• # question_answer Minimize $z=\sum\limits_{j=1}^{n}{{}}\sum\limits_{i=1}^{m}{{{c}_{ij}}\,{{x}_{ij}}}$                 Subject to : $\sum\limits_{j=1}^{n}{{{x}_{ij}}\le {{a}_{i}},\ i=1,.......,m}$                                   $\sum\limits_{i=1}^{m}{{{x}_{ij}}={{b}_{j}},\ j=1,......,n}$                 is a (L.P.P.) with number of constraints                    [MP PET 1999] A)                 $m+n$               B)                 $m-n$ C)                 mn         D)                 $\frac{m}{n}$

Condition (i),                    $i=1,{{x}_{11}}+{{x}_{12}}+{{x}_{13}}+.....+{{x}_{1n}}$                                 $i=2,{{x}_{21}}+{{x}_{22}}+{{x}_{23}}+......+{{x}_{2n}}$                    $i=3,{{x}_{31}}+{{x}_{32}}+{{x}_{33}}+......+{{x}_{3n}}$                    ....................                    $i=m,{{x}_{m1}}+{{x}_{m2}}+{{x}_{m3}}+.....{{x}_{mn}}\to$constraints                    Condition (ii),                    $j=1,\,{{x}_{11}}+{{x}_{21}}+{{x}_{31}}+......+{{x}_{m1}}$                    $j=2,{{x}_{12}}+{{x}_{22}}+{{x}_{32}}+......+{{x}_{m1}}$                    ....................                    $j=n,{{x}_{1n}}+{{x}_{2n}}+{{x}_{3n}}+......+{{x}_{mn}}\to n$ constraints                                        \ Total constraints = $m+n$.