Problem 9-9 Galaxy Cloud Services operates several data centers across the Unite

Question

Problem 9-9

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 Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in Michigan, Ohio, and California and two in New York.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.

Data Center Cost ($ millions) Secure Servers Super Secure Servers Michigan 3.5 40 30 New York 1 3 80 30 New York 2 3.5 40 50 Ohio 3 90 70 California 3 40 50 (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. Min x1 + x2 + x3 + x4 + x5 s.t. x1 + x2 + x3 + x4 + x5 Secure x1 + x2 + x3 + x4 + x5 Super Secure x1, x2, x3, x4, x5 = 0,1

Explanation / Answer

Decision Variables

There are five data centers and two types of servers, therefore total number of decision variables representation needs double subscripts, one for center and other for type of server. Let us use Xij as decision variable representing ith type of server ( i=1 for secure and i=2 for super secure type ) at jth center ( j=1 for Michigan, j=2 for New York1, j=3 for NewYork2, j=4 for Ohio and j=5 for California.

As mentioned in the question, Xij are binary that is Xij = 1 if the server Xij selected for upgradation otherwise Xij = 0

Objective function

Given in the question as Minimization of total cost of Upgradation/ addition, therefore we define cost coefficients corresponding to decision variables as Cij representing upgradation of ith type server at jth center., therefore objective function is defined as :

Minimization of Double summation (sigma over i and sigma over j) Cij * Xij for which Cij are as follows:

C11 = C21 = 3.5,   C12=C22=3, C13=C23=3.5, C14=C24=3, C15=C25=3

Constraints:

Constraints are in terms total number of servers and super servers, given as 80 each to be upgraded, represented as Sigma X1j = 80 and Sigma X2j = 80 or in expanded form as X11+X12+X13+X14+X15 = 80, X21+X22+X23+X24+X25 = 80

Second type of constraints are about number of each type of servers at each center as follows:

X11 <= 40, X21 <= 30 (Michigan) X12 <=80, X22 <= 30 (Newyork1), X13 <= 40, X23 <= 50 (Newyork2)

X14 <= 40, X24 <= 70 (Ohio) X15 <= 40, X25 <= 50 (California)

This completes the formulation of the problem.

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