Quality design by multiobjective analysis
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| Title: | Quality design by multiobjective analysis |
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| Authors: | Vörös, József, Szidarovszky, Ferenc |
| Publisher Information: | De Gruyter (Sciendo), Warsaw; Corvinus University of Budapest, Department of Mathematics ; University of Siena, Department of Mathematics; SAAS Ltd., Budapest |
| Subject Terms: | multiobjective programming problem, scalarization, Applications of mathematical programming, Multi-objective and goal programming, \(\ell_p\)-metric |
| Description: | For the classical multiobjective programming problem \[ \begin{gathered} \text{minimize}\quad | f_j(x_1,\dots,x_n)-q_j| \quad (j=1,\dots,J)\\ \text{s.t.}\quad g_i(x_1,\dots,x_n)=0\quad (i=1,\dots,I) \end{gathered} \] the authors discuss some simple scalarization models. Using different \(\ell_p\)-metrics the problem is transferred to the form \[ \begin{gathered} \text{minimize}\quad \sum_j \alpha_j| f_j(x_1,\dots,x_n) -q_j| ^p\\ \text{s.t.}\quad g_i(x_1,\dots,x_n)=0\quad (i=1,\dots,I) \end{gathered} \] which becomes \[ \begin{gathered} \text{minimize} \max_j\{\alpha_j| f_j(x_1,\dots,x_n) -q_j| \} \\ \text{s.t.} g_i(x_1,\dots,x_n)=0\quad (i=1,\dots,I) \end{gathered} \] in case of \(p=\infty\). Assuming linear functions \(f_j\) and \(g_i\) it is pointed out that in case of \(p=1\) and \(p=\infty\) the problem can be rewritten as linear programming problem and in case of \(p=2\) as quadratic programming problem which can be solved by standard algorithms. |
| Document Type: | Article |
| File Description: | application/xml |
| Access URL: | https://zbmath.org/1244278 |
| Accession Number: | edsair.c2b0b933574d..2e881ed5655496eb2efdd3544a8e90d5 |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | For the classical multiobjective programming problem \[ \begin{gathered} \text{minimize}\quad | f_j(x_1,\dots,x_n)-q_j| \quad (j=1,\dots,J)\\ \text{s.t.}\quad g_i(x_1,\dots,x_n)=0\quad (i=1,\dots,I) \end{gathered} \] the authors discuss some simple scalarization models. Using different \(\ell_p\)-metrics the problem is transferred to the form \[ \begin{gathered} \text{minimize}\quad \sum_j \alpha_j| f_j(x_1,\dots,x_n) -q_j| ^p\\ \text{s.t.}\quad g_i(x_1,\dots,x_n)=0\quad (i=1,\dots,I) \end{gathered} \] which becomes \[ \begin{gathered} \text{minimize} \max_j\{\alpha_j| f_j(x_1,\dots,x_n) -q_j| \} \\ \text{s.t.} g_i(x_1,\dots,x_n)=0\quad (i=1,\dots,I) \end{gathered} \] in case of \(p=\infty\). Assuming linear functions \(f_j\) and \(g_i\) it is pointed out that in case of \(p=1\) and \(p=\infty\) the problem can be rewritten as linear programming problem and in case of \(p=2\) as quadratic programming problem which can be solved by standard algorithms. |
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