A firm is considering projects A, B, . . . , H. Using binary variables x_a, ..., x_h and linear constraints, model the following conditions on the projects to be undertaken

Three alternatives are being considered for an engineering project. Their cash-flow estimates are shown in the accompanying table. A and B are mutually exclusive, and C is an optional add-on feature to alternative A. Investment funds are limited to $5,000,000. Another constraint on this project is the engineering personnel needed to design and implement the solution. No more than 10,000 person-hours of engineering time can be committed to the project. Set up a linear integer programming formulation of this resource allocation problem.

Exercise 2. Here is a classic geometric problem, in the same application domain as the Center Selection problem: Given n points with Cartesian coordinates (₁, 3) in the plane, and positive weights w, find one(!) point (x, y) that minimizes the weighted sum of the Euclidean distances to the given points. For formal clarity: We want to minimize n Σwi i-1 n •√(x- i-1 (x − x)² + (y - y₁)². This problem is at least not very easy to solve exactly. In the following we therefore propose a rough but rather quick and simple approximation algorithm: Instead of the Euclidean distance, take the Manhattan distance and minimize 1 w₁(x − x₂|+|y-yil). 2.1. Explain how the point that minimizes the weigthed sum of Manhattan distances can be found in polynonial time. That is: Sketch an algorithm, argue why it is correct, and explain your time bound. Try to keep the time bound as low as possible. 2.2. Show that this algorithm has an approximation ratio √2. More specifi- cally: The point found by the…

Suppose that a manufacturing company builds n different types of robots, sayrobots 1, 2, . . . , n. These robots are made from a common set of m types of materials, saymaterials 1, 2, . . . , m. The company has only a limited supply of materials for each year,the amount of materials 1, 2, . . . , m are limited by the numbers b1, b2, . . . , bm, respectively.Building robot i requires an aij amount from material j. For example, building robot 1requires a11 from material 1, a12 from material 2, etc. Suppose the profit made by sellingrobot i is pi. Write an integer linear program for maximizing the annual profit for thecompany

1. Let S = {1,2...n} be a set activities, where activity i € S has a start time si, a finish time f; and a profit p, asSociated with it. Design an dynamic programming algorithm for computing a set of compatible activities of maximum profit. Argue that the time complexity of you algorithm is O(n log(n)). (You must start with the correct definitions, and then write a recurrence relation.)

8. A school is creating class schedules for its students. The students submit their requested courses and then a program will be designed to find the optimal schedule for all students. The school has determined that finding the absolute best schedule cannot be solved in a reasonablo time. Instead they have decided to use a simpler algorithm that produces a good but non-optimal schedule in a more reasonable amount of time. Which principle does this decision best demonstrate? O A Unreasonable algorithms may sometimes also be undecidable O B. Heuristics can be used to solve some problems for which no reasonable algorithm exsts O C. efficiency Two algorithms that solve the same problem must also have the same O B. Approximate solutions are often identical to optimal solutions

If you can make an optimal solution for a problem by making optimal solutions for its subproblems, then the problem has the quality in question. a) Subproblems that overlap; b) best substructure; c) memory; d) greedy

This characteristic is met by a problem if it is feasible to develop an optimum solution for it by first creating optimal solutions for its subproblems. a) Subproblems that overlap; b) substructure that's optimum; c) memorization; d) greedy

In this problem we have n jobs j1, j2, ..., jn, each has an associated deadline d1, d2, ..., dn and profit p1, p2, ..., pn. Profit will only be awarded or earned if the job is completed before the deadline. We assume that each job takes 1 unit of time to complete. The objective is to earn maximum profit when only one job can be scheduled or processed at any given time.Provide the pseudocode of an algorithm to find the sequence of jobs to do with the maximum total profit. Also describe the main idea of your algorithm using plain language.[Hint: You can select the jobs in a greedy way. You can use the following example to help your analysis.] Job J1 J2 J3 J4 J5 Deadline 2 1 3 2 1 Profit 60 100 20 40 20 The best job sequence would be J2 →J1 →J3.

A school is creating class schedules for its students. The students submit their requested courses and then a program will be designed to find the optimal schedule for all students. The school has determined that finding the absolute best schedule cannot be solved in a reasonable time. Instead they have decided to use a simpler algorithm that produces a good but non-optimal schedule in a more reasonable amount of time. Which principle does this decision best demonstrate?