I have a region defined by following inequalities $$y\leq \min(a,b)\tag{1}$$ $$x\leq\frac{cd}{c+d-1}\tag{2}$$ $$\frac{c+d-1}{c}x+\frac{d}{\min(a,e)}y\leq d\tag{3}$$where $a,b,c,d,e$ are positive constants. In this case how can we show that if $$\min(a,b)<\min(a,e)\\\min(a,e)\leq \frac{cd}{c+d-1}$$ then the region associated with (1)-(3) is given by the following Figure (1) where the optimal point is shown by the circle (the optimal point is the point which maximizes the left hand side of (3))
On the other hand if $$\min(a,b)<\min(a,e)\\ \min(a,e)>\frac{cd}{c+d-1}$$ then the associated region and the optimal point are given by the following figure 
I do not understand how the optimal point in second case is as shown in the Figure. Any explanation in this regard will be much appreciated. Thanks in advance.
Here is the image that I saw in a research paper. In this figure we have $$a=K_u,b=M_R,c=N_d,d=K_d,e=N_b$$. The optimal points are mentioned with dark circles on the graph.
In the paper (found via Google Scholar), the objective value is LDoF$_{sum}$ (see either the description or the caption of Fig. 4), defined in formula (6) as LDoF$_{u}$+LDoF$_{d}$. So, the optimal point is not the one that maximizes (3).
Which point is optimal is determined by the slope of the diagonal line. If it is steeper than 45 degrees, the bottom point is optimal. If it is more horizontal, the top corner point is optimal.
For fig. (4d), the two conditions under the figure oppose each other: the first one makes the line more horizontal, while the second one makes the line less horizontal. This case is therefore not as simple as the other ones. Let's check the objective value at the circled point: $$\begin{align} LDoF_u + LDoF_d &= \min(K_u,M_r) + \frac{N_d}{N_d+K_d-1} \left(K_d - \frac{K_d}{\min(K_u,N_b)} \min(K_u,M_r)\right)\\ &= \frac{N_dK_d}{N_d+K_d-1} + \min(K_u,M_r) - \frac{N_dK_d}{N_d+K_d-1} \frac{\min(K_u,M_r)}{\min(K_u,N_b)}\\ \end{align}$$ The second term on the first line is obtained by solving (23) for LDoF$_d$. This equals the right hand side of the inequality at the left bottom of page 7202, which according to Theorem 1 is optimal.