Adv. Appl. Prob. 24, 506-508 (1992) Printed in N. Ireland © Applied Probability Trust 1992 EXTENSION OF THE BIVARIATE CHARACTERIZATION FOR STOCHASTIC ORDERS RHONDA RIGHTER, * Santa Clara University J. GEORGE SHANTHIKUMAR,** University of California, Berkeley Abstract The bivariate characterization of stochastic ordering relations given by Shanthikumar and Yao (1991) is based on collections of bivariate functions g(x, y), where g(x, y) and g(y, x) satisfy certain properties. We give an alternate characterization based on collections of pairs of bivariate func- tions, gt(x, y) and g2(X, y), satisfying certain properties. This characteriza- tion allows us to extend results for single machine scheduling of jobs that are identical except for their processing times, to jobs that may have different costs associated with them. LIKELIHOOD RATIO ORDERING; HAZARD RATE ORDERING; STOCHASTIC SCHEDULING AMS 1991 SUBJECf CLASSIFICATION: PRIMARY 6OE05 SECONDARY 9OB35 1. Preliminaries For convenience we list the following results for the bivariate characterization of likelihood ratio, hazard rate, and stochastically ordered random variables (Shanthikumar and Yao (1991)). Throughout we assume X and Yare independent random variables. For any bivariate function, g(x, y), define y):= g(x, y) - g(y, x). Also define {g(x, y): g(x, y) x) 'Ix {g(x, y): y) is increasing in x 'Ix {g(x, y): y) is increasing in x 'Ix}. Lemma 1. Y Y)] X)] ve E <§a, for a = lr, hr, st respectively. 2. Main result Let gt(x, y) and g2(X, y) be two bivariate functions, and let y) = gt(x, y) - g2(X, y). We consider the following set of conditions on gt and g2: (a) y) - x), i.e. gt(x, y) - g2(X, y) x) - gt(y, x), for all x (b) y) 0, i.e. gt(x, y) y) for all x (c) gt(x, y) x) for all x (d) gt(y, x) y) for all x (e) gt(x, y) increasing in x for all x (f) gt(x, y) decreasing in y for all y (g) t2(X, y) increasing in x for all x y. (h) y) decreasing in y for all y x. Received 14 October 1991; revision received 17 December 1991. *Postal address: Department of Decision and Information Sciences, Santa Clara University, Santa Clara, CA 95053, USA. **Postal address: Walter A. Haas School of Business, University of California, Berkeley, CA 94720, USA. Supported in part by NSF grant ECS-8811234. 506 available at https://www.cambridge.org/core/terms. https://doi.org/10.2307/1427705 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 10 Aug 2019 at 17:43:34, subject to the Cambridge Core terms of use,