線性規劃的標準型與規範型 (Standard and Canonical Forms)

線性規劃的標準型與規範型 (Standard and Canonical Forms)

Form	Minimization Problem	Maximization Problem
Standard	mins.t.∑j=1ncjxj∑j=1naijxj=bi,i=1,⋯,mxj≥0,j=1,⋯,n$\begin{array}{ll}min& \sum _{j=1}^n{c}_j{x}_j\\ \text{s.t.}& \sum _{j=1}^n{a}_{ij}{x}_j=b_i,i=1,\cdots ,m\\ & x_j\ge 0,j=1,\cdots ,n\end{array}$	maxs.t.∑j=1ncjxj∑j=1naijxj=bi,i=1,⋯,mxj≥0,j=1,⋯,n$\begin{array}{ll}max& \sum _{j=1}^n{c}_j{x}_j\\ \text{s.t.}& \sum _{j=1}^n{a}_{ij}{x}_j=b_i,i=1,\cdots ,m\\ & x_j\ge 0,j=1,\cdots ,n\end{array}$
Canonical	mins.t.∑j=1ncjxj∑j=1naijxj≥bi,i=1,⋯,mxj≥0,j=1,⋯,n$\begin{array}{ll}min& \sum _{j=1}^n{c}_j{x}_j\\ \text{s.t.}& \sum _{j=1}^n{a}_{ij}{x}_j\ge b_i,i=1,\cdots ,m\\ & x_j\ge 0,j=1,\cdots ,n\end{array}$	maxs.t.∑j=1ncjxj∑j=1naijxj≤bi,i=1,⋯,mxj≥0,j=1,⋯,n$\begin{array}{ll}max& \sum _{j=1}^n{c}_j{x}_j\\ \text{s.t.}& \sum _{j=1}^n{a}_{ij}{x}_j\le b_i,i=1,\cdots ,m\\ & x_j\ge 0,j=1,\cdots ,n\end{array}$