Homework #6
Study Guide 4
By: Abby Almonte
November 8, 2011
IE 327/L- Systems Engineering
Industrial Engineering Department
California 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
6
12
5
8
7
11
11
7
5
8
X2
80
210
50
180
150
190
198
171
108
154
X3
210
200
230
210
290
220
280
270
210
220
X4
400
455
431
499
350
455
390
330
460
420
15
38
12
25
20
40
34
17
20
23
2.) Natural Logarithm of Historical Data
Ln (Z)
1.791759
2.484907
1.609438
2.079442
Ln (X1)
4.382027
5.347108
3.912023
5.192957
Ln (X2)
5.347108
5.298317
5.438079
5.347108
Ln (X3)
5.991465
6.120297
6.066108
6.212606
Ln (X4)
2.70805
3.637586
2.484907
3.218876
2
1.94591
2.397895
2.397895
1.94591
1.609438
2.079442
5.010635
5.247024
5.288267
5.141664
4.682131
5.036953
5.669881
5.393628
5.63479
5.598422
5.347108
5.393628
5.857933
6.120297
5.966147
5.799093
6.131226
6.040255
2.995732
3.688879
3.526361
2.833213
2.995732
3.135494
3.) Regression Analysis
Regression Statistics
Multiple R
0.939760582
R Square
0.883149952
Adjusted R Square
0.789669913
Standard Error
0.145589863
Observations
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Regression
Residual
Total
ANOVA
df
SS
MS
F
Significance F
4 0.801009808 0.200252452 9.447470996 0.014972339
5 0.105982041 0.021196408
9 0.90699185
3
Intercept
X Variable 1
X Variable 2
X Variable 3
X Variable 4
Coefficients
5.7550
-0.0176
-0.2638
-0.7874
0.8171
Standard
Error
7.8449
0.2372
0.6555
0.7939
0.2952
t Stat
0.7336
-0.0742
-0.4025
-0.9918
2.7675
P-value
0.4961
0.9437
0.7039
0.3668
0.0395
4.) Extract coefficients from regression analysis
In a0 =
a1 = -0.018
a2 = -0.264
a3 = -0.787
a4 = 0.817
5.755 >> a0 =
315.7655475
Z = 315.766 * X1-0.018 * X2-0.264 * X3-0.787 * X40.817
4
Lower
95%
-14.4110
-0.6274
-1.9488
-2.8281
0.0581
Upper
95%
25.9210
0.5922
1.4211
1.2533
1.5761
Lower
95.0%
-14.4110
-0.6274
-1.9488
-2.8281
0.0581
Upper
95.0%
25.9210
0.5922
1.4211
1.2533
1.5761
If Z = 400
X1
1
2
3
4
5
X2
7
10
4
8
5
9
2
3.715
6
5
X3
1.8
4
4
1.21
5
X4
5
6.68
8
3
10.97
5
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
Labor (hour)
Parts
Packing Material
Profit
Problem:
Find out the number of toys for X1 and X2 that should
be produced per hour in order to maximize profit.
X1Model:
X2
Total Available
6
2
36
5 Objective
5 Function:
40
Z = 5X1 + 3X2
2 Subject
4 to: 28
$5
$3C1: 6X1 + 2X2 =< 36
C2: 5X1 + 5X2 =< 40
C3: 2X1 + 4X2 =< 28
X1 and X2 => 0
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Output Information:




Max Z
Number Xi
If any resources are left
over
Has multi optimal solutions
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 *X2a2 *X3a3 * X4a4
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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