Recommandé
Contenu connexe
Tendances
Tendances (20)
Similaire à Linkedin_PowerPoint
Similaire à Linkedin_PowerPoint (20)
Linkedin_PowerPoint
- 2. Why Reliability is important in Shortest Path Problems Reliable Shortest Path Formulation Outer Approximation Solution Algorithm Computational Study Conclusions and Other Applications
- 3. s t 35 45 STD DEV MEAN TRAVEL COST , 5 , 20 ORIGIN DESTINATION Reliable and risk averse routing suggest taking the path with least mean+std cost TC=55 TC=50
- 4. C A s I B D T 5,4 7,0 5,0 5,0 5,4 5,0 5,0 10,0 COST =20 + 32 BELLMAN PRINCIPLE OF OPTIMALITY DOES NOT HOLD ORIGIN DESTINATION MEAN TRAVEL COST STD DEV
- 5. Directed Graph G=(V,E) Origin Node r, Destination Node s aij is the link between node i and node j cov(aij, alk) is the covariance matrix Given Assumptions Link Travel Cost Distribution is Available One path with finite variance
- 6. 𝒁 = 𝐌𝐢𝐧 𝒘 𝒄𝒊𝒋 𝒙𝒊𝒋 𝒓𝒔 + (𝟏 − 𝒘) 𝝈𝒊𝒋 𝟐 𝒙𝒊𝒋 𝒓𝒔 𝟐 + 𝒄𝒐𝒗 𝒂𝒊𝒋, 𝒂𝒍𝒌 𝒙𝒊𝒋 𝒓𝒔 𝒙𝒍𝒌 𝒓𝒔 𝒂 𝒍𝒌 ∈ 𝑬 𝒍,𝒌 ≠ 𝒊,𝒋 𝒂 𝒊𝒋 ∈ 𝑬𝒂 𝒊𝒋 ∈ 𝑬𝒔∈𝑺𝒂 𝒊𝒋 ∈ 𝑬 𝒔∈𝑺 𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒋𝒊 𝒓𝒔 = 𝟏 ∀ 𝒊 = 𝒓 𝟎 ∀ 𝒊 ≠ 𝒓, 𝒊 ≠ 𝒔 −𝟏 ∀ 𝒊 = 𝒔𝒋:𝒂 𝒋𝒊∈ 𝑬𝒋:𝒂 𝒊𝒋 ∈ 𝑬 𝒙𝒊𝒋 𝒓𝒔 ∈ 𝟎, 𝟏 ∀𝒂𝒊𝒋 ∈ 𝑬, 𝒔 ∈ 𝑺 Mean Path Cost Standard Deviation of Path Cost Given a transportation network, determine a path with minimum mean plus standard deviation costs Flow Balance Constraint
- 7. 𝒁 = 𝐌𝐢𝐧 𝒘 𝒄𝒊𝒋 𝒙𝒊𝒋 𝒓𝒔 + (𝟏 − 𝒘) 𝒕 𝒓𝒔 𝒔∈𝑺𝒂 𝒊𝒋 ∈ 𝑬 𝒔∈𝑺 𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒋𝒊 𝒓𝒔 = 𝟏 ∀ 𝒊 = 𝒓 𝟎 ∀ 𝒊 ≠ 𝒓, 𝒊 ≠ 𝒔 −𝟏 ∀ 𝒊 = 𝒔𝒋:𝒂 𝒋𝒊∈ 𝑬𝒋:𝒂 𝒊𝒋 ∈ 𝑬 𝝈𝒊𝒋 𝟐 𝒙𝒊𝒋 𝒓𝒔 𝟐 + 𝒄𝒐𝒗 𝒂𝒊𝒋, 𝒂𝒍𝒌 𝒙𝒊𝒋 𝒓𝒔 𝒙𝒍𝒌 𝒓𝒔 𝒂 𝒍𝒌 ∈ 𝑬 𝒍,𝒌 ≠ 𝒊,𝒋 𝒂 𝒊𝒋 ∈ 𝑬𝒂 𝒊𝒋 ∈ 𝑬 ≤ 𝒕 𝒓𝒔 𝒙𝒊𝒋 𝒓𝒔 ∈ 𝟎, 𝟏 , 𝒕 𝒓𝒔 ≥ 𝟎 ∀𝒂𝒊𝒋 ∈ 𝑬, 𝒔 ∈ 𝑺 Mixed Integer Conic Quadratic Formulation Cone Constraint
- 8. 𝝈𝒊𝒋 𝟐 𝒙𝒊𝒋 𝒓𝒔 𝟐 + 𝒄𝒐𝒗 𝒂𝒊𝒋, 𝒂𝒍𝒌 𝒙𝒊𝒋 𝒓𝒔 𝒙𝒍𝒌 𝒓𝒔 𝒂 𝒍𝒌 ∈ 𝑬 𝒍,𝒌 ≠ 𝒊,𝒋 𝒂 𝒊𝒋 ∈ 𝑬𝒂 𝒊𝒋 ∈ 𝑬 ≤ 𝒕 𝒓𝒔 Conic Formulation inefficient for Large Networks Consider a network with 1000 links 1 million quadratic integer terms
- 9. CQP computationally tractable due to their special Structure Can be solved by polynomial time interior point algorithms Solvers like CPLEX and MOSEK offering this capability X2 + Y2 <= Z2 . Minimize Subject to ii T iiii dxcbxa i T i xf
- 10. Outer Approximation (OA)-Linearization and Decomposition Method Delivers optimal solution for convex MINLP (Bonami et al. 2008) ZyRx yxg yxf yx , 0),(toSubject Minimize ),( , f and g are convex function Decompose integer and nonlinear part Solve alternating sequence of Master Problem (MILP) & Subproblem (NLP)
- 11. Rx yxg y y j j yx j xf 0),() toSubject Minimize ),( NLP( , ZRx yy xx yxfyxf yy xx yxgyxg y j j j j jjjj jjjj yx , ),(),( 0),(),(toSubject Minimize , MILP Nonlinear Sub- Problem (SP) MILP Master Problem (MP) OA Steps
- 12. ),( yx SP Solution Linear Approximation OA Cuts are stronger cuts compare to other cuts such as Benders
- 13. Lemma 1. Function is convex𝒁 𝒓𝒔 𝒙, 𝒕 = 𝒙𝒊𝒋 𝒓𝒔 𝒄𝒐𝒗(𝒂𝒊𝒋, 𝒂𝒍𝒌) 𝒍,𝒌 ∈ 𝑬 𝒙𝒍𝒌 𝒓𝒔 𝒊,𝒋 ∈ 𝑬 − 𝒕 𝒓𝒔 Implication Outer Approximation Strategy guarantees globally optimal solution
- 14. Given at iteration h, OA SP solutions are: 𝒕 𝒓𝒔 𝒉 = 𝒙𝒊𝒋 𝒓𝒔 𝒉 𝒄𝒐𝒗(𝒂𝒊𝒋, 𝒂𝒍𝒌) 𝒂𝒍𝒌 ∈ 𝑬 𝒙𝒍𝒌 𝒓𝒔 𝒉 𝒂𝒊𝒋∈ 𝑬 𝒁 𝒉 = 𝒘 𝒄𝒊𝒋 𝒙𝒊𝒋 𝒓𝒔 𝒉 + (𝟏 − 𝒘) 𝒕 𝒓𝒔 𝒉 𝒔∈𝑺𝒂𝒊𝒋 ∈ 𝑬 𝒔∈𝑺 ∀𝒔 ∈ 𝑺 ∀𝒔 ∈ 𝑺 𝒙𝒊𝒋 𝒓𝒔 𝒉 The solution to the subproblem is obtained analytically No need to solve a NLP Implication
- 15. Theorem1. Given and at iteration h, the following cut can linearly approximate the conic constraints: 𝒙𝒊𝒋 𝒓𝒔 𝒉 𝒕 𝒓𝒔 𝒉 𝝈𝒊𝒋 𝟐 𝒙𝒊𝒋 𝒓𝒔 (𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒊𝒋 𝒓𝒔 𝒉 ) 𝒂 𝒊𝒋∈ 𝑬 + 𝒄𝒐𝒗 𝒂𝒊𝒋, 𝒂𝒍𝒌 𝒙𝒊𝒋 𝒓𝒔 𝒉 (𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒊𝒋 𝒓𝒔 𝒉 ) 𝒂 𝒍𝒌 ∈ 𝑬 𝒍,𝒌 ≠ 𝒊,𝒋 𝒂 𝒊𝒋∈ 𝑬 − 𝒕 𝒓𝒔 𝒉 𝒕 𝒓𝒔 − 𝒕 𝒓𝒔 𝒉 ≤ 𝟎 Implication Only those elements of the covariance matrix which are constructing the shortest paths are included
- 16. OA-MP FORMULATION 𝒁 = 𝐌𝐢𝐧 𝒘 𝒄𝒊𝒋 𝒙𝒊𝒋 𝒓𝒔 + (𝟏 − 𝒘) 𝒕 𝒓𝒔 𝒔∈𝑺𝒂 𝒊𝒋 ∈ 𝑬 𝒔∈𝑺 𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒋𝒊 𝒓𝒔 = 𝟏 ∀ 𝒊 = 𝒓 𝟎 ∀ 𝒊 ≠ 𝒓, 𝒊 ≠ 𝒔 −𝟏 ∀ 𝒊 = 𝒔𝒋:𝒂 𝒋𝒊∈ 𝑬𝒋:𝒂 𝒊𝒋 ∈ 𝑬 𝒙𝒊𝒋 𝒓𝒔 ∈ 𝟎, 𝟏 , 𝒕 𝒓𝒔 ≥ 𝟎 ∀𝒂𝒊𝒋 ∈ 𝑬, 𝒔 ∈ 𝑺 𝝈𝒊𝒋 𝟐 𝒙𝒊𝒋 𝒓𝒔 (𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒊𝒋 𝒓𝒔 ) 𝒂 𝒊𝒋∈ 𝑬 + 𝒄𝒐𝒗 𝒂𝒊𝒋, 𝒂𝒍𝒌 𝒙𝒍𝒌 𝒓𝒔 (𝒙𝒊𝒋 𝒓𝒔 − 𝒙𝒊𝒋 𝒓𝒔 ) 𝒂 𝒍𝒌 ∈ 𝑬 𝒍,𝒌 ≠ 𝒊,𝒋 𝒂 𝒊𝒋∈ 𝑬 − 𝒕 𝒓𝒔 𝒕 𝒓𝒔 − 𝒕 𝒓𝒔 ≤ 𝟎 ∀𝒔 ∈ 𝑺, 𝒉 = 𝟏, . , 𝑯
- 17. Input: Convergence tolerance ε, Maximum Number of Iteration H, Upperbound(UB)=+inf, Lowerbound (LB) =- inf, h=1 Initialization: Solve One-all Shortest path problem 1: If (UB-LB) <=ε or then go to step 6 2: Solve the SP (Closed Form Equations) 3: If ( SP<UB), Update the UP and the best point 4: Add the OA inequalities and solve the MP (No need to solve MP to optimality, Fletcher and Leyffer; 1994) 5: Update the LB, h=h+1, go to step 1 6:Report the Solutions and Algorithm Stops
- 18. Networks Number of Links(A) Number of Nodes (N) SiouxFalls 76 24 Anaheim 914 416 Barcelona 2522 1020 Chicago Sketch 2250 933 Experiments Set up OA-is coded in GAMS-C++/CPLEX MILP MP is solved via CPLEX solver MP is solved within 1% optimality gap Covariance matrix is randomly generated
- 19. 0 5 10 15 20 25 30 35 40 45 50 1 2 3 ConvergenceTolerance(%) Number of Iterations Sioux Falls Anaheim Chicago Sketch Barcelona Maximum number of iterations is three for large networks
- 20. 0 20 40 60 80 100 120 140 160 180 0.5 0.6 0.8 1 1.5 2 RunningTime(Sec) Master Problem Relative Optimality Gap(%) Anaheim Sioux Falls Chicago Sketch Barcelona Solution time is stable w.r.t. MP optimality gap
- 21. 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 RunningTime(Sec) Weight Associated with Mean Tavel Cost Sioux Falls Anaheim Chicago Sketch Barcelona Increase in weight on STD term increases the solution time for larger networks
- 22. 0.45 4.23 57.36 154.5 0.7 20.5 845.6 1235.6 0 200 400 600 800 1000 1200 1400 Sioux Falls Anaheim Chicago Sketch Barcelona RunningTime(Sec) Network Outer Approximation MIQCP OA outperforms the CPLEX solver
- 23. General Reliability Measures [Shahabi, Unnikrishnan, and Boyles, 2014, 2015] show application of OA to other forms of convex reliability metrics Joint Inventory Location Problems [Shahabi, Unnikrishnan, and Boyles, 2014] show that OA is efficient in solving multi-echelon joint inventory location problems with correlations in retailer demand
- 24. •A general efficient methodology for binary convex mixed integer programs was presented •Given the convexity assumption the method guarantees the global optimality •The sub-problem is analytically solved •Minimizes the impact of large correlation matrices and lead to savings in computational time
Notes de l'éditeur
- There are different ways to characterize reliability and risk taking behavior…One such metric is mean-std dev..this ppt is mainly going to focus on mean+std dev shortest path..
- CQP computationally tractable due to their special Structure Can be solved by polynomial time interior point algorithms Solvers like CPLEX and MOSEK offering this capability
- Nonlinear terms increase by number of O-D pairs
- Separates out the nonlinear and integer part…