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2. (50 points) Three suppliers (A, B and C) have submitted bids for satisfying the flu vaccine re- quirements of four facilities (1, 2, 3 and 4) of Northwestern Medicine. The table below displays the vaccine requirements of each facility, the maximum amount each supplier can provide to all facilities, and the bid each company has made for supplying each facility. Supply managers of Northwestern Medicine would like to determine the amount of vaccine to purchase from each supplier to satisfy the requirement of each facility at minimum overall cost of vaccine acquisition. (a) (5 points) Represent this problem as a network flow problem using a graph. Clearly label the nodes and arcs of the graph. (b) (15 points) Write a linear programming formulation of this problem. Clearly describe your decision variables, objective function and constraints. 1 2 Company A Company B Company C Requirement Facility 1 500 450 550 45,000 Bids ($s per 1000 units) Facility 2 Facility 3 Facility 4 200 650 450 300 800 150 450 700 250 50,000 25,000 15,000 Max Supply 50,000 40,000 45,000 (c) (10 points) Company A is offering an increase of 5,000 units in the amount of maximum supply, if Northwestern agrees to pay a fixed cost $5,000. Describe how you would update your model in part (b) as a linear mixed-integer program to determine whether to accept Company A's offer and the amount of vaccine to purchase from each company to satisfy the requirement of each facility at minimum fixed and variable cost of vaccine acquisition. Clearly define any new decision variables you need and any changes to the objective function and constraints. (d) (10 points) Company B has specified for each facility a minimum acceptable contract of 10,000 units; that is, if company B is awarded a contract to supply a particular facility, then the size of the contract must be at least 10,000 units. Describe how you would update your model in part (c) as a linear mixed-integer program to account for this specification. Clearly define any new decision variables you need and any changes to the objective function and constraints. (e) (10 points) As a fierce competitor of Company B, Company C requires that if it supplies any amount to a facility, then that facility cannot receive supplies from Company B. Describe how you would update your model in part (d) as a linear mixed-integer program to account for this specification. Clearly define any new decision variables you need and any changes to the objective function and constraints. 2. (50 points) Three suppliers (A, B and C) have submitted bids for satisfying the flu vaccine re- quirements of four facilities (1, 2, 3 and 4) of Northwestern Medicine. The table below displays the vaccine requirements of each facility, the maximum amount each supplier can provide to all facilities, and the bid each company has made for supplying each facility. Supply managers of Northwestern Medicine would like to determine the amount of vaccine to purchase from each supplier to satisfy the requirement of each facility at minimum overall cost of vaccine acquisition. (a) (5 points) Represent this problem as a network flow problem using a graph. Clearly label the nodes and arcs of the graph. (b) (15 points) Write a linear programming formulation of this problem. Clearly describe your decision variables, objective function and constraints. 1 2 Company A Company B Company C Requirement Facility 1 500 450 550 45,000 Bids ($s per 1000 units) Facility 2 Facility 3 Facility 4 200 650 450 300 800 150 450 700 250 50,000 25,000 15,000 Max Supply 50,000 40,000 45,000 (c) (10 points) Company A is offering an increase of 5,000 units in the amount of maximum supply, if Northwestern agrees to pay a fixed cost $5,000. Describe how you would update your model in part (b) as a linear mixed-integer program to determine whether to accept Company A's offer and the amount of vaccine to purchase from each company to satisfy the requirement of each facility at minimum fixed and variable cost of vaccine acquisition. Clearly define any new decision variables you need and any changes to the objective function and constraints. (d) (10 points) Company B has specified for each facility a minimum acceptable contract of 10,000 units; that is, if company B is awarded a contract to supply a particular facility, then the size of the contract must be at least 10,000 units. Describe how you would update your model in part (c) as a linear mixed-integer program to account for this specification. Clearly define any new decision variables you need and any changes to the objective function and constraints. (e) (10 points) As a fierce competitor of Company B, Company C requires that if it supplies any amount to a facility, then that facility cannot receive supplies from Company B. Describe how you would update your model in part (d) as a linear mixed-integer program to account for this specification. Clearly define any new decision variables you need and any changes to the objective function and constraints