The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows: Plant Number Proposed Plant Annual Fixed Cost Annual Capacity 1 Detroit $150,000 10,000 2 Toledo $275,000 20,000 3 Denver $400,000 30,000 4 Kansas City $475,000 40,000 The company's long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows. Distribution Center Number Distribution Center Annual Demand 1 Boston 20,000 2 Atlanta 30,000 3 Houston 30,000 The shipping cost per unit from each plant to each distribution center is as follows. Distribution Centers Plant Site Boston Atlanta Houston Detroit 5 2 3 Toledo 4 3 4 Denver 9 7 5 Kansas City 10 4 2 St. Louis 8 4 3 (a) Formulate a mixed-integer programming model that could be used to help Martin-Beck determine which new plant or plants to open in order to satisfy anticipated demand and minimize total cost (in thousands of dollars). (Let Xij units shipped in thousands from plant i to distribution center j with the existing plant in St. Louis being plant number 5. Let y1 = 1 if a plant is constructed in Detroit and O if = 1 if a plant is constructed in Toledo and 0 if not, Y3 = 1 if a plant is constructed in Denver and 0 if not, and Y4 = 1 if a plant is constructed in Kansas City and 0 if not.) not, Y2 Min s.t. Detroit Capacity Toledo Capacity Denver Capacity Kansas City Capacity St. Louis Capacity Boston Demand Atlanta Demand Houston Demand Xij 20 for all i and j and y11 Y 21 Y3. Y 4 binary (b) Solve the model you formulated in part (a). What is the optimal cost (in $)? $ What is the optimal set of plants to open? (Select all that apply.) Detroit Toledo Denver Kansas City (c) Using equation (13.1), (Sum of variables in the set 0) (sum of variables in the set z) = (number of variables in the set o) - 1 where O is the set of binary variables in our original optimal solution set to one and Z is the set of those set to zero, find a second-best solution. What is the increase in cost (in $) versus the best solution from part (b)? $ The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows: Plant Number Proposed Plant Annual Fixed Cost Annual Capacity 1 Detroit $150,000 10,000 2 Toledo $275,000 20,000 3 Denver $400,000 30,000 4 Kansas City $475,000 40,000 The company's long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows. Distribution Center Number Distribution Center Annual Demand 1 Boston 20,000 2 Atlanta 30,000 3 Houston 30,000 The shipping cost per unit from each plant to each distribution center is as follows. Distribution Centers Plant Site Boston Atlanta Houston Detroit 5 2 3 Toledo 4 3 4 Denver 9 7 5 Kansas City 10 4 2 St. Louis 8 4 3 (a) Formulate a mixed-integer programming model that could be used to help Martin-Beck determine which new plant or plants to open in order to satisfy anticipated demand and minimize total cost (in thousands of dollars). (Let Xij units shipped in thousands from plant i to distribution center j with the existing plant in St. Louis being plant number 5. Let y1 = 1 if a plant is constructed in Detroit and O if = 1 if a plant is constructed in Toledo and 0 if not, Y3 = 1 if a plant is constructed in Denver and 0 if not, and Y4 = 1 if a plant is constructed in Kansas City and 0 if not.) not, Y2 Min s.t. Detroit Capacity Toledo Capacity Denver Capacity Kansas City Capacity St. Louis Capacity Boston Demand Atlanta Demand Houston Demand Xij 20 for all i and j and y11 Y 21 Y3. Y 4 binary (b) Solve the model you formulated in part (a). What is the optimal cost (in $)? $ What is the optimal set of plants to open? (Select all that apply.) Detroit Toledo Denver Kansas City (c) Using equation (13.1), (Sum of variables in the set 0) (sum of variables in the set z) = (number of variables in the set o) - 1 where O is the set of binary variables in our original optimal solution set to one and Z is the set of those set to zero, find a second-best solution. What is the increase in cost (in $) versus the best solution from part (b)? $
