Basic Profit Models
Chapter 3
Part 1 – Influence Diagram
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In building spreadsheets for deterministic
models, we will look at:
ways to translate the black box representation
into a spreadsheet model.
recommendations for good spreadsheet model
design and layout
suggestions for documenting your models
useful features of Excel for modeling and
analysis
2
Example 1: Simon Pie
Two ingredients combine to make Apple Pies:
Fruit and frozen dough
The Pies are then processed and sold to local grocery
stores in order to generate a profit.
Follow the three steps of model building.
Step 1: Study the Environment and Frame the
Situation
Critical Decision: Setting the wholesale pie price
Decision Variable: Price of the apple pies
(this plus cost parameters will determine profits)
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Step 2: Formulation
Using “Black Box” diagram, specify cost parameters
Pie Price
Unit Cost, Filling
Unit Cost, Dough
Unit Pie Processing Cost
Fixed Cost
Model
profit
The next step is to develop the relationships inside
the black box. A good way to approach this is to create an
Influence Diagram.
An Influence Diagram pictures the connections between the
model’s exogenous variables and a performance measure4
(e.g., profit).
To create an Influence Diagram:
start with a performance measure variable.
Decompose this variable into two or more intermediate
variables that combine mathematically to define the
value of the performance measure.
Further decompose each of the intermediate variables
into more related intermediate variables.
Continue this process until an exogenous variable is
defined (i.e., until you define an input decision variable
or a parameter).
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Start here:
Profit
performance
measure
variable
Decompose this variable into the intermediate variables
Revenue and Total Cost
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Profit
Revenue
Total Cost
Now, further decompose each of these intermediate
variables into more related intermediate variables ...
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Profit
Total Cost
Revenue
Processing
Cost
Ingredient
Cost
Required
Ingredient
Quantities
Pies Demanded
Pie Price
Unit Pie
Processing Cost
Unit Cost
Filling
Unit Cost
Dough
Fixed Cost
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Step 3: Model Construction
Based on the previous Influence Diagram, create the
equations relating the variables to be specified in the
spreadsheet.
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Profit
Revenue
Total Cost
Profit = Revenue – Total Cost
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Profit
Revenue
Revenue = Pie Price * Pies Demanded
Pies Demanded
Pie Price
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Profit
Total Cost
Processing
Cost
Ingredient
Cost
Total Cost =
Processing Cost + Ingredients Cost + Fixed Cost
Fixed Cost
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Profit
Total Cost
Processing
Cost
Pies Demanded
Processing Cost =
Pies Demanded *
Unit Pie Processing Cost
Unit Pie
Processing Cost
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Profit
Total Cost
Ingredients Cost =
Qty Filling * Unit Cost Filling +
Qty Dough * Unit Cost Dough
Ingredient
Cost
Required
Ingredient
Quantities
Unit Cost
Filling
Unit Cost
Dough
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Simon’s Initial Model Input Values
Pie Price
$8.00
Pies Demanded and sold
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Unit Pie Processing Cost ($ per pie)
$2.05
Unit Cost, Fruit Filling ($ per pie)
$3.48
Unit Cost, Dough ($ per pie)
$0.30
Fixed Cost ($000’s per week)
$12
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Chapter 3
Part 2
Break-Even and Cross-Over
Analysis
MGS 3100
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Background
• The Generalized Profit Model:
– A decision-maker will break-even when profit
is zero.
– Set the generalized profit model equal to zero,
and then solve for the quantity (Q).
– For simplicity, assume that the quantity
produced is equal to the quantity sold. This
assumption will be relaxed in the module on
decision analysis.
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Basic Relationships
• Profit (π) = Revenue (R) - Cost (C)
• Revenue (R) = Selling price (SP) x Quantity
(Q)
• Cost (C) = [Variable cost (VC) x Quantity (Q)]
+ Fixed Cost (FC)
• Remember quantity produced = quantity sold
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Basic Relationships con’t
• By substitution:
• π = (SP x Q) – ((VC x Q) + FC)
• π = SP*Q - VC*Q – FC
Notice sign reversal
when parentheses are
removed!
• π = (SP-VC)*Q - FC
Just a bit of algebraic
reorganization…
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Contribution Margin
• If Contribution Margin (CM) = SP-VC,
then by substitution…
• π = CM*Q – FC
• In case you want to figure the quantity at
break-even, you just need to rearrange
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Break-Even Quantity
•
•
•
•
•
•
π = CM*Q – FC
π + FC = CM*Q
(π + FC)/CM = (CM*Q)/CM
(π + FC)/CM = Q
Q = (π + FC)/CM
In the case of break-even, where π =0, the
formula boils down to:
• Q = FC/CM
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Quantity and Profit Example
• Again, Q = (FC + π)/CM
• If fixed cost is $150,000 per year, selling price
per unit (SP) is $400, and variable cost per unit
(VC) is $250, what quantity (Q) will produce a
profit of $300,000?
• Q = ($150,000+$300,000)/($400-$250)
• Q = $450,000/$150
• Q = 3000
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Cross-Over Point
• The cross-over point (or indifference point)
is found when we are indifferent between
two plans.
• In other words, the quantity when profit is
the same for each of two plans.
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Cross-Over Point, con’t
• To find the cross-over point for Plan A and
B, set the profit formulas for each plan
equal to each other:
• πplanA = πplanB, so
• (CM*Q – FC) planA = (CM*Q – FC)planB
• QAtoB = (FCA - FCB)/(CMA – CMB)
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Cross-Over Point, con’t
• So all you need are the fixed costs and
contribution margins (selling price and variable
cost) to solve.
• For example, here are three plans
FC
VC
SP
Plan A
150,000
250
400
}
150
Plan B
450,000
150
400 250
}
Plan C
2,850,000
100
400 300
}
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Cross-Over Point, con’t
Breakeven Points for each plan are:
QBE =
Plan A
150,000/(400-250)
Plan B
450,000/(400-150)
= 1000 units
= 1800 units
Plan C
2,850,000/(400-100)
= 9500 units
What is the profit at each of these points?
Cross-Over Points
QCO
A to B
B to C
(150,000-450,000)/(150-250)
(450,000-2,850,000)/(250-300)
= 3000 units
= 48,000 units
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Calculating Profit at the Cross-Over
• After calculating cross-over, we have a quantity
that can be plugged back into the formula to find
profit at the cross-over point
πA = CMA*Q – FCA
= 150(3000) - 150,000
= $300,000, or
πB = 250(3000) - 450,000
= $300,000
πB = CMB*Q - FCB
= 250(48,000) - 450,000
= $11,550,000, or
πC = 300(48,000) - 2,850,000
= $11,550,000
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