Managing concurrency in cyclical projects under stochastic task environments: Vaccine development projects during pandemics

Abstract

Aggressive overlapping of stochastic activities during phases of vaccine development has been critical to making effective vaccines for COVID‐19 available to the public, at “pandemic” speed. In cyclical projects wherein activities can be overlapped, downstream tasks may need rework on account of having commenced prior to receiving requisite information that is only available upon completion of upstream task(s). We provide a framework to understand the interplay between stochastic overlap duration and rework due to overlap, and its impact on minimizing expected completion time for a cyclical project. We motivate the problem using the new paradigm for planning vaccine development projects. It best exemplifies features and scenarios in our model that were not considered and are also not apparent in the examples for cyclical development projects in the literature focused on engineered and manufactured products. We find that planning overlapping in scenarios that may be deemed ineffective with an assumption of deterministic tasks, can actually be beneficial when analyzed using stochastic task duration. We determine optimal planned start times for stochastic tasks as a function of a parameter that proxies for the extent of net gain/loss from overlap to minimize expected completion time for the project. We show that in situations with a net gain from overlap it is optimal to start the downstream task concurrently unless the downstream task does not stochastically dominate the upstream task and the net gain from overlap is not low enough. However, in situations with a net loss from overlap it is always optimal to have some degree of overlap in a stochastic task environment. We find that project rescheduling flexibility is always beneficial in a scenario with net loss from overlap and only beneficial in a scenario with net gain from overlap when the downstream task does not stochastically dominate the upstream task and the net gain from overlap is high enough. Our results on overlapping in 1‐to‐1, 1‐to‐n, and n‐to‐1 stochastic task configurations guide the development of an effective heuristic. Our heuristic offers good solution quality and is scalable to large networks as its computational complexity is linear in the number of tasks.

1. INTRODUCTION

The worldwide outbreak and exacerbation of the COVID‐19 pandemic has made even the stakeholders in the therapeutics and vaccine industry to adopt a more radical shift to institute concurrency (i.e., overlapping of phases or activities within phases) to varying extent during the vaccine development process (see Figure 1), in an unprecedentedly uncertain environment (Howard & Wright, 2021; Lurie et al., 2020). This industry is traditionally more sequential in its development process due to stringent regulations, safety, risk, efficacy, and cost considerations. Given the urgency to develop solutions at “pandemic speed,” as part of their disaster management strategy, governmental agencies too have offered funds and revised regulatory guidance to facilitate and benefit from overlapping of phases during development, in making an effective and safe vaccine available to public at large, at the earliest (Cornish, 2020; Hopkins & Wack, 2020). Keskinocak and Savva (2020) provide a brief review of research addressing challenges in vaccine development in their review of modeling research in healthcare management in operations management. Research in vaccine development has focused on identifying challenges and facilitating decision making during clinical trials and vaccine production that impact effectiveness and timely availability of vaccines (Cho, 2010; Dai et al., 2016; Deo & Corbett, 2009; DiMasi et al., 2016; Savva & Scholtes, 2014). Dai et al. (2016) study a supply chain contracting problem wherein a vaccine manufacturer contracts with a retailer for on‐time delivery of the influenza vaccine while also examining the implications of initiating at‐risk production undertaken prior to the design freeze for the vaccine. However, previous research in operations management has not analyzed the potential benefits of adopting concurrent engineering to plan for minimizing expected completion time for vaccine development projects during epidemics and pandemics.

FIGURE 1.

Illustration of vaccine development in traditional paradigm versus outbreak paradigm with overlapping of phases, adopted from Lurie et al. (2020)

Traditional research in project management has largely focused on project networks wherein the activities have precedence relationships (e.g., finish‐to‐start). These projects do not consider any overlapping of tasks due to strict finish‐to‐start precedence relationships. In contrast, product development projects across a variety of industries are composed of activities wherein specifications for a downstream task get progressively clarified during the execution of its upstream task(s). However, unlike finish‐to‐start relationship in traditional projects, in such environments, one can start a downstream task prior to the completion of an upstream task, albeit being constrained by a finish‐to‐finish relationship (Chakravarty, 2001; Krishnan et al., 1997; Lin et al., 2010; Loch & Terwiesch, 1998; Roemer et al., 2000; Roemer & Ahmadi, 2004; Wang & Lin, 2009). To manage this overlap effectively, firms develop information exchange mechanisms comprising frequent cycles or iterations to coordinate between the upstream and downstream tasks. Traditional projects that are devoid of any cycles for managing overlapping have been termed as acyclical projects (Meier et al., 2015). Hence, we refer to projects with an opportunity for overlapping tasks as cyclical projects. Stakeholders for vaccine development projects for COVID‐19 pandemic have recognized the opportunity and need to plan these development projects as cyclical projects instead, wherein one could pursue judicious overlapping of phases during vaccine development to reduce project time.

An overwhelming majority of research in project management has focused on acyclical projects. Even a great majority of the limited previous research in planning cyclical projects that does consider overlapping of activities in projects to minimize project completion time, has assumed task duration to be deterministic (Blackburn, 1991; Chakravarty, 2001; Eppinger, 1991; Hoedemaker et al., 1999; Ha & Porteus, 1995; Joglekar et al., 2001; Krishnan, 1996; Lin et al., 2008; Loch & Terwiesch, 1998; Roemer et al., 2000; Wang & Lin, 2009; Yang, 1991; Yassine et al., 1999). The need to consider task duration to be stochastic is apparent for activities in product development projects, for example, vaccine development projects, wherein activities and phases have expectations for both efficacy and safety that render task duration to be unpredictable. Several important research questions arise as one examines the problem of managing concurrency in project planning for cyclical projects with stochastic task duration. To what extent are the insights offered in the limited extant literature related to optimal degree of overlapping of activities in cyclical projects (predominantly focused on physical, discrete, engineered, and manufactured products) robust or enriched when one is no longer restricted by the assumption of task duration to be deterministic? How is the optimal degree of overlapping of tasks and ensuing expected completion time affected by the extent of net gain or loss from stochastic overlap, after accounting for added rework for the downstream task on account of having commenced prior to receiving full and finalized information from upstream task that is only available upon completion of the upstream task? Would it be ever beneficial to overlap in scenarios that can result in net loss from overlap? When and how does the nature of stochastic dominance of the downstream task relative to upstream task affect the extent of optimal overlap? When and to what extent is project rescheduling flexibility valuable in managing concurrency in cyclical projects with stochastic task duration? What is a suitable framework to shed light on these basic questions without unduly complicating the analysis with other contextual details? Finally, how can one incorporate the insights developed to optimize overlapping of activities in 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations to develop a heuristic that is effective and scalable, and offers high solution quality to manage concurrency in planning cyclical projects?

We develop a parsimonious framework to facilitate project planning and scheduling by offering technical insights to manage the degree of concurrency in a cyclical project network via judicious overlapping of activities to minimize project completion time in stochastic task environments. Note that in traditional acyclical projects with stochastic task duration, one cannot start a downstream activity at a planned start time if all its upstream activities that are its immediate predecessors are not completed by then. However, in cyclical projects, one can start a downstream activity at a planned start time even prior to completion of its upstream task, albeit with the risk of added rework in the downstream task due to performing the task with incomplete information.

Our modeling framework considers key issues that are well‐illustrated by the manner in which the ambitious paradigm for development of therapeutics and vaccines at pandemic speed has unfolded in the pharmaceutical industry worldwide to deal with the COVID‐19 pandemic (Cornish, 2020; Hopkins & Wack, 2020). One can view planning projects during phases of vaccine development and gearing up for manufacturing of vaccines, at three levels. At the macro level, planning for vaccine development entails major steps (e.g., preclinical trials, phase 1, phase 2, and phase 3), wherein stakeholders engaged in project management include governments, pharmaceutical and biotech firms, nonprofits, and universities (Cornish, 2020). At the intermediary level, project planning is focused on managing overlapping between two major consecutive stages. These projects can be viewed as sub‐projects when planning at the macro level. At a more micro level, project planning can be focused within a phase or stage (e.g., escalation trials in phase 1, 2, or 3). These can be viewed as sub‐sub‐projects associated with vaccine development that are typically under the purview/responsibility of one major stakeholder (i.e., a partner entity). Note that there is opportunity to consider overlapping of activities even within a phase or stage, and may entail 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations (Howard & Wright, 2021). Hence, we develop a framework comprising basic building blocks in a project network focused on planning at any of the above‐mentioned levels to glean insights from optimizing overlapping of activities in a stochastic project environment. Due to page limitations, for interested readers, we relegate more details on the problem context regarding vaccine development for COVID‐19 to Appendix B in the Supporting Information.

We analyze overlapping of 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations that represent the basic building blocks in a project network wherein all the requisite information to complete a downstream task is only available upon completion of its preceding upstream task (i.e., has a finish‐to‐finish precedence relationship). For example, due to target specifications for efficacy and safety for all tasks, the downstream tasks (with any rework on account of overlap) in vaccine development projects cannot finish prior to the completion of its upstream task (Cornish, 2020; Howard & Wright, 2021; IFPMA, 2019). Hence, a downstream task cannot be completed till its upstream task is completed. The potential saving in expected project completion time from starting a downstream task prior to completion of the upstream task may be offset by the added rework for downstream task for having worked with incomplete information. This rework for downstream task depends on evolution of information in the upstream task for the downstream task, sensitivity of downstream task to information from upstream task, and the effectiveness with which communications, reviews, and iterations are managed for information exchange between upstream and downstream task(s) (Krishnan et al., 1999). Thus, we introduce a parameter that proxies the net benefit or loss in time that is incurred per unit overlap on account of overlapping. In the absence of project rescheduling flexibility, all activities in a project are planned to start at their respective optimal planned start times. However, in the presence of full project rescheduling flexibility, activities can start earlier than planned start times. Further, unlike in deterministic task environments, in stochastic project environments (with finish‐to‐finish precedence relationships), it will become apparent later on, we also need to consider whether downstream task(s) stochastically dominate upstream task(s) in order to determine the optimal degree of concurrency.

We determine optimal overlaps as a function of a net gain/loss from overlap. We show that in the “net gain from overlap” scenario, when the downstream task stochastically dominates the upstream task, it is optimal to start the downstream task concurrently with the upstream task. In contrast, in this scenario when the downstream task does not stochastically dominate upstream task, we find that it is no longer optimal to start concurrently until the net gain from overlap is low enough. This counterintuitive result arises due to the constraint that manifests in cyclical projects with a finish‐to‐finish precedence relationship. However, in the “net loss from overlap” scenario, the optimal overlap is independent of the relative dominance in stochastic task duration. Unlike in a deterministic task environment, we show that it is always optimal to have some degree of overlap in this scenario in a stochastic task environment. At the same time, in this “net loss from overlap” scenario, we find that it is never optimal to start a downstream task concurrently. Further, we find that project rescheduling flexibility is always beneficial in “net loss from overlap” scenario and only beneficial in “net gain from overlap” scenario when the downstream task does not stochastically dominate upstream task and the net gain from overlap is high enough. In addition to building these insights, we determine optimal starts in 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations in order to develop an effective heuristic for optimizing degree of overlap in cyclical project networks with stochastic tasks. We estimate the benefit of optimal planned start time for a downstream activity in stochastic task environment vis‐à‐vis a benchmark that considers starting the downstream task at the expected completion time of the upstream task. We further estimate the benefit of project rescheduling flexibility on minimizing expected completion time vis‐à‐vis when one is constrained to only start at the optimal planned start time, due to lack of project rescheduling flexibility.

Overall, this research contributes on three fronts. First, to the best of our knowledge, we are the first to develop a parsimonious framework to analyze overlapping of stochastic tasks in cyclical projects. Second, we analytically determine the optimal planned start times for activities to minimize the expected completion time and offer managerial insights for project planning in cyclical projects. It is interesting to note that these results do not manifest when one limits the analysis to deterministic task duration. Our analytical results are also independent of the type of the distribution of tasks and also do not require any assumptions of independence. We also show that our analysis can be extended to address time–risk trade‐offs and minimize the expected completion time in budget‐constrained environments. The optimal degree of concurrency decreases when budgets get more constrained or when one wants a higher chance of finishing a project within a specified due date. Third, we incorporate our analytical results as building blocks to develop an effective heuristic for planning overlapping of stochastic tasks in cyclical project networks.

The rest of the paper is organized as follows. In Section 2, we review the relevant literature and highlight the contribution of this research vis‐à‐vis literature on optimizing planned start times to manage concurrency in cyclical projects. In Section 3, we introduce a model for overlapping two stochastic tasks to determine optimal planned start time for the downstream task in situations that may experience a net gain or net loss from overlap and further analyze their implications for overlapping in environments that may or may not have project rescheduling flexibility to start the downstream task earlier than planned if the upstream task were to finish prior to the planned start time for downstream task. Next in Section 4, we examine the optimal policies for multistage/multitask projects. We determine the optimal planned start time for downstream task(s) in 1‐to‐n and n‐to‐1 task configurations when faced with net gain or net loss from overlap. We close this section with a tabular overview of the manner in which overlapping has been optimized in this research as a function of net gain/loss from overlap (i.e., c) and nature of relative stochastic dominance of downstream tasks in 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations. In Section 5, we incorporate insights from the analysis of the basic building blocks in previous sections and develop a heuristic for overlapping activities in any general stochastic project network to minimize expected project completion time. The heuristic provides planned start time for activities in the project network. We examine the efficacy of our heuristic vis‐à‐vis two benchmark policies. The first benchmark policy represents project planning for cyclical projects in the absence of project rescheduling flexibility. The second benchmark policy considers a planning policy in the presence of project rescheduling flexibility, wherein downstream tasks are planned to start based on our heuristic but can be started earlier than planned (i.e., soon after completion of its immediately preceding upstream task) if the upstream task were to finish earlier than the planned start time for the downstream task. We also analyze the solution quality and the efficacy of our heuristic at two levels of arborescence in project networks (i.e., a serial project and a multistage arborescent project). Prior to moving to the conclusion section, for the sake of improved readability, we summarize the purpose of key figures and tables (that specify values of net gain or loss from overlap, and distributions of tasks considered for illustration) in the paper. We summarize key results and provide concluding remarks in Section 6.

2. LITERATURE REVIEW

The extant literature in project planning has largely focused on project networks wherein activities are assumed to be acyclical (i.e., devoid of any notion of iterations or overlapping of activities). In contrast, cyclical projects (e.g., product development projects) are composed of activities that can benefit from judicious overlapping of activities. To manage this overlap effectively, firms develop information exchange mechanisms comprising frequent cycles or iterations to coordinate between the upstream and downstream tasks. Optimizing the benefit of overlapping in cyclical projects with deterministic task duration while only facing a net gain from overlap have been illustrated in a variety of contexts, for example, automobile door panel and door handle (Krishnan et al., 1997), brake system design (Chakravarty, 2001), development of TV, medical devices, PC, and telephone (Terwiesch & Loch, 1999). On a close examination of the motivating examples of cyclical projects considered in the literature, one can see that models considered in the literature and the framework proposed in the ensuing section for overlapping activities in planning cyclical projects is relevant to all development projects for physical, discrete, engineered, and manufactured products, wherein specifications for downstream tasks get progressively clarified during the execution of their upstream task(s) and get fully defined upon completion of the upstream task(s). Given the focus of this paper, we only consider the literature on managing overlapping of activities in cyclical projects.

In the research stream focused on managing overlapping between two tasks with deterministic task duration, Krishnan et al. (1997) consider an upstream and downstream task, and identify the number of downstream iterations to minimize the development cycle time considering upstream evolution and downstream sensitivity. The overlapping results in rework for downstream activity that is assumed to be linear in the duration of overlap. Additionally, they assume the duration of rework to be a fraction of the overlap duration. Loch and Terwiesch (1998) study a scenario wherein an upstream and downstream task are overlapped with a goal to minimize the time‐to‐market. They offer insights for managing the extent of overlap, level of precommunication intensity, and communication frequency for design reviews under a variety of operating conditions. Lin et al. (2009) build on the above‐mentioned framework that considers an upstream and downstream task and develop a model to investigate the efficacy of overlapping tasks in conjunction with communication for functional integration. Subsequently, Lin et al. (2010) investigate the time–cost trade‐offs to determine the optimal degree of overlap and communication strategies. Tyagi et al. (2013) consider a more general functional form to estimate the rework duration on account of overlap. Hence, they use a meta‐heuristic (i.e., nondiscrete ant colony optimization) to minimize development cycle time and cost.

Another stream of research is focused on managing overlapping of activities with deterministic task duration in multistage serial or network projects to address time–cost trade‐offs. Roemer et al. (2000) determine an overlapping strategy based on the rate of evolution of information in the upstream stage and the sensitivity of the downstream stage to information in the upstream stage. Chakravarty (2001) examines the trade‐off between the added reconciliation work due to task overlap and project time saved from overlapping. Roemer and Ahmadi (2004) consider the interdependencies between overlapping and crashing to investigate the efficacy of overlapping and crashing in managing time–cost trade‐offs in product development projects. Meier et al. (2015) analyze the implications of five prespecified work policies for crashing and overlapping activities in product development projects. Meier et al. (2016) adopt a multi‐objective evolutionary algorithm to optimize time–cost trade‐off in iterative product development projects that entail crashing and overlapping of activities.

A related stream of research is focused on issues such as managing information exchange between upstream and downstream tasks, examining implications of overlapping without endogenizing it, and considering resource and deadline constraints. Ha and Porteus (1995) investigate the trade‐off between information exchange during frequent reviews of product design and setup cost/penalty time for each review. They do not study any added benefit from the overlapping between the upstream and downstream activities. Joglekar et al. (2001) develop a performance model wherein performance is defined as a measure of the product's fidelity with respect to its requirements. They find the optimal product design strategies in terms of being sequential, overlapping, or concurrent to manage product design process while the process is subject to resource and deadline constraints. Yassine et al. (1999) consider a decision analytic framework to decide whether to perform the downstream stage sequentially, in parallel, or with an overlap, based on the nature of dependence or interdependence between tasks. However, the extent of overlap is not endogenized. Lin et al. (2008) adopt a system dynamics approach to simulate the continuous upstream information evolution, evaluate its effect on downstream rework, and guide the extent to which overlapping be performed in various stages of development. Wang and Lin (2009) develop discrete‐event simulation‐based algorithm to analyze the effect of overlapping coupled tasks on product development time. Yassine et al. (2013) consider optimal information exchange between an upstream and downstream task using a dynamic programming formulation and analyze the behavior of the policy using Monte Carlo simulation.

Another stream of research related to managing cyclical product development projects is focused on addressing risk in project network environment. However, the modeling frameworks either do not entail overlapping of tasks or entail overlapping of tasks with deterministic task duration. Researchers in the first subset of this stream rely on a Markov modeling set‐up for a project network to analyze the effect of product architecture on product development cost, time, or schedule risk. However, the research does not study these effects under optimal degree of overlap in activities or the extent of communication. Browning and Eppinger (2002) examine the effects of varying product architectures on project cost and schedule risk. Jun et al. (2005) develop an analytic algorithm to estimate project cycle time based on the nature of information dependency, degree of overlap, and types of collaboration and communication to share information between tasks. Nasr et al. (2016) use a reward Markov method to estimate the expected duration and variance for iterative product development projects. They subsequently focus on resource‐constrained project management, albeit under deterministic task environments. Berthaut et al. (2011), Grèze et al. (2014a), and Grèze et al. (2014b) develop heuristics to investigate the effect of overlapping tasks on completion time of a project.

We classify the contribution of this research vis‐à‐vis literature on optimizing planned start times to manage concurrency in cyclical projects in Table 1. Unlike previous research, we consider a more realistic environment wherein the task duration for both upstream and downstream tasks are stochastic, and may be faced with scenarios of net gain or net loss from overlap. We motivate our framework and analysis in the context of vaccine development for pandemics, wherein challenging questions may have been raised when one was considering adoption of concurrency for the first time despite great uncertainty in the environment. We are the first to develop a parsimonious framework to plan for overlapping stochastic activities with a focus to minimize expected completion time for cyclical projects under a variety of scenarios (i.e., situations of net gain/loss from overlap, relative dominance of stochastic tasks, presence or absence of project rescheduling flexibility) that can capture the implications for overlapping effectiveness. To the best of our knowledge, we are the first to consider these issues in the stream of research that endogenizes overlapping of activities with stochastic duration, as these issues or insights would not manifest in projects if one considered overlapping with task duration to be deterministic. Product/manufacturing process development projects associated with vaccines and therapeutics for the pandemic best exemplify some features in our modeling framework for planning cyclical projects that are not apparent in the previous examples for cyclical projects considered in the literature. For example, the case of ineffective overlap that manifested during the product/manufacturing process development and emergency authorization of J&J vaccine for COVID‐19 or the emergency authorization for the use of hydroxychloroquine that was later determined to be ineffective (Grady, 2020; Hopkins, 2020; Whyte & Loftus, 2022). These also illustrate projects in an industry wherein they were traditionally undertaken by planning activities in a sequential manner, as though they were in a finish‐to‐start precedence relationship, as normally seen in acyclical development projects. The urgency of the pandemic forced the governmental agencies to recognize the opportunity to undertake cyclical project development for vaccines and therapeutics, with overlapping of phases. Nevertheless, the implications of this research for managing concurrency in cyclical projects with stochastic tasks extend beyond the vaccine and therapeutics industry, to cyclical projects associated with the product‐manufacturing process development of physical, discrete, engineered, and manufactured products. Next, we present our model setup for two overlapping stochastic tasks.

TABLE 1.

Literature on optimizing planned start times in managing overlapping of tasks in cyclical projects

		Task duration characteristics
		Deterministic	Stochastic	Stochastic dominance of tasks	Project rescheduling flexibility
Net gain/loss from overlap	Net gain scenario	Krishnan et al. (1997); Loch and Terwiesch (1998); Roemer et al. (2000); Chakravarty (2001); Roemer and Ahmadi (2004); Lin et al. (2009); Lin et al. (2010); Tyagi et al. (2013); Meier et al. (2015); Meier et al. (2016)	This research	This research	This research
	Net loss scenario	–	This research	This research	This research
Project scheduling heuristic		Roemer et al. (2000); Chakravarty (2001); Roemer and Ahmadi (2004); Meier et al. (2015); Meier et al. (2016)	This research	–	This research

3. MODEL FOR TWO OVERLAPPING STOCHASTIC TASKS

Let T ₁ be the random variable to represent the task time for the upstream task (i.e., Task 1). Let T ₂ be the random variable that denotes the task time for downstream task (i.e., Task 2), when the downstream task is started only after the completion of the upstream task (i.e., specifications for the downstream task only get fully defined and finalized upon completion of the upstream task). Note that Tasks 1 and 2 can represent any two overlapping tasks or phases during vaccine development (e.g., phase 1 and phase 2 trials; clinical phase and manufacturing process development; two consecutive escalation trials within a phase). Further, T ₁ and T ₂ during vaccine development represent task duration needed to meet their respective standard for efficacy and safety when the downstream task has full information available from its preceding task. Let $F_1(t)$ and $F_2(t)$ represent the cumulative distribution functions (cdf) given by $F_i(t)=P(T_i⩽t)$ where $i=1,2$ and $f_1(t)$ and $f_2(t)$ are the associated density functions, respectively. Further, let μ₁, μ₂ and σ₁, σ₂ be the means and standard deviations of T ₁ and T ₂, respectively. Let $[T_i^{\mathrm{Min}},T_i^{\mathrm{Max}}]$ denote the support of the cdf $F_i$, where $T_i^{\mathrm{Min}}>0$, and $T_i^{\mathrm{Max}}$ is finite. We assume $f_i(t)>0$ in $[T_i^{\mathrm{Min}},T_i^{\mathrm{Max}}]$. Finally, we define the project rescheduling flexibility to be the ability to start a downstream task even earlier than planned, in case the upstream task were to finish before the planned start time of the downstream task.

We first consider vaccine development environments wherein the resources in a project are shared and lack project rescheduling flexibility. Hence, the downstream task can start only at the planned start time. This prevents the ability to bring resources to start the downstream task soon upon completion of the upstream task in a stochastic task environment. We assume that Task 1 starts at time $t=0$ without loss of generality. Let $\theta ⩾0$ be the planned start time for Task 2 in the absence of project rescheduling flexibility. This results in a stochastic overlap with Task 1 (see Figure 2). Using the above‐mentioned definitions, the stochastic overlap duration of Task 2 with Task 1 is denoted by $W_1(\theta )$, where

$$W_1(\theta )=\max (0,T_1-\theta ).$$ (1)

FIGURE 2.

Illustration of a model for two overlapping stochastic tasks

Final specifications for the downstream task get progressively clarified during the execution of its upstream task. Note that in Figure 2, the vertical arrows represent information exchange taking place between the upstream and downstream tasks. To not clutter the figures in the rest of the manuscript, we will not show these vertical arrows that represent such information exchange. Table 2 summarizes mathematical notations that we use in this model.

TABLE 2.

List of mathematical notations along with the definitions

Notation	Definition	Notation	Definition
$T_i$	Random variable represents task time for Task i	c	Overlap effectiveness coefficient
$F_i(t)$	Cumulative distribution function for Task i	θ	Planned start time for downstream task
$f_i(t)$	Density function for Task i	$W(\theta )$	Stochastic overlap duration of two tasks
$[T_i^{\mathrm{Min}},T_i^{\mathrm{Max}}]$	Support of the cdf $F_i(t)$	$T(\theta )$	Project completion time
$G_i(p)$	100p th percentile of the distribution of $T_i$	$\mu (\theta )$	Expected completion time for the project

With a focus to develop an analytically tractable abstraction to bring additional insights on account of uncertainty in task duration, we determine the optimal planned start time for the downstream task and its ensuing benefits as a function of a parameter that proxies the opportunity to gain from overlap. This parameter represents the effectiveness with which communications, reviews, and iterations are managed for information exchange between upstream and downstream task(s). As we show in our analysis in ensuing subsection (Subsection 3.1), overlapping of tasks in stochastic task environments can be beneficial even in situations wherein the amount of time that gets added to the duration of downstream task to manage information exchange during overlap and perform any rework, is larger than the actual overlap with upstream task.

Let c be the ratio of “added time to manage information exchange during overlap and any rework on account of overlap” to “the overlap duration” with the upstream task. We term c as the Overlap Effectiveness Coefficient. Thus, a low value of the Overlap Effectiveness Coefficient represents situations wherein any detrimental effect of overlapping on the downstream task is low. We can see that $c<1$ represents the case of “effective overlap with net gain” and $c>1$ represents the case of “ineffective overlap with net loss.” Thus, the net gain from overlapping decreases in c and becomes a net loss situation when $c>1$. We fully characterize the optimal planned start time for the downstream task as a function of this overlap effectiveness parameter (i.e., c for all $c⩾0$). Note that as more data associated with development of cyclical projects become available, one could estimate this coefficient in a given context.

We illustrate this measure for effectiveness of overlapping activities in the context of vaccine development for COVID‐19 under the new paradigm wherein overlapping is being pursued aggressively. For example, $c=0.4$ implies that 40% of the time saving resulting from an overlap of Task 2 with Task 1 is lost due to an increase in the duration of Task 2 on account of added time to manage information exchange during overlap and perform any rework, resulting in a net saving of 60% of the duration of overlap. A value of $c=1.2$ implies that savings due to overlapping activities is more than offset by the degradation (due to any dis‐economies and severely disruptive effect on performing the downstream task on account of overlap) in the time performance of Task 2, resulting in an overall loss equal to 20% of the duration of overlap. In the context of vaccine development, such ineffectiveness in overlapping would manifest if the vaccine firm had to discard large quantities of vaccines that were produced prior to completion of phase 3, in anticipation of its success in efficacy and safety. It would entail potential rework in design and formulation of the vaccine or the process design for manufacturing (e.g., situation experienced by J&J vaccine developed for COVID‐19). It could also manifest in individual phases of vaccine development (as seen in Figure 1) that entail standard dose escalation trials (i.e., wherein subjects receive multiple doses over the duration of trial) in a given phase. One may consider a trial design protocol wherein a given dose is administered to all groups in parallel in order to save time, in anticipation of success in efficacy and safety of a dose. However, upon observing any serious adverse events one may have to revisit the protocol and do it all over again while also incurring time to find new subjects (Saul, 2005). The anticipated saving in time may be more than offset by the delay from rework, resulting in an overall loss.

The actual duration of Task 2 is given by $T_2(\theta )$, where $T_2(\theta )=T_2+c\times \max (0,T_1-\theta )$. The second term represents the added duration on account of stochastic overlap. Finally, we denote the project completion time by $T(\theta )$, where

$$T(\theta )=\max [T_1,\theta +T_2+c{W}_1(\theta )].$$ (2)

We rearrange Equation (2) as follows:

$$\begin{array}{cc}\hfill T(\theta )& =\theta +\max [T_1-\theta ,T_2+c{W}_1(\theta )]\hfill \\ & =\theta +\max [W_1(\theta ),T_2+c{W}_1(\theta )].\hfill \end{array}$$ (3)

Clearly, on one hand an increase in the extent of overlap $W_1(\theta )$ tends to decrease the project completion time. However, this gain may be more than offset by the increase in the duration of Task 2 due to rework from stochastic overlap. The planned start time for Task 2 denoted by θ is the decision variable of interest. Similar to Krishnan et al. (1997), Loch and Terwiesch (1998), Roemer et al. (2000), and the extant literature, we consider that overlapping results in rework for downstream activity that is linear in the level of overlap. Although, unlike previous research we consider stochastic overlap and also do not restrict the rework to be always less than the overlap. As the downstream task relies on information from upstream task, we also ensure that the downstream task after accounting for activities for facilitating information exchange during overlap and any rework due to overlap cannot finish prior to the completion of the upstream task (i.e., has a finish‐to‐finish precedence relationship). Hence, we first impose the condition that the downstream task is the last one to complete, that is, the event

$$A=\{\theta +T_2+c\max (0,T_1-\theta )>T_1\},$$ (4)

has occurred. This is in the same vein as the finish‐to‐finish precedence relationship considered in previous research on cyclical projects: Krishnan et al. (1997), Chakravarty (2001), Loch and Terwiesch (1998), Roemer et al. (2000), Roemer and Ahmadi (2004), Wang and Lin (2009), and Lin et al. (2010) wherein the last iteration starts at the time of completion of the upstream task, albeit in a deterministic task environment. In this stochastic environment, it also ensures that c remains as a parameter in our modeling framework and the net gain or loss per unit overlap is not affected by the extent of stochastic overlap. Mean and variance of project completion times are used as performance measures for determining the optimal planned start time. In the absence of project rescheduling flexibility in vaccine development environment, we first develop overlapping policies for two cases of overlap effectiveness coefficients, that is, $c⩾1$ (in Subsection 3.1) and $c<1$ (in Subsection 3.2), wherein the downstream task is planned to start at θ. Subsequently in Subsection 3.3, in the presence of project rescheduling flexibility, we develop policies for vaccine development environment, wherein the downstream task can be started earlier, in case the upstream task were to finish prior to the planned start time for the downstream task.

3.1. Planning in the absence of project rescheduling flexibility in a net loss from overlap scenario (i.e., $c⩾1$)

Our primary goal is to minimize the expected completion time of the project by choosing a planned start time of the downstream task while ensuring that event A has occurred to satisfy the finish‐to‐finish precedence relationship. When $c⩾1$, we can see that event A given by (4) is a sure event (i.e., P(A)=1), thus ensuring that downstream task cannot finish prior to completion of upstream task for any planned start time ($\theta ⩾0$). Hence,

$$T(\theta )\equiv \theta +T_2+c{W}_1(\theta ).$$ (5)

Its mean, that is, $\mu (\theta )$, is given by

$$\mu (\theta )=\theta +{\mu }_2+c{\int }_{\max \{T_1^{\mathrm{Min}},\theta \}}^{T_1^{\mathrm{Max}}}(t-\theta )f_1(t)dt,$$ (6)

and the objective function is:

$$\underset{0\le \theta }{\min }\mu (\theta ).$$ (7)

Let $G_1(p)$ denote the $100p\text{th}$ percentile of the distribution of T ₁, that is, $P(T_1⩽G_1(p))=p$.

Proposition 1

When $c=1$, the mean $\mu (\theta )={\mu }_1+{\mu }_2$ for θ in $[0,T_1^{\mathrm{Min}}]$ and it increases for $\theta >T_1^{\mathrm{Min}}$. When $c>1$, $\mu (0)={\mu }_2+c{\mu }_1$, and $\mu (\theta )$ decreases for θ in [0, θ₀] and then increases, where ${\theta }_0=G_1(1-\frac{1}{c})$ is the $100{(1-\frac{1}{c})}^{th}$ percentile of F ₁. The minimum value for $\mu (\theta )$ is ${\mu }_2+E(T_1|T_1>{\theta }_0)$.

All proofs are relegated to Appendix A in the Supporting Information.$\square$

In Appendix C in the Supporting Information, we show that one can obtain closed‐form expression of θ₀ for certain distributions of T ₁ and T ₂. As per Proposition 1, when $c=1$, in order to minimize the expected completion time for the project, it is optimal to plan the start time for the downstream task anywhere in the interval of $[0,T_1^{\mathrm{Min}}]$. However, one would be only wasting resources by planning to start at any time prior to $T_1^{\mathrm{Min}}$. Hence, in the absence of project rescheduling flexibility for vaccine development, it is not optimal to begin the downstream task concurrently (i.e., with θ₀ = 0) even when $c=1$. When $c>1$, and $T_1^{\mathrm{Min}}>0$, it is clearly not optimal to begin the downstream task concurrently, as the earliest optimal planned start time (i.e., ${\theta }_0=G_1(1-\frac{1}{c})$) would be greater than $T_1^{\mathrm{Min}}$.

From Proposition 1 we conclude that the mean project completion time is minimized when $\theta ={\theta }_0$. Note that for a given value of $c⩾1$, θ₀ only depends on the single percentile of the distribution of the upstream task. The result in Proposition 1 is reached under rather minimal assumptions on T ₁ and T ₂. The result is independent of the type of distribution of the two tasks and also does not require any assumptions of independence. For example, the conclusion is valid even when T ₁ and T ₂ are dependent random variables. This proposition offers a valuable guidance to a practitioner in deciding on the planned start time for Task 2 regardless of the nature of the distribution of the two tasks. We next examine the effect of change in overlap effectiveness coefficient (c) on the value of the optimal planned time (i.e., θ₀) for scheduling the start of the second task. Noting that $\frac{\partial G_1(u)}{\partial u}=\frac{1}{f_1(G_1(u))}$ (Arnold et al., 1992, p. 131), we obtain: $\frac{\partial {\theta }_0}{\partial c}=\frac{1}{f_1(G_1(1-\frac{1}{c}))c^2}<\frac{1}{f_1(G_1(1-\frac{1}{c}))}$. This derivative would be small whenever f ₁ is bounded away from 0 at $G_1(1-\frac{1}{c})$, the $100{(1-\frac{1}{c})}^{th}$ percentile of the cdf F ₁.

Next, we use an example to illustrate the efficacy of our policy to minimize the expected completion time for the project vis‐à‐vis a benchmark policy that is adopted in practice. The benchmark policy plans to start the downstream task at the expected completion time of the upstream task. In the following example, completion times of Tasks 1 and 2 have the distributions $T_1\sim Triangular(5,8,15)$ and $T_2\sim Triangular(6,8,11)$, respectively. Table 3 summarizes the performance of our policy vis‐à‐vis the benchmark policy. We use four different levels of overlap effectiveness coefficient to examine the performance of our policy. Table 3 illustrates that our policy can provide a significant reduction in expected project completion time when $c⩾1$. Note that the savings can be substantial even when overlap effectiveness coefficient (c) is close to one.

TABLE 3.

Comparing the project completion time of our policy versus the benchmark policy in the absence of project rescheduling flexibility, in case of $c⩾1$, where $T_1\sim Triangular(5,8,15)$ and $T_2\sim Triangular(6,8,11)$.

$T_1\sim Triangular(5,8,15)and{T}_2\sim Triangular(6,8,11)$
	Our policy: Optimal planned start time for Task 2		Benchmark policy: Planning start time for Task 2 at the expected time of T 1
c	$\theta ={\theta }_0$	$E[T({\theta }_0)]$	$E[T(\theta )]$ when $\theta =E[T_1]=9.33$	Reduction (%)
$c=1$	5	17.67	18.54	4.93%
$c=1.1$	6.65	17.99	18.61	3.45%
$c=2.5$	9.71	19.80	19.83	0.15%
$c=5$	11.25	20.84	21.99	5.52%

Additional questions of interest to managers would be, (a) whether the optimal planned start time would be robust for a variety of distributions that are nearly identical in their means and variances; (b) whether the respective expected completion times for these distributions are close; (c) whether the expected completion times are robust around the respective optimal planned start times. Our numerical analysis in Appendix D in the Supporting Information shows that it is indeed the case for all above‐mentioned questions.

We next focus on variance analysis and assume that T ₁ and T ₂ are independent. From Equation (5), we observe that:

$$\begin{array}{c}\hfill \text{Var}[T(\theta )]={\sigma }_2^2+c^2\text{Var}[W_1(\theta )].\end{array}$$ (8)

Proposition 2

$\mathrm{Var}[W_1(\theta )]$ is a decreasing function of θ.

Thus, from Equation (8) and Proposition 2, we note that $\text{Var}[T(\theta )]$ decreases asymptotically to ${\sigma }_2^2$ as θ increases to $T_1^{\mathrm{Max}}$. We use this proposition to further investigate the project time–schedule risk trade‐off with and without budget constraint. The results and insights associated with this trade‐off analysis are available in Appendix E in the Supporting Information.

3.2. Planning in the absence of project rescheduling flexibility in a net gain from overlap scenario (i.e., $c<1$)

We next determine the optimal planned start time for the downstream task in the absence of project rescheduling flexibility for vaccine development environment when the overlap effectiveness coefficient is less than one. During vaccine development, we need to ensure that the downstream task does not finish before an upstream task is completed as per its target specifications for efficacy and safety, and has provided all the requisite information for the downstream task. Hence, we ensure that the downstream task after accounting for any rework cannot finish prior to the completion of the upstream task. We first obtain an expression for the corresponding conditional cdf of $T(\theta )$ when $c<1$, and it is given by Proposition 3.

Proposition 3

When $0\le c<1$ and A is the event $\{\theta +T_2+c{W}_1(\theta )>T_1\}$, the conditional cdf of $T(\theta )$ given A exists whenever $\theta >T_1^{\mathrm{Min}}-{(1-c)}^{-1}{T}_2^{\mathrm{Max}}$. Further, $P(T(\theta )\le t|A)$ is the ratio of $P(\{T(\theta )\le t\}\cap A)$ and $P(A)$, where the numerator has one of the three forms given in (17a)–(17c) and the denominator has the corresponding form given in (16a)–(16c). The support of $T(\theta )$ has one of the three forms given in Remarks (a)–(c) in Appendix A in the Supporting Information.

We next examine the behavior of the completion time for the project as a function of the planned start time.

Proposition 4

When $0⩽c<1$, let ${\theta }^{\prime }$ be the smallest $\theta ⩾0$ for which P(A) = 1. If $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$, then ${\theta }^{\prime }=0$. If $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$, then ${\theta }^{\prime }=max(0,T_1^{\mathrm{Max}}-\frac{T_2^{\mathrm{Min}}}{1-c})$. $T(\theta )$ increases stochastically as θ increases over $[{\theta }^{\prime },\infty )$.

As a direct consequence of Proposition 4, we conclude that $E[T(\theta )]$ increases with θ. Thus the expected project completion time is minimized by planning to start the downstream task at the smallest θ which ensures that the second task will always finish only after the completion of upstream task after accounting for any rework.

In Figure 3, we first illustrate the optimal planned start time for Task 2 and the associated optimal expected completion time for the project as a function of the overlap effectiveness coefficient c, when $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$. Note that when $c<1$ and $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$, $P(A)=1$. Hence, as per Proposition 4, it is optimal to start Task 2 concurrently with Task 1 and the optimal expected project completion time ($E[T_2]+c\times E[T_1]$) is linear in c. Hence, the minimum expected completion time when $c\to 1^{-}$ is ${\lim }_{c\to 1^{-}}(E[T_2]+c\times E[T_1])=E[T_2]+E[T_1]$. As per Proposition 1, the minimum expected completion time when $c=1$ is $E[T_2]+E[T_1]$. Thus, while the optimal start time for the downstream task moves from being concurrent when $c<1$ to $T_1^{\mathrm{Min}}$ when $c=1$ (i.e., no longer performed concurrently), the optimal expected completion time for the project is still continuous in c.

FIGURE 3.

Optimal planned start time of Task 2 and optimal completion time of project in the absence of project rescheduling flexibility for $c<1$ and $c⩾1$ when $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$. In this example $T_1\sim Triangular(15,18,20)$ and $T_2\sim Triangular(22,24,27)$.

Now if $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$ and $c<1$, it is no longer guaranteed that the downstream task including rework would automatically finish only after the upstream task is done for all $\theta ⩾0$. Hence, the start time of the downstream task needs to be delayed in a manner which ensures that it is achieved. Recall that $T(\theta )$ still increases stochastically with θ and $E[T(\theta )]$ increases with θ. Hence, the earliest start time for downstream task at which the constraint is satisfied becomes the optimal start time for the downstream task. One can determine this earliest start time for the downstream task in cases where both the upstream and downstream tasks have a bounded distribution. The minimum θ which ensures that the downstream task including rework finishes no earlier than the upstream task is $\theta =T_1^{\mathrm{Max}}-\frac{T_2^{\mathrm{Min}}}{1-c}$. Additionally, for values of $1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}⩽c<1$, the constraint is not a concern. Hence, the optimal start time for the downstream task is zero (i.e., performed concurrently), when $1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}⩽c<1$. This provides a rather counterintuitive insight for planning vaccine development projects wherein the downstream task can be started concurrently only when the overlap effectiveness coefficient (i.e., rework duration as a fraction of overlap) is high enough and not when it is low, even when $c<1$. Thus, in the absence of project rescheduling flexibility, when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$ and the objective is to minimize expected completion time, concurrency should be pursued even when one is not as effective in realizing gains from overlapping. These results are independent of the type of distribution of task times.

In Figure 4, we illustrate the optimal planned start time for the downstream task and optimal expected completion time for the project as a function of the overlap effectiveness coefficient when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$. We use a simulation to estimate the optimal expected completion time for the project. In the simulation, we ensure that the downstream task including rework finishes only at or after the upstream task is done.

FIGURE 4.

Optimal planned start time of Task 2 and optimal completion time of project in the absence of project rescheduling flexibility for $c<1$ and $c⩾1$ when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$. In this example, $T_1\sim Triangular(15,18,20)$ and $T_2\sim Triangular(2,3,4)$.

Unlike the results in Figure 3 wherein it is optimal to plan to perform the downstream task concurrently for all values of the overlap effectiveness coefficient less than 1, it is now no longer optimal to perform the downstream task concurrently with the upstream task for all values of $c<1$. In contrast to the results in Figure 3, when $T_2^{\mathrm{Min}}$ is small enough, it is optimal to plan the start time for the downstream task in a nearly sequential manner (i.e., with a low stochastic overlap), when the overlap effectiveness coefficient is small and close to zero. Recall that a low value of c represents situations wherein any detrimental effect of managing overlapping is low. As the overlap effectiveness coefficient increases over the range of $0⩽c⩽1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}$, the optimal planned start time for the downstream task decreases in a nonlinear manner and approaches to concurrency at $c=1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}$. Further, it is optimal to perform the downstream task concurrently with the upstream task when $1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}⩽c<1$. Thus, we find an interesting insight for planning vaccine development projects in the absence of project rescheduling flexibility wherein overlapping is deemed effective (i.e., $c<1$). In this scenario, the expected project completion time decreases with an increase in the overlap effectiveness coefficient over the range of $0⩽c⩽1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}$, and is the lowest when the overlap effectiveness coefficient is $c=1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}$. For $1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}⩽c<1$, when it is optimal to start the downstream task concurrently, the expected project completion time increases in c in the interval of $[1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}},1]$. It is also interesting to note that the lowest expected completion time occurs when the overlap effectiveness coefficient is rather high, albeit less than one.

3.3. Planning in the presence of project rescheduling flexibility in a net gain (i.e., $c<1$) or net loss (i.e., $c⩾1$) from overlap scenarios

In Subsections 3.1 and 3.2, we have considered a project management environment for development of vaccines wherein one has to plan the start of downstream task in the absence of project rescheduling flexibility. In such an environment, one may not be able to start the downstream task soon after the completion of the upstream task, even in case the upstream task were to finish prior to the planned start time for the downstream task. We now consider an environment with project rescheduling flexibility, wherein the downstream task can be started earlier, in case the upstream task were to finish prior to the planned start time for the downstream task. Such an environment with presence of project rescheduling flexibility has been prevalent during the development of vaccines for COVID‐19 at several pharmaceutical/bio‐tech companies. The government agencies have provided ample funding to enable vaccine firms to aggressively pursue overlapping in planning and rescheduling projects in a stochastic task duration environment.

We now begin Task 2 at time θ or on completion of Task 1 whichever is earlier. Therefore,

$$\begin{array}{cc}\hfill T(\theta )& =\max (T_1,\min (T_1,\theta )+T_2+c\max (0,T_1-\theta ))\hfill \\ & =\max (T_1,T_1+T_2+(c-1)W_1(\theta )).\hfill \end{array}$$ (9)

Recall that $W_1(\theta )$ decreases stochastically with θ. The completion time for the project, $T(\theta )$, increases stochastically in overlap effectiveness coefficient (c) when $c>1$, and decreases stochastically when $c<1$. Further, if $c>1$, $\mu (\theta )$ is the smallest when θ is $T_1^{\mathrm{Max}}$ . In other words, if overlap effectiveness coefficient is greater than or equal to 1 ($c⩾1$), the expected completion time for the project is minimized when Task 2 is started immediately after completion of Task 1 (i.e., tasks will be performed sequentially). We can also examine $\text{Var}[T(\theta )]$ when $c>1$; in that case, $T(\theta )=T_1+T_2+(c-1)W_1(\theta )$. If T ₁ and T ₂ are assumed to be independent,

$$\begin{array}{ccc}\hfill \text{Var}[T(\theta )]& =& \text{Var}(T_1)+\text{Var}(T_2)+{(c-1)}^2\text{Var}[W_1(\theta )]\hfill \\ & & +2(c-1)\text{Cov}[T_1,W_1(\theta )].\hfill \end{array}$$ (10)

In Proposition 2, we have shown that $\text{Var}[W_1(\theta )]$ decreases. Similarly, we can show that $\text{Cov}(T_1,W_1(\theta ))$ decreases as well, by observing that its derivative with respect to θ (i.e., $[1-F_1(\theta )]\{E(T_1)-E(T_1|T_1>t)\}$) is negative. Hence, $\text{Var}[T(\theta )]$ decreases to $\text{Var}(T_1)+\text{Var}(T_2)$ as $\theta \to \infty$, from its value of $\text{Var}(T_2)+c^2\text{Var}(T_1)$ attained at $\theta =0$.

Recall that when $c<1$ we need to ensure that the downstream task along with any rework gets completed only at or after the completion of the upstream task. This would be a concern when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$. For the ease of exposition, we first discuss the case when $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$. If $c<1$, the expected completion time for the project, $\mu (\theta )$, is smallest when $\theta =0$. If overlap effectiveness coefficient is less than one, the completion time for the project is minimized when Tasks 1 and 2 are started together (i.e., tasks will be performed concurrently). Figure 5 shows that the optimal planned start time for Task 2 and the ensuing optimal expected completion time for the project are identical for both our overlapping policies (i.e., in the presence versus absence of project rescheduling flexibility). Thus, having started concurrently in situations wherein the downstream task never finishes prior to the upstream task (i.e., $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$) and $c<1$, project rescheduling flexibility is not valuable for overlapping activities during vaccine development.

FIGURE 5.

Optimal planned start time of Task 2 and optimal completion time of project for $c<1$ and $c⩾1$ when $T_2^{\mathrm{Min}}>T_1^{\mathrm{Max}}$ in the presence of project rescheduling flexibility. In this example, $T_1\sim Triangular(15,18,20)$ and $T_2\sim Triangular(22,24,27)$.

When $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$ and $c<1$, it is no longer guaranteed that the downstream task including rework would automatically finish only after the upstream task is done. Hence, the start time of the downstream task needs to be delayed in a manner which ensures that it is achieved. $T(\theta )$ increases stochastically with θ and consequently $E[T(\theta )]$ also increases with θ. Hence, the earliest start time for downstream task at which the constraint is satisfied becomes the optimal planned start time for the downstream task.

In Figure 6, we illustrate the optimal planned start time for the downstream task and optimal expected completion time for the project as a function of the overlap effectiveness coefficient when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$. We use a simulation to estimate the optimal expected completion time for the project. Note that the optimal planned start time for Task 2 (as seen in Figure 6) is identical to that in our overlapping policy in the absence of project rescheduling flexibility (as seen in Figure 4). Now, the downstream task can be started earlier, in case the upstream task were to finish prior to the planned start time for the downstream task. Thus, our policy with the presence of project rescheduling flexibility is only beneficial vis‐à‐vis policy with the absence of project rescheduling flexibility in situations wherein the upstream task completes prior to the planned start time for the downstream task, that is, when $T_1^{\mathrm{Min}}<\theta$.

FIGURE 6.

Optimal planned start time of Task 2 and optimal completion time of project for $c<1$ and $c⩾1$ when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$ in the presence of project rescheduling flexibility. In this example, $T_1\sim Triangular(15,18,20)$ and $T_2\sim Triangular(2,3,4)$.

When $c=0$, the optimal planned start time of the downstream task is $T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}$. The optimal planned start time for the policy with the presence of project rescheduling flexibility is in the interval of $[0,T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}]$ when $0⩽c<1$. Clearly, if $T_1^{\mathrm{Min}}⩾T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}$, the policy in the presence of project rescheduling flexibility performs identically vis‐à‐vis policy without project rescheduling flexibility over $0⩽c<1$. In this case the upstream task can never finish prior to the optimal planned start time for the downstream task to realize any added benefit from the project rescheduling flexibility. When $T_1^{\mathrm{Min}}<[T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}]$, the policy with project rescheduling flexibility again performs identically vis‐à‐vis policy without project rescheduling flexibility, when the optimal planned start time is in the interval $[0,T_1^{\mathrm{Min}}]$. The optimal planned start time in the presence of project rescheduling flexibility is in the interval $(0,T_1^{\mathrm{Min}}]$ when overlap effectiveness coefficient is such that $1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}-T_1^{\mathrm{Min}}}⩽c<1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}$. The optimal planned start time in the presence of project rescheduling flexibility is zero (i.e., tasks are performed concurrently) when overlap effectiveness coefficient is such that $1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}}⩽c<1$. When $T_1^{\mathrm{Min}}<[T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}]$, the optimal planned start time in the presence of project rescheduling flexibility is in the interval $[T_1^{\mathrm{Min}},T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}]$ when the overlap effectiveness coefficient is such that $0⩽c⩽[1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}-T_1^{\mathrm{Min}}}]$. Thus, the presence of project rescheduling flexibility is only beneficial vis‐à‐vis the absence of it when its optimal planned start time is in the interval $(T_1^{\mathrm{Min}},T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}]$ that occurs when $0⩽c<[1-\frac{T_2^{\mathrm{Min}}}{T_1^{\mathrm{Max}}-T_1^{\mathrm{Min}}}]$.

It is interesting to see that overlapping policy in the presence of project rescheduling flexibility for vaccine development is beneficial only when the overlap effectiveness coefficient is sufficiently low (i.e., when the net gain from overlapping after accounting for added time for managing information exchange and any ensuing rework is high enough). The overlapping policy in the absence of project rescheduling flexibility performs as good as the policy in the presence of project rescheduling flexibility when either $T_1^{\mathrm{Min}}$ is large enough, or the overlap effectiveness coefficient is beyond a threshold and results in an optimal planned start time for the downstream task that is less than $T_1^{\mathrm{Min}}$. Table 4 illustrates any potential savings in optimal expected completion time for the project in presence of project rescheduling flexibility vis‐à‐vis an environment without project rescheduling flexibility, when the optimal planned start time is in the interval $[T_1^{\mathrm{Min}},T_1^{\mathrm{Max}}-T_2^{\mathrm{Min}}]$. The simulations comprise 400 random project trials using task time distributions of all tasks associated with 1‐to‐1 task configurations (in Tables 3 and 4, and Figures 3, 4, 5, 6). These runs provide the requisite precision in estimation of expected completion times to assess the relative performance across all policies.

TABLE 4.

Savings in completion time for the project from adoption of our policy in the presence of project rescheduling flexibility vis‐à‐vis absence of project rescheduling flexibility, for a range of overlap effectiveness coefficients, when $T_2^{\mathrm{Min}}⩽T_1^{\mathrm{Max}}$. In this example, $T_1\sim Triangular(15,18,20)$ and $T_2\sim Triangular(2,3,4)$.

	Our policy in the absence of project rescheduling flexibility		Our policy in the presence of project rescheduling flexibility
Overlap effectiveness coefficient (c)	Planned Start Time for Task 2	Expected Completion Time for Project	Planned Start Time for Task 2	Expected Completion Time for Project	% Potential savings from adoption of our policy in the presence of rescheduling flexibility vis‐à‐vis the absence of rescheduling flexibility
0.01	17.30	20.276	17.30	20.020	1.26%
0.1	17.04	20.089	17.04	19.915	0.87%
0.2	16.68	19.862	16.68	19.773	0.45%
0.3	16.22	19.628	16.22	19.601	0.14%
0.4	15.68	19.440	15.68	19.438	0.01%
0.5	15	19.297	15	19.297	0%
0.6	13.97	19.150	13.97	19.150	0%
0.8	8.85	18.863	8.85	18.863	0%
0.9	0	18.858	0	18.858	0%
0.92	0	19.211	0	19.211	0%
0.95	0	19.741	0	19.741	0%
0.99	0	20.447	0	20.447	0%
1.1	16.17	20.855	16.17	20.744	0.11%
2.5	18	21.667	18	20.977	2.70%
5	18.58	22.058	18.58	20.913	4.56%

The savings from the policy in the presence of project rescheduling flexibility increase in the overlap effectiveness coefficient when $c⩾1$. However, the savings are low when c is higher than one, but close to one. Overall, we find that that our policy in the presence of project rescheduling flexibility vis‐à‐vis that in the absence of project rescheduling flexibility is beneficial when loss on account of added time for managing information exchange during overlap and any ensuing rework in the downstream task is below a threshold (i.e., c is quite low) or beyond a threshold (i.e., $c⩾1$). The result is independent of the type of distribution of the upstream and downstream tasks and also does not require any assumptions of independence. Thus, our results have significant implications for stakeholders engaged in project planning for vaccine development in evaluating the value of investment in resources that allow for project rescheduling flexibility that can enable start of the downstream task earlier than planned if the upstream task were to finish prior to the planned start time for the downstream task. In the next section, we investigate overlapping policies for environments wherein we have n tasks in an upstream or downstream stage.

4. OVERLAPPING STOCHASTIC TASKS IN 1‐to‐n and n‐to‐1 TASK CONFIGURATIONS

In this section, we analyze two additional basic building blocks that are important to plan for overlapping of activities in projects (e.g., vaccine development for COVID‐19) that are typically composed of multiple stages, with multiple activities in each stage. (Howard and Wright, 2021, p. 25) provide an overview of the Operation Warp Speed (OWS) vaccine development initiative in the United States. The report provides the status of major phases in vaccine development process at various OWS vaccine firms (i.e., Moderna, Pfizer/BionTech, AstraZeneca, Sanofi/GSK, Novavax, Janssen, and Novavax). One can see that these firms have planned to undertake the requisite phases of trials across different target populations in specific countries in a time‐phased manner. The report also indicates that as of January 2021, five of the six OWS vaccine firms had also started commercial scale manufacturing. At a macro level, vaccine firms can plan their timeline for overlapping in phases of vaccine development across several countries. This overlapping can entail 1‐to‐1, 1‐to‐n, or n‐to‐1 task configurations. For example, moving from preclinical trials to phase 1 trials in different countries entails a 1‐to‐n task configuration to manage degree of overlap. Further, moving from phase 3 trials in some countries to licensure in a country can entail overlapping in a n‐to‐1 task configuration. Additionally, gearing up for large‐scale production prior to completion of phase 3 trials in different countries can also entail overlapping in a n‐to‐1 task configuration. Even when planning projects at a micro level within a clinical phase, designing and managing escalation trials for different target populations provides opportunities for overlapping via 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations (Saul, 2005). In this section, we first look into 1‐to‐n task configuration and then n‐to‐1 task configuration.

4.1. 1‐to‐n task configuration

In situations wherein an upstream task can be followed by n downstream tasks that are independent of each other, our policies in Section 3 provide the individual optimal start times for each downstream task. However, there are situations wherein managers may plan to start n downstream tasks all together at the same time. This imposes an additional constraint to our setting in Section 3. We next develop an optimal overlapping policy for this scenario.

Let Stages 1 and 2 be the upstream and downstream stages, respectively. Stage 1 has a single task with T ₁ as the random variable to represent the task duration. As in Section 3, let $[T_1^{\mathrm{Min}},T_1^{\mathrm{Max}}]$ denote the support of the cdf F ₁, where $T_1^{\mathrm{Min}}>0$, and $T_1^{\mathrm{Max}}$ is finite. We assume $f_1(t)>0$ in $[T_1^{\mathrm{Min}},T_1^{\mathrm{Max}}]$. Stage 2 has n tasks that are independent of each other. Let $c_1,\text{…},c_n$ be the overlap effectiveness coefficients associated with n tasks in Stage 2 for overlapping with the single task in Stage 1, rank ordered such that $c_1⩽c_2⩽\cdots ⩽c_n$. Let $T_{21},T_{22},\text{…},T_{2n}$ be the distributions of the associated task completion times in Stage 2, when each is performed after completion of the single task in Stage 1. Further, $[T_{2i}^{\mathrm{Min}},T_{2i}^{\mathrm{Max}}]$ denote the support of the cdf $F_{2i}$, where $T_{2i}^{\mathrm{Min}}>0$, and $T_{2i}^{\mathrm{Max}}$ is finite. We assume $f_{2i}(t)>0$ in $[T_{2i}^{\mathrm{Min}},T_{2i}^{\mathrm{Max}}]$. Figure 7 illustrates this scenario, wherein all tasks in Stage 2 start at time θ. In this scenario, the project completion time is:

$$\begin{array}{cc}\hfill T(\theta )=& max(T_1,\theta +T_{21}+c_1{W}_1(\theta ),\theta +T_{22}+c_2{W}_1(\theta ),\text{…},\theta \hfill \\ & +T_{2n}+c_n{W}_1(\theta ))\hfill \\ \hfill =& \theta +max(W_1(\theta ),T_{21}+c_1{W}_1(\theta ),T_{22}+c_2{W}_1(\theta ),\text{…},T_{2n}\hfill \\ & +c_n{W}_1(\theta )).\hfill \end{array}$$ (11)

If $c_n\ge 1$, the expression for $T(\theta )$ can be simplified as below:

$$T(\theta )=\theta +\max (T_{21}+c_1{W}_1(\theta ),\text{…},T_{2n}+c_n{W}_1(\theta )).$$ (12)

We impose the condition that each downstream task completes only after the completion of the upstream task, that is, the event

$$A_{1\to n}^i=\{\theta +T_{2i}+c_i\max (0,T_1-\theta )>T_1\},$$ (13)

has occurred for $i=1,2,\text{…},n$. In case, c ₁ is at least one, $P(A_{1\to n}^i)=1$ for all $i=1,2,\text{…},n$. Conditioning on the value of T ₁, one can provide an expression for the cdf of $T(\theta )$ as follows:

$$\begin{array}{ccc}\hfill P(T(\theta )\le t)& =& F_1(\theta )\prod _{i=1}^n{F}_{2i}(t-\theta )\hfill \\ & & +{\int }_{t_1=\theta }^{\theta +(t-\theta )/c_n}\prod _{i=1}^n{F}_{2i}(t-\theta -c_i(t_1-\theta ))f_1(t_1)d{t}_1,t\ge \theta .\hfill \end{array}$$ (14)

When $n=1$, this reduces to Equations  (22a)–(22c). We can write:

$$\begin{array}{ccc}& & max(T_{21},\text{…},T_{2n})+c_1{W}_1(\theta )+\theta ⩽T(\theta )⩽max(T_{21},\text{…},T_{2n})\hfill \\ & & +c_n{W}_1(\theta )+\theta .\hfill \end{array}$$ (15)

Hence, $\mu (\theta )$ is always between $\theta +{\mu }_2+c_{1}E(W_1(\theta ))$ and $\theta +{\mu }_2+c_{n}E(W_1(\theta ))$ where μ₂ is the mean of $\max \{T_{21},\text{…},T_{2n}\}$. However, if $c_i$'s are all the same, that is, $c_1=c_2=\cdots =c_n=c$, and $c⩾1$, then:

$$\begin{array}{cc}\hfill T(\theta )& =\theta +c{W}_1(\theta )+max(T_{21},T_{22},\text{…},T_{2n}).\hfill \end{array}$$ (16)

From Expression (5) and Proposition 1, it follows that $\mu (\theta )$, that is, $E[T(\theta )]$, decreases as θ increases to θ₀ and then increases, where θ₀ is ${(1-\frac{1}{c})}^{th}$ percentile of the cdf F ₁. Thus, while we sought to develop an optimal policy that constrained all downstream tasks to start at the same time, this result shows that when the overlap effectiveness coefficients for all downstream tasks are identical and greater than or equal to 1, it is indeed optimal to start all downstream tasks at the same time, regardless. The implication for practitioners is that if overlap effectiveness coefficients for downstream tasks are large and close to each other, then it is optimal to start downstream tasks at the same time (i.e., with the identical overlap).

FIGURE 7.

Illustration of model for 1‐to‐n tasks scheduling

Note that when $c_i<1$ for all $i=1,2,\text{…},n$, it is not tractable to analyze the scenario when $T_{2i}^{\mathrm{Min}}<T_1^{\mathrm{Max}}$ for all $i=1,2,3,\text{…},n$. Hence, we next analyze scenarios wherein $T_{2i}^{\mathrm{Min}}⩾T_1^{\mathrm{Max}}$ for all $i=1,2,3,\text{…},n$. When $c_i<1$ for all $i=1,2,\text{…},n$, $\mu (\theta )$ increases in θ. Hence, it is optimal to start all the downstream tasks in Stage 2 concurrently with the task in the upstream stage.

Recall that $c_i$'s are rank ordered such that $c_1⩽c_2⩽\cdots ⩽c_n$. Now, if some $c_i$'s are less than one with the others being greater than or equal to 1, that is, $c_1⩽\cdots <1<c_k⩽\cdots ⩽c_n$, again, easy bounds for the project completion time can be obtained, when $T_{2i}^{\mathrm{Min}}⩾T_1^{\mathrm{Max}}$ for all $i=1,2,3,\text{…},k-1$. We can write: $max(T_{21},\text{…},T_{2n})+c_1{W}_1(\theta )+\theta ⩽T(\theta )⩽max(T_{21},\text{…},T_{2n})+c_n{W}_1(\theta )+\theta$. Hence, $\mu (\theta )$ would be the smallest somewhere between $\theta =0$ and ${(1-\frac{1}{c_n})}^{th}$ percentile of the cdf F ₁. In the next section, we present the scenario wherein n independent and parallel tasks in an upstream stage are followed by a single task in the downstream stage.

4.2. n‐to‐1 task configuration

Let $T_{11},\text{…},T_{1n}$ be the associated distributions task completion times for the n tasks in Stage 1. The downstream stage consists of a single task, which has a task time of T ₂, when performed after completion of all the n tasks in the upstream stage. Figure 8 illustrates this scenario. As in Section 3, let $[T_2^{\mathrm{Min}},T_2^{\mathrm{Max}}]$ denote the support of the cdf F ₂, where $T_2^{\mathrm{Min}}>0$, and $T_2^{\mathrm{Max}}$ is finite. We assume $f_2(t)>0$ in $[T_2^{\mathrm{Min}},T_2^{\mathrm{Max}}]$. Let $c_i$, that is, the marginal overlap effectiveness coefficient, be the rework required for Task 2 per unit time overlap of Task 2 with Task 1i in the upstream stage. The total rework in the downstream stage is assumed to be additive in marginal reworks on account of overlap with the upstream tasks in Stage 1. Further, $[T_{1i}^{\mathrm{Min}},T_{1i}^{\mathrm{Max}}]$ denote the support of the cdf $F_{1i}$, where $T_{1i}^{\mathrm{Min}}>0$, and $T_{1i}^{\mathrm{Max}}$ is finite. We assume $f_{1i}(t)>0$ in $[T_{1i}^{\mathrm{Min}},T_{1i}^{\mathrm{Max}}]$. For the sake of convenience in exposition, we consider that all upstream tasks are starting at the same time, without loss of generality. We relax this stipulation in Appendix G.3 in the Supporting Information (Figure 21).

FIGURE 8.

Illustration of model for n‐to‐1 task scheduling

The project completion time can be expressed as:

$$\begin{array}{cc}\hfill T(\theta )=& max(T_{11},T_{12},\text{…},T_{1n},\theta +T_2+\sum _{i=1}^n{c}_{i}max(T_{1i}-\theta ,0))\hfill \\ \hfill =& \theta +max(T_{11}-\theta ,T_{12}-\theta ,\text{…},T_{1n}-\theta ,T_2+\sum _{i=1}^n{c}_i{W}_{1i}(\theta )),\hfill \end{array}$$ (17)

where $W_{1i}(\theta )=max(T_{1i}-\theta ,0)$, $i=1,2,3,\text{…},n$. As seen before in Expression (3), we may replace $T_{1i}-\theta$ by $W_{1i}(\theta )$ in Expression (17) above to get the following result:

$$\begin{array}{c}\hfill T(\theta )=\theta +max(W_{11}(\theta ),W_{12}(\theta ),\text{…},W_{1n}(\theta ),T_2+\sum _{i=1}^n{c}_i{W}_{1i}(\theta )).\end{array}$$ (18)

We impose the condition that the downstream task after accounting for any rework due to overlap only completes after the completion of all upstream tasks, that is, the event $A_{n\to 1}=\{\theta +T_2+{\sum }_{i=1}^n{c}_{i}max(T_{1i}-\theta ,0)>max(T_{11},T_{12},\text{…},T_{1n})\}$ has occurred. The values of $c_i$'s affect the stochastic behavior of $T(\theta )$. If all the $c_i$'s are at least 1 then $P(A_{n\to 1})=1$. Also, the last component of the right hand side of Expression (17) dominates and consequently, with probability 1, we get

$$\begin{array}{c}\hfill T(\theta )=\theta +T_2+\sum _{i=1}^n{c}_i{W}_{1i}(\theta ).\end{array}$$ (19)

This result is a generalization of Expression (5). Thus proceeding in a manner similar to the section discussing the two‐task scenario, we arrive at the following result.

Proposition 5

When $c_i⩾1$, $i=1,2,\text{…},n$, $E[T(\theta )]$ decreases as θ increases to θ₁, and then increases, where θ₁ is $100{(1-\frac{1}{c})}^{th}$ percentile of the cdf $F_1(t)={\sum }_{i=1}^n{p}_i{F}_{1i}(t)$, with $c={\sum }_{i=1}^n{c}_i$ and $p_i=\frac{c_i}{c}$, and $i=1,2,\text{…},n$.

Now, if $T_2^{\mathrm{Min}}>max(T_{11}^{\mathrm{Max}},T_{12}^{\mathrm{Max}},\text{…},T_{1n}^{\mathrm{Max}})$ and ${\sum }_{i=1}^n{c}_i⩽1$, then:

$$\begin{array}{ccc}\hfill T(\theta )& =& max[max(T_{11},\theta ),max(T_{12},\theta ),\text{…},max(T_{1n},\theta ),\hfill \\ & & \theta (1-\sum _{i=1}^n{c}_i)+T_2+\sum _{i=1}^n{c}_{i}max(T_{1i},\theta )].\hfill \end{array}$$ (20)

We observe that each component in the right hand term increases (stochastically) with θ. Thus, $T(\theta )$ also increases with θ. Consequently, the mean project completion time $E[T(\theta )]$ is the smallest when $\theta =0$. Above results are independent of the type of distribution of tasks and also do not require any assumptions of independence. Further, these results have practical implications when the marginal overlap effectiveness coefficients are small enough or large enough.

When either the condition (1) $c_i⩾1$, for $i=1,2,\text{…},n$ or (2) $T_2^{\mathrm{Min}}>max(T_{11}^{\mathrm{Max}},T_{12}^{\mathrm{Max}},\text{…},T_{1n}^{\mathrm{Max}})$ and ${\sum }_{i=1}^n{c}_i⩽1$ fails to hold, it appears that actual values of the $c_i$'s and the properties of the cdfs $F_{1i}$'s need to be considered for the investigation of the properties of $E[T(\theta )]$.

Figure 9 summarizes the manner in which overlapping has been optimized as a function of net gain/loss from overlap (i.e., c) in 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations. We next use these building blocks developed thus far to create a heuristic to determine optimal overlap between stages of tasks in several representative networks.

FIGURE 9.

Overview of optimization of planned start times in different scenarios

5. HEURISTIC: PLANNING OVERLAPPING IN GENERAL CYCLICAL STOCHASTIC PROJECT NETWORK

In practice, product development projects comprise a network of tasks that are stochastic in nature and experience dependencies between upstream and downstream tasks. Recall that the entire process of vaccine development for COVID‐19 as seen in Figure 1 can be visualized as a mega project that can have sub‐projects nested within each stage/phase of vaccine development. In this section, we use the building blocks developed thus far (i.e., 1‐to‐1, 1‐to‐n, and n‐to‐1 stochastic task configurations) to develop a heuristic to determine optimal overlap between stages of tasks in a network. In Subsection 5.1, we first present a pseudocode for the heuristic developed for determining planned start time for downstream tasks in any general cyclical project network. In absence of project rescheduling flexibility, one would plan the project with the start times determined by the heuristic. In the presence of project rescheduling flexibility, activities are still planned to start based on the start times determined by the heuristic. However, now a downstream task can be started earlier than planned (i.e., soon after completion of its immediately preceding upstream task) if the upstream task(s) were to finish earlier than the planned start time for the downstream task.

5.1. Pseudocode for heuristic for a general cyclical project network

Consider a general network with N tasks ($N⩾2$) and specified finish‐to‐finish precedence relationships (that permit overlapping of activities). We need to determine at most $N-1$ starting times (i.e., θ) for downstream task(s). This happens when we have N back‐to‐back tasks in a serial network. Otherwise, you need to identify a smaller number of planned start times based on the precedence relationship structure of the tasks in the network.

Step (0) Initialize Set S to comprise all activities in the network, and sets $SU=S{U}^{\prime }=SD=\varphi$. Let the set P capture all the precedence relationships pairs for activities in the network.

Step (1) Identify all activities of S that have no predecessors. Set $SU$ to be this subset of upstream activities of S that has no predecessors.

Step (2) Identify immediate successors for all activities in $SU$. If there is no successor, stop the algorithm. Else, identify activities in $SU$ for whom the immediate predecessor for their immediate successors only belong to $SU$. Let $S{U}^{\prime }$ be the set of these upstream activities in $SU$. Let $SD$ be the set of immediate downstream activities for upstream activities in $S{U}^{\prime }$. Note that there could be one or more activities in $SU$ that have their immediate successor(s) whose immediate predecessor may not belong to $SU$. Hence, this downstream activity (or activities) would not belong to $SD$ at this time, and its planned start time can only be determined in a later iteration. Now, activities currently in $SD$ could be in one of the following task configurations with their upstream activities in $S{U}^{\prime }$, namely, 1‐to‐1, 1‐to‐n, or n‐to‐1. Determine the planned start time for downstream activities in $SD$ using one of the following as appropriate. Move to Step 3 when planned start time for all activities in $SD$ has been determined.

• (a)For each downstream activity in $SD$ that is part of 1‐to‐1 overlapping scenario perform the following to find the optimal planned start time for the associated downstream activity:
  • (i)
    Use Proposition 1 (i.e., 1‐to‐1 overlapping policy) to find the optimal planned start time for the downstream task:

    $$\begin{array}{ccc}\hfill \theta & =& G\left(1-\frac{1}{c}\right)\text{is}\text{the}100{\left(1-\frac{1}{c}\right)}^{th}\text{percentile}\text{of}\hfill \\ & & F\text{(i.e.,}\text{upstream}\text{task}\text{in}S{U}^{\prime }).\hfill \end{array}$$ (21)
  • (ii)
    Update the stochastic duration for the downstream task as below:

    $$\begin{array}{ccc}\hfill T_{\mathrm{Downstream}}^{\prime }({\theta }^{\ast })& =& T_{\mathrm{Downstream}}+cW({\theta }^{\ast })\hfill \\ & =& T_{\mathrm{Downstream}}+cmax\{0,T_{\mathrm{Upstream}}-{\theta }^{\ast }\}.\hfill \end{array}$$ (22)
• (b)For downstream activities in $SD$ that are part of 1‐to‐n overlapping scenario perform the following to find the optimal planned start time for these downstream activities:
  • (i)
    Use the overlapping effectiveness coefficient and Expression (14) to find an optimal planned start time for all these downstream tasks from the expression below:

    $$\begin{array}{ccc}\hfill \theta & =& G\left(1-\frac{1}{c}\right)\text{is}\text{the}100{\left(1-\frac{1}{c}\right)}^{th}\text{percentile}\text{of}\hfill \\ & & F\text{(i.e.,}\text{upstream}\text{task}\text{in}S{U}^{\prime }).\hfill \end{array}$$ (23)
  • (ii)
    Update the stochastic duration for each of the associated downstream tasks as below:

    $$\begin{array}{ccc}\hfill T_{\mathrm{Downstream}}^{\prime }& =& T_{\mathrm{Downstream}}+cW({\theta }^{\ast })\hfill \\ & =& T_{\mathrm{Downstream}}+cmax\{0,T_{\mathrm{Upstream}}-{\theta }^{\ast }\}.\hfill \end{array}$$ (24)
• (c)For each downstream activity in $SD$ that is part of n‐to‐1 overlapping scenario perform the following to find the optimal planned start time for the associated downstream activity:
  • (i)Use the overlapping effectiveness coefficient to find an optimal planned start time for the downstream task from Proposition 5 and the expression below:

    $\theta =$ $100{(1-\frac{1}{c})}^{th}$ percentile of the cdf $F(t)={\sum }_{i=1}^n{p}_i{F}_{1i}(t)$, with $c={\sum }_{i=1}^n{c}_i$ and $p_i=\frac{c_i}{c}$, and $i=1,2,\text{…},n$.

  • (ii)
    Update the stochastic duration for the downstream task as below:

    $$\begin{array}{c}\hfill T_{\mathrm{Downstream}}^{\prime }=T_{\mathrm{Downstream}}+\sum _{i=1}^n{c}_i{W}_i({\theta }^{\ast }).\end{array}$$ (25)

Step (3) Set S equal to $S-S{U}^{\prime }$ (i.e., remove all the upstream tasks with no predecessors at this stage whose planned start for all its immediate successors has been determined). Next, repeat the above‐mentioned steps from Step 1 until this set of activities (i.e., S) becomes empty.

Our heuristic for planning (or re‐planning), that relies on analytical results, is scalable to large networks as its computational complexity is linear in N. Any approach taking recourse to endogenizing planned start times numerically using simulations would be faced with computational complexity that grows exponentially in N and the number of time increments for computations (based on grid size). The efficacy of our heuristic becomes even more pronounced and pertinent when one has to consider frequent re‐planning in projects over time as seen in a stochastic task environment.

In the next section, we examine the efficacy of our heuristic vis‐à‐vis two benchmark policies. We illustrate the efficacy of our heuristic for two levels of arborescence in project networks.

5.2. Comparisons of solution quality and performance of the heuristic

We examine two project networks: (1) an arborescent multistage project (see Figure 10) that has all the building blocks developed in Sections 3 and 4; and (2) a serial project with tasks in series in 1‐to‐1 task configuration (see Figure 11). We keep the number of tasks constant and examine cases with different levels of arborescence in the project network.

FIGURE 10.

Illustration of an arborescent multistage project comprising tasks in 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations

FIGURE 11.

Illustration of a serial project comprising tasks in 1‐to‐1 task configurations

We utilize the pseudocode presented in Subsection 5.1 to develop details for implementing the mentioned heuristic. Due to space limitation, we relegate additional details associated with the implementation of the heuristic for the arborescent multistage project and the serial project to Appendix G in Appendices G.1 and G.2 in the Supporting Information, respectively. For an interested reader, we also present the implementation details of our heuristic for a more complex project network in Appendix G.3 in the Supporting Information. Next, we introduce the comparisons of solution quality and performance of our heuristic vis‐à‐vis two benchmark policies.

We evaluate the solution quality of our heuristic (i.e., percentage gap) for both multistage project and serial project in Table 5. We focus on values of overlap effectiveness coefficients that are close to one, to represent environments without project rescheduling flexibility, wherein overlapping activities can lead to inefficiencies and dis‐economies, albeit at small levels. If overlap effectiveness coefficients were to get higher due to severe dis‐economies resulting from the stochastic overlap, based on Proposition 1 and Proposition 5, the start times for downstream tasks would approach the ${(1-\frac{1}{c})}^{th}$ percentile of the distribution of upstream task(s). Consequently, the time performance of planning the project using the expected completion times of upstream tasks would approach the performance of our heuristic, when the overlap effectiveness coefficient gets close to two (i.e., when the rework time is twice the overlap).

TABLE 5.

Illustration of the solution quality for the heuristic and performance of the heuristic vis‐à‐vis benchmarks in multistage project (i.e., comprising 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations) and serial project, using following task distributions: $T_1\sim \text{Triangular}(4,20,25)$, $T_2\sim \text{Triangular}(6,20,30)$, $T_3\sim \text{Triangular}(9,25,35)$, $T_4\sim \text{Triangular}(7,12,14)$, and $T_5\sim \text{Triangular}(7,8,12)$

Solution quality of heuristic vis‐à‐vis numerical optimization (% gap)
Level of $c_{i}s$ (i.e., Overlap inefficiency)	Multi‐stage project	Serial project
Low	0.01%	1.61%
Moderate	1.24%	2.16%
High	1.61%	2.57%

Performance of heuristic vis‐à‐vis benchmarks
Comparisons	Multi‐stage project	Serial project
Saving from Heuristic versus Benchmark 1 (%)	4.12%	5.78%
Saving from Benchmark 2 versus Heuristic (%)	3.49%	4.12%

Note: The associated overlap effectiveness coefficients are $c_1=1.02$ and $c_2=c_3=c_4=1.03$ (for low level and for benchmark comparisons); $c_1=1.1$, $c_2=c_3=c_4=1.2$ (for moderate level), $c_1=1.2$, $c_2=c_3=c_4=1.3$ (for high level).

We next illustrate the efficacy of the proposed heuristic procedure vis‐à‐vis two benchmark policies in Table 5. The first benchmark policy (i.e., benchmark 1) represents project planning in the absence of project rescheduling flexibility wherein activities have to start at their planned start times. This benchmark policy uses expected completion times for upstream tasks to determine the planned start times for downstream tasks in a network. This benchmark enables us to estimate the percentage savings in expected completion time from our heuristic in the absence of project rescheduling flexibility. We consider the second benchmark (i.e., benchmark 2) as a planning policy in the presence of project rescheduling flexibility. In this benchmark policy the downstream task can be started earlier than its planned start time based on our heuristic (i.e., soon after completion of its immediately preceding upstream task) if the upstream task were to finish earlier than the planned start time for the downstream task. This can help estimate the cost of lack of project rescheduling flexibility in a stochastic cyclical project environment.

Note that the computational complexity for determining optimal overlaps using incremental enumeration with simulation grows exponentially in the number of stages in a project. Our heuristic provides reasonable quality solutions in polynomial time. We find that the solution quality of our heuristic (i.e., percentage gap) is good and ranges from 0.01% to 2.57% across a range of problem scenarios. The above‐mentioned simulations include hundred random project trials using task time distributions of the associated project (see Table 5). These runs provide the requisite precision in estimation of expected completion times to assess the relative performance across all policies. We refer the interested readers to Appendix G in the Supporting Information for further details.

Finally, prior to moving to the conclusion section, we summarize the purpose of key figures and tables (that specify values of c and distributions of tasks considered for illustration) in the paper. They highlight one or more of the following:

• (1)show optimal planned start time for downstream activity and expected completion time for a given task configuration (Figures 3, 4, 5, 6);
• (2)show the benefit of our optimal project planning policy vis‐à‐vis benchmark policy (Tables 3 and 5);
• (3)show the benefit of our optimal project planning policy in the presence of project rescheduling flexibility vis‐à‐vis in the absence of project rescheduling flexibility (Tables 4 and 5);
• (4)show the solution quality of our heuristic for multistage projects comprising combinations of 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations (Table 5).

6. CONCLUSION

The urgency faced during vaccine development initiatives for COVID‐19 pandemic have exemplified the need to deviate from a traditional serial process and consider judicious overlapping of tasks (i.e., phases) in these projects to minimize completion time. There is significant uncertainty in the time to complete each phase of the vaccine trial to meet the standard for efficacy and safety. It is also important to consider task duration to be stochastic in planning these projects, particularly when the product or process technology (e.g., mRNA vaccines) is also new to the world. In this environment, one may also experience a net loss in time due to excessive rework in downstream stages on account of overlapping. Thus, in contrast to a limited literature on endogenizing planned start times for tasks in cyclical projects focused on overlapping of activities with deterministic task duration (vis‐à‐vis a vast literature on acyclical projects that is devoid of any overlapping of tasks), we consider an environment wherein all task duration are stochastic. We propose a parsimonious framework to consider new issues (i.e., net gain/loss from overlap, stochastic dominance of tasks, and project rescheduling flexibility), and provide analytics to offer insights for managing the degree of concurrency in planning cyclical projects with stochastic tasks, that has hitherto not been considered in the literature.

We determine optimal planned start times for tasks in 1‐to‐1, 1‐to‐n, and n‐to‐1 task configurations. We characterize the optimal planned start times as a function of a net gain or loss from overlap. We show that in situations with a net gain from overlap, when the downstream task stochastically dominates the upstream task, it is optimal to start the downstream task concurrently with the upstream task. In contrast, when the downstream task does not stochastically dominate upstream task, our results show that it is no longer optimal to start the downstream task concurrently until the net gain from overlap is low enough. However, in the situation with a net loss from overlap, the optimal start time for downstream task is independent of the relative dominance in duration of tasks. Unlike in a deterministic task environment, we find that it is always optimal to have some degree of overlap in a net loss scenario in a stochastic task environment. At the same time, in this scenario, our results show that it is never optimal to start concurrently. The optimal degree of concurrency decreases when budgets get more constrained or when one wants a higher chance of finishing a project within a specified due date. We find that project rescheduling flexibility is always beneficial in a situation with net loss from overlap and only beneficial in a situation with a net gain from overlap when the downstream task does not stochastically dominate upstream task and the net gain from overlap is high enough. Finally, we incorporate our technical results to develop an effective heuristic for optimizing degree of overlap in cyclical project networks with stochastic tasks. We first estimate the benefit of our heuristic in the absence of project rescheduling flexibility vis‐à‐vis a benchmark that considers starting the downstream task at the expected completion time of the upstream task. We also estimate the benefit of project rescheduling flexibility on minimizing expected completion time vis‐à‐vis when one is constrained to only start at the planned start times, due to lack of project rescheduling flexibility. It is interesting to note that all the above‐mentioned results do not manifest for cyclical projects when one limits the analysis to deterministic task duration.

The implications of this research extend to managing overlapping in development projects, in sectors already considered in the literature, and well beyond the vaccine and therapeutics development industry. The emerging landscape of information‐intensive and disruptive core product or process technologies too are foreboding a major challenge for managing concurrency of stochastic tasks during development of new products and services. Finally, we close with some directions for future research on cyclical projects. Researchers can address issues of interplay between performance‐loss and time‐to‐market, and time–cost trade‐off associated with overlapping of activities in a stochastic project environment. Data rich environments can facilitate estimation of key parameters in planning models for future research.

Supporting information

Supporting Information.

Murthy, N. N. , Nagaraja, H. N. , & Rikhtehgar Berenji, H. (2022). Managing concurrency in cyclical projects under stochastic task environments: Vaccine development projects during pandemics. Production and Operations Management, 00, 1–21. 10.1111/poms.13907