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Set_7_TP_b - Dr Maddah ENMG 500 Engineering Management I...

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1 Dr. Maddah ENMG 500 Engineering Management I 12/20/05 The Transportation Problem (2) Balance Condition and Number of Basic Variables ¾ Consider the LP formulation for the TP 1 1 1 1 (TP) min subject to , 1, , , 1, , 0 m n ij ij i j n ij i i j m ij j j i ij Z c x x s i m u x d j n v x = = = = = = = = = ∑∑ ¾ A necessary condition for (TP) to have feasible condition is “balance.” That is, 1 1 m n i j i j s d = = = . ¾ If balance is violated, then an artificial balance could be established by adding “dummy” nodes. ¾ For example, if 1 1 , m n i j i j s d = = > then adding a dummy destination, n +1, with demand 1 1 1 m n n i j i j d s d + = = = and c i , n +1 = 0, i = 1, …, m , restores balance.
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2 ¾ The balance condition implies that any of the m + n constraints can be removed without affecting the optimal solution. 1 ¾ This implies that the number of basic variables in m + n 1 . Duality Results ¾ Let u i , i = 1, …, m , and v j , j = 1, …, m , be the dual variables associated with supply and demand constraints respectively.
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