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# Entire Survey of Math 2623 Course MATH 2623

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This 18 page Bundle was uploaded by ysu34 on Wednesday January 13, 2016. The Bundle belongs to MATH 2623 at Youngstown State University taught by Steven L. Kent in Fall 2015. Since its upload, it has received 78 views. For similar materials see Survey of Mathematics in Math at Youngstown State University.

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Date Created: 01/13/16

Survey of Mathematics Math 2623 CH 1-3 “Operations Research” GOAL: optimize a result subject to constraints • Optimize- to make best of • Constraints- additional conditions which must be satisfied by something Typically real life conditions • Example: working workers over 40 hours means more money but by law you cannot work workers over 40 hours (constraint) CHAPTER 1: Urban Services • Example: shoveling snow, checking parking permits, etc. Parking Control Officer Problem #1 = at least one parking meter Pool Park Find a route for parking control officer using sides of street with parking meters so: 1. all meters are checked 2. fewest retraced streets 3. officer starts/ends at same spot Deadheading- retracing steps Mathematical Modeling: Real Life Problem è Math Problem è Math Solution è Real Life Solution Transform Solve w/ math Transform Graph- finite collection of “dots” together with “links”, each connecting pair of “dots” Finite- has a limit Vertex- “dot” Edge- “link”, doesn’t have to be straight Examples: 5 dots, 4 edges Not a graph, “loop” A loop doesn’t have a path so it can’t be a circuit Path- connected sequence of edges beginning/ending at a vertex Circuit- path starts/ends at same vertex Euler Circuit for a graph- circuit uses every edge of graph and only once Examples: Circuit but not an Euler circuit Euler circuit because each edge is included once Graph Theory- mathematical theory of the properties and applications of graphs Pool Park Pool Park Edges represent parking meters. Euler circuit A math problem can have more than one solution Theorem- a mathematical fact, which needs to be proven using mathematical logic Connected Graph- at least one path between any pair of vertices Valence of a Vertex- number of edges which meet at a vertex Examples: Connected, can get to any vertex from one vertex Not connected, no path to the line vertices Euler’s Theorem • graph must be connected • every vertex of the graph must have an even valence Uniqueness Theorem • When a given problem has at most one solution Edge Walker Theorem • Only use with rectangular street network • Makes each valence in the network even Eulerizing a graph • Add edges to duplicate existing edges to a connected graph to make all valences even Chapter 1 Review Vocabulary • Leonard Euler- showed it was impossible to stroll a route visiting the seven bridges of a German town exactly once, V-E+F=2 • Chinese Postman Problem- problem of finding a circuit on a graph that covers every edge of the graph at least once and has shortest possible length • Circuit- path that starts and ends at the same vertex • Connected Graph- can reach any vertex from another by traversing edges • Graph- finite set of dots and connecting links • Vertex- point in a graph where edges end, dot • Edge- link, joins 2 vertices in a graph • Path- connected sequence showing a route that starts and ends at a vertex • Circuit- path that starts and ends at same vertex • Euler circuit- a circuit that covers each edge of a graph and only once • Valence of a Vertex- number of edges meeting at the vertex • Euler Circuit Theorem- if g has an euler circuit then g must be connected and all valences are even • Eulerizing- adding edges to existing edges to make even valances • Rectangular Street Network- series of rectangular blocks that form a rectangle • Digraph- each edge has an arrow indicating direct of edge • Management Science- operations research • Optimal Solution- problem has numerical solutions, the best is top ranked HW: Chapter 1: Skills Check 1-5, 7-11, 14-17, 19, 21, 25, 27 and do Exercises 2, 4, 5, 7-10, 13, 15, 16, 23-25, 27, 28, 30, 31, 32, 34, 38, 39, 40, 43, 46, 49, 50a CHAPTER 2: Business Efficiency CL C M S CL - 349 774 541 C 349 - 425 300 M 774 425 - 562 S 541 300 562 - CL- Cleveland, C- Chicago, M- Minneapolis, S- St. Louis The traveling salesman problem: find a route for the TSP which begins and ends in Cleveland, visits Chicago, St. Louis, and Minneapolis, and has smallest total distance. Complete Graph- there is exactly one edge between every pair of vertices Hamiltonian Circuit- for a graph is a path which begins/ends at same vertex, which includes every other vertex of graph but once. Transition into Math Problem 774 349 562 300 425 Consider all Hamiltonian circuits starting at CL. Make a tree graph, let the branches represent the circuits. Nearest Neighbor Algorithm • Algorithm: step by step process On the graph start at CL, pick the edge with the smallest weight to go to and continue choosing the smallest • Circuit: CL-C-S-M-CL Sorted Edges Algorithm • Make a list of the graphs weights in order from least to greatest • In order of smallest redraw the graph using the weights until you create an abstract representation Start at CL on abstract graph and pick edge with smallest weight to go to and continue • Circuit: CL-C-S-M-CL or reversed Kruskal’s Algorithm • Simple circuit with at least three distinct vertices in a graph and doesn’t revisit any vertex except start/end point Starts/ ends at one point, doesn’t revisit any vertex Tree- connected graph which has no simple circuits Spanning Tree for a Graph- sub graph of original tree and includes every vertex of original graph Minimum Cost Spanning Tree- has smallest total weight of edges selected from original graph Turning a Plane Around Problem Task A Unload passengers 13 mins Task B Unload cargo 25 mins Task C Clean cabin 15 mins Task D Load new cargo 22 mins Task E Load new passengers 27 mins Task A must be done before Task C starts Task B must be done before Task D starts Task B and C must be done before Task E Critical Path Analysis • Given an order requirement digraph find the length of the critical path, which is also the path with largest time • Then list all possible directed paths • Find length of each Counter Intuitive- picking biggest answer for shortest answer Chapter 2 Review Vocabulary • Hamiltonian Circuit- tour starts at a vertex and visits each vertex once and only once, returns where it started • Algorithm- step by step • Method of trees- visual method carrying out principle of counting • Tree- connected graph with no circuits • Complete Graph- exactly one edge between each pair of vertices • Fundamental Principle of Counting- counting outcomes of multistage processes • Brute Force- solves TSP, enumerates all Hamiltonian Circuits and selects minimum cost • Nearest Neighbor Algorithm- starting at home and visiting nearest city and then the next nearest until no other choices • Greedy Algorithm- doesn’t lead to optimal solution, cheapest action taken • Sorted Edges Algorithm- never uses 3 edges, uses all vertices, distances arranged numerically and shortest action taken first • Spanning Tree- subgraph of connected graph that is a tree • Kruskal’s Algorithm- add links in order of cheapest cost so every vertex belongs to a link • Critical Path- longest directed path in order requirement graph • Order Requirement Digraph- shows which tasks go first in directed path • Weight- number assigned, such as costs, distances, time HW: Skills Check 2-6, 8-13, 15, 16, 18-21, 24, 27-19 and Exercises 8, 10a, 14, 26-31, 33a, 34, 35, 38-40, 43, 44a-c, 45, 46a, 48, 50, 53, 55, 57a, 74- 76 CHAPTER 3: Planning and Scheduling Scheduling exams without conflict • A school needs to schedule finals in French-F, Math-M, History-H, Philosophy-P, English-E, Italian-I, Spanish-S, and Chemistry-C without conflict of exam times. • Each room only fits one class at a time and only two rooms are available for testing • Find a schedule that works Each ‘x’ represents one student taking both of two courses: F M H P E I S C F X X X X X M X X X H X X X P X X E X X X I X X X X X S X X C X X X Symmetry like M+F and F+M are the same student Draw a graph combining all outcomes of the chart and use the vertex coloring problem to label. • Assign a color to each vertex of the graph in such a way that no two vertices are connected. Vertex Chromatic Number of a Graph- minimum number of different colors required to solve the vertex coloring problem Abstract vs. Concrete • Closer the symbols are resembling the model, the more concrete it is • Further the symbols are resembling the model, the more abstract it is Priority List- order in ranking of the tasks according to importance Chapter 3 Review Vocabulary • Heuristic Algorithm- fast but no optimal solution • Independent Task- no edges connected • Priority List- ordering tasks for purpose of attaining a certain scheduling goal • Processor- person, machine, robot • Vertex Coloring- colors assigned to vertices that aren’t connected by a single edge PART 13.2: Fair Division Knaster Inheritance Procedure • Two participants • One asset in dispute • One participant has a higher monetary bid for each asset Example: Bob, Carol, Ted, and Alice place bids on their parents house Bob Carol Ted Alice $120,000 $200,000 $140,000 $180,000 The house is given to highest bidder and the highest bidder then gives everyone else cash. Carol won and pays everyone cash. Perceived Fair Share Bob Carol Ted Alice 120,000/4= 200,000/4= 140,000/4= 180,000/4= 30,000 50,000 35,000 45,000 Since Carol got house she gets her amount in equity of the house. Carol must put 200,000-50,000= $150,000 in a “temporary kitty” Each participant get money from the kitty. • $150,000-30,000-35,000-45,000-40,000= $40,000 to split • $40,000/4= $10,000 o Since Carol gets house this is in her equity Bob Carol Ted Alice 30,000+10,000= House 35,000+10,000= 45,000+10,000= 40,000 45,000 55,00 Strategic Bid- bid high enough to get exactly what you want Sincere Bid- what you honestly feel its worth HW: Skills Check 7, 9, 12, 13, 14 and Exercises 1-4,10, 11, 13, 14 PART 13.1 Adjusted Winner Procedure • two people • one asset in dispute • each participant has 100 points to assign The Trump Divorce (Homework Problem) Donald Ivana Conn. Estate 10 38 Palm Beach Mansion 40 20 Trump Plaza Apt. 10 30 Trump Tower Triplex 38 10 Cash and Jewelry 2 2 Donald gets the Mansion and Triplex Ivana gets Estate and Apartment Tie for cash and jewelry If the cash and jewelry are given to Ivana she wont have more than Trump so she gets it. Donald 78 points and Ivana 70 points Adjustment Phase • Mansion Ratio: 40/20 • Triplex Ratio: 38/10 Mansion is considered first for splitting 78-40x=70+20x 8=60x 2/15 Ivana gets 2/15 of the mansion Equitable- fair division Envy free- strategy, can guarantee player what they want Pareto optimal- cant better or worsen the situation Theorem- Adjusted winner procedure has all three properties above. HW: Skills Check 1-4, 6,10 CHAPTER 5: Exploring Data: Distributions Mean- average Concept- if all data is the same then the mean of data is equal to the number Outliers- numbers far away from data set Work: five siblings are 6, 6, 7, and 8 years old. Show this in a histogram. Five Number Summary • Min # - Q1 – Mean – Q3 – Max # Work: create a box and whisker plot and a stem and leaf plot with the five number summary of 10-20-30-70-90 Chapter 5 Review Vocabulary • Boxplot- graph of five number summary • Five Number Summary- smallest observation, Q1, median, Q3, largest observation • Histogram- distribution of outcomes • Individuals- described by data set, people, animals, etc • Mean- average • Median- middle number in set • Most- frequent • Range- lg-sm • Normal Distribution- symmetric bell shaped curve • Relative Frequency Distribution- gives all variables and % they occur • 68-95-99.7 Rule- 68% in one standard deviation, 95% in two standard deviation, 99.7% in three standard deviation of the mean • Standard Deviation- measure of variability of distribution about its mean as center, its square root of the average standard deviation of observations from their mean • Stemplot- stem and leaf, display variables, attaches final digits as leaves • Variable- characteristics take on different values for individuals • Correlation- -1 and 1 with same sign as regression line slopes • Explanatory Variable- X • Response Variable- Y HW: Skills Check 1-9, 11-16, 20-26, 29, 30 and Exercises 1, 4, 6-8, 10a, 14a, 15, 21 CHAPTER 6: Exploring Data: Relationships Review Vocabulary • Regression Line- describe response and explanatory variable • Scatterplot- graph of variables as points on plane • Linear- equation of line, y=mx+b • Hooke's Law- F=KX No assigned HW CHAPTER 7: Data for Decisions Statistics: The Science of Data Three Step Method • Label • Table • Translate back into real life situation Review Vocabulary: Summarizes chapter • Population- set of objects we want info about • Sample- info we collect from subset of population • Sampling- choosing sample from population • Convenience Sample- you pick samples • Voluntary Response- people chose to respond • Simple Random Sample- SRS, sample chosen by chance, equal • Table of Random Digits- 1/10 likely to be zero, entries on table labeled 0-9 • Double Blind- neither subject or persons know who received treatment • Margin of Error- 95% confidence, number to right of +- signs • Nonresponse- individuals chose not to respond • Observational Study- sample survey • Placebo Effect- dummy treatment, pill • Undercoverage- may leave some groups out HW: Skills Check 1-27 and Exercises 1, 3, 4, 6, 7, 9, 11, 12, 27, 29, 35, 48 CHAPTER 9: Social Choice The impossible dream #14, pg. 351 Example 3 1 1 1 1 1 1 A A B B C C D D B C C B D C B C D A D B B C D A D A A A Plurality Voting- Number of 1 place positions Borda Count- each row has point rankings and you add each letters total points up Hare System- put into rounds and each letter is ranked in total of how many times they are above others, lowest letter is eliminated each round Sequential Pairwise Voting with an Agenda- use majority role and use the agenda to pair each letter with another to compete against other Review Vocabulary • Agenda- ordering candidates • Approval Voting- vote however many they want • Kenneth J. Arrow- discovered any voting system can leave undesirable outcomes • Borda Count- points assigned by preferences • Hare System- eliminated based on number of first place votes • IIA- can go from losing one election to winning next by change in votes • Manipulation- submit ballot misrepresenting true preferences • Majority Rule- preferred by more than half • May’s Theorem- majority rule is only one satisfying three properties • Monotonicity- ballot changes favorable to one candidate without hurting another • Pareto Condition- latter candidate • Plurality Runoff- assuming no ties, runoff between two candidate with most first place votes • Plurality Voting- most first place votes • Preference List Ballot- most preferred to least HW: Skills Check 2-6, 8, 14, 16, 27 and Exercises 11, 14, 15, 17-20 CHAPTER 11: Weighted Voting Systems Vocabulary is all needed to understand the voting systems • Banzhaf Power Index- count of all voting combinations • Coalition- set of participants favor a given motion • Critical Voter- can reverse outcome by changing vote • Dictator- all control • Dummy voter- no power or meaning • Quota- minimum votes necessary • Veto power- issue cannot pass without them HW: Skills Check 1, 2, 11, 18 and Exercises 3, 14 1/13/16 11:37 PM 1/13/16 11:37 PM

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