By St Maxime le Confesseur
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Additional resources for Ambigua Introduction par Jean-Claude Larchet (Maximus the Confessor)
B" ß B# ! 1-9. (a) Let B" be the number of units on special risk insurance and B# be the number of units on mortgages. maximize subject to D œ &B" #B# $B" #B# Ÿ #%!! B# Ÿ )!! #B" Ÿ "#!! , B# ! Ñ and ^ ‡ œ $'!! (c) The relevant two equations are $B" #B# œ #%!! , so B" œ '!! and B# œ "# Ð#%!! D œ &B" #B# œ $'!! 1-10. Þ"F Ÿ #!! Þ#&L Ÿ )!! $L #F Ÿ "#ß !!! Lß F ! Ñ and T ‡ œ #*#! 1-11. (a) Let B3 be the number of units of product 3 produced for 3 œ "ß #ß $. maximize ^ œ &0B" #0B# #&B$ subject to (b) *B" $B# &B$ Ÿ &!!
2-2. 2-3. (a) As in the W yndor Glass Co. problem , we want to find the optim al levels of two activities that com pete for limited resources. Let B" and B# be the f raction purchased of the partnership in the first and second friends venture respectively. Resource Fraction of partnership in 1st Fraction of partnership in 2nd Money Summer work hours Unit Profit (b) maximize subject to Resource Usage per Unit of Activity 1 2 " ! " $&!!! $%!!! %!! &!! $%&!! $%&!! B# B" , B# Ÿ Ÿ Ÿ Ÿ " " '!!! '!! 3-11 Available Amount " " $'!!!
B" #B$ Ÿ "&! B$ Ÿ #! B" , B# , B$ ! 1-12. 1-13. , Ð#ß $Ñ is always f easible for any value of 5 . Moreover, the third constraint is always binding at Ð#ß $Ñ, 5B" B# œ #5 $. To check if Ð#ß $Ñ is optimal, observe that ch anging 5 simply rotates the line that always passes through Ð#ß $Ñ. Rewriting this e quation as B# œ 5B" Ð#5 $Ñ, we see that the slope of the line is 5 , and therefore, the slope ranges from ! to _. As we can see, Ð#ß $Ñ is optimal as long as the slope of the third constraint is less than the slope of the objective line, which is "# .