Transportation problem Factories Customers Requirement for goods Production capacity... Minimum cost of transportation satisfying the demand of customers. aiai aiai bjbj bjbj i-th factory delivers to j-th customer at cost c ij a1a1 a2a2 anan bmbm b1b1 b2b2

Transportation tableau 7 x 11 3 x 12 4 x 13 u18u18 4 x 21 2 x 22 2 x 23 u26u26 2 x 31 1 x 32 5 x 33 u33u33 v14v14 v22v22 v33v33 cost c ij of delivering from ith factory to jth customer supply a i of ith factory demand b j of jth customer shadow customer’s “price” shadow factory “price” shadow prices are relative to some baseline amount transported

Transportation problem a1a1 b1b1 a2a2 b2b2 b3b3 a3a3 a4a4 b4b4 7 x 11 3 x 12 4 x 13 u18u18 4 x 21 2 x 22 2 x 23 u26u26 2 x 31 1 x 32 5 x 33 u33u33 v14v14 v22v22 v33v33 7 x 11 3 x 12 4 x 13 0 x 14 u18u18 4 x 21 2 x 22 2 x 23 0 x 24 u26u26 2 x 31 1 x 32 5 x 33 0 x 34 u33u33 v14v14 v22v22 v33v33 v48v48 7 x 11 3 x 12 4 x 13 0 x 14 u18u18 4 x 21 2 x 22 2 x 23 0 x 24 u26u26 2 x 31 1 x 32 5 x 33 0 x 34 u33u33 9 x 41 8 x x 43 0 x 44 u40u40 v14v14 v22v22 v33v33 v48v48

7 x 11 3 x 12 4 x 13 0 x 14 u18u18 4 x 21 2 x 22 2 x 23 0 x 24 u26u26 2 x 31 1 x 32 5 x 33 0 x 34 u33u33 v14v14 v22v22 v33v33 v48v48 Transportation Simplex Applying the Simplex method to the problem Basic solution – min-cost method Pivoting – shadow prices set u 1 = 0, then u i + v j =c ij – reduced cost pivot if u i + v j > c ij Finding a loop take the smaller of the two must mark exactly m + n – 1 = 6 cells cost = 3×7 + 3×4 + 2×0 + 1×2 + 2×1 + 6×0 = 37 z = 37

7 x 11 3 x 12 4 x 13 0 x 14 u18u18 4 x 21 2 x 22 2 x 23 0 x 24 u26u26 2 x 31 1 x 32 5 x 33 0 x 34 u33u33 v14v14 v22v22 v33v33 v48v48 Transportation Simplex Applying the Simplex method to the problem Basic solution – min-cost method Pivoting – shadow prices set u 1 = 0, then u i + v j =c ij – reduced cost pivot if u i + v j > c ij Finding a loop z = u i =0 and v j must sum up to c ij = 4v j =

7 x 11 3 x 12 4 x 13 0 x 14 u18u18 4 x 21 2 x 22 2 x 23 0 x 24 u26u26 2 x 31 1 x 32 5 x 33 0 x 34 u33u33 v14v14 v22v22 v33v33 v48v48 Transportation Simplex Applying the Simplex method to the problem Basic solution – min-cost method Pivoting – shadow prices set u 1 = 0, then u i + v j =c ij – reduced cost pivot if u i + v j > c ij Finding a loop z = calculate u i + v j > > > > ≤ ≤

7 x 11 3 x 12 4 x 13 0 x 14 u18u18 4 x 21 2 x 22 2 x 23 0 x 24 u26u26 2 x 31 1 x 32 5 x 33 0 x 34 u33u33 v14v14 v22v22 v33v33 v48v48 Transportation Simplex Applying the Simplex method to the problem Basic solution – min-cost method Pivoting – shadow prices set u 1 = 0, then u i + v j =c ij – reduced cost pivot if u i + v j > c ij Finding a loop New basis Δ2+Δ 11+Δ1+Δ 22-Δ2-Δ 33-Δ3-Δ 3 z = > > > > ≤ ≤ +Δ+Δ 6-Δ6-Δ Largest feasible Δ = z = 29