≤ i ≤ nሽare the instancesof the variable A୧.ϕ, ψ denote
propositional formulas (corresponding toevents, i.e., subsets of ) constituted from the variables
inࣰ.
2.1.1. Possibility distribution
A possibility distribution describes knowledge about the unknown value taken by one or several
attributes used to describe states of affairs. For instance it may refer to the age of a man, the size
of a building, the temperature of a room, etc. Here it will refer to the ill-known consequence of a

3
decision. A possibility distribution can represent a state of knowledge (about the state of affairs)
distinguishing what is plausible from what is less plausible, what is the normal course of things
from what is not, what is surprising from what is expected.
The possibility distribution is denoted by π and it is a mapping from the universe of discourse
Ω to a bounded linearly ordered scale L exemplified by the unit interval ሾ0, 1ሿ, i.e. ߨ ∶ Ω →
ሾ0, 1ሿ.
The particularity of the possibilistic scale ‫ܮ‬ is that it can be interpreted in twofold: in an ordinal
manner, i.e. when the possibility degree reflects only an ordering between the possible values and
in a numerical interpretation, i.e. when possibility distributions are related to upper bounds of
imprecise probability distributions.
This distribution encodes available knowledge on real world: ߨ(ω) = 1 means that ω is
completely possible and ߨ(ω) = 0 means that it is impossible to ω to be the real world. A
possibility distribution ߨ is said to be ߙ − ݊‫,݀݁ݖ݈݅݉ݎ݋‬ if its normalization degree ℎ(ߨ) is equal
to ߙ, namely:
ℎ(π) = maxπ(ω) = α (1)
If ߙ = 1, then ϕ is said to be normalized.
2.1.2. Possibility and necessity measures
In probability theory, the quantity ܲ(¬ϕ)is fully determined by ܲ(ϕ)sinceܲ(ϕ) = 1 −
ܲ(¬ϕ).Hence, if ϕisnotprobable, then ¬ϕ is necessarily probable. However, the expression “it is
not possible that ϕ is true” not only implies that “¬ϕ is possible” but it also leads to a stronger
conclusion i.e. “it is necessary that ¬ϕ”.
Moreover, the expression “it is possible that ϕ is true” does not entail anything about the
possibility nor the impossibility ofϕ. Thus, the description of uncertainty about the occurrence of
ϕ needs two dual measures: the possibility measure Π(ϕ)and the necessity measure N(ϕ) = 1 −
Π(¬ϕ ) due to the weak relationship existing between these two quantities.
- Possibility measure: this measure evaluates to what extent ϕ is consistent with our
knowledge. It is given by:
Π(ϕ) = ሼπ(ω): ω ∈ ϕ ሽ(2)
Π(ϕ) is called the possibility degree ofϕ,it corresponds to the possibility to have one of the
models of ψ as the real world[21].The degree of possibility satisfies the following properties:
Π(⊥) = 0.
Π(⊺) = 1.
Π(ϕ) = 0. ϕis impossible.
Π(ϕ) = 1. ϕis completely possible.
Π(ϕ) = Π(¬ϕ) = 1. Φand¬ϕ are totally possible (case of total ignorance).
max (Π(ϕ), Π(¬ϕ)) = 1. Φor¬ϕmust be possible(criterion of normalisation).
- Necessity measure: it is the dual of the possibility measure. The necessity measure evaluates
at which level ϕ iscertainly implied by our knowledge. It is given by:
N(ϕ) = 1 − Π(¬ϕ)(3)

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Necessity measure corresponds to the certainty degree to have one of the models of ϕ as
the real world. This measure evaluates at which level ϕ is certainly implied by our
knowledge represented by π.
The dual relationship between Πand N expresses the fact that more is less certain: becomes
consistent with available beliefs.The degree of necessity has the following properties:
N(ϕ) = 0 ܽ݊݀ N(¬ϕ) = 0ϕis total ignorance.
N(ϕ) = 1 ܽ݊݀ N(¬ϕ) = 0ϕis certainly true.
N(ϕ) ∈ ሿ0,1ሾ ܽ݊݀ N(¬ϕ) = 0. ϕis somewhat certain.
N (ψ ∧ ϕ) = min(N (ψ), N (ϕ)). Conjonction axiom.
N (ψ ∨ ϕ) ≥ max(N (ψ), N (ϕ)). Disjunction axiom.

2.1.3. Possibilistic conditioning
The possibilistic conditioning consists in the revision of our initial knowledge, encoded by a
possibility distributionΠ, after the arrival of a new certain informationϕ ⊆ Ω. The initial
distribution Π is then replaced by another one, denotedΠ′
= Π(. |ϕ). The two interpretations of
the possibilistic scale (qualitative and quantitative) induce two definitions of possibilistic
conditioning [8]. In this paper, we focus on min-based conditioning (qualitative one) defined by:
Π(ω|ϕ) = ൞
1 if π(ω) = Π(ϕ) and ω ⊨ ϕi
π(ω) if π(ω) < ߎ(߶) ܽ݊݀ ߱ ⊨ ߶݅
0 otherwise
(4)
The distribution of possibilistic conditioning as defined above means that ifω falsifies, then it
becomes completely impossible. Otherwise, we distinguish two cases: Ifω is a bettermodel, then
it becomes totally possible otherwise it keeps the same level of ability.
We also use a so-called min-based independence relation, as a non-interactivity relation [23]. This
relation is obtained by using the min-based conditioning Equation 4 and it is defined by:
∀‫,ݔ‬ ‫,ݕ‬ ‫ݔ(ߎ ݖ‬ ∧ ‫)ݖ | ݕ‬ = ݉݅݊(ߎ(‫,)ݖ | ݔ‬ ߎ(‫.))ݖ | ݕ‬ (5)
2.2.Min-based possibilistic network
2.2.1. Preliminaries
There are two ways of knowledge representation: a logical representation and a graphical
representation. In this paper we are interested to the graphical representation. It is qualitative
network. A possibilistic network is an adaptation of the probabilistic (Bayesian) network, in the
sense where we use the same graphic structure which is the direct acyclic graph (DAG)
A min-based possibilistic network [14] over a set of variables V denoted by Π ‫ܩ‬௠௜௡ =
(‫,ܩ‬ ߨ௠௜௡)is characterized by:
• A graphical component: which is represented by a Directed Acyclic Graph (DAG) where
nodes correspond to variables and arcs represent dependence relations between variables.
• Numerical components: these components quantify different links in the DAG by using
local possibility distributions for each node A in the context of its parents denoted by ܷ஺.
More precisely:
- For every root node ‫ܷ(ܣ‬஺ = ∅), uncertainty is represented by the a priori possibility
degreeߨ(ܽ), for each instance a ∈ D୅, such thatmax ߨ(ܽ) = 1.

5
- For the rest of the nodes ‫ܷ(ܣ‬஺ ≠ ∅), uncertainty is represented by the conditional
possibility degree ߨ(ܽ| ܷ஺), for each instance a ∈ D୅, and ܷ஺ ∈ D୅, such
that maxߨ(ܽ| ܷ஺) = 1, for any ܷ஺.
The set of a priori and conditional possibility degrees induces a unique joint possibility
distribution defined by:
Definition 1: Let Π ‫ܩ‬௠௜௡ = (‫,ܩ‬ ߨୋ) be a min-based possibilistic network. Given the a priori and
conditional possibility distribution, the joint distribution denoted by ߨୋ, is expressed by the
following quantitative chain rule:
πୋ(Aଵ, … . , A୬) = min (Π(A୧| U୅୧)) (6)
Example 1:Let the graphic model of a possibilistic network illustrated in figure 1 and table of
initial opportunities distributions is given by the following table:
Figure 1. Model of a DAG
A
π୅(A)
A B
π୅(A)
A B C π୅(A) A B C π୅(A)
a1 0.1
a2 1.0
a1 b1 0.6
a1 b2 1.0
a2 b1 0.4
a2 b2 1.0
a1 b1 c1 0.5
a1 b1 c2 1.0
a1 b2 c1 1.0
a1 b2 c2 0.2
a2 b1 c1 0.3
a2 b1 c2 1.0
a2 b2 c1 0.0
a2 b2 c2 1.0
Table 1. Initial possibility distributions
A B C π୅(A) A B C π୅(A)
a1 b1 c1 0.5
a1 b1 c2 1.0
a1 b2 c1 1.0
a1 b2 c2 0.2
a2 b1 c1 0.3
a2 b1 c2 1.0
a2 b2 c1 0.0
a2 b2 c2 1.0
Table 2. Distributions de possibilité jointe
2.2.2. Fusion of min-based possibilistic networks
Merging uncertain information [10] is important to exploit complementarities between sources. It
provides thus a global and complete point of view. In this paper, we are interested in conjunctive

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mode which makes sense if all sources are considered as equally and fully reliable. One of the
basic conjunctive operators is the minimum operation (min).
Given two min-based possibilistic networksΠ‫ܩ‬௠௜௡ = (‫,ܩ‬ ߨீ)and Π‫ܩ‬ᇱ
௠௜௡ = (‫ܩ‬ᇱ
, ߨீᇲ),the
result of merging Π‫ܩ‬ and Π‫ܩ‬ᇱ
is the possibilistic network Π‫ܩ‬⨁ = (‫ܩ‬⨁, ߨ⊕)[22], such that:
∀ ω, π⊕(ω) = min (πୋ(ω), πୋᇲ(ω)) (7)
The syntactic counterpart of the fusion of two possibility distributions, associated to two
possibilistic networks, using the min operator is a new min-based possibilistic network whose
definition depends on the union of the two initial ones. In [22], the authors propose two principal
classes for merging min-based possibilistic networks:
• Fusion of two possibilistic networks મࡳ and મࡳᇱ
having the same network
structure. The resulting network Π‫ܩ‬⨁ retains the same structure: ‫ܩ‬⨁ = ‫ܩ‬ = ‫ܩ‬ᇱ
.The
possibility degrees are computed as follows: for each variable A, ߨ⊕(‫ܷ |ܣ‬஺) =
min(ߨீ(‫ܷ |ܣ‬஺), ߨீᇲ(‫ܷ |ܣ‬஺)).
• Fusion of two possibilistic networks મࡳ andમࡳᇱ
withdifferent structures. Two
cases are distinguishable:
- The union of graphs is acyclic. In this case, Π‫ܩ‬ and Π‫ܩ‬ᇱ
are ﬁrst expanded, with
additional variables and links, to ߎ‫ 1ܩ‬ = (‫ܩ‬⊕ = ‫ ܩ‬ ∪ ‫ܩ‬ᇱ
, ߨଵ ) and ߎ‫′ܩ‬ଵ = (‫ܩ‬ ⊕
= ‫ ܩ‬ ∪ ‫, ܩ‬ ߨଵ
ᇱ
) without affecting possibility distribution. The extension process of
a min-based possibilistic network can be performed either by adding variables or
arcs, namely,
Adding variable. The extension of Π‫ ܩ‬ = (G, π) by adding a new variable
X∉ ࣰ provides a new min-based possibilistic network Π‫ܩ‬௑ = (G௑, π௑)
whichinduces a joint possibility distribution πீ೉
, such that :
- GX = G ∪{X},
- The additional node X will represent the total ignorance, namely
∀‫ ݔ‬ ∈ ‫ܦ‬௑, π௑(‫)ݔ‬ = 1,
- The remaining variables preserve the same possibility distributions,
∀ܻ ∈ ܸ, ܻ ≠ ܺ, π௑ (ܻ | ܲ ܽ‫)) ܻ(ݎ‬ = ߨ(ܻ | ܲ ܽ‫.)) ܻ(ݎ‬
Then, ∀߱ ∈ Ω, ߨீ (߱) = ݉ܽ‫ ݔ‬πீ೉
(‫.)߱ݔ‬
Adding link. The extension of Π‫ ܩ‬ = (G, π) by adding a link from X to Y
(ܺ ∉ Par(Y )) provides a new min-based possibilistic network Π‫ܩ‬௅ =
(G௅, π௅) which induces a joint possibility distribution πGL , such that:
∀‫ݕ‬ ∈ ‫ܦ‬ܻ , ‫ݔ‬ ∈ ‫ܦ‬ܺ , ‫) ܻ(ݎܽ݌‬ ∈ ‫ܦ‬௉௔௥(௒ )π‫ܮ‬(‫)ݔ) ܻ(ݎܽ݌ | ݕ‬ = ߨ(‫.)) ܻ(ݎܽ݌ | ݕ‬
∀ܼ, ܼ ≠ ܻ, ∀‫ ݖ‬ ∈ ‫, ܼܦ‬ ‫)ܼ(ݎܽ݌‬ ∈ ‫, )ܼ(ݎܽ ܲܦ‬ π‫ܮ‬ (‫))ܼ(ݎܽ݌ | ݖ‬ = ߨ(‫ .))ܼ(ݎܽ݌ | ݖ‬
Then, ∀ ∈ω Ω, ߨீ (ω) = πீಽ
(ω).
Since, both initial min-based possibilistic networks have the same structure (by
adding nodes and links), then the fusion of same-structure networks is applied.
- The union of graphs is cyclic. In this case, additional variables are added to
eliminate cycles. Indeed, the resulting possibilistic network ΠG ⊕ = (G ⊕ , π ⊕ ) is
obtained by:
-

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Initially, let ‫ܩ‬⨁ = G′
Renaming all variables in G⊕ , such that ∀ܺ௜ ∈ ܸ, rename ܺ௜ to ܺ௜
ᇱ
,
Adding G to ‫ܩ‬⨁ , such that ‫ܩ‬⨁= G ∪ G′,
Linking each old variable ܺ௜ to the new correspondent variable ܺ௜
ᇱ
,
For each old variableܺ݅ , ∀‫ݔ‬௜ ∈ ‫ܦ‬௫௜ , ߨ⊕ (‫ݔ‬௜ | ܲܽ‫ܺ(ݎ‬௜ )) = ߨ(‫ݔ‬௜ | ܲܽ‫ܺ(ݎ‬௜ ))
For each new variable ܺ௜
ߨ(‫ݔ‬௜′| ‫ܺ( ′ݎܽ ܲ ݆ݔ‬௜ )) = ൜
ߨ (‫ݔ‬௜| ܲ ܽ‫ ݅ ݂ܫ )) ݅ܺ( ′ݎ‬ = ݆
0 otherwise
The new conditional distributions relative to the new variables ensure the equivalence
between new and old variables.
For more details on the fusion of possibilistic networks see [22].
3. DECISION MAKING UNDER UNCERTAINTY USING POSSIBILISTIC
NETWORKS
In a problem of decision under uncertainty, knowledge of the decision-maker is generally not
very informative. In other words, the agent does not know the real state of the world, but he
knows only that this state belongs to a finite set of possible states. A decision system is defined by
a finite set of states ܵ = ሼ‫ݏ‬ଵ, ‫ݏ‬ଶ, … . , ‫ݏ‬௡ሽ, a finite set of consequences X, a set of decisions
noted ‫ܦ‬ = ሼ݀ଵ, ݀ଶ, … . , ݀௠ሽ, and a set of preferences among the consequences. Each decision
݀௜ : ܵ → ܺ is a function that associates to every possible state of the world a consequence. The
preferences among the consequences are encoded by the utility function ߭: ܺ → ܷwhere U is
apreferably ordinal scale.
The theory of possibility allows one to express the uncertain knowledge on different states of the
world by using a possibility distribution. Indeed, it allows one to represent uncertain knowledge
by distinguishing what is plausible to what is less plausible. It provides also a suitable mean to
represent preferences on the consequences of decisions in order to distinguish the desirable
consequences from the less desirable ones [12].
The uncertainty on the possible states of the world is represented by a normalized possibility
distribution πthatassociates to a set of state variables a value in the interval [0, 1]. Likewise, the
preferences of the agent are represented by a different possibility distribution ߤ that associates to
a setof consequences a value in an ordinal scale U , represented by the interval [0,1] [12]. We
assume that the uncertainties and preferences are commeasurable[9].
In the context of decision theory under uncertainty proposed by Savage, uncertainty of the agent
is modeled by a probability distribution π on the set of possible states of the world and
itspreferences by a utility function ߤ with real values on the setX of the possible consequences of
his/her actions.
In contrast, in the possibilistic framework, knowledge of the agent is modeled by a normalized
possibilistic distributionπ which is a function from states to a simply ordered scale‫ܮ‬ of
plausibility: for a world ω, ߨ(ω) ∈ ‫ܮ‬ ∶ represents thedegree of likelihood that ωis the real state of
the world. If we consider that the information possessed by the agent on the decision problem is
purely ordinal, it is reasonable to think that not only his/her knowledge can be expressed by a
possibilistic distribution but also his/her preferences [6][16]. A distribution of possibilities can be
then seen as a utility [16] function.

International Journal of Artificial Intelligence &
Let ߤ be the possibility distribution representing the agent’s
simply orderly scale in [0, 1]. As in Savage theory, an action is represented by a
associates to a world an element of X
whose consequence is ݀(ω) ∈ ܺ
the utilities ߤ(݀(ω))in anappropriate manner for all
Two evaluation criteria have been proposed to achieve
commensurability between the scales of plausibility and utility
• Pessimistic criterion (Minimax
decision is that having the largest minimum utility :
U
• Optimistic criterion (Maximin):
decision is that having the largest maximum utility :
U
In this work, we are interested in the optimistic criterion for the calculation of the decision.
Example 1: Let us consider the problem of deciding whether we should or not take an umbrella,
knowing that it would rain. The two min
preferences of the agent are denoted
possibilistic graphs, let us first present the set of nodes used in the networks
- R: It’s raining.
- W: The grass is wet.
- UM: Take the umbrella.
- C: Cloudy atmosphere.
Figure 2. The possibilistic networks of
• Agent’s knowledge: described by the min
(‫ܩ‬௄, ߨ௄), where the graphical c
possible states of the world R, one decision va
The initial possibility distributions associated with
We suppose that the variables are binary
be the possibility distribution representing the agent’s preferences. ߤtakes its values in a
1]. As in Savage theory, an action is represented by a function d that
es to a world an element of X [16].The utility of an action (decision) d in a state
ܺ can be evaluated by combiningthe possibility degrees
in anappropriate manner for all the possible states of world [16].
Two evaluation criteria have been proposed to achieve such combinations assuming some form of
between the scales of plausibility and utility [6][16]:
Pessimistic criterion (Minimax). Criterion of a pessimistic decision maker:
decision is that having the largest minimum utility :
U∗(d) = minmaxன∈ஐ(1 − π୏ୢ(ω), μ(ω)) (8)
Optimistic criterion (Maximin): Criterion of an optimistic decision maker: the chosen
decision is that having the largest maximum utility :
U∗(d) = maxminன∈ஐ(π୏ୢ(ω), μ(ω))
: Let us consider the problem of deciding whether we should or not take an umbrella,
knowing that it would rain. The two min-based possibilistic networks representing knowledge and
preferences of the agent are denoted મࡷ࢓࢏࢔and મࡼ࢓࢏࢔ respectively. Before presenting the
possibilistic graphs, let us first present the set of nodes used in the networks and their
. The possibilistic networks of knowledge and preference of an agent
described by the min-based possibilistic network
, where the graphical component GK is given by Figure 2 (a). It contains one
possible states of the world R, one decision variable UM and two consequences {
The initial possibility distributions associated with Π‫ܭ‬௠௜௡ are given byTables 3
that the variables are binary.
Applications (IJAIA) Vol. 6, No. 2, March 2015
8
takes its values in a
function d that
.The utility of an action (decision) d in a state ωand
can be evaluated by combiningthe possibility degrees ߨ(ω)and
.
such combinations assuming some form of
). Criterion of a pessimistic decision maker: the chosen
Criterion of an optimistic decision maker: the chosen
(9)
: Let us consider the problem of deciding whether we should or not take an umbrella,
based possibilistic networks representing knowledge and
Before presenting the
and their meanings.
based possibilistic networkΠ‫ܭ‬௠௜௡ =
(a). It contains one
UM and two consequences {W, C}.
Tables 3and 4.

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R π୏(R)
r1 0.9
r2 1.0
Table 3. Initial possibility distributions relative to ΠK୫୧୬
W R ߨ௄(ܹ|ܴ) C R ߨ௄(‫)ܴ|ܥ‬ UM C ߨ௄(ܷ‫)ܥ|ܯ‬
w1 r1 0.4
w1 r2 1.0
w2 r1 1.0
w2 r2 0.0
c1 r1 1.0
c1 r2 0.2
c2 r1 0.3
c2 r2 1.0
bio1 of2 0.8
bio2 of1 0
um1 c1 1.0
um1 c2 1.0
um2 c1 1.0
um2 c2 1.0
Table 4. Initial possibility distributions relative to ΠK୫୧୬
• Agent’s preferences: expressed by the min-based possibilistic network ΠP୫୧୬ =
(G୔, μ),, where the graphical component G୔,is given by Figure 2 (b). It contains one
decision variable UM and two consequences {W, C}.The initial possibility distributions
associated with Πܲ௠௜௡are given by Tables 5 and6.
W μ (W) UM μ (UM)
w1 1.0
w2 1.0
um1 1.0
um2 1.0
Table5. Initial possibility distributions relative to ΠP୫୧୬
UM W R μ (UM|W R) UM W R μ (UM|W R)
um1 w1 r1 1.0
um1 w1 r2 1.0
um1 w2 r1 1.0
um1 w2 r2 1.0
um2 w1 r1 1.0
um2 w1 r2 0.0
um2w2 r1 0.8
um2 w2 r2 1.0
Table 6: Initial possibility distributions relative to ΠP୫୧୬
4. ON THE COMPUTATION OF OPTIMAL OPTIMISTIC
DECISIONS BASED ON MIN-BASED FUSION
Given our graphical model for representing decision making under uncertainty, in this section, we
propose an algorithm for dealing with decisions evaluation. In fact, at the semantic level
qualitative possibilistic decisions can be viewed as a data fusion problem of two particular
possibility distributions: one representing agent’s beliefs and the second representing the
qualitative utility.
We recall that agent’s knowledge and preferences are both represented by two separated min-
based possibilistic networks. The first represents agent’s beliefs and the second represents the

qualitative utility. Knowledge and preferences of the agent are both represented by two separated
min-based possibilistic networks, namely
respectively.
In what follows, we propose a method for computing optimal optimistic decisions based on the
fusion of ߨ௄ and ߤ (or Π‫ܭ‬௠௜௡
optimal optimistic decisions, it should be noted that each set of decision d induces a possibility
distribution ߨ௄ௗ in the following way
π
4.1.Describing optimistic decisions as a fusion process
We recall that making a decision comes down to choosing a subset d of the decision set D which
maximizes the optimistic qualitative utility by:
U
Where,
ߨ
Using equation (12), the optimistic utility decision
U
Using technical merging of two min
U
where π⨁(ω) = min (π୏(ω), µ(ω
Example 2: The two DAGs (‫ܩ‬௄
Their union is acycles, the result of merging
min-based possibilistic network
the union of the two graphs of Figure 1 and 2. The resulted min
Π‫ܩ‬⨁induces a unique possibility distribution
based possibilistic networks, namely Π‫ܭ‬௠௜௡ = (‫ܩ‬௄, ߨ௄)and Πܲ௠௜௡
andΠܲ௠௜௡). Before describing a fusion process for computing
in the following way [10]:
π୩ౚ
(ω) = min(π୩ (ω), πୢ (ω)) (10)
Describing optimistic decisions as a fusion process
making a decision comes down to choosing a subset d of the decision set D which
maximizes the optimistic qualitative utility by:
U∗(d) = maxmin
ன∈ஐ
(π୩ౚ
(ω), μ (ω))(11)
ߨௗ (߱) = ቊ
1 ݂݅ ߱ ⊨ ߶݅
0 ‫ݐ݋‬ℎ݁‫݁ݏ݅ݓݎ‬
(12)
), the optimistic utility decision U∗(d)becomes:
U∗(d) = maxmin
ன∈ஐ
(min (π୏(ω), μ(ω)) , πୢ (ω))
Using technical merging of two min-based possibilistic networks, this Equation (13) down to:
U∗(d) = maxmin
ன∈ஐ
(min (π⨁(ω) , πୢ (ω))(14)
(ω)).
௄and ‫ܩ‬௉) given in Example 1, Figure 1 havea different structure
cycles, the result of merging Π‫ܭ‬௠௜௡ = (‫ܩ‬௄, ߨ௄)and Πܲ௠௜௡ = (
based possibilistic network Π‫ܩ‬⨁ = (‫ܩ‬⨁, ߨ⊕ )where G⨁, isgiven in Figure 3
the union of the two graphs of Figure 1 and 2. The resulted min-based possibilistic network
induces a unique possibility distribution ߨ⊕.
Figure3. The DAG G⨁
10
௠௜௡ = (‫ܩ‬௉, ߤ),
Before describing a fusion process for computing
making a decision comes down to choosing a subset d of the decision set D which
) (13)
) down to:
erent structures.
(‫ܩ‬௉, ߤ),is the
3.G⨁is simply
based possibilistic network

11
The initial possibility distributions are given by Tables 7 and 8.
R π୏(R) W R π୏(W|R) C R π୏(C|R)
r1 0.9
r2 1.0
w1 r1 0.4
w1 r2 1.0
w2 r1 1.0
w2 r2 0.0
c1 r1 1.0
c1 r2 0.2
c2 r1 0.3
c2 r2 1.0
Table 7. Initial possibility distributions relative toΠG⨁
UMW C μ (UM|W C) UM W C μ (UM|W C)
um1 w1 c1 1.0
um1 w1 c2 0.0
um1 w2 c1 0.8
um1 w2 c2 1.0
um2 w1 c1 1.0
um2 w1 c2 1.0
um2 w2 c1 1.0
um2 w2 c2 1.0
bio2 of1 0
bio2 of2 1
Table 8.Initial possibility distributions relative toΠG⨁.
4.2.Computing optimal decisions using moral graph
In this section, we present the propagation process based on moral graph to compute optimal
decisions. It is composed of several steps, which progressively get close to exact possibility
degrees (i.e. converges to exact values). The first step consists in transforming the initial
possibilistic graph in to an equivalent undirected graph, called here for simplicity moral graph,
where each node (called cluster) contains a variable from the initial graph and its parents. The
clusters are quantified by local joint possibility distributions instead of the initial conditional
ones. Then, several stability procedures are used in order to guarantee that joint distribution
relative to any cluster is in agreement with those of its adjacent clusters.
Computing the optimistic optimal decisions amounts to find the normalization degree of the
moral graph resulting from the merging of the two possibilistic networks, the first min-based
possibilistic network encodes a joint possibility distribution representing available knowledge and
the second one encodes the qualitative utility, without going through the junction tree.Note that
the construction of the moral graph is done only once and has a polynomial complexity. However,
the simple stabilization procedure, multiple stabilization procedure and initialization (see below)
(which are all three polynomials) are repeated for each decisiond∗
.
4.2.1. Building the moral graph.
The first step is to transform the initial network in to an equivalent secondary structure, called
moral graph for simplicity of notation, and denoted by ℳ࣡. Each node in the moral graph MG is
called a cluster and it is constructed by adding to each node (variable) from the initial network its
parent set. This construction way insures that for any variable A corresponds only one cluster in

ℳ࣡ denoted by C௜. Between any two clusters
edge labeled with a separator, denoted
The construction of the possibilistic moral graph,
follows:
- For each variable Ai ,form a cluster
- For each edge connecting two nodes
between thecluster C௜ and the cluster
intersection.
Example 3: Let us continue with Example 2. We need to compute the optimal optimistic decision
UM = {um1; um2}. First, we start by constructing the moral graph (see Figure4) associated with
the graph ‫ܩ‬⨁ (Figure 3) representing the fusion of
contains four cluster C1 = {R}, C2 = {R, W}, C3= {R, C} and C4 = {R, W, UM} and their
separator S12 = {R}, S12 = {R}, S13 = {R}, S23 = {R}, S24 = {w} and S34 = {c}.
Figure.
4.2.2. Initialization.
For a given decision d, once the moral graph is built,
transformed into local joints in order to quantify it. Namely, for each cluster
local joint distribution relative to its
The quantification is proceeds by taking into account the decision
- For each cluster C௜, (resp.
- For each variable
min( ߨ஼೔
, ߨ⊕(‫ܣ‬௜|ܷ஺)).
- Encode the evidence ‫ܦ‬ =
Λ
- Identify a cluster C௜containing D
Between any two clusters C௜ and C௝ with a non-empty intersection exits an
edge labeled with a separator, denoted by S௜௝, containing the common variables in C௜
The construction of the possibilistic moral graph, noted ℳ࣡,from the initial graph is done as
For each variable Ai ,form a clusterC௜ = ‫ܣ‬௜ ∪ ܷ஺
each edge connecting two nodes ‫ܣ‬௜and ‫ܣ‬௝ : forman undirected edge in the moral graph
and the cluster C௝labeled with a separatorS௜௝corresponding
Let us continue with Example 2. We need to compute the optimal optimistic decision
(Figure 3) representing the fusion of Π‫ܭ‬௠௜௡ and Πܲ௠௜௡. The resulted moral graph
.4. Moral Graph MG of th DAG in Figure 3
For a given decision d, once the moral graph is built, the initial conditional distributions are
transformed into local joints in order to quantify it. Namely, for each cluster C ofℳ࣡
local joint distribution relative to its variables, called potential and denoted πେ୧
୍
.
by taking into account the decision ݀ as follows:
(resp.S௜௝) ߨେ୧
୍
← 1. (resp.S௜௝ ← 1)
variable ‫ܣ‬௜ , choose a cluster C௜containing‫ܣ‬
= ݀௜as likelihood Λ஽(݀):
Λ஽(݀): ቊ
1 ‫ ݀ ݏܽ ݀݁ݐܽ݅ܿ݊ܽݐݏ݊݅ ݏ݅ ܦ‬
0 ‫݀ ݁ݑ݈ܽݒ ܽ ݏܽ ݀݁ݐܽ݅ܿ݊ܽݐݏ݊݅ ݏ݅ ܦ‬ᇱ
≠ ݀
containing D:
12
empty intersection exits an
andC௝.
graph is done as
: forman undirected edge in the moral graph
corresponding to their
Let us continue with Example 2. We need to compute the optimal optimistic decision
. The resulted moral graph
the initial conditional distributions are
࣡, we assign a
‫ܣ‬௜ ∪ ܷ஺ߨ஼೔
←

݀
(16)

13
ߨେ୧
୍
← min( ߨେ୧
୍
, Λ஽). (17)
It is clear that the joint distribution encoded by the initialized moral graph is equivalent to the one
encoded by the initial networks since the local joint distributions in clusters are equal to the initial
local conditional distributions.
Note that Equations 16 and 17 do not appear in standard initialization of moral graph associated
with standard min-based possibilistic networks. It is proper to possibilistic decision problem. By
entering the fact D = d, the moral graph ℳ࣡ encodes π
ߨℳ࣡(ω) = min(π⊕(ω), πௗ(ω)), (18)
whereߨℳ࣡can be redefined from ℳ࣡ as follows:
Definition 1: The joint distribution associated with the moral graph ℳ࣡ is expressed by:
ߨℳ࣡ = min
୧:ଵ,୫
ߨ஼೔
. (19)
where m is the number of clusters in ℳ࣡ .
Then the qualitative utility associated with a decision d is summarized by the following
proposition:
Proposition 1: LetΠ‫ܭ‬௠௜௡ = (‫ܩ‬௄, ߨ௄),be a min-based possibilistic network representing agent’s
beliefs and Πܲ௠௜௡ = (‫ܩ‬௉, ߤ)be a min-based possibilistic network representing agent’s
preferences. Let Π‫ܩ‬⨁ = (‫ܩ‬⨁, ߨ⊕) be the result of merging Π‫ܭ‬௠௜௡ and Πܲ௠௜௡ using the min
operator. Let ℳ࣡, be the moral graph corresponding to Π‫ܩ‬⨁ generated using the above
initialization procedure. Then,
U∗(d) = max
ன∈ஐ
(ℳ࣡(ω)) (20)
Where U∗(d) is given in Equation 14.
Hence, after the initialization step, the moral graph really encodes the possibilistic optimistic
decision. The next step is used to apply the simple stabilization procedure and multiple
stabilization procedure in order to efficiently determine the value of U∗(d).
4.2.3. Simple Stability Procedure.
The simple stabilization procedure ensures that the potential of each clique is in agreement with
that of its neighbors. This procedure is applied through a mechanism of passage of messages
between different cliques. Indeed, each separator collects information from its corresponding
cliques in order to distribute it then to each of them in order to update them.
The potentials of any adjacent clusters C௜ and C௝ (with separator S௜௝) are updated as follows:
• Collect evidence(Update separator) :
ܵ௜௝
௧ାଵ
← min( maxେ୧ ୗ୧୨⁄ ߨେ୧
௧
, maxେ୨ ୗ୧୨⁄ ߨେ୧
୲
) (21)
• Distribute evidence (Update clusters) :

14
ߨ஼௜
௧ାଵ
← min( ߨେ୧
௧
, ߨୗ୧୨
௧ାଵ
) (22)
ߨ஼௝
௧ାଵ
← min( ߨେ୨
௧
, ߨୗ୧୨
௧ାଵ
) (23)
These two steps are repeated until reaching the stability of all clusters. This procedure is defined
as follows:
Definition 2: Let C௜and C௝ be two adjacent clusters in a moral graph ℳ࣡, and let S௜௝ be their
separator. The separator S௜௝is said to be stable if:
maxେ୧ ୗ୧୨⁄ ߨେ୧
୍
= maxେ୨ ୗ୧୨⁄ ߨେ୧
୍
(24)
Where maxେ୧ ୗ୧୨⁄ πେ౟
is the marginal distribution of S௜௝ defined from ߨେ୧
୍
(resp. ߨେ୨
୍
).
A moral graph ℳ࣡ is stable if all its separators are stable.
Proposition 2: Let ℳ࣡ be a stabilized moral graph, let ߨℳ࣡ be the joint distribution encoded by
MG after the initialization procedure. Then,
∀ C௜ , max ߨℳ࣡
ௌ
≥ ߙ (25)
Where,ߙ is the maximum value in all clusters.
- The theoretical complexity.
In this procedure, a separator will be treated if one of its corresponding clusters has been
modified. Moreover, a cluster will be treated if one of its corresponding separators has been
modified. Thus, the moral graph is considered as stable if none of its clusters has been modified.
It can be shown that the simplest ability is reached after a finite number of message passes, which
can be evaluated as follows:
Let ܰ be the number of clusters,ℳbethenumberofseparatorsandܲthenumberofvaluesinthe
possibilistic scale relative to all the clusters. The iteration in the simplest ability procedure is
repeated until there is no modification in the clusters. The maximal number of iterations occurs
when a degree is modified in one cluster during one iteration, thus we can have atmost ܰ ∗
ܲiterations.Each iteration runsܱ(ℳ)timestheCollet-Distribute evidence. Thus, the theoretical
complexity is ܱ(ℳ ∗ ܰ ∗ ܲ)and hence the stability is a polynomial procedure.
Algorithm1: Simple stability procedure
Begin
Whileℳ࣡ is not stable do
For each separator S௜௝do
-Collect evidence in S௜௝ from C௜and C௝
-Distribute evidence
End

15
4.2.4. Multiple Stability Procedure.
[3]Proved that the simple stabilization procedure does not always guarantee accurate marginal.
One needs to stabilize each clique with respect to all of its adjacent cliques but this can turns out
to be very costly in terms of calculation if the number of cliques is important. For that, [3] has
proposed to follows several steps in stabilizing the possibilistic moral graph over subsets of its
adjacent cliques. Authors of [3] have proposed several progress ivestabilization procedures based
on n parents, n children, n parents children and n neighbors by varying the value of n from
2 up to the cardinality of the considered subset. To illustrate the multiple stabilization procedure,
we consider the case of two parent’s stabilization. The principle of this procedure is to ensure for
each clique, with at least two parents, its stabilization over each pair of parents. The updating of
any cluster C௜ with respect to two of its parents C௝ and C௞ is performed as follows:
• Compute the potential of C௜ using C௝ and C௞:
ߨ஼ ← min( ߨେ୨
௧
, ߨେ୩
௧
)(26)
• Distribute the potential of ܵ using ‫:ܥ‬
ߨௌ ← max ߨ஼
஼ೄ
(27)
• Compute the potential of C௜ using C௝ and C௞:
ߨେ୧
௧ାଵ
← min( ߨେ୧
௧
, ߨௌ)
(28)
Once stability has been reached, the calculation of qualitative utility over a decision d will be
obtained as follows:
Proposition 3. Let Π‫ܭ‬௠௜௡ = (‫ܩ‬௄, ߨ௄)be a min-based possibilistic network representing agent’s
beliefs and Πܲ௠௜௡ = (‫ܩ‬௉, ߤ)a min-based possibilistic network representing agent’s preferences.
Π‫ܩ‬⨁is the result of merging of Π‫ܭ‬௠௜௡and Πܲ௠௜௡by using the min operator. Let ℳ࣡ be the
moral graph of Π‫ܩ‬⨁. The computation of optimistic decisionsreturns to calculate the
normalization degree of MG:
U∗(d) = max
஼೔
൫ߨ஼೔
൯
(29)
4.3.Algorithm
The computation of the optimal optimistic decisions is obtained using the following algorithm.
Algorithm : Computation of optimal optimistic decision
Data: મࡷ࢓࢏࢔ = (ࡳࡷ, ࣊ࡷ): Knowledge possibilistic network
મࡼ࢓࢏࢔ = (ࡳࡼ, ࣆ): Preferences possibilistic network
D = {D1,…,Dn}: Set of decisions,
Result:decisions ,ࣆ∗
Begin:
મࡳ⨁ = (ࡳ⨁, ࣊⊕)Fusion of મࡷ࢓࢏࢔ and મࡼ࢓࢏࢔
गऑ = MoralGraph(મࡳ⨁);
ࣆ∗
← ૙
࢏ ← ૚
‫ ܕܚܗۼ‬ ← ૙
۲‫ ܖܗܑܛܑ܋܍‬ ← ∅
For i = 1...n do
ࡵࡺࡵࢀ(गऑ , ۲࢏)

16
‫ ܕܚܗۼ‬ ← ‫( ۾܁܁‬गऑ , ۲࢏)
‫ ܕܚܗۼ‬ ← ‫( ۾܁ۻ‬गऑ , ۲࢏)
IF‫ ܕܚܗۼ‬ > ߤ∗
Then ۲‫ ܖܗܑܛܑ܋܍‬ ← ۲࢏
ࣆ∗
← ‫ ܕܚܗۼ‬
Else
IF‫ ܕܚܗۼ‬ = ࣆ∗
Then ۲‫ ܖܗܑܛܑ܋܍‬ ← ۲‫ ܖܗܑܛܑ܋܍‬ ∪ ۲࢏
EndIf
EndIf
Endfor
Return<Decision >
End
The result of the fusion steps of the two initial min-based networks Π‫ܭ‬௠௜௡and Πܲ௠௜௡is the min-
based networkΠ‫ܩ‬⨁ = (‫ܩ‬⨁, ߨ⊕). The construction of the moral graph ℳ࣡ associated with the
resulted fusion min-based network Π‫ܩ‬⨁ = (‫ܩ‬⨁, ߨ⊕)is ensured by calling the function moral
graph MoralGraph(ࢰࡳ⨁ ). In addition, the function Init(गऑ ; d) corresponds to the
initialization step. Similarly, the procedures‫(۾܁܁‬गऑ , ۲࢏)corresponds to the simple stabilization
procedure,MSP (ℳ࣡ , D௜) corresponds to the multiple stabilization and returns a normalization
degree relative to ℳ࣡ .
As it was already stated, the construction of the moral graph is only done once but the
initialization, the simple stabilization procedure and the multiple stability procedure step sare
repeated for each decision. More precisely, for each decision di= {d1,…,dn} a call to the
initialization, the simple stabilization procedure and the multiple stability procedure occurred.
The initialization function Init(गऑ ; d)allows the parameterization of the moral graph by the
decision di.
As for the simple stabilization procedure‫(۾܁܁‬गऑ , ۲࢏) and the multiple stability
procedure‫(۾܁ۻ‬गऑ , ۲࢏), they allows the computation of a normalization degree associated to
the parameterized moral graph. Finally, the algorithm returns optimal decisions, those that
maximize the normalization degree relative to the moral graph.
The interesting feature of our approach is that the decision calculation process has a polynomial
complexity: the complexity of the merge process is polynomial, in particular when the union of
the two graphs is free of cycles. In this case, the complexity of the fusion process is linear with
respect to the number of variables and parameters of the two graphs, as well as the complexity of
two stabilization procedures are polynomial.
Example 4: Let us continue with Example 3. We need to compute the optimal optimistic decision
the graph ‫ܩ‬⨁(Figure 3) representing the fusion of Π‫ܭ‬௠௜௡and Πܲ௠௜௡. The resulted moral graph
contains four cluster C1 = {R},C2 = {R, W}, C3= {R, C}and C4 = {R, W,UM} and their
Then, for each decision value in UM = {um1; um2}, we must run the algorithm in order to
compute the normalization degree associated with the moral graph.

17
ω πℳ࣡ ω πℳ࣡
r1 w1 c1 um1 0.3
r1 w1 c1 um2 0.3
r1 w1 c2 um1 0.0
r1 w1 c2 um2 1.0
r1 w2 c1 um1 0.3
r1 w2 c1 um2 0.3
r1 w2 c2 um1 0.4
r1 w2 c2 um2 0.4
r2 w1 c1 um1 0.0
r2 w1 c1 um2 0.0
r2 w1 c2 um1 0.0
r2 w1 c2 um2 0.0
r2 w2 c1 um1 0.8
r2 w2 c1 um2 0.9
r2 w2 c2 um1 0.2
r2 w2 c2 um2 0.2
Table 9. Joint distribution πℳ࣡
Step 1: UM= um1
In this case, the fact that UM = um1 is encoded as follows:
Λ௎ெ(‫:)1݉ݑ‬ ቊ
1 ܷ‫1݉ݑ ݏܽ ݀݁ݐܽ݅ܿ݊ܽݐݏ݊݅ ݏ݅ ܯ‬
The table 10 represents the joint distribution encoded by ℳ࣡ after the initialization procedure us.
ω πℳ࣡ ω πℳ࣡
r1 w1 c1 um1 0.3
r1 w1 c1 um2 0.0
r1 w1 c2 um1 0.0
r1 w1 c2 um2 0.0
r1 w2 c1 um1 0.3
r1 w2 c1 um2 0.0
r1 w2 c2 um1 0.4
r1 w2 c2 um2 0.0
r2 w1 c1 um1 0.0
r2 w1 c1 um2 0.0
r2 w1 c2 um1 0.0
r2 w1 c2 um2 0.0
r2 w2 c1 um1 0.8
r2 w2 c1 um2 0.0
r2 w2 c2 um1 0.2
r2 w2 c2 um2 0.0
Table 10. Joint distributions πℳ࣡ after the initialization procedure
Once the moral graph is quantified, then the simple stabilization procedure allows us to compute
the normalization degree of the moral graph which corresponds to the normalization degree of
any cluster. Using this procedure, we obtain:
R πେଵ R W πେଶ R C πେଷ W C UM πେସ W C UM πେସ
r1 0.9
r2 0.9
r1 w1 0.4
r1 w2 0.9
r2 w1 0.9
r2 w2 0.0
r1 w1 0.9
r1 w2 0.3
r2 w1 0.2
r2 w2 0.9
w1 c1 um1 0.9
w1 c1 um2 0.0
w1 c2 um1 0.0
w1 c2 um2 0.0
w2 c1 um1 0.8
w2 c1 um2 0.0
w2 c2 um1 0.9
w2 c2 um2 0.0
Table 11. Normalized potentials with UM=um1

18
݉ܽ‫ݔ‬గ஼ଵ = ݉ܽ‫ݔ‬గ஼ଶ = ݉ܽ‫ݔ‬గ஼ଷ = ݉ܽ‫ݔ‬గ஼ସ = 0.9
From the table 8 we can check that: h൫πℳ࣡ ൯ = 0.8 ≠ 0.9, which means that the moral graph is
not consistent. So we must to re-stabilize the moral graph using the multiple.stability procedure.
Using this procedure, we obtain:
R πେଵ R W πେଶ R C πେଷ W C UM πେସ W C UM πେସ
r1 0.8
r2 0.4
r1 w1 0.4
r1 w2 0.3
r2 w1 0.8
r2 w2 0.0
r1 w1 0.4
r1 w2 0.3
r2 w1 0.2
r2 w2 0.8
w1 c1 um1 0.3
w1 c1 um2 0.0
w1 c2 um1 0.0
w1 c2 um2 0.0
w2 c1 um1 0.8
w2 c1 um2 0.0
w2 c2 um1 0.4
w2 c2 um2 0.0
Table 12. Normalized potentials with UM=um1
݉ܽ‫ݔ‬గ஼ଵ = ݉ܽ‫ݔ‬గ஼ଶ = ݉ܽ‫ݔ‬గ஼ଷ = ݉ܽ‫ݔ‬గ஼ସ = 0.8
The normalization degree of the moral graph is: U∗(um1) = 0.8
Step 2: UM= um2
We repeat the same procedure described in the previous step, with:
Λ௎ெ(‫:)2݉ݑ‬ ቊ
The table 13 represents the joint distribution encoded by ℳ࣡ after the initialization procedure us.
ω ߨℳ࣡ ω ߨℳ࣡
r1 w1 c1 um1 0.0
r1 w1 c1 um2 0.3
r1 w1 c2 um1 0.0
r1 w1 c2 um2 1.0
r1 w2 c1 um1 0.0
r1 w2 c1 um2 0.3
r1 w2 c2 um1 0.0
r1 w2 c2 um2 0.4
r2 w1 c1 um1 0.0
r2 w1 c1 um2 0.0
r2 w1 c2 um1 0.0
r2 w1 c2 um2 0.0
r2 w2 c1 um1 0.0
r2 w2 c1 um2 0.9
r2 w2 c2 um1 0.0
r2 w2 c2 um2 0.2
Table 13. Joint distributions πℳ࣡ after the initialization procedure
In the same way, once the moral graph is quantified, using the simple stabilization procedure and
the multiple stabilization procedure, we obtain:
U∗(um2) = ݉ܽ‫ݔ‬గ஼ଵ = ݉ܽ‫ݔ‬గ஼ଶ = ݉ܽ‫ݔ‬గ஼ଷ = ݉ܽ‫ݔ‬గ஼ସ = 1.0
Thus, we can conclude that the optimal optimistic decision is UM =um2 with the maximal
qualitative utility which equals 1.0

19
5. CONCLUSION
In this paper, we proposed a new approximate approach for the computation of the qualitative
possibilistic optimal optimistic decision in a graphical context. Our approach first merges
possibilistic networks associated with available uncertain knowledge and possibilistic networks
associated with agent's preferences. We then showed that computing optimistic decisions comes
down to computing a normalization degree of the moral graph associated to the result graph of
merging agent's beliefs and preferences networks. This allows an efficient computation of optimal
decisions.
This approach allows one to avoid the transformation of the initial graph into a junction tree
which is known as a difficult problem. This approach is interesting when accurate approaches
fail, i.e., when the generation of the local possibility distributions by the standard algorithm is
impossible or takes a too long response time. In such case, our approach provides answers that are
very close to the exact marginal distributions.
As a future work, we plan to apply our solution to deal with the pessimistic decisions for
possibilistic decision problems.
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