Suppose that there are three Suppliers (Supplier A, Supplier B, Supplier C) and we are interested to evaluate the performance of the suppliers by using AHP method. Some of the criteria for supplier selection process can be defined as follows:

Cost, Quality, and Timeliness.

1. We would like to add more factors as criteria to the three above mentioned criteria. What else factor can be defined as criteria? Define at least two more criteria.
2. By considering only the above three criteria (Cost, Quality, and Timeliness), you first introduce three different suppliers that you may know and further rank these three alternative suppliers and select the best one using AHP method. The pairwise comparison matrix for each of the three suppliers for each criterion and the comparison matrix for the three criteria can be given by you based on your judgment and experiences.   

Make up an example to show that Dijkstra’s algorithm fails if negative edge lengths are allowed.

Group theory

Consider the group GL₂(Z_p) of invertible 2X2 matrices with entries in the field Zp, where p is an odd prime.

Zp is an abelian group under addition, the group of unites of Zp is Zp^x, which is an abelian group under multiplication. We say (Z_p , +, ·) is a field.

1. 1. Show that the subset D₂(Z_p) of diagonal matrices in GL₂(Z_p) is an abelian subgroup of order (p - 1)².
2. 2. For A, B ∈GL₂(Z_p), show that A and B are in the same right D₂(Z_p)-coset if and only if there are non-zero elements λ, µ ∈ Z_p such that A can be obtained from B by multiplying the first column by λ and the second column by µ.
   3. For A, B∈GL₂(Z_p), show that A and B are in the same left D₂(Z_p)-coset if and only if there are non-zero elements λ, µ ∈Z_p such that A can be obtained from B by multiplying the first column by λ and the second column by µ.
   4. Find a matrix   A such that the left and right D₂(Z_p) cosets containing A are different, that is, so that D₂(Z_p)A ≠ AD₂(Z_p).
   5. Find a non-identity matrix B such that the left and right D₂(Z_p) cosets containing B are the same, that is, so that D₂(Z_p)A = AD₂(Z_p).

3. The effect of financial leverage on ROE

Companies that use debt in their capital structure are said to be using financial leverage. Using leverage can increase shareholder returns, but leverage also increases the risk that shareholders bear.

Consider the following case:

Water and Power Co. is a small company and is considering a project that will require $500,000 in assets. The project will be financed with 100% equity. The company faces a tax rate of 25%. What will be the ROE (return on equity) for this project if it produces an EBIT (earnings before interest and taxes) of $140,000?

23.10%

21.00%

15.75%

22.05%

Determine what the project’s ROE will be if its EBIT is –$40,000. When calculating the tax effects, assume that Water and Power Co. as a whole will have a large, positive income this year.

-6.0%

-7.20%

-6.90%

-5.70%

Water and Power Co. is also considering financing the project with 50% equity and 50% debt. The interest rate on the company’s debt will be 13%. What will be the project’s ROE if it produces an EBIT of $140,000?

33.86%

32.25%

35.48%

27.41%

What will be the project’s ROE if it produces an EBIT of –$40,000 and it finances 50% of the project with equity and 50% with debt? When calculating the tax effects, assume that Water and Power Co. as a whole will have a large, positive income this year.

-28.27%

-27.19%

-23.92%

-21.75%

The use of financial leverage _______ the expected ROE, _________ the probability of a large loss, and consequently __________ the risk borne by stockholders. The greater the firm’s chance of bankruptcy, the__________    its optimal debt ratio will be.___________    manager is more likely to use debt in an effort to boost profits.

The CPI was 214.537 in 2009 and 232.957 in 2013 and the Tuition per credit hour was $99.65 in 2009 and $116.15 in 2013

(i) Find the absolute change for the CPI from 2009 to 2013.
Express your answer rounded to the nearest thousandth.

(ii) Find the relative change for the CPI from 2009 to 2013.
Express your answer rounded correctly to the nearest tenth of a percent.
%

(iii) Find the absolute change for the Tuition from 2009 to 2013.
Express your answer rounded to the nearest cent.
$

(iv) Find the relative change for the Tuition from 2009 to 2013.
Express your answer rounded correctly to the nearest tenth of a percent.
%

Which has increased more in this time period?  ? CPI Tuition

If you haven't answered the question correctly in 3 attempts, you can get a hint

Let A and B be two non empty bounded subsets of R:

1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A + sup B

2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup A

hint:( show c supA is a U.B for cA and show if l < csupA then l is not U.B)

(Connected Spaces)
(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k is connected.

Prove: An (n × n) matrix A is not invertible ⇐⇒ one of the eigenvalues of A is λ = 0

. Determine whether K4 (the complete graph on 4 vertices contains the following: i) A walk that is not a trail. ii) A trail that is not closed and is not a path. iii) A closed trail that is not a cycle.

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E.
b) Prove that every line segment between two points in R^k
is connected, that is Ep,q = {tp + (1 − t)q |
t ∈ [0, 1]} for any p not equal to q in R^k.
C). Prove that every convex subset of R^k is connected.

Real Mathematical Analysis, Pugh, 5.29 : Prove Corollary18 that r^th-order differentiability implies symmetry of D^rf, r ≥ 3. Use induction to show that (D^rf)_p (v₁,.....,v_r) is symmetric with respect to permutations of v₁,...,v_r−1 and of v₂,...,v_r. Then take advantage of the fact that r is strictly greater than 2. (Please provide a formal proof. Thanks)

Corollary 18: The r^th derivative, if it exists, is symmetric: Permutation of the vectors v₁,...,v_r does not affect the value of (D^rf)_p(v₁,...,v_r). Corresponding mixed higher-order partials are equal.

Problem 1.      Smoke Sensors, Inc. (SSI), is experiencing a tremendous growth in demand for its household smoke detectors. SSI produces both an AC model and a battery-operated model. It has an opportunity to be the exclusive supplier for a major department store chain, The Seers Company. Seers wishes to receive at least 20,000 AC models and 10,000 battery-operated models each week.

SSI's unanticipated prosperity has left it short of sufficient capacity to satisfy the Seer's contract over the short run. However, there is a subcontractor who can assist SSI by supplying the same types of smoke detectors. SSI must decide how many units it will make of each detector and how many units it will buy from the subcontractor. Data below summarize the production, price, and cost parameters.

Model

(hours per unit)

The subcontractor can supply any combination of battery or AC models up to 20,000 units total each week. The cost per unit to SSI is $21.50 and $20.00, respectively, for the AC and battery models. The contract with Seers calls for SSI to receive $25.00 for each AC model and $29.50 for each battery model.

Hint: Table below shows the possibilities.

1. Formulate the LP model which would allow SSI to determine the number of units of each type to produce and to buy to maximize total profit.
2. Use solver to solve the problem. Give the values of the decision variables, slacks, and the value of the objective function.

Solve the following initial value problem, showing all work. Verify the solution you obtain. y^''-2y^'+y=0; y(0)=1,y^' (0)=-2.