5-31. In the transportation model, one of the dual variables assumes an arbitrary value. Thismeans…

5-31. In the transportation model, one of the dual variables assumes an arbitrary value. Thismeans that for the same basic solution, the values of the associated dual variables arenot unique. The result appears to contradict the theory of linear programming, where thedual values are determined as the product of the vector of the objective coefficients forthe basic variables and the associated inverse basic matrix (see Method 2, Section 4.2.3).Show that for the transportation model, although the inverse basis is unique, the vectorof basic objective coefficients need not be so. Specifically, show that if cij is changed tocij + k for all i and j, where k is a constant, then the optimal values of xij will remain thesame. Hence, the use of an arbitrary value for a dual variable is implicitly equivalent toassuming that a specific constant k is added to all cij