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DOCUMENT DE TRAVAIL 2004-030

DECOMPOSITION ALGORITHM FOR MULTI-CRITERIA

DECISION TREES FRINI, Anissa GUITOUNI, Adel MARTEL, Jean-Marc

Version originale : Original manuscript : Version original :

ISBN – 2-89524-220-8

Série électronique mise à jour : On-line publication updated : Seria electrónica, puesta al dia

12-2004

Abstract: Making decision under uncertainty in dynamic situations consists on choosing, at each period of time, a decision that maximizes at the end the decision-maker’s outcomes. In several situations, these outcomes should be measured against a set of heterogeneous and conflicting criteria. Such situations could be represented using multi-criteria decision trees. The common approach considered to solve this problem consists on generating a set of non-dominated solutions. But, generating the set of non-dominated solutions becomes a very challenging endeavor for large problems. In this paper, we present a methodology to solve multi-criteria decision trees without generating the set of all non-dominated solutions. The proposed methodology is based on the decomposition principle. We present a decomposition theorem that generalizes the Bellman decomposition principle to the multi-criteria discrete decision trees. We propose then an algorithm that aggregates, for each criterion and period, the evaluations of each action over the decision horizon. Then, for each decision node, the recursive algorithm uses the multi-criteria decision aid (MCDA) method to choose among partial strategies. From one period to another, only partial strategies of best compromise are retained for the following iterations. This paper describes the proposed methodology and illustrates it with an academic example.

Keyword: Multi-Criteria Decision Tree, Decomposition, Best Compromise Strategies, MCDA Characterization.

I. INTRODUCTION

Dynamic and sequential decision-making situations involves that a series of decisions should be made at different periods of time in order to achieve one or several objectives. These decisions are interdependent and are made considering their immediate and anticipated consequences or outcomes. In general, uncertainty in dynamic problems is modeled by probabilistic states of the nature or by random consequences of decisions. Moreover, decisions at each period can be evaluated considering multiple and conflicting criteria. Such situations may represent several real-world applications such as tasks scheduling and resources allocation in military operations, choice of portfolio and management of financial assets, production regulation, investment project, etc. For instance, the management of financial assets is a multi-criteria dynamic decision making problem. At each period of time, a decision about buying or selling of financial securities is made. The states of the nature correspond to market behavior that determines securities performance. The decision criteria are the portfolio expected performance and variance. Resource allocation in military operations is another dynamic and multi-criteria decision-making problem. For example, in naval defense, at each decision period, a commander has to make a defensive action

This work has been funded by Defence R&D Canada – Valcartier.

DECOMPOSITION ALGORITHM FOR MULTI-CRITERIA DECISION TREES

Anissa Frini1, Adel Guitouni1,2 and Jean-Marc Martel1 1. Faculté des sciences de l’administration, Université Laval, Québec (Québec), G1K 7P4, Canada

anissa.frini@fsa.ulaval.ca , adel.guitouni@fsa.ulaval.ca , jean-marc.martel@fsa.ulaval.ca 2. Defense R&D Canada - Valcartier, 2459 Pie-XI Nord,

Val-Bélair, Quebec (Quebec), G3J 1X5 Canada, adel.guitouni@drdc-rddc.gc.ca

2

(missile or gun engagement, etc.) against anti-ship missiles targeting the frigate. These actions should be chosen considering a set of criteria such as the risk, the effectiveness and the optimum use of resources. The states of the nature correspond to the kill assessment results: the anti-ship missile is killed or not killed.

The following aspects characterizing dynamic decision problems under uncertainty are treated in this paper:

• Dynamic aspect. Decision problem is a small size dynamic problem with finite horizon. At each period, a decision is required. The consequences of each decision made at period t can change the characteristics of the problem at period t+1.

• Uncertainty aspect. With dynamic decision problems, uncertainty generally holds. We suppose for this problem that states of nature are defined for each period and are modeled by probabilistic distributions.

• Multi-criteria aspect. Several conflicting criteria intervene in decision process. We suppose that multi-criteria modeling is suitable for this problem and that a preliminary work was made in order to define the set of alternatives, the criteria family and the evaluations at each period. The set of alternatives is discrete (defined in an explicit way) and it is possible to enumerate all its content at each decision period. The criteria family (in its composition or its dimension) remains stable from one period to another. Preference modeling elements (thresholds, relative importance of criteria, value functions, etc.) are also stable. Alternative evaluations are measured by cardinal scales and are supposed to be known a priori at each period.

In this paper, we try to solve multi-criteria decision trees. In particular, this paper answers the following questions:

- Is it possible to generalize a decomposition principle for multi-criteria decision trees? What are the application conditions for this principle to provide best compromise strategies?

- What properties are required to solve dynamic multiple criteria problems facing uncertainty?

This paper is organized as follows. In section II, we present the problem formulation and introduce the notations. Section III reviews previous related works about dynamic methods and multi-criteria decision trees. In section IV, we state the principle of decomposition that will provide strategies of best compromise. The decomposition theorem is shown in section V. Then, in section VI, we characterize some MCDA methods according to decomposition properties and other desirable properties. Finally, section VII presents an illustration of this algorithm using some MCDA methods.

II. DECISION PROBLEM MODELING

A. Decision trees

Decision trees are generally perceived as useful tool to formulate and solve sequential decision problems under uncertainty (Keeney and Raiffa, 1976, von Winterfeld and Edwards, 1986, Lootsma, 1997, Covaliu and Olivier, 1995). They explicitly depict all scenarios of a problem, including timing and events (Covaliu and Olivier, 1995) and give a chronological and fully detailed view of the structure of the decision problem. The order, in which nodes are visited, is the chronological order. At each decision node, decisions are made and at each uncertainty node outcomes are revealed to the decision maker (Bielza and Shenoy, 1999). This explicitness facilitates the understanding and the formulation of the decision problem but contributes in the same time to the exponential growth of the decision tree representation. In fact, decision trees become very large for large-sized problems. The number of terminal nodes often measures decision tree size. This size increases roughly exponentially with the number of variables considered in the problem (Kirkwood, 1993). Thus, representation by decision tree is suitable only for relatively small-size sequential decision problems.

3

Despite these limitations, we have chosen to illustrate the decomposition principle using decision tree representation because it is an important tool for analyzing and communicating sequential decision problems. Similar to single criteria decision tree, the multi-criteria decision tree is composed of decision nodes and uncertainty nodes. Branches emerging from each decision node correspond to feasible immediate actions and branches emerging from uncertainty nodes correspond to outcomes related to specific occurrence of a state of nature (see figure 1). All possible strategies are illustrated on the decision tree.

Figure 1 : Multi-Criteria Decision Tree

B. Notation and Definition

Let D t =d t1 ,…d t

k …d tnt

be the set of decision nodes of period t. For each decision node d tk , let

A tk =a t

k )(1 ,…a tki )( ,…a t

kn )( be the set of alternatives at node k of period t. For each alternative a tki )( , let

E tki, =e t

ki ),(1 …e tkij ),( …e t

kin ),( ) be the set of emerging state of the nature from this alternative. Let f t be a

transition function, let k t and k 1+t the indexes of two successive nodes respectively of period t and t+1.

∀ k t ∈ 1,2…. tn , ∀ )( tki ∈ 1,2….n(k t ) , ∀ ),( tkij ∈ 1,2….n(i, k t )

f t : [ tk , )( tki , )),(( tt kkij ]→ k 1+t = tf [ tk , )( tki , )),(( tt kkij ]∈1,2…. 1+tn

(1)

Each decision node d tk of period t is related to a decision node d 1−t

k of period t-1 by:

d tkt

= d [ ]t

,kk) , j(i , i(k k f t-tt-t-t- ))( 1111

1−

(2)

When there is no confusion between two periods, we note for simplicity k instead of k t . Figure 2 illustrates all notations introduced above.

Let F = Mm ggg ,...,,...,1 be the criteria family. ∀ k ∈ 1,2…. Tn , ∀ i ∈ 1,2….n(k), ∀ j

∈1,2….n(i, k) , let V Tmkij ),,( (or for simplicity V T

mj , ) be the evaluation of branch j according to criteria m

Best compromise strategy

Period T-2

−

−

−

3,2

2,2

1,2

T

T

T

CCC

Period t+1

At+1

Period 1 Period t ……

3,

2,

1,

t

t

t

CCC

AT

3

2

1

T

T

T

CCC

( )

( )( )

AT-2AT-1

Period T-1

−

−

−

3,1

2,1

1,1

T

T

T

CCC

( )

( )( )

( )

……

4

at terminal node. ∀ k ∈ 1,2…. tn , ∀ i ∈ 1,2….n(k), ∀ j∈1,2….n(i, k) , let p tkij ),( (or for simplicity

p tj ) be the probability of state of nature e t

kij ),( . Let MCDA[X] be the result of application of Multi-Criteria Decision Aid (MCDA) method on the set X of discrete alternatives x taking into consideration the criteria family F.

Definition 1: A strategy is defined as the sub-tree describing a decision for each period and for each observable state of the nature. Each strategy is composed of partial strategies.

∀ k∈1,2,…n t , let S tk be the set of partial strategies starting from decision node d t

k of period t. N tk is

the cardinality of S tk . ∀ i∈1,2,.., n(k), let S t

ki )( be the set of partial strategies starting with action a tki )(

of decision node d tk of period t. N t

ki )( is the cardinality of S tki )( .

N tk = Card(S t

k ) = ∑=

)(

1)(

kn

i

tkiN (3)

∀ k∈1,2,…n t , a strategy emerging from node k corresponds to the qth element of S tki )( which is

expressed by:

s )()( qt

ki = a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

(4)

Where a tki )( ∈ A t

k , q∈1, 2,…, N tki )( is the index of partial strategies starting by action a t

ki )( , ⊗ is an

operator that extends a partial strategy by an action and

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

∈ S 1)1,,(

+tikf t × S 1

)2,,(+t

ikf t ×…×

S 1)),(,,(

+tkinikf t is a vector of partial strategies for each state of the nature e t

kij ),( ∀ j ∈ 1,2,… n(i, k). Using

this strategy formulation, the set S tk will be:

S tk =

)(

1

kn

i=∪

tkiN

q

)(

1=∪ a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

=)(

1

kn

i=∪

tkiN

q

)(

1=∪ s )()( q

tki

(5)

5

Definition 2: In a multi-criteria context, where optimal strategy does not exist, we look for best compromise strategies. A best compromise strategy is a strategy that achieves the best compromise between conflicting criteria family. Let us note that more than one strategy could

represent a best compromise.

Figure 2 : Designation of Nodes, Alternatives and States of the Nature in Multi-Criteria Decision Tree

C. Illustration

Let’s recall the academic example presented by Haimes (1998). This example will be reconsidered at section VII to illustrate the decomposition algorithm. The decision problem represents a drugs manufacturer facing a serious situation. An independent study has shown that its most popular drug causes sudden death. In the first period, a test is performed in order to evaluate the quality of the product: good or bad. In the second period, the final results of the complete study are revealed and the product will be

Period t Period t+1

d tkijkikf ttt

t )),(),(,( 1111

−−−−

a tkt )(1

a tki t )(

e tkt ),1(1

e tkt ),1(2

d 1

)2,1,(+t

kf tt

e tkkn tt )),((1

e tkkn tt )),((2

d 1)1)),(,(

+ttnkf t

t

d 1)2),(,(

+ttnkf t

t

e tki t ),(1

e tkin t ),(

.

.

.

.

.

.

.

.

.

.

a tkn t )(

d 1)),(,(

+tjkikf tt

t e t

kij t ),(

.

.

.

.

d 1)1,1,(

+tkf t

t

d 1)1),(,(

+tkikf tt

t

d 1)),(),(,(

+tkinkikf ttt

t

6

declared inherently of good or bad quality. The final results will come out in 9 months. In the mean time, the manufacturer has three options:

• Do nothing and hope that the results of the study are false and the drug is safe. • Run an advertising campaign warning its customers of the danger of taking the drug. • Recall the drug.

The results for expected lives lost and expected cost are given in the decision tree (see figure 3). Table 1 presents all strategies of the tree. Each strategy is composed of an immediate action for period 1 and an action for period 2 if the test result is good and another action if the test result is bad.

Figure 3: Example of Decision Problem Modeling

Recall

D3

D1

D2

(10,0)

(0,0) (0,1000)

0.9950.045

Good

Bad

0.9950.045

Good

Bad

0.9950.045

Good

Bad

(2,0) (2,100)

(9,0) (9,50)

(0,0) (0,1000)

0.2060.794

Good

Bad

0.2060.794

Good

Bad

0.2060.794

Good

Bad

(2,0) (2,100)

(9,0) (9,50)

D5

D4

(2,0) (2,75)

0.9950.045

Good

Bad

0.9950.045

Good

Bad

0.9950.045

Good

Bad

(3,0) (3,50)

(11,0) (11,25)

(2,0) (2,75)

0.2060.794

Good

Bad

0.2060.794

Good

Bad

0.2060.794

Good Bad

(3,0) (3,50)

(11,0) (11,25)

Advertise

Do nothing

Do nothing

Recall

Advertise

Do nothing

Recall

Advertise

Do nothing

Recall

Advertise

Do nothing

Recall

Advertise

Good

Good

Bad

Bad

0.66

0.34

0.66

0.34

7

Table 1: Strategies for the First-Period Node Second-Period decision Evaluation First-

Period decision good Bad Cost

($million) Lives lost

s1 Do nothing Do nothing Do nothing 0 299.66 s2 Do nothing Do nothing Advertise 0.68 56.696 s3 Do nothing Do nothing Recall 3.06 43.198 s4 Do nothing Advertise Do nothing 1.32 272.93 s5 Do nothing Advertise Advertise 2 29.966 s6 Do nothing Advertise Recall 4.38 16.468 s7 Do nothing Recall Do nothing 5.94 271.445 s8 Do nothing Recall Advertise 6.62 28.481 s9 Do nothing Recall Recall 9 14.983 s10 Advertise Do nothing Do nothing 2 22.4745 s11 Advertise Do nothing Advertise 2.34 15.7255 s12 Advertise Do nothing Recall 5.06 8.9765 s13 Advertise Advertise Do nothing 2.66 21.732 s14 Advertise Advertise Advertise 3 14.983 s15 Advertise Advertise Recall 5.72 8.234 s16 Advertise Recall Do nothing 7.94 20.9895 s17 Advertise Recall Advertise 8.28 14.2405 s18 Advertise Recall Recall 11 7.4915 s19 Recall - - 10 0

III. REVIEW OF DYNAMIC DECISION PROBLEM SOLVING APPROACHES

In previous works, most solving methods for dynamic decision problems are based on Bellman’s principle of optimality. In the beginning, determinist and stochastic dynamic programming were developed for the mono-objective mathematical programming problems (Bellman, 1957; Howard, 1960; Bellman and Dreyfus, 1965; Hadley, 1964; Krautkraemer, van Kooten and Young, 1992; Ranatunga, 1995, Bertsekas, 1976, Bertsekas and Shreve 1978, 1995). They consist in decomposing the problem using the Bellman principle of optimality. This principle is the cornerstone upon which dynamic programming is built. Under monotonicity and separability conditions, dynamic programming gives optimal solution. This solution is obtained recursively by folding back (or forward) the periods.

Afterwards, there has been an increasing concern to integrate multiple objectives considerations within dynamic programming, which led to the emergence of multi-objectives dynamic programming (MODP). In this area, the focus has been on formulating necessary conditions for the existence of solutions for multi-objectives dynamic optimization problems (Brown and Strauch, 1965; Chang, 1966; Chyung, 1967; Cunha and Polak, 1967) and on the extension of the principle of optimality to multi-objectives dynamic problems (Yu and Seiford, 1981; Li and Haimes, 1987; Trzaskalik, 1998). Under the conditions of backward separability and backward monotonicity, Li and Haimes (1987) state a principle of optimality for MODP. It is similar to the one adopted in single objective dynamic programming.

Dynamic programming has been applied to search the shortest path in a network with a finite number of states and actions. The algorithm consists in solving functional equations based on Bellman’s principle of optimality. Later research works (Climaco and Martin, 1982; Henig, 1985, 1994; Carraway, Morin and Moskowitz, 1990; White, Stewart and Carraway, 1992) have examined the case of bi-criteria decision problems. However, the two criteria have been aggregated in a single objective under the assumption of the existence of a monotone and continuous utility function. Brown and Strauch (1965) have examined

8

Bellman’s model to find the set of non-dominated paths. They showed that transitivity and strict persistence are sufficient conditions for the equations to generate the required set of non-dominated paths. Strict persistence means that for every node j and every two paths p and q from j to the destination,

qp f ⇒ ),(),( qipi f for every arc ),( ji where f is a strict preference relation.

However, Mitten (1974) and Sobel (1975) showed that in case of weak order (asymmetric and negatively transitive), every solution of Bellman’s equations is an optimal path under condition of weak persistence. Weak persistence means that for every node j and every two paths p and q from j to the

destination, ),(),( qipiqp ff ⇒ for every arc ),( ji where f is a preference relation. Moreover, Henig (1994) showed for multi-attribute path problem that the Bellman principle of optimality is valid under conditions of transitivity and persistence (strict and weak) if and only if the preferences can be represented by a linear (value) function over the attributes.

When the optimal solution doesn’t exist, the common used approach consists to generate the set of non-dominated solutions and to use decision-maker’s preferences to choose among them. For example, Haimes and al. (1990), and Haimes (1998) propose to resolve multi-criteria decision trees by recursion by folding back the tree. The dominance is used at decision nodes to identify the non-dominated partial strategies. The expected value is used at uncertainty nodes to aggregate distributional evaluations into punctual ones. Thus, Haimes’ algorithm generates the non-dominated strategies in the tree. The weakness of this approach is that the non-dominated set may be very large and consequently very hard to generate. In fact, the number of Pareto optimal paths can grow exponentially with the number of nodes (Hansen, 1980). Furthermore, the decision maker is still left with the problem of selecting the best path among the non-dominated ones.

The paper’s objective is to propose resolution methodology for multi-criteria decision trees that will be able to find the set of best compromise solutions without generating all non-dominated solutions. Compared to the generation of non-dominated solutions, the proposed methodology should reduce calculation time as well as cardinality of the solution set. One way to solve multi-criteria decision tree consists on applying a Multi-Criteria Decision Aid (MCDA) method on the set of all possible strategies at the first period. However, difficulties appear as soon as the number of strategies increases. To overcome these difficulties, the proposed methodology combines the use of MCDA method with the concept of decomposition. The MCDA method will be applied recursively at each decision node on the set of partial strategies of best compromise.

The following section focuses on the principle of decomposition proposed in our resolution methodology. This principle is valid (gives the best compromise solutions) under sufficient conditions. A theorem of decomposition, which specifies these conditions, will be stated in section V.

IV. PRINCIPLE OF DECOMPOSITION OF MULTI-CRITERIA DECISION TREES

Definition 3: The principle of decomposition: Each best compromise strategy has the property that all its partial strategies must also constitute best compromise strategies.

When the principle of decomposition is applied, every strategy of best compromise obtained at each decision node d t

k is necessary composed of partial strategies of best compromise. So, only the strategies composed of partial strategies of best compromise are considered. Instead of applying the MCDA method on the set S t

k , it will be applied on the set Γ tk ⊂ S t

k defined by the equations (6) and (7).

9

Tk

Tk S=Γ ∀k∈1,2… Tn (6)

for t = T-1 to t = 1,

Γ tk =

)(

1

kn

i=∪

tkiN

q

)(

1=∪ a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

+

+

+

1)),(,,(

1),,(

1)1,,(

γ

γ

γ

(7)

where a tki )( ∈ A t

k , tkΓ =MCDA[ t

kΓ ], tkS =MCDA[ t

kS ] ∀t∈1,…,T and γ 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t

∀j∈1,2….n(i, k).

Thus, the recursive equation deduced from the principle of decomposition is: tkS =

tkΓ ∀ t∈1,2….T, ∀ k∈1,2, …, n t (8)

V. MULTI-CRITERIA DECISION PROBLEM DECOMPOSITION

A. The theorem of decomposition

Let consider a MCDA method that verify the following conditions:

Condition 1: The relational preference system (r.p.s) of the multi-criteria decision aid (MCDA) method is a total preorder (P, I) or a partial preorder (P, I, R).

We use the symbol f to designate the relation P, ~ to designate the relation I, f to designate P∪I. P∪I is equivalent to the outranking relation S that means “at least as good as” and R is the incomparability relation. The relations P, I and R are deduced from S as follows:

s tk S r t

k and Not(r tk S s t

k ) ⇔ s tk P r t

k (9)

s tk S r t

k and r tk S s t

k ⇔ s tk I r t

k (10)

Not(s tk S r t

k ) and Not(r tk S s t

k ) ⇔ s tk R r t

k (11)

Condition 2: The relational preference system (r.p.s) verify the temporal consistence (strong and weak version).

The strong version (resp. weak) of temporal consistance means that if at a given period, the result of the multi-criteria decision aid method indicates a strict preference of a given partial strategy compared to another (resp. indifference between two strategies), this result remains the same if we prolong the two strategies by the same action at a previous period.

10

Strong version of temporal consistence:

If s 1),,( 0

+tjikf t f r 1

),,( 0

+tjikf t then a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

0f a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

r

s

+

+

+

1)),(,,(

1),,(

1)1,,(

0

(12)

Weak version of temporal consistence:

If s 1),,( 0

+tjikf t f r 1

),,( 0

+tjikf t then a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

0f a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

r

s

+

+

+

1)),(,,(

1),,(

1)1,,(

0

(13)

Condition 3: Resulting preference relations P and I from a MCDA (after exploitation) are transitive.

The relations P and I obtained by the MCDA method (after exploitation) are transitive. For d tk ∈D t ,

let s )()( 1qt

ki , s )()( 2qt

ki and s )()( 3qt

ki partial strategies in S tk . Transitivity is expressed as follows:

s )()( 1qt

ki f s )()( 2qt

ki and s )()( 2qt

ki f s )()( 3qt

ki ⇒ s )()( 1qt

ki f s )()( 3qt

ki (14)

s )()( 1qt

ki ∼s )()( 2qt

ki and s )()( 2qt

ki ∼s )()( 3qt

ki ⇒ s )()( 1qt

ki ∼s )()( 3qt

ki (15)

s )()( 1qt

ki f s )()( 2qt

ki and s )()( 2qt

ki ∼s )()( 3qt

ki ⇒ s )()( 1qt

ki f s )()( 3qt

ki (16)

s )()( 1qt

ki ∼s )()( 2qt

ki and s )()( 2qt

ki f s )()( 3qt

ki ⇒ s )()( 1qt

ki f s )()( 3qt

ki (17)

Condition 4: The set of best compromise solutions is not empty at each period t.

If these conditions are verified, it is possible to formulate the following decomposition theorem.

Theorem 1: If conditions 1, 2, 3 and 4 are verified ⇒ Each best compromise strategy has the property that all its partial strategies are also best compromise strategies.

i.e. : tkS =

tkΓ ∀ t∈1,2….T, ∀ k∈1,2….n t , where

tkΓ =MCDA[ t

kΓ ] and tkS =MCDA[ t

kS ] See appendix 1 for proof.

B. Resolution Methodology

We summarize in what follows the methodology steps for the resolution of a multi-criteria decision trees:

Step 1: Model the decision problem by a multi-criteria decision tree.

Step 2: Assign probabilities to states of the nature at every uncertainty node. Evaluate according to the criteria family all alternatives emerging from each decision node.

11

Step 3: Assign the path evaluation to every terminal branch.

Step 4: For any period t = T, T-1,…,1 and starting by last period,

• At each decision node of period t, apply the MCDA method on the set of partial strategies leaving this node. These strategies are composed of an immediate alternative prolonged by a partial strategy of best compromise in period t+1 for every state of the nature.

Note that the application of the MCDA method can lead to more than one best compromise strategy. If the r.p.s is a total preorder, the set of best compromise strategies contains ex-aequo strategies. In that case, we retain one representative strategy among indifferent ones. However, if the r.p.s is a partial preorder, we retain in the set of best compromise strategies one representative strategy among indifferent ones so that we keep only incomparable strategies.

• At each uncertainty node, the expected value (by criterion) is used to evaluate partial strategies.

• Eliminate from further analysis the partial strategies that are not retained by the MCDA method.

Step 4 is repeated until the set of best compromise solutions at the starting point of the tree is obtained. The strategies obtained at period 1 are all of best compromise and constitute “the solution” of the problem.

12

Algorithm

(1) Initialisation t = T For k=1 to k = n T

For i=1 to i = n(k) For m=1 to m = M

g Tm (a Tki )( )= ∑

=

),(

1

kin

jp T

j .V Tmj ,

End For End For

Γ Tk = a T

ki )( , i∈1,2….n(k) ), F = g 1 ,…,g m ,….g M , V Tk = g Tm (a T

ki )( ), i∈1,2….n(k), m= 1…M.

MCDA [Γ Tk ]=

TkΓ

End For (2) For t=T-1 to t=1

For k=1 to k=n t For i=1 to i=n(k)

For q=1 to q= N tki )(

For m=1 to m=M

g tm (

q

tkinikf

tjikf

tikf

t

t

t

+

+

+

1)),(,,(

1),,(

1)1,,(

γ

γ

γ

)= ∑=

),(

1

kin

jp t

j . g 1+tm (1

),,(+t

jikf tγ )

where g 1+tm (1

),,(+t

jikf tγ ) is the evaluation of the partial strategy 1

),,(+t

jikf tγ

according to criteria g 1+tm and p tj is the probability assigned to state of the

nature j. End for.

End for End for

Γ tk =

)(

1

kn

i=∪

tkiN

q

)(

1=∪ a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

+

+

+

1)),(,,(

1),,(

1)1,,(

γ

γ

γ

, F = g 1 ,…,g m ,….g M , V tk =g tm (γ t

k ),

γ tk ∈Γ t

k , m=1… M.

MCDA [Γ tk ]=

tkΓ

End for. End for.

13

The set of best compromise strategies (solution of the problem) corresponds to the result of the application of the MCDA method at decision node d 1

1 of period 1 on the set

)1(

1

n

i=∪

tiN

q

)1(

1=∪ a t

i )1( ⊗

qinif

jif

if

t

t

t

2))1,(,,1(

2),,1(

2)1,,1(

γ

γ

γ

.

VI. CHARACTERIZATION OF SELECTED MCDA METHODS

A. The properties

MCDA methods that will be applied to decision nodes must verify temporal consistence and transitivity (see the theorem 1). These two properties are conditions 2 and 3 of theorem 1. They guarantee that the decomposition of the tree leads to best compromise strategies. However, there are other properties that don’t appear as requirements to the decomposition but which are desirable to insure a good quality of the obtained results. The literature review reveals a series of properties, which characterize MCDA methods (Vincke, 1992; Pérez, 1994; Pirlot, 1997; Al-Shemmeri, Al-Kloub and Pearman 1997; Ozernoy, 1992; Bouyssou, 1986; Guitouni, 1998). From this literature, we retain the following desirable properties in order to characterize MCDA methods: neutrality, anonymous, fidelity, dominance and independence. These properties are drawn from the list of properties proposed by Guitouni (1998) as a comparison basis for MCDA methods. Among the original list, we don’t consider properties with subjective judgement and we limited our choice to the properties that affect the final result of the methodology. We define here retained properties:

Neutrality: The MCDA method is neutral if the final result is not affected by the denomination of the strategies. Anonymous: The MCDA method is anonymous if the final result is not affected by the denomination of the criteria. Fidelity: Fidelity means that the result given by the MCDA method reproduces the partial results on which all the criteria are unanimous. Dominance: The MCDA method respect the dominance principle if it concludes that strategy s t

k is preferred to strategy r t

k when the strategy s tk dominates strategy r t

k . Independence: The MCDA method verify the independence property if the preference relation between two strategies s t

k and r tk depends only on preferential information relative to compared strategies s t

k and r t

k .

These properties are important for the quality of results. In fact, the set of best compromise solutions is given by the MCDA method and its quality depends on the quality of the method used at decision nodes. For example, if the MCDA method doesn’t verify neutrality, the solution given by our methodology will depend on the nomination of the alternatives.

B. MCDA Methods

We consider four MCDA methods: weighted sum, dominance, lexicographic, and TOPSIS. The first three are elementary MCDA methods with different compensation level whereas TOPSIS is a single criterion synthesis method. We try to characterize them toward the previous discussed properties. We present here briefly these methods:

14

Weighted sum: s tk f r t

k ⇒ V(s tk )> V(s t

k ) where V is the aggregative function defined by

V(s tk ) = )(.

1

tkm

M

mm sg∑

=

π .

Dominance: s tk f r t

k ⇒ s tk D r t

k ⇒ mg (s tk ) ≥ mg (r t

k ) ∀ m∈1…M and ∃ 0m tel que

0mg (s tk ) >

0mg (r tk ).

Lexicographic: Let )1(g ,… )( 0mg ,… )(Mg the ordered list of criteria according to their weight.

s tk f r t

k ⇒ ∃ 0m such as )( 0mg (s tk )> )( 0mg (r t

k ) and )(mg (s tk )= )(mg (r t

k ) for each criteria m more

important than 0m .

TOPSIS: s tk f r t

k ⇒ C * (s tk )>C * (r t

k ) where C * (s tk )=

)()()(**

*tk

tk

tk

sDsDsD

+, )(*

tksD is the Euclidian distance

of s tk to the anti-ideal strategy, )(* t

ksD is the Euclidian distance of s tk to the ideal strategy.

C. Characterization

Let s tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

0 and r t

k = a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

r

s

+

+

+

1)),(,,(

1),,(

1)1,,(

0 the strategies s 1

),,( 0

+tjikf t and

r 1),,( 0

+tjikf t prolonged by a t

ki )( . Strategies s tk and r t

k are evaluated by equations (18) and (19):

∑≠

++ +=0

00)(.)(.)( 1

)..(1

)..(jj

tjikfmj

tjikfmj

tkm tt sgpsgpsg ∀ m∈1…M (18)

∑≠

++ +=0

00)(.)(.)( 1

)..(1

)..(jj

tjikfmj

tjikfmj

tkm tt sgprgprg ∀ m∈1…M (19)

Property 1: The weighted sum verifies the strong and weak version of temporal consistence.

Let mπ the weight of criteria m and V the aggregation function.

V(s tk )= )(.

1

tkm

M

mm sg∑

=

π =∑ ∑= ≠

++ +M

m jj

tjikfmj

tjikfmjm tt sgpsgp

1

1),,(

1),,(

000

)](.)(..[π

∑ ∑ ∑= ≠ =

++ +=M

m jj

M

m

tjikfmmj

tjikfmmj tt sgpsgp

1 1

1),,(

1),,(

000

)(..)(. ππ

V(s tk ) )(.)(. 1

),,(1

),,( 00

00

+

≠

+ ∑+= tjikf

jjj

tjikfj tt sVpsVp (20)

V(r tk ) )(.)(. 1

),,(1

),,( 00

00

+

≠

+ ∑+= tjikf

jjj

tjikfj tt sVprVp (21)

15

Using equations (20) and (21), we obtain: s 1

),,( 0

+tjikf tf r 1

),,( 0

+tjikf t ⇒ )()( 1

),,(1

),,( 00

++ > tjikf

tjikf tt rVsV ⇒ )()( t

ktk rVsV > ⇒ s t

k f r tk .

s 1),,( 0

+tjikf t ~r 1

),,( 0

+tjikf t ⇒ )()( 1

),,(1

),,( 00

++ = tjikf

tjikf tt rVsV ⇒ )()( t

ktk rVsV = ⇒ s t

k ~r tk .

Thus, the weighted sum verifies the strong version of temporal consistence.

Property 2: The dominance verifies the strong and weak version of temporal consistence.

Let consider the dominance method and let verify the temporal consistence property. s 1

),,( 0

+tjikf t f r 1

),,( 0

+tjikf t ⇒ s 1

),,( 0

+tjikf t D r 1

),,( 0

+tjikf t ⇒ mg (s 1

),,( 0

+tjikf t ) ≥ mg (r 1

),,( 0

+tjikf t ) ∀ m∈1…M et

∃ 0m such as 0mg (s 1

),,( 0

+tjikf t ) >

0mg (r 1),,( 0

+tjikf t ).

Equations (18) and (19) imply that s tk D r t

k since mg (s tk ) ≥ mg (r t

k ) ∀ m∈1…M et 0mg (s t

k ) >

0mg (r tk ).

Thus, s tk D r t

k ⇒ s tk f r t

k .

Since dominance method cannot conclude to indifference between strategies, the weak version of temporal consistence cannot be verified.

Property 3: The lexicographic method verifies the strong and weak version of temporal consistence.

Let consider the lexicographic method and let verify the temporal consistence property. Let )1(g ,… )( 0mg ,… )(Mg the ordered criteria list according to criteria weight.

s 1),,( 0

+tjikf t f r 1

),,( 0

+tjikf t ⇒ ∃ 0m such that )( 0mg (s 1

),,( 0

+tjikf t )> )( 0mg (r 1

),,( 0

+tjikf t ) and

)(mg (s 1),,( 0

+tjikf t )= )(mg (r 1

),,( 0

+tjikf t ) for each criteria m which weight is higher than criteria 0m .

Equations (18) and (19) imply that )(mg (s tk ) = )(mg (r t

k ) for each criterion which weight is higher than

criterion 0m and )( 0mg (s tk )> )( 0mg (r t

k ). Thus, s tk f r t

k .

The same reasoning imply that, s 1),,( 0

+tjikf t ~r 1

),,( 0

+tjikf t ⇒ )(mg (s 1

),,( 0

+tjikf t )= )(mg (r 1

),,( 0

+tjikf t ) ∀ m∈1…M

⇒ )(mg (s tk )= )(mg (r t

k ) ⇒ s tk ~r t

k .

Thus, lexicographic method verify strong and weak version of temporal consistence.

Property 4: TOPSIS doesn’t verify the strong version of temporal consistence.

Consider the example presented in section II-C and let show by counter examples that TOPSIS doesn’t verify the strong version of temporal consistence. On node D2, when applying TOPSIS, the decision “Do Nothing” is of best compromise that decision “Advertise”. But, when applied on node D1 considering all strategies, TOPSIS gives that the strategy s4 is preferred to strategy s1. So, when we prolong decisions “Do Nothing” and “Advertise” the preference order between s4 and s1 isn’t respected. Thus, the decomposition principle used with TOPSIS doesn’t guarantee to give as solution the best compromise strategies.

Properties 1, 2, 3 and 4 express that weighted sum, dominance and lexicographic methods verify temporal consistence, whereas TOPSIS doesn’t. However, all cited methods are transitive since the preorder [P,I] obtained is transitive. Also, Guitouni (1998) shows for these methods that neutrality; anonymous, fidelity, dominance and independence are verified. Table 2 summarizes the characterization results of the four considered MCDA methods.

16

Table 2: Some MCDA Methods Characterization Procedure/Property Temporal

consistence Transitivity Neutrality Anonymous Fidelity Dominance Independence

Weighted sum Yes Yes Yes Yes Yes Yes Yes Dominance Yes Yes Yes Yes Yes Yes Yes Lexicographic Yes Yes Yes Yes Yes Yes Yes TOPSIS No Yes Yes Yes Yes Yes Yes

Characterization’s Results on these MCDA methods show that when the MCDA method verifies the temporal consistence property, it verifies the independence property. Besides, the counter-example of TOPSIS permits to conclude that the implication independence ⇒ temporal consistence is not true.

VII. ILLUSTRATION & DISCUSSION

We resolve in this section a two-period decision tree using the methodology described above. The example we consider is presented in section 2.3. Table 3 summarizes the results obtained at period 2 when we use the recursion with the weighted sum, dominance, lexicographic and TOPSIS. Table 4 presents the results at period 1.

Remark that the strategy(ies) obtained by applying respectively the weighted sum, the dominance and the lexicographic method considering all strategies at period 1 is the same as the result obtained by applying the recursive procedure. However, it’s not the case for TOPSIS. Concerning TOPSIS, the strategy s2 obtained when considering all strategies at period one is different from the strategy s1 obtained with the recursive procedure. This is due to the fact that TOPSIS doesn’t verify the temporal consistence property.

Table 3: Results Obtained for the Second-Period Decision Nodes Node Decision obtained by

the weighted sum1 Decisions obtained by the Dominance

Decision obtained by the lexicographic method2

Decision obtained by

TOPSIS D2 Adv DN, Adv, R R DN D3 Adv DN, Adv, R R DN D4 DN DN, Adv, R R Adv D5 DN DN, Adv, R R Adv

DN: Do Nothing; Adv: Advertise; R: Recall.

Table 4: Results Obtained for the First-Period Decision Node

Solutions considering all strategies at period 1

Solutions by recursion

Weighted sum s10 s10 Dominance s1,s2,s10,s11,s12,s14,s15,s19 s1,s2,s10,s11,s12,s14,s15,s19 Lexicographic S19 S19 TOPSIS S2 S1

1 The normalization method is:

**

*

jxjx

ijxjxijr

−

−= where ijx

iMaxjx =* and ijx

iMinjx =* , criteria j has to be maximized.

Cost weight = 0.2 and lives lost weight = 0.8 2 Lexicographic method supposes that criteria lives lost is more important than cost.

17

Not all MCDA methods are suited to be used to solve multiple criteria decision trees. The result obtained above confirms the theoretical results suggested by theorem 1. When conditions stated by theorem 1 are verified by a MCDA method, the decomposition of the tree is efficient and leads to the best compromise strategies. It’s the case of the weighted sum, dominance and lexicographic method. In these cases, the set of best compromise strategies obtained by applying the MCDA method on the complete set of strategies at period 1 is the same that the set of best compromise strategies obtained by recursion.

However, TOPSIS doesn’t verify temporal consistence. Therefore, applying TOPSIS to solve multiple criteria decision trees do not guarantee that applying the decomposition principle will lead to valid results. Numerical results imply that the set of best compromise strategies obtained by recursion at period 1 is different from the set of best compromise strategies obtained by applying TOPSIS directly on the complete set of strategies. TOPSIS assign to each strategy a score that depends on the distance of the strategy to relative ideal and relative anti-ideal strategies. Intuitively, we suspect that TOPSIS doesn’t verify temporal consistence because of the definition of the relative ideal and relative anti-ideal points. Since evaluations of the ideal and anti-ideal depend on the set of strategies, ideal and anti-ideal can differ from one period to another. Thus, the distance of a partial strategy to the ideal at one period isn’t necessarily preserved when prolonging this partial strategy by an action at the previous period.

This illustrative example shows that if the MCDA method verify temporal consistence and transitivity, the decomposition of the tree is efficient. The TOPSIS counter example shows that when the MCDA method doesn’t verify temporal consistence, the decomposition of the tree doesn’t lead for all cases to best compromise strategies. In future research works, we can explore if this result can be generalized for each MCDA method that doesn’t verify temporal consistence.

VIII. CONCLUSION

Solving multiple criteria discrete dynamic or multi-periods problems is a very challenging endeavor. Recognizing that importance, this paper states a decomposition principle to facilitate solving multi-criteria decision trees while applying well known multiple criteria procedures. A decomposition theorem is formulated and gives the conditions for application of the decomposition principle. Instead of seeking the optimal (best) solution, we look for the solutions that achieve the best compromise considering the conflicting decision criteria. When the principle of decomposition is applied, every strategy of best compromise obtained at each decision node is necessary composed of partial strategies of best compromise. The decomposition principle constitutes a generalization of Bellman’s principle of optimality to the multiple criteria decision trees.

Based on the decomposition principle, a resolution methodology for the multiple criteria decision trees has been developed. The methodology consists on applying recursively a MCDA method at each decision node of the tree and keeping only the partial best compromise strategies for further analysis. This methodology is valid (gives the best compromise solutions) under sufficient conditions stated by the decomposition theorem. We showed that if the MCDA method verifies strong and weak temporal consistence and transitivity (after the exploitation phase), the decomposition principle leads to the best compromise strategies.

Results given by this methodology depend on the MCDA method properties. Thus, the choice of MCDA method to be applied at decision nodes must be made carefully. In order to help in this choice, we illustrated how we think MCDA methods should be characterized. Proposed properties of MCDA methods include the decomposition conditions as well as other desirable properties such as: neutrality, anonymous, fidelity, dominance and independence. MCDA methods considered in this paper are: the weighted sum, dominance, lexicographic and TOPSIS. Only TOPSIS doesn’t respect the conditions for the multiple criteria decomposition principle. A counter example illustrates that TOPSIS does not verify temporal

18

consistence. In addition, by examining characterization’s results, the counter-example of TOPSIS shows that independence property does not imply temporal consistence.

An academic example (Haimes, 1998) is used to illustrate the proposed method. Numerical results imply that the set of best compromise strategies obtained by recursion at period 1 is different from the set of best compromise strategies obtained by applying TOPSIS directly on the complete set of strategies. Hence, the principle of decomposition cannot be used efficiently with TOPSIS. However, results confirm the theoretical conclusion driven about the weighted sum, dominance and lexicographic method. These methods verify decomposition conditions and therefore, they give the best compromise strategies when applied with the recursive procedure.

This work constitutes a good contribution to generalize the dynamic programming work to multi-criteria decision analysis. It also helps applying discrete multi-criteria to solve dynamic decision problems. This work established a set of sufficient conditions to be verified before applying any MCDA methods. The relationship between independence and temporal consistency properties is to be further investigated.

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R. Planche, Dunod, Paris. Bertsekas D.P, Shreve S.E. 1978. Stochastic Optimal Control : The discrete Time Case. Academic Press:

New York, N.Y. Bertsekas D.P. 1976. Dynamic Programming and Stochastic Control. Academic Press: New York. Bertsekas D.P. 1995. Dynamic programming and optimal control. Athena scientific: Belmont, M.A. Bielza C, Shenoy P.P. 1999. A comparison of Graphical Techniques for Asymmetric Decision Problems.

Management science 45(11): 1552-1569 Bouyssou D. 1986. Some Remarks on the Notion of Compensation in MCDM. European Journal of

Operational Research 26(1): 150-160. Brown T.A, Strauch R.E. 1965, Dynamic programming in multiplicative lattices. Math. Analysis and

Applications 12(2): 364-370. Carraway R.L, Morin T.L, Moskowitz H. 1990. Generalized Dynamic Programming for Multi-criteria

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Département Opération et systèmes de décision, Faculté des sciences de l’administration, Université Laval.

Hadley G. 1964. Nonlinear and Dynamic Programming, Addison Wesley, Reading, Mass. Haimes Y, Li D, Tulsiani V. 1990. Multiobjective decision tree method. Risk Analysis 10(1): 111-129. Haimes Y.Y. 1998. Risk Modeling Assessment and Management, John Wiley & Sons Inc., 726 p.

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Hansen P. 1980. Bicriterion path problems. In Fandel G. et Gal T. (1980) Multiple Criteria Decision Making Theory and Application, Proceedings 1979, 109-127.

Henig M.I. 1985. The shortest path problem with two objective functions. EJOR 25: 281-291. Henig M.I. 1994. Efficient Interactive Methods for a Class of Multiattribute Shortest Path Problems.

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X. APPENDIX 1: THEOREM PROOF

MCDA method that are applied to decision nodes might be structured into two phases: the aggregation and exploitation phase. Relational preference system (r.p.s) obtained after exploitation could consider incomparability R. Depending on this, the r.p.s is either a total preorder [P, I] or a partial preorder [P, I, R]. The theorem demonstration is presented when the r.p.s is a total preorder and then when it’s a partial preorder.

1) Case of a total preorder

For each period t, and for each decision node d tk ∈ D t , the MCDA method applied on the set S t

k leads

to a total preorder. S tk can be partitioned into orderd sets (see figure 4). Within the same set, strategies are

20

ex-aequo (indifference class). Two strategies s tl and s t

h such as s tl P s t

h belong to two different classes l and h where Cl P Ch. The set of best compromise strategies (class of rank zero C0) contains strategies that are strictly preferred to any strategy of another class (C≠C0) and that is indifferent to other strategies of C0.

Figure 4: Partition of S tk (Case of Total Preorder)

Proof by recursion. Let d tk ∈ D t a decision node of period t. For t=T, it’s easy to verify that

∀k∈1,2,…n T , S Tk = Γ T

k . Suppose ∀k∈1,2,…n 1+t , S 1+tk = Γ 1+t

k and show that ∀ k∈1,2,…n t ,

S tk = Γ t

k . Let show that S tk ⊆ Γ t

k . This inclusion shows that each retained strategy obtained by

applying the MCDA method on the set S tk is necessarily composed of partial strategies of best

compromise. Consider s tk ∈ S t

k , there exists i∈1,2,…n(k), q∈1,2,…N tki )( , such as:

s tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

(22)

where s 1),,(

+tjikf t ∈ S 1

),,(+t

jikf t ∀ j∈ 1,2,…, n(i, k). Suppose there is j∈1,2,…, n(i, k) such as

s 1),,(

+tjikf t ∉ Γ 1

),,(+t

jikf t , then necessarily s 1),,(

+tjikf t ∉ S 1

),,(+t

jikf t and there is r 1),,(

+tjikf t ∈ S 1

),,(+t

jikf t such as

r 1),,(

+tjikf t f s 1

),,(+t

jikf t . Strong version of temporal consistence implies that:

Class of rank 0

Class of rank

Class of rank k

Class of rank h

~ f

f

f

C0

~

~

~

21

a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

r

s

+

+

+

1)),(,,(

1),,(

1)1,,(

f a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

= s tk

(23)

This strict preference comes in contradiction with s tk ∈ S t

k . Thus, whatever the state of the nature j,

s 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t and so s tk ∈ Γ t

k . In addition, s tk ∈ Γ t

k otherwise ∃tkγ ∈ Γ t

k (⊆ tkS ) such as

tkγ f s t

k , what comes in contradiction with s tk ∈ S t

k .

Let show that S tk ⊇ Γ t

k . This inclusion shows that every strategy γ tk ∈ Γ t

k appears as best

compromise strategy when we apply the MCDA method on the set S tk .

γ tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

+

+

+

1)),(,,(

1),,(

1)1,,(

γ

γ

γ

(24)

where i∈1,2,…, n(k), q∈1,2,…, N tki )( and γ 1

),,(+t

jikf t ∈ Γ 1),,(

+tjikf t ∀ j ∈ 1,2,…, n(i, k).

γ tk ∈ Γ t

k ⇒ γ tk is at least as good as any strategy β t

k ∈ Γ tk

⇒γ tk f β t

k where β tk =a t

ki )(' ⊗

'

1)),'(,',(

1),',(

1)1,',(

q

tkinikf

tjikf

tikf

t

t

t

+

+

+

γ

γ

γ

∀ i’∈1,2,…, n(k), ∀ q’∈1,2,…, N tki )(' , γ 1

),',(+t

jikf t ∈ Γ 1),',(

+tjikf t ∀ j ∈ 1,2,…n(i’,k). For every

j, γ 1),',(

+tjikf t ∈ Γ 1

),',(+t

jikf t ⊆ S 1),',(

+tjikf t so γ 1

),',(+t

jikf t f s 1),',(

+tjikf t

∀ s 1),',(

+tjikf t ∈S 1

),',(+t

jikf t .

Weak version of temporal consistence implies that:

22

a tki )(' ⊗

q

tkinikf

tjikf

tikf

t

t

t

+

+

+

1)),'(,',(

1),',(

1)1,',(

γ

γ

γ

f a tki )(' ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

+

+

+

1)),'(,',(

1),',(

1)1,',(

γ

γ

f ...f s tk =a t

ki )(' ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),'(,',(

1),',(

1)1,',(

∀i’∈1,2,…, n(k). By transitivity we have β t

k f s tk .

γ tk f β t

k and β tk f s t

k , ⇒γ tk f s t

k where s tk ∈S t

k . So, γ tk is at least as good as any strategy of S t

k . Let

note also that there is no strategy of S tk (not of best compromise) that is at least as good as γ t

k . In fact, let

s tk ∈ S t

k such as s tk f γ t

k then s tk ~γ t

k . So γ tk ⊂ S t

k .

2) Case of partial preorder

Let consider the case where the r.p.s obtained after exploitation is [P, I, R] with S=P∪I a transitive outranking relation that means “at least as good as”. S t

k is partitioned into classes that can be ranked by a partial order >c, characterized by asymmetric and transitive relation. There is one or many classes of rank zero that contain strategies that are not outranked by any strategy of another class of S t

k (see figure 5).

These strategies are called S-efficient and are the best compromise solutions. In S tk , we keep only one

representative strategy among ex-aequo ones so that elements of S tk are incomparable.

Figure 5 : Partition of S tk (Case of Partial Preorder)

According to Roy and Bouyssou (1993), when relation S is transitive (designated by f ), the set S tk

verify:

>c

>c

>c

>c

~ ~

~

~

~ ~

R

23

[1] S tk is such that, for any s t

k that doesn’t belong to any class of rank zero, there is at least one

strategy r tk S-efficient such as r t

k f s tk . S t

k is called an outranking set.

[2] No sub-set of S tk verifies condition (1). When S t

k verify conditions 1 and 2, it’s called minimal outranking set.

[3] Two strategies in S tk are incomparable.

[4] ∀ r tk ∈ S t

k , there is no strategy s tk ∈ t

kS that verifies s tk f r t

k . The strategies of S tk are S-

efficient. [5] S t

k is unique with substitution of ex-aequo strategies

These properties characterize the set of best compromise strategies. Thus, we need to show that the set Γ t

k verify [1] and [2] and [4]. Conditions [3] and [5] are verified by definition of S tk .

a) Γ tk is an outranking set of S t

k

∀k∈1,2,…n T , Γ Tk = S T

k that is an outranking set of S Tk . Suppose ∀k∈1,2,…n 1+t , Γ 1+t

k is an

outranking set of S 1+tk and let show that ∀k∈1,2,…n t , Γ t

k is an outranking set of S tk . Let

s tk ∈S t

k \ Γ tk , there is i∈1,2,…n(k), q∈1,2,…N t

ki )( such as s tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

.

1st case: ∀ j∈ 1,2,… n(i, k), s 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t , If ∀ j, s 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t then s tk ∈ Γ t

k

and so there is r tk ∈ Γ t

k such as r tk f s t

k .

2d case: There is j∈ 1,2,… n(i, k), s 1),,(

+tjikf t ∉ Γ 1

),,(+t

jikf t . If s 1),,(

+tjikf t ∉ Γ 1

),,(+t

jikf t , then there is

r 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t such as r 1),,(

+tjikf t f s 1

),,(+t

jikf t because Γ 1),,(

+tjikf t is an outranking set of

S 1),,(

+tjikf t .

Weak version of temporal consistence imply that:

a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

r

s

+

+

+

1)),(,,(

1),,(

1)1,,(

f s tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

(25)

24

Repeat the same thing for all j ∈ 1,2,…n(i, k) such as s 1),,(

+tjikf t ∉ Γ 1

),,(+t

jikf t . By transitivity, we obtain

r tk ∈ Γ t

k such as:

r tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

r

r

r

+

+

+

1)),(,,(

1),,(

1)1,,(

f s tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

s

s

s

+

+

+

1)),(,,(

1),,(

1)1,,(

where r 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t ∀j.

• If r tk

∈ Γ tk , we conclude that there is r t

k ∈ Γ t

k such as r tk

f s tk .

• If r tk

∉ Γ tk , then there is u t

k ∈ Γ t

k such as u tk f r t

k but r tk f s t

k so by transitivity, we conclude that

there is u tk

∈ Γ tk such as u t

k f s tk .

b) Γ tk is a minimal outranking set

Suppose there is an outranking set Ω tk ⊂ Γ t

k , whatever the strategy in Γ tk /Ω t

k , there is an element in

Ω tk that outranks this strategy. This is impossible because all strategies in Γ t

k are incomparable (because we retain only one representative strategy for each sub-class of rank 0).

c) Γ tk is S-efficient in S t

k

It’s easy to verify that ∀k∈1,2,…n T , each strategy of Γ Tk is S-efficient in S T

k because Γ Tk = S T

k .

Suppose ∀k∈1,2,…n 1+t , each strategy of Γ 1+tk is S-efficient in S 1+t

k and let show that ∀k∈1,2,…n t ,

each strategy of Γ tk is S-efficient in S t

k . Let γ tk ∈ Γ t

k , γ tk is S-efficient in Γ t

k , so there is no β tk ∈ Γ t

k

such as β tk f γ t

k . Let show that there is no strategy in S tk / Γ t

k that is strictly preferred to any strategy of

Γ tk . Suppose there is u t

k ∈S tk / Γ t

k such as u tk f γ t

k .

u tk = a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

u

u

u

+

+

+

1)),(,,(

1),,(

1)1,,(

(26)

where there is at least one j∈1,2,… n(i, k), u 1),,(

+tjikf t ∉ Γ 1

),,(+t

jikf t .

Γ 1),,(

+tjikf t is an outranking set of S 1

),,(+t

jikf t , then there is w 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t such as

w 1),,(

+tjikf t f u 1

),,(+t

jikf t . Weak version of temporal consistence imply that:

25

a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

u

w

u

+

+

+

1)),(,,(

1),,(

1)1,,(

f a tki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

u

u

u

+

+

+

1)),(,,(

1),,(

1)1,,(

Repeat the same thing to j∈1,2,…n(i, k) such as u 1),,(

+tjikf t ∉ Γ 1

),,(+t

jikf t . By transitivity, we have:

w tk = a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

w

w

w

+

+

+

1)),(,,(

1),,(

1)1,,(

f u tk =a t

ki )( ⊗

q

tkinikf

tjikf

tikf

t

t

t

u

u

u

+

+

+

1)),(,,(

1),,(

1)1,,(

where w 1),,(

+tjikf t ∈ Γ 1

),,(+t

jikf t ∀j.

Thus, w tk f u t

k and u tk f γ t

k ⇒ w tk f γ t

k .

So, we have a strategy w tk ∈ Γ t

k that is strictly preferred to γ tk ∈ Γ t

k , what comes in contradiction

with the definition of Γ tk . Thus, Γ t

k is S efficient in S tk .