# A Weakly Pareto Compliant Quality Indicator

^{*}

## Abstract

**:**

## 1. Introduction

- minimize the APF distance from the POF;
- obtain a good (usually uniform) distribution of the solutions found;
- maximize the APF extension i.e., for each objective the non-dominated solutions should cover a wide range of values (best case: the global optimum of each objective function must be found);
- maximize the APF density, i.e., high cardinality for the approximation set is desirable.

- A is closer to the POF than B;
- the solutions in A are better distributed than the ones in B;
- A is more extended than B;
- the size of A is greater than the size of B,

## 2. Definitions and Terminology

#### 2.1. Multi- and Many-Objective Optimization Problem

_{1},x

_{2},...,x

_{m}) and is a point in the decision space(X). An objective vector(y = y

_{1},y

_{2},...,y

_{m}), that is a point in the objective space (Y), is linked to each decision vector by means of evaluating function f. So, a MOOP, with m decision variables (parameters to be set), n targets (objective functions to be optimized), and c constraints (ℓ equality and c-ℓ are inequality constraints), can be mathematically represented as follows.

#### 2.2. Pareto Dominance

^{1}dominates another decision vector x

^{2}iff:

^{1}≺ x

^{2}. When one or more of these relations is not satisfied, x

^{1}does not dominate x

^{2}; this condition is denoted as x

^{1}⊀ x

^{2}. It is worth noticing that, for a single objective function, the standard relation less than is generally used to define the corresponding minimization problem, while the symbol ≺ represents a natural extension of < in the case of MOOPs [26].

^{1}strictly dominates another decision vector x

^{2}(denoted as x

^{1}≺≺ x

^{2}) iff:

^{1}weakly dominates the decision vector x

^{2}(denoted as x

^{1}≼ x

^{2}) iff:

^{1}is better than x

^{2}with respect to a subset of objective functions but x

^{2}is better than x

^{1}with respect to another subset, the two solutions are said incomparable, denoted as x

^{1}||x

^{2}(or x

^{2}||x

^{1}):

^{1}≺≺ x

^{2}⇒ x

^{1}≺ x

^{2}⇒ x

^{1}≼ x

^{2}

^{1}⋠ x

^{2}⇒ x

^{1}⊀ x

^{2}⇒ x

^{1}⊀⊀ x

^{2}.

## 3. Quality Indicator

#### 3.1. Definitions

#### 3.2. Comparison Methods

^{q}→ Bool, where q depends on the size of the QI set. Figure 2 shows some examples of interpretation functions (A and B are two APFs).

_{I,E}: C

_{I}

_{,E}(A,B) = E(I(A),I(B)).

#### 3.3. Compatibility and Completeness

_{I,E}is said ►-compatible if for each possible pair of APFs A and B:

_{I,E}is said ►-complete if for each possible pair of APFs A and B:

- A is better than B (A⊲B);
- A and B are incomparable and A outperforms B with respect to closeness, distribution, extension and cardinality.

#### 3.4. Closeness, Distribution, Extension, and Cardinality

- close to the POF; Figure 3 represents the extreme cases: an APF exhibiting good closeness only, and an APF with all good features but not close to the POF;
- well distributed (usually uniform); Figure 4 shows an APF exhibiting a uniform distribution only and an APF with all good features but not uniformly distributed;
- very extended (in the best case the global optimum of each objective function belongs to the APF); Figure 5 shows an APF with only a good extension and one with all good features but not extended;
- of high cardinality; Figure 6 shows an APF with good cardinality only and an APF with all good features but poor cardinality.

_{p}indicator [39] can be considered as the combination of slight variations of the Generational Distance and the IGD indicator [25]. Therefore, it presents the advantage and limitations of these indicators, that is it accounts for all the features but it is ≻≻-complete only.

^{m}), where n is the number of objectives and m is the number of APF points. The fastest methods for calculating Hypervolume (e.g., LebMeasure [45], Hypervolume by Slicing Objectives algorithm [46]) lead to a O(m

^{2}n

^{3})complexity. The DOA indicator has a lower computational cost, presenting a O(nMm) complexity, where M is the number of POF points.

## 4. The Weakly Pareto Compliant Quality Indicator

_{i}

_{,A}, is determined from the sub-set of A containing the solutions dominated by i (Figure 8). Hence, if the number of solutions belonging to D

_{i}

_{,A}is not null, for each approximated solution a∊D

_{i}

_{,A}the Euclidean distance df

_{i}

_{,a}between a and i is computed as:

- n
- number of objective functions,
- f
_{k}_{,a} - value of k-th objective function of the approximated solution a,
- f
_{k}_{,i} - value of k-th objective function of optimal solution i.

_{i}

_{,A}(Figure 9) between i and the nearest approximated solution belonging to D

_{i}

_{,A}is computed in the objective function space as:

_{i}

_{,A}| is the number of solutions belonging to D

_{i}

_{,A}.

_{i}

_{,A}(similarly to d

_{i}

_{,A}) is computed for i considering the solutions of A not dominated by i (i.e., A\D

_{i}

_{,A}):

_{i}

_{,a}is a reduced distance (Figure 10) between i and a non-dominated solution a of A (i.e., ∀a ∊ A: i || a), i.e., computed only for objectives k with f

_{k}

_{,a}≥ f

_{k}

_{,i}:

_{i}

_{,a}is equal to df

_{i}

_{,a}when a $\in $ D

_{i}

_{,A}. Moreover, defining n

_{a}(n

_{a}< n) as the number of functions for which the f

_{k}

_{,a}− f

_{k}

_{,i}≥ 0 (f

_{k}

_{,a}≥ f

_{k}

_{,i}, k = 1,..,n

_{a}and f

_{k}

_{,a}< f

_{k}

_{,i}, k = n

_{a}+ 1,..,n) expression (16) can be rewritten as:

_{i}

_{,A}= min(d

_{i}

_{,A},r

_{i}

_{,A})

_{i}

_{,A}, Equation (14), is computed considering all solutions in A regardless of the dominance relation with i, and only this quantity is considered then the IGD indicator is obtained, that is:

_{i}

_{,A}is the enabling key for DOA ≻-completeness.

## 5. ≻≻-Completeness

_{i}

_{,A}is always lesser than s

_{i}

_{,B}for each point i ∊ POF. In other words,

- ifs
_{i,A}< s_{i,B}∀ i ∊ POF, - then$$DOA(A)=\frac{1}{\left|POF\right|}{\displaystyle \sum _{i=1}^{\left|POF\right|}{s}_{i,A}}<\frac{1}{\left|POF\right|}{\displaystyle \sum _{i=1}^{\left|POF\right|}{s}_{i,B}}=DOA(B).$$

_{i}

_{,B}and a a solution of A that strictly dominates b (a≺≺b); only four scenarios are possible (see Figure 12):

- A1. i≺≺b Λ i≼a
- B1. i≺≺b Λ i||a
- C1. i≺b Λ i||a
- D1. i||b Λ i||a

**Remark 1**.

_{i,A}≤ df

_{i,a}, in detail:

- s
_{i,A}= df_{i,a}iff s_{i,A}= d_{i,A}Λ d_{i,A}= df_{i,a}; - s
_{i,A}< df_{i,a}either if s_{i,A}= d_{i,A}Λ d_{i,A}= df_{i,a*}< df_{i,a}(where a*∊A and a*≠a) or if s_{i,A}= r_{i,A}(this implies that r_{i,A}< d_{i,A}≤ df_{i,a}).

**Remark**

**2.**

_{i,A}≤ rf

_{i,a}, in detail:

- s
_{i,A}= rf_{i,a}iff s_{i,A}= r_{i,A}Λ r_{i,A}= rf_{i,a}; - s
_{i,A}< rf_{i,a}either if s_{i,A}= r_{i,A}Λ r_{i,A}= rf_{i,a*}< rf_{i,a}(where a*∊A and a*≠a) or if s_{i,A}= d_{i,A}(this implies that d_{i,A}< r_{i,A}≤ rf_{i,a}).

_{i}

_{,A}< s

_{i}

_{,B}will be proved for the four scenarios A1–D1: this inequality naturally implies the ≻≻-completeness of the DOA indicator.

#### 5.1. A1. i≺≺b Λ i≼a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i≼a implies that s

_{i}

_{,A}≤ df

_{i}

_{,a}(see Remark 1).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that df

_{i}

_{,a}< df

_{i}

_{,b}.

- i≼a ⇒
- f
_{k}_{,i}≤ f_{k}_{,a}, ∀ k = 1,…, n - a≺≺b ⇒
- f
_{k}_{,a}< f_{k}_{,b}, ∀ k = 1,…, n

#### 5.2. B1. i≺≺b Λ i||a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i||a implies that s

_{i}

_{,A}≤ rf

_{i}

_{,a}(see Remark 2).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that rf

_{i}

_{,a}< df

_{i}

_{,b}. Proof is given in the next section because scenario C encompasses scenario B.

#### 5.3. C1. i≺b Λ i||a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i || a implies that s

_{i}

_{,A}≤ rf

_{i}

_{,a}(see Remark 2).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that rf

_{i}

_{,a}< df

_{i}

_{,b}. Ordering the n objective functions of solution a in such a way that the first n

_{a}(with n

_{a}< n) are greater than those of i and recalling that:

- i||a ⇒
- f
_{k}_{,i}< f_{k}_{,a}, ∀ k = 1,…, n_{a}f_{k}_{,i}≥ f_{k}_{,a}, ∀ k = n_{a}+ 1,…, n - i≺b ⇒
- f
_{k}_{,i}≤ f_{k}_{,b}, ∀ k = 1,…, n - a≺≺b ⇒
- f
_{k}_{,a}< f_{k}_{,b}, ∀ k = 1,…, n

#### 5.4. D1. i||b Λ i||a

_{i}

_{,B}= r

_{i}

_{,B}= rf

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i||a implies that s

_{i}

_{,A}≤ rf

_{i}

_{,a}(see Remark 2).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that rf

_{i}

_{,a}< rf

_{i}

_{,b}.

_{a}are greater than those of i, ordering the n objective functions of solution b in such a way that the first n

_{b}are greater than those of i (with n

_{a}≤ n

_{b}< n, since a≻≻b Λ i||a implies n

_{a}≤ n

_{b}, while i||b implies n

_{b}< n) and recalling that:

- i||a ⇒
- f
_{k}_{,i}< f_{k}_{,a}, ∀ k = 1,…, n_{a}f_{k}_{,i}≥ f_{k}_{,a}, ∀ k = n_{a}+ 1,…, n - i||b ⇒
- f
_{k}_{,i}< f_{k}_{,b}, ∀ k = 1,…, n_{b}f_{k}_{,i}≥ f_{k}_{,b}, ∀ k = n_{b}+ 1,…, n - a≺≺b ⇒
- f
_{k}_{,a}< f_{k}_{,b}, ∀ k = 1,..., n

## 6. ≻-Completeness

_{i}

_{,A}is never greater than s

_{i}

_{,B}(for each point i∊POF) and always exists a point i*∊ POF for which s

_{i*}

_{,A}is lesser than s

_{i*}

_{,B}:

- ifs
_{i,A}≤ s_{i,B}∀ i ∊ POF Λ ∃ i*∊POF: s_{i*}_{,A}< s_{i*,B} - then$$\mathrm{DOA}(\mathrm{A})=\frac{1}{\left|POF\right|}\left({s}_{i*,A}+{\displaystyle \sum _{i=2}^{\left|POF\right|}{s}_{i,A}}\right)<\frac{1}{\left|POF\right|}\left({s}_{i*,B}+{\displaystyle \sum _{i=2}^{\left|POF\right|}{s}_{i,B}}\right)=\mathrm{DOA}(\mathrm{B}).$$

_{i}

_{,B}for a point i ∊ POF and a a solution of A that dominates b (a≺b). Moreover, the ≻-completeness of DOA is proved in the worst and most general case, i.e., when ∀ b∊B ∄ a∊A: a≺≺b (i.e., a≺b Λ a⊀⊀b, limit case); only five scenarios are possible (see Figure 13):

- A2. i≺≺b Λ i≺a
- B2. i≺≺b Λ i||a
- C2. i≺b Λ i≼a
- D2. i≺b Λ i||a
- E2. i||b Λ i||a

_{i,A}< s

_{i}

_{,B}is verified ∀i for the three scenarios A2, B2 and C2. While for the remaining two scenarios D2 and E2 we will prove that the following two sufficient conditions hold:

- α.
- s
_{i}_{,A}≤ s_{i}_{,B} - β.
- ∃ i* ∊ POF: s
_{i*}_{,A}< s_{i*}_{,B}.

#### 6.1. A2. i≺≺b Λ i≺a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i≺a implies that s

_{i}

_{,A}≤ df

_{i}

_{,a}(see Remark 1).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that df

_{i}

_{,a}< df

_{i}

_{,b}.

- i≺a ⇒
- f
_{k}_{,i}≤ f_{k}_{,a}, ∀ k = 1,…, n - a≺b ⇒
- f
_{k}_{,a}≤ f_{k}_{,b}, ∀ k = 1,…, n Λ ∃ j: f_{j}_{,a}< f_{j}_{,b}

#### 6.2. B2. i≺≺b Λ i||a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i||a implies that s

_{i}

_{,A}≤ rf

_{i}

_{,a}(see Remark 2).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that rf

_{i}

_{,a}< df

_{i}

_{,b}.

_{a}(with n

_{a}< n) are greater than those of i and recalling that:

- i||a ⇒
- f
_{k}_{,i}< f_{k}_{,a}, ∀ k = 1,…, n_{a}f_{k}_{,i}≥ f_{k}_{,a}, ∀ k = n_{a}+ 1,…, n - i≺≺b ⇒
- f
_{k}_{,i}< f_{k}_{,b}, ∀ k = 1,..., n - a≺b ⇒
- f
_{k}_{,a}≤ f_{k}_{,b}, ∀ k = 1,…, n

#### 6.3. C2. i≺b Λ i≼a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i≼a implies that s

_{i}

_{,A}≤ df

_{i}

_{,a}(see Remark 1).

_{i}

_{,A}< s

_{i}

_{,B}it is sufficient to demonstrate that df

_{i}

_{,a}< df

_{i}

_{,b}.

- i≼a ⇒
- f
_{k}_{,i}≤ f_{k}_{,a}, ∀ k = 1,…, n - a≺b ⇒
- f
_{k}_{,a}≤ f_{k}_{,b}, ∀ k = 1,…, n Λ ∃ j: f_{j}_{,a}< f_{j}_{,b}

#### 6.4. D2. i≺b Λ i||a

_{i}

_{,B}= d

_{i}

_{,B}= df

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i||a implies that s

_{i}

_{,A}≤ rf

_{i}

_{,a}(see Remark 2).

_{i}

_{,A}≤ s

_{i}

_{,B}it is sufficient to demonstrate that rf

_{i}

_{,a}≤ df

_{i}

_{,b}.

_{a}(with n

_{a}< n) are greater than those of i and recalling that:

- i||a ⇒
- f
_{k}_{,i}< f_{k}_{,a}, ∀ k = 1,…, n_{a}f_{k}_{,i}≥ f_{k}_{,a}, ∀ k = n_{a}+ 1,…, n - i≺b ⇒
- f
_{k}_{,i}≤ f_{k}_{,b}, ∀ k = 1,…, n Λ ∃ h: f_{h}_{,i}< f_{h}_{,b} - a≺b ⇒
- f
_{k}_{,a}≤ f_{k}_{,b}, ∀ k = 1,…, n Λ ∃ j: f_{j}_{,a}< f_{j}_{,b}

_{i}

_{,a}is strictly lesser than df

_{i}

_{,b}when 1 ≤ j ≤ n

_{a}, since

_{i}

_{,A}≤ s

_{i}

_{,B}. In particular, rf

_{i}

_{,a}= df

_{i}

_{,b}iff f

_{k}

_{,a}= f

_{k}

_{,b}∀ k = 1,…,n

_{a}Λ f

_{k}

_{,i}= f

_{k}

_{,b}∀ k = n

_{a}+ 1,…,n. In this case, obviously, f

_{h}

_{,i}< f

_{h}

_{,a}= f

_{h}

_{,b}with 1 ≤ h ≤ n

_{a}and f

_{j}

_{,a}< f

_{j}

_{,i}= f

_{j}

_{,b}with j > n

_{a}. As said before, the proof that ∃ i*∊POF: s

_{i*}

_{,A}< s

_{i*}

_{,B}has been reported in Appendix A.

#### 6.5. E2. i||b Λ i||a

_{i}

_{,B}= r

_{i}

_{,B}= rf

_{i}

_{,b}, because b is the solution which provides s

_{i}

_{,B}. Moreover, i||a implies that s

_{i}

_{,A}≤ rf

_{i}

_{,a}(see Remark 2).

_{i}

_{,A}≤ s

_{i}

_{,B,}it is sufficient to demonstrate that rf

_{i}

_{,a}≤ rf

_{i}

_{,b}. Ordering the n objectives f of solution a in such a way that the first n

_{a}are greater than those of i, ordering the n objectives f of solution b in such a way that the first n

_{b}are greater than those of i (with n

_{a}≤ n

_{b}< n, since a≺b Λ i||a implies n

_{a}≤ n

_{b}, while i||b implies n

_{b}< n) and recalling that:

- i||a ⇒
- f
_{k}_{,i}< f_{k}_{,a}, ∀ k = 1,…, n_{a}f_{k}_{,i}≥ f_{k}_{,a}, ∀ k = n_{a}+ 1,…, n - i||b ⇒
- f
_{k}_{,i}< f_{k}_{,b}, ∀ k = 1,…, n_{b}f_{k}_{,i}≥ f_{k}_{,b}, ∀ k = n_{b}+ 1,..,n - a≺b ⇒
- f
_{k}_{,a}≤ f_{k}_{,b}, ∀ k = 1,…, n Λ ∃ j: f_{j}_{,a}< f_{j}_{,b}

- rf
_{i}_{,a}< rf_{i}_{,b}if n_{b}≠ n_{a}; - rf
_{i}_{,a}≤ rf_{i}_{,b}if n_{b}= n_{a}.

_{i}

_{,a}= rf

_{i}

_{,b}iff f

_{k}

_{,a}= f

_{k}

_{,b}∀ k = 1,..,n

_{a}. In this case, obviously, f

_{j}

_{,a}< f

_{j}

_{,b}≤ f

_{j}

_{,i}whit j > n

_{a}. As said before, the proof that ∃ i*∊POF: s

_{i*}

_{,A}< s

_{i*}

_{,B}has been reported in Appendix A.

## 7. ⋫-Compatibility

**Remark 3**.

- s
_{i,A}< df_{i,a’}when a’ is dominated by i; - s
_{i,A}< rf_{i,a’}when a’ is not dominated by i.

_{i}

_{,A}≠ df

_{i}

_{,a’}and s

_{i}

_{,A}≠ rf

_{i}

_{,a’}, by which follows that DOA(A) does not change its value if a’ is moved off from A. These means that an APF B = A\{a’} has the same quality indicator value, i.e., DOA(A) = DOA(B). Using DOA seems B equivalent to A while A, having one more solution, is better than B [11]. Then the proposed method is not a ⊳-complete quality indicator. On the other hand, DOA together with its interpretation function is a ≻-complete comparison method, as it is demonstrated before.

**Remark 4**.

**Remark 5**.

## 8. DOA Validation

- convex and connected;
- non-convex and connected;
- convex and disconnected.

## 9. Conclusions and Future Work

## Author Contributions

## Conflicts of Interest

## Appendix A.

**Hypothesis.**

**Thesis.**

_{i*,A}= d

_{i*,A}< s

_{i*,B}.

**Proof by Reductio ad absurdum.**

- df
_{i*}_{,a*}= d_{i*}_{,A}⇒ df_{i*}_{,a*}≤ df_{i*}_{,a’}∀ a’∊D_{i*}_{,A} - df
_{i*}_{,a*}< r_{i*}_{,A}⇒ df_{i*}_{,a*}< rf_{i*}_{,a″}∀ a″∊A\D_{i*}_{,A} - when i≼a≺b ⇒ df
_{i}_{,a}< df_{i}_{,b}(see Section C2) - when i||a ∧ i≺b ∧ a≺b ⇒ rf
_{i}_{,a}≤ df_{i}_{,b}(see Section D2) - when i||a ∧ i||b ∧ a≺b ⇒ rf
_{i}_{,a}≤ rf_{i}_{,b}(see Section E2) - ∃ i*∊ POF and ∃ a*∊A: s
_{i*}_{,A}= d_{i*}_{,A}= df_{i*}_{,a*}< r_{i*}_{,A}(proved in Appendix B).

_{i*,B}≤ s

_{i*,A};

_{i*,B}: s

_{i*,B}= df

_{i*,b’}(≤ df

_{i*,a*}= s

_{i*,A})

_{i*,B}: s

_{i*,B}= rf

_{i*,b″}(≤ df

_{i*,a*}= s

_{i*,A}).

- (a)
- a*≺b’
- (b)
- a’≺b’, where a’∊D
_{i*}_{,A} - (c)
- a″≺b’, where a″∊A\D
_{i*}_{,A}.

_{i*}

_{,B}= rf

_{i*}

_{,b″}implies i*||b″, i.e., ∄ a∊D

_{i*}

_{,A}: a≺b″. Hence only one scenario has to be analyzed: a″≺b″, where a″∊A\D

_{i*}

_{,A}.

## Appendix B.

**Hypothesis.**

**Thesis.**

_{i,A}= d

_{i,A}= df

_{i,a}< r

_{i,A}.

**Proof.**

#### B.1. Proof by Induction with n = 2

**Hypothesis.**

**Proof.**

_{i,A}and |A\D

_{i,A}| = 0. This implies that:

**Hypothesis.**

**Proof.**

_{1},a

_{2}}, if ∃ i∊POF: i≼a

_{1}∧ i≼a

_{2}, then a

_{1}and a

_{2}belong to D

_{i,A}and |A\D

_{i,A}| = 0:

_{i,A}between a

_{1}and a

_{2}.

_{1}, i

_{2}belonging to the POF for which i

_{1}≼a

_{1}and i

_{2}≼a

_{2}. Obviously, i

_{1}||a

_{2}and i

_{2}||a

_{1}. Hence, without loss of generality it is assumed that:

_{i}

_{1,A}= d

_{i}

_{1,A}= df

_{i}

_{1,a1}< rf

_{i}

_{1,a2}= r

_{i}

_{1,A}∨ s

_{i}

_{2,A}= d

_{i}

_{2,A}= df

_{i}

_{2,a2}< rf

_{i}

_{2,a1}= r

_{i}

_{2,A}. The proof is by reductio ad absurdum. Supposing that:

_{i}

_{1,A}= r

_{i}

_{1,A}= rf

_{i}

_{1,a2}< df

_{i}

_{1,a1}= d

_{i}

_{1,A}∧ s

_{i}

_{2,A}= r

_{i}

_{2,A}= rf

_{i}

_{2,a1}< df

_{i}

_{2,a2}= d

_{i}

_{2,A},

#### B.2. Proof by Induction with n = 3

**Hypothesis.**

**Proof.**

**Hypothesis.**

**Proof.**

_{1},a

_{2}}, if ∃ i∊POF: i≼a

_{1}Λ i≼a

_{2}, then a

_{1}and a

_{2}belong to D

_{i,A}and |A\D

_{i,A}| = 0 ⇒ The proof is the same provided in (A9).

_{1}, i

_{2}belonging to the POF for which i

_{1}≼a

_{1}and i

_{2}≼a

_{2}. Obviously, i

_{1}||a

_{2}and i

_{2}||a

_{1}. Hence, without loss of generality it is assumed that:

_{i}

_{1,A}= d

_{i}

_{1,A}= df

_{i}

_{1,a1}< rf

_{i}

_{1,a2}= r

_{i}

_{1,A}∨ s

_{i}

_{2,A}= d

_{i}

_{2,A}= df

_{i}

_{2,a2}< rf

_{i}

_{2,a1}= r

_{i}

_{2,A}.

_{i}

_{1,A}= r

_{i}

_{1,A}= rf

_{i}

_{1,a2}< df

_{i}

_{1,a1}= d

_{i}

_{1,A}∧ s

_{i}

_{2,A}= r

_{i}

_{2,A}= rf

_{i}

_{2,a1}< df

_{i}

_{2,a2}= d

_{i}

_{2,A},

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**Figure 1.**In this case A and B are incomparable in terms of Pareto dominance, but A is preferable to B because A is closer to the Pareto-optimal front (POF), more extended, more populated, and better distributed.

**Figure 3.**An APF (blue dots) with all good features but not close to the POF and another (red triangles) that is only close to the POF.

**Figure 4.**An APF (blue dots) with all good features but not uniformly distributed and another (red triangles) that is only uniformly distributed.

**Figure 5.**An APF blue dots) with all good features but not extended and another (red triangles) that is only extended.

**Figure 6.**An APF (blue dots) with all good features but with poor cardinality and another (red triangles) that has only high cardinality.

**Figure 12.**Possible scenarios for A≻≻B.(

**a**) scenario A1: most relevant case; (

**b**) scenario B1 or C1; (

**c**) scenario C1: limit case; (

**d**) scenario D1.

**Figure 13.**Possible scenarios for A≻B. (

**a**) scenario A2 or C2; (

**b**) scenario B2 or D2; (

**c**) scenario C2: limit case; (

**d**) scenario D2: limit case; (

**e**) scenario E2.

**Figure 14.**Convex and connected POF (solid line), APF1 (◊), APF2 (+), APF3 (○), APF4 (□) and APF5 (●).

**Figure 15.**Non-convex and connected POF (solid line), APF1 (◊), APF2 (+), APF3 (○), APF4 (□) and APF5 (●).

**Figure 16.**Convex and disconnected POF (solid line), APF1 (◊), APF2 (+), APF3 (○), APF4 (□) and APF5 (●).

**Table 1.**Dominance relations between two solutions [11].

Symbol | Relation | Description |
---|---|---|

x^{1}≺≺x^{2} | strictly dominance x^{1} strictly dominates x^{2} | x^{1} is better than x^{2} with respect to each objective function |

x^{1}≺x^{2} | dominance x^{1} dominates x^{2} | x^{1} is not worse than x^{2} with respect to each objective function and x^{1} is better than x^{2} by at least one objective function |

x^{1}≼x^{2} | weakly dominance x^{1} weakly dominates x^{2} | x^{1} is not worse than x^{2} with respect to each objective function |

x^{1}||x^{2} | incomparability x^{1} and x^{2} are incomparable | x^{1} and x^{2} do not weakly dominate each other |

**Table 2.**Dominance relations between two APFs [11].

Symbol | Relation | Description |
---|---|---|

A≺≺B | A strictly dominates B | each solution belonging to B is strictly dominated by a solution belonging to A |

A≺B | A dominates B | each solution belonging to B is dominated by a solution belonging to A |

A⊲B | A is better than B | each solution belonging to B is weakly dominated by a solution belonging to A, and A≠B |

A≼B | A weakly dominates B | each solution belonging to B is weakly dominated by a solution belonging to A |

A||B | A and B are incomparable | A and B do not weakly dominate each other |

Indicator | Closeness | Distribution | Extension | Cardinality |
---|---|---|---|---|

Average Distance from Reference Set [30] | ⚫ | ⚫ | ⚫ | ⚫ |

Chi-Square-Like Deviation Measure [33] | ⚫ | |||

Completeness Indicator [31,32] | ⚫ | ⚫ | ⚫ | ⚫ |

Enclosing Hypercube [11] | ⚫ | |||

Generational Distance [34] | ⚫ | |||

Hypervolume [1] | ⚫ | ⚫ | ⚫ | ⚫ |

Inverted Generational Distance [29] | ⚫ | ⚫ | ⚫ | ⚫ |

M_{1}* [13] | ⚫ | |||

M_{2}* [13] | ⚫ | |||

M_{3}* [13] | ⚫ | |||

Maximum Pareto Front Error [34] | ⚫ | |||

Outer Diameter [26] | ⚫ | |||

Overall Nondominated Vector Generation [34] | ⚫ | |||

Overall Pareto Spread [35] | ⚫ | |||

Potential Function [27] | ⚫ | ⚫ | ⚫ | ⚫ |

Seven Points Average Distance [36] | ⚫ | ⚫ | ||

Spacing [37] | ⚫ | |||

Unary ε-Indicator [11,26] | ⚫ | |||

Uniform Distribution [38] | ⚫ | |||

Worst Distance from Reference Set [30] | ⚫ | |||

Δ [8] | ⚫ | ⚫ | ||

Δ_{p} [39,40] | ⚫ | ⚫ | ⚫ | ⚫ |

POF | APF | Closeness | Distribution | Extension | Cardinality | DOA |
---|---|---|---|---|---|---|

Convex and connected (see Figure 14) | APF1(◊) | poor | poor | poor | poor | 0.71040 |

APF2(+) | good | poor | poor | poor | 0.16940 | |

APF3(○) | good | good | poor | poor | 0.16287 | |

APF4(□) | good | good | good | poor | 0.10253 | |

APF5(●) | good | good | good | good | 0.06431 | |

Non-convex and connected (see Figure 15) | APF1(◊) | poor | poor | poor | poor | 0.79167 |

APF2(+) | good | poor | poor | poor | 0.24303 | |

APF3(○) | good | good | poor | poor | 0.23465 | |

APF4(□) | good | good | good | poor | 0.09301 | |

APF5(●) | good | good | good | good | 0.06993 | |

Convex and disconnected (see Figure 16) | APF1(◊) | poor | poor | poor | poor | 0.69510 |

APF2(+) | good | poor | poor | poor | 0.16866 | |

APF3(○) | good | good | poor | poor | 0.16810 | |

APF4(□) | good | good | good | poor | 0.07254 | |

APF5(●) | good | good | good | good | 0.06331 |

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## Share and Cite

**MDPI and ACS Style**

Dilettoso, E.; Rizzo, S.A.; Salerno, N. A Weakly Pareto Compliant Quality Indicator. *Math. Comput. Appl.* **2017**, *22*, 25.
https://doi.org/10.3390/mca22010025

**AMA Style**

Dilettoso E, Rizzo SA, Salerno N. A Weakly Pareto Compliant Quality Indicator. *Mathematical and Computational Applications*. 2017; 22(1):25.
https://doi.org/10.3390/mca22010025

**Chicago/Turabian Style**

Dilettoso, Emanuele, Santi Agatino Rizzo, and Nunzio Salerno. 2017. "A Weakly Pareto Compliant Quality Indicator" *Mathematical and Computational Applications* 22, no. 1: 25.
https://doi.org/10.3390/mca22010025