Graduate Midterm Examination

Transcription

1 MIE 532 UMass J.MacGregor Smith Fall 2012 Graduate Midterm Examination Please Place your name of the BACK of the LAST PAGE of the examination! Question #1: Question #2: Question #3: Question #4: Grand Total : 1

2 MIE 532 UMass J.MacGregor Smith Fall 2012 Graduate Midterm Examination Please Place your name of the BACK of the LAST PAGE of the examination! There are four questions on this examination. The first question is a series of (truefalse) questions, each worth 3 points each for a total of 30 points. The other three questions are open ended with point values as indicated. Please answer all questions and show all work on this examination paper. Please answer all questions and show all work on your examination paper. You may use the back of the sheets if necessary Question I 30 points: In the following true-false questions, answer all questions, there is no penalty for guessing.please circle the appropriate response! a) In a tree, any two vertices are connected by exactly one path. b) An undirected graph has a strongly connected orientation if and only if it possesses no bridges. c) If T is a tree, then the number of edges/arcs of the tree e(t) equals the number of vertices of the tree minus one, i.e. e(t) = v(t) 1. d) A decision problem is a question whose answer is either yes or no. Such a problem belongs to the class P if there is a polynomial-time algorithm that solves an instance of the problem in polynomial time. e) While the existence of an Eulerian tour can be checked in a graph by simply examining the degree of each node in a connected graph, the presence of a Hamilton Circuit in a G(N, A) has no simple property to guarantee its existence. f) A maximum branching final solution of a network may include loops. 2

3 g) In the number of iterations of the double-sweep algorithm for the K-Shortest path problem, the total number of double sweeps is bounded above by K N 2 iterations where N is the number of nodes in the graph. h) In the tracing procedure of the K-Shortest paths algorithm the t in H tj refers to the t m th shortest path passing through the intermediate node j. i) Dijkstra s algorithm can be directly used to detect negative cycles in a graph without any modifications. j) Linear programming is the fundamental mathematical model underlying most of the algorithms studied so far in this course. Question II: 25 points An assembly consists of two parts A and B. These parts go through the following operations in order: Forging, Drilling, Grinding, Painting and Assembly. The duration of these operations in days are summarized below: Part Forging Drilling Grinding Painting A B Upon Painting part A is assembled in two days, then A and B are assembled together in one day. It is desired to find the least time required for the total assembly (this problem is called the critical path problem). Be sure to carefully think through what is happening in the assembly process. i) Develop the network representation of the problem. ii) Formulate the objective and determine an appropriate algorithm for its solution. iii) Solve the problem with an algorithm we have studied this semester. 3

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6 Question III: 25 points A company is considering an investment plan with an investment horizon of 5 years. The investment opportunities available to them include: ı) One year bonds that yield 7% per year. ii) Three year bonds that yield 9% per year. iii) Five year bonds that yield 8% per year. Please do the following: a) Draw a network representation of the problem properly accounting for the beginning of the investment process and the year end maturity of the different bond investments. Be sure to account for compound interest. b) Which algorithm is most suited for this problem and how should it be modified to solve this problem? c) Find the optimal investment strategy. 6

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9 Question IV: 20 points Suppose forest F consists of t trees and contains v vertices. How many edges are in the forest F? 9

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