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hmwk2sol

# hmwk2sol - D = 130 the way to calculate is the same as in...

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ISyE 3232 Stochastic Manufacturing and Service Systems Spring 2011 Solutions to Homework 2 1. a) min( D, 7) has the pmf given by P (min( D, 7) = k ) = 1 / 10 if k = 5 3 / 10 if k = 6 6 / 10 if k = 7 0 otherwise Hence, E [min( D, 7)] = 1 10 * 5 + 3 10 * 6 + 6 10 * 7 = 6 . 5 . b) (7 - D ) + has the pmf given by P ((7 - D ) + = k ) = 1 / 10 if k = 2 3 / 10 if k = 1 6 / 10 if k = 0 0 otherwise Hence, E [(7 - D ) + ] = 1 10 * 2 + 3 10 * 1 = 0 . 5 . 2. a) E [max( D, 8)] = 8 * P ( D 8) + Z 10 8 s 5 ds = 8 * 3 / 5 + s 2 10 | 10 8 = 24 5 + 36 10 = 8 . 4 . b) E [(8 - D ) + ] = Z 8 5 8 - s 5 ds = 8 * 3 5 - s 2 10 | 8 5 = 24 5 - 39 10 = 0 . 9 . 3. p = 30, c v = 10, s v = 5. From the newsvendor class notes Proﬁt(7) = ( p - c v ) min { D, 7 } - ( c v - s v )(7 - D ) + . Hence E [Proﬁt(7)] = 20 E [min { D, 7 } ] - 5 E [(7 - D ) + ] = 127 . 5 using the values found in question 1. 4. Similar to the previous question E [Proﬁt(7)] = 20 E [min { D, 7 } ] - 5 E [(7 -

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Unformatted text preview: D ) + ] = 130 the way to calculate is the same as in question 2. 1 5. If D has an exponential distribution with mean 7 then the pdf of D is given by f D ( x ) = 1 7 e-1 7 x . As in the last 2 questions we need to ﬁnd E [min { D, 7 } ] and E [(7-D ) + ]. E [min( D, 7)] = 7 * P ( D ≥ 7) + Z 7 s 7 e-s/ 7 ds = 7 e-1 + (7-14 e-1 ) ≈ 4 . 43 E [(7-D ) + ] = Z 7 7-s 7 e-s/ 7 ds = 7-Z 7 s 7 e-s/ 7 ds = 7-(7-7 e-1 ) ≈ 2 . 575 E [Proﬁt(7)] = 20 E [min { D, 7 } ]-5 E [(7-D ) + ] = 75 . 725 . 2...
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hmwk2sol - D = 130 the way to calculate is the same as in...

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