Robust Optimisatin Tutorial -...

Tutorial on Robust Optimisation -modelling aspects and

applicationsDr. Fabricio Oliveira – RMIT University

[email protected] OptGroup Meetings – 16/03/2016

Dr. Fabricio Oliveira ([email protected]) – OptGroup Meeting, 14/03/2016

2

Outline for today...

Tentative agenda

¡ Optimisation under Uncertainty

¡ Robust Optimisation (RO)

¡ Small example

¡ Applications

¡ Pre-location of supply for disaster relief using RO


3

Optimisation underUncertaintyTraditional optimisation methods are not able to consider dynamics in their evaluation

¡ Presumes fixed input data;

¡ For each input, one solution;

max

x

c

T

x

s.t.: Ax b

x 2 X

(c1, A1, b1, X1)

(c2, A2, b2, X2)

(cn, An, bn, Xn)

...

x1

x2

xn

...


4

Optimisation underUncertaintyIn the eyes of the decision maker:

¡ Each set of fixed input data means a scenario;¡ This is often call scenario analysis (what if? analysis, sensitivity analysis…

you name it.)

¡ For each scenario, one strategy (set of actions);


max

x

c

T

x

s.t.: Ax b

x 2 X

(c1, A1, b1, X1)

(c2, A2, b2, X2)

(cn, An, bn, Xn)

...

x1

x2

xn

...

5


¡ Do you see a problem here? ¡ What are we supposed to do with a set of strategies at hand? (if the first idea

was to get a single optimal one!)

¡ What good does it make to know the best strategy after the future unveils?


max

x

c

T

x

s.t.: Ax b

x 2 X

(c1, A1, b1, X1)

(c2, A2, b2, X2)

(cn, An, bn, Xn)

...

x1

x2

xn

...

6


¡ What we have:

¡ But what we want looks more like this:

max

x

c

T

x

s.t.: Ax b

x 2 X

(c1, A1, b1, X1)

(c2, A2, b2, X2)

(cn, An, bn, Xn)

...

x1

x2

xn

...

(c1, A1, b1, X1)

(c2, A2, b2, X2)

(cn, An, bn, Xn)

...

x

⇤max

x

c

T

x

s.t.:

˜

Ax ˜

b

x 2 ˜

X

7

Optimisation underUncertaintyCan you spot the difference?

¡ Decisions have to be made today!¡ We can’t wait to see the future and use our time machine to go back and make the

right choice (yet).¡ Your decision-maker needs a single strategy

¡ How can we bring this uncertainty about the future to our optimisation problems?

7

(c1, A1, b1, X1)

(c2, A2, b2, X2)

(cn, An, bn, Xn)

...

x

⇤

Time flow

max

x

c

T

x

s.t.:

˜

Ax ˜

b

x 2 ˜

X


8

Robust Optimisation

When feasibility matters most…¡ In some cases, making decisions under uncertainty is hard not because of

performance only, but because of feasibility¡ An infeasible strategy is a useless strategy. A DSS that provides useless

strategies is a useless DSS. What is the modeller in this case?

¡ There is a great deal of confusion in the literature concerning Robust Optimisation¡ Penalty-based methods (Mulvey et. al, 1995)

¡ Min-max regret approaches¡ Chance-constraints (Nemirovski & Shapiro, 2006)¡ Worst-case feasibility methods

¡ Today we are going to focus on this last stream...


9

Robust Optimisation

Worst-case feasibility ¡ Robust solutions means solutions that are immunised (to some extent) to

estimation errors and variability of input data.

¡ Robustness concept is strongly related with the definition of uncertainty sets¡ Probability support set (PSS, Ξ), per se!¡ What extent of the PSS are you trying to cover with your protection level (uncertainty

set U).¡ The tractability of the Robust Counterpart is affected by the geometry of such

uncertainty sets.

DETERM

INISTIC

UNCERTAIN

ROBUST

max

x

c

T

x

s.t.: Ax b

x 2 X

max

x

c

T

x

s.t.: max

⌘2U✓⌅A(⌘)x b

x 2 X

max

x

c

T

x

s.t.: A(⌘)x b

x 2 X

8⌘ 2 U ✓ ⌅


10

max

x

c

T

x

s.t.: a

ij

x

j

+max

⌘2U

8<

:X

j2Ji

⌘

ij

a

ij

x

j

9=

; b

i

, 8i

x

j

� 0, 8j

Robust Optimisation

Robust Counterparts¡ Each aij in Ji from A becomes a

random variable ãij with support set defined as a symmetric and limited interval given by[aij – âij, aij + âij ].

¡ Let us define ηij = (ãij - aij)/âij, that follows an unknown (also symmetric) probability distribution distribution and take values in [-1, 1] ;

¡ Our Robust Counterpart is, thus, a bi-level problem, and dependent of the geometry of the uncertainty set U

Worst case

Uncertainty set


11

Robust Optimisation

Typical Geometries (1/3)1. Box representation(Soyster, 1973)

¡ Maximum protection level that is possible.

¡ Considers that all parameters will take the worst possible value.

¡ Pros: simplicity;

¡ Cons: high level of conservatism (high deterioration of objective function value)

a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

max

⌘2U

8<

:X

j2Ji

⌘ij aijxj : |⌘j | 1, 8j 2 Ji

9=

;

=

X

j2Ji

aijxj

U = {⌘ | k⌘k1 1} = {|⌘j | 1, 8j 2 Ji}

◼- U◼- Ξ


12

Robust Optimisation

max

x

c

T

x

s.t.: a

ij

x

j

+max

⌘2U

8<

:X

j2Ji

⌘

ij

a

ij

x

j

9=

; b

i

, 8i

x

j

� 0, 8j

a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2◼- U◼- Ξ







13

Robust Optimisation






a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

max

x

c

T

x

s.t.: a

ij

x

j

+

X

j2Ji

a

ij

x

j

b

i

, 8i

x

j

� 0, 8j

◼- U◼- Ξ


14

Robust Optimisation

Typical Geometries (2/3)2. Ellipsoidal set (Ben-Tal & Nemirovski, 1998)

¡ Main idea: corners tend to be unlikely to happen.

¡ This lead to a circular-like uncertainty set.

¡ Diameters can be parametrically controlled.

¡ Pros: controlled conservatism;

¡ Cons: leads to computationally complex robust counterparts (SOCPs)

max

⌘2U

8<

:X

j2Ji

⌘ij aijxj :

X

j2Ji

⌘

2ij ⌦

2

9=

;

=max

⌘2U

8>><

>>:

vuuut

0

@X

j2Ji

⌘ij aijxj

1

A2

:

X

j2Ji

⌘

2ij ⌦

2

9>>=

>>;

=max

⌘2U

8>><

>>:

vuuut

0

@X

j2Ji

⌘ij

1

A2 0

@X

j2Ji

aijxj

1

A2

:

X

j2Ji

⌘

2ij ⌦

2

9>>=

>>;

=⌦

sX

j2Ji

a

2ijx

2j

U = {⌘ | k⌘k2 ⌦} =

8<

:X

j2Ji

⌘2j ⌦2

9=

;


15

Robust Optimisation






¡ Cons: leads to computationally complex robust counterparts (SOCPs) a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

max

x

c

T

x

s.t.: a

ij

x

j

+max

⌘2U

8<

:X

j2Ji

⌘

ij

a

ij

x

j

9=

; b

i

, 8i

x

j

� 0, 8j

◼- U◼- Ξ


16

Robust Optimisation







a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

max

x

c

T

x

s.t.: a

ij

x

j

+ ⌦

sX

j2Ji

a

2x

2ij

b

i

, 8i

x

j

� 0, 8j

◼- U◼- Ξ

Ω= 1


17

Robust Optimisation







a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

max

x

c

T

x

s.t.: a

ij

x

j

+ ⌦

sX

j2Ji

a

2x

2ij

b

i

, 8i

x

j

� 0, 8j

◼- U◼- Ξ

Ω= √2


18

Robust Optimisation

Typical Geometries (3/3)3. Polyhedral set

(Bertsimas & Sim, 2004)

¡ Main idea: control conservatism while preserving computational tractability.

¡ We control, by means of an uncertainty budget Γ, how many dimensions can assume their worst value.

¡ Pros: controlled conservatism, numerical simplicity;

¡ Cons: interpretation of parameter Γ.

max

⌘2U

8<

:X

j2Ji

⌘ij aijxj :

X

j2Ji

|⌘j | �

9=

;

max

z

X

j2Ji

aijxjzij

s.a:

X

j2Ji

zij �i

0 zij 1, 8j 2 Ji

U = {⌘ | k⌘k1 �} = {X

j2Ji

|⌘j | �}


19

Robust Optimisation







U = {⌘ | k⌘k1 �} = {X

j2Ji

|⌘j | �}

min⇡,p

X

j2Ji

pij + �i⇡i

s.a: ⇡i + pij � aijxj

pij � 0, 8j 2 Ji

⇡i � 0

max

z

X

j2Ji

aijxjzij

s.a:

X

j2Ji

zij �i

0 zij 1, 8j 2 Ji

20

Robust Optimisation






¡ Cons: interpretation of parameter Γ. a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

max

x

c

T

x

s.t.: a

ij

x

j

+max

⌘2U

8<

:X

j2Ji

⌘

ij

a

ij

x

j

9=

; b

i

, 8i

x

j

� 0, 8j

◼- U◼- Ξ

21

Robust Optimisation







max

x

c

T

x

s.t.: a

ij

x

j

+ ⇡

i

�

i

+

X

j2Ji

p

ij

b

i

, 8i

⇡

i

+ p

ij

� a

ij

y

j

, 8i, j 2 J

i

p

ij

� 0, 8i, j, ⇡

i

� 0, 8ix

j

� 0, 8j

a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

◼- U◼- Ξ

Γ = 1

22

Robust Optimisation







max

x

c

T

x

s.t.: a

ij

x

j

+ ⇡

i

�

i

+

X

j2Ji

p

ij

b

i

, 8i

⇡

i

+ p

ij

� a

ij

y

j

, 8i, j 2 J

i

p

ij

� 0, 8i, j, ⇡

i

� 0, 8ix

j

� 0, 8j

a1

a2

[a1 � a1] [a1 + a1]

[a2 � a2]

[a2 + a2]

a1

a2

◼- U◼- Ξ

Γ = 2

23

Robust Optimisation

Trading-off conservatism and performance

¡ The concept of robustness budget is not obvious for decision-makers.¡ Typical question: what am I gaining in exchange for deteriorating my

performance?

¡ It is imperative to assess how the consideration of robust aspects affects solution robustness.¡ Assessment of the trade-off obj. function value vs. violation probability

P

vio = Pr

8<

:X

j

aijxj +X

j2Ji

⌘j aijxj > bi

9=

;


24

Robust Optimisation

Violation probability¡ Being the violation probability defined as

¡ Some theoretical bounds for Pvio are available in literature.¡ Li et al. (2012, Thm. 3.1): assuming ηj symmetric iid, we have:

¡ For more complex stochastic processes: simulate!

P

vio = Pr

8<

:X

j

aijxj +X

j2Ji

⌘j aijxj > bi

9=

;

P vio e��2

2|Ji|

where:

� = = 1 (Box)

� = ⌦ (Ellipsoidal)

� = � (Polyhedral)


25

Robust Optimisation

Relevant developments¡ Static decision-making vs. flexible decision making

¡ The consideration of post-observation measures has been considered in what is called Adjustable Robust Counterparts (Ben-Tal et al., 2004)

¡ More flexible, less conservative, but only (computationally) tractable if such adjustable decisions are affine in regards to the uncertain data.

¡ Considering available information for defining set U

¡ A major criticism to RO approaches is that any information on stochastic process is completely disregarded

¡ Very recently, Bertsimas et al. (2013) discuss the idea of using data to define U sets in what they call Data-driven RO.


26

Summarising up to here…

Main take-aways

¡ Robust optimisation plays an important role when feasibility, and not performance, becomes the main issue.

¡ There are many different streams when it comes to robust optimisation. In this case in particular, notice that:¡ We have not considered any special assumption about the distribution

probability. (Can you live with that?)

¡ Even symmetry considerations are only necessary for theoretical probability bounds.

¡ Also a very active field of research. Modern streams:¡ Stochastic robust optimisation

¡ Multi-stage robust optimisation

¡ Distributionally robust counterparts.


27

Practical Example

Knapsack problem

¡ Instance: ¡ 50 items, cj = U[1, 200], qj = U[1, 100];

¡ Random (Gaussian) deviations of 50% for qj ;

max

x

X

j

c

j

x

j

s.a:

X

j

q

j

x

j

V

0 x

j

1, 8j


28

Practical Example

Knapsack problem – Robust counterparts

max

x

X

j

c

j

x

j

s.a:

X

j

q

j

x

j

+

X

j

q

j

x

j

V

0 x

j

1, 8j

max

x

X

j

c

j

x

j

s.a:

X

j

q

j

x

j

+ ⌦

sX

j

q

2j

x

2j

V

0 x

j

1, 8j

max

x

X

j

c

j

x

j

s.a:

X

j

q

j

x

j

+ ⇡�+

X

j

p

j

V

⇡ + p

j

� q

j

x

j

, 8j0 x

j

1, 8j

Soyster

Ben-Tal&Nemirovski

Bertsimas&Sim

29

Pre-positioning of emergency supply using robust optimisationReal case-study

¡ Mega disaster in Rio de Janeiro (2011)¡ Biggest climate disaster in Brazilian history, one of the

10 worst in recent global history, according to ONU.

¡ More than 850 casualties, 304k affected somehow;

¡ 12 cities;

¡ R$25 million only for disaster relief operations.

¡ Objective: analyse what would be the impacts of pre-positioning in terms of disaster relief.¡ Supplies analysed: food, water, medicines, mattresses,

clothes, and cleaning supplies.

30

Pre-positioning of emergency supply using robust optimisationPrepositioning and resilient networks

Resilience: (ecology) the ability of an ecosystem to return to its original state after being disturbed (British Dict.)

¡ Similar to traditional location problems, however with three-stage dynamics¡ Locating facilities è Operate è Operate under contingency

¡ Suitable for planning systems operation when those are subject to severe impacts.

CONTINGENCY

31

Pre-positioning of emergency supply using robust optimisationPrepositioning and resilient networks

How can we make a distribution network resilient?

¡ Resilience = well-planned inventories¡ The concept of pre-positioning comes to place. How do we prepare stocks, as

efficiently as possible, to overcome difficult times?

¡ In the context of disaster relief, “as efficiently as possible” means both cost and service-level effective.

CONTINGENCY

32

Pre-positioning of emergency supply using robust optimisationTraditional scenario-based models¡ To guarantee feasibility under

contingency, traditional formulations would rely on scenarios¡ Contingency = having K facilities

unavailable after the disaster happens.

¡ The Aic matrix is responsible for mapping who is/is not available under scenario c

¡ Must be feasible for all scenarios, then it is feasible for any contingency;

¡ Umbrella contingency: the one that is worse than all others;

¡ Difficulties:

minX

i

Gixi +X

i

Sisi

QD X

i

xi QD

LIixi si LIixi, 8iX

j

t

cij si, 8i, c

X

j

t

cij siA

ci , 8i, c

X

i

t

cij = Dj , 8j, c

si � 0, tcij � 0, 8i, j, cxi 2 {0, 1}, 8i

✓n

K

◆=

n!

K!(n�K)!


33

Pre-positioning of emergency supply using robust optimisationWhat we proposeNotice that¡ If it holds for all scenarios c, it also holds for the worst-case (our umbrella

contingency).

¡ To find our worst-case scenario, we can use:

¡ Leading us to the following robust counterpart bi-level formulation…

X

i

siAci �

X

j

Dj , 8c

dWC⇤ = mina

siai

s.t.:X

i

ai K

ai 2 {0, 1}, 8i

dWC⇤ �X

j

Dj


34

Pre-positioning of emergency supply using robust optimisationRobust bi-level counterpartsmin

X

i

Gixi +X

i

Sisi

QD X

i

xi QD

LIixi si LIixi, 8iX

j

t

cij si, 8i, c

X

i

t

cij = Dj , 8j, c

d

WC⇤ �X

j

Dj , 8c

d

WC⇤ = mina

siai

s.t.:X

i

a K

ai 2 {0, 1}si � 0, tcij � 0, 8i, j, cxi 2 {0, 1}, 8i

minX

i

Gixi +X

i

Sisi

QD X

i

xi QD

LIixi si LIixi, 8iX

j

t

cij si, 8i, c

X

i

t

cij = Dj , 8j, c

(n�K)y �X

i

zi �X

j

Dj , 8c

y � zi si, 8iy, zi � 0, 8isi � 0, tcij � 0, 8i, j, cxi 2 {0, 1}, 8i

Strong Duality

35

Pre-positioning of emergency supply using robust optimisationSome facts about preliminary validations…

¡ We have used the following reference:A two-stage stochastic programming framework for transportation planning in disaster response(G Barbarosoglu and Y Arda, Journal of the Operational Research Society (2004) 55, 43–53)

¡ Case study:

KScenario-based Robust

#VAR. #CONST. $ #VAR. #CONST. $

0 71 (10) 32 99.643.200 77 (10) 38 99.643.200

1 271 (10) 92 228.572.800 77 (10) 38 228.572.800 2 521 (10) 167 453.572.800 77 (10) 38 453.572.800

3 521 (10) 167 711.019.600 77 (10) 38 711.019.600

4 I I I I I I

5 I I I I I I


36

Cost withoutsecurity

Cost increase due to security (%)K

1 2 3 4 5 6 7

[5,10] 126,572,800 112,98 I I I I I I

[10,20] 239,715,200 46,66 124,91 I I I I I

[15,30] 347,770,000 32,21 72,71 132,54 I I I I

[20,40] 461,110,000 24,29 52,48 85,27 131,24 I I I

[25,50] 580,200,000 19,30 38,61 63,54 93,88 130,24 I I

[30,60] 693,540,000 16,15 32,30 51,04 72,08 101,49 131,48 I

Some facts about preliminary validations…

¡ Cost increase due security

Pre-positioning of emergency supply using robust optimisation

37

-

200,000,000

400,000,000

600,000,000

800,000,000

1,000,000,000

1,200,000,000

1,400,000,000

1,600,000,000

1,800,000,000

0 1 2 3 4 5 6 7

Obj

ecti

ve F

unct

ion

Val

ue (

$)

K

Objective Function Value vs. K

[5,10][10,20][15,30][20,40][25,50][30,60]


¡ Trade-off cost vs. resilience



38


0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Inve

ntor

y

Locations


¡ Inventory distribution behaviour


39

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Inve

ntor

y

Locations

Pre-positioning of emergency supply using robust optimisationSome facts about preliminary validations…

¡ Inventory distribution behaviour


40


Results – main comparison

Comparison of the Deterministic and Robust Total Cost (R$)

UDC Deterministic Model Robust Model Percentage Difference

PTP 25,902,629.29 5,718,423.00 77.92%

TRS 2,672,609.47 1,656,414.00 38.02%

NFB 10,923,936.94 3,717,124.00 65.97%

RJ 2,497,760.00 1,629,070.00 34.78%

PTP and TRS 26,094,513.74 7,411,748.00 71.60%

PTP and NFB 34,349,476.21 8,890,707.00 74.12%

PTP and RJ 25,902,629.27 5,876,854.00 77.31%

TRS and NFB 11,104,368.41 5,506,272.40 50.41%

TRS and RJ 2,672,609.47 1,656,414.00 38.02%

NFB and RJ 10,923,936.94 3,717,124.00 65.97%

Average Cost 15,304,446.97 4,578,015.04 70.09%


41

Pre-positioning of emergency supply using robust optimisationMain results from this research

§ Results has shown that planning taking into consideration network resilience considerations implicates in considerable reductions in the total operational cost in case of contingencies, not to mention highest service levels, which are, in this case, absolute priority.

§ Acknowledgment: Giovanna Goes (PUC-Rio) for implementing models and sharing ideas. Currently submitted to JORS.

Other successful applications that I’ve worked on ¡ Street, A., Oliveira, F. and Arroyo, J.M., 2011. Contingency-constrained unit commitment with

security criterion: A robust optimization approach. Power Systems, IEEE Transactions on, 26(3), pp.1581-1590.

¡ Regina, A., Oliveira, F. and Scavarda L.F., 2016. Tactical capacity planning in a real-world ETO industry case: a robust optimization approach. To appear in International Journal of Production Economics

3/16/16Dr. Fabricio Oliveira ([email protected]) – OptGroup Meeting, 14/03/2016

42

1. Bertsimas, D. and Sim, M., 2004. The price of robustness. Operations research, 52(1), pp.35-53.

2. Soyster, A.L., 1973. Technical note—convex programming with set-inclusive constraints and applications to inexact linear programming. Operations research, 21(5), pp.1154-1157.

3. Ben-Tal, A. and Nemirovski, A., 1998. Robust convex optimization. Mathematics ofoperations research, 23(4), pp.769-805.

4. Li, Z., Tang, Q. and Floudas, C.A., 2012. A comparative theoretical and computationalstudy on robust counterpart optimization: II. Probabilistic guarantees on constraintsatisfaction. Industrial & engineering chemistry research, 51(19), pp.6769-6788.

5. Ben-Tal, A., Goryashko, A., Guslitzer, E. and Nemirovski, A., 2004. Adjustable robustsolutions of uncertain linear programs. Mathematical Programming, 99(2), pp.351-376.

6. Bertsimas, D., Gupta, V. and Kallus, N., 2013. Data-driven robust optimization. arXivpreprint arXiv:1401.0212.

7. Mulvey, J.M., Vanderbei, R.J. and Zenios, S.A., 1995. Robust optimization of large-scalesystems. Operations research, 43(2), pp.264-281.

8. Nemirovski, A. and Shapiro, A., 2006. Scenario approximations of chance constraints. InProbabilistic and randomized methods for design under uncertainty (pp. 3-47). SpringerLondon.

References

Date post:	12-Jul-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Robust Optimisatin Tutorial -...

Documents