A Spare Part Inventory Management Model for Better Maintenance of Intelligent Transportation Systems

Transcription

1 A Spare Part Inventory Management Model for Better Maintenance of Intelligent Transportation Systems Kaan Ozbay, Ph.D. Professor, Department of Civil and Environmental Engineering, Rtgers, The State University of New Jersey, 623 Bowser Road, Piscataway, NJ USA, Tel: (732) kaan@rci.rtgers.ed Eren Erman Ozgven, M.Sc. Gradate Research Assistant, Department of Civil and Environmental Engineering, Rtgers, The State University of New Jersey, 623 Bowser Road, Piscataway, NJ USA, Tel: (732) ozgvene@rci.rtgers.ed Sami Demirolk, M.Sc. Gradate Research Assistant, Department of Civil and Environmental Engineering, Rtgers, The State University of New Jersey, 623 Bowser Road, Piscataway, NJ USA, Tel: (732) dsami@rci.rtgers.ed Word cont: Figres + 2 Tables = 7493 Abstract: 238 Sbmission Date: Agst 1, 2011 Paper Sbmitted for Presentation and Pblication at the Transportation Research Board s 91 th Annal, Washington, D.C., 2012 Downloaded TRB 2012 from Annal amonline.trb.org Paper revised from original sbmittal.

2 ABSTRACT New Jersey s highway transportation system is highly dependent on the performance of its roadways in order to maximize operational capability and to minimize travel time and congestion. Long-term sstainability of the highways performance is directly related to the efficiency of the Intelligent Transportation Systems (ITS) that need to be maintained and operated properly. One of the most important concerns within the inspection and maintenance procedres of ITS eqipment is timely availability of the spare parts of essential components of ITS. Long-term down time of ITS eqipment de to the navailability of spare parts will not only increase personnel and repair time reqirements, costs of replacement parts bt also might lead to increased delays, poor air qality and fel consmption. In this paper, we propose an efficient spare parts inventory control model that can determine the optimm levels of the safety stocks nder probabilistic failre and availability assmptions for components of varios ITS eqipment. When this inventory control model is flly integrated into Rtgers Intelligent Transportation Systems Inspection and Maintenance Software (RITSIMS) (1), it will allow its sers to efficiently manage DOT s ITS spare parts inventory sing historic maintenance and inspection data that is being collected by respective databases of RITSIMS. This inventory control model will also be able to accont for the worst case scenarios that DOT can experience in terms of nexpectedly high level of eqipment failres de to natral or other disasters sch as hrricanes Downloaded TRB 2012 from Annal amonline.trb.org 2 Paper revised from original sbmittal.

3 INTRODUCTION Highly dense and complex roadway infrastrctre of New Jersey carries heavy traffic of cars and trcks. It is ths important to ensre that deployed ITS eqipment works properly and efficiently. Unnecessary ITS eqipment failres and malfnctions de to inefficiencies of inspection or maintenance can create many problems (1): Increase in maintenance expenditres (time and money), shortening of the sefl life of the ITS eqipment, Increase in reqirements for personnel time and spare part inventory, Waste of time and fel and a possible increase in polltant emissions de to additional congestion that can cased by the longer than expected down-time of ITS eqipment, One of the significant components of any maintenance system is being wellprepared for failres by ensring availability of spare parts. Crrently, many DOTs do not se a system to keep an active inventory of spare parts for their ITS devices. However, it is clearly stated in (2) that a lack of spare parts inventories and difficlties in procring critical spare parts in a timely manner are two key concerns. Patel (2) indicates that if spare parts are not available, repairs may take longer time to complete and may even create safety hazards and can reslt in liability and/or damages. First of all, it shold be noted that some of these spare parts can be only replaced by the contractor, original manfactrer or a qalified distribtor. Anecdotal evidence has sggested that the ability to get spare parts qickly is one of the most common reasons to rely on contract maintenance, reflecting on the governmental procedres that might tend to delay the reestablishment of ITS operations (2). However, many traffic operations nits have the responsibility of maintaining ITS eqipment sch as closed circit television systems (CCTV), variable message signs (VMS), sensors, cabinets, hbs, roadway information and commnications systems, etc. Actally, regardless of the contracting method sed, (by in-hose staff or a contractor) sfficient qantities of key parts shold be available for maintenance. "Some agencies rotinely keep on-hand a percentage of the original qantity as spare parts withot sing any mathematical model, bt based on experience. Sch an approach is considered to reslt in time-saving and redces the downtime of expensive ITS devices" (2). Moreover, over the past decade, there have been several manfactrers that have gone ot-of-bsiness all arond USA, leaving agencies with no immediate sorces for spares (3). To mitigate these problems, a maintenance plan of a DOT shold incorporate scrapping items and ensre that spares and replacements are anticipated. As the spare parts can be expensive and timely spply of spare parts in terms of replacements is critical, it is also often a significant economical problem to hold sfficient stocks. A trade-off between minimizing the cost for spare part inventories and maximizing the efficiency of the spare parts shold be achieved by careflly modeling an inventory management system specifically designed for spare parts. This model shold solve an optimization problem concerning the optimal investment allocation to both spare-part inventories and effective sage and maintenance of those spare parts. In a recent stdy condcted for NJDOT (4, 5), Rtgers Intelligent Transportation Systems (RITS) researchers sccessflly developed a state-of-the-art Intelligent Transportation Systems inspection and maintenance manal (ITSIMM) and Rtgers ITS Downloaded TRB 2012 from Annal amonline.trb.org 3 Paper revised from original sbmittal.

4 inspection and maintenance software (RITSIMS) based on ITSIMM. RITSIMS is crrently being tested by the DOT. Over time, the research team expects to obtain sbstantial amont of data abot the sage and performance of the ITS eqipment so that this information can be sed as an inpt for the spare parts inventory management model presented in this paper. (6). The inventory management model proposed in the following sections is an idea that was conceived dring this implementation phase of RITSIMS. Integration of the type of an inventory management system similar to the one described in this paper with the eqipment history featres of RITSIMS (4) will help traffic operations and maintenance personnel to improve their inspection and maintenance activities. Ths, the primary objective of this paper is first to design a theoretically robst inventory management system that wold minimize stock-ots and at the same time help in maintaining optimm inventory levels according to the criticality of the eqipment. The research team expects that the proposed spare part inventory management model will provide a redction in maintenance expenditres (time and monetary) and inventory costs de to efficient repairs and replacement of falty parts, better recording of the failre rates, monitoring the performance eqipment, and increasing the sefl life of the eqipment throgh better and timely maintenance. First, an overview of the research methodology is given inclding the details abot the proposed inventory management model. This is followed by the case stdies and the details of an application of the inventory management system within RITSIMS. Finally, conclsions and ftre recommendations are provided. THE SPARE PARTS INVENTORY MANAGEMENT PROBLEM In a technical report prepared for NJDOT (2), it is strongly recommended to have a spare parts inventory and management system for ITS components of NJDOT de to the following reasons: "An efficient management of a spare parts inventory integrated with software systems and efficient network management can spport transportation operations centers (TOCs) and statewide ITS operations of NJDOT. ITS, by definition, is made whole by the sm of its parts, and procring parts, repairs, management of warranties, etc. is a necessity that cannot be mixed with other activities or generalized and allowed to deteriorate. ITS technologies are complex, and swapping a new part for a defective one can be a qick and simple, yet a soltion that will keep ITS benefits flowing. TOCs across the contry are reporting percent spare parts on hand." These isses create the need for defining and careflly stdying the spare parts inventory management problem for ITS systems. First of all, it is important to realize that spare parts inventories are not intermediate or final ITS prodcts, therefore the policies that govern spare parts inventories are different from those which govern classical inventories. Uniqe aspects of spare parts inventory management for transportation can be listed as follows (adapted from (7) where they present a general spare parts inventory problem): The initial driver for spare parts demand is the ITS component failre. However, maintenance policies, rather than the sage of component within the ITS Downloaded TRB 2012 from Annal amonline.trb.org 4 Paper revised from original sbmittal.

5 eqipment, may dictate the need for spare parts inventories. For example, ways to restore the fnctionality of an ITS eqipment that has a broken part inclde to repair o replace the part. The decision of whether to repair or! replace has significant implications on spare parts! inventory levels.! Moreover, failre of a spare part may lead to the end-se and therefore replacement of the entire ITS eqipment. A reliability based information is generally not available within the DOT to the degree needed for the prediction of failre times, particlarly in the case of a new installed ITS eqipment. Part failres may create a problem, particlarly if the dependence relation is not known. Demands for parts are sometimes met throgh cannibalism of other parts or nits. Costs of being ot of a part generally inclde qality as well as lost prodction, and these costs are difficlt to qantify. Increased risk to DOT personnel may also be a factor, and costs associated with sch risks are not easy to calclate.! Obsolescence may be a problem as the eqipments for which the spare parts were designed become obsolete and are replaced. It is difficlt to determine how many nits of a part for an obsolescent eqipment to stock, and it may be difficlt to replace a part that no one still keeps in stock. Components of ITS eqipment are more likely to be stocked than complete nits if the major nit of ITS eqipment is expensive, and repair may be preferred to replacement if it is possible. Among all these, cost is apparently a determining factor for managing spare parts inventories for transportation systems optimally. An estimated cost analysis for the preventive maintenance of an ITS device for the Maryland State Highway Athority was provided in a report by Institte of Transportation Engineers (ITE) (8). Based on this analysis, the conclsion reached was that in-hose maintenance was more cost effective than otsorcing the maintenance tasks. Another related stdy provided gidelines for fnding operations and maintenance of ITS systems where spare parts inventory costs were considered as critical (9). For instance, among the operations and maintenance costs (O&C costs) for ITS systems, given by Research and Innovative Technology Administration of U.S. Department of Transportation, dynamic message signs O&C costs range was between $ in terms of 2005 dollars per year, whereas CCTV O&C costs range was between $ in terms of 2004 dollars per year, both having a life time of 10 years (7). Considering that NJDOT Traffic Operations North has more than 500 CCTV's all arond North Jersey (1) and observing the cost of maintaining and operating one CCTV device gives a rogh idea abot the importance of being prepared for failres in terms of having spare part stocks. To sm p, these cost-related stdies bring ot the isse of effectively managing the spare parts inventories, a key isse for the maintenance of ITS eqipment. Another significant isse to consider while modeling the spare parts inventories is that the lifetime of ITS components is shown to vary between 4 and 20 years. Althogh technological changes happen qickly, the procrement cycles for most DOTs do not. In fact, there are hardware components in the field -e.g., electromechanical controllers - that have been operational for many decades (3). The actal rate or time for replacement of Downloaded TRB 2012 from Annal amonline.trb.org 5 Paper revised from original sbmittal.

6 ITS field components depends on the mean time between failres for each component, which normally shold be obtained from the DOT database accmlated over years. It is recommended in (5) that the maximm nmber of spare parts that may be needed for ITS eqipment shold be procred, whether or not they are provided by the agency or via an otsorce contractor. In this paper, we focs on the optimal amont of spare parts needed for efficient ITS operations and maintenance procedres rather than having maximm nmber mentioned in (8), which is generally not known, or decided based on experience. Mathematical Formlation There are a nmber of objectives that are soght by the DOT personnel in order to develop an efficient spare parts inventory management methodology. These objectives mainly differ from the objectives of a commercial inventory management since spare parts inventory management in the context of a transportation system is niqe and significantly different from classical inventory and manfactring framework. In or spare parts inventory management problem, we inclde the following objectives: Minimization of failres and maximization of the performance of ITS eqipment, Maintaining a bffer or safety stock to accont for and respond fast for ITS component failres, Tracking the sage of spare parts and/or store a list of parts reqired for each piece of ITS eqipment. Optimally determining the amont of spare parts and their corresponding costs within the inventory (shortage, srpls, storage, etc.) that can be sed effectively in a DOT maintenance plan and within the allocations of fnding, The constraints for the spare parts inventory management problem, on the other hand, inclde: the storage space constraints, the minimm tolerable disrption level (possibly de to transportation, extreme failre rates, and spplier related disrptions) constraints. Figre 1 illstrates the overall problem with objective fnction and its constraints in the transportation and ITS operations concept. Downloaded TRB 2012 from Annal amonline.trb.org 6 Paper revised from original sbmittal.

7 Space& Constraints& Disrp0on& Constraints& Maintaining&a& bffer&or&safety& stock&to& accont&for&and& respond&fast&for& ITS&component& failres& Minimiza0on&of& failres&and& maximiza0on&of& the&performance& of&its&eqipment& Objec0ves& Tracking&the& sage&of&spare& parts&and/or& store&a&list&of& parts&reqired&for& each&piece&of&its& eqipment& Op0mally& determining&the& amont&of&spare& parts&and&their& corresponding& costs&within&the& inventory&that& can&be&sed& effec0vely&in&a& DOT& maintenance& plan&and&within& alloca0ons&of& fnding& FIGURE 1 Overview of the Spare Parts Inventory Management Problem To solve this spare parts inventory management problem, we propose a stochastic 239 spare parts inventory control (SSIC) model based on Hngarian Inventory Control Model 240 introdced in (10). In or model, demand is defined as the need for replacements of spare 241 parts de to failres, whereas delivery represents delivering the spare parts that are not 242 readily on hand in the spare parts inventory, if it exists. Note that maintenance activities 243 sally follow a schedled scheme, and therefore delivery for the spare part may not be 244 immediately condcted if the failre does not have high priority. 245 We will first mathematically describe a mlti-spare part model in the context of 246 spare parts inventory logistics. The stochastic spare parts inventory control (SSIC) 247 problem, as defined in this stdy, is to find the amont of safety stock in the spare parts 248 inventories with a probability 1 ε, so that independent delivery and demand processes 249 go on withot disrption at minimm cost. For instance, if the vale of ε, the probability 250 of disrption, is 0.1, the aim is that disrption will not occr 90% of the time. 251 In or model, we assme that deliveries, fixed and designated by n, take place 252 according to some random process at discrete times within a finite time interval [0, T ]. 253 These random times have joint probability distribtions the same as that of n random 254 points chosen independently from the interval [, ττ+ T] according to a niform 255 distribtion. A minimal nmber of spare parts, δ, is delivered with each delivery n. If 256 the total amont of delivery is D for some failre rate, U, U being a finite set of 257 different failre rate vales, there is also a random amont of delivery obtained by 258 choosing a random sample of size n 1 from a poplation niformly distribted in the 259 interval [0,1 nδ ]. The demand process for spare parts is defined similarly, with 260 parameters C for some, U, as the total amont of demand, γ as the minimal 261 amont of failres, and s as the nmber of failres creating the need for spare parts. We Downloaded TRB 2012 from Annal amonline.trb.org 7 Paper revised from original sbmittal.

8 assme that delivery and demand processes are independent, and there will be a sperscript l for each spare part. In each time interval, a minimal nmber of spare parts eqal to δ 0 is delivered. Then, a sample size of L ( L n) is taken from the niformly distribted poplation in 266 * * * the interval [0, D nδ ] and given by x1 x2... xn. Then, n 1 positive integers are 267 taken as j1 < j2 <... < jn 1 L. The delivered qantities of the spare part in the n time 268 intervals are assmed to be 269 * * δ + x, δ + x * * x,..., δ + x * x *, δ + ( D nδ) x j j j j j j n 1 n 2 n 1 The model for the demand process is similar. A sample size of N is taken from the poplation niformly distribted in the interval [0, C nγ ], and designated by * * * y1 y2... yn. Then, n 1 positive integers are taken as k1 < k2 <... < kn 1 N. The failed spare parts of a certain type are assmed to be * * * * * * γ + y, γ + y y,..., γ + y y, γ + ( C nγ) y. k k k k k k n 1 n 2 n 1 Assming that the delivery and demand processes are independent, we let * * * * * * X1 = xj, X 1 2 = xj x 2 j,..., X 1 n 1 = xj x, ( ) n 1 j X n 2 n = D nδ x jn 1. * * * * * * Y = y, Y = y y,..., Y = y y, Y = ( C nγ ) y 1 k 2 k k n 1 k k n k n 1 n 2 n 1 Let S denote the initial safety stock of spare parts for some, U. Another assmption is that we always want to have a safety stock for the parts, shown as S + D C. Then, the condition of no disrption for each, U is formlated as follows: S + δ + X γ + Y 1 1 S + 2δ + X + X 2γ + Y + Y S + ( n 1) δ + X + X X ( n 1) γ + Y + Y Y 1 2 n n 1 S + nδ + X1+ X Xn nγ + Y1+ Y Yn 282 The last ineqality S + D C is simply removed since it is aimed to have a model 283 withot disrption. 284 The problem has two stages, hence there are first and second stage decision 285 variables: a sbscript is given to each second stage variable. The first stage decision 286 variables in the model are M, the storage capacity (which is more important if we store 287 a large component sch as power spplies) of each spare part ll, = 1,..., r. The second 288 stage variables are m 0, the additional safety stock for each spare part ll, = 1,..., r, 289 and for each, U. We have an initial safety stock in the interval [0, T ] and to satisfy 290 the needs for spare parts needed for ITS eqipment, we are trying to find the optimm ()* additional safety stocks, m l ()* l 291 vales and the optimm storage capacities M. The M convex cost fnctions of storage capacities M, l = 1,..., r are g ( x), l = 1,..., r. The (1) ( r) second stage problem comes p after the delivery vales D = ( D,..., D ) are observed. The corresponding probabilities for each { D, C, U} are given by p. We Downloaded TRB 2012 from Annal amonline.trb.org 8 Paper revised from original sbmittal.

9 prescribe that no disrption occrs in any of the spare parts demand in the time intervals kt + τ;( k + 1) T + τ, l = 1,..., r, with probability 1 ε. This parameter, ε, represents the ( ) probability of disrption de to the navailability of the reqired item or disrptions in the transportation system, etc. The optimal vales of the second stage variables m 0 are the adjstment vales of the safety stocks. If at time kt + τ, the safety stock levels are m 0, the new stock levels calclated are m + m 0, U, l = 1,..., r. Here, the adjstments to stock levels incr some additional costs. Ths, the adjstment cost fnction of emergency spare part l is denoted by f ( x), l = 1,..., r. We approximate the joint distribtion of the random demand and delivery variables sing an approximate mltivariate normal distribtion with the random () variable, W l for each spare part l = 1,..., r, for i = 1,..., n, and for each spport, U. i () Therefore, W l i simply represent the vales of the probability distribtion of the spare parts in terms of demand mins delivery for any time step: () () () () () () () W l l l 1... l l l 1... l i = iγ + Y + + Yn iδ X Xn, l = 1,..., r, i = 1,..., n for each, U (1) This allows s to calclate the total cost where the highest and lowest vales have the lowest probabilities according to a pre-determined discretized normal distribtion of delivery and demands. The expectations, variances and elements of the covariance matrix for the random variable W i, l = 1,..., r, i = 1,..., n for each, U are calclated following the normal approximation gidelines given in (10). With this information, an overview of the model that shows the inpts and otpts can be seen as follows: Inpts n : Nmber of deliveries l m : Initial safety stock for spare parts D : Total amont of delivery l C : Total amont of consmption l W : Approximate normal l distribtion variable of the random consmption and delivery distribtions l l + () l g, f, q, q : Associated costs M : Total capacity ε : Probability of disrption SSIC Model Otpts l m : Additional amont of spare parts safety stock reqired to satisfy the needs for the vital spplies l M : Storage capacity for each spare part Downloaded TRB 2012 from Annal amonline.trb.org 9 Paper revised from original sbmittal.

10 Constraints It is significant to consider the probabilistic natre of the demand and delivery processes given the high stochasticities of the problem domain dealing with the maintenance and repair of ITS eqipment. Ths, or model shold take these stochasticities into accont in terms of probabilistic constraints. There are two types of constraints in the model, the probabilistic constraints, and the capacity constraints. As the probabilistic constraints ensre the minimal disrption of the sage of spare parts for the ITS eqipment with a given probability, the sm of initial stocks and deliveries has to be greater than or eqal to the demand for any time step. By replacing W i 's with their expectations and variances of the approximate normal distribtion, or probabilistic constraint is defined as r m + m µ i PW ( i m + m ) 1 Φ, Σ () ( Wi ) 1 ε l l= 1 σ (2) i Other constraints in the SSIC model are the capacity constraints. At any time step, the initial safety stock pls the optimal additional stock mst be smaller than the storage capacity for that spare part, and the sm of storage capacities for each spare part ( a, space occpied by each spare part l, l = 1,..., r, mltiplied by the storage capacity for each spare part, M ) mst be smaller than the overall capacity, M. Objective Fnction The objective cost fnction is the sm of individal costs listed below: Cost of Storage, g : It is obvios that there is a cost for storing each spare part l, l = 1,..., r. In case of ITS maintenance and operations, it is important to consider storage costs since the occrrence of an ITS eqipment failre is not known a priori. Cost of Srpls, q + i : This is incrred for each spare part if there is more inventory than demand. It can be modeled as a fixed cost or as a step fnction that allows very low or no cost for a certain srpls level and then a steep increase for higher levels of srpls. Cost of Shortage, q i : This cost is incrred for each spare part if there is not sfficient spare part inventory to satisfy the demand for the ITS eqipment. This is the most important cost component as the shortage of spare part spplies can case loss of time, fel and lead to congestion and accidents de to the inefficiently working or broken ITS eqipment. Cost of Adjstment, f : This cost is incrred by the natre of the twostage model. Sppose we have an initial amont of safety stock, bt to satisfy the probability constraint, we need more. This adjstment can be de to the nexpected factors sch as the increased nmber of demand for spare parts of a specific ITS eqipment de to a sdden failre. Of corse, this has to be penalized. It can be chosen as a linear fnction of the additional stock. The objective fnction incldes the demands for the spare parts mltiplied by their corresponding probabilities. This allows s to calclate the total cost where the highest and lowest demands have the lowest probabilities according to a pre-determined Downloaded TRB 2012 from Annal amonline.trb.org 10 Paper revised from original sbmittal.

11 discretized normal distribtion. Therefore, the model is minimized according to this total cost that is simply sm of the costs for different demand scenarios. The total cost of an eqipment failre de to the navailability of a spare part may be higher than others de to the additional safety stocks reqired, however, the probability associated with this high demand will be smaller than lower levels of demand closer to the mean (normal distribtion assmption). This self-controlling mechanism gives the DOT personnel the chance to determine the safety stocks more accrately. With this mathematical information, the formlation of the SSIC model is as follows: r n () () 1 l l + () l z µ i min g ( M ) + p f ( m ) + ( qi + qi ) 1 dz l= 1 T Φ U i= 1 ( l) ( l) σ m + m i Sbject to r l= 1 m + m µ i Φ, i= 1,..., n 1, Σ( g) 1 ε σ i m + m M, U, l = 1,..., r m 0, U, l = 1,..., r r l= 1 a ( M ) M Soltion Approach: Prékopa-Vizvari-Badics Algorithm (11) Solving the nonlinear SSIC problem with an exact soltion techniqe reqires extensive programming and optimization knowledge. pleps method (12), however, serves as a practical and easily applicable approximate method so that DOT personnel can se the reslts to decide on the inventory levels for the spare parts needed for ITS eqipment. For this prpose, the Prékopa-Vizvari-Badics algorithm will be sed to generate the plep sets for the relaxed disjnctive programming problem. () The algorithm is based on the discrete random variable ξ l i where Z = Z... 1 x xzr is the prodct set containing the spport of ξ, the vector of the discrete random variable () ξ l i. For the sake of illstration, we consider the r-dimensional vector as ( ) The algorithm is as follows: Step 0. Initialize k 0. Step 1. Determine z,..., z sch that 1, j1 r, j r { ( + ) r + ε} ( + + ) z = arg min y F y, z,..., z 1 1, j 2, k 1 r, k { ε r } z = arg min y F z, y, z,..., z 1 2, j 1, j 3, k 1 r, k { ( ) 1 ε r 1+ } z = arg min y F z,..., z, y 1 r, j 1, j r 1, k 1 r M (3),..., T ξ = ξ1 ξ r. Downloaded TRB 2012 from Annal amonline.trb.org 11 Paper revised from original sbmittal.

12 and let E { z1,,..., z, }. j1 r j r Step 2. Let k k+ 1. If j1+ k > k1+ 1, then go to Step 4. Otherwise, go to Step 3. F z1, j k, y, y R + and eliminate r 1 Step 3. Enmerate all the pleps of the fnction ( ) those which dominate at least one element in E ( y dominates z if y z, y z). If H is the set of the remaining pleps, then let E E H. Go to Step 2. Step 4. Stop, E is the set of all pleps of the CDF F() z = P( ξ z). This methodology provides discretized set of points, which gives the lower bond of a specific probability distribtion (13). These are sed to create the deterministic eqivalent of the probabilistic constraints, and they assre that the constraints will satisfy the given reliability level p = 1 ε. However, first of all, before applying the algorithm, continos distribtion fnctions in or model have to be converted into approximate discrete distribtions (14). For that prpose, for each demand, U, we approximate () the random variable W l () by a discrete variable ξ l with possible vales i 401 ϖ1 < ϖ2 <... < ϖl, where the distribtion fnction is: ( l )( ), 1,..., 1 F ϖ ϑ ϑ = L ξ i 402 PW ( i m + m ) Φ( Wi ) F ( l )( ϖ ), l 1,..., r W ϑ = = (4) i 1, ϑ = L 403 where ϖ1 < ϖ2 <... < ϖl are chosen to be eqidistant on some interval 0, B where () 404 F ( l )( B ) = 1 ζ for a prescribed small tolerance ζ. Here, B l is the selected pper W i () bondary of the interval of ϖ l vales for each commodity i l = 1,..., r, and for each demand, U. Using this process, we obtain the following probabilistic constraint in () the form of mltiplication of the cmlative distribtion fnctions of ξ l : r l= 1 F ( ϖ ϑ ) 1 ε, ϑ = 1,..., L 1, l = 1,..., r for each, U ξ ( l ) i That is, we discretize or continos cmlative distribtion fnction associated with W on its entire domain. Dring the analysis, we try to keep a sbstantial amont i 411 of accracy while choosing the possible vales of ξi in the interval 0, B selecting 412 B and N accordingly. With this idea, we try to preserve most of the information 413 related with the original fnction in the discretized one. The important point is that we 414 focs on the pper regions of the distribtion fnction that contribte to the calclation of 415 pleps with respect to the selected disrption probability. j i i CASE STUDIES First, a base case two-spare parts scenario is created where independent distribtions of power spplies and LEDs (light-emitting diode) are considered. The reason to choose these components is that we want to demonstrate the behavior of the proposed model for spare parts with different costs and distinct characteristics. According to the nit cost Downloaded TRB 2012 from Annal amonline.trb.org 12 Paper revised from original sbmittal.

13 database of U.S. Department of Transportation, Research and Innovative Technology Administration (15) and the LED vendors Dialight ( and Ecolx ( typical average nit cost for LEDs is selected as $2-3, and they are parts of signals and dynamic message signs basically. Power spplies, on the other hand, have nit costs that differ for varios ITS eqipments, and their range is from $30 to $350 (See for more details). In or base case scenario, depending on the severity of the failres or conditions casing those failres, there are five different failre rate scenarios given in an ascending order, where type 1 (=1) represents the lowest failre rate (best case), and type 5 (=5) is the highest failre rate (worst case). The idea is that the demand for LEDs starts lower than the demand for power spplies, bt increases more rapidly as the failre rate increases. Cost figres are selected as dollars for power spplies and LEDs. For the base case scenario, the following vales are chosen: ε is 0.1, ζ is 0.01, N is 40 and B is 200 for LEDs and 1000 for power spplies. Nmber of deliveries in the time interval (i.e., a month) is n = 4. (1) (2) Amonts of initial safety stock, m and m are 1 and 3 nits, respectively. (1) Srpls costs are q + (2) = 5/nit, q + = 1/nit, and shortage costs are (1) q (2) = 100 / nit, q = 10 / nit. (1) (2) (1) δ, δ, γ and (2) γ are calclated for each spare part and scenario separately. Cost of adjstment fnction is selected as f( x) = 2x. Costs of storage for each spare part are 5/nit and 1/nit, respectively. Total storage capacity is 100 nits. Spaces occpied by commodities are 1/nit. Expected total delivery and expected total demand vales are taken as 1 1 D = [1,3,5, 7,9] C = [5,10,15, 20, 25]. 2 2 D = [0,5,10,20,30] C = [20,30,40,50,60] Probabilities of the discrete spports of demand and delivery vales are p = [0.08,0.25,0.34,0.25,0.08]. The reslts are given in Table 1. To satisfy the spare part demand 90% of the time, initial safety stock mst be at least more than 30% of the total expected demand for the power spplies. That is, having more than 30% of the spare parts in the inventory before the failres will prevent disrption 90% of the time. For the LEDs, the initial stock mst be more than 40% of the total expected demand. This indicates the importance of correctly determining the initial safety stocks in the inventory as having this percentage of stocks will prevent disrption 90% of the time. When the total expected demand is eqal to the initial safety stock, the model finds the optimal additional safety stock vale as zero. Moreover, it may not be possible to make 4 deliveries in a given period of time, ths additional amont of safety stock for spare parts has to increase. Downloaded TRB 2012 from Annal amonline.trb.org 13 Paper revised from original sbmittal.

14 461 TABLE 1 Reslts for the Two Spare Parts Analysis with Low Demand Demand =1 =2 =3 =4 = Initial Stock (Power spply) Optimal Additional Stock (Power spply) Total Initial Stock (Power spply) Total Expected Demand (Power spply) Proportion of Initial Safety Stock to Total Expected Demand (Power spply) % 30.0% 26.7% 30.0% 28.0% Initial Stock (LED) Optimal Additional Stock (LED) Total Initial Stock (LED) Total Expected Demand (LED) Proportion of Initial Safety Stock to Total Expected Demand 40.0% 36.7% 37.5% 36.0% 36.7% (LED) Total Cost $855 It is reported in (3) that with regard to planning for knock-downs, lightning, floods, tornados, snow storms, major power failres, and other nforeseen events, some allowance needs to be made in terms of spare parts. For instance, this report states that one ITS system in Virginia with over 1000 roadside devices sffers from approximately one knockdown per year. Lightning is extremely variable and, despite the best attempts towards protection, electromagnetic plses (EMP) can damage the electronics, even when the devices are not directly hit. In fact, most damage to eqipment is not cased by direct lightening strikes, bt by indced voltages on condctors from nearby strikes. That is, ITS devices are often electronically sensitive devices placed in open areas on top of electrically condcting metal poles. To make things worse, these devices are often connected to both power and commnication systems via long-condcting copper wires. There have been examples of ITS camera installations in Florida where all of the PTZ nits (pan-tilt-zoom cameras) were rendered inoperative by a single storm where the damage came throgh the power spply (3). Flooding too can case major problems to ITS devices. In Bombay, India, the controller bases are for feet high to protect them from the monsoons (3). Obviosly, these risks to ITS components will vary significantly in different geographies and climates across the US. Therefore, or model shold also incorporate these worst-case scenarios. This can be achieved by changing the relevant parameters to be able to make this type of worstcase analysis. Five demand scenarios are analyzed as before. All other parameters being (1) (2) kept same as the base case (other than these vales: m and m are 4 and 10 nits, Downloaded TRB 2012 from Annal amonline.trb.org 14 Paper revised from original sbmittal.

15 respectively, N is selected as 100, and B is 1000 for LED's and 5000 for power spplies), we expect an increase in the total delivery and demand vales for spare parts as: 1 1 D = [3,5, 7,9,11] C = [16, 20, 24, 28,32]. 2 2 D = [5,10,15, 20, 25] C = [35, 45,55, 65, 75] With these vales, we change the mean, variance and the covariance matrices of the distribtion of the spare parts. Especially for LEDs, the distribtion is changed sbstantially to accont for the severity of the lightning affects. As observed from Table 2, to satisfy the needs for the spare parts 90% of the time, initial stock mst be more than 40% of the total demand for power spplies. That is, having more than 40% of the spare parts in the inventory before the failre will prevent disrption 90% of the time. However, for LEDs, the initial stock mst be almost 50% of the total demand. Of corse, 50% safety stock in the inventories may seem to be high for ITS operations, however these are the worst case scenarios as mentioned before. Traffic operations and maintenance personnel of DOT shold be aware of this risk, and plan the spare parts inventory accordingly. For instance, for a high risk roadway or region in terms of ITS eqipment failre, the spare part inventory can be handled via this type of high risk scenario. TABLE 2 Reslts for the Two Spare Parts Analysis with High Demand Demand =1 =2 =3 =4 = Initial Stock (Power spply) Optimal Additional Stock (Power spply) Total Initial Stock (Power spply) Total Expected Demand (Power spply) Proportion of Initial Safety Stock to Total Expected Demand (Power spply) 37.5% 40.0% 37.5% 39.3% 40.6% Initial Stock (LED) Optimal Additional Stock (LED) Total Initial Stock (LED) Total Expected Demand (LED) Proportion of Initial Safety Stock to Total Expected Demand 48.6% 46.7% 49.1% 47.7% 49.3% (LED) Total Cost $2256 As the failre rates of ITS components increases de to nexpected or nforeseen events, the demand for spare parts increases in a short amont of time to ensre Downloaded TRB 2012 from Annal amonline.trb.org 15 Paper revised from original sbmittal.

16 efficiency of ITS operations. Moreover, in the analysis, the demand for LEDs increases more rapidly than is the demand for the power spplies. The model reacts to this scenario by increasing the additional safety stock vales more rapidly for LEDs (Figre 2). The most severe condition of the high failre rate case having the largest demand reqires the largest safety stocks for both spare parts FIGURE 2 Changes in the Additional Safety Stock Vales verss Changes in the Demand Scenarios Software Implementation of the Spare Inventory management System within RITSIMS Preliminary ser interface screenshots proposed for inventory management sing RITSIMS are shown in Figre 3 and Figre 4 where an inventory check is being performed for a Variable Message Sign (VMS) when there are broken LEDs, and another check is being performed for a Closed Circit Television System (CCTV) with a failed power spply. There are two demand scenarios sed in the modle: low (best case) and high (worst case) demands. Each demand has five different failre rates (which will actally be obtained from the spare part failre data via RITSIMS at the end of the implementation stage). With this tool, traffic operations personnel from DOT not only can check the availability of the relevant inventory, bt also corresponding figres will help them to create a safety stock for ITS spare parts for ftre se. Downloaded TRB 2012 from Annal amonline.trb.org 16 Paper revised from original sbmittal.

17 FIGURE 3 Spare Parts Inventory Modle Downloaded TRB 2012 from Annal amonline.trb.org 17 Paper revised from original sbmittal.

18 (a) Downloaded TRB 2012 from Annal amonline.trb.org 18 Paper revised from original sbmittal.

19 (b) FIGURE 4 Spare Part Stocks Modle for LEDs with High Demand (FIGURE 4.a) and for Power Spplies with Low Demand (FIGURE 4.b) for Different Failre Rates CONCLUSIONS The spare parts inventory management for DOT s represents a very complex problem de to the difficlties of ITS eqipment failres and malfnctions cased by the lack of technical knowledge, inadeqate inspection dring installation and improper maintenance practices. To the best of or knowledge, many of the DOTs do not have a spare part inventory management model to assist inspectors, ITS design, traffic operations and maintenance personnel. To overcome the aforementioned problems, this paper proposes a Downloaded TRB 2012 from Annal amonline.trb.org 19 Paper revised from original sbmittal.

20 robst spare parts inventory management model that is being integrated within RITSIMS (1). This integrated tool is expected to assist the DOT personnel to improve the efficiency of their activities related to the maintenance and repair of varios ITS eqipment. Case stdies show that the proposed stochastic inventory management model can handle varios levels of demand ncertainty, inclding worst case scenarios, by adjsting planning of safety stock levels. It is expected that this system will help create costeffective soltions to many ITS related maintenance and repair problems by: monitoring the performance of eqipment, increasing the sefl life of the eqipment throgh improved and timely maintenance actions, keeping inventory costs lower, and effectively schedling maintenance activities. It is evident that a ftre stdy, namely the integration of the failre, inventory and cost data obtained from RITSIMS with the proposed inventory management model will reslt in an improvement for the inspection and maintenance of ITS eqipment. Addition of new fnctionalities related to the warranty, factory testing information, as well as the record of spare part costs are also possible ftre works that can improve the proposed model. Application of a prototype for stressfl conditions when nexpected events occr, and sing different distribtions for spare parts are also of interest to frther develop and validate or model. ACKNOWLEDGEMENTS This stdy was partially fnded by a grant from New Jersey Department of Transportation (NJDOT) and administered by Rtgers Center for Advanced Infrastrctre and Transportation (CAIT). Their spport is both acknowledged and appreciated. The contents of this paper reflect the views of the athors who are responsible for the facts and the accracy of the data presented herein. The contents of the paper do not necessarily reflect the official views or policies of the fnding agencies. REFERENCES 1. Ozbay, K., Ozgven, E. E., and Sertel, T., Manal of Gidelines for Inspection of ITS Eqipment and Facilities, Technical Report, No: FHWA-NJ , September 2008 ( NJ pdf). 2. Patel, R.K., Intelligent Transportation Systems (ITS) Operational Spport Contracts Implementation Plan, Technical Report, No: FHWA-NJ , Janary Gidelines for Transportation Management Systems Maintenance Concept and Plans, Technical Federal Highway Administration Report, No: FHWA-OP , PB Farradyne, Ozbay, K., Ozgven, E. E., Sertel, T., Aboobaker, N., Littleton, B., and Caglar, K., Manal of Gidelines for Inspection and Maintenance of Intelligent Transportation Systems, Transportation Research Record, 2129, pp , December Ozgven, E. E., Ozbay, K., and Sertel, T., Manal of Gidelines for Inspection and Maintenance of Intelligent Transportation Systems, Proceedings of the 15th World Congress on Intelligent Transport Systems, New York, November 16-20, Downloaded TRB 2012 from Annal amonline.trb.org 20 Paper revised from original sbmittal.

21 Ozbay, K., Ozgven, E. E., Sertel, T., and Demirolk, S., Implementation of Manal of Gidelines for Inspection of ITS Eqipment and Facilities, Technical Report, No: FHWA-NJ , Jne Kennedy, W.J., Patterson, J.W., and Fredendall, L.D., An overview of recent literatre on spare parts inventories, International Jornal of Prodction Economics, 76, pp , March Traffic Control System Operations, Installation, Management and Maintenance, Institte of Transportation Engineers (ITE), Daniels, G., and Starr, T., Gidelines for Fnding Operations and Maintenance of Intelligent Transportation Systems/Advanced Traffic Management Systems, Transportation Research Record, 1588, pp , Prékopa, A., On the Hngarian inventory control model, Eropean Jornal of Operational Research, 171, pp , Jne Prékopa, A., Mltivariate vale at risk and other topics, Annals of Operations Research, online first, DOI /s , Prékopa, A., Dal method for a one-stage stochastic programming problem with random RHS obeying a discrete probability distribtion, ZOR-Methods and Models of Operations Research, 34, pp , Prékopa, A., Probabilistic programming, in: A. Rszczynski and A. Shapiro, eds., Handbooks in operations research and management science, 10, Elsevier, New York, USA, pp , Noyan, N., and Prékopa A., A variant of the hngarian inventory control model, International Jornal of Prodction Economics, 103-2, pp , October ITS Unit Costs Database, U.S. Department of Transportation, Research and Innovative Technology Administration, Downloaded TRB 2012 from Annal amonline.trb.org 21 Paper revised from original sbmittal.