Dynamic programming and minimum risk paths

This paper addresses the problem of computing minimum risk paths by taking as objective the expected accident cost. The computation is based on a dynamic programming formulation which can be considered an extension of usual dynamic programming models: path costs are recursively computed via functions which are assumed to be monotonic. A large part of the paper is devoted to analyze in detail this formulation and provide some new results. Based on the dynamic programming model a linear programming model is also presented to compute minimum risk paths. This formulation turns out to be useful in solving a biobjective version of the problem, in which also expected travel length is taken into consideration. This leads to define nondominated mixed strategies. Finally it is shown how to extend the basic updating device of dynamic programming in order to enumerate all nondominated paths

The development of a weighted directed graph model for dynamic systems and application of Dijkstra’s algorithm to solve optimal control problems.

Master of Science (Chemical Engineering). University of KwaZulu-Natal. Durban, 2017.Optimal control problems are frequently encountered in chemical engineering process control applications as a result of the drive for more regulatory compliant, efficient and economical operation of chemical processes. Despite the significant advancements that have been made in Optimal Control Theory and the development of methods to solve this class of optimization problems, limitations in their applicability to non-linear systems inherent in chemical process unit operations still remains a challenge, particularly in determining a globally optimal solution and solutions to systems that contain state constraints. The objective of this thesis was to develop a method for modelling a chemical process based dynamic system as a graph so that an optimal control problem based on the system can be solved as a shortest path graph search problem by applying Dijkstra’s Algorithm. Dijkstra’s algorithm was selected as it is proven to be a robust and global optimal solution based algorithm for solving the shortest path graph search problem in various applications. In the developed approach, the chemical process dynamic system was modelled as a weighted directed graph and the continuous optimal control problem was reformulated as graph search problem by applying appropriate finite discretization and graph theoretic modelling techniques. The objective functional and constraints of an optimal control problem were successfully incorporated into the developed weighted directed graph model and the graph was optimized to represent the optimal transitions between the states of the dynamic system, resulting in an Optimal State Transition Graph (OST Graph). The optimal control solution for shifting the system from an initial state to every other achievable state for the dynamic system was determined by applying Dijkstra’s Algorithm to the OST Graph. The developed OST Graph-Dijkstra’s Algorithm optimal control solution approach successfully solved optimal control problems for a linear nuclear reactor system, a non-linear jacketed continuous stirred tank reactor system and a non-linear non-adiabatic batch reactor system. The optimal control solutions obtained by the developed approach were compared with solutions obtained by the variational calculus, Iterative Dynamic Programming and the globally optimal value-iteration based Dynamic Programming optimal control solution approaches. Results revealed that the developed OST Graph-Dijkstra’s Algorithm approach provided a 14.74% improvement in the optimality of the optimal control solution compared to the variational calculus solution approach, a 0.39% improvement compared to the Iterative Dynamic Programming approach and the exact same solution as the value–iteration Dynamic Programming approach. The computational runtimes for optimal control solutions determined by the OST Graph-Dijkstra’s Algorithm approach were 1 hr 58 min 33.19 s for the nuclear reactor system, 2 min 25.81s for the jacketed reactor system and 8.91s for the batch reactor system. It was concluded from this work that the proposed method is a promising approach for solving optimal control problems for chemical process-based dynamic systems

COMPARATIVE ANALYSIS AND IMPLEMENTATION OF DIJKSTRA'S SHORTEST PATH ALGORITHM FOR EMERGENCY RESPONSE AND LOGISTIC PLANNING

TransRoute: a web-based vehicle route planning application is proposed in this paper. This application leverages existing input-output (I/O) efficient implementations of shortest path algorithms (SPAs) to implement the proposed system that will fundamentally address the problems experienced in moving people, goods and services from one location to another. A number of SPAs are evaluated using landau notations. Main functionalities of the system will be implemented as a web-enabled geographic information system (GIS) application based on open-source technologies and object-oriented software development methodology using unified modeling language. Pilot implementation is done based on spatial data of three selected states in Nigeria, pulled from web-based mapping tools like Google Maps and Microsoft Bings respectively. In conclusion, the Dijkstra's algorithm implemented with double bucket dynamic data structure is selected for implementing the proposed route planning system, as past research efforts has proven that it is the fastest with run-time improvements from O(m + n/log C) to O(m) respectively. http://dx.doi.org/10.4314/njt.v36i3.3

Stochastic Route Planning in Public Transport

Journey planning is a key process in public transport, where travelers get informed how to make the best use of a given public transport system for their individual travel needs. A common trait of most available journey planners is that they assume deterministic travel times, but vehicles in public transport often deviate from their schedule. The present paper investigates the problem of finding journey plans in a stochastic environment. To fully exploit the flexibility inherent in multi-service public transport systems, we propose to use the concept of a routing policy instead of a linear journey plan. A policy is a state-dependent routing advice which specifies a set of services at each location from which the traveler is recommended to take the one that arrives first. We consider current time dependent policies, that is, when the routing advice at a given location is based solely on the current time. We propose two heuristic solutions that find routing policies that perform better than deterministic journey plans. A numerical comparison shows the achievable gains when applying the different heuristic policies based on extensive simulations on the public transport network of Budapest. The results show that the probability of arriving on time to a given destination can be significantly improved by following a policy instead of a linear travel plan

Routing Metrics Depending on Previous Edges: The Mn Taxonomy and its Corresponding Solutions

The routing algorithms used by current operators aim at coping with the demanded QoS requirements while optimizing the use of their network resources. These algorithms rely on the optimal substructure property (OSP), which states that an optimal path contains other optimal paths within it. However, we show that QoS metrics such as queuing delay and buffer consumption do not satisfy this property, which implies that the used algorithms lose their optimality and/or completeness. This negatively impacts the operator economy by causing a waste of network resources and/or violating Service Level Agreements (SLAs). In this paper, we propose a new so-called Mn taxonomy defining new metric classes. An Mn metric corresponds to a metric which requires the knowledge of the n previously traversed edges to compute its value at a given edge. Based on this taxonomy, we present three solutions for solving routing problems with the newly defined classes of metrics. We show that state-of-the-art algorithms based on the OSP indeed lose their original optimality and/or completeness properties while our proposed solutions do not, at the price of an increased computation time.Comment: 2018 International Conference on Communications (ICC 2018), Kansas City, MO (USA