### THE GENERALIZED LINEAR SEARCH PROBLEM, EXISTENCE OF OPTIMAL SEARCH PATHS

Zaid T. Balkhi
1987 Journal of the Operations Research Society of Japan
PROBLEM, PATHS Abstract A target x is a point on the real line given by the value of a random variable X, which has some distribution function F. A searcher starts looking for x from some point on the line, using a continuous path. He makes for x with un upper bound on his speed till he fmds it. The target being sought for might be in either direc· tion from the starting point, so the searcher has in general to retrace his steps many times before he attains his goal. It is desired to search in
more » ... sired to search in an optimal manner so as to minimize the expected cost of the search. All previous papers treated this problem using the origin as the starting point of the search. They have been proved that one can minimize the expected cost if the underlying distribution satisfies certain conditions. In this paper, the problem will be treated in the "General Case", which means that the sear.;h may start from any point on the real line. Conditions under which we can minimize the expected cost in the general case will be given. n 2 I n=l (see, , ,  and [11J). Thus, our aim is to minimize D(a,F) (or equivalently t.O(a,F) for fixed F). Define (1. 7) then the main problem is to find a search path a i~O} from class Q O a 00 -21f °lx!dF(x)1la o ! + 2 L o i=1 la ·Ix ~ {1-sign(a.) [F(a.)-F(a. 1)]} ~ ~ ~-One can easily see from (2.22) and (2.23) that class 00 might be embedded in the other classes by taking a O = 0, because then a k _ 2 = 0 for k = 3,4. Theorem 2.5: If a = {a i ; i~O} is an (O.S.P.) from class Ok' then la i + 2 1 > lail for all i ~ k-l; k = 0,1,2,3 & 4. Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited. Q D • .E •. Remark 2.6: We have, so far, proved that if a = {a.; i~O} is an (a.s.p.) ~ from class Q k then la. 21 > la.1 for all i~-l, k=O.l,2,3 & 4. Therefore, one ~+ ~ can restrict his attention to such kind of search paths. But one should keep in mind that la i + 2