In the following, solution a and solution b represent any two Pareto-optimal solutions selected for comparison. An element in the concordance matrix is given by If solution a is equal or preferred to i. The individual concordance index, C a , b measures the strength of the argument that, when solution a is compared to solution b for a given objective j, the value of F j a is at least as good as F j b. The discordance matrix element measures the strength of the argument that, when solution a is compared to solution b for a given objective j , the value of F j a is significantly worse than F j b.
Make the selection. After the concordance and discordance indices i. Once the outranking relationship is decided, mark S F at the corresponding position; here, S F stands for strong outranking relation. For instance, if C 1,2 satisfies the above rule 1, then mark C 1,2 as S F. In the end, count the number of S F in each row, which represents the extent of one objective outranking other objectives.
Then, by the rule of intersection, 17 the Pareto-optimal solution with the most S F in the horizontal row i. This method requires the user to input weightage for each objective, indifference threshold Q j , preference threshold P j , veto threshold V j , and a cutoff value 0—1.
As before, solutions a and b represent any two selected solutions to be compared. Indifference threshold Q j is the range of variation of each objective, for which it is not possible for the decision maker to favor the objective of one solution over that of another solution. It therefore represents the range of values over which the particular objective of two solutions is indiscernible.
If the difference between two values for a given objective exceeds preference threshold P j , then preference is given to the solution with the better value. Veto threshold V j serves to reject a solution relative to another solution if the difference between the values of a particular objective is too large to be acceptable. An element of the concordance matrix of m rows and columns is computed by 27 where The individual concordance index, C a, b , measures the strength of the argument that, when solution a is compared to solution b for a given objective j , the value of F j a is at least as good as F j b.
Construct elements of discordance matrix by Finally, make the selection based on the credibility matrix by calculating the difference between the strength sum of row and weakness sum of column of each solution, with or without cutoff values. If applying the cutoff value, every element of the credibility matrix with a value greater than the cutoff value is rounded up to 1.
The principle of NFM, developed from ELECTRE methods, is to rank the difference of the extent of one objective outranking other objectives and the extent of other objectives outranking this objective. It too requires the user to input weightage for each objective, indifference threshold Q j , preference threshold P j , and veto threshold V j. Construct discordance matrix of m rows and n columns by 31 where The discordance matrix element, D a , b , measures the strength of the argument that, when solution a is compared to solution b for the j th objective, the value of F j a is significantly worse than F j b.
Construct the credibility matrix by multiplying the concordance element and discordance element at the corresponding location. Make the selection to calculate the ranking score of each solution, i , as follows: The first term evaluates the extent to which solution i performs relative to all other solutions, while the second term evaluates the performance of all other solutions relative to solution i.
The solution with the highest ranking score S i is the recommended optimal solution. The reason for choosing MS Excel is that it is probably the most readily available and most commonly used computational tool in both academia and industry. Hence, researchers and engineers familiar with MS Office can utilize the program developed in this study. Furthermore, the selection methods program described here can be used in conjunction with our programs for MOO, namely, Excel-based MOO 21 and integrated multiobjective differential evolution.
The user interface of selection methods program is organized in a number of worksheets in Excel.
The overview worksheet provides brief guidelines on using the program. Pareto-optimal solutions i. Then, the user can execute selection methods, one by one or all by clicking on the corresponding icon Figure 1. Figure 1. Main interface of the selection methods program in Excel-VBA. High Resolution Image. The results of executing the selection methods are presented in both numbers and plots, with one worksheet for each method. In the worksheet for the results of a selection method, the last column provides the ranking score of each optimal solution; the selected solution row is shaded for ready reference, and the selected solution is shown as a red dot on plots of one objective against another objective.
Implemented code of each method was validated by application to at least one example available in the literature, as summarized in Table 1. Each method in the developed program recommended the same optimal solution as in the literature. Table 1. The selection methods program has been applied to choosing one optimal solution from the nondominated solutions of 12 mathematical and 13 chemical engineering problems, as summarized in Tables 2 and 3.
The mathematical problems have 2 or 3 objectives and 31— nondominated solutions, and the chemical engineering problems have 2—5 objectives and 9— nondominated solutions. Results from the application of selection methods to one mathematical problem and for two chemical engineering applications are presented and discussed in this section. These include the effect of weightage to be provided by the user on the optimal solution chosen by each method; see Tables 2 and 3. Table 2. Table 3. DTLZ3 is a mathematical problem with three objectives to minimize. To decrease the computational effort, one set of objective values is selected from every 10 sets, and finally the objective matrix with nondominated solutions is generated.
Four sets of weightage values are used to assess the effect of weightage on the recommended optimal solution by selection methods. The four cases are a equal weightage for each objective; b 0. The last three sets have very large weight on one objective in turn and smaller weight on other objectives.
Similarly, for all the problems in Tables 2 and 3 , case a is always for equal weightage for each objective, and the following cases have very large weight 0. In DTLZ3, there will be many plots because there are three objectives. To present them for all 10 methods in a single figure, only one of plot of F3 versus F1 for each method is chosen Figure 2. The bar chart in Figure 2 shows fractional objective values from the respective minimum eq 35 of the solution recommended by each of the 10 methods.
Fractional objective values are given by The normalization factor i. This allows presenting objective values of significantly different magnitude concisely in a single plot. Equation 35 can be used for both minimization and maximization objective. In the case of minimization objective which is common in many examples tested , the value of that objective of the recommended solution by a selection method is closer to the minimum if the bar is negligible; otherwise, it is far from the minimum of that objective.
On the other hand, in the case of maximization objective, the value of that objective of the recommended solution by a selection method is closer to the maximum if the bar is long i. In the DTLZ3 function, the objectives are to be minimized; hence, in Figures 2 and 3 , the particular objective is optimized greatly if its bar is very short or absent. As shown in these two figures, the bar height of an objective generally decreases with increasing weightage of that objective.
Recall that GRA does not require any weight input from the user; hence, the recommended solution by it is the same in all cases and plots in Figure 3. In effect, change in weightage has negligible effect in these scenarios. These results indicate the nonlinear effect of weightage in these methods on the recommended solution. This is a dividing-wall column design problem with two objectives to minimize utility cost and total capital cost and 55 nondominated solutions.
Three sets of weightages, namely, a equal weightage; b 0. As shown in these figures, the bar height of an objective decreases with increasing weightage of that objective. This is an industrial water network design problem with two objectives to minimize fresh water flow rate FW and total flow rate through regeneration units TRU. The general trend in Figure 7 is that bar height of an objective decreases with increasing weight of that objective. The only difference is that the number of nondominated solutions is small in this example.
This is similar to many examples in Table 1 used for validating the selection methods program , which have just 3—6 nondominated solutions or alternatives or candidates for selection. Based on the principle and algorithm described in section 2 as well as recommended solutions in the applications studied section 4 , 8 of the 10 methods studied can be put into 3 groups. All four groups are discussed in the following sections.
Sometimes, VIKOR also recommends a solution similar to that of the other two, but in the majority of cases, it chooses a different optimal solution. When solution A m , which is the midpoint between positive and negative ideal points, and solution A i in Figure 8 are compared, A i is always preferred to A m as long as the distance from A i to A — i. This will be ridiculous when A i is very far from both positive and negative ideal points.
Figure 8. Greater than 0.
Multi-objective Optimization for Materials Discovery via Adaptive Design
It is found that the objective with higher weightage is the one to be optimized with the increase of weight of strategy. In addition, the objective with higher absolute difference between adjacent solutions e. These two methods share many similarities and usually recommend very similar solutions, as Figures 3 and 5 reflect. Among all the problems and weightages considered, only for the Osyczka function are the solutions recommended by SAW and MEW very different see Figure 9.
For using either SAW or MEW, the minimization objective value of any optimal solution and the maximum value of the maximization objective cannot be 0 because they are in the denominator in the normalization equations eqs 17 and In addition, MEW cannot be employed for problems with a negative value in the objective matrix.
There are some situations e. These three methods have a similar fundamental concept of pairwise comparison between every objective of each pair of solutions. This makes them computationally intensive, particularly for problems with numerous nondominated solutions.
Hence, it is not easy for the user to decide these threshold values. Therefore, NFM, which requires fewer and physically meaningful inputs, is recommended from this group of similar methods. The advantage of FUCA as compared with other methods is that its principle is relatively simple. However, it still requires weightage values from the user, which makes it less attractive than GRA. Among the 10 methods tested, GRA is the only method that does not require any inputs from decision makers. It is found that the result generated by GRA is usually quite close to sometimes exactly the same as those obtained by other methods under the case of equal weightage for objectives involved.
Therefore, to reduce the subjectivity of selecting an optimal solution, whenever the user inputs are required, GRA is recommended from this group. MOO and selection of an optimal solution have been studied in the field of economics, management, and strategy planning for several decades. In the chemical engineering field, MOO is employed increasingly in the past 15 years, but selection of an optimal solution is not yet commonly studied.
Application of 10 selection methods presented above, with the exception of GRA, for selecting one optimal solution from the set of nondominated solutions, requires inputs from the user s , often referred to as decision makers. The most common inputs are weight for each of the objectives involved; other inputs may be required depending on the selection method. Although significance of weight for each objective is obvious, there will be some subjectivity in giving a suitable value to it. Moreover, the effect of weight on the selected optimal solution depends on the selection method.
On the other hand, significance of other inputs may not be as simple to understand as weights. All these mean that it is better to choose a method whose principle and the required inputs are easy to understand. The selected optimal solution will depend not only on required inputs but also on the selection method. Hence, choosing a selection method and its inputs is itself an optimization problem involving some subjectivity. For this and for minimizing the subjectivity, we propose the following approach.
It can be modified suitably according to the complexity and level of MOO application under consideration. In this study on methods for selecting one of the nondominated solutions obtained from MOO, principles and algorithms of 10 selection methods are elaborated using consistent terminology. All these methods are implemented in an MS Excel-based software package, which is verified against at least one solved example from the literature.
The 10 selection methods with different objective weightages are applied to 12 benchmark mathematical problems and 13 chemical engineering problems. Results show that objective weightages have nonlinear effect on the optimal solution recommended by a method. Based on the similarities and results, the 10 methods tested are put into four groups, and one method from each group is recommended as the most suitable method.
Choosing a selection method and giving required user inputs are somewhat subjective. Hence, domain knowledge and optimal values of decision variables should be combined with one or more of these methods for selecting an optimal solution. For this, an approach involving several decision makers is outlined. The authors declare no competing financial interest. Applications of multi-objective optimization in chemical engineering Rev. Freund Publishing House Ltd. A review with many refs.
Multiobjective optimization involves the simultaneous optimization of more than one objective function.
Multicriteria Design: Optimization and Identification (Applied Optimization) | KSA | Souq
This is quite commonly encountered in chem. A considerable amt. The general background of this area is presented at the beginning, followed by a description of how the results can be described in terms of Pareto sets. The several methods available for generating these optimal solns.
Applications of optimization in chem. Some comments are also made on possible directions that future research may take in this area. Multi-objective optimization applications in chemical engineering. Google Scholar There is no corresponding record for this reference. An improved multi-objective differential evolution with a termination criterion for optimizing chemical processes Comput. Elsevier B. Application problems have conflicting objectives and constraints, and max.
This study develops a termination criterion using the non-dominated solns. For this, several performance metrics are modified, and their variation with generations has been assessed on many test functions. Based on this anal. This study uses taboo list with multi-objective differential evolution to avoid re-visits and for better exploration of search space. Benefits of the termination criterion and taboo list are assessed on constrained benchmark problems.
While circuits in Biology are complex and subject to natural tradeoffs, most synthetic circuits are simple in terms of the number of regulatory regions, and have been designed to meet a single design criterion. In this contribution we introduce a multiobjective formulation for the design of biocircuits. We set up the basis for an advanced optimization tool for the modular and systematic design of biocircuits capable of handling high levels of complexity and multiple design criteria.
Our methodology combines the efficiency of global Mixed Integer Nonlinear Programming solvers with multiobjective optimization techniques. Through a number of examples we show the capability of the method to generate non intuitive designs with a desired functionality setting up a priori the desired level of complexity.
The methodology presented here can be used for biocircuit design and also to explore and identify different design principles for synthetic gene circuits. The presence of more than one competing objective provides a realistic design setting where every solution represents an optimal trade-off between different criteria. A hallmark of Synthetic Biology is, quoting Arkin, the ambition to formalize the process of designing cellular systems in the way that traditional engineering disciplines have formalized design and manufacture, so that complex behaviours can be achieved for practical ends [ 1 ].
In formalizing the design process, as it is the case in more traditional engineering disciplines, mathematical modeling and optimization play a central role. Over the past ten years, many advances have been achieved in the field, from the first bacterial toggle switches [ 2 ] and biological oscillators [ 3 ], to the recent mammalian cell to cell communication devices [ 4 ].
In a so called first wave of Synthetic Biology basic elements and small biological modules were successfully implemented and characterized. One of the challenges of the second wave in progress is the integration of modules to create circuits of increasing complexity [ 5 ].
However, as reported by Purnick and Weiss [ 5 ], the level of complexity achieved in synthetic circuits, measured by the number of regulatory regions, is relatively low. While circuits in Biology are complex, subject to natural tradeoffs and playing multiple roles [ 6 ], most synthetic designs are simple and perform a single task. Reported reasons for the current limited complexity in synthetic circuits include too simplistic engineering design principles [ 5 ], difficulty to independently control multiple cellular processes in parallel [ 7 ] and increasing problems to construct and test circuits as they get larger [ 8 ].
Efforts are necessary to overcome these difficulties and, quoting Lu et al. In parallel, new computational tools need to be developed to support these efforts [ 10 ]. In this contribution, our goal is to set up the basis of an advanced optimization tool for the modular and systematic design of biocircuits capable of handling high levels of complexity and multiple design criteria. Modular design requires the previous definition of standardized functional objects and interfaces [ 11 ]. From the foundations of Synthetic Biology, efforts have been held in order to characterize standard biological parts , i.
DNA sequences encoding a function that can be assembled with other standard parts.
The abstraction hierarchy proposed by Endy [ 12 ] classifies standard parts in three different layers: parts , which are defined as sequences with basic biological functions like for example DNA-binding proteins , devices which are combinations of parts with a particular function and systems which are combinations of devices.
An emerging catalogue of standard parts is available at the registry supported by the BioBricks Foundation [ 13 ]. Systematic design relies on mathematical models describing the circuit dynamics. In this regard, modular modeling tools are advancing to facilitate the mathematical representation of biological parts and their combinations [ 14 ], providing the description of the reactions taking place inside the different parts and the interfaces to connect them.
Inspired by the BioBrick registry of standard parts, Marchisio and Stelling [ 15 ] developed a formal modeling framework based on the ordinary differential equations ODE formalism which permits modular model composition and has been recently extended for the modeling of more complex eukaryotic systems [ 14 ]. Some remarkable advances have been also achieved regarding synthetic biology computer aided design tools [ 16 ]. The systematic design of circuits combining components or parts from a list or library can be formulated as an optimization problem [ 16 ]-[ 18 ] where the circuit model structure is manipulable through decision variables, and the desired behaviour of the circuit is encoded in the objective function to optimize.
Dasika and Maranas [ 17 ] developed an optimization framework for the design of biocircuits, based on the circuit modeling formulation by Hasty [ 19 ] and a multistart local outer approximation method for the optimization. A number of design problems were successfully solved within this framework including a circuit with inducer specific response, a genetic decoder and a concentration band detector. In this work, we advance the optimization-based design of biocircuits with two contributions: increasing the computation efficiency in order to handle higher levels of complexity and introducing multiple criteria in the design.
To this purpose, we first introduce a set of global MINLP solvers that reduce drastically the computation time for the monoobjective design problem in comparison with other published methods. Then we formulate a general multiobjective optimization framework that combines the efficiency of the global MINLP solvers with the ability to tackle multiple design criteria. The inducer specific response circuit design by Dasika and Maranas [ 17 ] is used to illustrate the efficiency of the MINLP methods presented and further reformulated with additional design criteria to discuss the advantages of a multiobjective formulation in the design of genetic circuits.
Table of contents
Optimization based design of biocircuits requires the integration of tools for modular modeling, simulation and optimization. As reported in the Background section, modular tools for modeling in Synthetic Biology are advancing fast as well as repositories of biological parts. Searching for a generic optimization framework, the methods presented next do not bound to a specific modeling tool, but accommodate to any ODE based modeling framework such that the circuit's model structure can be obtained from the starting list of parts by giving values to a set of integer variables.
The design problem consists of finding the best solution or solutions among the set of all possible alternatives according to a number of criteria. In this first part, we focus on problems with one unique design objective. Under these assumptions, the design of biocircuits can be formulated as a Mixed Integer Nonlinear Programming Problem [ 17 ],[ 18 ], where the model structure can be encoded by integer variables and the constraints are the dynamics of the system in form of ODEs.
Tunable kinetic parameters are real decision variables in the optimization model. For a complete formulation we refer to [ 17 ], where the single objective MINLP problem is formalized for a particular modeling framework [ 19 ]. Next, our focus is on the computational challenges of the resultant MINLP, since some features inherent to biological circuit models make it particularly difficult to solve.
In first instance, the dynamics of biocircuits are highly nonlinear, and the resultant optimization problem is non convex and multi-modal. In this type of problems, local methods lead to suboptimal solutions unless we start close to the global optimum. A number of approaches have been proposed in previous works to find the global optimum in monoobjective biocircuit design.
Dasika and Maranas [ 17 ] implemented a multistart local outer approximation algorithm where a convergence sequence of upper and lower bounds to the original problem is generated and a local optimum solution is identified at each iteration. In this way, a local deterministic search is performed from several points. Rodrigo et al. On the other hand, the design of gene circuits involves in general large search spaces that combine a high number of integer variables with the presence of real variables.
Global deterministic methods ensure convergence to the global optimum within a desired tolerance, but the computational burden is in general very high for non convex systems with large search spaces. Therefore, we have decided to employ global stochastic methods, which offer no guarantee of convergence to the global minimum in a finite number of iterations but showed excellent results solving complex process optimization problems in reasonable computation time [ 23 ]. These methods have been shown to be efficient metaheuristics in solving complex-process optimization problems from different fields, providing a good compromise between diversification exploration by global search and intensification local search.
MITS MITS uses a combinatorial component, based on Tabu Search [ 27 ], to guide the search into promising areas, where the local solver is activated to precisely approximate local minima. Exler et al. ACOmi ACOmi extends ant colony optimization meta-heuristic [ 28 ] to handle mixed integer search domains.
Schlueter et al. Finally, eSS eSS is an enhanced version of the scatter search for mixed integer search domain. Egea et al. In this contribution, we evaluate the efficiency of these methods in the context of Synthetic Biology and in particular for the systematic design of genetic circuits. For illustrative purposes we chose a representative design example from Ref. Starting from a list of components, the goal is to build a circuit with a specific response upon stimulation by two different inducers.
The dynamic model of the overall reaction network is constituted by a set of ordinary differential equations of the form:. The expression rates for the transcripts are known and they read:. We define the vector of binary variables y as the vector obtained by converting the matrix Y to a vector by columns. The tunable parameters are contained in a vector of real variables denoted by x. As mentioned, the goal is to achieve a specific response upon induction. This design goal is encoded in the following objective function Z to be maximized:.
The design problem is formulated as a MINLP where the decision variables are contained in the vectors y and x and the objective function to maximize is Z in 3 , subject to the system's dynamics 1. The following constraint on the maximum number of active pairs M max is also imposed:. First we use the original formulation of the problem by Dasika and Maranas [ 17 ] with a maximum of two promoter-transcript pairs, and compare the performance of the methods with the published results.
Afterwards, we gradually increase the network complexity to evaluate how the methods proposed scale with the increasing problem size.
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The results obtained are included in the Results and discussion section. In traditional engineering disciplines design problems are often multicriteria, where a number of design objectives are conflicting typically production and cost since we cannot increase one without decreasing the other. Problems with multiple and conflicting design criteria do not have a unique optimal solution, but a trade-off front between the competing objectives, also known as Pareto optimal front of solutions.
In biological systems, trade-offs between robustness, fragility, performance, and resource demands have been conjectured [ 6 ],[ 29 ]-[ 32 ]. We know that living organisms allocate limited resources to various competing traits, and arising tradeoffs are central to evolutionary biology. Furthermore molecular pathways have been shown in many cases to play diverse and complex roles. Multicritria Design: Optimization and Identification. Dordrecht: Kluwer Academic Publishers. Singapore: World Scientific.
DEB, K. Multi-Objective Optimization using Evolutionary Algorithms. GEN, M. Genetic Algorithms and Engineering Optimization. New York: Wiley Interscience. Dissertation, University of Michigan. Reading, MA: Addison-Wesley. LI, J. Berlin: Springer-Verlag. Berlin: Physica-Verlag. Fuzzy Set Theory — and Its Applications. Boston: Kluwer Academic.