Optimal resource allocation among transit agencies for fleet management

Transcription

1 Optial resource allocation aong transit agencies for fleet anageent To V Mathew a, Snehaay Khasnabis b, Sabyasachee Mishra b a Departent of Civil Engineering, Indian Institute of Technology Bobay, Powai, Mubai , India b Departent of Civil and Environental Engineering, Wayne State University, Detroit, Abstract Michigan-48202, United States Most transit agencies require governent support for the replaceent of their aging fleet. A procedure for equitable resource allocation aong copeting transit agencies for the purpose of transit fleet anageent is presented in this study. The proposed procedure is a 3-diensional odel that includes the choice of a fleet iproveent progra, agencies that ay receive the, and the tiing of investents. Earlier efforts to solve this proble involved the application of one or 2-diensional odels for each year of the planning period. These ay have resulted in suboptial solution as the odels are blind to the ipact of the fleet anageent progra of the subsequent years. Therefore, a new odel to address a long-ter planning horizon is proposed. The odel is forulated as a non-linear optiization proble of axiizing the total weighted average reaining life of the fleet subjected to iproveent progra and budgetary constraints. Two variants of the proble, one with an annual budget constraint and the other with a single budget constraint for the entire planning period, are forulated. Two independent approaches, naely, branch and bound algorith and genetic algorith are used to obtain the solution. An exaple proble is solved and results are discussed in details. Finally, the odel is applied to a large scale real-world proble and a detailed analysis of the results is presented. Keywords: transit fleet anageent, resource allocation, genetic algorith, branch and bound algorith Corresponding author.: Tel: , Fax: E-ail address: vto@civil.iitb.ac.in (To V Mathew), skhas@wayne.edu (Snehaay Khasnabis), hisabya@wayne.edu (Sabyasachee Mishra)

2 1. Introduction Transit planners are faced with the task of allocating funds for fleet aintenance and procureent for various transit agencies in the region. An iportant issue in transit fleet anageent is the decision regarding the replaceent and rehabilitation of a fleet. Procureent of new vehicles to retire old vehicles is a capital-intensive process. Transit agencies with liited resources depend on governental support for fleet anageent purposes. Historically, up to 80 percent of the capital cost of procuring transit buses in the United States has been borne by the federal governent, with the reainder shared by the state and local governents (FTA 1992). These funds are to be used to eet the dual purpose of replacing aged vehicles with new ones and rebuilding the older fleet. The proble addressed in this paper is typical to ost state Departents of Transportation (DOT) in the U.S. that support the fleet anageent of transit agencies in the state. The federal support to the fleet anageent progra is typically routed through the state. A bus that copletes its service life should ideally be replaced. States that do not have enough funds to procure new buses for their constituent agencies, have several rebuilding alternatives available to the. It is to be noted that the rebuild option is not a peranent solution; it only postpones the replaceent of a bus. Although any rebuild options exist, the two generic categories are bus rehabilitation and bus reanufacturing (Khasnabis and Naseer 2000). Bus rebuild progras usually have soe policy constraints associated with the, such as liiting the rehabilitation of a bus to no ore than two ties, liiting the reanufacturing to one tie and so forth. Based upon an analysis of repair and aintenance data of transit fleet in the state of Michigan, U.S., it was found that up to certain liits, it is cost-effective to rebuild an existing bus (Khasnabis and Naseer 2000). With a fraction of the procureent cost for a new bus, it ay be possible to extend the life of an existing bus by a few ore years. This strategy will require the state DOT to allocate funds partially for the purchase of new buses and partially for the rebuilding of existing buses. In short, the rebuild progra will help the state DOT in eeting federal standards, and aintaining the fleet strength with liited budget. Several issues need to be addressed on such rebuilding progras for a state DOT. The first issue is to decide whether or not a bus that has satisfied its service requireents should be replaced or rebuilt. The rebuilding itself can have ultiple schees that ay vary in cost and benefit. The second issue is to allocate resources to the constituent agencies in an equitable anner. Finally, 2

3 the allocation of funds over several years ust also be properly prograed. Norally, funds are allocated on an annual basis according to an approved budget. However, the decision ade in a given year will have an ipact on the health of the fleet in the subsequent years. Therefore, a decision that considers the future ipact of bus replaceent actions is preferred. The authors contend that, this is an asset anageent proble that should be properly interfaced with the state s long ter strategic plans. A procedure is not currently available that addresses all these three diensions, i.e. choice of rebuild/replace option, proportion of funds for each agency and proportion of funds over a planning period. This paper is aied at the developent of a odel for state DOT s for optial resource allocations for transit fleet anageent. 2. Review A review of literature in the area of transit asset anageent is attepted here. Although this ter can be very broad in its scope, here we focus on the strategic resource allocation for fleet replaceent and rebuilding progras. Exaples of resource allocation probles characterized by a fixed budget and copeting requireents are widely reported in literature spanning diverse areas such as operations research, anufacturing, finance, transport infrastructure projects (Ross 2000, Sheu 2006, Melachrinoudis and Kozanidis 2002, Ahed 1983). Transit planning is also rich in resources allocation and asset anageent probles: resource allocation to various agencies (Forkenbrock and Dueker 1979, Forkenbrock 1981), deterination of location, size and nuber of transit centers (Uyeno, and Willoughby 1995), allocating fleets in transportation networks (Diana, et al. 2006), and the dilea of purchasing new buses or retiring old buses (Sis et al. 1984, Khasnabis et al. 2002). Fleet anageent solutions for transit operators can be broadly classified into two groups. The first is fleet aintenance fro the perspective of the operator who is concerned with the day to day aintenance for an efficient fleet operation (Etschaier and Anagnostopoulos 1984, Etschaier 1985, Pake et. al 1985, Pake et.al 1986, Dutta 1989, Maze and Cook 1987). The second is a closely related proble, addressing the needs of a state transit planner in the replaceent and/or rebuilding of buses (Blazer et. al 1980, Rueda 1983, Davenport 2005). Each of these probles is characterized by a very specific forulation, stated objectives and constraints, as opposed to a standard forulation and solution ethodology. These probles deonstrate the benefit derived fro a proper atheatical odeling approach. They usually involve either the axiization or iniization of an objective function coprising a set of decision variables, subjected to various constraints expressed in the for of equations or inequalities. 3

4 Depending on the nature of the proble, these ethods can be forulated as linear prograing or non-linear prograing, integer, ixed integer and dynaic prograing odels. (Ahed 1983, Ariaratna and MacLeod 2002, Srour et al. 2006, Uyeno and Willoughby 1995, Melachrinoudis and Kozanidis 2002, Kozanidis and Melachrinoudis 2004). Traditionally, such allocation probles are solved by various fors of gradient search ethods (Deb 2001). These ethods assue the search space to be unifor and uniodal, to ensure a unique solution. However, seldo would one encounter such convex probles in real world. There are several ways to address such non-convex probles; notable aong the are Genetic Algorith (GA), a general purpose robust solution algorith (Mathew and Mohan, 2003, Karlaftis et al. 2007, Deb 2001), and Branch and Bound Algorith (BBA) to deal with integer variables and constraints (PSP 2007, Haggag 1981, Pillai 1998, Horn 2004). In this paper, a coprehensive atheatical prograing odel for fund allocation for transit fleet anageent is forulated. We propose two solution ethodologies: Genetic Algorith (GA) and Branch and Bound Algorith (BBA). We then present the results of an exaple proble followed by a full scale case study. First, the background of the proble is highlighted. 3. Background The resource allocation proble addressed in this paper is otivated by the structure of the US federal support to transit agencies. Current federal policies are designed to ensure that buses purchased with federal funds are properly aintained and reain in productive operation for a iniu noral service life, (MNSL). Buses that have achieved their MNSL are qualified to receive federal funds for replaceent. The federal funding is copliented in any cases by state DOT s through atching funds to local transit agencies for replacing transit buses. When the state DOT s do not have enough funds to provide necessary atching support to procure new buses for their constituent agencies, they ay allocate capital funds partly for the purchase of new buses, and partly for the rebuilding of existing buses. This strategy ay help the DOT in eeting the requireents of all the agencies, as set forth in the MNSL criteria. Bus rebuilding practices have been studied extensively focusing on the needs and experience of the transit industry (Balzer et al. 1980, Felicetti 1985, Khasnabis and Naseer 2000). Aong several iproveent progras, the ost intensive level is reanufacturing the engine and transission, installing new bulkheads and rebuilding structural coponents. The service life of 4

5 a bus can be extended by about four to ten years depending on the size of the bus and intensity of the progra. The second level is rebuilding coponents and systes to original specification for added life of about three to seven years depending on the size of the bus and intensity of the progra, however, at a lower cost. The third level includes repairs as required, with life about two to five years at a uch lower cost. The above studies clearly showed that the reanufacturing/rehabilitating of buses, if done properly, can be a cost-effective option (Khasnabis et al 2003). For the purpose of this paper, the three ters, naely replaceent, rehabilitation and reanufacturing are adapted fro the literature (Khasnabis et al. 2004). Replaceent (denoted as REPL) is the process of retiring an existing vehicle and procuring a copletely new vehicle. Buses replaced using federal funds ust have copleted their MNSL requireents. The ter Reaining Life (RL) is used to designate the difference between the MNSL and the age of the vehicle. By rehabilitation, (denoted as REHAB) an existing bus is rebuilt to the original anufacturer s specification, with priary focus on the vehicle s interior and echanical syste. Reanufacturing (denoted as REMANF) is the process by which the structural integrity of the bus is restored to original design standards. In this paper, four options were considered for iproving the life of the ageing fleet. REHAB1 and REHAB2 increase the expected life of fleet by two and three years respectively. REMANF and REPL increase the life of fleet by four and seven years respectively. There are two policy constraints adopted; (1) A vehicle can be rehabilitated only two ties (REHAB1 or REHAB2) before it ust be replaced (REPL) with a new bus, and (2) A vehicle can be reanufactured only one tie before it ust be replaced (REPL) with a new bus. (Khasnabis et al 2003). The policy directives represent a consensus opinion of transit operators in Michigan (Khasnabis and Naseer 2000). In the reainder of this paper, the generic ter rebuild has been used to ean rehabilitation and/or reanufacturing. Decisions on replaceent versus rebuilding of a large fleet, coprising various sizes and operating characteristics clearly require a coprehensive asset anageent strategy. The salient features of the earlier attepts to odel and solve this proble are also addressed. 4. Model The odel presented in this paper is an outgrowth of a sequence of earlier odeling efforts initiated by Khasnabis et al. (Khasnabis et al. 2004). A brief synopsis of this developent is 5

6 presented below: 4.1. Two-stage LP odel: Khasnabis et al. (2004) developed a two-stage, Linear Prograing (LP) based optiization odel (hereinafter referred to as the Two-stage LP odel) for the purpose of resource allocation. In the first stage, the budget available is allocated for the dual purpose of purchasing new buses and rebuilding existing buses. The odel is forulated as an optiization proble that axiizes the weighted life of the fleet being replaced, subjected to budget, deand, and the non-negativity constraints. In the second stage, this budget is distributed to the constituent agencies in such a anner that the weighted reaining life of the entire fleet for all constituent agencies is axiized. This two stage approach is based upon linear optiization, and the output fro the first stage serves as an input to the second stage. While each of the two stages is directed toward local optiization, the solution ay not necessarily reflect the global optiu. This perception otivated the developent of a single-stage odel described below Single-stage GA/BBA Model As a part of continued research, the two-stage LP odel was re-forulated as a single-stage optiization odel that can be used to allocate resources aong the constituent agencies directly for the replaceent and/or rebuilding of existing buses for a given year subjected to the usual budget, deand, and non-negativity constraint (Khasnabis and Mathew 2006 and Khasnabis et al. 2007). Note that the single-stage odel, unlike its predecessor, copletely bypasses the interediate step of allocating resources aong new buses and rebuilt buses. Rather, it is a direct and siultaneous allocation of resources aong the different fleet life iproveent progra and aong the constituent agencies. The single-stage odel is forulated so as to axiize the total weighted average reaining life (TWARL) for all the agencies. Two solution approaches using Genetic Algorith (GA) (Khasnabis and Mathew 2006) and Branch and Bound algorith (BBA) were proposed (Khasnabis et al. 2007). Both the approaches, as one would expect, yielded better solutions than the two-stage LP odel. The single-stage GA/BBA odels deonstrated the fraework of a anageent strategy over a one-year, short-ter planning horizon (Khasnabis and Mathew 2006; Khasnabis et al. 2007). The two-stage LP odel (2004) solved a ulti-year fund allocation proble by successive application of the two-stage LP odels over each year in the planning period. In this approach, 6

7 certain policy constraints (for instance, the restriction of a bus being reanufactured no ore than once) were treated anually fro one year to the other. In addition, it was felt that optiization being blind to future scenarios, ay have resulted in an inefficient allocation of resources. This realization led to the developent of the proposed odel, tered as the Single-stage 3-diensional odel, that incorporates the choice of different fleet life iproveent progras, allocation of funds to different agencies, and distribution of funds over the planning period. Thus, this paper is an effort to further expand the fraework by integrating the third diension i.e. tie frae into the single-stage odel. Two variants of the Single-stage 3-diensional odel are proposed in the paper: an annual budget odel (ABM) and a planning budget odel (PBM). In ABM, the total budget for the planning period as well as budgets for individual years is considered fixed. In PBM, a single budget is considered for the entire planning period. PBM is based on the assuption that the agency has the flexibility of borrowing onies fro future years allocation (and reducing the future years allocation accordingly). 5. Methodology A odified transit fund allocation odel is proposed that has three levels of decision aking: first, the choice of fleet iproveent progra; second, the allocation of funds aong the constituent transit agencies; and third, the allocation of funds over the years in the planning period. In addition, the odel siultaneously addresses all fleet rebuilding constraints (unlike the two-stage LP odel, where these policy options were treated anually). The odel is forulated as a atheatical progra that axiizes the total syste weighted average reaining life of the fleet, subject to budget, deand, rebuild, and non-negativity constraints. The notations used in the study along with an explanation of the terinology used in describing various odels are shown in Table Assuptions The following assuptions were ade while forulating the odel: (i) Buses that have copleted their MNSL requireents will be iproved either by REPL or by REBUILD. Further, REBUILD includes three options: REHAB1, REHAB2, and REMANF; (ii) Iproveent options REHAB1, REHAB2, REMANF, and REPL will result in additional reaining life of 2, 3, 7

8 4, and 7 years respectively; (iii) Buses rehabilitated two ties (REHAB1, REHAB2 or cobination) ust be replaced with new buses; (iv) Buses reanufactured once (REMANF), ust be replaced with new buses; (v) Any reaining surplus in a given year can be carried over to the next year; and, (vi) Deficits, although not desirable, are allowed, when policy constraints ake viable solutions infeasible; Note: assuptions (iii) and (iv) are referred to as policy constraints in the reainder of the paper Forulation The odel is forulated as an optiization proble where the objective is to axiize the total syste weighted average reaining life (TSWARL) of the fleet for all the agencies over the entire planning period, subject to budget, deand, rebuild, and non-negativity constraints. This forulation is given below: Maxiize : Subjected to: Z x Y ( ) N A rij xij j j1, (1) Y 1 i1 ( rij xij ) j1 A P yik ck b,, (2a) i1 k1 N A P N ik k 1 i1 k1 1 y c b B, (2b) P yik ri 0 i,, (3) k1 y,,,( ),( ) 2,3 4 i i i,, (4) where, y 0, (5) x ik ij yik if j l k,lk 2,3,4,7 0 otherwise i,( ), (6) ( ) in yi, yi if,, (7) 0 otherwise 8

9 i,( ) yi 0 otherwise. (8) The objective function (1) represents the su total of the weighted average reaining life of the fleet of all the constituent agencies for the whole planning period, designated as TSWARL or Z(x). The decision variable x ij is defined in equation (6) with the help of an auxiliary variable y ik. This definitional constraint ensures that the life of the buses is iproved by either two, three, or four years for a re-built bus and by 7 year for a new bus. Other buses in the syste will have no additional years added. The equations (2a) and (2b) represents the budget constraints for the annual budget odel (ABM) and the planning budget odel (PBM) respectively. The forer (2a) represents a fixed budget (b ) for each year that cannot be exceeded; and the later (2b) represents a fixed budget (B) for the seven-year planning horizon and the planner has the budget flexibility across the years. The equation (3) ensures that all the buses that have copleted their MNSL requireents will be iproved by either re-building or replacing. Equation (4) represents policy constraints which ensure that the buses that have been rehabilitated twice or reanufactured once will be replaced. The two ters in this constraint are defined in equations (7) and (8). These three constraints are specific to the case study presented in this paper (as discussed in section 3), and can be revised at the discretion of the user. Thus, the equations (4) and (7) ensure that a bus that was rebuilt twice (each tie its life is increased by or years) is replaced. Obviously, this policy is applicable only after + years. Siilarly, a bus that is reanufactured resulting in an increase in life by years ust be replaced (equations 4 and 8) and is applicable only after years. This constraint represented by equations (4, 7, and 8) is specific to the case study presented in this paper (as discussed in section 3), and can be revised at the discretion of the user. Equation (5) is a non-negativity constraint, which ensures that the nuber of buses chosen for iproveent is never negative. The forulation involves non-linear functions, non-differentiable functions, step functions, and integer variables. Although the step function can be generalized to linear fors, the forulation will require additional variables which ay result in variable explosion rendering the odel unsuitable for large/real world probles. 6. Solution Approach In order to solve the Single-stage 3-diensional odels efficiently, particularly for large 9

10 probles, two solution approaches are proposed, a Genetic Algorith (GA) and a Branch and Bound algorith (BBA). The non-linear non-continuous and non-differentiable for of the proble akes the traditional ethods (e.g. LP) ineffective (PSP, 2007). Both GA and BBA have been successfully used in such probles by several researchers. Further, each ethod has been successfully used over a single year horizon (Khasnabis and Mathew 2006, Khasnabis et al. 2007). The fundaental principles of these two ethods are described below Genetic Algorith Genetic algorith is a robust optiization process suitable for large non-linear, non-convex, discontinuous, or non-structured proble. It is based on the principles of natural genetics in which coplex inforation is stored and transferred by basic building blocks called genes. A siple ipleentation of the genetic algorith first converts the real variable into binary codes using upper and lower bounds, and the required precision. An instance of the solution is randoly generated for each variable and concatenated to for an individual. Siilarly, a population of such individuals is generated and evaluated. The evaluation is done by finding the objective function value, while three genetic operators (reproduction, cross-over, and utation) are applied to derive a better solution in the next generation. This process is repeated until convergence. GA is applicable to non-convex and non-linear probles because of its superiority over other search techniques which are liited by the continuity, differentiability, and uniodality of the evaluated functions. GA treats these liitations by (i) operating with codes of paraeter set and not with the paraeter theselves; (ii) searching for a population of points and not a single point; (iii) using the objective function inforation and not the derivative of the function; (iv) using probabilistic transition rules as opposed to deterinistic ones. All these features ake GA an attractive choice in solving the transit allocation proble. A canonical for of genetic algorith with binary coding, two point crossover, fixed population and copletion of a pre-specified nuber of iterations as the convergence criteria is used in this study. A stochastic sapling without replaceent is used for the reproduction operator. The GA/operators adopted include reproduction, crossover, utation, and elite operator. 10

11 Coding The first step in the GA solution procedure is coding the decision variables. This enables GA s to perfor genetic operators on the variables without violating any feasibility constraints. In the current proble, the decision variables are the nuber of buses chosen for each of the iproveent progras for a given agency i and year, naely x 2, x 3, x 4, and x 7. The variables x 2 and x 3 refer to the nuber of buses that will go for rehabilitation of first or second type (REHAB1 or REHAB2), thus enhancing their lives by two or three years respectively. The variable x 4 denotes the nuber of buses that will be reanufactured (REMANF), adding a life of four years to the fleet. The final decision variable x 7 denotes the nuber of buses that will be replaced (REPL), adding a life of seven years to the fleet. The binary coding schee adopted is shown in Figure 1. The first sub string represents the iproveent options for the first agency. Then the variables are coded for every agency and concatenated to for a sub-string representing the iproveent options for all agencies in the first year. Siilarly, sub-strings are coded for each year and concatenated to a string (chroosoe) to represent an instance of the solution Treatent of Constraints The proposed GA axiizes the weighted reaining life of the fleet subjected to the budgetary and iproveent option constraints. Since the traditional GA operators are blind to constraints that are typical of optiization probles, special purpose constraint treatent ethods are usually required to ensure the feasibility of the solution (Deb 2001). The budget constraints and iproveent option constraints are treated separately. The budget constraints specified by equation (2a) and (2b) are solved by the penalty ethod that converts the constrained proble to an unconstrained one by including a penalty value to the objective function (Ukkusuri et al. 2007). If the budget is violated by any candidate solution, its evaluation value is penalized. An exterior penalty function ethod is adopted which transfors the constrained objective function Z(x) into an unconstrained function (x). The unconstrained objective function for the PBM odel is given as: Maxiize ( x ) Z( x ) ax0, y c B and for the ABM odel the is given as: Maxiize ( x ) Z( x ) ax 0, y c b ik k (9) i k ik k (10) i k where ρ is a penalty ter. The constraints given by equation (3-5) are addressed by the coding schee (section 6.1.1). These steps ensure that GA perfors the search within the feasible 11

12 region. GA, now axiizes the unconstrained functions (9) or (10). Although several sophisticated ethods to address constraints are available, there is no precise way of fixing the penalty ter, penalty function ethod is very siple and has given good results in wide rage of probles (Deb 2001). Based on the fitness value, GA coputes the next trial solution using genetic operators (reproduction, crossover, and utation). Application of these genetic operators is expected to yield better off-springs and the process is repeated till convergence Stopping criteria There are several strategies for stopping the evolution process of the GA. Because it is difficult to define the optial solution, usually two procedures are adopted as convergence criterion: (1) the GA procedure is stopped when the variation in the fitness level aong generations is within a user defined range; and (2) the iteration is stopped when the nuber of generations has accuulated to a predeterined level. In this research, the GA was stopped when it reached a predefined nuber of generations (Ukkusuri et al. 2007). The solver used for the GA algorith is a coputer progra written in C language and ipleented on Linux platfor by one of the authors. For details refer Mathew and Mohan (2003) Branch and Bound Algorith Branch and Bound Algorith (BBA) is an enueration approach for solving various probles especially in discrete and cobinatorial situations. This approach has been applied to a variety of operations research probles, including integer prograing (IP). The BBA approach is applied in three steps. The first step involves coding of the decision variable cells. The second step involves odel initialization, where the convexity and the size of the proble in ters of nuber of variables, integers, bounds and surface nature are deterined. A diagnosis of the odel is perfored to check the nature of the desired odel (linear, quadratic, conic, non-linear, etc.). Finally, the third step involves the developent of constraint coded cells. Budget constraints (Equation 2(a) and 2(b)), andatory replaceent constraints (Equation (3)), and REBUILD constraints are coded. The principle used for branch and bound algorith is explained below 12

13 Let y ik is the nuber of buses to be added to a fleet when it reaches a zero reaining life for k type of iproveent for agency i, on th year. If y ik is not an integer, we can always find an integer [ y ik ] such that: [y ] y [y ] 1. (12) ik ik ik Equation (12) results in the forulation of two sub probles, with an additional upper bound constraint y [y ], (13) ik ik and another with lower bound constraint y [y ] 1. (14) ik ik If the decision variables with integer constraints already have integer solutions, no further action is required. If one or ore integer variables have non-integer solutions, the Branch & Bound ethod chooses one such variable and creates two new sub-probles where the value of that variable is ore tightly constrained. These sub probles are solved and the process is repeated, until a solution is found where all of the integer variables have integer values (to within a sall tolerance). The optiization proble used in the case study is large in ters of the nuber of variables and is solved using Preiu Solver Platfor (PSP, 2006 and PSP 2007). 7. Nuerical exaple In order to illustrate the feasibility of the proposed odel and solution approaches, a nuerical exaple of fund allocation aong three transit agencies is investigated over a planning duration of eight years (including the base year). The input data to the proble showing the base year distribution of the reaining life for three agencies before allocation of resources is shown in Table 2. Agency 1 has five buses with zero years of reaining life (RL), 10 buses with one year of RL and so on, for a total fleet size of 47. Siilar inforation for the other two agencies is presented in Table 2. The total fleet size of all the agencies is 111; The last colun of the Table 2 gives the weighted average reaining life (WARLi) for each agency, coputed fro the distribution of RL 13

14 for the agency. For exaple, the WARLi of the first agency is calculated as (0x5+1x10+ +7x8)/47 =3.74. The total weighted reaining life of all the agencies (TWARL) is The budgets available for each year and the unit cost for each iproveent options are shown in Table 3. For the base year, 18 buses have zero years of RL (Table 2). Replacing all the buses would require $1,487,720, (18x $81,540) which exceeds the first year budget. Siilarly, in the second year, 20 buses which had one year of RL in the base year will be qualified for iproveent. Replacing all these buses would require $1,630,800 (20x$81,540), which also exceeds the second year budget. In the third year all the 9 buses which had 3 years of RL in the base year, plus the nuber of buses in the base year that ay have been chosen for REHAB1, will be qualified for replaceent. Siilarly, buses chosen for REHAB1, REHAB2, and REMANF, in previous years, will qualify for REPL in the subsequent years, thus increasing the coplexity of the allocation proble. The proble now is to identify the iproveent options for each agency for every year, so that the total reaining life of the entire fleet is axiu subjected to the budget constraints Solution by Single-stage 2-diensional Model: First, the proble is solved by the Single-stage odel for the base year of 2002, and the results serve as the input for the year 2003, and the process is to be continued until the year Fro the year 2006 onwards, the policy constraints are anually incorporated into the solution. The solution using BBA is given in Table 4. The base year weighted average reaining life of the fleet is increased fro 9.99 in the base year (Table 2) to in 2002 due to the iproveent. Siilar TWARL values for each year are presented in Table 4. The TSWARL for all agencies for the planning period is It can also be observed that budget constraints were et in all years and a net surplus of $51,111 resulted at the end of the 8 th year (2009). This solution is intended as a benchark for coparing the perforance of the proposed odels Solution by Single-stage 3-diensional Models Suarized versions of the solutions of the exaple proble by four variants of the 3-diensional odel (ABM solved using GA and BBA, PBM solved using GA and BBA) are 14

15 presented in Table 5 along with the corresponding result of the two stage odel (Table 4). The TSWARL values are higher than the Single-stage 2-diensional for both the GA and BBA approaches. Although BBA approach has resulted in large surplus, GA gave better objective function value. The exaple proble is also solved using the 3-diensional PBM forulation (which assues there is only a single budget constraint for the planning duration) using both GA and BBA. Here, both approaches resulted in a uch better TSWARL than the ABM forulation (where there is no budget flexibility). This is possible because of better utilization of the budget. The relative perforances of GA and BBA are consistent in both ABM and PBM forulations. One can also observe that with the iproveent in the TSWARL, the total nuber of fleet that has gone for iproveent is reduced priarily due to the increased bus replaceent options. This suggests that although rebuilding policies help in eeting yearly budget, its benefit diinishes with a long planning horizon. 8. Case study A coprehensive case study is deonstrated by utilizing actual fleet data derived fro the Public Transportation Manageent Syste (PTMS), developed by the Michigan Departent of Transportation (MDOT). This database is used because it perits a direct coparison of results with the two stage LP odel proposed by Khasnabis et al. (2004). The basic odel paraeters (e.g. budgets, policy constraints, extended life values associated with different iproveent options, etc.) were kept unchanged in the case studies for Two-stage and Single-stage odel to ensure copatibility between the results. The data consist of ediu sized-ediu duty fleet of total size of 720 fro 93 agencies in Michigan that receive capital assistance fro MDOT. The distribution of the Reaining Life (RL) in years of the fleet for a few of the 93 agencies for the base year (2002) is shown in Table 6 along with the WARLi for each agency (last colun). Fro the last row of the table, one can observe that aong the total fleet of 720 buses, 235 buses have zero years of RL, and needing iediate replaceent. The base year total weighted average reaining life of the entire fleet (TWARL) is years. 15

16 8.1. Results of Two-stage LP Model: For the purpose of this deonstration, the sae four possible iproveent progra areas, REHAB1, REHAB2, REMANF, and REPL used in the exaple proble were used in the case study. Hence, the unit cost of iproveent option is adopted fro the Table 3 (except the budget). The above proble was originally solved by the two stage LP odel (Khasnabis et al. 2004) and results are presented in Table 7, along with the available budget Table 7 shows that for the year 2002, 107 REHAB1 buses for 2 years of extended life (to be eligible for REHAB again in 2004), and 128 REMANF buses for 4 years of extended life (that ust be replaced in 2006) represent the optiu solution, with no new buses purchased. Table 7 shows that this cobination results in a TWARL of years in replacing the 235 buses within the allocated budget of $5.789 illion. In the year 2003, the need for 122 vehicles is et by 106 new (REPL) buses (to need replaceent in 2010) and 16 REMANF buses (to be replaced in 2007), at the available budget of $9.13 illion, for a TWARL of years. Siilarly fleets coing through natural aging process are subjected to various iproveent progras for the subsequent years as presented in Table 7. Deficits in the years 2006 and 2008 are caused by a need to replace a very large nuber of buses, because of policy constraints that a bus can only be rehabilitated twice, and be reanufactured only once. The LP odel cannot allocate resources in these two cases, as feasible solutions do not exist. Hence, allocation was done heuristically, with the conscious knowledge that deficit will result (see Khasnabis et.al 2004 for details). Table 7 also presents a coparative suary of budgetary allocation, and the cost incurred along with surplus/deficits by the proposed approach as discussed above. Table 7 shows a net deficit of $ illion (constant dollars) for the seven year planning horizon. As indicated earlier, this solution requires anual intervention and judgent to address the policy constraints. The 3-diensional odel discussed below incorporates these policy constraints into the odel, thereby eliinating the need for a anual intervention Results of Single-stage 3-diensional Model: In this section, we present the results of the Single-stage 3-diensional PBM. The ABM approach is not used, as it is clear (fro Table 7) that there is no feasible solution to the proble when there are annual budget constraints. For instance, the budget available in 2006 is not sufficient for the andatory replaceent. Therefore, we solved the proble using the Single-stage 3-diensional PBM with two budget levels: first a total budget sae as the total aount spent by the two-stage 16

17 LP odel, tered as larger budget (i.e. $65.05 illion) and then, the total aount available for two stage LP odel, tered as saller budget (i.e. $ 52.0 illion). The results are presented below Case 1: Larger Budget Resource allocation by the Single-stage 3-diensional PBM odel (solved using both GA and BBA) is shown in Table 8. As explained in the forulation, the odel attepts to axiize TSWARL with flexibility in the budget to vary within the planning period, keeping the allocated aount within the available budget ($ illion). Note that the annual budget has no significance for the PBM as the total seven year budget of $65,054,053 serves as the only constraint. Solution by GA fro Table 8 shows that the 2002 allocation of 109 REPL buses, 61 REHAB1 buses, 30 REHAB2 buses, and 35 REMANF buses, resulted in a TWARL of years at an expense of $ illion. Siilar data for each succeeding year up to 2009 is presented in Table 8. For the planning period considered, a total of 1144 buses are treated by the REBUILD strategies copared to 1153 buses in the two stage LP odel. TSWARL is increased fro to years (an increase of years), where the aount coitted is less than that of the two stage LP odel (Table 7 versus Table 8). Solution by BBA fro Table 8 shows that in 2002, the need for 235 buses with zero years of RL are et by 150 REPL buses and 85 REHAB1 buses. Siilar allocations for various years are presented in the Table 8. A total of 969 buses are addressed, resulting in a TSWARL value of years. Coparing Tables 7 and 8, we find that the BBA approach results in an increase of TSWARL of years with a arginally saller investent. The BBA odel copared to the GA approach yielded a slightly higher TSWARL value at the expense of arginally higher spending Case 2: Saller Budget Solutions by GA and BBA with the available saller budget of $ illion are presented in Tables 9. The GA odel results in a TSWARL value of years over the seven year period, while exceeding the budget of $52.89 illion by $2.1 illion. Clearly this is not a feasible solution as the budgetary constraint over the seven year planning horizon has been violated. Note 17

18 that the present study uses generic constraint handling using penalty ethod which ay not be sufficient to handle such large probles having especially equality constraints which ay require sophisticated techniques such as odified selection and controlled utation (Deb 2001). Table 9 also shows that the BBA odel attains a TSWARL value of years, within the allowable (seven-year) budget of $52.8 illion. Coparing Table 7 with Table 9, one can conclude that the Single-stage 3-diensional BBA odel clearly yields a better solution than the two-stage LP odel. Even though the TSWARL is iproved arginally (fro 2874 to 2822), the capital savings of over $12 illion ($65.05 illion versus illion) is significant and clearly deonstrates the superiority of the 3-diensional approach. Lastly, when coparing the GA versus BBA approach, the later provides a better solution under the constrained budget scenario by yielding a higher TSWARL value (2822 yrs. versus 2730 yrs.) at a lower budget ($53.8 illion versus illion), as evident fro Tables 9. Agency-wide distribution of fleet and resources for the planning period is presented for a few saple agencies in Table 10 for the Larger Budget BBA approach. Year by year allocation of funds (aong four iproveent progras) aong each of the constituent agencies, along with corresponding costs for a total of $64,853,214 is shown in Table Coparison of GA and BBA A suarized version of the results of the Two-stage LP odel, and the proposed Single- stage 3-diensional GA and BBA odels are presented in Table 11. Clearly Single-stage odels provided a better solution than its Two-stage counterpart, both in ters of the TSWRL values (objective function), and the aounts coitted (budget constraint). When coparing the GA versus BBA approach, under the general fraework of Single-stage 3-diensional odels, the BBA approach appears to provide a better solution under the constrained budget scenario by yielding a higher TSWARL value at a lower budget for both larger and saller budget. However, a careful exaination of the results shows that BBA arrived at this by adopting large REPL option in the initial years requiring large investent in the first few years. On the other hand, the GA results that tend to use all the available options at soewhat of a constant spending rate over the years, ay be the preferred alternative for a planner. Overall, the Single-stage 3-diensional approach provides a uch better planning odel that the earlier ulti-stage forulations 18

19 9. Conclusion An integrated approach tered as the Single-stage 3-diensional odel for equitable allocation of governental resources aong copeting transit agencies is presented in this paper. The three entities, naely, progra, agency and tie frae constitute the three diensions ebedded in the odel. This odel is the result of continuing research on this topic by the authors and represents an iproveent of the earlier two generations of odels: the Two-stage LP odel and the Single-stage 2-diensional odel. The two stage LP odel has two optiization routines for each year, to be applied sequentially. The Single-stage 2-diensional odel integrates the above two; however the odel, like its predecessor, has to be applied for each year. In addition, certain policy constraints had to be incorporated anually in the earlier two applications. The proposed Single-stage 3-diensional odel has three diensions of decision aking, naely the choice of fleet iproveent progra, allocation of funds aong the constituent transit agencies, and annual allocation of funds over the planning period. In addition, all the policy constraints are incorporated in the odel alleviating the need for anual intervention. The odel is forulated as a atheatical progra that axiizes the total weighted average of reaining life of the fleet, subject to budget, deand, rebuild and non-negativity constraints. Two variants of the odel are forulated: an Annual Budget Model (ABM) and a Planning Budget Model (PBM). The budget for each year is fixed in the forer case, while in the later case; the planner has flexibility in spending the oney across years, within the confines of a total ulti-year budget. Two solution approaches are proposed, a Genetic Algorith (GA) and a Branch and Bound algorith (BBA). An exaple proble for three agencies for a planning period of eight years is solved to deonstrate the odel. The Single-stage 3-diensional odel perfored uch better than its predecessors in ters of optiizing the objective function within the confines of the constraints specified. The GA and BBA odels produced very siilar results. Finally, a case study is conducted using the real data fro Michigan DOT of the USA. The solution highlighted the efficiency of the 3-diensional odel for solving transit fund allocation. The proposed odel is copact and gives flexibility for a both short-ter and long-ter planning. This study clearly deonstrates the benefit of the new odel in providing a better solution. Further, this odel is expected to be a rational tool in deciding the governental fund allocation to copeting transit agencies. The odel could be extended to heterogeneous fleet, (sall, ediu and large) even though the case study deals with ediu sized buses, and can 19

20 serve as an asset anageent tool at the state level, encopassing transit fleet of various sizes and operating characteristics. 10. Acknowledgeent The work on the topic was initiated by Khasnabis and his colleagues at WSU in 2002 that resulted in the developent of the Two-stage LP odel. This research was supported by the USDOT through the UTC Progra at the University of Wisconsin Madison. The Single-stage 2-diensional odel was developed by Khasnabis and Mathew during Fall 2004, when Khasnabis, on sabbatical leave fro Wayne State University (WSU), served as a Fulbright Research Scholar at Indian Institute of Technology Bobay (IITB), India. The Single-stage 3-diensional odel reported in this paper represents continued and collaborative research effort between faculty ebers and a graduate student fro two institutions, WSU and IITB. The authors would like to express their sincere appreciation to: (1) USDOT and the University of Wisconsin for the initial funding for the precursor work, (2) WSU for granting sabbatical leave to Khasnabis, (3) the Fulbright Foundation for providing opportunities for research on asset anageent and (4) IITB the host institution for Khasnabis in India, providing a foru for exchange of ideas with its faculty ebers. The assistance of WSU and IITB in providing coputing facilities and the support of MDOT for the PTMS database in the sequence of odeling exercises are thankfully acknowledged. The opinions and viewpoints expressed in the paper are entirely those of the authors, and do not necessarily the policies and progras of the agencies entioned. 20

21 Table 1. Notations Used in the Forulation Variables Explanation x ij r ij l k y ik c k B A Y N P b i j k i,( ) nuber of buses which received reaining life of j years for an agency i on th planning year due to the iproveent progra nuber of existing buses with reaining life of j years for an agency i on th planning year additional year added to the life of the bus due to iproveent progra k, l 2,3,4,7 k nuber of buses chosen for the iproveent progra k adopted for an agency i on th planning year cost of ipleentation of the iproveent progra k on th year total budget available for the project for all planning years total nuber of agencies iniu service life of buses nuber of years in the planning period nuber of iproveent progras budget available for th planning year 1, 2,,A, the subscript for a transit agency 1, 2,,Y, the subscript for reaining life 1, 2,,N, the subscript used planning year 1,2,., P the subscript used for iproveent progra nuber of buses already iproved by, due to rehabilitation in the th planning i,( ) year for agency i, (, {2,3}) nuber of buses already iproved by due to reanufacture in the th planning year REHAB1 REHAB2 for agency i, ( {4}) the first iproveent progra- rehabilitation of bus yielding (=2) additional years the second iproveent progra- rehabilitation of bus yielding (=3) additional years REMANF the third iproveent progra- rehabilitation of bus yielding (=4)additional years REPL the last iproveent progra-replaceent of bus yielding 7 additional years 21

22 WARLi Weighted Average Reaining Life for agency i = Y ( r x ) j ij j1 Y j1 ij ( r x ) ij ij TWARL Total Weighted Average Reaining Life= TWARL WARLi = i Total Syste Weighted Average Reaining Life = Y A ij j1 Y i1 rij j1 ( r x ) j ij ( x ) TSWARL TWARL ij TSWARL PBM ABM GA BBA = Y N A ij j1 Y 1 i1 rij j1 ( r x ) j ij ( x ) Planning budget odel Annual budget odel Genetic algorith Branch and bound algorith ij 22

23 Table 2: The Base Year Fleet Data for the Exaple Proble Year Agency Distribution of Reaining Life (years) Fleet Size WARLi (years) 2002-Before Total * Note: *: TWARL 23

24 Table 3: Annual Budget, Iproveent Options and Unit Costs for the Exaple Proble Iproveent Options and Unit Cost ($) Year Budget REPL REHAB1 REHAB2 REMANF (7 Years) (2 Years) (3 Years) (4 Years) ,200,000 81,540 17,800 24,500 30, ,445,000 81,540 17,800 24,500 30, ,000 88,063 19,220 26,400 32, ,000 88,063 19,220 26,400 32, ,000 95,108 20,740 28,500 35, ,000 95,108 20,740 28,500 35, , ,720 22,400 30,780 38, , ,720 22,400 30,780 38,200 Total 7,386,000 24

25 Table 4: Resource Allocation for Multiple Years for The Exaple Proble: Single-stage 2-diensional BBA Model Assigned Assigned Assigned Assigned Total TWARL* Available Aount Year Fleet by Fleet by Fleet by Fleet by Assigned (years) Budget Coitted Surplus($) REPL REHAB1 REHAB2 REMANF Fleet ($) ($) ,200,000 1,199, ,445,000 1,425,920 19, , ,004 14, , ,693 1, , ,760 5, , , , ,760 4, , ,600 4,400 Total ** 7,386,000 7,334,889 51,111 Note: * : TSWARL=82.43 years. ** : TWARL= Object of optiization each year Note: ** : TWARL= Object of optiization each year ** : TSWARL=82.43 years. 25