9512.net
甜梦文库
当前位置:首页 >> 电力/水利 >>

A note on scenario reduction for two-stage stochastic programs两阶段随机规划注意减少场景


Operations Research Letters 35 (2007) 731 – 738

Operations Research Letters
www.elsevier.com/locate/orl

A note on scenario reduction for two-stage stochastic programs
Holger Heitsch, Werner R?misch?
Institute of Mathematics, Humboldt-University Berlin, 10099 Berlin, Germany Received 30 May 2006; accepted 7 December 2006 Available online 11 January 2007

Abstract We extend earlier work on scenario reduction by relying directly on Fortet–Mourier metrics instead of using upper bounds given in terms of mass transportation problems. The importance of Fortet–Mourier metrics for quantitative stability of twostage models is reviewed and some numerical results are also provided. ? 2007 Elsevier B.V. All rights reserved.
Keywords: Stochastic programming; Mass transportation; Probability metric; Two-stage; Scenario reduction

1. Introduction In the papers [2,5] a stability-based methodology is developed for reducing the set of scenarios in convex stochastic programming models. Such a reduction may be desirable in some situations when the underlying optimization models already happen to be large scale and the incorporation of a large number of scenarios might lead to huge programs and, hence, to high computation times. The idea of the scenario reduction framework in [2,5] is to compute the (nearly) best approximation of the underlying discrete probability distribution by a measure with smaller support in terms of a probability metric which is associated to the stochastic program in a natural way. Such “natural” (or canonical) metrics for probability measures are known

? Corresponding author.

E-mail address: romisch@math.hu-berlin.de (W. R?misch). 0167-6377/$ - see front matter ? 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2006.12.008

for (linear) two-stage stochastic programs: the rth order Fortet–Mourier metrics, where the choice of r 1 depends on the speci?c structure of the programs (see Section 3 and [10,11]). However, the strategies for scenario reduction developed in [2,5] are not based on Fortet–Mourier metrics, but on their upper bounds in form of certain mass transportation problems which enjoy speci?c properties and representations. In the present note we remove this drawback and develop scenario reduction algorithms that are rigorously based on Fortet–Mourier metrics. The key step in this direction is that we do no longer use the (generalized) distances c for scenarios as in [2,5], but so-called reduced distances (or costs) c ? which, indeed, are distances in the ?nite-dimensional scenario space and represent in?ma of certain optimization problems. Our paper is organized as follows. In Section 2 we discuss distances of (multivariate) probability measures that are based on mass transportation problems.

732

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

We review some of their topological properties, duality results and representations that are needed in the sequel. Section 3 reviews stability properties of multiperiod two-stage stochastic programs with respect to the distances introduced in the previous section. In Section 4 we extend our earlier theory and heuristic algorithms for optimal scenario reduction to the relevant metrics. Finally, we present some numerical experience for the new forward algorithm of scenario reduction. It is tested on realistic data from electricity portfolio management.

If P and Q are discrete probability measures having ?nitely many scenarios i (with probabilities pi ), i = 1, . . . , N , and ? j =: N +j (with probabilities qj ), j = 1, . . . , M , respectively, we obtain ? ?N M ? ? c (P , Q) = inf ij c( i , j ) : ij 0, ? i =1 j =1 ? N M ? , = q = p , i j ij ij ?
j =1 i =1

2. Distances of probability distributions A variety of distances of multivariate probability distributions are related to mass transportation problems. If P and Q belong to the set P( ) of all (Borel) probability measures on a closed subset of Rs and c : × → R is a nonnegative, symmetric and continuous cost function for transporting P to Q, the minimal transportation cost is given by ? c (P , Q) := inf
×

i.e. ? c (P , Q) is the optimal value of a linear transportation problem, and ? ?N +M ? c( i , j ) ij : ij 0, c (P , Q) = inf ? i,j =1 ? N +M N +M ? ? = P ( { } ) ? Q( { } ) , i i ij ji ?
j =1 j =1

c( , ? ) (d , d ? ) :
1

∈ P( × ),

= P,

2

=Q ,

(1)

i.e. c (P , Q) is the optimal value of a minimum cost ?ow problem. Hence, for discrete probability measures with ?nite support both functionals are computationally accessible. The most important cost functions in the context of the present paper are cr ( , ? ) := max{1, · ?? ? 0 r ?1 , ? ? ( , ? ∈ ),
0 r ?1

?

where 1 and 2 denote the projections onto the ?rst and second components, respectively. A minimizer ? ∈ P( × ) of (1) is called optimal transportation plan and ? c de?ned on P( ) × P( ) is a so-called Monge–Kantorovich functional. A variant of (1) is the mass transshipment problem given by
? c (P , Q)

} (3)

for some r 1 and 0 ∈ . In this case, both func? tionals ? c (P , Q) and c (P , Q) are ?nite if P and Q belong to the set Pr ( ) of all probability measures having absolute moments of order r. We will use the ? ? notation ? r and r for ? cr and cr , respectively. The Kantorovich–Rubinstein functional r is a metric on Pr ( ), called the Fortet–Mourier metric of order r [3]. It satis?es the estimate
r ?

:= inf

×

c( , ? ) (d , d ? ) :
1

∈ M( × ),

?

2

=P ?Q ,

(2)

where M( × ) denotes the set of all ?nite measures ? on × and c de?ned on P( ) × P( ) is called Kantorovich–Rubinstein functional. We refer to [7,9] for a comprehensive presentation of theory and applications of mass transportation problems.

P (d ) ?

r

Q(d )

r

?

r (P , Q)

(4)

for all P , Q ∈ Pr ( ) [7, Theorem 6.2.5]. Moreover, convergence of a sequence (Pn ) of probability mea? sures in the metric space (Pr ( ), r ) to some limit P

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

733

is equivalent to ( ? r (Pn , P )) tending to 0 as n → ∞ and to the weak convergence of (Pn ) to P and the convergence of rth order absolute moments of Pn to those of P [7, Theorems 6.3.1]. The following dual representation and characterization are of special interest here. The corresponding results are derived in [7, Theorem 5.3.2] and [9, Section 4.3]. Proposition 2.1. For all probability measures P , Q ∈ ? Pr ( ) the Kantorovich–Rubinstein functional r admits the dual representation
? r (P , Q) =

3. A review of stability for two-stage models If the second stage of a linear stochastic program with recourse models a (stochastic) dynamical decision process, as is the case in a variety of applications, the two-stage problem takes on the form min f0 ( , x)P (d ) : x ∈ X , (8)

a closed where X is a polyhedral subset of Rm , subset of Rs , P is a Borel probability measure on and the integrand f0 is of the form f0 ( , x) = c, x + inf ? ? ?
j =1

f ∈Fr

sup

f ( )P (d ) ?

f ( )Q(d ) , (5)

qj ( ), yj :

where Fr is the class of functions f : → R satisfying f ( ) ? f ( ? ) cr ( , ? ), ? , ? ∈ . Proposition 2.2. Let be compact and r 1. Then the Kantorovich–Rubinstein functional with cost function cr coincides with a Monge–Kantorovich functional with reduced cost c ?r . More precisely, it holds
? r (P , Q) = ? c ?r (P , Q) =

Wj yj = hj ( ) ? Tj ( )yj ?1 , ? ? yj ∈ Yj , j = 1, . . . , , ?

(9)

?c ?r (P , Q)

? r (P , Q), (6) × is given by

where the real-valued function c ?r on c ?r ( , ? ) := inf
n?1

cr ( i ,
i =1

i +1 )

: n ∈ N, (7) cr and

i

∈ ,

1

= ,

n

=? . with c ?r

The function c ?r is a metric on coincides with cr if r = 1.

The compactness assumption in Proposition 2.2 is not restrictive here since it will be used for probability measures with ?nite support. The importance of Proposition 2.2 in the present context is due to the fact that Kantorovich–Rubinstein functionals are appropriate for stability issues (see Section 3), but Monge–Kantorovich functionals, i.e., mass transportation problems, allow for special representations (see Section 4).

with c ∈ Rm , polyhedral subsets Yj of Rmj , recourse costs qj ( ) ∈ Rmj , right-hand sides hj ( ) ∈ Rrj , technology matrices Tj ( ) ∈ Rrj ×mj ?1 and recourse matrices Wj ∈ Rrj ×mj for j = 1, . . . , and some ∈ N; the vectors qj (·), hj (·) and the matrices Tj (·) are (potentially) stochastic and af?ne functions of . Then the second stage program has separable block structure and the recourse variable y has the form y = (y1 , . . . , y ). When rewriting the model as a two-stage stochastic programming model with recourse decision y = (y1 , . . . , y ), the recourse matrix has separable block structure with W1 , . . . , W and the matrices T1 ( ), . . . , T ( ) appearing as its main and lower diagonal blocks. The following stability result for optimal values v(P ) and -approximate ?rst-stage solution sets S (P ) of (8), (9) is derived in the recent paper [11]. Proposition 3.1. Let P ∈ P +1 ( ) and the solution set S(P ) of (8), (9) be nonempty and bounded. Assume that hj ( ) ? Tj ( )x ∈ Wj (Yj ) holds for each j = 1, . . . , and all pairs ( , x) ∈ × X (relatively complete recourse). Moreover, assume ker (Wj )∩Yj∞ ={0} for j = 1, . . . , ? 1, where Yj∞ denotes the (polyhedral) horizon cone to Yj .

734

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

Then there exist constants L > 0 and ? > 0 such that for any ∈ (0, ?) the estimates |v(P ) ? v(Q)| L d∞ (S (P ), S (Q))
? +1 (P , Q), L? +1 (P , Q), ?

hold whenever Q ∈ P +1 ( ) and +1 (P , Q) < . Here, d∞ denotes the Pompeiu–Hausdorff distance on compact subsets of Rm . We note that the horizon cone Yj∞ contains all elements xj ∈ Rmj such that x + xj ∈ Yj for all x ∈ Yj and ∈ R+ . The condition ker (Wj ) ∩ Yj∞ = {0} implies the boundedness of the constraint set {yj ∈ Yj : Wj yj = uj } for all right-hand sides uj . The case = 1 corresponds to the situation of linear two-stage models with ?xed recourse (see [10, Theorem 24]). Hence, together with the results in [8,10], the number r should be selected as r = 1 if either costs or right-hand sides in (8), (9) are random, r = 2 if only costs and right-hand sides are random in (8), (9) and r = + 1 if, in addition, all technology matrices are random in (8) and (9). Since the (approximate) optimal second stage decisions are compact with respect to the weak topology in some space Lr ( , F, P; Rm ) with m = j =1 mj , some probability space ( , F, P) and some r related to r [6], a choice of r larger than suggested may lead to stronger properties of the second stage decisions. 4. Optimal scenario reduction Let P be a discrete probability distribution with scenarios i and probabilities pi , i = 1, . . . , n. If the number n of scenarios is large, one might wish to delete scenarios of P in a best possible way, i.e., such that the original problem or, more precisely, its optimal value admits minimal changes. To make this requirement precise, we denote by QJ a discrete distribution whose support consists of a subset of scenarios j , j ∈ {1, . . . , n}\J , of P having probabilities qj , j∈ / J . Hence, it is of interest to determine a subset J of {1, . . . , n} and probabilities qj , j ∈ / J , such that the distance |v(P ) ? v(QJ )| of optimal values is minimal with respect to all subsets of given cardinality. But, in general, this distance is dif?cult to handle. According to Proposition 3.1 we know, however, that, for

two-stage models, |v(P ) ? v(QJ )| can be estimated by a multiple of some metric or functional of P and QJ . Hence, one might consider (P , QJ ) instead and arrives at the principle of optimal scenario reduction: Fix k ∈ N, k < n, and determine a solution of the minimization problem ? ? min (P , QJ ) : J ? {1, . . . , n}, ? ? ? qj = 1 . (10) #J = n ? k, qj 0, ?
j∈ /J

In a ?rst step, it is of interest to ?x J and to determine the optimal weights qj , j ∈ / J , such that QJ is a probability measure, i.e., to solve the best approximation problem. ? ? ? ? min (P , QJ ) : qj 0, qj = 1 . (11) ? ?
j∈ /J

The next result asserts that the latter problem (11) is solvable and provides an explicit representation of the ? in?mum in case = r . Theorem 4.1. For given nonempty subset J of {1, . . . , n} problem (11) has a solution Q? J = ? q and it holds j∈ /J j j DJ :=
?

, Q? r (P? J) ?? ?
r (P , QJ )

= min =
i ∈J

: qj
j)

0,
j∈ /J

qj = 1

? ? ?

pi min c ?r ( i ,
j∈ /J m?1

=
i ∈J

pi min
=1

cr (

l

,

l

+1

) : m ∈ N, (12)

l ∈ {1, . . . , n}, l1 = i, lm = j ∈ /J ,

? = p + where qj / J , with Jj := j i ∈Jj pi , ?j ∈ {i ∈ J |j = j (i)} and the index j (i) belonging to arg minj ∈ ?r ( i , j ), ?i ∈ J , i.e., the optimal re/J c distribution consists in adding each deleted scenario

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

735

8000 7000 6000 5000 4000 3000 0 24 48 72 96 120 144 168

1000

500

0

-500

-1000 0 24 48 72 96 120 144 168

Fig. 1. Load scenarios for one week and mean shifted initial load scenario tree.

weight to that of some of those scenarios being closest w.r.t. c ?. Proof. Due to Proposition 2.2 we have the identity ? r (P , QJ )= ? c ?r (P , QJ ), where the reduced cost function c ?r is a metric on the support of P. Since [2, Theorem 2] is established for the Monge–Kantorovich functional, it implies the desired representation ? ? ? ? (P , Q ) : q 0 , q = 1 min ? c J j j ? ? ?r
j∈ /J

approximation algorithms for the metric k-median problem in [1] and [12, Chapter 25] achieve guarantees of 6 2 3 and 6 times the optimal. Simple heuristics may be derived by extending the two extremal cases k = n ? 1 and k = 1 of problem (13). These problems correspond to solving
l ∈{1,...,n}

min

pl min c ?r ( l ,
j =l

j)

(k = n ? 1)

and
n u∈{1,...,n}

min

=
i ∈J

pi min c ?r ( i ,
j∈ /J

j ),

pi c ?r ( u , i )
i =1 i =u

(k = 1).

together with the asserted redistribution rule. The preceding result coincides with [2, Theorem 2] if cr is a metric, i.e., r = 1. Using the explicit formula (12), the problem (10) of optimal scenario reduction is of the form min DJ =
i ∈J

pi min c ?r ( i ,
j∈ /J

j)

: (13)

Their solutions are the index sets J = {l1 } and {1, . . . , n}\{u1 }, respectively. The two sets arise from different algorithmic ideas: backward reduction and forward selection. Both ideas can be extended and lead to backward and forward heuristics for ?nding approximate solutions of (13). For example, the forward selection procedure determines an index set J [k ] of deleted scenarios having cardinality n ? k . Algorithm 4.2 (Forward selection). Step[0] : Step[i] : J [0] := {1, . . . , n}. ui ∈ arg min pk

J ? {1, . . . , n}, #J = n ? k ,

i.e., it represents a metric k-median problem in the metric space ( , c ?r ). The problem is known to be NP-hard, hence, (polynomial-time) approximation algorithms and heuristics become important. The

Step[k + 1] : Optimal redistribution.

min c ?r ( k , j ), j∈ / J [i ?1] \{u} [ i ] [ i ? 1 ] J := J \{ui }.

u∈J [i ?1] k ∈J [i ?1] \{u}

736

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

This algorithm was ?rst studied in [5] for the case c ?r = cr . There it is shown that the algorithm requires O(k n2 ) operations. Although the algorithm does not lead to optimality in general, the performance evaluation of its implementation in [5] is very encouraging.

5. Numerical experience We consider the scenario tree in [2,5] representing the increasing uncertainty of electrical load in a stochastic electrical power production model for a
Table 1 Numerical results for optimal scenario reduction based on
Number of scenarios Relative r =1 1 5 10 20 50 100 150 200 300 400 500 600 1.000 0.522 0.419 0.323 0.230 0.169 0.137 0.117 0.094 0.072 0.050 0.028
? r -distances

weekly time horizon (see [4] for further information). The scenario tree is obtained by calibrating a time series model for the electrical load, by simulating a large number of load realizations, and by constructing an initial ternary load scenario tree based on sample means and standard deviations of the simulated realizations. The initial load scenario tree represents a discrete probability distribution P that consists of 36 = 729 uniformly distributed scenarios and enters a 7-period two-stage stochastic programming model (Fig. 1). Table 1 presents our computational results for optimal scenario reduction of the initial load

?

r

r =2 1.000 0.646 0.536 0.420 0.305 0.220 0.178 0.148 0.102 0.067 0.039 0.018

r =3 1.000 0.684 0.589 0.469 0.335 0.242 0.185 0.143 0.085 0.049 0.024 0.009

r =4 1.000 0.696 0.577 0.472 0.337 0.222 0.156 0.112 0.057 0.028 0.012 0.004

r =5 1.000 0.687 0.582 0.466 0.301 0.180 0.114 0.076 0.032 0.013 0.005 0.001

r =6 1.000 0.682 0.556 0.431 0.256 0.133 0.077 0.045 0.016 0.006 0.002 0.000

r =7 1.000 0.668 0.535 0.395 0.210 0.094 0.049 0.025 0.008 0.002 0.001 0.000

Table 2 Numerical results for optimal scenario reduction based on ? r
Number of scenarios Relative ? r -distances r =1 1 5 10 20 50 100 150 200 300 400 500 600 1.000 0.522 0.419 0.323 0.230 0.169 0.137 0.117 0.094 0.072 0.050 0.028 r =2 1.609 0.738 0.574 0.448 0.308 0.221 0.179 0.149 0.102 0.067 0.039 0.018 r =3 2.354 0.940 0.713 0.538 0.359 0.253 0.192 0.147 0.088 0.050 0.025 0.009 r =4 3.146 1.079 0.787 0.600 0.378 0.248 0.171 0.121 0.062 0.030 0.012 0.004 r =5 3.910 1.209 0.820 0.617 0.369 0.211 0.134 0.088 0.037 0.015 0.005 0.001 r =6 4.627 1.217 0.803 0.601 0.331 0.168 0.097 0.058 0.021 0.007 0.002 0.000 r =7 5.302 1.257 0.794 0.565 0.286 0.130 0.066 0.035 0.011 0.003 0.001 0.000

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

737

500

500

0

0

-500

-500

0

24

48

72

96

120

144

168

0

24

48

72

96

120

144

168

500

500

0

0

-500

-500

0 1000 500 0 -500 -1000 0 1000 500 0 -500 -1000 0

24

48

72

96

120

144

168 1000 500 0 -500 -1000

0

24

48

72

96

120

144

168

24

48

72

96

120

144

168 1000 500 0 -500 -1000

0

24

48

72

96

120

144

168

24

48

72

96

120

144

168
?

0

24

48

72

96

120

144

168

Fig. 2. Reduced trees containing k = 20 scenarios obtained by using r (left column) and ? r (right column) for r = 1, 2, 4, 7.

738

H. Heitsch, W. R?misch / Operations Research Letters 35 (2007) 731 – 738

scenario tree by using Algorithm 4.2. A comparison ? with Table 2 shows the improvement of using r instead of ? r . Both tables display the relative distances between the original load tree and some of the reduced ones, and the effects of varying the order r ? of the Fortet–Mourier metrics r and the functionals ? r , respectively. The relative distances are computed by dividing all distances by the Fortet–Mourier distance between the initial load distribution P and the Dirac measure at the scenario obtained in the ?rst for? ward selection step, i.e., by r (P , u ). To compute 1 a reduced tree for r = 1, the running time on a PC equipped with a 3 GHz processor is less than 10 s including about 4 s to compute the scenario distances cr (·, ·). For r > 1about 9 s are needed in addition to compute the reduced cost c ?r (·, ·). Fig. 2 illustrate the structure of the reduced scenario trees consisting of 20 scenarios for varying order r. As approximations ? of probability distributions with respect to r approximately recover rth order absolute moments (see (4)), different scenarios for different r are selected with a tendency to outer scenarios for growing r. Acknowledgements This work was supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin, the BMBF and a grant of EDF—Electricité de France. The authors wish to thank the referee for his suggestions to improve the presentation and A. Martin (University of Darmstadt) and J. Rambau (University of Bayreuth) for helpful discussions.

References
[1] M. Charikar, S. Guha, E. Tardos, D.B. Shmoys, A constantfactor approximation algorithm for the k-median problem, J. Comput. Syst. Sci. 65 (2002) 129–149. [2] J. Dupaˇ cová, N. Gr?we-Kuska, W. R?misch, Scenario reduction in stochastic programming: an approach using probability metrics, Math. Program. 95 (2003) 493–511. [3] R. Fortet, E. Mourier, Convergence de la répartition empirique vers la répartition théorique, Ann. Sci. Ecole Norm. Sup. 70 (1953) 266–285. [4] N. Gr?we-Kuska, W. R?misch, Stochastic unit commitment in hydro-thermal power production planning, in: S.W. Wallace, W.T. Ziemba (Eds.), Applications of Stochastic Programming, MPS-SIAM Series in Optimization, 2005, pp. 633–653. [5] H. Heitsch, W. R?misch, Scenario reduction algorithms in stochastic programming, Comput. Optim. Appl. 24 (2003) 187–206. [6] H. Heitsch, W. R?misch, C. Strugarek, Stability of multistage stochastic programs, SIAM J. Optim. 17 (2006) 511–525. [7] S.T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, New York, 1991. [8] S.T. Rachev, W. R?misch, Quantitative stability in stochastic programming: the method of probability metrics, Math. Oper. Res. 27 (2002) 792–818. [9] S.T. Rachev, L. Rüschendorf, Mass Transportation Problems, vols. I and II, Springer, Berlin, 1998. [10] W. R?misch, Stability of stochastic programming problems, in: A. Ruszczy? nski, A. Shapiro (Eds.), Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10, Elsevier, Amsterdam, 2003, pp. 483–554. [11] W. R?misch, R.J.-B. Wets, Stability of -approximate solutions to convex stochastic programs, preprint 325, DFG Research Center Matheon “Mathematics for key technologies”, SIAM J. Optim. (2006), submitted. [12] V.V. Vazirani, Approximation Algorithms, Springer, Berlin, 2001.



更多相关文章:
...scenario reduction for two-stage stochastic prog....pdf
A note on scenario reduction for two-stage stochastic programs两阶段随机规划注意减少场景 - Operations Research...
基于场景分析的含风电系统机组组合的两阶段随机优化_图文.pdf
关键词:电力系统;风力发电;机组组合;两阶段随机规划;场景束约束 A two-stage stochastic optimization of unit commitment considering wind power based on scenario ...
各种优化算法及能源模型在能源系统中的应用(林明113220....doc
两阶段随机规划(Two-stage stochastic programming(TSP))是将随机规划转化 为相对确定形式的一种方法. 当需要获得政策分析方案但数据不确定时,两阶段 随机规划可以...
1_6237190_两阶段随机规划的若干算法及应用研+究.pdf
1_6237190_两阶段随机规划的若干算法及应用研+究_...FOR TWO-STAGE STOCHASTIC PROGRAM AND ITS ...on production and supply plan, we give a two-...
基于场景分析的含风电系统机组组合的两阶段随机优化.pdf
关键词:电力系统;风力发电;机组组合;两阶段随机规划;场景束约束 A two-stage stochastic optimization of unit commitment considering wind power based on scenario ...
基于两阶段机会约束随机规划的含风电机组组合问题_图文.pdf
基于两阶段机会约束随机规划的含风电机组组合问题_数学_自然科学_专业资料。...A Two-Stage Chance-Constrained Stochastic Program for Unit Commitment with ...
线性二阶锥两阶段随机规划问题的统计推断.pdf
for a two-stage program whose second stage ...的两阶段随机规划问题,该问题的全部参数都是随机 ...(2009) Lectures on Stochastic Programming Modeling ...
含风电场的机组组合的两阶段随机规划模型_图文.pdf
含风电场的机组组合的两阶段随机规划模型_机械/仪表...ATwo-stageStochasticOptimizationConsidering Modelof....Thescenarioreductionmethodisintroducedforenhancing ...
不确定环境下等待时间受限的混合流水车间调度问题研究.pdf
通过探索这类问题的特征,建立了两阶段随机规划模型,...atwo-stagestochastican programmingmodelwithuncertain...on the steady state accommodate practical ...
基于两阶段随机规划方法的灌区水资源优化配置_付银环.pdf
第5期 付银环等:基于两阶段随机规划方法的灌区水资源优化配置 77? 由计算结果...A two-stage inexact-stochastic programming model for planning carbon dioxide ...
Solving two-stage robust optimization problems usin....pdf
for an identified scenario, which is very ...those in a twostage stochastic programming model....(2) Note that the resulting problem in (2) ...
基于CVaR的两阶段第四方逆向物流网络设计模型.pdf
的需求量为随机变量,以总利润的期望值最大为目标建立两阶段随机规划模型;最后,...andstagestochasticdemandofreprocessedproductsasstochasticvariables,proposeatwoThe...
应急储备库选址与资源配置随机规划模型研究(论文)_图文.pdf
weproposea Technology,Wuhan,China) scenario...basedtwostagestochasticprogrammingresources modelfor...1.4模型构建 构建两阶段随机规划模型P如下: 1509 ?...
漳卫南灌区农业水资源优化配置研究.doc
两阶段随机规划 中图分类号:TV21 文献标志码:A ...twostage stochastic programming (TSP)and linear ...scenario balanced the system benefit and irrigation...
阶段随机规划的若干算法及应用研究_图文.pdf
10424 学位论文 多阶段随机规划的若干算法及应用研究...FOR MULTI-STAGE STOCHASTIC PROGRAM AND ITS ...on production and supply pla n, we give a ...
集装箱港口堆场箱位数的二阶段随机规划方法.pdf
集装箱港口堆场箱位数的二阶段随机规划方法_电子/...Whilefocusingonthedeterminationofthenumberofflat...atwo-stagestochasticprogrammingmodelwithcompensation...
基于场景树及二阶段规划的风电-梯级水电站联合调度模型....pdf
随机规划 Scenario-Tree and Two-Stage Approach for...Keywords: scenario tree; stochastic program; ...Romisch W. Scenario reduction and scenario tree ...
基于带补偿随机规划的蒸汽动力系统优化设计.pdf
基于本研究建立的多 周期带补偿的二阶段随机规划 ...on stochastic programming with recourse GAI Limei1...stage to avoid a conservative design based on ...
基于不确定蒸汽需求和设备故障的锅炉系统随机规划设计.pdf
关键词:不确定性;两阶段随机规划;系统工程;模拟;...A 文章编号:04381157(2014)09351207 Boiler...on stochastic programming under uncertain steam ...
A-457_基于平均近似估计SAA的电容器组优化配置方法_图文.pdf
a two-stage model base on Quasi-Monte Carlo Halton...scenario reduction techniques was proposed and the ...( x, Pwk )] 为二阶段随机规划模型的目标函数,...
更多相关标签:

All rights reserved Powered by 甜梦文库 9512.net

copyright ©right 2010-2021。
甜梦文库内容来自网络,如有侵犯请联系客服。zhit325@126.com|网站地图