Review Article
1) Article Reference
Pan TC, Kao JJ (2009) GA-QP Model to Optimize Sewer System Design, JOURNAL OF ENVIRONMENTAL ENGINEERING, 135(1) 17-24
2) Summary
This paper presents an application of genetic algorithms to optimize the design of a sewer system. A case study is presented to illustrate the methodology using system composed of 79 links, 56 nodes and a drainage area of 260 ha.
The author’s presents a discussion of the complexity of designing a sewer system targeting the minimum cost, maintaining the system hydraulic and construction constraints. A description of the concepts of genetic algorithm is also presented including all its elements and behavior. For this problem the decision variable of the sewer are coded as genes and each chromosome represents one design. The first constraint assures that the pipe has enough flow capacity, the second constraint takes care of ensuring the downstream pipe has diameter equal or greater then the upstream neighbor. It is also considered the possibility of installing a pump in each node. The fitness function is used considering the construction cost of the network. A description of the selection, crossover and mutation methods is provided. For this particular study, the author used also a quadratic programming (QP) model, to improve efficiency, the nonlinear cost fitness function was approximated to a QP. The design parameters for this case study were the maximum velocity, minimum velocity, minimum slope, maximum proportional water depth and minimum cover depth. To avoid finding a solution which was not the true best, due to simplifications or factors that are hard to formulate into the model, it was also used a MGA function.
Different results for the case study were presented achieving a minimum cost of approximate 1.7 million dollars. According to the author the GA and quadratic model approach were satisfactory to find a good solution for the problem.
3) Discussion
It really surprised me to see that this paper was published in 2009 and nobody had ever tried such approach for solving sewer systems network. After all our classes, I felt quite confident in understating all the terminology and the methodology for solving the GA presented in this paper. Actually I think the paper had even too much of GA theory instead of new contributions for science. The best part, I think, is that I feel very encouraged to apply GA optimization methods for my research area.
Tuesday, April 28, 2009
Assignment #10a - Optimization in public sector
Review Article
1) Article Reference
BRILL ED (1979) USE OF OPTIMIZATION MODELS IN PUBLIC-SECTOR PLANNING, MANAGEMENT SCIENCE, 25
2) Summary
This paper dates from the seventies and deals with the use of optimization models applied to public sector problem solving. According to the authors, many optimization models were concentrated in finding the best economic solution, but failure to consider equity and have empirical shortcoming in estimating benefits and costs. The author also suggests that multi objective programming can examine tradeoffs between different objectives and mention the use of goal programming as well. Many limitations to the use of optimization techniques are pointed, specially related to complete and incomplete multi objective programming, with some examples as the planning of a lakeside park with solutions favoring boaters or swimmers and the location of regional facilities.
An important issue, as pointed by the author, is that usually optimization models have been used to find the “answer” for the public sector, considering the economic efficiency and social optimality. As empirical problems arise, new approaches had to be developed. Among many reasons pointed by the author, the fact that many of the problems are wicked, demonstrated the difficulty to find one best approach for solving this problems.
A general flow chart on how to use optimization techniques for the planning process is presented and some alternatives discussed in more detail such as the join use of models: optimization and simulation; analytical and optimization; and using a toolbox of models. Several examples are presented, including one study developed in the Netherlands to analyze the implementation of an estuary dam, considering several factors such as flood protection, ecological aspects, costs and social impacts providing elements for the parliament ultimate decision.
Two main advantages by that time are pointed by the author as the capability of generating alternatives and facilitating evaluations and generating alternative solutions that are different from each other.
The author finally concludes that although multiojective planning in the way it was developed by his time, “as the second generation of optimization techniques”, had improved the way of solving optimization problems, but was still limited to the wicked organization of public sector problems. He also argues that such techniques should be used to gain insights about the problem itself, develop alternative scenarios and support human creativity to find the best solutions of a problem.
3) Discussion
I personally think it is great to have a historical overview on how the author, and scientific community, is approaching optimization problems. Besides the historical importance, I think an important issue that puts his discussion a little out of date is the considerable advance of computer potential and capacity to solve mathematical problems nowadays.
1) Article Reference
BRILL ED (1979) USE OF OPTIMIZATION MODELS IN PUBLIC-SECTOR PLANNING, MANAGEMENT SCIENCE, 25
2) Summary
This paper dates from the seventies and deals with the use of optimization models applied to public sector problem solving. According to the authors, many optimization models were concentrated in finding the best economic solution, but failure to consider equity and have empirical shortcoming in estimating benefits and costs. The author also suggests that multi objective programming can examine tradeoffs between different objectives and mention the use of goal programming as well. Many limitations to the use of optimization techniques are pointed, specially related to complete and incomplete multi objective programming, with some examples as the planning of a lakeside park with solutions favoring boaters or swimmers and the location of regional facilities.
An important issue, as pointed by the author, is that usually optimization models have been used to find the “answer” for the public sector, considering the economic efficiency and social optimality. As empirical problems arise, new approaches had to be developed. Among many reasons pointed by the author, the fact that many of the problems are wicked, demonstrated the difficulty to find one best approach for solving this problems.
A general flow chart on how to use optimization techniques for the planning process is presented and some alternatives discussed in more detail such as the join use of models: optimization and simulation; analytical and optimization; and using a toolbox of models. Several examples are presented, including one study developed in the Netherlands to analyze the implementation of an estuary dam, considering several factors such as flood protection, ecological aspects, costs and social impacts providing elements for the parliament ultimate decision.
Two main advantages by that time are pointed by the author as the capability of generating alternatives and facilitating evaluations and generating alternative solutions that are different from each other.
The author finally concludes that although multiojective planning in the way it was developed by his time, “as the second generation of optimization techniques”, had improved the way of solving optimization problems, but was still limited to the wicked organization of public sector problems. He also argues that such techniques should be used to gain insights about the problem itself, develop alternative scenarios and support human creativity to find the best solutions of a problem.
3) Discussion
I personally think it is great to have a historical overview on how the author, and scientific community, is approaching optimization problems. Besides the historical importance, I think an important issue that puts his discussion a little out of date is the considerable advance of computer potential and capacity to solve mathematical problems nowadays.
Assignment #9 - Compromise programming - instream flow - multiobjective
1) Article Reference
Shiau JT, Wu FC (2006) Compromise programming methodology for determining instream flow under multiobjective water allocation criteria, JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION, 42(5), pp. 1179-1191
2) Summary
This paper presents a multiobjective approach for evaluating instream flow water allocation considering increasing water uses within the basin. This approach uses the concept of Range of Variability (RVA) Approach and the Indicators of Hydrologic Alterations (IHAs).
A case study is presented for the Kaoping creek in southwestern Taiwan. Extensive description of the hydrologic and water use characteristics of the basin is provided. Special concerns are related to endangered and endemic species. By date, instream flows are been provide to support the ecological health of the systems, but is believed that those releases are incapable of guarantying sufficient flow variation required for the sustainability of the aquatic biota. Tables presented monthly average inflow and water uses for agricultural and municipal uses within the basin.
For the optimization analyses of the water allocation schemes, a description of the methodology used is provide, including a short review of the Range of Variability Approach concepts and the overall degree of hydrologic alteration. A detailed table is provided to describe the IHAs used in the RVA. The IHAs are grouped in 5 different specifications which represent different hydrologic parameters.
A description of the weir (main diversion point) operation model is also provided. Within this description the operation of the weir and the different water uses are described.
The main goal of the weir is to supply the registered agricultural demand , the projected municipal water supply and the instream flow conditions. Considering the municipal water supply and the agricultural use, the objective is to minimize the shortages periods, represented by a shortage ratio.
To solve the minimization function, the authors used a Multiobjective compromise programming approach. The results evaluated the current schedule operation impacts on water shortages and hydrologic alterations, the effects of different instream flow releases ,the effects of weighting factors and the Ecological Effects of proposed instream flow release
The main conclusions of the paper were that the inclusion of the individual degrees of alteration associated with the 32 IHAs made possible to optimize the weir operation scheme through compromise programming and showed that the current instream releases do not meet minimum requirements for guarantee ecological health downstream.
3) Discussion
I enjoyed reading this paper as it gave me a better sense on the practical application of multi objective problem solving. I think this kind of optimization is very useful, as it really approaches real life problems. Usually there are more than one objective that almost always differ from each other. Although, I think the evaluation of the intream flow necessities could have been better evaluated.
Shiau JT, Wu FC (2006) Compromise programming methodology for determining instream flow under multiobjective water allocation criteria, JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION, 42(5), pp. 1179-1191
2) Summary
This paper presents a multiobjective approach for evaluating instream flow water allocation considering increasing water uses within the basin. This approach uses the concept of Range of Variability (RVA) Approach and the Indicators of Hydrologic Alterations (IHAs).
A case study is presented for the Kaoping creek in southwestern Taiwan. Extensive description of the hydrologic and water use characteristics of the basin is provided. Special concerns are related to endangered and endemic species. By date, instream flows are been provide to support the ecological health of the systems, but is believed that those releases are incapable of guarantying sufficient flow variation required for the sustainability of the aquatic biota. Tables presented monthly average inflow and water uses for agricultural and municipal uses within the basin.
For the optimization analyses of the water allocation schemes, a description of the methodology used is provide, including a short review of the Range of Variability Approach concepts and the overall degree of hydrologic alteration. A detailed table is provided to describe the IHAs used in the RVA. The IHAs are grouped in 5 different specifications which represent different hydrologic parameters.
A description of the weir (main diversion point) operation model is also provided. Within this description the operation of the weir and the different water uses are described.
The main goal of the weir is to supply the registered agricultural demand , the projected municipal water supply and the instream flow conditions. Considering the municipal water supply and the agricultural use, the objective is to minimize the shortages periods, represented by a shortage ratio.
To solve the minimization function, the authors used a Multiobjective compromise programming approach. The results evaluated the current schedule operation impacts on water shortages and hydrologic alterations, the effects of different instream flow releases ,the effects of weighting factors and the Ecological Effects of proposed instream flow release
The main conclusions of the paper were that the inclusion of the individual degrees of alteration associated with the 32 IHAs made possible to optimize the weir operation scheme through compromise programming and showed that the current instream releases do not meet minimum requirements for guarantee ecological health downstream.
3) Discussion
I enjoyed reading this paper as it gave me a better sense on the practical application of multi objective problem solving. I think this kind of optimization is very useful, as it really approaches real life problems. Usually there are more than one objective that almost always differ from each other. Although, I think the evaluation of the intream flow necessities could have been better evaluated.
Wednesday, April 1, 2009
Assignment #8 - Neural Network / Reservoir Optimization
Review Article
1) Article Reference
Neelakantan TR, Pundarikanthan NV (2000) “Neural network-based simulation-optimization model for reservoir operation,” JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT, 126(2) pp. 57-64
2) Summary
This paper presents an application of neural network-based simulation optimization model for reservoir operation. A case study is presented for Chennai city, in India. The general framework of the study presented by the author follows: 1) Develop a conventional simulation model to obtain results from different operation policies; 2) Train a back propagation neural network model using those results; 3) Link the neural network model as a sub model with direct search non-linear programming optimization models; 4)Find the optimal policy and near optimal policies using neural network –optimization model; 5)Refine the optimal policies obtained using conventional simulation optimization model and determine the optimal policy.
The optimization is done using the Hookes and Jeeves direct search method. For this case it was considered better to have small water supply restrictions along the time from having one big drought. Then the optimization goal was to minimize the minimum sum of the deficits. For that a deficit index was defined. The optimization procedure started at an initial point and progress using Hookes and Jeeves method until an optimal solution. Several initial points were used as this algorithm can get stacked in local minimums.
A neural network simulation model was used in order to enhance the speed of the process. For this case the authors used a back propagation approach. The network was trained using pairs of input and output vectors.
The decision set was composed of 18 decision variables. Based in inflow pattern the year was divided into six time periods. Four out of five supply levels were considered at each time. The results are presented in two different scenarios. For each scenario, the authors presents the results for 4 different policies including the storage for each time period, the release and the deficit. The inclusion of two additional reservoirs is also analyzed.
As the major result, the authors concluded that the neural network based simulation-optimization model performed satisfactory. Also the authors mention that reservoir operation problems considering several and more complicated networks could be handled by this method.
3) Discussion
The paper presents a very interesting approach to solve a very common reservoir operation problem. I believe several attempts have been made to find methodologies for optimal operation rules for reservoir. I think this is an emerging issue as water demand is increasing significantly in large metropolitan areas and the reservoirs systems have to be used optimally.
1) Article Reference
Neelakantan TR, Pundarikanthan NV (2000) “Neural network-based simulation-optimization model for reservoir operation,” JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT, 126(2) pp. 57-64
2) Summary
This paper presents an application of neural network-based simulation optimization model for reservoir operation. A case study is presented for Chennai city, in India. The general framework of the study presented by the author follows: 1) Develop a conventional simulation model to obtain results from different operation policies; 2) Train a back propagation neural network model using those results; 3) Link the neural network model as a sub model with direct search non-linear programming optimization models; 4)Find the optimal policy and near optimal policies using neural network –optimization model; 5)Refine the optimal policies obtained using conventional simulation optimization model and determine the optimal policy.
The optimization is done using the Hookes and Jeeves direct search method. For this case it was considered better to have small water supply restrictions along the time from having one big drought. Then the optimization goal was to minimize the minimum sum of the deficits. For that a deficit index was defined. The optimization procedure started at an initial point and progress using Hookes and Jeeves method until an optimal solution. Several initial points were used as this algorithm can get stacked in local minimums.
A neural network simulation model was used in order to enhance the speed of the process. For this case the authors used a back propagation approach. The network was trained using pairs of input and output vectors.
The decision set was composed of 18 decision variables. Based in inflow pattern the year was divided into six time periods. Four out of five supply levels were considered at each time. The results are presented in two different scenarios. For each scenario, the authors presents the results for 4 different policies including the storage for each time period, the release and the deficit. The inclusion of two additional reservoirs is also analyzed.
As the major result, the authors concluded that the neural network based simulation-optimization model performed satisfactory. Also the authors mention that reservoir operation problems considering several and more complicated networks could be handled by this method.
3) Discussion
The paper presents a very interesting approach to solve a very common reservoir operation problem. I believe several attempts have been made to find methodologies for optimal operation rules for reservoir. I think this is an emerging issue as water demand is increasing significantly in large metropolitan areas and the reservoirs systems have to be used optimally.
Tuesday, March 24, 2009
Assignment #7 - Infiltration Based BMP Optimization
Review Article
1) Article Reference
Perez-Pedini C, Limbrunner JF, Vogel RM (2005) “Optimal location of infiltration-based best management practices for storm water management,” JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT, 131(6) pp. 441-448
2) Summary
This paper presents an approach to optimize the location of infiltration based best management practices to reduce peak flow during storm events. According to the author many optimization applications were previously developed to find optimal location, design and operation of detention ponds in a watershed to reduce peak flow.
The infiltration based best management practices considered are infiltration basins, rain gardens and pervious pavements. The approach to integrate a wide variety of distributed storage and infiltration storm water controls acting in combination have been called as low impact development (LID).
According to the author the major goal of this study is to introduce a methodology to determine the optimal number and location of infiltration based practices to reduce peak flow. It was used a fully distributed model based on the SCS curve number approach. The model was applied to the Aberjona River watershed. This model was programmed in excel and VBA with a system of 4533 square HRUs that have a side length of 120 m. It was used the D8 algorithm for runoff routing. A detailed description of all mathematical formulas used to model the water movement within the distributed model was presented. This model was then calibrated for a storm event using 15 min storm data from two rain gages and compared with a flow gage at the watershed outlet.
The optimization routines were developed using excel and a commercial available Genetic Algorithm optimizer called Evolver. The overall goal of the optimization was to locate the HRUs which if BMPs were applied, would lead to a maximum reduction of peak flow. Some restrictions were applied in order to reduce the feasible space. The author also provided some details of the constraints used in the Genetic Algorithm programming.
As the main results the authors showed a Trade off curve between the reduction of peak flow and the number of best management practices to be implemented. The optimum locations were also presented in maps showing its location within the watershed. The authors concluded that a GA algorithm with a distributed hydrologic model presented satisfactory results for finding the optimal locations and quantity of BMPs to reduce peak flow.
3) Discussion
These papers are getting better and better to read. This one was especially nice as it is easy to understand and I am more familiar with the concepts used throughout the research. I think the authors had a good idea developing this application as it seems a new approach using consolidated methods. This is quite encouraging as I am feeling that the amount of my understanding of the article is increasing significantly.
Tuesday, March 3, 2009
Assignment #6
Review Article
1) Article Reference
Behera, P, Papa, F., Adams, B (1999) “Optimization of Regional Storm-Water Management Systems” Journal of Water Resources Planning and Management, 125(2) pp. 107-114
2) Summary
The paper discusses the application of optimization methods for storm water management systems. As new lands are being occupied, land developers and municipalities have to deal with runoff quantity and quality control. One of the most common solutions adopted are the implementation of detention ponds followed by a number of Best Management Practices available nowadays. In this case, the detention ponds referred as storm water management (SWM) ponds are considered to control both quantity and quality for a given catchment.
To support finding the best alternative for this projects, optimization using dynamic programming is used to find the best design parameters such as storage volume, release rate and pond depth. The objective is to minimize the costs of implementing the SWM ponds in each of the catchments. The authors consider costs related to the value of land, construction, and operation, maintenance and repair. The decision variables are the active storage volume of the pond, controlled release rate of the pond, and pond depth.
The system constraints are based on two major categories: Runoff Control Performance and Pollution Control Performance. The runoff model inputs statistical meteorological data and transform to runoff considering catchment hydrology and control systems hydraulics. The pollution control is modeled considering the average annual fraction of suspended solids removed from the SWM pond.
The constrains and the conceptual models for the optimization of SWM pond design for single catchment are explained as well as the previous work done in the area and the assumptions made. For the multiple parallel catchments the authors also present an specific model, with a optimization function and constrains for pollution control and runoff. Some details about the computations are also provided.
The authors concludes that is possible to optimize SWM ponds using dynamic programming and to achieve optimal design criteria’s considering single catchment and multi-catchment systems with water quantity and quality aspects. They also suggest different uses such as planning activities, preliminary design and scenario analyses.
3) Discussion
I personally enjoyed more this paper than the other previous two. Maybe because I am more familiar with the conceptual models of storm water systems and I actually find this area more interesting. One thing that really surprised me was to realize that the first author of the paper is a graduate student of the civil engineering department like us. As this is the first of this kind we are dealing I still haven’t completely understood the paper. I´m hoping to clarify it better on the class discussion on Wednesday. See you guys there…
1) Article Reference
Behera, P, Papa, F., Adams, B (1999) “Optimization of Regional Storm-Water Management Systems” Journal of Water Resources Planning and Management, 125(2) pp. 107-114
2) Summary
The paper discusses the application of optimization methods for storm water management systems. As new lands are being occupied, land developers and municipalities have to deal with runoff quantity and quality control. One of the most common solutions adopted are the implementation of detention ponds followed by a number of Best Management Practices available nowadays. In this case, the detention ponds referred as storm water management (SWM) ponds are considered to control both quantity and quality for a given catchment.
To support finding the best alternative for this projects, optimization using dynamic programming is used to find the best design parameters such as storage volume, release rate and pond depth. The objective is to minimize the costs of implementing the SWM ponds in each of the catchments. The authors consider costs related to the value of land, construction, and operation, maintenance and repair. The decision variables are the active storage volume of the pond, controlled release rate of the pond, and pond depth.
The system constraints are based on two major categories: Runoff Control Performance and Pollution Control Performance. The runoff model inputs statistical meteorological data and transform to runoff considering catchment hydrology and control systems hydraulics. The pollution control is modeled considering the average annual fraction of suspended solids removed from the SWM pond.
The constrains and the conceptual models for the optimization of SWM pond design for single catchment are explained as well as the previous work done in the area and the assumptions made. For the multiple parallel catchments the authors also present an specific model, with a optimization function and constrains for pollution control and runoff. Some details about the computations are also provided.
The authors concludes that is possible to optimize SWM ponds using dynamic programming and to achieve optimal design criteria’s considering single catchment and multi-catchment systems with water quantity and quality aspects. They also suggest different uses such as planning activities, preliminary design and scenario analyses.
3) Discussion
I personally enjoyed more this paper than the other previous two. Maybe because I am more familiar with the conceptual models of storm water systems and I actually find this area more interesting. One thing that really surprised me was to realize that the first author of the paper is a graduate student of the civil engineering department like us. As this is the first of this kind we are dealing I still haven’t completely understood the paper. I´m hoping to clarify it better on the class discussion on Wednesday. See you guys there…
Sunday, February 22, 2009
Assignment #5 - Sensor Placement in Municipal Water Networks
Review Article
1) Article Reference
Berry, J., Fleisher, L, Hart, W. Phillips, C. and Watson, J.-P. (2005) “Sensor Placement in Municipal Water Networks”, Journal of Water Resources Planning and Management, 131(3) pp. 237-243
2) Summary
This paper presents an approach for determining the placement of contaminant sensors within a municipal water network. To increase the protection of water supply systems, the use of real time early warning systems (EWS), have been widely adopted. Usually, utilities wish to place online sensors so deployment cost is minimized and the level protection maximized.
Considering a variety of approaches, the paper assumes that a attack occurs only in a single point; consider the total population exposed; the sensors protect downstream populations; and transitions between time periods are ignored. With this assumptions the authors simplify health impacts, ignore concentration and temporal effects.
The objective of the model was to minimize the expected fraction of the population that is at risk for some attack. The attack is modeled as a release of contaminant at a single point of the network. It is assumed that any point downstream of the attack can be contaminated. The EPANET water network simulator was used to determine water flow in the network. The attack scenarios were defined by a probability distribution over all pairs of population weighted flow and attacks points (from experts and scenario development).
To solve the optimization problem it was used mixed integer programming were the objective was to minimize the expected number of exposed people. The constraint considered that if a node is directly attacked it is directly contaminated. The second relate that a sensor can cover flow in a pipe in both directions. The third propagates the contaminant along the flow. One constraint is used to limit the maximum number of sensors.
To evaluate the model it was used three data sets. Two dataset were develop in EPANET and one is real. As all three dataset have some missing data, synthetic data was used to complete the datasets. The EPANET was used to determine flow patterns during four six-hour period with a 24 hour time period. The dataset 1 was adapted from Example Network provided by EPANET 2.0 with 36 nodes, 40 pipes and 1 pump station. The dataset 2 was adapted from Example Network 3 from EPANET 2.0 with 97 nodes, 117 pipes, two reservoirs and three tanks. The dataset 3 was adapted from a real world data with 470 nodes, 621 pipes, three pumps and four tanks.
The results are presented numerically in three tables. For dataset 1 and 2 the maximum number of sensors was 7 while for dataset 3 the maximum was 300. The authors present a comprehensive discussion about the numerical results and some explanations about the unlikely or unexpected outcomes. The conclusions showed that mixed integer programming can be used effectively to solve large scale sensor placement problems. Some ideas of the model generalization are presented relating the temporal effects, placement locations, sensor costs, and performance objective.
3) Discussion
I found this article useful to better understand optimization problems and it s applications and more specifically the integer programming method. I found it very similar to the previous paper. I couldn’t find any problems with the problem proposed. Further research suggestions would be to apply this methodology to real real life problems.
The objective of the model was to minimize the expected fraction of the population that is at risk for some attack. The attack is modeled as a release of contaminant at a single point of the network. It is assumed that any point downstream of the attack can be contaminated. The EPANET water network simulator was used to determine water flow in the network. The attack scenarios were defined by a probability distribution over all pairs of population weighted flow and attacks points (from experts and scenario development).
To solve the optimization problem it was used mixed integer programming were the objective was to minimize the expected number of exposed people. The constraint considered that if a node is directly attacked it is directly contaminated. The second relate that a sensor can cover flow in a pipe in both directions. The third propagates the contaminant along the flow. One constraint is used to limit the maximum number of sensors.
To evaluate the model it was used three data sets. Two dataset were develop in EPANET and one is real. As all three dataset have some missing data, synthetic data was used to complete the datasets. The EPANET was used to determine flow patterns during four six-hour period with a 24 hour time period. The dataset 1 was adapted from Example Network provided by EPANET 2.0 with 36 nodes, 40 pipes and 1 pump station. The dataset 2 was adapted from Example Network 3 from EPANET 2.0 with 97 nodes, 117 pipes, two reservoirs and three tanks. The dataset 3 was adapted from a real world data with 470 nodes, 621 pipes, three pumps and four tanks.
The results are presented numerically in three tables. For dataset 1 and 2 the maximum number of sensors was 7 while for dataset 3 the maximum was 300. The authors present a comprehensive discussion about the numerical results and some explanations about the unlikely or unexpected outcomes. The conclusions showed that mixed integer programming can be used effectively to solve large scale sensor placement problems. Some ideas of the model generalization are presented relating the temporal effects, placement locations, sensor costs, and performance objective.
3) Discussion
I found this article useful to better understand optimization problems and it s applications and more specifically the integer programming method. I found it very similar to the previous paper. I couldn’t find any problems with the problem proposed. Further research suggestions would be to apply this methodology to real real life problems.
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