Tom Cruise, Freddy Prinze Jr., Harrison Ford, and MattLeBlanc are marooned on a desert island with JenniferAniston, Courteney Cox, Gwyneth Paltrow, and JuliaRoberts. The "compatibility measures" in Table 52 indicatehow much happiness each couple would experience if theyspent all their time together. The happiness earned by acouple is proportional to the fraction of time they spendtogether. For example, if Freddie and Gwyneth spend halftheir time together, they earn happiness of tC9) = 4.5.a Let xu be the fraction of time that the ith man spendswith the jth woman. The goal of the eight people is tomaximize the total happiness of the people on the island. Formulate an LP whose optimal solution will yieldthe optimal values of the xy's.b Explain why the optimal solution in part (a) willhave four xu = I and twelve xy = 0. The optimal solution requires that each person spend all his or her timewith one person of the opposite sex, so this result is often referred to as the Marriage Theorem.c Determine the marriage partner for each person.d Do you think the Proportionality Assumption of linear programming is valid in this situation
Tom Cruise, Freddy Prinze Jr., Harrison Ford, and Matt
LeBlanc are marooned on a desert island with Jennifer
Aniston, Courteney 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. For example, if Freddie and Gwyneth spend half
their time together, they earn happiness of tC9) = 4.5.
a Let xu 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 xy's.
b Explain why the optimal solution in part (a) will
have four xu = I and twelve xy = 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.
c Determine the marriage partner for each person.
d Do you think the Proportionality Assumption of linear programming is valid in this situation
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