Taking the first-order derivative of the left hand side of Eq. (13)
with respect to t2, it yields ½dðs c þ c3Þ þ c2ð1 dt2Þedt2 . Hence
the left hand side of Eq. (13) is a non-negative and continuous
function which increases strictly in the interval ½0;~t2 and
decreases strictly in the interval ð~t2;1Þ, where ~t2 ¼ c2þdðscþc3Þ c2d . As
a result, the left hand side of Eq. (13) has a maximum at the point
t2 ¼ ~t2, and is s c þ c3 þ c2
d exphc2þdðsc2 cþc3Þi. To find the value of
(t1, t2) such that TP(t1, t2) in Eq. (8) has a maximum value, we consider
the following two cases: (i) bs (i + b + h)cP0 and (ii)
bs (i + b + h)c < 0.