Midterm Review AOS

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University of Pittsburgh *

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1090

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Electrical Engineering

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Dec 6, 2023

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#aos#prep23 10 questions on midterm

The Rappaport Communications Company provides cellular telephone services in several mid-western states. The company is planning to expand its customer base by offering cellular service in northeastern Ohio to the cities of Cleveland, Akron, Canton, and Youngstown. The company will install the hardware necessary to service customers in each city on pre existing communications towers in each city. The locations of these towers are summarized below: The company also needs to construct a new communications tower some-where between these cities to handle intercity calls/ This tower will also allow cellular calls to be routed onto the satellite system for worldwide calling service. The tower the company is planning to build can cover areas within a 40-mile radius. Thus, the tower needs to be located within 40 miles of each of these cities. Where should we build the communications tower to minimize the sum of the distances to the given cities? Formulate the problem Tower location -> x,y cord Constr. Within 40 miles from each city Obj. min sum of distance , Tower C1, C2, C3 D.V x,y -> coordinates of tower Assume Cleveland -> xc, yc Akron -> xa, ya Canton -> xca, yca youngstown , xy, yy Obj. min distance

Min [sqrt ((x-xc) ^2 + (y-yc)^2) sqrt ((x-xa) ^2 + (y-ya)^2) sqrt ((x-xca) ^2 + (y-yca)^2) sqrt ((x-xy) ^2 + (y-yy)^2)] Constraint: [sqrt ((x-xc) ^2 + (y-yc)^2) sqrt ((x-xa) ^2 + (y-ya)^2) sqrt ((x-xca) ^2 + (y-yca)^2) sqrt ((x-xy) ^2 + (y-yy)^2)] <= 40

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This would be 23 if it was expected Consider the following MOLP Max: 4x1 + 2x2 Min: x1 + 3x2 Subject to 2x1 + x2 <= 18 X1 + 4x2 <= 12 X1 + x2 >= 4 x1,x2 >=0 What are the steps that you would follow to end up with the pareto front? Here is an approach for the pareto problems: Evaluate Objectives Separately: Start by optimizing each objective independently while keeping the other objective constant. This involves: Optimizing Objective A while fixing the value of Objective B. Optimizing Objective B while fixing the value of Objective A. Obtain Endpoints: As a result of the separate optimizations in step 1, you will obtain two endpoints. These endpoints represent the best achievable values for each objective when the other is restricted. Let's say you obtain:

For Objective A: A=number1, B=number2 For Objective B: A=number3, B=number4 Determine Intermediate Points: To find intermediate points along the Pareto front, restrict one objective while varying the other within the ranges established by the endpoints. Specifically: To find intermediate points along the Pareto front for Objective A, limit Objective B within the range [number2, number4]. To find intermediate points along the Pareto front for Objective B, limit Objective A within the range [number1, number3]. This approach ensures that you explore the trade-offs between the two objectives effectively and identify all the Pareto optimal solutions in your multi-objective optimization problem. Objective = Min Cost 13x1a + 9x1b + 11x2a + 12x2b + 55y1a + 93y1b + 65y2a + 58y2b X integers, y binary Run time Constraint

0.4x1a + 1.1x1b <= 8 0.5x2a +1.2x2b <= 6 Demand Constraint X1a + x2a >= 3 X1b + x2b >= 7 Linking Constraint X1a - (8/.4)*y1a <= 0 X2a - (6/.5)*y2a <= 0 X1b - (8/1.1)*y1b <0 X2b - (6/1.2)*y2b <=0

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