# 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 11 1 1 (TP) min subject to , 1, , , , 0 mn ij ij ij n ij i i j m ij j j i ij Zc x xs i m u xd j nv x == = = = ∑∑ ¾ A necessary condition for (TP) to have feasible condition is “balance.” That is, s d = . ¾ If balance is violated, then an artificial balance could be established by adding “dummy” nodes. ¾ For example, if , s d > then adding a dummy destination, n +1, with demand 1 ni j ds 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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## This note was uploaded on 12/23/2010 for the course ENMG 500 taught by Professor Drbacelmaddah during the Fall '09 term at American University of Beirut.

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

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