Revision Modeling Of Two-Stage Stochastic Programming Problem

(1)

REVISION MODELING OF TWO-STAGE STOCHASTIC

PROGRAMMING PROBLEM

Herman Mawengkang, Saib Suwilo, and Opim S. Sitompul

Mathematics Department, Faculty of Mathematics and Natural Sciences

University of North Sumatera

Abstract_{: Stochastic programming is an important tool in medium to long term planning where there are} uncertainties in the data. In this paper, we consider two-stage stochastic programming problem. The model is not well defined, since there are random vectors imposed in the model to present the uncertainties of the model parameter. Therefore a revision of the modeling process is necessary, leading to so-called deterministic equivalents of the original model. This paper discusses about how to get the deterministic equivalent model.

1. INTRODUCTION

Medium to long term planning is essential to the success of business and project management. In these applications, the problem can be divided into multiple stages, usually over time. Dynamic programming [1, 2], bilevel programming [7] and mathematical programming with equilibrium constraints [5] are useful modeling and solution techniques for problems with two or more stages. As a lot of data is not available at planning stages, the decisions need to be flexible enough to cope with different eventualities. Stochastic Programming [3, 4, 6] is an increasingly important problem class for long term planning. Instead of treating the future as a certainty with known data as in classical optimization, stochastic programs incorporate information from spectrum of possible future events. This gives decision makers the ability to quantify the risk in different scenarios.

Stochastic programming began in the mids 1950s, and was one of the motivations for Dantzig’s seminal work on linear programming. Early work concentrated on the two-stage linear programs. For examples, Van Slyke and Wets [ 8 ] developed the L-shaped method which is the basis of many algorithms used today.

This paper discusses about modifying the two-stage stochatic programming into deterministic equivalents model, in such a way that we could solve the original stochastic programming problem more easily. Deterministic equivalent formulation of two-stage stochastic programming model can also be found in [ 9,10 ]. However, in this paper we address several alternatives to get such deterministic equivalent model.

2. STOCHASTIC PROGRAMS: GENERAL FORMULATION

We define the stochastic (linear) program as the following model

0 min ( , )

s.t. ( , ) 0, 1, , , , i

n

g x

g x i m

x X

ξ ξ

⎫ ⎪⎪

≤ = _⎬

⎪

∈ ⊂ _⎪⎭ %

% _K

฀

(1)

where

ξ

%

is a random vector varying over a set k

Ξ ⊂

฀

. More precisely, we assume throughout that a family F of “events”, i.e. subsets of Ξ, and the probability distribution P on F are given. Hence for every subset A⊂Ξ that is an events, i.e. A∈F, the probability P(A) is known. Furthermore, we assume that the functions

g x

_i

( , ) :

⋅ Ξ →

฀

∀

x i

,

are random variables themselves and that the probability distribution P is independent of x.

However, problem (1) is not well define since the meanings of “min” as well as of the constraints are not cleat at all, if we think of taking a decision on

x before knowing the realization of

ξ

%

. Therefore a revision of the modeling process is necessary, leading to so-called deterministic equivalents for (1), which can be introduced in various ways.

3. DETERMINISTIC EQUIVALENT FORMULATION

Let us now come back to deterministic equivalents for (1). For instance, in analogy to the particular stochastic linear program with recourse, for problem (1) we may proceed as follows. With

0

if

( , )

0,

( , )

otherwise,

i i

i

g x

g

x

g x

ξ

+

⎧

≤

=

⎨

⎩

the ith constraint of (1) is violated if and only if

( , )

0

i

g

+

x

ξ

>

for a given decision x and realization

ξ

of

ξ

%

. Hence we could provide for each constraint a recourse or second-stage activity

y

_i

( )

ξ

that, after observing the realization

ξ

, is chosen such as to compensate its constraint’s violation - if there is one - by satisfying

g x

_i

( , )

ξ

−

y

_i

( )

ξ

≤

0

. This extra effort is assumed to cause an extra cost or penalty of

qi per unit, i.e. our additional costs (called the

(2)

1

( , ) min ( ) | ( ) ( , ), 1, ,

m

i i i i

y i

Q xξ q y ξ y ξ g+ xξ i m

=

⎧ ⎫

= _⎨ ≥ = _⎬

⎩

∑

L ⎭ (2)

yielding a total cost-first-stage and recourse cost-of

0

( , )

0

( , )

f x

ξ

=

g x

ξ

+

Q x

ξ

(3)

Instead of (2), we might think of a more general linear recourse program with a recourse vector

( )

n

y

ξ

∈ ⊂

Y

฀

, (Y is some given polyhedral set, such as {y | y ≥ 0}), an arbitrary fixed

m n

×

matrix

W (the recourse matrix) and a corresponding unit cost vector

q

∈

฀

n, yielding for (3) the recourse function

{

}

( , )

min

T

|

( , ),

y

Q x

ξ

=

q y Wy

≥

g

+

x

ξ

y

∈

Y

(4)

where

( , )

(

₁

( , ),

,

( , )

)

T m

g

+

x

ξ

g

+

x

ξ

g

+

x

ξ

=

L

.

If we think of a factory producing m products,

( , )

i

g x

ξ

could be understood as the difference {demand}-{output} of a product i. Then

( , )

0

i

g

+

x

ξ

>

means that there is a shortage in product I, relative to the demand. Assuming that the factory is committed to cover the demands, problem (2) could for instance be interpreted as buying the shortage of products at the market. Problem (4) instead could result from a second-stage or

emergency production program, carried through with the factor input y and a technology represented by the matrix W. Choosing W=I, m×m identity matrix, (2) turns out to be a special case of (4).

Finally we also could think of a nonlinear recourse program to define the recourse function for (3); for instance,

Q x

( , )

ξ

could be chosen as

{

}

( , ) min ( ) | ( ) ( , ), 1, , ; n ,

i i

Q xξ q y H y g+ xξ i m y Y

= ≥ = L ∈ ⊂฀ (5)

where

q

:

฀

n

→

฀

and

H

_i

:

฀

n

→

฀

are supposed to be given.

In any case, if it is meaningful and acceptable to the decision maker to minimize the expected value of the total costs (i.e. first-stage and recourse costs), instead of problem (1) we could consider its deterministic equivalent, the (two-stage) stochastic program with recourse

{

}

0 0

min ( , ) min ( , ) ( , ) .

x X∈ E f xξ%

ξ

= x X∈ Eξ% g x

ξ

+Q x

ξ

% % % ₍₆₎

The above two-stage problem is immediately extended to the multistage recourse program as follows: instead of the two decisions x and y, to be

sequential decisions

x x

₀

, ,

₁

,

x

_K

(

x

nτ

)

τ

∈

L

฀

, to

be taken at the subsequent stages

τ

=

0,1,

L

,

K

. The term “stages” can, but need not, be interpreted as “times periods”.

Assume for simplicity that the objective of (1) is deterministic, i.e.

g x

₀

( , )

ξ

=

g x

₀

( )

. At stage

(

1)

τ τ

≥

we know the realizations

ξ

₁

,

L

,

ξ

_τ of the

random vectors

ξ

%

₁

,

L

,

ξ

%

_τ as well as the previous decisions

x

₀

,

L

,

x

_τ₋₁, and we have to decide on

x

_τ such that the constraints(s) (with vector valued

constraint functions

g

_τ)

0 1

( ,

,

, ,

,

0)

g x

_τ

L

x

_τ

ξ

L

ξ

_τ

≤

are satisfied, which - as stated - at this stage can only be achieved by the proper choice of

x

_τ, based on the knowledge of the previous decisions and realizations. Hence, assuming a cost function

q

_τ(

x

_τ), at stage

τ

≥

1

we have a recourse function

{ }

0 1 1 1 0 1 1 1

( , , , , , , ) min ( ) | ( , , , , , , ) 0

x

Q x x x q x g x x x

τ

τ= K τ− ξ Kξτ = τ τ τ K τ− ξ Kξτ ≤

indicating that the optimal recourse action

x

ˆ

_τ at time

Ĳ depends on the previous decisions and the realizations observed until stage Ĳ , i.e.

0 1 1

ˆ

ˆ ( ,

,

, ,

,

),

1

x

_τ

=

x x

_τ

L

x

_τ−

ξ

L

ξ τ

_τ

≥

Hence, taking into account the multiple stages, we get as total costs for the multistage problem

1

0 0 1 0 0 , , 0 1 1 1

1

ˆ ˆ

( , , , ) ( ) ( , , , , , , )

K K

f x g x E Q x x x

τ τ τ τ

ξ ξ τ

ξ ξ − ξ ξ

= = +

∑

%L%

L L L (7)

yielding the deterministic equivalent for the described dynamic decision problem, the multistage stochastic program with recourse

1 0

0 0 , , 0 1 1 1

1

ˆ ˆ

min ( ) ( , , , , , , )

K

x X g x Eξ ξτQ x xτ xτ τ τ

ξ ξ

− ∈

=

⎡ ₊ ⎤

⎢ ⎥

⎣

∑

%L% L % L % ⎦ (8)

obviously a straight generalization of our former (two-stage) stochastic program with recourse (6).

For the two-stage case, in view of their practical relevance it is worthwhile to describe briefly some variants of recourse problems in the

stochastic linear programming setting. Assume that we are given the following stochastic linear program

"min"

s.t. ,

( ) ( ),

T

c x

Ax b

T ξ x hξ

⎫ ⎪

= _⎪

⎬

= _⎪

% %

(3)

Comparing this with the general stochastic program (1), we see that the set

X

⊂

฀

n is specified as

{

n

|

,

0

}

X

=

x

∈

฀

Ax

=

b x

≥

where the

m

₀

×

n

matrix A and the vector b are

assumed to be deterministic. In contrast, the

m

₁

×

n

matrix

T

( )

⋅

and vector

h

( )

⋅

are allowed to depend

on the random vector

ξ

%

, and therefore to have random entries themselves. In general, we assume that this dependence on

ξ

∈ Ξ ⊂

฀

k is given as

( )

0 1

1

0 1

1

ˆ ˆ _, _, ˆ _,

k K

T T T T

h h h h

ξ

⎫

= + _⎪

⎬

= + _⎪⎭

L

(10)

with deterministic matrices

T

ˆ

0

,

L

,

T

ˆ

k and vectors

0

ˆ

,

ˆ

k

h

L

h

. Observing that the stochastic constraints in (9) are equalities (instead of inequalities, as in the general problem formulation (1)), it seems meaningful to equate their deficiencies, which, using linear recourse and assuming that

{

n

|

0

}

Y

=

y

∈

฀

y

≥

, according to (4) yields the

stochastic linear program with fixed recourse

{

}

{

}

min ( , )

s.t. ,

0. where

( , ) min | ( ) ( ) , 0 T

x

T

E c x Q x

Ax b x

Q x q y Wy h T x y

ξ ξ

ξ ξ ξ

⎫

+

⎪ ⎪

=

⎪

≥ _⎬

⎪ ⎪ ⎪

= = − ≥ _⎭

% %

(11)

In particular, we speak of complete fixed recourse if the fixed

m n

×

recourse matrix W satisfies

{

z z

|

=

Wy y

,

≥

₀

}

=

฀

m1₍₁₂₎

This implies that, whatever the first-stage decision x

and the realizations

ξ

of

ξ

%

turn out to be, the second-stage program

{

}

( , )

min

T

|

( )

( ) ,

0

Q x

ξ

=

q y Wy

=

h

ξ

−

T

ξ

x y

≥

will always be feasible. A special case of complete fixed recourse is simple recourse, where with the identity matrix I of order m1:

W =(I,-I) (13)

Then the second-stage program reads as

{

}

( , ) min ( )T ( )T | ( ) ( ) , 0, 0

Q xξ ₌ q+ y+₊ q− y− y+₋y−₌hξ ₋Tξx y+_≥ y−_≥

i.e. for

q

+

q

−

≥

0

, the recourse variables

y

+ and

y

− can be chosen to measure (positively) the absolute deficiencies in the stochastic constraints.

Generally, we may put all the above problems into the following form:

0 min ( , )

s.t. ( , ) 0, 1, , ( , ) 0, 1, , , i

i

n

E f x

E f x i s

E f x i s m

x X

ξ ξ ξ

⎫ ⎪ ⎪

≤ = _⎪

⎬

= = + ⎪

⎪

∈ ⊂ ⎪⎭ %

%

% %

% _L

฀

(14)

where the fi are constructed from the objective and

the constraints in (1) or (9) respectively. So far, f0

represented the total costs (see(3) or (7)) and

1

,

m

f

L

f

could be used to describe the first-stage feasible set X. However, depending on the way the functions fi are derived from the problem functions gj

in (1), this general formulation also includes other types of deterministic equivalents for the stochastic program (1).

To give just two examples showing how other deterministic equivalent problems for (1) may be generated, let us choose first

α

∈

[ ]

0,1

and define a “payoff” function for all constraints as

1

if

( , )

0,

1,

, ,

( , )

otherwise

i

g x

i

m

x

α

ξ

ϕ

ξ

α

−

≤

=

⎧

=

⎨

−

⎩

L

Consequently, for x infeasible at

ξ

we have an

absolute loss of

α

, whereas for x feasible at

ξ

we

have a return of

1

−

α

. It seems natural to aim for decisions on x that, at least in the mean (i.e. on average), avoid an absolute loss. This is equivalent to the requirement.

( , )

0

E

_ξ

ϕ

x

ξ

ϕ

x

ξ

dP

Ξ

=

∫

≥

%

Defining

0

( , )

0

( , ) and ( , )

1

( , )

f x

ξ

=

g x

ξ

f x

ξ

= −

ϕ

x

ξ

, we get

0 0

1

( , ) ( , )

1 if ( , ) 0, 1, , , ( , )

otherwise

i

f x g x

g x i m

f x

ξ

α

ξ

α

= ⎫

⎪

− ≤ =

⎧ ⎬

=_⎨ _⎪

⎩ ⎭

(4)

implying

1

( , )

0

E f x

_ξ%

ξ

%

= −

E

_ξ%

ϕ

x

ξ

%

≤

where, with the vector-valued function

(

1

)

( , )

( , ),

,

_m

( , )

T

g x

ξ

=

g x

ξ

L

g

x

ξ

,

{ } { }

{

}

(

)

(

{

}

)

{

}

(

)

1 1

( , ) 0 ( , ) 0

1

( , ) ( , )

( 1)

( 1) | ( , ) 0 | ( , ) 0

| ( , ) 0

g x g x

E f x f x dP

dP dP

P g x P g x

P g x

ξ

ξ ξ

α α

α ξ ξ ξ ξ

ξ ξ

Ξ

≤ ≤/

= =

= − +

= − ≤ + ≤/

− ≤

∫

% %

144444444424444444443

Therefore the constraint

E f x

_ξ% ₁

( , )

ξ

%

≤

0

is

equivalent to

P

(

{

ξ

| ( , )

g x

ξ

≤

0

}

)

≥

α

. Hence, under these assumptions, (14) reads as

{

}

(

)

0

min ( , )

s.t. | ( , ) 0 , 1, ,

x X E g x

P g x i m

ξ

α

∈ ⎫⎪

⎬

≤ = ≥ _⎪⎭

%

L (16)

Problem (16) is called a probability constrained or

chance constrained program (or a problem with joint probabilistic constraints).

If instead of (15) we define

[ ]

0,1 ,

1,

,

i

m

α

∈

=

L

and analogous “payoffs” for every single constraint, resulting in

0 0

1

( , )

1 if

( , )

0

( , )

1,

, ,

otherwise

i i

i

f x

g x

f x

i

m

ξ

α

ξ

α

=

−

≤

⎧

=

⎨

=

⎩

L

then we get from (14) the problem with single (or

separate) probabilistic constraints:

{

}

(

)

0

min ( , )

s.t. | ( , ) 0 , 1, ,

x X

i i

E g x

P g x i m

ξ ξ

ξ ξ α α

∈ ⎫⎪

⎬

≤ ≥ = ≥ _⎪⎭

%

L

(17)

If, in particular, we have that the functions

g x

_i

( , )

ξ

are linear in x and if furthermore the set X is convex polyhedral, i.e. we have the stochastic linear program

"min" ( ) s.t. ,

( ) ( )

0

T

c x

Ax b

T x h x

x

ξ

ξ ξ

⎫ ⎪

= _⎪

⎬

≥ _⎪

⎪

≥ _⎭

%

% %

then problems (16) and (17) become

{

}

(

)

min ( )

s.t. | ( ) ( )

T x X E c x

P T x h

ξ

α

∈ ⎫⎪

⎬

≥ ≥ _⎪⎭

% %

(18)

and, with

T

_i

( ) and

⋅

h

_i

( )

⋅

denoting the ith row and

ith component of

T

_i

( ) and

⋅

h

_i

( )

⋅

respectively,

{

}

(

)

min ( )

s.t. | ( ) ( ) , 1, ,

T x X

i i i

E c x

P T x h i m

ξ

α

∈ ⎫⎪

⎬

≥ ≥ = _⎪⎭

% %

L

(19)

the stochastic linear programs with joint and with single chance constraints respectively.

Obviously there are many other possibilities to generate types of deterministic equivalents for (1) by constructing the fi in different ways out of the

objective and the constraints of (1).

Formally, all problems derived, i.e. all the above deterministic equivalents, are mathematical programs. Another interesting topic to be explored is, whether or under which assumptions do they have properties like convexity and smoothness such that we have any reasonable chance to deal with them computationally using the toolkit of mathematical programming methods.

4. CONCLUSION

The model of stochastic programming problem needs to be revisioned into a deterministic equivalent model such that the original problem would be well defined and solvable. This paper has described some possibilities to generate types of deterministic equivalent for model of two-stage stochastic program.

5. REFERENCES

R. Bellman, Dynamic Programming, Princeton University Press, New Jersey, 1957.

D.P. Bertsekas, Dynamic Programming and Optimal Control, Prentice Hall, Englewood Cliffs, NJ, 1995.

J.R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer-Verlag, New York, 1997. P. Kall and S.W Wallace, Stochastic Programming,

John Wiley, Chicester and New York, 1994. Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical

Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1996. S. Ross, Introduction to Stochastic Dynamic

Programming, Academic Press, New York and London, 1983.

K. Shimizu, Y. Ishizuka, and J.F. Bard,

(5)

R.Van Slyke and R.J.B. Wets, L-shaped Linear Programs with application to Optimal Control, SIAM Journal on Applied Mathematics, 17 (1969), pp. 638-663.

S. Takriti and S. Ahmed, On Robust Optimization of Two-Stage Systems, Working paper of Georgia Inst. Of Tech. 2001.