https://en.wikipedia.org/wiki/Pareto_front Jump to content [ ] Main menu Main menu move to sidebar hide Navigation * Main page * Contents * Current events * Random article * About Wikipedia * Contact us * Donate Contribute * Help * Learn to edit * Community portal * Recent changes * Upload file [wikipe] Wikipedia The Free Encyclopedia Search [ ] Search * Create account * Log in [ ] Personal tools * Create account * Log in Pages for logged out editors learn more * Contributions * Talk Contents move to sidebar hide * (Top) * 1Definition * 2Marginal rate of substitution * 3Computation * 4Approximations * 5References [ ] Toggle the table of contents Pareto front [ ] Add languages Add links * Article * Talk [ ] English * Read * Edit * View history [ ] Tools Tools move to sidebar hide Actions * Read * Edit * View history General * What links here * Related changes * Upload file * Special pages * Permanent link * Page information * Cite this page * Get shortened URL * Download QR code * Wikidata item Print/export * Download as PDF * Printable version From Wikipedia, the free encyclopedia Set of all Pareto efficient situations In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions.^[1] The concept is widely used in engineering.^[2]^ : 111-148 It allows the designer to restrict attention to the set of efficient choices, and to make tradeoffs within this set, rather than considering the full range of every parameter.^[3]^: 63-65 ^[4]^ : 399-412 [300px-Front_pareto]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. [256px-Pareto_Efficient_Frontier_102]A 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. Definition[edit] The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function f : X - R m {\displaystyle f:X\ rightarrow \mathbb {R} ^{m}} {\displaystyle f:X\rightarrow \mathbb {R} ^{m}}, where X is a compact set of feasible decisions in the metric space R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \ mathbb {R} ^{n}}, and Y is the feasible set of criterion vectors in R m {\displaystyle \mathbb {R} ^{m}} {\displaystyle \mathbb {R} ^{m}}, such that Y = { y [?] R m : y = f ( x ) , x [?] X } {\displaystyle Y=\{y\ in \mathbb {R} ^{m}:\;y=f(x),x\in X\;\}} {\displaystyle Y=\{y\in \ mathbb {R} ^{m}:\;y=f(x),x\in X\;\}}. We assume that the preferred directions of criteria values are known. A point y ' ' [?] R m {\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}} {\displaystyle y^{\prime \prime }\in \mathbb {R} ^{m}} is preferred to (strictly dominates) another point y ' [?] R m {\ displaystyle y^{\prime }\in \mathbb {R} ^{m}} {\displaystyle y^{\ prime }\in \mathbb {R} ^{m}}, written as y ' ' [?] y ' {\displaystyle y ^{\prime \prime }\succ y^{\prime }} {\displaystyle y^{\prime \prime } \succ y^{\prime }}. The Pareto frontier is thus written as: P ( Y ) = { y ' [?] Y : { y ' ' [?] Y : y ' ' [?] y ' , y ' [?] y ' ' } = [?] } . {\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 \}.} {\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 \}.} Marginal rate of substitution[edit] A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers.^[5] A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as z i = f i ( x i ) {\displaystyle z_{i}=f ^{i}(x^{i})} {\displaystyle z_{i}=f^{i}(x^{i})} where x i = ( x 1 i , x 2 i , ... , x n i ) {\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^{i})} {\displaystyle x^{i}=(x_{1}^{i},x_{2}^{i},\ldots ,x_{n}^ {i})} is the vector of goods, both for all i. The feasibility constraint is [?] i = 1 m x j i = b j {\displaystyle \sum _{i=1}^{m}x_ {j}^{i}=b_{j}} {\displaystyle \sum _{i=1}^{m}x_{j}^{i}=b_{j}} for j = 1 , ... , n {\displaystyle j=1,\ldots ,n} {\displaystyle j=1,\ldots ,n} . To find the Pareto optimal allocation, we maximize the Lagrangian: L i ( ( x j k ) k , j , ( l k ) k , ( m j ) j ) = f i ( x i ) + [?] k = 2 m l k ( z k - f k ( x k ) ) + [?] j = 1 n m j ( b j - [?] k = 1 m x j k ) {\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)} {\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)} where ( l k ) k {\displaystyle (\lambda _{k})_{k}} {\displaystyle (\ lambda _{k})_{k}} and ( m j ) j {\displaystyle (\mu _{j})_{j}} {\ displaystyle (\mu _{j})_{j}} are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good x j k {\displaystyle x_{j}^{k}} {\displaystyle x_{j}^{k}} for j = 1 , ... , n {\displaystyle j=1,\ldots ,n} {\displaystyle j=1,\ldots ,n} and k = 1 , ... , m {\displaystyle k=1,\ldots ,m} {\displaystyle k=1,\ldots ,m} gives the following system of first-order conditions: [?] L i [?] x j i = f x j i 1 - m j = 0 for j = 1 , ... , n , {\ displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{i}}}=f_{x_ {j}^{i}}^{1}-\mu _{j}=0{\text{ for }}j=1,\ldots ,n,} {\ displaystyle {\frac {\partial L_{i}}{\partial x_{j}^{i}}}=f_{x_ {j}^{i}}^{1}-\mu _{j}=0{\text{ for }}j=1,\ldots ,n,} [?] L i [?] x j k = - l k f x j k i - m j = 0 for k = 2 , ... , m and j = 1 , ... , n , {\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,} {\ 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,} where f x j i {\displaystyle f_{x_{j}^{i}}} {\displaystyle f_{x_{j}^ {i}}} denotes the partial derivative of f {\displaystyle f} {\ displaystyle f} with respect to x j i {\displaystyle x_{j}^{i}} {\ displaystyle x_{j}^{i}}. Now, fix any k [?] i {\displaystyle k\neq i} {\displaystyle k\neq i} and j , s [?] { 1 , ... , n } {\displaystyle j,s\ in \{1,\ldots ,n\}} {\displaystyle j,s\in \{1,\ldots ,n\}}. The above first-order condition imply that f x j i i f x s i i = m j m s = f x j k k f x s k k . {\ 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}}}.} {\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}}}.} Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.^[citation needed] Computation[edit] Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering.^[6] They include: * "The maxima of a point set" * "The maximum vector problem" or the skyline query^[7]^[8]^[9] * "The scalarization algorithm" or the method of weighted sums^[10] ^[11] * "The [?] {\displaystyle \epsilon } {\displaystyle \epsilon } -constraints method"^[12]^[13] * Multi-objective Evolutionary Algorithms Approximations[edit] Since generating the entire Pareto front is often computationally-hard, there are algorithms for computing an approximate Pareto-front. For example, Legriel et al.^[14] call a set S an e-approximation of the Pareto-front P, if the directed Hausdorff distance between S and P is at most e. They observe that an e -approximation of any Pareto front P in d dimensions can be found using (1/e)^d queries. Zitzler, Knowles and Thiele^[15] compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity. References[edit] 1. ^ proximedia. "Pareto Front". www.cenaero.be. Retrieved 2018-10-08. 2. ^ Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., Introduction to Optimization Analysis in Hydrosystem Engineering (Berlin/ Heidelberg: Springer, 2014), pp. 111-148. 3. ^ Jahan, A., Edwards, K. L., & Bahraminasab, M., Multi-criteria Decision Analysis, 2nd ed. (Amsterdam: Elsevier, 2013), pp. 63-65 . 4. ^ Costa, N. R., & Lourenco, 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. 5. ^ 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 1-84542-157-4. OCLC 58538348. 6. ^ Tomoiaga, Bogdan; Chindris, 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". Energies. 6 (3): 1439-55. doi:10.3390/en6031439. hdl:2117/18257. 7. ^ Nielsen, Frank (1996). "Output-sensitive peeling of convex and maximal layers". Information Processing Letters. 59 (5): 255-9. CiteSeerX 10.1.1.259.1042. doi:10.1016/0020-0190(96)00116-0. 8. ^ Kung, H. T.; Luccio, F.; Preparata, F.P. (1975). "On finding the maxima of a set of vectors". Journal of the ACM. 22 (4): 469-76. doi:10.1145/321906.321910. S2CID 2698043. 9. ^ Godfrey, P.; Shipley, R.; Gryz, J. (2006). "Algorithms and Analyses for Maximal Vector Computation". VLDB Journal. 16: 5-28. CiteSeerX 10.1.1.73.6344. doi:10.1007/s00778-006-0029-7. S2CID 7374749. 10. ^ 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:10.1007/s00158-005-0557-6. ISSN 1615-147X. S2CID 18237050. 11. ^ 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:10.1007/s00158-009-0460-7. ISSN 1615-147X. S2CID 122325484. 12. ^ "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: 10.1109/TSMC.1971.4308298. ISSN 0018-9472. 13. ^ Mavrotas, George (2009). "Effective implementation of the e-constraint method in Multi-Objective Mathematical Programming problems". Applied Mathematics and Computation. 213 (2): 455-465. doi:10.1016/j.amc.2009.03.037. ISSN 0096-3003. 14. ^ Legriel, Julien; Le Guernic, Colas; Cotton, Scott; Maler, Oded (2010). Esparza, Javier; Majumdar, Rupak (eds.). "Approximating the Pareto Front of Multi-criteria Optimization Problems". Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science. 6015. Berlin, Heidelberg: Springer: 69-83. doi:10.1007/978-3-642-12002-2_6. ISBN 978-3-642-12002-2. 15. ^ Zitzler, Eckart; Knowles, Joshua; Thiele, Lothar (2008), Branke, Jurgen; Deb, Kalyanmoy; Miettinen, Kaisa; Slowinski, Roman (eds.), "Quality Assessment of Pareto Set Approximations", Multiobjective Optimization: Interactive and Evolutionary Approaches, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer, pp. 373-404, doi:10.1007/ 978-3-540-88908-3_14, ISBN 978-3-540-88908-3, retrieved 2021-10-08 * Retrieved from "https://en.wikipedia.org/w/index.php?title= Pareto_front&oldid=1184190980" Categories: * Power engineering * Pareto efficiency Hidden categories: * CS1: long volume value * Articles with short description * Short description is different from Wikidata * All articles with unsourced statements * Articles with unsourced statements from July 2020 * This page was last edited on 8 November 2023, at 22:14 (UTC). * Text is available under the Creative Commons Attribution-ShareAlike License 4.0 ; additional terms may apply. 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