# MCDM, EMO and Hybrid Approaches: Tutorial and Review

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## Abstract

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## 1. Introduction

## 2. Multi-Objective Optimization

- $\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)=\mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)$⇔${f}_{i}\left({\mathbf{x}}^{\left(1\right)}\right)={f}_{i}\left({\mathbf{x}}^{\left(2\right)}\right)$ : $i\in \{1,2,\dots ,p\}$
- $\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\ge \mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)$⇔${f}_{i}\left({\mathbf{x}}^{\left(1\right)}\right)\ge {f}_{i}\left({\mathbf{x}}^{\left(2\right)}\right)$ : $i\in \{1,2,\dots ,p\}$
- $\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)>\mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)$⇔${f}_{i}\left({\mathbf{x}}^{\left(1\right)}\right)>{f}_{i}\left({\mathbf{x}}^{\left(2\right)}\right)$ : $i\in \{1,2,\dots ,p\}$

#### 2.1. Dominance Concept

- ${\mathbf{x}}^{\left(1\right)}$ strongly dominates ${\mathbf{x}}^{\left(2\right)}$⇔$\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)>\mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)$;
- ${\mathbf{x}}^{\left(1\right)}$ (weakly) dominates ${\mathbf{x}}^{\left(2\right)}$⇔$\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\ge \mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)\wedge \mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\ne \mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)$;
- ${\mathbf{x}}^{\left(1\right)}$ and ${\mathbf{x}}^{\left(2\right)}$ are non-dominated with respect to each other ⇔$\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\ngeqq \mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)\wedge \mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)\ngeqq \mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)$.

#### 2.2. Decision Making

- ${\mathbf{x}}^{\left(1\right)}\succ {\mathbf{x}}^{\left(2\right)}\iff $${\mathbf{x}}^{\left(1\right)}$ is preferred over ${\mathbf{x}}^{\left(2\right)}$;
- ${\mathbf{x}}^{\left(1\right)}\sim {\mathbf{x}}^{\left(2\right)}\iff $${\mathbf{x}}^{\left(1\right)}$ and ${\mathbf{x}}^{\left(2\right)}$ are equally preferable;
- ${\mathbf{x}}^{\left(1\right)}\Vert {\mathbf{x}}^{\left(2\right)}\iff $${\mathbf{x}}^{\left(1\right)}$ and ${\mathbf{x}}^{\left(2\right)}$ are incomparable;

- ${\mathbf{x}}^{\left(1\right)}\u2ab0{\mathbf{x}}^{\left(2\right)}\iff $${\mathbf{x}}^{\left(1\right)}$ is either preferred over ${\mathbf{x}}^{\left(2\right)}$ or they are equally preferred.

- If ${\mathbf{x}}^{\left(1\right)}$ dominates ${\mathbf{x}}^{\left(2\right)}$⇒${\mathbf{x}}^{\left(1\right)}\succ {\mathbf{x}}^{\left(2\right)}$.

- If ${\mathbf{x}}^{\left(1\right)}\succ {\mathbf{x}}^{\left(2\right)}\iff V\left(\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\right)>V\left(\mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)\right)$;
- If ${\mathbf{x}}^{\left(1\right)}\sim {\mathbf{x}}^{\left(2\right)}\iff V\left(\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\right)=V\left(\mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)\right)$;
- If ${\mathbf{x}}^{\left(1\right)}\u2ab0{\mathbf{x}}^{\left(2\right)}\iff V\left(\mathbf{f}\left({\mathbf{x}}^{\left(1\right)}\right)\right)\ge V\left(\mathbf{f}\left({\mathbf{x}}^{\left(2\right)}\right)\right)$.

#### 2.3. Preference Eliciting and Modeling

- Asking about goals or aspiration levels for the objectives;
- Pairwise comparisons of solutions in objective space;
- Asking the DM which objectives and by how much they would be willing to worsen to allow improvements in other objectives;
- Asking the DM to specify exact marginal rates of substitution between objectives and a reference objective (trade-offs);
- Directly asking for the search direction;
- Directly asking the importance of each objective to get an idea of weights or to rank the objectives;
- Yes–no questions, for instance: Do you like this search direction?

## 3. Incorporating Decision Maker’s Preferences

#### 3.1. A Priori Approach

#### 3.2. A Posteriori Approach

#### 3.3. Interactive Approach

## 4. MCDM Interactive Techniques

- Phases of computing and decision making would alternate: the human would guide the computer (algorithm) towards the most preferred solution;
- The human and the computer were performing tasks that they were good at;
- Learning (of one’s preferences) was possible;
- The ideas were based on using linear programming or non-linear programming;
- Systematic progress towards the most preferred solution would take place;
- The methods would generally operate with non-dominated solutions, in other words, allow exploration of the Pareto-optimal (non-dominated) frontier.

- STEP method due to Benayoun et al. (1971) [45];
- GDF method due to Geoffrion, Dyer, and Feinberg (1972) [36];
- ZW method due to Zionts and Wallenius (1976) [37];
- Reference point method due to Wierzbicki (1980) [20];
- Reference direction method due to Korhonen and Laakso (1986) [46];
- Pareto Race due to Korhonen and Wallenius (1988) [47];

#### 4.1. STEP Method (Benayoun et al., 1971) [45]

#### 4.2. GDF Algorithm (Geoffrion, Dyer, and Feinberg, 1972) [36]

- Choose an initial solution ${\mathbf{x}}^{\left(1\right)}\in \mathbf{X}$. Set $k=1$ (iteration counter).
- Determine an optimal solution ${\mathbf{y}}^{\left(k\right)}$ of the direction-finding problem$$\begin{array}{cc}\text{Maximize}\hfill & {\nabla}_{\mathbf{x}}U({f}_{1}\left(\mathbf{x}\right),\dots ,{f}_{p}\left(\mathbf{x}\right))\mathbf{y}\hfill \\ \text{subject to}\hfill & \mathbf{y}\in \mathbf{X}.\hfill \end{array}$$
- Set ${\mathbf{d}}^{\left(k\right)}={\mathbf{y}}^{\left(k\right)}-{\mathbf{x}}^{\left(k\right)}$. This step determines the “best” search direction based on a linear (first-order Taylor expansion) approximation of U.
- Next, solve the step-size problem for an optimal t:$$\begin{array}{cc}\text{Maximize}\hfill & U({f}_{1}({\mathbf{x}}^{\left(k\right)}+t{\mathbf{d}}^{\left(k\right)}),\dots ,{f}_{p}({\mathbf{x}}^{\left(k\right)}+t{\mathbf{d}}^{\left(k\right)}))\hfill \\ \text{subject to}\hfill & 0\le t\le 1.\hfill \end{array}$$
- Set ${\mathbf{x}}^{(k+1)}={\mathbf{x}}^{\left(k\right)}+{t}^{k}{\mathbf{d}}^{\left(k\right)},k=k+1$, and return to the direction-finding problem. Theoretical termination criterion is satisfied if ${\mathbf{x}}^{\left(k\right)}$ and ${\mathbf{x}}^{(k+1)}$ are equal.

#### 4.3. ZW Method (Zionts and Wallenius, 1976) [37]

#### 4.4. Reference Point Method (Wierzbicki, 1980) [20]

#### 4.5. Reference Direction Approach (Korhonen and Laakso, 1986) [46]

#### 4.6. Pareto Race (Korhonen and Wallenius, 1988) [47]

## 5. EMO Introduction and History

- Step 1: Create a random initial population (i.e., a set of solution points in the decision space).
- Step 2: Evaluate the individuals (i.e., the solution points) in the population with respect to objective(s) and constraints, if present, and assign fitness (i.e., quality measure).
- Step 3: Repeat the generations (i.e., iterations of the evolutionary algorithm) until termination.
- Substep 1: Select the fitter individuals (referred to as parents) from the population for reproduction (i.e., producing new solution points through genetic operators of crossover and mutation).
- Substep 2: Produce new individuals (referred to as offspring) through crossover and mutation operators.
- Substep 3: Evaluate the new individuals and assign fitness.
- Substep 4: Replace the low-fitness individuals in the population with high-fitness individuals that may have been generated through crossover and mutation.

- Step 4: Report the highest fitness individual as the output.

## 6. Hybrid Methods

- Stage at which preference is incorporated;
- Manner in which preference information is elicited;
- Type of preference modeling performed;
- Integration of preference model with EMO search;
- Choice of the EMO, i.e., Pareto-based, indicator-based or decomposition-based.

#### 6.1. Phelps and Köksalan (2003) [38]

#### 6.2. Branke, Greco, Słowiński, and Zielniewicz (2009) [41]

#### 6.3. Deb, Sinha, Korhonen, and Wallenius (2010) [42]

#### 6.4. Sinha, Deb, Korhonen, and Wallenius (2014) [43]

## 7. Interaction Styles, Behavioral Considerations, and Future Work

- DM providing erroneous preferences;
- DM providing conflicting or inconsistent preferences;
- DM preference structure changing in different regions of the objective space;
- DM preference structure changing as a function of learning;
- DM becoming biased (anchored) based on the initial set of options presented;
- DM unable to compare two options in terms of either dominance or indifference.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Dominance concept for a maximization problem where A dominates B and C; A, D, and E are non-dominated.

**Figure 2.**Non-dominated set from a discrete set of points and a Pareto-optimal front that dominates the entire search space.

**Figure 11.**A light beam approach integrated within an EMO that finds a crowded set of points close to the Pareto-frontier based on the aspirations of the DM.

**Figure 12.**Projection of a feasible and infeasible reference point on the Pareto-optimal frontier within an EMO.

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Sinha, A.; Wallenius, J.
MCDM, EMO and Hybrid Approaches: Tutorial and Review. *Math. Comput. Appl.* **2022**, *27*, 112.
https://doi.org/10.3390/mca27060112

**AMA Style**

Sinha A, Wallenius J.
MCDM, EMO and Hybrid Approaches: Tutorial and Review. *Mathematical and Computational Applications*. 2022; 27(6):112.
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**Chicago/Turabian Style**

Sinha, Ankur, and Jyrki Wallenius.
2022. "MCDM, EMO and Hybrid Approaches: Tutorial and Review" *Mathematical and Computational Applications* 27, no. 6: 112.
https://doi.org/10.3390/mca27060112