Uncertainty is rampant in military expeditionary operations spanning high-intensity combat to humanitarian operations. These missions require rapid planning and decision-support tools to address the logistical challenges involved in provid-ing support in often austere environments. The US Army’s adoption of an enterprise resource planning system provides an opportunity to develop automated decision-support tools and other analytical models designed to take advantage of newly available logistical data. This research presents a tool that runs in near-real time to assess risk while conductingcapacity planning and performance analysis designed for inclusion in a suite of applications dubbed the Military Logistics Network Planning System, which previously only evaluated the mean sample path. Logistical data from combat opera- tions during Operation Iraqi Freedom drive supply requisition forecasts for a contingency scenario in a similar geo-graphic environment. A nonstationary queueing network model is linked with a heuristic logistics scheduling methodology to provide a stochastic framework to account for uncertainty and assess risk. Keywords Logistics, capacity planning, queueing, networks, forecasting, risk analysis, nonstationary arrival process, time-dependentarrival rate, dispersion ratio, index of dispersion for counts 1. Introduction This paper presents a decision-support tool to support mili- tary logistics associated with sustaining an expeditionaryforce in response to the call to action from the 2005RAND study 1on logistical issues during the invasion of Iraq. The tool is designed to run in near-real time todevelop feasible plans, rapidly assess alternatives, identifylogistical capacity requirements to support expeditionary operations, and assess the associated uncertainty and risks relevant to military planners and decision-makers. Themodel is restricted to a general expeditionary scenario but not limited to an invasion. Modeling efforts focus on repair parts (US Class of Supply Nine, CL IX) but alsoinclude food and water (CL I) and ammunition (CL V), as they share similar sustainment resources. 2By design, the tool is focused on sustainment and excludes the time phased force deployment data (TPFDD) problem of get-ting a deploying unit’s personnel and equipment into thetheater of operations. The paper extends Rogers et al. 3by incorporating uncertainty and the use of a time-dependentvariance correction to enable risk analysis. The rest of this paper is organized as follows: Section 2 motivates the paper; Section 3 introduces related work; 1Department of Industrial & Systems Engineering, North Carolina State University, USA 2Center for Additive Manufacturing and Logistics, North Carolina State University, USA 3Department of T extile and Apparel, T echnology and Management, North Carolina State University, USA Corresponding author: Brandon M McConnell, Center for Additive Manufacturing and Logistics, Campus Box 7906, Daniels Hall 400, North Carolina State University, Raleigh, NC 27695-7906, USA. Email: bmmcconn@ncsu.edu First published online: 10 July 2019McConnell, B.M., Hodgson, T.J., Kay, M.G., King, R.E., Liu, Y., Parlier, G.H., Thoney-Barletta, K., & Wilson, J.R. 2021. Assessing uncertainty and risk in an expeditionary military logistics network, Journal of Defense Modeling and Simulation: Applications, Methodology, Technology , Vol 18, No 2, 135–156. https://doi.org/10.1177/1548512919860595 Section 4 establishes this research’s perspective on risk and presents a stochastic framework for risk analysis;Section 5 outlines the requisition demand forecastingmethodology; and Section 6 demonstrates the model on a notional operation. Section 7 provides closing thoughts. 2. Motivation In March 2003 during Operation Iraqi Freedom (OIF), within a week of the fall of Baghdad, the 3rd InfantryDivision experienced equipment readiness for key ground combat systems dropping from 90% to under 70% due to distribution problems for CL IX; the 2005 RAND studyexamining these problems concluded that these distribu-tion problems stemmed from a lack of automated decision- support tools capable of generating analysis fast enough to support the rapid pace of operations. 1 More recently, in 2018, the Defense Science Board Task Force on Survivable Logistics concluded that‘‘[l]ogistics data is neither as accessible nor used as effi-ciently as it should be’’ and provided recommendations that included increasing the ability to wargame logistics with sufficient ‘‘fidelity to identify logistics constraints tooperations.’’ 4 3. Related work Rogers5proposes a Military Logistics Network Planning System (MLNPS) that harnesses both the Army’s new enter- prise resource planning (ERP) system called the Global Combat Support System—Army (GCSS-A) and mission-based forecasting (MBF) to assist decision-makers and plan-ners with several aspects of military logistics. 6As opposed to using historical average usage rates for repair parts, MBF provides a tailored forecast using stratified sampling7that considers force composition, environmental conditions, operation duration, and pla nned mission profiles.8–13MBF incorporates regression methods using these explanatory fac- tors, a new technique to improve forecast accuracy for inter- mittent demand referred to as the ‘‘Markov-Bootstrap Method,’’14–16and both predictive and prognostic methods using sensor-derived data ass ociated with Condition-based Maintenance (CBM).8,17–19Some key MLNPS functions include identifying required capacities across the logistics network to support an expeditiona ry operation, anticipating bottlenecks, conducting what- if analysis, and course of action (COA) analysis. The MLNPS is a significant contri-bution as it is the only known decision-support tool designed to use the Army’s ERP data. The MLNPS fundamentally models the logistics net- work as a large factory with each logistics node repre- sented as a machine in the factory. Supply requisitions (e.g., CL IX) act as jobs that must be processed throughthis factory. With this approach, the MLNPS exploits an engine known as the Virtual Factory (VF) to optimally (or near-optimally) schedule these requisitions across the net-work in near-real time to minimize the maximum lateness; lateness is the requisition completion time minus the due date. Hodgson et al. 20introduced the VF in 1998, and it has sustained a number of improvements: most notably, Thoney et al.’s21addition of batch processing. Rogers5 finds this heuristic optimization provides a good forecast for real system performance and demonstrates significant insights that would have impacted planners in 2003. The ability to handle batch processing allows the VF to schedule for multiple network locations while also accounting for transportation between locations, since a truck may be modeled as a batch processor itself. Trainor22 and Melendez23use batch processing in conjunction with the VF to address military deployments (also see Hodgson et al.24). Rogers et al.3extend this idea by focusing on repair parts transiting the military logistics network by using nested batch processors to model requested repair parts being loaded into pallets, which are loaded into con-tainers then shipped via surface (ocean) vessels. Using the VF to process this data along with a MBF for future demand, the MLNPS ter minates with a near-optimal sequence to maximize customer ( requesting unit) satisfaction by minimizing the maximum lateness. Rather than focusing on the sequencing, Rogers et al. 3demonstrate that the VF provides a forecast of when, where, and how much queuing will occur at various nodes across the logistics network. Validating their model against real performance data fromt h e2 0 0 3i n v a s i o no fI r a qd u r i n g OIF demonstrates that the MLNPS can accurately approxi mate the performance of a real logistics network. Usi ng this model and mean value parameters, they develop a trial-and-error method to test drive a logistics plan and assess how well it will support a planned expeditionary operatio n using deterministic outputs from the VF. Their model assesses the logistics network both for the invasion of Iraq and for a notional intervention sce- nario set in Africa. With two analysts, model setup took 1–2days from scratch (hours if already created) with run times taking minutes to hours depending on the time horizon and the size of the network studied; to our knowledge, this isfaster than contemporary models. The MLNPS has tremendous potential as a military logistics model for several key reasons: it supports end-to-end analysis in near-real time (a capability deemed critical by Army leadership) and is designed to incorporate both GCSS-A data and MBF. 25,26As illustrated by Figure 1, the GCSS-A can provide the model with the current location and status of all requisitions currently in the system. The model requires the planned logistics network and necessaryconstraints as an input from the planner. The MLNPS then forecasts how well the network will perform, where signif- icant queuing will exceed thresholds causing sustainment2 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) problems, and provides the decision-maker with informa- tion to adjust the plan or notify subordinate units of whatto expect as the operation unfolds. The Army’s new ERP system (GCSS-A), with its abil- ity to provide data on demand, provides unprecedentedaccess to data. The Army must harness that data and trans-form it into useful, actionable risk information for deci-sion-makers. While the uncertainty cannot be altogethereliminated, GCSS-A makes it possible for tools to accountfor this uncertainty. 4. A stochastic framework that permits risk analysis 4.1. Risk in the military logistics context As a deterministic tool, the MLNPS does not adequately assess uncertainty or permit analysis of risk. The term risk is ubiquitous in both military and nonmilitary applicationsbut often lacks precision or quantification. 27This paper defines risk as a set of possible outcomes with negativeconsequences. For this definition, assessing risk requiresestimation of both the severity and likelihood of those out-comes. 28,29For a more detailed review of the supply chain, military, government, and transportation infrastructure risk, see McConnell30(Section 3.3). This view of risk is justified to assess operational risk faced by the military logistics network in the form of unan-ticipated requirements and disruptions from events causedby enemy action, terrain, and weather. 8Beyond these con- siderations, a commander faces risks that address timing, performance, or other concerns. These may be measuredby any number of qualitative or quantitative metrics thatframe how the commander views risk. The framework forrisk analysis must be flexible to accommodate differentnotions of risk and address various sources of risk, even ifonly through what-if functionality. Further complicatingthe analysis is the fact that risk has both time and location components that greatly increase the dimensionality—risk can evolve over time as the operational plan unfoldsand might be concentrated in different logistic nodes as afunction of time.This research’s perspective on risk analysis extends Alderson et al.’s 31definition of operational resilience : ‘‘the ability of a system to adapt its behavior to maintain continuity of function (or operations) in the presence ofdisruptions.’’ Alderson et al. 31link infrastructure resilience to system operation (function) and focus on disruptions.Our model must enable what-if analysis by capturing bothseverity and likelihood. 4.2. Relevant advances in queueing theory Queuing theory provides a rich and convenient source oftools to apply to this problem as it can account for uncer-tainty in both arriving requisitions and processing times through a network. Recent advances also integrate time- varying properties while relaxing the mathematically con-venient but not always realistic Markovian assumptionsfrom classical textbooks. Queuing theorists largely con-cern themselves with two basic questions that align withour stated research objectives: (1) given a stochastic sys-tem’s properties, estimate its performance (performanceanalysis) and (2) given a performance target, estimate the system properties required to achieve the desired perfor- mance (capacity planning). Recognizing that the number of servers at a queue is analogous to the capacity of a logistics node, it is clearthat queuing theory can assist the MLNPS in setting capa-cities (or estimating required capacities) to achieve better performance. Figure 2 (mimicking Mandelbaum and Momc ˇilovic´ 32) demonstrates that by increasing the system scale, it is possible to increase the quality of service with-out loss of efficiency—in other words, properly setting thecapacity can benefit both quality and efficiency. We focus on queuing theory that supports staffing time-dependent (nonstationary) queueing systems and non-Markovian (non-exponential) properties, particularlyarrival and service processes. Many studies present evi-dence of time-varying system properties in everythingfrom hospitals to call centers and even trucks at a sea-port. 33–38It is well known that Markovian approximations to non-Markovian systems can perform poorly, particu- larly if sufficient variation exists in the arrival process.39– 41Jennings et al.42use an infinite server (IS) and Normal approximation to choose the time-dependent staffing func-tion, s(t), to stabilize the probability of delay for a G t=GIt=stmodel with nonstationary non-Markovian arri- vals (the first Gt) and nonstationary identical and indepen- dent general service times (the GIt). Liu and Whitt43use IS models to develop offered-load (OL) and modified offered-load (MOL) approximations for the time- dependent staffing required to stabilize the expected delayand abandonment probabilities for the M t=GI=st+GI queue. This approach is extended to a feed-forward net-work structure of M t=GI=st+GIqueues.44He et al.41 Figure 1. Conceptual overview of the Military Logistics Network Planning System (MLNPS).3DP: decision point; ERP: enterprise resource planning.McConnell et al. 3 extend established staffing procedures to allow for non- Markovian arrival processes using a heavy-traffic limitthat applies to systems where some delay is expected. Their study of the impact and interplay between arrival pro- cess variability and service tim e distributions demonstrates that the variance correction used by Jennings et al. 42is robust for nonstationary models. Rea ders wanting a more thorough treatment should see Defraeye and Nieuwenhuyse.45 4.3. Modeling the military logistics network The military logistics network stretching from US depotsto the expeditionary theater of operations can be modeledas a queueing network where logistical nodes (or pro- cesses) are queues and the requisitions are arriving as orders via GCSS-A. The simplified model in Figure 3illustrates how orders are sourced, picked, and packed,then shipped to the ordering unit. The model is a feed-forward queuing network as supply requisitions movefrom the sourcing depot across the network to the orderingunit and do not require rework or depart the networkthrough a lateral exit. In the simplified network shown, requisitions are arrivals in the queuing theory terminology and arrive according to the requisition forecast discussedin Section 5. There are multiple classes of arrivals basedon the mode of transportation to be used, whether military Figure 2. Performance of M=M=cqueues (for c∈f1,4,16,64g) demonstrating that it is possible to increase both the efficiency and quality through staffing; utilization ( ρ) is the arrival rate ( l) divided by the total service rate ( cμ),ρ=l=cμ. Figure 3. Military logistics network as a queueing network (simplified for illustration).4 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) air, a contracted point-to-point service denoted world-wide express (WWX), or surface shipment (ocean freight). Within each arrival class are many subclasses defined bythe specific route that the requisition must take through the network, largely defined within a class by the sourcing node and the ordering unit. We obtain this feed-forward structure through several important assumptions. Firstly, we assume that all requisi- tions are appropriately handled, routed, and moved alongthe required path from the point of supply to the point of consumption—in other words, we do not model lost, misrouted, or frustrated cargo. This is reasonable due tothe sheer size of the network and the immense requisition flow volume. In addition, the military is actively embra- cing interconnected technologies, such as radio-frequencyidentification, two-dimensional bar codes, satellite-based communications, and others that help prevent these logisti- cal mishaps. 46–50This paper models repair parts (CL IX) along with food and water (CL I) and ammunition (CL V); other classes of supply are beyond the scope of this paper. The feed-forward structure implies that by estimating the performance or capacity required at an upstream loca- tion, if the departure process is obtainable it also provides the arrival process to the downstream nodes so the analysiscan be repeated for downstream locations. Consider the simplified military logistics network depicted as a feed- forward queuing network in Figure 3; supply requisitionsarrive at either Defense Depot Susquehanna, Pennsylvania (DDSP) or to an alternative depot acting as a sourcing point (SP1 or SP2). The requisitions from alternativedepots still move to DDSP as it is the primary consolida- tion and containerization point (CCP) 51for palletizing and containerizing shipments. As Rogers5discusses, most requisitions travel via military aircraft, WWX, or surface shipment. Shipments traveling on military aircraft (ocean freight) are palletized (containerized) at DDSP then loadedonto military aircraft (container ships) at the airport (sea- port) of embarkation, or APOE (SPOE) for movement into theater where they are offloaded at the airport (seaport) ofdebarkation, or APOD (SPOD). 51These shipments are then transported to the theater distribution center (TDC). Requisitions being shipped via WWX travel directly to theTDC. At the TDC, pallets and containers are broken down and moved forward via military transportation to the order- ing units. This final process is often referred to as the lasttactical mile (LTM) due to the transportation occurring through a designated combat zone. Using MBF and the data from GCSS-A, it is possible to estimate the arrival processes to the network shown in Figure 3. Since MBF is not available for all unit types, Section 5 presents a data-driven approach to forecastingrequisitions for a given scenario. This forecast provides an estimated nominal time-varying arrival rate to the upstream sourcing nodes.4.4. Virtual Factory delayed-infinite server feed- forward model 4.4.1. Network perspective. To assess risk and answer the research objectives, this paper uses a tandem Virtual Factory delayed infinite server (VF-DIS) OL approximation for each location; the model alternates between using theVF and the DIS approximations for each location movingacross the network from upstream to downstream. The appeal lies in leveraging each s ubmodel’s strengths. The VF is efficient and can handle nested batch processing (e.g., amultipack inside a pallet on a truck) as well as a number of realistic constraints, and with small refinements, the DIS model 43provides a computationally efficient method to evaluate performance and predict capacity requirements while communicating a sense of the uncertainty in the pre- dictions. This tandem approach uses each model to the max-imum potential while avoiding each model’s weaknesses. The approach starts with the data-driven requisition forecast and the performance target of average delay desired for each of the network locations. These perfor- mance targets are obtained from stakeholders, proposed forthe sake of continued analysis, or derived from seniorleader interactions. The forecast conforms to the require- ments described in Section 5 but in general is both nonsta- tionary and non-Markovian. We assume that an empiricalor fitted theoretical distribution is available for the process- ing time at each location in the network to account for sto- chastic variation; these may be general (non-exponential). Figure 4 offers a visualization of the VF-DIS process with a small portion of the network. It depicts the networkas two complementary models, the DIS and the VF approaches, with the bold (blue) line representing the VF- DIS hybrid solution approach for an analyst conductingrisk analysis and military logistics planning. The dashed line represents the model workflow if only using a single model in isolation from the other. Taking the logistics net-work, processing logic, and requisition forecast, l(t),a s inputs, the VF-DIS first uses the VF at the upstream nodes to assess the arrival process and the resulting departure process given the default time-dependent capacity plan, s t, which can be initialized via a simple constant capacity. With the average delay targets for the upstream nodes, the DIS model calculates the time-dependent capacity requiredto meet the targets by providing both a time-dependent average ( s t) and the stochastic variability around that aver- age. These capacity plans (functions over time) are inputsto the VF, which accounts for location-specific policies, logic, and schedules; the VF then returns the time- dependent departure process, represented as σ(t). Since the network is feed-forward, the departure process is thedownstream arrival process so the model advances down- stream once the upstream planning is complete and repeats the process until it reaches the most downstream nodes.McConnell et al. 5 The VF can model location-specific policies via its pro- cessor logic. These are important because some locations do not work weekends, and resources vary according tospecific schedules driven by real-world considerations.Another example stems from a working policy used by a container packing location (one of several batch pro- cesses): a container is considered full when it reaches itseffective capacity, reaches the minimum capacity to sendand there are no orders left to pack, or when it has been sitting open (and partially filled) for at least three days. The VF’s processor logic conveniently incorporates thesenuances. 3 Planners may obtain the performance targets for each node from senior leader guidance, experts, or simply a staffestimate as part of the planning process. If not derivedfrom successful historical performance or leader guidance, staff planners may choose to use a range of targets they believe to be feasible and present them with their resultingimpacts to the staff and/or leadership for analysis and deci-sion. The calculations this choice permits is the science, but choosing an appropriate performance target is clearly part of the art of the planning method described here. The VF-DIS performs the same steps at every location. If that location is not of interest or cannot be impacted byavailable decisions, there is no need to linger performinganalysis after obtaining the departure process. However, when the VF-DIS model reaches a node of interest such as the TDC or the LTM trucks, it may be desirable to identifya logistics plan for that location to meet senior leader per-formance guidance and assess the associated risk. Section 4.4.2 provides the technical details for this process. 4.4.2. Node perspective. Consider an arbitrary node in the logistics network. Let wbe the average delay taken as the performance target for this node and denote the average arrival rate on day tasl(t); the VF provides this nonsta- tionary arrival rate using the data-driven requisitionforecast discussed in Section 5. The logistics node is repre- sented as a G t=GI=stqueueing model where the processing time, S, at this location is independent and identically dis- tributed according to the general distribution FS. Figure 5 labels this Model 1. Define σ(t)as the average departure rate on day tfor the associated departure process. Denote Q(t)as the queue length at time t. Let B(t)be the number of busy servers at time twith mean m(t)=E½B(t)/C138and var- iance v(t);B(t)is approximately as follows: B(t)∼Normal (m(t)+1=2,z(t)m(t)), ð1Þ with E½B(t)/C138=ð(t/C0w)+ 0l(t/C0w/C0x)/C22FS(x)dx, ð2Þ where the notation (x)+=maxfx,0gand the service time complementary cumulative distribution function /C22FS(x)=1/C0FS(x). Equation (2) calculates the average capacity over time required to maintain the performance target w>0. Define fA(t),tø0gas the counting process that tracks the number of arrivals (events) by time t, then the arrival process index of dispersion (for counts), I(t), is the variance-to-mean ratio of the cumulative number of arri- vals (events) as given by Equation (3). If the node sees Mt arrivals according to a nonhomogeneous Poisson process (NHPP),52I(t)=1,t>0: I(t)=Var (A(t)) E½A(t)/C138,t>0: ð3Þ The arrival process is overdispersed ifI(t)>1 and underdispersed ifI(t)<1. If the arrival process to Model 1 is significantly overdispersed then a naı ¨ve implementation of the DIS OL approximation will underestimate risk to the decision-maker because, while the predicted average will be true, the model will underestimate the variance. Figure 4. Visualization of the Virtual Factory delayed infinite server (VF-DIS) offered-load network model. The models complement each other—our unified approach is denoted in bold (blue). The dashed lines represents using the two single modelapproaches. Note: w 0represents the average elapsed time from a requisition establishment to the release to the source depot. DDSP: Defense Depot Susquehanna, Pennsylvania; CCP: consolidation and containerization point. (Color online only.)6 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) Using the Sudan scenario from Rogers et al.,3Figure 6 pro- vides an estimated dispersion for the LTM trucks revealing that the LTM arrivals are over five times more variable than a NHPP. In other applications, healthcare clinics oftensee underdispersion ( ≈0.4–0.6) in appointment-based sys- tems but overdispersion ( ≈1.5–2.5) in emergency depart- ments or at call centers. 53–56 This research assumes that with GCSS-A data an ana- lyst can estimate the dispersion for a given location (Section 5 enables estimating the dispersion using multiple sample paths for the requisition forecast). If arrivals occur according to a NHPP, then I(t)=1,8t, and for a fixed t,B(t)is a Poisson random variable with mean E ½B(t)/C138(2).43,57This implies m(t)=v(t). If the arri- val process is not a NHPP, then m(t)6¼v(t)and the heuris- tic risk correction factor (RCF), ~z(t), enables the following approximation: v(t)≈~z(t)m(t), ð4Þ with the correction factor ~z(t)=maxfz(t),1g, ð5Þ z(t)=1+(c2 a(t)/C01) E½S/C138ð∞ 0½1/C0FS(x)/C1382dx, ð6Þwhere c2 a(t)≈Var (A(t/C0w)/C0A(t/C0w/C0η))Ðt t/C0ηl(u/C0w)du, ð7Þ for a chosen η>0. The capacity recommendation follows the square root staffing (SRS) rule: sγ(t)=E½B(t)/C138+δγﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Var (B(t))p lm , ð8Þ establishing a buffer to hedge against stochastic variability by establishing a probability γand associated quality of service parameter δγsuch that P(N(0,1)>δγ)=γ, which exploits the Normal approximation to the Poisson. 4.4.3. Discussion. Figure 5 graphically depicts the node specific approximation. Model 2 ( Mt=GI=stqueue) can approximate Model 1 (see Figure 5) as it captures thetime-dependent fluctuations in the arrival process with the same deterministic mean function while retaining Model 1’s non-exponential service time distribution and time-varying capacity. The only difference is in the arrival pro-cess variability. Convenience motivates the transition toModel 2 and requires a correction to account for the Figure 5. Queueing models for a location. Model 2 approximates Model 1 by focusing on the time-dependent behavior. Model 3 is the delayed infinite server (DIS) offered-load approximation for Mt=GI=st; Liu and Whitt43show that Model 3 approximates Model 1. Within Model 3, the contents of the first two queues, Q(t)and B(t), respectively, are independent Poisson random variables for a fixed t. See Table 1 for the departure rate, ~σ(t). NHPP: nonhomogeneous Poisson process.McConnell et al. 7 arrival variability. There is no reliable method to obtain analytical approximations for the capacity required over time to meet performance targets for Model 1 and the useof simulation will not permit the analysis to be performed in near-real time. From the network perspective, Model 2 enables Poisson superposition at a downstream node thatreceives requisitions from multiple upstream nodes; theaggregate arrival process is then a NHPP. To identify the capacity to achieve the average delay target w>0 for Model 2, Liu and Whitt 43show that the DIS OL approximation works well for systems such as the military logistics system where requisitions may be expected to wait for some (even small) amount of timebefore processing. The DIS model, depicted as Model 3 inFigure 5, approximates Model 2 by using two infinite capacity queues in series. This presentation of the DIS model omits the abandonment process implying theassumption that there is no lost, misrouted, or frustratedcargo. The first queue represents the waiting space in Model 2; the second queue represents the service facility of Model 2.The infinite capacity implies that the departure process from each queue in Model 3 is also a NHPP, which is computationally and mathematically critical in the feed-forward network. The idea behind Model 3 is simple.Requisitions arrive to the first queue (waiting area) andwait a deterministic amount of time equal to the target average delay, w, before continuing to the second queue. This implies an average arrival rate of β(t)=l(t/C0w), tøwat the second queue, which is just a deterministic time shift. At the second queue, requisitions immediately begin processing according to the general service time distribu-tionG=F S. The objective is to determine the number in the second queue at time t,B(t), which serves as the first- order approximation of the number of busy servers in Model 2 while maintaining an average delay of w. In sim- pler terms, Model 3 approximates Model 2 by having allrequisitions wait the desired average delay then simplyobserves how many busy servers would be in the secondqueue over time. Based on the mathematics of the IS queue, this is obtained via direct calculation using Equations (1) and (2), applying known results for theM t=GI=∞queue (see Theorem 1, Eick et al.57).43 While Equation (2) calculates the average capacity over time required to maintain the performance target, themathematics of Model 3 fully specify the approximate dis-tribution of this predicted capacity requirement. Section 6exploits this idea and uses Monte Carlo methods to esti- mate what is not available analytically. Table 1 sum- marizes the remaining analytical formulas for the DIS OLapproximation. The DIS approximation prediction for Model 2 with M t arrivals implies B(t)∼Poisson (E½B(t)/C138). Correcting for the variability of the Gtarrival process requires application of a result from the stationary G=G=∞queueing model as a heuristic, which is consistent with Jennings et al.42(see Section 6) and He et al.41(see Section 3). This paper is the first to use a time-shifted variance correction to integratethe DIS approach. The function ~z(t)(5) does not allow a reduction in var- iance (an optional modeling assumption due to lack of dis-persion data—this avoids a false sense of certainty) andTable 1. Model 3 delayed infinite server (DIS) approximations for Model 2 (assumes Mtarrivals). Performance feature DIS approximation (for a fixed t) Queue length, Q(t) ∼Poisson with mean E ½Q(t)/C138=Ðt^w 0l(t/C0x)dx Number of busy servers, B(t)∼Poisson with mean E ½B(t)/C138=Ð(t/C0w)+ 0l(t/C0w/C0x)/C22Fs(x)dx Departure rate, ~σ(t)∼NHPP with time-varying rate ~σ(t)=Ð(t/C0w)+ 0l(t/C0w/C0x)dFS(x) T otal number in systema,X(t) X(t)=Q(t)+B(t) aSystem refers to a specific node; t^w=minft,wg;(t/C0w)+=maxft/C0w,0g. NHPP: nonhomogeneous Poisson process. Figure 6. The last tactical mile (LTM) arrival dispersion for the Sudan scenario is over five times more variable as anonhomogeneous Poisson process (dashed line). The graphshows arrival dispersion in 463L pallet equivalent units fromEquation (9).8 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) can increase the variance for B(t)using the time-dependent generalization of the heavy-traffic peakedness (6) taken from Whitt’s58treatment of the stationary G=G=∞model, which characterizes the variance-to-mean ratio of the steady-state number of busy servers. Since Model 1 is non- stationary, (6) is a heuristic. Equation (6) assumes a sta-tionary service time distribution, but this can be relaxed. Equation (7) is a time-dependent generalization of the asymptotic variability parameter and is similar in form to(3); this paper time-shifts the asymptotic variability para- meter by wto account for the DIS approximation. The parameter ηdetermines the dispersion estimate in the local interval ½t/C0η,t/C138; this paper uses a timestep of one day and numerical evaluations confirmed that η=1 is a good choice for this application. The intuition is that both thearrival variability (7) and the service time variability (6) affect the variance of the capacity prediction (number of busy servers). Specifically, the tail of the service time dis-tribution drives the impact on the variance with the termÐ ∞ 0½1/C0FSt(x)/C1382dxin (6). Since we estimate the arrival variability parameter (7) empirically, it can be undefined whenÐt t/C0ηβ(u)du=0, which occurs during lulls in arrivals on ηconsecutive days. When this occurs, we define c2 a(t)=0 for this special case such that the RCF z(t)=1, which aligns with engineering intuition. This RCF (5) corrects the variance in the approximation for Model 2 so the final results may serve as an approxi- mation for Model 1. The Normal approximation to the Poisson implies that instead of B(t)∼Poisson (m(t)), which underestimates risk, the required capacity is actually approximated by B(t)∼Normal (m(t)+1=2,z(t)m(t)). Adding a half to the mean function corrects for the conver-sion between a discrete distribution to a continuous one, but may be omitted in practice if desired. This continuity correction may lead to a positive bias in the capacity fore-cast when the nominal requirements are relatively low and the continuity correction has a larger relative impact on the mean. In Section 6, which demonstrates this technique,B(t)is a Normal distribution truncated on the interval ½0,+∞)to prevent the sampled z(t)from pushing the probability below zero; this is equivalent to (B(t)jB(t)>0). Figure 5 displays an overview of the entire process to pre- dict capacity requirements for a single location. The VF is responsible for both the arrival rate to each node, l(t),a n d determining the departure rate, σ(t), as it is designed to handle location-specific packing policies, work schedules, and other realism constraints. Much of the queueing literature cited is motivated by staffing requirements for call centers and often employs (8) over a discretized time horizon. In the case of expedi-tionary military logistics such continuous control is not likely to be possible, even over long subintervals. Section 6 addresses this concern and also demonstrates a techniqueto use the VF-DIS model structure to generate possible capacity plans for a location given the time-varying risk information. Combined with the tandem VF-DIS approach for the network, this approximation provides a framework to describe the average requirements that fluctuate over time as well as the stochastic variation around that deterministicprediction. In short, this includes both severity and likeli- hood in the predictions. While the feed-forward structure imparts computational efficiency and suggests a simple sequential approach, the time-varying (nonstationary) property and presence ofnon-Markovian (non-exponential) arrival and service pro- cesses greatly complicate the analysis. The reader will notice that the fundamental modeling unit traversing thenetwork is changing as well and there is no common unit for capacity (number of servers); Rogers 5uses number of requisitions per day for DDSP, 463L pallets59per day for the APOE, and 40 foot container equivalents (FEU) perday for surface freight, and even the number of medium truck companies as a capacity unit. Queueing theory has no easy way of accounting for these unit changes. To simplify the analysis, the model employs a 463L pallet equivalent unit (PEU) as the base unit of capacity tobe used throughout the network. For reference, Figure 7 portrays airmen loading a 463L pallet holding multipack containers into a military aircraft. This unit is reasonableas it readily converts to the 20 foot container (TEU) and FEU equivalent units widely used in the logistics practi- tioner community. The CCP packs requisitions and multi-pack boxes into containers. The TDC requires an Figure 7. Depiction of a 463L pallet with multipack containers in Al Taqqadum, Iraq (9 January 2009). Seabees assigned toNaval Mobile Construction Battalion 7 and airmen of the AirForce Expeditionary Logistics Readiness Squadron Detachment4 maneuver 463L pallets into an Air Force C-17 fortransportation. 59McConnell et al. 9 equivalency unit as it breaks down both pallets and con- tainers and organizes them with individual requisitions foronward transport. Given a particular Army truck company equipment list, known as the modified table of organiza- tion and equipment (MTOE), 51it is possible to convert between PEU and a collection of logistics distribution resources. Establishing the PEU unit also provides an opportunity to account for the tare weight. One can calculate the num- ber of PEUs arriving (or departing) location mon day tby taking the total weight and volume for that day and loca- tion and dividing by the effective maximum capacity of the 463L pallet given by Equation (9) below. The larger of the two determines the 463L pallet equivalent capacityrequired; note that for resources the smaller of the two is the offered resource capacity. Table 2 lists the 463L maxi- mum and effective capacity. The use of cubic inches pre- serves integrality in the model for computational efficiency and adequately models the many requisitions with less than a unit cubic foot volume: Pmt=maxTotalWT mt 95%PEU WT,TotalCU mt 85%PEU CU/C26/C27 _ ð9Þ Using 95% of the maximum weight accounts for the tare weight. Using 85% of the maximum volume accounts for the inevitable voids between contents that prevent using all of the physical container volume. Whether a trac-tor trailer or a shipping container, spaces are considered full at the 85% volume utilization point. 5. Generating a data-driven demand forecast The MLNPS provides a convenient set of tools for analyz- ing the performance of logistical COAs as well as a means of identifying the capacities needed to hit performance tar- gets. Both of these capabilities require forecasted demand from certain classes of supply, such as food and water, ammunition, and repair parts, over the studied time hori-zon. Since MBF, driven by consumption data, is not avail- able for platforms outside of Army aviation, we use available modern combat data as a surrogate to generate the demand forecast. Acknowledging this is supply-side data and therefore a faulty signal for true demand, thisdata-driven process mimics some MBF techniques, such as stratifying on unit type and mission intensity, in orderto generate the best demand forecast possible without trueMBF. 5.1. Data-driven approach It is critical to use modern combat data to get a representa-tive picture of modern combat repair part demand. Toforecast demand for different missions in varying opera-tional environments, the process presented here would bereplicated using data generated under those conditions. While order data may not be a perfect demand signal,when properly characterized this data provides an initial approximation of the demand required in absence ofconsumption-driven MBF. The model focuses on foodand water, ammunition, and repair parts primarilybecause the data describe repair parts and they all share common resources across the distribution network. An author from the 2005 RAND study 1provided OIF repair part requisitions and the US TransportationCommand (TRANSCOM) provided data on all requisitionsthat DDSP processed in 2003. Together these datasets pro-vide the weight, volume, sourcing depot, requisition date,requesting unit and location, and other factors for all repairpart orders processed by DDSP and those destined for unitsin Kuwait and Iraq. For specific operational characteristics,the process relies on data from the first 87 days of OIF,which consists of 647,189 individual requisitions (11.6million parts) spanning the two weeks prior to crossing theline of departure (LD) through to the end of May 2003.The DDSP-specific data contain the 7.7 million requisi-tions from 2003. Both datasets have a time precision of days, which aligns with the chosen time step for this work. Due to similar terrain and environmental factors the OIFdataset is sufficient for the analysis, which considers anotional operation in Sudan. 5.2. Process overview—a sample path approach Generating a demand forecast requires three key inputs–the task organization, concept of the operation, and thetimeline–which are estimates produced during the militaryplanning process. The task organization lists what unitsare conducting the operation and may change over time.The specific tasks (missions) for these units are found inthe concept of the operation. These products provide the analyst with the specific details (who, what, when, where, and why) for the operation. The timeline is essential to breaking down the time hor- izon of interest into distinct missions for the units. Theforecast must account for the fact that repair part demandis different by both unit type (think infantry versus aviationunits) and type of operation (preparing for combat versusTable 2. Maximum and effective capacity of the 463L pallet. Weight (lbs) Volume (cu in) 100% 463L capacity 10,000 838,080 Effective PEU 9500 (95%) 712,368 (85%) PEU: pallet equivalent unit.10 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) combat). This is accomplished by defining three opera- tional intensity levels (OILs) then using the OIF invasion data to characterize each type of unit’s demand under those operational descriptions to describe pre-combat (OIL 1),steady-state operations out of an established base (OIL 2),and direct (major) combat operations or high-intensity con- flict (OIL 3). Since these levels are qualitative in nature, Table 3 provides recommended guidelines for identifyingthese levels using historical data. Figure 8 graphically shows the forecasting process, which begins by generating requisitions for the food, water, and ammunition supplies that must sustain the unitsin the model. Then a specific workflow addresses repair part demand for every unit for every OIL. This workflow determines the number and timing of requisitions, how those requisitions are routed to the ordering unit, therequired delivery date, order weight and volume, and thesourcing depot. Once this has occurred for every unit forevery OIL, the concatenation of these requisitions consti-tutes the forecast. Due to how the forecast employs prob-ability distributions, the resulting forecast is a projectedsample path for that time horizon. 5.3. Modeling the demand process 5.3.1. Food, water (CL I), and ammunition (CL V). Unit size plays an obvious role in estimating CL I requirements. Since bottled water was the primary source of potablewater during OIF and this research focuses on initial expe- ditionary operations, we generate pallets of bottled water to supply the units originating at the TDC. 1,5This assump- tion may vary across the time horizon as may the startinglocations of those pallets, depending on the logistics plan.Army Tactics, Techniques, and Procedures (ATTP) 4-41,‘‘Army Field Feeding and Class I Operations’’ recom-mends ration cycles and feeding plan guidelines (seeSection 5.3 of Rogers 5) to identify the capacity required to haul CL I to the units so the remaining capacity may beallocated to other classes of supply. Assuming an average Brigade Combat Team (BCT) strength of 4500 soldiers with attached enablers thatTable 3. Guidelines to identify operational intensity levels. Operational intensityDistinguishing features Level 1 APrior to crossing line of departure ANot conducting combat operations APreparation for combat operations Level 2 AOperations conducted from established and secured base or fixed location AOperations take on a steady-state, routine nature during this period Level 3 AConducting invasion AMajor combat operations/ high-intensity conflict Figure 8. Process overview to generate a demand forecast. More details available in McConnell.30The irregularly shaped (blue) box indicates the steps required for every unit type and operational intensity level (OIL). (Color online only.)McConnell et al. 11 increase the troop count by 10% yields approximately 4950 personnel per BCT. Consistent with doctrinal waterplanning guides, assume 7.27 gallons of water per personper day is required in an arid environment to account forhydration, hygiene, and feeding purposes. The standardplanning factor estimates that a pallet may hold up to 228 gallons of bottled water. 60Using a 10% breakage planning factor, this yields a daily water requirement of 173 palletsof water per day for a BCT. 5A pallet holds 576 Meals- Ready-to-Eat (MREs) and, assuming initial rations arethree MREs per day, the daily food requirement is approx-imately 26 pallets per day. These rations are nonperish- able. Based on planning factors provided by the Army’s Training & Doctrine Command Analysis Center–Fort Lee(TRAC-LEE, a logistical analysis center), ammunitionrequirements would be approximately 60% of the waterweight and 40% of the water volume. 5 5.3.2. Generating repair part (CL IX) requisitions. This section describes the workflow depicted in Figure 8 by the irregu- lar shaded (blue) box. This process occurs for each unit,with generated requisitions being collected into a large listalong with the CL I and CL V requisitions. Depending on multiple factors, such as the time horizon, task organiza- tion, and mission, this process generates a significant num-ber of requisitions. For reference, the Sudan scenario inSection 6 generates requisitions for 61 days prior to the LDand 90 days of operations after. In addition to the almost2.3 million requisitions that competed for DDSP resources during the OIF invasion used as a surrogate for global, non-Sudan demand, the random requisitions generated forjust the Sudan operation are of the order of 294,149 requi-sitions on average (20 samples with sample standard devia-tion of 11,538). Analysis of the OIF invasion data clearly identified that the number of daily requisitions varies both with the typeof unit and based on that unit’s OIL. For a given unit, themodel assumes each day as independent and identicallydistributed within an OIL, as the data showed weak auto- correlations. This assumption is not limiting and can be relaxed. The OIF invasion data permits modeling the eightunit types listed in Table 4. For each unit, the sub-timelineof each OIL determines the number of requisitions perday. If an Infantry Brigade Combat Team (IBCT) has anOIL schedule as depicted in Figure 9, the number of requi- sitions released on days 1–14 requires the appropriate esti- mated probability distribution for OIL 1. In Figure 9,Nrel (j),j=1,2,3 is the daily number of requisitions ordered (released by GCSS-A) for OIL level j. Table 4 lists the unit types the model can support using the OIFdata taken from 6 March–31 May 2003; interested readers may refer to McConnell 30(pp.92–94) for distribution and sample size details.After generating a unit’s requisition volume across the timeline of interest, the process randomly selects eachrequisition’s mode of transportation according to a speci-fied distribution, which identifies the route. The Sudan operation in Section 6 employs the distribution from Table 5, taken from the OIF invasion data. The allotted time to deliver the requisition in days, known as the standard delivery time ( SDT), is generated based on a requisition’s transportation mode ( TransM ). This time is added to the release time ( RT) to calculate the due date ( DD) according to Equation (10), where iindexes each requisition. In Equation (11), each standard deliverytime (conditional on transportation mode) is modeled with a discrete uniform distribution (denoted DU). Since stan- dard delivery times vary by requisition priority andservice-specific processes, assuming that the actual stan-dard delivery times found in Table 6 from Department of the Army Pamphlet (DA PAM) 710-2-1 (Using Unit Supply System) are the upper bounds ( b TransM ) and allow- ing up to an approximate 70% reduction ( aTransM )t oTable 4. Unit types supported by model. Relies on historical data taken from Operation Iraqi Freedom invasion 6 March–31May 2003. Unit Operational intensity level (OIL) IBCT 1, 2, 3 HBCT a1, 2, 3 AVN BN 1, 2, 3 EN BN Single level only DIV HQ Single level onlySustainment BDE Single level only Misc Enabler BN Single level only 3×Truck CO Single level only aNow called ABCT , also used as surrogate for Stryker Brigade Combat T eam (SBCT). IBCT: Infantry Brigade Combat T eam; AVN BN: Aviation battalion; EN BN: engineer battalion; DIV HQ: division headquarters; BDE: brigade; BN: battalion; CO: company. Figure 9. Example for generating the number of requisitions by day for an Infantry Brigade Combat T eam (IBCT). Note:F OIL level unit type is the estimated probability distribution for the number of requisitions required by day for a particular unit type at aspecific operational intensity level (OIL).12 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) account for varying priority designations helps to account for individual prioritization.61For more details on requisi- tion priorities see Rogers5(p.31) and DA PAM 710-2-161 (Chapter 2 and Table 2.2): DD i=(SDT ijTransM i)+RTi, ð10Þ (SDT ijTransM i)∼iidDU (aTransM ,bTransM ): ð11Þ The VF requires orders to have both a weight and vol- ume to properly capture the details of processes such asmovement by truck, packing a 463L pallet, and breaking down and sorting packages in a 40 foot container. The OIF invasion data provides an estimate of these marginaldistributions. Reality requires them to be correlated. Usinga Gaussian copula (see Ross 7) allows using a different tar- get correlation based on the unit type from Table 4 whilerespecting the correct marginal distributions for weightand volume (see pp.94–95 of McConnell 30). Sampled weights or volumes that exceed that requisition’s transpor- tation mode get reassigned to the maximum capacity for the offending weight or volume; this is reasonable as thisonly occurs less than one-fifth of 1% of the time in thedata. Each requisition is assigned to a sourcing depot condi- tional on the requesting unit according to the empiricalprobability mass function found for each unit type’s source depot location. These empirical distributions can vary based on order content; since the model does not specifyindividual types of repair parts, this approach permits cap-turing the unit-type variation of supply depots while keep-ing the level of detail at overall requisition attributes. Asexplained by Rogers, 5certain Defense Logistics Agency(DLA) distribution centers have specialized stock—such as communications equipment at Tobyhanna, Pennsylvania—and these specializations affect these sour- cing distributions. After specifying the unit (demand node), transportation mode, and sourcing depot (supply node), it is a simple lookup from the appropriate route matrix that stores the route for a requisition going from that depot to that unit with that transportation mode. Collectively these route matrices comprise the route library, which is an output of the logistical network modeling process. After identifying each requisitions route, the entire requisition forecast is completed by aggregating the CL I/CL V requisitions, each unit’s forecast, and the forecast for global requisitions that will compete for DDSP resources. A comparable set of requisitions that are routed through DDSP in OIF pro- vides a start point for the non-expeditionary requisitions that share continental US (CONUS) resources upstream.5 This procedure (Figure 8) provides a data-driven approach to generating requisition forecasts, since MBF is not yet available for all Army units and platforms. The result is a sample path approach that requires replication to assess the uncertainty in the forecast itself, which is the arrival process to the queueing network described inSection 4. 6. Risk-based expeditionary logistics planning for a notional operation Armed with the methodology from Section 4, this section applies the VF-DIS framework on a notional operation. The section uses the same fictional scenario used by Rogers,5where an IBCT and a Stryker Brigade Combat Team (SBCT) are conducting operations from South Sudan into Sudan against the self-styled Islamic State in Iraq and Syria (ISIS). The SBCT will operate in the outly- ing Darfur region, while the IBCT operates in the capital of Khartoum.5A Division Headquarters conducts com- mand and control for the operation and the units are pro- vided with enablers that include engineer and aviation units; Table 7 presents the Task Organization used to gen- erate the forecast for supply requisitions using theTable 5. Transportation mode distribution taken from Operation Iraqi Freedom data. Transportation Mode Probability Military air 0.7340 World-wide express (WWX) 0.1292 Surface (ocean) 0.1368 Table 6. Bounds for standard delivery times (SDTs) in days modeled by the discrete uniform (DU) distribution in Equation(11). Transportation mode distribution taken from OperationIraqi Freedom invasion data. Transportation mode (a TransM ,bTransM ) Probability Military air (12, 18) 0.734 World-wide express (WWX) (10, 14) 0.129 Surface (ocean) (52, 75) 0.137Table 7. Task organization for Sudan mission.5 Task organization 1×Infantry Brigade Combat T eam (IBCT) 1×Stryker Brigade Combat T eam (SBCT) 1×Engineer battalion (construction effects) 1×Aviation battalion (attack & lift) 1×Sustainment brigade 1×Division headquarters 3×Battalions of miscellaneous enablersMcConnell et al. 13 procedure outlined in Section 5. In his analysis, Rogers5 evaluates different COAs based on potential locations for the TDC. While the techniques from this section could assist with evaluating the COAs presented by Rogers, this section focuses on Rogers’ selected option, SudanCOA 1, locating the TDC in Juba, the capital of South Sudan. 6.1. Focusing on the last tactical mile Applying the techniques from Section 4 to the final plan recommended by Rogers5illustrates the contribution of this methodology. The LTM trucks—the ground units that transport supplies from the TDC to the BCT Supply Support Activities (SSAs)—are the ideal candidate for thisdemonstration for several reasons. The LTM trucks are the logistical link to the units, which implies this may be a location where a theater commander has the most controlas they are not necessarily constrained by ties to airports, seaports, or other infrastructure or process restrictions,including CONUS effects. Practically, their geographical proximity to the units also incurs more risk. Given thefeed-forward structure of the network, the LTM trucks are the final node to analyze and identify the required capacity for operations; the process to analyze other nodes is almostidentical except that no other node is the furthest down- stream resource. This property creates a few technical challenges that are readily overcome but not present atany other place in the feed-forward network. In simpler terms, if analysis of LTM trucks is possible, it is possible to do this for any node upstream. 6.2. Overview of the notional operation The operational details and timeline are identical toRogers’ 5scenario; Figure 10 provides a visual summary. The SPOD is located in the port of Mombasa, Kenya. The TDC is located in Juba, South Sudan, with the APODnearby. Although not depicted in Figure 10, the CONUS network is also identical to Rogers’ scenario. Figure 10. Sudan course of action 1 overview with the main supply route (MSR) IRISH annotated.62PH: phase; IBCT: Infantry Brigade Combat T eam; SBCT: Stryker Brigade Combat T eam.14 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) The logistics network in Figure 10 is identical to the one used by Rogers to enable direct comparison with keydetails summarized here. For more details on the logistics network, the reader is encouraged to see Chapter 3 of Rogers. 5To model the logistics network, we make the fol- lowing assumptions. 1. CONUS dedicated truck routes depart six days a week (Monday–Saturday). Trucks travel seven days a week but may not deliver on a Saturday orSunday. 2. CONUS dedicated trucks have unlimited capacity as DLA can quickly acquire additional trucks.Similarly, there are always enough trucks to move requisitions from source depots to CCPs or from a CCP to the APODs/SPODs. 3. Restocking the sourcing depots does not require the same resources used by the distribution system. 4. A CONUS depot sources all orders then ships them to their destination. 5. Once a multipack, pallet, or container has started loading, it remains at the location until full or the maximum waiting period has been met (which- ever is soonest). Pallets wait up to three days withcontainers waiting up to 15. Multipacks, pallets, and containers are considered full when they reach 95% of maximum weight capacity or 85%maximum volume capacity. If at least one of these criteria are met, prior to sending, any requi- sitions that could fit in the remaining space arepulled from the queue and packed (on a revised slack basis) to ship forward to the next location. 6. All pallets and containers contain repair parts for different units unless the route to a unit does not support break bulk operations. This ensures the model does not ship near-empty containers. 7. If a requisition can fill an entire multipack, pallet, or container, the logistics node builds it unit-pure (no other unit orders included). Unit-pure multi-packs, pallets, and containers do not require breakdown and sorting operations at the TDC and advance directly to the LTM trucks. 8. All air shipments use 463L pallets and all ocean freight employ 40 foot containers. These pallets and containers are comprised of multipack boxesand individual parts that are too large to fit in the multipacks. 9. All surface (ocean) freight travels on commercial ships. This is not a limiting assumption but fits the Sudan scenario. 10. Orders for the Sudan scenario follow the same distribution of transportation modes as occurredin OIF. This is not a limiting assumption, as this is an input to the forecasting procedure. Unlike Rogers, 5times to process requisitions are stochas- tic and are akin to service time distributions. The meanroundtrip times from Rogers 5are kept fixed (10 days for IBCT, 8 for SBCT), but this research assumes LTM con- voys may sometimes return up to one day early if thereare good conditions, but may be delayed significantly if adversely impacted by weather or mechanical failures. Without data to estimate these deviations, we assume theLTM roundtrip time is distributed via a generalized Beta distribution having the form a+(b/C0a)Beta (α 1,α2)with the first shape parameter α1=1:2 and second shape para- meter α2=12.63The generalized lower and upper bounds result from the assumption that roundtrips to support the maneuver units take a minimum of 90% of the meanroundtrip time from Rogers 5and no more than twice the mean time. 6.3. Forecasting the required capability to achieve the performance target at location With the network capacities determined by Rogers et al.3 for this scenario, the VF readily provides the sample aver- age arrival rate in PEU to the LTM trucks using 55 samplepaths. The reader is reminded that multiple sample pathsare necessary due to our method of forecasting repair part demand; with a different and perhaps more direct forecast- ing method–such as consumption data-driven MBF–theaverage scenario demand over time might be more accessi- ble, which would require only one run of the VF instead of the multiple runs required in this work. The analysis pre-sented used 55 sample forecasts. The DIS model requires a performance target for this node. Based on Rogers’ 5findings, this section uses a per- formance target that requires the capacity needed to achieve a requisition average (not time average) delay ofseven days. The sample dispersion at the LTM is both time-varying and greater than 1 ( ≈5), which requires the RCF to adjust the forecasted variance. Without this correc-tion, the model would underestimate risk. The forecasted requirements presented in Figure 11 include the risk correction for the LTM required capacitygiven in PEU. The solid bold (blue) line marks the aver- age, with shaded regions denoting the probability that the required capacity is in that range on any given day. Thegraph is not smooth for good reasons. The LTM node is located furthest downstream and is subject to the accumu- lated effects of every node’s schedule nuances (e.g., somelogistics nodes in CONUS do not operate on Saturdays and Sundays). With the timeline fixed, ship schedules,McConnell et al. 15 dedicated trucking routes, and long convoy roundtrip times create a very jagged forecast. This forecast provides the framework that allows ana- lyzing potential outcomes in a stochastic (probabilistic) sense because it provides a fully specified distribution forrequired capacity for every day. With this in place, it ispossible to rapidly compute probabilities, calculate expec- tations, or even generate realizations via Monte Carlo methods. This implies that if something can be calculatedor generated then it is possible to get stochastic descrip-tions for any metrics of interest. 6.4. Generating courses of action The queueing theory that permits construction of the fore-cast depicted in Figure 11 implies a continuous or near-continuous control of that logistics node, but that is hardlypossible in the military logistics context, especially under expeditionary conditions. It may be possible to plan for significant capacity changes once per phase. Perhaps acommander can adjust resources once in the planning hori-zon or maybe not at all. The forecast generated with the VF-DIS model is useful for generating some default capac- ity options for a specific location. Developing multiple options is attractive, as the compu- tational efficiency of these models permits rapid detailedanalysis of each option and permits comparing them over time. One approach is to simply look at the requirements forecasted by Figure 11 and visually set the capacities withintuition or external knowledge about the plan; this tech- nique might develop a plan to ensure the LTM trucks have 70 PEU for Phases I and II then only 20 PEU for PhasesIII and IV (presumably freeing up some capacity for othermissions, including a reserve). A more detached technique would be to simply use the forecast from Figure 11 to cal- culate the daily value-at-risk (VaR 0:95), also known as the 95th quantile, and plan each phase to receive the capacity set to the phase’s time-averaged VaR 0:95. Alternatively, a constant capacity throughout the planning horizon may beappropriate. The analysis proceeds with these three plans, although one can evaluate any given plan. Figure 12 provides a visualization of these competing options over time with the constant option set to 78 PEU, consistent with Rogers’ 5final recommended plan for the LTM trucks. The chart overlays the three LTM options against the backdrop of average required capacity, the 75th quantile for required capacity, and the95th quantile for required capacity to convey a sense ofthe stochastic variation that exists about the average. These options serve to demonstrate the flexibility of this approach and the capability to evaluate any given logisti-cal capacity plan. Figure 11. Forecasted capacity required at the last tactical mile (LTM) in 463L pallet equivalent units (PEUs) to achieve the target performance of seven-day average delay (55 sample paths). Dashed vertical lines denote phases of the operation. Brigade CombatT eam operational intensity level timelines are displayed below the graph for reference. OIL: operational intensity level; IBCT:Infantry Brigade Combat T eam; SBCT: Stryker Brigade Combat T eam. (Color online only.)16 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) 6.5. Evaluating multiple courses of action Regardless of the complexity or the number of capacity plans, the visualization provided by Figure 12 is not enough. Commanders want to understand the impacts ofthese plans in meaningful terms that include both expected performance and an understanding of the uncertainty involved. With the VF and the DIS models, an analyst canevaluate backlogs, delay, lateness, utilization, and other measures by location, by day, by requesting unit, transpor- tation mode, or any combination of these. The model equips the analyst to dig for insights. We present some examples of initial insights that may be constructed bydefault to inform the decision-maker. Figure 13 shows average backlog over time for the three options as an example.This approach extends Rogers et al. 3by not only evalu- ating average delay across the time horizon, but also by making stochastic information available either directly cal- culated analytically using the distributions (by day) or viaMonte Carlo methods, 64,65which are both quick and acces- sible with modern computers. Although Figure 13 currentlyshows an average backlog only, it is just as easy to presentconfidence intervals, quantile bands, or other stochasticvisualizations for a chosen metric of interest. Presentingrisk-based information that depicts the uncertainty coupledwith delay predictions provides a more complete under-standing of the tradeoffs between multiple COAs. Seniorleaders seek to understand the risks faced and how to miti-gate them—to that end it is critical to estimate how badthings can actually get both by location and over time. 6.6. Further evaluation of a specific plan After evaluating a set of plans over time, we arbitrarilyselect the constant capacity plan taken from Rogers et al. 3 for further analysis. With the outputs of the VF and DISmodels, an analyst can evaluate backlog, delay, lateness,and other constructed metrics over time for a specific planand under multiple what-if scenarios. Motivated by themassive sandstorm that resulted in a seven-day disruption to CL IX part resupply early in the invasion of Iraq in 2003, Rogers 5evaluates the impact of a complete disrup- tion of the resupply vehicles for this Sudan scenario. It ispossible to perform a risk-based analysis of the constant78 PEU capacity plan for LTM trucks from Rogers 5under a complete disruption of the resupply vehicles. Figure 12. Generating options for the last tactical mile (LTM) trucks in the Sudan scenario. PEU: pallet equivalent unit. (Color online only.) Figure 13. Average backlog in pallet equivalent unit (PEU) at last tactical mile (LTM) trucks for the LTM capacity plans fromFigure 12. (Color online only.)McConnell et al. 17 Rogers5shows the value of performing what-if analysis on a given plan to assess how it performs under different potential outcomes. This VF-DIS model not only permits the same analysis but also shows information beyond theaverage by connecting potential outcomes with their likeli-hoods. Figure 14 illustrates this by showing the averagebacklog at the LTM trucks with and without a seven-daydisruption starting at D +11; the figure also depicts how bad the backlog can get by showing the 75th and 95thquantiles. The peak backlog with a seven-day disruptionwill be almost 480 PEU (7.4 times the no disruption peak).By taking into account the variation around the average,an analyst can forecast that there is a 75% chance the peakbacklog with this seven-day disruption would be less than550 PEU (8.7 times the no disruption peak) if the LTMtrucks have a 78 PEU capacity. Similarly, there is a 95%chance that the peak backlog would be no more than 685PEU (11 times the peak when there is no disruption peak).The recovery times are also available from Figure 14. Because of the underlying distribution forecast by day, Monte Carlo methods are readily available to assess notonly average behavior over time for a specified logisticsplan but also the full distribution of worst-case behavior.Traditional average worst-case metrics used in finance,such as conditional value at risk, can be readily computed, but the computational speeds enable looking at more than the conditional expectation of the worst-case scenarios(e.g., worst 5%). Further, the worst-case distribution of aparticular metric is available. 30 The utility of risk-based measures for what-if analysis cannot be overstated. These tools enhance the MLNPSand extend the possible depth of analysis. Using multipledemand forecasts (sample paths) required multiple runs ofthe VF to estimate the sample arrival rate as well as thesample dispersion. If the US Army continues to developMBF beyond aviation units, this analysis would require only one run with the VF if GCSS-A data provided thesample dispersion for the variance correction. Multiplesample paths are currently required to obtain the requiredcapacity forecast (Figure 11), but with an approach suchas MBF that does not rely on sample paths, this is obtain-able with a single run of the VF. By exploiting the strengths of both the DIS model and the VF as well as the feed-forward network structure, thisapproach maximizes its computational advantages. Afterobtaining the forecasted requirements to meet a logisticstarget (Figure 11), the subsequent analysis is computation-ally efficient using either analytical results, VF output, orsimple Monte Carlo methods, which are extraordinarilyfast in our experience just working with MATLAB. 66 7. Conclusion This research contributes to the military expeditionarylogistics planning problem in several ways. Firstly, sinceMBF is not operational for the majority of Army plat-forms, formations, missions, or operational environments,we use supply-side modern combat data taken from OIFand apply MBF-style techniques, namely stratified sam-pling, to generate the best forecast possible for the demandsignal. This forecast accounts for unit type and operationalmission. Based on an operational scenario, we improve theMLNPS capabilities by adding techniques to account foruncertainty and assess risk; we achieve this improvement using a sample path-based forecasting approach, incorpor- ating recent advances in time-varying queueing networks,such as the DIS approximation, and fusing these capabil-ities with the strengths of the VF using the tandemapproach. The VF excels at realistic tasks such as properlypacking multipacks, pallets, and containers, timing theshipments, and accommodating real-world schedules; thequeueing model integrates both time-varying propertiesand overdispersion. This is the first paper to use a time-shifted variance correction to account for overdispersionin a DIS setting. These enhancements to the MLNPS center on two fun- damental tasks: (1) given a plan, estimate the plan’s per-formance, and (2) given a target performance, find therequired plan. This research provides a data-to-decisionsupport process that yields a framework for assessing riskas both the severity and likelihood of possible outcomesbecome available via analytical calculation or MonteCarlo methods. Our intent is to highlight the utility GCSS-A data can provide by enabling decision-support and plan-ning models in ways not previously possible. Efforts to capture readiness-based performance metrics and model a logistical plan while including the collectionof contingency (branch) plans are already underway. Figure 14. Daily last tactical mile (LTM) backlog with and without a seven-day disruption (starting D +11) for the Sudan course of action (COA) 1 scenario in Rogers5for the 78 pallet equivalent unit (PEU) LTM plan. PH: phase.18 Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 00(0) Current efforts are also working toward integrating alter- native routing and decision points into the modelingframework. Future efforts should expand this emphasis on readiness and provide prescriptive decision-support capa- bility rather than the predictive outputs this paper illus-trates. Outputs should include identification or visualization of the readiness-versus-cost tradeoffs. We envision these ongoing efforts resulting in a risk mitigation design matrix to provide insights on how to ‘‘optimize’’ various resilient, robust, adaptive, or flexiblelogistics network mitigation strategies against potentiallydisruptive conditions or catastrophic events within an operational risk landscape (defined by event probability versus consequence). The methods in this paper apply to other contexts that employ an underlying stochastic queueing network modelthat exhibits nonstationary and/or non-Markovian (non- exponential) arrival processes. These applications include disaster relief, humanitarian operations, and understandinginternational commercial global supply chain disruptions. Acknowledgments The authors would like to thank Marc Robbins from RAND, Joseph Faris from DLA Distribution, and LTCJohn Hiltz & Team from TRANSCOM; this paper would not have been possible without their assistance. We are indebted to the skilled professionals at ERDC, TRAC-LEE, and TRAC-MRY for taking time to showcase their analytical tools as well as provide invaluable feedback for both this paper and our military logistics research. Theauthors also thank several members of CASCOM for pro- viding insights into GCSS-A and its potential. A special thanks to Walt DeGrange for sharing his military logisticsexpertise. Disclaimer The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Army, the Department of Defense, or the United States Government. Declaration of conflicting interests The authors declare that there is no conflict of interest. Funding This work was supported, in part, by a grant from the Army Research Office (grant # W911NF1910055). ORCID iD Brandon M McConnell https://orcid.org/0000-0003- 0091-215XAppendix: list of acronyms Readers seeking definitions of military acronyms will find Joint Publication (JP) 1-02 Department of Defense Dictionary of Military and Associated Terms a valuable resource.51