The following table gives the average response time

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The following table gives the average response time (in minutes) to a fire in each tract (the columns) if that tract is served by a station in a given tract (the rows). The bottom row gives the forecasted average number of fires per day that will occur in each of the tracts. Assigned Station Located in Tract Response Times to Fire in Tract 1 2 3 4 5 1 5 12 30 20 15 2 20 4 15 10 25 3 15 20 6 15 12 4 25 15 25 4 10 5 10 25 15 12 5 Average frequency of fires 2 1 3 1 3 Formulate and solve a binary integer programming model for this problem. Which two tracts receive stations and which fire stations serve the other three tracts? 4. (20 pts) Solve the following problem using the branch-and-bound methodology (by hand). Show the branch-and-bound tree and the sub-problems at each node of the tree. When solving the LP relaxation at each node, you may use whatever technique that you like, e.g. solve using AMPL, or the problem may be simple enough to deduce the answer. maximize 2 x 1 + 5 x 2 subject to 10 x 1 + 30 x 2 30 95 x 1 - 30 x 2 75 x 1 ,x 2 ∈ { 0 , 1 } 5. (20 pts) Three tanks fight a three-way duel. Tank A has probability 1/2 of destroying the tank at which it fires, tank B has probability 1/3 of destroying the tank at which it fires, and tank C has probability 1/6 of destroying the tank at which it fires. The tanks fire together and each tank fires at the strongest opponent not yet destroyed. Form a Markov chain by taking as the state space the set of remaining tanks. Fill in the one-step transition matrix. References [1] Robert Fourer, David M. Gay, and Brian W. Kernighan. AMPL: A Modeling Language for Mathematical Programming . Brooks/Cole Publishing Company, second edition, 2003. [2] Charles M. Grinstead and J. Laurie Snell. Introduction to Probability . American Mathematical Society, 1997.
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