bounds in mulstistage stochastic programmingdinamico2.unibg.it/icsp2013/doc/ps/icsp2013...

Basic Facts Lower bounds Upper bounds Numerical Results Conclusions

Bounds in Mulstistage Stochastic Programming

Francesca Maggioni(∗)Dept. of Management, Economics and Quantitative Methods,

University of Bergamo (ITALY)

(∗) joint work withElisabetta Allevi & Marida Bertocchi

XIII International Conference on Stochastic ProgrammingBergamo, 8–12 July 2013

F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 1 / 23


Introduction

The Overall Goal

Evaluation if it is worth the additional computations for the stochastic programversus simplified approaches.

Deeper understanding of expected value solution and relation to the stochastic one.

Useful because it could help to predict how the stochastic model will perform whenis not solvable.

If the gap between lower and upper bounds for the objective value is acceptably small,one may even stop and never fully optimize.

The general idea behind construction of bounds we adopt, is that for every optimizationproblem of minimization type:

lower bound to the optimal solution can be found by relaxation of constraints;

upper bound to the optimal solution can be found by inserting feasible solutions.



Introduction

Information - Quality - Rolling Horizon

We introduce three classes of measures:

1 Measures of Information in MLSP (MSPEV, MEPEV, MEVRS);

2 Measures of the Quality of EV Solution in MLSP (MLUSS, MLUDS);

3 Rolling Horizon Measures in MLSP.

QUESTIONS

1 What is the value of different level of information in multistage SP?2 Does the stochastic optimal solution inherit properties from the deterministic

solution or are they totally different?What is the reason of the badness of the deterministic solution in multistage SP?Can we update the stochastic solution from the deterministic one in multistage SP?

3 What is the advantage of a Rolling Horizon Procedure?



Basic Facts and Notations


Let us consider a random process

ξH−1 = (ξ1, ξ2, . . . , ξH−1) ,

where ξt is defined on a probability space (Ξt,A t, p), such that:Ξ = ΠtΞt is the support;p is a given probability distribution;A t (with A t−1 ⊆ A t) is a sigma-algebra describing the evolving information setgenerated by the process over time;

EξH−1 : mathematical expectation with respect to ξH−1;

x = (x1, x2, . . . , xH): decision vector with decision xt at time t is At−1−

measurable and depending by the history up to t:

decision(x1) → observation(ξ1) → decision(x2) → observation(ξ2) → . . .

. . . → decision(xt−1) → observation(ξt−1) → decision(xt).




Multistage Linear Stochastic Program (MLSP )

RP := minx

EξH−1z(x, ξ

H−1)

:= minx1

c1x1+Eξ1

[

minx2

c2x2(ξ1)+ Eξ2

[

· · ·+ EξH−1

[

minxH

cHxH(

ξH−1

)

]]]

Ax1 = h1 ,

T 1(ξ1)x1 +W 2(ξ1)x2(ξ1) = h2(ξ1) ,

...

TH−1(ξH−1)xH−1(ξH−1) +WH(ξH−1)xH(ξH−1) = hH(ξH−1) ,

x1≥ 0 ; xt(ξt−1) ≥ 0 , t = 2, . . . , H;




Scenario Tree

ξ1, . . . , ξS : possible realizations (or scenarios) of future outcomes ξH−1;

πs := probability of scenario s;

πa(j),j := conditional probability of the random process in node j given its historyup to the ancestor node a(j);

every node in the tree carries an optimal decision problem.




Scenario Tree

ξ1, . . . , ξS : possible realizations (or scenarios) of future outcomes ξH−1;

πs := probability of scenario s;

πa(j),j := conditional probability of the random process in node j given its historyup to the ancestor node a(j);

every node in the tree carries an optimal decision problem;



WS - EV

Multistage Wait-and-see Problem (WS)

WS := EξH−1 min

x1(ξH−1),...,xH (ξH−1

)

c1x1(ξH−1) +. . .+ cHxH(ξH−1)

s.t. Ax1 = h1 ,

T 1(ξ1)x1(ξH−1) +W 2(ξ1)x2(ξH−1) = h2(ξ1) ,

...

TH−1(ξH−1)xH−1(ξH−1) +WH(ξH−1)xH(ξH−1)=hH(ξH−1) ,

x1≥ 0 ; xt(ξH−1) ≥ 0 , t = 2, . . . , H .



WS - EV

The Expected Value Problem EV

The Expected Value problem EV (with ξ=(Eξ1, Eξ2, . . . , EξH−1)):

EV := minx

z(x, ξ)

:= minx1,...,xH

c1x1 + · · ·+ cHxH

s.t. Ax1 = h1 ,

T 1(ξ1)x1 +W 2(ξ1)x2 = h2(ξ1) ,

...

TH−1(ξH−1)xH−1 +WH(ξH−1)xH = hH(ξH−1) ,

xt≥ 0 , t = 1, . . . , H.

Theorem WS ≤ RP ≤ EEV , Madansky (1960)

EEV : the solution of RP model, having 1st−stage variables fixed from EV .F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 9 / 23


MSPEV

SPEV - Sum of Pairs Expected Values (2-stage case)

Let the PAIRS subproblem of scenarios ξr and ξk (k = 1, . . . , S) be defined as follows:

min zP (x, ξr, ξk) := min(c1x1 + c2πrx2(ξr) + (1− πr)c

2x2(ξk))

s.t. Ax1 = h1 ,

Trx1 +Wrx

2(ξr) = ξr ,

Tkx1 +Wkx

2(ξk) = ξk ,

x1≥ 0 ; x2(ξr) ≥ 0; x2(ξk) ≥ 0 .

The Sum of Pairs Expected Values (SPEV ) (Birge & Louveaux (1997)) is defined as:

SPEV :=1

1− πr

S∑

k=1,k 6=r

πk min zP (x, ξr, ξk) .



MSPEV

MSPEV - Multistage Sum of Pairs Expected Values

We fix an auxiliary scenario a with the following characteristics:1 If H is the first stage where scenarios a and k branch then:

πjt,jt+1 =

{

πr if t = H

1 if t 6= H

2 πa = πr, since πa = 1 · 1 · 1 · . . . · πr · 1 · . . . · 1.F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 11 / 23


MSPEV

We solve the following pair subproblems defined as:

min zP (x, ξa, ξk) :=min(c1x1+

H−1∑

t=2

ctxta(ξa) +

H∑

t=H

[

πactxt

a(ξa) + (1− πa)ctxt

k(ξk)]

)

s.t. Ax1 = h1 ,

T t−1a xt−1

a (ξa) +W tax

ta(ξa) = ht

a(ξa) ,

T t−1k xt−1

k (ξk) +W tkx

tk(ξk) = ht

k(ξk) ,

x1≥ 0 ; xt

a(ξa) ≥ 0 , xtk(ξk) ≥ 0 , t = 2, . . . , H.

The Multistage Sum of Pairs Expected Values (MSPEV ) is defined as follows:

MSPEV :=1

1− πa

nH∑

k=1,k 6=r

πk min zP (x, ξa, ξk)

Proposition (Maggioni F. , Allevi E. & Bertocchi M. (submitted to JOTA, 2012))

If ξr /∈ Ξ, then MSPEV = WS.

If ξr ∈ Ξ then WS ≤ MSPEV ≤ RP.



MSTEV - MSQEV . . .

Lower bounds by changing the probability measure

The method may be refined by solving sub-problems with 3, 4 . . . , etc. scenarios at atime.

Proposition (Maggioni F. & Pflug G. (in preparation))

WS ≤ MSPEV ≤ MSTEV ≤ MSQEV ≤ · · · ≤ RP .

Notice that the higher is the reference index, the fewer problems have to be solved butwith an increasing number of scenarios.

Figure : (a) two-scenarios subtree (b) three scenarios subtree (c) four scenarios subtree.



MEPEV - MEV RS

Upper bounds by inserting feasible solutions

Let x1k be optimal first stage solution to the pair subproblems of ξa and ξk,

k = 1, . . . , nH , k 6= r. The Multistage Expectation of Pairs Expected Value is:

MEPEV := mink=1,...,nH∪{r}

(EξH−1 min

x(2,H)

z(x1k,x

(2,H), ξH−1)) .

Let xtr be the optimal solution until stage t of the deterministic problem under

scenario r; the Multistage Expected Value of the Reference Scenario is:

MEV RSt := EξH−1 min

x(t+1,H)

z(xtr,x

(t+1,H), ξH−1) ∀t = 1, . . . , H − 1


RP ≤ MEPEV ≤ MEV RS1 ≤ . . . ≤ MEV RSH−1



MLUSS - MLUDS

Measures of the Quality of Deterministic Solution in MLSP

V SSt: Classical evaluation of the expected value solution xt = (x1, . . . , xt).

Calculate EEV t := Eξ

(

z(

xt(

ξ)

, ξ))

and compare it with RP using

V SSt := EEV t−RP , [ Escudero et al. (2007)]

MLUSSt: Fix at zero all the variables which are at zero in the EV solution(skeleton) until stage t and then solve the SP.

MLUSSt := MESSV t−RP , t = 1, . . . , H − 1 .

MLUDSt: Consider the EV solution xt as a starting point (deterministic plus)

to the SP. Test if it is upgradeable:

MLUDSt := MEIV t−RP , t = 1, . . . , H − 1 .

[For the two-stage case see Maggioni & Wallace AOR (2012).]



MLUSS - MLUDS

Properties of MLUSS-MLUDS


MLUSSt+1≥ MLUSSt, ∀ t = 1, . . . , H − 2 ,

MLUDSt+1≥ MLUDSt, ∀ t = 1, . . . , H − 2 ,

EEV t≥ MEIV t

≥ RP, ∀ t = 1, . . . , H − 1 ,

EEV t− EV ≥ V SSt

≥ MLUDSt≥ 0, ∀ t = 1, . . . , H − 1 .



Rolling Horizon Measures in MLSP

Rolling Horizon Measures in Multistage Linear Stochastic Programming



Rolling Horizon Measures in MLSP

Rolling Horizon Measures in Multistage Stochastic Programming

The Rolling Horizon Value of the Reference Scenario can be expressed as:

RHV RS := EξH−1 min

xHz(xH−1

r,ev , xH , ξH−1) ,

where the solution xH−1r,ev is obtained with a rolling horizon procedure as described before.

The Rolling Horizon Value of Stochastic Solution with Respect to Reference Scenario is:

RHV SS := RHV RS −RP .

Advantages of the Rolling Time Horizon Procedure:

it updates the estimations of the solution at each stage taking into account thearrival of new information.

the computational complexity decreases since the number of scenarios is reduced byconsidering the stochasticity only one stage at a time.



% deviation from RP for increasing cost values

[TP,MEIV 2] ≈ 13.64% best range with respect to RP ;

[EV,EEV 3] ≈ 45.53% worst range with respect to RP ;



% deviation from RP versus complexity of calculation



% deviation from RP versus complexity of calculation([EV,RHESSV ] = [0, 0.1] CPU s.)



Conclusions - Future Works

We have analyzed lower and upper bounds for linear multistage programs by using:

Different Levels of Information: MSPEV,MEPEV,MEV RS;

Quality of Deterministic Solution: MESSV,MEIV ;

Rolling Horizon Measures: RHEEV,RHV RS,RHESSV,RHEIV .

Differences among the values allow to evaluate:

if it is worth the additional computations for the stochastic program versus thesimplified approaches proposed.

give insight of what is potentially wrong with a solution coming from thedeterministic/approximated method considered.

Future worksExtend the analysis to:

non linear case (in progress)

mixed-integer linear case;

real sized problem.



References

1 Birge: The value of the stochastic solution in stochastic linear programs with fixedrecourse, Math. Prog., 24, 314–325 (1982).

2 Birge, Louveaux : Introduction to stochastic programming, Springer-Verlag, NewYork, (1997).

3 Escudero, Garın, Merino, Perez: The value of the stochastic solution in multistageproblems, Top, 15, 48–64 (2007).

4 Madansky : Inequalities for stochastic linear programming problems. Manag. Sci.,6, 197–204 (1960).

5 Maggioni, Wallace: Analyzing the quality of the expected value solution instochastic programming. AOR 200, 37-54 (2012).

6 Maggioni, Allevi, Bertocchi : Bounds in multistage linear stochastic programming.submitted to JOTA (2012) (also on SPEPS, 2 (2012)).

7 Maggioni, Kaut, Bertazzi : Stochastic optimization models for a single sinktransportation problem. Comput. Manag. Sci. 6, 251–267 (2009).


bounds in mulstistage stochastic programmingdinamico2.unibg.it/icsp2013/doc/ps/icsp2013...

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