An Economic Decision Model for Determining the
Appropriate Level of Business Process Standardization (Appendix)
A
A.1
BPS and Profit Margin
Linear Approximation of the Experience Curve
The process costs 𝐶0 at the decision point can be calculated by inserting the cumulated demand
𝐷0,cum up to the decision point into the function of Henderson’s Law from Equation (5):
−𝑎
𝐶0 = 𝐶(𝐷0,cum , 𝑎) = 𝐾(𝐷0,cum )
(Eq. A.1.1)
We apply the tangent of Equation (A.1.1) to linearly approximate the development of the process costs
over the planning horizon. To derive the tangent, we must determine its slope and its tangent point. The
slope of the tangent is determined as follows:
𝐶 ′ (𝐷0,cum , 𝑎) =
for 𝑎̃: =
𝜕𝐶(𝐷0,cum , 𝑎)
−𝑎−1
= −𝑎𝐾(𝐷0,cum )
= −𝑎̃𝐾𝐷0,cum −𝑎
𝜕𝐷0,cum
(Eq. A.1.2)
𝑎
𝐷0,cum
The tangent point equals the costs at the decision point and can be directly observed:
𝐶(𝐷0,cum , 𝑎) = 𝐾𝐷0,cum −𝑎 = 𝐶0
(Eq. A.1.3)
In the approximated version of Henderson’s Law, the process costs linearly decrease in the cumulated
std
demand 𝐷𝑡,cum
that has been reached starting from the decision point as shown in Equation (A.1.4)
std
std
std
𝐶(𝐷𝑡,cum
, 𝑎) = 𝐾𝐷0,cum −𝑎 −𝑎̃𝐾𝐷0,cum −𝑎 ∙ 𝐷𝑡,cum
= 𝐶0 − 𝐶0 𝑎̃𝐷𝑡,cum
A.2
(Eq. A.1.4)
Demand-Weighted ProfitMargin
The profit margin 𝑀𝑣,𝑡 of a process variant 𝑣 in period 𝑡 equals the difference between the sales price
𝑃𝑣 and the process costs 𝐶𝑣,𝑡 in that period, as shown in Equation (A.2.1).
𝑀𝑣,𝑡 = 𝑃𝑣 − 𝐶𝑣,𝑡
(Eq. A.2.1)
For the process costs, we can use the linearly approximated experience curve from Equation (6). Of
std
course, the process costs depend only on the cumulated demand 𝐷𝑡,𝑣,cum
covered by that process variant
std
under consideration and not on the complete cumulated demand 𝐷𝑡,cum
.
std
𝑀𝑣,𝑡 = 𝑃𝑣 − 𝐶𝑣,0 + 𝐶𝑣,0 𝑎̃𝐷𝑡,𝑣,cum
1
(Eq. A.2.2)
We now apply the assumption (A.2) of the constant demand weights and replace the cumulated demand
std
𝐷𝑡,𝑣,cum
that was covered by process variant 𝑣 starting from the decision point by the complete cumu-
lated demand adjusted by the respective demand weight 𝑤𝑣std.
std
𝑀𝑣,𝑡 = 𝑃𝑣 − 𝐶𝑣,0 + 𝐶𝑣,0 𝑎̃ 𝑤𝑣std 𝐷𝑡,cum
(Eq. A.2.3)
The demand-weighted periodic profit margin 𝑀𝑡 equals the demand-weighted variant-specific profit
margins 𝑀𝑣,𝑡 in period 𝑡, as shown in Equation (A.2.4).
𝑛
𝑛
std
𝑀𝑡 = ∑ 𝑤𝑣std 𝑀𝑣,𝑡 = ∑ 𝑤𝑣std (𝑃𝑣 − 𝐶𝑣,0 + 𝐶𝑣,0 𝑎̃𝑤𝑣std 𝐷𝑡,cum
)
𝑣=0
𝑣=0
𝑛
2
std
= ∑ [𝑤𝑣std (𝑃𝑣 − 𝐶𝑣,0 ) + 𝐶𝑣,0 𝑎̃𝐷𝑡,cum
(𝑤𝑣std ) ]
𝑣=0
𝑛
=∑
𝑤𝑣std (𝑃𝑣
− 𝐶𝑣,0 ) +
𝑣=0
𝑛
2
std
𝑎̃𝐷𝑡,cum ∑(𝑤𝑣std ) 𝐶𝑣,0
𝑣=0
= 𝑀0
(Eq. A.2.4)
std
+ 𝑎̃𝐷𝑡,cum
𝐺cost
𝑛
for 𝑀0 ≔ ∑
𝑛
𝑤𝑣std
𝑣=0
B
B.1
2
(𝑃𝑣 − 𝐶𝑣,0 ) and 𝐺cost ≔ ∑ (𝑤𝑣std ) 𝐶𝑣,0
𝑣=0
Application of Anderson’s Model (1994)
Linear Extrapolation for Process Quality (as an example)
As the two reference points (𝑥1 ; 𝑦1 ) and (𝑥2 ; 𝑦2 ) required to set up a linear extrapolation, we use the
status quo and the case of complete standardization.
Reference point 1: 𝑦1 = 𝑄, 𝑥1 = 𝐺
Reference point 2: 𝑦2 = (1 + 𝑠𝑄 ) ∙ 𝑄, 𝑥2 = 1
Based on these reference points, we can set up the linear extrapolation as shown in Equation (B.1.1).
𝑦(𝑥) =
𝑦2 − 𝑦1
(𝑥 − 𝑥1 ) + 𝑦1
𝑥2 − 𝑥1
(Eq. B.1.1)
The relationship between the Gini coefficient of a distinct process variant profile after BPS 𝐺 std (= 𝑥)
and the associated process quality 𝑄(𝐺 std )) (= 𝑦(𝑥)) can be determined by inserting the reference
points into Equation (B.1.1) as shown in Equation (B.1.2):
𝑄(𝐺 std ) =
𝑄 ∙ (1 + 𝑠𝑄 ) − 𝑄
𝑄 ∙ 𝑠𝑄
∆𝐺 + 𝑄,
(𝐺 std − 𝐺) + 𝑄 =
1−𝐺
1−𝐺
for ∆𝐺 ≔ (𝐺 std − G)
The change in process quality for a given Gini coefficient 𝐺 std after BPS equals:
2
(Eq. B.1.2)
𝑄(𝐺 std ) − 𝑄 =
𝑄 ∙ 𝑠𝑄
𝑄 ∙ 𝑠𝑄
∆𝐺 + 𝑄 − 𝑄 =
∆𝐺
1−𝐺
1−𝐺
(Eq. B.1.3)
The resulting relative change of process quality for a given Gini coefficient 𝐺 std after BPS compared to
the status quo prior to BPS is shown in Equation (B.1.4).
𝑄 ∙ 𝑠𝑄
𝑠𝑄
𝑄(𝐺 std ) − 𝑄 1 − 𝐺 ∆𝐺
∆𝑄(𝐺) =
=
=
∆𝐺
𝑄
𝑄
1−𝐺
B.2
(Eq. B.1.4)
Changes in Customer Satisfaction for a Given Gini Coefficient
According to Anderson’s model of customer satisfaction and retention, the customer satisfaction 𝑆𝐴𝑇
prior to BPS and after BPS can be expressed as shown in Equations (B.2.1) and (B.2.2).
𝑆𝐴𝑇 = 𝛼𝑆𝐴𝑇 + 𝛽𝑄 𝑄 + 𝛽𝐸𝑋𝑃 𝐸𝑋𝑃 + 𝛽𝑁𝐶𝐷 𝑁𝐶𝐷 + 𝛽𝑃𝐶𝐷 𝑃𝐶𝐷 + 𝜀
(Eq. B.2.1)
𝑆𝐴𝑇(𝐺 std ) = 𝛼𝑆𝐴𝑇 + 𝛽𝑄 𝑄(𝐺 std ) + 𝛽𝐸𝑋𝑃 𝐸𝑋𝑃 + 𝛽𝑁𝐶𝐷 𝑁𝐶𝐷(𝐺 std ) + 𝛽𝑃𝐶𝐷 𝑃𝐶𝐷(𝐺 std )
(Eq. B.2.2)
+𝜀
We can now insert the derived functions for the model parameters given the Gini coefficient 𝐺 std as
shown in Equation (B.2.3).
𝑠𝑄
𝑆𝐴𝑇(𝐺 std ) = 𝛼𝑆𝐴𝑇 + 𝛽𝑄 𝑄 (
∆𝐺 + 1) + 𝛽𝐸𝑋𝑃 𝐸𝑋𝑃
1−𝐺
𝑠𝑄 + 𝑠𝑇
𝑠𝑄 + 𝑠𝑇
+ 𝛽𝑁𝐶𝐷 𝑁𝐶𝐷 (−
∆𝐺 + 1) + 𝛽𝑃𝐶𝐷 𝑃𝐶𝐷 (
∆𝐺 + 1)
1−𝐺
1−𝐺
(Eq. B.2.3)
+𝜀
Based on these intermediate results, we can calculate the changes in customer satisfaction ∆𝑆𝐴𝑇(𝐺 std )
for a given Gini coefficient 𝐺 std after BPS as shown in Equation (B.2.4). The result can be found in
Equation (13) in the manuscript.
∆𝑆𝐴𝑇(𝐺 std ) = 𝑆𝐴𝑇(𝐺 std ) − 𝑆𝐴𝑇
𝑠𝑄
= 𝛼𝑆𝐴𝑇 − 𝛼𝑆𝐴𝑇 + 𝛽𝑄 𝑄 (
∆𝐺 + 1) − 𝛽𝑄 𝑄 + 𝛽𝐸𝑋𝑃 𝐸𝑋𝑃
1−𝐺
𝑠𝑇 + 𝑠𝑄
− 𝛽𝐸𝑋𝑃 𝐸𝑋𝑃 + 𝛽𝑁𝐶𝐷 𝑁𝐶𝐷 (−
∆𝐺 + 1) − 𝛽𝑁𝐶𝐷 𝑁𝐶𝐷
1−𝐺
𝑠𝑇 + 𝑠𝑄
+ 𝛽𝑃𝐶𝐷 𝑃𝐶𝐷 (
∆𝐺 + 1) − 𝛽𝑃𝐶𝐷 𝑃𝐶𝐷 + 𝜀 − 𝜀
1−𝐺
𝑠𝑄
𝑠𝑇 + 𝑠𝑄
= 𝛽𝑄 (𝑄
∆𝐺) + 𝛽𝑁𝐶𝐷 (−𝑁𝐶𝐷
∆𝐺)
1−𝐺
1−𝐺
𝑠𝑇 + 𝑠𝑄
+ 𝛽𝑃𝐶𝐷 (𝑃𝐶𝐷
∆𝐺)
1−𝐺
3
(Eq. B.2.4)
B.3
Changes in the Retention Rate for a Given Gini Coefficient
According to Anderson’s model, the retention rate 𝑟 prior to BPS and after BPS can be expressed as
shown in Equations (B.3.1) and (B.3.2).
𝑟 = 𝛼𝑟 + 𝛽𝑆𝐴𝑇 (𝑆𝐴𝑇) + 𝜀
(Eq. B.3.1)
𝑟(𝐺 std ) = 𝛼𝑟 + 𝛽𝑆𝐴𝑇 𝑆𝐴𝑇(𝐺 std ) + 𝜀
(Eq. B.3.2)
Based on this intermediate result, we can calculate the changes in retention rate ∆𝑟(𝐺 std ) for a given
Gini coefficient 𝐺 std after BPS as shown in Equation (B.3.3).
∆𝑟(𝐺 std ) = 𝑟(𝐺 std ) − 𝑟 = 𝛼𝑟 − 𝛼𝑟 + 𝛽𝑆𝐴𝑇 (𝑆𝐴𝑇(𝐺 std ) − 𝑆𝐴𝑇) + 𝜀 − 𝜀
= 𝛽𝑆𝐴𝑇 ∆𝑆𝐴𝑇(𝐺 std )
= 𝛽𝑆𝐴𝑇 (𝛽𝑄 (Q
+ 𝛽𝑃𝐶𝐷 (𝑃𝐶𝐷
𝑠𝑄
𝑠𝑇 + 𝑠𝑄
∆𝐺) + 𝛽𝑁𝐶𝐷 (−𝑁𝐶𝐷
∆𝐺)
1−𝐺
1−𝐺
𝑠𝑇 + 𝑠𝑄
∆𝐺))
1−𝐺
(Eq. B.3.3)
𝑠𝑄
𝑠𝑇 + 𝑠𝑄
∆𝐺) + 𝛽𝑆𝐴𝑇 𝛽𝑁𝐶𝐷 (−𝑁𝐶𝐷
∆𝐺)
1−𝐺
1−𝐺
𝑠𝑇 + 𝑠𝑄
+ 𝛽𝑆𝐴𝑇 𝛽𝑃𝐶𝐷 (𝑃𝐶𝐷
∆𝐺)
1−𝐺
= 𝛽𝑆𝐴𝑇 𝛽𝑄 (Q
C
C.1
Objective Function
Simplification of the Cumulated Process Demand
std
The cumulated process demand 𝐷𝑡,cum
after BPS in period 𝑡 can be defined as the sum of the periodic
process demands 𝐷𝑡std up to period 𝑡 as shown in Equation (C.1.1).
𝑡
std
𝐷𝑡,cum
= ∑ 𝐷𝑗std
(Eq. C.1.1)
𝑗=0
𝑗
With 𝐷𝑗 = 𝐷0 (1 + 𝜇𝐷std ) + 𝜎𝑍𝑗 for 𝑍𝑗 ~𝑁(0,1) and 𝐷𝑗std = 𝛿𝐷𝑗 based on Equations (1) and (3) from
the manuscript, we can insert the general demand model for the periodic process demands:
𝑡
std
𝐷𝑡,cum
=
∑ 𝐷𝑗std
𝑗=0
𝑡
𝑗
= ∑ [𝛿𝐷0 (1 + 𝜇𝐷std ) + 𝛿𝜎𝑍𝑗 ] .
(Eq. C.1.2)
𝑗=0
𝑗
In a next step, we divide Equation (C.1.2) into its deterministic part, i.e., ∑𝑡𝑗=0 𝛿𝐷0 (1 + 𝜇𝐷std ) , and its
stochastic part, i.e., ∑𝑡𝑗=0 𝛿𝜎𝑍𝑗 = 𝛿𝜎 ∑𝑡𝑗=0 𝑍𝑗 . This leads to Equation (C.1.3).
4
𝑡
𝑡
∑ [𝛿𝐷0 (1 +
𝑗
𝜇𝐷std )
𝑡
+ 𝛿𝜎𝑍𝑗 ] = ∑ 𝛿𝐷0 (1 +
𝑗=0
𝑗
𝜇𝐷std )
+ 𝛿𝜎 ∑ 𝑍𝑗
𝑗=0
(Eq. C.1.3)
𝑗=0
Now we can analyze both parts in detail. The deterministic part is a geometric sequence. Therefore, we
can apply the law of the partial sum of a geometric sequence to simplify the expression. The law for the
partial sum of geometric sequence is defined as shown in Equation (C.1.4).
𝑡
∑ 𝑎𝑞 𝑗 = 𝑎
𝑗=0
1 − 𝑞 𝑡+1
1−𝑞
(Eq. C.1.4)
If we set the BPS-adjusted process demand 𝛿𝐷0 =: 𝑎 and the demand drift 1 + 𝜇𝐷std =: 𝑞, we can simplify the deterministic part of Equation (C.1.3) as shown in Equation (C.1.5).
𝑡
𝑡+1
∑ 𝛿𝐷0 (1 +
𝑗
𝜇𝐷std )
= 𝛿𝐷0
𝑗=0
1 − (1 + 𝜇𝐷std )
1 − (1 + 𝜇𝐷std )
𝑡+1
= 𝛿𝐷0
1 − (1 + 𝜇𝐷std )
−𝜇𝐷std
(Eq. C.1.5)
According to assumption (A1), the stochastic part of Equation (C.1.3) equals the sum of 𝑡 independent
and identically normally distributed random variables with a mean of zero and a standard deviation of 1.
Because of the reproduction property of the normal distribution, we know that the sum of normal distributions is again normally distributed. Therefore, 𝑍𝑡sum follows the distribution shown in Equation
(C.1.6).
𝑡
𝛿𝜎 ∑ 𝑍𝑗 =
𝑗=0
𝑡
𝑡
𝑡
𝛿𝜎𝑍𝑡sum ~𝑁 (𝛿𝜎 ∑ 𝜇(𝑍𝑗 ) ; 𝛿 2 𝜎 2 ∑ ∑ 𝜎( 𝑍𝑗 )𝜎(𝑍𝑖 )𝜌𝑖,𝑗 ) ;
𝑗=0
𝑗=0 𝑖=0
(Eq. C.1.6)
for 𝜌𝑖,𝑗 𝑏e defined as the correlation coefficient
As the periodic process demand prior to BPS is observable, the deviation 𝑍0 at the decision point equals
zero. Therefore, we can start with 𝑗 = 1 as shown in Equation (C.1.7).
𝑡
𝑡
𝛿𝜎 ∑ 𝑍𝑗 = 𝛿𝜎 ∑ 𝑍𝑗 = 𝛿𝜎𝑍𝑡sum
𝑗=0
(Eq. C.1.7)
𝑗=1
Additionally, it is known that the expected values for all periodic demand deviations equal zero, meaning
that the expected value of the sum of all periodic deviations up to period 𝑡 equals zero, too.
𝑡
𝜇(𝛿𝜎𝑍𝑡sum )
𝑡
= 𝛿𝜎 ∑ 𝜇(𝑍𝑗 ) = 𝛿𝜎 ∑ 0 = 0
𝑗=1
(Eq. C.1.8)
𝑗=1
Furthermore, we can use the independence between the periodic demand deviations from assumption
(A1) and set their correlation coefficients equal to zero (𝜌𝑖,𝑗 = 0 ∀𝑖, 𝑗 ⋀ 𝑖 ≠ 𝑗). Thus, the variance of the
5
sum of the periodic demand deviations 𝜎 2 (𝑍𝑡sum ) equals the sum of their variances as shown in Equation
(C.1.9).
𝑡
𝜎 2 (𝛿𝜎𝑍𝑡sum )
𝑡
𝑡
2 2
𝑡
2 2
2
2 2
= 𝛿 𝜎 ∑ ∑ 𝜎( 𝑍𝑗 )𝜎(𝑍𝑖 )𝜌𝑖,𝑗 = 𝛿 𝜎 ∑ 𝜎 (𝑍𝑗 ) = 𝛿 𝜎 ∑ 1
𝑗=1 𝑖=1
𝑗=1
𝑗=1
(Eq. C.1.9)
= 𝛿 2 𝜎 2𝑡
Consequently, we can represent the stochastic part of Equation (C.1.3), i.e., 𝛿𝜎𝑍𝑡sum , by a normally
distributed random variable with a mean of zero and a variance of 𝛿 2 𝜎 2 𝑡. Recombining the stochastic
and the deterministic part of Equation (C.1.3), we finally get Equation (C.1.10).
𝑡
𝐷
std
t,cum
𝑡+1
= ∑ [𝛿𝐷0 (1 +
𝑗
𝜇𝐷std )
𝑗=0
C.2
1 − (1 + 𝜇𝐷std )
+ 𝛿𝜎𝑍𝑗 ] = 𝛿𝐷0
− 𝜇𝐷
+ 𝛿𝜎𝑍𝑡sum
(Eq. C.1.10)
Expected Value of the Periodic Cash Flows
As the periodic cash flows have stochastic and deterministic parts, we first expand the Equation (17)
from the manuscript to facilitate the calculation of its expected value.
𝑡
𝐶𝐹𝑡std = (𝛿𝐷0 (1 + 𝜇𝐷std ) + 𝛿𝜎𝑍𝑡 ) (𝑀0
+
𝑎̃𝐺cost (𝛿𝜎𝑍𝑡sum
+ 𝛿𝐷0
1 − (1 + 𝜇𝐷std )
𝑡+1
))
−𝜇𝐷std
𝑡
(Eq. C.2.1)
𝑡
= 𝛿𝐷0 (1 + 𝜇𝐷std ) 𝑀0 + 𝛿 2 𝐷0 (1 + 𝜇𝐷std ) 𝑎̃𝐺cost 𝜎𝑍𝑡sum
𝑡+1
𝑡
+ 𝛿 2 𝐷0 2 (1 + 𝜇𝐷std ) 𝑎̃𝐺cost
2 2
+𝛿 𝜎
𝑍𝑡 𝑎̃𝐺cost 𝑍𝑡sum
1 − (1 + 𝜇𝐷std )
+ 𝛿𝜎𝑍𝑡 𝑀0
−𝜇𝐷std
2
+ 𝛿 𝜎𝑍𝑡 𝐷0
1 − (1 + 𝜇𝐷std )
𝑡+1
−𝜇𝐷std
Now, we can determine the expected value of the periodic process cash flows 𝐸(𝐶𝐹𝑡std ) as shown in
Equation (C.2.2).
𝑡
𝑡
𝐸(𝐶𝐹𝑡std ) = 𝐸 (𝛿𝐷0 (1 + 𝜇𝐷std ) 𝑀0 ) + 𝐸 (𝛿 2 𝐷0 (1 + 𝜇𝐷std ) 𝑎̃𝐺cost 𝜎𝑍𝑡sum )
2
2
+ 𝐸 (𝛿 𝐷0 (1 +
1
𝑡
𝜇𝐷std ) 𝑎̃𝐺cost
𝑡+1
− (1 + 𝜇𝐷std )
−𝜇𝐷std
) + 𝐸(𝛿𝜎𝑍𝑡 𝑀0 )
𝑡+1
2 2
+ 𝐸(𝛿 𝜎
6
𝑍𝑡 𝑎̃𝐺cost 𝑍𝑡sum ) +
2
𝐸 (𝛿 𝜎𝑍𝑡 𝐷0
1 − (1 + 𝜇𝐷std )
−𝜇𝐷std
)
(Eq. C.2.2)
In a next step, we eliminate all components whose expected value equals zero and replace the expected
values of deterministic terms by their values. The result in shown in Equation (C.2.3)
𝐸(𝐶𝐹𝑡std )
= 𝛿𝐷0 (1 +
𝑡
𝜇𝐷std ) 𝑀0
2
2
+ 𝛿 𝐷0 (1 +
1−
𝑡
𝜇𝐷std ) 𝑎̃𝐺cost
𝑡+1
(1 + 𝜇𝐷std )
−𝜇𝐷std
(Eq. C.2.3)
+ 𝐸(𝛿 2 𝜎 2 𝑍𝑡 𝑎̃𝐺cost 𝑍𝑡sum )
Now, we calculate the expected value 𝐸(𝛿 2 𝜎 2 𝑍𝑡 𝑎̃𝐺cost 𝑍𝑡sum ). First of all, the deterministic variables
can be put outside of the expected value operator as shown in Equation (C.2.4).
𝐸(𝛿 2 𝜎 2 𝑍𝑡 𝑎̃𝐺cost 𝑍𝑡sum ) = 𝛿 2 𝜎 2 𝑎̃𝐺cost 𝐸(𝑍𝑡 𝑍𝑡sum )
(Eq. C.2.4)
What remains is the expected value of a product of two random variables. Determining the expected
value of a product of two random variables requires applying the covariance formula from Equation
(C.2.5). The result is shown in Equation (C.2.6).
𝐶𝑂𝑉(𝑋, 𝑌) = 𝐸(𝑋𝑌) − 𝐸(𝑋)𝐸(𝑌) ⇔ 𝐸(𝑋𝑌) = 𝐶𝑂𝑉(𝑋, 𝑌) + 𝐸(𝑋)𝐸(𝑌)
(Eq. C.2.5)
𝐸(𝑍𝑡 𝑍𝑡sum ) = 𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡sum ) + 𝐸(𝑍𝑡 )𝐸(𝑍𝑡sum ) = 𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡sum )
(Eq. C.2.6)
Considering the definition of the term 𝑍𝑡sum as the sum of the independent random deviations from the
sum
demand trend, we can divide it up into the cumulated deviations up to the period 𝑡 − 1, 𝑍𝑡−1
, and the
deviation in period 𝑡, which is 𝑍𝑡 .
sum
𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡sum ) = 𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡−1
+ 𝑍𝑡 )
(Eq. C.2.7)
On this foundation, we can use the linearity of the covariance to simplify Equation (C.2.6) as follows.
sum
sum )
𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡−1
+ 𝑍𝑡 ) = 𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡−1
+ 𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡 )
(Eq. C.2.8)
Considering that the covariance of a random variable with itself equals its variance and that the periodic
deviation 𝑍𝑡 is independent from the cumulated deviations, i.e., 𝐶𝑂𝑉(𝑍(0,1); 𝑍(0, 𝑡 − 1)) = 0, we can
simplify Equation (C.2.8) as shown in Equation (C.2.9).
sum )
𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡−1
+ 𝐶𝑂𝑉(𝑍𝑡 , 𝑍𝑡 ) = 𝜎 2 (𝑍𝑡 ) = 1
(Eq. C.2.9)
Now we can determine the expected periodic cash flows as shown in Equation (C.2.10).
𝐸(𝐶𝐹𝑡std )
= 𝛿𝐷0 (1 +
𝑡
𝜇𝐷std ) 𝑀0
2
2
+ 𝛿 𝐷0 (1 +
1−
𝑡
𝜇𝐷std ) 𝑎̃𝐺cost
+ 𝛿 2 𝜎 2 𝑎̃𝐺cost
C.3
Present Value of the Expected Periodic Cash Flows
The present value of the expected periodic cash flows equals:
7
𝑡+1
(1 + 𝜇𝐷std )
−𝜇𝐷std
(Eq. C.2.10)
𝜏
𝑃𝑉 = ∑
𝑡=0
1
𝑡
[𝛿𝐷0 (1 + 𝜇𝐷std ) 𝑀0
(1 + 𝑖)𝑡
(Eq. C.3.1)
𝑡+1
𝑡
+ 𝛿 2 𝐷0 2 (1 + 𝜇𝐷std ) 𝑎̃𝐺cost
1 − (1 + 𝜇𝐷std )
+ 𝛿 2 𝜎 2 𝑎̃𝐺cost ]
−𝜇𝐷std
The first step to simplify Equation (C.3.1) is to separate the total sum into different summands as shown
in Equation (C.3.2).
𝜏
𝑡
(1 + 𝜇𝐷std )
𝑃𝑉 = ∑ 𝛿𝐷0 𝑀0
(1 + 𝑖)𝑡
𝑡=0
𝜏
𝑡
𝛿 2 𝐷02 𝑎̃𝐺cost (1 + 𝜇𝐷std )
+∑
(1 + 𝑖)𝑡
−𝜇𝐷std
𝑡=0
𝜏
−∑
𝑡=0
𝜏
𝛿
2
𝐷02 𝑎̃𝐺cost
(1 +
−𝜇𝐷std
+ ∑ 𝛿 2 𝜎 2 𝑎̃𝐺cost
𝑡=0
2𝑡
(1 + 𝜇𝐷std )
std
𝜇𝐷 )(
(1 + 𝑖)𝑡
(Eq. C.3.2)
1
(1 + 𝑖)𝑡
Each of the summands is a geometric sequence. Consequently, the law for the partial sum of the geometric sequence can be applied. The present value of each summand can be obtained by inserting the
starting point of each geometric sequence and its growth factor as shown in Equation (C.3.3).
𝜏+1
𝑃𝑉 = (𝛿𝐷0 𝑀0 +
𝛿 2 𝐷02 𝑎̃𝐺cost
−𝜇𝐷std
(1 + 𝜇𝐷std )
(1 + 𝑖)𝜏+1
1 + 𝜇𝐷std
1−
1+𝑖
1−
)
2 𝜏+1
−
𝛿 2 𝐷02 𝑎̃𝐺cost
−𝜇𝐷std
(1 + 𝜇𝐷std )
(1 + 𝜇𝐷std )
1−[
]
(1 + 𝑖)
1−
(Eq. C.3.3)
2
𝜇𝐷std )
(1 +
(1 + 𝑖)
1
(1 + 𝑖)𝜏+1
1
1−1+𝑖
1−
+ 𝛿 2 𝜎 2 𝑎̃𝐺cost
C.4
Final Objective Function including the Investment Outflows
Taking all intermediate results and the investment outflows from Equation (20) in the manuscript together, leads to the following final definition of objective function:
8
MAX:
𝑁𝑃𝑉 = 𝑃𝑉 − 𝐼 =
(𝛿𝐷0 𝑀𝑜 +
𝜏+1
(1 + 𝜇𝐷std )
(1 + 𝑖)𝜏+1
𝑎̃𝐺cost 𝛿 2 𝐷02 1 −
)
− 𝜇𝐷std
1 + 𝜇std
1 − 1 + 𝐷𝑖
2 𝜏+1
−
𝑎̃𝐺cost 𝛿 2 𝐷02
(1 + 𝜇𝐷std )
− 𝜇𝐷𝑠
(1 + 𝜇𝐷std )
1−[
]
(1 + 𝑖)
2
(1 + 𝜇𝐷std )
1−
(1 + 𝑖)
1
(1 + 𝑖)𝜏+1
−𝐼
1
1−1+𝑖
1−
+ 𝑎̃𝛿 2 𝐺cost 𝜎 2
where:
𝑛
𝑛
𝛿 = ∑ 𝑤𝑐 [(1 − 𝑓𝑐
std
)(𝑥𝑐 −𝑥𝑐 )
𝑀0 = ∑ 𝑤𝑣std (𝑃𝑣 − 𝐶𝑣,0 )
]
𝑐=1
v=0
std )
𝑤𝑐std =
𝑤𝑐 [(1 − 𝑓𝑐 )(𝑥𝑐 −𝑥𝑐
v=0
𝑤𝑐 [(1 − 𝑓𝑐 )(𝑥𝑐−𝑥𝑐
𝛿
𝑛
𝑤0std
2
𝐺cost = ∑ ( 𝑤𝑣std ) 𝐶𝑣,0
𝛿
std )
𝑤𝑣std = 𝑥𝑐
𝑚
]
=1−∑
]
𝑛
2
𝐺 = ∑ ( 𝑤𝑣std )
v=0
𝑛
𝑤𝑣std
𝑣=1
𝐼 = ∑ |𝑥𝑐 − 𝑥𝑐std |𝐼𝑐
𝑐=1
𝜇𝐷 (𝐺 std ) = 𝜇𝐷 + ∆𝑟(𝐺 std )
= 𝜇𝐷 + 𝛽𝑆𝐴𝑇 [𝛽𝑄 (Q
subject to:
𝑥𝑐std ∈ {0; 1} and 𝑅
9
𝑠𝑄
𝑠𝑇 + 𝑠𝑄
𝑠𝑇 + 𝑠𝑄
) ∆𝐺 + 𝛽𝑁𝐶𝐷 (−𝑁𝐶𝐷
) ∆𝐺 + 𝛽𝑃𝐶𝐷 (𝑃𝐶𝐷
) ∆𝐺]
1−𝐺
1−𝐺
1−𝐺
D
Questionnaire and Responses for the Real-World Case
Demand of the coverage switching processes
How many process instances where executed in the last period?
9,875
How will the periodic demand relatively increase or decrease over the planning horizon?
+10% per year
What is the standard deviation of the periodic demand?
1,200 per year
Execution options of the coverage switching processes
What is the average revenue of the integration of one contract?
90.00 EUR
Execution option
Fraction of the demand
covered by this
execution option
What fraction of the currently conCosts
nected brokers would leave if this
per execution
execution option were eliminated?
Submission of
end-customer
information in
electronic form
30%
20.00 EUR
5%
Submission of
end-customer
information in
paper form
70%
25.00 EUR
5%
Broker updates
information
70%
11.25 EUR
25%
Call center updates
information
30%
37.50 EUR
0%
Broker changes
contract
80%
3.75 EUR
25%
Call center
changes contract
20%
12.50 EUR
0%
Experience curve effects
How high were the average costs per execution in the last period?
48.25 EUR
How did the average execution costs change in the last period?
- 2.50 EUR
Process quality and time from Anderson’s model
How do you rate the current process quality on a 10 point scale? (1 = very low,…, 10 = very high)
10
8
By what percentage would the process quality improve due to BPS?
+12.50%
By what percentage would the process time improve due to BPS?
+63.33%
Company-specific adjustment factors from Anderson’s model
Adjustment factor
Derivation
Values
Concentration
(CONC)
The inverse of the number of competitors comprising 70 percent of the sales
in the industry
20
Ease of evaluating
quality (QEVAL)
How difficult or easy is it to evaluate
quality (1 = very difficult,…, 10 = very
easy)?
8
Differentiation
(DIFF)
How strongly do you differ from your
competitors on a scale from 1 to 10 (1
= very weak,…, 10 = very strong)?
4
Involvement (INVOLV)
How would you rate the involvement
of your customers on a scale from 1 to
10 (1 = very low,…, 10 = very high)?
3
Frequency of usage
(USAGE)
How would you rate the frequency of
your customers’ usage of the integration process on a scale from 1 to 10 (1
= very low,…, 10 = very high)?
5
Switching costs
(SC)
How would you rate your customers’
switching costs on a scale from 1 to 10
(1 = very low,…, 10 = very high)?
3
Difficulty of standardization (DSTD)
How would you rate the standardization difficulty within your industry on
a scale from 1 to 10 (1 = very low,…,
10 = very high)?
9
Further parameters
What is the planning horizon for investment decisions within your company?
7 years
What is the risk-adjusted discount rate for investment decisions within your company?
4% per year
11
Process quality and time from Anderson’s model (using process variant 3 as master process)
By what percentage would the process quality improve due to BPS?
-10.00%
By what percentage would the process time improve due to BPS?
-30.00%
Process quality and time from Anderson’s model (using process variant 4 as master process)
By what percentage would the process quality improve due to BPS?
+11.25%
By what percentage would the process time improve due to BPS?
+57.00%
Figure 1: The master process (basic scenario)
E
E.1
Sensitivity Analysis (Basic Scenario)
Adjusted Values of the Objective Function
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
Opt. Profile
12
Quality Effect
Time Effect
Demand Drift
Demand
4,380,780
4,380,912
4,381,043
4,381,175
4,381,306
4,381,569
4,381,701
4,381,832
4,381,964
4,382,096
no changes
4,380,927
4,381,029
4,381,131
4,381,233
4,381,336
4,381,540
4,381,642
4,381,745
4,381,847
4,381,949
no changes
3,650,498
3,784,457
3,924,260
4,070,170
4,222,465
4,547,392
4,720,648
4,901,542
5,090,427
5,287,671
no changes
2,115,078
2,556,206
3,003,406
3,456,678
3,916,022
4,852,926
5,330,486
5,814,119
6,303,823
6,799,599
no changes
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
Opt.
Profile
E.2
Learning Curve
Planning Horizon
Quality
4,229,378
4,259,790
4,290,202
4,320,614
4,351,026
4,411,850
4,442,262
4,472,674
4,503,086
4,533,498
2,146,961
2,547,911
2,970,308
3,415,614
3,885,416
4,905,559
5,459,823
6,046,462
6,667,909
7,326,822
4,380,881
4,380,993
4,381,104
4,381,215
4,381,327
4,381,549
4,381,661
-
no changes
no changes
no changes
no changes
Delta compared to the Status Quo
-0.5
-0.4
-0.3
-0.2
-0.1
+0.1
+0.2
+0.3
+0.4
+0.5
Opt. Profile
-0.5
-0.4
-0.3
-0.2
-0.1
+0.1
+0.2
+0.3
+0.4
+0.5
Opt.
Profile
13
Demand
Variance
4,381,179
4,381,231
4,381,282
4,381,334
4,381,386
4,381,490
4,381,542
4,381,593
4,381,645
4,381,697
Quality Effect
Time Effect
Demand Drift
Demand
20,361
20,492
20,624
20,755
20,886
21,150
21,281
21,413
21,544
21,676
no changes
20,507
20,609
20,711
20,813
20,916
21,120
21,223
21,325
21,427
21,529
no changes
17,296
17,974
18,683
19,426
20,203
21,872
22,766
23,703
24,686
25,716
no changes
9,835
11,963
14,146
16,382
18,673
23,417
25,870
28,378
30,939
33,554
no changes
Demand
Variance
21,016
21,016
21,017
21,017
21,018
21,018
21,019
21,019
21,020
21,020
Learning Curve
Planning Horizon
Quality
19,663
19,934
20,205
20,476
20,747
21,289
21,560
21,831
22,102
22,373
9,426
11,381
13,501
15,803
18,302
23,972
27,185
30,685
34,499
38,658
20,461
20,573
20,684
20,795
20,907
21,129
21,241
-
no changes
no changes
no changes
no changes
F
Glossary
BPS-specific variables
Superscript indicating a variable’s value after BPS
𝐬𝐭𝐝
𝑮𝐜𝐨𝐬𝐭
𝑮
Cost-weighted Gini Coefficient
Gini Coefficient
Process Variants and Contexts Variables
A distinct process variant
𝒗
𝒄
A distinct process context
𝒘𝒄
Demand weight of a process context
𝒏
Total number of process contexts
𝑴𝒕
Profit margin in period t
Demand Model
Periodic process demand
𝑫𝒕
𝝁𝑫
Demand trend
𝒁𝒕
Periodic demand deviation
𝝈
Standard deviation of the periodic demand deviations
Demand Effects of BPS
Fraction of demand for process context c that can only be tapped by the corresponding
𝒇𝒄
process variant v
𝜹
Total relative change in the process demand due to BPS
𝒘𝐬𝐭𝐝
𝒗
Demand weight covered by process variant v
𝒘𝐬𝐭𝐝
𝟎
Demand weight covered by the master process
Learning Curve
Cumulated demand
𝑫𝐜𝐮𝐦
𝒂
Elasticity of the process costs regarding the cumulated demand
𝑲
Process costs of for the first output
𝐂
Process costs
𝑫𝟎,𝐜𝐮𝐦
̃
𝒂
𝑫𝐬𝐭𝐝
𝒕,𝐜𝐮𝐦
Cumulated process demand up to the decision point
Adjusted elasticity of the process costs regarding the cumulated demand
Cumulated process demand that has been reached starting from the decision point up to
period t
Quality and Time Effects
14
𝑸
Process quality
𝑻
Process time
𝒔𝑸
Relative increase in process quality in case of complete standardization compared to
the status prior to BPS
𝒔𝑻
Relative increase in process time in case of complete standardization compared to the
status prior to BPS
𝑺𝑨𝑻
Customer satisfaction
𝑬𝑿𝑷
Customer expectation
𝑵𝑪𝑫
Negative Confirmation/Disconfirmation
𝑷𝑪𝑫
Positive Confirmation/Disconfirmation
𝜷𝑸
Sensitivity of customer satisfaction w.r.t. process quality
𝜷𝑵𝑪𝑫
Sensitivity of customer satisfaction w.r.t. negative confirmation/-disconfirmation
𝜷𝑷𝑪𝑫
Sensitivity of customer satisfaction w.r.t. positive confirmation/disconfirmation
𝒓
𝜷𝑺𝑨𝑻
Retention rate
Sensitivity of retention rate w.r.t. customer satisfaction
Objective Function
Decision variable indicating that process context c is covered by the respective process
𝒙𝒄
variant (𝑥𝑐 = 1) or the master process (𝑥𝑐 = 0) prior to BPS
𝒙𝐬𝐭𝐝
𝒄
Decision variable indicating that process context c is covered by the respective process
variant (𝑥𝑐 = 1) or the master process (𝑥𝑐 = 0) after to BPS
𝑪𝑭𝒕
Periodic process cash flows in period t
𝒕
A distinct period within the planning horizon
𝝉
Total planning horizon
𝒊
Risk-adjusted interest rate
𝑰
Overall investment outflows
𝑰𝒄
Investment outflows for process context c
𝑷𝑽
𝑵𝑷𝑽
𝑹
15
Risk-adjusted expected present value
Risk-adjusted expected net present value
Set of constraints regarding admissible values of 𝑥𝑐std