Revision Modeling Of Two-Stage Stochastic Programming Problem

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 Lshaped method which is the basis of many algorithms
used today.
This paper discusses about modifying the twostage 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 twostage 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
⎫
min g 0 ( x, ξ% )
⎪⎪
(1)
s.t. gi ( x, ξ% ) ≤ 0, i = 1,K , m, ⎬
⎪
x∈ X ⊂ ฀ n,
⎪⎭

6

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 gi ( x, ⋅) : 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

if gi ( x, ξ ) ≤ 0,
⎧ 0
gi+ ( x, ξ ) = ⎨
otherwise,
⎩ g i ( x, ξ )
the ith constraint of (1) is violated if and only if

gi+ ( x, ξ ) > 0 for a given decision x and realization
ξ of ξ% . Hence we could provide for each constraint
a recourse or second-stage activity yi (ξ ) that, after
observing the realization ξ , is chosen such as to
compensate its constraint’s violation - if there is one by satisfying gi ( x, ξ ) − yi (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
recourse function) amount to

Revision Modeling of Two-Stage Stochastic Programming Problem
Herman Mawengkang, Saib Suwilo, and Opim S. Sitompul

⎧m
⎫
Q( x, ξ ) = min ⎨∑ qi yi (ξ ) | yi (ξ ) ≥ gi+ ( x, ξ ), i = 1,L , m ⎬ (2)
y
⎩ i =1
⎭

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

f 0 ( x, ξ ) = g 0 ( x, ξ ) + Q ( x, ξ )

(3)

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

y (ξ ) ∈ Y ⊂ ฀ n , (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
n

cost vector q ∈ ฀ , yielding for (3) the recourse
function

{

Q( x, ξ ) = min qT y | Wy ≥ g + ( x, ξ ), y ∈ Y

y

(

+

+

+

where g ( x, ξ ) = g1 ( x, ξ ),L , g m ( x, ξ )

}

(4)

T

)

.

If we think of a factory producing m products,
gi ( x, ξ ) could be understood as the difference
{demand}-{output}

of

a

product

i.

Then

+
i

g ( x, ξ ) > 0 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
Q( x, ξ ) = min q( y) | H i ( y ) ≥ gi+ ( x, ξ ), i = 1,L , m; y ∈ Y ⊂ ฀ n , (5)

{

where

}

q:฀ n → ฀

and

Hi : ฀ 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

{

}

min Eξ% f 0 ( x, ξ% ) = min Eξ% g0 ( x, ξ% ) + Q( x, ξ% ) . (6)
x∈X

x∈ X

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
taken at stages 1 and 2, we are now faced with K+1

x0 , x1 ,L , xK ( xτ ∈ ฀ nτ ) , to
be taken at the subsequent stages τ = 0,1,L , K .
sequential decisions

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 0 ( x, ξ ) = g 0 ( x ) . At stage

τ (τ ≥ 1)

we know the realizations

ξ1 ,L , ξτ

of the

random vectors ξ%1 ,L , ξ%τ as well as the previous
decisions x0 ,L , xτ −1 , and we have to decide on

xτ such that the constraints(s) (with vector valued
constraint functions gτ )
gτ ( x0 ,L , xτ , ξ1 ,L , ξτ ≤ 0)
are satisfied, which - as stated - at this stage can only
be achieved by the proper choice of xτ , based on
the knowledge of the previous decisions and
realizations. Hence, assuming a cost function
qτ ( xτ ), at stage τ ≥ 1 we have a recourse function
Qτ = ( x0 , x1 ,K, xτ −1 , ξ1 ,K, ξτ ) = min {qτ ( xτ ) | gτ ( x0 , x1 ,K, xτ −1 , ξ1 ,K, ξτ ) ≤ 0}
xτ

indicating that the optimal recourse action xˆτ at time
IJ depends on the previous decisions and the
realizations observed until stage IJ , i.e.

xˆτ = xˆτ ( x0 ,L , xτ −1 , ξ1 ,L , ξτ ),τ ≥ 1
Hence, taking into account the multiple stages, we
get as total costs for the multistage problem
f 0 ( x0 , ξ1 ,L , ξ K ) = g 0 ( x0 ) + ∑ Eξ% ,L,ξ% Qτ ( x0 , xˆ1 ,L , xˆτ −1 , ξ1 ,L , ξτ )
K

τ =1

1

(7)

τ

yielding the deterministic equivalent for the
described dynamic decision problem, the multistage
stochastic program with recourse
K
⎡
⎤
min ⎢ g0 ( x0 ) + ∑ Eξ%1 ,L,ξ%τ Qτ ( x0 , xˆ1 ,L , xˆτ −1 , ξ%1 ,L , ξ%τ ) ⎥ (8)
x0 ∈X
τ =1
⎣
⎦

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

⎫
⎪
s.t.
Ax = b, ⎪
⎬
T (ξ% ) x = h(ξ% ), ⎪
x ≥ 0. ⎪⎭

"min" cT x

(9)

7

Jurnal Sistem Teknik Industri Volume 7, No. 4 Oktober 2006

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

{

n

is specified as

{

+

y − can be chosen to measure (positively) the

m0 × n matrix A and the vector b are
assumed to be deterministic. In contrast, the m1 × n
matrix T (⋅) and vector h(⋅) are allowed to depend
on the random vector ξ% , and therefore to have
where the

random entries themselves. In general, we assume
that this dependence on

ξ ∈Ξ ⊂ ฀

is given as

T (ξ ) = Tˆ 0 + ξ1Tˆ 1 ,L , ξ K Tˆ k , ⎫⎪
⎬
h (ξ ) = hˆ0 + ξ1hˆ1 ,L , ξ K hˆ k , ⎪⎭
with deterministic matrices

(10)

Tˆ 0 ,L , Tˆ k and vectors

hˆ0 ,L , hˆ k . 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

{

}

Y = y ∈ ฀ n | y ≥ 0 , according to (4) yields the
⎫
⎪
⎪
s.t.
Ax = b,
⎪
x ≥ 0.
⎬
⎪
where
⎪
Q( x, ξ ) = min qT y | Wy = h(ξ ) − T (ξ ) x, y ≥ 0 ⎪
⎭

{

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

ϕ ( x, ξ ) = ⎨

(11)

(12)

and the realizations ξ of ξ% turn out to be, the
second-stage program

ξ

we have an
we

for
have a return
decisions on x that, at least in the mean (i.e. on
average), avoid an absolute loss. This is equivalent to
the requirement.

Eξ%ϕ ( x, ξ% ) = ∫ ϕ ( x, ξ )dP ≥ 0
Ξ

Defining

f 0 ( x, ξ ) = g 0 ( x, ξ ) and f1 ( x, ξ ) = −ϕ ( x, ξ ) , we

}

Q( x, ξ ) = min qT y | Wy = h(ξ ) − T (ξ ) x, y ≥ 0

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

8

if gi ( x, ξ ) ≤ 0, i = 1,L , m,
otherwise

α , whereas for x feasible at ξ
of 1 − α . It seems natural to aim

This implies that, whatever the first-stage decision x

W =(I,-I)

and define a

absolute loss of

In particular, we speak of complete fixed recourse if
the fixed m × n recourse matrix W satisfies

{

α ∈ [ 0,1]

Consequently, for x infeasible at

}

y ≥ 0} = ฀ m1

(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
f1 ,L , f m could be used to describe the first-stage

⎧1 − α
⎩ −α

}

{ z | z = Wy,

⎫
⎪
s.t. Eξ% fi ( x, ξ% ) ≤ 0, i = 1,L , s
⎪⎪
⎬
%
Eξ% f i ( x, ξ ) = 0, i = s + 1,L , m, ⎪
⎪
x∈ X ⊂ ฀ n
⎪⎭
min Eξ% f 0 ( x, ξ% )

“payoff” function for all constraints as

min x Eξ% c x + Q( x, ξ% )

{

absolute deficiencies in the stochastic constraints.
Generally, we may put all the above problems
into the following form:

generated, let us choose first

stochastic linear program with fixed recourse
T

+

−

i.e. for q + q ≥ 0 , the recourse variables y and

X = x ∈ ฀ | Ax = b, x ≥ 0

k

}

Q( x, ξ ) = min (q+ )T y + + (q− )T y − | y + − y − = h(ξ ) − T (ξ ) x, y + ≥ 0, y − ≥ 0

}

n

Then the second-stage program reads as

(13)

get

⎫
⎪ (15)
⎧α − 1 if gi ( x, ξ ) ≤ 0, i = 1,L , m, ⎬
f1 ( x, ξ ) = ⎨
⎪
otherwise
⎩ α
⎭
f 0 ( x, ξ ) = g 0 ( x, ξ )

Revision Modeling of Two-Stage Stochastic Programming Problem
Herman Mawengkang, Saib Suwilo, and Opim S. Sitompul

implying

then problems (16) and (17) become

Eξ% f1 ( x, ξ% ) = − Eξ%ϕ ( x, ξ% ) ≤ 0
where,

with

the

vector-valued

function

T

g ( x, ξ ) = ( g1 ( x, ξ ),L , g m ( x, ξ ) ) ,
Ξ

{ g ( x ,ξ )≤ 0}

(α − 1)dP + ∫

{ g ( x ,ξ )≤/ 0}

and, with Ti (⋅) and hi (⋅) denoting the ith row and
ith component of Ti (⋅) and hi (⋅) respectively,

Eξ% f1 ( x, ξ% ) = ∫ f1 ( x, ξ )dP
=∫

⎫⎪
⎬ (18)
s.t. P ({ξ | T (ξ ) x ≥ h(ξ )}) ≥ α ⎪⎭
min x∈X Eξ% cT (ξ% ) x

⎫⎪
⎬ (19)
s.t. P ({ξ | Ti (ξ ) x ≥ hi (ξ )}) ≥ α i , i = 1,L , m ⎪⎭
min x∈X Eξ% cT (ξ% ) x

α dP

= (α − 1) P ({ξ | g ( x, ξ ) ≤ 0}) + P ({ξ | g ( x, ξ ) ≤/ 0})
144444444
42444444444
3
=1

− P ({ξ | g ( x, ξ ) ≤ 0})

Therefore

the

constraint

equivalent to P

Eξ% f1 ( x, ξ% ) ≤ 0

({ξ | g ( x, ξ ) ≤ 0}) ≥ α .

is

Hence,

under these assumptions, (14) reads as

⎫⎪
⎬ (16)
s.t. P ({ξ | g ( x, ξ ) ≤ 0}) , i = 1,L , m ≥ α ⎪⎭
min x∈X Eξ% g 0 ( x, ξ )

Problem (16) is called a probability constrained or
chance constrained program (or a problem with joint
probabilistic constraints).
If
instead
of
(15)
we
define

α i ∈ [ 0,1] , i = 1,L , m

and analogous “payoffs”

for every single constraint, resulting in

f 0 ( x, ξ ) = g 0 ( x, ξ )

⎧α − 1 if gi ( x, ξ ) ≤ 0
f1 ( x, ξ ) = ⎨ i
i = 1,L , m,
otherwise
⎩ αi

then we get from (14) the problem with single (or
separate) probabilistic constraints:

⎫⎪
⎬ (17)
s.t. P ({ξ | gi ( x, ξ ) ≤ 0}) ≥ α i , i = 1,L , m ≥ α ⎪⎭
min x∈X Eξ% g 0 ( x, ξ )

If, in particular, we have that the functions gi ( x, ξ )
are linear in x and if furthermore the set X is convex
polyhedral, i.e. we have the stochastic linear program

⎫
⎪
Ax = b,
s.t.
⎪
⎬
%
%
T (ξ ) x ≥ h(ξ ) x ⎪
⎪
x≥0
⎭

"min" cT (ξ% ) x

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
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D.P. Bertsekas, Dynamic Programming and Optimal
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J.R. Birge and F. Louveaux, Introduction to Stochastic
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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,
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S. Ross, Introduction to Stochastic Dynamic
Programming, Academic Press, New York and
London, 1983.
K. Shimizu, Y. Ishizuka, and J.F. Bard,
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Programming, Kluwer Academic Publishers,
Boston, MA, 1997.

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Jurnal Sistem Teknik Industri Volume 7, No. 4 Oktober 2006

R.Van Slyke and R.J.B. Wets, L-shaped Linear
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10

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