the assignment needs to use Excel to solve all the problems. The original Excel file is not reuquired, but every step with formula needs to take screenshot. Every graph needs to be labelled and explained. Make sure you know how to solve all the problems and capable to finish them before the due time…

assignment_5.docx

aly6050___module_1_project_1_.pdf

phstat_4.zip

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Assignment 5

(2) The textile mill model •

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Produces 3 types of fabric (F1, F2, F3) on 2 types of machines (Regular, Special). Or Buy.

Time to delivery is 13 weeks.

15 regular machines (F2, F3) + 3 special machines (F1, F2, F3) + Buy (F1, F2, F3)

Demand for F1, F2, F3 is 45000, 76500, 10000 yards.

Special loom capacity in yards/hour – 4.7, 5.2, 4.4

Regular loom capacity in yards/hour – 0, 5.2, 4.4

Manufacture cost in $/yard – 0.65, 0.61, 0.50

Outsource cost in $/yard – 0.85, 0.75, 0.65

2.1. Minimize total cost to meet demand when mill operates 24×7. Hint: If your model is right, total

cost to meet demand = $83756.12.

2.2. Minimize total cost to meet demand when mill closes during weekends and special loom capacity

reduces by 50%. Clearly mention the change in total cost.

(3) Modelling and Optimization.

3.1. Chapter 13 – Solve problem 23.

assumptions?

Did you make any assumptions?

If yes, what are your

3.2. Repeat the problem when “the per-pound cost of holding inventory each month is estimated to be

2.2% of the cost of the product”. Clearly mention the total cost, (production + inventory) in both cases.

Assignment 5 is continued on next page…

(4) Chapter 13 – Solve problem 23 continued…

4.1. This is a continuation of the previous problem. In addition to the assumption “the per-pound cost

of holding inventory each month is estimated to be 2.2% of the cost of the product”, also assume that

the factory now closes during weekends. Run the solver to optimize model to meet demand and

minimize total cost. Clearly mention the total cost (production + inventory) of your model.

4.2. Can you explain a scenario that is beyond the limits/assumptions of your model?

(5) Assignment 4 – Problem 4 continued… You should already have columns Bin center, PDF and CDF

from frequency calculations. See slides. Calculate and plot the

5.1. PDF of Cost Difference = Outsourcing Cost – Manufacturing Cost and

5.2. CDF of Cost Difference = Outsourcing Cost – Manufacturing Cost.

using the envelop method. Use solver to fit the curves as required (make sure area under PDF = 1).

Explain solver settings in detail.

6. Job scheduling model

6.1. Chapter 14 – Solve problem 21.

6.2. Resolve the problem if Time to complete project 4 is 13 and deadline is 25.

7. Travelling salesman model

7.1. Chapter 14 – Solve problem 22.

7.2. Resolve the problem if you must start from location 8 and end at location 8.

Note: For all assignment problems, show all work. Show all equations in your model. Use CTRL+~ to

display all formulae in your model. Show constraints in solver. Solutions without model equations and

solver setup will not earn points.

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ALY-6050

Week 1 Project

1. This Week 1 Project has 100 points.

2. Solutions should consist of an Excel workbook and a Word document. Please attach both files

when submitting the project.

3. Perform all calculations and analysis In the Excel workbook.

4. In the Word document, write a report summarizing the results obtained in your Excel

workbook.

5. The project will be graded according to the following rubric:

The submission of each weekly project will consist of an Excel workbook and a Word document (an R

script file is optional)– a minimum of two submissions that have been submitted as attachments. For

each weekly project, students should complete their analytic work in an Excel workbook, and write a

minimum of 1000 words in a Word document describing their findings. The Word document should

consist of a title page (including student’s name, assignment title, course number and title, the current

academic term, instructor’s name, and the assignment completion date), and a reference page. The

Word submission of each project will consist of three sections:

(i)

(ii)

(iii)

Introduction

Analysis

Conclusion

The weekly projects will be graded by using the following criteria:

(a) Excel Workbook (60 points)

(i)

Problem set-up and modeling (40%)

(ii)

Problem solution and accuracy (60%)

(b) Word Document (40 points)

(i)

Description of the problem introduction (20%)

(ii)

Description of the problem analysis (30%)

(iii)

Description of conclusion (30%)

(iv)

Writing Mechanics, Title page, and References (20%)

2

Project:

The project consists of four sections. Please complete each section in a separate

worksheet, but in the same workbook. Name your Excel workbook as:

ALY6050-Week1Project-Your Last Name-First initial.xlsx.

For example: ALY6050-Week1Project-Behboudi-R.xlsx

If using R, you may include your solutions to all four problems in the same R script file. Make

sure that you separate your codes written for each individual problem by writing comments in

the R script file. Save your R script file as:

ALY6050-Week1Project-Your Last Name-First initial.R

In the Word document, explain the experiments and their respective conclusions, and

additional information as indicated in each problem below:

In the following set of problems, the notation R is used to refer to a standard uniform random

variable (a continuous random value between 0 and 1) and r is its numerical value.

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Problem 1:

Generate 10000 random values r. For each r generated, calculate the random value by:

= − ( ),

where “ “ is the natural logarithm function.

Investigate the probability distribution of the random variable by doing the following:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Create a relative frequency histogram of .

Select a probability distribution that, in your judgement, is the best fit for .

Support your assertion above by creating a probability plot for .

Support your assertion above by performing a Chi-squared test of best fit with a 0.05

level of significance.

In the word document, describe your methodologies and conclusions.

In the word document, explain what you have learned from this experiment.

Hints and Theoretical Background:

A popular method for generating random values according to a certain probability distribution is to use

the inverse transform method. In this method, the cumulative function of the distribution ( ( )) is

used for such a random number generation. More specifically, a standard uniform random value is

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generated first. Most software environments are capable of generating such a value. In Excel and R,

functions “=RAND()” and “runif()” generate such a value respectively. After has been created, it

then replaces ( ) in the expression of the cumulative function and the resulting equation is solved for

the variable .

For example, suppose we wish to generate a random value according to the exponential distribution

with a certain mean (say ). The cumulative function for the exponential distribution is:

( ) = −

−

(The quantity in the above description is called the rate of the exponential random variable and is

denoted by .)

Therefore, to generate a random value that belongs to the exponential distribution with a mean of .

We first generate a standard uniform value , then replace ( ) by in the above expression, and solve

the resulting equation for the variable :

= −

−

−

−

= −

= ( − )

= − ( − )

The formula above means that if is a standard uniform random variable, then the random variable

obtained by the expression = − ( − ) will belong to the exponential distribution with an

average which is equal to the value of . This formula can be simplified as:

= − ( )

(Note that If is a standard uniform random variable, then (1 − ) is also standard uniform.)

A special case of the above formula is when = . This means that a random variable generated by

the formula = − ( ) is an exponential random variable with an average of 1 (or, rate=1).

**************************************************************************************************

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Problem 2:

Generate three sets of standard uniform random values , and , each consisting of

10,000 values. Next, Calculate the random value according to the following formula:

= − ( ).

Investigate the probability distribution of the random variable by doing the following:

(i)

(ii)

(iii)

Create a relative frequency histogram of .

Select a probability distribution that, in your judgement, is the best fit for .

Support your assertion above by creating a probability plot for .

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(iv)

(i)

(ii)

Support your assertion above by performing a Chi-squared test of best fit with a 0.05

level of significance.

In the word document, describe your methodologies and conclusions.

In the word document, explain what you have learned from this experiment.

Hints and Theoretical Background:

This problem is related to a theorem in the probability theory. The theorem states that:

If , , … , are identical and independent exponential random variables each with a mean of , then

the random variable obtained by their sum, that is + + … + , will have a ( , )

probability distribution, where is the shape parameter of the Gamma distribution and = .

From the Hints and Theoretical Background of Problem 1, we know that if is a standard uniform

random variable, then = − ( ) is an exponential random variable with an average of 1. Therefore,

if , , and are three independent standard uniform random variables, then = − ( ) ,

= − ( ) , and = − ( ) are three independent and identical (each with a mean of 1)

exponential random variables. Thus, according to the theorem above, the random variable formed by

their sum, that is (− ( )) + (− ( )) + (− ( )) , will belong to the ( , )

probability distribution.

However algebraically,

(− ( )) + (− ( )) + (− ( )) = −( ( ) + ( ) + ( )) = − ( ).

Therefore, if , , and are three independent standard uniform random variables between zero

and 1, then the random variable formed by the formula = − ( ) will belong to the

( , ) probability distribution.

***************************************************************************************************

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Problem 3:

Generate a set of 10,000 pairs of standard uniform random values and . Then perform the

following algorithm for each of these 10000 pairs: Let the output of this algorithm be denoted

by .

Step 1: Generate random values = − ( ) and = − ( ).

Step 2: Calculate =

( − )

. If ≤ , then generate a random number . If > . accept

as (that is, let = ); otherwise if ≤ . , accept − as (that is, let = − ).

If > , no result is obtained, and the algorithm returns to step 1. This means that the algorithm

skips the pair and for which < without generating any result and moves to the next pair
and .
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After repeating the above algorithm 10,000 times, a number of the values will be generated.
Obviously ≤ , since there will be instances when a pair and would not generate
any result, and consequently that pair would be wasted.
Investigate the probability distribution of by doing the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Create a relative frequency histogram of .
Select a probability distribution that, in your judgement, is the best fit for .
Support your assertion above by creating a probability plot for .
Support your assertion above by performing a Chi-squared test of best fit with a 0.05
level of significance.
In the word document, describe your methodologies and conclusions.
In the word document, explain what you have learned from this experiment.
Hints and Theoretical Background:
Other than the inverse transform method used for generating random values that are according to a
certain particular probability distribution, a second applied method for generating random values is the
Rejection algorithm. The details of this algorithm are explained below:
Suppose we wish to generate random values that is according to a certain probability distribution
with ( ) as its probability density function (pdf). Also suppose that the following two conditions are
satisfied (i) we are able to generate random values that belong to a probability distribution whose
probability density function is ( ), (ii) there exists a positive constant such that
( )
( )
≤ for all
( )
values (this means that the ratio ( ) is always bounded and does not grow indefinitely. This condition
is almost always satisfied for any two probability density functions ( ) and ( )).
The rejection algorithm can now be implemented as follows:
Step 1: Generate a random value that belongs to the probability distribution with ( ) as its pdf and
generate a standard uniform random value .
( )
Step 2: Evaluate = ( ). If ≤ , then accept as the random variable (that is, let = ); otherwise
return to Step1 and try another pair of ( , ) values.
A few remarks about the Rejection algorithm is worth noting:
1. The probability that the generated value will be accepted as , is:
( )
.
( )
This is the reason
why the algorithm uses a standard uniform value and accepts as if ≤
( )
.
( )
2. Each iteration of the algorithm will independently result in an accepted value with a probability
( )
equal to: ( ≤ ( )) = . Therefore, the number of iterations needed to generate one
accepted value follows a geometric probability distribution with mean .
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Relevancy of Problem 3 to the Rejection Algorithms:
In problem 3, the random variable , selected from an exponential probability distribution with =
and a pdf of ( ) = − , is used to first generate the absolute value of a standard normal random
variable (| | has the pdf: ( ) =
√
− ), and then assign positive or negative signs to this
value (through a standard uniform variable ) in order to obtain a standard normal random
value. It can be shown algebraically that
−( − )
( )
( )
=√
− ( − )
≤√
for all values (note that
≤ for all values). Therefore, the constant in the assumptions of the algorithm can be
chosen to be: = √
≈ . . Therefore,
( )
( )
=
− ( − )
. Hence the following algorithm can be
used to generate the absolute value of a standard normal random variable:
Step 1: Generate random variables and ; with being exponential with ate=1, and being uniform on
( , )
− ( − )
Step 2: If ≤ , then accept as the random variable (that is, set = ); otherwise return to
Step1 and try another pair of ( , ) values.
Note that in step 2 of the above algorithm, the condition ≤
− ( ) ≤
( − )
.
− ( − )
is mathematically equivalent to:
However, we have already seen in the Hints and Theoretical Backgrounds of the
earlier problems that if is standard uniform, then − ( ) is exponential with rate=1. Therefore, the
algorithm for generating the absolute value of the standard normal random variable can be modified as
follows:
Step 1: Generate independent exponential random variables and ; each with ate=1.
Step 2: Evaluate =
( − )
.
If ≤ , then accept as the random variable (that is, set = );
otherwise return to Step1 and try another pair of ( , ) values.
In fact, it is the above version of the Rejection algorithm that is being implemented in Problem 3.
However, in order to obtain a standard normal random value (instead of its absolute value), the step 2
of the above algorithm has been modified as follows:
Step 2: Evaluate =
( − )
.
If ≤ , then generate a standard uniform variable . If ≥ . , set
= , otherwise set = − . If > , return to step 1 and try another pair of ( , ) values.

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Note: The standard normal random value generated by the Rejection algorithm can be used to generate

any normal random value with a mean and a standard deviation . Once a standard normal variable

has been generated, it suffices to evaluate + to generate the desired normal variable.

•

Problem 4:

In the algorithm of problem #3 above, there are instances when the generated random values

do not satisfy the condition ≥ In order to obtain an acceptable value for . In such cases,

the algorithm returns to step 1 and generates another two values to check for acceptance. Let

be the number of iterations needed to generate of the values ( ≤ ). Let =

.

(For example, suppose that the algorithm has produced 7000 of the values ( = ) after

10,000 iterations ( = , ). Then =

= . . This means that it takes the

algorithm 1.43 iterations to produce one output. In fact, itself is a random variable.

Theoretically, ( ) – the expected value (i.e., average) of – of an algorithm is a measure of

the efficiency of that algorithm.)

Investigate by the following sequence of exploratory data analytic methods:

(i)

(ii)

Estimate the expected value and the standard deviation of .

As the number of iterations becomes larger, the values will approach a certain

limiting value. Investigate this limiting value of by completing the following table and

plotting versus . What value do you propose for the limiting value that

approaches to?

M

10

20

30

40

50

60

70

80

90

100

200

300

400

500

W

8

600

700

800

900

1000

(iii)

(iv)

In the word document, communicate to the reader your findings about .

In the word document, explain what you have learned from this experiment.

*****************************************************************************************

For Problems 1-4,

In the word document, write a summary of the results that you have obtained in the above

three experiments. Try to conceptualize those results and state whether the results of your

experiments are consistent with the theoretical facts that you have reviewed in the Hints and

Theoretical Background sections. In particular, address the following:

(i)

(ii)

(iii)

(iv)

(v)

If is a standard uniform random variable, then − ( ) has the _________

probability distribution.

The sum of th …

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