Why basic feasible solution of a transportation problem always involves not more than $m+n-1$ allocations?












0












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I was reading Balanced transportation problem. I got stuck in the conditions of Basic Feasible solution.



If we suppose a transportation problem where $m$ number of supply points and $n$ number of demand points then why the basic feasible solution always involves not more than $m+n-1$ allocations.



My confusion --- If we consider this transportation problem as L.P.P problem then it will have $m+n$ equations and $mn$ variable. So we can get $m+n$ basic variable. i.e there can be $m+n$ allocation .



But it does not happen. Whenever we see there is $m+n$ allocations we can be confident that there is a loop.



Can anyone please help me to understand where I am missunderstanding?



Can anyone please help me ?






linear-algebra linear-programming







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  • $begingroup$
    did I answer your question?
    $endgroup$
    – LinAlg
    Feb 25 at 16:29
















0












$begingroup$


I was reading Balanced transportation problem. I got stuck in the conditions of Basic Feasible solution.



If we suppose a transportation problem where $m$ number of supply points and $n$ number of demand points then why the basic feasible solution always involves not more than $m+n-1$ allocations.



My confusion --- If we consider this transportation problem as L.P.P problem then it will have $m+n$ equations and $mn$ variable. So we can get $m+n$ basic variable. i.e there can be $m+n$ allocation .



But it does not happen. Whenever we see there is $m+n$ allocations we can be confident that there is a loop.



Can anyone please help me to understand where I am missunderstanding?



Can anyone please help me ?






linear-algebra linear-programming







share|cite|improve this question









$endgroup$












  • $begingroup$
    did I answer your question?
    $endgroup$
    – LinAlg
    Feb 25 at 16:29














0












0








0





$begingroup$


I was reading Balanced transportation problem. I got stuck in the conditions of Basic Feasible solution.



If we suppose a transportation problem where $m$ number of supply points and $n$ number of demand points then why the basic feasible solution always involves not more than $m+n-1$ allocations.



My confusion --- If we consider this transportation problem as L.P.P problem then it will have $m+n$ equations and $mn$ variable. So we can get $m+n$ basic variable. i.e there can be $m+n$ allocation .



But it does not happen. Whenever we see there is $m+n$ allocations we can be confident that there is a loop.



Can anyone please help me to understand where I am missunderstanding?



Can anyone please help me ?






linear-algebra linear-programming







share|cite|improve this question









$endgroup$




I was reading Balanced transportation problem. I got stuck in the conditions of Basic Feasible solution.



If we suppose a transportation problem where $m$ number of supply points and $n$ number of demand points then why the basic feasible solution always involves not more than $m+n-1$ allocations.



My confusion --- If we consider this transportation problem as L.P.P problem then it will have $m+n$ equations and $mn$ variable. So we can get $m+n$ basic variable. i.e there can be $m+n$ allocation .



But it does not happen. Whenever we see there is $m+n$ allocations we can be confident that there is a loop.



Can anyone please help me to understand where I am missunderstanding?



Can anyone please help me ?






linear-algebra linear-programming




linear-algebra linear-programming






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share|cite|improve this question











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asked Jan 31 at 16:45









cmicmi

1,141312




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  • $begingroup$
    did I answer your question?
    $endgroup$
    – LinAlg
    Feb 25 at 16:29


















  • $begingroup$
    did I answer your question?
    $endgroup$
    – LinAlg
    Feb 25 at 16:29
















$begingroup$
did I answer your question?
$endgroup$
– LinAlg
Feb 25 at 16:29




$begingroup$
did I answer your question?
$endgroup$
– LinAlg
Feb 25 at 16:29










1 Answer
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$begingroup$

The transportation problem only has $m+n-1$ independent constraints, see this answer. That means that there cannot be $m+n$ basic variables, but only $n+m-1$.






share|cite|improve this answer









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  • $begingroup$
    I think this question it's actually a duplicate of my question. Isn't it?
    $endgroup$
    – Al t.
    Feb 3 at 2:13






  • 1




    $begingroup$
    @Alt. yes, although you seemed more appreciative of the answer :)
    $endgroup$
    – LinAlg
    Feb 3 at 2:19












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

The transportation problem only has $m+n-1$ independent constraints, see this answer. That means that there cannot be $m+n$ basic variables, but only $n+m-1$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I think this question it's actually a duplicate of my question. Isn't it?
    $endgroup$
    – Al t.
    Feb 3 at 2:13






  • 1




    $begingroup$
    @Alt. yes, although you seemed more appreciative of the answer :)
    $endgroup$
    – LinAlg
    Feb 3 at 2:19
















1












$begingroup$

The transportation problem only has $m+n-1$ independent constraints, see this answer. That means that there cannot be $m+n$ basic variables, but only $n+m-1$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I think this question it's actually a duplicate of my question. Isn't it?
    $endgroup$
    – Al t.
    Feb 3 at 2:13






  • 1




    $begingroup$
    @Alt. yes, although you seemed more appreciative of the answer :)
    $endgroup$
    – LinAlg
    Feb 3 at 2:19














1












1








1





$begingroup$

The transportation problem only has $m+n-1$ independent constraints, see this answer. That means that there cannot be $m+n$ basic variables, but only $n+m-1$.






share|cite|improve this answer









$endgroup$



The transportation problem only has $m+n-1$ independent constraints, see this answer. That means that there cannot be $m+n$ basic variables, but only $n+m-1$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 31 at 16:51









LinAlgLinAlg

10.1k1521




10.1k1521












  • $begingroup$
    I think this question it's actually a duplicate of my question. Isn't it?
    $endgroup$
    – Al t.
    Feb 3 at 2:13






  • 1




    $begingroup$
    @Alt. yes, although you seemed more appreciative of the answer :)
    $endgroup$
    – LinAlg
    Feb 3 at 2:19


















  • $begingroup$
    I think this question it's actually a duplicate of my question. Isn't it?
    $endgroup$
    – Al t.
    Feb 3 at 2:13






  • 1




    $begingroup$
    @Alt. yes, although you seemed more appreciative of the answer :)
    $endgroup$
    – LinAlg
    Feb 3 at 2:19
















$begingroup$
I think this question it's actually a duplicate of my question. Isn't it?
$endgroup$
– Al t.
Feb 3 at 2:13




$begingroup$
I think this question it's actually a duplicate of my question. Isn't it?
$endgroup$
– Al t.
Feb 3 at 2:13




1




1




$begingroup$
@Alt. yes, although you seemed more appreciative of the answer :)
$endgroup$
– LinAlg
Feb 3 at 2:19




$begingroup$
@Alt. yes, although you seemed more appreciative of the answer :)
$endgroup$
– LinAlg
Feb 3 at 2:19


















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