# Complementary slackness and the transportation problem

1. Aug 22, 2007

### catcherintherye

hello,I have been given the transportation problem (T)

defined by the cost matrix
$$\left(\begin{array}{ccccccc}5&3&9&3&8&2\\5&6&3&15&7&16\\9&20&10&22&17&25\\3&7&3&14&9&14\end{array}\right)$$

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

$$\left(\begin{array}{ccccccc}2&1&0&0&0&0\\0&7&6&0&0&0\\0&0&3&3&0&0\\0&0&0&1&6&2\end{array}\right)$$

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

...this i found to be

(T*)

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

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

$$\left(\begin{array}{ccccccc}0&0&0&1&0&2\\0&7&0&0&6&0\\2&0&4&0&0&0\\0&1&5&3&0&0\end{array}\right)$$

solves (T)

now since i have
$$x_i_j > 0$$

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 $$y_1$$ 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 $$y_1, ...y_4, z_1,.....,z_6$$ to be zero will lead to another dual vaiable in the system being negative. What am I missing
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 22, 2007

### EnumaElish

What is the interpretation or the units of the y's and the z's? Miles? Tons?

3. Aug 23, 2007

### catcherintherye

i don't think so, the variables $$y_i, z_j$$ are dual variables, i don't think they have units as such. The primal problem was to minimize cost subject to some constraints, more specifically,
$$minimize \sum_{i=1}^{4}\sum_{j=1}^{6}c_ij x_ij$$

subject to
$$x_ij \geq 0 \sum_{j=1}^{6}x_ij \leq p_i, i= 1,...,4 \sum_{i=1}^{4}x_ij \geq q_j, j=1,...,6$$

Last edited: Aug 23, 2007
4. Sep 13, 2007

### EnumaElish

How do you figure demand = supply? Is that part of the problem, or are you assuming it? Why can't demand < supply, or demand > supply?

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