Problem 11-13 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 shown in table below. SHIPPING COST PER UNIT FOR THE MARTIN-BECK SUPPLY CHAIN Boston Houston Plant Site Detroit Toledo Denver Kansas City St. Louis Distribution Centers Atlanta 2 3 7 4 A network representation of the potential Martin-Beck supply chain is shown in figure below. THE NETWORK REPRESENTATION OF THE MARTIN-BECK COMPANY SUPPLY CHAIN DESIGN PROBLEM Plants 10 Detroit Distribution Centers Boston 30 20 Toledo 4 9 20 Denver 7 5 20 Atlanta 10 14 40 Kons City Hosen 20 30 St. Louis 1 Demands Capacities Distributive Routes Each potential plant location is shown; capacities and demands are shown in thousands of units. This network representation is for a transportation problem with a plant at St. Louis and at all four proposed sites. However, the decision has not yet been made as to which new plant or plants will be constructed. b. Formulate a model that could be used for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that no more than two plants can be located in Denver, Kansas City, and St. Louis. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300) Let y1 = 1 if a plant is constructed in Detroit; 0 if not y2 = 1 if a plant is constructed in Toledo; o if not y3 = 1 if a plant is constructed in Denver; O if not y4 = 1 if a plant is constructed in Kansas City; o if not xij = the units shipped in thousands from plant i to distribution center i i= 1,2,3,4,5, and j = 1,2,3 Min *11+ 13+ X21+ 22+ *23 X311 Jx12 + *43+ *5 + X52 + X53 Yi Y2 s.t K11 + *12 + X13 Vi 21 + *22 + X23 31 + X32 + X33 441 + *42 X43 X51 + + 53 x51 *11 + M21 + 31 + bxa1 + K22 + X32 + X42 + x52 X13 + K23 + + 43 3) + *53 xij 2 for all i and j; y1 y2 31+ 32+ *331+ X4+ X42 2+ 3+ y4 2 74 Detriot capacity Toledo capacity Denver capacity Kansas City capacity St. Louis capacity Boston demand KRK S Atlanta demand Houston demand Y3 Y3 + y4
