Question 1 (Network design optimization). [20 points] ABC Limited Corp. has won contracts to supply a high-tech product to customers in Asia, North America, Europe, and South America. The company has determined three potential locations for building its manufacturing plants. The fixed costs of opening a plant at each one of the three potential locations, the variable production costs per unit and the production capacity at each potential plant are shown in the following table: According to the signed contracts, the demand from customers in Asia, North America, Europe, and South America are 13,10,15, and 7 units, respectively. The shipping costs (in $100 per unit) from each potential plant to each customer region are given in the following table: For notation consistency, use index i=1,2,3 to represent plant locations in Brazil, Philippines, and Mexico, respectively. Similarly, use index j=1,2,3,4 to respectively represent customer regions in Asia, North America, Europe, and South America. (1) Assume that all the demands must be satisfied, and the company needs to determine the optimal facility location decision and demand allocation plan in order to minimize the total production and shipping costs. Please help the firm formulate an optimization model (with linear objective function and constraints) to achieve the minimum total costs. For simplicity, assume that the shipping quantities from a plant to a demand region does not necessarily have to be an integer. In your model, you can either use the concrete numbers (e.g., use 50,000 in your model for fixed cost of the plant in Brazil), or alternatively, you can clearly define a variable to represent each parameter (e.g., define fi as the fixed cost of plant in site i ) and use them in your model