A decision support methodology to support military asset and resource planning

ABSTRACT

This paper presents a decision-support methodology to support the development and assessment of military asset and resource strategies. The methodology is built around a system dynamics model that allows users to examine the performance of a strategy over time. The novelty of the model lies in its flexibility and ability to address questions about asset planning from both holistic and lifecycle viewpoints. From the user perspective, the decision-support methodology is structured around three phases: (1) Design an asset management strategy, (2) Evaluate strategy using system dynamics simulation model, and (3) Generate performance indicators and analyse results. The methodology is developed and used in a real case study to support the modernisation of the Australian Defence Force. This paper demonstrates that system dynamics offers useful methods to study the dynamics of supply-demand, and support the development of systemic asset and resource management strategies.

KEYWORDS:

• Defence
• system dynamics
• asset and resource planning
• decision analysis

1. Introduction

This paper focuses on the important topic of strategic decision making in the area of military asset and resource planning. This topic is crucial for the present and future of defence-force performance (i.e., operational, costs). International Organisation for Standardisation defines asset management and planning as an integrated activity to produce value from a system of assets (ISO, 2014). A wide range of decision problems (related to asset management and planning) such as fleet size and mix, workforce planning, maintenance, resource planning and allocation, and lifecycle analysis can be investigated. We use the term “asset” to describe a collection of systems to be considered for decision analysis. As the scope of the studied assets, we particularly focus on “strategic military assets”. The key feature for strategic assets is that they should be significant enough to influence strategic decisions and stakeholders’ values (Größler, 2007). Similarly, we use the term “resource” to be a strategic resource, defined as a key determinant for the present and future performance of assets, not merely as a thing needed for routine operations (Größler, 2007). In the process of managing assets to achieve particular goals, organisations build, deplete, and manage resources (Amit & Schoemaker, 1993).

A strategic military asset usually has a high-value and a long lifespan (over a couple of decades). Further, these strategic assets rely on building and utilising several resource types. For example, a submarine is considered a strategic asset with over 30 years lifespan that depends on several resources, such as maintenance facilities and personnel, to operate and to maintain.

Strategy development for military asset and resource planning can be viewed as an unstructured, complex, and dynamic problem, as it exhibits several challenging features for decision making (Sterman, 2010):

• Path-dependence: decisions are long-term and largely irreversible. The decision to buy a particular asset is a large investment with long-term operational and economic impacts. For example, Australia’s new fleet of 12 submarines will cost taxpayers $50 billion (Department of Defence, 2016).

• Dynamic complexity: the performance of any selected strategy is influenced by feedback interactions among decisions made by dispersed defence functions and divisions, including: acquisition policies, resourcing decisions, workforce planning decisions, and supply-chain management.

• Characterised by delays and trade-offs: planners are faced with the challenge to make decisions now whose outcomes can vary at different temporal scales and performance dimensions. For example, the pressure to cut recruitment rates will lead to a short-term reduction in costs, however, can have detrimental consequences on the sustainability of training pipelines and the availability of future workforce.

• Tightly coupled subsystems: sharing resources among different entities (e.g., spatial regions, assets) and activities (e.g., operations and training) creates tightly coupled relationships among assets and their constituent systems. Reductionist thinking that fails to account for these interdependencies may lead to suboptimal outcomes of improving the performance of one asset or a function on the expense of another. For example, military assets are often used for the dual purposes of operations and training, creating interdependency and inevitable trade-off decisions between operational availability and satisfying training requirements.

In this paper, we are motivated by the premise that devising effective military asset management strategies needs to be informed by methodologies that provide the following capabilities:

1. Capturing the interconnections among the problem elements to identify the flow-on consequences of decisions. This includes the need for a systemic and hierarchical representation of the strategy, in terms of the resources, activities, interrelationships, and decisions that influence the strategy performance (K. Warren, 2002);

2. Analysis and assessment of the strategy dynamic behaviour, including multi-level (asset, activities, resources) view of key performance indicators to allow for linking high-level and low-level performance (Damle, 2003); and

3. The flexibility of designing and examining new strategies by modelling the system “as it can be” not only “as it is” (Bisschop & Meeraus, 1982).

The paper’s primary objective is to present a new decision-support methodology that applies to a wide range of strategic planning in military asset and resource planning (such as workforce planning, and fleet-mix strategies). The methodology is also capable of handling challenges associated with unstructured complex and dynamic problems mentioned above. In this direction, we look for answers for following research questions throughout this paper: (i) what are the class of decision problems covered in strategy development for asset management and resource planning?, (ii) what are the current research gaps in the military asset management and resource planning?, and (iii) what are the essential properties and features of appropriate decision-support methodologies to address wide range of (military) asset management problems?

The rest of the paper is organised as follows: Section 2 provides a thorough description of the research areas related to the proposed methodology, from both application and methodological perspectives. This section addresses the research questions (i) and (ii). Section 3 presents a real-world case study where we have applied the decision-support methodology to analyse what-if questions to support strategy development for the new submarine fleet in the Australian Defence Force. Section 4 presents the proposed decision support methodology. Section 3 and 4 together answer the last research question by explaining the features of the methodology. In Section 5, we demonstrate the applicability and usefulness of the proposed methodology through a hypothetical scenario where the user would like to use different functionalities of the decision-support model and to test the dynamic effects of different decisions on the fleet performance over its lifetime. In Section 6, we conclude with some comments on the strengths and weaknesses of the proposed methodology and we identify a number of future research directions.

2. Research background

This section sets the stage for the reader from both application and methodological perspectives in order to help position the paper in the relevant body of knowledge on the use of models and methods to support strategy development in military asset and resource planning. In Section 2.1, we start by characterising the class of problems that the proposed methodology is designed to tackle. In Section 2.2, we first explain the role of simulation modelling for military assets and resource planning. Next, we outline a broad classification of how simulation models have been used from the level of their system view (micro-level, macro-level) and the type of decisions they are intended to support (e.g., operations management, strategy development). In Section 2.3, we briefly introduce system dynamics (SD) as a modelling approach and present a literature review on the use of SD to support military asset and resource planning.

2.1. Strategy development for asset and resource planning

When devising strategies, decision makers and planners are not faced with isolated decisions (Ackoff, 1967), but need to handle a set of problems that are connected through space and time. Strategy development for asset and resource planning is a decision analysis area that spans a variety of problems and decisions because of the strong interplay between different objectives, resources, and activities. These decision problems include (See Figure 1): fleet size and mix, fleet renewal and lifetime extension, fleet lifecycle analysis, workforce planning, including training, resource provision and planning including supply network design and facility location, maintenance strategies.

Figure 1. The class of decision problems covered in strategy development for asset planning.

The fleet size and mix problem aims to determine the composition of the fleet (i.e., number and type of assets) that can achieve certain strategic objectives while complying with constraints such as the available budget (Mathew et al., 2010; New, 1975). In many real-world situations, the old fleet is not just scrapped, but is operated or extended for a period while introducing the new fleet in order to minimise performance shortfalls (e.g., capability gaps). In such a situation, planners are faced with a set of interactive decisions about the replacement schedule, including: the timing of fleet renewals, the numbers of assets to renew, the timing of fleet retirement, and the length of the extension period (Khasnabis et al., 2002; Simms et al., 1984). A related planning problem is the lifecycle assessment (and total cost of ownership), where the problem’s scope is extended to include considerations related to activities in the different phases (i.e., operations, support, and retirement) during the asset’s life from “cradle to grave” (e.g., D’Souza et al. (2016)). Depending on the study scope, multiple dimensions can be used to assess and evaluate a portfolio of options, including: carrying capacity (e.g., fleet availability and utilisation), economic (e.g., total cost of ownership), and environmental (e.g., carbon emissions).

Taking a lifecycle viewpoint, maintenance is a crucial activity in the asset’s life as it influences the asset’s performance (Pinjala et al., 2006). Formulating a maintenance strategy is a non-trivial task involving interdependent decisions, such as the type and specification of the maintenance regime (i.e., calendar-based vs condition-based) and the amount of resources required to perform maintenance (spares, equipment, workforce). These decisions cannot be made in isolation without considering the long-term effects on the asset performance and assumptions about the availability of required resources. This leads to resource planning problems which aim to find the best strategy (i.e., amount of resources, timing, location) for investing and allocating resources among a number of assets in order to optimise several objectives (e.g., maximise availability and minimise cost), sometimes simultaneously, over a long planning period (Mathew et al., 2010; Mishra et al., 2013). Many of the resources required to sustain assets are spatially distributed, which means that planners are faced with decisions related to the spatial dimension when allocating and prioritising resources. This includes decisions regarding which assets will be assigned to which location or region (i.e., the homeport assignment problem), and which areas will be able to serve which resource requirements (i.e., facilities’ capacities and locations). Depending on the definition of the asset, this problem can be extended to decisions about the design of the whole supply network (e.g., Snyder (2006)). Workforce (or personnel) planning aims to close the gap between workforce supply and demand, and minimise the effects of any imbalances (Edwards, 1983). This includes devising decisions about the number of recruits, attrition rates, training policies, and promotion policies in order to meet requirements.

Although it is evident these strategic planning decisions are interrelated, they are often treated independently to reduce the complexity of the problem. For example, on deciding the optimal fleet size and mix, one assumes that the supporting workforce is readily available or is a constraint that will be dealt with through another study or decision tool. This gives planners a partial view of the problem, and does not allow for investigating the effects arising from the interactions of multiple decisions (Größler et al., 2015). However, the decision about fleet size has direct and indirect implications to workforce planning in terms of the workforce required to operate the fleet and the training capacity of the fleet if it is to be used as a training platform. Part of the complexity of the problem is capturing the interconnections among the problem elements to identify the flow-on consequences of decisions.

There is a need for a decision-support methodology that allows users to develop and evaluate military asset and resource planning strategies in a holistic way (Bernstein, 2017). This research gap is the focus of this paper. The paper contributes by proposing an SD-based decision-support methodology. We first outline the methodological context of the paper by presenting a broad outline of the use of simulation to address problems that are similar to ours. Next, we articulate the use of SD.

2.2. Use of simulation modelling for asset and resource planning

Analytical approaches, including optimisation models, are widely used and proven to be useful for analysing the problem classes described in Section 2.1. While such analytical approaches are useful (Mathew et al., 2010), they can be too simplistic to address the complex and dynamic features of decision making. On the other hand, simulation models are powerful for generating dynamic and systemic behaviour, and examining system performance under alternative scenarios.

The military is a big user of simulation models with a diverse and wide range of roles and applications (Hill & Tolk, 2017). Broadly speaking, military simulation models fall into three main categories. These categories include live,¹ virtual,² and constructive simulation (Hill et al., 2001). Simulation models developed for military assets and resource planning fall into the constructive simulation category. Constructive simulations are considered to be contained within the computer with the ability to take some human input. It is also possible to classify constructive simulations into subcategories based on a number of dimensions. For example, a constructive simulation model may be dynamic or static depending on whether the passage of time is explicitly considered or not. Constructive simulations can be also continuous or discrete depending on whether state variables within the model change at any time in the model or at discrete points.

The primary role of military asset and resource planning simulation models is “descriptive”, which is meant to (statically) describe processes and activities involved in asset and resource planning and explain how the system will look in the future. Nevertheless, military asset and resource planning simulation models might also be used for “prescriptive” purposes/roles such as in simulation-optimisation approaches where the outputs of the analysis are expected to provide a set of “optimal policies”, or the simulation results assist the decision making process.

Lastly, descriptive military asset and resource planning simulation models can be divided into two categories based on their system view, namely microscopic and macroscopic models (Bowers et al., 2017). Microscopic models are developed to address the operational side of asset planning and therefore can be used for “mission rehearsal” roles (Hill et al., 2001). Further, these types of models capture the system with great detail. The high level of detail is useful when detailed operational plans are needed to manage systems. Nevertheless, to develop these types of models, several assumptions, and a huge amount of data to develop, test, and validate are essential. In such situations in which limited time and data exist, the usefulness of microscopic models may diminish. Further, the runtime of microscopic models can be relatively high which puts a burden on their use in supporting strategic conversations between different stakeholders in group decision support discussions. For instance, a Monte Carlo discrete event simulation model, which is built for the Canadian force design, has a runtime close to several weeks (Eisler & Allen, 2012). Another Monte Carlo-based simulation model SaFe, which is also used by the Canadian Defence Forces, may run for 2.5 hours (Eisler et al., 2014). Macroscopic models are placed at the other end of the spectrum in simulation modelling such that they are developed for supporting and enhancing strategic analysis rather than operational and tactical planning of missions. Different from microscopic models, macroscopic models capture the studied system at an aggregate level. Hence, they require less time (to run and develop) and data compared to microscopic models. Besides, they are easy to maintain and update. Microscopic models can be also useful to provide decision insights. For instance, they can be utilised to check if available resources can fulfill the requested demand and assist to determine the amount of needed resources without developing a detailed plan regarding how requested demand can be supplied on an operational or tactical basis (Wesolkowski & Eisler, 2014).

Macroscopic models address strategic issues and thus can be used for policy-making purposes. Macroscopic models specifically fit decision support need that Bisschop and Meeraus (1982) have clearly articulated in asset management as: “[Models] are used as a framework for analysis, for data collection, and for discussion. They are created to improve one’s conceptual understanding of the problem. If several decision makers are involved in a final decision or set of recommendations, models can be used as neutral moderators to guide the discussions. Different viewpoints can be tested and examined. In such an environment the actual values of model results are not so important, but the relative values resulting from testing different scenarios are of interest. The model is a learning device, and should never be expected to produce final decisions”. According to According to Bhargava (1993), in these situations where there is limited available data and a wide range of viewpoints and assumptions about the future, models can be useful to support reasoning about different assumptions, their implications, and to support the decision maker’s conceptual understanding of the problem. Our proposed methodology is centred on the use of macroscopic SD simulation models in military asset and resource planning.

In the next section, we introduce SD modelling in the context of strategic problems in military planning of assets and resources.

2.3. Use of SD modelling for strategic problems in military planning

SD has several powerful features that make it well suited to support strategy development. First, the strategy or problem structure, as captured in the SD model, focuses attention on the cause-and-effect relationships among the strategy elements (i.e., decisions, resource states, and performance indicators) that derive the dynamic performance of the strategy. For example, an SD model can capture the feedback interactions and coupling that arise from resource sharing (Größler et al., 2015), or the delay in building and retaining resources. This causal view of the strategy allows planners to understand and explain why strategies exhibit a particular behaviour, and why particular decisions fail to achieve objectives. The resource-based view of strategies can be easily captured by the stock-and-flow constructs of a SD model (K. Warren, 2005). For example, assets and resources can be modelled by stocks to represent different stages in the processes which cause these resources to change their state lifecycle during the planning period. Further, the processes that lead to a change in the state of assets and resources can be modelled as flows. This approach also aligns with the objectives of lifecycle analysis by providing a focused view on the asset at different states throughout the planning period and taking into account the possibility of delayed effects (Onat et al., 2016). Second, SD is a scenario-based approach, which allows users to set up “what-if” questions about possible changes in the management decisions and external drivers. The ability to represent and simulate the strategy, in the form of an SD model, allows decision makers to make assumptions, and examine their implications. Third, SD models allow for designing and experimenting with different assumptions underpinning strategies without requiring detailed and precise data because the emphasis is on simulating the behavioural patterns and identifying the conditions which determine the strategy’s performance.

Whereas SD has been used for strategy development in a wide range of industry and business applications (Cosenz & Noto, 2016), its adoption and use in military strategy development has been slower in pace and scope. A closer look on the body of knowledge on the use of SD to support military asset planning and strategy development reveals that SD has been used for strategy development within a specific class of problems, and limited use as an integrated methodology for looking at the interactions among these sub-problems, and therefore investigating the strategic asset planning problem in a holistic and life-cycle oriented way.

In Table 1, we present an overview of studies utilising SD to approach strategic asset planning in the military. We map these studies to the classes of problems as described in Section 2.2. The review is not intended to be comprehensive, however, it should provide a sufficient overview of the existing works in the strategic asset planning domain. The review shows SD applications have been predominantly used for addressing a narrow class of problems by focusing on a particular subsystem (e.g., workforce planning or maintenance problems). Nevertheless, literature is scarce on the use of SD as a decision-support methodology for holistic asset management across its lifecycle, from acquisition to retirement. This is the area where this paper contributes. It is also important to mention that SD modelling is a “top – down” approach, and therefore, it is convenient to analyse problems from a macro and holistic-thinking perspective (Ding et al., 2018; Swanson, 2002). Whereas “bottom – up” approaches such as agent-based models are more suitable to develop microscopic models rather than macroscopic models (Ding et al., 2018; Railsback & Grimm, 2019). Further, microscopic (e.g., agent-based) models get too detailed to simulate the asset lifecycle, from acquisition to retirement because of the large number of parameters and rules, which makes model parameter identification difficult. Due to these reasons, we propose a decision-support methodology, where the foundation of the methodology is an SD model which captures the interactions of decisions in military asset and resource strategic planning.

Table 1. A literature review of studies utilising SD for strategy development in military applications.

Paper	Model purpose	Class of problems
Andersen and Emmerichs (1982)	Examine the relationships among the retirement, separation, and recruitment decisions on the workforce mix and costs	Workforce planning
H. H. Turan, S. Elsawah, and M. J. Ryan (2020)	Examine the fleet renewal schedule on fleet availability	Fleet renewal
Coyle and Gardiner (1991)	Test the effect of different delay orders and lead time on the completion rate of submarines.	Fleet schedule
Thomas et al. (1997)	Examine the combined effects of training, retirement, and recruitment decisions on the workforce size	Workforce planning
Linard et al. (1999)	Examine the effect of different employment decisions (e.g., contract-based vs permanent) on workforce size and profile	Workforce planning
H. H. Turan, S. Elsawah, and M. J. Ryan (2019)	Examine the effects of separation, promotion, recruitment and fleet renewal decisions on the workforce size and cost	Workforce planning
Jalalvand et al. (2020)	Examine the effects of separation, promotion, recruitment and fleet renewal decisions on the workforce size, cost, fleet availability and capability	Fleet renewal and workforce planning
Adamides et al. (2004)	Assess the long-term performance of military aircraft engine lifecycle maintenance systems, while considering feedback relationships with the dynamics of the human resource system	Maintenance, Workforce planning
McLucas et al. (2006)	Examine the effects of the interrelationships between aircraft availability, maintenance, and available workforce on the fleet readiness.	Maintenance, Workforce planning
Garza et al. (2014)	Examine the effects of different promotion, recruitment, and training decisions on workforce levels by rank	Workforce planning
Markham (2008)	Examine interactions amongst training capacity and the uncertainty and volatility of separation rates	Workforce planning
Fan et al. (2010)	Investigate the causes for the bullwhip effect caused in the weapons maintenance supply system.	Maintenance, supply networks
Armenia et al. (2012)	Examine the effects of different promotion, recruitment, and training decisions on workforce levels by rank	Workforce planning
Jiang (2013)	Examine the feedback effects between introducing new technologies on training and manpower requirements.	Workforce planning, training
Jnitova et al. (2017)	Examine the effects of different training pipeline design on the training delays.	Workforce planning, training
Hudson et al. (2017)	Examine the effects of ordering delays on the capacity to fulfill ammunition demand during missions.	Supply network
Ahram et al. (2017)	Examine the interaction between technology selection and human resource decisions on training and flow on effect on total cost of ownership.	Workforce planning, training, total cost of ownership

3. Project and case study context

In this section, we describe the case study context in which the decision-support methodology has been developed and applied. First, we start by an overview of asset planning context in the Australian Defence Force, with a particular focus on decision-making issues and challenges, which motivate the use of decision-support methodologies. Then, we focus on the design of the submarine capability support system as a case study while following the modelling steps from Sterman (2010) (see Section 4 for details).

The Australian Defence Force (ADF) is currently undergoing a significant redesign process, with massive investments in expanding and renewing a number of key strategic assets, including the expansion of Australia’s submarine fleet (Department of Defence, 2016). Through this modernisation process, decision makers are faced with the challenge of designing asset and resource planning strategies including the issues and problem classes discussed in Sections 1 and 2. To devise these strategies, planners need a decision-support methodology that allows for examining the asset planning problem from a holistic and lifecycle viewpoint, taking into consideration the interactions among the different resources and systems (such as support, maintenance, and workforce planning). This need has led to the initiation of this ongoing modelling project, where the research-client team (i.e., papers’ co-authors and software developers) has been engaged by the ADF to develop this decision-support methodology. Multiple case studies for devising strategies for different classes of assets are used as a testbed to support the development, testing, and refinement of the methodology. This research problem can be approached as a design thinking task, where the aim is to use the models to support the development of a strategy to design the support system rather than investigate an existing problematic situation. According to Gault et al. (1987), a “design” approach has two meanings. The first is to use a set of model building blocks or components to design and assemble a model the way an engineer builds a system (e.g., car) from its parts. The second approach is to build and evaluate alternative futures (i.e., alternative “to be” system designs). This is different from the traditional view of investigating an existing system to predict its future behaviour. In this paper, we use the submarine fleet as a case study to illustrate the development and use of the proposed methodology.

3.1. Case study: Design of the support system for the submarine fleet

Submarines are strategic and complex defence assets with a long lifespan. For example, it is expected that the existing submarine fleet of the Royal Australian Navy (i.e., Collins Class Submarines) will be decommissioned after over 40 years of service (Hellyer, 2018). Further, Goldrick (2016) states that four Los Angeles class nuclear-powered attack submarines of the United States Navy are scheduled for decommissioning in 2017 after 36 years’ service, while the hull life of the Ohio class nuclear-powered ballistic missile submarines has been extended to 42 years. Submarine support systems is a term used to describe the enterprise (i.e., geographically-distributed resources, processes, governance) that manage the underpinning enabling Fundamental Inputs to Capability (FIC) that sustain the fleet. The design of an effective support system for the existing and submarine fleet spans across the complex decision problems described in Section 2.2. Decision makers are faced with the need to make investments decision now that will influence the fleet’s performance significantly for future decades. Although some decisions can have short-term consequences, there might be a long delay until the effects of these decisions are realised and traced back to their actual root causes. For instance, the length and intensity of crew training programs can directly effect both the availability and amount of crew to operate the fleet. In the meantime, the training programs also effect the number of future instructors to train personnel leading to a sequential effect on the fleet availability. The methodology, and the underpinning SD model, were developed iteratively and interactively through close engagement with experts from the various aspects of the submarine support system. The model development process followed standard SD practice (Bowers et al., 2017; Elsawah, Ryan, et al., 2018; Jnitova et al., 2017), based on examples published in the field classics (e.g., Sterman (2010)). The project was designed as an agile development process composed of iterative cycles of three phases: (1) deliverable scoping and business rules elicitation where the aim was to identify the class of problems underpinning the strategy development, interrelationships, and how these are contextualised in the case study context (see Elsawah, Ryan, et al. (2018)). This has led to identifying modelling requirements and functionalities, as well as the business rules underpinning the model’s logic; (2) model implementation or the technical development and testing; and (3) strategy set up and analysis, where the methodology was used by the client to examine strategies, leading into adjustments and refinements.

4. Decision-support methodology

In this section, we propose a decision-support methodology for strategy development in dynamic asset and resource planning in a defence context, as discussed in Section 2.1. The main idea behind the methodology is to leverage the power of SD simulation models. Table 2 presents an overview of the modelling process in SD projects in general together with how our process has been carried out, following the modelling steps from Sterman (2010). From the user viewpoint, the decision-support methodology is organised into three steps (See Figure 2): (1) Design an asset management strategy, (2) Evaluate strategy using an SD simulation model, and (3) Generate performance indicators and analyse. In Sections 4.1 to 4.3, we describe each step in terms of the employed concepts and tools, using the submarine capability support system as a case study.

Figure 2. The proposed decision support methodology.

Table 2. Overview of the modelling process.

Problem articulation	Understanding the problem and its sub-problems and articulating them and selecting the problem/system boundary	SD modelling approach was selected after several meetings with stakeholder and clients who are also experts and managers from ADF. The modelling goal was to investigate high-value strategic military asset behaviour to answer questions including what are the factors effecting the military asset availability (i.e., operational readiness) in the long and the short-term? And, the question: what are the optimum levels of resources (workforce, maintenance,etc.) to maximise asset availability without leading excessive resource under utilisation?
Dynamic hypothesis	Formulating a dynamic hypothesis related to the problem dynamics that are endogenously generated from the feedback structure within the selected boundaries (Linnéusson et al., 2018). The exploring the relevant literature on problem domain and studying the situation at the ADF are essential activities to undertake and to formulate a dynamic hypothesis	The asset model has been divided into submodels including maintenance, workforce, crew and fleet management. The focus was the interrelation between submodels and their effect to final key performance indicator mainly asset availability. Further, to investigate adequate level of resources (e.g., docks) to maintain high asset availability, and trade-off performance of different resources were secondary focus.
Formulation	Developing and constructing a simulation model based on the dynamic problem and test the hypotheses. According to Sterman (2010), this is generally an intuitive process; and therefore, there are no universally accepted descriptions of how to implement system dynamics projects. The built model consists of specifications of the structure and business rules, input parameter estimations, behavioural relationships between submodules of system, initial conditions and settings, and tests for model stability and consistency.	The asset management and resource planning simulation model was created by interacting decision-makers (who are also clients) with dual purposes in mind first to generate learning of effect between different aspects of the modelled system, and to able to analyse the existing system. Ideas and techniques from existing literature were used during model development stage. Moreover, real-world definitions based on meetings with client and interview with ADF personnel (one of whom is the co-author of this paper) were used to build the model. Literature review led to identifying sub-structures and their relationship with each other. Review stage also result in simplification of some aspects of the model such as different workforce trades not included in the final model.
Testing	Investigating if the model sufficiently reproduces with respect to its objectives. SD models are considered as causal-descriptive models. Hence, the validation of the internal structure of the model, and its potential to explain how the system behaves are crucial. The usefulness of the model in explaining problem situation determines model’s validity (Sterman, 2010).	During model development stage, tests for consistency were utilised. In these tests, formulation for each equation and each sub-structure thoroughly checked and processed. All feedback and reinforcement loops were closed and units were checked. The manual and automated parameter testing (via Java code templates) spotted improvements for consistency. Walk-through workshops were organised with client and experts from ADF to confirm structure and parameter estimations. Furthermore, extreme condition and behaviour sensitivity tests were conducted in the Anylogic software.
Policy formulation and evaluation	Using the SD model to investigate realistic scenarios, also utilising it to explore possible “what if” scenarios. Testing sensitivity of important input parameters and observing their effect on strategic policies for asset management.	Fleet transition policies suggested by Hellyer (2018) are tested to demonstrate the applicability of the model. Further, a set of “what if” scenarios were tested on several important decision parameters and their effect on key performance indicators were investigated.

4.1. Design an asset management strategy(ies)

4.1.1. Concepts: Strategy, strategy components and decision variables

In the first step, the user designs a strategy by determining those strategy components, and constituent decision variables, to be evaluated and compared. The strategy concept covers the decision variables related to the problems described in Section 2.1. Here, by strategy, we mean a set of the input variables (also called factors or parameters) that determines the “design” or “configuration” of the strategy in question. Essentially, a strategy is a vector composed of all input variables. Strategies are made of six interlinked components, where each component constitutes a set of variables for the user to set values for (See Table 3). These strategy components are:

1. Asset composition component covers decision variables related to the mix and types or classes, such as the number of submarine classes and size.

2. Regions component covers decision variables related to resource planning and supply network design. A region is a management unit where assets and resources exit. For example, a region which has training facilities can serve the training activity for workforce. Regions serve functions in the system by undertaking activities in the asset lifecycle. Regions vary in their capacity to undertake different activities for different assets (i.e., Regions-Activity-Capacity). For example, a region may have the capacity to undertake a particular maintenance activity for a particular class. Region can share resources according to the resource sharing rules determined by the user (i.e., Regions-Preference- Matrix). This gives the user the flexibility to determine their management units, as well as to examine various assumptions about the how resources are supplied and requirements are met.

3. Asset service schedule component covers decision variables related to the rate by which assets enter and exit service, and their home region.

4. Asset lifecycle component covers variables related to the rates by which assets move through various activities or states during its lifespan.

5. Asset maintenance component covers variables related to the maintenance strategy, including resource requirements and maintenance regime (i.e., condition-based or schedule-based).

6. Workforce planning component covers variables related to the building up of the workforce, including recruitment and training. Workforce is divided into two types: workforce (in uniform), and workforce (industry).

Table 3. Model input parameters for the design of an asset (submarine) management strategy.

Strategy component	Input parameters	Decision problems to be addressed
Asset composition	Number of submarine classes	Fleet size and mix
	Number of submarines per class
Regions	Number of regions in the support system	Resource planning,
	Regions-Activity-Capacity matrix	and supply network
	Regions-Preference-Matrix	design
Asset service schedule	Rate of submarines entry to service, for each region	Fleet transition, and
	Rate of submarines exit of service, for each region	lifetime extension
Asset lifecycle	Number of activity types	Lifecycle planning
	Sequence of activities
	Expected duration (for each activity type)
Asset maintenance	Number of maintenance activity types	Maintenance regime
	Maintenance regime (for each maintenance activity)	design
	Condition-based maintenance: threshold for maintenance triggers)
	Docking facilities required (for each maintenance activity type)	Maintenance,
	Docking facilities building rate (for each region)	resource planning
	Docking facilities decommissioning rate (for each region)	facilities location
	Workforce (industry) required (for each maintenance activity type)
Workforce planning	Initial (in uniform) workforce levels, at each stage in the pipeline: recruits, trainees, trained crew. (for each region)	Workforce planning and training
	Minimum number of trained crew
	Recruitment rate (for each region)
	Attrition rate (for each region)
	Expected lead time for trainees to become crew
	Capacity of off-shore training
	Workforce (industry) recruitment rate (for each region)
	Workforce (industry) attrition rate (for each region)

Figure 3 is a causal loop diagram (CLD) of the causal structure deriving the availability of the military asset management and resource planning support system. The CLD presents a high view of the key causal variables, the interlocking feedback loops, and delays that drive the system performance, in particular, operational readiness (i.e., fleet availability).

Figure 3. The causal loop diagram (CLD) for military asset management and resource planning system.

We would like to highlight the key feedback loops that exist in the system. First is the reinforcing dynamics (R1) created by using the assets for both operational and training purposes. Operational readiness drives the number of operational days that the asset could be utilised. Given that the asset is used for both operations and training, the number of operational days drives the rate by which personnel are able to complete their training requirements. The increase in the rate of generating trainees results in an increase in the availability of crew, which in effect drives operational readiness. This implies that the decrease in operational readiness does not only influence immediate capability gaps, but it has a delayed consequence in terms of the effect on training capacity, and therefore crew readiness. Other reinforcing loops (R2 and R3) capture the relationships between the availability of resources (i.e., facilities, equipment, personnel) for the two critical lifecycle activities (maintenance and licencing) and resulting delays. For example, the availability of additional docks results in shorter maintenance delays, while the latter results in an increase in the availability of docks (as more docks are less busy with).

Balancing feedback loops (denoted by B) present the control mechanisms that exist in the system. B1 captures the decision-making process of increasing maintenance resource capacity through tapping to external resources, which adds to the inter-region transaction cost. This presents a trade-off decision for balancing capability gaps with inter-region transactions. B2 captures another control mechanism, through which the decision to withdraw the old fleet is driven by the capability gap. Whereas this may seem effective in increasing operational readiness, it affects the maintenance delays (i.e., more assets to be maintained).

The exact military asset system dynamics depend on the setting of external parameters associated with the elements of the system, for example, availability requirements as determined by the Government and crew separation rates. To analyse the dynamic behaviour of this system in more detail, and in order to address the effectiveness of different decision options, a simulation model is developed and used to analyse options.

4.1.2. Tools: Simulation and strategy configuration

The user can design a set of management strategies to be evaluated using two MS Excel files: Simulation Configuration File and Strategy Configuration File. The Simulation Configuration File allows the user to determine parameters related to the simulation setup:

• The planning horizon for evaluating strategies (i.e., simulation run length).

• The number of strategies to be examined.

• The file paths for the Strategy Configuration Files of strategies to be examined.

The Strategy Configuration File allows the user to determine the parameters related to the strategy design. The files have macro functionalities that walk the user easily through the process of populating the strategy components.

4.2. Simulate strategy performance using SD model

4.2.1. Concepts: Hierarchical process model and modules structure

In the second step, the user evaluates the designed strategy(ies) using the SD model. The model is automatically configured to run simulations for each strategy. Assets move through several stages (e.g., processes and/or activities) during their lifecycle. This observation is taken into account while constructing the model’s hierarchy as shown in Figure 4. The abstract concepts such as the asset lifecycle correspond to the high-level in the model’s hierarchy. On the contrary, the physical elements of the system such as flow rates correspond to the low-levels in the hierarchy. The parametric values of flows are determined by the decision maker depending on a particular strategy that wanted to be investigated. Resources are consumed by activities based on some rules and the user-defined input values to produce effects. Rules consist of: (i) decision/business rules such as the resource allocation rule to determine priorities among activities and assets, and (ii) rules used to capture the physical constraints such as the non-negativity restriction on stock values. It should be also noted that the proposed hierarchical view of the asset resembles the conceptualisation of the system that the submarine support system designers utilise. Further, the hierarchical design enables decision makers to perform a top-down analysis by paying special attention to elements and behaviour that may have an effect on performance (R. G. Coyle, 1992).

Figure 4. The hierarchical model design (green box is java functions, brown boxes denote input from the strategy components).

Figure 5 provides the stock-and-flow diagram (SFD) of the developed SD model.³ The most essential is to calculate the number of vessels/submarines are ready for operations at time $\mathit{t}$ which is calculated via Eq (1(1) $\begin{array}{rl}& \mathrm{New}\mathrm{vessels}\mathrm{waiting}\mathrm{to}\mathrm{start}\mathrm{operations}(\mathit{t})\text{break}={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{vessels}\mathrm{entering}\mathrm{service}-\mathrm{new}\mathrm{vessels}\mathrm{starting}\mathrm{operations}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{Initial}\mathrm{number}\mathrm{of}\mathrm{new}\mathrm{vessels}\text{break}\mathrm{waiting}\mathrm{to}\mathrm{start}\mathrm{operations}\end{array}$(1) ).

Figure 5. The stock-and-flow structure and modules in the developed model. Colours show the information exchange among modules.

(1) $\begin{array}{rl}& \mathrm{New}\mathrm{vessels}\mathrm{waiting}\mathrm{to}\mathrm{start}\mathrm{operations}(\mathit{t})\text{break}={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{vessels}\mathrm{entering}\mathrm{service}-\mathrm{new}\mathrm{vessels}\mathrm{starting}\mathrm{operations}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{Initial}\mathrm{number}\mathrm{of}\mathrm{new}\mathrm{vessels}\text{break}\mathrm{waiting}\mathrm{to}\mathrm{start}\mathrm{operations}\end{array}$(1)

When a submarine starts the operation, it is require to allocated sufficient amount of workforce depending on crewing structure of the particular time of submarine. This is ensured by Eq (2(2) $\mathrm{New}\mathrm{vessels}\mathrm{starting}\mathrm{operations}(\mathit{t})\text{break}=\mathit{f}(\mathrm{Allocated}\mathrm{work}\mathrm{force})$(2) ). Further, the number of submarine in operation at time $\mathit{t}$ is calculated by Eq (3(3) $\begin{array}{rl}& \mathrm{Vessels}\mathrm{in}\mathrm{operations}(\mathit{t})\text{break}={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{new}\mathrm{vessels}\mathrm{starting}\mathrm{operations}-\mathrm{vessels}\mathrm{exiting}\mathrm{licensing}\\ & -\mathrm{vessels}\mathrm{exiting}\mathrm{operation}]\mathrm{\Delta }\mathit{t}+\mathrm{Initial}\mathrm{number}\mathrm{of}\mathrm{vessels}\mathrm{in}\mathrm{operations}\end{array}$(3) ) which is a function of initial number of submarines in operation at time 0 ($\mathit{t}=0$) plus new submarines starting operations at time $\mathit{t}$ and minus sum of the number of submarines exiting operations and licencing activity.

(2) $\mathrm{New}\mathrm{vessels}\mathrm{starting}\mathrm{operations}(\mathit{t})\text{break}=\mathit{f}(\mathrm{Allocated}\mathrm{work}\mathrm{force})$(2)

(3) $\begin{array}{rl}& \mathrm{Vessels}\mathrm{in}\mathrm{operations}(\mathit{t})\text{break}={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{new}\mathrm{vessels}\mathrm{starting}\mathrm{operations}-\mathrm{vessels}\mathrm{exiting}\mathrm{licensing}\\ & -\mathrm{vessels}\mathrm{exiting}\mathrm{operation}]\mathrm{\Delta }\mathit{t}+\mathrm{Initial}\mathrm{number}\mathrm{of}\mathrm{vessels}\mathrm{in}\mathrm{operations}\end{array}$(3)

Another important component of the model is workforce in particular uniform (crew) and non-uniform workforce. Non-uniform workforce (crew) related stock and flow equations are presented between Eqs (4(4) $\begin{array}{rl}& \mathrm{Recruits}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{recruitment}\mathrm{rate}-\mathrm{attrition}\mathrm{rate}\text{break}-\mathrm{off}-\mathrm{shore}\mathrm{trainees}\mathrm{generating}\mathrm{rate}]\mathrm{\Delta }\mathit{t}\\ & \text{break}+\mathrm{initial}\mathrm{number}\mathrm{of}\mathrm{recruits}\end{array}$(4) )-(9(9) $\begin{array}{rl}& \mathrm{Trained}\mathrm{becoming}\mathrm{crew}(\mathit{t})\\ & =\mathrm{Off}-\mathrm{shore}\mathrm{trainees}\mathrm{generating}\mathrm{rate}\\ & (\mathit{t}-\mathrm{expected}\mathrm{lead}\mathrm{time}\mathrm{for}\text{break}\mathrm{trainees}\mathrm{to}\mathrm{become}\mathrm{crew})\end{array}$(9) ). These equations model the non-uniform workforce career progression stages from recruitment to trained crew on submarines. To illustrate, the number of trained crew (that can be allocated to a submarine) at time $\mathit{t}$ is given in Eq (7(7) $\begin{array}{rl}& \mathrm{Trained}\mathrm{crew}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{trainees}\mathrm{becoming}\mathrm{crew}\\ & -\mathrm{allocated}\mathrm{workforce}-\mathrm{attrition}\mathrm{rate}\\ & +\mathrm{released}\mathrm{crew}]\mathrm{\Delta }\mathit{t}+\mathrm{initial}\mathrm{trained}\mathrm{crew}\end{array}$(7) ).

(4) $\begin{array}{rl}& \mathrm{Recruits}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{recruitment}\mathrm{rate}-\mathrm{attrition}\mathrm{rate}\text{break}-\mathrm{off}-\mathrm{shore}\mathrm{trainees}\mathrm{generating}\mathrm{rate}]\mathrm{\Delta }\mathit{t}\\ & \text{break}+\mathrm{initial}\mathrm{number}\mathrm{of}\mathrm{recruits}\end{array}$(4)

(5) $\mathrm{Recruits}\mathrm{attrition}\mathrm{rate}(\mathit{t})=\mathrm{Recruits}(\mathit{t})\times -\mathrm{attrition}\mathrm{fraction}$(5)

(6) $\begin{array}{rl}& \mathrm{Trainees}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{off}-\mathrm{shore}\mathrm{trainees}\\ & -\mathrm{trainees}\mathrm{becoming}\mathrm{crew}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{initial}\mathrm{number}\mathrm{of}\mathrm{trainees}\end{array}$(6)

(7) $\begin{array}{rl}& \mathrm{Trained}\mathrm{crew}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{trainees}\mathrm{becoming}\mathrm{crew}\\ & -\mathrm{allocated}\mathrm{workforce}-\mathrm{attrition}\mathrm{rate}\\ & +\mathrm{released}\mathrm{crew}]\mathrm{\Delta }\mathit{t}+\mathrm{initial}\mathrm{trained}\mathrm{crew}\end{array}$(7)

(8) $\mathrm{Trained}\mathrm{crew}\mathrm{attrition}\mathrm{rate}=\mathrm{Trained}\mathrm{crew}\times -\mathrm{attrition}\mathrm{fraction}$(8)

(9) $\begin{array}{rl}& \mathrm{Trained}\mathrm{becoming}\mathrm{crew}(\mathit{t})\\ & =\mathrm{Off}-\mathrm{shore}\mathrm{trainees}\mathrm{generating}\mathrm{rate}\\ & (\mathit{t}-\mathrm{expected}\mathrm{lead}\mathrm{time}\mathrm{for}\text{break}\mathrm{trainees}\mathrm{to}\mathrm{become}\mathrm{crew})\end{array}$(9)

Another workforce type in our model is non-uniform workforce which can be considered as maintenance technician workforce. This workforce type is required to perform maintenance activities and they can be hired outside of defence workforce and they don’t need to go through career progression stages unlike uniform-workforce. Eq (10(10) $\begin{array}{rl}& \mathrm{Work}\mathrm{force}\mathrm{industry}\mathrm{available}(\mathit{t})\text{break}={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{recruitment}\mathrm{rate}+\mathrm{released}\mathrm{workforce}-\mathrm{allocated}\mathrm{workforce}\\ & -\mathrm{attrition}\mathrm{rate}]\mathrm{\Delta }\mathit{t}+\mathrm{initial}\mathrm{industry}\mathrm{workforce}\mathrm{available}\end{array}$(10) ) shows the number of non-uniform (industry) workforce available at time $\mathit{t}$.

(10) $\begin{array}{rl}& \mathrm{Work}\mathrm{force}\mathrm{industry}\mathrm{available}(\mathit{t})\text{break}={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{recruitment}\mathrm{rate}+\mathrm{released}\mathrm{workforce}-\mathrm{allocated}\mathrm{workforce}\\ & -\mathrm{attrition}\mathrm{rate}]\mathrm{\Delta }\mathit{t}+\mathrm{initial}\mathrm{industry}\mathrm{workforce}\mathrm{available}\end{array}$(10)

Appendix A Section contains equations used under each module of the model.

4.2.2. Tools: AnyLogic java library

The model is implemented as a Java library using AnyLogic${ }^8$ Professional Licence. The library includes three types of modules: (1) Java scripts functions that execute particular functions (or utilities) in the model, such as configuring modules based on the configuration files, (2) SD modules, and (3) datasets as input from the Strategy Configuration File. The model runs as a Java applet in the user’s browser, a crucial software requirement for the defence users who have limited access to software installation.

4.3. Generate performance indicators and analyse results

4.3.1. Concepts: Multi-level performance indicators

Whereas the output from the SD modules generates behaviour of the simulated effects, performance indicators focus on those aspects of behaviour that are relevant for objectives (Hall et al., 2004). Performance indicators represent a subset of the model’s state variables that are relevant to the stakeholders’ objectives. Therefore, a library of functions is developed and used to calculate a set of performance indicators for each strategy to allow for objective assessment and comparison of strategies. Performance indicators are designed around three levels:

• Fleet or class level (e.g., number of assets in operations over time, capability gap calculated as the number of weeks where the number of assets in operations is below than a user-defined benchmark).

• Activity level (e.g., maintenance delays by maintenance type and region, number of inter-region transactions).

• Resource level (e.g., resource utilisation rates by resource type and region).

This hierarchical view of the strategy performance gives users the ability to diagnose different aspects of the strategic outcomes in a holistic way. On one hand, it enables an aggregate or high level view of the strategy performance. On the other hand, it enables drilling into the granularity needed to explain the generated dynamics (See Van Looy and Shafagatova (2016) and references therein). For example, the user can use the fleet level indicators to examine the overall performance against requirements, and the risk of shortfalls. However, this information on its own, does not give insight into the causes of the shortfall period. Overlaying this with information about activity performance, the user can see that this shortfall is caused by delays in maintenance activities. A more detailed interrogation of the performance indicators of this activity’s input resources will highlight those constraining resources that limit the capacity to meet requirements. This allows for linking low-level performance to high-level performance, which is an essential need for strategy development and implementation.

4.3.2. Tools: Strategy evaluation and comparison output files

The model generates two MS Excel files: Simulation Run Output file and Simulation Comparison Output file. The Simulation Run Output file stores output data from a single strategy. If the user uploads more than one strategy, the Simulation Comparison Output file gives a comparison of strategies’ performance indicators. Both files have built-in data charts for visualising indicators.

4.4. Validation of the simulation model

Validation of the simulation models, in particular the SD model, is essential in order to increase the confidence the developed model. In this direction, we apply the formal validation tests and steps for SD models as described in Y. Barlas (1996, 2016):

• First, we conducted direct structure tests. Directed model structure tests are mostly qualitative comparisons with the literature and knowledge about the real-world system provided by the defence experts from the ADF. The experts were satisfied with the final results since the model structure and results obtained were consistent with real system.

• Second, we performed several structure-oriented behaviour tests. Structure-oriented behaviour tests are qualitative and use the developed SD simulation model to evaluate the capability of the structures to represent the expected feedback behaviours (Linnéusson et al., 2018). For example, we first applied zero workforce recruitment and observed the workforce shortage through the planning period (which behaved as expected). Second, we tested very high maintenance duration time which led to high waiting times and almost zero fleet availability as time progress (this was also consistent with our expectations). Further, we (extremely) increased the recruitment and we observed a delayed effect in workforce surplus which was also consistent with career progression modelled in the SD. We also applied behaviour sensitivity tests (e.g., the effect of changes in separation and recruitment rates to availability of assets), and boundary adequacy tests which proved validity of the model since all results were consistent with the initial expectations.

• Lastly, it should be noted that behaviour pattern tests do not yield added value to validate the model structure (Y. Barlas, 1996), but validate the generated behaviour of the model. Nevertheless, the conceptual model does not contain parameter values that require an application case study where input data is retrieved from the real-world system (Linnéusson et al., 2018).

The performed validations tests have (with some confidence) confirmed the overall model behaviour. We also include tests to justify the modelling assumptions, with the help of defence experts one of whom is the co-authors of this paper. Moreover, the application of a real-life inspired case study (see Section 5) has resulted in strengthening the structure-oriented behaviour test, by exploring errors in the SD model. As a result, model equations have been updated by adding some parameters and new structures.

5. Methodology demonstration

To demonstrate the applicability and usefulness of the proposed methodology, we present a hypothetical scenario(s) where the user would like to use different functionalities of the SD model and to test the dynamic effects of different decisions on the fleet (i.e., submarines) performance over its lifetime (i.e., life-cycle dynamics). Because of the sensitive nature of the real-world application, hypothetical scenarios and data are used to illustrate the methodology. The tested scenarios can be undertaken in any order, but we presented them in a particular order in the following description to provide a holistic view of different aspects of the methodology. The initial base scenario is based on the following assumptions (unless explicitly adjusted in the following discussion):

(i) Planning horizon length is chosen as 50 years to capture the entire lifecycle of submarines and to observe the delayed effects of decisions taken during submarines’ lifetime after the retirements. The simulation model is run with weekly time steps.

(ii) There are two types of submarines; i.e., Collins Class Submarine (CCSM) and Future Submarine (FSM). The old fleet consists of only CCSM.

(iii) The old fleet size; i.e., the number of CCSM, is equal to six.

(iv) A capability gap occurs if the number of total operational submarines (CCSM and FSM) in a particular week is less than the predefined threshold value. The total capability gap is defined as the cumulative sum of the capability gaps until the end of the planning period.

(v) The predefined threshold value for total fleet capability is six submarines for any week during the planning horizon.

(vi) The primary objective of the decision maker is to replace the old fleet with the new fleet by achieving the minimum capability gap and providing maximum fleet availability.

5.1. Evaluation of different options for fleet transition

As a first step, the user is interested to examine how the fleet availability and capability (i.e., the number of operational submarines) would change during the planning horizon if different fleet-size, fleet-mix, and fleet-transition (i.e., transition from the old fleet to the new fleet) strategies are chosen. In this direction, the users must populate/parametrise the Fleet Schedule modules (Service Entry and Service Exit Modules described in Subsection 4.2.1) with the number of new submarines to be acquired, the timing of acquisitions and the timing of retirements for the old fleet for each option/strategy wanted to be examined. In addition, the Resource modules (Workforce and Capacity Allocation Modules described in Subsection 4.2.1) have to be populated with the current and anticipated future number of resources (e.g., docks, maintenance manpower, crew, trainees, etc.), and the Activity modules (Licencing, Maintenance and Operations Module described in Subsection 4.2.1) have to include the list and order of activities performed on or by each submarine. Moreover, the licencing, maintenance and operations modules have to also contain the duration of these activities together with the required amount of resources to perform these activities during the life of the fleet. The initial forecast of the ADF is the commissioning of 12 FSMs during the 50 years period (Department of Defence, 2016; Hellyer, 2018). Thus, in the first option, the user aims to retire the entire old fleet as early as possible (within the first 10 years of the planning period) without any lifetime extension and commencing all-new fleet acquisitions within the first 15 years. In the second option, the lifespan of all of the old fleet is extended for an additional 20 years while again all of the new fleet is acquired within the first 15 years.⁴

Figure 6 presents availability and capability gap metrics for Options 1 and 2. Interestingly, an extension of lifespan for 20 years does not significantly increase the availability; e.g., the number of operational submarines almost follow the same pattern (as in Figure 6(a)) and, further, the submarines are underutilised compared to no-extension option (Figure 6(b)). Thus, in the third option, instead of spending resources for extending lifetime of the old fleet, the user wishes to examine the option of acquiring six new additional submarines in the second half of the planning period (around 25 years later).

Figure 6. Comparison of the performance (availability) of Option 1 and 2.

Figure 7 demonstrates that introducing new submarines leads to a reduction in the capability gap (Figure 7(d)) and a higher accumulated total time spent in operations (i.e., useful time or utilised) as shown in Figure 7(c). However, the user will notice an increase in the capability gap and a decrease in the percent availability in the last 10 years (after week 2000 in Figures 7(b) and 7(d)). To solve this issue without any need for an additional (funding) acquisition, the commissioning of six new submarines in Option 3 is split into two equal acquisitions (three submarines per acquisition) in the fourth option.

Figure 7. Comparison of Option 1, Option 2 and Option 3 in terms of fleet performance (availability).

In the fourth option, the commissioning of the second lot of three new submarines is deferred till the last 10 years of the planning period. Figure 8 shows that Option 4; i.e., splitting/spreading the acquisitions over the planning period instead of one large batch of commissioning, positively affects the capability of the fleet (see Figure 8(d)). Similarly, the number of operational submarines and the percent of available submarines in the last 10 years of the planning horizon improve with this option (Figure 8(a,b)). Moreover, Figure 8(c) presents that postponed acquisition of the new fleet does not lead to any diminishment in operational utilisation.

Figure 8. Comparison of Option 3 and Option 4 in terms of fleet performance (availability).

To get a complete overview of options investigated so far, the users may utilise a visualisation toolbox of the proposed decision-support system. To illustrate, the toolbox produces box-plot graphs as in Figure 9 to enable decision makers to investigate fleet availability metrics during the planning horizon. Results conclude that Option 2 is inferior compared to the rest of the options studied. Furthermore, even though Option 3 and Option 4 have a higher number of operational submarines compared to Option 1, all of these three options have almost identical fleet availability percentage distribution, which indirectly indicates both Option 3 and Option 4 have some underutilised submarines due to the lack of vital resources.

Figure 9. The distribution of fleet availability metrics during the planning horizon for each option.

Figure 10 is another visualisation presenting the sensitiveness of options concerning capability threshold value as cumulative histograms. In the figure, y-axes indicate the total number of weeks that the number of operational submarines is under a given capability threshold. Figure 10 shows that Option 1 and Option 2 never achieve nine operational submarines. Moreover, in Option 3, the number of operational submarines can reach up to 11 which is the highest value among the four options compared.

Figure 10. The sensitivity of options (1–4) with respect to capability threshold.

5.2. Improving the fleet availability by activity level analysis functionality

It may not be economically or politically feasible to acquire more submarines than the initially decided number, which means that Option 3 and Option 4 may be undesirable. In the previous section, the user observed that lifespan extension of the old fleet by 20-years does not increase the fleet capability (see Figure 6(d)). Further, the lifespan extension causes a decrease in operational availability (see Figure 9(b)) compared to no extension strategy; i.e., Option 1. To understand the underlying reasons behind the poor performance, decision makers may want to conduct additional analysis on the life cycle of the submarines to identify bottleneck activities by using the activity module of the developed decision support methodology.

A submarine goes through several stages during its life cycle. In each stage, an activity (such as licencing and maintenance) is performed by or on a submarine. Thus, time spent (including waiting times) in these activities has to be investigated to identify the root cause. Figure 11 compares times spent for licencing, waiting for maintenance, waiting for industry manpower that are needed for maintenance and waiting for docks to become available for Option 1 and Option 2. The maintenance activity is causing low operational availability and a high capability gap for Option 2 compared to Option 1.

Figure 11. Waiting time for the activities involved in Option 1 and Option 2.

It can be also observed that the total waiting times for docks and industry manpower start increasing sharply around week 500. In Option 1, all the old fleet is retired by year 10 (around week 500) which decreases the total number of active submarines in the fleet that needs maintenance. Decision makers may initially decide to build a dock around week 200 to improve the performance of Option 2. Option 5 includes a 20-year life extension of old submarines and the construction of an additional dock in week 200. Figure 12 shows how additional dock capacity affects the waiting times. After construction of the additional dock, the total waiting time for available docks reduces drastically for Option 5. On the other hand, the bottleneck shifts to industry manpower, and it becomes a scarce resource in Option 5.

Figure 12. Effect of an additional dock capacity on Option 1 and 2.

The user may therefore introduce Option 6 to examine the simultaneous effect of both increasing dock and industry manpower capacity. The manpower capacity is doubled at the commissioning date of the additional dock (around week 200) in Option 6. Figure 13 shows that manpower capacity increase solves the long waiting time problem. However, an increase in waiting time for docks is observed compared to the previous option (e.g., Option 5).

Figure 13. Effect of an additional manpower capacity on Option 2.

5.3. Investigating resources to identify the under-utilised capacity

In the previous subsection, users observed that the doubling industry manpower capacity in the Option 6 resulted in almost zero waiting time for manpower capacity needed for maintenance. Hence, it is possible to have some idle manpower capacity in this option. In the next step, the users invoke resource modules (workforce and capacity allocation modules) of the decision support tool to find under- or over-utilised resources in the system. Figure 14 shows the fluctuations in industry manpower utilisation, crew utilisation, dock utilisation, and total crew available for Option 5 and Option 6.

Figure 14. Resource utilisation for Options 5 and 6.

The figure presents that doubling the manpower capacity causes under-utilisation of capacity especially early and late in the planning period. Also, the current (employed in all options so far) crew recruitment strategy leads to an almost exponential decrease in the utilisation of crew, and crew redundancy. In this direction, decision makers may use the what-if analysis toolbox integrated into the resource module to find the optimal configuration of manpower and capacity.

Figure 15 demonstrates a two-way sensitivity analysis showing how the capability gap for Option 6 alters with a different combination of crew recruitment and manpower separation (retirement) strategies. In the crew recruitment rate dimension, the value of 1 denotes the current rate of recruitment. When the only crew recruitment rate is considered, the rate of 0.3 achieves the minimum capability gap. This means that a 70% reduction in crew recruitment is possible without causing any capability gap increase. In the manpower separation dimension, the value of 0 denotes the current level of industry manpower in Option 6; i.e., after doubling manpower capacity. Figure 15 shows that the minimum capability gap is attained between 0.0 and 0.2 manpower separation rates. It concludes that reducing the current level of manpower by up to 20% would not result in any increase in the capability gap.

Figure 15. Optimising resource utilisation via what-if toolbox for Option 6.

After conducting resource level analysis, decision makers decide to set crew recruitment rate as 0.3 and manpower separation multiplier as 0.2 in the new resource-optimised strategy called Option 7. Figure 16 presents a comparison of all options with respect to the capability gap. It is obvious that resource optimised Option 7 behaves as well as Option 6, which reveals the power of resource level analysis.

Figure 16. Comparison of Options (1–7) performance (capability gap).

5.4. Summary of the demonstration

In this section, we demonstrated how the proposed framework improves the quality of decisions on fleet life-cycle management by discussing and evaluating several options proposed by decision makers. Figure 17 shows how fleet sizes changes under each transition option throughout the planning period. In addition to fleet size information, acquisition and retirement times of submarines can be retrieved from this figure.

Figure 17. Change of fleet sizes for each transition option.

We summarise simulation outputs and our findings for all transition options investigated in Tables 4 and 5. Table 4 provides a brief description of each option and presents primary performance indicators (e.g., fleet availability, capability gap, and average number of submarines in each week), and Table 5 includes secondary performance metrics such as waiting times for activities (e.g., maintenance and licencing) and resource utilisation (e.g., crew, manpower, and docks) for each option investigated. The obtained results show that it is possible to reach the goals (e.g., high fleet availability and low capability gap) by acquiring a fewer number of submarines. To illustrate, both Option 3 and Option 4 acquire 18 new submarines (FSM Class) and obtain only 47% and 49% fleet availability and provide 5.99 and 6.01 available submarines per week, respectively. On the other hand, Option 7 acquires only 12 new submarines and achieves almost 60% fleet availability, and provides 7.68 available submarines per week. Further, the proper and systematic use of the different modules embedded in decision-support tool enables optimisation of resource capacities without compromising the availability of assets. For example, Option 7 uses 70% less crew and 20% less industry manpower compared to Option 6 (see Subsection 5.3) but still achieves very close fleet performance as in Option 6.

Table 4. Summary of different fleet transition options and primary performance indicators.

Option	Short description	% Average availability of fleet	Average submarines in operation	Total capability gap	Total time in operation
			(submarine per week)	(submarines)	(week)
Option 1	No lifetime extension	49	5.0	1694	6565
Option 2	20-years lifetime extension	40	5.0	1760	6549
Option 3	No lifetime extension and 6 additional FSM submarines	48	6.0	1042	7813
Option 4	No lifetime extension and split acquisition of 6 additional FSM submarines	49	6.0	972	7813
Option 5	20-years lifetime extension and additional dock capacity	41	5.3	1561	6924
Option 6	20-years lifetime extension and additional dock and manpower capacity	61	7.9	298	10356
Option 7	Resource optimised policy	59	7.6	409	10022

Table 5. Summary of secondary performance indicators and resource utilisation.

Option	Average Waiting Time	Total Time	Total Time in Waiting	Total Time in	Average Industry	Average Crew	Average Dock
	to Start Maintenance (week)	in Licensing (week)	for Industry Manpower (week)	Waiting for Docks (week)	Manpower Utilisation (%)	Utilisation (%)	Utilisation (%)
Option 1	50	938	3916	4580	51	16	31
Option 2	102	924	6106	8690	49	17	31
Option 3	43	1076	7670	5362	54	18	34
Option 4	44	1076	6533	4933	54	18	34
Option 5	86	982	14588	294	52	17	14
Option 6	47	1606	89	2299	48	22	23
Option 7	47	1527	1375	2299	53	22	22

6. Concluding remarks: Methodology strengths, limitations, and future research

In this paper, we show that asset management and resource planning consist of a wide class of decision problems ranging from fleet mix to maintenance planning (please see Figure 1). We also conclude that existing studies in the current literature, particularly in the military domain, have only touched on a few aspects of asset planning by focusing on a particular subsystem (e.g., workforce planning or maintenance problems). Therefore, we develop a decision support methodology for holistic military asset management across the asset’s lifecycle, from acquisition to retirement. We argue that holistic methodologies are best handled by macroscopic models since macroscopic models capture the studied system at an aggregate level. In addition to providing a holistic system view, the developed decision support methodology has to be capable of addressing challenges (such as path-dependence and tightly coupled subsystems) associated with unstructured complex and dynamic decision problems encountered in military asset and resource planning. Throughout the paper, we explain that system dynamics (SD) is well-suited to deal with mentioned complexities due to its capability of capturing the feedback inter-dependencies between different parts of the system. As a consequence, the decision support methodology is built on an SD model for evaluating strategies for asset and resource planning throughout the asset’s lifecycle.

The methodology presented is developed and applied to support asset and resource planning in the context of the ADF, which is in the process of massive investments and modernisation decisions and challenges that are ideally suited to quantitative decision analysis methodologies that can look at the interactive and long-term effects of different options.

The developed methodology has several strengths that have the potential to improve decision making and strategy development in military asset and resource planning. First, the methodology is built on a hierarchical view of the military asset, its resources, and its performance indicators. From a decision-making perspective, this enables the linking of low-level and high-level performances, and sets focus on those critical aspects for strategic performance. In addition, the alignment between the stocks and flows and resource-based views provides a transparent trail of evidence to support decisions, and ease of communication about findings by collapsing and expanding parts of the model. Second, the methodology is built on a holistic view of the strategy, connecting various decision problems that are often treated in isolation. The interactions of decisions made by decision makers in different parts of the submarine enterprise can be examined; for example, illustrating why particular resource allocation options need to be made. The addition of the spatial feature of defining regions to the methodology adds the power of testing complicated strategic questions (Größler et al., 2015). Third, the methodology offers users the flexibility to construct and examine multiple strategies in an easy user-friendly MS Excel tool, with the view that each strategy represents a possible hypothesis. This flexibility promotes “assumption thinking” by modelling the system “as it can be” not only “as it is”.

Through our development and application of the methodology, we have identified limitations that warrant precaution. First, the use of the methodology requires a fair level of modelling awareness and assumption thinking that may not be possessed by some users. For example, our experience shows that some users have been pushing to add unnecessary details to the model to make it more “real”. This can defeat the purpose of the model as a macroscopic decision-analysis tool, add unnecessary complexity, and therefore, limit its flexibility. Having said that, this challenge is not unique to our methodology but shared across other modelling studies especially those with a strategic lens (Elsawah, Pierce, et al., 2017). Second, the generation of performance indicators and analysis of results is conducted through trial-and-error experimentation, where the user populates multiple Strategy Configuration Files to seek a desirable behaviour. Although these experiments should be conducted by an informed user, there is always the risk that another set of experiments could have led to better results (R. G. Coyle, 1985). This opens opportunities for enriching the methodology with techniques for pruning the solution space (i.e., optimisation and data-mining) and methods for guided experimentation (e.g., Chakladar (2016)). Moreover, the graphical user-interface and visualisation capability of the model is modest, opening opportunities for visual analytics techniques especially with a focus on uncertainty communication and a hierarchical view of results.

There are few opportunities to extend the methodology. The first extension is the further (vertical) development of the SD model. As the project progresses through other case studies, there is an opportunity to extend the model to include other resources and functions, such as spare management strategies. If it is needed, the modular and structured design enables decision makers to add more complexity and detail to the existing model. The second extension is the further (horizontal) development of the SD model. In order to meet operational objectives, different assets in military fleets such as frigates, aircraft carriers must interact with each other. This holistic point of view on force design adds new complexity that has to be addressed in the model. In this case, resource management gets challenging (i.e., resource allocation among assets) since some crucial resources are shared between multiple fleets. From a methodological point of view, subsystems of the military asset management model may be captured with other modelling approaches rather than SD such as a discrete event simulation can be used to model the maintenance subsystem and similarly, workforce (uniform) subsystem can be designed by an agent-based model. In this context, as a future research direction, a hybrid/multi-method asset management model, in which subsystems are modelled via discrete event and/or agent-based and aggregated with an SD model would be an interesting contribution to the literature.

Acknowledgment

We qualify that the analyses in this paper are not a reflection of the position, intent or opinions of the Royal Australian Navy or any defense organization.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

 This work is funded by the Capability Systems Centre, supported by the Australian Defence Force.

Notes

1. Live simulations consist of real people using real systems. Field exercises fall into this category.

2. Virtual simulations involve real people using simulated systems. These can be thought of as flight simulators or virtual environments. These types of simulations also include combined exercises where real people, using real systems, interact with and react to the actions of simulated people or systems (Hill et al., 2001).

3. We provide implementation screenshots of stock and flow diagrams in AnyLogic Software in the Appendix A Section in Figure A1, A2, A3. Further, the web page (https://unswcsc.bitbucket.io/) contains several supplementary materials for the manuscript including a video demo containing some implementation insights.

4. The fleet transition options discussed in this subsection are mainly motivated from the technical report written by Hellyer (2018).

Appendix A:

Anylogic implementation details

Figure A1. Anylogic state chart to model a submarine’s status throughout its lifecycle.

Figure A2. Anylogic stock and flow diagram to model crew’s career progression.

Figure A3. Anylogic stock and flow diagram to model submarine’s operational availability.

Model equations

Class level

(11) $\mathrm{New}\mathrm{vessels}\mathrm{waiting}\mathrm{to}\mathrm{start}\mathrm{operations}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}\left[\mathrm{vessels}\mathrm{entering}\mathrm{service}-\mathrm{new}\mathrm{vessels}\mathrm{starting}\mathrm{operations}\right]\mathrm{\Delta }\mathit{t}+\mathrm{Initial}\mathrm{number}\mathrm{of}\mathrm{new}\mathrm{vessels}\mathrm{waiting}\mathrm{to}\mathrm{start}\mathrm{operations}$(11)

(12) $\mathrm{New}\mathrm{vessels}\mathrm{starting}\mathrm{operations}(\mathit{t})=\mathit{f}(\mathrm{Allocated}\mathrm{workforce})$(12)

(13) $\begin{array}{rl}& \mathrm{Vessels}\mathrm{in}\mathrm{operations}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{new}\mathrm{vessels}\mathrm{starting}\mathrm{operations}-\mathrm{vessels}\mathrm{exiting}\mathrm{licensing}\\ & -\mathrm{vessels}\mathrm{exiting}\mathrm{operation}]\mathrm{\Delta }\mathit{t}+\mathrm{Initial}\mathrm{number}\mathrm{of}\mathrm{vessels}\mathrm{in}\mathrm{operations}\end{array}$(13)

(14) $\begin{array}{rl}& \mathrm{Class}\mathrm{total}\mathrm{operations}\mathrm{week}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}\mathrm{Adding}\mathrm{weeks}\mathit{t}\mathrm{in}\mathrm{operations}\mathrm{\Delta }\mathit{t}\\ & +\mathrm{initial}\mathrm{number}\mathrm{of}\mathrm{class}\mathrm{total}\mathrm{operations}\mathrm{weeks}\end{array}$(14)

(15) $\mathrm{Vessels}\mathrm{exiting}\mathrm{operations}(\mathit{t})=\mathrm{vessel}\mathrm{exiting}\mathrm{licensing}(\mathit{t}-\mathrm{Expected}\mathrm{lead}\mathrm{time}\mathrm{in}\mathrm{operations})$(15)

(16) $\mathrm{Class}\mathrm{total}\mathrm{seaweeks}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}\mathrm{Adding}\mathrm{weeks}\mathrm{\Delta }\mathit{t}+\mathrm{initial}\mathrm{number}\mathrm{o}\mathrm{class}\mathrm{total}\mathrm{seaweeks}$(16)

Vessel level

(17) $\mathrm{weeks}\mathrm{in}\mathrm{operations}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}\mathrm{Adding}\mathrm{weeks}\mathrm{in}\mathrm{operations}\mathrm{\Delta }\mathit{t}$(17)

(18) $\mathrm{Adding}\mathrm{seaweeks}=\mathrm{Adding}\mathrm{weeks}\times \mathrm{footnote}\mathrm{size}\mathrm{weeks}\mathrm{to}\mathrm{seaweeks}\mathrm{conversion}$(18)

Service exit

(19) $\mathrm{Vessels}\mathrm{in}\mathrm{retirement}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}\mathrm{vessels}\mathrm{retiring}\mathrm{\Delta }\mathit{t}$(19)

Workforce uniform capacity

(20) $\begin{array}{rl}& \mathrm{Recruits}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{recruitment}\mathrm{rate}-\mathrm{attrition}\mathrm{rate}-\mathrm{off}-\mathrm{shore}\mathrm{trainees}\mathrm{generating}\mathrm{rate}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{initial}\mathrm{number}\mathrm{of}\mathrm{recruits}\end{array}$(20)

(21) $\mathrm{Recruits}\mathrm{attrition}\mathrm{rate}(\mathit{t})=\mathrm{Recruits}(\mathit{t})\times -\mathrm{attrition}\mathrm{fraction}$(21)

(22) $\begin{array}{rl}& \mathrm{Trainees}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{off}-\mathrm{shore}\mathrm{traniess}-\mathrm{trainees}\mathrm{becoming}\mathrm{crew}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{initial}\mathrm{number}\mathrm{of}\mathrm{trainees}\end{array}$(22)

(23) $\begin{array}{rl}& \mathrm{Traineed}\mathrm{crew}(\mathit{t})={\int }_t^{t=\mathrm{\Delta }t}[\mathrm{trainees}\mathrm{becoming}\mathrm{crew}-\mathrm{allocated}\mathrm{workforce}-\mathrm{attrition}\mathrm{rate}+\mathrm{released}\mathrm{crew}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{initial}\mathrm{trained}\mathrm{crew}\end{array}$(23)

(24) $\mathrm{Trained}\mathrm{crew}\mathrm{attrition}\mathrm{rate}=\mathrm{Trained}\mathrm{crew}\times -\mathrm{attrition}\mathrm{fraction}$(24)

(25) $\begin{array}{rl}& \mathrm{Trained}\mathrm{becoming}\mathrm{crew}(\mathit{t})=\mathrm{Off}-\mathrm{shore}\mathrm{trainess}\mathrm{generating}\mathrm{rate}\\ & (\mathit{t}-\mathrm{expected}\mathrm{lead}\mathrm{time}\mathrm{for}\mathrm{trainees}\mathrm{to}\mathrm{become}\mathrm{crew})\end{array}$(25)

Capacity docks allocation

$\begin{array}{rl}& \mathrm{Docks}\mathrm{availability}(\mathit{t})=\mathit{f}(\mathrm{vessels}\mathrm{existing}\mathrm{maintenance}(\mathit{t}),\mathrm{docks}\mathrm{decomission}\mathrm{schedule},\\ & \mathrm{vessels}\mathrm{waiting}\mathrm{for}\mathrm{maintenance}(\mathit{t}))\end{array}$

Licencing

(27) $\begin{array}{rl}& \mathrm{Vessels}\mathrm{waiting}\mathrm{for}\mathrm{licensing}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{vessels}\mathrm{exiting}\mathrm{licensing}-\mathrm{vessels}\mathrm{starting}\mathrm{licensing}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{inital}\mathrm{number}\mathrm{of}\mathrm{vessels}\mathrm{waiting}\mathrm{for}\mathrm{licensing}\end{array}$(27)

(28) $\mathrm{Vessels}\mathrm{starting}\mathrm{licensing}(\mathit{t})=\mathit{f}(\mathrm{allocated}\mathrm{workforce})$(28)

(29) $\begin{array}{rl}& \mathrm{Vessels}\mathrm{in}\mathrm{licensing}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{vessels}\mathrm{starting}\mathrm{licensing}-\mathrm{vessels}\mathrm{exiting}\mathrm{licensing}]\mathrm{\Delta }\mathit{t}\\ & +\mathrm{inital}\mathrm{number}\mathrm{of}\mathrm{vessels}\mathrm{in}\mathrm{licensing}\end{array}$(29)

(30) $\mathrm{Vessels}\mathrm{exiting}\mathrm{licensing}(\mathit{t})=(\mathrm{vessels}\mathrm{starting}\mathrm{licensing}-\mathrm{expected}\mathrm{duration}\mathrm{for}\mathrm{licensing})$(30)

(31) $\mathrm{Vessels}\mathrm{starting}\mathrm{licensing}(\mathit{t})=\mathit{f}(\mathrm{allocated}\mathrm{docking}\mathrm{facility})$(31)

Capacity workforce non-uniform

(32) $\begin{array}{rl}& \mathrm{Allocated}\mathrm{workforce}\mathrm{non}-\mathrm{uniform}(\mathit{t})=\mathit{f}(\mathrm{workforce}\mathrm{industry}\mathrm{requirement}(\mathit{t}),\\ & \mathrm{vessels}\mathrm{waiting}\mathrm{for}\mathrm{maintenance}(\mathit{t}))\end{array}$(32)

Workforce non-uniform

(33) $\begin{array}{rl}& \mathrm{Work}\mathrm{force}\mathrm{industry}\mathrm{available}(\mathit{t})={\int }_t^{t+\mathrm{\Delta }t}[\mathrm{recruitment}\mathrm{rate}+\mathrm{released}\mathrm{workforce}-\mathrm{allocated}\mathrm{workforce}\\ & -\mathrm{attrition}\mathrm{rate}]\mathrm{\Delta }\mathit{t}+\mathrm{inital}\mathrm{industry}\mathrm{workforce}\mathrm{available}\end{array}$(33)