Question

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: Proposed Plant Annual Fixed Cost Annual Capacity Detroit $175,000 10,000 Toledo $300,000 20,000 Denver $375,000 30,000 Kansas City $500,000 40,000 The company’s long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows: Distribution Center Annual Demand Boston 30,000 Atlanta 20,000 Houston 20,000 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) Develop 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. Solve the model and answer the following questions. What is the optimal cost? $ 860000 What is the optimal set of plants to open? (b) Using equation 12.1, find a second-best solution. What is the optimal set of plants to open? What is the increase in cost versus the best solution from part (a)?

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: Annual Fixed Cost Annual Capacity Proposed Plant Detroit Toledo $175,000 10,000 $300,000 20,000 Denver $375,000 30,000 Kansas City $500,000 40,000 The company's long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows: Distribution Center Annual Demand Boston 30,000 Atlanta 20,000 Houston 20,000 The shipping cost per unit from each plant to each distribution center is as follows: Distribution Centers Plant Site Boston Atlanta Houston Detroit Toledo Denver Kansas City Is Louc

Answer 1

Min Z= 5x11+2x12+3x13+4x21+3x22+4x23+9x31+7x32+5x33+10x41+4x42+2x43+8x51+4x52+3x53+175000y¹+300000y²+375000y³+500000y⁴

y1=1 if a plant is constructed in Detroit, 0 if not

y2=1 if a plant is constructed in Toledo, 0 if not

y3=1 if a plant is constructed in Denver, 0 if not

y4=1 if a plant is constructed in Kansas City, 0 if not

Subject to:

x11+x12+x13<=10y¹

x21+x22+x23<=20y²

x31+x32+x33<=30y³

x41+x42+x43<=40y⁴

x51+x52+x53<=30

x11+x21+x31+x41+x51=30

x12+x22+x32+x42+x52=20

x13+x23+x33+x43+x53=20

Solution:

The best solution is to build a plant in Kansas City (y4=1), shipping 20,000 units from Kansas City to Atlanta, shipping 20,000 units from Kansas City to Houston and shipping 30,000 units from St. Louis.