多目标规划帕累托解算例

Pareto:
In the single objective case, one attempts to obtain the best solution, which is absolutely superior to all other
alternatives.
在单目标的情况下，一个试图以获得最佳的解决方案，这是绝对优于所有其他的替代品。

In the multiple objective case, there does not necessarily exist a solution that is best with respect to all objectives
because of incommensurability and conflict among objectives.
在多个目标的情况下，不存在必然存在着一个解决方案，最好是不可通约性和目标之间的的冲突，因为所
有的目标。

There usually exist a set of solutions; nondominated solutions or Pareto optimal solutions, for the multiple
objective case which cannot simply be compared with each other.
通常存在的一整套解决方案;非支配的解决方案或帕累托最优的解决方案，为多个目标的情况下，不能简单
地互相比较。

For a given nondominated point in the criterion space Z, its image point in the decision space S is called efficient
or noninferior. A point in S is efficient if and only if its image in Z is nondominated.
对于一个给定的的标准空间z的非支配点，其形象在决定空间S点是所谓的效率或劣。

非支配当且仅当其
在Z的形象是一个S点是有效的。

Definition 1: For a given point z0€Z, it is nondominated if and only if there does not exist another point z€Z
such that, for the maximization case,where, z0 is a dominated point in the criterion space Z.
Definition 2: For a given point x0€S, it is efficient if and only if there does not exist another point x€S such
that, for the maximization case,where, x0 is inefficient.定义1：对于一个给定的点Z0属于Z，它非支配当
且仅当不存在另一点于属于z的，最大化的情况下，其中，Z0是在标准空间Z.的主导点
定义2：对于一个给定的点x0属于S，它是有效的当且仅当不存在另一点x属于S，最大化的情况下，其
中，X0是低效的。

Example 1: Two-objective (bicriteria) linear programming
例1：两个目标（bicriteria）线性规划
m ax
We can observe that both regions are convex and the extreme points of Z are the images of extreme points of S.
我们可以观察到，这两个地区是凸的并且极端点的Z是极值点S的的图像。

The extreme points in the feasible region S of the decision space are shown in Fig. 4.1:
在可行区域的决策空间小号的极端点如图.4.1：
图4.1在决策空间的可行域和有效的解决方案，
The corresponding extreme points in the feasible region Z of the criterion space are shown in Fig. 4.2:
在标准空间的可行区域ž相应的极值点如图. 4.2：
图4.2在标准空间的可行区域和非支配的解决方案
The slashed border of the feasible region Z is identified as the set of nondominated solutions because it is noted that as z1(x1, x2) increases from 0 to 3, z2(x1, x2) decreases from 0 to -1, and accordingly, all points between z2, z3 and z4 are nondominated points.
The corresponding efficient points in the decision space are the segment between points x2, x3 and x4 . Ideal point or positive ideal point or solution (PIP or PIS) is denoted by z* = [z1* z2* … z q*]，where z k* = sup { fk(x)| x€S}, k=1, 2, …, q
Negative ideal point or solution (NIP or NIS) is denoted by z - = [z1-z2-… z q-]，where z k- = inf {fk(x)| x€S}, k=1,2,…,q
可行区域ž削减边界是确定的非支配解集，因为它增加Z1（X1，X2）从0到3，Z2（X1，X2）的跌幅从0到-1，并据此指出，Z2，Z3和Z4之间的所有点都是非支配点。

相应的决策空间的有效点之间的点是X2，X3和X4之间的线段。

理想点或积极的理想点或解决方案（PIP或PIS）表示z* = [z1* z2* … z q*]，where z k* = sup { fk(x)| x€S}, k=1, 2, …, q
消极的理想点或解决方案(NIP or NIS) 表示z - = [z1-z2-… z q-]，where z k- = inf {fk(x)| x€S}, k=1,2,…,q。