OPTIMIZATION MODELING FOR TRADEOFF ANALYSIS OF HIGHWAY INVESTMENT ALTERNATIVES

Transcription

1 IIT Networks and Optimization Seminar OPTIMIZATION MODEING FOR TRADEOFF ANAYSIS OF HIGHWAY INVESTMENT ATERNATIVES Dr. Zongzhi i, Assistant Professor Dept. of Civil, Architectural and Environmental Engineering Illinois Institute of Technology, Chicago, Illinois 6066 Phone: (32) , Fax: (32) Chicago, Illinois February 2, 2009

2 Contents of This Presentation Introduction to Transportation Asset Management Optimization Formulations for Project Selection A Computational Study Work in Progress Concluding Remarks 2

3 Dimensions of a Highway Transportation System Operational Functions System Goals Infrastructure (Facilities) 3

4 Existing Analytical Tools for Investment Decision-Making Pavement Management Systems Bridge Management Systems Maintenance Management Systems Safety Management Systems Congestion Management Systems 4

5 Need for Overall Highway Asset Management Interdependency of System Components Increasing System Demand Budget Pressure Accountability Requirements Technological Advancements 5

6 Highway Asset Management System Components Feedback System Goals and Objectives Asset Inventory Asset Valuation Updated System Performance Measures Performance Modeling ist of Candidate Projects from Needs Assessment Project-evel ife-cycle Benefit-Cost Analysis Project Selection Budget Project Implementation 6

7 Optimization Formulation for Systemwide Highway Project Selection As the 0- Multi-Choice Multidimensional Knapsack Problem Multi-choice corresponds to multiple categories of budgets designated for different highway management programs Multi-dimension refers to a multiyear project implementation period, and The objective is to select a subset from all economically feasible candidate projects to achieve maximized total benefits under various constraints. Basic Model Maximize A T.X Subject to C T.X B where A is the vector of benefits of N projects, A = [a, a 2,, a N ] T, X is the decision vector for all decision variables, X= [x, x 2,, x N ] T, C is the vector of costs of N projects using budget 7

8 st round decisio n Addressing Budget Uncertainty in Project Selection Issues of Budget Uncertainty in Project Selection 2 nd round decisio ns 3 rd round decisio ns Multi-Year Project Implementation Period Estimated budgets for all years Years with accurate budget information Years with accurate budget information Years with accurate budget information Year 0 Year T Using Recourse Decisions to Address Budget Uncertainty Year to t t + to t 2 t (-2) + to t (-) t (-) + to t t + to t (+) Budget possibility s 2 possibilities s (-) possibilities s possibilities s (+) possibilities Stage : Deterministic (Initially estimated budgets) Stage 2: Determinis Stochastic (p 2 = s 2.s 3 s -.s.s + combinations) tic Stage - Deterministic Stochastic (p - = s -.s.s + combinations) : Stage : Deterministic Stochastic (p = s.s + combinations) 8

9 A Stochastic Model with Ω-stage Budget Recourse Decisions Maximize A T.X + E ξ2 [Q 2 (X 2 (p), ξ 2 )] E ξω [Q Ω (X Ω (p), ξ Ω )] () Stage : Subject to C T.X E(B ) ( ) ( ) ( ) T 2 2 A. X p B p = EB (3) 2 Stage 2: E ξ2 [Q 2 (X 2 (p), ξ 2 )]= max { } (4) Subject to C T.X 2 (p) B 2 (p) (5) X + X 2 (p) A T. X ( p) B ( p) = EB ( ) Stage : E ξ [Q (X (p), ξ )]= max { } Subject to C T.X (p) B (p) X + X 2 (p) + + X (p) A T. X Ω Stage Ω: E ξω [Q Ω (X Ω (p), ξ Ω )]= max { } () Subject to C T.X Ω (p) B Ω (p) where A is the vector X of + benefits X 2 (p)+ + of XN (p)+ +X projects, Ω A (p) = [a, a 2,, a N ] T, C is the vector of Ω Ω ( ) ( ) ( ) p B costs of N projects using budget from management program k in year t, C = [c, c 2,, c N ] T, X (p) is the decision vector using budget B (p) at stage, X (p)= [x, x 2,, x N ] T, a i is benefits of project i, i =, 2,, N, c i is costs of project i using budgets from p = EB management program k in year t, x i is the decision variable for project i, ξ is randomness associated with budgets at stage and decision space, Q(X (p), ξ ) is the p recourse function at stage, E ξ2 [Q(X (p), ξ )] is the mathematical E(B expectation of the ) = [ P( B ( p) ) B ( p) ] recourse function at stage, B (p) is the p= pth possibility of budget for management program k in year t at stage, p(b (p)) is the probability of having budget scenario (2) (6) (7) (8) (9) (0) 9

10 Budget for Stage Computation Criterion to determine Budget for Stage K Computation M - For yearly constrained budget scenario: Minimize ΔB (p)= [( ( ) ( )) ] K M M 2 B p - E B k= t= - For cumulative budget scenario: Minimize ΔBp ( ) t= (p)= = [ P ( p) ( p) ] B. B An Example Budget Possibility (0% Chance) k = t = EB Budget Possibility 2 (25% Chance) t No change 25% ower t No change No change k k , Budget Possibility 3 (65% Chance) t No change 25% Higher t Expected Budget k k [( ( ) ( )) ] B p - E B p= where 0

11 Incorporate Deferment-Based Project Implementation Option Enhanced Stochastic Model Incorporate Segment-Based Project Implementation Option Tie-ins of multiple projects within one highway segment or across multiple highway segments for actual implementation Benefits of all constituent projects of a segment-based project group added together The constituent projects may request budgets from different programs in multiple years The size of the decision vector in the stochastic model is reduced. Incorporate Corridor-Based Project Implementation Option As an extension of segment-based project implementation option, the tie-ins of multiple projects within one or more highway segments is extended to a freeway corridor or a major urban arterial corridor Benefits of all constituent projects of a corridor-based grand project group combined The constituent projects may request budgets from different programs in multiple years The size of the decision vector in the stochastic model is further reduced.

12 Theorem of agrange Multipliers Redefined the optimization model for Stage Objective Max z(y ) =A T.Y Subject to C T.Y B, where Y is stage decision vector with 0/ integer elements. For non-negative agrange K MMultipliers, λ, the agrangian K M relaxation K M of the model can be written as Objective z R (λ ) = max { [ ( )]} A T. Y. - C T + λ B. Y A T - ( λ. C T ) k= t= ( ) K M T T ( ) Y A - λ. C Subject to Y with 0/ integer elements.. k= t= { } ( ). Y (. B + λ ) k= t= * Y = k = t= = max ( ( ) ) A K M T - λ C., if k=t= 0, otherwise > 0 The unconstrained solution to z R (λ ) = max is The solution algorithm T for * the a) Y original optimization model needs to focus on Feasible toconstraint sc. Y B determining agrange Multipliers λ to satisfy the following conditions: b) * * whe re Y K M k= t= ( ( )) K M A T - λ. C [ λ.( B - C T. Y )] = 0 tomaintain optimality tooriginal optimizati. =,if k= t= 0,otherwise >0 onmodel 2

13 Proposed Algorithm for Stage Computations Step 0 (Initialization and Normalize) Determine budget B (p) for Stage such that ΔB (p) = minimum { B (), B (2),, B (p )} ' c' i=ci/b (p), B p =, andc = c Select all projects and sore projects by benefits (A i ) in descending order Normalize contract costs and budget for each (k, t): Step (Determine the Most Violated Constraint k, t) Set C = maximum {C } for all k, t K M A _ i (λ.c k= t= ( ) n i= Step θ i2 = (Compute K M C the forall Increase or, otherwis of agrange Multiplier Value λ ) { } k,t (c i. ) mimc k= t= ) i ' C λ = λ + θ i.( ' C ) Step 3 (Increase λ by and Reset X i the Value Zero) et and C = C - c i for all k, t Remove project i and reset decision variable x i = 0 If C for all k, t, go to Step 4. Otherwise, go to Step. Step 4 (Improve Solution) Check whether the projects with zero-variable values can have the value one without violating the constraints C. 3

14 Proposed Algorithm for Stage Operations (Con t) Step 5 (Further Improve Solution with Budget Carryover) A small amount of budget might be left after project selection and it could be carried over to the immediate following year one year at a time to repeat Steps to 4 to further improve the solution. B k (p) B k2 (p) B k,t- (p) B (p) B k, t+ (p)... B km (p) Before After B k, t+ (p) + ΔB (p)... B km (p) One-period budget carryover for remaining budget from year t to year t+: Increase budget B k,t+ (p) by ΔB (p) = B (p) - C and this leaves B (p) = 0 after budget carryover. - Hold solution for the preceding years from to t - Re-optimize for the remaining years from t+ to M - Repeat until reaching the last year M. 4

15 Computational Complexity of the Proposed Algorithm Steps -4: Computational complexity is O(M.N 2 ) Step 5: Budget carryover requires M iterations Ω-Stage recourses needs at most M interactions This gives an overall complexity of O(M 3 N 2 ). Since M<<N, the algorithm remains a complexity of O(N 2 ). 5

16 A Computational Study for Model Application Candidate Project Data - Preparation Eleven-year data on 7,380 candidate projects proposed for Indiana state highway programming during were used to apply the proposed heuristic approach for systemwide project selection Examples of estimated project-level life-cycle benefits: Project No. et Year ength anes (Miles) AADT Work Type Project Cost Project Benefit Items (%) AC VOC Mobility Safety Env. Total Benefits ,200 Bridge widening 2,29, ,703, ,630 Pavement resurfacing 4,620, ,365, ,70 Pavement resurfacing 3,000, ,545, ,770 Added travel lanes 750, ,806, ,90 Pavement resurfacing,573, ,943, ,50 Pavement rehabilitation 5, ,505, ,664 Rigid pavement replace 96, , ,00 Pavement rehabilitation 3, , ,29 Bridge widening 08, , ,994 Pavement resurfacing 2,757, ,702,627 Budget Data The annual average budgets designated for new construction, pavement preservation, bridge preservation, maintenance, safety improvements, roadside improvements, ITS installations, and miscenaneous programs were approximately 700 million dollars with 4 percent increment per year The initial budget estimates were updated three times, providing 4-stage budget recourse decisions. Considerations of Project Implementation Options Segment-based project implementation option: selecting projects by roadway segment Corridor-based project implementation option: selecting projects in corridors I-64, I-65, I- 6

17 Computational Study Results - Comparison of Total Benefits and Matching Rates of Selected Comparison of Total Benefits of Selected Projects Projects Total Benefits (in 990, Billion Dollars) Project Benefits by Highway System Goal Budget Project Implementation Option Agency Cost VOC Mobility Safety Environment Total Deterministic Segment-based Corridor-based Deferment-based Stochastic Deterministic Stochastic Average Segment-based Corridor-based Deferment-based Average Segment-based Corridor-based Deferment-based Comparison of Consistency Matching Rates of Selected Projects Deterministic Stochastic Deterministic budget Stochastic budget Average Budget Comparison Method Average Match with Indiana Number of DOT Authorization Project Implementation Projects Number Percent Option Selected Segment-based 6,06 5, % Corridor-based 5,964 4, % Deferment-based 6,038 5, % Segment-based 6,023 5, % Corridor-based 6,05 5, % Deferment-based 6,024 5, % Average 6,006 5, % 6,02 5, % Segment-based 6,020 5, % Corridor-based 5,990 4, % Deferment-based 6,03 5, % Projects Authorized by Indiana DOT 6,34 Projects Matched for All Project Section Strategies 4, % 7

18 Needed Model Enhancements The proposed stochastic model addressing budget constraints by program category and by year, project tie-ins, and budget uncertainty is discussed. Model enhancements are needed for: Adding chance constraints for expected infrastructure conditions and system operations service levels after project implementation Incorporating constraints for maximum allowable risks in the benefits of interdependent projects that would facilitate tradeoff analysis across different types of assets. This will help answer the following critical questions: - What happens if there is an across the board x percent decrease in both pavement and bridge investment levels? - 8

19 Addressing Risks of Project Benefits in Trade-off Analysis High Probability ow High Risk ow Risk Probability Difference between risk and uncertainty Risk involves objective probabilities and measurable quantities Uncertainty involves subjective probabilities and immeasurable quantities Financial analysts and engineers have long dealt with the problems of managing, mitigating, and minimizing risk. Among the techniques used are mean-variance analysis, Value at Risk (VaR) and Stochastic Dominance Selecting projects for transportation asset management is similar to selecting stocks for a portfolio. Instead of stocks, 9

20 Augmenting the Stochastic Model into Two-Phase Optimization Phase I Optimization: Find Minimum of Risks of Project Benefits Markowitz mean-variance n n model formulation Min Subject to i= n i= x j= i x i x j cov(b, b Β = 00%, i j ) x i 0,and E(b i ) B i where x i is the proportion of our budget in dollar that are invested in project i, b i is the benefits of project i, B i is the threshold benefits of project i, B is budget constraint, and i =, 2,, n. Phase II Optimization: Use Optimal Value of the Objective Function from Phase I as Upper Bound Constraint of Risks of Project Benefits added to the Proposed Stochastic Model. 20

21 Project Benefits, Costs, and Covariance in the Markowitz Model Proportion of Budget to be Used by a Project Project Benefits Costs Proportion of Obtainable Budget b C X = C /B 2 b 2 C 2 X 2 = C 2 /B 3 b 3 C 3 X 3 = C3/B N 3 3 COV(b,b i j) = E(b,b i j) -E(b i)e(b j) = bi,s bj,tp(b i,s,b j,t )-[ bi,s P(b i,s )][ bj,tp(b j,t )] S= T= N i= N b >> i B X i B = N C N X N = C N /B Covariance of Benefits for Each Pair of Projects 3 S= i= 3 T= P j P i b i, b i,m b i,h b j, P(b i, b j ) P(b im,b j ) P(b ih, b j ) P(b j, ) b j,m P(b i, b jm ) P(b im,b jm ) P(b ih, b jm ) P(b j,m ) b j,h P(b i, b jh ) P(b im,b jh ) P(b ih, b jh ) P(b j,h ) P(b i, ) P(b i,m ) P(b i,h ) 2

22 Wolfe s P Formulation for Solving the Markowitz mean-variance Markowitz model Model can be re-written in its general form: Objective Min z(x) = -cx + (/2)x T Qx Subject to Ax b, x 0. where c = coefficient vector of the decision vector x, x = [x, x 2,, x N ] T, Q = positive definite matrix for the coefficients of the quadratic terms, A = vector of expected benefits of N projects, A = [a, a 2,, a N ] T, b = threshold benefits of N projects, b = [b, b 2,, b N ] T. As all variables x, x 2,, x N are nonnegative, the Wolfe s method could be adopted for solving a P formulation derived from the Markowitz mean-variance model as follows: Objective: min w = a + a a k Subject to Qx e +A T y = c T Ax - e = b x 0. where a, a, a 2, k = Non-negative artificial variables, e, e = 22

23 The Wolfe s Modified Simplex Algorithm Step : Modify the constraints so that the right-hand side of each constraint is non-negative. This requires that each constraint with a negative right-hand side be multiplied through by - Step 2: Identify each constraint that is now an = or constraint Step 3: Cover each inequality constraint to the standard form. If constraint i is a constraint, add a slack variable s i. If constraint i is a constraint, add an excessive variable e i Step 4: For each = or constraint identified in Step 2, add an artificial variable a k Step 5: Solve for the P by satisfying the complementary slackness requirements: ye = 0 and ex = 0 If the optimal value w > 0, the P has no feasible solution. The solution x to which w = 0 is the optimal solution to the original Markowitz mean- variance model. 23

24 Concluding Remarks An improved stochastic model, along with an efficient heuristic algorithm, is introduced to address budget uncertainty and project implementation option issues in systemwide highway project selection Computational study reveals that the stochastic model is able to determine the best project implementation option aimed to achieve the highest overall return on investments The stochastic model needs to be further enhanced as two-phase optimization by addressing risks of project benefits to rigorously carry out cross-asset trade-off analysis The Markowitz mean-variance model could be employed to find the upper bound of the 24

25 Bio-Sketch of Zongzhi i Education Chang an University (B.E.) Purdue University - M.S.C.E. and Ph.D. in Transportation (Advisor: Kumares C. Sinha, U.S. NAE Member) - M.S.I.E. in Optimization (Advisor: Thomas. Morin) Professional Experience Two World Bank Financed Highway Projects From 2006, nine major research projects funded by ASCE, FHWA, Illinois DOT, Indiana DOT, U of Wisc MRUTC/CFIRE, Purdue JTRP, and Galvin Congestion Initiative (over $.8 million grants) Research Interests Transportation systems analysis, evaluation, and asset management Statistical and econometric methods for transportation infrastructure performance modeling and safety analysis Optimization, and risk and uncertainty modeling for transportation infrastructure systems and dynamic traffic networks. 25