Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complementary slackness and the transportation problem

  1. Aug 20, 2007 #1
    hello,I have been given the transportation problem (T)

    defined by the cost matrix

    the demand vector q=(2,8,9,4,6,2)

    the supply vector p=(3,13,6,9)

    the problem is as follows

    a) use the northwest vertex rule to find a basic feasible solution of (T)

    ...this i found to be


    (b) write down the linear program for the dual program (T*)

    ...this i found to be


    maximize \sum_{i=1}^{4}p_i y_i + \sum_{j=1}^{6}q_j z_j

    subject to
    \\ y_i + z_j \leq c_ij, y_1,....,y_4,z_1,.....z_6,

    (c) use the complementary slackness conditions to show that the shipping matrix


    solves (T)

    now since i have
    [tex] x_i_j > 0 [/tex]

    for (i,j) = (1,4), (1,6), (2,2), (2,5), (3,1), (3,3), (4,2), (4,3), (4,4) complementary slackness implies that the associated dual constraints are binding and we have a system of nine equations in 10 unknowns. Now here is the rub. I would usually progress by substituting the above values (from the shipping matrix) into the primal constraints and derive that one of them is non binding, then deduce that the associated dual variable is zero by complementary slackness. However since we have here supply=demand, no such constraint will occur. I have done a similar problem where the next progression was to assume that one of the dual variables say [tex] y_1 [/tex] is 0 and then the system of nine equalities from above are solvable, however if as an example i take y_1 =0, this implies that z_3 =-8 which is infeasible. Infact as it turned out assuming any dual variable [tex] y_1, ...y_4, z_1,.....,z_6 [/tex] to be zero will lead to another dual vaiable in the system being negative. What am I missing:confused:
  2. jcsd
  3. Aug 21, 2007 #2
    btw, i really need a prompt answer if at all possible, this is for my referal which has to be in next week!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Complementary slackness transportation Date
Finding self complementary graphs Jan 29, 2012
Testing optimality via complementary slackness Jul 25, 2011
Equivalence relation in complementary subspace Sep 11, 2008
Slack variables Aug 1, 2007