Roger J-B Wets

Dealing with Uncertainty

in Decision Making Models

Roger J-B Wets

[email protected]

University of California, Davis

Dealing with Uncertainty – p. 1/33

I. A product mix problem

Dealing with Uncertainty – p. 2/33

A formulation

A furniture manufacturer must choose xj ≥ 0, how many
dressers of type j = 1, . . . , 4 to manufacture so as to
maximize profit

4

cjxj = 12x1 + 25x2 + 21x3 + 40x4

j=1

The constraints: Madir, Izmir, Turkey, Spring 2010
tc1x1 + tc2x2 + tc3x3 + tc4x4 ≤ dc
tf1x1 + tf2x1 + tf3x1 + tf4x1 ≤ df

tcj (tfj) carpentry (finishing) man-hours: dresser type j
dc (df ) = total time available for carpentry (finishing)

Dealing with Uncertainty – p. 3/33

Product mix problem (2)

Solution via linear programming:

max c, x so that T x ≤ d, x ∈ IRn+.

With

     

T = tc1 tc2 tc3 tc4 = 4 9 7 10 dc = 6000
  ,  
 

tf1 tf2 tf3 tf4 1 1 3 40 df 4000

Optimal: xd = (4000/3, 0, 0, 200/3)
Value: $ 18,667.

Dealing with Uncertainty – p. 4/33

Product mix problem (3)

But . . . “reality” can’t be ignored!
tcj = tcj + ηcj, tfj = tfj + ηfj

entry possible values
dc + ζc: 5,873 5,967 6,033 6,127
df + ζf : 3,936 3,984 4,016 4,064

10 random variables, say, 4 possible values each

L = 1, 048, 576 possible pairs (T l, dl)

Dealing with Uncertainty – p. 5/33

Product mix problem (4)

What if j4=1(tcj + ηcj)xj > dc + ζc ? =⇒ overtime
With ξ = (η{·,·}, ζ{·}), recourse: (yc(ξ), yf (ξ)) @ cost (qc, qf ).

max c, x −p1 q, y1 −p2 q, y2 · · · − pL q, yL
−y2
s.t. T 1x −y1 ... ≤ d1
y2 ≥ 0, −yL
T 2x ≤ d2
... · · · yL ≥ 0. ...

T Lx ≤ dL

x ≥ 0, y1 ≥ 0,

Structured large scale l.p. (L ≈ 106)

Dealing with Uncertainty – p. 6/33

Product mix problem (5)

Define Ξ = ξ = (η, ζ) , pξ = prob [ξ = ξ]
Q(ξ, x) = max −q, y Tξx − y ≥ dξ, y ≥ 0
EQ(x) = E{Q(ξ, x)} = pξQ(ξ, x)

ξ∈Ξ

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. 7/33

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 ≈ 106
possibilities.

Dealing with Uncertainty – p. 8/33

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 ≈ 106
possibilities.

Recall: xd = (1, 333.33, 0, 0, 66.67)
expected “profit” relying on xd = $ 16,942.

Dealing with Uncertainty – p. 8/33

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 ≈ 106
possibilities.

Recall: xd = (1, 333.33, 0, 0, 66.67)
expected “profit” relying on xd = $ 16,942.

• xd is not close to optimal
• xd isn’t pointing in the right direction

Dealing with Uncertainty – p. 8/33

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. 9/33

II. Modeling, modeling &
modeling!

Dealing with Uncertainty – p. 10/33

Uncertain parameters

Deterministic Optimization problem:
min f0(x) so that x ∈ S ⊂ IRn

Dealing with Uncertainty – p. 11/33

Uncertain parameters

Deterministic Optimization problem:
min f0(x) so that x ∈ S ⊂ IRn

Uncertain parameters: ξ ∈ Ξ ⊂ IRN ,
min f0(ξ, x) so that x ∈ S(ξ) ⊂ IRn

Dealing with Uncertainty – p. 11/33

Uncertain parameters

Deterministic Optimization problem:
min f0(x) so that x ∈ S ⊂ IRn

Uncertain parameters: ξ ∈ Ξ ⊂ IRN ,
min f0(ξ, x) so that x ∈ S(ξ) ⊂ IRn

Wait-and-see solution ??
x(ξ) ∈ argmin f0(ξ, x) x ∈ S(ξ)

What’s needed: a here-and-now solution.

Dealing with Uncertainty – p. 11/33

The NewsVendor Problem

ξ ∈ Ξ ⊂ IR+ demand for a (perishable) good
e.g., plant capacity, overbooking, etc.

x ≥ 0 quantity ordered @ 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. 12/33

The “deterministic” approach

Pick ξˆ ∈ Ξ (guessing the future) and solve
min f0(ξˆ, x) so that x ∈ S(ξˆ) ⊂ IRn

Dealing with Uncertainty – p. 13/33

The “deterministic” approach

Pick ξˆ ∈ Ξ (guessing the future) and solve
min f0(ξˆ, x) so that x ∈ S(ξˆ) ⊂ IRn

NewsVendor: Ξ = [0, 150], pick ξˆ = 75,
max −cx + (c + r)y
x ≥ 0, 0 ≤ y ≤ min{x, ξˆ}

Solution: xo = yo = ξˆ, obj. value = rξˆ = 1125
But doesn’t tell much about “profit” if ξ = 75!

Dealing with Uncertainty – p. 13/33

Scenario Analysis

Pick ξ1, . . . , ξL (scenarios), and for each ξl find:
xl ∈ argmin f0(ξl, x) x ∈ S(ξl)

and “reconcile” the solutions to obtain xo.

Dealing with Uncertainty – p. 14/33

Scenario Analysis

Pick ξ1, . . . , ξL (scenarios), and for each ξl find:
xl ∈ argmin f0(ξl, x) x ∈ S(ξl)

and “reconcile” the solutions to obtain xo.
NewsVendor: pick ξ1 = 10, ξ2 = 20, . . . , ξ15 = 150,
(xl, yl) ∈ argmax{−cx + (c + r)y y ≤ min[ξl, x]}

x≥0,y≥0

Wait-and-see sol’ns: xl = ξl. “Reconciliation”?
No help in choosing xo the quantity to order.

Dealing with Uncertainty – p. 14/33

ξ: Estimated Density h

ξ log-normal: h(z) = (zτ √2π)−1 e− (ln z−θ)2
2τ 2

θ = 4.43, τ = 0.38; H(z) = z h(s) ds
0

from data, expert(s), all information available

0.014

0.012

0.01 lognormal density

Expectation = 90

Strd.Dev. = 36

0.008

0.006

0.004

0.002

0
0 20 40 60 80 100 120 140 160 180 200

might affect choice of ξˆ, scenarios: ξ1, . . .

Dealing with Uncertainty – p. 15/33

Maximize Expected Return

max − cx + E{(c + r)yξ}
so that x ≥ 0, 0 ≤ yξ ≤ min [ ξ, x ]

The equivalent deterministic program:
max −cx + EQ(x), EQ(x) = E{Q(ξ, x)}

x≥0

where Q(ξ, x) = (c + r)ξ if ξ ≤ x,
(c + r)x if ξ ≥ x

EQ(x) = (c + r) x∞

ξ H(dξ) + x H(dξ)

0x

Dealing with Uncertainty – p. 16/33

Optimal: Expected Profit

x∗ = H−1 r = H−1(0.6) = 99.2
c+r

for c = 10, r = 15.

1

0.9 Cumulative Distribution
lognormal

0.8 Expect.=90, Std.Dev.=36

0.7

0.6

0.5

0.4

0.3

H0.2

0.1

92.2

0
0 20 40 60 80 100 120 140 160 180 200

Dealing with Uncertainty – p. 17/33

NewsVendor’s Objective

1100

1050

1000

950

900

850 Ew = 90, Stdev=36
Cost=10, Revenue=15

Opt. sol’n: 99.2, E[profit] = 1,084

800 Sol’n guess: 75, E[profit] = 1,000

750

700

650

600
40 60 80 100 120 140 160

Dealing with Uncertainty – p. 18/33

. . . but is maximum expected
return the “real” objective?

Dealing with Uncertainty – p. 19/33

The Returns’ Densities

x 10−3
8

7

6 Return Densities

5

x=72.2

4

3

x=92.2

2

1 x=152.2 x=112.2
0 x=132.2
−1000 −500
0 500 1000 1500 2000 2500 3000

Dealing with Uncertainty – p. 20/33

Choosing the Returns’ Distribution

1

0.9 x=112.2
x=92.2
Distribution
0.8 Functions

0.7

x=132.2

0.6

0.5

0.4

0.3 x=72.2

x=152.2

0.2

0.1

0

−1000 −500 0 500 1000 1500 2000 2500

Dealing with Uncertainty – p. 21/33

Decision Criteria

Reducing the choice of a distribution function
to the choice of a “number”

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. 22/33

Maximizing Expected Utility

“generic” stochastic optimization problem:
max E{f0(ξ, x)} such that fi(ξ, x) ≤ 0, i = 1, . . . , m,

Risk-averse or risk-seeking =⇒ utility function
von Neuman-Morgenstern: under “rationality”
(axiomatics) there exists a utility function u such
that x¯ ∈ argmax E{u f0(ξ, x) } (subject to the
constraints) identifies the preferred return’s
distribution

Modeling hurdle: no blueprint for u’s design!

Dealing with Uncertainty – p. 23/33

Robust Optimization

“generic” optimization problem: max γ
so that γ − f0(ξ, x) ≤ 0,
fi(ξ, x) ≤ 0, i = 1, . . . , m,

“robust” counterpart: max γ
so that γ − f0(ξ, x) ≤ 0, ∀ ξ ∈ U ⊂ Ξ
fi(ξ, x) ≤ 0, i = 1, . . . , m, ∀ ξ ∈ U ,

Challenges:

formulate a computationally tractable robust
counterpart
specify reasonable uncertainty for set U

Dealing with Uncertainty – p. 24/33

Reliability: Chance Constraints

Satisfy constraints with probability α ∈ (0, 1]
min f0(x) so that prob. [ x ∈ S(ξ) ] ≥ α

Variant:
min f0(x)

so that prob. [ fi(ξ, x) ≤ 0 ] ≥ αi, i ∈ I
αi dictated by

contractual obligations
company policy, guess, etc.

Dealing with Uncertainty – p. 25/33

NewsVendor: Chance C. Model

max −cx + (c + r)y so that x ≥ 0, y ∈ [H−1(α), x]

1

0.9

0.8

0.7

0.6 Cumulative Distribution
lognormal

0.5 Expect.=90, Std.Dev.=36

0.4

0.3

H0.2

0.1

0 108.1 135.87

0 20 40 60 80 100 120 140 160 180 200

Infeasible if x < H−1(α). Profit? later
When α = 0.9 : xˆ = 135.9; α = 0.75 : xˆ = 108.1.

Dealing with Uncertainty – p. 26/33

VaR: Value-at-Risk

Let F (s; x) = prob [ −cx + Q(ξ, x) ≤ s ]
Value-at-Risk (VaR) for α ∈ (0, 1):
VaR(α; x) = F −1(α; x) (= sup{v v ∈ F −1(α; x)})
Objective: find x that maximizes VaR(α; x)

Dealing with Uncertainty – p. 27/33

VaR: Value-at-Risk

Let F (s; x) = prob [ −cx + Q(ξ, x) ≤ s ]
Value-at-Risk (VaR) for α ∈ (0, 1):
VaR(α; x) = F −1(α; x) (= sup{v v ∈ F −1(α; x)})
Objective: find x that maximizes VaR(α; x)
Challenge: x → VaR(α; x) isn’t concave.
Heuristic: F is N (µ(x), σ(x)2) and

VaR(α; x) = N −1(α; µ(x), σ(x)2)

Dealing with Uncertainty – p. 27/33

VaR: NewsVendor Problem

0.2

0.18 x=112.2
x=92.2
VaR (Value @ Risk)

0.16

0.14

x=132.2

0.12

0.1

10%

0.08

0.06 x=72.2

x=152.2

0.04

0.02

0

−400 −200 0 200 400 600

Dealing with Uncertainty – p. 28/33

Stochastic Dominance

1

0.9 x=112.2
x=92.2
Distribution
0.8 Functions

0.7

x=132.2

0.6

0.5

0.4

0.3 x=72.2

x=152.2

0.2

0.1

0

−1000 −500 0 500 1000 1500 2000 2500

Stochastic Dominance: Dx(s) ≤ Dxˆ(s), ∀ s
=⇒ probability of the return to be ≤ s

always smaller when choosing x rather than xˆ

unfortunately unusual

‘

Dealing with Uncertainty – p. 29/33

Second order Stochastic Dominance

s

D2(s) = D(ξ) dξ = E{(s − ξ)+}

−∞

D2: the expected shortfall function

3500 Expected ShortFall x = 72.2
3000 x = 112.2
2500
2000 x = 92.2
1500
1000 152.2 132.2

500 0 500 1000 1500 2000 2500
0

−500

Dealing with Uncertainty – p. 30/33

Stochastic Dominance Constraint

NewsVendor problem
max rx, x ≥ 0

such that Dx2(s) ≤ G2(s), s ∈ [α, β]
given a “desirable” distribution function G
Dx: distribution of actual return, decision x

rx when ξ ≤ x and (c + r)ξ − cx when ξ < x

leads to a semi-infinite optimization problem
used in portfolio optimization, for example

Dealing with Uncertainty – p. 31/33

Perturbing the Probability Measure

900

850

800

102.4 Optimal sol’n
750 Ew = 100, Stdev= 40

"loss": 0.15%
Newsboy problem
700 Objective function
Ew = 90, Stdev=36
650 Cost=10, Revenue=15
Optimal sol’n: 99.2

600

550
40 60 80 100 120 140 160

stress testing via distribution contamination

Dealing with Uncertainty – p. 32/33

A few references

1. A. Shapiro, D. Dentcheva & A. Ruszczynski, Lectures
on Stochastic Programming, MPS-SIAM Series. 2009.

2. S. Choi, A. Ruszczynski & Y. Zhao, A multi-product
risk-averse newsvendor with law invariant coherent
measures of risk, optimization-online, 2009

3. G. Pflug & W. Römisch, Modeling, Measuring and
Managing Risk, World Scientific, 2007

4. J. Dupacˇová, Stress testing via contamination. Coping
with Uncertainty: Modeling and Policy Issues, LNEMS
581, Springer. 2006. pp. 29-46.

5. R. Wets, An Optimization Primer, 2011 “soon”

Dealing with Uncertainty – p. 33/33