Intro to Decomposition:Creating a Three Factor Model

(Cups/Server) (Servers/Hour) (Hours)

Ted Mitchell

We have seen that both

• Number of Servers as an Inputand

• Number of Café Hours as an Input• Have meaningful impacts on the Outputs of• 1) Number of Cups Sold, Q• 2) Dollar of Sales Revenue, R• 3) Dollars of Gross Profit, G

When we consider the

• Explicit impact of hours of operation, H, we leave the number servers implicit and constant

• When we consider the number of servers, S, we leave the number of hours implicit and constant

We wish to have both made Explicit

• Number of Servers, S, and the Number of Store Hours, H, made explicit in the same analysis

Creating a Three Factor Model

• Requires • A process of expansion• A process of aggregation• A process of decomposition

Step 1

• Identify the Two Factor Model you wish to expand into a Three Factor Model and which previously implicit variable is to be made explicit

• Cups sold, Q = cups per hour x hours, HCups Sold, Q = (Q/H) x H• Expand the cups per hour machine to make

the number of servers explicit, S

Step 2

• Introduce the variable to be made explicit as unity into the Two Factor model

• (Number of Servers, S) /(Number of Servers, S) = Unity

• S/S = 1• Cups Sold, Q = (Q/H) x 1 x Hours, H• Cups Sold, Q = (Q/H) x (S/S) x Hours, H

Step 3 Aggregate the Expansion Factors

• Cups sold, Q = (Q/H) x (S/S) x Hours open, H• Cups Sold, Q = [(QxS) / (HxS)] x Hours open, H• r = [(QxS) / (HxS)] is an ugly and large

conversion factor

Step 4 Decompose the Ugly Aggregated Conversion Factor

• Into two conversion rates• [(QxS) / (HxS)] = (Q/S) x (S/H)• Cups Sold, Q = (Q/S) x (S/H) x Hours, H• where• (Q/S) = Conversion Factor #1 =

(cups sold per server)

• (S/H) = Conversion Factor #2 = (number of servers per

hour)• H = Input Factor = (Number of Hours, H

Three Factor Marketing Machine

• Model that makes the number of servers, and the number of operating hours explicit elements in the Marketing Machine

Three Factor MachineInput Factor: H Number of HoursConversion factor #2, S/H Servers per HourConversion factor #1, Q/S Cups Sold per ServerOutput: Q Number of Cups Sold

Note: The original rate of Cups per Hour is lost and has been replaced by two new rates: Servers per Hour and Cups per Server.

Many people think of this as decomposing the original rate

• The Rate of (Cups per Hour) into• (Cups sold per Server) x (Servers per Hour)• But this is inaccurate• Reorganize the 3 Factor Machine as an identity

of rates• Q = (Q/S) x (S/H) x H• Divide both sides by H• (Q/H) = (Q/S) x (S/H)

The other Outputs of the Two-Factor Models can be expended as well

• 1) Output: Dollars of Sales RevenueRevenue, R = (dollar sales per server) x (servers per hour) x number of hoursR = (R/S) x (S/R) x H

• 2) Output: Dollars of Gross ProfitGross Profit, G = (gross profit per server) x (servers per hour) x (number of hours, H)G = (G/S) x (S/H) x H