A linear-programming-based method for determining whether or not n demand points are on a hemisphere

Whenever n demand points are located on a hemisphere, spherical location problems can be solved easily using geometrical methods or mathematical programming. A method based on a linear programming formulation with four constraints is presented to determine whether n demand points are on a hemisphere. The formulation is derived from a modified minimax spherical location problem whose Karush-Kuhn-Tucker conditions are the constraints of the linear program. © 1993 John Wiley & Sons, Inc.     

On estimating survival for reliability models based on age-related stochastic comparisons

A new class of nonparametric reliability models is introduced and studied. A distribution is said to be better at age s than at age t (sBt) if the residual lifetime at age s is stochastically greater than or equal to the residual lifetime at age t. Applications to various forms of replacement policies, including the cannibalization of failed systems, are noted. For fixed s < t, the problem of estimating a survival curve assumed to belong to the sBt class is addressed using recursive methods. An sBt estimator is derived in closed form, and its uniform strong consistency at an optimal rate of convergence is demonstrated. A simulation study strongly supports the claim that the sBt estimator tends to outperform the empirical survivor function in small- and moderate-size samples. © 1993 John Wiley & Sons, Inc.     

Solving time/cost trade-off problems with discounted cash flows using generalized benders decomposition

In this article we consider a project scheduling problem where there are cash flows throughout the life of the project and where shorter activity durations can be attained by incurring greater direct costs. In particular, the objective of this problem is to determine the activity durations and a schedule of activity start times so that the net present value of cash flows is maximized. We formulate this problem as a mixed-integer nonlinear program which is amenable to solution using the generalized Benders decomposition technique developed by Geoffrion. We test the algorithm on 140 project scheduling problems, the largest of which contains 30 nodes and 64 activities. Our computational results are quite encouraging inasmuch as 123 of the 140 problems require less than 1 CPU second of solution time. © 1993 John Wiley & Sons, Inc.     

Asymptotically sufficient grouping

Under certain conditions, a sample from a multivariate distribution can be grouped in such a way that asymptotically (as the sample size increases) no information is lost by grouping, and the joint distribution of the observed frequencies can be approximated by a multivariate normal distribution. © 1993 John Wiley & Sons, Inc.     

A multi-run interactive method for bicriterion optimization problems

This article introduces a new conceptual and methodological framework for the use of decision makers and their interactions with the computer in bicriterion decision making. The new method, called the multirun interactive method, attempts to estimate the prior of the decision maker on his uncertain preference nature using the minimum cross-entropy principle. A computational study is performed with four hypothesized prior distributions under various interaction conditions. Other important aspects related to the method, such as implementation of the method, decision making under certainty, decision making with multiple decision makers, and bicriterion integer programming, are also discussed.     

Optimal resource extraction with uncertain reserves

We consider the problem of finding a plan that maximizes the expected discounted return when extracting a nonrenewable resource having uncertain reserves. An extraction plan specifies the rate at which the resource is extracted as a function of time until the resource is exhausted or the time horizon is reached. The return per unit of resource extracted may depend on the rate of extraction, time, and the amount of resource previously extracted. We apply a new method called the generalized search optimization technique to find qualitative features of optimal plans and to devise algorithms for the numerical calculation of optimal plans.     

Scheduling on a single machine with a single breakdown to minimize stochastically the number of tardy jobs

Jobs with known processing times and due dates have to be processed on a machine which is subject to a single breakdown. The moment of breakdown and the repair time are independent random variables. Two cases are distinguished with reference to the processing time preempted by the breakdown (no other preemptions are allowed): (i) resumption without time losses and (ii) restart from the beginning. Under certain compatible conditions, we find the policies which minimize stochastically the number of tardy jobs.     

A proof for the longest-job-first policy in one-machine scheduling

We consider a one-machine scheduling problem with earliness and tardiness penalties. All jobs are assigned a common due date and the objective is to minimize the total penalty due to job earliness and tardiness. We are interested in finding the optimal combination of the common due-date value and the job sequence. Despite the fact that this problem in general is very hard to solve, we prove that there exists at least a common property for all optimal solutions: The first job in an optimal sequence is one of the longest jobs. We also prove that this property holds for a general class of unimodal penalty functions.