Bringing value-based business process management to the operational process level

Abstract

For years, improving processes has been a prominent business priority for Chief Information Officers. As expressed by the popular saying, “If you can’t measure it, you can’t manage it,” process measures are an important instrument for managing processes and corresponding change projects. Companies have been using a value-based management approach since the 1990s in a constant endeavor to increase their value. Value-based business process management introduces value-based management principles to business process management and uses a risk-adjusted expected net present value as the process measure. However, existing analyses of this issue operate at a high (i.e., corporate) level, hampering the use of value-based business process management at an operational process level in both research and practice. Therefore, this paper proposes a valuation calculus that brings value-based business process management to the operational process level by showing how the risk-adjusted expected net present value of a process can be determined. We demonstrate that the valuation calculus provides insights into the theoretical foundations of processes and helps improve the calculation capabilities of an existing process-modeling tool.

Appendices

Appendix 1: Determination of path probabilities

To determine the expected value of a process cash flow and its variance, it is essential to determine the path probabilities $p_{k}$. This is presented in the following. During a process improvement project, a process is presented as a process model with a process-modeling tool. With the help of this formal presentation, it is possible to formally describe, how a path probability $p_{k}$ is determined. In order to do so, the process model [as defined in Hollingsworth and WfMC (2003, p. 266)] of a process P is defined as a graph G.

The process model of a process P is a graph, because a process model is a set of nodes (vertices) that are interconnected by arrows (edges) (Gibbons 1985). The set of vertices is denoted by V and the set of edges by E and we write G = (V, E). Because the edges are arrows, a process is a directed graph (Gibbons 1985). More precisely, we assume that a process model of a process P is defined as a graph G as followed:

• (D1) A process model of a process P is a directed graph G = (V, E) with one root vertex $a_{0}$ and one final vertex $a_{D + 1}$, toward which all edges are directed. It is V the set of vertices and E the set of edges.

• (D2) The set V consists of the set of actions A united with the set RC of the routing constructs (van der Aalst et al. 2003) to denote control flow patterns of P, $a_{0}$ and $a_{D + 1}$, i.e., $V\text{ := }A \cup RC \cup a_{0} \cup a_{D + 1}$.

• (D3) A contains all D actions of P, numbered from 1 to D.

• (D4) RC is the set of the routing constructs to denote the control flow patterns, e.g., XOR-split, XOR-join, AND-split and AND-join (van der Aalst et al. 2003). Each element has one distinct index. For example, in Fig. 2 (left process) these vertices are XOR-join ₁, AND-split ₂, AND-join ₃ and XOR-split ₄.

• (D5) The edge-set E contains all the directed edges between the vertices. The directed edge $(v_{i} , v_{j} , p_{ij} )$ is a member of the set E if and only if there is an arrow between vertex $v_{i} \in V$ and $v_{j} \in V$, pointing from $v_{i}$ to $v_{j}$ and having a probability for this transition (Hollingsworth and WfMC 2003, p. 282) of $p_{ij}$, with $0 < p_{ij} \le 1$, as weight. Each vertex in A has exactly one edge pointing toward it and exactly one edge pointing away from it.

The actions and routing constructs of a process path $pp_{k}$ plus $a_{0}$ and $a_{D + 1}$ form (in a first step) a path multiset $PS_{k}$, whose elements are out of V. The fact that it is a multiset is important to consider loops, as then the same vertices of G can occur several times. Each vertex $v_{i}$ in $PS_{k}$ that occurs more than once (in a second step) is given an index $n \, \in \, {\mathbb{N}}$ in the form $v_{i}^{(1)} , v_{i}^{(2)} , \ldots ,v_{i}^{\left( n \right)} , \ldots$. The index indicates the number of the iteration of a loop that the vertex is assigned to. This is to distinguish the vertices from one another because each of them is from different iterations that have different probabilities of being executed. In the left process in Fig. 2, there are the path sets.

PS ₁ = {a ₀ , a ₁ , XOR-join ⁽¹⁾₁ , AND-split ⁽¹⁾₂ , a ⁽¹⁾₂ , a ⁽¹⁾₃ , AND-join ⁽¹⁾₃ , XOR-split ⁽¹⁾₄ , a ₅ },

PS ₂ = {a ₀ , a ₁ , XOR-join ⁽¹⁾₁ , AND-split ⁽¹⁾₂ , a ⁽¹⁾₂ , a ⁽¹⁾₃ , AND-join ⁽¹⁾₃ , XOR-split ⁽¹⁾₄ , a ⁽¹⁾₄ , XOR-join ⁽²⁾₁ , AND-split ⁽²⁾₂ , a ⁽²⁾₂ , a ⁽²⁾₃ , AND-join ⁽²⁾₃ , XOR-split ⁽²⁾₄ , a ₅ } and so on, with v ₁: = a ₀ , v ₂: = a ₁ , v ⁽¹⁾₃ : = XOR-join ⁽¹⁾₁ , v ⁽²⁾ ₃ : = XOR-join ⁽²⁾₁ , …, v ⁽¹⁾₄ : = AND-split ⁽¹⁾₂ , v ⁽²⁾ ₄ : = AND-split ⁽²⁾₂ , …, v ⁽¹⁾₅ : = a ⁽¹⁾₂ , v ⁽²⁾₅ : = a ⁽²⁾₂ , …, v ⁽¹⁾₆ : = a ⁽¹⁾₃ , v ⁽²⁾₆ : = a ⁽²⁾₃ , …, v ⁽¹⁾₇ : = AND-join ⁽¹⁾₃ , v ⁽²⁾₇ : = AND-join ⁽²⁾₃ , …, v ⁽¹⁾₈ : = XOR-split ⁽¹⁾₄ , v ⁽²⁾₈ : = XOR-split ⁽²⁾₄ , …, v ⁽¹⁾₉ : = a ⁽¹⁾₄ , …, and v ₁₀: = a ₅.

Every process path $pp_{k}$ is executed with a certain path probability $p_{k}$ that is the product of the transition probabilities of process path $pp_{k}$:

$$p_{k} = \mathop \prod \limits_{{v_{i}^{\left( m \right)} ,v_{j}^{\left( n \right)} \in PS_{k} }} p_{{i^{\left( m \right)} j^{\left( n \right)} }} \,{\text{ for all }}\, p_{{i^{\left( m \right)} j^{\left( n \right)} }} > 0.$$

(22)

The transition probability $p_{{i^{\left( m \right)} j^{\left( n \right)} }}$ that $v_{i}^{\left( m \right)}$ is followed by $v_{j}^{\left( n \right)}$ can be estimated and is fixed. These transition probabilities could be estimated by an expert (Hubbard 2007) or by analyzing process log files (zur Muehlen and Shapiro 2010) using, for example, a process-mining framework like ProM (Rubin et al. 2007). In the left process in Fig. 2, for example, for the process path pp ₁ there are the (non-zero) transition probabilities $p_{12} = 1, p_{{23^{\left( 1 \right)} }} = 1, p_{{3^{\left( 1 \right)} 4^{\left( 1 \right)} }} = 1, p_{{4^{\left( 1 \right)} 5^{\left( 1 \right)} }} = 1, p_{{4^{\left( 1 \right)} 6^{\left( 1 \right)} }} = 1, p_{{5^{\left( 1 \right)} 7^{\left( 1 \right)} }} = 1, p_{{6^{\left( 1 \right)} 7^{\left( 1 \right)} }} = 1, p_{{7^{\left( 1 \right)} 8^{\left( 1 \right)} }} = 1$ and $p_{{8^{\left( 1 \right)} ,10}} = 0.9$. All other transition probabilities are zero. Then it is

$$\begin{gathered} p_{1} = \prod\limits_{{v_{i}^{\left( m \right)} ,v_{j}^{\left( n \right)} \in PS_{1} }} {p_{{i^{\left( m \right)} j^{\left( n \right)} }} } \hfill \\ = \underbrace {1}_{{a_{0} \,to\,a_{1} }} \cdot \underbrace {1}_{{a_{1} \,to\,XOR - join_{1} }} \cdot \underbrace {1}_{{XOR - join_{1} \,to\,AND - split_{2} }} \cdot \underbrace {1}_{{AND - split_{2} \,to\,a_{2} }} \cdot \underbrace {1}_{{AND - split\,to\,a_{3} }} \hfill \\ \cdot \underbrace {1}_{{a_{2} \,to\,AND - join_{3} }} \cdot \underbrace {1}_{{a_{3} \,to\,AND - join_{3} }} \cdot \underbrace {1}_{{AND - join_{3}\,\,to\,XOR - split_{4} }} \cdot \underbrace {0.9}_{{XOR - split_{4}\,to\,a_{5} }}=0.9 \hfill \\ \end{gathered}$$

and

$$p_{2} = \mathop \prod \limits_{{v_{i}^{\left( m \right)} ,v_{j}^{\left( n \right)} \in PS_{2} }} p_{{i^{\left( m \right)} j^{\left( n \right)} }} = 0.09,{\text{ etc}}.$$

Expression (22) is not true in the event that a process model contains an OR-split (van der Aalst et al. 2003). This fact is important in Sect. 5.3, when showing how this valuation calculus helped to improve the calculation capabilities of a process-modeling tool. However, every OR-split can formally be transformed into a composition of XOR-splits and AND-splits, which allows the use of expression (22). Otherwise, the path probabilities need to be estimated.

Appendix 2: Process-probability-space

In probability theory, “a probability space is a triple (Ω, ${\mathcal{F}}$, P) of a sample space Ω, a [sigma]-algebra ${\mathcal{F}}$ and a probability measure P on ${\mathcal{F}}$” (Feller 1971, p. 116). The sample space Ω is the set of all possibilities that the object under consideration can take; it is thus the set of all possible process paths, as these represent all possibilities of a process execution. A sigma-algebra has properties such that:

• (1) “If a set A is in ${\mathcal{F}}$ so is its complement [$A^{C} = {\Omega }\backslash A$].

• (2) If $\left\{ {A_{n} } \right\}$ is any countable collection of sets in ${\mathcal{F}}$, then also their union $\mathop {\bigcup }\nolimits A_{n}$ and intersection $\mathop {\bigcap }\nolimits A_{n}$ belong to ${\mathcal{F}}$” (Feller 1971, p. 112).

That the sigma-algebra in Definition 2 is the power set of the set of all process paths means that (i) and (ii) are fulfilled.

“A probability measure P on a [sigma]-algebra ${\mathcal{F}}$ of sets in Ω is a function assigning a value P{A} ≥ 0 to each set A in ${\mathcal{F}}$ such that P{Ω} = 1 and that for every countable collection of non-overlapping sets A _n in ${\mathcal{F}}$ [it is] ${\bf P}\left\{ { \cup \,A_{n} } \right\} = \mathop \sum \nolimits_{n} {\bf P}\left\{ {A_{n} } \right\}$” (Feller 1971, p. 115).

All process paths are mutually exclusive, and they represent all possibilities how a process can be executed. Every process path $pp_{k}$ is executed with a certain path probability $p_{k} > 0$. Given that there is exactly one process path taken if a process is executed and that they are mutual exclusive, the probabilities $p_{k}$ sum up to 1, fulfilling P{Ω} = 1. The property ${\mathbf{P}}\left\{ {\mathop {\bigcup }\nolimits A_{n} } \right\} = \mathop \sum \nolimits_{n} {\mathbf{P}}\left\{ {A_{n} } \right\}$ also holds for every countable collection of non-overlapping sets A _n in ${\mathcal{F}}$ since ${\mathcal{F}}$ is the power set of Ω.

Appendix 3: Expected value of the process cash flow

Let the probability that an action $a_{d} \in AS,$ with $AS:= \mathop {\bigcup }\nolimits_{k = 1}^{{\left| {\Omega } \right|}} AS_{k}$, is executed when executing a process be

$$Pr\left( {a_{d} } \right): = {\mathbf{P}}_{P} \left\{ {PP_{{a_{d} }} } \right\} = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k}\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right)$$

with the indicator function

$${\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) = \left\{ \begin{array}{ll} 1,& a_{d} \in AS_{k} \\ 0,& a_{d} \notin AS_{k} \\ \end{array} \right.$$

and the set $PP_{{a_{d} }}$ of process paths in which the action $a_{d}$ is

$$PP_{{a_{d} }} = \left\{ {\left. {pp_{k} \in {\Omega }} \right|a_{d} \in AS_{k} } \right\}.$$

Then it is:

$$\begin{aligned} E\left[ {CF_{P} } \right] & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} E\left[ {\left. {CF_{P} } \right|PI = k} \right] \cdot Prob\left( {PI = k} \right) = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} E\left[ {CF_{{pp_{k} }} } \right] \cdot p_{k} \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \left( {p_{k} \cdot E\left[ {\mathop \sum \limits_{{a_{d} \in AS_{k} }} CF_{{a_{d} }} + \mathop \sum \limits_{s = 1}^{S} CF_{{pa_{s} }} } \right]} \right) = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \left( {p_{k} \cdot \left( {\mathop \sum \limits_{{a_{d} \in AS_{k} }} E\left[ {CF_{{a_{d} }} } \right] + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right]} \right)} \right) \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \left( {\mathop \sum \limits_{{a_{d} \in AS_{k} }} p_{k} \cdot E\left[ {CF_{{a_{d} }} } \right]} \right) + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \left( {p_{k} \cdot \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right]} \right) \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \left( {\mathop \sum \limits_{{a_{d} \in AS_{k} }}\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) \cdot p_{k} \cdot E\left[ {CF_{{a_{d} }} } \right]} \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right] \cdot \underbrace {{\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} }}_{ = 1} \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \left( {\mathop \sum \limits_{{a_{d} \in AS}} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) \cdot p_{k} \cdot E\left[ {CF_{{a_{d} }} } \right]} \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right] \\ & = \mathop \sum \limits_{{a_{d} \in AS}} \left( {\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) \cdot p_{k} \cdot E\left[ {CF_{{a_{d} }} } \right]} \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right] \\ & = \mathop \sum \limits_{{a_{d} \in AS}} E\left[ {CF_{{a_{d} }} } \right]\left( {\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k}\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) } \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right] \\ & = \mathop \sum \limits_{{a_{d} \in AS}} E\left[ {CF_{{a_{d} }} } \right] \cdot Pr\left( {a_{d} } \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }} } \right] \\ \end{aligned}$$

Appendix 4: Variance of the process cash flow

In the following first step, it is shown that $Var\left[ {CF_{P} } \right] = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \nolimits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} } \right]$ in two ways. The first way is similar to the beginning of the calculation for the expected value in Appendix 3. The second way is more detailed and includes $\mathop \sum \nolimits_{k = 1}^{{\left| {\Omega } \right|}} E\left[ {\left( {CF_{{pp_{k} }} - E\left[ {CF_{P} } \right]} \right)^{2} } \right]\cdot p_{k}$, a more intuitive expression for $Var\left[ {CF_{P} } \right]$. This is why both ways are presented.

4.1 Way 1

$$\begin{aligned} Var\left[ {CF_{P} } \right] & = E\left[ {CF_{P}^{2} } \right] - E\left[ {CF_{P} } \right]^{2} = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} E [CF_{P}^{2} |PI = k] \cdot Prob\left( {PI = k} \right) \\ & \quad = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} } \right] \\ \end{aligned}$$

4.2 Way 2

$$\begin{aligned} Var\left[ {CF_{P} } \right] & = E\left[ {\left( {CF_{P} - E\left[ {CF_{P} } \right]} \right)^{2} } \right] \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} E [\left( {CF_{P} - E\left[ {CF_{P} } \right]} \right)^{2} | PI = k] \cdot Prob\left( {PI = k} \right) \\ & = \mathop \sum \limits_{{\varvec{k} {\bf = 1}}}^{{\left| {\Omega } \right|}} \varvec{E}\left[ {\left( {\varvec{CF}_{{\varvec{pp}_{\varvec{k}} }} - \varvec{E}\left[ {\varvec{CF}_{\varvec{P}} } \right]} \right)^{2} } \right] \cdot \varvec{p}_{\varvec{k}} \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} - 2 \cdot CF_{{pp_{k} }}^{2} \cdot E\left[ {CF_{P} } \right] + E\left[ {CF_{P} } \right]^{2} } \right] \\ & = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} } \right] - 2 \cdot E\left[ {CF_{P} } \right]\underbrace {{\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }} } \right]}}_{{ = E\left[ {FQ_{P} } \right]}} + E\left[ {CF_{P} } \right]^{2} \underbrace {{\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} }}_{ = 1} \\ & = - 2 \cdot E\left[ {CF_{P} } \right]^{2} + E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} } \right] = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} } \right] \\ \end{aligned}$$

In the following second step, it is shown how $Var\left[ {CF_{P} } \right]$ can be calculated only by using the expected values and variances of the cash flows of the actions of a process.

Let the probability that both actions $a_{d} \in AS$ and $a_{j} \in AS$, with $AS\text{ := }\mathop {\bigcup }\nolimits_{k = 1}^{{\left| {\Omega } \right|}} AS_{k}$, are executed when executing a process be

$$Pr\left( {a_{d} ,a_{j} } \right): = {\mathbf{P}}_{P} \left\{ {PP_{{a_{d} ,a_{j} }} } \right\} = \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{j} } \right)$$

with the set $PP_{{a_{d} ,a_{j} }}$ of process paths, which contains the action $a_{d}$ as well as the action $a_{j}$:

$$PP_{{a_{d} ,a_{j} }} = \left\{ {\left. {pp_{k} \in {\Omega }} \right|a_{d} \in AS_{k} , a_{j} \in AS_{k} } \right\}.$$

Then it is:

$$\begin{aligned} Var\left[ {CF_{P} } \right] & \overbrace { = }^{first step} - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {CF_{{pp_{k} }}^{2} } \right] = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {\left( {\mathop \sum \limits_{{a_{d} \in AS_{k} }} CF_{{a_{d} }} + \mathop \sum \limits_{s = 1}^{S} CF_{{pa_{s} }} } \right)^{2} } \right] \\ & = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot E\left[ {\left( {\mathop \sum \limits_{{a_{d} \in AS}} CF_{{a_{d} }} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) + \mathop \sum \limits_{s = 1}^{S} CF_{{pa_{s} }} } \right)^{2} } \right] \\ & = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \\ & \quad \cdot E\left[ {\left( {\mathop \sum \limits_{{a_{d} \in AS}} CF_{{{\text{a}}_{\text{d}} }} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right)} \right)^{2} + 2\left( {\mathop \sum \limits_{{a_{d} \in AS}} CF_{{a_{d} }} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right)} \right)\left( {\mathop \sum \limits_{s = 1}^{S} CF_{{pa_{s} }} } \right) + \left( {\mathop \sum \limits_{s = 1}^{S} CF_{{pa_{s} }} } \right)^{2} } \right] \\ & = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \\ & \quad \cdot E\left[ \begin{gathered} \mathop \sum \limits_{{a_{d} \in AS}} CF_{{a_{d} }}^{2}\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) + \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} CF_{{a_{d} }} \cdot CF_{{a_{j} }} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right)\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{j} } \right) \hfill \\ \quad + 2\mathop \sum \limits_{{a_{d} \in AS}} \mathop \sum \limits_{s = 1}^{S} CF_{{a_{d} }} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) \cdot CF_{{pa_{s} }} + \mathop \sum \limits_{s = 1}^{S} CF_{{pa_{s} }}^{2} + 2\mathop \sum \limits_{s = 1}^{S - 1} \mathop \sum \limits_{j = s + 1}^{S} CF_{{pa_{s} }} \cdot CF_{{pa_{j} }} \hfill \\ \end{gathered} \right] \\ & = - E\left[ {CF_{P} } \right]^{2} \\ & \quad + \mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \left( {\mathop \sum \limits_{{a_{d} \in AS}} E\left[ {CF_{{a_{d} }}^{2} } \right]\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }}^{2} } \right]} \right. \\ & \quad + \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} E\left[ {CF_{{a_{d} }} \cdot CF_{{a_{j} }} } \right]\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right)\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{j} } \right) + 2\mathop \sum \limits_{{a_{d} \in AS}} \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{a_{d} }} \cdot CF_{{pa_{s} }} } \right]\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) \\ & \quad \left. { + 2\mathop \sum \limits_{s = 1}^{S - 1} \mathop \sum \limits_{j = s + 1}^{S} E\left[ {CF_{{pa_{s} }} \cdot CF_{{pa_{j} }} } \right]} \right) \\ & = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{{a_{d} \in AS}} E\left[ {CF_{{a_{d} }}^{2} } \right] \cdot \left( {\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) } \right) + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }}^{2} } \right] \cdot \underbrace {{\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} }}_{ = 1} \\ & \quad + \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} E\left[ {CF_{{a_{d} }} \cdot CF_{{a_{j} }} } \right] \cdot \left( {\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right)\cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{j} } \right) } \right) \\ & \quad + 2\mathop \sum \limits_{{a_{d} \in AS}} \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{a_{d} }} \cdot CF_{{pa_{s} }} } \right] \cdot \left( {\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} \cdot {\hbox{I\!I}}_{{AS_{k} }} \left( {a_{d} } \right) } \right) + 2\mathop \sum \limits_{s = 1}^{S - 1} \mathop \sum \limits_{j = s + 1}^{S} E\left[ {CF_{{pa_{s} }} \cdot CF_{{pa_{j} }} } \right] \cdot \underbrace {{\mathop \sum \limits_{k = 1}^{{\left| {\Omega } \right|}} p_{k} }}_{ = 1} \\ &\quad = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{{a_{d} \in AS}} E\left[ {CF_{{a_{d} }}^{2} } \right] \cdot Pr\left( {a_{d} } \right) \\ & \quad + \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{pa_{s} }}^{2} } \right] + \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} E\left[ {CF_{{a_{d} }} \cdot CF_{{a_{j} }} } \right] \cdot Pr\left( {a_{d} ,a_{j} } \right) + 2\mathop \sum \limits_{{a_{d} \in AS}} \mathop \sum \limits_{s = 1}^{S} E\left[ {CF_{{a_{d} }} \cdot CF_{{pa_{s} }} } \right] \cdot Pr\left( {a_{d} } \right) \\ & \quad + 2\mathop \sum \limits_{s = 1}^{S - 1} \mathop \sum \limits_{j = s + 1}^{S} E\left[ {CF_{{pa_{s} }} \cdot F_{{pa_{j} }} } \right] \\ & = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{{a_{d} \in AS}} \left( {Var\left[ {CF_{{a_{d} }} } \right] + E\left[ {CF_{{a_{d} }} } \right]^{2} } \right) \cdot Pr\left( {a_{d} } \right) + \mathop \sum \limits_{s = 1}^{S} \left( {Var\left[ {CF_{{pa_{s} }} } \right] + E\left[ {CF_{{pa_{s} }} } \right]^{2} } \right) \\ & \quad + \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} \left( {Cov\left[ {CF_{{a_{d} }} ,CF_{{a_{j} }} } \right] + E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{a_{j} }} } \right]} \right) \cdot Pr\left( {a_{d} ,a_{j} } \right) \\ & \quad + 2\mathop \sum \limits_{{a_{d} \in AS}} \mathop \sum \limits_{s = 1}^{S} \left( {Cov\left[ {CF_{{a_{d} }} ,CF_{{pa_{s} }} } \right] + E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{pa_{s} }} } \right]} \right) \cdot Pr\left( {a_{d} } \right) \\ & \quad + 2\mathop \sum \limits_{s = 1}^{S - 1} \mathop \sum \limits_{j = s + 1}^{S} \left( {Cov\left[ {CF_{{pa_{s} }} ,CF_{{pa_{j} }} } \right] + E\left[ {CF_{{pa_{s} }} } \right]E\left[ {CF_{{pa_{j} }} } \right]} \right) \\ & = - E\left[ {CF_{P} } \right]^{2} + \mathop \sum \limits_{{a_{d} \in AS}} \left( {Var\left[ {CF_{{a_{d} }} } \right] + E\left[ {CF_{{a_{d} }} } \right]^{2} } \right) \cdot Pr\left( {a_{d} } \right) + \mathop \sum \limits_{s = 1}^{S} \left( {Var\left[ {CF_{{pa_{s} }} } \right] + E\left[ {CF_{{pa_{s} }} } \right]^{2} } \right) \\ & \quad + \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} \left( {\rho_{{a_{d} ,a_{j} }} \cdot \sigma_{{a_{d} }} \cdot \sigma_{{a_{j} }} + E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{a_{j} }} } \right]} \right) \cdot Pr\left( {a_{d} ,a_{j} } \right) \\ & \quad + 2\mathop \sum \limits_{{a_{d} \in AS}} \mathop \sum \limits_{s = 1}^{S} \left( {\rho_{{a_{d} ,pa_{s} }} \cdot \sigma_{{a_{d} }} \cdot \sigma_{{pa_{s} }} + E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{pa_{s} }} } \right]} \right) \cdot Pr\left( {a_{d} } \right) \\ & \quad + 2\mathop \sum \limits_{s = 1}^{S - 1} \mathop \sum \limits_{j = s + 1}^{S} \left( {\rho_{{pa_{s} ,pa_{j} }} \cdot \sigma_{{pa_{s} }} \cdot \sigma_{{pa_{j} }} + E\left[ {CF_{{pa_{s} }} } \right]E\left[ {CF_{{pa_{j} }} } \right]} \right) \\ \end{aligned}$$

Appendix 5: Probability of each action in process PR

In order to determine the expected value of $CF_{PR}$ we first need to determine the probability of each action. This is:

$$\begin{aligned} Pr\left( {a_{1} } \right) &= 0.9 + 0.09 + 0.009 + \cdots = 0.9 \cdot \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = 1, \hfill \\ Pr\left( {a_{2}^{\left( 1 \right)} } \right) &= 0.9 + 0.09 + 0.009 + \cdots = 0.9 \cdot \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = 1, \hfill \\ Pr\left( {a_{2}^{\left( 2 \right)} } \right) &= 0.09 + 0.009 + 0.0009 + \cdots = 0.09 \cdot \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = 0.1, \hfill \\ &\qquad \ldots , \hfill \\ Pr\left( {a_{3}^{\left( 1 \right)} } \right) &= 0.9 + 0.09 + 0.009 + \cdots = 0.9 \cdot \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = 1, \hfill \\ Pr\left( {a_{3}^{\left( 2 \right)} } \right) &= 0.09 + 0.009 + 0.0009 + \cdots = 0.09 \cdot \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = 0.1, \hfill \\ &\qquad \ldots , \hfill \\ Pr\left( {a_{4}^{\left( 1 \right)} } \right) &= 0.09 + 0.009 + 0.0009 + \cdots = 0.09 \cdot \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = 0.1, \hfill \\ &\qquad \ldots . \hfill \\ \end{aligned}$$

Thus, it is for example

$$\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{2}^{\left( i \right)} } \right) = \mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} = \frac{1}{1 - 0.1} = \frac{10}{9},$$

which is multiplied with $E\left[ {CF_{{a_{2} }} } \right]$ since it is $E\left[ {CF_{{a_{2}^{\left( i \right)} }} } \right] = E\left[ {CF_{{a_{2} }} } \right]$ for all $i \in {\mathbb{N}}$.

Appendix 6: Details to determine the variance of $\varvec{CF}_{{\varvec{PR}}}$

In order to determine the variance of $CF_{PR}$ with expression (21) it is necessary to calculate $\mathop \sum \nolimits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{a_{j} }} } \right] \cdot Pr\left( {a_{d} ,a_{j} } \right)$. Hence, we need to determine the probabilities $Pr\left( {a_{d} ,a_{j} } \right)$. According to expression (15) the process paths and the respective path probabilities need to be calculated. For example there are the process paths

$$\begin{aligned}& pp_{1} : a_{1} , a_{2}^{\left( 1 \right)} , a_{3}^{\left( 1 \right)} ;\hfill \\ &pp_{2} : a_{1} , a_{2}^{\left( 1 \right)} , a_{3}^{\left( 1 \right)} , a_{4}^{\left( 1 \right)} , a_{2}^{\left( 2 \right)} , a_{3}^{\left( 2 \right)}; \hfill \\ & pp_{3} : a_{1} , a_{2}^{\left( 1 \right)} , a_{3}^{\left( 1 \right)} , a_{4}^{\left( 1 \right)} , a_{2}^{\left( 2 \right)} , a_{3}^{\left( 2 \right)} , a_{4}^{\left( 2 \right)} , a_{2}^{\left( 3 \right)} , a_{3}^{\left( 3 \right)}; \hfill \\ & pp_{4} : a_{1} , a_{2}^{\left( 1 \right)} , a_{3}^{\left( 1 \right)} , a_{4}^{\left( 1 \right)} , a_{2}^{\left( 2 \right)} , a_{3}^{\left( 2 \right)} , a_{4}^{\left( 2 \right)} , a_{2}^{\left( 3 \right)} , a_{3}^{\left( 3 \right)} , a_{4}^{\left( 3 \right)} , a_{2}^{\left( 4 \right)} , a_{3}^{\left( 4 \right)}, \,{\text{and}} \hfill \\ & pp_{5} : a_{1} , a_{2}^{\left( 1 \right)} , a_{3}^{\left( 1 \right)} , a_{4}^{\left( 1 \right)} , a_{2}^{\left( 2 \right)} , a_{3}^{\left( 2 \right)} , a_{4}^{\left( 2 \right)} , a_{2}^{\left( 3 \right)} , a_{3}^{\left( 3 \right)} , a_{4}^{\left( 3 \right)} , a_{2}^{\left( 4 \right)} , a_{3}^{\left( 4 \right)} , a_{4}^{\left( 4 \right)} , a_{2}^{\left( 5 \right)} , a_{3}^{\left( 5 \right)} , \hfill \\ \end{aligned}$$

with $p_{1} = 0.9$; $p_{2} = 0.09$; $p_{3} = 0.009$; $p_{4} = 0.0009$, and $p_{5} = 0.00009.$ Considering this five paths Table 5 shows the probabilities $Pr\left( {a_{d} ,a_{j} } \right)$. For example, the cell in row $a_{2}^{\left( 1 \right)}$ and column $a_{1}$ gives $Pr\left( {a_{2}^{\left( 1 \right)} ,a_{1} } \right)$. Due to the fact that $Pr\left( {a_{d} ,a_{j} } \right) = Pr\left( {a_{j} ,a_{d} } \right)$ it is enough to determine values of the lower triangular table. Since it is $a_{d} \ne a_{j}$ in expression (21) and $Pr\left( {a_{d} ,a_{d} } \right) = Pr\left( {a_{d} } \right)$ the values on the diagonal do not need to be determined. The process has potentially an infinite number of paths, which means that this table does not contain all relevant probabilities. However, it displays the structure how the values change, which makes it easy to consider all probabilities $Pr\left( {a_{d} ,a_{j} } \right)$.

Table 5 Probabilities (PR a _d , a _j ) in Process PR

In Table 5, the values $Pr\left( {a_{d} ,a_{j} } \right)$ for the same actions $a_{d}$ and $a_{j}$ are encircled. For example, the values in the cells of rows $a_{3}^{\left( 1 \right)}$to $a_{3}^{\left( 5 \right)}$ and column $a_{2}^{\left( 1 \right)}$to $a_{2}^{\left( 5 \right)}$contain the values for $Pr\left( {a_{d} ,a_{j} } \right)$ considering the appearance of the actions $a_{2}$ and $a_{3}$ in the process paths $pp_{1}$ to $pp_{5}$. All of these values have to be considered when calculating $E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{a_{j} }} } \right] \cdot Pr\left( {a_{d} ,a_{j} } \right)$ in expression (21) for the actions $a_{2}$ and $a_{3}$. The different colors show areas with the same structure of the values, to know how to use the formula for a geometric series. With this it is possible to determine $\mathop \sum \nolimits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{a_{j} }} } \right]\cdot Pr\left( {a_{d} ,a_{j} } \right)$ in expression (21).

Overall it is

$$\begin{aligned} & \mathop \sum \limits_{{a_{d} ,a_{j} \in AS,a_{d} \ne a_{j} }} E\left[ {CF_{{a_{d} }} } \right]E\left[ {CF_{{a_{j} }} } \right] \cdot Pr\left( {a_{d} ,a_{j} } \right)\end{aligned}$$

$$ \begin{aligned} =\underbrace {2}_{{due\, to\, Pr\left( {a_{d} ,a_{j} } \right) = Pr\left( {a_{j} ,a_{d} } \right)}} \cdot \left[ {\underbrace {{\mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{2}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{1} }} } \right] \cdot Pr\left( {a_{2}^{\left( i \right)} ,a_{1} } \right)}}_{grey\,dashed}} \right. + \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} E\left[ {CF_{{a_{2}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{2}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{dark \,grey} \\ & \underbrace {{ + \mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{3}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{1} }} } \right] \cdot Pr\left( {a_{3}^{\left( i \right)} ,a_{1} } \right)}}_{grey\,dashed} + \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{3}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( 1 \right)} }} } \right] \cdot Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( 1 \right)} } \right)}}_{grey\,dashed} \\ & \quad+ \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 2}^{i} E\left[ {CF_{{a_{3}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{dark \,grey} \\ & \quad+ \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } \mathop \sum \limits_{j = i + 1}^{\infty } E\left[ {CF_{{a_{3}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{black} \\ & \quad+ \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} E\left[ {CF_{{a_{3}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{3}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{3}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{dark \,grey} \\ & \quad + \underbrace {{\sum\limits_{i = 1}^{\infty } {E\left[ {CF_{{a_{4}^{(i)} }} } \right]E\left[ {CF_{{a_{1} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{1} } \right)} }}_{white} + \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( 1 \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( 1 \right)} } \right)}}_{white} \\ & \quad+ \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( 2 \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( 2 \right)} } \right)}}_{white} + \underbrace {{\sum\limits_{i = 2}^{\infty } {\mathop \sum \limits_{j = 3}^{i + 1} E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)} }}_{midium\,dark\,grey} \\ &\quad + \underbrace {{\sum\limits_{i = 2}^{\infty } {\mathop \sum \limits_{j = i + 2}^{\infty } E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{2}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)} }}_{light\,grey} + \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{3}^{\left( 1 \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( 1 \right)} } \right)}}_{white} \\ & \quad+ \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{3}^{\left( 2 \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( 2 \right)} } \right)}}_{white} + \underbrace {{\sum\limits_{i = 2}^{\infty } {\mathop \sum \limits_{j = 3}^{i + 1} E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{3}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)} }}_{midium\,dark\,grey} \\ &\quad + \underbrace {{\sum\limits_{i = 2}^{\infty } {\mathop \sum \limits_{j = i + 2}^{i + 1} E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{3}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)} }}_{light\,grey} + \underbrace {{\sum\limits_{i = 2}^{\infty } {\mathop \sum \limits_{j = 3}^{i - 1} E\left[ {CF_{{a_{4}^{\left( i \right)} }} } \right]E\left[ {CF_{{a_{4}^{\left( j \right)} }} } \right] \cdot Pr\left( {a_{4}^{\left( i \right)} ,a_{4}^{\left( j \right)} } \right)} }}_{midium\,dark\,grey} \end{aligned}$$

$$ \begin{aligned} = 2 \cdot \left[ {\underbrace {{E\left[ {CF_{{a_{2} }} } \right]E\left[ {CF_{{a_{1} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{2}^{\left( i \right)} ,a_{1} } \right)}}_{grey\,dashed}} \right. + \underbrace {{E\left[ {CF_{{a_{2} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} Pr\left( {a_{2}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{dark \,grey} \\ & \quad+ \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{1} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{3}^{\left( i \right)} ,a_{1} } \right)}}_{grey\,dashed} + \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( 1 \right)} } \right)}}_{grey\,dashed} \\ & \quad+ \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 2}^{i} Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{dark \,grey} + \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } \mathop \sum \limits_{j = i + 1}^{\infty } Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{black} \\ &\quad + \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} Pr\left( {a_{3}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{dark \,grey} \\ & \quad + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{1} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{1} } \right)}}_{white} + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( 1 \right)} } \right)}}_{white} \\ & \quad + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( 2 \right)} } \right)}}_{white} + \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 3}^{i + 1} Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{midium\,dark \,grey} \\ & \quad + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = i + 2}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{light\,grey} + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( 1 \right)} } \right)}}_{white} \\ & \quad + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( 2 \right)} } \right)}}_{white} + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{4} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 3}^{i +1} Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{midium\,dark \,grey} \\ &\quad + \left. \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = i + 2}^{\infty } Pr\left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{light \,grey} + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{4} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = i + 2}^{i - 1} Pr\left( {a_{4}^{\left( i \right)} ,a_{4}^{\left( j \right)} } \right)}}_{midium\,dark \,grey} \right]\\ & \qquad = 2 \cdot \left[ {\underbrace {{E\left[ {CF_{{a_{2} }} } \right]E\left[ {CF_{{a_{1} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{2}^{\left( i \right)} ,a_{1} } \right)}}_{grey\,dashed}} \right. + \underbrace {{E\left[ {CF_{{a_{2} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} Pr\left( {a_{2}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{dark \,grey} \\ &\quad+ \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{1} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{3}^{\left( i \right)} ,a_{1} } \right)}}_{grey\,dashed} \\ &\quad + E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\left( {\underbrace {{\mathop \sum \limits_{i = 1}^{\infty } Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( 1 \right)} } \right)}}_{grey\,dashed} + \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 2}^{i} Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{dark\,grey} + \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } \mathop \sum \limits_{j = i + 1}^{\infty } Pr\left( {a_{3}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{black}} \right) \\ & \quad+ \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} Pr\left( {a_{3}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{dark\,grey} \\ &\quad + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{1} }} } \right]\mathop \sum \limits_{i = 1}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{1} } \right)}}_{white} \\ & \quad + E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\left( {\underbrace {{\mathop \sum \limits_{i = 1}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( 1 \right)} } \right)}}_{white} + \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( 2 \right)} } \right)}}_{white} + \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 3}^{i + 1} Pr \left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{midium\,dark\,grey} + \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = i + 2}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{2}^{\left( j \right)} } \right)}}_{light\,grey}} \right) \\ &\quad+ E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\left( {\underbrace {{\mathop \sum \limits_{i = 1}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( 1 \right)} } \right)}}_{white} + \underbrace {{\mathop \sum \limits_{i = 1}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( 2 \right)} } \right)}}_{white} + \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 3}^{i + 1} Pr \left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{midium\,dark\,grey} + \underbrace {{\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = i + 2}^{\infty } Pr \left( {a_{4}^{\left( i \right)} ,a_{3}^{\left( j \right)} } \right)}}_{light\,grey}} \right) \\ & \left. { + \underbrace{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{4} }} } \right] {\mathop \sum \limits_{i = 2}^{\infty } \mathop \sum \limits_{j = 1}^{i - 1} Pr \left( {a_{4}^{\left( i \right)} ,a_{4}^{\left( j \right)} } \right)}}_{midium\,dark\,grey}} \right]\end{aligned}$$

$$\begin{aligned} = 2 \cdot \left[ {\underbrace {{E\left[ {CF_{{a_{2} }} } \right]E\left[ {CF_{{a_{1} }} } \right] \cdot \frac{10}{9}}}_{greydashed}} \right. + \underbrace {{E\left[ {CF_{{a_{2} }} } \right]E\left[ {CF_{{a_{2} }} } \right] \cdot \frac{1}{9}\underbrace {{\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{\frac{10}{9}}}}_{dark\,grey} \\ &\quad + \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{1} }} } \right] \cdot \frac{10}{9}}}_{grey\,dashed} + E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\left( {\underbrace {\frac{10}{9}}_{grey\,dashed} + \underbrace {{\frac{1}{9}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{dark \,grey} + \underbrace {{\frac{1}{9}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{black}} \right) \\ & \quad+ \underbrace {{E\left[ {CF_{{a_{3} }} } \right]E\left[ {CF_{{a_{3} }} } \right] \cdot \frac{1}{9}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{dark \,grey} \\ & \quad+ \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{1} }} } \right] \cdot \frac{1}{9}}}_{white} + E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{2} }} } \right]\left( {\underbrace {\frac{1}{9}}_{white} + \underbrace {\frac{1}{9}}_{white} + \underbrace {{\frac{1}{90}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{midium\,dark \,grey} + \underbrace {{\frac{1}{90}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{light \,grey}} \right) \\ &\quad + E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{3} }} } \right]\left( {\underbrace {\frac{1}{9}}_{white} + \underbrace {\frac{1}{9}}_{white} + \underbrace {{\frac{1}{90}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{midium\,dark \,grey} + \underbrace {{\frac{1}{90}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{light \,grey}} \right)\left. { + \underbrace {{E\left[ {CF_{{a_{4} }} } \right]E\left[ {CF_{{a_{4} }} } \right] \cdot \frac{1}{90}\mathop \sum \limits_{i = 0}^{\infty } 0.1^{i} }}_{midium\,dark \,grey}} \right] \end{aligned}$$

$$\begin{aligned}= 2 \cdot \left[ {\underbrace {{500 \cdot 1{,}000 \cdot \frac{10}{9}}}_{grey\,dashed}} \right. + \underbrace {{500 \cdot 500 \cdot \frac{10}{81}}}_{dark \,grey} \\ & \quad+ \underbrace {{500 \cdot 1{,}000 \cdot \frac{10}{9}}}_{grey\,dashed} + 500 \cdot 500\left( {\underbrace {\frac{10}{9}}_{grey\,dashed} + \underbrace {\frac{10}{81}}_{dark \,grey} + \underbrace {\frac{10}{81}}_{black}} \right) + \underbrace {{500 \cdot 500 \cdot \frac{10}{81}}}_{dark \,grey} \\ & \quad+ \underbrace {{5{,}000 \cdot 1{,}000 \cdot \frac{1}{9}}}_{white} + 5{,}000 \cdot 500\left( {\underbrace {\frac{1}{9}}_{white} + \underbrace {\frac{1}{9}}_{white} + \underbrace {\frac{1}{81}}_{midium\,dark \,grey} + \underbrace {\frac{1}{81}}_{light \,grey}} \right) + 5{,}000 \\ &\quad \cdot 500\left( {\underbrace {\frac{1}{9}}_{white} + \underbrace {\frac{1}{9}}_{white} + \underbrace {\frac{1}{81}}_{midium\,dark \,grey} + \underbrace {\frac{1}{81}}_{light \,grey}} \right)\left. { + \underbrace {{5{,}000 \cdot 5{,}000 \cdot \frac{1}{81}}}_{midium\, dark \,grey}} \right] \\ & = 2 \cdot \left[ {500 \cdot 1{,}000 \cdot \frac{10}{9}} \right. + 500 \cdot 500 \cdot \frac{10}{81} \\ & \quad+ 500 \cdot 1{,}000 \cdot \frac{10}{9} + 500 \cdot 500 \cdot \frac{110}{81} + 500 \cdot 500 \cdot \frac{10}{81} \\ &\quad + 5{,}000 \cdot 1{,}000 \cdot \frac{1}{9} + 5{,}000 \cdot 500 \cdot \frac{20}{81} + 5{,}000 \cdot 500 \cdot \frac{20}{81}\left. { + 5{,}000 \cdot 5{,}000 \cdot \frac{1}{81}} \right] \\ \end{aligned}$$