Introduction

The adequate sizing of logistics support is essential to obtain a desired military capability while optimizing resource use. However, the extremely complex, stochastic, and nonlinear nature of a typical air-base aircraft maintenance system has led analysts to achieve only limited success using optimization methods to study and improve the system's performance. They have instead turned to simulations such as the Logistics Composite Model (http://www.dtic.mil/matris/ddsm/srch/ddsm0086.html, accessed 28 December 2004) or the Airfield Simulation Tool (Cusick, 2004) to analyse current capabilities and to seek system improvements. However, such models are often too expensive, too complex, or too difficult to operate to benefit smaller or more narrowly focused organizations.

The Argentine Air Force (AAF) is reviewing its logistics doctrine with the aim of improving its logistics support structure. However, the current AAF environment is characterized by resource constraints that affect logistics decision-makers twofold. First, resources are scarce: limited human and physical means will lead to relatively few options to materialize the logistics support. Second, a lack of skilled human resources, limited computer systems and low compatibility of existing databases bound the planning process itself. Therefore, our goal was to develop a user-friendly, mathematically accessible method for the AAF to estimate the capacity of a given logistics support system to service combat aircraft between successive sorties. The user would re-run the model using different combinations of resources and planned activity levels until the desired system performance is obtained. We chose a spreadsheet environment because of its widespread use within the AAF, and because of its proven ability to handle stochastic capacity and inventory planning problems (see, eg, Beversluis and Jordan, 1995).

Related work includes Granger et al (2005) who use parametric decomposition and two-moment approximation methods with queuing networks to improve steady-state network performance, and present an airfield operations example to motivate their method. Desa and Christer (2001) discuss a case study of inter-city bus fleet maintenance in a developing country when maintenance data is lacking. Finally, Adamides et al (2004) use system dynamics and simulation to assess the long-term performance of military jet engine maintenance systems to support investment decisions in engine reliability and maintenance infrastructure.

Method description

Our objective was to develop a spreadsheet-based method to quickly evaluate the mean number of aircraft that can be serviced in a specified time interval between consecutive sorties, for a given resources mix and air-base physical geometry. This maintenance resources evaluation technique (MRET) uses an analytic-simulation heuristic to estimate the mean and variance of a failed aircraft's downtime for unplanned maintenance. These parameters are then used as the input to a Monte Carlo simulation of a user-defined network of scheduled tasks (eg cargo loading/unloading actions, routine maintenance, refueling, and documentation preparation) and unscheduled tasks (repair or replacement of failed components) necessary to prepare aircraft for subsequent sorties. Scheduled and unscheduled tasks together constitute the actions necessary to recover an aircraft (prepare it for use) prior to its next sortie. The MRET probabilistically computes resources usage. Usage depends on whether only scheduled tasks, or both scheduled and unscheduled tasks are required, as shown in Figure 1. The method's spatial layout is designed around a central facility and one or more dispersed sites (a dispersed site is a collection of maintenance resources needed to prepare aircraft for sorties). Resources are grouped into one of two types: flexible (type-A) items that can be applied to different aircraft, such as personnel, munitions handling equipment or test equipment; and type-B items that can only be assigned to one aircraft, such as a replacement part. Cannibalization of serviceable parts from an operative aircraft is not modelled. Each site may have a different number of aircraft to recover, as well as a different amount of each resource type on hand. When a resource is required at a particular dispersed site, it can be obtained from either another dispersed site or from the central facility; although a time delay is incurred to move the resource from its original location to the requesting site. The MRET requires the following input data:

• aircraft fleet and planned activity (eg aircraft types and numbers, planned sorties)
• required scheduled tasks and associated sequencing necessary to recover the aircraft
• resources needed to perform the scheduled and unscheduled tasks
• critical aircraft failure modes and their associated failure rates
• time needed to perform scheduled and unscheduled tasks
• total number of resources available of each type, and their physical distribution among the dispersed sites and the central facility

Figure 1.

Aircraft recovery time.

Full figure and legend (20K)

The MRET then computes a 95% confidence interval for the mean number of aircraft that can be recovered within a given time interval. If this computed number is less than a user-specified minimum, then the resources or the planned level of activity can be changed and the method recomputed until desired results are obtained.

Computational method

The MRET's computation follows four steps:

Step 1: The scheduled task network is defined, and the minimum, most likely, and maximum task times are specified for each task. These three task times are a function of the level of resources assigned to that task.

Step 2: An analytical method is used to compute the mean and variance of the unscheduled maintenance time (UDT) for each aircraft.

Step 3: The completion of the conditional network of aircraft recovery activities defined in Step 1 is simulated by randomly generating completion times for each task. Scheduled task times are drawn from triangular distributions, characterized by their minimum, most likely, and maximum times. (We use triangular distributions because the minimum, maximum, and most likely values are typically obtainable. Furthermore, the triangular cdf accommodates a wide variety of symmetrical and asymmetrical cdf forms. A disadvantage is that the true extreme values may not be known.) The need for any unscheduled tasks for each aircraft is determined by simulating the aircraft's failure time (accomplished by drawing an exponentially distributed random variate with rate equal to the sum of the aircraft's critical failure mode rates). If this simulated failure time is less than or equal to the aircraft's last sortie time, then the aircraft would have experienced a critical failure and the unscheduled task time UDT is then drawn from a lognormal distribution, characterized in Step 2.

Step 4: Step 3 is replicated until a 95% confidence interval of the mean number of aircraft that can be recovered for pre-established time intervals is calculable. We specify the number of replications to assure that the confidence interval width is less than or equal to one aircraft. By repeating Steps 1–4 for different time intervals, a logistics planner can create a plot of the mean number of aircraft recovered versus time, for a given level of resources.

Analytical method for UDT

UDT is defined as the transit time needed to gather any type-A and type-B resources needed, plus the time necessary to perform the unscheduled task itself. The method assumes that all resources must be obtained before the unscheduled task can start. The MRET accommodates the existence of m=1, 2, ..., M maintenance sites, with P_m identical aircraft per site, each of which has N critical failure modes (refer to Figure 2). A critical failure mode is one whose repair cannot be deferred (other failures could also occur during a given sortie, but they are assumed to be noncritical, and their repairs can thus be postponed as a future scheduled task). We also assume that all failure modes are independent, their failure rates are constant, and that only one critical failure can occur per aircraft per sortie. Each failure mode n (for n=1, 2, ..., N), has a particular maintenance time UDT_n, which depends not only on the repair time (UTT_n) distribution but also on its associated resource transit time (TT_n) distribution (we define UDT_n as the sum of TT_n and UTT_n). Each failure mode n requires one type-B resource, and up to I_n different type-A resources. Resource transit times TT and repair times UTT are assumed to be independent. Since only one of N possible critical failure modes can occur per aircraft, thus UDT₁, UDT₂, ..., UDT_N are mutually exclusive random variables. UDT's actual probability density function is the convolution of the TT and UTT densities, which are themselves distributed as mixture probability density functions of the n=1, 2, ..., N respective TT_n and UTT_n random variable densities (see, eg, Leemis, 1995). The mixture densities and the convolutions are not calculated directly because the TT_n moments and the TT_n and UTT_n densities are all unknown. Furthermore, we felt that computing the mixture densities and the convolution would require mathematical expertise beyond that of a typical military planner. Instead, we assume that UDT's density function is lognormal, characterized by the estimated UDT mean and variance (see, eg, Blanchard et al, 1995) for a justification of the lognormal form). We also assume that the mean and variance of each unscheduled maintenance time random variable UTT_n, for all failure modes n=1, 2, ..., N, is known. The mean and variance of each transit time random variable TT_n is estimated using a heuristic analytic-simulation method. Figure 3 presents the probabilistic network of the process of gathering resources to fix a particular failure mode. Node (7) denotes the completion of transit time which can only be realized if nodes (5) and (6) are both realized, that is, all resource types (A and B) must arrive at the aircraft location before the unscheduled maintenance task can start.

Figure 2.

Unscheduled maintenance time. t_p is the aircraft p's total time for its last trip.

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Figure 3.

Transit time (TT) computation.

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The network between nodes (2) and (5) represents the time delay necessary to obtain a type-B resource item (associated with an aircraft's failure mode n) from a storage location. Two sources of type-B supply are considered for the MRET: either from the dispersed site where the aircraft is located or from the central facility. There is a time delay STB_n associated with the site and a time delay CTB_n with the central facility.

The probability of obtaining a particular type-B resource item from the site supply depends on the number of items kept there and the simultaneous demand from all aircraft serviced at that site. The constant failure rate assumption allows resource demand to be a Poisson process; therefore, the conditional probability that the time delay is based on the resource being available at site, written as P(STB_n), is

where NF_n is the number of mode n failures at site m, SS_n is the quantity of type-B resources stored at the site, associated with failure mode n, _n is the failure rate of mode n, t_p is the usage time (eg, mileage or hours) by a particular vehicle p in the trip just completed, and is the Poisson cdf parameter for NF_n.

The probability that the time delay is associated with obtaining the resource from the central location is the complement of P(STB_n); therefore, P(CTB_n) is computed as

We now use a result by Yaspan (1968) and an identity by Bronstein and Semedian (1976), which collectively state that if h=1, 2, ..., H random variables Z_h are mutually exclusive, then the overall mean () and variance (VAR(Z)) can be computed as

where p_h is the probability of occurrence (weighting factor) of event Z_h. Since the events of resource supply from the site or from the central facility are mutually exclusive, then (1) and (2) can be used to compute the mean and variance of the transit time of this type-B resource (ie the time from node (2) to node (5)), by using P(STB_n) and P(CTB_n) as weighting factors.

The network between node (3) and node (6) represents the gathering time from various locations for any of the i=1, 2, ..., I_n different type-A resources needed to repair an aircraft's type n failure mode. For the MRET, three locations are considered: from the local site where the aircraft is being serviced, or from the central facility, or from another dispersed site after a delay due to their own use of the resource. There is a time delay STA_i associated with delivery from the local site, a time delay CTA_i from the central facility and a time DTA_i from the other dispersed site.

Since an aircraft's critical mode failures are assumed to represent a Poisson process, we can define the probability of supplying a type-A resource i from the local site, P(STA_i), as

where RA_i is the quantity of required i resources at the local site m, SA_i is the quantity of available i resources at the local site m, F_i=_n=1^r_in is the frequency of use of resource i for the rN failure modes that require a type-A resource i for repair, and t_p is the usage time (eg, mileage or hours) by a particular aircraft p in the trip just completed.

The probability of being forced to obtain a type-A resource i from either a central location or from another site, P(CDTA_i), is the complement of P(STA_i); therefore, P(CDTA_i) is

The network between nodes (4) and (6) represents the options of obtaining a type-A resource from the central facility or from another dispersed site. The probability of obtaining this resource from the central facility depends on the number of resources available there and the simultaneous demand from all sites. This demand is that which cannot be satisfied by the site's resources. Since aircraft failures represent a Poisson process, therefore we can define the probability of supplying a type-A resource i from the central facility, P(CTA_i), as

where RAC_i is the number of required i resources from the central facility, SAC_i is the number of available i resources at the central facility, F_i is the frequency of use of resource i, P(CDTA_i)_m is the probability of obtaining the i resource from either the central facility or from another site, for use at site m, t_p is the usage time by a particular aircraft p in the trip just completed, is the total requirement of an i resource from all sites.

The probability of obtaining an i resource from another dispersed site (after they have finished using it) is the complement of P(CTA_i); therefore, P(DTA_i) is computed as follows:

Since supply from the local site, the central facility, or another dispersed site are mutually exclusive events, then (1) and (2) can be used to compute the mean and variance of the transit time of a type-A resource (time delay from node (3) to (6)). In this case, P(STA_i) is used as the weight for the distribution of times from the local site, the product P(CDTA_i)P(CTA_i) as the weight for the transit time from the central facility and the product P(CDTA_i)P(DTA_i) as the weight for the transit time from another dispersed site.

The solution for the mean and variance of the time delay across the network between nodes (2)–(5) and nodes (3)–(6) is now complete. To find the mean and variance of transit times across the whole network (between nodes (1) and (7)) imposes a different challenge due to the logic associated with node (7). For each unscheduled maintenance action, the overall transit time is jointly distributed in terms of the transit times required to gather the type-B resource and all type-A resources needed for the repair. In graphical evaluation and review technique (GERT) terminology, this particular node within a conditional network is an AND node because all the arriving tasks must be performed in order for the node to be realized (Pritsker and Happ, 1966). When times in the network are random variables, no feasible analytical method is available (Pritsker and Happ, 1966; Whitehouse, 1973). Using Pritsker and Happ's (1966) suggestion, we use an analytic-simulation heuristic for the MRET's Step 2 to estimate the mean and variance for the assumed lognormal joint distribution, as follows: we first use a triangular distribution for a type-B resource, and exponential transit time distributions for type-A resources. (Our rationale for using exponential distributions is that right-skewed, high-variance distributions are typical of the time needed to obtain these resources.) We then build an AweSim model and simulate the network in Figure 3 for a range of resource levels and transit time parameter values, and finally perform a regression analysis of the simulation results to estimate the transit time mean and variance as a function of these variables. The regression enables us to quickly re-run the MRET for a variety of scenarios without having to re-accomplish the Step 2 analytic-simulation heuristic for each scenario.

If UTT_n and TT_n are independent for all n=1, 2, ..., N, then the UDT_n mean and variance are

The final step is to describe the UDT mean and variance for each aircraft. Since UDT₁, UDT₂, ..., UDT_N are mutually exclusive random variables, then (1) and (2) lead to

Equations (3) and (4) represent the mean and variance of an aircraft's unscheduled maintenance time as weighted by each mode's relative failure rate. Finally, (3) and (4) determine the parameter values for the lognormal probability distribution used to generate unscheduled maintenance times in the MRET's spreadsheet Monte Carlo simulation.

Verification

An AAF scenario consisting of 12 aircraft was modelled, based on two dispersed sites (each supporting six identical aircraft) and one central facility used to store and distribute type-A and type-B resources. Six different scheduled tasks were defined, including weapons loading and unloading, pilot debriefing, inspection, refueling, and predeparture briefing, as shown in Figure 4. Each site contained two maintenance teams, with each team responsible for maintaining three aircraft. To capture the effect of resource contention, three triangular distributions were defined for each scheduled activity, conditioned on whether the resource levels at each site allowed only one, two, or three assigned aircraft to be simultaneously maintained. We refer to these three respective levels as low, medium, and high. The analytic-simulation UDT heuristic regression was computed for failure mode repairs requiring one type-B resource and up to five different type-A resources. From the regression, the main predictors of the transit time mean and variance were found from the variable values observed from simulation. The following analytical expressions were designed:

where RM is the average of the transit time means for type-A resources, in minutes, ND is the number of type-A resources, and SM is the transit time mean for type-B resource, in minutes.

where CV is the coefficient of variation (standard deviation ), and CV_RA is the maximum CV of type-A resources.

Figure 4.

Tasks and precedence relationships.

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These formulas were developed using data within the following range:

MRET verification first involved checking the method's (mean recovery time) response reasonableness to changes in aircraft reliability, sortie length, type-A resource physical distribution, and geographical resource dispersion.

Aircraft reliability

We would expect that an increase in reliability would result in decreased mean recovery time, and that the effect would be stronger for lower resource levels. We varied the MTBF of each failure mode from 50 to 1000 h, and simultaneously set all resource levels first at low, then medium, and finally at high. Figure 5 supports our conclusion that the MRET's response follows the expected general trend for changes to aircraft reliability.

Figure 5.

Reliability effect.

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Sortie length

We anticipated that an increase in the previous sortie's duration would cause an increase in the mean recovery time due to a greater probability of failures. This effect was also predicted to be more noticeable as the aircraft reliability decreases. To check the method's response, the sortie length was varied from 1 to 4 h, and the mean recovery time computed using a parameter of 1000, 100, and 50 h for the MTBF of each failure mode. Figure 6 shows the results.

Figure 6.

Effect of sortie length.

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Type-A resource physical distribution

We predicted that as more resources are stored at the local site (near the aircraft), the mean recovery time would decrease. This effect was also predicted to be more noticeable as the aircraft reliability decreases. To check the method's response, for each type-A resource a total number of nine units were assigned to the overall system. The number of these resources stored at each site was varied from zero to 4 and the mean recovery time computed using a parameter of 1000, 100, and 50 h for the MTBF of each failure mode. Figure 7 depicts the results, which suggest that the method's response to changes in the distribution of type-A resources among the central facility and the maintenance site conforms to reasonable expectations.

Figure 7.

Effect of type A resources distributions.

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Resource Spatial Separation

We believed that the greater the distance between the central facility and each dispersed site, the greater the mean recovery time would be, due to an increasing delay in availability of resources. This effect was also predicted to be more noticeable as the aircraft reliability decreases. To check the method's response, the mean transit time needed to move a type-A or -B resource from the central facility to both maintenance sites was varied from 40 to 50 min (while keeping its variance constant) and the mean recovery time computed using a parameter of 100 and 50 h for the MTBF of each failure mode. The results in Figure 8 support our prediction.

Figure 8.

Effect of resources dispersion.

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Analytic-simulation heuristic analysis

The next step was to compare the MRET to a discrete-event (AweSim) simulation model. Our goal was not to validate the MRET, but rather to determine whether the MRET's combination of an analytic-simulation heuristic and a Monte Carlo simulation could produce results comparable to the more realistic queuing representation of dynamic discrete event simulation. If it did, then the better suitability for spreadsheet programming and the advantage in response speed obtainable by MRET would justify further development of this class of method. The two models were designed to differ only in the way unscheduled maintenance is modelled. (Note that the discrete event simulation model required about 1000 lines of code to describe the 12-aircraft two-site scenario used to verify the MRET.) Resource levels were varied from low (constrained, with only one aircraft serviced at a time, and one unit of each type-A resource available at each site) to high (relatively unconstrained, with three aircraft serviced simultaneously, and three units of each type-A resource available at each site). The MTBF for all failure modes were varied from 50 to 400 h. The output from both models were tabulated and compared in absolute and percentage terms, as shown in Table 1. The MRET's heuristic UDT approach performed reasonably well (5% error) for medium or high resource levels and low to moderate demand (0.78 probability or higher of obtaining resources at site). The error increased with decreasing resource levels. This divergence behaviour has been noted in similar research (see, eg, Dietz and Jenkins, 1997).

Table 1 - Recovery time mean and standard deviation.

Full table

Comparison of mean number of recovered aircraft

This measurement is the one that logistics and operational planners would most likely use in their decision making process. Figures 9, 10 and 11 compare the MRET and the discrete event simulation results for different resource levels and P(STA) values. From Figures 9, 10 and 11, the MRET and the simulation model tend to behave in a similar manner for all resource availabilities and unscheduled maintenance action demands considered in this experiment. For each resource level, when the probability of obtaining resources at the local site is high (ie, a high P(STA) value) the concordance between the two models is very good. The MRET behaves in a pessimistic (conservative) manner, by underestimating the mean number of recovered aircraft. Note that the compared values are point estimates. To establish the statistical significance of differences between the point estimates, we tested a null hypothesis that the mean quantities of recovered aircraft were identical, with 95% confidence. In all cases the null hypothesis was rejected—the differences are statistically significant at =0.05. In practical terms those differences were found to be less conclusive. If we accept that for operational planning purposes fractional aircraft have no practical meaning and that, adopting a risk-averse behaviour, decision-makers would round down the model response, then the maximum practical difference observable with this experiment was one aircraft at a high resource level and two aircraft at low or medium resource levels. The greater divergence between the models was always observed for high-demand situations. The MRET's conservative nature suggests that it would tend to understate the number of aircraft that a maintenance system is actually capable of recovering in a given time interval. Conversely, it would overstate required resources to support a desired sortie generation rate.

Figure 9.

MRET versus SIM: RL=high, P(STA)=0.996.

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Figure 10.

MRET versus SIM: RL=low, P(STA)=0.32.

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Figure 11.

MRET versus SIM: RL=medium, P(STA)=0.569.

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Conclusions and future research

The MRET's embedded heuristic-simulation transit time heuristic represents a potentially significant savings in modelling effort. Translating a user's particular Figure 3 network into a discrete-event simulation model is straightforward but can be done centrally by more experienced analysts, and a typical spreadsheet's built-in regression tools can then be used to develop the regression model for estimating the resource transit time parameters for a given scenario. Once done, the MRET can be quickly re-run by a planner with more modest analytical skills for any combination of input parameter values within the regression's range.

Overall, the MRET verification efforts were sufficiently successful for AAF planners to begin the next step of validating the method's results within their environment. The payoff will be a robust, yet simple spreadsheet-based logistics method that can quickly run on existing, widely available computer hardware and software—greatly increasing the chance of model acceptance by the AAF. This simple, fast tool will assist fleet operations planners in assessing the feasibility of logistics support under various conditions.

Our current research seeks to validate the MRET by using AAF operational data, and by comparing MRET results to output from an accepted air-base logistics support model. The MRET must also be scaled to accommodate more of the data that may be needed to model an actual operating environment. The stability of the spreadsheet software and computational time for large-scale scenarios must also be assessed. Once done, we believe that this modelling approach would be useful for any military or civilian fleet vehicle support organization that seeks to adequately size their logistics support structure while optimizing resource use.

Statement of contribution

The Argentine Air Force (AAF) seeks to improve its logistics support structure, but faces significant competition for material resources. Furthermore, a lack of skilled human resources, limited computer systems and low compatibility of existing databases bound the planning process. Therefore, our goal was to develop a user-friendly, mathematically accessible method for the AAF to estimate the capacity of a given logistics support system to service combat aircraft.

We introduce a method for the AAF to estimate the mean number of aircraft that can be restored in a given time between consecutive sorties, given specified maintenance resources and base spatial layout. Our method uses an analytical approach to estimate the mean and variance of aircraft unscheduled downtime. These parameters are then used in a Monte Carlo simulation of scheduled and unscheduled maintenance tasks necessary to prepare aircraft for the next sortie. Programmed in a spreadsheet, our model achieves the appropriate blend of realism and tractability for an AAF planner.

Existing military aviation logistics analyses are typically based either on large-scale simulations that are expensive and require significant data input and expertise to run, or on analytical methods that require advanced queuing concepts and steady-state assumptions that limit analysis to long-run behaviours. Instead, we embed a regression-based metamodel into our Monte Carlo simulation of airbase activity. The regression approximates the resource queuing and transit time delays found in more complex dynamic discrete-event simulations, while sparing the military planner the cost and expertise needed to build and interpret such models. It also enables the military planner to study both transient and long-run maintenance performance. Finally, while we focused on the AAF environment, we believe that our approach will support any fleet operations support planner who wishes to 'right size' their maintenance structure.

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