Citation
Chang, S. Y., E. D. Brill Jr., & L. D. Hopkins. 1982. Use of mathematical models to generate alternative solutions to water resources planning problems. Water Resources Research, 18(1), 58-64.
Summary
Due to the complexity of many water resources planning problems, it is often impossible to fully know all the attributes of the objectives, or the trade-offs between attributes. Problems often have a large set of feasible solutions, and many of these are nearly as optimal as the ideal solution. Un-modeled considerations in problem formulations may influence the results of the optimal solution. When bearing this in mind, it is often prudent to generate multiple alternatives. The article presents three methods for alternative generation (AG): 1) Hop, Skip, and Jump (HSJ), 2) random method, and 3) branch and bound screening (BBS).
The AG methods were tested on a case study of a wastewater treatment system with 15 waste sources, 12 potential plant sites, and 15 potential pipe links. The problem was solved using a mixed integer programming model.
The HSJ method then generated multiple solutions by minimizing the sum of the non-zero variables in the original answer, subject to a limit on total annual cost. The cost constraint was set 10% higher than the optimal cost. The random method works by randomly selecting decision variables and maximizing the sum. Again, feasible solutions are limited by a total annual cost constraint. The BBS method generates a set of feasible solutions meeting the constraints of the objective function, and then screens them to maximize difference in solutions.
Three criteria are used to evaluate the AG methods: 1) performance against the objective function standard (i.e. total annual cost), 2) difference between alternatives produced, and 3) computation time. The results demonstrated that BBS was best at meeting the cost constraint, followed by the random method, and HSJ. On the other hand, BBS generates the smallest amount of difference between alternatives. Each type of method produced distinct types of difference. HSJ varied in degree of centralization of plants. Random tended to produce solutions with different locations, but the same number of plants and interceptors. BBS resulted in solutions where some parts remained constant, but others varied in arrangement. BBS required the least computing time, followed by the random method, and then HSJ. However, computing time was very small overall. Required time will be highly problem dependent.
Considering that each method can be computed quickly, and each result in solutions that are distinctly different, it may be best in practice to use all three methods in alternative exploration.
Discussion
I think that this is an important article with more technical suggestions on AG methods than the Brill 1979 article. I agree that a pure economic efficiency objective is limited in public-sector planning. As a future topic of research, I am curious how sensitive the solutions are to changes in the limit of the cost constraint. In this case, the author used 10%, but he stated that setting the number is a matter of judgment. I wonder how much variation in this number affects the results.
Sunday, March 6, 2011
Brill 1979
Citation
Brill, E. D. Jr. 1979. The use of optimization models in public-sector planning. Management Science, 25(5), 413-422.
Summary
Brill proposed that optimization, instead of being considered the end product, should be considered as a tool to facilitate the planning process. He contends that is many situations, it is not true that optimization can produce the one-true-and-right answer. This is because pure optimization 1) does not consider equity, and 2) it is often difficult to put empirical values on costs and benefits.
Brill presents three ways that optimization can be constructively incorporated into the planning process, as an aid to human creativity and decision making. First, simulation and optimization can be used jointly. In this case, an optimization model is used to determine a solution, and a simulation is performed to evaluate that solution. This has been put to use in designing outpatient healthcare facilities. Secondly, optimization can be paired with analytical models. This method has been applied to the timing of police car schedules. In this case, the optimization model was solved multiple times while changing constraints. The solutions were then analyzed using a time-dependent queuing model. Finally, planners can use a toolbox of models. This entails using multiple models to “develop, evaluate, and elaborate alternative solutions.”
Brill described how optimization models can be used in the planning process to spark the creativity of the analyst, and help develop innovative solutions. This occurs when there is human-machine interaction. In this situation, the computer would produce a first solution, which the analyst would then evaluate, with the intention of identifying “attractive features” and making model refinements. Running the model through computer optimization again will create a cycle in which the computer assists the person in arriving at the best answer.
For this process to work, optimization results must satisfy two criteria: 1) they must meet minimal constraints, and 2) they must be different. The author proposed that difference be quantified in an objective function, where the goal is to minimize the sum of the variables that are non-zero in the original design. In the first iteration, all non-zero values go to zero, and the model is re-solved. This is the “maximum difference.” In subsequent iterations, Lagrange multipliers are used to produce “more different” solutions, based on the relative effect of changes in problem constraints.
Discussion
I agree with Brill that optimization tools are more effective in planning when coupled with human judgment. I think that the human aspect allows for more subjective and “political” concerns, which a computer is unable to know, to be incorporated into the model.
Brill acknowledged that his approach for maximizing difference was in the early development and implementation stages when this article was published. One question that I have is how many Lagrange multipliers would need to be determined in order to accurately assess the trade-off of changes in constraints? Does it depend on the type of problem (linear, piece-wise, etc.)?
Brill, E. D. Jr. 1979. The use of optimization models in public-sector planning. Management Science, 25(5), 413-422.
Summary
Brill proposed that optimization, instead of being considered the end product, should be considered as a tool to facilitate the planning process. He contends that is many situations, it is not true that optimization can produce the one-true-and-right answer. This is because pure optimization 1) does not consider equity, and 2) it is often difficult to put empirical values on costs and benefits.
Brill presents three ways that optimization can be constructively incorporated into the planning process, as an aid to human creativity and decision making. First, simulation and optimization can be used jointly. In this case, an optimization model is used to determine a solution, and a simulation is performed to evaluate that solution. This has been put to use in designing outpatient healthcare facilities. Secondly, optimization can be paired with analytical models. This method has been applied to the timing of police car schedules. In this case, the optimization model was solved multiple times while changing constraints. The solutions were then analyzed using a time-dependent queuing model. Finally, planners can use a toolbox of models. This entails using multiple models to “develop, evaluate, and elaborate alternative solutions.”
Brill described how optimization models can be used in the planning process to spark the creativity of the analyst, and help develop innovative solutions. This occurs when there is human-machine interaction. In this situation, the computer would produce a first solution, which the analyst would then evaluate, with the intention of identifying “attractive features” and making model refinements. Running the model through computer optimization again will create a cycle in which the computer assists the person in arriving at the best answer.
For this process to work, optimization results must satisfy two criteria: 1) they must meet minimal constraints, and 2) they must be different. The author proposed that difference be quantified in an objective function, where the goal is to minimize the sum of the variables that are non-zero in the original design. In the first iteration, all non-zero values go to zero, and the model is re-solved. This is the “maximum difference.” In subsequent iterations, Lagrange multipliers are used to produce “more different” solutions, based on the relative effect of changes in problem constraints.
Discussion
I agree with Brill that optimization tools are more effective in planning when coupled with human judgment. I think that the human aspect allows for more subjective and “political” concerns, which a computer is unable to know, to be incorporated into the model.
Brill acknowledged that his approach for maximizing difference was in the early development and implementation stages when this article was published. One question that I have is how many Lagrange multipliers would need to be determined in order to accurately assess the trade-off of changes in constraints? Does it depend on the type of problem (linear, piece-wise, etc.)?
Sunday, February 20, 2011
Summary and Discussion of Lee (1992)
Citation
Lee, HB & Deininger, RA. (1992). Optimal locations of monitoring stations in water distribution system. J. of Env. Eng. 118, 4-16.
Summary
Monitoring drinking water quality is mandated by the USEPA Clean Water Act, to ensure that water is safe for public consumption. In determining where samples should be taken in a distribution system, the goal should be to pick locations with the highest representativeness. This is assessed by the coverage of the node. Coverage is the percentage of the demand at node (di) to the overall demand of the system (D). Integer programming can be used to determine the location for water quality sampling that achieves the greatest overall coverage.
In distribution systems, water can travel through different routes and nodes. The water quality measured at a downstream node is a function of the water quality of the nodes contributing to its flow. Thus, water quality at upstream nodes can be inferred by considering flow patterns. These relationships are represented in a water fraction matrix. The water fraction is:
W(i,j) = (flow from node j)/(inflow to sampling station i)
The result is a number between 0 and 1. A decision is required for what fraction of water is required to call a node “covered.” Based on that threshold, the water fraction matrix can be converted into a coverage matrix of binary variables (0 and 1), which represents nodes that are “covered” and “not-covered” by a sampling location.
The integer optimization problem is then takes the following form:
Max ∑diyi
s.t.
∑ xi < NS
∑ ai,jxi – yi > 0 j = 1,2,….,n
Where xi, yi are 0 or 1; NS is number of sampling stations; dj is demand at node I; ai,j is the transposed coverage matrix, and n is the number of nodes
The result of a case study on Flint, Michigan, showed that integer programming was able to arrive at a sampling scheme which covered 54% of demand. This was a significant improvement over an existing coverage of 18.5%. The authors also showed that integer programming could account for multiple flow scenarios by developing multiple coverage matrices.
Discussion
I think that this type of problem is an ideal application for optimization analysis, since it avoids many of the complications of “wicked” water problems. In this case, all the choices are known, the results have little impact on human values, and the water managers have control over the distribution system. Given these conditions, the benefits expected from optimization analysis should be able to be fully realized.
I would be interested to see how this type of approach works when the problems become more “messy.” For example, what if the objective is to select water quality sampling locations across an expansive and diverse watershed? How can “upstream” considerations be included when water is traveling overland, as opposed to in pipes? I expect that elevation could be the new measure of “upstream.” It would be interesting to see if the disciplines of GIS and systems analysis could be combined to solve such a complex problem.
Wednesday, February 2, 2011
Summary and Discussion of Nishikawa (1998)
Nishikawa Tracy (1998) “Water-resources optimization model for Santa Barbara, California,” Journal of Water Resources Planning and Management 124(5) pp. 252-263.
Summary
The author used a linear programming model to determine the optimal management scenario for the city of Santa Barbara during a 5-year drought. Santa Barbara’s greatest water supply source is the Santa Ynez River. Groundwater contributes 20% of demand. Two large reservoirs, the Gibralter and Cachuma Reservoir, also provide surface water. A desalination plant is another minor water source (253-4).
The analysis objective is minimizing operating cost, subject to four constraints:
1) Capacity constraints of surface and groundwater. Supply cannot exceed capacity.
2) Hydraulic head constraints to minimize coastal saltwater intrusion
3) Demand constraints
4) Temporal balance constraints to reflect yearly carryover in the Cachuma Reservoir (256-7)
The decision variables are the quantities of water delivered monthly from surface and groundwater sources.
The author tested and compared three scenarios: 1) a base case of average monthly releases, 2) current actual monthly releases, and 3) proposed diversion distribution (257). The results showed that the water resource objectives can be optimally met at a cost of approximately $5,560,000. In the current system, delivering water over a 5-year time period would result in a cost of about $7,555,000, or 36% higher than the base case. The proposed distribution, which corresponds with demand, would cost around $5,070,000, or 9% less than the base case. However, to realize these savings, it will be very important that timing corresponds with demand.
Through sensitivity analysis, the author discovered that 1) water carried over in the Cachuma reservoir has a higher value than desalinated water, 2) lowering peak annual demand results in lower cost, 3) accepting higher saltwater intrusion can result in significant savings, 4)that desalination plants may not be needed.
Discussion
I thought it was interesting how the author accounted for saltwater intrusion in the optimization model. The author notes that the minimum head value (sea level, or 0m) still allows for saltwater intrusion, due to the higher density of saltwater. Future work needs to be done to quantify how much saltwater is moving landwards with a head value of 0m. This is especially relevant because, as the author discovered during sensitivity analysis, there is a negative correlation between salt water intrusion and water supply cost – allowing increased intrusion lowers cost. Setting a hydraulic head boundary along coastal regions of -9.14 meters or higher will allow corresponds to a savings of about one million dollars over a five year drought period. Considering that the total base cost is $5,560,000, this is a significant savings.
I also thought that it was interesting how little of a benefit the desalination plant had on operational cost. Eliminating the desalination water source in the model resulted in an increase in only $100,000 over five years, or 2% per year. Additionally, the model did not account for the capital costs of the plant, which, when included, may make desalination economically infeasible.
Summary
The author used a linear programming model to determine the optimal management scenario for the city of Santa Barbara during a 5-year drought. Santa Barbara’s greatest water supply source is the Santa Ynez River. Groundwater contributes 20% of demand. Two large reservoirs, the Gibralter and Cachuma Reservoir, also provide surface water. A desalination plant is another minor water source (253-4).
The analysis objective is minimizing operating cost, subject to four constraints:
1) Capacity constraints of surface and groundwater. Supply cannot exceed capacity.
2) Hydraulic head constraints to minimize coastal saltwater intrusion
3) Demand constraints
4) Temporal balance constraints to reflect yearly carryover in the Cachuma Reservoir (256-7)
The decision variables are the quantities of water delivered monthly from surface and groundwater sources.
The author tested and compared three scenarios: 1) a base case of average monthly releases, 2) current actual monthly releases, and 3) proposed diversion distribution (257). The results showed that the water resource objectives can be optimally met at a cost of approximately $5,560,000. In the current system, delivering water over a 5-year time period would result in a cost of about $7,555,000, or 36% higher than the base case. The proposed distribution, which corresponds with demand, would cost around $5,070,000, or 9% less than the base case. However, to realize these savings, it will be very important that timing corresponds with demand.
Through sensitivity analysis, the author discovered that 1) water carried over in the Cachuma reservoir has a higher value than desalinated water, 2) lowering peak annual demand results in lower cost, 3) accepting higher saltwater intrusion can result in significant savings, 4)that desalination plants may not be needed.
Discussion
I thought it was interesting how the author accounted for saltwater intrusion in the optimization model. The author notes that the minimum head value (sea level, or 0m) still allows for saltwater intrusion, due to the higher density of saltwater. Future work needs to be done to quantify how much saltwater is moving landwards with a head value of 0m. This is especially relevant because, as the author discovered during sensitivity analysis, there is a negative correlation between salt water intrusion and water supply cost – allowing increased intrusion lowers cost. Setting a hydraulic head boundary along coastal regions of -9.14 meters or higher will allow corresponds to a savings of about one million dollars over a five year drought period. Considering that the total base cost is $5,560,000, this is a significant savings.
I also thought that it was interesting how little of a benefit the desalination plant had on operational cost. Eliminating the desalination water source in the model resulted in an increase in only $100,000 over five years, or 2% per year. Additionally, the model did not account for the capital costs of the plant, which, when included, may make desalination economically infeasible.
Saturday, January 22, 2011
Summary and Discussion of Reed (2009)
Reference
Reed, P.M., and Kasprzyk, J.R., “Water Resources Management: The Myth, the Wicked, & The Future.” ASCE Journal of Water Resources Planning & Management, v135, n6, pp. 411-413, 2009.
Summary
Reed and Kasprzyk provide a modern update to Liebman’s 1976 critique of water resources planning and management. They seek to “elucidate what water management science should be in the future,” by revisiting important challenges discussed in the past. The editorial discussed three main supposed flaws, and places for improvement, in our thinking on water management:
1) Optimality is a myth
2) Water is a “wicked” problem
3) “The future requires transparency and constructive decision aiding”
The author’s criticize optimality analysis as overly simplistic, with too many invalid assumptions. For example, Reed and Kasprzyk content that the complexity of water management means scientists can never have perfect knowledge of water systems. Therefore, they believe, no model can be completely valid. This reflects the “wicked” nature of water resources. Wicked, in this context, identifies five traits described by Rittel and Webber in 1973, including 1) no consensus on what constitutes optional conditions, 2) no objective judgments, 3)problems that are not “decomposable” into separate distinct issues, 4) often irreversible decisions, and 5) a wide range of highly uncertain consequences.
The authors suggest that to work with wicked problems, water managers will need to overcome their traditional thinking about water. This means recognizing the limitations of traditional methods for classifying, decomposing, and solving water problems. It means bridging the artificial distinction between “engineering,” “science,” and “policy,” and uniting disciplines of water research, such as water management and water cycle science.
Ultimately, the authors recommend that water managers embrace a system called “constructive decision-aiding science.” They recommend collaboration and negotiation to assess diverse perspectives and identify decision alternatives. They view this as a “mechanism for discovering system dependencies and/or trade offs.” They place this kind of analysis above optimization, concluding that it can no longer be assume that there is one single correct solution.
Discussion
This 2009 editorial does a good job of updating Liebman’s paper, which is now over thirty years old. It is interesting to see how issues which were raised decades ago are still pertinent today.
I agree that it is always important to recognize that models are projections only. They are based on facts, but often require assumptions. The result is that any time assumptions are factored in to models, model certainty declines. Thus, it is important to recognize what models can and cannot do, and not misapply the results.
I found a big short-falling of the article to be the lack of compelling examples of what exactly is deleterious about “decomposing” water management problems into component parts. If models are flawed and myopic, what is an example of their failure? The authors discussed the Chesapeake Watershed models as exemplifying the challenge of “how to formulate, parameterize, and use our models in a manner that does not invalidate our ability to falsify important hypotheses.” However, they also admit that the model has had a very positive impact on policy. In addition, the models in use are subject to constant improvement, including increased monitoring stations, increased parameter acceptance, and longer time projections (
Summary and Discussion of Liebman (1976)
Reference
Liebman, Jon (1976) “Some simple-minded observations on the role of optimization in public systems decision making,” Interfaces 6(4) pp. 102-108.
Summary
Liebman addresses the efficacy of applying optimization tools to public decision making regarding “wicked” problems. He explains that optimization developed to solve problems that were simple, not controversial, the constraints were apparent, the client was clear, and when the result was in place gained, the client benefit and no one was hurt. He does not think that a decision-making paradigm, designed under these conditions, is applicable to public policy where there is little agreement on public goals, constraints, and evaluation criteria. Additionally, Liebman views decision making as conflict resolution, in which a moral choice on who should be benefitted, and who should be harmed, has to be made. To him, this choice must always be a human choice, no matter how flawed or irrational.
Liebman makes three “simple-minded” recommendations for how to improve our used of modeling:
1) That decision makers utilize more fully the “understanding of interrelationships within the modeled system” that the modeler gains from developing the model. This places the emphasis less on the outcome of the model, and more of the experience gained by the person designing the model.
2) Since problems can be modeled in multiple ways, decision makers should have access to multiple models, based on diverse points of view.
3) Models should be simplified and used as tools to increase public understanding and awareness.
Discussion
Liebman presents several very interesting and insightful distinctions. I thought it was interesting to compare the decision making hierarchy of private and public entities. I also liked the examples he compared of wicked and non-wicked problems. It was interesting that optimization analysis evolved in the military during WWII to solve very specific, clear objectives. It was then applied to the private sector, where goals are fairly uniform (e.x. maximizing profit) and decisions are often authoritarian (e.x. owner or CEO). It makes sense that optimization would be very effective in these situations.
I agree with several of his points. I think that a modeler should use her understanding of a system to develop simple representations of complex systems that can be used for public education. I also agree that an important function of modeling can be in determining policy alternatives.
However, there is one significant point of contention I have with Liebman’s position. Liebman states that models should be used to identify alternatives, but not to make decisions regarding wicked problems. He advocates this because wicked problems are complex, controversial, and supposedly there is no consensus on what criteria define the “best” solution. He advocates instead that policy makers should decide, no matter how “imperfect and irrational” they may be.
I think that if we should not rely on modeling, the author should propose a better tool. Decision makers are often elected officials, such as lawyers or politicians, with little to no technical expertise. I do not think their judgment is more highly qualified than the analyst, or even the computer that models the problem. Additionally, I think that it is possible to objectively reach a state of “highest benefit.” Highest benefit to me means greatest utility. The philosophy of utilitarianism states that a moral choice is one that provides the greatest good for the greatest number. By using multi-objective analysis, and defining the greatest good as “minimum cost,” “minimum environmental disruption,” and “equity” (or some combination thereof), I think it is feasible to say that an optimization model could provide, to a better degree than a politician, the “best” solution to a water resources problem.
Future work…?
I think that it would be interesting to examine several policy decision case studies. It would be especially interesting to find case studies solved under various circumstances, such as with or without models, and by degree of public participation. By modeling these case studies retroactively, I would like to compare how the chosen alternative measures up to what the model projects as the optimal solution. It would be very interesting to see if one type of decision making strategy appeared to be more effective than the others.
Liebman, Jon (1976) “Some simple-minded observations on the role of optimization in public systems decision making,” Interfaces 6(4) pp. 102-108.
Summary
Liebman addresses the efficacy of applying optimization tools to public decision making regarding “wicked” problems. He explains that optimization developed to solve problems that were simple, not controversial, the constraints were apparent, the client was clear, and when the result was in place gained, the client benefit and no one was hurt. He does not think that a decision-making paradigm, designed under these conditions, is applicable to public policy where there is little agreement on public goals, constraints, and evaluation criteria. Additionally, Liebman views decision making as conflict resolution, in which a moral choice on who should be benefitted, and who should be harmed, has to be made. To him, this choice must always be a human choice, no matter how flawed or irrational.
Liebman makes three “simple-minded” recommendations for how to improve our used of modeling:
1) That decision makers utilize more fully the “understanding of interrelationships within the modeled system” that the modeler gains from developing the model. This places the emphasis less on the outcome of the model, and more of the experience gained by the person designing the model.
2) Since problems can be modeled in multiple ways, decision makers should have access to multiple models, based on diverse points of view.
3) Models should be simplified and used as tools to increase public understanding and awareness.
Discussion
Liebman presents several very interesting and insightful distinctions. I thought it was interesting to compare the decision making hierarchy of private and public entities. I also liked the examples he compared of wicked and non-wicked problems. It was interesting that optimization analysis evolved in the military during WWII to solve very specific, clear objectives. It was then applied to the private sector, where goals are fairly uniform (e.x. maximizing profit) and decisions are often authoritarian (e.x. owner or CEO). It makes sense that optimization would be very effective in these situations.
I agree with several of his points. I think that a modeler should use her understanding of a system to develop simple representations of complex systems that can be used for public education. I also agree that an important function of modeling can be in determining policy alternatives.
However, there is one significant point of contention I have with Liebman’s position. Liebman states that models should be used to identify alternatives, but not to make decisions regarding wicked problems. He advocates this because wicked problems are complex, controversial, and supposedly there is no consensus on what criteria define the “best” solution. He advocates instead that policy makers should decide, no matter how “imperfect and irrational” they may be.
I think that if we should not rely on modeling, the author should propose a better tool. Decision makers are often elected officials, such as lawyers or politicians, with little to no technical expertise. I do not think their judgment is more highly qualified than the analyst, or even the computer that models the problem. Additionally, I think that it is possible to objectively reach a state of “highest benefit.” Highest benefit to me means greatest utility. The philosophy of utilitarianism states that a moral choice is one that provides the greatest good for the greatest number. By using multi-objective analysis, and defining the greatest good as “minimum cost,” “minimum environmental disruption,” and “equity” (or some combination thereof), I think it is feasible to say that an optimization model could provide, to a better degree than a politician, the “best” solution to a water resources problem.
Future work…?
I think that it would be interesting to examine several policy decision case studies. It would be especially interesting to find case studies solved under various circumstances, such as with or without models, and by degree of public participation. By modeling these case studies retroactively, I would like to compare how the chosen alternative measures up to what the model projects as the optimal solution. It would be very interesting to see if one type of decision making strategy appeared to be more effective than the others.
Subscribe to:
Posts (Atom)