For a commercial flight, the payload it carries is the weight of passengers and their baggage. Therefore fixed by seat configuration. Fuel weight is determined by the range of the flight. MTOW or maximum structural take-off weight.
So what exactly are engineers trying to maximize when they design for take-off at MTOW?
MTOW = OEW + PL + FW. - This expression seems to indicate that the only free variable is the operating empty weight, and that's what they are trying to maximize, but obviously that doesn't make sense.
My question is why most planes (and most airlines do it this way) are designed to take off on the MTOW line in this PL-R diagram (2-3), and not on the maximum payload line (1-2) or on the maximum fuel line (3-4).