Dealing with Uncertainty

Transcription

1 Dealing with Uncertainty in Decision Making Models Roger J-B Wets University of California, Davis Dealing with Uncertainty p.1/??

2 A product mix problem Dealing with Uncertainty p.2/??

3 Formulation A furniture manufacturer must choose x j 0, how many dressers of type j = 1,..., 4 to manufacture so as to maximize profit 4 j=1 c j x j = 12x x x x 4 The constraints: t c1 x 1 + t c2 x 2 + t c3 x 3 + t c4 x 4 d c t f1 x 1 + t f2 x 1 + t f3 x 1 + t f4 x 1 d f t cj (t fj ) carpentry (finishing) man-hours: dresser type j d c (d f ) = total time available for carpentry (finishing) Dealing with Uncertainty p.3/??

4 Product mix problem (2) Solution via linear programming: max c, x so that T x d, x IR n +. With T = t c1 t c2 t c3 t c4 = t f1 t f2 t f3 t f , d c d f = Optimal: x d = (4000/3, 0, 0, 200/3) Value: $ 18,667. Dealing with Uncertainty p.4/??

5 Product mix problem (3) But... reality can t be ignored! t cj = t cj + η cj, t fj = t fj + η fj entry possible values d c + ζ c : 5,873 5,967 6,033 6,127 d f + ζ f : 3,936 3,984 4,016 4, random variables, say, 4 possible values each L = 1, 048, 576 possible pairs (T l, d l ) Dealing with Uncertainty p.5/??

6 Product mix problem (4) What if 4 j=1 (t cj + η cj )x j > d c + ζ c? = overtime With ξ = (η {, }, ζ { } ), recourse: (y c (ξ), y f cost (q c, q f ). max c, x p 1 q, y 1 p 2 q, y 2 p L q, y L s.t. T 1 x y 1 d 1 T 2 x y 2 d T L x y L d L x 0, y 1 0, y 2 0, y L 0. Structured large scale l.p. (L 10 6 ) Dealing with Uncertainty p.6/??

7 Product mix problem (5) Define Ξ = { ξ = (η, ζ) }, p ξ = prob [ξ = ξ] Q(ξ, x) = max { q, y y Tξ x d ξ, y 0 } EQ(x) = E{Q(ξ, x)} = ξ Ξ p ξ Q(ξ, x) the equivalent deterministic program (DEP): max c, x + EQ(x) so that x IR n +. a non-smooth convex optimization problem: EQ concave. Dealing with Uncertainty p.7/??

8 Product mix problem (6) Solution of DEP, or large scale l.p.,: Optimal: x = (257, 0, 665.2, 33.8) expected Profit: $ 18,051 The solution x is robust : it considered all 10 6 possibilities. Recall: x d = (1, , 0, 0, 66.67) expected profit = $ 16,942. x d is not close to optimal x d isn t pointing in the right direction Dealing with Uncertainty p.8/??

9 A Conversation with G.B. Dantzig G.B.D. [Q] When will pratitioners/optimizers adopt stochastic programming? G.B.D. [A] The day they get really interested in getting good solutions! Dealing with Uncertainty p.9/??

10 Mathematics & Numerics Stochastic Programming relies on: linear, non-linear, mixed-integer programming large scale: decomposition methods, structured programs, grid computing Variational Analysis: non-smooth, duality, epi-convergence (approximations), etc. Probability: stochastic processes, asymptotic laws Statistics: estimation, lack of data issues Functional Analysis, Combinatorial Geometry, etc. Dealing with Uncertainty p.10/??

11 Modeling, modeling & modeling! Dealing with Uncertainty p.11/??

12 Uncertain parameters Deterministic Optimization problem: max f 0 (x) so that x S IR n Uncertain parameters: ξ Ξ IR N, max f 0 (ξ, x) so that x S(ξ) IR n Wait-and-see solution?? x(ξ) argmax { f 0 (ξ, x) } x S(ξ) What s needed: a here-and-now solution. Dealing with Uncertainty p.12/??

13 The Newsboy Problem ξ Ξ IR + demand for a (perishable) good. x 0 quantity unit cost: c = 10 y 0 quantity sold, per unit profit r = 15 Total revenue (possibly negative): cx + (c + r)y where 0 x, 0 y min {x, ξ} Find optimal x! Dealing with Uncertainty p.13/??

14 The deterministic approach Pick ˆξ Ξ (guessing the future) and solve max f 0 (ˆξ, x) so that x S(ˆξ) IR n Newsboy: Ξ = [0, 150], pick ˆξ = 75, max cx + (c + r)y x 0, 0 y min{x, ˆξ} Solution: x o = y o = ˆξ, obj. value = r ˆξ = 1125 But doesn t tell much about profit if ξ 75! Dealing with Uncertainty p.14/??

15 Scenario Analysis Pick ξ 1,..., ξ L (scenarios), and for each ξ l find: x l argmax { f 0 (ξ l, x) x S(ξ l ) } and reconcile the solutions to obtain x o. Newsboy: pick ξ 1 = 10, ξ 2 = 20,..., ξ 15 = 150, (x l, y l ) argmax{ cx + (c + r)y y min[ξ l, x]} x 0,y 0 x l = ξ l : Wait-and-see sol ns! Reconciliation? No help in choosing x o the quantity to order. Dealing with Uncertainty p.15/??

16 ξ: Estimated Density h ξ log-normal: h(z) = (zτ 2π) 1 (ln z θ)2 e 2τ 2 θ = 4.43, τ = 0.38; H(z) = z 0 h(s) ds from data, expert(s), all information available lognormal density Expectation = 90 Strd.Dev. = might affect choice of ˆξ, scenarios: ξ 1,... Dealing with Uncertainty p.16/??

17 Maximize Expected Return max cx + E{(c + r)y ξ } so that x 0, 0 y ξ min [ ξ, x ] The equivalent deterministic program: max x 0 where Q(ξ, x) = cx + EQ(x), EQ(x) = E{Q(ξ, x)} EQ(x) = (c + r) { (c + r)ξ if ξ x, (c + r)x if ξ x ( x 0 ξ H(dξ) + x ) x H(dξ) Dealing with Uncertainty p.17/??

18 Newsboy s Objective: cx + EQ(x) Newsboy problem Objective function Ew = 90, Stdev=36 Cost=10, Revenue=15 Optimal sol n: Dealing with Uncertainty p.18/??

19 Optimal: Expected Profit ( r ) x = H 1 = H 1 (0.6) = 92.2 c + r via differentiation (when c = 10, r = 15) Cumulative Distribution lognormal Expect.=90, Std.Dev.= H Dealing with Uncertainty p.19/??

20 Eutrophication Management I II III IV Treatment plant Reed basin Strategic question: Include or not the (costly) reed basins in the pollution management plan xj size of plant or reed basin in location j Uncertainty: ξ (weather + ) Dealing with Uncertainty p.20/??

21 Management Problem min c, x x, Dx + E { IV i=i so that Ax b, 0 x υ, } θ i (d i (ξ) T i (ξ, x ) T (ξ)x d(ξ) = water quality in sections I,..., IV c, x x, Dx construction amortization Ax b budgetary/technological constraints, υ j upper bounds on plant or basin size θ i compliance measure to standard for basin i Dealing with Uncertainty p.21/??

22 Monitoring Function θ i ρ i excellent acceptable unacceptable γ i θ i (τ) = 0 if τ < 0, (excellent) ρ i τ 2 /2γ i if τ [0, γ i ] (acceptable) ρ i τ ρ i γ i /2 if τ > γ i (unacceptable) Dealing with Uncertainty p.22/??

23 Scenario Analysis vs. SP-Sol n let S = { x Ax b, 0 x υ }, (D = 0) Scenarios: ξ 1,. [.., ξ L atmospheric conditions x l = argmin x S c, x + ] IV i=i θ i(d i (ξ l ) T i (ξ l ), x ) Analysis: reed basins never enter solutions x l { IV } x = argmin x S c, x + E i=i θ i(d i (ξ) T i (ξ), x ) Analysis: x includes the reed basins as soon as the budgetary constraint allows for it. Moral: beware of decisions based on scenario analysis! Dealing with Uncertainty p.23/??

24 ... but is maximum expected return our real objective? Dealing with Uncertainty p.24/??

25 Returns: a Comprehensive View The actual return is a random variable: { f 0 (ξ, x) if x S(ξ) ζ x = otherwise its distribution function, F (s; x) = prob[ζ x s] and density: (d/ds)f (s; x). Newsboy: ζ x = cx + Q(ξ, x) if x 0 with F (s; x) = prob[ revenue = ζ x s] Dealing with Uncertainty p.25/??

26 Newsboy: Returns Densities 8 x Return Densities 5 x= x= x=112.2 x=152.2 x= Dealing with Uncertainty p.26/??

27 Choosing the Returns Distribution Distribution Functions 0.7 x= x= x= x=152.2 x= Dealing with Uncertainty p.27/??

28 Decision Criteria Reducing the choice of a distribution function to the choice of a number reliability first! maximize expected return (scaled?), max. E{return} & minimize customers lost, minimize Value-at-Risk (VaR), minimize the probability of any loss, minimizing a Safeguarding Measure variants & combinations of the above Dealing with Uncertainty p.28/??

29 Utility Theory FUND. THM. With F ( ; x) the returns distribution of f(ξ, x), suppose that for any feasible x 1, x 2 : F ( ; x 1 ) is either preferred or not (possibly indifferent) to F ( ; x 2 ), then there exists a continuous, real-valued utility function u : IR IR such that F ( ; x 1 ) F ( ; x 2 ) E{u(f(ξ, x 1 ))} > E{u(f(ξ, x 2 ))} knowing or discovering u u continuous, real-valued (safeguarding issues) Practically: θ : IR IR an appraisal function Dealing with Uncertainty p.29/??

30 Appraisal via Risk Measures Dealing with Uncertainty p.30/??

31 Minimal Loss Let f(ξ, x) = return: decision x, event ξ min x prob{ξ f(ξ, x) α} Newsboy with α = 0 [ prob min x 0 ] cx + Q(ξ, x) 0 Solution: x = 0! if ξ x, return = 15x if ξ x, return = cx + (c + r)ξ 0 implies x 2.5 ξ ξ Dealing with Uncertainty p.31/??

32 Reliability: Chance Constraints satisfy constraints with probability α (0, 1] min f 0 (x) so that prob. [ x S(ξ) ] α Variant: min f 0 (x) so that prob. [ f i (ξ, x) 0 ] α i, i I α i dictated by contractual obligations company policy, guess, etc. Dealing with Uncertainty p.32/??

33 Newsboy: Chance C. Model max cx + (c + r)y so that x 0, y [H 1 (α), x] Cumulative Distribution lognormal Expect.=90, Std.Dev.= H Infeasible if x < H 1 (α). Profit? not measured: When α = 0.9 : ˆx = 135.9; α = 0.75 : ˆx = Dealing with Uncertainty p.33/??

34 Chance Constraint: Appraisal Fcn θ r 1 H ( α ) x θ(f(ξ, x)) = rx 8 if x [0, H 1 (α)] otherwise highly discontinuous, doesn t take value of ξ into account Dealing with Uncertainty p.34/??

35 VaR: Value-at-Risk Let F (s; x) = prob [ cx + Q(ξ, x) s ] Value-at-Risk VaR(α; x): returns threshold. VaR(α; x) = F 1 (α; x) (= sup{v v F 1 (α; x)}) prob[ Returns VaR(α; x)] = 1 α Objective: find x that maximizes VaR(α; x) Newsboy: α = 0.10, prob[ returns VaR(0.1, )] = 90% VaR(.1; 72.2) = 555,, VaR(.1; 152.2) = 226 Dealing with Uncertainty p.35/??

36 VaR: Newsboy Problem VaR Risk) x= x=132.2 x= % 0.06 x= x= Dealing with Uncertainty p.36/??

37 VaR: Properties Challenge: x VaR(α; x) isn t concave (comp. unfriendly!) & unstable Appraisal fcn: θ(f(ξ, x)) = VaR(α, x) Unstable: α max VaR(α; ) discontinuous and independent of ξ! Dealing with Uncertainty p.37/??

38 Rescuing Risk? Heuristic: F is N (µ(x), σ(x) 2 ) and VaR(α; x) = N 1 (α; µ(x), σ(x) 2 ) Requires only estimating (µ(x), σ(x)), but even then: no (reliable) algorithm to find x opt. Exceptional case: returns distribution is gaussian (or elliptic) Dealing with Uncertainty p.38/??

39 However... α F(.,x 1 ) F(.,x 2 ) VaR( α,.) VaR(α, x 1 ) = VaR(α, x 2 ) = indifferent Dealing with Uncertainty p.39/??

40 A Safeguarding Measure: CVaR CVaR( ;x) α VaR( ;x) α α α F( ;x) Dealing with Uncertainty p.40/??

41 Conditional Value-at-Risk Let ζ x = f(ξ, x) = cx + Q(ξ, x), [t] = min{0, t} CVaR(α; x) = E{ζ ζ VaR(α; x)} G(z; α, x) = z α E{[f(ξ, x) + z] } THM. For f(ξ, ) concave, CVaR(α; ) finite, concave, cont., i.e., argmax comp. friendly! Proof. CVaR(α; x) = max z G(z; x, α) and (z, x) G(z; α, x) is jointly concave (definition) Dealing with Uncertainty p.41/??

42 CVaR: Properties x = argmax x CVaR(α; x) so that (z, x ) argmax G(z; α, x) Stability. α CVaR(α; x) continuous with leftand right-hand derivatives [sup-projection argument] Appraisal fcn: continuous, concave ) max z ([ z α [ cx + Q(ξ, x) + z] VaR(α; x) = max { argmax z G(x; α, x) } Dealing with Uncertainty p.42/??

43 Concave Risk Measures Set ζ = f(ξ, x) random returns, Z = { ζ } θ : Z IR (convex) risk function if Concavity: ζ, η Z, λ [0, 1] θ(λζ + (1 λ)η) λθ(ζ) + (1 λ)θ(η) Monotonicity: ζ η = θ(ζ) θ(η) Translation Equivariance: α IR, ζ Z, θ(α + ζ) = α + θ(ζ) Dealing with Uncertainty p.43/??

44 Robust Optimization Dealing with Uncertainty p.44/??

45 Robust Optimization Paradigm generic optimization problem (parameters ξ): max f 0 (ξ, x) so that f i (ξ, x) 0, i = 1,..., m assume uncertainty model is known (e.g. bounds) a robust feasible solution is feasible for all values of the uncertain parameters optimize the objective over all robust feasible solutions or and inner approximation to it Dealing with Uncertainty p.45/??

46 Robust Counterpart for U Ξ max γ so that γ f 0 (ξ, x) 0, ξ U Challenges: f i (ξ, x) 0, i = 1,..., m, ξ U formulate a computationally tractable robust counterpart specify reasonable uncertainty for set U Dealing with Uncertainty p.46/??

47 Ellipsoidal Uncertainty max f 0 (x) so that (a i + ξ i ), x (a i 0 + ξ i 0) 0, i = 1,..., m Ellipsoidal uncertainty: S i = Σ 1/2 i (positive definite) U i (ξ i 0, ξi ) U i = { S i u u 2 1 } Robust counterpart (with x 0 = 1) max f 0 (x) so that (a i 0, a i ), x + x, S i x 1/2 0, i = 1,..., m a SOCP (Second-Order Cone Program) marginally more difficult to solve than LP when f 0 is linear (via interior pt). Dealing with Uncertainty p.47/??

48 Robust Newsboy Problem U = [0, ˆξ] U = { ξ k, k = 1,..., ν } samples from ξ similar, or related, to Chance Constraints U = { ξ prob(q(ξ, x) β) α } or U = { ξ prob(q(ξ, x) β) α } or U = { ξ Q(ξ, x) κ } all similar, or related, to VaR. Dealing with Uncertainty p.48/??

49 Appraisal via Penalization Dealing with Uncertainty p.49/??

50 Penalties for Shortcomings ξ random demand for a (perishable) good x 0 quantity unit cost: c = 10 r = 15 unit profit per sale ρ unit cost per lost-sale ( customer ) γ = 1 unit disposal cost ( oversupply ) Expected Revenue: cx + E{Q(ξ, x)} Q(ξ, x) =(c + r)ξ γ(x ξ) if ξ x, (c + r)x ρ(ξ x) if ξ x Dealing with Uncertainty p.50/??

51 Sol n: Fcn of Penalty Parameters x = H 1( r + ρ c + r + ρ + γ ) = H 1( 15 + ρ ) 26 + ρ Optimal Solution as a function of rho c=10, r=15, gamma= rho Dealing with Uncertainty p.51/??

52 with Penalty Param.: Properties Appraisal fcn: continuous, concave { rx + γ(ξ x) θ(f(ξ, x)) = rx + (c + r ρ)(ξ x) if ξ x if ξ x r θ r γ c+ρ c+ρ ξ x ρ x (ρ) = argmax cx + E{Q(ξ, x; ρ)} continuous. Dealing with Uncertainty p.52/??

53 with Penalty Param.: Newsboy x opt (ρ) = H 1 ( r+ρ c+r+γ+ρ ) ( ) = H 1 15+ρ 26+ρ 3 x Return Densities Fcn of "shortage" penalty 2 rho=0 xopt=90.1 rho=4 xopt= rho=8 xopt= Dealing with Uncertainty p.53/??

54 Penalties & Chance Constraints Newsboy: with C.C.: x CC = H 1 (α) max rx so that x max [ 0, [H 1 (α) ] ( ) with Penalty ρ: x P = H 1 r+ρ r+c+ρ max cx + x 0 (c + r)ξ H(dξ) + (c + r + ρ)x H(x) Given α corresponding ρ & vice-versa. with c = 10, r = 15: α =.75 = ρ = 15 (α =.5, ρ = 5) when ρ = 10 = α = 5/7.71 Dealing with Uncertainty p.54/??

55 MORAL Model with Reliability criterion: evaluate implied costs with Appraisal function: evaluate reliability Dealing with Uncertainty p.55/??

56 Risk Aversion Function: q > 1 θ(t) = qt if t 0 ( 2 pq 2 2(q 1) q 1 2p t q 1) qp if t [0, p] t + p(q 1)/2 if t p θ p 1 t q Dealing with Uncertainty p.56/??

57 Newsboy: Reformulation Properties { ( max E θ x 0 )} cx + Q(ξ, x) takes into account risk aversion/risk seeking description via only two parameters: p, q q aversion to risk, p desirable return level discovering θ u by vallibrating p and q convexity is preserved, i.e. optimization friendly Dealing with Uncertainty p.57/??

58 Newsboy with Risk Aversion x = parameters: q = 4 ~ Risk Aversion p = 400 ~ Returns Threshold Dealing with Uncertainty p.58/??

59 Newsboy: Higher Risk Aversion x = parameters: q = 10 ~ risk aversion p = 800 ~ revenue threshold Dealing with Uncertainty p.59/??

60 Risk Aversion: Densities 8 x Revenues Densities q=10, p=800 x = 72 q=4, p=400 x = Dealing with Uncertainty p.60/??

61 Risk Seeking: 0 q < 1 θ(t) = qt if t 0 ( 2 pq 2 2(q 1) q 1 2p t q 1) qp if t [0, p] t + p(q 1)/2 if t p x = q θ p 1 t Risk Seeking parameters: q =0.2 ~ risk seeking p =10 ~ revenye threshold Dealing with Uncertainty p.61/??

62 Appraisal Fcn & Safeguarding max E { θ(f 0 (ξ, x)) } so that CVaR(α; x) β f i (x) 0, i I!! Constraints (CC, CVaR, etc.) = discontinuous appraisal (utility) function Dealing with Uncertainty p.62/??

63 Stability w.r.t. Information Solution Robustness Dealing with Uncertainty p.63/??

64 Prob. Measure Dependence Correct prob. measure is G instead of H? x H argmax E H {f(ξ, x)} x G argmax E G {f(ξ, x)}. THM. dist(g, H) small implies dist(x G, x H ) small. [Lipschitz continuity of ε-approximate solutions.] Proof. Approximation theory for (stochastic) optim. problems (via epi-convergence). In practice, highly robust solutions! Dealing with Uncertainty p.64/??

65 Perturbing the Probability Measure Optimal sol n Ew = 100, Stdev= 40 "loss": 0.15% Newsboy problem Objective function Ew = 90, Stdev=36 Cost=10, Revenue=15 Optimal sol n: Dealing with Uncertainty p.65/??