Bounds in Mulstistage Stochastic Programmingdinamico2.unibg.it/icsp2013/doc/ps/ICSP2013...
Transcript of 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
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 2 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 2 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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?
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 3 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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?
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 3 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
Basic Facts and Notations
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).
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 4 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
Basic Facts and Notations
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;
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 5 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
Basic Facts and Notations
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 6 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
Basic Facts and Notations
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;
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 7 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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 .
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 8 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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) .
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 10 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 12 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 13 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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
Proposition (Maggioni F. , Allevi E. & Bertocchi M. (submitted to JOTA, 2012))
RP ≤ MEPEV ≤ MEV RS1 ≤ . . . ≤ MEV RSH−1
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 14 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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
Proposition (Maggioni F. , Allevi E. & Bertocchi M. (submitted to JOTA, 2012))
RP ≤ MEPEV ≤ MEV RS1 ≤ . . . ≤ MEV RSH−1
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 14 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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).]
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 15 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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).]
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 15 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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).]
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 15 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
MLUSS - MLUDS
Properties of MLUSS-MLUDS
Proposition (Maggioni F. , Allevi E. & Bertocchi M. (submitted to JOTA, 2012))
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 .
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 16 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
Rolling Horizon Measures in MLSP
Rolling Horizon Measures in Multistage Linear Stochastic Programming
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 17 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 18 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
% 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 ;
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 19 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
% deviation from RP versus complexity of calculation
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 20 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
% deviation from RP versus complexity of calculation([EV,RHESSV ] = [0, 0.1] CPU s.)
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 21 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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.
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 22 / 23
Basic Facts Lower bounds Upper bounds Numerical Results Conclusions
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).
F. Maggioni, E. Allevi, M. Bertocchi Bounds in MSP ICSP2013 Bergamo 23 / 23