Principles of Mathematics in Operations Research

Solutions 289

=>an = 2n+3n.

13.4

a) The left hand side of the following constraint represents the complementary

survival probability of a threat,

l-\\{l-Pji)Xii>du\/i.

3

Then,

1 - dk > H(l-Pji)Xji <* Clog(l - dj) > X)[Clog(l -Pji)}xji,VC > 0-

3 3

With a suitable choice of £, and let —6; = (,log(l — di), —Oji = £log(l —Pj%),

we will have

2_Ja 0 iX 3 i - ^»> ^*'

Let ajj = [fljij and /3j = [6«J (with a suitable choice of £ > 0), yielding

^a,^ > /3;, Vi.

3

b) The first three objective functions are equivalent to each other, so are the
last two. The flaw lies in the equivalence of the third and the fourth objective
functions: max/3; 2 ^k min(l — /3,z). In particular,

maxyx + y 2 + 2/3 = min(l - yx) + (1 - y 2 ) + (1 - 2/3)

is true. However,

maxj/ij/22/3 = min(l - 2/i)(l - y 2 )(l - y 3 ) = 1 2/12/22/3

is false because of the cross terms.