Pareto front explained
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] 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] [4]
Definition
The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function
, where
X is a
compact set of feasible decisions in the
metric space
, and
Y is the feasible set of criterion vectors in
, such that
Y=\{y\inRm: y=f(x),x\inX \}
.
We assume that the preferred directions of criteria values are known. A point
is preferred to (strictly dominates) another point
, written as
y\prime\prime\succy\prime
. The Pareto frontier is thus written as:
P(Y)=\{y\prime\inY: \{y\prime\prime\inY: y\prime\prime\succy\prime,y\prime ≠ y\prime\prime \}=\empty\}.
Marginal rate of substitution
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
where
is the vector of goods, both for all
i. The feasibility constraint is
for
. To find the Pareto optimal allocation, we maximize the
Lagrangian:
Li((x
,(λk)k,(\muj)
λk(zk-fk(x
\muj\left(bj-\sum
\right)
where
and
are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good
for
and
gives the following system of first-order conditions:
=
| 1-\mu |
f | |
| j=0forj=1,\ldots,n, |
=-λk
fork=2,\ldots,mandj=1,\ldots,n,
where
denotes the partial derivative of
with respect to
. Now, fix any
and
. The above first-order condition imply that
Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.
Computation
Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering.[6] They include:
-constraints method"
[12] [13] - Multi-objective Evolutionary Algorithms
Approximations
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 ε-approximation of the Pareto-front P, if the directed 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 compare several algorithms for Pareto-set approximations on various criteria, such as invariance to scaling, monotonicity, and computational complexity.
References
- Web site: proximedia. Pareto Front. 2018-10-08. www.cenaero.be. https://web.archive.org/web/20200226003108/https://www.cenaero.be/Page.asp?docid=27103&. 2020-02-26. dead.
- Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., Introduction to Optimization Analysis in Hydrosystem Engineering (Berlin/Heidelberg: Springer, 2014), pp. 111–148.
- Jahan, A., Edwards, K. L., & Bahraminasab, M., Multi-criteria Decision Analysis, 2nd ed. (Amsterdam: Elsevier, 2013), pp. 63–65.
- 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.
- Book: Just, Richard E.. The welfare economics of public policy : a practical approach to project and policy evaluation. E. Elgar. Hueth, Darrell L., Schmitz, Andrew.. 2004. 1-84542-157-4. Cheltenham, UK. 18–21. 58538348.
- 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. Energies. 6. 3. 1439–55. 10.3390/en6031439. free. 2117/18257. free.
- Nielsen. Frank. 1996. Output-sensitive peeling of convex and maximal layers. Information Processing Letters. 59. 5. 255–9. 10.1.1.259.1042. 10.1016/0020-0190(96)00116-0.
- 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. 10.1145/321906.321910. 2698043. free.
- Godfrey. P.. Shipley. R.. Gryz. J.. 2006. Algorithms and Analyses for Maximal Vector Computation. VLDB Journal. 16. 5–28. 10.1.1.73.6344. 10.1007/s00778-006-0029-7. 7374749.
- 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. 10.1007/s00158-005-0557-6. 1615-147X. 18237050.
- 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. 10.1007/s00158-009-0460-7. 1615-147X. 122325484.
- 1971. 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. 10.1109/TSMC.1971.4308298. 0018-9472.
- Mavrotas. George. 2009. Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems. Applied Mathematics and Computation. 213. 2. 455–465. 10.1016/j.amc.2009.03.037. 0096-3003.
- Legriel. Julien. Le Guernic. Colas. Cotton. Scott. Maler. Oded. 2010. Esparza. Javier. Majumdar. Rupak. 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 . en. Berlin, Heidelberg. Springer. 69–83. 10.1007/978-3-642-12002-2_6. 978-3-642-12002-2. free.