Tamara MacEwen
IE 416
Page 398 Problem 3
By: Ms. Tamara MacEwen
Parisay’s comment is in red.
Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a desert island
with Jennifer Aniston, Courtney Cox, Gwyneth Paltrow, and Julia Roberts. The “compatibility
measures” in Table 52 indicate how much happiness each couple would experience if they spent
all their time together. The happiness earned by a couple is proportional to the fraction of time
they spend together.
Table 52
TC
FP
HF
ML
JA
7
7
3
5
CC
5
8
5
5
GP
8
9
7
6
JR
2
4
9
7
a) Let Xij be the fraction of time that the ith man spends with the jth woman. The goal of the
eight people is to maximize the total happiness of the people on the island. Formulate an LP
whose optimal solution will yield the optimal values of the xij’s.
Input
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Tamara MacEwen
IE 416
Output
a) Explain why the optimal solution in part (a) will have four xij = 1 and twelve xij = 0. The
optimal solution requires that each person spend all his or her time with one person of the
opposite sex, so this result is often referred to as the Marriage Theorem.
In order to maximize happiness, each man will spend time with the most compatible female
that is still available. Each person will stay with the same partner the entire day to maximize
happiness. If they were to change partners for a fraction of the day, the new partners would
have lower happiness ratings and would cause the happiness rating to be lower than the
maximum.
If for a fraction of time we have combination of 4 couples that maximizes happiness,
then it can be true for all day. That is why all Xij are zero but 4 couples that are 1
(whole day).
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Tamara MacEwen
IE 416
b) Solve the problem using assignment problem
Input
Output
Report to Manager:
If Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt LeBlanc are marooned on a desert
island with Jennifer Aniston, Courtney Cox, Gwyneth Paltrow, and Julia Roberts, the maximum
total happiness would be achieved if the following men spent their time with only one woman.
Tom Cruise – Gwyneth Paltrow
Freddy Prinze Jr. – Courtney Cox
Harrison Ford – Julia Roberts
Matt LeBlanc – Jennifer Aniston
This combination would give the maximum compatibility measure of 30.
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