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a first course in mathematical modeling
A First Course In Mathematical Modeling 5th Edition Frank R. Giordano, William P. Fox, Steven B. Horton - Solutions
Consider the following game, called the Prisoner's Dilemma. The game goes as follows: We assume that the police have arrested two suspects for a crime. Two prisoners have the option to confess or not to confess to a crime they committed. The prosecutor has only enough evidence to convict both
For each game in this problem, plot the payoffs and use the plot to determine if the game is a total conflict or partial conflict game.a.b.c. d.
Use the definition provided in Section 10.1 for dominance to determine if any row or column strategies should be eliminated in the games described in problem 1ai. State the row and columns that can be eliminated.Data from problem 1a- iUsing the definition provided for the movement diagram,
Consider whether to join a social network, such as Facebook. List several decision criteria and the variables associated with each.
The NBC TV network earns an average of $400,000 from a hit show and loses an average of $100,000 on a flop (a show that cannot hold its rating and must be canceled). If the network airs a show without a market review, 25% turn out to be hits, and 75% are flops.For $40,000, a market research firm
Help the President and Congress consider balancing the budget and reducing the deficit. What variables are important?
Retirement and Social Security. Should U.S. citizens build their own retirement through 401Ks or use the current Social Security program? Build models to be able to compare these systems and provide decisions that can help someone to plan a better retirement.
Many colleges face the problem of drug testing for their athletes. Assume we are concerned with the following costs:c1 = cost if an athlete is falsely accused of using a drug.c2 = cost if an athlete cannot be accurately identified as a drug user.c3 = cost due to invasion of privacy if a nonuser is
Golf Smart sells a particular brand of driver for $200 each. During the next year, it estimates that it will sell 15, 25, 35, or 45 drivers with respective probabilities of 0.35, 0.25, 0.20, and 0.20. They can buy drivers only in lots of 10 from the manufacturer. Batches of 10, 20, 30, 40, and 50
ESPN is trying to decide which of three football games to televise in the upcoming season in the southern region of the United States: Alabama versus Auburn, Florida versus Florida State, or Texas A&M versus LSU. The estimated viewers (in millions of homes) of the games differ according to the
For a new development area, a local investor is considering three alternative real estate investments: a hotel, a restaurant, and a convenience store. The hotel and the convenience store will be adversely or favorably affected depending on their closeness to the location of gasoline stations, which
Given the following payoff matrix:Determine the best plan by each of the following criteria and show your work:a. Laplace.b. Maximin.c. Maximax.d. Coefficient of optimism (assume that x = 0.55)e. Regret (minimax).
We have a choice of two investment strategies: stocks and bonds. The returns for each under two possible economic conditions are as follows:a. If we assume the probability of Condition 1 is p1 = 0.75 and Condition 2 is p2 = 0.25, compute the expected values and select the best alternative.b. What
We are considering one of three alternatives A, B, or C under uncertain conditions. The payoff matrix is as follows:Determine the best plan by each of the following criteria and show your work:a. Laplace.b. Maximin.c. Maximax.d. Coefficient of optimism (assume that x = 0.65).e. Regret (minimax).
Given the following payoff matrix, show all work to answer parts (a) and (b).a. Which alternative do we choose if our criterion is to maximize the expected value?b. Find the opportunity loss (regret) table and compute the expected opportunity loss (regret) for each alternative. What decision do
Consider the steroid testing in baseball scenarios.a. Under what changes would the commissioner be able to justify his 100% testing results?b. Add a second drug test and now determine the results of the decision tree. Is the baseball commissioner justified for 100% testing now after adding a
Consider the All-Green energy company in Problem 5, Section 9.2. All Green has considered asking a marketing research group to perform a market research study. Within one month, this group can report on the study whether the results were good (G) or discouraging (D) to pursue this new line of
Consider the steroid testing in baseball of Problem 7. Assume new data for the expanded roster has been collected, and our table now is as follows. Build a new decision tree and interpret the results:Data from problem 7Testing for Steroids in Baseball. Baseball has dramatically changed the
Consider Problem 8. Modify the proportionality results as shown in the following and determine the new results of the decision tree:a. b. Data from problem 8The worst-case scenario for a baseball player is a lifetime ban. We initially assume there are only two options: test everyone or test no one.
The worst-case scenario for a baseball player is a lifetime ban. We initially assume there are only two options: test everyone or test no one. We assume monetary assets or costs in our analysis. We define the following:B = benefit for correctly identifying a steroid user and banning this user from
Testing for Steroids in Baseball. Baseball has dramatically changed the penalties associated with failing steroid drug testing. The new penalties that Commissioner Bud Selig has proposed are an approach of ``three strikes and you're out," which goes as follows:The first positive test would result
A local TV studio is deciding on a possible new TV show. A successful TV show earns the station about $450,000, but if it is not successful, the station loses about $150,000. Of the previous 100 shows reviewed by the local TV station, 25 turned out to be successful TV shows, and 75 turned out to be
A nuclear power company in California is deciding on two possible locations for constructing a new nuclear power plant. Let's call the possible locations A and B. The nuclear disaster in Japan has made the nuclear power company very aware of the potential damage to nuclear power plants due to
Consider a scenario in which state colleges must actively recruit students. California Sci has $750,000 in assets available. Its Board of Regents has to consider several options. The board may decide to do nothing and put the $750,000 back into the college operating budget. They may directly
A big private oil company must decide whether to drill in the Gulf of Mexico. It costs $1 million to drill, and if oil is found its value is estimated at $6 million. At present, the oil company believes that there is a 45% chance that oil is present. Before drilling begins, the big private oil
Consider the Las Vegas wheel from Example 1 with the following new payoffs and probabilities:Consider three spins and decide your optimal strategy.Data from example 1You are in a Las Vegas Casino and have encountered the following game board, which is to be spun randomly (electronically):Assuming
Consider the Las Vegas wheel problem in Example 1 where the wheel no longer has equal probabilities but probabilities for values $10, $6, $4, $0 as follows:Consider three spins and decide your optimal strategy.Data from example 1You are in a Las Vegas Casino and have encountered the following game
The local TV station has $150,000 available for research and wants to decide whether to market a new advertising strategy for the station. The station is located in the city, but its viewers are statewide. The station management has developed three alternatives that need to be analyzed:Alternative
An oil company is considering making a bid on a new alternative energy contract to be awarded by the government. The company has decided to bid $2.10 billion. The oil company has a good reputation, and it estimates that it has a 70% chance of winning the contract bid. If the oil company wins the
Assume the following probability distribution of daily demand for bushels of strawberries:Further assume that unit cost is $3 per bushel, selling price is $5 per bushel, and salvage value on unsold units is $2. We can stock 0, 1, 2, or 3 units. Assume that units from any single day cannot be sold
A new energy company, All Green, has developed a new line of energy products. Top management is attempting to decide on both marketing and production strategies. Three strategies are considered and are referred to as A (aggressive), B (basic), and C (cautious). The conditions under which the study
Consider Problem 3. List some considerations that would cause the decision maker not to choose the larger expected value.Data from problem 3The financial success of a ski resort in Squaw Valley is dependent on the amount of early snowfall in the fall and winter months. If the snowfall is greater
The financial success of a ski resort in Squaw Valley is dependent on the amount of early snowfall in the fall and winter months. If the snowfall is greater than 40 inches, the resort always has a successful ski season. If the snow is between 30 and 40 inches, the resort has a moderate season, and
Problem 9, Section 9.1.For the following two problems from Section 9.1, complete a tree diagram and solve each problem.Data from problem 9Consider a construction firm that is deciding to specialize in building high schools or elementary schools or a combination of both over the long haul. The
Problem 6, Section 9.1.For the following two problems from Section 9.1, complete a tree diagram and solve each problem.Data from problem 6Consider a firm handling concessions for a sporting event. The firm's manager needs to know whether to stock up with coffee or cola and is formulating policies
Consider the scenario in Exercise 9. Under what probabilities would the construction firm be indifferent to the type of contract it could receive?Data from exercise 9Consider a construction firm that is deciding to specialize in building high schools or elementary schools or a combination of both
Consider a construction firm that is deciding to specialize in building high schools or elementary schools or a combination of both over the long haul. The construction company must submit a bid proposal, which costs money to prepare, and there are no guarantees that it will be awarded the
Refer to the rolling of a pair of dice example. Determine the probability of rolling a 7 or an 11. If you roll a 7 or 11, you win $5, but if you roll any other number, you lose $1. Determine the expected value of the game.
For the firm handling concessions in Exercise 6, find and interpret the break point of the decision based on the weather probability. Discuss under what probabilistic conditions for weather the concession rm should sell cola or coffee.Data from exercise 6Consider a firm handling concessions for a
A term life insurance policy will pay a beneficiary a certain sum of money on the death of the policyholder. These policies have premiums that must be paid annually. Suppose the company is considering selling 1-year term life insurance for $550,000 with a cost of $1050 to either a 59-year-old male
We have engaged in a business venture. Assume the probability of success is P(s) = 2/5; further assume that if we are successful, we make $55,000, and if we are unsuccessful, we lose $1750. Find the expected value of the business venture.
The number of attempts to use an ATM per person per month and their probabilities are listed in the following table. Compute the expected value and interpret that value. Discuss how similar calculations could be used to determine the number of ATMs needed.
Let's assume in a class there were 8 scores of 100, 5 scores of 95, 3 scores of 90, 2 scores of 80, and 1 score of 75. Compute the average grade.
Let's assume you have the following numerical grades in a course: 100, 90, 80, 95, and 100. Compute your average grade.
Express the softball manager's problem as a linear or integer program and solve it with computer software.
Write down the linear program associated with solving maximum flow from s to t in the graph in Figure 8.37.Figure 8.37
Suppose a computer procedure needs to check all of 2100 possibilities to solve a problem. Assume the computer can check 1;000;000 possibilities each second.a. How long will it take the computer to solve this problem this way?b. Suppose that the computer company comes out with a new computer that
Consider again the graph in Figure 8.36. Now suppose that the cost of placing a vertex in S varies. Suppose the cost of placing vertex i in S is g(i) = (–i2 + 6i –5)3 for i є {1, 2, 3, 4, 5}. Repeat parts (a), (b), and (c) of the previous problem for this new version of the problem. This is
Does Dijkstra's Algorithm work when there might be arcs with negative weights?
In the sport of orienteering, contestants (``orienteers'') are given a list of locations on a map (``points'') that they need to visit. Orienteering courses are typically set up in natural areas such as forests or parks. The goal is to visit each point and return to the starting location as quickly
Explain why the procedure for using a maximum flow algorithm to find the size of a maximum matching in a bipartite graph works. Can it be used to find a maximum matching (as opposed to the size of one)? Can it be used for graphs that are not bipartite? Why or why not?
Use our maximum flow algorithm to find the maximum flow from s to t in the graph of Figure 8.31.Figure 8.31
Asmall suburban city is experimenting with a newway to keep its main park clean. From May through October, a crew is needed every day to pick up and remove trash. Rather than contracting with one company for the entire period, the city manager takes bids from firms on its website. Firms submit bids
Find a shortest path from node a to node j in the graph in Figure 8.33 with edge weights shown on the graph.Figure 8.33
Consider a modification of the softball manager's problem where we are interested in the best starting lineup. How can our mathematical model be modified to solve this problem? What new techniques are needed to solve models of this type?
Given a graph G = (V(G), E(G)), consider the following strategy for finding a minimum vertex cover in a graph. Step 0: Start with S =∅. Step 1: Find a vertex v of maximum degree (one that has the greatest number of incident edges). Add this vertex to S. Step 2: Delete v and all its
Write a computer program that takes integers m, n, ri for 1 ≤ i ≤ m and sj for 1≤ j≤ n as input and that either outputs a 0–1 matrix with m rows and n columns with row sums ri and column sums sj , or says that no such matrix can exist (some programming experience required).
Investigate a social network that is of interest to you. Carefully define what the vertices represent and what the edges represent. Are there any new modeling techniques that you had to employ?
Here we consider the weighted vertex cover problem. Suppose the graph in Figure 8.23 represents an instance of vertex cover in which the cost of having vertex i in S is w(i) = (i –2)2 + 1 for i = 1, 2, 3, 4, 5. For example, if v4 is in S, we must use w (4) = (4 –2)2 + 1 = 5 units of our
A path on n vertices, Pn, is a graph with vertices that can be labeled v1, v2, v3,.... vn, so that there is an edge between v1 and v2, between v2 and v3, between v3 and v4, . . . , and between vn–1 and vn. For example, the graph P5 appears in Figure 8.22. Compute β(P5). Compute β(P6).
Explain, in your own words, why a maximum flow algorithm can solve the matrix problem from this section.
Considering the following values, determine whether there is a 0–1 matrix with m rows and n columns, with row sums ri and column sums sj. If there is such a matrix, write it down.
Considering the values below, determine whether there is a 0–1 matrix with m rows and n columns, with row sums ri and column sums sj. If there is such a matrix, write it down.
Will graphs formed with the procedure used to make the one in Figure 8.18 always be bipartite, regardless of the data? Why or why not?Figure 8.18
Find a maximum matching in the graph in Figure 8.21. How many edges are in the maximum matching? Now suppose we add the edge bh to the graph. Can you find a larger matching?Figure 8.21
In the text for this section, there is the sentence ``When G is bipartite with bipartition 〈A,B〉, it is clear that no matching can be bigger than IAI, and no matching can be bigger than IBI.'' Explain why this is true.
Write computer software that finds the best piecewise linear function given a data set S along with α and β .
Consider the data set S = {(0,0), (2,9), (4,7), (6,10), (8,20)}. Using α = 5 and β = 1, determine the best piecewise linear function for S.
Using the same data set from the example in the text,recompute Table 8.5 with α = 2 and β = 1.Table 8.5
For the example in the text above, explain why, in Table 8.5, the entries in row 3, column 7, and row 4, column 7 are the same. Plot the data and draw a line segment from point 3 to point 7, and another from point 4 to point 7.Table 8.5
Can you think of other relations that one could consider?
Just as actors have their Bacon numbers, there is a relation defined between authors of scholarly papers and the prolific Hungarian mathematician Paul Erdös. Use the Internet to find out what you can about Paul Erdös and Erdös numbers. Consider your favorite mathematics professor (no jokes here,
Suppose you are an actor, and your Bacon number is 3. Can future events ever cause your Bacon number to rise above 3? What, in general, can you say about an actor's Bacon number in terms of how it can change over time?
At a large meeting of business executives, lots of people shake hands. Everyone at the meeting is asked to keep track of the number of times she or he shook hands, and as the meeting ends, these data are collected. Explain why you will obtain an even number if you add up all the individual
Suppose r1 = 4, r2 = 3, and r3 = 7.a. b. c.
Consider the graph in Figure 8.11.Figure 8.11a. Write down the set of edges E(G).b. Which edges are incident with vertex b?c. Which vertices are adjacent to vertex c?d. Compute deg (a).e. Compute IE(G)I.
Following a major storm, an inspector must walk down every street in a region to check for damaged power lines. Suppose the inspector's region can be modeled with the following graph. Vertices represent intersections, and edges represent streets. The numbers on the edges are called edge weights;
The Mathematics Department at a small college plans to schedule final exams. The class rosters for all the upper-class math courses are listed in Table 8.3. Find an exam schedule that minimizes the number of time periods used.Table 8.3
Consider the graph of Figure 8.8.Figure 8.8a. Color the graph with three colors. b. Now suppose that vertices 1 and 6 must be colored red. Can you still color the graph with three colors (including red)?
Consider the two political maps of Australia described in Problem 3. What is the smallest number of colors needed to color these maps?Data from problem 3Find a political map of Australia. Create a graph model where there is a vertex for each of the six mainland states (Victoria, South Australia,
Can you think of other real-world problems that can be solved using techniques from the section about the bridges of Königsberg?
Find a political map of Australia. Create a graph model where there is a vertex for each of the six mainland states (Victoria, South Australia, Western Australia, Northern Territory, Queensland, and New South Wales) and an edge between two vertices if the corresponding states have a common border.
The bridges and land masses of a certain city can be modeled with graph G in Figure 8.7.Figure 8.7a. Is G Eulerian? Why or why not? b. Suppose we relax the requirement of the walk so that the walker need not start and end at the same land mass but still must traverse every bridge exactly once. Is
One of the best-known interpolation methods is Newton's Method, which exploits a quadratic approximation to the function f (x) at a given point x1. The quadratic approximation q is given by:a. Starting with x = 4 and a tolerance of ε = 0.01, use Newton's Method to minimize f (x) = x2 + 2x, over
One of the more interesting search techniques uses the Fibonacci sequence. This search method can be employed even if the function is not continuous. The method uses the Fibonacci numbers to place test points for the search. These Fibonacci numbers are defined as follows: F0=F1=1 and
For Example, 3, show that the optimal value of x is x∗ = 1.Data from Example 3and we will search for an optimal value of c in the closed interval [0, 3]. We choose a tolerance t = 0.2. We apply the Golden Section Search Method until an interval of uncertainty is less than 0.2. The results are
For Example, 2, show that the optimal value of c is c∗= 8/ 9. Apply the definition of the absolute value to obtain a piecewise continuous function. Then find the minimum value of the function over the interval [0, 3].Data from example 2Because f (x1) > f (x2), we discard all values in [x2,
Use the curve fitting criterion to minimize the sum of the absolute deviations for the following models and data set:a. y = ax
Use the Golden Section Search Method with a tolerance of t = 0.2.a. b.
Use the Dichotomous Search Method with a tolerance of t = 0.2 and " ε = 0.01.a. b.
Firestone, headquartered in Akron, Ohio, has a plant in Florence, South Carolina, that manufactures two types of tires: SUV 225 radials and SUV 205 radials. Demand is high because of the recent recall of tires. Each batch of 100 SUV 225 radial tires requires 100 gal of synthetic plastic and 5 lb of
Afarmer has 30 acres on which to growtomatoes and corn. Each 100 bushels of tomatoes require 1000 gallons of water and 5 acres of land. Each 100 bushels of corn require 6000 gallons of water and 2.5 acres of land. Labor costs are $1 per bushel for both corn and tomatoes. The farmer has available
With the rising cost of gasoline and increasing prices to consumers, the use of additives to enhance performance of gasoline may be considered. Suppose there are two additives, Additive 1 and Additive 2, and several restrictions must hold for their use: First, the quantity of Additive 2 plus twice
Why is sensitivity analysis important in linear programming?
For the example problem in this section, determine the sensitivity of the optimal solution to a change in c2 using the objective function 25x1 + c2x2.
Optimize 5x + 3y subject to:Use the Simplex Method to find both the maximum solution and the minimum solution to Problems 8–12. Assume x ≥ 0 and y ≥ 0 for each problem.
Optimize x – y subject to:Use the Simplex Method to find both the maximum solution and the minimum solution to Problems 8–12. Assume x ≥ 0 and y ≥ 0 for each problem.
Optimize 6x + 5y subject to:Use the Simplex Method to find both the maximum solution and the minimum solution to Problems 8–12. Assume x ≥ 0 and y ≥ 0 for each problem.
Optimize 6x + 4y subject to:Use the Simplex Method to find both the maximum solution and the minimum solution to Problems 8–12. Assume x ≥ 0 and y ≥ 0 for each problem.
Optimize 2x + 3y subject to:Use the Simplex Method to find both the maximum solution and the minimum solution to Problems 8–12. Assume x ≥ 0 and y ≥ 0 for each problem.
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