Demonstrating the Applicability of PAINT to Computationally Expensive Real-life Multiobjective Optimization
DDemonstrating the Applicability of PAINT toComputationally Expensive Real-life Multiob jectiveOptimization
Markus Hartikainen and Vesa OjalehtoOctober 5, 2018
Abstract
We demonstrate the applicability of a new PAINT method to speed upiterations of interactive methods in multiobjective optimization. As our testcase, we solve a computationally expensive non-linear, five-objective problemof designing and operating a wastewater treatment plant. The PAINT methodinterpolates between a given set of Pareto optimal outcomes and constructsa computationally inexpensive mixed integer linear surrogate problem for theoriginal problem. We develop an IND-NIMBUS R (cid:13) PAINT module to combinethe interactive NIMBUS method and the PAINT method and to find a preferredsolution to the original problem. With the PAINT method, the solution processwith the NIMBUS method take a comparatively short time even though theoriginal problem is computationally expensive.
In this section, we give background for our study and a brief overview of thispaper. First, in Section 1.1, we describe the aim of this study and the structureof this paper. In Section 1.2, we introduce the basic concepts of multiobjectiveoptimization that are used in this paper. In Section 1.3, we consider the mainissues related to computationally expensive multiobjective optimization. Fi-nally, in Section 1.4, we describe our test case, i.e., the multiobjective problemof designing and operating a wastewater treatment plant.
In this paper, we demonstrate how the interpolation method PAINT (intro-duced in [12–14]) can be used to speed up the iterations of an interactivemethod when solving computationally expensive multiobjective optimizationproblems. For this, we revisit a computationally expensive five-objective opti-mization problem from [37] that models designing and operating a wastewatertreatment plant. a r X i v : . [ c s . NA ] S e p n our study of the wastewater problem, we used the interactive NIMBUSmethod, as was used in [37]. Compared to the previous study of [37], theiterations of the interactive method NIMBUS were much faster because of asurrogate problem constructed with the PAINT method. In our study, thedecision maker was Mr. Kristian Sahlstedt as in [37] and, thus, we were able toask the decision maker to compare his experiences of using the NIMBUS methodto solve the problem with and without the PAINT method. This comparisonof the two approaches gave a unique perspective to our study.The structure of this paper is as follows: After describing the background ofour study in this section, we outline the PAINT approach to solving computa-tionally expensive problems and the PAINT method in Section 2. We describethe new IND-NIMBUS R (cid:13) PAINT module (also called the PAINT module inthis paper for short) in Section 3. In Section 4, we illustrate how we used thePAINT method to construct a Pareto front approximation and a mixed integerlinear surrogate problem for the wastewater treatment problem. In Section 4,we also describe our decision maker’s involvement in solving the problem withthe PAINT module. In Section 5, we further analyze the decision making pro-cess of Section 4. Finally, in Section 6, we give our conclusions and ideas forfurther research.
Multiobjective optimization concerns simultaneously optimizing multiple con-flicting objectives. A general formulation for a multiobjective optimizationproblem with k objectives ismin ( f ( x ) , . . . , f k ( x ))s.t. x ∈ S, (1)where f i are the objective functions and S is the feasible set. A vector x ∈ S is called a (feasible) solution. For these problems, instead of a single optimalsolution there typically exist many Pareto optimal solutions. A solution x ∈ S is said to (Pareto) dominate another solution y ∈ S if f i ( x ) ≤ f i ( y ) for all i = 1 , . . . , k and f j ( x ) < f j ( y ) for at least one j ∈ { , . . . , k } . A solution x ∗ ∈ S is Pareto optimal, if there does not exist a solution x ∈ S that dominates it. Avector z = ( f ( x ) , . . . , f k ( x )) with x ∈ S is called an outcome, and an outcomeis called Pareto optimal if it is given by a Pareto optimal solution. The set ofPareto optimal outcomes is called the Pareto front.Although many Pareto optimal solutions typically exist, only one has to bechosen for implementation. Distinguishing between Pareto optimal solutionsrequires preference information about the objectives of the problem. In multi-objective optimization, it is often assumed that there exists a decision makerwho is an expert in the application area and who is prepared to answer ques-tions concerning those preferences. In this paper, this whole process of choosinga single solution for implementation is called solving the problem and, whenwe want to emphasize the decision maker’s involvement, it is also referred toas the decision making process.The type of information that is asked from the decision maker depends onthe multiobjective optimization method that is used to solve the problem. Dif- erent types of multiobjective optimization methods (as categorized in [22, 38])are no-preference methods, a priori methods, a posteriori methods and inter-active methods. In no-preference methods the decision maker is not asked anyquestions. No-preference methods are applicable to problems, where the de-cision maker is not available or does not want to get involved. In a priorimethods, the decision maker is first asked for preference information and thenthe best solution according to those preferences is found. The difficulty with apriori methods is that the decision maker may find it hard to define preferenceswithout ever seeing feasible or Pareto optimal solutions. In a posteriori meth-ods, a representative set of the Pareto optimal solutions is found from whichthe decision maker is allowed to choose a preferred one. The difficulty witha posteriori methods is that generating a representative solution set may betime-consuming and choosing a preferred solution from a large set of solutionsmay be hard (see e.g., [19]).In this paper, we follow the ideology of interactive methods in solving mul-tiobjective optimization problems. In interactive methods, it is understoodthat any preference information given by the decision maker is only partialand perhaps flawed. Thus, the decision maker is allowed to explore the Paretooptimal solutions by guiding the interactive method. This allows the decisionmaker to learn about the problem (as argued e.g., in [27]) and find a preferredsolution without examining too many solutions. For more information aboutinteractive methods, see e.g., [22, 27]. More specifically, in this paper, we usethe interactive synchronous NIMBUS method, introduced in [24–26]. Some multiobjective optimization problems are computationally expensive (seee.g., [2, 10, 15, 37, 43]). This may be caused e.g., by the need to use compu-tationally expensive simulations for evaluating the objective functions. Inter-active methods have an advantage to a posteriori methods in solving computa-tionally expensive problems, because the decision maker may guide the searchin interactive methods and, thus, fewer solutions need to be computed. Thereis, however, a drawback. When using interactive methods, the decision makerhas to wait while new solutions are computed with respect to his/her updatedpreferences. For computationally expensive problems, this may take a longtime, which may be frustrating for the decision maker (as argued e.g., in [18]).In order to compute new solutions faster within the interactive method,one can use approximation. Two different approximation schemes can be iden-tified: approximating the objective functions and approximating the Paretofront. The objective functions may be approximated e.g., with meta-modelslike the response surface methodologies, Support Vector Machines or RadialBasis Functions (see e.g., [30]). These have been used in multiobjective opti-mization e.g., in [30, 42]. This is not, however, a straightforward task, becauseas the number of decision variables and objectives increases, the approxima-tion itself becomes a very computationally expensive task. Another approachis approximating the Pareto front. Pareto front approximations can be found .g., in [1, 4, 6, 21, 29, 36, 44], where [1, 6, 21, 29] include decision makingaspects connected to these. Note that in this paper, we distinguish between aPareto front representation (a discrete set of Pareto optimal outcomes) and aPareto front approximation (something more approximate that possibly con-tains vectors that are not outcomes of the problem, but merely approximatethem).In this paper, we use the Pareto front approximation approach introduced in[12–14]. In those papers, a new Pareto front approximation method PAINT isintroduced and details on decision making with the produced approximation arecovered. The PAINT method uses a novel way to integrate the knowledge aboutPareto dominance into the approximation. The PAINT method interpolatesbetween a given set of Pareto optimal outcomes to construct a Pareto frontapproximation. The approach differs from the other approaches for decisionmaking with Pareto front approximations (mentioned above) because it is ableto approximate also nonconvex Pareto fronts. Furthermore, the Pareto frontapproximation constructed with PAINT implies a multiobjective mixed integerlinear surrogate problem (for the original problem) that can be solved with anyinteractive method. The other approaches are either applicable only to convexmultiobjective optimization problems or use only a custom-made procedure forchoosing a preferred point on the approximation. Further details on the PAINTmethod are covered in Section 2. Designing and operating a wastewater treatment plant is a complex problemwith many conflicting criteria that have to be considered at the same time. Inthis paper, we consider a plant using so-called activated sludge process, whichis globally the most common method of wastewater treatment. We model theproblem as a five-objective optimization problem, which was previously studiedalso in [37]. The five-objective problem is an extension of the three-objectiveproblem treated in [10]. The approach of this paper differs from the approachof [37] because we use the PAINT method to approximate the Pareto front andto construct a surrogate problem for the original problem. In this way, the timethat the decision maker has to wait while using an interactive method becomesshorter.Figure 1 shows the schematic layout of the wastewater treatment plant thatwas designed in [37]. The wastewater treatment begins with grit removal. Af-ter the grit removal, solids are separated by a gravitational settling. Raw andmixed sludge removed from the primary settlers is fermented in a separate reac-tor and partly recycled back to the water line to provide readily biodegradablecarbon source for denitrification. The bioreactor consists of four anoxic zones,three aerobic zones and one deoxygenation zone. Nitrate-rich activated sludgeis recycled from zone 8 of the bioreactor to zone 1. Return sludge and primaryeffluent are directed to zone 1. Methanol is injected to zone 2 to support denitri-fication. Excess sludge is pumped from zone 8 of the bioreactor to the beginningof the water process, from which it is removed in the primary settlers together ith raw sludge. Raw and mixed sludge is thickened gravitationally into ap-proximately 4.5% total solids prior to anaerobic digestion. Anaerobic digestionproduces biogas and the produced biogas can be converted into electrical orthermal energy. The digested sludge is dewatered by centrifuges into approx-imately 28% total solids. The reject water from sludge treatment is pumpedto the beginning of the plant. The wastewater treatment process is simulatedwith the commercial GPS-X simulator (see [8]) and the model is based on thefindings of P¨oyry Engineering Ltd. For more information about the wastewatertreatment plants using activated sludge process, see e.g., [10, 32, 37].The objectives of the optimization problem are the amount of nitrogen inthe effluent ( g/m , grams per a cubic meter of effluent), aeration power con-sumption in the activated sludge process ( kW ), chemical consumption ( g/m ,grams per a cubic meter of effluent), excess sludge production ( kg/d , kilogramsper day) and biogas production ( m /d , cubic meters per day). The first oneis the main goal of activated sludge process and the four others are connectedto the operational costs. This multiobjective optimization problem allows thesimultaneous consideration of the performance of the plant (through the nitro-gen removal rate) and different aspects of the operational costs. Naturally, thelast objective is maximized and the others are minimized. Decision variables ofthe problem are the percentage of inflow pumped to fermentation, the amountof excess sludge removed, the dissolved oxygen setpoint in the last aerobic zoneand the methanol dose. Thus, the methanol dose is both a decision variableand an objective. Grit Removal PrimaryGravitationalSettlingFermenterGrit SludgeAnaerobicDigestionDewateringbyCentrifuges Thickening SecondaryGravitationalSettlingBioreactorMethanolInfluent PurifiedWaterExcess sludgeFermentedSludgefluid fluid waterwater 4.5%sludge28%sludge BiogasAeration Sludge
Figure 1: A schematic layout of the wastewater treatment plant
Each simulation of the wastewater treatment plant of [37] took about 11seconds on the GPS-X simulator. This made the problem computationallyexpensive. In addition, one could notice from the Pareto optimal outcomescomputed for the problem that the problem is nonconvex. During the analysisin [37], 200 simulations were run to optimize the scalarizations (i.e., singleobjective optimization problems, whose optimal solutions are Pareto optimal olutions to the multiobjective optimization problem) given by the interactiveNIMBUS method that was used to solve the problem. This means that eachiteration of the interactive method took more than half an hour. Even thoughinteresting solutions to the problem were found in [37], the computational timeof iterations was an inconvenience to the decision maker (according to personalcommunications with the authors of [37]). This means that there was room forimprovement using the PAINT method. In this section, we describe the PAINT approach to solving computationallyexpensive problems. The applicability of the PAINT approach is then demon-strated in Section 4 by solving a computationally expensive multiobjective op-timization problem of wastewater treatment plant design and operation.The PAINT approach is based on the Pareto front approximation con-structed by the PAINT method. The PAINT method was proposed in [13],and it is based on the concept of an inherently nondominated Pareto frontapproximation introduced in [14] and the mathematical concepts of [12]. ThePAINT method interpolates between a given set of Pareto optimal outcomesin a way that the interpolants neither dominate nor are dominated by the setof given Pareto optimal outcomes and, in addition, they are not dominated byeach other (i.e., the interpolation is an inherently nondominated Pareto frontapproximation, as defined in [14]). In this paper, a vector on the Pareto frontapproximation is called an approximate (Pareto optimal) outcome.The general functionality of the PAINT method is as follows: The PAINTmethod first constructs the Delaunay triangulation of the given set of Paretooptimal outcomes and then chooses the appropriate polytopes from it to thePareto front approximation. In this paper, this is realized with the Octave-based (see [5, 31]) implementation that was developed during the research of[13].The Pareto front approximation constructed with the PAINT method im-plies a computationally inexpensive mixed integer linear surrogate problem forthe original problem, as described in [13]. The Pareto front of the surrogateproblem is exactly the Pareto front approximation and, thus, a preferred so-lution to the surrogate problem implies a preferred vector on the Pareto frontapproximation, which is also called a preferred approximate outcome in thispaper. The algorithm of the PAINT method and more exact details can befound in [13].Decision making in the PAINT approach is described in Figure 2. In thePAINT approach to solving computationally expensive problems, we assumethat there exists a set of Pareto optimal solutions to the computationally ex-pensive problem. This set may have been generated with any a posteriorimethod. The set of the related outcomes is inputted into the PAINT method.PAINT then interpolates between the set of given Pareto optimal outcomes andoutputs the interpolation that implies a mixed integer linear surrogate problem pproximatePareto frontwith PAINT andconstruct thesurrogate problem Find a preferredsolution to the surrogate problemwith an interactive method Project the solutionto the surrogate problem on the Pareto front of theoriginal problemDecision maker satisfied with the solution found?Add the solution found to the setof given solutions,reapproximatethe Pareto frontwith PAINT andreconstruct the surrogate problem Stop Decision maker'sinvolvement yesno
Figure 2: A flowchart of the decision making process for the original problem.After the mixed integer linear surrogate problem has been formulated, thedecision maker gets involved and uses an interactive method of his/her choiceto find a preferred solution to the surrogate problem. The outcomes given byPareto optimal solutions to the surrogate problem are vectors on the Paretofront approximation, which is in the same space as the original Pareto front.Thus, the decision maker is able to give his/her preferences on them. Thepreferred approximate outcome is projected on the actual Pareto front of theoriginal problem by solving achievement scalarizing problem (see [40, 41]) withthe approximate outcome as a reference point. More details on the projectioncan be found in [13]. Projecting the solution may take time, depending on thecomputational costs of the problem. If the problem is very computationallyexpensive, the projection can be done without the involvement of the decisionmaker.The projection of the preferred approximate outcome (i.e., a Pareto optimalsolution to the original problem) is shown to the decision maker and, if he/sheis satisfied, the decision making process stops, because a preferred solution hasbeen found. If the decision maker is not satisfied, it is possible to update thePareto front approximation by adding the new Pareto optimal outcome to thegiven set of Pareto optimal outcomes and by recomputing the approximationwith the PAINT method. This yields a more accurate approximation and wecan again use an interactive method to find a preferred solution to the new(more accurate) surrogate problem. This process can be repeated as manytimes as necessary.The PAINT method is a powerful tool as it can interpolate between anygiven set of Pareto optimal outcomes, i.e., the way that the outcomes havebeen generated does not affect the functionality of the method. In addition,since it is based on the concept of inherent nondominance (see [14]), it will not rovide interpolants that would mislead the decision maker. Finally, the mixedinteger linear surrogate problem implied by the approximation allows one touse any interactive method for finding a preferred approximate outcome on thePareto front approximation.The PAINT method has a couple of shortcomings, already noted in [13].First, the PAINT method does not provide any information about the preim-age of the Pareto front approximation in the decision space. This means thatthe decision maker has to project the approximate outcome (i.e., the solutionto the surrogate problem) on the Pareto front of the original problem in orderto find out the values of the decision variables. Second, the PAINT methodcannot detect any disconnectedness in the Pareto front, but always interpolatesbetween the outcomes whenever the interpolation is inherently nondominated.Thus, the approximation might be inaccurate if e.g., the decision space is dis-connected or the objective functions are highly nonconvex. R (cid:13) PAINT Module
IND-NIMBUS R (cid:13) (see [23]) is a multi-platform desktop software framework, cur-rently available for Windows and Linux operating systems, intended to providea flexible tool-set for implementation of multiobjective optimization methods.So far, the IND-NIMBUS framework has been used to implement the syn-chronous NIMBUS [24–26] and the Pareto Navigator [6] methods. The IND-NIMBUS R (cid:13) software can be connected to an external sources that model theproblem, such as the GPS-X simulator used for modeling the wastewater treat-ment plant. For this paper, the IND-NIMBUS R (cid:13) software framework has beenused to develop a so-called IND-NIMBUS R (cid:13) PAINT module that combines thePAINT and NIMBUS methods for computationally expensive multiobjectiveoptimization. The PAINT module implements most of the functionalities de-scribed in Figure 2.The synchronous NIMBUS method [24–26] is an interactive multiobjectiveoptimization method. The NIMBUS method uses classification of objectivesas the preference information. Given a Pareto optimal solution to the multi-objective optimization problem, the decision maker can classify the objectivesinto classes I < , I ≤ , I = , I ≥ and I <> defined, respectively, as classes of objectivefunctions that the decision maker wants to improve as much as possible, wantsto improve to a given aspiration level z i , allows to remain unchanged, allowsto deteriorate until a given bound (cid:15) i and allows to change freely for a while.This preference information is converted into several different single objectivesubproblems with the help of different scalarization functions as proposed in[26]. These subproblems are solved to generate different Pareto optimal solu-tions, which are shown to the decision maker who can then see how well thedesired preferences could be attained. The decision maker can choose any ofthese solutions as the starting point of the next iteration, i.e., classification.This iterative procedure can either start with a solution given by the decisionmaker or from a so-called neutral compromise solution and it is repeated untilthe decision maker is satisfied with the solution at hand. Further informationabout the synchronous NIMBUS with other means to direct the search process s given in [26].The NIMBUS method has been successfully applied to shape design of ul-trasonic transducers [16], designing a paper machine headbox [11], optimalcontrol in continuous casting of steel [28], separation of glucose and fructose[9], intensity modulated radiotherapy treatment planning [34], brachytherapy[35] and optimizing heat exchanger network synthesis [20], among others. Inaddition, it uses classification of objectives that has been found cognitively just[19]. These facts make the NIMBUS method an ideal choice as the interactivemethod for solving the PAINT surrogate problem of the wastewater treatmentplant model. Figure 3: A screen shot of the IND-NIMBUS R (cid:13) PAINT module
Figure 3 shows the screen shot of the IND-NIMBUS R (cid:13) PAINT module. Inthe PAINT module, the decision maker can give his/her preferences concerningthe surrogate problem by classification of the objective functions. The givenclassification information is used to formulate a single objective subproblemof the surrogate problem. The subproblem is modeled using the OptimizingProgramming Language (OPL, see [39]), and this (mixed integer linear model)is solved using CPLEX (see [17]). An optimal solution to the subproblem givesa new approximate Pareto optimal outcome, corresponding to the preferencesgiven by the decision maker. This approximate outcome is shown to the decisionmaker. If the decision maker so wishes, he reclassify the objectives of the newapproximate outcome which yields another approximate outcome.As described in Section 2, approximate Pareto optimal outcomes can beprojected on the Pareto front of the original problem using the PAINT mod-ule (using the Project Solution button near the bottom of the screen). Theprojection of the approximate outcome, that is, a Pareto optimal solution to he original problem is shown to the decision make. As mentioned, for a com-putationally expensive problem this may take time, but fortunately projectingan approximate outcome can be done without the involvement of the decisionmaker.The approximate Pareto optimal outcomes and the actual Pareto optimalsolutions that have been found during the decision making process are visualizedon the right side of the PAINT module. The decision maker can choose anyof the approximate Pareto optimal outcomes as the starting point of the nextNIMBUS iteration (i.e., as the basis for a new classification of objectives). Theprocess stops when the decision maker has found a preferred solution to theoriginal problem through projecting an approximate Pareto optimal outcome.In the current version of the PAINT module, only one of the scalarizationsof the synchronous NIMBUS method (i.e., the achievement scalarizing sub-problem) has been implemented. That is, unlike in the synchronous NIMBUSmethod, the decision maker can see only one approximate Pareto optimal out-come for given preferences. It should also be noted that any solver capable ofsolving the surrogate subproblem (e.g., GLPK, see [7]) could be used insteadof CPLEX.The current version of the PAINT module does not implement the construc-tion or updating of the surrogate problem. If one wishes to update the surrogateproblem using Pareto optimal outcomes obtained by e.g., projection, the de-cision making process must be stopped, and the surrogate problem must bemanually updated using Octave. In future versions, updating the Pareto frontapproximation should be implemented under a third button in the PAINT mod-ule that would then automatically update the approximation and the surrogateproblem. In this section, we demonstrate how the PAINT method and the IND-NIMBUS R (cid:13) PAINT module were used to solve the wastewater treatment problem, describedin Section 1.4. First, in Section 4.1, we describe the construction of the Paretofront approximation with PAINT before the involvement of the decision maker.Then, in Section 4.2, we describe how the decision maker used the PAINTmodule to solve the wastewater treatment problem.
First, a set of 200 mutually nondominated solutions to the wastewater problemwas found with the evolutionary UPS-EMO algorithm (introduced in [3]) andthe GPS-X simulator. To study the optimality of these solutions, each one waslocally improved using an achievement scalarizing problem [40, 41], which wasoptimized with Matlab fmincon-function with finite differences approximatedgradients. This resulted in 195 mutually nondominated solutions. The maxi-mal improvement in the values of the achievement scalarizing problem was atmost 3% so the local improvement did not cause much change. This built ourconfidence that the final solutions were close to Pareto optimal. We took the utcomes given by these solutions as the set of given Pareto optimal outcomesfor the PAINT method. The whole process of producing this set took aboutthree days on a standard laptop.After this, we computed a Pareto front approximation based on the givenset of Pareto optimal outcomes with the PAINT method (see Section 2). ThePAINT method chose 4272 polytopes for interpolation in the Pareto front. Inorder to reduce the computational complexity of the implied mixed integerlinear surrogate problem, we removed polytopes that were subsets of largerpolytopes from the approximation. This resulted in a collection of 608 polytopeswhose union covered the same space in R as that of the larger collection. Inaddition, all sets of vertices of the polytopes in the collection were affinelyindependent and, thus, the number of vertices of all the polytopes was five orless, as shown in [12]. Using the PAINT method to construct the Pareto frontapproximation took approximately 19 hours on Intel R (cid:13) Xeon R (cid:13) E5410 CPU.The mixed integer linear surrogate problem implied by the smaller collectionwas equivalent to that implied by the larger collection, but it was computation-ally less expensive. As described in [13], the surrogate problem could be writtenas min ( z , . . . , z )s.t. (cid:80) j =1 (cid:80) l =1 λ j,l = 1 (cid:80) l =1 λ j,l ≤ y j , for all j = 1 , . . . , (cid:80) j =1 y j = 1where λ ∈ [0 , × y ∈ { , } z i = (cid:80) j =1 (cid:80) l =1 λ j,l p A l,j i for all i = 1 , . . . , , (2)where each row of the matrix A ∈ R × contained the indices of the verticesof one polytope in the smaller collection of polytopes. The component λ j,l ofthe matrix variable λ ∈ R × was for all j = 1 , . . . ,
608 and l = 1 , . . . , l of the polytope given by row j in the matrix A . Thevariable y determined which of the rows of the matrix λ was nonzero. By thethird constraint, only one row in the matrix λ had nonzero elements.Problem (2) had 608 × Using the PAINT module, our decision maker (Mr. Kristian Sahlstedt fromP¨oyry Environment Ltd) was able to examine the approximate outcomes andto project any of them on the Pareto front of the original problem. This entiredecision making process was done within a couple of hours and the decisionmaker’s involvement was only about an hour, which could not have been pos-sible by merely using the original computationally expensive problem. gN/m ](min) Aerationpower[ kW ](min) Chemicalconsump-tion [ g/m ](min) Excesssludge[ kg/d ](min) Biogasproduction[ m /d ](max) s s s s p p Before the decision maker started using the PAINT module, we gave him abrief overview of the methods from the user’s perspective. We told him thata set of Pareto optimal outcomes has been computed and that a new PAINTmethod has been used to interpolate between those outcomes. We also informedhim that the outcomes given by PAINT are only approximate Pareto optimaloutcomes and, thus, more computation has to be done to find the closest realPareto optimal outcome. In addition, we told him that the PAINT methoddoes not unfortunately provide any information about the decision variablesand those values can only be known after the real Pareto optimal solution isfound. Since our decision maker had previous experiences with the NIMBUSmethod, all of this was very clear to him. In addition, he did not find any ofthis too inconvenient.Table 1 shows the approximate Pareto optimal outcomes generated (approx-imate outcomes s , . . . , s ) and the outcomes given by actual Pareto optimalsolutions to the wastewater treatment problem (outcomes p , p ) that were in-spected by the decision maker. The decision making process started from theapproximate outcome s in Table 1. The outcome s was given by the neutralcompromise solution to the surrogate problem.The decision maker wanted to see further (approximate) Pareto optimaloutcomes. After a classification of objectives in the PAINT module, the op-timal solution to the new subproblem for the surrogate problem gave the ap-proximate Pareto optimal outcome s . The approximate outcome s has morebiogas production than the approximate outcome s , but is worse in all theother objectives. Especially, the chemical consumption is very large. Let usemphasize that finding the approximate outcome s was especially smooth,since the mixed integer linear problem was computationally inexpensive.Because the decision maker was not completely satisfied with the approx-imate outcome s , he decided to continue and find another approximate out-come. This yielded the approximate outcome s , which has much lower chemicalconsumption and a slightly lower excess sludge production than both approxi-mate outcomes s and s . Unfortunately, the approximate outcome s is worse han both approximate outcomes s and s in all the other objectives.A classification of objectives of the approximate outcome s and solving thenew subproblem for the surrogate problem led to the approximate outcome s .This new approximate outcome is roughly the same as the approximate out-come s in both excess sludge production and biogas production. However, theamount of nitrogen for the approximate outcome s is slightly higher than forthe approximate outcome s , but this is compensated by chemical consumptionand aeration power that are considerably smaller.After having inspected the four approximate Pareto optimal outcomes, thedecision maker felt that he had learned enough about the surrogate problem.First, the decision maker decided to project the approximate outcome s onthe Pareto front of the original wastewater treatment problem. The projectionof the approximate outcome s (i.e., solution to the achievement scalarizingproblem with the approximate outcome s as the reference point) took a littleover half an hour. The projection was done using the GPS-X simulator andthe Controlled Random Search algorithm. The projection was outcome p inTable 1. According to our decision maker’s assessment, the outcome p wasfairly close to the approximate outcome s in all objectives. However, he feltthat there might still be more preferred solutions to the problem.Because the approximate outcome s and the actual Pareto optimal outcome p were close to each other, it was decided not to update the Pareto frontapproximation. Instead, the decision maker wanted to project the approximateoutcome s and obtained the outcome p in Table 1. The Pareto optimaloutcome p has slightly higher amount of nitrogen in the effluent than thePareto optimal outcome p , but it has considerably lower aeration power andthe amount of chemical consumption. The Pareto optimal outcome p wasvery preferred by the decision maker and he chose it as the final solution to theproblem.The final solution (including a way to design and operate a wastewatertreatment plant) will be further inspected by more accurate simulators beforeimplementing. However, it will act as a guideline for the design of the wastew-ater treatment plant. We filmed our decision maker Mr. Sahlstedt during the decision making processand asked him some additional questions regarding the usability of the methods.The purpose of the video was to reveal any issues that he might have had whileusing the PAINT module and to find out whether any aspects of the PAINTmethod that were hard to understand.When analyzing the video, it seems that the key point in the usability ofthe PAINT method and the PAINT module is informing the decision makerabout the approximate nature of the method. In order to make the PAINTmethod more usable, it would be a good idea to produce an introductory videointroducing the key points of the method. Since our decision maker was already amiliar with the IND-NIMBUS R (cid:13) software, there was no need to introduce it.This may not always be the case and, thus, the video should also include ashort introduction to the NIMBUS method and the IND-NIMBUS R (cid:13) software.The decision maker thought that the PAINT method and the PAINT mod-ule were easy and intuitive to use. In addition, he said that the PAINT methodprovided a definite improvement to merely using the NIMBUS method becauseof the faster computational times (a couple of seconds against half an hour)between the iterations of the interactive method. He thought that neither theapproximate nature nor the fact that the preferences had to be based only onthe objective function values were big drawbacks. In addition, no approximateoutcomes that the decision maker would have assessed implausible were foundduring the decision making process.For both projected approximate outcomes s and s , our decision makerassessed that the actual Pareto optimal solutions were close enough, taking intoaccount the uncertainties in the model, and there was no need to recomputethe Pareto front approximation. Thus, the inability of the PAINT module inreconstructing the Pareto front approximation (as mentioned in Section 3) wasnot an issue.Except for correcting a couple of minor bugs from the software, the decisionmaker did not offer any improvements. He did, however, agree with us that thePAINT method should be able to detect disconnectedness in the Pareto frontand that the decision variables should be somehow approximated, too. How-ever, detecting disconnectedness was not an issue on this occasion, because allthe approximate Pareto optimal outcomes found seemed plausible to the deci-sion maker and the approximate Pareto optimal outcomes that were projectedwere rather close to their projections. Finally, since one of the decision vari-ables was also an objective (i.e., the methanol dose), that was approximated inour problem, although the PAINT method does not in general do this. In this paper, we described how the PAINT method can be used to speed upiterations of interactive methods when solving computationally expensive mul-tiobjective optimization problems. The PAINT method was used to construct aPareto front approximation that then implied a mixed integer linear surrogateproblem for the original problem. As our case problem, we studied a five-objective optimization problem of designing and operating a wastewater treat-ment plant. In addition, we introduced a new IND-NIMBUS R (cid:13) PAINT modulethat combines the PAINT method and the interactive NIMBUS method.The PAINT method and the PAINT module worked well in this problem.The decision maker found it easy and intuitive to use the interactive NIMBUSmethod to find a preferred approximate outcome on the Pareto front approx-imation. The low computational cost of using the interactive method withthe surrogate problem was a definite improvement to using interactive methoddirectly to solve the computationally expensive wastewater treatment problem.The experimental design in this paper was unique: Because our decisionmaker had already used the IND-NIMBUS R (cid:13) to study the same wastewater reatment problem, he was able to compare the experiences of using the IND-NIMBUS software with and without the PAINT method. According to thedecision maker’s opinion, the PAINT method provided a significant improve-ment. This implies that the PAINT method should be also applicable to othercomputationally expensive problems.The IND-NIMBUS R (cid:13) PAINT module is still in the development phase and,thus, it lacks some essential functionality (like the implementations of the otherscalarizations of the synchronous NIMBUS method) and, also, it still has somebugs. If there had been no bugs in the software, the investigation of the problemwith the PAINT module would have been even more fluent. Further effort hasto be put in correcting these bugs.The PAINT method requires one to use additional methods and software togenerate the given set of Pareto optimal outcomes and to solve the mixed integerlinear surrogate problem. In this paper, we used the UPS-EMO algorithm togenerate the Pareto optimal outcomes and the IND-NIMBUS software with thePAINT module and the CPLEX solver to solve the surrogate problem. In futureresearch, other applicable methods and software can be also used together withthe PAINT method.
The authors of this paper were financially supported by the Academy of Finland(grant number 128495). The authors are indebted to Mr. Kristian Sahlstedtfrom P¨oyry Environment Ltd for his expertise on the problem and for acting asthe decision maker. In addition, the authors wish to thank Drs. Jussi Hakanenand Timo Aittokoski from the Research Group in Industrial Optimization at theUniversity of Jyv¨askyl¨a, Department of Mathematical Information Technologyfor helping to solve the problem. During the writing of this paper, Prof. Mi-ettinen provided her helpful comments that helped the authors improve thispaper.
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