The dual problem to the primal equation form of the transportation problem can be stated as follows:
Here the (m + n) set of dual variables u i and v j are unrestricted (free) variables. Note that the primal has a redundant equation due to the equality of the total supply and demand. Thus, a feasible basis matrix to the transportation problem is of order (m + n − 1) × (m + n − 1). It can be shown any feasible basis matrix can be arranged into a triangular form. For a given basis, the simplex method requires that the corresponding dual constraints must hold at equality, that is, we must have u i + v j = c ij for all variables x ij in the basis. This (m + n − 1) × (m + n) set of dual equations can be reduced to an (m + n − 1) × (m + n − 1) system by arbitrarily setting one of the dual variables, say u 1= 0. This corresponds to removing, as a redundant constraint, the first equation of the transportation problem. The resulting dual square set of equations also has a triangular form that...
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© 2001 Kluwer Academic Publishers
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Gass, S.I., Harris, C.M. (2001). Transportation simplex (primal-dual) method . In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_1065
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DOI: https://doi.org/10.1007/1-4020-0611-X_1065
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