IEOR150F10_hw05_sol

# IEOR150F10_hw05_sol - IEOR 150 Fall 2010 Homework 5 1(a Let...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: IEOR 150, Fall 2010 Homework 5 1. (a) Let ¯ q 1 and ¯ q 2 be the optimal order quantities of retailers 1 and 2, we have P ( D 1 ≤ ¯ q 1 ) = ¯ q 1- 1- = 1- w 1 r 1 = 0 . 5 ⇒ ¯ q 1 = 0 . 5 . Similarly, ¯ q 2 = 1- w 2 r 2 = 0 . 4. (b) If retailer 2 has no stock-out ( D 2 ≤ q 2 ), then R 1 = D 1 . On the other hand, if retailer 2 does has stock out ( D 2 < q 2 ), then R 1 = D 1 + D 2- q 2 . Therefore, P ( R 1 ≤ q 1 ) = P ( R 1 ≤ q 1 ,D 2 ≤ q 2 ) + P ( R 1 ≤ q 1 ,D 2 > q 2 ) = P ( D 1 ≤ q 1 ,D 2 ≤ q 2 ) + P ( D 1 + D 2- q 2 ≤ q 1 ,D 2 > q 2 ) Note that P ( D 1 ≤ q 1 ,D 2 ≤ q 2 ) = P ( D 1 ≤ q 1 ) P ( D 2 ≤ q 2 ) = q 1 q 2 because D 1 and D 2 are independent. Also note that P ( D 1 + D 2- q 2 ≤ q 1 ,D 2 > q 2 ) = Z 1 q 2 P ( D 1 + y- q 2 ≤ q 1 ) 1 1- dx (integrating over all possible realization of D 2 ) = Z 1 q 2 P ( D 1 ≤ q 1 + q 2- y ) dx = Z 1 q 2 ( q 1 + q 2- y ) dx = ( q 1 + q 2 )(1- q 2 )- 1 2 (1- q 2 2 ) ....
View Full Document

## This document was uploaded on 01/13/2011.

### Page1 / 2

IEOR150F10_hw05_sol - IEOR 150 Fall 2010 Homework 5 1(a Let...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online