An enterprise consisting of three coal mines transport coal to four different municipal power plants. Suppose the mines, which we label as Mine1, 2, and 3, have respective annual productions of 500,000; 500,000; and 400,000 tons. The four municipalities, Cities1, 2, 3, and 4, have corresponding annual demands of 400,000; 300,000; 500,000; and 200,000 tons. To each combination of a mine and city, there is a cost per unit amount associated with transporting coal from the mine to the city. Table 3 summarizes costs, in units of millions of dollars per hundred thousand tons, for each different mine-city combination. We need to construct an LP whose objective is to minimize the total cost of transporting the coal from the mines to the cities. a) Build the mathematical model of the problem. b) Solve it with the help of any computer software, e.g. Maple, and interpret the numbers you will find in the considered-story context (do not present any detailed transformations and/or calculations, just the solution) Remark: Because the total coal amount produced by the mines equals the combined demand of the cities, this LP constitutes a balanced transportation problem.

Consider the previous problem above in its unbalanced form. For this purpose, assume that the annual productions are the following: 550 000; 600 000; and 450 000 tons. Additionally take into account the mines' storage costs that amount, respectively, to 0.25, 0.35, 0.15 in units of millions of dollars per hundred thousand tons. So now we need to construct an LP model whose objective is to minimize the sum of the total cost of transporting the coal from the mines to the cities plus the storage cost for the coal that eventually remains in the mines. a) Build the mathematical model of the problem. b) Solve it with the help of any computer software, e.g. Maple, and interpret the numbers you will find in the considered-story context (do not present any detailed transformations and/or calculations, just the solution)