An interesting problem--Ask for help

[liuerwu]

Given a1,a2,a3,b1,b2,b3,Ca,Cb, Let:
A1=min(Ka.Ca, a1), A2=min(Ka.Ca, a2), A3=min(Ka.Ca, a3),
B1=min(Kb.Cb, b1), A2=min(Kb.Cb, b2), A3=min(Kb.Cb, b3),

If:
Ca - A1 >= B1+B2+B3, and
Cb - B1 >= A1+A2+A3

then how to choose Ka, Kb, to maximize: A1+A2+A3+B1+B2+B3

note: Ka and Kb must in the range [0, 1].

Thanks for all possible hints or solutions.
Br

 

[steve10]

I assume Ka.Ca is a product of Ka and Ca, and Kb.Cb a product of Kb and Cb.
Don't have a solution, but thought the following might help.

(1). For any two number x and y, the following is true:

min(x,y)=(x+y)/2-|x-y|/2.

(2) The objective quantity A₁+A₂+A₃+B₁+B₂+B₃ (assume to be f) can therefore be written as

f = (Ka.Ca+a₁)/2-|Ka.Ca-a₁|/2
+(Ka.Ca+a₂)/2-|Ka.Ca-a₂|/2
+(Ka.Ca+a₃)/2-|Ka.Ca-a₃|/2
+(Kb.Cb+b₁)/2-|Kb.Cb-b₁|/2
+(Kb.Cb+b₂)/2-|Kb.Cb-b₂|/2
+(Kb.Cb+b₃)/2-|Kb.Cb-b₃|/2
= (a₁+a₂+a₃+b₁+b₂+b₃)
+((Ka.Ca-a₁)-|Ka.Ca-a₁|)/2
+((Ka.Ca-a₂)-|Ka.Ca-a₂|)/2
+((Ka.Ca-a₃)-|Ka.Ca-a₃|)/2
+((Kb.Cb-b₁)-|Kb.Cb-b₁|)/2
+((Kb.Cb-b₂)-|Kb.Cb-b₂|)/2
+((Kb.Cb-b₃)-|Kb.Cb-b₃|)/2.

(3) Notice that each term (except the first) is in the form

(x-|x|)/2

which has only two values, 0 and x. It reach maximum 0 when x>=0.
(4) This looks like a linear programming problem, which means, among other things, that it reaches extremes (maximum or minimum) when Ka and Kb are 0 or 1.