Planning and Scheduling Business Processes under Control-Flow Uncertainty

# Planning and Scheduling Business Processes under Control-Flow Uncertainty
Source: [https://arxiv.org/html/2609.05578](https://arxiv.org/html/2609.05578)
Stefanie Rinderle\-Ma[https://orcid.org/0000-0001-5656-6108](https://orcid.org/0000-0001-5656-6108)Affiliation:Technical University of Munich, Germany TUM School of Computation, Information, and TechnologyE\-mail[\{michel\.kunkler,stefanie\.rinderle\-ma\}@tum\.de](mailto:{michel.kunkler,stefanie.rinderle-ma}@tum.de)

###### Abstract

Scheduling activities in business processes can improve efficiency \(e\.g\., reduce makespan\), but is challenging because the exact sequence of activities required to complete a case is often uncertain due to decisions based on data that emerges during execution\. Nevertheless, probabilistic information regarding such decisions can often be estimated or derived from historical execution logs, and can help anticipate which execution paths are likely to lead to successful completion\. Planning with particular execution paths affects feasibility, i\.e\., the probability of successful completion, and the expected number of superfluous activities that are planned but never executed\. We frame the problem as a chance\-constrained optimization problem and present two formulations: A decomposed approach with two stages, a planning stage that minimizes the expected number of superfluous activities subject to a feasibility constraint, and a scheduling stage that minimizes the makespan over the planned activities; and an integrated approach that combines planning and scheduling into a single formulation\. Evaluation on two real\-world and one synthetic dataset shows that the integrated approach yields superior makespans but is intractable at scale, while the decomposed approach scales to large settings\.

###### Keywords:

Business Processes Chance Constraints Scheduling

## 1Introduction

Scheduling the execution of activities in business processes can improve operational efficiency, e\.g\., by reducing the makespan of cases and improving resource utilization\. Moreover, schedules provide visibility into the future, which can help resources prepare for their upcoming activities and identify capacity bottlenecks, thereby supporting management decisions regarding business process improvements\[[1](https://arxiv.org/html/2609.05578#bib.bib1)\]\. Formalizing and solving an optimization problem for scheduling the execution of business process activities is challenging for several reasons: high computational complexity, modeling complexity from control\-flow or resource constraints, and the degree of uncertainty in business process execution\[[9](https://arxiv.org/html/2609.05578#bib.bib5)\]\. Current work on scheduling activities of business processes has primarily addressed setting up computationally tractable optimization models and capturing a rich set of control\-flow and resource constraints\[[13](https://arxiv.org/html/2609.05578#bib.bib6),[12](https://arxiv.org/html/2609.05578#bib.bib7)\], or addressed uncertain activity durations\[[6](https://arxiv.org/html/2609.05578#bib.bib2)\]and resource availability\[[8](https://arxiv.org/html/2609.05578#bib.bib3),[11](https://arxiv.org/html/2609.05578#bib.bib4)\]\.

A key driver of uncertainty in business process execution is that the sequence of activities required to successfully complete a case is often not known with certainty beforehand, e\.g\., due to runtime constraints or decisions based on data revealed during execution\. Consequently, the exact set of activities that must be scheduled for a case is not known in advance\. While this issue has been acknowledged\[[9](https://arxiv.org/html/2609.05578#bib.bib5),[6](https://arxiv.org/html/2609.05578#bib.bib2)\]and termedcontrol\-flow induced uncertainty\[[9](https://arxiv.org/html/2609.05578#bib.bib5)\], existing work has either focused on scheduling only until the course of the business process can be determined with high certainty\[[6](https://arxiv.org/html/2609.05578#bib.bib2)\]or adopted online resource allocation approaches\[[8](https://arxiv.org/html/2609.05578#bib.bib3),[11](https://arxiv.org/html/2609.05578#bib.bib4)\]\. Both strategies undermine some of the benefits of scheduling, e\.g\., by preventing resources from preparing for their upcoming activities\.

In this work, we address the problem of planning and scheduling the activities of cases under control\-flow induced uncertainty until case completion\. We assume that a schedule for a case is feasible if the scheduled activities are sufficient to complete it\. Moreover, we assume that any unnecessarily scheduled activity can be skipped\. We refer to such activities assuperfluous\. The key idea of this work is to formulate activity selection as achance\-constrained optimization problem, in which activities are selected such that a predefined case feasibility threshold is satisfied while minimizing an optimization objective\. We present two approaches\. In thei\) decomposed approach, we first determine the set of activities to be scheduled by solving a planning problem that minimizes the expected number of superfluous activities subject to a feasibility constraint; in the second step, the planned activities are scheduled to minimize the makespan, i\.e\., the completion time of the final activity\. In theii\) integrated approach, planning and scheduling are combined into a single formulation that directly minimizes the makespan subject to a feasibility constraint\. The decomposed approach allows exclusive activities to overlap, since at most one will actually execute\. The integrated approach also allows non\-exclusive activities to overlap, which can be beneficial when their joint execution probability is low, keeping the impact on feasibility minor\.

An exemplary business process with control\-flow uncertainty is shown in Fig\.[1](https://arxiv.org/html/2609.05578#S1.F1)\. It depicts a repair center with an exclusive choice \(with two branches and their branching probabilitiesp1,1p\_\{1,1\}andp1,2p\_\{1,2\}\) and a loop \(with redo probabilityq1q\_\{1\}\)\. Below the process model, an example schedule is shown for two cases and three resources \(one per swim lane\)\. In this schedule, theRepairandQuality Control \(QC\)activities are planned twice for each case\. Since the probability that both cases require a second loop iteration is low, these activities are scheduled to overlap, which can be achieved by the proposed integrated approach\.

Figure 1:Exemplary schedule for two cases with overbooking of unlikely activities with a feasibility ofpcf≈0\.9295p\_\{\\text\{cf\}\}\\approx 0\.9295and an expected number of feasible cases𝔼fc≈1\.9283\\mathbb\{E\}\_\{\\text\{fc\}\}\\approx 1\.9283\.We formalize both approaches, provide constraint programming implementations and evaluate them on two real\-world and one synthetic dataset\. The results show that the integrated approach yields superior makespans but is intractable at scale, while the decomposed approach scales to large settings\.

The remainder of this paper is structured as follows\. Sects\.[2](https://arxiv.org/html/2609.05578#S2)and[3](https://arxiv.org/html/2609.05578#S3)present the decomposed and integrated approaches, respectively\. Sect\.[4](https://arxiv.org/html/2609.05578#S4)evaluates both approaches, Sect\.[5](https://arxiv.org/html/2609.05578#S5)discusses related work, and Sect\.[6](https://arxiv.org/html/2609.05578#S6)concludes\.

## 2Decomposed Planning and Scheduling

In this section, we formalize the decomposed approach for planning and scheduling activities in business processes under control\-flow induced uncertainty\. The approach employs stochastic process trees \(SPT\) as introduced in\[[2](https://arxiv.org/html/2609.05578#bib.bib8)\]due to the decomposition property of the underlying process trees\[[14](https://arxiv.org/html/2609.05578#bib.bib12)\]and the augmentation with stochastic information\. We reformulate the notation by\[[2](https://arxiv.org/html/2609.05578#bib.bib8)\]because our approach relies on a tree traversal, which requires 1\) a unique identifier for each node and 2\) a top node\.

###### Definition 1\(Stochastic Process Tree\)

LetTTbe a stochastic process tree which consists of sets of operators \(sequential→\\rightarrow, parallel\+\+, exclusive choices×\\times, redo loops↻\\circlearrowright\), observable activitiesAct\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}, silent transitionsτ\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\tau$\}\}\}and a designated root⊤\\top\. Let⋆\\stardenote the tree’s nodes, i\.e\.,⋆:=\+∪×∪↻∪→∪Act∪τ∪\{⊤\}\\star:=\\hbox to12\.34pt\{\\vbox to12\.34pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.16995pt\\lower\-6\.16995pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.26 0 L 0 8\.26 L \-8\.26 0 L 0 \-8\.26 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-5\.96996pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.26\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\\cup\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}~\\cup\\circlearrowright\\cup\\rightarrow\\cup~\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}~\\cup~\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\tau$\}\}\}~\\cup\\\{\\top\\\}\. The root⊤\\tophas exactly one child,↻\\circlearrowrightnodes have exactly two ordered children \(do\-partc​h1ch\_\{1\}, redo\-partc​h2ch\_\{2\}\), the other operator nodes \(→\\rightarrow,\+\+,×\\times\) have at least two children and the activity nodes \(Act\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}andτ\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\tau$\}\}\}\) are leaf nodes\. Letpi,j∈\[0,1\]p\_\{i,j\}\\in\[0,1\]denote the transition probability of branchjjfrom an exclusive choice×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}, with∑jpi,j=1\\sum\_\{j\}p\_\{i,j\}=1, andqi∈\[0,1\)q\_\{i\}\\in\[0,1\)the redo probability of a redo loop↻i\\circlearrowright\_\{i\}\. We assume stationarity, i\.e\.,pi,jp\_\{i,j\}andqiq\_\{i\}remain constant over loop iterations\.

We define the following selector functions on a node⋆i\\star\_\{i\}ofTT\. SinceTTis an ordered tree, children of every node are indexed from left to right, starting with 1: Letch\(⋆i\)ch\(\\star\_\{i\}\)denote the set of children of⋆i\\star\_\{i\}; letchj\(⋆i\)ch\_\{j\}\(\\star\_\{i\}\)denote thejj\-th child of⋆i\\star\_\{i\}, letpar\(⋆i\)par\(\\star\_\{i\}\)denote the parent node of⋆i\\star\_\{i\}; letbr\(⋆i\)br\(\\star\_\{i\}\)denote the branch index of⋆i\\star\_\{i\}within its parent; letanc\(⋆i\)anc\(\\star\_\{i\}\)denote the set of strict ancestors of⋆i\\star\_\{i\}, letdesc\(⋆i\)desc\(\\star\_\{i\}\)denote the descendants of⋆i\\star\_\{i\}\(including⋆i\\star\_\{i\}itself\); and letls\(⋆i\)ls\(\\star\_\{i\}\)denote the left siblings of⋆i\\star\_\{i\}\. For two activity nodesActi\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{i\}andActj\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{j\}, we writeActi<Actj\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{i\}<\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{j\}ifActi\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{i\}is to the left ofActj\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{j\}in the process tree, i\.e\.,Acti<Actj⇔Acti∈⋃k∈a​n​c​\(Actj\)∪\{Actj\}⋃m∈l​s​\(k\)d​e​s​c​\(m\)\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{i\}<\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{j\}\\iff\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{i\}\\in\\bigcup\_\{k\\in anc\(\\mathord\{\\framebox\{\\makebox\[14\.35005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{j\}\)\\cup\\\{\\mathord\{\\framebox\{\\makebox\[14\.35005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\_\{j\}\\\}\}\\bigcup\_\{m\\in ls\(k\)\}desc\(m\)\.

The stochastic process tree for the business process in Fig\.[1](https://arxiv.org/html/2609.05578#S1.F1)is shown in Fig\.[2](https://arxiv.org/html/2609.05578#S2.F2)\.

\{forest\}Figure 2:Stochastic process tree for the repair process\.Given a stochastic process tree, we first solve a chance\-constrained planning problem that selects exclusive\-choice branches and loop iteration counts subject to a predefined case feasibility threshold\. Because superfluous activities occupy resources at runtime without contributing to the process outcome, we minimize their expected number to reduce unnecessary resource contention in the subsequent scheduling stage\. The planned activities are then scheduled to directly minimize the makespan, exploiting mutual exclusivity to allow overlapping resource assignments\. We develop both stages first for a single case and then extend them to the multi\-case setting\.

### 2\.1The Planning Problem

Given a stochastic process tree, we define the decision variables for branch selection and loop unrolling, derive the resulting case feasibility and expected number of superfluous activities, and formulate the planning problem as a chance\-constrained optimization problem\.

###### Definition 2\(Decision Variables Planning Problem\)

For each exclusive choice node×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}, letxi,j∈\{0,1\}x\_\{i,j\}\\in\\\{0,1\\\}denote whether branchjjis to be planned\. For each loop node↻i\\circlearrowright\_\{i\}, letri∈ℕ\+r\_\{i\}\\in\\mathbb\{N\}^\{\+\}denote the number of planned loop iterations, yieldingrir\_\{i\}executions of the do\-part andri−1r\_\{i\}\-1executions of the redo\-part\.

###### Definition 3\(Case Feasibility\)

Let T be a stochastic process tree,x,rx,rthe decision variables, and⋆i\\star\_\{i\}a node in T\. Then the node feasibilitypnf\(x,r\)\(⋆i\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\star\_\{i\}\)of⋆i\\star\_\{i\}is defined as:

pnf\(x,r\)\(⋆i\)\\displaystyle p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\star\_\{i\}\):=\{pnf\(x,r\)\(ch1\(⋆i\)\),if⋆i=⊤,∏jpnf\(x,r\)\(chj\(⋆i\)\),if⋆i∈→∪\+,∑jxi,jpi,jpnf\(x,r\)\(chj\(⋆i\)\),if⋆i∈×,\(1−qi\)pnf\(x,r\)\(ch1\(⋆i\)\)⋅1−\(qipnf\(x,r\)\(ch2\(⋆i\)\)pnf\(x,r\)\(ch1\(⋆i\)\)\)ri1−qipnf\(x,r\)\(ch2\(⋆i\)\)pnf\(x,r\)\(ch1\(⋆i\)\),if⋆i∈↻,1,if⋆i∈τ∪Act\.\\displaystyle:=\\begin\{cases\}p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\text\{ch\}\_\{1\}\(\\star\_\{i\}\)\),&\\text\{if \}\\star\_\{i\}=\\top,\\\\ \\prod\_\{j\}p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\text\{ch\}\_\{j\}\(\\star\_\{i\}\)\),&\\text\{if \}\\star\_\{i\}\\in~\\rightarrow\\cup~\\hbox to12\.34pt\{\\vbox to12\.34pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.16995pt\\lower\-6\.16995pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.26 0 L 0 8\.26 L \-8\.26 0 L 0 \-8\.26 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.06946pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.25 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\},\\\\ \\sum\_\{j\}x\_\{i,j\}\\,p\_\{i,j\}\\,p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\text\{ch\}\_\{j\}\(\\star\_\{i\}\)\),&\\text\{if \}\\star\_\{i\}\\in\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.125pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.32 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\},\\\\ \\begin\{aligned\} &\\bigl\(1\-q\_\{i\}\\bigr\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{1\}\(\\star\_\{i\}\)\)\\cdot\\\\ &\\frac\{1\-\\bigl\(q\_\{i\}p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{2\}\(\\star\_\{i\}\)\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{1\}\(\\star\_\{i\}\)\)\\bigr\)^\{r\_\{i\}\}\}\{1\-q\_\{i\}p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{2\}\(\\star\_\{i\}\)\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{1\}\(\\star\_\{i\}\)\)\}\\end\{aligned\},&\\text\{if \}\\star\_\{i\}\\in\\circlearrowright,\\\\ 1,&\\text\{if \}\\star\_\{i\}\\in\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\tau$\}\}\}\\cup\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\.\\end\{cases\}\(1\)Based on the node feasibilitypnf\(x,r\)\(⋆i\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\star\_\{i\}\), we define the case feasibilitypcf\(x,r\)p\_\{\\text\{cf\}\}^\{\(x,r\)\}as:

pcf\(x,r\):=pnf\(x,r\)​\(⊤\)p\_\{\\text\{cf\}\}^\{\(x,r\)\}:=p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\top\)\(2\)

We computepcf\(x,r\)p\_\{\\text\{cf\}\}^\{\(x,r\)\}top\-down from the root⊤\\topvia the node feasibilitypnf\(x,r\)\(⋆i\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\star\_\{i\}\), the probability that execution of⋆i\\star\_\{i\}is feasible given\(x,r\)\(x,r\)\. For an exclusive choice×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}, only selected branches contribute, since taking an unselected branch at runtime renders the execution infeasible\. For a loop↻i\\circlearrowright\_\{i\}, the do\-part executes at least once and each further iterationkkcontributes with probabilityqik−1​\(1−qi\)q\_\{i\}^\{k\-1\}\(1\-q\_\{i\}\), depending on both the do\- and redo\-parts, yieldingpnf\(x,r\)\(↻i\)=\(1−qi\)pnf\(x,r\)\(ch1\(↻i\)\)\+qipnf\(x,r\)\(ch1\(↻i\)\)∑k=2ri\(qik−2\(pnf\(x,r\)\(ch1\(↻i\)\)pnf\(x,r\)\(ch2\(↻i\)\)\)k−1\(1−qi\)\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(\\circlearrowright\_\{i\}\)=\(1\-q\_\{i\}\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{1\}\(\\circlearrowright\_\{i\}\)\)\+q\_\{i\}p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{1\}\(\\circlearrowright\_\{i\}\)\)\\sum^\{r\_\{i\}\}\_\{k=2\}\(q\_\{i\}^\{k\-2\}\(p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{1\}\(\\circlearrowright\_\{i\}\)\)p\_\{\\text\{nf\}\}^\{\(x,r\)\}\(ch\_\{2\}\(\\circlearrowright\_\{i\}\)\)\)^\{k\-1\}\(1\-q\_\{i\}\)\), whose closed form appears in Eq\.[1](https://arxiv.org/html/2609.05578#S2.E1)\.

###### Definition 4\(Superfluous Activities\)

Let T be a stochastic process tree andx,rx,rthe decision variables\. The planning countν\(x,r\)​\(p​a​r​\(a\),a\)\\nu^\{\(x,r\)\}\(par\(a\),a\)gives the number of times activityaais planned under selection\(x,r\)\(x,r\)\(wherep​a​r​\(a\)par\(a\)denotes the parent ofaa, cf\. Def\.[1](https://arxiv.org/html/2609.05578#Thmdefinition1)\)\. The expected execution countλ\(x,r\)​\(p​a​r​\(a\),a\)\\lambda^\{\(x,r\)\}\(par\(a\),a\)gives the expected number of runtime executions ofaa\. The expected number of superfluous activitiess\(x,r\)s^\{\(x,r\)\}sums the differenceν\(x,r\)−λ\(x,r\)\\nu^\{\(x,r\)\}\-\\lambda^\{\(x,r\)\}over activitiesa∈Acta\\in\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\.

ν\(x,r\)\(⋆i,⋆j\)\\displaystyle\\nu^\{\(x,r\)\}\(\\star\_\{i\},\\star\_\{j\}\):=\{1,if⋆i=⊤,xi,br\(⋆j\)ν\(x,r\)\(par\(⋆i\),⋆i\),if⋆i∈×,riν\(x,r\)\(par\(⋆i\),⋆i\),if⋆i∈↻∧br\(⋆j\)=1,\(ri−1\)ν\(x,r\)\(par\(⋆i\),⋆i\),if⋆i∈↻∧br\(⋆j\)=2,ν\(x,r\)\(par\(⋆i\),⋆i\),else\.\\displaystyle:=\\begin\{cases\}1,&\\text\{if \}\\star\_\{i\}=\\top,\\\\ x\_\{i,br\(\\star\_\{j\}\)\}\\nu^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{if \}\\star\_\{i\}\\in\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.125pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.32 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\},\\\\ r\_\{i\}\\nu^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{if \}\\star\_\{i\}\\in\\circlearrowright\\land~br\(\\star\_\{j\}\)=1,\\\\ \(r\_\{i\}\-1\)\\nu^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{if \}\\star\_\{i\}\\in\\circlearrowright\\land~br\(\\star\_\{j\}\)=2,\\\\ \\nu^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{else\.\}\\end\{cases\}\(3\)λ\(x,r\)\(⋆i,⋆j\)\\displaystyle\\lambda^\{\(x,r\)\}\(\\star\_\{i\},\\star\_\{j\}\):=\{1,if⋆i=⊤,xi,br\(⋆j\)pi,br\(⋆j\)λ\(x,r\)\(par\(⋆i\),⋆i\),if⋆i∈×,1−qiri1−qiλ\(x,r\)\(par\(⋆i\),⋆i\),if⋆i∈↻∧br\(⋆j\)=1,qi−qiri1−qiλ\(x,r\)\(par\(⋆i\),⋆i\),if⋆i∈↻∧br\(⋆j\)=2,λ\(x,r\)\(par\(⋆i\),⋆i\),else\.\\displaystyle:=\\begin\{cases\}1,&\\text\{if \}\\star\_\{i\}=\\top,\\\\ x\_\{i,br\(\\star\_\{j\}\)\}p\_\{i,br\(\\star\_\{j\}\)\}\\lambda^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{if \}\\star\_\{i\}\\in\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.125pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.32 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\},\\\\ \\frac\{1\-q\_\{i\}^\{r\_\{i\}\}\}\{1\-q\_\{i\}\}\\lambda^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{if \}\\star\_\{i\}\\in\\circlearrowright\\land~br\(\\star\_\{j\}\)=1,\\\\ \\frac\{q\_\{i\}\-q\_\{i\}^\{r\_\{i\}\}\}\{1\-q\_\{i\}\}\\lambda^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{if \}\\star\_\{i\}\\in\\circlearrowright\\land~br\(\\star\_\{j\}\)=2,\\\\ \\lambda^\{\(x,r\)\}\(par\(\\star\_\{i\}\),\\star\_\{i\}\),&\\text\{else\.\}\\end\{cases\}\(4\)s\(x,r\)\\displaystyle s^\{\(x,r\)\}:=∑a∈Act\(ν\(x,r\)​\(p​a​r​\(a\),a\)−λ\(x,r\)​\(p​a​r​\(a\),a\)\)\\displaystyle:=\\sum\_\{a\\in\\mathord\{\\framebox\{\\makebox\[13\.85985pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\}\\bigl\(\\nu^\{\(x,r\)\}\(par\(a\),a\)\-\\lambda^\{\(x,r\)\}\(par\(a\),a\)\\bigr\)\(5\)

We computeν\(x,r\)\\nu^\{\(x,r\)\}andλ\(x,r\)\\lambda^\{\(x,r\)\}bottom\-up; both depend on the activity’s exclusive choice and loop ancestors\. Forν\(x,r\)\\nu^\{\(x,r\)\}, the activity counts as planned only if every×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}ancestor is selected\. Its base value11is multiplied byrir\_\{i\}for each loop ancestor in whose do\-part it lies, and byri−1r\_\{i\}\-1for each in whose redo\-part it lies\. Forλ\(x,r\)\\lambda^\{\(x,r\)\}, the base value11is multiplied bypi,j​xi,jp\_\{i,j\}x\_\{i,j\}at each×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}ancestor and by the expected iteration count∑k=1riqik−1\\sum\_\{k=1\}^\{r\_\{i\}\}q\_\{i\}^\{k\-1\}\(do\-part\) or∑k=2riqik−1\\sum\_\{k=2\}^\{r\_\{i\}\}q\_\{i\}^\{k\-1\}\(redo\-part\) for each loop ancestor; both sums admit closed forms used in Eq\.[4](https://arxiv.org/html/2609.05578#S2.E4)\.

Fig\.[3](https://arxiv.org/html/2609.05578#S2.F3)illustrates the node feasibilities for the running example as well as the planned and expected execution counts\.

\{forest\}Figure 3:Stochastic process tree with selection\(x,r\)\(x,r\), node feasibilities inred, planned \(ν\\nu\) and expected executed \(λ\\lambda\) counts inblue\.#### Solving the Planning Problem

Based on the above, we can write the planning problem as a mixed\-integer nonlinear program \(MINLP\) \(Eq\.[6](https://arxiv.org/html/2609.05578#S2.E6)\) that minimizes the expected number of superfluous activities while satisfying a case feasibility thresholdθ∈\(0,1\]\\theta\\in\(0,1\]\.

minx,r\\displaystyle\\min\_\{x,r\}s\(x,r\)\\displaystyle s^\{\(x,r\)\}\(6\)s\.t\.\\displaystyle\\text\{s\.t\.\}pcf\(x,r\)≥θ\\displaystyle p\_\{\\text\{cf\}\}^\{\(x,r\)\}\\geq\\thetaxi,j∈\{0,1\}\\displaystyle x\_\{i,j\}\\in\\\{0,1\\\}∀×i,∀j\\displaystyle\\forall\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.125pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.32 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\},\\forall jri∈ℕ\+\\displaystyle r\_\{i\}\\in\\mathbb\{N\}^\{\+\}∀↻i\\displaystyle\\forall\\circlearrowright\_\{i\}

### 2\.2The Scheduling Problem

Given a selection\(x,r\)\(x,r\)from the planning problem, we construct a scheduling problem that minimizes the makespan of the planned activities, i\.e\., the time until the end of the last activity in a case, e\.g\., makespan=7=7for the blue case in Fig\.[1](https://arxiv.org/html/2609.05578#S1.F1)\. To this end, we unroll the process tree and remove non\-selected branches to obtain the set of activities and their precedence constraints and their mutual exclusivity\. We then identify activities that are mutually exclusive; since at most one of them will be executed at runtime, they can be assigned to the same resource concurrently\. Finally, we formulate the scheduling problem as a MINLP\.

#### Unrolled and Branch Selected Process Tree

Given a selection\(x,r\)\(x,r\), we transform the process tree by removing non\-selected branches and unrolling loops\. For everyxi,j=0x\_\{i,j\}=0, the nodec​hj​\(×i\)ch\_\{j\}\(\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}\)and its descendants are removed\. Each loop node↻i\\circlearrowright\_\{i\}is unrolled according torir\_\{i\}: ifri=1r\_\{i\}=1, it is replaced by its do\-partch1\(↻i\)ch\_\{1\}\(\\circlearrowright\_\{i\}\)\. Forri\>1r\_\{i\}\>1, it is replaced by a sequential composition beginning with the do\-part, where each subsequent iteration is an exclusive choice between a silent transitionτ\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\tau$\}\}\}\(loop exit\) and the redo\-part followed by the do\-part and the same choice recursively, terminating afterrir\_\{i\}iterations\. Any schedule must respect the precedence orders set out by the original process tree; this equally applies to the unrolled and branch\-selected process tree\. Otherwise, important functional or compliance ordering constraints would be violated, e\.g\., examining the patient before surgery\. Hence, an activity⋆i\\star\_\{i\}can only start after all its predecessorspr\(⋆i\)pr\(\\star\_\{i\}\)have completed wherepr\(⋆i\)=desc\(⋆j\)∩Actpr\(\\star\_\{i\}\)=desc\(\\star\_\{j\}\)\\cap\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}, i\.e\., all descendants of all left siblings⋆j∈ls\(⋆k\)\\star\_\{j\}\\in ls\(\\star\_\{k\}\), where⋆k\\star\_\{k\}is⋆i\\star\_\{i\}itself or any ancestor of⋆i\\star\_\{i\}withpar\(⋆k\)∈→par\(\\star\_\{k\}\)\\in\\rightarrow\.

Since at most one branch of an exclusive choice is executed at runtime, mutually exclusive activities can share a resource in overlapping time slots\.

###### Definition 5\(Mutual Exclusivity\)

Two activitiesai,aja\_\{i\},a\_\{j\}are mutually exclusive \(×​\(ai,aj\)=1\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\_\{i\},a\_\{j\}\)=1\) if there exists a common exclusive\-choice ancestor×k∈a​n​c​\(ai\)∩a​n​c​\(aj\)∩×\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{k\}\\in anc\(a\_\{i\}\)\\cap anc\(a\_\{j\}\)\\cap\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}with distinct childrenm≠nm\\neq nsuch thatai∈d​e​s​c​\(c​hm​\(×k\)\)a\_\{i\}\\in desc\(ch\_\{m\}\(\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{k\}\)\)andaj∈d​e​s​c​\(c​hn​\(×k\)\)a\_\{j\}\\in desc\(ch\_\{n\}\(\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{k\}\)\)\. The set of activities exclusive toaia\_\{i\}is×​\(ai\):=\{aj∈Act∣×​\(ai,aj\)=1\}\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\_\{i\}\):=\\\{a\_\{j\}\\in\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\\mid\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\_\{i\},a\_\{j\}\)=1\\\}\.

#### Scheduling Formulation

We formalize the scheduling problem as a joint resource\-assignment and activity\-start\-time decision problem to minimize the makespan\. For each activitya∈Acta\\in\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}, letIaI\_\{a\}be an interval variable with accessorsstart​\(Ia\)\\text\{start\}\(I\_\{a\}\),end​\(Ia\)\\text\{end\}\(I\_\{a\}\),dur​\(Ia\)\\text\{dur\}\(I\_\{a\}\), andoverlapping​\(Ia,Ia′\)∈\{0,1\}\\text\{overlapping\}\(I\_\{a\},I\_\{a^\{\\prime\}\}\)\\in\\\{0,1\\\}which equals11iffIaI\_\{a\}andIa′I\_\{a^\{\\prime\}\}share a common time point\. Letℛ\\mathcal\{R\}be the set of resources, and for each\(a,ρ\)∈Act×ℛ\(a,\\rho\)\\in\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\\times\\mathcal\{R\}, letda,ρd\_\{a,\\rho\}be the processing time ofaaonρ\\rhoandza,ρ∈\{0,1\}z\_\{a,\\rho\}\\in\\\{0,1\\\}the allocation variable\. The formulation in Eq\.[7](https://arxiv.org/html/2609.05578#S2.E7)minimizes the makespan \([7a](https://arxiv.org/html/2609.05578#S2.E7.1)\), sets the processing time per resource assignment \([7b](https://arxiv.org/html/2609.05578#S2.E7.2)\), enforces precedence \([7c](https://arxiv.org/html/2609.05578#S2.E7.3)\), ensures each activity is assigned to exactly one resource \([7d](https://arxiv.org/html/2609.05578#S2.E7.4)\), and prevents temporal overlap of non\-mutually\-exclusive activities on the same resource \([7e](https://arxiv.org/html/2609.05578#S2.E7.5)\)\.

minz,I\\displaystyle\\min\_\{z,I\}\\hskip 9\.24994ptmax⁡\(\{end​\(Ia\)∣a∈Act\}\)\\displaystyle\\max\(\\left\\\{\\text\{end\}\(I\_\{a\}\)\\mid a\\in\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\\right\\\}\)\(7a\)s\.t\.dur​\(Ia\)=∑ρ∈ℛza,ρ​da,ρ\\displaystyle\\text\{dur\}\(I\_\{a\}\)=\\sum\_\{\\rho\\in\\mathcal\{R\}\}z\_\{a,\\rho\}d\_\{a,\\rho\}∀a∈Act\\displaystyle\\forall a\\in\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\(7b\)start​\(Ia\)≥end​\(Ia′\)\\displaystyle\\text\{start\}\(I\_\{a\}\)\\geq\\text\{end\}\(I\_\{a^\{\\prime\}\}\)∀a∈Act,∀a′∈p​r​\(a\)\\displaystyle\\forall a\\in\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\},\\forall a^\{\\prime\}\\in pr\(a\)\(7c\)∑ρ∈ℛza,ρ=1\\displaystyle\\sum\_\{\\rho\\in\\mathcal\{R\}\}z\_\{a,\\rho\}=1∀a∈Act\\displaystyle\\forall a\\in\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\(7d\)za,ρ⋅za′,ρ⋅overlapping​\(Ia,Ia′\)=0\\displaystyle z\_\{a,\\rho\}\\cdot z\_\{a^\{\\prime\},\\rho\}\\cdot\\text\{overlapping\}\(I\_\{a\},I\_\{a^\{\\prime\}\}\)=0∀a,a′∈Act:a≠a′,×​\(a,a′\)=0,∀ρ∈ℛ\\displaystyle\\begin\{aligned\} &\\forall a,a^\{\\prime\}\\in\\mathord\{\\framebox\{\\makebox\[16\.64992pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}:a\\neq a^\{\\prime\},\\\\ &\\hskip 9\.24994pt\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.125pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.32 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a,a^\{\\prime\}\)=0,\\;\\;\\forall\\rho\\in\\mathcal\{R\}\\end\{aligned\}\(7e\)

### 2\.3Multi\-Case Planning and Scheduling

In practice, multiple cases must be scheduled concurrently, so we extend the decomposed approach tonncases\. The per\-case feasibilitypcfp\_\{\\text\{cf\}\}does not directly convey how many cases are expected to complete successfully; we therefore adopt the expected number of feasible cases𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}as the chance constraint\. We reuse the MINLP \(Eq\.[6](https://arxiv.org/html/2609.05578#S2.E6)\) with per\-case thresholdθ=E/n\\theta=E/n, whereEEdenotes the required expected number of feasible cases, and apply the resulting selection\(x,r\)\(x,r\)uniformly to allnncases\. Since precedence and mutual exclusivity apply within each case, the scheduling formulation extends tonncases by taking the union of thennunrolled process trees\. A schedule from the decomposed approach for the two cases from the running example is shown in Fig\.[4](https://arxiv.org/html/2609.05578#S2.F4)\.

Figure 4:Schedule generated by the decomposed approach for the running example with two cases\.A lower number of superfluous activities could in principle be achieved by per\-case planning, i\.e\., composingnncopies of the process tree in parallel, but this renders the planning problem intractable for largenn; the integrated approach in Sect\.[3](https://arxiv.org/html/2609.05578#S3)addresses this by jointly optimizing planning and scheduling\.

## 3Integrated Approach

The decomposed approach has two limitations that can affect the optimality of the resulting schedule, which we address in the integrated approach: First, minimizing superfluous activities in the planning stage does not necessarily lead to a minimal makespan in the scheduling stage, as selecting a branch with more \(superfluous\) activities can be beneficial for the makespan, e\.g\., when this avoids scheduling on a bottleneck resource\. Second, the decomposed approach only allows exclusive activities to be scheduled concurrently on the same resource\. Additionally allowing non\-exclusive activities to overlap on a resource can further shorten the makespan\. This holds especially in the multi\-case setting where it can be meaningful to, e\.g\., schedule two activities from two different cases that both have low execution probability concurrently as this will only lead to a minor decrease of𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}\(see Fig\.[1](https://arxiv.org/html/2609.05578#S1.F1)\)\. We therefore present an integrated approach that combines planning and scheduling into a single formulation, allowing per\-case selections and the overlapping scheduling of non\-exclusive activities\.

###### Definition 6\(Decision Variables Integrated Problem\)

For each caseκ∈K\\kappa\\in Kand each exclusive choice node×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}, letxi,jκ∈\{0,1\}x^\{\\kappa\}\_\{i,j\}\\in\\\{0,1\\\}indicate whether branchjjis selected forκ\\kappa\. For each loop node↻m\\circlearrowright\_\{m\}, letrmκ∈ℕ\+r^\{\\kappa\}\_\{m\}\\in\\mathbb\{N\}^\{\+\}denote the number of planned iterations, yieldingrmκr^\{\\kappa\}\_\{m\}executions of the do\-part andrmκ−1r^\{\\kappa\}\_\{m\}\-1executions of the redo\-part\.

We obtain unrolled\-and\-selected process trees, from which we directly derive precedence and mutual exclusivity relationships and compute the set of activities for a scenario and its probability\.

###### Definition 7\(Unrolled\-and\-Selected Process Tree\)

For caseκ\\kappaand selection\(xκ,rκ\)\(x^\{\\kappa\},r^\{\\kappa\}\), the unrolled\-and\-selected process treeT\(xκ,rκ\)T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}is obtained fromTTby: \(i\) deleting, for every×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}and every branchjjwithxi,jκ=0x^\{\\kappa\}\_\{i,j\}=0the subtree rooted atc​hj​\(×i\)ch\_\{j\}\(\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}\); if only one branch of×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}remains,×i\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\_\{i\}is replaced by that branch; \(ii\) replacing the do\-subtree of each loop↻m\\circlearrowright\_\{m\}by a sequential composition ofrmκr^\{\\kappa\}\_\{m\}unrolled do\-copies interleaved withrmκ−1r^\{\\kappa\}\_\{m\}\-1unrolled redo\-copies \(do, redo, do,…\\dots, redo, do\), and its redo\-subtree by a placeholderτ\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\tau$\}\}\}that is never visited at runtime\. We writev^\\hat\{v\}for nodes ofT\(xκ,rκ\)T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}andt⁡\(v^\)t\(\\hat\{v\}\)for the original\-tree node thatv^\\hat\{v\}instantiates \(undefined on the placeholderτ\\tau\)\.

Unlike the unrolling in Sect\.[2](https://arxiv.org/html/2609.05578#S2), we flatten the loop and retain↻m\\circlearrowright\_\{m\}rather than introducing exit\-×\\timess, so that every×\\timesand↻\\circlearrowrightinT\(xκ,rκ\)T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}carries the originalpi,jp\_\{i,j\}orqmq\_\{m\}\.

###### Definition 8\(Planned Activities\)

The set of planned activities under selection\(xκ,rκ\)\(x^\{\\kappa\},r^\{\\kappa\}\)is

A\(xκ,rκ\):=\{aκ,v^∣v^∈T\(xκ,rκ\),t\(v^\)∈Act\},A\(x,r\):=⋃κ∈KA\(xκ,rκ\),A^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}:=\\\{a\_\{\\kappa,\\hat\{v\}\}\\mid\\hat\{v\}\\in T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\},\\,t\(\\hat\{v\}\)\\in\\mathord\{\\framebox\{\\makebox\[18\.00005pt\]\[c\]\{\\scriptsize$\\text\{Act\}$\}\}\}\\\},\\quad A^\{\(x,r\)\}:=\\bigcup\_\{\\kappa\\in K\}A^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\},\(8\)witht⁡\(aκ,v^\):=t⁡\(v^\)t\(a\_\{\\kappa,\\hat\{v\}\}\):=t\(\\hat\{v\}\)\. For convenience, we writeaαa\_\{\\alpha\}foraκ,v^a\_\{\\kappa,\\hat\{v\}\}withα=\(κ,v^\)\\alpha=\(\\kappa,\\hat\{v\}\)\. We lift the precedence constraint in Eq\.[7c](https://arxiv.org/html/2609.05578#S2.E7.3)to planned activities byp​r​\(aκ,v^\):=\{aκ,v^′∣v^′∈p​r​\(v^\)\}pr\(a\_\{\\kappa,\\hat\{v\}\}\):=\\\{a\_\{\\kappa,\\hat\{v\}^\{\\prime\}\}\\mid\\hat\{v\}^\{\\prime\}\\in pr\(\\hat\{v\}\)\\\}, withp​rprapplied toT\(xκ,rκ\)T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}\.

#### Expected Case Feasibility with Overlapping Activities

Since the integrated formulation permits non\-exclusive activities to overlap on the same resource, case feasibility no longer depends solely on branch selection and loop unrolling: for non\-exclusive overlapping activities, multiple activities may need to execute, reducing𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}\. We formalize this by defining \(i\) an activity priority order that determines which activity runs when overlapping allocations conflict, \(ii\) the execution probabilities of overlapping activities, and \(iii\) a lower bound on the expected number of feasible cases\.

###### Definition 9\(Activity Priority Order\)

When two planned activitiesaα=aκ,v^a\_\{\\alpha\}=a\_\{\\kappa,\\hat\{v\}\}andaα′=aκ′,v^′a\_\{\\alpha^\{\\prime\}\}=a\_\{\\kappa^\{\\prime\},\\hat\{v\}^\{\\prime\}\}are allocated to overlap on the same resource, only the higher\-priority one can execute; we sayaαa\_\{\\alpha\}has higher priority thanaα′a\_\{\\alpha^\{\\prime\}\}iffaα<aα′a\_\{\\alpha\}<a\_\{\\alpha^\{\\prime\}\}, with:

aα<aα′⇔\\displaystyle a\_\{\\alpha\}<a\_\{\\alpha^\{\\prime\}\}\\iff\(κ<κ′\)∨\(κ=κ′∧aα∈p​r​\(aα′\)\)∨\\displaystyle\\Bigl\(\\kappa<\\kappa^\{\\prime\}\\Bigr\)\\;\\lor\\;\\Bigl\(\\kappa=\\kappa^\{\\prime\}\\land a\_\{\\alpha\}\\in pr\(a\_\{\\alpha^\{\\prime\}\}\)\\Bigr\)\\;\\lor\\;\(9\)\(κ=κ′∧aα′∉p​r​\(aα\)∧aα∉p​r​\(aα′\)∧v^<v^′\)\\displaystyle\\Bigl\(\\kappa=\\kappa^\{\\prime\}\\land a\_\{\\alpha^\{\\prime\}\}\\notin pr\(a\_\{\\alpha\}\)\\land a\_\{\\alpha\}\\notin pr\(a\_\{\\alpha^\{\\prime\}\}\)\\land\\hat\{v\}<\\hat\{v\}^\{\\prime\}\\Bigr\)

###### Definition 10\(Overlapping Activities\)

Two activitiesaα,aα′a\_\{\\alpha\},a\_\{\\alpha^\{\\prime\}\}are overlapping if they are allocated to the same resource and have overlapping time intervals\. The set of activities overlapping withaαa\_\{\\alpha\}given intervalsIIand resource assignmentszzis denotedO\(x,r,I,z\)​\(aα\)O^\{\(x,r,I,z\)\}\(a\_\{\\alpha\}\)\.

O\(x,r,I,z\)​\(aα\):=\{aα′∈A\(x,r\)∖\{aα\}\|overlapping​\(Iaα,Iaα′\)∧∑ρ∈ℛ\(zaα,ρ​zaα′,ρ\)=1\}\\displaystyle O^\{\(x,r,I,z\)\}\(a\_\{\\alpha\}\):=\\left\\\{a\_\{\\alpha^\{\\prime\}\}\\in A^\{\(x,r\)\}\\setminus\\\{a\_\{\\alpha\}\\\}\\;\\middle\|\\;\\begin\{aligned\} &\\text\{overlapping\}\(I\_\{a\_\{\\alpha\}\},I\_\{a\_\{\\alpha^\{\\prime\}\}\}\)\\land\\\\ &\\sum\_\{\\rho\\in\\mathcal\{R\}\}\(z\_\{a\_\{\\alpha\},\\rho\}z\_\{a\_\{\\alpha^\{\\prime\}\},\\rho\}\)=1\\end\{aligned\}\\right\\\}\(10\)

Since an activity can overlap with multiple activities from another case, and those activities may have correlated executions, we can no longer treat each overlap as an independent threat to case feasibility\. To capture the joint execution probabilities of overlapping activities, we introduce a per\-case scenario\-based approach which exploits three structural facts: \(i\) scenarios across cases are independent; \(ii\) exclusive planned activities can never be executed together; \(iii\) within each scenario, activity executions are deterministic, resolving any remaining intra\-case dependencies\.

###### Definition 11\(Execution Scenario\)

An execution scenarioω\\omegafor caseκ∈K\\kappa\\in Kis a runtime realization drawing, independently at each reached visit, a branchbi^ωb\_\{\\hat\{i\}\}^\{\\omega\}at everyi^\\hat\{i\}witht⁡\(i^\)∈×t\(\\hat\{i\}\)\\in\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}and an iteration countnm^ω∈ℕ\+n\_\{\\hat\{m\}\}^\{\\omega\}\\in\\mathbb\{N\}^\{\+\}at everym^\\hat\{m\}witht⁡\(m^\)∈↻t\(\\hat\{m\}\)\\in\\circlearrowright; let𝒳^​\(ω\)\\hat\{\\mathcal\{X\}\}\(\\omega\)andℒ^​\(ω\)\\hat\{\\mathcal\{L\}\}\(\\omega\)denote the sets of×\\timesand↻\\circlearrowrightnodes visited inω\\omega\.ω\\omegais planning\-feasible,ω∈Ωκ\(x,r\)\\omega\\in\\Omega\_\{\\kappa\}^\{\(x,r\)\}, iff all draws fit the plan:xt⁡\(i^\),bi^ωκ=1x^\{\\kappa\}\_\{t\(\\hat\{i\}\),b\_\{\\hat\{i\}\}^\{\\omega\}\}=1andnm^ω≤rt⁡\(m^\)κn\_\{\\hat\{m\}\}^\{\\omega\}\\leq r^\{\\kappa\}\_\{t\(\\hat\{m\}\)\};Ωκ\(x,r\)\\Omega\_\{\\kappa\}^\{\(x,r\)\}is finite despite the infinite scenario space\. Forω∈Ωκ\(x,r\)\\omega\\in\\Omega\_\{\\kappa\}^\{\(x,r\)\}, its probability and visited activities are:

ps​\(ω\)\\displaystyle p\_\{s\}\(\\omega\):=∏i^∈𝒳^​\(ω\)pt⁡\(i^\),bi^ω⋅∏m^∈ℒ^​\(ω\)qt⁡\(m^\)nm^ω−1​\(1−qt⁡\(m^\)\)\\displaystyle:=\\prod\_\{\\hat\{i\}\\in\\hat\{\\mathcal\{X\}\}\(\\omega\)\}p\_\{t\(\\hat\{i\}\),\\,b\_\{\\hat\{i\}\}^\{\\omega\}\}\\cdot\\prod\_\{\\hat\{m\}\\in\\hat\{\\mathcal\{L\}\}\(\\omega\)\}q\_\{t\(\\hat\{m\}\)\}^\{n\_\{\\hat\{m\}\}^\{\\omega\}\-1\}\(1\-q\_\{t\(\\hat\{m\}\)\}\)\(11\)Aκ\(x,r\)​\(ω\)\\displaystyle A\_\{\\kappa\}^\{\(x,r\)\}\(\\omega\):=\{aκ,v^∈Aκ\(x,r\)∣v^​visited in​ω\}\\displaystyle:=\\\{a\_\{\\kappa,\\hat\{v\}\}\\in A\_\{\\kappa\}^\{\(x,r\)\}\\mid\\hat\{v\}\\text\{ visited in \}\\omega\\\}

###### Definition 12\(Exclusive Planned Activities\)

We lift the mutual exclusivity relation of Eq\.[5](https://arxiv.org/html/2609.05578#Thmdefinition5)\(applied toT\(xκ,rκ\)T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}\) to planned activities; two are exclusive only if they belong to the same case:

×\(aκ,v^,aκ′,v^′\):=𝟏\[κ=κ′\]⋅×\(v^,v^′\),×\(aα\):=\{aα′∣×\(aα,aα′\)=1\}\.\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\_\{\\kappa,\\hat\{v\}\},a\_\{\\kappa^\{\\prime\},\\hat\{v\}^\{\\prime\}\}\):=\\mathbf\{1\}\[\\kappa=\\kappa^\{\\prime\}\]\\cdot\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(\\hat\{v\},\\hat\{v\}^\{\\prime\}\),\\,\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\_\{\\alpha\}\):=\\\{a\_\{\\alpha^\{\\prime\}\}\\mid\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\\text\{\}\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{\-6\.0255pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 0 \-8\.34\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\\text\{\}\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\_\{\\alpha\},a\_\{\\alpha^\{\\prime\}\}\)=1\\\}\.\(12\)

###### Definition 13\(Threats and Intra\-Case Feasibility\)

For an activity setA′⊆A\(x,r\)A^\{\\prime\}\\subseteq A^\{\(x,r\)\}, let𝒯⁡\(A′\)⊆A\(x,r\)\\mathcal\{T\}\(A^\{\\prime\}\)\\subseteq A^\{\(x,r\)\}contain the threats to activities inA′A^\{\\prime\}, i\.e\., activities that overlap with somea∈A′a\\in A^\{\\prime\}, have higher priority thanaa, and are non\-exclusive toaa\. Further, let the intra\-case feasibility indicatorχ\(x,r,I,z\)​\(κ,ω\)\\chi^\{\(x,r,I,z\)\}\(\\kappa,\\omega\)be11iff scenarioω\\omegaof caseκ\\kappacontains no internal conflict, i\.e\., no activity ofκ\\kappais threatened by another non\-exclusive activity ofκ\\kappa\. Formally:

𝒯\(x,r,I,z\)​\(A′\)\\displaystyle\\mathcal\{T\}^\{\(x,r,I,z\)\}\(A^\{\\prime\}\):=⋃a∈A′\(\{f∈O\(x,r,I,z\)​\(a\)∣f<a\}∖×​\(a\)\)\\displaystyle:=\\bigcup\_\{a\\in A^\{\\prime\}\}\\bigl\(\\\{f\\in O^\{\(x,r,I,z\)\}\(a\)\\mid f<a\\\}\\setminus\\hbox to12\.45pt\{\\vbox to12\.45pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 6\.2255pt\\lower\-6\.2255pt\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@begingroup@\{stroke=\#000000\} \\lxSVG@begingroup@\{fill=\#000000\} \\lxSVG@setlinewidth\{\\the\\pgflinewidth\}\\lxSVG@begingroup@\{stroke\-width=0\.4pt\} \\lx@inpgf@ignorespaces\\nullfont\\hbox to0\.0pt\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{ \{\{\}\}\\lx@inpgf@ignorespaces\\hbox\{\\hbox\{\{\\lxSVG@begingroup@\{\_scopebegin=1\} \{\{\}\{\}\{\{\\lx@inpgf@ignorespaces\}\}\{\\lx@inpgf@ignorespaces\} \{\}\{\}\{\}\{\}\{\}\{\{\}\\lxSVG@stroke\\lxSVG@drawpath@unclipped\{M 8\.34 0 L 0 8\.34 L \-8\.34 0 L 0 \-8\.34 Z\}\{fill:none\} \\lx@inpgf@ignorespaces \}\{\{\{\{\}\}\\lxSVG@begingroup@\{\_scopebegin=1\} \\lxSVG@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-3\.125pt\}\{\-1\.75pt\}\\lxSVG@begingroup@\{transform=matrix\(1\.0 0\.0 0\.0 1\.0 \-4\.32 \-2\.42\)\} \\pgfsys@hbox\{58\}\\lxSVG@closescope \}\}\} \\lxSVG@closescope \}\}\} \} \\lxSVG@closescope \{\{\{\}\{\}\}\}\{\\lx@inpgf@ignorespaces\}\{\\lx@inpgf@ignorespaces\}\\hss\}\\lxSVG@discardpath\\lxSVG@closescope \\hss\}\}\\lxSVG@closescope\\endpgfpicture\}\}\(a\)\\bigr\)\(13\)χ\(x,r,I,z\)​\(κ,ω\)\\displaystyle\\chi^\{\(x,r,I,z\)\}\(\\kappa,\\omega\):=\[𝒯\(x,r,I,z\)\(Aκ\(x,r\)\(ω\)\)∩Aκ\(x,r\)\(ω\)=∅\]\\displaystyle:=\\mathbf\{1\}\\\!\\left\[\\mathcal\{T\}^\{\(x,r,I,z\)\}\(A\_\{\\kappa\}^\{\(x,r\)\}\(\\omega\)\)\\cap A\_\{\\kappa\}^\{\(x,r\)\}\(\\omega\)=\\emptyset\\right\]

Building on the above definitions, we can now compute the expected number of feasible cases\.

###### Definition 14\(Expected Feasible Cases\)

For a caseκ\\kappawith threat activitiesCC, the case threat probabilitypctp\(x,r\)​\(κ′,C\)p\_\{\\text\{ctp\}\}^\{\(x,r\)\}\\allowbreak\(\\kappa^\{\\prime\},C\)is the probability that another caseκ′\\kappa^\{\\prime\}executes at least one of its own activities inCC, i\.e\., realizes one ofκ\\kappa’s threats; it uses a tail\-absorbing scenario weightp~s\\tilde\{p\}\_\{s\}that differs frompsp\_\{s\}only at the maximum loop iteration\. The overlap\-adjusted case feasibilitypcfo​\(κ\)p\_\{\\text\{cfo\}\}\(\\kappa\)is the probability that caseκ\\kappacompletes without a realized threat; it sumsps​\(ω\)p\_\{s\}\(\\omega\)over planning\-feasible scenarios, withχ⁡\(κ,ω\)\\chi\(\\kappa,\\omega\)handling intra\-case threats and a product overκ′≠κ\\kappa^\{\\prime\}\\neq\\kappahandling cross\-case threats\. The expected number of feasible cases𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}sumspcfop\_\{\\text\{cfo\}\}over cases; it is a lower bound, since a cross\-case higher\-priority activity need not be executed when its case is already infeasible\.

p~s​\(ω\)\\displaystyle\\tilde\{p\}\_\{s\}\(\\omega\):=∏i^∈𝒳^​\(ω\)pt⁡\(i^\),bi^ω⋅∏m^∈ℒ^​\(ω\)qt⁡\(m^\)nm^ω−1\(1−qt⁡\(m^\)\)𝟏\[nm^ω<rκ′t⁡\(m^\)\]\\displaystyle:=\\prod\_\{\\hat\{i\}\\in\\hat\{\\mathcal\{X\}\}\(\\omega\)\}p\_\{t\(\\hat\{i\}\),\\,b\_\{\\hat\{i\}\}^\{\\omega\}\}\\cdot\\prod\_\{\\hat\{m\}\\in\\hat\{\\mathcal\{L\}\}\(\\omega\)\}q\_\{t\(\\hat\{m\}\)\}^\{\\,n\_\{\\hat\{m\}\}^\{\\omega\}\-1\}\\,\(1\-q\_\{t\(\\hat\{m\}\)\}\)^\{\\mathbf\{1\}\[n\_\{\\hat\{m\}\}^\{\\omega\}<r^\{\\kappa^\{\\prime\}\}\_\{t\(\\hat\{m\}\)\}\]\}\(14\)pctp\(x,r\)​\(κ′,C\)\\displaystyle p\_\{\\text\{ctp\}\}^\{\(x,r\)\}\(\\kappa^\{\\prime\},C\):=∑ω∈Ωκ′\(x,r\)p~s\(ω\)⋅\[C∩Aκ′\(x,r\)\(ω\)≠∅\]\\displaystyle:=\\sum\_\{\\omega\\in\\Omega\_\{\\kappa^\{\\prime\}\}^\{\(x,r\)\}\}\\tilde\{p\}\_\{s\}\(\\omega\)\\cdot\\mathbf\{1\}\\\!\\left\[C\\cap A\_\{\\kappa^\{\\prime\}\}^\{\(x,r\)\}\(\\omega\)\\neq\\emptyset\\right\]pcfo\(K,x,r,I,z\)​\(κ\)\\displaystyle p\_\{\\text\{cfo\}\}^\{\(K,x,r,I,z\)\}\(\\kappa\):=∑ω∈Ωκ\(x,r\)ps\(ω\)⋅χ\(x,r,I,z\)\(κ,ω\)⋅∏κ′∈K∖\{κ\}\(1−pctp\(x,r\)​\(κ′,𝒯\(x,r,I,z\)​\(Aκ\(x,r\)​\(ω\)\)\)\)\\displaystyle:=\\begin\{aligned\} \\sum\_\{\\omega\\in\\Omega\_\{\\kappa\}^\{\(x,r\)\}\}&p\_\{s\}\(\\omega\)\\cdot\\,\\chi^\{\(x,r,I,z\)\}\(\\kappa,\\omega\)\\cdot\{\}\\\\ &\\prod\_\{\\kappa^\{\\prime\}\\in K\\setminus\\\{\\kappa\\\}\}\\\!\\bigl\(1\-p\_\{\\text\{ctp\}\}^\{\(x,r\)\}\(\\kappa^\{\\prime\},\\mathcal\{T\}^\{\(x,r,I,z\)\}\(A\_\{\\kappa\}^\{\(x,r\)\}\(\\omega\)\)\)\\bigr\)\\end\{aligned\}𝔼fc\(K,x,r,I,z\)\\displaystyle\\mathbb\{E\}\_\{\\text\{fc\}\}^\{\(K,x,r,I,z\)\}:=∑κ∈Kpcfo\(K,x,r,I,z\)​\(κ\)\\displaystyle:=\\sum\_\{\\kappa\\in K\}p\_\{\\text\{cfo\}\}^\{\(K,x,r,I,z\)\}\(\\kappa\)

p~s\\tilde\{p\}\_\{s\}accounts for runtime realizations that exceed the plan: if a loop requires more than itsrt⁡\(m^\)κ′r^\{\\kappa^\{\\prime\}\}\_\{t\(\\hat\{m\}\)\}planned iterations, caseκ′\\kappa^\{\\prime\}still executes all planned iterations before the infeasibility becomes apparent, and the activities executed in them may threaten other cases\.p~s\\tilde\{p\}\_\{s\}therefore adds this tail probability \(qrq^\{r\}for a loop withrrplanned iterations\) to the scenario with the maximum iteration count by omitting its final\(1−q\)\(1\-q\)factor\. Withpsp\_\{s\}instead,pctpp\_\{\\text\{ctp\}\}would understate the threat probability and𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}would not be a lower bound\.

#### The Integrated Multi\-Case Scheduling Problem Formulation

The objective is again to minimize the makespan for a given expected number of feasible cases thresholdE∈\[0,\|K\|\)E\\in\[0,\|K\|\)\. Unlike the decomposed formulation, the decision variables and constraints are not fixed a priori, as the set of planned activitiesA\(x,r\)A^\{\(x,r\)\}, their precedence relations, and the interval and assignment variables all depend on the selection\(x,r\)\(x,r\)\. The formulation is therefore stated in a declarative way over the variable\-size sets induced by\(x,r\)\(x,r\); in practice, it can be instantiated by unrolling loops to a sufficient maximum bound with indicator variables disabling unused iterations, or solved directly via constraint\-programming solvers that natively support optional and variable\-length structures\. Each loop ancestor multiplies the number of XOR and loop occurrences inT\(xκ,rκ\)T^\{\(x^\{\\kappa\},r^\{\\kappa\}\)\}, so\|Ωκ\(x,r\)\|\|\\Omega\_\{\\kappa\}^\{\(x,r\)\}\|grows exponentially in loop\-nesting depth\. For each planned activity, we introduce an intervalIa,a∈A\(x,r\)I\_\{a\},\\ a\\in A^\{\(x,r\)\}\.

minx,r,I,z\\displaystyle\\min\_\{x,r,I,z\}\\hskip 9\.24994ptmax⁡\(\{end​\(Ia\)∣a∈A\(x,r\)\}\)\\displaystyle\\max\(\\left\\\{\\text\{end\}\(I\_\{a\}\)\\mid a\\in A^\{\(x,r\)\}\\right\\\}\)\(15a\)s\.t\.𝔼fc\(K,x,r,I,z\)≥E\\displaystyle\\mathbb\{E\}\_\{\\text\{fc\}\}^\{\(K,x,r,I,z\)\}\\geq E\(15b\)dur​\(Ia\)=∑ρ∈ℛza,ρ​dt⁡\(a\),ρ\\displaystyle\\text\{dur\}\(I\_\{a\}\)=\\sum\_\{\\rho\\in\\mathcal\{R\}\}z\_\{a,\\rho\}d\_\{t\(a\),\\rho\}∀a∈A\(x,r\)\\displaystyle\\forall a\\in A^\{\(x,r\)\}\(15c\)start​\(Ia\)≥end​\(Ia′\)\\displaystyle\\text\{start\}\(I\_\{a\}\)\\geq\\text\{end\}\(I\_\{a^\{\\prime\}\}\)∀a∈A\(x,r\),∀a′∈p​r​\(a\)\\displaystyle\\forall a\\in A^\{\(x,r\)\},\\;\\forall a^\{\\prime\}\\in pr\(a\)\(15d\)∑ρ∈ℛza,ρ=1\\displaystyle\\sum\_\{\\rho\\in\\mathcal\{R\}\}z\_\{a,\\rho\}=1∀a∈A\(x,r\)\\displaystyle\\forall a\\in A^\{\(x,r\)\}\(15e\)

## 4Evaluation

In this section, we evaluate the performance and the applicability of the proposed approaches\. We first describe the business processes used for evaluation, followed by the implementation details and the evaluation metrics used to compare the approaches\. Finally, we present and analyze the results of the evaluation\.

### 4\.1Business Processes

We select three business processes with varying characteristics to evaluate the approaches under different conditions\. The business processes are selected based on their complexity, resource requirements, and the availability of event logs that contain activity durations\. The selected business processes are:Repair Shop, an artificial repair center whose process model and branching/looping probabilities are shown in Fig\.[1](https://arxiv.org/html/2609.05578#S1.F1);BPIC\-14I, a real\-world process from the Business Process Intelligence Challenge \(BPIC\) 2014, based on IT service management data from Rabobank Group ICT; andBPIC\-17W, a real\-world process from BPIC 2017, containing event data from a loan application process at a Dutch financial institute: we selected only the workflow \(W\) activities, representing the internal work steps performed by bank employees\. The structural characteristics of the business processes are summarized in Tab\.[1](https://arxiv.org/html/2609.05578#S4.T1)\.

Table 1:Structural characteristics of the evaluation business processes\.We discovered the process trees for the two real\-world business processes using the inductive miner\[[10](https://arxiv.org/html/2609.05578#bib.bib11)\], with branching and looping probabilities obtained by replaying the event log on the discovered process tree\. Resource capabilities and processing times are derived from the event logs, where a resource is permitted for an activity if observed executing it at least once, and durations are estimated as medians\. To account for varying daily availability, we select a representative day closest to the mean active resource count, yielding 8 of 14 resources for BPIC\-14I, and 28 of 145 for BPIC\-17W\. For the Repair Shop business process, we generate 9 synthetic resources with varying capabilities\.

### 4\.2Implementation

##### Decomposed Approach

Since CP\-SAT does not natively support power operations, which are required for the activity planning part, therr\-dependent geometric\-series coefficients are precomputed for a sufficient set ofrrvalues \(rmax=10r\_\{\\max\}=10\) and implemented as lookup tables\. The total time budget is shared: the planning phase receives at most half of the budget, and the remaining wall\-clock time is passed to the scheduling phase\.

##### Integrated Approach

The integrated formulation is encoded as a single CP\-SAT model\. Since CP\-SAT does not support a dynamic number of variables, loop unrolling is limited to a maximum iteration bound \(rmax=10r\_\{\\max\}=10\)\.

##### Baseline Approach

A direct comparison with existing approaches is precluded: they either schedule only up to the next uncertain decision point\[[6](https://arxiv.org/html/2609.05578#bib.bib2)\]or allocate resources online during execution\[[8](https://arxiv.org/html/2609.05578#bib.bib3),[11](https://arxiv.org/html/2609.05578#bib.bib4)\], and thus yield no a\-priori schedule until case completion whose makespan and feasibility could be compared against ours\. We therefore compare our approaches with a baseline that selects the single most probable execution using the probabilities of exclusive choices and the expected loop length\. Hence, it does not consider the expected feasible cases constraintEE\. The planned activities from the baseline approach are scheduled using the same scheduling logic as for the decomposed approaches, so its scheduling solutions may not be optimal\.

### 4\.3Evaluation Results

We ran the evaluation on a machine with an AMD Ryzen 9 PRO 8945HS CPU and 32 GB of RAM, with a 20\-minute time limit per run\. The results are summarized in Tab\.[2](https://arxiv.org/html/2609.05578#S4.T2)\. We report the*expected number of feasible cases*of a solution, indicating how closely the solution meets the feasibility constraintEE, the*makespan*of a solution, the number of*scheduled activities*, reflecting the total resource workload including potentially superfluous activities, and the solver*duration*and*optimality*flag for each phase\. For those runs marked with “–”, no feasible solution was found within the time limit\.

Table 2:Evaluation results \(N : number of cases, E : expected feasible cases constraint,𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}: expected feasible cases, MS : makespan \(s\), SA : scheduled activities, PD : planning duration \(s\), PO : planning optimality, SD : scheduling duration \(s\), SO : scheduling optimality, ID : integration duration \(s\); baseline : single\-path selection \(argmax branch, expected loop length\), independent of E;bold: best value per row among the decomposed and integrated approaches \(𝔼fc\\mathbb\{E\}\_\{\\text\{fc\}\}closest toEEfrom above, i\.e\., meeting the feasibility constraint most tightly, lowest MS, fewest SA\);underlined: a baseline MS or SA that matches or beats this best, which the single\-path baseline may attain while violating the feasibility constraint, i\.e\.,𝔼fc<E\\mathbb\{E\}\_\{\\text\{fc\}\}<E\)For the decomposed approach, the planning phase is trivially fast and always optimal, while scheduling is the clear bottleneck\. Still, a feasible schedule is found for every configuration except BPIC\-17W withN=50N\{=\}50, the most complex process with the highest resource, case, and expected\-feasible\-case counts\. The integrated approach scales poorly, mostly failing to find a feasible solution for larger configurations within the time limit\. Where it does solve \(Repair Shop and BPIC\-14I withN=5N\{=\}5, BPIC\-17W withN=1N\{=\}1\), it yields shorter makespans than the decomposed approach, typically at the cost of more scheduled activities\. The baseline approach’s feasibility varies by process: for BPIC\-14I, with its single exclusive choice, it matches the decomposed approach at the60%60\\%feasibility constraint; for the more complex BPIC\-17W, feasibility is extremely low; for Repair Shop, feasibility is lower than the other approaches but with the same makespan as the decomposed approach\.

## 5Related Work

Planning and scheduling of activities has been addressed in the literature in different domains, including business process management \(BPM\), operations research \(OR\), and AI planning\.

##### Scheduling and Resource Allocation in Business Process Management\.

In our prior survey\[[9](https://arxiv.org/html/2609.05578#bib.bib5)\], we identified that existing work on scheduling business process activities often treats the branch taken at exclusive choices as a decision variable and formulates optimization problems that select a single best path \(e\.g\.\[[13](https://arxiv.org/html/2609.05578#bib.bib6)\]\)\. By contrast, in this work, the branch selection is a stochastic outcome necessitating a formulation that plans for sufficiently many paths to guarantee a target level of feasibility\. Some approaches explicitly address the control\-flow induced uncertainty by either scheduling as long as certainty is relatively high\[[6](https://arxiv.org/html/2609.05578#bib.bib2)\]or presenting online resource\-allocation approaches instead of scheduling the whole case\[[8](https://arxiv.org/html/2609.05578#bib.bib3),[11](https://arxiv.org/html/2609.05578#bib.bib4)\]\.

##### Resource\-Constrained Project Scheduling \(RCPSP\)\.

A related prominent problem in OR is the RCPSP, which only involves scheduling a fixed set of activities with known durations and precedence constraints, but does not consider uncertainty in the activity network structure\[[5](https://arxiv.org/html/2609.05578#bib.bib9)\]\. The RCPSP with alternative subgraphs \(RCPSP\-AS\)\[[7](https://arxiv.org/html/2609.05578#bib.bib10)\]is similar to our problem formulation: the project network contains alternative execution paths, resembling exclusive choices; however, the choice of alternatives is a decision variable \(as in\[[13](https://arxiv.org/html/2609.05578#bib.bib6)\]\)\.

##### AI Planning\.

AI planning approaches have also addressed the problem of planning and scheduling under uncertainty\. In particular, conformant probabilistic planning \(CPP\) seeks solutions that conform to uncertain outcomes, similar to our approach\[[4](https://arxiv.org/html/2609.05578#bib.bib13)\]\. Chance\-constrained CPP has also been studied: closest to our approach,\[[3](https://arxiv.org/html/2609.05578#bib.bib14)\]uses probabilistic simple temporal networks \(a graph model without uncertain control flow, but with uncertain activity durations\) and LP relaxation to compute a chance\-constrained schedule\.

## 6Discussion and Conclusion

In this work, we addressed planning and scheduling of business process activities under control\-flow induced uncertainty until case completion\. We presented a decomposed approach with sequential planning and scheduling stages, and an integrated approach combining both in a single formulation; both use chance\-constrained optimization to select activities such that a predefined case feasibility threshold is met while minimizing the makespan\. The evaluation reveals a trade\-off between solution quality and scalability: the integrated approach can yield superior makespans but scales poorly\. The decomposed approach, in contrast, reliably finds feasible schedules across nearly all tested configurations\. In practice, scheduling cases until completion rather than only up to the next decision point allows resources to prepare for upcoming activities and enables reliable commitments towards customers\. The feasibility threshold acts as a management lever balancing the risk of rescheduling against reserved but potentially unused capacity, comparable to overbooking\. Since all required inputs can be discovered from event logs, the approaches integrate into existing process\-aware information systems with little modeling effort\. A limitation of both formulations is that branch selection for exclusive choices within loops is uniform across iterations, which can lead to suboptimal plans, since later iterations with lower execution probability could benefit from planning fewer branches\. Future work could address iteration\-dependent branch selection and incorporate processing time uncertainty alongside control\-flow uncertainty\.

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