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1. Better Fitness, Inc. (BFI), manufactures exercise equipment at its plant in Freeport, Long Island. It recently designed two universal weight machines for the home exercise market. Both machines use BFI-patented technology that provides the user with an extremely wide range of motion capability for each type of exercise performed. Until now, such capabilities have been available only on expensive weight machines used primarily by physical therapists. At a recent trade show, demonstrations of the machines resulted in significant dealer interest. In fact, the number of orders that BFI received at the trade show far exceeded its manufacturing capabilities for the current production period. As a result, management decided to begin production of the two machines. The two machines, which BFI named the BodyPlus 100 and the BodyPlus 200, require different amounts of resources to produce. The BodyPlus 100 consists of a frame unit, a press station, and a pec-dec station. Each frame produced uses 4 hours of machining and welding time and 2 hours of painting and finishing time. Each press station requires 2 hours of machining and welding time and 1 hour of painting and finishing time, and each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing, and packaging each BodyPlus 100. The raw material costs are $450 for each frame, $300 for each press station, and $250 for each pec-dec station; packaging costs are estimated to be $50 per unit. The BodyPlus 200 consists of a frame unit, a press station, a pec-dec station, and a leg-press station. Each frame produced uses 5 hours of machining and welding time and 4 hours of painting and finishing time. Each press station requires 3 hours of machining and welding time and 2 hours of painting and finishing time, each pec-dec station uses 2 hours of machining and welding time and 2 hours of painting and finishing time, and each leg-press station requires 2 hours of machining and welding time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling, testing, and packaging each BodyPlus 200. The raw material costs are $650 for each frame, $400 for each press station, $250 for each pec-dec station, and $200 for each leg-press station; packaging costs are estimated to be $75 per unit. For the next production period, management estimates that 600 hours of machining and welding time, 450 hours of painting and finishing time, and 140 hours of assembly, testing, and packaging time will be available. Current labor costs are $20 per hour for machining and welding time, $15 per hour for painting and finishing time, and $12 per hour for assembly, testing, and packaging time. The market in which the two machines must compete suggests a retail price of $2400 for the BodyPlus 100 and $3500 for the BodyPlus 200, although some flexibility may be available to BFI because of the unique capabilities of the new machines. Authorized BFI dealers can purchase machines for 70% of the suggested retail price. BFI's president believes that the unique capabilities of the BodyPlus 200 can help position BFI as one of the leaders in high-end exercise equipment. Consequently, he has stated that the number of units of the BodyPlus 200 produced must be at least 25% of the total production. Managerial Report (a) What is the recommended number of BodyPlus 100 and BodyPlus 200 machines to produce? (b) How does the requirement that the number of units of the BodyPlus 200 produced be at least 25% of the total production affect profits? (c) Where should efforts be expended in order to increase profits?

2. Cloud Services Capacity Planning. Galaxy Cloud Services operates several data centers across the United States containing servers that store and process the data on the Internet. Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in Michigan, Ohio, and California and two in New York. Management is considering increasing the capacity of these data centers to keep up with increasing demand. Each data center contains servers that are dedicated to Secure data and to Super Secure data. The cost to update each data center and the resulting increase in server capacity for each type of server are as follows: Data center Cost($millions) Secure Servers Super Secure Servers Michigan 2.5 50 30 New York 1 3.5 80 40 New York 2 3.5 40 80 Ohio 4 90 60 California 2 20 30 The projected needs are for a total increase in capacity of 90 Secure servers and 90 Super Secure servers. Management wants to determine which data centers to update to meet projected needs and, at the same time, minimize the total cost of the added capacity. In particular you have been asked to (a) Formulate a binary integer programming model that could be used to determine the optimal solution to the capacity increase question facing management. (b) Solve the model formulated in part (a) to provide a recommendation for management.

3. Hub Location at Western Airlines Western Airlines has decided that it wants to design a hub system in the United States. Each hub is used for connecting flights to and from cities within 1000 miles of the hub. Western run flights among the following cities: Atlanta, Boston, Chicago, Denver, Houston, Los Angeles, New Orleans, New York, Pittsburgh, Salt Lake City, San Francisco, and Seattle. The company wants to determine the smallest number of hubs it will need to cover all of these cities, where a city is "covered" if it is within 1000 miles of at least one hub. The following table lists the cities that are within 1000 miles of other cities. Cities Cities within 1000 Miles) Atlanta(AT) AT,CH,HO,NO,NY,PI Boston(BO) BO, NY, PI Chicago(CH) AT,CH,NY,NO,PI Denver(DE) DE, SL Houston(HO) AT, HO, NO Los Angeles(LA) LA,SL,SF New Orleans(NO) AT,CH,HO,NO New York(NY) AT, BO, CH, NY, PI Pittsburgh(PI) AT, BO, CH, NY, PI Salt Lake City (SL) DE LA, SL, SF, SE San Francisco (SF) LA, SL, SF, SE Seattle(SE) SL, SF, SE Formulate and solve a linear binary model to find the minimum number of hub locations that can cover all cities. Please name those cities location which will cover all cities. Note Since all the cities are binary, binary is the requirement for this problem. So you are supposed to use @BIN(AT); @BIN(BO); ....@BIN(SE);

4. Distribution Plan to Minimize Cost. Consider the following network representation of a transportation problem:

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