Pareto Front

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An encyclopedic overview of the Pareto front, the set of Pareto-efficient solutions in multi-objective optimization, commonly used in engineering to evaluate trade-offs.

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# Pareto front Source: [https://en.wikipedia.org/wiki/Pareto_front](https://en.wikipedia.org/wiki/Pareto_front) From Wikipedia, the free encyclopedia In[multi\-objective optimization](https://en.wikipedia.org/wiki/Multi-objective_optimization), the**Pareto front**\(also called**Pareto frontier**or**Pareto curve**\) is the set of all[Pareto efficient](https://en.wikipedia.org/wiki/Pareto_efficient)solutions\.[\[1\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-1)Colloquially, this means when there are many distinct objectives to consider in an[optimization problem](https://en.wikipedia.org/wiki/Optimization_problem), a Pareto front represents the set of solutions where no solution outperforms any other solution in the set at every objective, and every solution not in the set is outperformed by at least one solution in the Pareto front in every objective\.[\[2\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-2)The concept is widely used in[engineering](https://en.wikipedia.org/wiki/Engineering)\.[\[3\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-3):111–148It allows the designer to restrict attention to the set of efficient choices, and to make[tradeoffs](https://en.wikipedia.org/wiki/Trade-off)within this set, rather than considering the full range of every parameter\.[\[4\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-4):63–65[\[5\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-5):399–412 [![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Front_pareto.svg/330px-Front_pareto.svg.png)](https://en.wikipedia.org/wiki/File:Front_pareto.svg)Example of a Pareto frontier\. The boxed points represent feasible choices, and smaller values are preferred to larger ones\. Point*C*is not on the Pareto frontier because it is dominated by both point*A*and point*B*\. Points*A*and*B*are not strictly dominated by any other, and hence lie on the frontier\.[![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Pareto_Efficient_Frontier_1024x1024.png/330px-Pareto_Efficient_Frontier_1024x1024.png)](https://en.wikipedia.org/wiki/File:Pareto_Efficient_Frontier_1024x1024.png)A[production\-possibility frontier](https://en.wikipedia.org/wiki/Production-possibility_frontier)\. The red line is an example of a Pareto\-efficient frontier, where the frontier and the area left and below it are a continuous set of choices\. The red points on the frontier are examples of Pareto\-optimal choices of production\. Points off the frontier, such as N and K, are not Pareto\-efficient, since there exist points on the frontier which Pareto\-dominate them\. The Pareto frontier,*P*\(*Y*\), may be more formally described as follows\. Consider a system with function![{\displaystyle f:X\rightarrow \mathbb {R} ^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fdf6cfa9adbc216a925f176f2982992068a793), where*X*is a[compact set](https://en.wikipedia.org/wiki/Compact_space)of feasible decisions in the[metric space](https://en.wikipedia.org/wiki/Metric_space)![{\displaystyle \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d), and*Y*is the feasible set of criterion vectors in![{\displaystyle \mathbb {R} ^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e), such that![{\displaystyle Y=\{y\in \mathbb {R} ^{m}:\;y=f(x),x\in X\;\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f810a83bb3bf54e6cbfd84db7ba304939d176d)\. We assume that the preferred directions of criteria values are known\. A point![{\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/024264047d824e59a6268b56fa71af223e151494)is preferred to \(strictly dominates\) another point![{\displaystyle y^{\prime }\in \mathbb {R} ^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74a4753538d4b9858264d2168fd0fbd3f20b2f07), written as![{\displaystyle y^{\prime \prime }\succ y^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0735c8618d9acd13ae084bc37bcbeb897b097a64)\. The Pareto frontier is thus written as: ![{\displaystyle P(Y)=\{y^{\prime }\in Y:\;\{y^{\prime \prime }\in Y:\;y^{\prime \prime }\succ y^{\prime },y^{\prime }\neq y^{\prime \prime }\;\}=\emptyset \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4158dd154d62a61bf0ac6c11067ee09dd965be0d) ## Marginal rate of substitution \[[edit](https://en.wikipedia.org/w/index.php?title=Pareto_front&action=edit&section=2)\] A significant aspect of the Pareto frontier in economics is that, at a Pareto\-efficient allocation, the[marginal rate of substitution](https://en.wikipedia.org/wiki/Marginal_rate_of_substitution)is the same for all consumers\.[\[6\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-6)A formal statement can be derived by considering a system with*m*consumers and*n*goods, and a utility function of each consumer as![{\displaystyle z_{i}=f^{i}(x^{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7129ceae0088b99225259998162cef9e22e177)where![{\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d81a12733d99bbc1ac978da2d77349b67a57c34)is the vector of goods, both for all*i*\. The feasibility constraint is![{\displaystyle \sum _{i=1}^{m}x_{j}^{i}=b_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a74beec39625f7e6b2914c6e70a7ce5495edbfb6)for![{\displaystyle j=1,\ldots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c39c89dce6309ea91bb90112a6babe359d6ff7d6)\. To find the Pareto optimal allocation, we maximize the[Lagrangian](https://en.wikipedia.org/wiki/Lagrange_multiplier): ![{\displaystyle L_{i}((x_{j}^{k})_{k,j},(\lambda _{k})_{k},(\mu _{j})_{j})=f^{i}(x^{i})+\sum _{k=2}^{m}\lambda _{k}(z_{k}-f^{k}(x^{k}))+\sum _{j=1}^{n}\mu _{j}\left(b_{j}-\sum _{k=1}^{m}x_{j}^{k}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b508b219cd951e3afce57d203ea2bd8b8b38f184) where![{\displaystyle (\lambda _{k})_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820134a186eadaec240e0320e851bde58e8ce30e)and![{\displaystyle (\mu _{j})_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22ac7ecfb09371c5d0d9b50f999c4f552d7575b4)are the vectors of multipliers\. Taking the[partial derivative](https://en.wikipedia.org/wiki/Partial_derivative)of the Lagrangian with respect to each good![{\displaystyle x_{j}^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01e8f07e344279d1d155074eb04440d51e2087ef)for![{\displaystyle j=1,\ldots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c39c89dce6309ea91bb90112a6babe359d6ff7d6)and![{\displaystyle k=1,\ldots ,m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09a3c5f7ce97004561fe563631144135783e8164)gives the following system of first\-order conditions: ![{\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{i}}}=f_{x_{j}^{i}}^{1}-\mu _{j}=0{\text{ for }}j=1,\ldots ,n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f566742a6b1e8ed19d9a4d2d96c937d55dd73dd)![{\displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{k}}}=-\lambda _{k}f_{x_{j}^{k}}^{i}-\mu _{j}=0{\text{ for }}k=2,\ldots ,m{\text{ and }}j=1,\ldots ,n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2e717b59c1005002518221417d594c2a388343)where![{\displaystyle f_{x_{j}^{i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0ad90ef32f56a56210af7f74db42df2f8dd749)denotes the partial derivative of![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)with respect to![{\displaystyle x_{j}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1383d67a632caa0da8ca8e7f970436f6472cc4)\. Now, fix any![{\displaystyle k\neq i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f65196f10cd17df530701fa98651b629d1756b4)and![{\displaystyle j,s\in \{1,\ldots ,n\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/618b09f65ed0efeeda3003c7c8a5287d5576ce89)\. The above first\-order condition imply that ![{\displaystyle {\frac {f_{x_{j}^{i}}^{i}}{f_{x_{s}^{i}}^{i}}}={\frac {\mu _{j}}{\mu _{s}}}={\frac {f_{x_{j}^{k}}^{k}}{f_{x_{s}^{k}}^{k}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7a491efde24f5d8a9bb3dc2e02c1551a6bfee7)Thus, in a Pareto\-optimal allocation, the marginal rate of substitution must be the same for all consumers\.[\[7\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-7) [Algorithms](https://en.wikipedia.org/wiki/Algorithm)for computing the Pareto frontier of a[finite set](https://en.wikipedia.org/wiki/Finite_set)of alternatives have been studied in[computer science](https://en.wikipedia.org/wiki/Computer_science)and power engineering\.[\[8\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-8)They include: - "The[maxima of a point set](https://en.wikipedia.org/wiki/Maxima_of_a_point_set)" - "The maximum vector problem" or the[skyline query](https://en.wikipedia.org/wiki/Skyline_operator)[\[9\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-9)[\[10\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-10)[\[11\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-11) - "The scalarization algorithm" or the method of weighted sums[\[12\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-Kimde_Weck2005-12)[\[13\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-MarlerArora2009-13) - "The![{\displaystyle \epsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)\-constraints method"[\[14\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-Mavrotas2009-15)[\[16\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-16) - Multi\-objective Evolutionary Algorithms[\[17\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-17)[\[18\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-18) Since generating the entire Pareto front is often computationally\-hard, there are algorithms for computing an approximate Pareto\-front\. For example, Legriel et al\.[\[19\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-19)call a set*S*an***ε*\-approximation**of the Pareto\-front*P*, if the directed[Hausdorff distance](https://en.wikipedia.org/wiki/Hausdorff_distance)between*S*and*P*is at most*ε*\. They observe that an*ε*\-approximation of any Pareto front*P*in*d*dimensions can be found using \(1/*ε*\)*d*queries\. Zitzler, Knowles and Thiele[\[20\]](https://en.wikipedia.org/wiki/Pareto_front#cite_note-20)compare several algorithms for Pareto\-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity\. 1. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-1)proximedia\.["Pareto Front"](https://web.archive.org/web/20200226003108/https://www.cenaero.be/Page.asp?docid=27103&)\.*www\.cenaero\.be*\. Archived from[the original](http://www.cenaero.be/Page.asp?docid=27103&)on 2020\-02\-26\. Retrieved2018\-10\-08\. 2. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-2)Kang, Shida; Li, Kaiwen; Wang, Rui \(2025\-06\-01\)\.["A survey on pareto front learning for multi\-objective optimization"](https://doi.org/10.1007/s41965-024-00170-z)\.*Journal of Membrane Computing*\.**7**\(2\):128–134\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/s41965\-024\-00170\-z](https://doi.org/10.1007%2Fs41965-024-00170-z)\.[ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier))[2523\-8914](https://search.worldcat.org/issn/2523-8914)\. 3. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-3)Goodarzi, E\., Ziaei, M\., & Hosseinipour, E\. Z\.,*Introduction to Optimization Analysis in Hydrosystem Engineering*\([Berlin](https://en.wikipedia.org/wiki/Berlin)/[Heidelberg](https://en.wikipedia.org/wiki/Heidelberg):[Springer](https://en.wikipedia.org/wiki/Springer_Science+Business_Media), 2014\),[pp\. 111–148](https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111)\. 4. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-4)Jahan, A\., Edwards, K\. L\., & Bahraminasab, M\.,*Multi\-criteria Decision Analysis*, 2nd ed\. \([Amsterdam](https://en.wikipedia.org/wiki/Amsterdam):[Elsevier](https://en.wikipedia.org/wiki/Elsevier), 2013\),[pp\. 63–65](https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63)\. 5. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-5)Costa, N\. R\., & Lourenço, J\. A\., "Exploring Pareto Frontiers in the Response Surface Methodology", in G\.\-C\. Yang, S\.\-I\. Ao, & L\. Gelman, eds\.,*Transactions on Engineering Technologies: World Congress on Engineering 2014*\(Berlin/Heidelberg: Springer, 2015\),[pp\. 399–412](https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398)\. 6. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-6)Just, Richard E\. \(2004\)\.*The welfare economics of public policy: a practical approach to project and policy evaluation*\. Hueth, Darrell L\., Schmitz, Andrew\. Cheltenham, UK: E\. Elgar\. pp\.18–21\.[ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier))[1\-84542\-157\-4](https://en.wikipedia.org/wiki/Special:BookSources/1-84542-157-4)\.[OCLC](https://en.wikipedia.org/wiki/OCLC_(identifier))[58538348](https://search.worldcat.org/oclc/58538348)\. 7. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-7)Just, Richard E\.; Hueth, Darrell L\.; Schmitz, Andrew \(2005\-01\-01\)\.[*The Welfare Economics of Public Policy: A Practical Approach to Project and Policy Evaluation*](https://books.google.es/books?hl=en&lr=&id=GXwAAgAAQBAJ&oi=fnd&pg=PR1&dq=Richard+E.+Just.+(2001).+The+Welfare+Economics+of+Public+Policy:+A+Practical+Approach+to+Project+and+Policy+Evaluation.+Edward+Elgar.&ots=Lh86TUlJmo&sig=5Sdx4eyYKjHzQgHrVexqjhDabAU&redir_esc=y#v=onepage&q&f=false)\. Edward Elgar Publishing\.[ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier))[978\-1\-84542\-157\-1](https://en.wikipedia.org/wiki/Special:BookSources/978-1-84542-157-1)\. 8. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-8)Tomoiagă, Bogdan; Chindriş, Mircea; Sumper, Andreas; Sudria\-Andreu, Antoni; Villafafila\-Robles, Roberto \(2013\)\.["Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA\-II"](https://doi.org/10.3390%2Fen6031439)\.*Energies*\.**6**\(3\):1439–55\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.3390/en6031439](https://doi.org/10.3390%2Fen6031439)\.[hdl](https://en.wikipedia.org/wiki/Hdl_(identifier)):[2117/18257](https://hdl.handle.net/2117%2F18257)\. 9. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-9)Nielsen, Frank \(1996\)\. "Output\-sensitive peeling of convex and maximal layers"\.*Information Processing Letters*\.**59**\(5\):255–9\.[CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_(identifier))[10\.1\.1\.259\.1042](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.259.1042)\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1016/0020\-0190\(96\)00116\-0](https://doi.org/10.1016%2F0020-0190%2896%2900116-0)\. 10. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-10)Kung, H\. T\.; Luccio, F\.; Preparata, F\.P\. \(1975\)\.["On finding the maxima of a set of vectors"](https://doi.org/10.1145%2F321906.321910)\.*Journal of the ACM*\.**22**\(4\):469–76\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1145/321906\.321910](https://doi.org/10.1145%2F321906.321910)\.[S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier))[2698043](https://api.semanticscholar.org/CorpusID:2698043)\. 11. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-11)Godfrey, P\.; Shipley, R\.; Gryz, J\. \(2006\)\. "Algorithms and Analyses for Maximal Vector Computation"\.*VLDB Journal*\.**16**:5–28\.[CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_(identifier))[10\.1\.1\.73\.6344](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.73.6344)\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/s00778\-006\-0029\-7](https://doi.org/10.1007%2Fs00778-006-0029-7)\.[S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier))[7374749](https://api.semanticscholar.org/CorpusID:7374749)\. 12. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-Kimde_Weck2005_12-0)Kim, I\. Y\.; de Weck, O\. L\. \(2005\)\. "Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation"\.*Structural and Multidisciplinary Optimization*\.**31**\(2\):105–116\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/s00158\-005\-0557\-6](https://doi.org/10.1007%2Fs00158-005-0557-6)\.[ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier))[1615\-147X](https://search.worldcat.org/issn/1615-147X)\.[S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier))[18237050](https://api.semanticscholar.org/CorpusID:18237050)\. 13. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-MarlerArora2009_13-0)Marler, R\. Timothy; Arora, Jasbir S\. \(2009\)\. "The weighted sum method for multi\-objective optimization: new insights"\.*Structural and Multidisciplinary Optimization*\.**41**\(6\):853–862\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/s00158\-009\-0460\-7](https://doi.org/10.1007%2Fs00158-009-0460-7)\.[ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier))[1615\-147X](https://search.worldcat.org/issn/1615-147X)\.[S2CID](https://en.wikipedia.org/wiki/S2CID_(identifier))[122325484](https://api.semanticscholar.org/CorpusID:122325484)\. 14. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-14)"On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization"\.*IEEE Transactions on Systems, Man, and Cybernetics*\. SMC\-1 \(3\):296–297\. 1971\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1109/TSMC\.1971\.4308298](https://doi.org/10.1109%2FTSMC.1971.4308298)\.[ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier))[0018\-9472](https://search.worldcat.org/issn/0018-9472)\. 15. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-Mavrotas2009_15-0)Mavrotas, George \(2009\)\. "Effective implementation of the ε\-constraint method in Multi\-Objective Mathematical Programming problems"\.*Applied Mathematics and Computation*\.**213**\(2\):455–465\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1016/j\.amc\.2009\.03\.037](https://doi.org/10.1016%2Fj.amc.2009.03.037)\.[ISSN](https://en.wikipedia.org/wiki/ISSN_(identifier))[0096\-3003](https://search.worldcat.org/issn/0096-3003)\. 16. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-16)Carvalho, Iago A\.; Coco, Amadeu A\. \(September 2023\)\. "On solving bi\-objective constrained minimum spanning tree problems"\.*Journal of Global Optimization*\.**87**\(1\):301–323\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/s10898\-023\-01295\-8](https://doi.org/10.1007%2Fs10898-023-01295-8)\. 17. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-17)Zhang, Qingfu; Hui, Li \(December 2007\)\. "MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition"\.*IEEE Transactions on Evolutionary Computation*\.**11**\(6\):712–731\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1109/TEVC\.2007\.892759](https://doi.org/10.1109%2FTEVC.2007.892759)\. 18. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-18)Carvalho, Iago A\.; Ribeiro, Marco A\. \(November 2019\)\. "A node\-depth phylogenetic\-based artificial immune system for multi\-objective Network Design Problems"\.*Swarm and Evolutionary Computation*\.**50**100491\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1016/j\.swevo\.2019\.01\.007](https://doi.org/10.1016%2Fj.swevo.2019.01.007)\. 19. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-19)Legriel, Julien; Le Guernic, Colas; Cotton, Scott; Maler, Oded \(2010\)\. "Approximating the Pareto Front of Multi\-criteria Optimization Problems"\. In Esparza, Javier; Majumdar, Rupak \(eds\.\)\.*Tools and Algorithms for the Construction and Analysis of Systems*\. Lecture Notes in Computer Science\. Vol\.6015\. Berlin, Heidelberg: Springer\. pp\.69–83\.[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/978\-3\-642\-12002\-2\_6](https://doi.org/10.1007%2F978-3-642-12002-2_6)\.[ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier))[978\-3\-642\-12002\-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-12002-2)\. 20. [↑](https://en.wikipedia.org/wiki/Pareto_front#cite_ref-20)Zitzler, Eckart; Knowles, Joshua; Thiele, Lothar \(2008\),["Quality Assessment of Pareto Set Approximations"](https://doi.org/10.1007/978-3-540-88908-3_14), in Branke, Jürgen; Deb, Kalyanmoy; Miettinen, Kaisa; Słowiński, Roman \(eds\.\),*Multiobjective Optimization: Interactive and Evolutionary Approaches*, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer, pp\.373–404,[doi](https://en.wikipedia.org/wiki/Doi_(identifier)):[10\.1007/978\-3\-540\-88908\-3\_14](https://doi.org/10.1007%2F978-3-540-88908-3_14),[ISBN](https://en.wikipedia.org/wiki/ISBN_(identifier))[978\-3\-540\-88908\-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-88908-3), retrieved2021\-10\-08

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