2. In the transportation model, one of the dual variables assumes an arbitrary value. This means that for the same basic solution, the values of the associated dual variables are not unique. The result appears to contradict the theory of linear programming, where the dual values are determined as the product of the vector of the objective coefficients for the 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 vector of basic objective coefficients need not be so. Specifically, show that if Cij is changed to cij + k for all i and j, where k is a constant, then the optimal values of Xij will remain the same. Hence, the use of an arbitrary value for a dual variable is implicitly equivalent to assuming that a specific constant k is added to all Cij"

L Solve the assignment models in Table 5.40.

(a) Solve by the Hungarian method.

(b) TORA Experiment. Express the problem as an LP and solve it with TORA.

(c) TORA Experiment. UseTORA to solve the problem as a transportation model.
226 Chapter 5 Transportation Model and Its Variants TABLE 5.40 Data for Problem 1 (0 (ii)
$3 $8 $2 $10 $3 $3 $9 $2 $3 $7 $8 $7 $2 $9 . $7 $6 $1 $5 $6 $6 $6 $4 $2 $7 $5 $9 $4 $7 $10 $3 $8 $4 $2 $3 $5 $2 $5 $4 $2 $1 $9 $10 $6 $9 $10 $9 $6 $2 $4 $5

(d) Solver Experiment. Modify Excel file solverEx5.3-l.xls to solve the problem.

(e) AMPL Experiment. Modify ampIEx5.3-1b.txt to solve the problem.