# An Improved SPEA2 Algorithm with Local Search for Multi-Objective Investment Decision-Making
## 3 Multi-Objective Decision-Making in Enterprise Investment  
In the traditional sense, profit oriented investment decisions can be expressed as following. The annual investment budget of one enterprise in one \(n\) year plan period is \(b_{i}\), \((i=1,2,...,n)\), and there are \(m\) different projects that can be invested. \(d_{ij}\) is known to represent the profit from the \(j\)th (\(j=1,2,...,m\)) per unit investment project in year \(i\). The goal is to maximize the total profit during the planned period. However, the profit based single objective decision often fails to take account of the complexity of investment. The evaluation of an investment is often carried out from many angles, such as investment profits, investment risks and the environmental cost of the project. In this work, investment profits, investment risks and environmental cost of the project are considered in the investment decision-making. In economics, environmental cost (EC) [44] can be understood as the consumption of energy corresponding to investment. Table 1 shows the investment plan to be determined, in which the variable \(I_{ij}\) represents the capital invested in the \(j\)th investment project in year \(i\), \(R_{i}\) is the investment risk of \(i\)th project, and \(E_{i}\) is the energy consumption per unit output of \(i\)th project. Thus, the optimization objective considering investment risk, investment profit and environmental cost can be expressed as:  
\[\max \text{Profit}=\sum_{i}\sum_{j}d_{ij}I_{ij},\] \[\min [\text{Risk}=\sum_{i}\sum_{j}R_{i}I_{ij},\text{EC}=\sum_{i}\sum_ {j}E_{j}d_{ij}I_{ij}].\]  
Among them, the total amount of investment in the \(i\)th year should not exceed the investment budget in the \(i\)th year, and the investment amount should not be negative. The above model is a general  
Figure 3: Pareto boundary of two test functions and comparison of convergence results of different methods. investment planning model, which can be slightly modified in practical application and extended to other economic activities (such as: selection of construction projects, utilization of foreign capital). In this model, the values of \(d_{ij}\), \(R_{j}\), and \(E_{j}\) are known as inputs of the investment optimization decision model, which are needed to solve the multi-objective optimization decision problem. \(I_{ij}\) is the optimization parameter, which is the optimized output of this paper. Profit, risk and environmental consumption are the optimization objectives. The purpose of investment decision-making is to seek the optimal investment decision (i.e., the optimal value of \(I_{ij}\)) to maximize investment profit and minimize investment risk and environmental consumption. A specific investment plan is used as a case to discuss the specific application of ant colony algorithm. During the three-year planning period, an enterprise needs to invest four development projects as follows: Project A can be invested at the beginning of the first year to the third year. It is estimated that every 10,000 dollars of investment can make a profit of 2000 dollars from the investment projects, and the annual capital and profits of that year can be reinvested into the investment plan, while project B needs to be invested at the beginning of the first year. After two years of investment, it can make a profit of 5000 dollars per 10,000 dollars, and it can be reinvested into the investment plan. However, the maximum investment for the project should not exceed 200,000 dollars, while project C needs to be invested at the beginning of the second year. After two years, every 10,000 dollars of investment can make a profit of 6000 dollars per year, but the amount of investment for this project should not exceed 100,000 dollars, while project D needs to be invested in the third year, with a profit of 4000 dollars per 10,000 dollars, but the amount of investment for the project is limited to 150,000 dollars. During the entire planning period, the amount of investment that the enterprise can invest is 300,000 dollars.