A Multi-Objective Optimization Model for Transit Fleet Resource Allocation

Transcription

1 Mishra et al A Multi-Objective Optiization Model for Transit Fleet Resource Allocation By Sabyasachee Mishra, Ph.D., P.E. Assistant Professor Departent of Civil Engineering University of Mephis Mephis, TN P: Eail: sishra@ephis.edu Sushant Shara, Ph.D. Research Associate NEXTRANS Center, Regional University Transportation Center Purdue University, West Lafayette P: +--- Eail: shara@purdue.edu To V Mathew, Ph.D. Professor Departent of Civil Engineering, Indian Institute of Technology Bobay, Powai, Mubai - 000, India. P: +-- Eail: tv@civil.iitb.ac.in Snehaay Khasnabis, Ph.D., P.E. Professor Eeritus Departent of Civil and Environental Engineering Wayne State University Detroit, MI 0 Phone: Eail: skhas@wayne.edu Word Count: Nuber of Tables: Nuber of Figures: Total Count: =, + ( x0) =, Date Subitted: Noveber 0 Subitted for Publication and Peer Review and for Copendiu of Papers CD-ROM at the Annual Meeting of the Transportation Research Board (TRB) in January 0

2 Mishra et al Abstract State and local transit agencies require governent support to preserve their aging transit fleet. With passage of tie, transit fleet gets older and requires aintenance cost to keep it operational. To provide services at a desired level, transit agencies require aintaining a iniu fleet size. Two iperative considerations fro transit planning viewpoint are () reaining life of the total fleet, and () cost required to aintain the fleet size. When the forer is a quality easure indicating the health of the fleet, the latter is an econoic easure requiring iniu expenditure levels. Ideally, the agencies would like to axiize the total reaining life of the fleet and iniize the total cost required to aintain the fleet size. In this paper, the authors propose a ulti-objective optiization odel (MO) to siultaneously incorporate both objectives when subjected to budget and a nuber of operational constraints. The MO proble is solved by using classical weight su approach by eploying Branch and Bound Algorith (BBA) that has proven to be better than other solution ethodologies. The MO resulted in pareto optial solutions with the possible trade-off between the two objectives. The odel is applied to a real large scale transit fleet syste in the state of Michigan, U.S. The case study results deonstrate, the proposed odel is copact, efficient, robust and suitable for long range planning with ultiple solutions to choose fro a pareto optial frontier. The correlation between decision variables and objective functions has been investigated in-depth and provides iportant insights. The proposed odel can act as a tool for resource allocation for transit fleet aong agencies for state and local agencies. Keywords: transit fleet, ulti-objective optiization, branch and bound algorith, pareto optial. Introduction A nuber of state Departents of Transportation (DOT) and their local transit agencies are concerned about the escalating costs of new buses and lack of funds to keep up with their replaceent needs. The cost of replacing the aging transit fleet in the US to aintain current perforance levels is estiated to exceed one billion dollars annually (, ). The addition of new buses to the existing fleet of any transit agency is a capital intensive process. In the US, the federal governent provides a bulk of the capital funds needed to replace the aging transit fleet, with the requireent of a iniu atching support (usually 0%) fro non-federal sources. A bus that copletes its service life should ideally be replaced. However, lack of capital funds often prevents state DOTs fro procuring new buses for their constituent agencies. Several rebuilding alternatives to bus replaceent are available to the transit industry that can be classified under two generic categories: bus rehabilitation and bus reanufacturing. A nuber of studies conducted between 0 and 000 explored the econoics of replaceent of buses versus rebuilding of existing buses. Most of these studies found that up to certain liits, it is cost-effective to rebuild an existing bus thereby extending the life by a few years with a fraction of the procureent cost of a new bus. The proble addressed is typical to a state DOT in the US that supports the fleet anageent of its constituent transit agencies. While replacing the aging fleet is the ost desirable option fro a quality point of view, budgetary constraints require transit agencies to use a cobination of new and old buses to provide services for their custoers. Thus the challenge before the agency lies in finding an optiu cobination of new and old buses by partially replacing and partially preserving the existing fleet. Two iperative and conflicting easures in the resource allocation are () fleet quality easure; and () econoic easure. These two easures are quantified in the following section. Fleet Quality Measure A quality easure can be considered as the reaining life of the fleet syste. An agency s objective is to axiize the quality easure. Recent studies attepted to iprove upon the original odel by suggesting both structural and ethodological changes (, ). These studies attepted to axiize the

3 Mishra et al. 0 0 quality of the bus fleet by optiizing different surrogates of Reaining Life (RL). In this context, Total Weighted Average Reaining Life (TWARL) for one agency can be defined as: TWARL = f ij j r ij () j f i ij where, f ij is the nuber of buses for an agency i with reaining life of j years on th planning year; r ij is the reaining life of j years for an agency i on th planning year for a corresponding bus; i is the agency, j is the reaining life, and is the planning year in consideration. Mathew et al. (00) reforulated the existing odel () by axiizing Total Syste Weighted Average Reaining Life (TSWARL) defined as the su of TWARL over the planning period in consideration (), i.e. TWARL, where: TSWARL = f ij j r ij () j f i ij Both TWARL and TSWARL can be looked upon as surrogates of the quality of the fleet and need to be axiized. In the reainder of this paper, we will be using abbreviations of these quality easures. Further, research presented in this paper is also based upon an alternative approach of cost iniization as another econoic easure. Econoic Measure In ters of econoic easure the agency has scarcity in funding to anage the fleet so that the fleet size is aintained with iniu cost. Net Present Cost (NPC) is used in a nuber of studies to easure the expenditure. For the proposed resource allocation proble, NPC is defined as the su of spending in iproveent of transit options when the dollar value of the expenditure is brought to its present value using appropriate interest rate throughout the planning period. If x i,k is the nuber of buses chosen for agency i, for iproveent option k, and for tie period ; c k is the cost of iproveent option k in year ; and is the interest rate/discount factor then NPC can be easured as NPC = x i,k c k ( + ) i k () 0 The agency s objective is to iniize the NPC while aintaining the quality easure to iniu standards. Measure Preference If the agency s objective is to axiize the quality easure, that axiu value ay not be attainable within a given budget constraint. Both of the above entioned easures are conflicting, and an agency would like to invest iniu aount (in ters of net present cost) over planning period to obtain the best fleet quality easure possible. In a true sense both objectives (to axiize fleet quality and to iniize the NPC) should be considered siultaneously. In this paper, we present a ulti-objective optiization (MO) fraework rather iniizing or axiizing each objective separately. MO results in a set of pareto optial solutions as opposed to just one solution, and it allows the agency to investigate a RL can be defined as the difference between the iniu noral service life (MNSL) and the age of the bus. The MNSL of a ediu-sized bus, the subject atter of this study is taken as seven years per guidelines of the U. S. Departent of Transportation.

4 Mishra et al. 0 0 trade-off between the two objectives. Pareto-optial solutions are those in which it is ipossible to ake one objective better off without necessarily aking the other objective(s) worse off (). The reainder of the paper is organized as follows. A literature review is presented in the next section along with the need of the research followed by ethodology. The case study section shows the structure of a real world data. Results and discussion section includes the findings of the research. Lastly, the suary of research and future steps are presented in the conclusion section.. Literature Review Literature review section is organized into three areas: () transit fleet iproveent options, () quantitative easures for fleet anageent, () ulti-objective optiization applications for transit. The review is not intended to be exhaustive, but to highlight soe of the general trends in addressing the allocation proble. Transit Fleet Iproveent Options Transit iproveent options drew significant attention in the 0s, and received renewed research interest in the late 0s. A nuber of studies found replaceent, rehabilitation, and reanufacturing are the preferred options (-). The literature review clearly showed that reanufacturing and rehabilitation of buses, if done properly, can be a cost-effective option. The studies entioned above stressed the iportance of proper preventive aintenance as a priary factor contributing to the success of rehabilitation progras. These studies ephasized that rehabilitation, if properly done, can be a successful strategy, clearly referring to the quality of aintenance and steps taken by the agency to prevent ajor breakdowns in achine coponents or bus body infrastructure. For the purpose of this paper, the following ters are adapted fro the literature. i. Replaceent (REPLACE): Process of retiring an existing vehicle and procuring a copletely new vehicle. Vehicles replaced using federal dollars ust have copleted their MNSL requireents. ii. iii. Rehabilitation (REHAB): Process by which an existing vehicle is rebuilt to the original anufacturer s specification, with priary focus on the vehicle interior and echanical syste. Reanufacturing (REMANF): Process by which the structural integrity of the vehicle is restored to original design standards. This includes reanufacturing the body, the chassis, the drive train, and the vehicle interior and echanical syste. 0 0 Note: The generic ter REBUILD has been used in this paper to ean Rehabilitation and or Reanufacturing. Quantitative Measures for Fleet Manageent Service life axiization is the ost coonly adopted easure of effectiveness (MOE) for the resource allocation in conjunction with budgetary constraints. Soe proinent bus aintenance anageent studies include: bus aintenance progras for cost-effective reliable transit service (0-), a generalized fraework for transit bus aintenance operation (), anpower allocation for transit bus aintenance progra (), fraework for evaluating a transit agency's aintenance progra (), a siulation odel for coparing a bus aintenance syste's perforance under various repair policies (), and perforance indicators for aintenance anageent (). These probles priarily cater to an operator, who is concerned with the day to day aintenance for an efficient fleet operation. Another coonly adopted MOE for transit resource allocation is in ters of onetary units. The perforance easures frequently used in literature are axiization of revenue, return, or profits,

5 Mishra et al. 0 0 benefit to cost ratio, internal rate of return, pay off period, cost effectiveness (-). NPC is a widely understood and used MOE in transportation decision aking. Miniization of NPC has been used as a MOE for evaluation of transit level of service (); for evaluation of rail transit investent priorities (); for finding the optial bus transit service coverage in an urban corridor (); for odeling the tiing of public infrastructure projects (); for a decision support tool for evaluating investents in transit systes fare collection(); for analyzing induced deand with introduction of new transit syste (); for analyzing transportation ipacts on econoic developent (0); for analyzing externalities associated with light rail investent (); for resource allocation aong transit agencies(-); and for project selection proble under uncertainty (). Multi-Objective Optiization in Transit Application Research on ulti-objective optiization application for transit has been liited in the literature. Lianbo et al. () studied the train planner for urban rail transit; they proposed a ulti-objective optiization odel for train foration, train counts as well as operation periods considering factors such as transport capacity, the requireents of traffic organization, corporation benefits, passenger deands, and passenger choice behavior under ulti-train-routing ode. Desai et al. () studied the fleet anageent for electric vehicles by iniizing fuel econoy and pollutants (HC; CO; NOx); Mauttone and Urquhart () analyzed a MO transit network design proble and obtained non-doinated solutions representing different trade-off levels between the conflicting objectives of users and operators. Wu et al. () used MO to obtain optial transit stops under Aerican Disabilities Act (ADA). Aong the solution ethodologies, Genetic Algorith (GA) is extensively used to odel ulti-objective probles. However, in this study we will be using branch and bound analysis (BBA) and not genetic algorith (GA), as Mathew et. al. (00) reports that GA provides inferior solution copared to BBA in solving transit resource allocation probles ().. Motivation While soe studies described in literature ade a contribution towards forulating and solving the transit resource/fleet allocation proble and others applied MO in transit, to the best of authors knowledge the above entioned objectives of transit fleet resource allocation have never been investigated siultaneously. Most of the siilar studies in literature have following liitations: 0 0 In literature only fleet quality easure (TWARL or TSWARL) has been axiized without considering any econoic easure (like Net Present Cost (NPC)) other than available budget. A key gap in literature has been in ters of inability to control NPC. The previous studies have only considered NPC as a byproduct of the process of axiizing TSWARL. In practice, while considering the investent for fleet iproveent, state DOTs are concerned about econoic easures such as NPC for spending in a ulti-year planning period, that ay becoe a critical factor in the final decision aking process. Thus, iniizing NPC rather obtaining it as a byproduct would have ore significant iplication to the transit agency. In literature there is a lack of understanding of relationship between both perforance easures TSWARL and NPC. Further, the possible relationship between different iproveent options such as REHAB, REMANF, and REPLACE and both easures (TSWARL and NPC) has not been studied explicitly. This paper investigates the correlation between each option and perforance easures. Moreover, in this paper we also show that in a ulti-year planning period how TWARL of a single year is related to either and both perforance easures.

6 Mishra et al.. Methodology As entioned earlier, in this study we forulate a ulti-objective optiization proble with the first objective of iniization of NPC of the total investent of the fleet for all the agencies over the entire planning period and the second objective of axiization of TSWARL, subjected to budget, deand, rebuild, and non-negativity constraints. For ease in understanding the coplete forulation there is a siple explanation provided in front of each equation. The forulation notations are given below followed by the forulation and their explanation: Variables Explanation b : budget available for th planning year c k : cost of ipleentation of the iproveent progra k on th year f i,j : Nuber of buses for an agency i with reaining life of j years on th planning year l k : additional year added to the life of the bus due to iproveent progra k, l k {,,,} r ij : nuber of existing buses with reaining life of j years for an agency i on th planning year x ij : nuber of buses which received reaining life of j years for an agency i on th planning year due to the iproveent progra y ik : nuber of buses chosen for the iproveent progra k adopted for an agency i on th planning year δ i,(α,β) : nuber of buses already iproved by, years due to rehabilitation in the th planning year for agency i, (α, β {,}) δ i,(γ) : nuber of buses already iproved by years due to reanufacture in the th planning year for agency i, (γ {}) The interest rate used for NPC A : total nuber of agencies B : total budget available for the project for all planning years i :,,,A, the subscript for a transit agency j :,,,Y, the subscript for reaining life k :,,., P the subscript used for iproveent progra :,,,N, the subscript used planning year N : nuber of years in the planning period P : nuber of iproveent progras REHAB : the first iproveent progra- rehabilitation of bus yielding (=) additional years REHAB : the second iproveent progra- rehabilitation of bus yielding (=) additional years REMANF : the third iproveent progra- rehabilitation of bus yielding (=)additional years REPLACE : the last iproveent progra-replaceent of bus yielding additional years TSWARL : Total Syste Weighted Average Reaining Life, TSWARL = TWARL TWARL : Total Weighted Average Reaining Life=TWARL = i WARL i WARL i : Weighted Average Reaining Life for agency i=warl i = f ij j r ij j f ij Y : iniu service life of buses Z x : The objective function as iniization of NPV for the resource allocation in the planning period ρ : Weight factor 0 ρ

7 Mishra et al. Matheatical Construct Explanation Eq.# Overall Objective: weighted su of noralized net present cost (NPC) and su of noralized total syste weighted average reaining life of Miniize Z = (- ρ) *Z x the fleet (TSWARL). As we are iniizing ρ *Z x ; 0 ρ NPC and axiizing TSWARL, the value of () the objective function TSWARL is to be negative to represent the overall proble as iniization proble. The weight (ρ) considered here is between 0 and. N A P Z x = x i,k c Objective function : net present cost of the () k transit fleet resource allocation (NPC) ( + ) = i= k= N A J j= J = i= j= Z x = Subjected to following constraints (r i,j + x i,j ) j ) (r i,j + x i,j Objective function : su total of the weighted average reaining life of the fleet of all the constituent agencies for the whole planning period (TSWARL) () N A P y ik c k < b, = i= k= N b = B = P y ik = r ij, i,, j k= y iγ = δ i,(αβ) αβ y iγ = δ i,(αβ) αβ α, β {,}, γ{} y iγ > 0 N = + δ i,(γ) i,, j + δ i,(γ) i,, j y ik, ifj = l k, l k {,,,} x ij = { 0, Otherwise Constraint: Total cost of iproving the buses () for different iproving schees, agencies and over a planning period should not exceed budget for the planning period Constraint: Planning period budget is equal to () the su available budget for each year, where budget is a priori. Constraint: The buses that are iproved under () iproveent schee k are the ones that have copleted their iniu noral service life and have reaining life j Constraint: All the buses that have copleted their iniu noral service life (0) Constraint: The buses that have been rehabilitated twice or reanufactured once will be replaced Constraint: Non-negativity constraint. Nuber of buses chosen for iproveent should be greater than 0 Constraint: The life of the buses is iproved by either two, three, or four years for a re-built bus and by seven years for a new bus () () () δ i,(αβ) (α+β) in {y iα, α yiβ } if α + β, > α + β, = { 0, Otherwise Constraint: Auxiliary constraint of Eq. (), represents replaceent option after α + β years (REHAB) ()

8 Mishra et al. y γ iα, > γ, δ i,(γ) = { 0, Otherwise Constraint: Auxiliary constraint of Eq. (), represents replaceent option after γ years (REMANF) () The overall objective function is shown in equation (), is a weighted su of noralized NPC and TSWARL. Since we are iniizing NPC and axiizing TSWARL, the value of the objective function TSWARL is negative to represent the overall proble as iniization proble. The weight (ρ) considered here is between 0 and. The objective functions shown in equations () and () represent NPC and the TSWARL respectively of the transit fleet resource allocation. The decision variable x is defined in equation () with the help of an auxiliary variable y. This definitional constraint in equation () ik ensures that the life of the buses is iproved by either two, three, or four years for a re-built bus and by seven years for a new bus. Other buses in the syste will have no additional years added. The constraint () 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, which is deterined previously. The choice of TSWARL is defined by the user. A lower value of TSWARL suggests low cost iproveent options are chosen, and vice versa. Equation () represents the constraint of a fixed budget for the sevenyear planning horizon with the planner having the budget flexibility across the years. Equation () represents the planning period budget being equal to the su available budget for each year. Equation (0) ensures that all the buses that have copleted their Miniu Noral Service Life (MNSL) requireents will be eligible for iproveent as per Federal Highway Adinistration (FHWA) standards. MNSL can be defined as the nuber of years or iles of service that the vehicle ust provide before it qualifies for federal funds for rehabilitation, reanufacturing and replaceent. Equation () 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 () and (). These three constraints are specific to the case study presented in this paper, and can be revised at the discretion of the user. Thus, equations () and () ensure that a bus that was rebuilt twice (each tie its life is increased by α or β years is replaced. This policy is applicable only after α + β years. Siilarly, a bus that is reanufactured resulting in an increase in life by γ years ust be replaced (equations and ) and is applicable only after γ years. This constraint presented in equations (,, and ) is specific to the case study presented in this paper, and can be revised at the discretion of the user. Equation () is a non-negativity constraint to ensure that the nuber of buses chosen for iproveent is never negative. The forulation involves non-linear functions, nondifferentiable 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.. Solution Approach The solution ethodology is presented in Figure.The first step is to initiate the solver to read the input and look up for the binary variable indices with lower bound to 0 and upper bound to for each binary decision variable. Please note that the objective function consists of both TSWARL and NPC in the proposed transit fleet resource allocation odel. An iportant consideration needs to be given to the overall objective function which cannot be a direct su of both the objectives as it is possible that agnitude of one objective ay be very high copared to other. In classical weighted soe approach this is overcoe by noralizing each objective function. The noralized objective function can be deterined by obtaining expectation of each objective function value divided by the expected value of the objective function. The next step is to construct one epty node and create a tree by setting an initial value of objective function. In the tree we try to solve for a node by setting the binary variable bounds, and fix ij

9 Mishra et al. binary value according to the two vectors in the node. The binary variables 0 or for the four iproveent options REHAB, REHAB, REMANF, and REPLACE are considered in the optiization proble. In the optiization proble all the constraints described in equation () through () are to be considered. The best value of the objective function is to be estiated considering initial weight starting fro 0. Start Initiate Solver Is tree epty No Yes Choose a Node Reset Bounds and Choose a Binary Variable Solve the Multi Objective Proble with Constraints (Iteration=n) Is Solution Feasible? No Yes Is Solution Iproved fro Iteration # n? Yes No Branch No Is Solution Integer? Yes Update Incubent Convergen ce Criteria Satisfied? Yes Record all feasible solutions for the Pareto Front No 0 FIGURE : Solution Algorith for BBA Based Multi-Objective Transit Fleet Resource Allocation Proble In the first optiization results, the solution algorith checks if the current result is feasible and satisfying all constraints with reasonable value of the objective function. Next, it copares the resulted objective value with the current best and check if all the binary variables are or 0. Solution algorith updates the current best objective value, i.e. if newly obtained objective is better than the current, then set current to the new one, otherwise keep the incubent. This process is repeated up to a change of objective function value between iterations achieves a precision of 0 for all weights. Suarize all results and draw the pareto front to visualize the ulti objective optiization results. End

10 Mishra et al Data A Public Transportation Manageent Syste (PTMS) database developed by Michigan Departent of Transportation (MDOT) containing actual fleet data is used for the case study deonstration. The distribution of the Reaining Life (RL) in years of the fleet for a few of the agencies for the base year (00) is shown in Table. Only a fraction of the table is presented for brevity. Table shows the distribution of fleet size by their reaining life (RL) for each agency. For exaple, agency has one bus with zero years of RL, buses with seven years of RL and so on, for a total fleet size of.the last row of the table shows that the total fleet is of 0 buses, of which buses have zero years of RL, and need replaceent. The last colun of the Table gives the weighted average reaining life (WARLi) for each agency, coputed fro the distribution of RL for the agency. For exaple, the WARLi of the first agency is calculated as (0x+x0+ +x)/ =.. The base year total weighted average reaining life of the entire fleet (TWARL) is. years. The following iproveent options are used in the case study: Replaceent (REPLACE) process of retiring an existing vehicle and procuring a copletely new vehicle. Buses proposed to be replaced using federal dollars are expected to be at the end of their MNSLs, as described above. (Life expectancy: seven years) Rehabilitation (REHAB) process by which an existing bus is rebuilt to the original anufacturer s specification. The focus of rehabilitation is on the vehicle interior and echanical systes, including rebuilding engines, transission, brakes, and so on. Two types of rehabilitation: REHAB and REHAB with oderate to higher levels of engine rebuilds are considered in this study (Life expectancy: to years) Reanufacturing (REMANF) process by which the structural integrity of the bus is restored to original design standards. This includes reanufacturing the bus chassis as well as the drivetrain, suspension syste, steering coponents, engine, transission, and differential with new and anufactured coponents and a new bus body. ( Life Expectancy: years) Further, it was assued that a vehicle ay be rehabilitated (REHAB or REHAB) only up to two consecutive ters, and then ust be replaced (REPL) with a new bus. A vehicle with REHAB and REHAB (or vice versa) in two consecutive ters also should be replaced. A vehicle ay be reanufactured (REMANF) only one tie, and then ust be replaced (REPL) with a new bus. A vehicle rehabilitated (REHAB and REHAB) once can be eligible for reanufacturing (REMANF) before it is replaced (REPLACE). TABLE Base year distribution of reaining life (RL), fleet size, and weighted average of reaining life of saple agencies before allocation of resources for the case study Agency Distribution of Reaining Life 0 Total Fleet Size WARLi (years) Total 0.

11 Mishra et al Case Study Proble The budgets available for each year and the unit cost for each iproveent options for each year are shown in Table. A seven year planning period is considered conforing to the MNSL requireent of ediu sized buses. Replacing all the buses with zero years of RL (last row and second colun of Table ) would require $,,00 ( x $,0) of investent which exceeds the first year budget. Siilarly, in the second year, buses which had one year of RL in the base year will qualify for iproveent. TABLE Available Budget and Cost of Iproveent Options Budget Iproveent Options and Costs Year REPLACE REHAB REHAB (X= Years) (X= Years) (X= Years) REMANF (X= Years) 00,,000,0,00,00 0,0 00,0,000,0,00,00 0,0 00,0,000,0,0,00,0 00,00,000,0,0,00,0 00,0,000,0 0,0,00,0 00,00,000,0 0,0,00,0 00,0,000 0,0,00 0,0,00 00,0,000 0,0,00 0,0,00 Total,,000 Replacing all these buses with reaining life year would require $,,0 (x$,0), which also exceeds the second year budget and so on for other years. Moreover, if the replaceent process is continued fro year 00 through 00, when the buses reach their MNSL, it will cost $,, (i.e. *,0+*,0+ +*0,0) to aintain the fleet size of 0 buses throughout the planning period. However the total available budget is only $,,000 (Table ). Therefore, there is a need for a echanis to identify iproveent options for each agency, so that the NPC is iniized with a user defined TSWARL. The case study proble is solved using Preiu Solver Platfor (-).. Results and Discussion The results fro the proposed odel are illustrated in the Table. We can see the values for a set of weights ranging fro 0 to one (the value of ρ) for each objective function. If the weight is 0 the forulation is equal to single objective iniization of NPC as an objective function; whereas, if the weight is it represents a case of axiization of TSWARL as the only objective. The weights in the case study were choosen to represent as any possible points in the coplete the solution space, hence 0 weights were generated between 0 and. Only a total of seven points (including the two extree points) are shown in Table. TABLE Best pareto optial solutions along with their weights REPLACE (X) ( Years) REHAB (X) ( Years) REHAB (X) ( Years) REMANF (X) ( Years) TSWARL NPC ($) Weight (ρ).,, ,0, ,, 0. 0.,, 0. 0.,,0 0..,,00 0..,,.00

12 Mishra et al. 0 Table list the best pareto optial solution along with the nuber of buses allocated for each iproveent option, value of objectives TSWARL and NPC and the corresponding weights. It should be noted due to space constraints we are not listing all the 0 solutions rather randoly picked solutions and extree solutions. Table shows the exteree solutions are the best solutions for each objective. However the least value of NPC objective ($,,) can also be attributed to large surplus, i.e. the budget is not being fully utilized and hence a total surplus of $,0, (see Table ) leads to this value. Although the budget constraint keeps the aount coitted for iproveents below budget, but does not liit the iniu aount to be spent leading to large surplus in this case. The algorith akes use of this feature leading to high surplus but at the sae tie iniu NPC. The Table lists the year-wise allocation of resources for iproveent, the two solutions listed are extree, one for weight (ρ) = 0 and another for weight (ρ) =. Each row shows yearly allocation of buses for each iproveent option and subsequently oney for those iproveents in that year. All of the 0 solutions of the proposed odel contain this inforation. TABLE Coplete solutions of the proposed odel at extree weights (ρ = 0; ρ = ) Year REPLACE (X) ( Years) ρ = 0 REHAB REHAB REMANF Total Aount (X) (X) (X) Nuber ( Years) ( Years) ( Years) of Buses TWARL NPC Coitted Surplus (No. of Buses) (years) ($) ($) ($) ,,0,,0 -,0, ,,,,0 -, ,,,0,,, ,00,0,0,,, ,,00,,0,, ,,,,00,0, ,,0,,00,0, ,,,,00,00 Total.,,,,,0, ρ = ,,00,,00 -,, ,,,,0 -, ,00,,0,,, ,,,,,, ,,0,,,0, ,,0,,00,, ,00,,,00,, ,,,,0 -,0,0 Total.,,,,,0 The Figure (a) is an illustration of the final pareto frontier solutions. We can observe diversity of solutions across the region and a pareto frontier representive quality of solutions. One can observe none of the points in the figure is the best solution representing the iniu value of NPC and the axiu value of TSWARL, these solutions are pareto optial solutions. The Figure (b) illustrates all the

13 Mishra et al. solutions at different weights and their corresponding value of TSWARL and NPC in onetary ters, a nice spread of solutions confirs better solution quality. Figure (c) shows that iniu value of NPC can be achieved at weight 0, which is intuitive as at ρ=0, proble becoes single objective iniization of NPC. The value of NPC at weight 0 is so low, it ay see like outlier. The reason for this is the aount coitted for iproveents is very low and a reasonable aount of surplus oney stays available for this particular weight (as seen in Table, last colun). Siilarly in Figure (d) we can observe highest value of TSWARL that we try to achieve in the proble (ρ=) and nice spread of solutions at different weights. Pareto Frontier (a) Pareto Front and Pareto Optial Solutions (b) Plot of TSWARL, Net Present Cost and weights (c) Objective Net Present Cost Values at different weights (d) Objective TSWARL values at different weights 0 FIGURE Value of each objective function with respect to the weights The result presented in Table leads to interesting questions about possible correlation between objectives (TSWARL and NPC) and nuber of buses iproved under each option and TWARL. Relationship between the objective function and decision variables (buses under each iproveent options) are shown in Figure. It ay be noted that the intent of the figures below is to understand the relationship between TSWARL and NPC with different objective functions and not to forecast or predict

14 Mishra et al. the results. The results shown in Figure are not intended to be used as a substitute for optiization in resource allocation probles rather has been shown to explore correlation. Goodness of fit: SSE:.0e+00 R-square: 0.0 Coefficients: p =. p = 0 (a) Linear Relationship between TSWARL and Fleet for Replaceent (REPLACE) option in the planning period Goodness of fit: SSE:.e+00 R-square: 0. Coefficients: p = -. p = 0 (b) Linear Relationship between TSWARL and Fleet for Rehabilitation (REHAB) option in the planning period Goodness of fit: SSE:.e+00 R-square: 0. Coefficients: p =-. p = 0 Goodness of fit: SSE:.e+0 R-square: Coefficients: p = -.00e+00 p =.e+00 0 (c) Linear Relationship between TSWARL and Total Fleet selected for all iproveents in the planning period (d) Linear Relationship between NPC and Total Fleet selected for all iproveents in the planning period FIGURE Relationship of iproveent options with objective functions during planning period for all generated solutions in a linear polynoial for as f(x) = p*x + p As can be observed fro Figure, only a linear relation between TSWARL and REHAB option has acceptable R-square value. Further, it is an inverse relationship, i.e. the higher the nuber of buses in REHAB, the lower the value of TSWARL. The relationship between TSWARL and REPLACE option is a positive one, but it does not show a strong correlation. Siilarly, there is a weak correlation between TSWARL and Total Fleet for iproveent, NPC, and REPLACE option (Figure (d)). Other coparisons are not listed, as the relationship betweeen decision variables and objective functions is

15 Mishra et al. weak. It can be inferred that TSWARL is inversely ipacted by buses going for rehabilitation for years and that is the reason we see very sall nuber of buses allocated for rehabilitation (REHAB) option in the Table. In fact the key to axiizing TSWARL is to rehabilitate the least nuber of buses in the fifth year. This further leads to question of exploring each year iproveents along with the objective function values. The Figure (a-d) illustrates soe insights in that direction. Goodness of fit: SSE:.e+00 R-square: 0. Coefficients: p =.0 p = 0 a) Linear correlation between TSWARL and TWARL in the year 00 Goodness of fit: SSE:.e+00 R-square: 0. Coefficients: p =. p =0 b) Linear correlation between TSWARL and TWARL in the year 00 Goodness of fit: SSE:.e+00 R-square: 0.0 Coefficients: p =. p = 0 c) Linear correlation between TSWARL and TWARL in the year 00 Goodness of fit: SSE:.0e+0 R-square: 0.0 Coefficients: p =.e+00 p =.e+00 d) Linear correlation between NPC and TWARL in the year 00 0 FIGURE Relationship of TWARL for year 00, 00, 00 with objective function values in a linear polynoial for as f(x) = p*x + p In the Figure (a), it can be observed that there is a strong correlation between TWARL for the year 00 and TSWARL. This iplies that the fleet chosen for the iproveent in the last year of planning period significantly contributes to the high value of TSWARL. However, it should be observed that the siilar relationship does not hold for TWARL values for the years 00, 00, years (Figures b

16 Mishra et al and c) and for other years Further, it is seen that the TWARL values do not have a significant effect on the value of Net Present Cost. Budget Sensitivity To understand the budget sensitivity, ultiple runs with different weights and lower budgets were perfored. It was observed that solver fails to obtain a solution without violating the budget constraints below % of the original budget value $,,000. Therefore, in the Table, we present solutions obtained at lower budget value $,0,0 (% reduction of $,,000) and the weights of ρ = 0, 0. and. It can be observed (Table ) that at a lower budget, the algorith chooses lower cost, and ediu age iproveent options (REHAB and REMANF) to achieve the iniu NPC value. At the original budget and for siilar weight, the algorith prefers extree iproveent options (REHAB and REPLACE) with higher cost and longer life (better quality). Another interesting observation is for ρ = (Maxiization of TSWARL), the total nuber of buses for iproveent between year reains the sae at both the budget levels (Table and ). The difference starts to appear after year 00 where the odel tries to adjust for the budget.. Synthesis of Results The ulti objective optiization approach presented for the transit resource allocation resulted in a pareto optiial solutions deostrating trade off between NPC and TSWARL. The optiization results show that appropriate iproveent options can be choosen to achieve a specific objective function. The relationship between NPC and TSWARL is non-linear in nature because of the incorporation of the interest factors in coputing NPC. When NPC is copared with individual year quality easure (TWARL), it is observed that initially, TWARL reains relatively constant with increase in NPC up to a certain point, beyond which TWARL increases in the later years. In all the solutions, a relationship between replaceent (REPLACE) option and rehabilitation option (REHAB) with TSWARL has been consistently observed. However, the relationship between TSWARL and REHAB is inversely proportional but strongly correlated. This represents the fleet size chosen for this iproveent governs the overall objective of TSWARL. The relationship between TSWARL and TWARL in the year 00 is very strongly correlated copared to any other relationship between decision variables and objectives. Thus suggesting that the last year s total expected weighted reaining life plays an iportant role in axiizing the TSWARL objective. A liitation of the forulation is that iniu NPC can be achieved by investing relatively less in the iproveents and obtaining alarge surplus (reducing expenditure fro budget) as the constraint is to spend less than a particular budget value. This can be overcoe by adding a constraint on iniu spending aount. A sensitivity analysis for a lower budget shows efficacy of the odel to work at percent lower than original budget and obtain results. A coparison between low budget and exact budget cases show variation in fleet selection for each iproveent option under different budget levels. Coputational Effort The average coputational tie to solve this proble for a single weight using the PSP solver platfor (;) is two inutes in a Windows bit operating syste, on i Quad Core Processor and GB RAM. The overall tie taken to obtain the all the 0 solutions is approxiately two hours. 0

17 Mishra et al. TABLE Coplete solutions of the proposed odel at lower budget and weights (ρ=0; ρ=0.; ρ=) Year REPLACE (X) ( Years) ρ = 0 (Budget=$,0,0) REHAB REMANF Total Aount (X) REHAB (X) (X) Nuber ( Years) ( Years) ( Years) of Buses TWARL NPC Coitted Surplus (No. of Buses) (years) ($) ($) ($) Total ,,,,0,,0 ρ = 0. (Budget=$,0,0) E Total 0.,0,,0,0,,0 ρ = (Budget=$,0,0) E Total 0.,,,0,,,

18 Mishra et al Conclusion A novel ulti-objective optiization odel for transit fleet resource allocation is proposed in this paper. Two conflicting objectives, axiization of TSWARL and iniization of NPC are used. TSWARL is a quality easure represents reaining life of the transit fleet that the agency would like to axiize. Further, it is equally iportant to deterine the cost required to achieve a certain TSWARL, this in ters of present value of the cost can be referred to NPC. It being an expenditure easure the transit agency would like to iniize NPC, a preise that conflicts with axiizing TSWARL. In the single objective optiization proble either TSWARL or NPC can be analyzed only one at a tie. Further, while analyzing TSWARL the single objective optiization proble is blind to the NPC, and vice versa; as each is assued as a by-product of other. The proposed ultiobjective optiization proble has the advantage of considering both objectives siultaneously and provides a series of solutions as a trade-off for the decision aker. Branch and bound algorith (BBA) is used to solve the ulti-objective forulation since it results in better optial solutions copared to GA for such probles. The transit fleet data over an eight year period fro the Michigan Departent of Transportation is used as the case study. As per FTA standards, four iproveent options are used to allocate the fleet approaching MNSL. The ulti objective transit fleet resource allocation odel has ultiple diensions of significant contribution to research and practice. First, the proposed odel provides a trade-off between two objectives TSWARL, the quality easure and NPC, the cost easure. An analysis of this trade-off has not been attepted in literature. Second, solutions to both objectives in a ulti-year planning period provide the decision akers with ultiple options. Third, the proposed ethod allows the decision akers to explore the trade-off solutions between the conflicting objectives like TSWARL and NPC to ake an infored decision. The research in transit resource allocation can be further enhanced in several ways. The classical technique of weighted su approach presented in the paper has been extensively applied in ulti-objective optiization research and practice. However, recent advanceent of evolutionary approaches for solving ulti-objective optiization can be considered in future research. The case study deonstrated in the paper is for the ediu duty, ediu sized transit fleet syste in Michigan. The ethodology can be applied to different fleet age types, policy, and budget constraints. Another factor is fleet uncertainty because of bus breakdown, accidents or other events, that can be odeled into the proble to build a robust fleet resource allocation.. References. FTA. (00a). Public Transportation in the U.S.: Perforance and Condition. Federal Transit Adinistration, A Report To Congress Volue, Chapter, Section 0.. FTA. (00b). Public Transportation in the U.S.: Perforance and Condition. Federal Transit Adinistration, A Report To Congress Volue, Chapter, Section 0.. Mishra, S., Mathew, T. V., and Khasnabis, S. (00). Single Stage Integer Prograing Model for Long Ter Transit Fleet Resource Allocation, in Journal of Transportation Engineering, Aerican Society of Civil Engineers (ASCE), vol. (). pp Mathew, T.V., Khasnabis, S. and Mishra, S. 00. Optial resource allocation aong transit agencies for fleet anageent. Transportation Research Part A: Policy and Practice, (), pp... Shara, S., and Mathew, T. V. (0) Multi-objective network design for eission and traveltie trade-off for a sustainable large urban transportation network, Environent and Planning B: Planning and Design, 0.. Blazer, B. B., Savage A. E., and Stark, R. C. (0). Survey and Analysis of Bus Rehabilitation in the Mass Transportation Industry. ATE Manageent and Service Copany, Cincinnati, Ohio.

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