[Mathematical Programming Formulation] A company has two plants (P), one distribution center (DC) and two warehouses (W/H). The following network graph shows how the company can ship products from plants to warehouses, production quantities, demands and unit shipping costs. Plants Distribution Center Warehouses Legend () ? 1 (iii) ? (v)? (xiii) ? (vi) ? (ix)? (xi)? j (xii) ? (vii) ? (x)? (viii) ? (ii)? 2 2 (iv)? Plant 1 (P1) Plant 2 (P2) Distribution Center (DC) Demand (units) Shipping cost/unit W/H 1 W/H2 DC Supply (units) 3 80 9 4 70 2 4 60 90 The company would like to formulate a minimum cost network flow model. Some further information is provided below: The company uses railway on direct routes (i.e., from a plant to a warehouse) and can send unlimited quantities on these direct routes. The company needs to use trucks on indirect routes (i.e., from a plant to the distribution center and then from the distribution center to a warehouse). Number of products that can be shipped from each plant to the distribution center by trucks is at most 50. Similarly, number of products that can be shipped from the distribution center to each warehouse by trucks is at most 50. (a) (18 points] Considering that you will formulate a minimum cost network flow model, put the necessary pa- rameter values on the network (shown with a question mark in () through (x) on the graph) and parame- ter explanations on the legend (shown with a question mark in (xi) through (xiii) on the figure). Remark: Explanations on the legend should NOT be symbols, they should be short explanations. (b) [3 points] Define the decision variables with all necessary information. (c) (4 points) Write the objective and objective function mathematically. (d) [11 points] Write the constraints mathematically