The maximal possibility degree is affected to the minimal utility

The third polynomial case of possibilistic Choquet integrals (denoted by Min-Class) con- cerns possibilistic lotteries where the maximal possibility degree namely 1 is affected to the minimal utility in the lottery. This class is defined as follows:

Definition 4.4 Let U = {u1, . . . un} be the set of possible utilities where uminis the minimal utility in a possibilistic lottery such that umin ≤ un. In the case of Min-Class, each lottery L ∈ L is as follows: L = h1/umin, . . . , λn/uni.

Proposition 4.10 DT-OPT-ChN is polynomial in the case of Min-Class.

Proof. [Proof of Proposition 4.10]

In what follows, we present necessary proof in numerical setting (the same principle is valid for the ordinal setting).

We will consider the case where α = 1

• Case 1: L, L0 and L” have the same minimal utility ui. L = h1/ui, . . . , λn/uni ⇒ ChN(L) = ui

L0 = h1/ui, . . . , λ0n/uni ⇒ ChN(L0) = uj L” = h1/ui, . . . , λ”n/uni,

L1 = h1/ui, . . . , max(λn, βλ”n)/uni, L2 = h1/ui, . . . , max(λ0n, βλ”n)/uni ⇒ ChN(L1) = ChN(L2) = ui.

• Case 2: L and L0 have the same minimal utility ui but not L”. L = h1/ui, . . . , λn/uni and L0 = h1/ui, . . . , λ0n/uni

⇒ Ch_N(L) = ChN(L0) = ui and L” = h1/uj, . . . , λ”n/uni.

– If L” = h1/uj, . . . , λ”n/uni (i.e. ui < uj) L1 = h1/ui, . . . , max(αλ0n, βλ”n)/uni, L2 = h1/ui, . . . , max(αλ0n, βλ”n)/uni ⇒ ChN(L1) = ChN(L2) = ui.

– If L” = h1/uj, . . . , λ”i/ui, . . . , λ”n/uni (i.e. ui > uj) L1 = hβ/uj, . . . , max(αλn, βλ”n)/un>,

L2 = hβ/uj, . . . , max(αλ0n, βλ”n)/un> ⇒ Ch_N(L1) = ChN(L2).

• Case 3: L and L0 don’t have the same minimal utility. L = h1/ui, . . . , λn/uni and L0 = h1/uj, . . . , λ0n/uni ⇒ Ch_N(L) = ui and ChN(L0) = uj.

– If L” = h1/ui, . . . , λ”n/uni (i.e. ui> uj) L1 = h1/ui, . . . , max(αλ0n, βλ”n)/uni,

L2 = h1/uj, . . . , max(αλ0_i, βλ”i)/ui, . . . , max(αλ0n, βλ”n)/uni = h1/uj, . . . , max(λ0i, β)/ui, . . . , max(αλ0n, βλ”n)/uni

⇒ Ch_N(L1) = ui > ChN(L2) = uj.

– If L” = h1/ui, . . . , λ”j/uj, . . . , λ”n/uni (i.e. ui< uj) L1 = h1/ui, . . . , max(αλ0n, βλ”n)/un>,

L2 = hβ/ui, βλ”i+1/ui+1, . . . , βλ”j−1/uj−1, 1/uj, . . . , max(αλ0_n, βλ”n)/un>

⇒ ChN(L1) = ui

ChN(L2) = ui+ (ui+1− ui)(1 − β) + (ui+2− ui+1)(1 − max(β, βλ”i+1)) + · · · + (uj− uj−1)(1 − max(β, . . . , βλ”j−1)) + (uj+1− uj)(1 − max(β, . . . , 1)) + · · · + (un− un−1)(1 − max(β, . . . , 1))

= ui+ (ui+1− ui)(1 − β) + (ui+2− ui+1)(1 − max(β, βλ”i+1)) + · · · + (uj − uj−1)(1 − max(β, . . . , βλ”j−1)) + (uj+1− uj)(1 − 1) + · · · + (un− un−1)(1 − 1) = ui + (ui+1− ui)(1 − β) + (ui+2− ui+1)(1 − max(β, βλ”i+1)) + · · · + (uj − uj−1)(1 − max(β, . . . , βλ”j−1))

Let X = (ui+1− ui)(1 − β) + (ui+2− ui+1)(1 − max(β, βL00[ui+1])) + · · · + (uj− uj−1)(1 − max(β, . . . , βλ”j−1))

X ≥ 0 since (ui+1− ui)(1 − β) ≥ 0 (ui+2− ui+1)(1 − max(β, βλ”i+1) ≥ 0 . . .

(uj− uj−1)(1 − max(β, . . . , βλ”j−1)) ≥ 0 ⇒ ChN(L2) = ui+ X (s.t. X ≥ 0) ⇒ Ch_N(L1) ≤ ChN(L2)

• Case 4: L and L0 _{have not the same minimal utility i.e.} L = h1/uj, . . . , λn/uni and

L0 = h1/ui, . . . , λ0n/uni

⇒ Ch_N(L) = uj and ChN(L0) = ui

– If L” = h1/ui, . . . , λ”n/un > (i.e. ui > uj): similar to the first item in the previous case

– If L” = h1/ui, . . . , λ”j/uj, . . . , λ”n/uni (i.e. ui < uj): similar to the second item in the previous case

• Case 5: The minimal utility in L (resp. L0_{, L”) is u}

i (resp. uj, uk). – If ui> uj > uk,

L0 = h1/uj, . . . , λ0i/ui, . . . , λ0n/uni

and L” = h1/uk, . . . , λ”j/uj, . . . , λ”i/ui, . . . , λ”n/uni ⇒ Ch_N(L) = ui and ChN(L0) = uj, ChN(L) > ChN(L0) L1 _{= hβ/u}

k, . . . , βλj/uj, . . . , 1/ui, . . . , max(λn, βλ”n)/uni L2 = hβ/uk, . . . , 1/uj, . . . , 1/ui, . . . , max(λ0n, βλ”n)/uni

ChN(L1) = uk+ (uk+1− uk)(1 − β) + · · · + (ui+1− ui)(1 − 1) ChN(L2) = uk+ · · · + (uj+1− uj)(1 − 1) < ChN(L1). – If ui> uk> uj L = h1/ui, . . . , λn/uni, L0 = h1/uj, . . . , λ0k/uk, . . . , λ0i/ui, . . . , λ0n/uni and L” = h1/uk, . . . , λ”i/ui, . . . , λ”n/uni ⇒ Ch_N(L) = ui and ChN(L0) = uj, ChN(L) > ChN(L0) L1 = hβ/uk, . . . , 1/ui, . . . , max(λn, βλ”n)/uni

L2 = h1/uj, . . . , max(λ0_k, βλ”k/uk, . . . , max(λ0n, βλ”n)/un>

ChN(L1) = uk+ · · · + (ui− ui−1)(1 − max(β, . . . , max(λi−1, βλ”i−1))) ChN(L2) = uj. ⇒ ChN(L1) > ChN(L2). – If uj > ui > uk L = h1/ui, . . . , λj/uj, . . . , λn/uni, L0 = h1/uj, . . . , λ0n/uni and L” = h1/uk, . . . , λ”i/ui, . . . , λ”j/uj, . . . , λ”n/uni ⇒ Ch_N(L) = ui and ChN(L0) = uj, ChN(L0) > ChN(L) L1 = hβ/uk, . . . , 1/ui, . . . , max(λn, βλ”n)/un>

L2 = hβ/uk, . . . , βλ”i/ui, . . . , 1/uj, . . . , max(λ0n, βλ”n)/un> ChN(L1) = uk+ · · · + (ui+1− ui)(1 − 1)

ChN(L2) = uk+ · · · + (ui+1−ui)(1 − max(β, . . . , max(λ0i, βλ”i))) + · · · + (uj+1− uj)(1 − 1)

ChN(L2) = ChN(L1) + (ui+1− ui)(1 − max(β, . . . , max(λ0i, βλ”i))) + · · · + (uj+1− uj)(1 − 1) ⇒ Ch_N(L2) > ChN(L1). – If uk> ui > uj L = h1/ui, . . . , λk/uk, . . . , λn/uni, L0 = h1/uj, . . . , λ0i/ui, . . . , λ0_k/uk, . . . , λ0n/uni and L” = h1/uk, . . . , λ”n/uni ⇒ Ch_N(L) = ui and ChN(L0) = uj, ChN(L) > ChN(L0)

L1 = h1/ui, . . . , max(λk, β)/uk, . . . , max(λn, βλ”n)/uni L2 = h1/uj, . . . , max(λ0n, βλ”n)/uni

ChN(L1) = ui ChN(L2) = uj ⇒ ChN(L1) > ChN(L2). – If uk> uj > ui L = h1/ui, . . . , λj/uj, . . . , λk/uk, . . . , λn/uni, L0 = h1/uj, . . . ,0k/uk, . . . , λ0n/uni and L” = h1/uk, . . . , λ”n/uni ⇒ Ch_N(L) = ui and ChN(L0) = uj, ChN(L0) > ChN(L) L1 = h1/ui, . . . , λj/uj, . . . , max(λn, βλ”n)/uni

L2 = h1/uj, . . . , max(λ0n, βλ”n)/uni ChN(L1) = ui

ChN(L2) = uj

⇒ Ch_N(L1) > ChN(L2).

4.8

Conclusion

In this chapter, we have developed possibilistic decision trees where possibilistic decision criteria presented in Chapter 2 are used. We have proposed a full theoretical study of the complexity of the problem of finding an optimal strategy in possibilistic decision trees. Table 4.8 summarizes the results of this study.

Upes Uopt P U LΠ LN OM EU ChN ChΠ

P P P P P P NP-hard NP-hard

Table 4.2: Results about the Complexity of ΠT ree − OP T for the different possibilistic criteria

In fact, we have developed necessary proofs for each decision criterion in order to show if the monotonicity property is verified (to apply dynamic programming in order to find the optimal strategy) or not. Then, we have shown that strategy optimization in possibilistic decision trees is a polynomial problem for most of possibilistic decision criteria except for possibilistic Choquet integrals. Indeed, we have shown that the problem of finding a

strategy optimal w.r.t possibility-based or necessity-based Choquet integrals is NP-hard via a reduction from a 3SAT problem. Nevertheless, we have identified three particular cases when these criteria satisfy the monotonicity property.

In next chapter, we develop an alternative solving approach for possibilistic decision tree with Choquet integrals since dynamic programming cannot be applied. More precisely, we propose an implicit enumeration approach via a Branch and Bound algorithm.

Solving algorithm for

Choquet-based possibilistic

decision trees

5.1

Introduction

In the previous chapter, we have shown that the problem of finding an optimal strategy w.r.t possibilistic Choquet integrals is NP-hard. As a consequence, the application of dynamic programming may lead to suboptimal strategies.

As an alternative, we propose to proceed by implicit enumeration via a Branch and Bound algorithm based on an optimistic evaluation of the Choquet value of possibilistic decision trees.

In order to study the feasibility of our proposed solutions for finding the optimal strategy w.r.t possibilistic decision criteria in decision trees, we propose also an experimental study. In what follows, Section 5.1 presents the Branch and Bound algorithm and Section 5.2 gives our experimental results.

The main results of this chapter are published in [7, 8].

5.2

Solving algorithm for non polynomial possibilistic Cho-