[Type the company name]
Operations Management
Lab Exercises and Examples
Howard J. Weiss
August 20, 2013
Table of Contents
Productivity ..................................................................................................................................... 1
Project Management ...................................................................................................................... 6
Forecasting .................................................................................................................................... 13
Statistical Process Control............................................................................................................. 25
Product Design and Decision Making ........................................................................................... 46
Capacity Planning .......................................................................................................................... 61
Location & Transportation ............................................................................................................ 72
Layout............................................................................................................................................ 81
Waiting Lines ................................................................................................................................. 87
Inventory ....................................................................................................................................... 91
Aggregate Planning ....................................................................................................................... 98
MRP & Lot Sizing ......................................................................................................................... 103
Scheduling ................................................................................................................................... 111
Productivity
Lab data file: productivity.lab.xlsx
Objectives
 To introduce you to the two software packages
 To explain productivity measures
Terminology & Concepts



Productivity
Input
Output
Excel Functions & Options
 Data Sort
 Filters
 Copy from and Paste to
Preparation: The preparation each week will be the same.
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Productivity is defined very simply as
𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 =
𝑂𝑢𝑡𝑝𝑢𝑡
𝐼𝑛𝑝𝑢𝑡
Often times, but not always, the output is the number of units produced or customers served.
There can be many different inputs. Sometimes we use only one input and other times we
collapse several inputs into one measure which is typically Total Input Dollars.
Exercise 1: Computing the single-factor productivity. (Excel)
The table below and the spreadsheet named Exercise 1 contains the total salaries and the
number of wins for all of the Major League Baseball teams during the decade of 2000-2009. The
author has taken the ratio of Salary (input) to number of wins (output) and computed a Cost
per Win measure. This, of course, is the opposite (inverse) of productivity which is output/input
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rather than input/output. Let us compute the 30 productivity measures, one for each team, and
sort the data from most productive team to least productive team.
Payroll
CPW
Payroll
CPW
Team
($000,000) Wins
($000)
Team
($000,000) Wins
($000)
Angels
$887.20
900
98.5
Nationals
$464.30
711
65.3
Astros
$771.60
832
92.7
Orioles
$717.20
698
102.7
Athletics
$522.60
890
58.7
Padres
$546.70
769
71.1
Blue Jays
$679.00
805
84.3
Phillies
$794.60
850
93.5
Braves
$927.00
892
104
Pirates
$437.10
681
64.2
Brewers
$527.20
741
71.1
Rangers
$710.80
776
91.6
Cardinals
$843.10
913
92.3
Rays
$400.80
694
57.8
Cubs
$906.80
807
112.4
Red Sox
$1,168.10
920
127
D-Backs
$730.10
805
90.7
Reds
$583.20
751
77.7
Dodgers
$1,002.00
862
116.2
Rockies
$612.40
769
79.6
Giants
$789.10
855
92.3
Royals
$473.60
672
70.5
Indians
$650.50
816
79.7
Tigers
$761.30
729
104.4
Mariners
$880.80
837
105.2
Twins
$501.90
863
58.2
White
Marlins
$349.00
811
43
Sox
$773.50
857
90.3
Mets
$1,086.50
815
133.3
Yankees
$1,685.30
965
174.6
http://sportsillustrated.cnn.com/2010/writers/tom_verducci/01/19/verducci.efficiency/index.h
tml?eref=sihp
In order to preserve the original data we will make a copy of the original worksheet.




Right-click on the Exercise 1 tab at the bottom of the sheet.
Select Move or Copy
Check the Create a Copy box at the bottom
Click OK.
Now we will begin our (simple) work. The first part is to compute the 32 productivity measures.
 Label Cell E5 as “Productivity”
 In cell E6 place the formula for productivity – “=C6/B6”. This should yield a productivity
of 1.01. The number may be formatted as $ when it should not be since it is wins per
(million) dollars. Reformat from $ to number if necessary.
 Copy Cell E6 down to the last row of the data.
The last step is to sort the results from the most productive team to the least productive teams.
There are two ways we can accomplish this.
 Select all cells, including the column labels. That is, select cells A5:E35.
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




Click on the Data tab.
Click on Sort
Under column change Sort by to Productivity.
Change Order to Largest to Smallest
Click OK
Place the answers to the following questions in the cells indicated in the spreadsheet.
1. Which team was the most productive during the decade?
2. Which team was the least productive during the decade?
There is another very useful, easy way to perform the sort. Since the data has already been
sorted by productivity we will demonstrate sorting the data by number of wins.





Select row 5, the row of column labels.
Click on the Data tab.
Click on Filter. Notice that the column headers now have drop down boxes.
Click on the drop-down box in column C, wins.
Select Sort Largest to Smallest
3. Which team won the most games from 2000-2009?
4. Which team won the least games from 2000-2009?
Exercise 2: Using POM to compute a single factor productivity.
Either software package can be used to compute a single factor productivity measure. Let’s
consider the example in the textbook. A company produced 1000 units using 250 labor-hours. It
is easier to compute this by hand (1000/250) but let us use POM for Windows simply to be
exposed to the software and the steps involved in using POM for Windows to solve a problem.
 From the module menu of POM for Windows select Productivity. Note that each week
we will be working with one or two of the modules.
 Select File, New. Note that for some modules when you select New there will be a list of
submodels and for others, like productivity, you will be directed right to the creation
screen.
 Enter 1 for the number of inputs.
 Enter 1 for the number of time periods.
 Click OK
 In the Output row under Period 1 place 1000.
 In the Input row, in the far left column, place the name “Labor-hours”. It is always a
good idea to label the rows or columns when given a chance rather than leaving the
default values in.
 In the input row under Period 1 place 250.
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Notice that we are leaving empty the cell in the column labeled “$/input (optional)”. This cell
could be used to convert labor-hours to dollars as will be the case in the second example.
At this point the data is entered so we are ready to solve the problem. The easiest way to solve
a problem is to press the Solve icon on the far right hand side of the first tool-bar. Alternatively,
you may use File, Solve or the F9 function key. After solving, the main solution window will
appear. For some models, additional output windows may be found and can be seen under the
Window menu item.
1. What is the productivity as measured in units/labor-hour?
Tables can be copied from POM for Windows to Excel and this is a good way to have all of the
results from any lab in one workbook.
 Right-click on the solution table in POM for Windows.
Notice that the choices are to Copy the Table, Copy the Row or Copy the Column. You can not
copy an arbitrary region (rectangle) of cells as you can in Excel.




Select Copy Table
Go to the Exercise 2 sheet in the lab file
Right click on cell A1
Select Paste
Exercise 3: Multifactor productivity and productivity comparisons (Excel OM)
Often times, there may be more than one input factor. In these cases, we combine the factors
into one Input measure by converting the factors to dollars. We will consider Example 2 from
the textbook. The table below contains the information from the problem. In this case we have
two input factors. The first is labor-hours while the second is overhead. We need to convert
these to dollar amounts. The labor hour rate is $20 per hour while overhead is already given in
dollars so its conversion factor is simply 1.
Output
Labor hours
Overhead
Period 1
8
32
400
Period 2
14
32
800
$ per unit
20
1
 Click on the Excel OM tab in Excel.
 Click on the Alphabetical menu from the Excel OM tab. This will display the drop down
list of modules
 Click on Productivity
 Enter 2 factors
 Enter 2 time periods
 Click OK
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A new spreadsheet has been added to the workbook. The spreadsheet is setup with the
formulas that we need for the computations. All we need to do is to enter the data into the
orange cells.
 Copy the table from the spreadsheet named Exercise 3 into the data area (orange) on
the spreadsheet created by Excel OM.
Check that the results in the Excel OM spreadsheet match the results given in the text for
Example 2. This problem is not terribly hard but it is easier to determine all of the results using
the software rather than doing it by hand. As a rule, we are more interested in the changes in
productivity from period to period than we are in the productivity measure itself. For example,
the multifactor productivity for the old system and the new system have both been computed.
The most important number is the percentage change in productivity of a 26% increase when
productivity goes from .0077 to .0097. Note that it is easier to enter two periods into one
model rather than solving the two models separately. That is, we could have run a one period
productivity model for the old system and a separate one period productivity model for the
new system but then we would have had to compute the percentage changes by hand rather
than letting the software do it for us.
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Project Management
Lab data file: project.lab.xlsx
Objectives
 To introduce you to the two software packages we will be using this semester
 To explain the 3 different project management models we will be examining this
semester
Terminology & Concepts
 Critical path
 Project completion time
 Project completion time standard deviation
 Project completion time standard variance
 Activity time
 Activity time standard deviation
 Activity time variance
 Early start
 Early finish
 Late start
 Late finish
 Optimistic time
 Most likely time
 Pessimistic time
 Slack
 Activity normal time
 Activity crash time
 Activity normal cost
 Activity crash cost
Excel Functions & Options
 Conditional formatting
 Normal Distribution functions
NORMDIST; NORMINV; NORMSDIST; NORMSINV
 Paste Special
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
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Exercise 1: Single time estimate model using POM for Windows.
Consider the seven activity project given by the table below.
Precedence Precedence
1
2
Days
A
B
C
D
E
F
G
21
5
7
3
5
8
2
A
A
B
C
C
E
d
d
f
We want to determine how long the project will take, which activities are critical (to meeting
this deadline) and how much slack the non-critical activities have.
Open POM for Windows. Go to
Modules, Project Management
File, New, Single time estimate
In order to open the create data set window.
Enter 7 for the number of tasks and be sure that the check box for precedence list is checked at
the bottom left and click OK at the bottom right. This will create a blank data screen with 7
rows.
Copy and Paste
We do not want to enter each data item by hand, one-at-a-time wherever possible so we will
copy and paste from the Excel workbook to POM for Windows. This copy and paste operation
will be essential for the exams.
Sometimes when we copy and paste, we will include the header row of the table and
sometimes we will not. It depends on whether or not POM for Windows allows the header row
to be changed or not. If you look at the POM for Windows data screen you will see that the
column headers are filled in (the blue labels with the gray backgrounds). Thus when we copy
we do NOT want to copy the column headers.
1. Go to the sheet named Exercise 1 in the Excel lab workbook and copy cells A2.D8.
2. Go to POM for Windows and right-click on the upper left hand cell that has the word “A”
in it. Click Paste.
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At this point you should have the data table filled with the task names, times and precedences
for all 7 tasks. It is a good idea to save the file (File, Save) at this point. Please note that POM for
Windows will append “.pro” to whatever name you use. This way, if you save a file named
example1 this week and save a file named example1 next week they will have two different
names. One will be example1.pro while the other will be example1.for (for forecasting).
Solving problems in POM for Windows
The easiest way to solve a problem is to press the Solve icon on the far right hand side of the
first tool-bar. Alternatively, you may use File, Solve or the F9 function key. After solving the
main solution window will appear. Other windows may be found under the Window menu
item.
After solving the problem, please copy the solution table to the Excel spreadsheet that has the
data. (Right-click on the POM for Windows solution table, select copy table and then paste in
Excel.) Place the answers to the following questions in the cells indicated on the Exercise 1
spreadsheet.
1.
2.
3.
4.
How many days will this project take to complete?
Which are the critical activities?
Activity E should start sometime between day ________ and day _______.
If activity B takes 2 days longer than scheduled (7 days instead of 5) then how many
days longer will the project take to complete? Note that you can check your answer by
changing the time for B from 5 to 7 in POM and then looking at the new result.
5. If activity C takes 2 days longer than scheduled (9 days instead of 7) then how many
days longer will the project take to complete?
6. If activity E takes 2 days longer than scheduled (7 days instead of 5) then how many days
longer will the project take to complete?
7. Using the charts (Window, Chart from the POM for Windows menu), which activities
appear to be critical?
Exercise 2: The three-time-estimate example – computation of times and standard deviations
using Excel.
Consider the example given by the table below.
A
B
C
D
Optimistic
time (a)
10
4
4
2
Most
Likely
Pessimistic Precedence Precedence
Time (m) Time (b)
1
2
22
28
4
10 A
6
14 A
3
4 B
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E
F
G
1
7
2
5
8
2
9 C
9 C
2 E
d
d
f
The expected time for an activity is given by t = (a + 4m + b)/6. The standard deviation for each
activity is given by [(b-a)/6]. The variance for each activity time is given by
[(b-a)/6]2.
Excel Exercise: Use Excel to add three columns (columns G, H and I) that compute the expected
time, standard deviation and variance for each activity. Hint: Put the formulas in row 2 for
Activity A and then copy the three columns down for the remaining rows. The computed times
in Column G should be the same as for Exercise 1.
Use Exercise 4 to compare the times, standard deviations and variances that you just
computed.
Exercise 3: The three-time-estimate example using Excel OM.
Exercise 4 contains the same data as Exercise 3. We will do a complete analysis using Excel OM.
From the Excel main menu use
OM, Project Management, Predecessor list(AON).
to create the spreadsheet initialization sheet. Be sure to select the 3 time estimate option in
the lower left of the creation form. The spreadsheet is very similar to the sheet we saw as the
problem creation window in POM for Windows.
The example has 7 activities with a maximum of 2 precedences. Check the graph box in order
that the Gantt Chart will be displayed. Also, do NOT put in a title because this will insert two
extra rows and your row numbers will not agree with mine.
Again, we will copy and then we will paste special the data rather than entering each cell
individually. The structure of the spreadsheet that was created in Excel OM requires 2 copy and
paste operations.
 Copy the times from cells A1.D8 of the Exercise 4 sheet to cell A5 of the sheet created
by Excel OM. If you use Paste Special instead of Paste then the Excel OM formatting
(Orange background) will be maintained.
 Copy the precedences from cells E1.F8 to cell C14.
The formulas have been created already so that when you are finished with the pasting the
solutions will appear. The time and standard deviation and variance computations should agree
with the computations performed in Exercise 3.
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On the sheet that was created by Excel OM, highlight (give the cell a background color) the
answers to the following questions:
1. How long will the project take?
2. What is the standard deviation of the length of time of the project?
3. What is the variance of the length of time of the project?
Conditional formatting in Excel: Notice in the spreadsheet that in the cases where the
slack is zero it is formatted in red. Excel’s Conditional Formatting option has been
used to create this format. In order to see the rule that is used (in Excel
2007/2010/2013), click on the Home Tab, Conditional Formatting, Manage rules.
Also, look at all of the options above the rules when you click on Conditional
Formatting.
Let’s try to format the Activity name in bold red if the task is critical (slack is zero).
1.
2.
3.
4.
5.
6.
7.
8.
9.
Select the cells that have the task names.
From the Conditional Formatting menu select New Rule.
Select “Use a formula to determine which cells to format”.
In the rule description location click on the slack for A (which is 0).
Delete the $ signs from the formula (or else the row names will all refer to the
slack of the first activity rather than the slack in their individual row.)
Type “=0”
Click on the Format button
Change the Color to red, click on Bold and click OK.
Exit the conditional formatting menu by clicking OK as many times as required.
Probabilistic analysis of projects
Since we have a mean and standard deviation, we can address probabilistic questions regarding
the completion time. In order to do so, we will use the Normal distribution calculator that is in
POM for Windows.
Go back to POM for Windows and click on the blue icon on the toolbar that is 4 th from the right
and looks like a Normal Distribution. This calculator can be used to either compute cutoffs given
a probability or to compute probabilities given a cutoff (one tailed) or cutoffs (two-tailed). Let
us compute a 95% confidence interval for project completion time. That is, we want to compute
cutoffs given a probability (of 95%).
The middle box should already contain the mean of 39 from previous work. We need to enter
the standard deviation into the appropriate box. Enter the standard deviation value from the
Excel OM spreadsheet for this exercise. Press compute and view the results. Copy the graph
(right-click) from POM for Windows to the Excel spreadsheet for Exercise 4.
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Normal Distribution Exercise: In project management it is more typical to answer a one tailed
question rather than to find a (two-tailed) confidence interval. Use the Normal Distribution
Calculator to answer the following two questions. Copy the graphs from POM for Windows to
the Excel spreadsheet for Exercise 4. Place the answers to the following two questions next to
the graphs.
4. What is the probability that the project will be completed within 42 days?
5. How many days must be allowed in order to be 99% certain that the project will be
completed by this number of days allowed?
Optional Excel Exercise: Excel has Normal distribution functions. Help from Microsoft
on these functions is contained in the links below.
NORMDIST
function
NORMINV function
NORMSDIST
function
NORMSINV
function
Use the appropriate Normal Distribution functions in Excel to answer the two questions
above. This exercise is optional because it is much easier to answer the questions using
POM for Window’s Normal Distribution Calculator rather than Excel.
Exercise 4 – Crashing
Consider the project given below:
regular
crash
time
regular
time
crash
Activity Predecessors (days)
cost
(days)
cost
a
10
11000
9
15000
b
15
20000
13
25000
c
a
10
9000
6
20000
d
a
20
25000
18
30000
e
c
15
20000
10
35000
f
b
17
20000
15
30000
g
b
12
15000
10
25000
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The regular times and costs for each activity are given. However, we have the option for
accelerating the activities. This option is termed crashing. For example, Activity b will normally
be performed in 15 days at a cost of $20,000. However, if we need to we can get it done in as
little time as 13 days. If we choose to do it in 13 days we will pay $25,000 rather than $20,000.
We assume that we can get it done in 14 days at a prorated cost of $22,500. The cost per day of
accelerating activity B is ($25,000 - $20,000)/(15-13) = $2,500 per day saved.
Excel Exercise: In the spreadsheet for Exercise 5, compute the crash cost per day for each of the
activities.
Depending on which activities are crashed we can save different numbers of days for the
project. We would like to save project time in the least costly fashion. POM for Windows will
accomplish this for us.
To set up the data table in POM for Windows use File, New, Crashing, 7 activities. Copy the
data, save the file, press solve and copy the results to the designated cell on the Exercise 5
sheet.
1. What is the (normal) length of time of the project?
2. What is the minimum (crashed) time for length of the project?
3. How much extra will it cost to reduce the project time to its minimum?
The main solution window only gives the two options of the regular and minimum times. Open
the second window (“Crash schedule”) and copy the results to the designated spot on the
Exercise 5 spreadsheet.
4. If we want to reduce the project to 30 days, which activities should be crashed and by
how many days?
5. What will be the total cost of crashing to reduce the project to 30 days?
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Forecasting
Lab data file: forecast.lab.xlsx
Objectives
 To explain the error measures that are used to assess the quality of the forecasts
 To explain the different time-series model
o To demonstrate the use of regression for forecasting
o To demonstrate the use of the Normal distribution for creating a forecast range
(or interval)
 To demonstrate the use of Excel’s Data Analysis Tool for Regression
Terminology & Concepts
 Bias
 MAD (Mean Absolute Deviation)
 MSE (Mean Squared Error)
 MAPE (Mean Absolute Percentage Error)
 Standard error
 Point forecast
 Interval forecast
Excel Functions used
 ABS
 SUM
 AVERAGE
 COUNT
 INTERCEPT
 SLOPE




SQRT
STEYX
TREND
FORECAST


Excel’s Data Analysis Tool
Excel’s Graph
o Trendline option
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
We will be using the average Phillies attendance at Veterans stadium from 1971 to 2003 as our
data set. Even though this data set is old we will be able to make forecasts using these data for
2004 - 2009 and compare the forecasts to the actual attendance. Presumably, much of the
difference is the effect of having a new stadium.
Exercise 1: Computing errors in Excel
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On the Excel spreadsheet for exercise 1, we display three columns of information for Phillies
attendance from 1971, their first year in Veteran’s stadium, to 2003, the last year that the
Phillies were in the Vet. The first column has the year, the second column has the past data
(average attendance per game) and the third column has the forecasts for average attendance
that were made for the same period of time. In the worksheet the attendance is listed in a
column before the forecasts but, of course, the forecasts for each year were made prior to
knowing the average attendance. It does not matter what method was being used to make
these forecasts. It does matter that the same method was used to make all of the forecasts
over the 33 years. Using Excel we will compute the error measures.
The Forecast error is defined as the actual demand (attendance) minus the forecast value. That
is,
Error in period t = Actual demand in period t – Forecast value for period t
 In cell D8 place the Excel formula for the error. As explained in the lecture we also are
interested in the absolute value of the error, the error squared and the absolute value
of the percentage error.
 The Excel function for absolute value is ABS(number). In cell E8 place “=ABS(D8)”
(without the quotations) in order to compute the absolute values.
 In cell F8 place the formula for multiplying D8 (or E8) by itself in order to square the
number.
 In cell G8 place the formula for the absolute percentage error. That is the absolute value
of the error divided by the actual demand (E8/B8).
 Now select the four cells in D8:G8, grab the handle on the lower right of the selection
and copy the four cells down to the bottom of the data at the year 2003.
 In cell D42 place the Excel formula for the AVERAGE of the 33 errors.
 Copy cell D42 to the next 3 columns on the right.
Row 42 gives the values for the Bias, Mean Absolute Deviation (MAD), Mean Squared Error
(MSE) and Mean Absolute Percentage Errors (MAPE). We will check the computations using
Excel OM in a short while but we need to compute one more error measure – the standard
error. The formula for the standard error is given by the square root of the sum of the errors
squared divided by n - 2 where n is the number of forecasts. (see equation 14 in the forecasting
chapter).
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 In cell F41 use Excel’s SUM function to enter the formula for the sum of the errors
squared that are in F8:F40. Before completing the formula please note the value for the
MSE in cell F42.
When you entered the SUM formula, you should have noticed something strange about
cell F42. The value changed even though we did not try to change it! This is bad
behavior on the part of Excel and not due to any mistake that we have made.
Continuing with the computation of the Standard error we need to know how many forecasts
were made.
 In cell F45 use enter “=COUNT(F8:F40) to count the number of forecasts that were
made. (It should be 33).
 Finally, in cell F46 we need the formula for the standard error. This is given by the
square root of the sum of the errors squared divided by n - 2 (see equation 14 in the
forecasting chapter). Enter “=SQRT(f41/(F45-2)).
Check your work
Excel OM and POM for Windows each has a model for performing error analysis. Let us use
Excel OM’s model to check our Excel work.
 Go to the Excel OM menu on the Excel tab in Excel.
 Select the Forecasting Module
 Select the Error Analysis Model.
 At the creation screen enter 33 because there are data and forecasts for 33 years (19712003). An Excel spreadsheet for error analysis will be added to your workbook.
 Copy the data (cells A7:C40) from the Exercise sheet
 Right click in the upper left-hand cell of the new sheet and Paste Special, Values to
preserve the formatting. The bias, MAD, MSE, MAPE and standard error on this Excel
OM spreadsheet should all match the ones on the spreadsheet we developed by hand.
Questions
Place the answers to these questions in the cells provided in the exercise 1 spreadsheet.
1. On average, have we overforecasted (forecasts > actual) or underforecasted (forecasts <
actual) and by how much?
2. Suppose method A has a bias of 10 and method B for the same data has a bias of 20. Is it
true that method A must be more accurate than method B?
3. What is (the numerical value of) the MAD?
4. Why do we care about the MAD?
5. What is (the numerical value of) the standard error?
6. In what two ways is the standard error used?
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7. What is (the numerical value of) the MAPE?
8. What is the advantage of knowing the MAPE rather than the MAD or standard error?
Exercise 2: The naïve method – using POM for Windows
In POM for Windows, select Forecasting, File, New, Time series analysis and create a blank data
set with 33 past periods of data. Notice that there is a box that says “Method” above the data.
This box will enable us to easily move from method to method. The default method is the naïve
method. For the naïve method we simply forecast that next period’s demand will be equal to
this period’s demand. Thus, we would forecast for 2004 that attendance will average 27,901
since that is what attendance averaged in 2003. We don’t need software to get the forecast but
we do need software to perform the error analysis. Copy the data from the Exercise 2
worksheet of lab workbook to POM for Windows. Press the solve button and copy the results to
the designated cell on the Exercise 2 spreadsheet.
On the spreadsheet, point to the answers to the following questions. By point, I mean,
wherever possible place “=” into the cell and then the cell name of the answer. For example
enter “=B8” (without quotation marks) if the answer to the question can be found in cell B8.
1. What is the forecast for 2004?
2. Does the naïve method tend on average to overforecast or underforecast for this data
set? By how much?
3. What is (the numerical value of) the MAD?
4. What is (the numerical value of) the MSE?
5. What is (the numerical value of) the standard error?
6. What is (the numerical value of) the MAPE?
Note: Some myomlab questions ask for the forecasts not only for the future but for years for
which you have data. In POM for Windows these “forecasts” can be found on a separate screen
named “Details and Error Analysis”. The screen can be accessed from the Window menu in
POM as shown below
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The bottom portion of the screen is displayed below where you can see the column named
“Forecast”.
In Excel OM the “Forecasts are readily visible” since Excel OM displays the details and error
analysis. A portion of the screen is shown below.
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Exercise 3: Moving averages
Another method is to forecast the next period based on the average of the previous n periods.
For example, a 2-period moving average would use (20,231 +27,901)/2 = 24,066 as the forecast
for 2004 since these were the demands in the most recent two years.
Use the drop down box to change the method to Moving Averages. Notice that a second box
appears above the data. Change the n for the moving average to 2 and press the SOLVE button.
1. Does the 2-period moving average forecast agree with the 24,066 computed by hand
above?
I have asked you to compute 2-period moving averages but one of the decisions we must make
is what value to use for n. We typically base this decision on one of the error measures. Open
the Window that gives the forecasting errors as a function of n.
2. Which value of n yields the lowest standard error?
3. What is this standard error value?
Open the Graph Window. Use the scrollbar to change the value of n.
4. Describe what happens to the graph of the forecasts as n increases?
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Exercise 4: Weighted moving averages
There is no law that says that we must give equal weight to the previous n periods. Therefore,
we could take a two period moving average with weights of .6 and .4 (instead of .5 and .5) or
we could take a three period moving average with weights of .7, .2 and .1 instead of .33, .33
and .33.
In POM for Windows use the drop down box to change the method to weighted moving
average. Notice that we still have a box above the data asking for a value for n. Place a 3 in this
box. Now, there is a place for the weights on the right hand side of the table. Enter the
numbers .7, .2 and .1, press Solve, copy the results to the spreadsheet and answer the following
questions. Notice that in POM for Windows the first weight is for the most recent data (and will
be the highest). Unfortunately, due to the nature of Excel’s SUMPRODUCT function, in Excel
OM the first weight is for the oldest data (and will be the lowest). This is one of the very rare
instances where POM for Windows and Excel OM are not consistent.
1. What is the forecast for 2004?
2. Based on the standard error is this method better or worse than the naïve method?
Exercise 5: Exponential smoothing
Exponential smoothing is a method that works in a fashion that is similar to moving averages.
The formula for exponential smoothing is given by equation 4 in the chapter.
Ft = Ft-1
At-1 – Ft-1)
In words, “The forecast for this period (period t) is equal to the forecast for last period (period
t(Actual-Forecast) that we made in
the last period.”
1.
Ft ) if
2.
Ft ) if
In POM for Windows, change the method to exponential smoothing and place a .5 in the box
for alpha. Examine and copy the details sheet rather than the summary output?
3. Was the forecast for the year 2000 higher than the actual demand or lower and by how
much?
4. Is the forecast for the year 2001 higher or lower than the forecast for the year 2000 and
by how much?
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5. What is the ratio of the change in the forecast from year 2000 to year 2001 (answer to
question 4) to the error made in the year 2000 (answer to question 3)? Does this match
ne of the decisions we must make is to
6.
Ope
7.
Exercise 6: Trend Analysis
Trend analysis is simply regression where the independent variable is time. Because the
independent variable is time, the slope of the regression line represents the trend in the data.
In POM for Windows, change the method to trend analysis, solve the problem, copy the results
into the spreadsheet and answer the following questions.
1. Looking at the trend equation, what is the trend in attendance? That is, on average, over
the 33 years has attendance been rising or falling and by how many persons per game
per year?
2. What is the forecasted attendance for 2004?
3. What is the forecasted attendance for 2005?
4. What is the standard error using trend analysis?
 In cell J7 enter “=SLOPE(B5:B37,A5:A37) and verify that the slope function yields
the same slope/trend as POM for Windows did.
 In cell J8 enter “=STEYX(B5:B37,A5:A37) and verify that the Standard error
function yields the same standard error of 5674 as POM for Windows did.
The worksheet for this exercise contains the actual attendance since 2004 when the Phillies
started playing in the new ballpark. We will use Excel’s TREND function in order to forecast the
attendance that would have occurred in Veteran’s Stadium since 2004.
 In cell N5 enter the formula =TREND($B$5:$B$37,$A$5:$A$37,L5)
 Copy this formula to cell N10
5. What is the effect of playing in a new ballpark? That is, what is the average of the
differences between the forecasts and the actual attendance from 2004 until now?
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Use the Normal distribution calculator in POM for Windows to answer the following questions.
Copy the graphs from POM for Windows to the worksheet for questions 6 through 8.
6. Give a 95% confidence interval for 2004 attendance? (This is an interval forecast. Our
previous forecasts have all been single points.)
7. Based on the trend equation how confident are you that attendance will be at least
27,000 persons per game?
8. For what number of persons per game would you be 95 confident that attendance will
at least reach this number? This is not the same as question 6!
 Change the method to linear regression.
 Look at the values for x.
 Press the Solve button and verify that the results are the same as when using trend
analysis.
 Use Excel to insert an xy graph.
 Right click on the graph line to insert a regression line and verify that the trend
(slope) matches the trend computed in POM and with Excel’s SLOPE function.
Exercise 7: Linear regression over time with time periods NOT equally spaced
The worksheet in the Excel file contains a data set from the Philadelphia Inquirer regarding
sales of homes. In this case, the years are not evenly spaced. We can use regression (rather
than trend analysis) to perform the forecasting where the independent variable is year and the
dependent variable is number of homes sold or average sales price.
Home Sales
Use Excel OM and open Forecasting, Simple Linear Regression/Least Squares with 7 past
periods. Copy the years and number of sales into the x and y columns respectively. Notice the
missing data between 1995 and 2000. Place the answers to the following questions in cells B21
through B24 of the spreadsheet generated by Excel OM.
1.
2.
3.
4.
What is the trend in terms of the number of sales per year?
What is the forecast for sales in 2012?
What is the forecast for sales in 2014?
What is the standard error of the trend line?
Repeat the exercise using the average price as the dependent variable. Place the answers to the
following questions in cells B25 through B27 of the spreadsheet generated by Excel OM.
5. What is the trend in terms of the number of sales per year
6. What is the forecast for the average price in 2006?
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7. What is the forecast for the average price in 2010?
Optional Excel Exercise
Use Excel’s SLOPE function to answer Question 1 above
Use Excel’s STEYX function to answer Question 4 above
Use Excel’s TREND function to create forecasts for 2012 through 2020
Use Excel’s FORECAST function to create forecasts for 2012 through 2020
Exercise 8: Multiple regression – POM for Windows
While in Exercises 2 through 6 we used time-series methods to forecast Phillies attendance this
was probably not the best of ideas. It is quite possible that attendance is a function of how
good (or dare I say, great) the team is and we have not yet included this information in our
analysis.
In POM for Windows, begin a new problem (File, New). For the model, use multiple regression
(rather than time-series analysis). There are 33 data points and 2 independent variables (time
and winning percentage). Copy the data (columns A through D) from the Exercise 8 spreadsheet
into POM for Windows and press Solve. Copy the results from POM for Windows into the Excel
worksheet.
1. What is the multiple regression equation?
2. What effect does winning percentage have on attendance per game? More specifically,
if the Phillies can increase their winning percentage by ten percentage points (say from
50% to 60%) by how much will average attendance change?
3. Has the standard error improved over the trend analysis model?
4. Does this improved equation, that includes winning percentage, help you to forecast for
future years? In other words, can you use the multiple regression equation to forecast
attendance in 2010, 2011?
Exercise 9: Multiple regression – Excel’s Data Analysis tool - Optional
Perform statistical and engineering analysis with the Analysis ToolPak
Excel has a multiple regression add-in that would be useful for you to see.
To access the Data Analysis tool do the following:
Excel 2003: Click on Tools, Data Analysis to get the regression tool. If Data
Analysis is not a menu item then click on Tools, Add-ins and check the 2 Analysis
checkboxes
Excel 2007/2010/2013: Click on the Data Tab and then Data Analysis (towards
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the right). If Data Analysis is not a menu item then click on
Office button at the top left,
Excel options (at bottom),
Add-ins (3rd from bottom on left),
Go (at bottom, near middle)
Check the 2 Analysis checkboxes.
After the Data Analysis window opens, select Regression. The regression tool appears
below.




Enter the attendance per game (A5:A38) for the Y range.
Enter the Time and Winning Percentages (B5:B38) for the X range.
Check the Labels box since row 5 contains the column labels.
Click on Output Range.
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 Click on the blank text box on the right of output range.
 Enter H12 in the textbox as the place to put the answers.
 Click OK.
The regression results should match the results generated by POM for Windows.
Exercise 9: Nonlinear regression – Excel
Look at the data on the associated spreadsheet. It does not look linear. Using Excel’s Graphing
capabilities it will be easy to see what equation has the best fit.
Optional Excel Exercise
Right click on the line in the graph
Select the Add trendline option
At the bottom, click on Display Equation on Chart and on Display R-squared value on
chart
What is the R2 displayed on the graph.
In the Format Trendline Window click on each of the other regression types (except for
moving average) one at a time and examine the R2.
What is the highest R2?
What is the equation that yields the highest R2?
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Decision Making
Objectives
 To explain the decision table analyses (note plural)
 To explain how to create decision tables
 To explain how to create and analyze decision trees
 To demonstrate Excel’s scenario tool
 To demonstrate Excel’s data table tool
Terminology & Concepts
 Alternative/option/decision
 State of nature/scenario
 Expected value/expected monetary value
 Maximin (worst case scenario analysis – think conservative or pessimistic)
 Maximax (best case scenario analysis – think optimistic or desperate)
 Expected Value with Perfect Information
 Expected Value of Perfect Information
 Regret or opportunity loss
 Decision node
 Chance node/Event node
 Branches
Excel Functions & Options
 SUMPRODUCT
 MIN
 MAX
 SMALL
 LARGE
Preparation: Print this lab document in Word.
1. Open Excel OM
a. Open the Excel lab file for the week
b. Place your start time on the Contents sheet of the excel workbook.
2. Open POM for Windows
Exercise 1: A decision table example using Excel
A company is considering modifying its capacity. There are four options available.
1. Overtime – this is very flexible but there is a limit to the amount of capacity which can
be added
2. Subcontract – this is also flexible and limited
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3. Hire part-timers – we can not hire and fire every day so this is not as flexible as either of
the two previous options but allows us to add more capacity than just using overtime or
subcontracting
4. Hire full timers – this is the least flexible but enables us to add the most capacity
The future is uncertain. We are not sure if demand will be low, medium or high. The profits
depend on the demand and our decision and are given in the table below.
Alternative\Probability
Overtime
Subcontract
Hire part time
Hire full time
Low
demand
0.2
5
3
-1
-7
Normal
demand
0.5
7
6
9
10
High
demand
0.3
10
12
14
15
On the sheet for exercise 1 we display the data. Excel OM or POM for Windows will solve this
problem for you but in this exercise we will get the results using Excel. Since we are using Excel
please highlight the answers to the questions on the spreadsheet.
Expected (Monetary) Value: The expected value is the sum of the products of the probabilities
multiplied by the profits. In cell F2 place the label EMV. In cell F3, use the SUMPRODUCT
function to enable Excel to multiply the probabilities in row 2 by the profits in row 3. When you
enter the values for the probabilities in the formula be sure to use $B$2:$D$2 rather than
B2.D2 so that when you copy this formula down to the other two rows it will NOT move the
probabilities out of the formula. Copy the formula and answer the following questions.
1. What decision leads to the highest expected value?
2. What is this expected value?
3. Is expected value a useful tool for this situation? That is, is this particular capacity
decision made repeatedly?
Maximin: In cell G2 place the label “Row min”, in cell G3 use the MIN function to determine the
minimum (worst case) for row 3. Copy the formula down to the other two rows.
4. What decision leads to the best of the worst case outcomes?
5. What is the value of the maximin (the best worst case outcome)? Note that the question
uses the term value, not expected value.
6. What is the major advantage of using this decision?
Excel’s SMALL function could also be used to find the minimum value in the row. It
would be entered as “=SMALL(B4.D4,1)”. While the syntax is more complicated than the
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MIN function, the SMALL function can be used to find the 2 nd smallest, 3rd smallest, etc.
by varying the second parameter of the function.
Maximax: In cell H2 place the label “Row max”, in cell H3 use the MAX function to determine
the maximum (best case) for row 3. Copy the formula down to the other two rows.
7. What decision leads to the best of the best case (maximax) outcomes?
8. What is the value of the maximax (the best of the best case outcome)?
9. Can you find this best case (maximax) by simply determining the best outcome of the 12
outcomes in the table in the table?
Excel’s LARGE function is analogous to the SMALL function.
Expected Value of Perfect Information: Perfect information assumes that we make the best
decision under any scenario. In cell A9 place the label “Best outcome”. In cell B9 use the MAX
function to determine the best outcome if demand is low. Copy this formula to the next two
columns on the right. In cell E9 use the SUMPRODUCT function in order to determine the
expected value under certainty – the weighted average of this row of best outcomes with the
row of probabilities.
10. What is the expected value under certainty?
11. How much would you be willing to pay for information about the future demand? That
is, by how much would the expected value increase if you always knew in advance what
scenario/state of nature would occur as opposed to not knowing?
Excel OM. Create a 4 by 3 decision table using Excel OM. At the creation screen be sure to
select the options for EVPI and regret/opportunity loss. Enter the data and check your results
on the Excel exercise. Check the values on the spreadsheet created by Excel OM against your
computations and answers to this exercise.
Regret/Opportunity Loss
Examine the regret/opportunity table at the bottom of the spreadsheet. Find the minimax
regret strategy by examining the maximum column (rather than the expected column).
12. What decision should be made using minimax regret?
Note that you should have four different answers to questions 1, 4, 6 and 11above. Performing
the analysis is easy but interpreting the results is not. The choice depends to some extent on
personality and the repeated nature of the decision!
OPTIONAL: Exercise 2: Excel’s Scenario Tool – Use Worksheet – 2.Excel Scenarios
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Excel has a useful tool that I would like to demonstrate next. In this spreadsheet we are
reverting to our original capacity planning example. The probabilities that were given for
low, medium and high demand were .2, .5 and .3. These probabilities refer to the future
so they are forecasts. Suppose that three different persons are studying this problem.
The first has given us the probabilities above while the second is an optimist who
believes the probabilities should be .1, .5 and .4 and the third is a pessimist who believes
the probabilities should be .5, .3 and .2.
We could copy the sheet two times and place the new probabilities into the new sheets
in order to get the results but Excel has a tool that makes the display and collection of
information a little better.
Please go to
Excel 2003: Tools, Scenarios and select Add.
Excel 2007: Data tab, Data Tools Section, What-if Analysis, Senarios.
This displays the following scenario manager.
We will use our scenario manager to allow us to easily change the probabilities. Our first
scenario will be our original scenario. In the Scenario name textbox place “Original”. In
the Changing cells location place B6.D6 since these are the probabilities that we will
change. The scenario values will appear with the original values in them so click OK.
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You will then be back at the Scenario Manager. Click Add, Make the scenario name
“Optimistic”, Click, leave the changing cells as they are, click OK and change the
probabilities to the optimistic values of .1, .5 and .4 as indicated above and click OK.
Create the pessimistic scenario in the same way. The probabilities in the pessimistic
scenario are .5, .3, .2.
To change the scenario that is displayed, click on the scenario name and then click show.
Thus, it becomes very easy to display what happens when the probabilities change.
We have one final aspect regarding the Scenario manager. In the scenario manager,
click on Summary. The following will appear and the cells listed as result cells will be
highlighted on the spreadsheet. We could change these, but to make it easy, simply click
OK.
A new summary scenario sheet will be created that gives the values of the results cells
for each of the scenarios!
Scenario Summary
Changing Cells:
$B$6
$C$6
$D$6
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Current
Values:
Normal
Optimistic
Pessimistic
0.2
0.5
0.3
0.2
0.5
0.3
0.1
0.5
0.4
0.5
0.3
0.2
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Result Cells:
$F$6
$F$7
7.5
7.5
8
$F$8
7.2
7.2
8.1
$F$9
8.5
8.5
10
$F$10
8.1
8.1
10.3
$F$14
10.5
10.5
11.5
$B$20
0.2
0.2
0.1
$F$21
3
3
3.5
$F$22
3.3
3.3
3.4
$F$23
2
2
1.5
$F$24
2.4
2.4
1.2
Notes: Current Values column represents values of changing cells at
time Scenario Summary Report was created. Changing cells for each
scenario are highlighted in gray.
6.6
5.7
5
2.5
8.5
0.5
1.9
2.8
3.5
6
Exercise 3: Creating a decision table using POM-QM for Windows
Consider the example from the lecture. Chip's Chocolate Chips sells fresh baked cookies at the
main terminal of the Tri-County Airport. The daily demand distribution is
Dozens Sold
Probability
24
25
26
27
28
29
30
.05
.10
.20
.25
.25
.10
.05
The cookies cost $1.10 per dozen to prepare and they sell for $2 per dozen. Cookies left over at
the end of the day are discarded.
Prior to performing any analysis please answer the following two questions.
1. How many dozens would a pessimistic decision-maker purchase?
2. How many dozens would an optimist purchase?
We will use POM-QM to create the table for us.
In POM-QM for Windows select Decision Analysis, File, New, One Period Inventory. On the
create screen select 7 for the number of demands. We need to enter three pieces of cost
information at the top and then enter the demand table that is given above.
Profit per unit: The profit per unit is given by the selling price per unit of $2.00 minus the cost
per unit of $1.10 for a value of .9
Profit per unit Excess: This refers to the profit when we make more units than are demanded. In
some cases, but not in this case, we may be able to sell the units for some amount. This is called
a salvage value. In this particular case if we have excess units we receive nothing. Those units
cost us $1.10 so our profit on the excess units is actually a loss of $1.10. Enter -1.10 as the value
for this row.
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Profit per unit short: In some cases if we are short we can make an emergency purchase of
additional units (usually at a premium price) and then sell the units. In this case we simply don’t
make the sale. Thus if we are short our profit is 0.
Next, enter the demand values of 24 through 30 into the demand column and then enter the
probabilities into the probability column. We are prepared to solve the problem.
3.
4.
5.
6.
7.
How many dozens should be purchased in order to maximize the expected revenue?
How many dozens should be purchased according to the maximin criterion?
How many dozens should be purchased according to the maximax criterion?
How much would you pay, in order to determine the demand in advance?
Is expected value a good criterion for this situation? Why or why not?
OPTIONAL Exercise 4: Using Excel’s Data Table to create the decision table
In order to use a data table in Excel the first step is to perform the computations for one
pair of numbers. We will do the computations for a supply of 26 and a demand of 29.
1. In cell B11 place the formula for the total cost based on the supply and the cost per
unit.
2. In cell B13 place the formula for the number of dozens sold based on the supply and
the demand. If the supply and the demand are equal then the number sold is
obvious. In the cases where the demand is less than the supply you need to think
about how many units are sold. You will want to use either Excel’s IF function or
Excel’s MAX function.
3. In cell B14 place the formula for the Total revenue based on the number sold and
the selling price
4. In cell B16 compute the total profit as a function of the total revenue and the total
cost.
Your result should be $23.40
Now exchange the numbers 26 and 29 in the supply and demand cells and verify that
the result you get is $20.10.
Creating the data table
We now are in a position to create the data table. The spreadsheet is color coded in
order to make the creation easy.
In cell B21 place the formula that points to the profit. That is, enter “=B16” (without the
quotes) since B16 is the cell that holds the profit.
5. Select the entire table, B21.I28
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6. From the Excel menu select Data, Table
Where it asks for the row, it is referring to the row of demands in row 21. The cell that
corresponds to this is the demand cell in B9 so place B9 in the entry.
Where it asks for the column, it is referring to the column of supplies. The cell that
corresponds to supply is the supply cell in B8 so place B8 in the entry.
Your input should appear as below. Click OK and the results should match our previous
results.

Exercise 5: Sequential decision tree using Excel OM
Consider the following example
Chris Ford is currently employed at a salary of $50,000 but is unhappy with his job. While Chris
is unhappy due to lack of support from his manager his major concern is with salary. Chris has
taken two immediate actions in order to try and alleviate the problem. He has applied for a job
with another firm (at a salary of $60,000) and he has applied for a new position in his current
firm (at a salary of $70,000). As life has it, Chris has been offered the new job with the new firm
and has 10 days to decide whether or not to take the offer. Unfortunately, he will not find out
about the internal transfer for at least a month. Chris feels that if he stays with his current firm
there is a 70% chance that he will get the new position. If he does not get the new position then
he can, of course, continue in his current position or he is considering quitting his current job
on principle in order that he may devote full time to searching for a new job (after a brief
vacation in Europe). If he looks for a new job then he figures there is a 20% chance he will land
a job with a salary of $65,000; a 40% chance that he will land a job with a salary of $55,000, a
30% chance that he will land a job with a salary of $45,000 and a 10% chance that he will not
land a job and therefore have to perform menial labor earning $20,000. Should Chris take the
job offer in hand or not?
We will use Excel OM to create the decision table for the following example. (The virtual lecture
on decision trees used POM-QM for Windows to explain the decision tree concepts.)
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Starting: From the OM menu, select Decision Analysis, Decision Trees. The screen will appear as
follows.
The form on the right is used to create the branches. The data will be entered directly into the
spreadsheet. Currently, the tree is showing a decision node. The value of the node will be
placed above the node and the decision will be placed below when the computations are done.
We need to add two decision branches relating to the choice of accepting the job or not. The
form on the right has a default value of 2 branches to add and is initially set at node 1 so simply
click on the “Add 2 DECISIONS from Node 1” button to generate the following screen.
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In the cell that says Decision 1, type “Accept job” and in the cell beneath that type the value of
$60000. In the cell that says Decision 2, type “Reject offer”. Now if we reject the offer we need
to see if we get the promotion at our current position. Click on Node 3. The Decision Table form
will appear. Click on the button to add two EVENTS to node 3. The screen will appear as follows.
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Change Event 1 to “gets promotion” and Event 2 to “No promotion”. When you type these two
terms, they will not fit in the cell. The tree will be automatically reformatted when more
branches are added. Change the .5 on top of gets promotion to .7 and change the .5 on top of
No promotion to .3. Finally, below gets promotion place the salary of $70,000.
Next click on Node 5 to add two decisions for deciding whether to stay or leave. Your sheet
should appear as follows.
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Excel MSOM has already put in formulas to perform some intermediate computations. For
example, the $49,000 at Node 3 is .7*70,000 + .3*0.
Change Decision 3 to Stay and place the current salary of $50,000 in below this. Change
Decision 4 to Leave and then use the Decision tree form to add 4 events to node 7. This leads to
the following.
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Finally, change events 3 through 6 to jobs 1 through 4 and place the 4 salaries and 4
probabilities into the appropriate cells. This leads to the last tree.
The tree is solved backwards. For example, the expected value of looking for a new job is the
$50,500 shown at node 7. Since $50,500 is better than $50,000 the choice at Node 5 is 7 (the
bottom node) with a value of $50,500. Thus, the expected value of node 3 is .7*70,000 + .3
*50,500 or $64,150. This is better than the offer of $60,000 so the choice in Node 1 is 3, the
bottom node, and the value of the tree is $64,150.
Notice, that decisions are based on expected value. For example, in node M26 rather than
taking the sure $50,000 we are willing to risk finding a job that pays only $20,000 or $45,000
since we might get a job that pays $55,000 or $65,000.
Exercise 6: Decision Tree Sensitivity – Excel Active Model May not work in Speakman lab The spreadsheet contains a decision tree for Example 3. It is laid out in a fashion different form
that in Figure 5.14 but yields the same result. Please use this spreadsheet to answer the
following questions.
1. For what range of probabilities of high sales should we purchase the CAD system
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2. "Favorable market sales" has been defined as 25,000 units. Suppose this is optimistic. At
what value would we change our decision and hire engineers?
3. "Unfavorable market sales" has been defined as 8,000 units. Suppose this is optimistic.
At what value would we change our decision and hire engineers?
4. How does the price affect our decision?
5. How sensitive is the decision to the manufacturing costs WITHOUT CAD? That is, as you
raise or lower the manufacturing costs without CAD, what happens to the decision at
the beginning (left) of the decision tree?
6. How sensitive is the decision to the manufacturing costs WITH CAD? That is, as you raise
or lower the manufacturing costs with CAD, what happens to the decision at the
beginning (left) of the decision tree?
Exercise 7: A minimization example
Marie Curtiss lives in Cheltenham, PA and works in Stratford, NJ. Marie has five possible options
(4 driving and 1 train) to take from home to work. The times for each of these driving routes
depend on the general traffic conditions in the Delaware Valley. The estimated times for each
route as a function of the traffic conditions are
given in the table below. For example, you can
see that under good conditions (no
congestion) Route 1 is the best but under bad
conditions Route 1 is the worst.
Route 1
Route 2
Route 3
Route 4
Trains
None
40
60
45
50
70
Traffic Congestion
Mild
Severe
70
90
65
75
65
85
70
75
70
70
PA 73 to Tacony Palmyra Bridge to NJ 73 to I295 South to Bellmawr
Broad St to I76 (Vine st) to B Franklin Br to I276 to I295 North to Bellmawr
Route 3: Broad St to US1 to I76 to B Franklin Br to I276 to I295 N to Bellmawr
Broad St to US1 to I76 to W Whitman Br to I276 to I295 North to Bellmawr
Route 5: Septa train to Lindenwold train
In the past 100 days there was no congestion on 30 days, mild congestion on 50 days and
severe congestion on 20 days.
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REMINDER: Travel times are costs not profits!
1. Suppose Marie wants to leave home at the last possible moment but absolutely
guarantee that she will not be late for work which begins at 9:00. When should Marie
leave? What option should Marie select?
2. Suppose that Marie is generally a very optimistic person but be on time for work. Which
option should Marie use? When should Marie leave for work?
3. Suppose that Marie is the type of person who is always second-guessing herself and
wondering, “What if I had done the right thing instead of what I actually did?” Which
option should Marie use? Why?
4. Is expected value a good decision rule for this problem? Why or why not?
5. If Marie is interested in minimizing the total transit time over a year then she should
take which of the 5 options? Why?
6. If Marie listens to KYW before she leaves for work and if KYW traffic reports are
accurate, then what is the number of minutes that could be saved per day?
Statistical Process Control
Lab data file: qc.lab.xlsx
Objectives
 To explain control charts
 To demonstrate acceptance sampling concepts through operating characteristic (OC)
curves
Terminology & Concepts
 p-chart
 x-bar chart, R-chart
 c-chart
 Upper control limit (UCL)
 Lower control limit (LCL)
 Center line


 Acceptable quality level (AQL)
 Lot tolerance percent defective (LTPD)
 Sampling plan (n,c)
sample size, n
critical value, c
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Excel Function
Sum
Preparation:
3. Open Excel OM
a. Open the Excel lab file for the week
b.
4. Open POM for Windows
Notes:
 Be careful when copying data to and from Excel and POM for Windows
 After copying, check that the data is reasonable.



In all cases the sample size, n must be specified.
In all cases, the confidence, which is usually 3-sigma (99.73%) must be specified
Sometimes control charts are created according to specifications while other times
charts are created according to data that has been given.
The key is to understand what type of chart is being requested and then to enter either the
sample data or the specification(s).
Specification
mean percentage of
defective units per sample
p-chart
x-bar chart
mean, std dev of some
measurement
or
mean, range of some
measurement
c-chart
Mean number of defects
per item
Data
list of samples with
number/percentage of
defectives per sample
list of actual sample means
and standard deviations
or
list of actual sample means
and ranges
or
data where mean and
range for each sample must
be computed
list of items with number of
defects on each item given
Exercise 1: Artillery shells (155mm) are produced in a forging process. The diameter of the shell
is a critical factor that must be controlled. From samples of size 4 of shells produced each day,
the mean and the range of the diameter have been found as given below. We will create and
examine a 3-sigma (99.7%) control chart.
Day
Mean
Range
MSOM Lab Exercises and Examples
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1
2
3
4
5
155.09
155.02
154.96
155.05
155.06
0.042
0.046
0.041
0.050
0.045
Open POM for Windows and select the quality control module. In this example measurements
are given (millimeters) so this requires an x-bar chart. Use File, New and select the x-bar chart.
There are 5 samples and at the bottom left of the creation screen use the checkbox that
indicates that the mean and range are given. Copy the data from the lab workbook and paste it
into POM for Windows. Above the data enter the sample size of 4! This is crucial as all control
charts are established by using a fixed sample size. Press the solve button and copy the results
from the results table to the workbook.
1.
2.
3.
4.
What is “x-bar-bar”, the average of the 5 daily averages?
What is the upper control limit for the x-bar chart?
What is the lower control limit for the x-bar chart?
With respect to the means, has the process been in control? That is, do all of the five
sample means fall between the upper control limit and the lower control limit? (The
graph in POM for Windows is VERY useful for answering this question.). Note: This is a
necessary but not sufficient condition for the process to be in control. See Figure S6.7.
While the means may be in control there could still be problems. Suppose we are filling
packages with 14 ounces of food and that one package has 13 ounces while another has 15
ounces. Then the average of these two packages is correct even though the packages are bad.
(This is similar to a positive forecasting error canceling out a negative forecasting error). Thus
we not only check the means but we also check the ranges of the samples.
5.
6.
7.
8.
What is “R-bar”, the average of the 5 daily ranges?
What is the upper control limit for the R-bar chart?
What is the lower control limit for the R-bar chart?
With respect to the ranges, has the process been in control? That is, do all of the five
ranges fall between the upper control limit and the lower control limit? (The graph in
POM for Windows is VERY useful for answering this question.)
If the process is in control with respect to both the means and the ranges then the process is in
control.
9. Has this process been in control?
Most textbooks, including yours set the center line as the overall average. This is the case
sometimes, but many more times the center line is specified. For example, in the example
above, in spite of the questions I asked we would NOT use the overall average but instead
MSOM Lab Exercises and Examples
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would set the center line to exactly 155 (rather than the overall average of 155.036) since the
shell is supposed to be 155 millimeters.
In POM for Windows return to the Data Screen and set the center line in POM for Windows to
155. The center line setting is above the table of data. Solve the revised model, copy the results
to the workbook and answer the following questions.
10. What is the upper control limit for the mean chart?
11. What is the lower control limit for the mean chart?
12. With respect to the means, has the process been in control?
Exercise 2: Samples of 25 staplers are taken every half hour. The average percent defective is
1.5 percent. Use POM for Windows to construct the 3 sigma (99.73%) control chart.
This problem has specifications given rather than data. In addition, it is giving the percentage of
defectives in the samples of 25. Thus, it is a p-chart. Use File, New and select p-chart. When
specifications are given, set the number of samples to 1. Enter the center line (.02) as the
specification in the data table (not in the area above the data as was done for exercise 1), enter
the sample size (25) above the data, press solve and copy the results to the workbook.
1. What is the upper control limit for the p-chart?
2. What is the lower control limit for the p-chart?
Exercise 3: Given below are the results of inspections of samples of light bulbs taken over the
past 10 hours. The sample size is 200. We will use a 3 sigma control chart to analyze the data.
Time
8:00
am
9:00
10:00
11:00
12:00
1:00
2:00
3:00
4:00
5:00
Number of
defective
lightbulbs
5
4
4
7
3
4
0
6
7
0
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This problem has data. Since the number of defective units is given it is a p-chart. The software
will convert the number of defectives to a percentage of defectives based on the sample size of
200.
1. Using Excel (not Excel OM) what is the total number of units that were inspected over
the 10 hours?
2. Using Excel’s SUM function on the data what is the total number of defective units
found.
3. Using Excel what is the percentage of defective units found between 8am and 5pm?
Use POM for Windows to create the p-chart for this example. Copy the data, set the sample size
(200) in the area above the data, press solve and copy the results to the workbook.
4. What is “p-bar”, the overall average percentage of defective units in the 10 samples?
(This should agree with your answer to question 3 above).
5. What is the upper control limit?
6. What is the lower control limit?
7. Has the process been in control?
Exercise 4: Ten newly produced pieces of furniture were examined and the number of scratches
found on the 10 items was as given in the table below. We will construct the 3-sigma control
chart using Excel OM in order to determine if the process is in control.
This is a c-chart since we are talking about defects (scratches) per item!
Piece
1
2
3
4
5
6
7
8
9
10
Scratches
3
6
2
1
0
0
4
2
7
8
1. Using Excel, what is the total number of defects on the 10 pieces?
2. What is the average number of defects per piece?
From the OM menu select Quality control, c-chart and on the creation screen enter the number
of samples as 10. Copy the data into the shaded area, copy (paste special, values) the answers
to the data sheet for this problem and answer the following standard questions.
MSOM Lab Exercises and Examples
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3. What is “c-bar”, the overall average number of defects per unit for the 10 samples? This
should agree with your answer to Q2.
4. What is the upper control limit?
5. What is the lower control limit?
6. Has the process been in control? That is, do any of the sample number of defectives fall
above the upper control limit or below the lower control limit? (The graph in Excel OM is
useful for answering this question.)
Exercise 5: For each of the last 12 hours, samples of bacteria counts in the water produced by a
local company have been taken. Each sample was of 6 bottles. Use POM for Windows to
construct the 3-sigma control chart based on the data below.
Bacteria count (parts per million)
Bottle
Bottle
Bottle
Bottle
Bottle
Bottle
Sample 1
2
3
4
5
6
1
26
22
26
23
29
29
2
24
28
29
29
22
26
3
28
22
26
28
21
20
4
29
26
25
26
24
24
5
25
28
21
25
28
27
6
28
18
22
35
10
14
7
24
25
21
28
29
21
8
27
22
22
24
21
21
9
22
27
28
28
29
20
10
21
25
21
21
23
22
11
24
23
21
23
21
21
12
29
28
28
22
22
26
Parts per million is a measurement just like inches or millimeters or pounds or minutes.
Therefore, we need to create an x-bar chart. On the creation screen, enter 12 samples and
select the raw data option from the lower left. When you click on raw data, another scrollbar
will be visible where you can select the sample size of 6. Copy the data to POM for Windows,
press solve and copy the results to the workbook.
1.
2.
3.
4.
What is “x-bar”, the overall average parts per million in the 72 bottles?
What is the upper control limit?
What is the lower control limit?
Does the process appear to have been in control?
5.
6.
7.
8.
What is “R-bar”, the average range?
What is the upper control limit for the Range chart?
What is the lower control limit for the Range chart?
Does the process appear to have been in control?
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Reminder: The process is in control only if both the mean chart and the range chart are in
control!
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Product Design and Decision Making
Lab data file: design.lab.xlsx
Objectives
 To explain the decision table analyses (note plural)
 To explain how to create decision tables
 To explain how to create and analyze decision trees
 To demonstrate Excel’s scenario tool
 To demonstrate Excel’s data table tool
Terminology & Concepts
 Alternative/option/decision
 State of nature/scenario
 Expected value/expected monetary value
 Maximin (worst case scenario analysis – think conservative or pessimistic)
 Maximax (best case scenario analysis – think optimistic or desperate)
 Expected Value with Perfect Information
 Expected Value of Perfect Information
 Regret or opportunity loss
 Decision node
 Chance node/Event node
 Branches
Excel Functions & Options
 SUMPRODUCT
 MIN
 MAX
 SMALL
 LARGE
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
MSOM Lab Exercises and Examples
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Exercise 1: A decision table example using Excel
A company is considering modifying its capacity. There are four options available.
5. Overtime – this is very flexible but there is a limit to the amount of capacity which can
be added
6. Subcontract – this is also flexible and limited
7. Hire part-timers – we can not hire and fire every day so this is not as flexible as either of
the two previous options but allows us to add more capacity than just using overtime or
subcontracting
8. Hire full timers – this is the least flexible but enables us to add the most capacity
The future is uncertain. We are not sure if demand will be low, medium or high. The profits
depend on the demand and our decision and are given in the table below.
Low
Normal High
demand demand demand
Alternative\Probability
0.2
0.5
0.3
Overtime
5
7
10
Subcontract
3
6
12
Hire part time
-1
9
14
Hire full time
-7
10
15
On the sheet for exercise 1 we display the data. Excel OM or POM for Windows will solve this
problem for you but in this exercise we will get the results using Excel. Since we are using Excel
please highlight the answers to the questions on the spreadsheet.
Expected (Monetary) Value: The expected value is the sum of the products of the probabilities
multiplied by the profits. In cell F2 place the label EMV. In cell F3, use the SUMPRODUCT
function to enable Excel to multiply the probabilities in row 2 by the profits in row 3. When you
enter the values for the probabilities in the formula be sure to use $B$2:$D$2 rather than
B2.D2 so that when you copy this formula down to the other two rows it will NOT move the
probabilities out of the formula. Copy the formula and answer the following questions.
13. What decision leads to the highest expected value?
14. What is this expected value?
15. Is expected value a useful tool for this situation? That is, is this particular capacity
decision made repeatedly?
Maximin: In cell G2 place the label “Row min”, in cell G3 use the MIN function to determine the
minimum (worst case) for row 3. Copy the formula down to the other two rows.
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16. What decision leads to the best of the worst case outcomes?
17. What is the value of the maximin (the best worst case outcome)? Note that the question
uses the term value, not expected value.
18. What is the major advantage of using this decision?
Excel’s SMALL function could also be used to find the minimum value in the row. It
would be entered as “=SMALL(B4.D4,1)”. While the syntax is more complicated than the
MIN function, the SMALL function can be used to find the 2 nd smallest, 3rd smallest, etc.
by varying the second parameter of the function.
Maximax: In cell H2 place the label “Row max”, in cell H3 use the MAX function to determine
the maximum (best case) for row 3. Copy the formula down to the other two rows.
19. What decision leads to the best of the best case (maximax) outcomes?
20. What is the value of the maximax (the best of the best case outcome)?
21. Can you find this best case (maximax) by simply determining the best outcome of the 12
outcomes in the table in the table?
Excel’s LARGE function is analogous to the SMALL function.
Expected Value of Perfect Information: Perfect information assumes that we make the best
decision under any scenario. In cell A9 place the label “Best outcome”. In cell B9 use the MAX
function to determine the best outcome if demand is low. Copy this formula to the next two
columns on the right. In cell E9 use the SUMPRODUCT function in order to determine the
expected value under certainty – the weighted average of this row of best outcomes with the
row of probabilities.
22. What is the expected value under certainty?
23. How much would you be willing to pay for information about the future demand? That
is, by how much would the expected value increase if you always knew in advance what
scenario/state of nature would occur as opposed to not knowing?
Excel OM. Create a 4 by 3 decision table using Excel OM. At the creation screen be sure to
select the options for EVPI and regret/opportunity loss. Enter the data and check your results
on the Excel exercise. Check the values on the spreadsheet created by Excel OM against your
computations and answers to this exercise.
POM for Windows note: If you enter an “=” sign, without the quotes, in the probability row
then POM for Windows will assign equal probabilities of 1/n to each of the n scenarios.
Regret/Opportunity Loss
Examine the regret/opportunity table at the bottom of the spreadsheet. Find the minimax
regret strategy by examining the maximum column (rather than the expected column).
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24. What decision should be made using minimax regret?
Note that you should have four different answers to questions 1, 4, 6 and 11above. Performing
the analysis is easy but interpreting the results is not. The choice depends to some extent on
personality and the repeated nature of the decision!
OPTIONAL: Exercise 2: Excel’s Scenario Tool – Use Worksheet – 2.Excel Scenarios
Excel has a useful tool that I would like to demonstrate next. In this spreadsheet we are
reverting to our original capacity planning example. The probabilities that were given for
low, medium and high demand were .2, .5 and .3. These probabilities refer to the future
so they are forecasts. Suppose that three different persons are studying this problem.
The first has given us the probabilities above while the second is an optimist who
believes the probabilities should be .1, .5 and .4 and the third is a pessimist who believes
the probabilities should be .5, .3 and .2.
We could copy the sheet two times and place the new probabilities into the new sheets
in order to get the results but Excel has a tool that makes the display and collection of
information a little better.
Please go to
Excel 2003: Tools, Scenarios and select Add.
Excel 2007/2010/2013:: Data tab, Data Tools Section, What-if Analysis, Senarios.
This displays the following scenario manager.
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We will use our scenario manager to allow us to easily change the probabilities. Our first
scenario will be our original scenario. In the Scenario name textbox place “Original”. In
the Changing cells location place B6.D6 since these are the probabilities that we will
change. The scenario values will appear with the original values in them so click OK.
You will then be back at the Scenario Manager. Click Add, Make the scenario name
“Optimistic”, Click, leave the changing cells as they are, click OK and change the
probabilities to the optimistic values of .1, .5 and .4 as indicated above and click OK.
Create the pessimistic scenario in the same way. The probabilities in the pessimistic
scenario are .5, .3, .2.
To change the scenario that is displayed, click on the scenario name and then click show.
Thus, it becomes very easy to display what happens when the probabilities change.
We have one final aspect regarding the Scenario manager. In the scenario manager,
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click on Summary. The following will appear and the cells listed as result cells will be
highlighted on the spreadsheet. We could change these, but to make it easy, simply click
OK.
A new summary scenario sheet will be created that gives the values of the results cells
for each of the scenarios!
Scenario Summary
Current
Values:
Normal
Optimistic Pessimistic
Changing Cells:
$B$6
0.2
0.2
0.1
0.5
$C$6
0.5
0.5
0.5
0.3
$D$6
0.3
0.3
0.4
0.2
Result Cells:
$F$6
$F$7
7.5
7.5
8
6.6
$F$8
7.2
7.2
8.1
5.7
$F$9
8.5
8.5
10
5
$F$10
8.1
8.1
10.3
2.5
$F$14
10.5
10.5
11.5
8.5
$B$20
0.2
0.2
0.1
0.5
$F$21
3
3
3.5
1.9
$F$22
3.3
3.3
3.4
2.8
$F$23
2
2
1.5
3.5
$F$24
2.4
2.4
1.2
6
Notes: Current Values column represents values of changing cells at
time Scenario Summary Report was created. Changing cells for each
scenario are highlighted in
gray.
Exercise 3: Creating a decision table using POM-QM for Windows
Consider the example from the lecture. Chip's Chocolate Chips sells fresh baked cookies at the
main terminal of the Tri-County Airport. The daily demand distribution is
MSOM Lab Exercises and Examples
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Dozens Sold
Probability
24
25
26
27
28
29
30
.05
.10
.20
.25
.25
.10
.05
The cookies cost $1.10 per dozen to prepare and they sell for $2 per dozen. Cookies left over at
the end of the day are discarded.
Prior to performing any analysis please answer the following two questions.
8. How many dozens would a pessimistic decision-maker purchase?
9. How many dozens would an optimist purchase?
We will use POM-QM to create the table for us.
In POM-QM for Windows select Decision Analysis, File, New, One Period Inventory. On the
create screen select 7 for the number of demands. We need to enter three pieces of cost
information at the top and then enter the demand table that is given above.
Profit per unit: The profit per unit is given by the selling price per unit of $2.00 minus the cost
per unit of $1.10 for a value of .9
Profit per unit Excess: This refers to the profit when we make more units than are demanded. In
some cases, but not in this case, we may be able to sell the units for some amount. This is called
a salvage value. In this particular case if we have excess units we receive nothing. Those units
cost us $1.10 so our profit on the excess units is actually a loss of $1.10. Enter -1.10 as the value
for this row.
Profit per unit short: In some cases if we are short we can make an emergency purchase of
additional units (usually at a premium price) and then sell the units. In this case we simply don’t
make the sale. Thus if we are short our profit is 0.
Next, enter the demand values of 24 through 30 into the demand column and then enter the
probabilities into the probability column. We are prepared to solve the problem.
10. How many dozens should be purchased in order to maximize the expected revenue?
11. How many dozens should be purchased according to the maximin criterion?
12. How many dozens should be purchased according to the maximax criterion?
13. How much would you pay, in order to determine the demand in advance?
14. Is expected value a good criterion for this situation? Why or why not?
OPTIONAL Exercise 4: Using Excel’s Data Table to create the decision table
In order to use a data table in Excel the first step is to perform the computations for one
MSOM Lab Exercises and Examples
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pair of numbers. We will do the computations for a supply of 26 and a demand of 29.
7. In cell B11 place the formula for the total cost based on the supply and the cost per
unit.
8. In cell B13 place the formula for the number of dozens sold based on the supply and
the demand. If the supply and the demand are equal then the number sold is
obvious. In the cases where the demand is less than the supply you need to think
about how many units are sold. You will want to use either Excel’s IF function or
Excel’s MAX function.
9. In cell B14 place the formula for the Total revenue based on the number sold and
the selling price
10. In cell B16 compute the total profit as a function of the total revenue and the total
cost.
Your result should be $23.40
Now exchange the numbers 26 and 29 in the supply and demand cells and verify that
the result you get is $20.10.
Creating the data table
We now are in a position to create the data table. The spreadsheet is color coded in
order to make the creation easy.
In cell B21 place the formula that points to the profit. That is, enter “=B16” (without the
quotes) since B16 is the cell that holds the profit.
11. Select the entire table, B21.I28
12. From the Excel menu select Data, Table
Where it asks for the row, it is referring to the row of demands in row 21. The cell that
corresponds to this is the demand cell in B9 so place B9 in the entry.
Where it asks for the column, it is referring to the column of supplies. The cell that
corresponds to supply is the supply cell in B8 so place B8 in the entry.
Your input should appear as below. Click OK and the results should match our previous
results.
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
Exercise 5: Sequential decision tree using Excel OM
Consider the following example
Chris Ford is currently employed at a salary of $50,000 but is unhappy with his job. While Chris
is unhappy due to lack of support from his manager his major concern is with salary. Chris has
taken two immediate actions in order to try and alleviate the problem. He has applied for a job
with another firm (at a salary of $60,000) and he has applied for a new position in his current
firm (at a salary of $70,000). As life has it, Chris has been offered the new job with the new firm
and has 10 days to decide whether or not to take the offer. Unfortunately, he will not find out
about the internal transfer for at least a month. Chris feels that if he stays with his current firm
there is a 70% chance that he will get the new position. If he does not get the new position then
he can, of course, continue in his current position or he is considering quitting his current job
on principle in order that he may devote full time to searching for a new job (after a brief
vacation in Europe). If he looks for a new job then he figures there is a 20% chance he will land
a job with a salary of $65,000; a 40% chance that he will land a job with a salary of $55,000, a
30% chance that he will land a job with a salary of $45,000 and a 10% chance that he will not
land a job and therefore have to perform menial labor earning $20,000. Should Chris take the
job offer in hand or not?
We will use Excel OM to create the decision table for the following example. (The virtual lecture
on decision trees used POM-QM for Windows to explain the decision tree concepts.)
Starting: From the OM menu, select Decision Analysis, Decision Trees. The screen will appear as
follows.
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The form on the right is used to create the branches. The data will be entered directly into the
spreadsheet. Currently, the tree is showing a decision node. The value of the node will be
placed above the node and the decision will be placed below when the computations are done.
We need to add two decision branches relating to the choice of accepting the job or not. The
form on the right has a default value of 2 branches to add and is initially set at node 1 so simply
click on the “Add 2 DECISIONS from Node 1” button to generate the following screen.
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In the cell that says Decision 1, type “Accept job” and in the cell beneath that type the value of
$60000. In the cell that says Decision 2, type “Reject offer”. Now if we reject the offer we need
to see if we get the promotion at our current position. Click on Node 3. The Decision Table form
will appear. Click on the button to add two EVENTS to node 3. The screen will appear as follows.
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Change Event 1 to “gets promotion” and Event 2 to “No promotion”. When you type these two
terms, they will not fit in the cell. The tree will be automatically reformatted when more
branches are added. Change the .5 on top of gets promotion to .7 and change the .5 on top of
No promotion to .3. Finally, below gets promotion place the salary of $70,000.
Next click on Node 5 to add two decisions for deciding whether to stay or leave. Your sheet
should appear as follows.
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Excel MSOM has already put in formulas to perform some intermediate computations. For
example, the $49,000 at Node 3 is .7*70,000 + .3*0.
Change Decision 3 to Stay and place the current salary of $50,000 in below this. Change
Decision 4 to Leave and then use the Decision tree form to add 4 events to node 7. This leads to
the following.
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Finally, change events 3 through 6 to jobs 1 through 4 and place the 4 salaries and 4
probabilities into the appropriate cells. This leads to the last tree.
The tree is solved backwards. For example, the expected value of looking for a new job is the
$50,500 shown at node 7. Since $50,500 is better than $50,000 the choice at Node 5 is 7 (the
bottom node) with a value of $50,500. Thus, the expected value of node 3 is .7*70,000 + .3
*50,500 or $64,150. This is better than the offer of $60,000 so the choice in Node 1 is 3, the
bottom node, and the value of the tree is $64,150.
Notice, that decisions are based on expected value. For example, in node M26 rather than
taking the sure $50,000 we are willing to risk finding a job that pays only $20,000 or $45,000
since we might get a job that pays $55,000 or $65,000.
Exercise 6: A minimization example
Marie Curtiss lives in Cheltenham, PA and works in Stratford, NJ. Marie has five possible options
(4 driving and 1 train) to take from home to work. The times for each of these driving routes
depend on the general traffic conditions in the Delaware Valley. The estimated times for each
route as a function of the traffic conditions are given in the table below. For example, you can
see that under good conditions (no congestion) Route 1 is the best but under bad conditions
Route 1 is the worst.
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Route
1
Route
2
Route
3
Route
4
Trains
Traffic Congestion
None
Mild
Severe
40
70
90
PA 73 to Tacony Palmyra Bridge to NJ 73 to I295 South to Bellmawr
60
65
75
Broad St to I76 (Vine st) to B Franklin Br to I276 to I295 North to Bellma
45
65
85
50
70
75
Route 3: Broad St to US1 to I76 to B Franklin Br to I276 to I295 N to Bel
70
70
70
Broad St to US1 to I76 to W Whitman Br to I276 to I295 North to Bellm
Route 5: Septa train to Lindenwold train
In the past 100 days there was no congestion on 30 days, mild congestion on 50 days and
severe congestion on 20 days.
REMINDER: Travel times are costs not profits!
7. Suppose Marie wants to leave home at the last possible moment but absolutely
guarantee that she will not be late for work which begins at 9:00. When should Marie
leave? What option should Marie select?
8. Suppose that Marie is generally a very optimistic person but be on time for work. Which
option should Marie use? When should Marie leave for work?
9. Suppose that Marie is the type of person who is always second-guessing herself and
wondering, “What if I had done the right thing instead of what I actually did?” Which
option should Marie use? Why?
10. Is expected value a good decision rule for this problem? Why or why not?
11. If Marie is interested in minimizing the total transit time over a year then she should
take which of the 5 options? Why?
12. If Marie listens to KYW before she leaves for work and if KYW traffic reports are
accurate, then what is the number of minutes that could be saved per day?
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Capacity Planning
Lab data file: capplan.lab.xlsx
Objectives
 To explain break even analysis (cost-volume analysis, crossover charts) and their
applications in .
Terminology & Concepts
 Break-even point
 Selling price or revenue per unit
 Fixed cost
 Variable cost per unit
 Buy/Make
Excel Tool
 Goal Seek
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Note: This lab is structured a little differently from previous labs. In this lab, I will present
examples and then present exercises. The solutions to the exercises are in the workbook. Do
not look at them until you have tried the problem yourself.
One of the tools commonly used to answer capacity questions is the breakeven model. The
most basic model is one that consists of three factors (fixed cost, variable cost and selling price)
that you should have seen in a couple of courses prior to this one. We will begin with this model
but then move on to breakeven models that are more appropriate for .
The basic breakeven model
Notation
P = selling price per unit
FC = Fixed cost. This is independent of the number of units served or produced.
VC = Variable cost per unit.
Q = quantity or volume in units
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Given these factors the total profit is given by
Profit =Total Revenue – Total Costs
= P*Q
- (FC + VC*Q)
In this model, the breakeven point is the volume at which there is neither a profit nor a loss.
Thus we are looking for the point at which
Total Revenue = Total Costs.
Example 1 (traditional break-even analysis): A company has (annual) fixed costs of $20,000.
Variable costs are $12 per unit and the selling price of the item is $20 per unit. Find the
breakeven point in both units and dollars.
In both software packages, when you use File, New you get the option of either a breakeven
analysis or a cost-volume analysis. The break-even analysis is designed for this particular model
that has a fixed cost, a variable cost and a selling price. For all other types of breakeven models
select cost-volume analysis from the submenu. Using Excel OM (Breakeven analysis, Breakeven)
our results are as displayed below. The complete spreadsheet is in our Excel lab workbook.
Notice that the breakeven point can be expressed as either 2500 units or as $50,000. At 2500,
the total costs are $50,000 and the total revenues are $50,000. There is neither a profit nor a
loss.
Breakeven Analysis
Data
Fixed cost
Variable cost
Revenue
Option 1
20000
12
20
Results
Breakeven points
Units
2500
Dollars $ 50,000.00
Graph: A graph is presented below. Notice that the revenues begin at 0 and climb at a steady
rate of $20 per unit while the costs begin at $20,000 and climb at a lesser rate of $12 per unit.
Thus for each unit produced we have a contribution per unit of $20-$12 = $8. In order to
recover the fixed costs at this rate we need
$20,000/($20-$12) = $20,000/$8 = 2500 units which is reported as the breakeven point. In
general the breakeven point is
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the amount of money to be recovered/the rate of recovery.
Also, notice that the breakeven point can be expressed in dollars if we look at the vertical ($)
axis rather than the horizontal (units) axis. The software reports a dollar value breakeven point
of $50,000. In general, in production we are more concerned with units rather than $ values.
Cost-volume analysis
120000
100000
$
80000
60000
40000
20000
0
0
2000
4000
6000
Units
Costs
Revenue
Semifixed costs
Breakeven analysis generally assumes that the fixed costs remain the same regardless of the
volume. This is not always true. Consider a medical clinic. There are fixed costs for operating
the clinic such as the lease or loan payments for the facility. In addition though there are other
“fixed” costs that depend on the clinic’s hours. If the clinic is open from 6am to 10pm then
there are fixed costs for the salary of the day (6am -2pm) supervisor and fixed costs for the
salary of the evening (2pm-10pm) supervisor. Thus the total fixed costs depend on whether
there is one shift or 2 shifts at the clinic. This is demonstrated graphically in the spreadsheet
named semifixed costs where we display both increases in the number of shifts but also
increases in the number of facilities.
Breakeven analysis for operations
In operations, our breakeven analysis is used for situations different from above. As a general
rule our concern in operations is NOT with revenue so the following models are not trying to
find a point of $0 profit but rather are trying to find a point where one option is equal to
another option.
Example 2: Equipment choice
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Gateway Delivery Service is purchasing a new delivery vehicle. Of the two under consideration,
one has a gasoline engine and the other has a diesel engine. The purchase prices are $20,000
for the regular car and $23,000 for the diesel car. The operating costs for the regular car are
$.32 per mile, while the operating costs for the diesel car are $.27 per mile. Find the breakeven
point (in units).
The breakeven point in this example is the volume at which we are indifferent between the two
cars because the total costs are the same. Let m= miles. Then, we are looking for m such that
Total Cost of regular car = Total cost of diesel car
$20,000 + .32m
= $23,000 + .27m
.32m - .27m = $23,000-$20,000
.05m=$3000
m=$3000/.05
m=60,000 miles.
The Excel OM analysis (Breakeven analysis, cost-volume) is given below along with the graph.
Notice that the regular car is less costly up to the breakeven point and then the diesel car
becomes less costly. Thus, if we think that the car will be used for less than 60,000 total miles
we should purchase the regular car otherwise we should purchase the diesel car. Notice, that
this problem contains only costs and has no revenue. This is the typical type of problem in
operations.
Cost-volume analysis
Data
Fixed cost
Variable cost
Regular Diesel
20000
0.32
Results
Breakeven points
Regular vs. Diesel
Units
Dollars
60000
39200
MSOM Lab Exercises and Examples
23000
0.27
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$
Cost-volume analysis
70000
60000
50000
40000
30000
20000
10000
0
0
50000
100000
150000
Units
Regular
Diesel
Example 3: Buy/Make
Another standard type of capacity problem in operations is deciding whether we should make
something ourselves or buy from an outside vendor. Consider the following:
Tiara Computer Company produces microcomputer systems for home use. Provided with the
system are joysticks. Currently, Tiara purchases the joysticks from Trackit Inc. for $10 per pair. A
feasibility study has indicated that Tiara could produce the joysticks itself. The cost to set up the
production would be $30,000 and the joysticks could be produced at a cost of $8 per pair. Find
the breakeven volume.
In this problem we must once again equate the costs of two options. Let j= pairs of joysticks.
Then we are looking for j such that
Total cost buy = Total cost make
10*j
= 30,000+8j
2j=30000
j=15000.
Again, we use Excel OM (Breakeven analysis, cost-volume) and the data, results and graph are
shown below. Notice that below the breakeven point, buying is less expensive while above the
breakeven point making is less expensive.
Cost-volume analysis
Data
Make
Buy
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Fixed cost
Variable cost
30000
8
Results
Breakeven points
Make vs. Buy
0
10
Units
Dollars
15000
150000
The breakeven volume can be seen to be 15,000 units or $150,000.
$
Cost-volume analysis
350000
300000
250000
200000
150000
100000
50000
0
0
10000
20000
30000
40000
Units
Make
Buy
Example 4: Multiple options
A company has three machines from which it can select. The purchase costs of the three
machines and the variable costs are given in the table below.
a. Find the economical volume for each of the machines.
b. If the anticipated demand is 50,000 units which machine should be selected?
c. At what volume will the total cost for Machine 1 be $1,000,000?
Machine
Purchase cost
1
2
3
$100,000
$180,000
$250,000
Variable cost per
unit
$20
$17
$16
Data
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Machine 1
Machine 2
Machine 3
100000
180000
250000
20
17
16
Fixed cost
Variable cost
Volume(optional)
50000
Results
Breakeven points
Units
Dollars
Machine 1 vs. Machine 2 26666.66667 633333.3333
Machine 1 vs. Machine 3
37500
850000
Machine 2 vs. Machine 3
70000
1370000
Volume analysis @50000 units
Machine 1
Machine 2
Machine 3
Total cost
$1,100,000.00 $1,030,000.00 $1,050,000.00
$
Cost-volume analysis
2000000
1800000
1600000
1400000
1200000
1000000
800000
600000
400000
200000
0
0
20000
40000
60000
80000
100000
Units
Machine 1
Machine 2
Machine 3
Examining both the table and the graph we see that machine 1 is the best choice for volumes
below 26667 units, machine 2 is best for volumes between 26,667 and 70,000 units and
machine 3 is best for volumes above 70,000 units. There is an extra break even point at 37,500
units. This is the comparison of machines 1 and machine 3. However, at 37,500 units, both
machine 1 and machine 3 are more expensive than machine 2 so we don’t care very much
about this particular breakeven point.
We will use Excel’s Goal Seek tool for question c. Look at the screen below. In Excel 2003
click on Tool, Goal Seek or in Excel 2007/2010/2013 click on the Data tab, What-if
Analysis dropdown in the Data Tools section, Goal Seek. The small goal seek window
appears as seen below. We indicate in this window that we want to set cell B21 to the
desired $1,000,000 by changing the volume in cell B11. After pressing okay we get the
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results as displayed on the next page. At a volume of 45,000 units, our total cost for
machine 1 is $1,000,000.
Cost-volume analysis
Data
Fixed cost
Variable cost
Machine 1
Machine 2
Machine 3
100000
180000
250000
20
17
16
Volume(optional)
Results
Breakeven points
Machine 1 vs.
Machine 2
Machine 1 vs.
Machine 3
Machine 2 vs.
Machine 3
45000
Units
Dollars
26666.66667 633333.3333
37500
850000
70000
1370000
Volume analysis @45000 units
Machine 1
Total cost
MSOM Lab Exercises and Examples
Machine 2
$
Machine 3
$
Page 68 of 118
$1,000,000.00 945,000.00
Example 5: Including Revenue as a process
970,000.00
A company is trying to decide between two processes. Process 1 would cost $300,000 to
implement and lead to a variable cost of $225 per customer. Process 2 would cost $400,000 to
implement but lead to a lower variable cost of $200 per customer. The revenue per customer is
$400.
a. What is the breakeven point in units for Process 1?
b. What is the breakeven point in units for process 2?
c. What is the breakeven point between Process 1 and Process 2?
d. If 8000 customers are expected to be processed which machine should be chosen and
what is the total profit (or loss)?
Data
Fixed cost
Variable cost
Volume(optional)
Process 1
Process 2
Revenue
300000
400000
0
225
200
400
8000
Results
Breakeven points
Units
Dollars
Process 1 vs. Process 2
4000
1200000
Process 1 vs. Revenue 1714.285714 685714.2857
Process 2 vs. Revenue
2000
800000
Volume analysis @8000 units
Process 1
Process 2
Revenue
Total cost
$2,100,000.00 $2,000,000.00 $3,200,000.00
The breakeven point for process 1 is the breakeven point between process 1 and revenue or
1714 units. The breakeven point for process 2 is the breakeven point between process 2 and
revenue or 2000 units. The breakeven point between process 1 and process 2 is 4000 units.
Thus, if the expected volume is below 4000 units we should select process 1 because it has
lower fixed costs. If the anticipated volume is above 4000 units we should select process 2
because it has lower variable costs. The graph should help to clarify the results.
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$
Cost-volume analysis
2000000
1800000
1600000
1400000
1200000
1000000
800000
600000
400000
200000
0
0
2000
4000
6000
8000
10000
Units
Process 1
Process 2
Revenue
Exercises. Solve exercises 1-4 below using Excel OM or POM for Windows. Use Excel for
Exercise 5. It demonstrates one of the principles from the lecture. The solutions to the exercises
are in the workbook. Do not look at them until you have tried the problem yourself. Be sure you
understand the meaning, use, interpretation and application of the breakeven point as well as
how to compute it.
1. A company pays $9200 per month in fixed costs. It costs $.70 per unit to manufacture the
product that sells for $.90 per unit.
a. What is the breakeven point (in units/month)
b. If the company makes and sells 75,000 units in a month what is the profit?
c. Use Excel’s goal seek tool in order to determine the volume required in order to have a profit
of $10,000
2. A company is trying to decide between the purchase of two machines. The basic machine
costs $30,000 to purchase and will lead to per unit costs of $12. The deluxe machine costs
$10,000 more but will save $2 per unit.
a. What is the breakeven point (in units)
b. If the company makes and sells 8,000 units in a month what is the total cost for each
machine?
3. A company is trying to decide whether to purchase components it needs or to make the
components itself. The price of the components from an outside vendor is $28. Producing them
in-house would cost $10,000 to set up the production and then would cost $20 per unit.
a. What is the breakeven point (in units)
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b. If the company makes and sells 1,000 units in a month what is the total cost for buying and
the total cost for making?
4. Three telephone calling plans are available. One costs $5 per month and $.20 per minute.
The second costs $25 per month and $.10 per minute while the third costs $40 per month and
$.05 per minute. Over what range of minutes should each plan be used?
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Location & Transportation
Lab data file: location.lab.xlsx
Objectives
 To explain location models
o Factor rating
o Center of gravity
o Use of the median in location
 One-dimensional location
 City block distance
 To explain the transportation problem
o Supply equals demand
o Supply greater than demand
o Application to location
Terminology & Concepts
 factor
 weight
 x-coordinate, y-coordinate
 distance
 center of gravity
 Shipping cost
o proportionality
o additivity
 Supply
 Demand
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Excel functions and tools
 Sum
 Sumproduct
 Average
 Median
MSOM Lab Exercises and Examples



Sort
Data Filter
Bubble Graph
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Exercise 1: Factor rating using Excel
A company is considering locating in one of three sites and has identified five factors that are
important and the relative importance (weight) for each of these factors. They have scored
each site on each factor and the scores are given below. Higher scores are preferable. The
company would like to know at which of the three sites it should locate.
FACTORS
Land space
Land costs
Traffic density
Neighborhood
income
Zoning laws
Weight Site A
Site B
Site C
0.30
60
70
80
0.25
40
80
30
0.20
50
80
60
0.15
0.10
50
80
70
20
40
90
Either POM for Windows or Excel OM will solve this problem for us but we will use Excel (not
Excel OM) to solve the problem and then you may use either software package to check your
answers.
We simply need to determine the weighted total or weighted average for each of the sites. This
is a straightforward application of the SUMPRODUCT function.
 In cell B8 use the SUM function to total the 5 weights in column B.
 In cell A9 place the name “Weighted Total” and in cell A10 place the name “Weighted
average”.
 In cell C9 place the SUMPRODUCT formula that multiplies the 5 weights in column b by the
5 scores in column C. Be sure to fix the weights by using the $ notation using the F4 key.
 Copy cell C9 to cells D9.E9.
 To get the weighted average we need the weighted total divided by the total weight.
Therefore, in cell C10 place the weighted total (C9) divided by the total of the weights (B8).
Be sure to fix the weight total by using the $ notation (F4 key).
 Copy the results in C10 to columns D and E.
Your spreadsheet should appear as follows.
FACTORS
Land space
Land costs
Traffic density
Neighborhood
Weight Site A
Site B
Site C
0.30
60
70
80
0.25
40
80
30
0.20
50
80
60
0.15
50
70
40
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income
Zoning laws
0.10
80
20
90
Total of Weights
1.00
Weighted Total
53.5
69.5
58.5
Weighted Average
53.5
69.5
58.5
Notice that since the total weight is 1, the weighted average and total are the same.
1. Which site should be chosen?
Exercise 2: Center of Gravity
A company has eight factories in eight different U.S. cities. The company would like to locate a
distribution hub in the “center” of these facilities. The table below contains the longitudes and
latitudes of each factory as well as the number of units to be shipped to the distribution center
(in millions). We will use Excel (not Excel OM) to find both the unweighted and weighted center
of gravity.
Site
Weight Longitude Latitude
Los Angeles
6
118
34
San Francisco
3
123
38
Houston
1
95
30
Dallas
4
97
33
St. Louis
1
90
38
Chicago
1
87
42
Pittsburgh
5
80
41
Atlanta
7
84
34
Even though this model is entirely different from factor rating, the steps are essentially the
same. We will compute the total for the columns, the average for the columns, the weighted
total for the columns and the weighted average for the columns.
 In cell B11 place the SUM function for the 9 elements in the column. Copy this to columns C
and D.
 In cell C12 place the AVERAGE function for the 9 elements in the column. Copy this to
column D.
 In cell C13 place the SUMPRODUCT function for the weights in column B multiplied by the
longitudes in column C. Be sure to fix the weights with $ using the F4 function key. Copy this
to column D.
 In cell C14 place the weighted total (C13) divided by the total of the weights ($B$11) and
copy this to column D.
 Format to two decimal places and your spreadsheet should appear as follows.
Site
Weight Longitude Latitude
Los Angeles
6
118
34
San Francisco
3
123
38
Houston
1
95
30
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Dallas
St. Louis
Chicago
Pittsburgh
Atlanta
Total
Average
Weighted Total
Weighted
Average
Questions:
4
1
1
5
7
97
90
87
80
84
33
38
42
41
34
28
774
96.75
2725
290
36.25
1003
97.32
35.82
1. What is the center of gravity if the weights are NOT considered?
2. What is the weighted center of gravity?
Excel Graph - Optional
We will create a bubble graph in Excel. This will display the locations but used different
sized bubbles to indicate the different weights of the locations. The bubble graph
requires the x coordinate, followed by the y coordinate followed by the weight. Thus, to
make charting easier
Step 1: Copy the Weight column (B) to the right hand side of the data (column E).
Step 2: Select the three columns (Longitude, Latitude, Weight) without their
headers.
Excel 2003:
Step 3: Click on the chart wizard.
Step 4: Select bubble chart and finish the chart
Excel 2007/2010/2013
Step 3: Insert tab, Other Charts, Bubble
In either case right-click on the chart to format it.
Beyond the textbook The Distance table
In POM for Windows select the Location Module, File, New, Center of gravity and create
a table with 8 possible locations. Copy and paste the data from Exercise 2 into POM for
Windows. Do not include the header row when you copy. Press the Solve button and
copy the results to the spreadsheet labeled “Center of Gravity”
First, notice that the results, that is, the average and weighted average, match the
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results we derived above when using Excel. Second, notice that POM for Windows also
reports
the median location. In this example it is longitude = 90, latitude = 34. The median is the
Center of Gravity when distances are measured by right-angles rather than by straight
line.
Please observe that POM for Windows has two other windows of outputs and a graph.
The extra windows display the distances from each location to every other location, the
total distance and the total weighted distance. One table is for straight line distance
while the other is for city-block movement.
Copy the straight-line distance table to the current spreadsheet. Use the straight-line
distance table to answer the following.
3. If you need to locate at one of the 7 existing sites, what city minimizes the total
movement (or total weighted distance).
Exercise 3: Transportation
A Texas company is shipping units from three factories to 5 distribution centers. The supply at
each of the factories and the demands at each of the centers are given. The total supply and
the total demand are equal. The cost to ship one unit from each factory to each distribution
center is given. The information is below.
Unit costs Dallas
Austin
$20
Beaumont
$70
Corpus Christi
$45
Demand
(units)
50
El Paso
$45
$65
$80
70
Fort
Worth Galveston Houston
$10
$30
$15
$55
$90
$70
$50
$30
$60
65
55
Supply
(units)
100
80
150
90
Assumptions:
The shipping costs along any origin-destination pair are proportional. For example, the
cost to ship the first unit from Austin to Dallas is $20, the cost to ship the second unit is $20, …,
the cost to ship the millionth unit from Austin to Dallas is the same $20. In other words, we do
not receive any quantity discounts for shipping a large number of units. Please note that we
have seen proportionality before when we covered breakeven analysis. Proportional is the
mathematical word and no quantity discount is the operational implication.
MSOM Lab Exercises and Examples
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The shipping costs along any origin-destination pair are additive. For example, if we
ship one unit from Austin to Dallas and one unit from Austin to El Paso the cost is $20 +$45 =
$65. Let me give an example of when this might not hold. Suppose we shipped one unit from
Philadelphia to Pittsburg and one unit from Philadelphia to Harrisburg. If we can put them on
the same truck and simply stop at Harrisburg on our way to Pittsburg then the total cost would
be less than the sum of the two individual costs of shipping the two units from Philadelphia to
the two destinations. Additive is the mathematical word and no geographical discount is the
operational implication.
In the table below we display a sample solution which is NOT the best possible solution.
Unit costs
Austin
Beaumont
Corpus Christi
Total shipped in
Dallas El Paso
10
20
10
10
30
40
50
70
Fort
Total shipped
Worth Galveston Houston
out
10
15
45
100
10
30
20
80
45
10
25
150
65
55
90
The total cost is the sum of the number of units shipped multiplied by the shipping cost per unit
for each of the 15 cells. In cell B20, use the SUMPRODUCT function to find the total shipping
cost for this solution.
1. What is the total cost?
Let’s find the best answer by using the software. Either POM for Windows or Excel OM will find
the shipments that should be made to minimize the total shipping cost but POM for Windows
will give us some extra useful information so we will use POM for Windows.
 In POM for Windows, from the Module menu select transportation
 On the creation screen select 3 origins and 5 destinations and be sure to use the default
value of minimizing costs
 Copy the data to the empty data table
 Press Solve
 Copy the results to the spreadsheet.
Questions:
2. What is the minimum total shipping cost?
3. By what percentage are the costs of the sample solution higher than the minimum
costs?
Notice that not all of the cells in the optimal solution have shipments. We have a rule that we
never need to use more than m+n-1 shipments where m is the number of rows and n is the
number of columns. In this case, m+n-1 = 3+5-1=7.
MSOM Lab Exercises and Examples
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4. There are 15 (3 by 5) possible shipments. How many of the cells have shipments in this
optimal solution found using POM for Windows?
Notice that there is a second window of output named Marginal Costs. Copy this table to the
spreadsheet.
Notice in the spreadsheet that for every cell in the original solution that does NOT have a
shipment there is a marginal cost and for cells with shipments there is NO marginal cost. In the
solution we should not ship any units from Austin to El Paso. The marginal cost is $25 for this
cell. This means that if we decide to ship a unit from Austin to El Paso our total cost will
increase by $25. Notice that this is not the same as the original shipping cost of $45 from Austin
to El Paso.
5. Examine the marginal cost table. What will it cost to ship a unit from Austin to El Paso?
Exercise 4: Supply > Demand
In most cases the supply would be greater than the demand. This causes a very minor change in
the output display. Consider the previous example except suppose that the demand at Houston
drops by 10 units to 80 units. Now the total supply is greater than the total demand.
 Copy the numbers into the POM for Windows transportation model.
 Press Solve
 Copy the results
Questions:
1. What is the minimum total shipping cost?
2. This problem had 10 more units than were demanded. Where do these units end up
sitting at the end of the problem?
Exercise 5: Supply > Demand
In some cases the supply would be less than the demand. This causes a very minor change in
the output display. Consider the original example except suppose that the demand at El Paso
rises by 10 units to 80 units. Now the total supply is greater than the total demand.
 Copy the numbers into the POM for Windows transportation model.
 Press Solve
 Copy the results
Questions:
MSOM Lab Exercises and Examples
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3. What is the minimum total shipping cost?
4. This problem had 10 fewer units than were demanded. What is the repercussion of this?
Mathematically, supply greater than demand and demand greater than supply are easy to
handle with the addition of dummy destinations or origins. Operationally, there is an enormous
difference between the two situations. If supply is greater than demand that is normal. If
demand is greater than supply then we have problems other than worrying about the
shipments. The real issue is to determine what we need to do to make up the shortages as
quickly as possible.
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Exercise 6: Transportation Costs and Location
A company currently has two factories. The demand has increased and they are going to add a
factory in either Philadelphia or Seattle. The supplies, demands and costs are given below.
L.A.
New
Orleans
Philadelphia
Seattle
Demand
Pittsburgh St Louis
$ 100
$
75
Denver
$
50
$
$
$
$
$
$
80
40
110
200
$
$
$
60
50
70
100
90
90
30
400
Supply
150
225
350
350
We need to analyze the total shipping costs using Philadelphia and compare these with the
total shipping costs using Seattle. Thus, two separate transportation problems need to be
solved. We will provide the easiest way. In addition, we will use Excel OM, just to demonstrate
Excel OM and transportation. In order for Excel OM to work you must have Solver installed as
one of the add-ins.




From the Excel OM menu select transportation
On the creation screen select 4 origins and 3 destinations
Copy the data into the table.
Now at this point, we need to set up the problem to analyze Philadelphia. In order to do
this, change the supply at Seattle from 350 to 0. This eliminates the Seattle factory!
 Click on Tools, Solver, Solve and okay
Question: What is the total cost for Philadelphia?
In order to find the total cost for Seattle
 Change the supply at Philadelphia to 0
 Change the supply at Seattle to 350
 Click on Tools, Solver , Solve and okay
Question: What is the total cost for Seattle?
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Layout
Lab data file: layout.lab.xlsx
Objectives
 To explain layout models
o Operations Layout
o Assembly Line Balancing
 To explain the difference between optimization and heuristic methods
Terminology & Concepts
 Work center/department
 Loads/weight
 Cost/distance
 Pairwise comparison
 Heuristic method
 Explicit enumeration




Cycle time
Minimum theoretical number of stations
Efficiency
Balance delay (inefficiency)
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Exercise 1: Process layout – number of layouts
Suppose two processes, A and B, need to be placed in two rooms, rooms 1 and 2. There are two
ways to accomplish this. Process A can be placed in Room 1 with Process B in room 2 or Process
B can be placed in Room 1 with Process A in Room 2.
Q1. On the worksheet, list all of the ways to place 3 processes, A, B and C into three Rooms, 1, 2
and 3.
Q2. On the worksheet, list all of the ways to place 4 processes, A, B, C and D into four Rooms, 1,
2, 3 and 4.
MSOM Lab Exercises and Examples
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Q3. In general, if there are n rooms and processes how many ways are there to place the n
processes in n rooms?
Exercise 2: Process layout – POM for Windows
A company needs to locate 7 processes into 7 work rooms. The flows (in trips per day) between
processes are as given below.
Flow Table
Broaching
Boring
Drilling
Grinding
Milling
Turning
Ship &
Receive
Broaching Boring Drilling Grinding Milling Turning Ship & Receive
0
25
0
20
0
10
50
0
0
15
0
30
0
45
0
0
0
0
0
70
65
0
0
15
0
35
0
10
0
20
0
25
0
10
25
50
45
0
0
0
0
20
55
45
60
60
15
25
0
The rooms into which the processes will be placed appear as below.
Room
1
Room
4
Room
7
Room
2
Room
5
Room
3
Room
6
The distances (in yards) between processing rooms are given in the table below.
Distance
Table
Room 1
Room 2
Room 3
Room 4
Room 5
Room 6
Room 7
Room
1
0
16
32
16
32
48
32
Room
Room
Room
Room
Room
2
3
4
5
6
Room 7
16
32
16
32
48
0
16
32
16
32
16
0
48
32
16
32
48
0
16
32
16
32
16
0
16
32
16
32
16
0
48
64
16
32
48
32
48
64
16
32
48
0
1. How many ways are there to place these 7 processes in the 7 rooms?
MSOM Lab Exercises and Examples
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Using POM for Windows’ Layout module, create a 7 by 7 problem. Since the distance from
room I to room J is the same as the distance from Room J to room I for every pair of rooms use
the symmetric distance option which is the default option on the creation screen.
 Copy the two tables of data, the flow table and the distance table, separately into the
data area of POM for Windows. Do not copy the header rows.
 The method above the data is set to explicit enumeration. Change this method to
pairwise comparison. Pairwise comparison is a heuristic method. A heuristic method is
a method that generates good solutions but not necessarily the best solution.
 Press the Solve button and copy the main results table to the spreadsheet in the area
for table 1.
2. What is the total movement?
3. What is the assigned area for each room?
 Return to the data and change the method to explicit enumeration. This method, finds
the total movement for every one of the 7! = 5040 possible layouts and selects the
layout with the minimum total movement.
 Press the Solve button and copy the main results table to the spreadsheet in the area
for table 2.
4.
5.
6.
7.
What is the total movement?
What is the assigned area for each room?
Did the heuristic method of pairwise comparison find the best possible layout?
What is the percentage increase in movement when using pairwise comparison
instead of explicit enumeration?
8. What is the percentage decrease in movement when comparing explicit
enumeration to pairwise comparison?
I want to make certain that you understand how the total movement is being computed. There
are 42 interactions of rooms when counting room I to room J as different from room J to room
I. Let us consider one of the interactions.
 Copy the department to department flow table results from the second POM output
Window to the Excel workbook.
9. How many trips are made from room 5 to room 3?
10. What is the distance from room 5 to room 3?
The movement (from room to room) is the number of trips multiplied by the distance.
MSOM Lab Exercises and Examples
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11. What is the total movement of the trips from room 5 to room 3?
Suppose that shipping and receiving must be placed in Room 7 because Room 7 is the loading
dock. In POM, click on the edit data button to return to the data. The Fixed Room column
feature at the right of the table enables us to fix a process in a specific room. If you click on a
cell in the Flow (table) column a drop-down box will appear with the room names.
 In the row for Shipping, click on the right-most cell and a drop-down box will appear.
Use the drop-down box in the right-most column to fix shipping and receiving in room 7,
press Solve and copy the results to the spreadsheet.
12. What is the total movement?
13. Is this better or worse than the solution found when shipping was not restricted
to Room 7?
Important: There is a lesson in the last question. When you add constraints or conditions to
ANY situation, the situation might stay the same or get worse but it can not improve!
A natural question should be, “If pairwise comparison is not guaranteed to find the optimal
solution whereas explicit enumeration is, why don’t we always use explicit enumeration?” In
order to address this question please create a new, empty operations layout data table that is
10 by 10. Leave all the data as 0s, set the method to explicit enumeration and press Solve.
(Notice that the Solve button becomes a Cancel button.)
14. What is the number of possible layouts for a problem with 10 rooms and 10
processes?
15. Why can’t we use explicit enumeration all of the time?
Exercise 3: Process layout – Excel OM
We will use the same example as in Exercise 2. In Excel OM, select the Operations Layout model
and create a 7 by 7 data table.
 Copy the original data into the table in the proper places.
Excel OM handles the room assignments differently than POM for Windows does. There is a
column into which the room assignments are made. Initially, Excel OM assigns the rooms in
order – Room 1, Room 2, etc. In other words, every process is assigned to a Room. Thus the
total movement is the movement for this particular layout.
1. What is the total movement?
MSOM Lab Exercises and Examples
Page 84 of 118
Notice that for the first time this semester, the Excel OM spreadsheet has a Solve button. It is
not possible to fix SOME of the rooms. Either, you enter the rooms for every process as done
above or you press the Solve button to find the best possible layout.
 Press the Solve button.
2. What is the total movement?
3. What method was used to get this solution?
Exercise 4: Assembly line balancing
A company manufactures a product. The demand for the product is 96 units per 24 hour day.
Use Excel (not Excel OM) to answer the following questions. You will be able to check the
answers to the first 4 questions after more information is given and you use the software to
solve the problem.
The cycle time is the length of time allowed to manufacture a product or perform a service. The
cycle time is the inverse of the demand rate, expressed in the production time units (minutes in
this example). For example, if the demand is 30 units per hour then the cycle time is 2 minutes.
1. What is the cycle time (in minutes) that will meet the demand?
We want to divide the work up and spread out into stations. Our goal is to assign tasks to
stations in such a way that the precedences are met and we don’t use more time at a station
than what the cycle time allows The minimum theoretical number of stations is given by
Min theoretical # stations = Total time to produce 1 unit/cycle time rounded UP
2. Suppose it takes 50 minutes to manufacture each unit. Using the cycle time from
question 1, what is the minimum theoretical number of stations?
The total amount of production time allocated during each cycle is given by
Total production time per cycle = Cycle time * number of stations
3. Using the answers to questions 1 and 2 above, how much work time will be allocated
during each cycle?
The amount of idle time per cycle is given by
Idle time per cycle = Time allocated per cycle – Total time to make 1 unit
4. How much idle time will accrue in each cycle?
MSOM Lab Exercises and Examples
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Our goal is to end up with the actual number of stations being equal to the minimum
(theoretical) number of stations. We don’t have any method available that is guaranteed to find
the optimal solution. POM for Windows has 5 heuristic methods to solve this problem. These
methods find good solutions but not necessarily the best possible solutions.
The precedence table for the product is given below.
 Use the Assembly line balancing module from POM for Windows and create a data table
for this 11 task problem. Be sure that in the lower left of the creation screen you select
Minutes as the time unit for the tasks.
TASK
A
B
C
D
E
F
G
H
I
J
K
Predecessor Predecessor Predecessor
Minutes 1
2
3
3
6
7 A
5 A
B
2 B
4 C
5 F
7 D
E
1 H
6 E
4 G
I
j
 Copy the data
 Leave the method at longest operation time.
 Set the demand rate (above the data table) to compute and then to 96 units per 24
hours
 Press Solve.
5. Do your answers to questions 1 through 4 match the results found by using the
software?
6. What is the actual (not theoretical) number of stations used?
Notice that there is a second window of output that has the minimum number of stations for
each of the 5 methods.
7. What is the actual (not theoretical) number of stations among all of the methods used?
MSOM Lab Exercises and Examples
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Waiting Lines
Lab data file: queue.lab.xlsx
Objectives
 To explain waiting line models
o Single server
o Multiple server
o Constant service time
o Finite population model
 To compare different deployments of capacity
 To explain costing of waiting lines
Terminology & Concepts
 Arrival rate
 Service rate
 Utilization
 Average waiting time
 Average system time
 Average number waiting
 Average number in the system
 Probability of exactly n customers in the system
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Notes: In waiting line models, the arrival and service rates must use the same time unit. In
example 1 the time unit is minutes. When using Excel it is a good idea to place “per minute” or
“per hour”, etc. in the cells to the right of the arrival and service rates. In POM for Windows the
time unit is in the area above the data table.
The results for average time waiting and average time in the system have the same time unit as
the input. For example, in example 1, the average waiting time (see results and question 4) is
given as 1.33. This is 1.33 minutes since the arrival and service rates are in minutes. Please note
that question 4 asks for the number of seconds so you need to use Excel to multiply the .67 by
60 seconds per minute. Several questions require small calculations to be performed. Some of
the calculations are multiplication by 60 while others are comparisons of two results. These
calculations should all be performed in Excel in the workbook rather than by using a calculator
or even doing the calculations in your head.
MSOM Lab Exercises and Examples
Page 87 of 118
Exercise 1: Characteristics of a telephone operator system
Using a single operator
Phone calls arrive at the Temple University switchboard at a rate of 2 per minute. It takes, on
average, 20 seconds to handle each call – that is, to direct the call to the proper extension.
There currently is a single operator.
 Even though this is a single server model, use Excel OM to create a multiple server
(M/M/s) model.
 Enter the arrival and service rates in the appropriate cells in column B.
 Next to these two cells, in column C, place the term “per minute” in order to remind us
of the time unit being used in this problem.
 Place a 1 in the number of servers and copy the answers to the summary sheet.
1.
2.
3.
4.
5.
What percentage of the time is the operator busy?
What is the average number of customers on hold?
What is the average number of customers either on hold or talking with the operator?
What is the average time (in seconds) that customers are on hold?
What is the average time (in seconds) that customers are on the phone before the
operator directs their call to the proper extension?
Using 2 operators
The university would like to improve its performance so it is considering adding another
operator.
 Change the number of servers from 1 to 2 and copy the new answers to the summary
sheet.
6. What percentage of time is each of the two operators busy?
7. How does this compare with the single operator answer (Q1) ?
8. What is the average number of customers on hold?
9. What percent of the corresponding single operator answer is this (Q2)?
10. What is the average number of customers either on hold or talking with the operator?
11. What percent of the single operator answer is this (Q3)?
12. What is the average time (in seconds) that customers are on hold?
13. What percent of the single operator answer is this (Q4)?
14. What is the average time (in seconds) that customers are on the phone before the
operator directs their call to the proper extension?
15. What percent of the single operator answer is this (Q5)?
Using a faster operator
MSOM Lab Exercises and Examples
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Another way for the university to improve the service would be to hire a more efficient
operator. Suppose the university could hire an operator that was twice as good (could service
customers in 10 seconds on average rather than 20 seconds on average) as the first.
 Change the number of servers back to 1, change the service rate and copy the new
answers to the summary sheet.
16. What percentage of the time would this new, improved operator be busy?
17. How does this compare with our original operator (Q1)?
18. How does this compare with the two operator system (Q6)?
19. What is the average time (in seconds) that customers are on hold in the new system?
20. Is this better or worse than the time on hold in the two operator system?
21. What is the average time (in seconds) that customers would spend before being
transferred to the correct department?
22. Is this better or worse than the total time customers spend in the two operator system?
Can you explain the importance of questions 20 and 22 in the context of using 2 checkers vs. 1
checker and 1 bagger in a supermarket checkout system?
Using a more consistent operator
Suppose that the university could automate the service and that each phone call would take
exactly 20 seconds rather than 20 seconds on average.
 From the OM menu, create a constant service (M/D/1) model, enter the arrival and
service rates and copy the results to the summary sheet.
23. What percentage of the time would this new, improved operator be busy?
24. How does this compare with our original operator?
25. What is the average time (in seconds) that customers are on hold in the new system?
26. What percent of the corresponding single operator answer is this?
27. What is the average time (in seconds) that customers would spend before being
transferred to the correct department?
28. What percent of the corresponding single operator answer is this?
Exercise 2 – The finite population model
A specialized computer lab in the School of Business has 30 computers. The lab is open 5 days
per week for 8 hours each day. On average, problems occur on each of these computers once
per 5 day week. On average, it takes two hours to repair these problems. Currently there is one
technician.
 Create a finite population model.
MSOM Lab Exercises and Examples
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 Enter the daily rate at which computers breakdown, enter the daily rate at which the
technician repairs computers, enter 1 repairperson and enter 30 customers.
1. What is the percentage of time that the technician is busy?
2. What is the average number of computers that are unusable?
3. Suppose a second technician is added. How will this affect the average number of
computers that are unusable?
Exercise 3 – costing
Customers arrive at Haircuts, Inc, at a rate of 12 per hour. Haircuts, Inc employs 5 stylists, each
of whom is paid $10 per hour (plus tips). A haircut takes 20 minutes. The HairStylists is a
competitor located in the same strip mall. The manager of Haircuts, Inc is concerned that if
customers wait too long they will visit the competition the next time. Therefore, she wants to
perform a cost analysis and is using $25 per hour of customer waiting time as the cost of lost
goodwill and future business.
 From POM for Windows, select the multiple server model and at problem creation
select to use costs.
 In the data table enter the arrival rate, service rate, number of serves, wages and cost of
waiting.
 Solve.
 Copy the table of probabilities to the Excel workbook and copy the table of cost vs.
number of servers to the workbook.
Examine the table of probabilities.
1. What is the probability that a customer will have to wait?
Examine the cost vs. number of servers results.
2. How many stylists should be working to minimize the total cost?
MSOM Lab Exercises and Examples
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Inventory
Lab data file: inventory.lab.xlsx
Objectives
 To explain independent demand inventory models
o ABC Analysis
o EOQ Based models
 The Basic EOQ model
 The POQ model
 Quantity Discounts
Terminology & Concepts
 Demand rate
 Holding cost rate
 Setup/Order cost
 Economic Order Quantity (EOQ)
 Average inventory
 Number of orders per year
o time between orders
 Reorder point
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Excel
 Data sort
 Data filter
 SQRT
Exercise 1: ABC Analysis
A company stocks 25 items. The annual item usage and the cost of the items is given in the
table on the next page and in the spreadsheet named “ABC”. We will compute the annual dollar
volume of the items and sort and classify them according to dollar volume usage and then use
POM for Windows to check our work.
MSOM Lab Exercises and Examples
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Begin by copying the spreadsheet to a second sheet in order so that we can move the original
data around.
 In the spreadsheet, in cell E1 place the title “Dollar Volume”.
 In cell E2 place the formula for multiplying item A’s usage in cell B2 times its unit cost in
cell C2.
 Copy this formula down column E for the remaining 24 items.
 In cell A28 place the label “total”
 In cell E28 place the formula for summing the column.
Now that we have computed the dollar volume, the next step is to sort the items according to
the dollar volume usage that we just computed.
 Select all of the data from cell A1 to cell E26.
 Click on Data, Sort or Use the Filters on the Data tab to sort according to dollar-volume
 Be sure that at the bottom of the sort dialog that the header row option button is
checked and at the top of the dialog sort by “Dollar volume”, be sure that the sort is set
to descending and click okay.
Item
A
B
C
D
E
F
G
H
I
J
K
L
M
Annual Unit
Usage cost
6850
62.27
5770
50.23
8582
5.18
3064
45.16
1373
36.86
9296
35.13
8622
3.35
1529
74.87
1075
28.45
3664
2.66
1665
13.21
5279
43.04
6245
60.38
Item
N
O
P
Q
R
S
T
U
V
W
X
Y
Annual Unit
Usage cost
2574
22.72
3196
49.96
2613
15.15
8468
77.8
5273
86.31
293
40.17
8902
7.33
3963
26.4
1301
76.53
6714
50.65
9407
16.65
1658
10.64
The last step is to classify the items into 3 categories named A. B and C. As a general rule, 20%
of the items should be A items, 30% of the items should be B items and the remaining 50% of
the items should be C items. Since we have 25 items in this example that means that 5 of the
items should be A items, 7 or 8 items should be B items and the remaining 12 or 13 items
should be C items.




Label Column F as “Category”.
In the first 5 cells place the letter A.
In the next 8 cells place the letter B.
In the next 12 cells place the letter C.
MSOM Lab Exercises and Examples
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At this point you can check your work by using the ABC analysis model that is in the POM for
Windows software. Create a problem with 25 items, copy the data into the table, and notice
that above the data the settings for A and B items are 20% and 30% respectively. Press the
solve button and compare the POM for Windows table with your table.
The last step is to see if we have separated the categories at the proper points. Review the
categories and be certain that the separation points make sense. The cutoff points of 20% and
30% are rules of thumb. The idea is to group the most important items into the A category and
the least important into the C category.
Exercise 2 – The EOQ model
Burrell Inc. stocks an item that costs $175 per unit. The holding cost is $25 for each unit held
one year.
1. What is the interest rate used to compute the holding cost?
The item has a demand of 8,000 units per year. The order cost is $75 per order. The company
works 250 days per year. There is a 3 day leadtime between the time the order is placed and
the order is received.
We will build an Excel spreadsheet that will answer several questions regarding this problem.
The two most basic results we are looking for are the Economic Order Quantity and the reorder
point.
We will use column A on the Exercise 2 spreadsheet for labels and column B on the spreadsheet
for values.














In cell A1 place the title for the spreadsheet – “Economic Order Quantity Model”.
In cell A5 place the word “Data”.
In cell A6 place the term “Demand rate, D”.
In cell B6 place the value for the demand of 8,000 as given above.
In cell A7 place the term “Setup cost, S”
In cell B7 place the value for the setup (order) cost of 75.
In cell A8 place the term “Holding cost, H”
In cell B8 place the value for the holding cost of 25.
In cell C8 place a note “The holding cost is a fixed dollar amount per unit”. (Some of our
models will have the holding cost expressed as a percentage of unit cost rather than as a
fixed amount per unit.)
In cell A9 place the term “Unit price, c”
In cell B9 place the value for the unit price of $175.
In cell A10 place the term “Days per year”.
In cell B10 place the value of 250 days per year.
In cell A11 place the term “Lead time in days”
MSOM Lab Exercises and Examples
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 In cell B11 place the value of 3 days.
Now we will enter the formulas for the results.
 In cell A13 place the word “Results”
 In cell A14 place the term, “Optimal Order Quantity, Q*)
The formula for the EOQ is given by formula (12-1) as
Q* 
2 DS
H
 In cell B14 place the formula for the EOQ that uses Excel’s square root formula and
refers to the values in column B above. The formula is “=SQRT(2*B6*B7/B8)”.
 In cell A15 place the term “Average inventory”
 The average inventory is half of the EOQ in this model. In cell B15 place the appropriate
formula.
 In cell A16 place the term “Number of orders per year”.
 In cell B16 place the formula for the number of orders per year. This is given by the
demand divided by the order quantity. (D/Q*)
 In cell A17 place the term “Days between orders”.
 In cell B17 place the formula for the number of days between orders. This is given by the
number of days per year divided by the number of orders per year. Refer to the
appropriate cells.
 In cell A19 place the term “Annual Holding Cost”
 In cell B19 place the formula for the annual holding cost. This is given as
QH
2
 In cell A20 place the term “Annual Setup Cost”
 In cell B20 place the formula for the annual setup cost. This is given by the number of
orders which is computed above, multiplied by the cost per order which is in the data
above.
 In cell A21 place the term “Total inventory costs”
 In cell B21 place the formula for adding the annual holding cost and the annual setup
cost.
 In cell A23 place the term “Annual unit costs”
 In cell B23 place the formula for multiplying the annual demand, D times the unit cost, c.
 In cell A24 place the term “Total cost”
 In cell B24 place the formula for the sum of the annual inventory costs and the unit
costs.
 In cell A26 place the term “Daily demand rate”
 In cell B26 place the formula for the annual demand rate (D) divided by the number of
days per year.
MSOM Lab Exercises and Examples
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 In cell A27 place the term “Reorder point”
 In cell B27 place the formula for the lead time multiplied by the daily demand rate.
We will use POM for Windows to check our work.
 Select the EOQ model from the Inventory module. Notice that the create screen does
not request any information regarding number of rows or number of columns.
 Enter the data into the blank data table.
 Press Solve and copy the results to the spreadsheet in cell D1. Check that the answers in
your spreadsheet match the answers from POM for Windows.
2.
3.
4.
5.
6.
7.
What is the optimal order quantity?
What is the average inventory?
What is the average annual holding cost?
What is the number of orders per year?
What is the average annual setup cost?
What is the average number of workdays between orders?
(Days per year / # orders per year)
8. What is the average annual cost?
9. What is the reorder point?
10. If the unit cost is $100 rather than $200 what is the EOQ? What does this demonstrate?
Suppose the company must order in lots that are multiples of size 12. That is, they must order
either 12 units or 24 units or 36 units or 48, etc. Determine the largest multiple of 12 that is
below the EOQ and the smallest multiple of 12 that is above the EOQ and compare the total
costs when using each of these two numbers.
11. How many units should be ordered?
In the Excel spreadsheet that we built for this model
 copy column B to column C
 replace cell C 14 with the number determined in Question 11.
12. What is the total of the holding and the setup costs using the new order quantity?
Burrell Inc, had been ordering weekly (every 5 days).
13. What order quantity had they been using?
We could simply replace C14 with the answer from question 13 but instead let’s use POM for
Windows. Return to the problem and to editing the data. Above the data area is a box that says
“Order Quantity (0=EOQ)”. Change the 0 to the answer to question 13 and solve.
MSOM Lab Exercises and Examples
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14. What is the total of the holding and the setup costs using Burrell’s order quantity?
15. By what percentage are the Burrell’s holding and setup cost greater than the optimal
holding and setup costs.
Important: In both POM for Windows and Excel OM the holding cost can either be a fixed
number, such as in this example, or it can be a percentage of the unit cost. In POM for
Windows, if you place a “%” after the number for the holding cost it will be treated as a
percentage of the unit cost. Otherwise it will be treated as a fixed dollar amount. In Excel OM,
the decision regarding % vs $ must be made at the time of the problem creation since the
spreadsheet must be setup accordingly.
Exercise 3 – Quantity discounts
Consider the data from Exercise 2 but suppose that the cost per unit is given by the following
schedule.
1-144 units per order; $200 each
145-288 units per order; $199 each (a .5% discount)
289 or more units per order; $198 each
First note, that there is no reason to order from 1 to 144 units because this would only increase
the price. That is, we know that if the price is fixed we would order 200 units. If we order 200
units the cost per unit is $199 which is less than the cost per unit of $200 if we order 1 to144
units.
Second, recall from Exercise 2 that if the cost per unit is fixed we would choose to order 200
units. Thus, if the price is fixed at $199 we would order 200. That is, if our choice is anywhere
between 145 and 288 units we would pick 200 since the price per unit is the same for all values
in that range.
Finally, we might be willing to order 289 units since this would reduce the per unit price. There
would be no reason to order 290 or more units since it is the 289th unit which brings us the
discount. In summary, we need to compare the total cost of ordering 200 units versus the total
cost of ordering 289 units. We will use the EOQ model to perform the analysis and use the
quantity discount model to check our work.
Use the EOQ model to compute the total cost of ordering 200 units per order at $199 per unit
and paste the answers into the exercise 4 spreadsheet. Modify the data to compute the total
cost of ordering 289 units per order at $198 per unit and paste the answers into the exercise 4
spreadsheet.
1. What is the total cost of ordering 200 units per order?
2. What is the total cost of ordering 289 units per order?
3. What is the least cost order quantity?
MSOM Lab Exercises and Examples
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Check your work by using the quantity discount model in POM for Windows.
 In the inventory module select File, New, Quantity Discount Model.
 On the problem creation screen select 3 price ranges since that is how many are in this
example.
 Enter the demand rate, setup cost and holding cost rate as before. For the 3 price
ranges enter lower and upper ranges of 1 to 144, 145 to 288 and 289 to 999999. In each
of these 3 ranges place the appropriate price. Press Solve.
Exercise 4 – The POQ
Consider the same data as in Exercise 2 but assume that we produce the units ourselves and
that the production rate is 15,000 units per year when production occurs. Recall that the
number of (production) days in a year in this set of exercises is 250 days.
1.
2.
3.
4.
What is the daily demand rate?
What is the daily production rate?
During the time that production is on, at what rate will be putting items into inventory?
Clearly we cannot produce during the entire year or we will produce too many units.
What percentage of the year should we be producing?
There are several ways that we can produce according to this percentage. We could be on for 2
days and off for 1 day or we could be on for 2 weeks and off for 1 week, etc. The question is
what should be the size of the production run.
Use Excel OM’s Inventory Module to create the POQ model. When you create the model, be
certain to select the Fixed holding cost option. Enter the six parameters requested. Remember
that there are 250 days in a year.
5.
6.
7.
8.
What is the optimal production quantity?
What is the total cost?
How many days will it take to produce the optimal production quantity?
How many days of demand will be satisfied by the optimal production quantity?
MSOM Lab Exercises and Examples
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Aggregate Planning
Lab data file: aggplan.lab.xlsx
Objectives
 To explain aggregate planning models
o Pure & mixed strategies
o The transportation model
Terminology & Concepts
 Demand
 Production
 Holding/Carrying
 Shortage
o Lost sales
o Backlog
 Hiring/Firing
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Exercise 1: Pure strategies/Mixed strategies/Trial and Error
It is the middle of April. The ABC Company has already completed its aggregate planning for
May through next April since this plan must be implemented in less than 15 days so materials
must already have been ordered and any hiring/firing for May 1 should already have been
started. Thus, the ABC Company is now considering its aggregate plan for June of this year
through May of next year. The demands for those 12 months are expected to be as given in the
table below. The company wants to decide how many units to produce in each month.
The company will produce 1300 units this May. The plan is to have no inventory remaining at
the end of May.
The holding cost for the item is $2 per unit-month. Unmet demands are backlogged and the
cost of backlogging is $4 per unit month. It costs $100 to hire and train a worker and $200 to
fire a worker. Each worker can produce 20 units per month. Subcontracting and overtime are
not options in this problem.
We want to do a couple of things.
MSOM Lab Exercises and Examples
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o Examine the two pure strategies and their costs
o Perform a small trial and error process to improve the costs
Month
June ‘ 10
July
August
September
October
November
December
January ‘11
February
March
April
May
Total
Average
Demands
1125
1150
1300
1350
1400
1600
1600
1550
1300
1150
1050
1025
15600
1300
In order to perform the cost analysis, we need the hiring and firing costs to be on a per-unit
basis rather than a per worker basis.
1. What is the hiring cost per unit?
2. What is the firing cost per unit?
From Excel OM create an Aggregate Planning spreadsheet for 12 periods with backlogging.
(Also, for readability, select dollars only rather than dollars and cents.)
 Ignore the three per unit production costs (regular time, overtime, subcontracting) since
they are not relevant to this problem since subcontracting and overtime are not available.
 Enter the holding, shortage (backlog), increase (hire) and decrease (fire) costs.
 Copy the demands (with month names) to the table
The first pure strategy we will try is to produce the same amount in each month.
 In cell C20 (the production quantity for June place the average demand ($B$33). Be sure to
use the $s for the next step.
 Copy this cell down for the remaining 11 months.
There should, of course, be no costs for hiring and firing but the inventory levels should be
changing each month.
3. What is the cost of holding inventory?
MSOM Lab Exercises and Examples
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4. What is the cost of shortages?
5. What is the total cost for this pure strategy?
Let us try our other pure strategy of meeting the demand exactly in every month.
 In cell C20 (the production quantity for June) place the June demand (B20). Be sure not to
use the $s for the next step.
 Copy this cell down for the remaining 11 months.
There should, of course, be no costs for inventory (holding and shortages) but there should be
hiring and firing costs.
6. What is the cost of increasing production (through hiring)?
7. What is the cost of decreasing production (through firing)?
8. What is the total cost for this pure strategy?
Let us now try a mixed strategy. Mixed strategies do not have to have structure but let us put
some structure in place. Let us break up the planning period into two halves, June through
November and December through May.
 In cell C20 (the production quantity for June) place the average demand for the first 6
months. (Use the average function and be certain to include $ for the next step.
 Copy this cell down for the next 5 months.
 In cell C26 (the production quantity for December) place the average demand for the last 6
months. (Use the average function and be certain to include $ for the next step.
 Copy this cell down for the next 5 months.
We no longer have a pure strategy so you should be seeing all four types of cost (holding,
shortage, increase, decrease) show up.
9. Has the total cost been improved over our previous two solutions?
Let us try one last mixed strategy. Repeat the process above except split the planning period
into 3, 4-month periods.
10. Has the total cost been improved over our previous two solutions?
Please note, that while we are stopping here, in reality, more trials would be attempted. Also,
please notice, that if we assume that the last plan is our answer, the only part that will
definitely be implemented is the production in June. The reason for this is that one month from
today (middle of May) we will update our information and perform a similar analysis for this
July through June of next year. The July answer that we derive in one month may be different
from the July answer derived in the original example.
Exercise 2: The transportation model
MSOM Lab Exercises and Examples
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Assume that today is November 20 and XYZ is performing aggregate planning over a 3 month
planning horizon. Presumably XYZ is already in the process of implementing the December plan
so XYZ now would be planning for January, February and March. The relevant information for
these three months is given in the table below.
Month
Demand
January
15000
February
12000
March
18000
RT
OT
Cost/unit Cost/unit
10
13
10
13
12
15
In addition, the daily capacity is 700 units. XYZ works 5 days per week except for federal
holidays. Overtime capacity is limited to 10% of regular time capacity, the holding cost is $2 per
unit per month, shortages are not permitted and the planned initial inventory is 50 units.
We are going to set this problem up as a transportation model. Recall that the transportation
model is given by a table with supplies in the rows and demands in the column. We only need
to make a slight adjustment for this example. The supplies are capacities, the origins are the
sources of the capacity and the destinations are the demands as given by time rather than
location.
In this example we have 7 sources of capacity. We have the initial inventory and for each month
we have regular time capacity and overtime capacity. Thus the transportation model for this
problem is a 7 by 3 model.
 From POM for Windows select Transportation and create a new 7 by 3 transportation
(minimize) model.
 Label the rows as Init Inv, Jan RT, Jan OT, Feb Rt, Feb OT, Mar RT and Mar OT.
 Label the columns as Jan, Feb, Mar.
 In the far right columns place the 7 capacities.
 In the bottom row place the 3 demands.
We now need to enter the costs in each of the 21 cells. The costs consist of two parts – the
production cost and the holding cost. Let us begin with row 2.
 In the first cell in row 2 (Jan RT –Jan) place the production cost of $10.
 In the second cell in row 2 (Jan RT – Feb) place the total cost of $12. This is $10 for
production in January plus $2 to hold the item until February.
 In the third cell in row 2 (Jan RT – Mar) place the total cost of $14. This is $10 for production
in January plus $4 to hold the item for two months until March.
Repeat the process for row 3 except note that the production cost is $13 rather than $10 since
this row is for overtime in January.
MSOM Lab Exercises and Examples
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When entering the cost for cell 1 in row 4 (Feb RT-Jan) we need to be careful. We are not
allowed to produce any units because this would mean we are producing them one month late.
The textbook uses an x to indicate this. We need to have numbers in these cells. Therefore, to
preclude anything from being produced we need to place a very high cost. POM for Windows
will put in a high number if you type an ‘x’ or you can do it yourself.
 In the second cell in row 4 (Feb RT –Feb) place the production cost of $10.
 In the third cell in row 4 (Feb RT – Mar) place the total cost of $12. This is $10 for
production in January plus $2 to hold the item for two months until March.
Fill in the remaining costs, being careful to select the correct cost from the cost table. The costs
depend on the month (they go up in March) and depend on regular time vs. overtime.
Fill in the first row, using production costs of 0. The costs along the row are $0, $2 and $4
respectively.
Press solve to minimize the total cost. Reading across the rows tells you when the units are
produced and reading down the column tells you when they are used or shipped.
1. What is the minimum total cost?
2. How many units will be produced in January during regular time?
3. How many units will be produced using overtime during the entire planning horizon?
Exercise 3: The transportation model – revisited
Both POM for Windows and Excel OM have a Transportation Model within Aggregate Planning.
Either package will create the model where costs don’t vary from month to month. In POM for
Windows, changes can not be made but in Excel OM it is possible to change the costs if they are
not the same from month to month.





In Excel OM select Aggregate Planning, Transportation
On the creation screen select 3 periods
Fill in the costs at the top and the capacities and demands as given.
Modify the appropriate costs for March that have changed.
Use Solver (Tools in Excel 2003, Data Tab in Excel 2007/2010/2013) to find the optimal
solution
Your total cost should match what you found in Exercise 3. The production quantities may vary.
MSOM Lab Exercises and Examples
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MRP & Lot Sizing
Lab data file: mrp.lab.xlsx
Lecture/Lab
Objectives
 To explain MRP
 To explain Lot sizing
Terminology & Concepts
Material Requirements Planning
Parts tree
Level
Parent
Child
Lead time
Indented Bill of materials
Lead time offsetting
Initial inventory
Scheduled receipts
Lot sizes
Minimum lot/order sizes
Lot sizing
Demand
Holding cost
Setup cost
Lead time
Methods
a. Lot-for-lot (evaluate by hand)
b. All at once (use POM for Windows to evaluate)
c. Wagner-Whitin (use POM for Windows)
d. Other
Preparation
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Preparation: Print the two blank tables so that you can fill them in by hand.
MSOM Lab Exercises and Examples
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Exercise 1 - MRP
Consider the parts tree given below for the end item named A. On the right of each item is the
number required for its parent. For example, each A requires 2 Bs and 3Ds. Each B requires 3Es
and 2 Fs. All lead times are 1 week except for item D which has a lead time of 2 weeks.
A
B (2)
222(ss
E (3)
F (2)
D (3)
E (1)
G (4)
Eventually the software will do all of the work for us but to ensure that you understand the
mechanics we will do several computations by hand. Prior to performing the computations we
want to display the indented bill of materials.
Indented Bill of Materials
A
B
E
F
D
E
G
(1)
2
3
2
3
1
4
Hopefully, you can see that the parts tree diagram and the indented bill of materials give us the
same information and represent the same product.
1.
2.
3.
4.
How many Gs are required to produce 1 unit of A?
How many Gs are required to produce 100 units of A?
How many Es are required to produce 1 unit of A?
How many Es are required to produce 100 units of A?
MSOM Lab Exercises and Examples
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MRP Tables and lead time offsetting – by hand
Suppose the demand for A is 100 units in week 7. We want to fill in the table below by hand.
The table has two rows for each of the items. The numbers that go in the rows are identical but
the location of the numbers will differ by the lead time for the product. This is termed lead time
offsetting. The row named as requirements contains the demand when it is needed. The row
named order release contains the demand when we need to start production due to the lead
time. For example, we have already filled in the demand of 100 in the requirements for A for
week 7. Since A has a lead time of 1 week, this means that you should place a 100 in the order
release row for A in week 6 (one week earlier).
Now the need to start 100 As in week 6 will generate demands for the Bs and the Ds. We stay in
the same column but multiply by the parts per unit.
 In the requirements row for item B, in week 6, place a 200 (100 * 2).
 In the order release row for item B, in week 5, place the same 200. This is lead time
offsetting by one week.
 In the requirements row for item D, in week 6, place a 300 (100 * 3).
 In the order release row for item D, in week 4, place the same 300. This is lead time
offsetting by two weeks.
Week 1
Week 2
Week 3
A - req
Week 4
Week 5
Week 6
Week 7
100
- ord
B - req
- ord
D - req
- ord
E - req
- ord
F - req
- ord
G - req
- ord
MSOM Lab Exercises and Examples
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The order releases for B and D will generate demands for E, F and G. F and G are
straightforward but E is slightly more complicated because it is used for both B and D. However,
because the order releases for B and D occur in two different weeks it will be easy to deal with
part E in this hand exercise.
 In the requirements row for item E, in week 5, place a 600 (200 * 3) based on B.
 In the order release row for item E, in week 4, place the same 600. This is lead time
offsetting by one week.
 In the requirements row for item E, in week 4, place a 300 (300 * 1) based on D
 In the order release row for item E, in week 3, place the same 300. This is lead time
offsetting by one week.
Complete items F and G on your own. Notice that the only operations performed were
multiplications and shifting numbers left! This is very easy but unfortunately there can be too
many computations to perform by hand.
We will check our work using POM for Windows. Select the MRP model and create a table with
7 periods and 7 lines in the indented Bill of Materials.
Fill in the table. The level is the same as the number of indents in the indented bill of materials.
There are 0 indents for A, 1 indent for B and D and 2 indents for E, F and G. In the row for A,
place the demand of 100 in period 7. Your POM for Windows table should appear as
Item
nam
e
A
B
E
F
D
E
G
Lev
el
0
1
2
2
1
2
2
Lea
d
tim
e
1
1
1
1
2
1
1
# per Onhand
paren invento
t
ry
1
2
3
2
3
1
4
0
0
0
0
0
0
0
Minimu
m
Quantit
y
Lot
size
0
0
0
0
0
0
0
0
0
0
0
0
0
0
pd
1
0
0
0
0
0
0
0
pd
2
0
0
0
0
0
0
0
pd
3
0
0
0
0
0
0
0
pd
4
0
0
0
0
0
0
0
pd
5
0
0
0
0
0
0
0
pd
6
0
0
0
0
0
0
0
pd
7
10
0
0
0
0
0
0
0
(This is a good time to use the “don’t display 0” option on the toolbar or in the Format menu.)
Press Solve. The results table has 6 lines per item rather than 2. We will explain all 6 in a little
while but for now the first and last lines for each item in POM for Windows should match the
table you have filled in.
MSOM Lab Exercises and Examples
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Exercise 2: Suppose that demand for item A still totals 100 but that it is split over 3 weeks. That
is, demand is for 30 units in week 5, 50 units in week 6 and 20 units in week 7. Fill in the table
below by hand. We have three times as much work to do as before.
Week 1
Week 2
Week 3
A – req
Week 4
Week 5
Week 6
Week 7
30
50
20
- ord
B – req
- ord
D – req
- ord
E - req
- ord
F – req
- ord
G – req
- ord
 Item A: Copy the 3 numbers to the order release row but shift them left one week in order
to lead time offset.
 Item B: Take each of the 3 order releases from item A, multiply by 2 and place the result in
the same week as the corresponding order release from A. Copy the 3 numbers to the order
release row but shift them left one week in order to lead time offset.
 Item D: Take each of the 3 order releases from item A, multiply by 3 and place the result in
the same week as the corresponding order release from A. Copy the 3 numbers to the order
release row but shift them left two weeks in order to lead time offset.
 Item E: This will be a little more difficult because item E is used for both B and D. We will do
this in two steps.
o Take each of the 3 order releases from item B, multiply by 3 and place the result in
the same week as the corresponding order release from B. After each number place
a comma.
o Take each of the 3 order releases from item D, multiply by 1 and place the result in
the same week as the corresponding order release from D (after the comma if there
is another number in the cell).
o Place the sum of the numbers in each cell into the order release row but shift them
MSOM Lab Exercises and Examples
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left one week in order to lead time offset.
 Perform the appropriate operations on items F and G.
Modify the demands in POM for Windows in order to check your work.
Fill in the order releases for item E in the table below.
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
E-ord rel
Other issues
MRP systems have other features that we want to examine. Even though the only operations to
this point have been addition, multiplication and shifting we will let POM for Windows do all of
our work for us.
Initial inventory: Suppose that B has an initial inventory of 70 units. Change the initial
inventory in POM for Windows, solve and fill in the order releases for item E in the table below.
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
E-ord rel
1. How is E affected by the inventory for B?
Scheduled receipts: Suppose that, in addition, 10 units of D are scheduled to arrive in week 2.
Place these 10 units in week 2 in the row for D. Since D is not an end (level 0) item, the program
will treat the 10 units as a scheduled receipt rather than a demand.
Solve and fill in the order releases for item E in the table below.
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
E-ord rel
2. How is E affected by the scheduled receipts for D?
Lot sizes: Suppose that, in addition, D must be ordered in multiples of 12. Change the lot size to
12, solve and fill in the order releases for item E in the table below.
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
E-ord rel
3. How is E affected by the lot sizes for D?
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Minimum order sizes: Suppose that, in addition, the minimum order quantity for E is 100.
Change the minimum quantity to 100 (in both lines for E in the software), solve and fill in the
order releases for item E in the table below.
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
E-ord rel
4. How is E affected by the minimum order quantity?
Exercise 3: Lot sizing
A company has the following 6 demands for the next 6 periods.
Demands
Week
1
2
3
4
5
6
50
20
80
40
90
30
The holding cost is $2 per unit-week and the setup cost is $100 per order. We want to
determine when orders should be placed (and for how many units) in order to minimize total
costs.
Ultimately, we will solve this using the Wagner-Whitin method in the software but in order to
understand the situation let us do some work by hand.
Lot for Lot: Suppose that we use a lot-for-lot policy. That is, in each week we produce exactly
what we need. (We saw this policy in a different context in aggregate planning and at the time
termed it “chase demand”).
1. What will be the total holding cost of this schedule?
2. What will be the total setup cost in this schedule?
3. What will be the total cost of this schedule?
Let us check our work using POM for Windows. Use the Lot sizing module in POM for Windows,
create a table with 6 periods, fill in the demand and cost data and select lot-for-lot as the
method in the area above the data. Press SOLVE and check your answers above against the
POM for Windows answers.
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Make all at once: Suppose that we make all 310 units in period 1. Fill in the following table by
hand.
Week
Demand
Produced
1
2
3
4
5
6
Total
50
20
80
40
90
30
310
310
End of week
inventory
310
4. What will be the total holding cost of this schedule?
5. What will be the total setup cost in this schedule?
6. What will be the total cost of this schedule?
Again, let us use POM for Windows to check our work. From the data screen, change the
method to User Defined. Enter 310 as the production quantity in the first period, press SOLVE
and compare your answers above to the answers in POM for Windows.
Wagner-Whitin: The optimal solution is given by the Wagner Whitin-method. Change the
method to Wagner-Whitin. You should be able to do this from the solution screen without
returning to the data editing screen.
Fill in the optimal production schedule in the table below.
Week
Demand
Produced
1
2
3
4
5
6
Total
50
20
80
40
90
30
310
310
End of week
inventory
7. What is the total cost of this schedule?
8. If Wagner-Whitin always finds the optimal solution why are the other methods
presented in the textbook and the software?
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Scheduling
Lab data file: sched.lab.xlsx
Objectives
 To explain Scheduling Models
o One machine scheduling
 Priority rules
o Two machine scheduling
o The assignment model
Terminology & Concepts
 Jobs
 Machines/Processes
Preparation:
1. Open this lab file (Word doc)
2. Open Excel OM
3. Open the Excel Lab file for the week
4. Open POM for Windows
5. Begin work
Exercise 1 – One machine scheduling
A company currently has 13 jobs that are being processed by one employee. The length of time
for each job and the due time for each job are given in the table below.
time
due
Customer (hours)
(hours)
A
28
217
B
32
206
C
10
119
D
20
237
E
18
11
F
24
173
G
20
21
H
17
163
I
20
87
J
16
292
K
10
173
L
11
227
M
13
292
We will use POM for Windows to examine the results when using the various scheduling rules.
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Create a data table for 13 jobs using the Job Shop Scheduling Module. On the creation screen
select one machine scheduling. Set the names to A, B, C, D…
Leave the Date Received as 0 for all the jobs and copy the two columns of times into the two
rightmost columns in the POM table. Leave the starting date above the data as 0. Leave the
method as Shortest Processing Time, press Solve and copy the results to the Excel spreadsheet.
1. What is the total processing time for the 13 jobs?
2. Examine the bottom of the results that is below the table. In which order should the
jobs be performed?
3. What is the total flow time for all 13 jobs?
4. What is the average flow time? That is, on average, how long will jobs be in the system
under this schedule?
5. How many jobs will be late under this schedule?
6. What is the average number of late hours per job?
7. What is the maximum number of late hours?
8. What is the average number of jobs in the system displayed in the results? Verify that
this is given by
Average number of jobs in system = total flow time/total processing time
9. What is the average utilization displayed in the results? Verify that the utilization is
given by
Utilization = 1/average number of jobs in the system
Note, you can resolve the problem simply by changing the rule. That is, you do not have to
return to the edit screen to resolve the problem. Also, notice that there is a second window of
output which presents a summary of results. Open this window and copy the results to the
workbook.
10. Examine the column of flowtimes. What rule (method) minimizes the average flow time
for this example?
11. Examine the column of average number of jobs in the system. What rule (method)
minimizes the average number of jobs for this example?
12. Examine the column of utilizations. What rule (method) maximizes the utilization for
this example?
Note: The Shortest Processing Time rule always minimizes the average flow time and number of
jobs in the system and always maximizes the utilization.
13. Examine the column of number of jobs late. What method minimizes the number of late
jobs?
Note: Moore’s method always minimizes the number of late jobs.
14. Examine the column of average late hours. What method minimizes the average
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number of late hours?
Note: We do not have a method guaranteed to always minimize the average lateness.
Exercise 2 – starting day
We continue with the same example. Change the starting day to 1, solve, and copy the results
to the worksheet named “1 machine - starting days” in the space with the 1 highlighted.
1. How do the results compare with the results above that use a starting day of 0?
Note: When the starting time is 0 the software assumes that it is the end of time 0. For any
other starting time the software assumes that it is the beginning of that time. Since the end of
hour 0 and the beginning of hour 1 are the same the results are the same.
Change the starting time to 5, solve, and copy the results to the worksheet named “1 machine starting days” in the space with the 5 highlighted.
2. How do the average flow times compare with the results obtained using a starting time
of 0?
3. How do the averages of number of jobs in the system and utilizations compare with the
results obtained using a starting time of 0?
4. How does the number of late jobs compare with the results obtained using a starting
time of 0?
5. How does the average lateness compare with the results obtained using a starting time
of 0?
Note: The end of day 0 and the beginning of day 5 differ by only 4 days so the average flow
times are 4 (rather than 5) hours more since we are starting our work 4 hours later. The
software computes the average number of jobs in the system and the utilization beginning with
the starting time rather than 0. Therefore, these do not change with a change in the starting
time. Finally, since the work is starting later more jobs are late and the average lateness
increases although not by 4 hours per job as did the flow time.
Exercise 4 – two machine scheduling
A company has 13 jobs that must first be processed on machine 1 and then processed on
machine 2. The times for the jobs are in the table below.
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machine machine
1
2
Customer (minutes) (minutes)
A
36
28
B
35
32
C
26
10
D
37
20
E
16
18
F
32
24
G
16
20
H
31
17
I
23
20
J
44
16
K
32
10
L
37
11
M
44
13
POM for Windows will find the optimal schedule for us but let us first build a table of times that
mimics a Gantt chart for the order A, B …M by using Excel.
Label columns D through G as “Machine 1 start”, Machine 1 Finish”, “Machine 2 start”,
“Machine 2 Finish”. The row for job A will be a little different than the row for jobs B through M
so we will need to set up the first two rows in the table.
 In D2 place a 0.
 In E2 place the formula for the start time (D2) + the processing time on machine 1 (B2)
 In F2 place the formula (=E2) since job A can start on machine 2 as soon as it is finished on
machine1
 In G2 place the formula for the start time (E2) + the processing time on machine 2 (C2)
 In D3 place the formula that points to the finish time for the preceding job (=E2).
 In E3 place the formula for the start time (D3) + the processing time on machine 1 (B3)
 In F3 place the formula (=max(E3,G2)) since job B can not start on machine 2 until both it is
done on machine 1 (E3) and the job before it is done on machine 2 (G2)
 In G3 place the formula for the start time (F3) + the processing time on machine 2 (C3)
 Copy row 3 down for the remaining jobs.
1. What is the completion time of this schedule?
We will now compare this to the best schedule as given by POM for Windows.
 In POM for Windows, create a new file for two machines with 13 jobs.
 Copy the data
 Press solve
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2. What is the minimum completion time for these 13 jobs?
3. What schedule yields this minimum time?
Exercise 5 – The assignment problem
The records of five clients must be examined by one of five accountants. Since the clients’
needs and the accountants’ skills are different, the lengths of time for any accountant to work
on the records for any client vary. The table below contains the time (in hours) for each
accountant working on each client.
Accountant Accountant Accountant Accountant Accountant
A
B
C
D
E
Client
1
16
28
36
18
20
Client
2
32
32
35
24
16
Client
3
16
10
26
20
14
Client
4
31
20
37
17
11
Client
5
32
44
23
37
13
By hand determine which accountant should work for each client in order to minimize the total
amount of hours that the work for the 5 clients need. Each client should get the accountant
who takes the least amount of time to perform the work.
1. Which clients will each accountant serve?
a. Accountant A serves client(s) _____________________
b. Accountant B serves client(s)_____________________
c. Accountant C serves client(s)_____________________
d. Accountant D serves client(s)_____________________
e. Accountant E serves client(s)_____________________
2. What is the total time that is needed?
Now in an assignment problem, each accountant must have one client and each client must
have one accountant. Thus, the problem above is NOT an assignment problem because we have
allowed the accountants to serve more than one client. This makes the problem somewhat
more difficult but we have software that solves this easily.
 In POM for Windows, select the Assignment Module and create a new 5 by 5 problem with
an objective of minimizing the costs.
 Copy the data into the table, Solve and copy the results to the spreadsheet.
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3. Which accountant should work for each client in order to minimize the total amount of
hours that the work for the 5 clients need?
4. What is the total time that is needed?
Suppose we relax the rules and allow any accountant to work on up to two clients. That is, our
total demand will be 5 jobs (clients) but our total supply will be 10 (account-jobs). This now is a
transportation problem.
 In POM for Windows, select the Transportation Module and create a new 5 by 5 problem
with an objective of minimizing the costs.
 Copy the data into the table.
 Enter demands of 1 for each client (in the rightmost column) but supplies of 2 for each
accountant (in the bottom row).
Reminder: While the transportation model is typically organized such that supplies are
in the rows and demands are in the columns this is not required. It is perfectly fine to
have supplies in the columns and demands in the rows. This is the case in this problem.
 Solve and copy the results to the spreadsheet.
5. Which accountant should work for each client in order to minimize the total amount of
hours that the work for the 5 clients need?
6. What is the total time that is needed?
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