Sinusoidal flapping and pitching

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With 𝑎_𝑘𝑗 = 1 for 𝑤_𝑘𝑑_𝑘𝑗 < r and 𝑎_𝑘𝑗 = 0 for 𝑤_𝑘𝑑_𝑘𝑗 ≥ r computes a set of at most p-centers with a radius smaller than r or shows that no such set exists.

The literature on covering problems is divided into two major parts: the location set covering problem (LSCP) and the maximal covering location problem (MCLP). LSCP is an earlier version facility location problem and it aims at locating the least number of facilities that are required to cover all demand points. Since all the demand points need to be covered in LSCP, regardless of their population, remoteness, and demand quantity, the resources required for facilities could be excessive. Recognizing this problem, the MCLP model that does not require full coverage to all demand points was developed. Instead, the model seeks the maximal coverage with a given number of facilities. The MCLP, and different variants of it, have been extensively used to solve various emergency service location problems.

2.11.3.3 Mixed-integer programming models

Starting with a given set of potential facility sites many location problems can be modelled as mixed integer programming models. Apparently, network location models differ only gradually from mixed integer programming models because the former ones can be stated as discrete optimization models. Yet network location models explicitly take the structure of the set of potential facilities and the distance metric into account while mixed-integer programming models just use input parameters without asking where they come from. A rough classification of discrete facility location models can be given as follows: (a) single- vs. Multistage models, (b) uncapacitated vs. capacitated models, (c) multiple- vs. single-sourcing, (d) single- vs. multi-product models, (e) static vs. dynamic models, and, last but not least, (f) models without and with routing options included.

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site selection is a spatial problem, mathematics and optimization techniques are often inadequate to offer acceptable solution because of their failure to incorporate all relevant aspects of the problem in the overall framework. An alternative framework that is capable of resolving site selection is Geographical Information Systems. After a comprehensive study of the existing literatures, a number of gaps have been observed in literatures. The gaps in the literatures are further detailed below:

i. The optimal locations for bio-refineries depend on a number of other issues that are difficult to quantify and model mathematically (Xie, 2009), some of the researchers determined their best locations as pure mathematical problem (Xiao-Hua, et al., 2014; Florese et al., 2008), this is inadequate, since location problem is a spatial problem. It can be best solved using spatially added tools or programs.

Hence this work will integrate GIS with mathematical location models in obtaining the optimal location.

ii. Literature review reveals that few research works carried out on suitability analysis of bio-energy plant omitted the economic aspect of their determined optimal location on the suppliers. Few works that researched on economic viability of the centralized biogas plant, did not consider that of location analysis.

Since profitability is the major drive in many ventures, economic availability study which is lacking in previous literatures will be incorporated in the location analysis, hence an integrated approach which is necessary will be the core of this study.

iii. One of the important tasks in the suitability analysis is the integration of different preference criteria by providing weightage factors to the criteria, this area has not been fully explored by many literatures in this field. Few that explored MCA were

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based on their local preference, since preference criteria are usually localized, this study will appropriate the preference criteria on the conditions of the study area.

iv. The available information in the current literature on combining socio-environmental suitability and economic optimality in local geospatial scale is not adequate. There is a requirement for further research on optimal size and location of biomass-based facilities considering socio-economic factors, hence this study.

v. In addition, majority of work on site suitability analysis available in literatures concluded their research by providing the suitability index of the area under study;

this is inadequate, since the most suitable area could be in thousands of kilometers square. There is need to optimize these suitable areas by appropriate location models to obtain specific locations. This work will therefore integrate suitability analysis with location modeling.

Presently there are no scientific works to the best of the author‘s knowledge available online in Africa as a case study on optimal location of biogas technology plant.

Considering the immense need for renewable energy source for Nigeria; the population density of Anambra state which is the second to highest in the country; the necessity to integrate sensitive projects like CBP in urban planning; the climax condition of environmental deterioration and population explosion of the study area. Strategic positioning and integration of environmental variables well suited for the study area will definitely enhance the standard of living in the study area.

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CHAPTER THREE

MATERIALS AND METHODS