Consider the following continuous probability density function for a certain type of goods required in a month: fr 8 >>>>>>>> < >>>>>>>>: 0 r

Consider the following continuous probability density function for a certain type of goods required in a month: fðrÞ ¼ 8 >>>>>>>>< >>>>>>>>: 0 r < 80 f2ðrÞ¼1 5 þ r 400 80  r < 100 f3ðrÞ ¼ 1:5  r 400 100  r  120 0 r > 120 Costs of inventory are as follows: c1 ¼ cost of surplus ¼ 200 c2 ¼ cost of shortage ¼ 9; 000 Like in Model II (slightly modified) of the chapter determine the optimal stock S*, so that the total inventory expected cost, described by the following integral, is minimized: CðSÞ ¼ Z 100 80 c1  S  r 2  f2ðrÞdr þ Z S 100 c1  S  r 2  f3ðrÞdr þ Z 120 S c1 S2 2r f3ðrÞdr þ Z 120 S c2 ðr  SÞ 2 2r f3ðrÞdr: