Integer Programming problems are difficult to solve. The goal is to find an optimal solution that minimizes cost. With the help of Groebner based algorithms the optimal solution can be found if it exists. The application of the Groebner based algorithm and how it works is the topic of research. The Algorithms are The Conti-Traverso Algorithm and the Original Conti-Traverso Algorithm. Examples are given as well as proofs that correspond to the algorithms. The latter algorithm is more efficient as well as user friendly. The algorithms are not necessarily the best way to solve and integer programming problem, but they do find the optimal solution if it exists.

This study deals with analysis of the agricultural company layout. The utilization of agricultural areas is optimized with such a sowing technique that leads to the highest harvest profit. All these issues are being solved by the conversion to the linear programming model which is the most resembling to the combination of production planning and allocation problem. A case study is based on actual data given by Agrofarma Týnec, s.r.o. company. The 48 sections of field in different proportion are taken in consideration as well as five different crops. The study tries to deal with more options taking in account different trade prices of set commodities and a hectare sowing limitation. The main model anticipates pessimistic trade prices expectations and does not restrict hectare sowing with single crops. The model is followed by the price coefficient analysis that gives the price level to be kept to let the computed solution be optimal. Lingo13.0 was used as an optimization software.

Real-world systems usually contain some degree of natural uncertainty, their parameters are more or less variable. When seeking optimal solution, optimization models often disregard this variability and consider parameters of the model to be constant. This thesis focuses on methods of post-optimization analysis. Thorough post-optimization analysis should be a part of every optimization process of systems with variable parameters. Post-optimization analysis can identify parameters whose variability poses the greatest threat to the systems performance. This thesis describes some of the basic post-optimization methods and then a new method based on interval arithmetics is formulated.

While the channel capacity of an isolated noise-limited wireless link is well-understood, the same is not true for the interference-limited wireless links that coexist in the same area and occupy the same frequency band(s). The performance of these wireless systems is coupled to each other due to the mutual interference. One such wireless scenario is modeled as a network of simultaneously communicating node pairs and is generally referred to as an interference channel (IC). The problem of characterizing the capacity of an IC is one of the most interesting and long-standing open problems in information theory. A popular way of characterizing the capacity of an IC is to maximize the achievable sum-rate by treating interference as Gaussian noise, which is considered optimal in low-interference scenarios. While the sum-rate of the single-band SISO IC is relatively well understood, it is not so when the users have multiple-bands and multiple-antennas for transmission. Therefore, the study of the optimal sum-rate of the multi-band MIMO IC is the main goal of this thesis. The sum-rate maximization problem for these ICs is formulated and is shown to be quite similar to the one already known for single-band MIMO ICs. This problem is reduced to the problem of finding the optimal fraction of power to be transmitted over each spatial channel in each frequency band. The underlying optimization problem, being non-linear and non-convex, is difficult to solve analytically or by employing local optimization techniques. Therefore, we develop a global optimization algorithm by extending the Reformulation and Linearization Technique (RLT) based Branch and Bound (BB) strategy to find the provably optimal solution to this problem. We further show that the spatial and spectral channels are surprisingly similar in a multi-band multi-antenna IC from a sum-rate maximization perspective. This result is especially interesting because of the dissimilarity in the way the spatial and frequency channels affect the perceived interference. As a part of this study, we also develop some rules-of-thumb regarding the optimal power allocation strategies in multi-band MIMO ICs in various interference regimes. Due to the recent popularity of Interference Alignment (IA) as a means of approaching capacity in an IC (in high-interference regime), we also compare the sum-rates achievable by our technique to the ones achievable by IA. The results indicate that the proposed power control technique performs better than IA in the low and intermediate interference regimes. Interestingly, the performance of the power control technique improves further relative to IA with an increase in the number of orthogonal spatial or frequency channels. / Master of Science

Many concrete problems, ranging from Portfolio selection to Water resource management, may be cast into a multiobjective programming framework. The simplistic way of superseding blindly conflictual goals by one objective function let no chance to the model but to churn out meaningless outcomes. Hence interest of discussing ways for tackling Multiobjective Programming Problems. More than this, in many real-life situations, uncertainty and imprecision are in the state of affairs. In this dissertation we discuss ways for solving Multiobjective Programming Problems with fixed and fuzzy coefficients. No preference, a priori, a posteriori, interactive and metaheuristic methods are discussed for the deterministic case. As far as the fuzzy case is concerned, two approaches based respectively on possibility measures and on Embedding Theorem for fuzzy numbers are described. A case study is also carried out for the sake of illustration. We end up with some concluding remarks along with lines for further development, in this field. / Operations Research / M. Sc. (Operations Research)

Many concrete problems, ranging from Portfolio selection to Water resource management, may be cast into a multiobjective programming framework. The simplistic way of superseding blindly conflictual goals by one objective function let no chance to the model but to churn out meaningless outcomes. Hence interest of discussing ways for tackling Multiobjective Programming Problems. More than this, in many real-life situations, uncertainty and imprecision are in the state of affairs. In this dissertation we discuss ways for solving Multiobjective Programming Problems with fixed and fuzzy coefficients. No preference, a priori, a posteriori, interactive and metaheuristic methods are discussed for the deterministic case. As far as the fuzzy case is concerned, two approaches based respectively on possibility measures and on Embedding Theorem for fuzzy numbers are described. A case study is also carried out for the sake of illustration. We end up with some concluding remarks along with lines for further development, in this field. / Operations Research / M. Sc. (Operations Research)

This thesis contains several parts - current state analysis, a solution draft based on theoretical reccommandations and last but not least results brought by the chosen strategy.

In the recent years there has been a great deal of attention on the optimal demand and supply side strategy. The increase in renewable energy sources and the expansion in demand response programmes has shown the need for a robust power system. These changes in power system require the control of the uncertain generation and load at the same time. Therefore, it is important to provide an optimal scheduling strategy that can meet an adequate energy mix under demand response without affecting the system reliability and economic performance. This thesis addresses the following four aspects to these changes. First, a renewable obligation model is proposed to maintain an adequate energy mix in the economic dispatch model while minimising the operational costs of the allocated spinning reserves. This method considers a minimum renewable penetration that must be achieved daily in the energy mix. If the renewable quota is not achieved, the generation companies are penalised by the system operator. The uncertainty of renewable energy sources are modelled using the probability density functions and these functions are used for scheduling output power from these generators. The overall problem is formulated as a security constrained economic dispatch problem. Second, a combined economic and demand response optimisation model under a renewable obligation is presented. Real data from a large-scale demand response programme are used in the model. The model finds an optimal power dispatch strategy which takes advantage of demand response to minimise generation cost and maximise renewable penetration. The optimisation model is applied to a South African large-scale demand response programme in which the system operator can directly control the participation of the electrical water heaters at a substation level. Actual load profile before and after demand reduction are used to assist the system operator in making optimal decisions on whether a substation should participate in the demand response programme. The application of these real demand response data avoids traditional approaches which assume arbitrary controllability of flexible loads. Third, a stochastic multi-objective economic dispatch model is presented under a renewable obligation. This approach minimises the total operating costs of generators and spinning reserves under renewable obligation while maximising renewable penetration. The intermittency nature of the renewable energy sources is modelled using dynamic scenarios and the proposed model shows the effectiveness of the renewable obligation policy framework. Due to the computational complexity of all possible scenarios, a scenario reduction method is applied to reduce the number of scenarios and solve the model. A Pareto optimal solution is presented for a renewable obligation and further decision making is conducted to assess the trade-offs associated with the Pareto front. Four, a combined risk constrained stochastic economic dispatch and demand response model is presented under renewable obligation. An incentive based optimal power dispatch strategy is implemented to minimise generation costs and maximise renewable penetration. In addition, a risk-constrained approach is used to control the financial risks of the generation company under demand response programme. The coordination strategy for the generation companies to dispatch power using thermal generators and renewable energy sources while maintaining an adequate spinning reserve is presented. The proposed model is robust and can achieve significant demand reduction while increasing renewable penetration and decreasing the financial risks for generation companies. / Thesis (PhD (Electrical Engineering))--University of Pretoria, 2020. / Electrical, Electronic and Computer Engineering / PhD (Electrical Engineering) / Unrestricted

Yes / Investigation of the environmental/economic optimal operation management of a microgrid (MG) as a case study for applying a novel modified multi-objective grey wolf optimizer (MMOGWO) algorithm is presented in this paper. MGs can be considered as a fundamental solution in order for distributed generators’ (DGs) management in future smart grids. In the multi-objective problems, since the objective functions are conflict, the best compromised solution should be extracted through an efficient approach. Accordingly, a proper method is applied for exploring the best compromised solution. Additionally, a novel distance-based method is proposed to control the size of the repository within an aimed limit which leads to a fast and precise convergence along with a well-distributed Pareto optimal front. The proposed method is implemented in a typical grid-connected MG with non-dispatchable units including renewable energy sources (RESs), along with a hybrid power source (micro-turbine, fuel-cell and battery) as dispatchable units, to accumulate excess energy or to equalize power mismatch, by optimal scheduling of DGs and the power exchange between the utility grid and storage system. The efficiency of the suggested algorithm in satisfying the load and optimizing the objective functions is validated through comparison with different methods, including PSO and the original GWO. / Supported in part by Royal Academy of Engineering Distinguished Visiting Fellowship under Grant DVF1617\6\45

This Thesis Work will concentrate on a very interesting problem, the Vehicle Routing Problem (VRP). In this problem, customers or cities have to be visited and packages have to be transported to each of them, starting from a basis point on the map. The goal is to solve the transportation problem, to be able to deliver the packages-on time for the customers,-enough package for each Customer,-using the available resources- and – of course - to be so effective as it is possible.Although this problem seems to be very easy to solve with a small number of cities or customers, it is not. In this problem the algorithm have to face with several constraints, for example opening hours, package delivery times, truck capacities, etc. This makes this problem a so called Multi Constraint Optimization Problem (MCOP). What’s more, this problem is intractable with current amount of computational power which is available for most of us. As the number of customers grow, the calculations to be done grows exponential fast, because all constraints have to be solved for each customers and it should not be forgotten that the goal is to find a solution, what is best enough, before the time for the calculation is up. This problem is introduced in the first chapter: form its basics, the Traveling Salesman Problem, using some theoretical and mathematical background it is shown, why is it so hard to optimize this problem, and although it is so hard, and there is no best algorithm known for huge number of customers, why is it a worth to deal with it. Just think about a huge transportation company with ten thousands of trucks, millions of customers: how much money could be saved if we would know the optimal path for all our packages.Although there is no best algorithm is known for this kind of optimization problems, we are trying to give an acceptable solution for it in the second and third chapter, where two algorithms are described: the Genetic Algorithm and the Simulated Annealing. Both of them are based on obtaining the processes of nature and material science. These algorithms will hardly ever be able to find the best solution for the problem, but they are able to give a very good solution in special cases within acceptable calculation time.In these chapters (2nd and 3rd) the Genetic Algorithm and Simulated Annealing is described in details, from their basis in the “real world” through their terminology and finally the basic implementation of them. The work will put a stress on the limits of these algorithms, their advantages and disadvantages, and also the comparison of them to each other.Finally, after all of these theories are shown, a simulation will be executed on an artificial environment of the VRP, with both Simulated Annealing and Genetic Algorithm. They will both solve the same problem in the same environment and are going to be compared to each other. The environment and the implementation are also described here, so as the test results obtained.Finally the possible improvements of these algorithms are discussed, and the work will try to answer the “big” question, “Which algorithm is better?”, if this question even exists.