B from your optimal solution in 2a what is the total

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(b) From your optimal solution in 2a, what is the total amount of waste produced (in inches of paper rolls)? (c) If you relax the integrality restriction and solve this problem as a Linear Programming problem, you may obtain dual values for each order width. In fact for this particular problem, the dual values for the order widths are (you can verify this): Width Dual value 20” 0.0 40” 0.5 50” 1.0 55” 0.0 75” 1.0 It is possible to interpret the dual value as the saving in wide rolls that you might achieve for each extra narrow roll that you obtain; for example, a value of 0.5 for a 40” roll indicates that you save 5 wide rolls if you could obtain 10 extra 40” rolls. (You will need to think about this.) It is natural to ask: Of all the patterns of narrow rolls that fit within one wide roll, which has the greatest total dual value? Formulate this problem as a knapsack problem. Show that the maximum-value pattern is not one of the ones given, and that adding it allows you to get a better solution.
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3. (20 pts) Despite the bad economy, a new real estate development project is underway in Arizona. The project is to develop a new retirement community, complete with shuffle board games and bingo parlors. The new community has been divided into five tracts. Two fire stations are to be located, with no more than one fire station in any particular tract. Each fire station is to respond to all fires that occur in the tract in which the station is located as well as in the other tracts that are assigned to it. The decisions to be made consist of (1) which two tracts will receive a fire station, and (2) the assignment of each of the other tracts to one of the fire stations. The objective is to minimize the overall average of the response times to fires.
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