Question

The parameter table given below shows the transportation problem formulation of Option 1 for the Better Products Co. problem presented in Sec. 9.3 of the textbook. As stated in the textbook, the optimal solution for this transportation problem has the following basic variables (allocations):

x12 = 30, x13 = 30, x15 = 15, x24 = 15, x25 = 60, x31 = 20, x34 = 25

Verify that this optimal solution actually is optimal by applying just the optimality test portion of the transportation simplex method to this solution.

Cost per Unit Distributed Destination (Product) 4 5(D) 4 5(D) Supply 41 2728 24 40 29 M 23 37 30 27 21 20 75 75 45 Source Demand 30 30 40 75

Answer 1

For basic solutions, ui + vj = cij ----(1)
For non-basic solutions, ui + vj - cij = wij ----(2)
For optimality to be realized all wij's has to be <= 0

Start with u1=0. From expression (1), calculate all the ui's and vj's. Then using (2), find the wij's in the non-basic cells.

u1+v2=27 => v2=27
u1+v3=28 => v3=28
u1+v5=0 => v5=0
u2+v5=0 => u2=0
u2+v4=23 => v4=23
u3+v4=21 => u3= -2
u3+v1=37 => v1=39

V4 39 27 28 23 0 u1- 0 w11-2 30 30 w14 -1 15 u2 0 w21- -1 w22 W23--Mİ 15 $37 21 u3- -2 20 w32- -5 w33 25 w35 -2

We find that all the values of wij are less than zero. So, the optimality has been reached.