scholarly journals Algebraic solution of a problem of optimal project scheduling in project management

2021 ◽  
Vol 8 (1) ◽  
pp. 73-87
Author(s):  
Nikolay K. Krivulin ◽  
◽  
Sergey A. Gubanov ◽  
Keyword(s):  
Project Management ◽  
Optimization Problem ◽  
Optimal Scheduling ◽  
Algebraic Solution ◽  
Maximum Deviation ◽  
Scheduling Problem ◽  
Minimax Optimization ◽  
Start Time ◽  
The Times ◽  
Constrained Minimax

A problem of optimal scheduling is considered for a project that consists of a certain set of works to be performed under given constraints on the times of start and finish of the works. As the optimality criterion for scheduling, the maximum deviation of the start time of works is taken to be minimized. Such problems arise in project management when it is required, according to technological, organizational, economic or other reasons, to provide, wherever possible, simultaneous start of all works. The scheduling problem under consideration is formulated as a constrained minimax optimization problem and then solved using methods of tropical (idempotent) mathematics which deals with the theory and applications of semirings with idempotent addition. First, a tropical optimization problem is investigated defined in terms of a general idempotent semifield (an idempotent semiring with invertible multiplication), and a complete analytical solution of the problem is derived. The result obtained is then applied to find a direct solution of the scheduling problem in a compact vector form ready for further analysis of solutions and straightforward computations. As an illustration, a numerical example of solving optimal scheduling problem is given for a project that consists of four works.

scholarly journals Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 303
Author(s):  
Nikolai Krivulin
Keyword(s):  
Decision Problem ◽  
Optimization Problem ◽  
Pareto Frontier ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Optimal Solution ◽  
Algebraic Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Idempotent Algebra

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

scholarly journals Solving optimal generation scheduling problem of microgrid using teaching learning based optimization algorithm

2020 ◽  
Vol 17 (3) ◽  
pp. 1632
Author(s):  
Surender Reddy Salkuti
Keyword(s):  
Demand Response ◽  
Operating Cost ◽  
Optimal Scheduling ◽  
Energy Resources ◽  
Successful Implementation ◽  
Scheduling Problem ◽  
Solar Pv ◽  
Teaching Learning Based Optimization ◽  
Diesel Generators ◽  
Teaching Learning

<p>This paper proposes a new optimal scheduling methodology for a Microgrid (MG) considering the energy resources such as diesel generators, solar photovoltaic (PV) plants, wind farms, battery energy storage systems (BESSs), electric vehicles (EVs) and demand response (DR). The penetration level of renewable and sustainable energy resources (i.e., wind, solar PV energy, geothermal and ocean energy) in power generation systems is increasing. In this work, the EVs and storage are used as flexible DR sources and they can be combined with DR to improve the flexibility of MG. Various uncertainties exist in the MGs due to the intermittent/uncertain nature of renewable energy resources (RERs) such as wind and solar PV power outputs. In this paper, these uncertainties are modeled by using the probability analysis. In this paper, the optimal scheduling problem of MG is solved by minimizing the total operating cost (TOC) of MG. The TOC minimization objective is formulated by considering the cost due to power exchange between main grid and MG, diesel generators, wind, solar PV units, EVs, BESSs, and DR. The successful implementation of optimal scheduling of MG requires the widespread use of demand response and EVs. In this paper, teaching-learning-based optimization (TLBO) algorithm is used to solve the proposed optimization problem. The simulation studies are performed on a test MG by considering all the components of MG.</p>

scholarly journals 3/2-approximation algorithm for a single machine scheduling problem

2021 ◽  
Vol 17 (3) ◽  
pp. 240-253
Author(s):  
Natalia S. Grigoreva ◽  
Keyword(s):  
Approximation Algorithm ◽  
Optimization Problem ◽  
Single Machine Scheduling ◽  
Release Time ◽  
Scheduling Problem ◽  
Delivery Time ◽  
Scheduling Problems ◽  
Delivery Times ◽  
Jobshop Scheduling ◽  
Single Processor

The problem of minimizing the maximum delivery times while scheduling tasks on a single processor is a classical combinatorial optimization problem. Each task ui must be processed without interruption for t(ui) time units on the machine, which can process at most one task at time. Each task uw; has a release time r(ui), when the task is ready for processing, and a delivery time g(ui). Its delivery begins immediately after processing has been completed. The objective is to minimize the time, by which all jobs are delivered. In the Graham notation this problem is denoted by 1|rj,qi|Cmax, it has many applications and it is NP-hard in a strong sense. The problem is useful in solving owshop and jobshop scheduling problems. The goal of this article is to propose a new 3/2-approximation algorithm, which runs in O(n log n) times for scheduling problem 1|rj.qi|Cmax. An example is provided which shows that the bound of 3/2 is accurate. To compare the effectiveness of proposed algorithms, random generated problems of up to 5000 tasks were tested.

Optimization of Makespan in Job Shop Scheduling Problem by Golden Ball Algorithm

2016 ◽  
Vol 4 (3) ◽  
pp. 542 ◽  
Author(s):  
Fatima Sayoti ◽  
Mohammed Essaid Riffi ◽  
Halima Labani
Keyword(s):  
Optimization Problem ◽  
Job Shop ◽  
Job Shop Scheduling ◽  
Combinatorial Optimization Problem ◽  
Scheduling Problem ◽  
Job Shop Scheduling Problem ◽  
Shop Scheduling ◽  
Problem Finding ◽  
Benchmark Instances ◽  
Golden Ball

<em>Job shop scheduling problem (JSSP) is considered to belong to the class of NP-hard combinatorial optimization problem. Finding a solution to this problem is equivalent to solving different problems of various fields such as industry and logistics. The objective of this work is to optimize the makespan in JSSP using Golden Ball algorithm. In this paper we propose an efficient adaptation of Golden Ball algorithm to the JSSP. Numerical results are presented for 36 instances of OR-Library. The computational results show that the proposed adaptation is competitive when compared with other existing methods in the literature; it can solve the most of the benchmark instances.</em>

Assessment of a Simpler Friction Factor in an Algebraic Solution for Adiabatic Coiled Capillary Tubes

2020 ◽  
Vol 28 (04) ◽  
pp. 2050033
Author(s):  
Thiago Torres Martins Rocha ◽  
Sara Isabel De Melo Resende ◽  
Hélio Augusto Goulart Diniz ◽  
Fernando Antônio Rodrigues Filho ◽  
Raphael Nunes De Oliveira
Keyword(s):  
Friction Factor ◽  
Algebraic Solution ◽  
Maximum Deviation ◽  
Empirical Correlations ◽  
Capillary Tubes ◽  
Average Deviation ◽  
Flow Rates ◽  
Mass Flow Rates ◽  
Rms Error ◽  
Dimensionless Correlations

In this work, the performance of an existing algebraic solution for adiabatic coiled capillary tubes, in subcritical cycles, is investigated. However, the C-M&N friction factor, commonly used, was replaced by Schmidt friction factor, which is less complex. Two existing dimensionless correlations were also evaluated for comparison. To assess the effect of altering the friction factor, experimental data collected in the literature were used as reference. Analyzing the present results and that with C-M&N friction factor, it was observed that adopting the Schmidt friction factor does not cause a relevant impact on the solution. The deviations of the predicted versus experimental mass flow rates were comprised in a range between –8% and 12%, with average deviation (AD), absolute average deviation (AAD) and root mean square (RMS) error of –0.1%, 2.7% and 3.4%, respectively. The empirical correlations presented unsatisfactory results, with maximum deviation around 40%. Therefore, it was concluded that using the Schmidt friction factor is adequate to reduce the complexity of the algebraic solution and to maintain the accuracy.

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