# An Improved SPEA2 Algorithm with Local Search for Multi-Objective Investment Decision-Making
## 3 Multi-Objective Decision-Making in Enterprise Investment
What needs to be solved is which kind of investment plan can make the enterprise obtain the maximum profit in the whole investment plan. However, there will be risks for every investment. Enterprises always expect the least risk when they invest. The risk parameters associated with each investment are \(R=[0.1,0.25,0.5,0.75]\). At the same time, enterprises expect the environmental cost of investment as low as possible. Here, the environmental cost coefficients are \(E=[0.15,0.3,0.2,0.4]\). This is a problem of determining the optimal investment plan. There are many feasible plans to satisfy this investment plan. Due to the limitations of various investment conditions, it is difficult to find the optimal investment plan by general methods. The optimization objective can be expressed as:  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Year**} & \multicolumn{6}{c}{**Projects**} \\ \cline{2-7}  & **1** & **2** & \(\cdots\) & \(j\) & \(\cdots\) & \(m\) \\ \hline
1 & \(I_{11}\) & \(I_{12}\) & \(\cdots\) & \(I_{1j}\) & \(\cdots\) & \(I_{1m}\) \\
2 & \(I_{21}\) & \(I_{22}  
\[ \begin{array}{ll}\max&\text{Profit}=0.2I_{1A}+0.2I_{2A}+0.2I_{3A}\\ &+0.5I_{1B}+0.6I_{2C}+0.4I_{2D},\\ \min&\text{Risk}=0.1(I_{1A}+I_{2A}+I_{3A})+0.25I_{1B}\\ &+0.5I_{2C}+0.75I_{2D},\\ \min&\text{EC}=0.15\cdot(0.2I_{1A}+0.2I_{2A}+0.2I_{3A})\\ &+0.3\cdot 0.5I_{1B}+0.2\cdot 0.6I_{2C}+0.9\cdot 0.4I_{2D},\\ \text{s.t.}&I_{1A}+I_{1B}\leqslant 30\\ &I_{2A}+I_{2C}\leqslant 30-I_{1B}+0.2I_{1A}\\ &I_{3A}+I_{3D}\leqslant 30+0.5I_{1B}+0.2I_{1A}+0.2I_{2A}-I_{2C}\\ &I_{2B}\leqslant 20;I_{2C}\leqslant 15;I_{3D}\leqslant 10\\ &I_{1A}\geqslant 0;I_{2A}\geqslant 0;I_{2A}\geqslant 0\\ &I_{1B}\geqslant 0;I_{2C}\geqslant 0;I_{3D}\geqslant 0.\end{array} \]  
The proposed method is used to solve this problem. The model parameters are as follows: the initial population size is 300, the external archive size is 50, the crossover probability is 0.6, and the mutation probability is 0.3. Firstly, we only consider investment profit and investment risk as optimization objectives. Figure 4a gives the Pareto optimal boundary considering profit and risk. It can be seen from Figure 4a that profits and risks are positively correlated. When the risk is less than 8, the profit and risk are almost linear, but when the profit is greater than 16 (within the dashed line of Figure 4a), the risk will accelerate. This conclusion can provide data support for decision-makers, who can maximize profits according to the risk range they can bear. Table 2 gives a strategy for investment only considering profit. Table 3 gives an investment strategy considering both profit and risk. The total profit of the investment strategy given in Table 2 is 275,000 dollars, but if the risk is taken into account, the risk of the investment strategy is 22.24. Table 3 gives an investment strategy considering both profit and risk. The total profit of the investment strategy is 23.88, but the risk is reduced to 15.64. Under Table 3 investment strategy, the profit is only 13.16% less than the investment strategy of Table 2, but the risk is reduced by 29.68%. From the above results, it can be seen that, while maximizing profits, minimizing investment risks is particularly important for enterprises. Figure 4b gives a convergence comparison between the improved SPEA2 method and the methods of NSGA-II and SPEA2. As can be seen from Figure 4b, for the complex solution space of investment decision, the method in this paper can converge to the Pareto optimal boundary faster than the other two methods, and the final convergence result is more stable, which indicates that the results obtained by improved SPEA2 method are more close to the Pareto optimal boundary than the other two methods. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Year** & **A** & **B** & **C** & **D** \\ \hline
1 & 12.5 & 17.5 & & \\
2 & 0 & & 15 & \\
3 & 16.25 & & & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Investment strategy (only profit is considered).