MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem

Abstract

Related to the safety of public lives and property in the lower area of reservoirs, flood control is a priority for most large reservoirs. Considering both dam safety and downstream flood control, reservoir flood control is a multi-objective problem (MOP). To meet the needs of irrigation and generating electricity after the flood, the decision maker usually has his/her preferred final scheduling water level. To deal with this kind of MOP with user-preference information, we incorporate user-preference information into the framework of MOEA/D (multi-objective evolutionary algorithm-based decomposition). The widely used preference information is mainly composed of reference points and preference directions. Compared with the Pareto dominance-based multi-objective evolutionary algorithms (MOEAs), MOEA/D can naturally include two kinds of preference information since MOEA/D is directly based on the reference point and the preference direction. The weight vector of a subproblem in MOEA/D is just its preference. Aiming to obtain uniformly distributed solutions on the objective space, one of innovation points in this paper is using modified Tchebycheff decomposition instead of Tchebycheff decomposition as the decomposition method. To focus the search on the interesting regions of decision maker, the other innovation point in this paper is to integrate biased subproblem (weight vector) adjustment into the framework of MOEA/D. The distribution of subproblems (weight vectors) are adjusted periodically so that the subproblems are re-distributed adaptively to search the interesting regions. Some subproblems, which are far away from the preference regions, are deleted. And then some new subproblems, which are expected to search the preference regions, are added into the current evolutionary population. The efficiency and the effectiveness of the proposed algorithm are assessed through multi-objective reservoir flood control problem and two- to ten-objective test problems.

Appendices

Appendix A: Analysis of the ability of modified Tchebycheff approach

Theorem 3

Let $\mathbf {x}^* \in \Omega$ be a Pareto optimal solution of an MOP (1). Then there exists a weighting vector $0<\mathbf {w} \in \mathbf {R}^m$ such that $\mathbf {x}^*$ is a optimal solution of modified Tchebycheff problem (6), where the ideal objective vector $\mathbf {z}^*$ is replaced by the utopian objective vector $\mathbf {z}^{**}=\mathbf {z}^{*}-\mathbf {\varepsilon }$, $\mathbf {\varepsilon }>\mathbf {0}$ is a relatively small but computationally significant vector. (This Theorem is similar to the Theorem 3.4.5 in Miettinen (1999).)

Proof

Let us suppose that there does not exist a weight vector $\mathbf {w}>\mathbf {0}$ such that $\mathbf {x}^*$ is a solution of modified Tchebycheff problem (6). Due to $f_i(\mathbf {x})>z_i^{**},i=1,\ldots , m, \forall ~\mathbf {x} \in \Omega$, we mark $w_i^*=\frac{f_i(\mathbf {x}^*)-z_i^{**}}{\alpha },i=1,\ldots ,m$ where $\alpha >0$ is some normalizing factor. If $\mathbf {x}^*$ is not a optimal solution of subproblem with weight vector $\mathbf {w}^*=(w_1^*,\ldots ,w_m^*)$, there exists another solution $\bar{\mathbf {x}} \in \Omega$ that is a optimal solution of this subproblem, meaning that $\max _{1 \le i \le m} \left\{ \frac{f_i(\bar{\mathbf {x}})-z_i^{**}}{w_i} \right\} < \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x}^*)-z_i^{**}}{w_i} \right\} = \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x}^*)-z_i^{**}}{\frac{f_i(\mathbf {x})-z^{**}}{\alpha }} \right\} = \alpha$. Thus $\frac{f_i(\mathbf {x}^*)-z_i^{**}}{w_i}<\alpha$. For each $i=1,\ldots ,m$, we have that $\frac{f_i(\bar{\mathbf {x}})-z_i^{**}}{w_i} =\frac{f_i(\mathbf {x}^*)-z_i^{**}}{\frac{f_i(\mathbf {x})-z^{**}}{\alpha }} =\alpha \times \frac{f_i(\bar{\mathbf {x}})-z_i^{**}}{f_i(\mathbf {x})-z^{**}} <\alpha$. Therefore, we can obtain $f_i(\bar{\mathbf {x}}) < f_i(\mathbf {x}^*)$. Here we have a contradiction with the Pareto optimality of $\mathbf {x}^*$, which completes the proof. $\square$

Theorem 4

The optimal solution of modified Tchebycheff approach (6) with $\mathbf {w}>\mathbf {0}$ is a Pareto optimal solution of an MOP (1), where $g^{mtch}(\mathbf {x} ~|~ \mathbf {w},\mathbf {z}^*) = \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x})-z_i^*}{w_i} \right\}$ is replaced by $g^{mtch}(\mathbf {x} \left| \right. \mathbf {w},\mathbf {z}^*) = \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x})-z_i^*}{w_i} \right\} + \rho \sum _{i=1}^m \left( \frac{f_i(\mathbf {x})-z_i^*}{w_i} \right)$, $\rho > 0$ is a relatively small but computationally significant scalar.

Proof

Let $\mathbf {x}^* \in \Omega$ be a solution to modified Tchebycheff problem (6). Let us assume that $\mathbf {x}^*$ is not Pareto optimal solution of MOP (1). Then there exists a point $\bar{\mathbf {x}} \in \Omega$ such that $\bar{\mathbf {x}}$ dominates $\mathbf {x}^*$ (i.e. $f_i(\bar{\mathbf {x}}) \le f_i(\mathbf {x}^*),i=1,\ldots ,m$ and $f_j(\bar{\mathbf {x}}) < f_j(\mathbf {x}^*)$ for at least one $j \in \{1,2,\ldots , m\}$). Now we have $f_i(\bar{\mathbf {x}})-z_i^* \le f_i(\mathbf {x}^*)-z_i^*,i=1,\ldots ,m$ and $f_j(\bar{\mathbf {x}})-z_j^* < f_j(\mathbf {x}^*)-z_j^*$ for at least one j. Due to $\mathbf {w}>\mathbf {0}$, we obtain $\frac{f_i(\bar{\mathbf {x}})-z_i^*}{w_i} \le \frac{f_i(\mathbf {x}^*)-z_i^*}{w_i},i=1,\ldots ,m$ and $\frac{f_j(\bar{\mathbf {x}})-z_j^*}{w_j} \le \frac{f_j(\mathbf {x}^*)-z_j^*}{w_j}$, at least one $j \in \{1,\ldots ,m\}$. Then we have $\max _{1 \le i \le m} \left\{ \frac{f_i(\bar{\mathbf {x}})-z_i^*}{w_i} \right\} \le \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x}^*)-z_i^*}{w_i} \right\}$ and $\sum _{1 \le i \le m} \left\{ \frac{f_i(\bar{\mathbf {x}})-z_i^*}{w_i} \right\} < \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x}^*)-z_i^*}{w_i} \right\}$. So we can get $g^{mtch}(\bar{\mathbf {x}} ~|~ \mathbf {w},\mathbf {z}^*) = \max _{1 \le i \le m} \left\{ \frac{f_i(\bar{\mathbf {x}})-z_i^*}{w_i} \right\} + \rho \sum _{i=1}^m \left( \frac{f_i(\bar{\mathbf {x}})-z_i^*}{w_i} \right) < \max _{1 \le i \le m} \left\{ \frac{f_i(\mathbf {x}^*)-z_i^*}{w_i} \right\} + \rho \sum _{i=1}^m \left( \frac{f_i(\mathbf {x}^*)-z_i^*}{w_i} \right) = g^{mtch}(\mathbf {x}^* ~\left| ~ \right. \mathbf {w},\mathbf {z}^*)$. Here we have a contradiction with the assumption that $\mathbf {x}^* \in \Omega$ is a solution of modified Tchebycheff problem (6), which completes the proof. $\square$

Appendix B: Proof of Theorem 1

Proof

Using reduction to absurdity, we will prove that the intersection point of the line $\frac{f_1-z_1^*}{w_1} = \frac{f_2-z_2^*}{w_2} = \cdots = \frac{f_m-z_m^*}{w_m} (w_i \ne 0,i=1,2,\ldots ,m)$ and the PF is the optimal solution of the subproblem with preference direction (weight vector) $\mathbf {w}=\left( w_1,w_2,\ldots ,w_m\right) \left( \sum _{i=1}^m w_i = 1,\lambda _i>0,i=1,2,\ldots ,m\right)$. Let us suppose that the optimal solution of the resultant subproblem with preference direction (weight vector) $\mathbf {w}=(w_1,w_2,\ldots ,w_m)$ is $\bar{\mathbf {f}}=(\bar{f_1},\ldots ,\bar{f_m})$ and $\bar{\mathbf {f}}=(\bar{f_1},\ldots ,\bar{f_m})$ is a Pareto optimal solution but not in the line $\frac{f_1-z_1^*}{w_1} = \frac{f_2-z_2^*}{w_2} = \ldots = \frac{f_m-z_m^*}{w_m} (w_i \ne 0,i=1,2,\ldots ,m)$. Then we have the two following non-empty sets $\bar{L} = \left\{ l \left| \right. \frac{\bar{f_l}-z_l^*}{w_l} < \max _{1 \le i \le m} \left\{ \frac{\bar{f_i}-z_i^*}{w_i} \right\} , l=1,\ldots ,m \right\}$ and $\bar{G} = \left\{ g \left| \right. \frac{\bar{f_g}-z_g^*}{w_g} = \max _{1 \le i \le m} \left\{ \frac{\bar{f_i}-z_i^*}{w_i} \right\} , g=1,\ldots ,m \right\}$. Therefore, we have $\frac{\bar{f_l}-z_l^*}{w_l} < \max _{1 \le i \le m} \left\{ \frac{\bar{f_i}-z_i^*}{w_i} \right\} = \frac{\bar{f_g}-z_g^*}{w_g} = g^{mtch} \left( \bar{\mathbf {f}}~|~\mathbf {w},\mathbf {z}^* \right) , \forall l \in \bar{L}, g \in \bar{G}$.

Let us suppose that $\bar{\mathbf {f}}=(\bar{f_1},\ldots ,\bar{f_m})$ is an internal point of the PF without loss of generality. Because of the PF is piecewise continuous, we can find the point $\hat{f}=(\hat{f_1},\ldots ,\hat{f_m})$ in the $\delta$-neighborhood of $\bar{\mathbf {f}}=(\bar{f_1},\ldots ,\bar{f_m})$ meeting the two following conditions:

• $\hat{f} = (\hat{f_1},\ldots ,\hat{f_m})$ is a Pareto optimal solution.

• $\hat{f_l}>\bar{f_l},l \in \bar{L}; \hat{f_g}>\bar{f_g}, \forall g \in \bar{G}$

As $\hat{f}=(\hat{f_1},\ldots ,\hat{f_m})$ is a $\delta$-neighbor to $\bar{\mathbf {f}}=(\bar{f_1},\ldots ,\bar{f_m})$, this implies that $\frac{\bar{f_l}-z_l^*}{w_l} < \frac{\hat{f_l}-z_l^*}{w_l} < \frac{\hat{f_g}-z_g^*}{w_g} < \frac{\bar{f_g}-z_g^*}{w_g}, \forall l \in \bar{L},g \in \bar{G}$. Therefore, $g^{mtch} \left( \hat{\mathbf {f}} | \mathbf {w},\mathbf {z}^* \right) = \max _{g \in \bar{G}} \left\{ \frac{\hat{f_g}-z_g^*}{w_g} \right\} < \max _{g \in \bar{G}} \left\{ \frac{\bar{f_g}-z_g^*}{w_i} \right\} = g^{mtch}(\bar{\mathbf {f}} ~|~ \mathbf {w},\mathbf {z}^*)$. This conclusion is clearly inconsistent with the assumption, so the Theorem 1 is proofed. $\square$

Appendix C: Proof of Theorem 2

Proof

Let us assume that $\acute{\mathbf {w}}=(\acute{w_1},\ldots ,\acute{w_m})$, rather than $\mathbf {w}^{opt}$, is the optimal weight vector to the preferred solution $\mathbf {F}=(f_1,\ldots ,f_m)$ based on the reference point $\mathbf {z}=(z_1,\ldots ,z_m)$, i.e. $h(\acute{\mathbf {w}}~|~\mathbf {F},\mathbf {z}) < h(\mathbf {w}^{opt}~|~\mathbf {F},\mathbf {z})$ and $\acute{\mathbf {w}} \ne \mathbf {w}^{opt}$. If we note that $L=\left\{ l~|~\acute{w_l}< w_l^{opt}\right\} , E=\left\{ e~|~\acute{w_e} = w_e^{opt}\right\}$, $G=\left\{ g~|~\acute{w_g}>w_g^{opt} \right\} .$ Let $W_m =\left\{ \mathbf {w}=(w_1,\ldots ,w_m) | \sum _{i=1}^{m} w_i =1, w_i\ge 0 \right\}$, then there exist $G \ne \emptyset$ and $L \ne \emptyset$ for $\acute{\mathbf {w}} \ne \mathbf {w}^{opt}$ and $\acute{\mathbf {w}}, \mathbf {w}^{opt} \in W_m$, we have $0<w_i^{opt}<1$ for each $i=1,\ldots ,m$.

Due to $\acute{w_l} < {w_l^{opt}}, \forall l \in L$, we have $\frac{f_l-z_l}{\acute{w_l}} > \frac{f_l-z_l}{w_l^{opt}}=\sum _{i=1}^m \left( f_i-z_i \right) , \forall l \in L$. Because $\acute{w_e} = {w_e^{opt}}, \forall e \in E$, we have $\frac{f_e-z_e}{\acute{w_e}} = \frac{f_e-z_e}{w_e^{opt}}=\sum _{i=1}^m \left( f_i-z_i \right) , \forall e \in E$. Owing to $0< {w_g^{opt}} < \acute{w_g}, \forall g \in G$, we have $\frac{f_g-z_g}{\acute{w_g}} < \frac{f_g-z_g}{w_g^{opt}}=\sum _{i=1}^m \left( f_i-z_i \right) , \forall g \in G$. Therefore, $\frac{f_l-z_l}{\acute{w_l}} > \sum _{i=1}^m \left( f_i-z_i \right) = \frac{f_e-z_e}{\acute{w_e}} > \frac{f_g-z_g}{\acute{w_g}}, \forall l \in L, \forall e \in E,\forall g \in G$. Because of $L \ne \emptyset$, we can get $h(\acute{\mathbf {w}} ~|~ \mathbf {F},\mathbf {z}) = \max _{1 \le i \le m} \left\{ \frac{f_i-z_i}{\acute{w_i}} \right\} > \sum _{i=1}^m \left( f_i-z_i \right) = \max _{1 \le i \le m} \left\{ \frac{f_i-z_i}{w_i^{opt}} \right\} = h(\mathbf {w}^{opt} ~|~ \mathbf {F},\mathbf {z})$. But this conclusion is inconsistent with the assumption, so Theorem 2 is proved. $\square$