Risk management in some types of location problems under uncertainty


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1 Risk management in some types of location problems under uncertainty Laureano F. Escudero Joint work with Antonio AlonsoAyuso and Celeste Pizarro Universidad Rey Juan Carlos, Móstoles (Madrid), Spain Exploratory Workshop on Location Analysis: Trends on Theory and Applications IMUS, Universidad de Sevilla, November 2830, 2011
2 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
3 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
4 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
5 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
6 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
7 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
8 Table of Contents 1 Location problems under uncertainty that can be treated 2 Basic notions Risk Neutral strategy Seminal papers in 3 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures
9 Location problems: Special mixed 01 programs máx s.t. ax + cy Ax + By = b x {0, 1} n, y 0, (1)
10 Our experience for large scale location problems Production planning dimensioning and location Plant location & sizing in Strategic Supply Chain Mgmnt Designing connected rapid transit networks Road construction selection in Forestry harvesting planning Tactical portfolio planning in the Natural Gas Supply Chain Prison Facility Site selection under Uncertainty Air traffic Flow Management planning. Flight routes selection Cluster location in multiperiod copper extraction planning in mining Multiperiod locationassignment Energy generation capacity expansion planning and sources location Combinatorial nonlocation most difficult problem: Stochastic SPP
11 Production planning dimensioning and location Aim: Determining product selection and plant location and capacity (i.e., sizing). Main Uncertainties: product price and demand and production cost. VaR and meanrisk averse strategies for twostage stochastic mixed 01 optimization problem for maximizing expected net profit. Method: A twostage BFC algorithmic approach. Ref. AlonsoAyuso, Escudero, Garín, Ortuño, Pérez. OMEGA 05.
12 Plant location & sizing in Strategic Supply Chain Mgmnt Aim: Strategic planning for supply chains and, so, deciding on production topology, plant location and capacity (i.e, sizing), product selection and allocation, and vendor selection for raw materials. Main Uncertainties: product net profit and demand, raw material cost and production cost along time horizon. A risk neutral twostage stochastic mixed 01 optimization problem for maximizing expected net profit. Ref. AlonsoAyuso, Escudero, Garín, Ortuño, Pérez. JOGO 03, and AlonsoAyuso, Escudero, Ortuño. SORT 07.
13 Designing connected rapid transit networks Aim: Location of stations to be constructed and the links between them for in 1st step maximizing the expected number of trips and minimizing the route length, and in 2nd step attempting to minimize an estimation of the total number of transfers that should be made by the users to arrive at their destinations. Uncertainty: Trip demand between origin and destination pairs of demand. Deterministic method. 1st step: Plain CPLEX for the pure combinatorial model network designing, 2nd step: ad.hoc heuristic. Ref. Escudero, Muñoz. 1st paper TOP 09; 2nd paper submitted 2nd evaluation 2011.
14 Road construction selection in Forestry harvesting planning Aim: Managing land designated for timber production and divided into harvest cells. High volatility; timber price scenarios over time. For each time period: Decision on decide which cells to cut and location of access roads to build. A risk neutral multistage stochastic mixed 01 optimization problem for maximizing expected net profit. Method: A specialization of the BFC algorithmic approach (AEO EJOR 03). Ref. AlonsoAyuso, Escudero, Guignard, Quinteros, Weintraub. ANOR 11.
15 Tactical portfolio planning in the Natural Gas Supply Chain Aim: Natural Gas SCM for selecting production fields, transportation capacity booking, bilateral contracts and spot markets. Uncertainty: Buyer s nomination and prices, as well a spot prices. A risk neutral multistage stochastic mixed 01 optimization problem for maximizing expected profit. Method: A nonsymmetric BFCMS algorithmic approach. Ref. Fodstad, Midhum, Romo, Tomasgard, in Bertocchi, Consigli, Dempster (eds.) book, Springer, Ref. Escudero, Garín, Merino, Pérez. COR 12.
16 Prison Facility Site selection under Uncertainty Aim: Solving the strategic problem of timing location and capacity (i.e., sizing) of prison facilities in the Chilean prison system for minimizing expected total cost. Other applications: locating public (hospitals) or private (malls) centers. Main uncertainty: demand for capacity along time horizon. A risk neutral multistage stochastic mixed 01 optimization problem for minimizing expected total cost. Method: A methaheuristic BCC algorithmic approach. based on BFC. Ref. Hernández, AlonsoAyuso, Bravo, Escudero, Guignard, Marianov, Weintraub. COR, accepted for publication 2011.
17 Air traffic Flow Management planning. Flight routes selection Aim: Air traffic flow management for deciding the flight scheduling and location of the routes for each flight in an air network. Uncertainties: airport arrival and departure capacity, the air sector capacity and the flight demand. A risk neutral multistage stochastic mixed 01 optimization problem for minimizing expected total cost. Method: Plain use of MIP solver CPLEX. Ref. Agustín, AlonsoAyuso, Escudero, Pizarro. Submitted, 2nd evaluation, 2011.
18 Cluster location in multiperiod copper extraction planning in mining Aim: Supporting decisions on sequencing and locating clusters for material extraction copper mines. Main uncertainty: Highly volatility of copper prices along a time horizon. A variety of risk averse strategies for multistage stochastic mixed 01 optimization problem for maximizing total profit. Plain use of MIP solver CPLEX. Ref. AlonsoAyuso, Carvallo, Escudero, Guignard, Pi, Puranmalka, Weintraub. To be submitted 2011.
19 Multiperiod locationassignment Aim: Solving the strategic problem of timing the location of facilities and assigning of customers to facilities in a multiperiod environment. Main uncertainties: facilities setup and maintenance costs, customers assignment cost to the facilities, timing at which the customers need to be assigned to the facilities. A risk neutral multistage stochastic pure combinatorial problem for minimizing expected total cost. Method: A methaheuristic FRC algorithmic approach. Ref. AlbaredaSambola, AlonsoAyuso, Escudero, Fernández, Pizarro. To be submitted.
20 Energy generation capacity expansion planning and sources location Aim: Deciding on the optimal mix of different electric power technologies and plant location and sizing along a time horizon for a price taker. Main uncertainty: Fuel prices. A CVaR and Stochastic Dominance Constraints risk averse strategies for multistage stochastic mixed 01 optimization problem for maximizing total profit. Plain use of MIP solver CPLEX. Ref. L.F. Escudero, M. Inorta, M.T. Vespucci. In preparation.
21 Combinatorial nonlocation most difficult problem: Stochastic SPP Aim: Maximizing the meanrisk averse strategy: Maximizing the objective function expected value of the Stochastic SSP minus the weighted risk of obtaining a scenario whose objective function value is worse than a given threshold. Uncertainty: Objective function coeffs. Method: A onestage DEM splitting variable decomposition approach by using a deterministic SPP and a 01 knapsack problem for each scenario at each Lagrangean iter. Ref. Escudero, Landete, RodriguezChia, EJOR 11.
22 Risk Mngment in above types of Location problems Deterministic approach on expected values of uncertain parameters. It can be a big fiasco. Stochasticity treatment via Scenario Analysis. Risk Management: Risk neutral and Risk averse strategies. Feasibility for each scenario. Simple, partial, full recourse. multistage (dynamic), twostage (dynamic), one stage (static). Risk measures: minmax; risk neutral: mean; risk averse: VaR, CVaR, meanrisk, stochastic dominance cons.
23 Basic notions Risk Neutral strategy Seminal papers in Scenario trees in A stage of a given time horizon is a set of consecutive time periods where the realization of the uncertain parameters takes place. A scenario is a realization of the uncertain parameters along the stages of a given time horizon. A scenario group for a given stage is the set of scenarios with the same realization of the uncertain parameters up to the stage.
24 t = 1 t = 2 t = 3 t = 4 Basic notions Risk Neutral strategy Seminal papers in Ω = Ω 1 = {10, 11,...,17}; Ω 2 = {10, 11, 12} G = {1,...,17}; G 2 = {2, 3, 4} 14 N 9 = {1, 4, 9}; σ(9) = Multistage nonsymmetric scenario tree
25 Basic notions Risk Neutral strategy Seminal papers in Let the following notation related to the scenario tree: T, set of the T stages along the time horizon. Ω, set of scenarios. G, set of scenario groups, so that we have a directed graph where G is the set of nodes. G t, set of scenario groups in stage t, for t T (G t G). Ω g, set of scenarios in group g, for g G (Ω g Ω). t(g), stage to which scenario group g belongs to, for g G. σ(g), immediate ancestor node of node g, for g G. w ω, likelihood assigned by the modeler to scenario ω Ω. N g, set of scenario groups {k} such that Ω g Ω k, for g G (N g G). That is, set of ancestor scenario groups to scenario group g, including itself.
26 Basic notions Risk Neutral strategy Seminal papers in Risk Neutral strategy. Max objective function expected value over scenarios máx Q E = g G w g (a g x g + c g y g ) s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g g G x g {0, 1} n t, y g 0 g G. (2) DEM, Nonanticipativity constraints (NAC) (Wets, SIAM Review 74 for twostage problems and many others after)
27 Basic notions Risk Neutral strategy Seminal papers in 6 earlier seminal papers in Stochastic Programming. Chronological order E.M.L. Beale. On minimizing a convex function subject to linear inequalities, Journal of the Royal Statistical Society, Ser. B, 17: , G.B. Dantzig. Linear programming under uncertainty, Management Science 1: , A. Charnes and W.W. Cooper. Chanceconstrained programming, Management Science 5:7379, R. JB. Wets. Programming under uncertainty: the equivalent convex program, SIAM Journal on Applied Mathematics 14:89105, R. Van Slyke and R.JB. Wets. Lshaped linear programs with application to optimal control and stochastic programming, SIAM Journal on Applied Mathematics 17: , R.JB Wets. Stochastic programs with fixed recourse: The equivalent Deterministic program, SIAM Review 16: , BUT WHAT ABOUT TWOSTAGE AND MULTISTAGE STOCHASTIC MIXED INTEGER OPTIMIZATION THEORY, MODELS AND ALGORITHMS?
28 Basic Risk Averse measures Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures max mean subject to an upper bound on semideviation max mean  weighted semideviation min semideviation subject to a lower bound on mean min expected deviation from scenarios optimal value max ValueatRisk (VaR) max Conditional VaR max mean  weighted probability of nondesired scenarios stochastic dominance cons strategies (SDC)
29 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #1: max mean Q E subject to an upper bound on semideviation σ 2 máx Q E = g G w g (a g x g + c g y g ) s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g g G σ 2 σ 2 x g {0, 1} n t, y g 0 g G. (3) Inspired in Markowitz, JoFinance 52 and 1959 book, Wiley, Ogryczak & Ruszczynski, EJOR 99 and Ahmed, MP 06 for twostages and continuous vars.
30 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Semideviation σ 2 σ 2 = ω Ω w ω (Q ω Q E ) 2 where Q E : max obj. fun. expected value is given in model (2) and Q ω = g N d (a gˆx g + c g ŷ g ) of the solution ˆx ω, ŷ ω of the variables x ω, y ω in that model, where ω Ω d, d G T.
31 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #2: min expected deviation from optimal obj. fun. value over scenarios Let the Scenario Immunization model (Dembo, ANOR 91 for twostages and continuous vars) D = mín ω Ω wω (Q ω Q ω ) l s.t. A g x σ(g) + A g x g + B g y σ(g) + B g y g = b g g G x g {0, 1} n t, y g 0 g G. (4) See in (Escudero, 1995, in Zenios (ed.), Quantitative Methods. AI and Supercomputers in Finance) a twostep strategy (, 1) for solving model (4).
32 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #3: max mean Q E & VaR for β success probability, a 01 γ parameter, a ρ weighting parameter and a big enough M ω parameter. máx γ g G w g (a g x g + c g y g )+ρα s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g G (a g x g + c g y g )+M ω ν ω α ω Ω g N d w ω ν ω 1 β ω Ω x g {0, 1} n t, y g 0 g G ν ω {0, 1} ω Ω. See Gaivoronski & Pflug, 1999, JoR 05; Charpentier & Oulidi MMOR 08, among others for two stages and continuous vars. (5)
33 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #4: max mean & weighted VaR & Conditional VaR (i.e., obj. fun expected shortfall on reaching VaR) máx γ g G w g (a g x g + c g y g )+ ( ( ρ α 1 β d G T w d α ) ) g N d (a g x g + c g y g ) + s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g G x g {0, 1} n t, y g 0 g G α R. Inspired in Rockafellar & Uryasev, JoR 00 and Ahmed, MP06 for twostages and continuous vars; Schultz & Tiedemann, MP 06 for twostages. (6)
34 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #5: max VaR for given success probability & weighted Conditional over VaR (i.e., obj fun expected excess on reaching VaR) máx α+ρ ( ) w d w g (a g x g + c g y g ) α + d G T g G d s.t. A g x σ(g) + A g x g + B g y σ(g) + B g y g = b g G (a g x g + c g y g )+M ω ν ω α ω Ω g N d w ω ν ω 1 β ω Ω x g {0, 1} n t, y g 0 g G ν ω {0, 1} ω Ω. (7)
35 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #6: max mean  weighted obj fun expected shortfall over given threshold φ ( ) máx w d w g (a g x g + c g y g ) ρ w ω v ω d G T g G d ω Ω s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g G w g (a g x g + c g y g )+v ω φ ω Ω g G d x g {0, 1} n t, y g 0 g G v ω 0 ω Ω. (8) Inspired in Eppen, Martin & Schrage, OR 89 for twostages. Related to ICC of Klein Haneveld book (1986).
36 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #7: max mean  weighted probability of nondesired scenario máx w g (a g x g + c g y g ) ρ w ω ν ω g G ω Ω s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g G (a g x g + c g y g )+M ω ν ω φ ω Ω g N d x g {0, 1} n t, y g 0 g G ν ω {0, 1} ω Ω. (9) See Schultz & Tiedemann, SIAM JoO 03 for twostages.
37 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #8: stochastic dominance cons (firstorder SDC) for threshold profile φ p with β p success probability p P máx w g (a g x g + c g y g ) g G s.t. A g x σ(g) + A g x g + B g y σ(g) + B g y g = b g G (a g x g + c g y g )+M ω ν ωp φ p ω Ω, p P g N d w ω ν ωp 1 β p p P ω Ω x g {0, 1} n t, y g 0 g G ν ωp {0, 1} ω Ω, p P. (10)
38 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #8: stochastic dominance cons (firstorder SDC) for threshold profile φ p with β p success probability p P (cont.) Note: The objective fun. value of scenario ω is not below the threshold φ p with β p probability, for p P, where P is the set of profiles under consideration. Inspired in Gollmer, Neise & Schultz, SIAM JoO 08 for twostages.
39 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #9: stochastic dominance cons (secondorder SDC) for obj fun expected shortfall e p bound on reaching threshold φ p, p P máx w g (a g x g + c g y g ) g G s.t. A gx σ(g) + A g x g + B gy σ(g) + B g y g = b g G φ p (a g x g + c g y g ) v ωp ω Ω, p P g N d w ω v ωp e p p P ω Ω x g {0, 1} n t, y g 0 g G v ωp 0 ω Ω, p P. (11)
40 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse alternative #9: stochastic dominance cons (secondorder SDC) for obj fun expected shortfall e p bound on reaching threshold φ p, p P (cont.) Inspired in Gollmer, Gotzes & Schultz, MP 11 for twostages. The concept of the obj fun expected shortfall on reaching a given threshold may have its roots in the ICC concept due to Klein Haneveld book (1986). See also Eppen, Martin & Schrage, OR 89 and Klein Haneveld & van der Vlerk, CMS 06 for twostages.
41 Solution considerations: Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures The strategies #3: Mean & VaR, #5: CVaRover, #8: SDC1 and #9: SDC2 have a computational disadvantage over the strategies #2: Scen Imm, #4: CVaR, #6: VAR & SHORTFALL and #7: MEAN & RISK, since they have constraints linking 01 variables from different scenarios. In any case, a decomposition approach must be used for problem solving of huge instances, e.g., exact BFCMS (Escudero, Garín, Merino & Pérez, COR 12) and inexact FRC (AlonsoAyuso, Escudero, Olaso & Pizarro, in preparation) for very large scale stochastic mixed 01 problems plus a device for treating those linking constraints at given steps in exact BFCMS and inexact FRC
42 Solution considerations: Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures The strategies #3: Mean & VaR, #5: CVaRover, #8: SDC1 and #9: SDC2 have a computational disadvantage over the strategies #2: Scen Imm, #4: CVaR, #6: VAR & SHORTFALL and #7: MEAN & RISK, since they have constraints linking 01 variables from different scenarios. In any case, a decomposition approach must be used for problem solving of huge instances, e.g., exact BFCMS (Escudero, Garín, Merino & Pérez, COR 12) and inexact FRC (AlonsoAyuso, Escudero, Olaso & Pizarro, in preparation) for very large scale stochastic mixed 01 problems plus a device for treating those linking constraints at given steps in exact BFCMS and inexact FRC
43 Solution considerations: Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures The strategies #3: Mean & VaR, #5: CVaRover, #8: SDC1 and #9: SDC2 have a computational disadvantage over the strategies #2: Scen Imm, #4: CVaR, #6: VAR & SHORTFALL and #7: MEAN & RISK, since they have constraints linking 01 variables from different scenarios. In any case, a decomposition approach must be used for problem solving of huge instances, e.g., exact BFCMS (Escudero, Garín, Merino & Pérez, COR 12) and inexact FRC (AlonsoAyuso, Escudero, Olaso & Pizarro, in preparation) for very large scale stochastic mixed 01 problems plus a device for treating those linking constraints at given steps in exact BFCMS and inexact FRC
44 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse bibliography. Chronological order H.M. Markowitz. Portfolio selection, Journal of Finance 7:7791, H.M. Markowitz. Portfolio selection, Wiley, P.C. Fishburn. Meanrisk analysis with risk associated with below target returns, The American Economic Review 67: , G.D. Eppen, R.K. Martin, L. Schrage. Scenario approach to capacity planning, Operations Research 34: , R. Dembo. Scenario immunization, Annals of Operations Research 30:6390, L.F. Escudero. Robust Portfolios for MortgageBacked Securities. In S.A. Zenios, editor, Quantitative Methods. AI and Supercomputers in Finance. Unicom, London, , P. Artzner, F. Delbaen, L. Eber, D. Health. Coherent measures of risk, Mathematical Finance 9: , A.A. Gaivoronski, G. Pflug. Finding optimal portfolios with constraints on valueatrisk. In B. Green, editor, Proceedings of the Third International Stockholm Seminar on Risk Behaviour and Risk Management. Stockholm University, W. Ogryczak, A. Ruszczynski. From stochastic dominance to meanrisk models: semideviations as risk measures, European Journal of Operations Research 116: 3350, R. T. Rockafellar, S. Uryasev. Optimization on conditional valueatrisk, Journal of Risk 2:2141, D. Dentcheva, A. Ruszczynski. Optimization with stochastic dominance constraints, SIAM Journal on Optimization 14: , W. Ogryczak, A. Ruszczynski. Dual stochastic dominance and related risk models, SIAM Journal on Optimization 13:6078, R. Schultz, S. Tiedemann. Risk Averse via excess probabilities in stochastic programs with mixedinteger recourse, SIAM Journal on Optimization 14: , 2003.
45 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse bibliography. Chronological order (cont.) D. Dentcheva, A. Ruszczynski. Optimality and duality theory for stochastic optimization with stochastic nonlinear dominance constraints, Mathematical Programming, Ser. B 99: , A.A. Gaivoronski, G. Plug. Valueatrisk in portfolio optimization: properties and computational approach, Journal of Risk 7:1131, W.K. Klein, M.H. van der Vlerk. Integrated chance constraints: reduced forms an algorithm, Computational Management Science 3: , R. Schultz, S. Tiedemann. Conditional valueatrisk in stochastic programs with mixed integer recourse, Mathematical Programming, Ser. B 105: , S. Ahmed. Convexity and decomposition of meanrisk stochastic programs, Mathematical Programming, Ser. B 106: , D. Dentcheva, R. Henrion, A. Ruszczynski. Stability and sensitivity of optimization problems with first order stochastic dominance constraints, SIAM Journal on Optimization 18: , A. Charpentier, A. Oulidi. Estimating allocations for ValueatRisk portfolio optimization, Mathematical Methods of Operations Research, doi /s , R. Gollmer, F. Neise, R. Schultz. Stochastic programs with firstorder stochastic dominance constraints induced by mixedinteger linear recourse, SIAM Journal on Optimization 19: , L.F. Escudero, On a mixture of the FixandRelax Coordination and Lagrangean Substitution schemes for multistage stochastic mixed integer programming, TOP 17:529, A. AlonsoAyuso, L.F. Escudero, C. Pizarro. On SIP algorithms for minimizing the meanrisk function in the MultiPeriod Single Source Problem under uncertainty, Annals of Operations Research 166: , 2009.
46 Risk Averse alternatives to Risk Neutral strategy Recommended bibliography for Risk Averse measures Risk Averse bibliography. Chronological order (cont.) the book A. Shapiro, D. Dentcheva, A. Ruszczynski. Lectures on Stochastic Programming, MPSSIAM Series on Optimization, , L.F. Escudero, M. Landete, A. RodriguezChia. Stochastic Set Packing Problem, European Journal of Operational Research 211: , R. Gollmer, U. Gotzes, R. Schultz. A note on secondorder stochastic dominance constraints induced by mixedinteger linear recourse, Mathematical Programming, Ser. A 126: , N. Miller, A. Ruszczynski. Riskaverse twostage stochastic linear programming: Modeling and decomposition, Operations Research 59: , L. Aranburu, L.F. Escudero, M.A. Garín, G. Pérez. Stochastic models for optimizing immunization strategies in fixedincome security portfolios under some sources of uncertainty, Submitted for publication P. Beraldi, M. Costabile, I. Massabo, E. Russo, F. de Simeone, A. Violi. A multistage stochastic programming approach for capital budgeting problems under uncertainty, Submitted for publication 2011.
47 Illustrative examples Note: COINOR, CPLEX v9 and v.12 failed to find a feas sol in several hours. Mixed integer model 1. Pilot case: Tactical portfolio planning in the Natural Gas Supply Chain. Risk neutral. Ω =1000 scenarios, m=98456 constraints, n01= vars, nc=22221 continuous vars, elapsed time = 182 secs, SUN WS, 2.6Ghz, 16Gb RAM, linux GAP=0 % (optimal soln). Escudero, Garín, Merino, Pérez, COR 12.
48 Illustrative examples (cont.) CPLEX v.12.0 did not fail to obtain optimal soln, but bigger cases require decomposition approaches. Mixed integer model 2. Pilot case: Air Traffic Flow Management under uncertainty. Randomly generated. Risk neutral. Ω =48 scenarios, m= constraints, n= (mostly, 01 vars), elapsed time = 415 secs, PC under Ubuntu, Inter Core Duo, 2Ghz, 4Gb RAM. GAP=0 % (optimal soln). Agustín, AlonsoAyuso Escudero, Pizarro, Submitted 2nd evaluation, 2011.
49 Illustrative examples (cont.) CPLEX v.12.0 has problems for obtaining optimal soln. Decomposition approaches are a must. Mixed integer model 3. Pilot case: Copper extraction planning under uncertainty in future cooper prices. Realistic case. Many risk averse measures. Ω =45 scenarios, m=480490, n01=167951, nc=823, elapsed time: from 398 to secs. Big HW/SW platform: 2 quadcore Xeon E Ghz 64bit processors with 6Mb of cache each, GAMS, CPLEX v12.2. GAP=0 % (optimal soln). AlonsoAyuso, Carvallo, Escudero, Guiganrd, Pi, Puranmalka, Weintraub. To be submitted 2011.
50 Illustrative examples (cont.) Note: COINOR, CPLEX v9 and v.12 failed to find a feas sol in 6 hours of elapsed time. Pure combinatorial model 1. Pilot case: Stochastic Set Packing Problem. Randomly generated. Risk adverse: meanrisk. Ω =100 scenarios, m=220 constraints, n01= vars, elapsed time = secs, SUN WS, 2.2Ghz, 4Gb RAM, GAP=6.82 %. Escudero, Landete, RodriguezChia, EJOR 11.
51 Illustrative examples (cont.) CPLEX v.12.3 has problems for obtaining optimal soln. Decomposition approaches are a must. Pure combinatorial model 2. Pilot case: On solving the multiperiod locationassignment problem under uncertainty. Randomly generated. Risk neutral. Ω =158 scenarios, m=121979, n01=99700, elapsed time: 313 secs, Intel Core 2 Duo, 2.60Ghz, 3Gb RAM, inexact FRC algorithm implemented in C++ code with CPLEX v12.3 as a MIP solver. GAP=0 % (optimal soln), Time GAP=44.6 %. AlbaredaSambola, AlonsoAyuso, Escudero, Fernández, Pizarro. To be submitted.
52 Illustrative examples (cont.) CPLEX v.12.3 has problems for obtaining optimal soln. Decomposition approaches are a must. Pure combinatorial model 2. Pilot case: On solving the multiperiod locationassignment problem under uncertainty. Randomly generated. Risk neutral. Ω =241 scenarios, m=188713, n01=154220, elapsed time: 241 secs, Intel Core 2 Duo, 2.60Ghz, 3Gb RAM, inexact FRC algorithm implemented in C++ code with CPLEX v12.3 as a MIP solver. GAP=0 % (optimal soln), Time GAP=60.8 %. AlbaredaSambola, AlonsoAyuso, Escudero, Fernández, Pizarro. To be submitted.
53 Illustrative examples (cont.) CPLEX v.12.3 doesn t obtained a soln in 8 hours of elaped time. Pure combinatorial model 2. Pilot case: On solving the multiperiod locationassignment problem under uncertainty. Randomly generated. Risk neutral. Ω =147 scenarios, m=18231, n01=116860, elapsed time: 2958 secs, Intel Core 2 Duo, 2.60Ghz, 3Gb RAM, inexact FRC algorithm implemented in C++ code with CPLEX v12.3 as a MIP solver. AlbaredaSambola, AlonsoAyuso, Escudero, Fernández, Pizarro. To be submitted.
54 Stochastic papers published in scientific journals A. AlonsoAyuso, L.F. Escudero, M.T. Ortuño. BFC, a BranchandFix Coordination algorithmic framework for solving some types of stochastic pure and mixed 01 programs, European Journal of Operational Research 151: , L.F. Escudero, J. Salmerón. On a FixandRelax framework for largescale resourceconstrainedproject scheduling, Annals of Operations Research 140: , L.F. Escudero, A. Garín, M. Merino, G. Pérez, A twostage stochastic integer programming approach as a mixture of BranchandFix Coordination and Benders Decomposition schemes, Annals of Operations Research 152: , L.F. Escudero, M.A. Garín, M. Merino, G. Pérez. The value of the stochastic solution in multistage problems, TOP 15:4864, A. AlonsoAyuso, L.F. Escudero, C. Pizarro. On SIP algorithms for minimizing the meanrisk function in the MultiPeriod Single Source Problem under uncertainty, Annals of Operations Research 166: , M. P. Cristóbal, L.F. Escudero, J.F. Monge. On Stochastic Dynamic Programming for solving largescale tactical production planning problems, Computers & Operations Research 36: , A. AlonsoAyuso, N. Domenica, L.F. Escudero, C. Pizarro. Structuring bilateral energy contract portfolios in competitive markets, in G. Consigli, M.I. Bertocchi, D.A.H. Dempster (eds.), Stochastic optimization methods in Finance and Energy, Springer, 2011, pp L.F. Escudero, M.A. Garín, M. Merino, G. Pérez, An algorithmic framework for solving large scale multistage stochastic mixed 01 problems with nonsymmetric scenario trees, Computers & Operations Research 39: , 2012.
Abstract. 1. Introduction. Caparica, Portugal b CEG, ISTUTL, Av. Rovisco Pais, 1049001 Lisboa, Portugal
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