We consider two-machine routing open shop problem on a tree. In this problem a transportation network with a tree-like structure is given, and each node contains some jobs to be processed by two mobile machines. Machines are initially located in the predefined node called the depot, have to traverse the network to perform their operations on each job (and each job has to be processed by both machines in arbitrary order), and machines have to return to the depot after performing all the operations. The goal is to construct a feasible schedule for machines to process all the jobs and to return to the depot in shortest time possible. This problem is known to be NP-hard even in the case when the transportation network consists of just two nodes.We propose an instance reduction procedure which allows to transform any instance of the problem to a simplified instance on a chain with limited number of jobs. The reduction considered preserves the standard lower bound on the optimum. We describe four possible outcomes of this procedure and prove that in three of them the initial instance can be solved to the optimum in linear time, thus introducing a wide polynomially solvable subclass of the problem considered. Our research can be used as a foundation to construct efficient approximation algorithms for the two-machine routing open shop on a tree.
KEYWORDSScheduling; open shop with delays; routing open shop; standard lower bound; instance reduction; polynomially solvable subclass; overloaded node; overloaded edge 1. If i 1 = i 2 or j 1 = j 2 (but not both) then2. If operation of job J j ∈ J (v) is the first to start by machine M i then s ji dist(I; v 0 , v).3. If machine M i processes operation O ji before the processing of an operation O
Sensitivity of the pressure decline curve during the hydraulic fracturing to poroelastic effects, Sib.Abstract. Computer simulators of hydraulic fracturing rely on known physical properties of the reservoir, in particular, the leak-off coefficient and the confining in situ stress. This information can be obtained from the solution of the inverse problem by analyzing of the pressure decline curve (PDC) in pump-in/shut-in tests. The goal of the present work is to demonstrate that poroelastic effect can have significant influence on the behavior of the PDC and, hence, to results of its analysis.For computer simulations we use the mathematical model of the hydraulic fracture in poroelastic medium developed in [6]. We investigate the dependence of the PDC on the closure stress and of the rock permeability. It is shown that classical methods for interpretation of PDC can lead to a significant error, in particular, in estimation of the minimal in situ confining stress.Keywords: hydraulic fracture, mathematical modeling, pressure decline curve, poroelastic effects.Lgotina, E.V., Baykin, A.N., Golovin, S.V., Krivtsov, A.M., Sensitivity of the pressure decline curve during the hydraulic fracturing to poroelastic effects.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.