Solving Multi-Objective Decision Problems: MATLAB's Power Tool for Linear Programming Multi-Objective Optimization

# Solving Multi-objective Decision Problems: MATLAB Linear Programming for Multi-objective Optimization

## 1. Overview of Multi-objective Decision Problems

A multi-objective decision problem (MDP) is a type of optimization problem where there are multiple, conflicting objectives. These objectives cannot be optimized simultaneously, necessitating trade-offs and compromises among them. MDPs are ubiquitous in the real world, such as in portfolio optimization, supply chain management, and sustainable development planning.

Characteristics of MDPs include:

- **Multiple objectives:** Existence of multiple, conflicting objectives that cannot be optimized at the same time.
- **Trade-offs and compromises:** Necessity to weigh objectives against each other to find an acceptable solution.
- **Decision variables:**可控 variables that affect the objective values.
- **Constraints:** Conditions that limit the range of values for decision variables.

## 2. MATLAB Linear Programming for Multi-objective Optimization

In this section, we will explore modeling and solving multi-objective optimization problems within linear programming in MATLAB. Multi-objective optimization involves simultaneously optimizing multiple objective functions, a common scenario in real-world problems. MATLAB offers various multi-objective optimization algorithms to address a range of linear programming issues.

### 2.1 Establishing a Linear Programming Model

#### 2.1.1 Decision Variables and Objective Functions

Linear programming multi-objective optimization models typically include multiple decision variables and multiple objective functions. Decision variables represent the controllable parameters to be optimized, while objective functions represent the objectives to be maximized or minimized.

For instance, consider a portfolio optimization problem where the goal is to maximize investment returns while minimizing risk. Decision variables could be the amounts invested in different assets, and objective functions could be the investment return and risk.

#### 2.1.2 Constraints

In addition to objective functions, linear programming models may also include constraints that limit the range of values for decision variables. Constraints can be in the form of equalities or inequalities.

For example, in a portfolio optimization problem, the total investment amount may be restricted, or investments in specific assets may be limited. These constraints ensure that the model's solutions are feasible.

### 2.2 Multi-objective Optimization Algorithms

MATLAB provides various multi-objective optimization algorithms to address linear programming problems. These algorithms work by aggregating multiple objective functions into a single objective function that can be optimized.

#### 2.2.1 Weighted Sum Method

The weighted sum method is a simple yet commonly used multi-objective optimization algorithm. It creates a single objective function by multiplying each objective function by a weight and summing them. Weights represent the relative importance of each objective function to others.

% Weighted sum method
weights = [0.5, 0.5];   % Objective function weights
f = @(x) weights(1) * f1(x) + weights(2) * f2(x);

#### 2.2.2 ε-constraint Method

The ε-constraint method is an algorithm that converts all objective functions into constraints. It works by setting a target value ε for each objective function and then optimizing one objective function while treating the others as constraints.

% ε-constraint method
epsilon = 0.5;   % Target value for objective function
f = @(x) f1(x);
constraints = @(x) [f2(x) <= epsilon];

#### 2.2.3 Goal Programming Method

The goal programming method is an algorithm that converts all objective functions into constraints. It works by setting a target value and weight for each objective function and then optimizing one objective function while treating the others as constraints.

% Goal programming method
target = [0.5, 0.5];   % Objective function target values
weights = [1, 1];   % Objective function weights
f = @(x) f1(x);
constraints = @(x) [f2(x) <= target(2), f1(x) >= target(1)];

## 3.1 Problem Modeling and Data Preparation

#### 3.1.1 Problem Description

In multi-objective linear programming problems, we typically need to optimize multiple conflicting objective functions simultaneously. These objective functions are often in competition with each other, meaning that optimizing one may cause the values of others to decrease. To solve this problem, we need to find a set of Pareto optimal solutions that balance all objective functions.

#### 3.1.2 Data Collection and Processing

Before modeling begins, data related to the problem needs to be collected and processed. This data includes:

- **Decision variables:** Variables to be optimized, usually repres