Question 3 (Mathematical Modelling) (25 points)(+ Bonus 5 points) A) (10 points) In p-hub median problem, p hubs are opened, origin-destination (o-d) flows travel via hubs and interhub transportation is discounted by a factor that is between 0 and 1 to reflect transportation economies of scale. Problem has the following setup: (1) n demand locations (origin/destinations) are given, (2) the flow for the n2 location (o-d) pairs is given, (3) the per unit transportation cost between all location (o-d) pairs and a hub-to-hub transportation discount factor is given, and (4) the total amount of flow between hub k and m cannot exceed capacity Ckm. (5)The objective is to locate p hubs to minimize the total transportation cost of the flows and the fixed cost of setting up the hubs. Write down a mixed integer mathematical optimization model of p-hub median problem and use the following decision variable in your model: Xijkm : the amount of flow from customer i to j via hubs k and m Notice that X is not the percentage of flow or a 01 variable, it denotes the amount total flow from i to j using hubs k and m and it is between 0 and hij, where hij : denotes the total amount of demand of customer j from customer i Define other variables and parameters clearly (if necessary) (problem must be linear) If you use a big M constant, specify a valid upper bound for the associated parameter. A-1) Each hub can serve at most C amount of flow. (5 points) A-2) Each hub can serve at most L customers. (5 points) B) Write down the mathematical optimization model of the travelling salesman problem. Define variables and constraints carefully (5 points) C) Suppose we adopt the following extension to the p-median problem: (5 points) - If more than K customers are assigned to a facility then % discount is applied for each customer assigned to that facility. - Yij is 1 if customer i is assigned to facility j