Availability of Weapon Systems with Logistic Delays: A Simulation Approach

Transcription

1 Avalablty of Weapon Systems wth Logstc Delays: A Smulaton Approach K Sadananda Upadhya and N K Srnvasan Centre for Aeronautcal System Studes and Analyses Defence Research and Development Organsaton Bangalore, Inda Abstract The avalablty of weapon systems such as fghter arcraft, battle tanks and warshps durng hgh ntensty conflcts becomes low. In ths paper the avalablty of fghter arcraft wth fve major subsystems (structures, engne, avoncs, electrcal and envronmental) are consdered. Ths depends manly on attrton factors (falure due to unrelablty and falure due to battle damage) and logstc delays, whch affect repar process. We develop a smulaton model consderng the fghter arcraft as the weapon system for arrvng at transent solutons for avalablty wth logstc delays. The methodology s based on dscrete event smulaton usng Monte Carlo technques. The falure tme dstrbuton (Webull) for dfferent subsystems, repar tme dstrbuton (exponental) and logstc delay tme dstrbuton (lognormal) were chosen wth sutable parameters. The results ndcate the pronounced decrease n avalablty (to as low as 20% n some cases) due to logstc delays. The results are however senstve to the relablty, mantanablty and logstc delay parameters. Key Words: Dscrete event smulaton, Monte Carlo technque, Weapon systems and Avalablty

2 1. Introducton Avalablty of weapon systems durng hgh ntensty conflcts s essental from the pont of vew of wnnng the battle. Mantanng a hgh level of avalablty of weapon systems such as arcraft, battle tanks and warshps becomes dffcult durng campagns. The avalablty of these weapon systems decreases to a low level wthn the frst few days of the battle [1]. Ths s manly due to logstc delays affectng the repar process and attrton factors such as falure due to system unrelablty and falure due to battle damage. Battle damage falures occur because of the nature of the conflct. In an earler paper we developed a smulaton model for avalablty of weapon systems under battlefeld condtons consderng only one repar level wthout logstc delays. Further the weapon system as a whole.e., the arcraft as an entty, was consdered. In a study of battle tanks, Kessler has analysed the loss rate due to battle damage of tanks and the mantenance requrements n wartme through a smulaton to arrve at logstc requrements usng exponental dstrbuton [2]. Emerson has dscussed the capablty of arbases to generate effectve combat sortes and some mprovement optons, whch could ncrease the combat capablty of arbases durng wartme [3]. The smulaton model developed here consders the fghter arcraft as the weapon system. The falure of the arcraft may be due to falure of any one of ts fve major subsystems namely structures, engne, avoncs, electrcal and envronmental. Snce the relablty parameters of the subsystems vary, the relablty related falures for these subsystems s assumed to occur n the proporton α 1 : α2 : α3 : α4 : α5 such that 5 = 1 α = 1. Here α 1 corresponds to the fracton of the system falure due to the unrelablty of the structures, α 2 that of engne, α 3 that of avoncs, α 4 that of electrcal subsystems and α 5 corresponds to the fracton of the system falure due to the unrelablty of envronmental subsystems. Consderng the vulnerablty of these subsystems n battlefeld condtons, the falures due to battle damage are modelled to

3 occur n the proporton β 1 : β2 : β3 : β4 : β5 wth β = 1. Here β 1 corresponds to the fracton of system falure due to the battle damage of the structures, β 2 that of engne, β 3 that of avoncs, β 4 that of electrcal subsystems and β 5 corresponds to the fracton of system falure due to the battle damage of the envronmental subsystems. (We assume that all arcraft are reparable, though n practce a few may be scrapped.) In practce, falures may be happenng as multple events caused by a sngle event such as fre n the engne. Multple events may cause damage to several subsystems such as engne, structures and electrcal subsystems. We however, treat the falure as separate events occurrng to these subsystems. We assume that the falure s due to one man subsystem, whch trggers the other events. In the present paper we are not consderng condtonal probabltes or Bayesan approach for falures snce the 5 = 1 requred data are not avalable to the present authors. In order to restore the faled arcraft they have to be repared. The repar process can be at dfferent levels such as feld level, ntermedate level and depot level. The faled subsystem may requre mnor repar, partal repar or major repar. If the repar work s of mnor nature then t may be performed at the feld level. The repar n some cases may be done at the ntermedate level. The major repar s performed at the depot level [4]. If the spare parts requred are not n stock, they may have to be transported from another source such as a nearby depot or a local suppler. Therefore the logstc requrement n terms of spares or mantenance crew or equpment, at all the repar levels may ental logstc delay for the repar process. Snce the paper addresses the logstc delay n general, as due to spares or mantenance crew or equpment, though n practce all the three factors may not play a large part n any gven stuaton, a logstc delay term s ncluded.

4 2. Methodology The smulaton methodology s based on dscrete event smulaton (DES) usng Monte Carlo technque [5]. Snce the structures, engne and envronmental subsystems are mechancal n nature, falure rates are not treated as constant but age related; therefore t s approprate to use Webull dstrbuton as the falure tme dstrbuton. For electrcal subsystems also the Webull dstrbuton s employed as the falure tme dstrbuton as electrcal subsystems often nclude mechancal components. Avoncs conssts bascally of electronc components; therefore exponental dstrbuton for falure tme s consdered approprate [6,7]. The repar process may follow ether exponental dstrbuton or lognormal dstrbuton but n the present paper we consder only exponental dstrbuton for the repar process [7]. Snce logstc delays are found to have long tal dstrbutons, a lognormal dstrbuton s appled for modellng logstc delays [4]. The probablty densty functons and the relablty functons of these dstrbutons are as follows: Exponental dstrbuton: Probablty densty functon f (t) = λ exp( λt), t > 0, λ > 0 (1) Relablty functon R( t) = exp( λt) (2) where λ s the falure rate. Webull dstrbuton: n t n 1 t n Probablty densty functon: f ( t) = ( ) exp( ( ) ), t0 t0 t0 t > 0, t 0, n > 0 (3) Relablty functon t n : R( t) = exp( ( ) ) t (4) where n s the shape parameter and t0 s the characterstc lfe or scale parameter. Log-normal dstrbuton: Probablty densty functon: 2 1 f (t) = 1 log( t) µ exp, t 0, < µ <, σ 2 > 0 (5) tσ 2π 2 σ where µ and σ are the mean and standard devaton of the normally dstrbuted random varable log(t ) (For lognormal dstrbuton the relablty functon do not have a closed form expresson.) 0

5 In most of the studes two states are consdered wth falure rate and repar rate and then falure tme and repar tme are modeled as exponental dstrbutons. In ths case analytcal expresson for avalablty can be obtaned. If the falure tme and/or the repar tme dstrbutons are not exponental, the analytcal expresson for the avalablty becomes dffcult. Therefore a smulaton technque s requred for estmatng the avalablty [8]. The major assumptons of the model are as follows. 1. The falure can be ether due to unrelablty or battle damage 2. The system fals when one of the fve subsystems fal and the falures are ndependent. 3. The model consders only sngle falures. Multple falures and that due to common cause falures are not consdered. 4. The logstc delay ncludes delay due to spares or crew or equpment but treated as a sngle random varable. The smulaton model s developed as follows. 1. Falure tme dstrbuton f (t), due to system unrelablty s consdered to be a mxture of Webull dstrbutons,.e., f ( t) = α 1 f1( t) + α2 f2( t) + α3 f3( t) + α4 f4( t) + α5 f5( t), such that α = 1, wth f ( ), f ( ), f ( ), f ( ) and f ( ) beng Webull dstrbutons for the fve 1 t 2 t 3 t 4 t 5 t subsystems wth parameters n, ), n, ), n, ), n, ) and ( n5, t05 ) ( 1 t01 ( 2 t02 respectvely, n s are the shape parameters and ( 3 t03 5 = 1 ( 4 t04 t 0 s are the characterstc tmes and α, = 1,2,3,4, 5 represents the fracton of the system falure due to the unrelablty of the th subsystem. 2. The fracton of the system falure due to battle damage of the th subsystem s β ( = 1,2,3,4,5 ) such that β = 1. The values of α s as well as 5 = 1 expert opnon avalable to the authors. β s are deduced from hstorcal records or based on

6 3. The fracton of the th faled subsystem requrng mnor repar at the feld repar shop s γ 1, the fracton requrng partal repar at ntermedate repar shop s γ 2 and that requrng major repar at depot repar shop s 3 3 j= 1 γ = 1 for all. Ths s shown n the decson tree [Fg. 1]. j γ such that 4. The repar tme dstrbuton s modelled as exponental. The mean tme for mnor repar (MTTR) s MTTR maj. MTTR, for partal repar MTTR par, and for major repar mn 5. The logstc delay n procurng spares or crew or equpment at the repar shops s modelled as lognormal wth dfferent set of parameters for the three repar shops. We treat the logstc delay as an aggregate model snce we do not have specfc data for delay due to spares, crew or equpment separately. The parameters of the lognormal dstrbuton wll be derved from the 50 th ( M 50 ) and 80 th ( M 80 ) percentles of the correspondng dstrbutons whch are estmates provded by experts [7]. In the smulaton, the random varates are generated usng nverse transform method [9]. Snce the relablty functon of the exponental dstrbuton s n the closed form, nverse transform method s appled drectly for generatng random varates from ths dstrbuton. As the lognormal dstrbuton does not have a closed form expresson for the relablty functon the drect applcaton of the nverse transform method s dffcult. Therefore the transformaton due to Marsagla and Bray s used to generate random varates from the lognormal dstrbuton [10]. 3. Structure of Smulaton The smulaton structure s based on the state transton dagram as gven n Fg. 2. The arcraft n servce and the arcraft n repar are consdered as two separate states. The falures due to unrelablty and battle damage are the transtons that take place for the arcraft n servce state to the arcraft n repar state. At the repar state the faled tem may requre mnor repar, partal repar or major repar. If the requrement

7 M n o r Repar F e ld Level S t ru c t u re P a rta l R epar In te rm e d a te Level M a jo r Repar D e pot Level M n o r Repar F e ld Level E ngne P a rta l R epar In te rm e d a te Level M a jo r Repar D e pot Level M n o r Repar F e ld Level A rc ra ft A v o n cs P a rta l R epar In te rm e d a te Level M a jo r Repar D e pot Level M n o r Repar F e ld Level E le c trc a l P a rta l R epar In te rm e d a te Level M a jo r Repar D e pot Level M n o rr e p a r F e ld Level E n v ro n m e n t a l P a rta l R epar In te rm e d a te Level M a jo r Repar D e pot Level F g. 1 : D e cson Tree for R epar Process s mnor repar then transton wll be to the feld repar shop, f t s partal then the transton wll be to the ntermedate repar shop and n the last case transton wll be to the depot repar shop. Therefore from the arcraft-n-repar state, transtons take place to one of arcraft-n- repar shop states. Transton may take place wth the subsystem moved from one repar shop to another due to lack of spares/crew/equpment at that shop.

8 A r c r a f t n S e r v c e F e ld Repar S h o p I n t e r m e d a t e R e p a r S ho p D e p o t R e p a r S ho p A r c r a f t u n d e r rep ar * M nor R ep ar * P artal R e p a r * M ajor R ep ar F g. 2 : S ta te Transton Dagram for S m u la to n 4. Implementaton The mplementaton s llustrated wth the flow chart n Fg. 3. The smulaton s mplemented wth an ntal fgure of three hundred fghter arcraft (whch can be vared) wth the assumpton that all of them are n n-servce condton on the frst day. Though the number of sortes to be generated would be need based, n the present smulaton three sortes per day for each arcraft were consdered. For llustratve purpose the sorte duraton (flyng hours) s taken to be three hours and the smulaton nterval s taken as one day,.e., twenty-four calendar hours (The number of arcraft, number of sortes per day and sorte duraton can be vared n any smulaton experment.) The battle damage rate s vared from frst day. It usually decreases wth number of days and here the rate of decrease s mplemented as user nput. In the smulaton, the arcraft s consdered to be faled (or not avalable) f any one of the subsystems fal. In a sorte, the possblty of falure of only one subsystem s consdered (However, n actual practce several subsystems can fal n one sorte due to common cause falures such as fre n the cockpt/fuselage.) Further, the falure can be ether due to battle damage or due to unrelablty.

9 Arcraft Avalable Arcraft Faled Mnor Repar No Partal Repar No Major Repar No Yes Yes Yes Feld Repar Shop Intermedate Repar Shop Depot Repar Shop Scrap No Spares/Crew/Equpment Avalable Yes Logstc Delay + Repar Repar Arcraft Ready Fg. 3: Flow Chart for Smulaton Usng smulaton run, the possblty of battle damage s evaluated frst. If ths does not occur, the smulaton run s contnued for fndng the possblty of falure due to unrelablty. For consderng the falure due to battle damage we generate a unform random varate u1 and compare t wth the battle damage rate assumed for that partcular day.

10 If u 1 s less than the battle damage rate for that day the falure s consdered to have occurred due to battle damage. Further, to fnd out whch subsystem s battle damaged we generate a unform random varate u 2 and verfy ts locaton n the ntervals 0, β ], ( β, ], ( β, β ], β, ] and β, 1]. If t les n ( β = = 1 = 1 4 ( 3 β = 1 = 1 ( 4 = 1 0, β ], we consder the falure s due to arcraft structures falure, f t les n ( β = 1 ( β, ], that due to engne falure, f t les n ( β, β ], that due to avoncs = 1 = 1 falure, f t les n β, ], that due to falure of electrcal subsystems ( 3 β = 1 = 1 otherwse the falure wll be due to falure of envronmental subsystems. If there s no falure due to battle damage then the possblty of relablty related falures s smulated by generatng a unform random varate u 3. If the relablty computed from equaton (4) wth approprate parameter values for the subsystem selected s less than u 3, we consder the falure s due to the unrelablty of the chosen subsystem. We assume that the preventve mantenance (PM) s carred out at tme ntervals denoted by TBO, the Tme Between Overhaul (T). The tme elapsed from the prevous overhaul or preventve mantenance acton s generated randomly n the nterval (0, T) for that subsystem. Ths s generated only once for the fve subsystems consdered for each arcraft on the frst day before the smulaton and ths s taken as the age (τ ) of the subsystem on the frst day before flyng. For the purpose of calculatng relablty at the end of each sorte, the age (n flyng hours) t of the subsystem s updated by addng sorte duraton toτ. To fnd out whch subsystem s faled due to unrelablty, a smlar exercse s carred out as n the case of battle damage falures wth β s replaced by α s and usng a new unform random varate u 4. When a partcular subsystem fals, t s smulated for the knd of repar requred. The faled subsystem s classfed as requrng mnor repar, partal repar and major repar. Ths s done by generatng a unform random varate u 5,.e., for nstance, when the

11 subsystem faled s engne, f u 5 les n ( 0, γ 21] we consder the repar to be mnor n nature, f u 5 les n γ, γ + ] the repar s partal and f u 5 les n ( γ + 22, 1] ( γ γ the repar wll be major. If the repar s mnor, t wll be performed at the feld repar shop, f t s partal, the repar wll be done at the ntermedate repar shop and the major repar wll be carred out at the depot repar shop. The repar tme s then generated from the exponental dstrbuton consdered for that type of repar. At each level of repar, the need for logstc requrement s smulated usng a unform random varate u 6. If u 6 s less than 0.5 (whch s arbtrary), we assume there s no logstc requrement at that repar shop, otherwse we assume that logstc requrement s needed. When requred, the logstc delay tme s generated from the lognormal dstrbuton consdered for that repar shop and s added to the repar tme to get the actual repar tme. The number of arcraft avalable at the end of the day s decded on the bass that the arcraft has faled or not, and f faled, whether t s repared and made avalable wthn that day (24 calendar hours). [Ths approach can be modfed by takng the smulaton nterval as 12 hours, 8 hours as requred]. Ths s done by comparng the actual repar tme wth the tme avalable on that day snce the tme of falure. If the actual repar tme exceeds the tme avalable on that day, t s consdered not avalable at the end of that day and we check to fnd the day on whch t becomes avalable and consder accordngly. The number of avalable arcraft at the end of each day s obtaned by subtractng the number of arcraft not avalable at the end of that day from the number of arcraft avalable on that day. Then the avalablty s calculated as the rato of the number of arcraft avalable at the end of the day to the ntal number of arcraft deployed for battle at the start of the smulaton. 5. Smulaton Results 5.1 Parameter Values Snce ths subject s hghly senstve and not much data are avalable n the open lterature or through other hstorcal sources to the present authors, the choce of

12 parameters for nput varatons s based on expert opnon, relevant reports and nteracton wth user groups [11,12]. The values used here are consdered typcal and ndcatve of the battlefeld condtons, whch could however vary wdely n dfferent conflcts. The purpose here s to llustrate the methodology and not to provde specfc results for certan battlefeld condtons. In Table 1, the set of TBO used n the smulaton, the correspondng α and β values are gven. The α s for unrelablty related falures were based on nternal reports [11]. The β s for battle damage falures were based on the battle damage occurred for the fve subsystems n the Vetnam War [12]. The γ values for all the subsystems at each repar level s based on expert opnon and s provded n Table 2. In Table 3, the Webull parameters for the fve subsystems are provded whch are consdered typcal for age related falures of these subsystems. The characterstc tmes for the Webull dstrbutons were consdered at two levels: mnmum and maxmum. The shape parameters were kept constant snce t depends manly on falure mode. For structures the shape parameter s takes as 3.4 n whch case the Webull dstrbuton s close to the normal dstrbuton [13]. For engne and envronmental subsystems t s taken as 2.0; (n ths case the Webull dstrbuton corresponds to the Ralegh dstrbuton). The shape parameter for avoncs s taken as 1.1, whch s close to the exponental dstrbuton. As a typcal example, the battle damage rate s taken as the number of arcraft damaged per one thousand sortes as 20 on the frst three days and 15 from fourth day to the end of the second week and 8 per thousand sortes from the second week to the end of the smulaton, 60 days. The repar tme dstrbuton parameter at the three repar levels was consdered n two cases, whch are labeled as optmstc and pessmstc (Table 4). The 50 th and 80 th percentles of the lognormal dstrbuton for logstc delay tmes are provded n Table 5.

13 Subsystem TBO α β ( hours) Structures Engne Avoncs Electrcal Envronmental Table 1: Parameter Values for TBO, α and β fractons for the subsystems Repar Type Subsystem Mnor Partal Major Structure Engne Avoncs Electrcal Envronmental Table 2: Parameter values of γ fractons for dfferent subsystems Subsystems Structures Engne Avoncs Electrcal Envronmental Webull Parameters t 01 n 1 t 02 n 2 t 03 n 3 t 04 n 4 t 05 n 5 Mnmum Maxmum Table 3: Webull parameters for unrelablty related falures Repar Type Mnor Partal Major Repar Tme MTTR mn MTTR par MTTR maj Optmstc Pessmstc Table 4: Exponental dstrbuton parameters for the repar tme Feld level Intermedate level Depot level M 80 M 50 M 80 M 50 M 80 M Table 5: Lognormal dstrbuton parameters for logstc delay tmes at the repar levels

14 5.2 Results The smulaton results are shown as graphs of avalablty vs. battle day for the data used n the smulaton. In the fgures the avalablty of the fghter arcraft wth and wthout logstc delay are shown (Fgures 4 to 7). To comprehend the effect of logstc delay on the avalablty, the avalablty fgures at the end of 1, 7, 14 and 60 th day were analysed (Table 6). Avalablty Days Wth Logstc Delay Wthout Logstc Delay Fg.4. Avalablty for Webull parameters beng mnmum and repar tme parameters beng optmstc Avalablty Days Wth Logstc Delay Wthout Logstc Delay Fg.5. Avalablty for Webull parameters beng maxmum and repar tme parameters beng optmstc

15 Avalablty Days Wth Logstc Delay Wthout Logstc Delay Fg.6.Avalablty for Webull parameters beng mnmum and repar tme parameters beng pessmstc Avalablty Days Wth Logstc Delay Wthout Logstc Delay Fg.7.Avalablty for Webull parameters beng maxmum and repar tme parameters beng pessmstc Repar Tme Parameters Webull Parameters Maxmum Mnmum Logstc Optmstc Pessmstc Delay (LD) Day Day Day Day Day Day Day Day Wthout LD Wth LD Wthout LD Wth LD Table 6. Avalablty of arcraft on dfferent days of the battle

16 It s seen from the avalablty graphs that assumng no logstc delay the avalablty decreases to nearly 40% of the ntal number deployed on the frst day. If logstc delays are ncluded at the repar levels then the avalablty s lkely to reduce to 20%, a dfference of nearly 20%. Further we see that the separaton between the two curves correspondng to wth and wthout logstc delay decreases when the repar tme dstrbuton parameters become pessmstc, n whch case the logstc delay has less effect on avalablty. From the avalablty fgures provded n Table 6, we see clearly the effect of logstc delay. The best avalablty fgures correspond to the case for whch the Webull parameters (characterstc tme) beng maxmum, the repar tme parameters s pessmstc and there s no logstc delay. The worst avalablty fgures correspond to the case of the Webull parameters beng mnmum, the repar tme parameter s pessmstc and there s no logstc delay. Comparng the best and worst possble stuatons, we see that the avalablty decreases from 71% to 41% at the end of frst week, a dfference of 30%. At the end of second week ths dfference comes down to 40%, whch could cause a serous concern for the mltary decson-makers. In the steady state (at the end of 60 days) the avalablty becomes 45% n the best case and 20% n the worst case. 5. Dscussons and Conclusons The smulaton model developed n the present paper enables estmaton of avalablty of arcraft (wth fve subsystems) at three repar levels under battlefeld condtons. The nput condtons can be vared to smulate specfc battle condtons and combat operatons as well as the number of subsystems consdered. The sgnfcance of the model s that logstc delays are ncluded n the repar process. It can be seen that logstc delay reduces the avalablty to a large extent. The smulaton results are senstve to the nput dstrbuton parameters and the model assumptons. Snce the data avalable to us s meagre and s also of senstve nature, we have tred to nput parameters, whch are typcal and gathered from experts n the feld. Therefore the results as such may not apply to all stuatons but are only

17 ndcatve and help to brng out the usefulness of ths smulaton approach. However, the smulaton results defntely ndcate that durng hgh ntensty conflcts the avalablty of weapon systems becomes low wthn the frst few days of the battle tself. Ths may lead to tactcal decsons such as abandonng or postponement of certan combat operatons or the need for renforcement of weapon. Snce specfc data on battlefeld condtons are sparsely avalable for recent conflcts, the results obtaned can only be valdated wth expert opnons. Ths lmtaton s always felt n ths feld of combat related smulatons. The smulaton model can be used to consder the effect of operatonal parameters such as reducng the logstc delay and mprovng the mantanablty whch wll vary the nput parameters. The model developed here can be sutably modfed and appled to other weapon systems such as battle tanks, artllery guns and warshps. References 1. Upadhya, K.S., and Srnvasan, N.K., "A Smulaton model for Avalablty under Battlefeld Condtons", Smulaton, Vol. 74, No. 6, pp , June Kessler, J., Loss Rates and Mantenance Requrements n Wartme, Systems Analyses and Modellng n Defence, Development Trends and Issues, Plenum Press, New York, Emerson D.E. Smulaton Models for Assessng Force Generaton and Logstcs Support n a Combat Envronment, Systems Analyses and Modellng n Defence, Development Trends and Issues, Plenum Press, New York, Blanchard, B. S., Logstc Engneerng & Management, Prentce Hall, New Jersey, Banks, J., Handbook of Smulaton, John Wley, New York, Ebelng, C.E., An Introducton to Relablty and Mantanablty Engneerng, McGraw Hll, New York, 1997

18 7. Carter, A. D.S., Mechancal Relablty, Second edton, Macmllan, London, Broln, A., Qualty and Relablty of Techncal Systems, Theory, Practce, Management, Second edton, Sprnger-Verlag, Berln, Gordon, G., System Smulaton, Second Edton, Prentce Hall, New Jersey, Shannon, R.E., Systems Smulaton the art and Scence, Prentce Hall, New Jersey, Internal Reports, (Centre for Aeronautcal Systems Studes Analyses) [DRDO], Bangalore, Inda. 12. IDA Paper P-2421, Support Costs and Relablty n Weapons Acquston: Approaches for Evaluatng New Systems, Insttute of Defence Analyses, Washngton D.C., O Connor P.D.T. Practcal Relablty Engneerng, Thrd edton, John Wley, 1991