Problem 6-23 (Algorithmic) Find the shortest route from node 1 to node 7 in the network shown. If the constant is "1" it must be entered in the box. If your answer is zero enter "0". For negative values enter "minus" sign (-). Min x12 + x13 + x14 + x23 + x25 + x32 + x35 + x46 + x52 + x53 + x56 + x57 + x65 + x67 s.t. Flow Out Flow In Node 1 x12 + x13 + x14 = Node 2 x23 + x25 + x12 + x32 + x52 = Node 3 x32 + x35 + x13 + x23 + x53 = Node 4 x46 + x14 = Node 5 x52 + x53 + x56 + x57 + x25 + x35 + x65 = Node 6 x65 + x67 + x46 + x56 = Node 7 + x57 + x67 = xij ≥ 0 for all i and j Optimal Solution: Variable Value x12 x13 x14 x23 x25 x32 x35 x46 x52 x53 x56 x57 x65 x67 Shortest Route: Length =

At the end of each of the next 8 years, you plan to put $25,000 in the bank. If the annual interest rate is 3%, what is the present value of this planned savings stream? What will the bank balance be at the end of the 8 year period? [Hint: once you know the present value of the savings stream, it's easy to work out the future value]

A toll-free phone number is available from 9 AM to 9 PM for your customers to register complaints about a product purchased from your company. Past history indicates that an average of 2.8 calls is received per minute.

What is the area of opportunity?

Find the center of S₃ and the centralizer of each element of S₃

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 5 2 cos(7x) x dx, n = 8 1 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule

The Scooter Company uses a process cost system for making scooter wheels. Materials are added at the beginning of the process and conversion costs are uniformly incurred. At the beginning of October the work-in-process is 40 percent complete and at the end of the month it is 60 percent complete. Spoilage is detected at the end of the process. Other data for the month include:

Required

a. Prepare a production cost worksheet assuming that spoilage is recognized and the weighted-average method is used.

b.Prepare journal entries to record transferring out of cost from the work-in-process accounts.

please explain calulations.

Two independent methods of forecasting based on judgment and experience have been prepared each month for the past 10 months. The forecasts and actual sales are as follows:

c. Prepare a naive forecast for periods 2 through 11 using the given sales data. Compute each of the following; (1) MSE, (2) MAD, (3) tracking signal at month 10, and (4) 2s control limits. (Round your answers to 2 decimal places.)

In this problem, you will illustrate a geometric fact using vectors associated with 2D regions in 3D space. We start with the four points given as:

A : (0, 0, 0) B : (1, 0, 0) C : (0, 1, 0) D : (0, 0, 1).

These four points define a 3D object which is a tetrahedron (not a regular one, however).

1. Find an equation for the plane through points A, B, C; call this plane P1. Using this, find a normal to P1 which points “outwards” (out of the 3D solid, that is). Find the area of the triangle ABC. Finally, find the “outwards” normal to P1 which has length equal to the area of triangle ABC; call this normal vector ~n1.

2. Now find an equation for the plane through the points A, B, D; call this plane P2. Using this equation, find a normal to P2 which points “outwards.” Find the area of the triangle ABD. Finally, find the “outwards” normal to P2 which has length equal to the area of the triangle ABD; call this normal vector ~n2.

3. Repeat step 2 with the points A, C, D; call the resulting plane P3 and normal ~n3.

4. Repeat step 2 with the points B, C, D; call the resulting plane P4 and normal ~n4.

5. Compute ~n1 + ~n2 + ~n3 + ~n4. You should get that the vector sum is equal to zero. Do you have any idea why this should be true?

How do you compute the area of one of those triangles? One possible way is to use the cross product. We know that k~u × ~vk is equal to the area of the parallelogram formed by ~u and ~v. So, the area of the “triangle” formed by these two vectors should be half of this.

You collect several thousand Drosophila melanogaster individuals from the UC Davis experimental orchard in Winters. Youuse1000 of these flies to establish a laboratory population, which you maintain at a census population size of 1000 each generation. You then establish from the remaining field-collected flies a series of replicated populations of size 10, 100, 200, and 500and maintain each at the starting size (10, 100, 200, 500) for several generations. After some time, you sequence each lab population.

a. If one plotted for each lab population, the frequency of each nucleotide variant vs. its true frequency in the UCD population, how would the correlations differ across lab populations?

b. Which lab populations do you think would provide the best estimate of the true UCD frequencies? Why?

c. Now imagine that one carried out the same type of correlation analysis of allele frequencies ,but instead of comparing each population to the true UCD frequencies you compare the allele frequency of the replicated populations to each other (e.g., the populations of size 10 are compared to one another, the populations of size 100 are compared to one another, etc.). How would the pairwise correlations of frequency vary from one population size to another?

d. What two aspects of the sampling of flies in this entire experiment would lead to allele frequency deviations from the true UCD frequencies for sites free of natural selection?

e. You measure sequence divergence between each lab population and the sibling species, Drosophila simulans. How will the expected divergence vary across replicated populations of different size? Why?

This week you learned about the addition and multiplication counting principles, permutations and combinations.  Identify a real life example of where one of these principles can be used.

Find the rook polynomial and an expression for the number of matchings of 5 men (rows) with 5 women (columns) given the following forbidden pairings:
(M1,W4), (M2,W2), (M3,W3), (M4,W2), (M4,W4), (M5,W1), (M5,W3), (M5,W45).

Answer is 5! - 8x4! + 21x3! - 20x2!+ 6x1!, please explain how to get it, thanks.

A) Suppose a group G has order 35 and acts on a set S consisting of four elements. What can you say about the action?

B) What happens if |G|=28? |G|=30?

Problem 6-09 (Algorithmic) The Ace Manufacturing Company has orders for three similar products: Product Order (Units) A 1950 B 400 C 1300 Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are as follows: Machine Capacity (Units) 1 1600 2 1400 3 1000 Product Machine A B C 1 $1.00 $1.30 $0.90 2 $1.30 $1.30 $1.30 3 $0.80 $0.90 $1.20 Use the transportation model to develop the minimum cost production schedule for the products and machines. Show the linear programming formulation. If the constant is "1" it must be entered in the box. If your answer is zero enter "0". The linear programming formulation and optimal solution are shown. Let xij = Units of product j on machine i. Min x1A + x1B + x1C + x2A + x2B + x2C + x3A + x3B + x3C s.t. x1A + x1B + x1C ≤ x2A + x2B + x2C ≤ x3A + x3B + x3C ≤ x1A + x2A + + x3A = x1B + x2B + x3B = x1C + x2C + x3C = xij ≥ 0 for all i, j If required, round your answers to the nearest whole number. Optimal Solution Units Cost 1-A $ 1-B $ 1-C $ 2-A $ 2-B $ 2-C $ 3-A $ 3-B $ 3-C $ Total $

Using your calculator, run a regression analysis on the following bivariate set of data with y as the response variable.

x y

71.2 23.1

90.8 122.9

88.8 82.9

57.7 18.3

71.4 8.6

60.4 25.9

60.2 -43.1

88.5 77.4

68.2 -5.9

41.1 -83

61.4 -2.5

87.1 39.4

Find the correlation coefficient and report it accurate to three decimal places. r =

What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.) r² = %

Based on the data, calculate the regression line (each value to three decimal places) y = x +

Predict what value (on average) for the response variable will be obtained from a value of 82.2 as the explanatory variable. Use a significance level of α = 0.05 to assess the strength of the linear correlation.

What is the predicted response value? (Report answer accurate to one decimal place.) y =

On March 1, 1974, a grand jury indicted seven former aides to U.S. President Richard Nixon for attempting to cover up White House involvement in a burglary of the Democratic National Committee at the Watergate complex in Washington. On April 18, the judge in the case, John Sirica, issued a subpoena for tapes of President Nixon’s conversations with the defendants. The President’s attorney, James St. Clair, attempted to delay responding to the subpoena. The prosecutor, Leon Jaworski, then used an unusual procedure to appeal directly to the Supreme Court and request that the Court order the President to supply the tapes. The Court heard oral arguments on July 8, and the justices met on July 9 to decide the case. One justice, William Rehnquist, withdrew from the case, probably because he had worked in President Nixon’s Justice Department. Of the remaining eight justices, six quickly agreed to uphold the prosecutor’s request. Two justices, Warren Burger and Harry Blackmun, were reluctant to uphold the prosecutor’s request, because they thought his direct appeal to the Supreme Court was improper. Also on July 9, President Nixon’s attorney said that the President had “not yet decided” whether he would supply the tapes if the Supreme Court ordered him to. This statement was probably intended to pressure the Court into backing down from the confrontation. At minimum, it was probably intended to encourage some justices to vote against upholding the prosecutor’s request. If the vote was split, the President could argue that it was not sufficiently definitive for a matter of this magnitude. Jaworski believed that in the event of a split vote, the President would “go on television and tell the people that the presidency should not be impaired by a divided Court.” We will regard this as a two-player game. Player 1 is Justices Burger and Blackmun, whom we assume will vote together; we therefore treat them as one player. Player 2 is President Nixon. First, Justices Burger and Blackmun decide how to vote. If they vote to uphold the prosecutor’s request, the result is an 8-0 Supreme Court decision in favor of the prosecutor. If they vote to reject the prosecutor’s request, the result is a 6-2 Supreme Court decision in favor of the prosecutor. After the Supreme Court has rendered its decision, President Nixon decides whether to comply by supplying the tapes, or to defy the decision.

President Nixon’s preferences are as follows: • Best outcome (payoff 4): 6-2 decision, President defies the decision. • Second-best outcome (payoff 3): 6-2 decision, President supplies the tapes. • Third-best outcome (payoff 2): 8-0 decision, President supplies the tapes. • Worst outcome (payoff 1): 8-0 decision, President defies the decision. Explanation: The President’s best outcome is a divided decision that he can defy while claiming the decision is not really definitive. His worst outcome is an 8-0 decision that he then defies; this would probably result in immediate impeachment. As for the two intermediate outcomes, the President is better off with the weaker vote, which should give him some wiggle room. Justices Burger and Blackmun’s preferences are as follows: • Best outcome (payoff 4): 6-2 decision, President supplies the tapes. • Second-best outcome (payoff 3): 8-0 decision, President supplies the tapes. • Third-best outcome (payoff 2): 8-0 decision, President defies the decision. • Worst outcome (payoff 1): 6-2 decision, President defies the decision. Explanation: In their best outcome, Burger and Blackmun get to vote their honest legal opinion that the prosecutor’s direct appeal to the Court was wrong, but a Constitutional crisis is averted because the President complies anyway. In their second-best outcome, they vote dishonestly, but they succeed in averting a major Constitutional crisis. In their third-best outcome, the crisis occurs, but because of the strong 8-0 vote, it will probably quickly end. In the worst outcome, the crisis occurs, and because of the weak vote, it may drag out. In addition, in the last outcome, the President may succeed in establishing the principle that a 6-2 Court decision need not be followed, which no member of the Court wants. 1. Draw an extensive form game tree for the situation described, providing clear labels and payoffs for each player. 2. Use backward induction to make a prediction about the outcome. 3. Find out what actually happened and write a brief summary.