Offer one example of an IT or computer application that can be modeled as the TSP problem. This must be at least one paragraph. (something other than scheduling a bunch of jobs on a single machine)

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.)

(a) Show that F is a σ-algebra.

(b) Show that the set function given by

μ(E)= 0 if E is countable ;

μ(E) = ∞ otherwise

is a measure, where E ∈ F.

The equation (sqrt(x^2+y^2) - R)^2 +z^2=r^2 represents a torus. (a) Find a suitable parametrization of torus. (b) Compute surface area of torus.

Consider the following quadratic forms

q(x1, x2) = 3x1^2 − 6x1x2 + 11x2^2 and

r(x1, x2, x3) = x1^2 − x2^2+x3^2+ 2x1x2 − 6x1x3+2x2x3,

on R 2 and R 3 , respectively. In both cases do the following.

(a) Find the symmetric matrix A representing the quadratic form.

(b) Find a corresponding orthogonal matrix P of eigenvectors of that matrix.

(c) Write down the maximum and minimum values of the quadratic form over the unit vectors (in R 2 and R 3 , respectively).

Please include steps and answer a), b) and c)

1A) Let S be the upward oriented surface of the box [-pi,pi]x[0,pi]x[0,pi]without the face xy plane. That is, in the standard view, the box has a front and back, a left and right face, a top face, but no bottom face. Let F(x,y,z)=< ycos(z),zcos(x),xcos(y) >. Find the flux of curl F across S directly,  without using stokes theorem.

1B)  Let S be the upward oriented surface of the box [-pi,pi]x[0,pi]x[0,pi]without the face xy plane. That is, in the standard view, the box has a front and back, a left and right face, a top face, but no bottom face \. Let F(x,y,z)= < ycos(z),zcos(x),xcos(y) >. Find the flux of curl F across S using line integrals (stokes theorem).

DOE

SIX SIGMA

Factorial Design (DOE)

An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results show the averages of the three replicates

1. Estimate the factor effects. (Please show steps)
2. Which effects appear to be large?

Identify historical uses of statistics and probability for great good and for great harm. How can ethics and values be applied in this field? Discuss A Modest Proposal, by Jonathan Swift as part of your response.

Step 1: Identify and Solve a Typical Problem
There are a number of typical models in the Operations Research field which can be applied to a wide range of supply chain problems. Select one of the following typical models:

• Travelling Salesperson Problem (TSP)
• Multiple Traveling Salesman Problem (mTSP)
• Knapsack Problem
• Vehicle Routing Problems (VRP)
• Job Shop Scheduling
• Parallel Machine Scheduling
• Christmas lunch problem
• Newsvendor problem
• Pickup and delivery
• Travelling thief problem
• Eight queens problem
• Minimum Spanning Tree
• Hamiltonian path problem

1.1. Background:
• Provide a detailed explanation of the selected problem.

1.2. Model
• Provide typical mathematical model of the selected problem and clearly explain different aspects of the model (e.g. decision variable, objective function, constraints, etc.)

1.3. Solving an Example
• Develop a mathematical model for a workable and reasonable size of the problem.
– For many typical problems, when size of the problem increases, it becomes NP-Hard. In other words, your computer will not be able to solve it mathematically. Therefore, ‘workable and reasonable size’ here means that size of the selected problem should not be too small or too large.
• Solve the problem in Excel and transfer your solution to Word. It is required that details and steps of getting the solution are provided in the Word document.
• Interpret the findings and discuss.

Step 2: LR on Application of Selected Typical Model in Design and Analysis of Supply Chain
• Identify at least 5 peer reviewed articles in which your selected typical problem has been employed to address knowledge gaps in supply chain field.
– At least one of the selected articles should be published after 2010.
• Write a comprehensive literature review on the application of “your selected” typical model in design and analysis of supply chain and address the following (but not limited to) points:
– What type of problems in supply chain can be addressed by the selected typical problem?
– Compare similarities and differences of selected articles.
– Discuss the suitability of using the selected typical model in design/analysis of various supply chains.
– What are the limitations of your selected typical problem?
– Undertaking any additional critical and/or content analysis on the application of selected typical problem in design and analysis of supply chain is highly recommended.

Step 3: Summary of Findings
• A summary of findings regarding the strengths and weaknesses of the selected typical problem in design and analysis of supply chain should be summarised in this section.
Note:
• From each article something unique should be explained in the report. • Word limit: 2500 ± 500 words

(a) Show that there are, up to isomorphism, exactly 8 matroids whose underlying set has three elements. Calling the elements a, b, c, exhibit, for each of these matroids, its bases, cycles and independent sets.

(b) Consider the matroid M on the set E = {a, b, c, d}, where the bases are the subsets of E having precisely two elements. Detrmine all the cycles of M, and show that there is no graph G such that M is the cycle matroid M(G).

8.54. Show that permutation programs are not always optimal for continuous production,
that is, provide a counterexample.

Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a nonnegative integer and prove that your formula is valid.

a) f(0) = 1, f(n) = -f(n-1) for n ≥ 1

b) f(0) = 1, f(1) = 0, f(2) = 2, (n) = f(n-3) for n ≥ 3.

c) f(0) = 0, f(1) = 1, f(n) = 2 f(n + 1) for n ≥ 2

d) f(0) = 0, f(1) = 1, f(n) = 2 f(n - 1) for n ≥ 1

e) f(0) = 2, f(n) = f(n – 1) if n is odd and n ≥ 1 and f(n) = 2f(n-2) if n ≥ 2.

Kindly solve this in 15 minutes, and kindly ignore it if i did not get answer on time.

Calculate mean, median and mode of the following   

class Interval:              0-9       10-19   20-29   30-39   40-49   50-59   60-69 70-79

No. of students:           9        13        27        57        64        46        31        9

1. In the UK sales of music CDs per year have declined from 60 million in 2013 to 48 million in 2016 and further to 32 million in 2018.

   1. (a) What was the average rate of change in CD sales from 2013 to 2016 and what was the average rate of change from 2016 to 2018.

   2. (b) Based on this data, is number of CDs sales decreasing linearly? Give a reason for your answer.

2. The cost of producing x solar panels is c(x) = 1000 + 500x − 20x2 (in dollars).

   1. (a) Using the definition of the derivative from lectures, compute the marginal cost when 10 solar panels have already been produced.

   2. (b) Compare this to the actual extra cost for producing an 11th solar panel.

   3. (c) Work out the equation for the tangent to the graph of c(x) at x = 10.