A cement company has 3 (three) factories in different locations, namely in A, B, and C. Production costs in A are $600,000 per ton, in B $400,000 per ton and in C $500,000,-per ton. Each factory currently only produces 100,000 tons per day according to market demand, even though the actual installed capacity of the three factories is capable of producing 120,000 tons per day. The company sends production from the three factories to 3 (three) distributor locations, namely X, Y, and Z. The demand for Distributor X is 100,045 tons per day, Distributor Y is 120,045 tons per day, and Distributor Z is 80,045 tons per day. The company wants to use warehouses located at J and K with a storage capacity of 125,000 tons and 175,000 tons per day respectively and shipping costs (in thousands of dollar) as shown in the table below. The value "M" means it is not possible to make a delivery.

The goal of the company is to minimize total costs. Therefore, help the company with:

a. Formulate this problem by drawing a conceptual model with a network model

b. Defining the decision variables, modeling the objective function mathematically and modeling the constraints

c. Find a solution (allocation from the factory to the buffer warehouse and allocation from the buffer warehouse to the distributor) of the problems above using the solver including the total cost!

\begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Warehouse } \\ \hline Factory & J & K \\ \hline A & 75 & M \\ \hline B & M & 85 \\ \hline C & 60 & 90 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|} \hline & \multicolumn{3}{|c|}{ Distributor } \\ \hline Warehouse & X & Y & Z \\ \hline J & M & 80 & 125 \\ \hline K & 80 & 90 & M \\ \hline \end{tabular}