Critical Path Problem Research Articles

A hybrid modified-NSGA-II VNS algorithm for the Multi-Objective Critical Disruption Path Problem

This paper considers a Multiple Objective variant of the Critical Disruption Path problem to extend its suitability in a range of security operations relying on path-based network interdiction, including flight pattern optimisation for surveillance. Given a pair of nodes s and t from the network to be monitored, the problem seeks for loopless s−t paths such that, within the induced subgraph obtained via deletion of the path, the size of the largest connected component is minimised, the number of connected components is maximised, while concurrently reducing as much as possible the cost of such disruption path. These three objectives are possibly in conflict with each other, and the scope of this work is to allow for an efficient and insightful approximation of the Pareto front, looking for a trade-off between costs and effectiveness to secure the most convenient paths for security and surveillance operations. We first introduce and formulate the Multi-Objective Critical Disruption Path Problem (Multi-Objs-CDP) as a mixed integer programming formulation (MO-CDP), then we propose an original evolutionary metaheuristic algorithm hybridising modified-NSGA-II and VNS for finding an approximation of the Pareto front, as well as a procedure securing the efficient generation of a high quality pool of initial solutions. The experimental performance of the proposed algorithm, as compared with a variety of competing approaches, proves to be fully satisfactory in terms of time efficiency and quality of the solutions obtained on a set of medium to large benchmark instances.

Realistic solution of fuzzy critical path problems, case study: the airport’s cargo ground operation systems

Realistic solution of fuzzy critical path problems, case study: the airport’s cargo ground operation systems

Network Analysis for Projects with High Risk Levels in Uncertain Environments

The critical path method is one of the oldest and most important techniques used for planning and scheduling projects. The main objective of project management science is to determine the critical path through a network representation of projects. The critical path through a network can be determined by many algorithms and is useful for managing, monitoring, and controlling the time and cost of an entire project. The essential problem in this case is that activity durations are uncertain; time presents considerable uncertainty because the time of an activity is not always easily or accurately estimated. This issue increases the need to use neutrosophic theory to solve the critical path problem. Real-world problems are characterized by a lack of precision, consistency, and completeness. The concept of neutrosophic sets has been introduced as a generalization of fuzzy, intuitionistic fuzzy, and crisp sets to overcome the ambiguity surrounding real-world problems. Truth-, falsity-, and indeterminacy-membership functions are used to express neutrosophic elements. This study was performed to examine a neutrosophic event-oriented algorithm for determining the critical path in activity-on-arc networks. The activity time estimates are presented as trapezoidal neutrosophic numbers, and score and accuracy functions are used to obtain a crisp model of the problem. An appropriate numerical example is then used to explain the proposed method.

Intelligent Critical Path Computation Algorithm Utilising Ant Colony Optimisation for Complex Project Scheduling

In large and complex project schedule networks, existing algorithms to determine the critical path are considerably slow. Therefore, an algorithm with a faster convergence is needed to improve the efficiency of the critical path computation. The ant colony algorithm was first applied to the travelling salesman problem to determine the shortest path. However, many problems require the longest path in practice; the critical path in the scheduling problem is the longest path in the scheduling network. In this study, an improved ant colony algorithm to determine the critical path by setting the path distance and time as negative, while the transition probability remains unchanged, is proposed. The case of a coal power plant engineering, procurement, and construction (EPC) project was considered. The results show that a peak number of optimal solutions appeared at approximately the 9th iteration; however, instabilities and continued fluctuations were observed even afterward, indicating that the algorithm has a certain randomness. Convergence is apparent at the 29th iteration; after the 34th iteration, a singular optimal solution, the longest or critical path, is obtained, indicating that the convergence rate can be controlled and that the critical path can be obtained by setting appropriate parameters in the solution method. This has been found to improve the efficiency of calculating the critical path. Case validation and algorithm performance testing confirmed that the improved ant colony algorithm can determine the critical path problem and make it computationally intelligent.

On optimizing interval data based critical path problem for the analysis of various fuzzy quantities using fuzzification and centroid based defuzzification techniques

On optimizing interval data based critical path problem for the analysis of various fuzzy quantities using fuzzification and centroid based defuzzification techniques

Multi Objective Multi Mode Project Management Problem in Triangular Fuzzy Environment

Execution of any project with optimum duration, cost, quality and risk is very significant for project administrators in recent very competitive commercial situation. Sometimes it is not possible to have detailed earlier statistics about project criteria. In such situations, estimation of different Decision makers are considered in linguistic variables and altered into triangular fuzzy numbers as fuzzy numbers have ability to deal with vagueness. In this paper, we frame a new multi-mode multi objective critical path problem and suggest a possibilistic methodology to find critical path for a project where three decision makers’ views are considered as three modes of execution in terms of linguistic variables. We have formulated model of multiple mode in project network problem and find its solution with fuzzy programming approach with exponential membership and linear membership function. The proposed approach is useful to solve multi-mode project management problem which calculates optimal critical path according to four criteria- time, cost, risk and quality with three activities modes of execution in fuzzy environment.

Possibilistic Distribution for Selection of Critical Path in Multi Objective Multi-Mode Problem with Trapezoidal Fuzzy Number

Execution of any project with optimum duration, cost, quality and risk is very significant for project administrators in recent very competitive commercial situation. Sometimes it is not possible to have detailed earlier statistics about project criteria. In such situations, estimation of different Decision makers are considered in linguistic variables and altered into triangular fuzzy numbers as fuzzy numbers have ability to deal with vagueness. In this paper, we frame a new multi-mode multi objective critical path problem and suggest a possibilistic methodology to find critical path for a project where three decision makers’ views are considered as three modes of execution in terms of linguistic variables. In this paper have formulated model of multiple mode in project network problem and find its solution with fuzzy programming approach with exponential membership and linear membership function. The proposed approach is useful to solve multi-mode project management problem which calculates optimal critical path according to four criteria- time, cost, risk and quality with three activities modes of execution in fuzzy environment.

A group TOPSIS-COPRAS methodology with Pythagorean fuzzy sets considering weights of experts for project critical path problem

A group TOPSIS-COPRAS methodology with Pythagorean fuzzy sets considering weights of experts for project critical path problem

Algorithm for Optimizing Construction and Assembling of Power Equipment with Low Environmental Impact

The main purpose of this paper is to elaborate an algorithm (followed by an adequate software), designed to solve power engineering optimization problems applying critical path method. All the shortcomings are cleared when using the representation of the program via a graph, where the values of the arcs are the durations of the component operations, and applying the critical path method (CPM), which mainly consists in determining a path of a maximum value between two peaks of that graph. For illustrating the algorithm and the computing program we propose an application from power engineering: a 400 kV electrical overhead line section realization. In the first part of the paper we present the application as a critical path problem. In the second part, we determine the critic path in a program graph and time reserves. In the third part, we present a representative numerical application.

Software Application and Algorithm Designed for Power Systems Equipment and Network Construction Optimization

The main goal of this article is to conceive an algorithm for a software application designed to solve many power engineering optimization problems applying the critical path method (CPM). The classical solution for these problems is found via the schedule graph (the Gantt method), which presents a series of shortcomings: it does not highlight clearly the interdependencies between operations, it does not explain the temporal coincidences, it does not indicate the alternatives of declaring the various operations, it does not have a rigorous mathematical foundation, it does not allow any studies of optimization. All the shortcomings are cleared when using the representation of the program via a graph, where the values of the arcs are the durations of the component operations, and applying the critical path method (CPM), which mainly consists in determining a path of a maximum value between two peaks of that graph. Using CPM offers a great number of advantages: it offers a clear image of the evolution in time of the program, it allows the decreasing of the total duration of implementation for the program without condensing the component operation, it highlights the operations that directly determine the duration of implementation of the program, as well as those that allow the redistribution of the resources and the reduction of cost; it offers the alternative of rapid evaluation of the consequences of certain delays in realizing certain operations (without totally rebuilding the graph and the calculations); it can be easily implemented on a computer, having a solid mathematical base. Applying CPM and optimizing the program is achieved via the following steps: a) setting the list of the component operations, of their characteristics and representing the program via graphs; b) setting the graphs in order; c) determining the critical path (CP) and its value; d) calculating the time stocks related to the realization of the component operation (using the Ford and the Bellman-Kalaba algorithms); e) optimizing the program as from the length, the cost and the necessary resources points of view. For illustrating the algorithm and the computing program we propose an application from power engineering: a 400 kV electrical overhead line section construction. In the first part of the paper we present the application as a critical path problem. In the second part, we determine the critic path in a program graph and time reserves. In the third part, we present a representative numerical application. In the fourth part, it is described the computing program.

Comparision of Critical Path Problem under Fuzzy Environment with Conventional Method

Comparision of Critical Path Problem under Fuzzy Environment with Conventional Method

Optimizing insuring critical path problem under uncertainty based on GP-BPSO algorithm

This study considers a novel class of bi-level fuzzy random programming problem about insuring critical path. In this study, each task duration is assumed as a fuzzy random variable and follows the known possibility and probability distributions. Because there doesn’t exist an effective way to solve the problem directly, we first reduce the chance constraint to two equivalent random subproblems under two kinds of different risk attitudes. Then, we may use sample average approximation (SAA) method for reformulating the equivalent random programming subproblems as their approximation problems. Since the approximation problems are also hard to be solved, we explore a hybrid genotype phenotype binary particle swarm optimization algorithm (GP-BPSO) for resolving two equivalent subproblems, where dynamic programming method (DPM) is used for finding the solution in the lower level programming. At last, a series of simulation examples are performed for demonstrating the validity of the hybrid GP-BPSO compared with the hybrid BPSO algorithm.

Utility of Genetic Algorithms for Solving Large-Scale Construction Time-Cost Trade-Off Problems

The Time/Cost Trade-off (TCT) problem has long been a popular optimization question for construction engineering and management researchers. The problem manifests itself as the opti- mization of total costs of construction projects that consist of indirect project costs and individual activity costs. The trade-off occurs as project duration and, as a result, indirect project costs de- crease with reduced individual activity duration. This reduction in individual activity duration is achieved by increasing resource allocation to individual activities, which increases their costs to completion. Historically, metaheuristic solutions have been applied to small scale problems due to computational complexities and requirements of larger networks. In this paper, we demonstrate that the metaheuristic approach is highly effective for solving large scale construction TCT problems. A custom Genetic Algorithm (GA) is developed and used to solve large benchmark networks of up to 630 variables with high levels of accuracy (<3% deviation) consistently using computational power of a personal computer in under ten minutes. The same method can also be used to solve larger net- works of up to 6,300 variables with reasonable accuracy (∼7% deviation) at the expense of longerprocessing times. A number of simple, yet effective, techniques that improve GA performance for TCT problems are demonstrated; the most effective of which is a novel problem encoding, based on weighted graphs, that enables the critical path problem to be partially solved for all candidate solutions a priori, thus significantly increasing fitness evaluation. Other improvements include parallel fitness evaluations, optimal algorithm parameters, and the addition of a stagnation criteria. We also present some guidelines of optimal algorithm parameter selection through a comprehensive parameter sweep and a computational demand profile analysis. Moreover, the methods proposed in this article are based on open source development projects that enable scalable solutions without significant development efforts. This information will be beneficial for other researchers in improving computational efficiency of their solution in addressing TCT problems.

Solving Critical Path Problem in Project Network by a New Enhanced Multi-objective Optimization of Simple Ratio Analysis Approach with Interval Type-2 Fuzzy Sets

Decision making is an important issue in business and project management that assists finding the optimal alternative from a number of feasible alternatives. Decision making requires adequate consideration of uncertainty in projects. In this paper, in order to address uncertainty of project environments, interval type-2 fuzzy sets (IT2FSs) are used. In other words, the rating of each alternative and weight of each criterion are expressed by IT2FSs. Moreover, for obtaining weights of criteria, interval type-2 fuzzy analytic hierarchy process (AHP) method is employed. In addition, a new enhanced model of multi-objective optimization on the basis of simple ratio analysis (MOOSRA) method is developed with IT2F-relative preference relation. Finally, to illustrate applicability of the introduced approach, an existing application from literature is adopted and solved.

Intuitionistic fuzzy multi-criteria group decision making with an application to critical path selection

In this paper, we develop a new fuzzy multi-criteria group decision making method using triangular intuitionistic fuzzy numbers (TIFNs) for determining critical path in a critical path problem (CPP). The CPP considered here involves both quantitative and qualitative assessments of the decision makers on multiple conflicting criteria. The intuitionistic fuzzy numbers are introduced since they consider both preferences and non-preferences simultaneously and are thus capable of representing qualitatively evaluated information more effectively than fuzzy sets. The proposed method involves fuzzy evaluation based on the extended preference relation of TIFNs using \((\alpha ,\beta )\)-cuts considered for preferences and non-preferences, respectively. The preference intensity function based on the extended preference relation of TIFNs leads to the strength and weakness index scores of the possible paths on given criteria. Furthermore, we define the total performance score of each project path using its strength and weakness index scores. The path that has the highest score is selected as the best alternative in terms of its criticality for the entire project to finish as per the chosen criteria. A numerical illustration is provided to demonstrate working of the proposed methodology.

Network Interdiction through Length-Bounded Critical Disruption Paths: a Bi-Objective Approach

In this paper the Bi-Objective k-Length-Bounded Critical Disruption Path (BO-kLB-CDP) optimization problem is proposed, aimed at maximizing the interdiction effects provided on a network by removing a simple path connecting a given source and destination whose length does not exceed a certain threshold. The BO-kLB-CDP problem extends the Critical Disruption Path (CDP) problem introduced by Granata et al. in [Granata, D. and Steeger, G. and Rebennack, S., Network interdiction via a Critical Disruption Path: Branch-and-Price algorithms, Computers & Operations Research, Volume 40, Issue 11, November 2013, Pages 2689–2702]. Several real applications of this class of optimization problems arise in the field of security, surveillance, transportation and evacuation operations. In order to overcome some limits of the original CDP problem and increase its suitability for practical purposes, first we consider a length limitation for Critical Disruption Paths. Second, we generalize the concept of network interdiction considered in the CDP: beside minimizing the cardinality of the maximal connected component after the removal of the CDP, now we are also interested in maximizing the number of connected components in the residual graph. A Mixed Integer Programming formulation for the BO-kLB-CDP problem is therefore proposed and discussed, presenting the results of a multiple objective analysis performed through a computational experience on a large set of instances.

A new probability model for insuring critical path problem with heuristic algorithm

Abstract In order to obtain an adequate description of risk aversion for insuring critical path problem, this paper develops a new class of two-stage minimum risk problems. The first-stage objective function is to minimize the probability of total costs exceeding a predetermined threshold value, while the second-stage objective function is to maximize the insured task durations. For general task duration distributions, we adapt sample average approximation (SAA) method to probability objective function. The resulting SAA problem is a two-stage integer programming model, in which the analytical expression of second-stage value function is unavailable, we cannot solve it by conventional optimization algorithms. To avoid this difficulty, we design a new hybrid algorithm by combining dynamic programming method (DPM) and genotype-phenotype-neighborhood based binary particle swarm optimization (GPN-BPSO), where the DPM is employed to find the critical path in the second-stage programming problem. We conduct some numerical experiments via a critical path problem with 30 nodes and 42 arcs, and discuss the proposed risk averse model and the experimental results obtained by hybrid GPN-BPSO, hybrid genetic algorithm (GA) and hybrid BPSO. The computational results show that hybrid GPN-BPSO achieves the better performance than hybrid GA and hybrid BPSO, and the proposed critical path model is important for risk averse decision makers.

Linear programming approach for solving fuzzy critical path problems with fuzzy parameters

To the best of our knowledge, there is no method in the literature to find the fuzzy optimal solution of fully fuzzy critical path (FFCP) problems i.e., critical path problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method is proposed for the same. Also, it is shown that it is better to use JMD representation of LR flat fuzzy numbers in the proposed method as compared to the other representation of LR flat fuzzy numbers.

Critical Path Problem under Fuzzy Environment

In this paper, a novel approach has been made to find the critical path in a directed acyclic graph, where for each arc lengths fuzzy numbers are assigned instead of crisp numbers. Procedures are designed to find the optimal path, and finally illustrative examples are provided to demonstrate the proposed approach.

Network interdiction via a Critical Disruption Path: Branch-and-Price algorithms

This paper presents an innovative approach to maximally disconnect a given network. More specifically, this work introduces the concept of a Critical Disruption Path, a path between a source and a destination vertex whose deletion minimizes the cardinality of the largest remaining connected component. Network interdiction models seek to optimally disrupt network operations. Existing interdiction models disrupt network operations by removing vertices or edges. We introduce the first problem and formulation that optimally fragments a network via interdicting a path. Areas of study in which this work can be applied include transportation and evacuation networks, surveillance and reconnaissance operations, anti-terrorism activities, drug interdiction, and counter human-trafficking operations. In this paper, we first address the complexity associated with the Critical Disruption Path problem, and then provide a Mixed-Integer Linear Programming formulation for finding its optimal solution. Further, we develop a tailored Branch-and-Price algorithm that efficiently solves the Critical Disruption Path problem. We demonstrate the superiority of the developed Branch-and-Price algorithm by comparing the results found via our algorithm with the results found via the monolith formulation. In more than half of the test instances that can be solved by both the monolith and our Branch-and-Price algorithm, we outperform the monolith by two orders of magnitude.