Optimal selection of stochastic intervals under a sum constraint

Abstract

We model a selection process arising in certain storage problems. A sequence (X 1, · ··, Xn ) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En (c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

An adaptive O(log n)‐optimal policy for the online selection of a monotone subsequence from a random sample

“…In this section, we prove the upper bound (9) by showing that it holds for all policies that are based on acceptance intervals. The adaptive policy π n and the unique optimal policy π * n both have this property.…”

Probabilistic analysis of an approximation algorithm for maximum subset sum using recurrence relations

“…In this case, the expected number of objects selected is asymptotically Into, i.e., Inn. As noted in [4], for the uniform distribuition U[0, 1], the first-bin problem for FF, (i.e., the cardinality of S' produced by Algorithm LP,) is equivalent to the record-breaking problem: in a sequence xl,x2,...,xn of i.i.d, samples uniform on [0, 1], how many times is a number encountered which is larger than all previous numbers? It is well known [12] that the expected number of record highs is Hn, and hence asymptotically In n. Now let us consider the case when m > r. Define 6m = Nm-Nm-l, for m > 1.…”

“…It is obvious that this problem is no easier than the subset sum problem, and hance, is NP-hard. When the cardinality of S' is the only objective function to maximize, our problem reduces to the selection problem [4], namely, to pack as many objects into S' as possible under the constraint that the sum of sizes of items in S ~ is no larger than capacity c. The selction problem can be generalized to the dual bin packing problem [5], i.e., to maximize the number of items packed into m bins of capacity c.…”

“…Coffman, Flatto, and Weber [4] showed that there exists an on-line selection algorithm which achieves the best En(c), i.e., the average numner of objects selected. Furthermore, if F(z) ~ Ax '~ as x ~ 0 for some A, a > 0, then for a fixed c,…”

Simple probability models for assessing the value of information in defense against missile attack

“…Bernoulli trial, then the number oftargets destroyed out of m, N ( T ) , will be distributed If it is assumed that each attempt to acquire and destroy a target is an independent ( : ) (4) The problem is to find the optimal value of m, that is, one that maximizes this expected value. Other criteria and controls are of course possible.…”