The original problem is: $$\max\limits_{x,y} ~ \min\{\min\{a(x,y),b(x,y)\}+\min\{c(x,y),d(x,y)\},e(x,y)\}$$
Can it be written as $$\max\limits_{x,y} ~~ T$$ with constraints $$t_1+t_2 \geq T; ~e(x,y) \geq T;~a(x,y) \geq t_1;~b(x,y) \geq t_1;~c(x,y) \geq t_2;~d(x,y) \geq t_2 $$ or there any easier way? To become a convex problem, what are the conditions for constraints?
Your reformulation is correct. For the problem to be convex, a,b,c,d,e each need to be jointly concave in (x,y).