Question

Super Products Inc. has three plants, located at Aville, Bville, and Cville, producing shipments of its new wonder trouser-press which have to be shipped to four retail centers. The plants at Aville, Bville, and Cville produce 12, 17, and 11 shipments of trouser-presses per week, respectively. Each retail center needs to receive 10 shipments per week. The distance from each plant to each retail center in kilometers is as follows:

                                                             Retail Center

                                               1              2              3              4

                          Aville         800         1300         400           700

Plant                  Bville       1100         1400         600         1000

                          Cville         600         1200         800          900

Shipping costs per shipment are $100 plus 25 cents per kilometer. Super Products Inc. wishes to design an optimal shipping plan to minimize costs.

i) Set up an appropriate transportation problem

ii) Determine an initial basic feasible solution using the Vogel Method and then solve the problem using the transportation simplex. Be sure to compute not only the required pattern of shipments but also the actual minimum cost of making them.

iii) How would your solution change if the cost of shipments between Bville and Retail Center 3 became prohibitively expensive? Explain.

Answer 1

(i)

Note that cost is computed as $100 + 0.25*Miles e.g. for 800 miles, Cost = $100 + 0.25*800 = $300.

(ii)

Application of VAM

Transportation Simplex method

Assign u_i for each i-th row and v_j for each j-th column.
Start with u₁=0 and compute the values of other u_i's and v_j's using the formula u_i+v_j=c_ij for the basic cells only. c_ij be the cost figures.
Compute w_ij = u_i + v_j - c_ij for the non-basic cells.
Perform loop pivoting for w_ij having the highest positive value.
If all w_ij's are less than or equal to zero, then the solution is an optimal solution.

Note that at the end of the second iteration itself, all the w_ij's are less than or equal to zero. Therefore, this is the optimal tableau.

Optimal solution

(iii)

The initial basic feasible solution will become optimal if the Bville - RC3 pair needs to remain non-basic. Since the cost c_ij will be too high, w₂₃ will always be negative and the tableau will be optimal.