Galaxy Cloud Services operates several data centers across the United States tha

Question

Galaxy Cloud Services operates several data centers across the United States that contain servers which 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 Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in Michigan, Ohio, and California and two in New York.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 is as follows: The projected needs are for a total increase in capacity of 80 Secure servers and 80 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. (a) Formulate a binary integer programming model that could be used to determine the optimal solution to the capacity increase question facing management. If required, round your answers to one decimal place Let x_1 ={1 if data center i is updated 0 if data center t is not updated (b) solve the model formulated in part (a) to provide a recommendation for management. optimal solution: Select your answer Total Cost: $ Millions

Explanation / Answer

The appropriate mathematical formulation of the given problem as LP model is

Min(total cost) = 3.5x1+3x2+3.5x3+3x4+3x5

As given in the question

Cost related to update data centres is given

also the required number of secure servers is 80 and that of super secure servers is also 80.Our constraints become as s. t.              40x1+80x2+40x3+90x4+40x5=80               ;     30x1+30x2+50x3+70x4+50x5=80

as number of both kind of servers to be updated is given in the problem. Also here x1, x2,x3,x4,x5={0,1} as defined in the binary integer programming in the given problem.

So the above 3 equations combined answer our first part.

PART b    To solve the model formed above to get a recommendation for management. We will solve the LP using Simplex method. Change the objective function Min(total cost=z{say}) to Maximize as ------    Max(z')=-3.5x1-3x2-3.5x3-3x4-3x5...   Find an initial basic feasible solution.      Since there are two constraints and 5 unknowns , for obtaining a solution , WE assign 0 value to any 3 of thexi's . We set x1=x2=x3=0, the basic solution corresponding to above substitution we get,     90x4+40x5=80      70x4+50x5=80. on solving for x4 and x5. we found both are positive hence the basic feasib;le solution is feasible and non degenerate.         We draw the first table of our simplex method

Where x4 ,x5 are basic variables and others are non basic. continuing solving the problem using simplex method, we will get our optimal solution

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