Explain what it means for an assignment model to be balanced. Give detailed mathematical examples.

In a job shop operation, four jobs may be performed on any of four machines. The hours required for each job on each machine are presented in the following table. The plant supervisor would like to assign jobs so that total time is minimized. Find the best solution. Which assignments should be made?

Explanation of the problem chapter 3.12 problem 49 of the book introduction to the mathematical programming 4th edition

Solution: Step # 1
Let Xij be the amount of money invested at the beginning of month i, for a period of j month.
Objective function: Step # 2
The objective is to maximize the available cash at the beginning of month 5.
X14 = collect the money invested at the beginning of month 1 of 4 months.
X23 = collect the money invested at the beginning of month 2 of 3 months.
X32 = collect the money invested at the beginning of month 3 of 2 months.
X41 = collect the money invested at the beginning of month 4 of 1 month.
Therefore the objective function is:
Maximize Z = 1.08 X14 + 1.03 X23 + 1.01 X32 + 1.001 X41
Restriction 1
It would be the money invested at the beginning of month 1 plus bills paid in month 1 which would be equal to the money available at month 1.

                       X11 + X12 + X13 + X14 + 600 = 400 + 400 (MONTH 1)
Restriction 2
It would be the money invested at the beginning of month 2 plus bills paid in month 2 which would be equal to the money available at month 2.

X21 + X22 + X23 + 500 = 1.001 X11 + 800 (MONTH 2)
Restriction 3 and 4
In the same way for the rest of the month we have:

X31 + X32 + 500 = 1.001 X12 + 1.001 X21 + 300 (MONTH 3)
X41 + 250 = 1.001 X13 + 1.001 X22 + 1.001 X31 + 300 (MONTH 4)

What I need is an explanation of the problem of how the data was extracted at each step

Now that we have seen Euler in action, let's return to examining the content of the M-File Euler.m. We have already explained the first line, where we defined the parameters our function takes. To see the meaning behind the third and fourth lines, type:

>> x = zeros(10,1);
y = zeros(10,1);
[x,y]

Thus we can see that the third and fourth lines of our M-File zero out the contents of our arrays x and y before we begin. Now, you'll continue examining our code.

Question: In your Word document, briefly explain what is happening in each remaining line of the M-File Euler.m.

The M-File is:

function [x,y] = Euler(h, x0, y0, interval_length, func)
nsteps = floor(interval_length/h) + 1;
x = zeros(nsteps,1);
y = zeros(nsteps,1);
x(1) = x0;
y(1) = y0;
for i=2:nsteps
    y(i) = y(i-1) + h* func(x(i-1), y(i-1));
    x(i) = x(i-1) + h;
end

Show all work

A ball is projected upward from the top of a 90 foot building at a velocity of 64 feet per second. The ball's height above the ground below the building is described by the function h(t)=-16t^2 + 64t + 90 , with t being the time in seconds after the ball is projected upward. a.) Determine the amount of the vertical intercept, and interpret what this means in the context of the problem (in terms of seconds and feet above the ground). b.) Determine the amount of all horizontal intercepts (if any) , and interpret what they mean in teh context of the problem (in terms of seconds and feet above the ground). c. Write the coordinates of the vertex. Interpret what these numbers mean in the context of the problem. d. If nothing stops the ball before, then how much time elapses until the ball hits the ground below the building?

In Exercises 3–6, find (a) the maximum value of Q(x) subject to
the constraint x^Tx = 1, (b) a unit vector u where this maximum is
attained, and (c) the maximum of Q(x) subject to the constraints
x^Tx = 1 and x^Tu = 0. Q(x) = 3x²₁ + 9x²₂ + 8x₁x₂

consider f(x) = ln(x)

a) Approximate f(0.9) and f(1.1)

b) Use Taylor remainder to find an error formula for Taylor polynomial.

Give error bounds for each of the two approximations in (a).

Which of the two approximations in part (a) is closer to correct value?

c) Compare an actual error in each case with error bound in part (b).

consider f(x) = ln(x)

a) Approximate f(0.9) and f(1.1)

b) Use Taylor remainder to find an error formula for Taylor polynomial.

Give error bounds for each of the two approximations in (a).

Which of the two approximations in part (a) is closer to correct value?

c) Compare an actual error in each case with error bound in part (b).

given the logistics equation y'=9(1-y)y
on the interval [0,5], w/ initial condition y(0)=1.25
1. what are the equilibrium Solutions? What is the long-term behavior of the solution?
2. solve the ode exactly and plot the solution on [0,5].

given the ivp y'=-y^2 on the interval [0,2] with initial condition y(0)=-1/2
1. solve the ode. what is the behavior of the true solution as x approaches 2?

Determine the solution of the following initial boundary-value problem using the method of separation of Variables

Uxx=4Utt 0<x<Pi t>0

U(x,0)=sinx 0<=x<Pi

Ut(x,0)=x 0<=x<Pi

U(0,t)=0 t>=0

U(pi,t)=0 t>=0

Find the biggest “*prime gap” (see definition) from the prime numbers between 1 and 1,000,000.

*Prime gap = the difference between two consecutive primes.

The exercise can be completed manually or with a computer program. Whichever seems easiest.

1.) A sheet of paper is cut into 6 same-size parts. Each of the parts is then cut into 6 ​same-size parts and so on.

a. After the 8th cut, how many of the smallest pieces of paper are​ there?

b. After the nth​ cut, how many of the smallest pieces of paper are​ there?

2.) Find the sum of the sequence 5+15+25+35+...65

3.)How many terms are there in each of the following sequences?

a. 1,2,2²,2³,...2²⁹⁹

b.9,13,17,21,...329

c.32,33,34,35,...432

The laser sight Jupiter uses for surveying is a little off. The mean error is 0.29m, meaning that it tends to provide measurements that are 0.29m too long. The standard deviation of the errors is 0.35m. She decides to recalibrate the device, but she wants to test it afterward to see if she made things better or worse. She collects a random sample of 43 measurements of a 100m distance. a) Identify the population of interest. b) Identify the variable of interest. What type of variable is it? c) ​​​​​​​If she measured the 100m distance BEFORE recalibrating, what would the mean of the measurements have been? d) ​​​​​​​If she wishes to assess how far off the sight is AFTER recalibrating, what parameter should she estimate? Mean error ? Std. Error .0676? e) ​​​​​​​Are the conditions for estimating the parameter you chose in part d met? What assumptions would you need to make? f)​​​​​​ Estimate the parameter you chose in part d with 99% confidence. Does her recalibration appear to have improved this situation? g)​​​​​​​ If she wishes to assess how reliable the sight is AFTER recalibrating, what parameter should she estimate? h) ​​​​​​​Are the conditions for estimating the parameter you chose in part g met? What assumptions would you need to make? i)​​​​​Estimate the parameter you chose in part g with 99% confidence. Does her recalibration appear to have improved this situation? j)​​​​​​​ Overall, do you think her recalibration made things better or worse?

sample data:

1.) Find the 100th and the nth term for each of the following sequences.

a. 1,3,5,7,...

b.70,100,130...

c.1,3,9...

d.8,8⁴,8⁷,8¹⁰,...

e. 200+6x2³¹, 200+7x2³¹, 200+8x2³¹

2. Find the first five terms in sequences with the following nth terms.

a. 2n²+6

b.4n+1

c. 10ⁿ-5

d.2n-1