Homework #6Study Guide 4

By: Abby Almonte

November 8, 2011

IE 327/L- Systems Engineering

Industrial Engineering DepartmentCalifornia State Polytechnic University, Pomona

Table of Contents

1.) Question 1: Cobb-Douglass Production Function……………………………………………………………………………………………………2

a. Historical Data…………………………………………………………………………………………………………………………………………………2

b. Natural Logarithm……………………………………………………………………………………………………………………………….…………..2

c. Regression Analysis………………………………………………………………………………………………………………………………….………3

d. Extract Coefficients………………………………………………………………………………………………………………………………….………4

2.) Question 2: Linear Programming………………………………………………………………………………………………….………………………..6

3.) Question 3: Regression Analysis…………………………………………………………………………………………………..…………………………8

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1- Use the handout on web page on using Excel to find Cobb-Douglas Production Function CDPF. Find CDPF for a function with four variables X1 to X4. Consider the following historical data. Try to come up with only one isoquant line (Perform trial and error, it may take some time). You can download the Excel file in relation to CDPF and make changes to it.

1.) Historical Data: Data on output (Z) and input (X1, X2, X3,X4)

Z X1 X2 X3 X46 80 210 400 15

12 210 200 455 385 50 230 431 128 180 210 499 257 150 290 350 20

11 190 220 455 4011 198 280 390 34

7 171 270 330 175 108 210 460 208 154 220 420 23

2.) Natural Logarithm of Historical Data

Ln (Z)Ln (X1)

Ln (X2)

Ln (X3)

Ln (X4)

1.791759

4.382027

5.347108

5.991465

2.70805

2.484 5.347 5.298 6.120 3.637

2

907 108 317 297 5861.609

4383.912

0235.438

0796.066

1082.484

9072.079

4425.192

9575.347

1086.212

6063.218

8761.945

915.010

6355.669

8815.857

9332.995

7322.397

8955.247

0245.393

6286.120

2973.688

8792.397

8955.288

2675.634

795.966

1473.526

3611.945

915.141

6645.598

4225.799

0932.833

2131.609

4384.682

1315.347

1086.131

2262.995

7322.079

4425.036

9535.393

6286.040

2553.135

494

3.) Regression Analysis

Regression Statistics

Multiple R0.939760

582R Square 0.883149

3

952Adjusted R Square

0.789669913

Standard Error0.145589

863Observations 10

ANOVA

df SS MS FSignificance

F

Regression 40.80100980

80.20025245

29.44747099

6 0.014972339

Residual 50.10598204

10.02119640

8 Total 9 0.90699185

CoefficientsStandard

Error t Stat P-valueLower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 5.7550 7.8449 0.7336 0.4961 -14.4110 25.9210 -14.4110 25.9210X Variable 1 -0.0176 0.2372 -0.0742 0.9437 -0.6274 0.5922 -0.6274 0.5922X Variable 2 -0.2638 0.6555 -0.4025 0.7039 -1.9488 1.4211 -1.9488 1.4211

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X Variable 3 -0.7874 0.7939 -0.9918 0.3668 -2.8281 1.2533 -2.8281 1.2533X Variable 4 0.8171 0.2952 2.7675 0.0395 0.0581 1.5761 0.0581 1.5761

4.) Extract coefficients from regression analysis

In a0 = 5.755 >> a0 =315.765547

5a1 = -0.018 a2 = -0.264 a3 = -0.787 a4 = 0.817

Z = 315.766 * X1-0.018 * X2

-0.264 * X3-0.787 * X4

0.817

5

If Z = 400

X1 X2 X3 X41 7 9 1.8 52 10 2 4 6.683 4 3.715 4 84 8 6 1.21 35 5 5 5 10.97

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2- We discussed a model in relation to LP in Study Guide 4. Write a summary of this problem in the form of a diagram. Diagram should consists of a rectangle that represents the problem and its model, one arrow towards the rectangle that represents “input information” for that model, and another arrow from the rectangle that represents “output information”. Under each arrow make a list of all input information used in that model and output information obtained from that model. Inside the rectangle write a sentence as what the problem is and copy/paste the model that represents that problem. For example for LP, copy the mathematical formulation. List the controllable and uncontrollable factors.

Input Information:

Coefficient of O.F. Limitations Usage of raw materials/labor

in each toy (not considered in this particular analysis)

Resources X1 X2 Total AvailableLabor (hour) 6 2 36Parts 5 5 40Packing Material 2 4 28Profit $5 $3

Output Information:

Max Z Number Xi If any resources are left

over Has multi optimal solutions

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Problem: Find out the number of toys for X1 and X2 that should be produced per hour in order to maximize profit.

Model:

Objective Function: Z = 5X1 + 3X2

Subject to: C1: 6X1 + 2X2 =< 36C2: 5X1 + 5X2 =< 40C3: 2X1 + 4X2 =< 28X1 and X2 => 0

Controllable Factors: Production Process Amount of hours workers work Schedule of workers Training for workers

Uncontrollable Factors: Demand Price of raw material Union price of workers (minimum wage of workers) Suppliers price

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3- Explain why and how we use regression analysis as a tool to find production function.

Regression analysis is a technique used to analyze relationship of two or more variables. It is used widely for forecasting and predicting, which means being able to make a rough estimate of the future through the use of constants and other variables. We use regression analysis to help us predict an isoquant line so that there is flexibility in decision making. An isoquant line is an unusual shaped line that shows all of the possible outcomes for the same value of Z. An example of the flexibility would be that the supplier that sells X1 products are going out of business, so they sell their products at half of the price. By referring to the isoquant line, the company can pick a point where more of X1 is used, and not a lot of X2 products are used.

To find product function using regression analysis, we can use Excel. First make a table that shows all of the Z and X values. Then, transform the values into natural logarithm or log10. It does not matter which one is used, as long as the same concept is used throughout the whole process. After finding Ln or Log10, we can perform a regression analysis. For Excel 2010, I had to insert the add-in in order to complete the function. I click on “File”, then “Options”, which are both circled below in purple in Figure 6.1.

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Figure 6.1

After clicking on “Options”, a new window opens up.

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Figure 6.2

After the window opens up, click on “Add-Ins” and click on “Go” beside the Manage drop down menu.

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Figure 6.3

An Add-Ins screen will then pop up. Click on “Analysis ToolPak”, and then press “OK”. By doing this, I am adding an add-in that will allow me to do regression analysis.

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Figure 6.4

I now have access to use analysis tools. It is shown under “Data” and in the analysis box, and is named “Data Analysis”, circled below in red.

Figure 6.5

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After clicking on data analysis, a box pops up. Scroll down to find regression, highlight regressing by clicking on it, and press “OK”.

Figure 6.6

Input the X and Y values by highlighting the values in the natural logarithm. Leave everything else as is, and press okay.

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Figure 6.7

After completing these steps, the summary output opens up in a new tab in the same excel file. After it opens up, you extract the coefficients from the X variables, and use the coefficients as the constants in the Z equation:

Z = a0 *X1a1 *X2

a2 *X3a3 * X4

a4

After you have the equation, you can predict the value of z using the x variables, or you can predict one of the x variables using the X and z variables.

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