Business Decision Models <click here to go to the podcast>
We develop a mathematical model of a real business decision problem. We analyze the model to
develop insights into the business problem and then find a solution for the business problem.
Steps: 1. Define the business problem – What are the input variables/also called parameters (known &
unknown or certain & uncertain), decision variables and output variables/also called results?
2. Develop a mathematical model in Excel – What are the mathematical
relationships between the variables?
3. Collect the input data. (i.e. what are the values of all the input variables?)
4. Develop the solution – Determine the best values of the decision variables.
5. Sensitivity analysis: Analyze how changing input variable values affects output variable values.
6. Implement the results.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 3
Example 1 – The Quality Sweater Company – breakeven analysis for a new catalog project
The Quality Sweaters Company the company wants to print a small catalog and mail it to selected
customers. The fixed cost of printing the catalog is $30,000. The variable cost of $0.80 per catalog if less
than one million catalogs are printed; otherwise the variable cost is $0.70 per catalog. The cost of
mailing each catalog is $1.10. The company estimates that the average size of a customer order to be
$120 and that its average cost (for product and labour) to complete a customer order is about 40% of
the order’s value or $120×40% = $48. Customers place orders and make payments over the internet.
The company estimates that the average cost to process a customer order is $1.55. The company is not
sure what the customer response rate will be; it thinks that the response rate could be 3 percent if it has
a good mailing list. The following probability distribution is a better estimate of the customer response rate:
Response rate:
2.0% 2.2% 2.4% 2.6% 2.8% 3.0% 3.2% 3.4%
Probability:
0.05 0.10 0.10 0.15 0.20 0.20 0.15 0.05
The company is not sure how many catalogs to print and mail; it thinks that one million catalogs may be
a good number. Develop an Excel model to answer the following questions.
i.
ii.
iii.
iv.
v.
If the company wants to break-even on profit, then what response rate is needed if the company
prints and mails 1,000,000 catalogs?
If the company wants to break-even on profit, then how many catalogs should it print and mail if
the response rate is 3 percent?
If the company prints and mails 1,000,000 catalogs, how does a change in the response rate affect profit?
If the response rate is 3 percent, how does a change in the number of catalogs printed and mailed affect
profit?
Incorporate the probability distribution for the customer response rate into your Excel model.
Step 1. Define the business problem
State the variables. In this course we state the variables in the order in which they occur in the problem.
Input variables (parameters):
Known (certain)
Fixed cost of printing the catalog
Variable printing cost per catalog
Cost of mailing a catalog
Average revenue of a customer order
Average cost of a customer order
Average cost of processing a customer order
Unknown (uncertain)
Customer response rate
Decision variables:
Number of catalogs to print and mail
Output variables (results):
Total profit = Total revenue - Total
cost Total revenue
Number of customer orders
Total cost
Total printing cost
Total mailing cost
Total customer order cost
Total processing cost
white colour
$30,000
$0.80 (< one million), $0.70 (≥ one million)
$1.10
$120
$48
In 3QA3 in all problems,
$1.55
exams, etc. you MUST
blue colour
always use these 4
3% plus a probability distribution
standard colours (white,
yellow colour
blue, yellow, green) for
one million ?
the 4 variable types.
green colour
Some output variables are stated
in the problem.
Most output variables are
intermediate calculations. We don’t
know what these variables are until
we develop the Excel model.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 4
Step 2. Develop a mathematical model in Excel / Step 3. Collect the input data
Known input variables or known parameters
- ALWAYS show in the order given in the question
- ALWAYS use detailed names
- ALWAYS bold the heading, use correct units and
decimal places, center the values, and use these borders
- ALWAYS show values in white
Unknown input variables or unknown parameters
- same as above except ALWAYS show values in (light) blue
Decision variables
- same as above except ALWAYS show values in yellow
Output variables or results
- same as above except ALWAYS show values in (light) green
- each cell must contain a formula
e.g. the formula in cell B29 is =B4+B25*IF(B25>=C6,C5,B5)
See file ‘Excel - Lectures 1-4 Notes - Business models, Ch 1.xlsx’ in Avenue > Content > Lectures 1-4…
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 5
Step 4. Develop the solution – determine the best values of the decision variables
<click here to go to the podcast>
Now that we have a mathematical model, the questions posed by the company can be answered.
We use Excel’s ‘Goal Seek’ tool to answer the break-even questions. These are questions i and ii.
i. If the company wants to break-even on profit, then what response rate is needed if the company
prints and mails 1,000,000 catalogs?
Click Data tab
Click ‘Goal Seek’
In the Goal Seek dialog box:
for ‘Set cell:’ click B35 – Total profit
for ‘To value:’ enter 0 -- zero
for ‘By changing cell:’ click B13 –
Customer response rate
In other words, Excel will find the
value of the Customer response
rate that makes the Total profit
value zero.
Click OK.
Solution: The ‘Target value’ in cell B35 is 0 when the Customer response rate is 2.60% In
other words, when the Number of catalogs to print and mail is 1,000,000 the company
breaks even when the Customer response rate is 2.60%.
Then Total cost = Total revenue = $3,117,104 and Total profit = $0.
Answer question ii in a similar way on your own.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 6
Step 5. Sensitivity analysis: Analyze how changing input variable values affects output variable values.
<click here to go to the podcast>
One-way Data Table
A one-way Data Table allows us to see how changing the value of one input variable affects the value(s) of
one (or more) output variables. The values of the one input variable can be arranged vertically in a column
or horizontally in a row.
We use a one-way Data Table to answer simple what-if questions. These are questions iii and iv.
iii. If the company prints and mails 1,000,000 catalogs, how does a change in the response rate affect profit?
one input variable one output variable
To create the Data Table:
- Enter the input variable values in a
column range, such as E8:E18.
- Enter a formula for the output variable
value in the cell one column to the right
and one row above the input variable
values.
This means, in cell F7 type =B35.
Colour this cell Green.
Now highlight the entire table, beginning
with the upper-left blank cell
(i.e. highlight E7:F18).
Select Data > What-If Analysis > Data Table.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 7
In the dialog box leave the Row Input
cell blank and enter B13 in the
Column Input cell*. This is the cell
where the original value of the input
variable is.
*Our input values are in a column.
Click OK.
Excel fills in the left column of the
Data Table. Each value is the Total
profit for a particular value of the
Customer response rate.
See file ‘Excel - Lectures 1-4 Notes - Business models, Ch 1.xlsx’ in Avenue > Content > Lectures 1-4…
Notes:
1. Data Tables can be one-way or two-way. One-way means we change one input variable, two-way
means we change two input variables. There are no three-way, four-way, etc. Data Tables. If we want
to change three or more variables then we must use something different (i.e. Scenario Manager. In the
Data Table dialog box we use either the row input cell or column input cell for a one-way Data Table,
and we use both for a two-way Data Table.
2. If you click anywhere in the body of the Data Table (F8:F18), Excel will tell you it has used the TABLE
function. You cannot make any changes in the body of the Data Table. If you try, Excel issues an error
message. You cannot delete part of a Data Table; you must delete the entire Data Table (F8:F18). Trying
to make changes in the body of a Data Table or deleting part of a Data Table can cause Excel to crash.
SO MAKE SURE YOU SAVE YOUR EXCEL FILE BEFORE WORKING WITH OR CHANGING A DATA
TABLE. BE VERY CAREFUL WITH YOUR DATA TABLES DURING THE EXAMS.
It is good practice and professional to display the data in a Data Table in a chart/graph.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 8
Highlight the data in E8:F18.
Click Insert > Charts > Scatter
Click on the Scatter Chart that
joins points with a line.
Tidy up the chart by right clicking
areas of the chart and making the
desired changes. See the Excel Tutorial
for information and practice.
See file ‘Excel - Lectures 1-4 Notes - Business models, Ch 1.xlsx’ in Avenue > Content > Lectures 1-4…
As the Data Table and chart indicate, Total profit increases in a linear manner as the Customer response
rate varies. Each 0.20% increase in the Customer response rate increases profit by $140,900.
Therefore the Company is willing to pay up to $140,900 for a customer mailing list that increases
the Customer response rate by 0.20%.
Now, on your own, create a one-way Data Table to answer question iv.
iv. If the response rate is 3 percent, how does a change in the number of catalogs printed and mailed
affect profit?
The Data Table for the Number of catalogs to print and mail shows that Total profit increases in a nonlinear
manner as the Number of catalogs increases.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 9
Two-way Data Table <click here to go to the podcast>
A two-way Data Table allows us to see how changing the values of two input variables (e.g. Number of
catalogs to print and mail and Customer response rate) affects the value(s) of one (or more) output
variables (e.g. Total profit). The values of one input variable must be arranged vertically in a column (e.g.
Number of catalogs to print and mail) and the values of the other input variable must be arranged
horizontally in an adjacent row (e.g. Customer response rate).
The company estimates the following probability distribution for the response rate:
Response rate:
2.0% 2.2% 2.4% 2.6% 2.8% 3.0% 3.2% 3.4%
Probability: 0.05 0.10 0.10 0.15 0.20 0.20 0.15 0.05 Incorporate this uncertainty
into the model.
To create the two-way Data Table
Enter the output cell (Total profit) in the top-left corner of the Data Table.
That is, in cell E15 type =B35. Colour this cell Green.
Type the possible values of the two inputs variables below and to the right of this corner
cell. That is, type the values of the Number of catalogs to print and mail in cells E16:E24, and
type the values of the Customer response rate in cells F15:M15. Colour the uncertain input variable
Blue, and the decision variable Yellow.
Since we have a probability distribution for the Customer response rate, type the probabilities along the
bottom of the data table. Colour these cells Blue (because they correspond unknown/uncertain input
variables).
Add a column for the Expected value of the Total profit. Use the SUMPRODUCT formula. For example,
the formula in cell N16 is = SUMPRODUCT(F16:M16,$F$25:$M$25). Colour these cells Green.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 10
Now highlight the entire table, beginning with the upper-left Green cell (i.e. highlight E15:M24).
Select Data > What-If Analysis > Data Table
In the Data Table dialog box
- the Row Input cell is Customer response rate,
- the Column Input cell is the Number of catalogs to print and sell.
Click cell B13. Cell $B$13 is where
the original value of Customer
response rate is.
Click cell B25. Cell $B$25 is where
the original value of Number of
catalogs to print and sell is.
Click OK.
Excel substitutes each pair of input values into these row and column input cells, recalculates the
spreadsheet, and enters the corresponding output value in the table.
(Hint: If the values are incorrect it is usually because you switched the row and column cells in the
dialog box.)
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 11
Printing and mailing 600,000 catalogs is not a good idea; the Expected value is negative—a loss. The
highest Expected value (of Total profit) is $287,724 per year; so printing and mailing 2,200,000 catalogs
seems to be best. However there is a large probability of a large loss. So printing and mailing this many
catalogs may be too risky for the company. Later in this course (Chapter 8) you will learn how to
account for risk (i.e. the probability of a large loss). Here is a preview of some of what you will learn.
Aside:
In Chapter 8, which we will study in the second half of the course, we will account for risk (i.e. the
probability of large losses) by using a utility function, u(x), to convert a profit value, x, to a utility value,
u.
We will see that our utility function will be of the form: ( ) = 1.0 − 1.0 × − /900,000
Utility values can be positive or negative. High utility values are good, low utility values are not good.
For example, if Customer response rate = 2.00% and Number of catalogs to print and sell = 600,000, then profit = x = -$324,600 and the corresponding utility = u = 1.0 − − /900,000 = 1.0 − −(−324,600)/900,000
= -0.4343
Profit, x
Utility, u
The highest Expected utility is 0.1072 per year, so printing and mailing 1,800,000 catalogs is best. If this
is the appropriate utility function for the company then this the best solution for the company.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 12
Setting variable names (also called range names) <click here to go to the podcast>
It is good practice to use variable names (also called range names) rather than cell addresses. This
makes it easier to check formulas and for others in the company to use and change your model.
Unfortunately the textbook does not use variable names. We will only use variable names once, in
Chapter 8, for utility functions in TreePlan. There is more information on variable names in your Excel
Tutorial file; in the file go to List of Topics > Working with Formulas > Range Names.
PC > Excel > Formulas > Defined Names
Mac > Excel 2016 … works like a PC
Mac > Excel 2011 > Insert > Name (use options: Define (name), Create (from selection), Paste (list of names)
i) Review existing variable names
First use Excel > Formulas > Name Manager to review and perhaps delete some existing variable names.
This excel file is for illustrative
purposes only. It is from
lectures much later in the
course. When we get to those
lectures we will study this
excel file more carefully.
The Name Manager dialog box states that cell E5 on Sheet 1 has the name ‘RT’. Notice that the current
value in this cell is 130.00.
We can highlight this variable name
and click ‘Delete’ to delete it. Then
we can name some other cell ‘RT’.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 13
ii) Setting variable names manually
- Highlight cell B4.
Move cursor to Name Box.
Type name, e.g. Fixed cost (no blank spaces, use _)
OR
- Right click cell B4
From the command box click ‘Define Name …’
A dialog box appears.
Write the new name: e.g. Fixed_cost. Click OK.
Now cell B4 has two names: B4 and Fixed_cost
iii) Setting variable names automatically
Highlight a range of variable names and values
Click Formulas tab *
Click ‘Create from Selection’ *
In the dialog box click ‘Left column’. Click OK.
This gives the names in the left cell to the
values in the right cell. Spaces in the names are
changed to underscore (_).
Example
Now cell A7 has two names: A7 and
Cost_of_mailing_a_catalog
* Mac > Excel 2011 > Insert > Name > Create (from selection)
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 14
iv) It is good practice to display the list of all variable names in a convenient location on the worksheet.
Select the cell where you want to
display the list of variable names: E8
Click ‘Formulas’ tab *
Click ‘Use in Formula’ *
Click ‘Paste Names’ *
In the dialog box click ‘Paste List’
The list of names appears. Add a
title, format the cells, and border in
the usual professional way.
Redo the Results’ formulas; notice
the changes. These formulas are
easier for others to use, and
easier for you to use later.
* Mac > Excel 2011 > Insert > Name > Paste (list of names)
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 15
Practice Problems
1. Consider the Bill Pritchett’s Shop example in Section 1.5 (p. 10) of the textbook
Bill’s Pritchett’s Precious Time Pieces, buys, sells, and repairs old clocks and clock parts. Bill sells rebuilt
springs for a unit price of $10. The fixed cost of the equipment to build the springs is $1,000. The
variable cost per unit is $5 for spring material. If we represent the number of springs (units) sold as the
variable X, we can restate the profit as follows:
Profit = $10X − $1,000 − $5X
Figure 1.4 shows the formulas used in developing the decision model for Bill Pritchett. Cells B4, B5, and
B6 show the known input variables—namely, revenue per unit, fixed cost, and variable cost per unit,
respectively. Cell B9 is the decision variable in the model, and it represents the number of units sold
(i.e., X). Using these entries, the total revenue, total variable cost, total cost, and profit are computed in
cells B12, B14, B15, and B16, respectively. For example, if we enter a value of 1,000 units for X in cell B9,
the profit is calculated as $4,000 in cell B16, as shown in Figure 1.5
.
In addition to computing the profit, decision makers are often interested in the break-even point (BEP).
The BEP is the number of units sold that will result in total revenue equaling total costs (i.e., profit is $0).
We can determine the BEP analytically by setting profit equal to $0 and solving for X in Bill Pritchett’s
profit expression. That is
0 = (Selling price per unit) × (Number of units) − (Fixed cost) − (Variable cost per unit) × (Number of units)
which can be rewritten as
Break-even point (BEP) = Fixed cost ⁄ (Selling price per unit − Variable cost per unit) For
Bill Pritchett’s example, we can compute the BEP as $1,000 / ($10 − $5) = 200 springs.
i. If the selling price cost is $10 per unit, how does a change in the number of units sold affect
profit? Use a one-way Data Table to consider the following number of units sold: 800, 850, 900, 950,
1,000, 1,050, and 1,100. Draw a chart/graph of your results.
ii. If the number of units sold is 1,000, how does a change in the selling price affect profit? Use a oneway Data Table to consider selling prices of $9.00, $9.50, $10.00, $10.50, and $11.00. Draw a
chart/graph of your results.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 16
iii. Bill Pritchett estimates the following probability distribution for the number of units sold:
Number of units sold:
800
850
900
950
1,000 1,050 1,100
Probability:
0.05 0.10 0.15 0.20 0.20 0.20 0.10
Use a two-way Data Table to consider this uncertainty and possible selling prices of $9.00, $9.50, $10.00,
$10.50, and $11.00. What is the best Selling price and what is the expected Profit for this Selling price?
2. Product Mix Problem
A company produces four varieties of ties: all-silk (S), all-polyester (P), and two blends of polyester and
cotton (B1 and B2). The company has contracts with several department store chains to supply ties
each month. The contracts specify the prices the company receives for each variety and the minimum
quantities (i.e. demand) of each variety of tie that the company must supply. The contract also allows
for larger, quantities to be supplied (up to a maximum quantity) if the company has sufficient capacity
and chooses to supply more than the minimum quantities. The company uses three materials: silk,
polyester and cotton, and labour to produce the ties. The quantities, availabilities and costs of the
materials are shown in the table below. The production process is fully automated, so the company
uses a standard labor cost of $0.75 per tie for each variety of tie. The relevant data are:
Use Excel to calculate the
materials cost from the given data.
Create a business decision model in Excel to help the company earn as much total profit as possible.
Use the model to help you select the number of ties to produce each month.
Lectures 1-4 Notes - Excel, Business models, Ch 1, App B, Excel tutorial page 17