Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol27.Issue3.2000:
Operations Research Letters 27 (2000) 135–141
www.elsevier.com/locate/dsw
Solving nonstationary in nite horizon stochastic production
planning problems
Alfredo Garcia ∗ , Robert L. Smith 1
Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Received 1 October 1998; received in revised form 1 June 2000
Abstract
Forecast horizons are de ned as long enough planning horizons that ensure agreement of rst period optimal production
decisions of nite and in nite horizon problems regardless of changes in future demand. In this paper, we prove forecast horizon
existence and provide computational procedures for production planning problems that satisfy the following monotonicity
property: for any xed nite planning horizon, there exist rst period optimal solutions that are monotone with respect to
c 2000 Elsevier Science B.V. All rights reserved.
monotone changes in stochastic demand.
Keywords: Production planning; In nite horizon optimal control; Decision and forecast horizons
1. Introduction
Production planners face substantial uncertainty in
the demand for their products that comes from a variety of sources. For instance, demand could be sensitive to varying economic conditions such as GNP
or interest rates, or alternatively, technical innovation
may imply unexpected early obsolescence. Moreover,
it is usually the decisions in the rst few periods that
are of immediate concern for the decision maker and
forecasting is more dicult and costlier for problem
parameters farther into the future.
∗ Correspondence address: The Brattle Group, 1133 20th Street,
Suite 800, Washington, DC 20036, USA.
1 This work was supported in part by NSF grants DDM-9214894
and DMI-9713723.
Motivated by these issues, the concept of a forecast
horizon (see for example, [1]) has long been studied
in the literature. The idea is that problem parameters
changes far enough o should not a ect the optimal
decisions of the rst few periods.
In this paper, we prove forecast horizon existence
and provide computational procedures for production planning problems that satisfy the following
monotonicity property: for any xed nite planning
horizon, there exist rst period optimal solutions that
are monotone with respect to monotone changes in
stochastic demand.
Monotonicity of optimal production plans under
stochastic demand has been established under different modeling assumptions by Karlin [4], Veinott
[12], Kleindorfer and Kunreuther [5], Morton [6],
Sethi and Cheng [7] among others. We will not
attempt in this paper to prove new monotonicity
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 4 9 - 3
136
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
results. Rather, by assuming that monotonicity of
optimal solutions is given, we show how the in nite horizon non-stationary versions of these production planning problems can be exactly solved
by rolling horizon procedures. Smith and Zhang
[8] have recently exploited the above-mentioned
monotonicity property to prove existence of forecast horizons in a deterministic setting. Their approach depends crucially on the monotonicity of
the production decisions at all time periods, a feature that may be too restrictive in many models
(e.g. set-up costs), since higher inventory levels
from earlier production decisions could substitute higher production decisions in later periods.
Moreover, the formula they provide to compute
a forecast horizon does not ensure that it is the
minimal forecast horizon, a highly desirable feature since longer forecasts are costlier and less
accurate.
In this paper, we show that monotonicity of only
the rst production decision is sucient to prove existence of forecast horizons. Moreover, our emphasis
on extreme monotone optimal solutions allows us
to e ectively detect the minimal forecast horizon by
means of a stopping rule with an embedded selection
procedure.
f(It ; xt ; Dt ) = It + xt − Dt . The one-stage total cost
incurred is
Ct (xt ; It ; Dt ) = ct (xt ) + ht (max{0; It+1 })
+pt (max{0; −It+1 });
where ct (·) is the production cost function, ht (·) inventory holding cost and pt (·) is either the opportunity cost for demand not satis ed or the backlogging
cost function.
We assume that demand at time period t, Dt is a
non-negative random variable with probability distribution Ft (·).
Hence, the expected t-stage cost-to-go incurred is
Z I +x
ht (I + x − u) dFt (u)
E[Ct (x; I; Dt )] = ct (x) +
Z ∞ 0
pt (−I − x + u) dFt (u):
+
I +x
For a planning horizon of T -periods, the production
planning problem is then
"T −1
#
X
t
min E
Ct (xt ; It ; Dt )
t=0
(PT )
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; 2; : : : ; T − 1;
M ¿xt ¿0
2. Problem formulation
Consider a single product rm where a decision for
production must be made at the beginning of each
period t, t = 0; 1; 2; : : : . Customer orders arrive during each time interval. We will agree to call the “demand” realized during time period t, the total number of customer orders received during time interval
[t; t + 1) and shall denote it by the random variable
Dt .
Thus, if at the beginning of time period t, there are It
units available of inventory and a production decision
of xt is made, the inventory–production system follows
the equation
integer;
where M is the maximal production capacity and ∈
(0; 1) is the discount factor.
2.1. Stochastic ordering
We say that a random variable X with probability distribution F(·) is stochastically larger than a
random variable Y with probability distribution G(·),
written X ¡ Y , if for every x ∈ R:
F(x)6G(x):
Let A be a poset. A parameterized family of random
variables {Xa }a∈A is stochastically increasing in a, if
a ¡ a′ implies Xa ¡ Xa′ .
It+1 = f(It ; xt ; Dt ):
2.2. Notation for parametric analysis
If Dt exceeds It + xt , the excess demand is either not satis ed (as in [4,6,12]) and f(It ; xt ; Dt ) =
max{It + xt − Dt ; 0} or backlogged (as in [5,8]) and
We assume that for any time period t, random demand Dt may be any of the random variables in the
stochastically increasing collection {Da }a∈A , where
137
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
A is a nite poset. To this collection we associate the
family of distribution functions {Fa }a∈A . Moreover,
we will denote by a and a, the supremum and the in mum of the set A, respectively, i.e.
a = sup {a};
a = inf {a}:
a∈A
a∈A
We shall refer to a as the “zero” index to which we associate the trivial distribution F(x)=1, for x ∈ [0; ∞).
In other words, to the “zero” index we associate zero
demand. Similarly, we shall denote by D and F(·),
the
random demand and distribution function associated
to index a.
Let us denote by DT the set of T -long sequences
of independent random demands from period 0 up to
period T − 1. Formally,
DT =
TY
−1
{Da }a∈A :
t=0
To every sequence of the form (D0 ; D1 ; : : : ; DT −1 ) ∈
DT we associate the production planning problem
(PT (D1 ; : : : ; DT −1 )) de ned as follows:
(PT (D0 ; D1 ; : : : ; DT −1 ))
#
"T −1
X
t
Ct (xt ; It ; Dt )
min E
t=0
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; 2; : : : ; T − 1;
M ¿xt ¿0 integer:
We endow the cartesian product set DT with the product ordering “¡T ”, i.e.
(D0 ; D1 ; : : : ; DT −1 ) ¡T (D0′ ; D1′ ; : : : DT′ −1 )
⇔ Dt ¡ Dt′ t = 0; 1; : : : ; T − 1:
Hence, for every in nite sequence (D0 ; D1 ; : : : ;
DT −1 ; : : :) ∈ D we associate the problem
(P(D0 ; D1 ; : : : ; DT −1 ; : : :))
" T
#
X
t
Ct (xt ; It ; Dt )
min
lim E
T →∞
s:t
t=0
It+1 = f(It ; xt ; Dt );
M ¿xt ¿0 integer:
t = 0; 1; 2; : : : ;
In order to relate in nite and nite horizon problems
we de ne the embedding ,→T : DT 7→ D as follows:
(D0 ; D1 ; : : : ; DT −1 ) 7→ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :):
In words, we extend the nite sequence by appending
“zero” demand for all time periods after the planning
horizon T .
2.4. Standing assumptions
Our rst assumption, requires that if it exists, the
limit of nite horizon optimal production plans is an
optimal solution to the in nite horizon production
planning problem.
Assumption 1. Let (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :) ∈
D and {x0∗ (T ); x1∗ (T ); : : : ; xT∗ −1 (T )}T be a collection of
optimal solutions to problems (PT (D0 ; D1 ; : : : ; DT −1 ; 0;
0; : : :)).
If there exists a limit plan, say {x0∗ ; x1∗ ;
: : : ; xT∗ −1 ; xT∗ ; : : :}, then it is an optimal solution to
the in nite horizon problem (P(D0 ; D1 ; : : : ; DT −1 ; DT ;
DT +1 ; : : :)).
This assumption is fairly standard (see for instance
[2]) and holds for example when costs functions are
uniformly bounded and decision spaces are compact.
We shall denote by x0∗ (D0 ; D1 ; : : : ; DT −1 ) the rst period production decision in a production plan that
solves problem (PT (D0 ; D1 ; : : : ; DT −1 )).
Assumption 2 (Monotonicity of optimal plans).
For every T , there exists an optimal production plan
to problem (PT (D0 ; D1 ; : : : ; DT −1 )) whose rst period
decision is monotone in (D0 ; D1 ; : : : ; DT −1 ); i.e.
2.3. In nite horizon production planning
(D0 ; D1 ; : : : ; DT −1 ) ¡T (D0′ ; D1′ ; : : : ; DT′ −1 )
We now introduce the in nite horizon production
planning problem as a suitable extension to problem
(PT ).
As above, in order to parametrize the in nite horizon production planning
Q∞ problem we de ne the in nite product D = t=0 {Da }a∈A .
⇒ x0∗ (D0 ; D1 ; : : : ; DT −1 )¿x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 ):
Example. It is well known that in production–
inventory models with setup costs the optimal policy
structure is of the form (st ; St ) for stage t (see [7]
and the references therein). These thresholds are
monotone in stochastically increasing demand. It is
138
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
straightforward to infer that rst period decisions are
also monotone given that initial inventory is xed.
This is not necessarily the case for production decisions at later stages where stochastically larger
incoming inventories may neutralize the e ect of
greater thresholds.
DT +1 ; : : :) ∈ D and for every T ¿T ∗ , we have
x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
for every (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :) ∈ D such
that Dt = Dt′ for 06t ¡ T .
3. Review of solution concepts
4. Forecast horizon existence and computation
The gains in modeling accuracy a orded by an in nite horizon are severely compromised by the technical diculties in accurately forecasting problem parameters. This consideration motivates the problem of
nding a nite horizon such that the rst optimal decision for such horizon coincide with the in nite horizon
counterpart. If such a horizon exists (which is called
a solution horizon), it not only provides a rationale to
set such horizon as the decision makers planning horizon, but interestingly enough motivates a nite algorithm to solve an in nite problem via a rolling horizon
procedure.
We begin our analysis with the next straightforward
observation:
x0∗ (D0 ; D1 ; : : : ; DT −1 ) = x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :)
In words, solving the production planning problem
with nite horizon T is equivalent to solving the in nite horizon problem whereby we append “zero” demand after T .
Let us de ne the map x0 : D 7→ {0; 1; 2; : : : ; M } as
follows; for (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :) ∈ D:
x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
De nition 1. Planning Horizon T ∗ is called a Solution Horizon for demand forecast (D0 ; D1 ; : : : ; DT −1 ; DT ;
DT +1 ; : : :) ∈ D i for every T ¿T ∗ , we have
x0∗ (D0 ; D1 ; : : : ; DT −1 )
=x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :);
where x0∗ (D0 ; D1 ; : : : ; DT −1 ) and x0∗ (D0 ; D1 ; : : : ; DT −1 ;
DT ; DT +1 ; : : :) are rst period optimal decisions for
the T -planning horizon production planning problem
and the in nite horizon production planning problem,
respectively.
However, the solution horizon concept is of little
practical interest, for its computation may potentially
require an in nite forecast of data. Thus, the concept
of a forecast horizon (see for example, [1]), that is, a
long enough planning horizon that entails the insensitivity of rst period optimal production decision with
respect to changes in demand distribution at the tail
is very attractive to practitioners. In brief, in order to
compute the rst period optimal production decision,
the planner needs only forecast demand distributions
for a nite number of periods and this decision is insensitive to changes in demand distribution at the tail.
De nition 2. Planning horizon T ∗ is called a forecast
horizon for D if and only if for (D0 ; D1 ; : : : ; DT −1 ; DT ;
= lim x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :)
T →∞
Lemma 1. The map x0 : D 7→ {0; 1; 2; : : : ; M } is
well-de ned; continuous and monotone in D.
Proof. By the monotonicity of the optimal plan (Assumption 2) we have that
x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT )
¿x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0)
since this inequality is preserved by the embedding
“,→T ”, we have
x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :)
¿x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; 0; : : :);
thus by this monotonicity property and the fact that
rst period production decisions are bounded the limit
exists. Moreover, the map x0 (:) can also be seen as
the uniform limit of the functions x0T () de ned as
x0T (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :);
which are trivially continuous (by niteness of DT )
then it follows the map x0 (:) inherits continuity.
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
As for monotonicity, let us pick (D0 ; D1 ; : : : ; DT −1 ;
DT ; DT +1 ; : : :) ∈ D and (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ;
DT′ +1 ; : : :) ∈ D such that
Dt ¡ Dt′ ;
By continuity,
D;
: : :)
lim x0 (D0 ; D1 ; : : : ; DT −1 ; D;
T →∞
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
t = 0; 1; 2; : : : :
By de nition of x0 (:) there exists a planning horizon
Ta such that for T ¿Ta we have
x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0 ; D1 ; : : : ; DT −1 ):
Similarly, there exists a nite planning horizon Tb such
that for T ¿Tb we have
x0 (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
and
lim x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :)
T →∞
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :):
Hence, there exists a nite T ∗ such that for every
T ¿T ∗ we have
D;
: : :)
x0 (D0 ; D1 ; : : : ; DT −1 ; D;
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 ):
By monotonicity of optimal plans:
x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0 ; D1 ; : : : ; DT −1 )
139
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :):
In other words, for any forecast (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ;
DT′ +1 ; : : :) ∈ D such that Dt = Dt′ for 06t ¡ T :
x0 (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
¿x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 )
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :):
=x0 (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
for every T ¿max{Ta ; Tb }.
5. Stopping rule
In view of Assumption 1, the rst period decision
map de ned above “inherits” optimality and motivates
the next straightforward result.
Proof. Let us consider forecasts (D0 ; D1 ; : : : ; DT −1 ; D;
: : :) and (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :):
D;
Then by Lemma 1, it follows that
The probable multiplicity of monotone optimal production plans leads to the existence of many forecast
horizons. The minimality of the forecast horizon identi ed in the above existence proof is thus of great
interest.
In his work, Topkis [9,10] and recently compiled in
[11] developed a general monotonicity theory of optimal solutions using lattice programming techniques
that not surprisingly encompasses many production
planning models. The application of this theory in
the context of deterministic production planning with
convex costs ensures the existence of a smallest and
a largest optimal solution that are monotone in demand. Recently, Hopenhayn and Prescott [3] have applied Topkis’s results to stochastic dynamic programming models such as problem (PT ). These results will
be the basis for our selection procedure that we now
introduce.
Assuming costs are uniformly bounded as follows:
D;
: : :)
x0 (D0 ; D1 ; : : : ; DT −1 ; D;
sup ct (·) 6 c(·);
Corollary. For every (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
∈ D; there exists a solution horizon.
4.1. Forecast horizon existence
By exploiting the monotonicity of the map x0 (:),
we prove the existence of a forecast horizon.
Theorem. Under assumptions 1 and 2; there exist
a forecast horizon for problem (P(D0 ; D1 ; : : : ; DT −1 ;
DT ; DT +1 ; : : :)).
¿x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
¿x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :)
t
sup ht (·) 6 h(·);
t
sup pt (·) 6 p(·);
t
140
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
one can construct a pessimistic scenario, in which production and inventory holding costs are at their maximal levels, and random demand is exactly the one
associated with the supremum index, as de ned in
Section 2.2, namely
N
−1
X
lim sup
s:t
It+1 = f(It ; xt ; D);
N →∞ t=0
M ¿xt ¿0;
M ¿xt ¿0 integer;
t = 0; 1; 2; : : : ;
where
Z I +x
+ x − u) d F(u)
I; D)]
= c(x)
h(I
E[C(x;
+
0
Z ∞
p(−I
− x + u) d F(u):
+
I +x
The above problem is easy to solve by means of the
functional equation:
I; D)]
(DP) V (I ) = min {E[C(x;
M ¿x¿0
+ E[V (f(I; x; D))]}:
Let us now consider the next simpler nite-dimensional
problem:
#
"T −1
X
t Ct (xt ; It ; Dt ) + T +1 · V (IT )
min E
(P T )
t=0
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; : : : ; T − 1:
M ¿xt ¿0 integer;
By the application of Topkis [12] to problem (P T )
as in [3], there exist optimal plans to problem (P T )
such that their rst period production decisions are
monotone in monotone changes in stochastic demand.
Let us pick xT0 with the property that xT0 is the smallest
of such decisions.
Similarly, if we solve
"T −1
#
X
t
min E
Ct (xt ; It ; Dt )
t=0
(P T )
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; : : : ; T − 1;
M ¿xt ¿0 integer;
xT0 +1 ¿xT0
and is also the largest of all such solutions. By the
Forecast Horizon Existence Theorem, we know that
these sequences must meet, in other words the algorithm we are to describe below must stop after a nite
number of steps:
I; D)]
E[C(x;
t
min
we know that there exists an optimal plan such that its
rst period action say, xT0 is monotonically increasing
in T , i.e.
Step 1: Solve functional equation (DP). T = 1.
Step 2: Solve (P T ) and (P T ) for xT0 and xT0 .
Step 3: If xT0 = xT0 then Stop.
Else T = T + 1; Go to Step 2.
Proposition 1. Let T ∗ be the forecast horizon detected by the above procedure. Then; T ∗ is also the
minimal forecast horizon.
Proof. By contradiction, let us assume there exists
T ¡ T ∗ such that T is the minimal forecast horizon.
By hypothesis,
xT0 ¿ xT0 :
But since xT0 is the rst period action of the smallest
optimal solution to problem (P T ) and xT0 is the rst period action of the largest optimal solution to problem
(P T ), this implies that the above inequality is valid
for any chosen pair of optimal solutions to problems
(P T ) and (P T ), but this contradicts T being a forecast
horizon.
References
[1] C. Bes, S. Sethi, Concepts of forecast and decision horizons:
applications to dynamic stochastic optimization problems,
Math. Oper. Res. 13 (1987) 295–310.
[2] D. Heyman, M. Sobel, Stochastic models in operations
research, Vol. II, McGraw-Hill, New York, 1984.
[3] H. Hopenhayn, E. Prescott, Stochastic monotonicity and
stationary distributions for dynamic economies, Econometrica
60 (1992) 1307–1406.
[4] S. Karlin, Dynamic inventory policy with varying stochastic
demands, Management Sci. 6 (3) (1960) 231–258.
[5] P. Kleindorfer, H. Kunreuther, Stochastic horizons for the
aggregate planning problem, Management Sci. 25 (5) (1978)
1020–1031.
[6] T. Morton, The nonstationary in nite horizon inventory
problem, Management Sci. 24 (14) (1978) 981–993.
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[7] S. Sethi, F. Cheng, Optimality of (s; S) policies in inventory
models with Markovian demand, Oper. Res. 45 (6) (1997)
931–939.
[8] R.L. Smith, R. Zhang, In nite horizon production planning in
time varying systems with convex production and inventory
costs, Management Sci. 44 (9) (1998) 1313–1320.
[9] D. Topkis, Ordered optimal solutions, Ph.D Thesis, Stanford
University, 1969.
141
[10] D. Topkis, Minimizing a submodular function on a lattice,
Oper. Res. 26 (2) (1978) 305–321.
[11] D. Topkis, Supermodularity and Complementarity, Princeton
University Press, Princeton, NJ, 1998.
[12] A. Veinott, Optimal policy in a dynamic single-product
nonstationary inventory model with several demand classes,
Oper. Res. 13 (5) (1965) 761–778.
www.elsevier.com/locate/dsw
Solving nonstationary in nite horizon stochastic production
planning problems
Alfredo Garcia ∗ , Robert L. Smith 1
Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Received 1 October 1998; received in revised form 1 June 2000
Abstract
Forecast horizons are de ned as long enough planning horizons that ensure agreement of rst period optimal production
decisions of nite and in nite horizon problems regardless of changes in future demand. In this paper, we prove forecast horizon
existence and provide computational procedures for production planning problems that satisfy the following monotonicity
property: for any xed nite planning horizon, there exist rst period optimal solutions that are monotone with respect to
c 2000 Elsevier Science B.V. All rights reserved.
monotone changes in stochastic demand.
Keywords: Production planning; In nite horizon optimal control; Decision and forecast horizons
1. Introduction
Production planners face substantial uncertainty in
the demand for their products that comes from a variety of sources. For instance, demand could be sensitive to varying economic conditions such as GNP
or interest rates, or alternatively, technical innovation
may imply unexpected early obsolescence. Moreover,
it is usually the decisions in the rst few periods that
are of immediate concern for the decision maker and
forecasting is more dicult and costlier for problem
parameters farther into the future.
∗ Correspondence address: The Brattle Group, 1133 20th Street,
Suite 800, Washington, DC 20036, USA.
1 This work was supported in part by NSF grants DDM-9214894
and DMI-9713723.
Motivated by these issues, the concept of a forecast
horizon (see for example, [1]) has long been studied
in the literature. The idea is that problem parameters
changes far enough o should not a ect the optimal
decisions of the rst few periods.
In this paper, we prove forecast horizon existence
and provide computational procedures for production planning problems that satisfy the following
monotonicity property: for any xed nite planning
horizon, there exist rst period optimal solutions that
are monotone with respect to monotone changes in
stochastic demand.
Monotonicity of optimal production plans under
stochastic demand has been established under different modeling assumptions by Karlin [4], Veinott
[12], Kleindorfer and Kunreuther [5], Morton [6],
Sethi and Cheng [7] among others. We will not
attempt in this paper to prove new monotonicity
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 4 9 - 3
136
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
results. Rather, by assuming that monotonicity of
optimal solutions is given, we show how the in nite horizon non-stationary versions of these production planning problems can be exactly solved
by rolling horizon procedures. Smith and Zhang
[8] have recently exploited the above-mentioned
monotonicity property to prove existence of forecast horizons in a deterministic setting. Their approach depends crucially on the monotonicity of
the production decisions at all time periods, a feature that may be too restrictive in many models
(e.g. set-up costs), since higher inventory levels
from earlier production decisions could substitute higher production decisions in later periods.
Moreover, the formula they provide to compute
a forecast horizon does not ensure that it is the
minimal forecast horizon, a highly desirable feature since longer forecasts are costlier and less
accurate.
In this paper, we show that monotonicity of only
the rst production decision is sucient to prove existence of forecast horizons. Moreover, our emphasis
on extreme monotone optimal solutions allows us
to e ectively detect the minimal forecast horizon by
means of a stopping rule with an embedded selection
procedure.
f(It ; xt ; Dt ) = It + xt − Dt . The one-stage total cost
incurred is
Ct (xt ; It ; Dt ) = ct (xt ) + ht (max{0; It+1 })
+pt (max{0; −It+1 });
where ct (·) is the production cost function, ht (·) inventory holding cost and pt (·) is either the opportunity cost for demand not satis ed or the backlogging
cost function.
We assume that demand at time period t, Dt is a
non-negative random variable with probability distribution Ft (·).
Hence, the expected t-stage cost-to-go incurred is
Z I +x
ht (I + x − u) dFt (u)
E[Ct (x; I; Dt )] = ct (x) +
Z ∞ 0
pt (−I − x + u) dFt (u):
+
I +x
For a planning horizon of T -periods, the production
planning problem is then
"T −1
#
X
t
min E
Ct (xt ; It ; Dt )
t=0
(PT )
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; 2; : : : ; T − 1;
M ¿xt ¿0
2. Problem formulation
Consider a single product rm where a decision for
production must be made at the beginning of each
period t, t = 0; 1; 2; : : : . Customer orders arrive during each time interval. We will agree to call the “demand” realized during time period t, the total number of customer orders received during time interval
[t; t + 1) and shall denote it by the random variable
Dt .
Thus, if at the beginning of time period t, there are It
units available of inventory and a production decision
of xt is made, the inventory–production system follows
the equation
integer;
where M is the maximal production capacity and ∈
(0; 1) is the discount factor.
2.1. Stochastic ordering
We say that a random variable X with probability distribution F(·) is stochastically larger than a
random variable Y with probability distribution G(·),
written X ¡ Y , if for every x ∈ R:
F(x)6G(x):
Let A be a poset. A parameterized family of random
variables {Xa }a∈A is stochastically increasing in a, if
a ¡ a′ implies Xa ¡ Xa′ .
It+1 = f(It ; xt ; Dt ):
2.2. Notation for parametric analysis
If Dt exceeds It + xt , the excess demand is either not satis ed (as in [4,6,12]) and f(It ; xt ; Dt ) =
max{It + xt − Dt ; 0} or backlogged (as in [5,8]) and
We assume that for any time period t, random demand Dt may be any of the random variables in the
stochastically increasing collection {Da }a∈A , where
137
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
A is a nite poset. To this collection we associate the
family of distribution functions {Fa }a∈A . Moreover,
we will denote by a and a, the supremum and the in mum of the set A, respectively, i.e.
a = sup {a};
a = inf {a}:
a∈A
a∈A
We shall refer to a as the “zero” index to which we associate the trivial distribution F(x)=1, for x ∈ [0; ∞).
In other words, to the “zero” index we associate zero
demand. Similarly, we shall denote by D and F(·),
the
random demand and distribution function associated
to index a.
Let us denote by DT the set of T -long sequences
of independent random demands from period 0 up to
period T − 1. Formally,
DT =
TY
−1
{Da }a∈A :
t=0
To every sequence of the form (D0 ; D1 ; : : : ; DT −1 ) ∈
DT we associate the production planning problem
(PT (D1 ; : : : ; DT −1 )) de ned as follows:
(PT (D0 ; D1 ; : : : ; DT −1 ))
#
"T −1
X
t
Ct (xt ; It ; Dt )
min E
t=0
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; 2; : : : ; T − 1;
M ¿xt ¿0 integer:
We endow the cartesian product set DT with the product ordering “¡T ”, i.e.
(D0 ; D1 ; : : : ; DT −1 ) ¡T (D0′ ; D1′ ; : : : DT′ −1 )
⇔ Dt ¡ Dt′ t = 0; 1; : : : ; T − 1:
Hence, for every in nite sequence (D0 ; D1 ; : : : ;
DT −1 ; : : :) ∈ D we associate the problem
(P(D0 ; D1 ; : : : ; DT −1 ; : : :))
" T
#
X
t
Ct (xt ; It ; Dt )
min
lim E
T →∞
s:t
t=0
It+1 = f(It ; xt ; Dt );
M ¿xt ¿0 integer:
t = 0; 1; 2; : : : ;
In order to relate in nite and nite horizon problems
we de ne the embedding ,→T : DT 7→ D as follows:
(D0 ; D1 ; : : : ; DT −1 ) 7→ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :):
In words, we extend the nite sequence by appending
“zero” demand for all time periods after the planning
horizon T .
2.4. Standing assumptions
Our rst assumption, requires that if it exists, the
limit of nite horizon optimal production plans is an
optimal solution to the in nite horizon production
planning problem.
Assumption 1. Let (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :) ∈
D and {x0∗ (T ); x1∗ (T ); : : : ; xT∗ −1 (T )}T be a collection of
optimal solutions to problems (PT (D0 ; D1 ; : : : ; DT −1 ; 0;
0; : : :)).
If there exists a limit plan, say {x0∗ ; x1∗ ;
: : : ; xT∗ −1 ; xT∗ ; : : :}, then it is an optimal solution to
the in nite horizon problem (P(D0 ; D1 ; : : : ; DT −1 ; DT ;
DT +1 ; : : :)).
This assumption is fairly standard (see for instance
[2]) and holds for example when costs functions are
uniformly bounded and decision spaces are compact.
We shall denote by x0∗ (D0 ; D1 ; : : : ; DT −1 ) the rst period production decision in a production plan that
solves problem (PT (D0 ; D1 ; : : : ; DT −1 )).
Assumption 2 (Monotonicity of optimal plans).
For every T , there exists an optimal production plan
to problem (PT (D0 ; D1 ; : : : ; DT −1 )) whose rst period
decision is monotone in (D0 ; D1 ; : : : ; DT −1 ); i.e.
2.3. In nite horizon production planning
(D0 ; D1 ; : : : ; DT −1 ) ¡T (D0′ ; D1′ ; : : : ; DT′ −1 )
We now introduce the in nite horizon production
planning problem as a suitable extension to problem
(PT ).
As above, in order to parametrize the in nite horizon production planning
Q∞ problem we de ne the in nite product D = t=0 {Da }a∈A .
⇒ x0∗ (D0 ; D1 ; : : : ; DT −1 )¿x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 ):
Example. It is well known that in production–
inventory models with setup costs the optimal policy
structure is of the form (st ; St ) for stage t (see [7]
and the references therein). These thresholds are
monotone in stochastically increasing demand. It is
138
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
straightforward to infer that rst period decisions are
also monotone given that initial inventory is xed.
This is not necessarily the case for production decisions at later stages where stochastically larger
incoming inventories may neutralize the e ect of
greater thresholds.
DT +1 ; : : :) ∈ D and for every T ¿T ∗ , we have
x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
for every (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :) ∈ D such
that Dt = Dt′ for 06t ¡ T .
3. Review of solution concepts
4. Forecast horizon existence and computation
The gains in modeling accuracy a orded by an in nite horizon are severely compromised by the technical diculties in accurately forecasting problem parameters. This consideration motivates the problem of
nding a nite horizon such that the rst optimal decision for such horizon coincide with the in nite horizon
counterpart. If such a horizon exists (which is called
a solution horizon), it not only provides a rationale to
set such horizon as the decision makers planning horizon, but interestingly enough motivates a nite algorithm to solve an in nite problem via a rolling horizon
procedure.
We begin our analysis with the next straightforward
observation:
x0∗ (D0 ; D1 ; : : : ; DT −1 ) = x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :)
In words, solving the production planning problem
with nite horizon T is equivalent to solving the in nite horizon problem whereby we append “zero” demand after T .
Let us de ne the map x0 : D 7→ {0; 1; 2; : : : ; M } as
follows; for (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :) ∈ D:
x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
De nition 1. Planning Horizon T ∗ is called a Solution Horizon for demand forecast (D0 ; D1 ; : : : ; DT −1 ; DT ;
DT +1 ; : : :) ∈ D i for every T ¿T ∗ , we have
x0∗ (D0 ; D1 ; : : : ; DT −1 )
=x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :);
where x0∗ (D0 ; D1 ; : : : ; DT −1 ) and x0∗ (D0 ; D1 ; : : : ; DT −1 ;
DT ; DT +1 ; : : :) are rst period optimal decisions for
the T -planning horizon production planning problem
and the in nite horizon production planning problem,
respectively.
However, the solution horizon concept is of little
practical interest, for its computation may potentially
require an in nite forecast of data. Thus, the concept
of a forecast horizon (see for example, [1]), that is, a
long enough planning horizon that entails the insensitivity of rst period optimal production decision with
respect to changes in demand distribution at the tail
is very attractive to practitioners. In brief, in order to
compute the rst period optimal production decision,
the planner needs only forecast demand distributions
for a nite number of periods and this decision is insensitive to changes in demand distribution at the tail.
De nition 2. Planning horizon T ∗ is called a forecast
horizon for D if and only if for (D0 ; D1 ; : : : ; DT −1 ; DT ;
= lim x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :)
T →∞
Lemma 1. The map x0 : D 7→ {0; 1; 2; : : : ; M } is
well-de ned; continuous and monotone in D.
Proof. By the monotonicity of the optimal plan (Assumption 2) we have that
x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT )
¿x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0)
since this inequality is preserved by the embedding
“,→T ”, we have
x0∗ (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :)
¿x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; 0; : : :);
thus by this monotonicity property and the fact that
rst period production decisions are bounded the limit
exists. Moreover, the map x0 (:) can also be seen as
the uniform limit of the functions x0T () de ned as
x0T (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0 ; D1 ; : : : ; DT −1 ; 0; 0; : : :);
which are trivially continuous (by niteness of DT )
then it follows the map x0 (:) inherits continuity.
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
As for monotonicity, let us pick (D0 ; D1 ; : : : ; DT −1 ;
DT ; DT +1 ; : : :) ∈ D and (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ;
DT′ +1 ; : : :) ∈ D such that
Dt ¡ Dt′ ;
By continuity,
D;
: : :)
lim x0 (D0 ; D1 ; : : : ; DT −1 ; D;
T →∞
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
t = 0; 1; 2; : : : :
By de nition of x0 (:) there exists a planning horizon
Ta such that for T ¿Ta we have
x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0 ; D1 ; : : : ; DT −1 ):
Similarly, there exists a nite planning horizon Tb such
that for T ¿Tb we have
x0 (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
and
lim x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :)
T →∞
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :):
Hence, there exists a nite T ∗ such that for every
T ¿T ∗ we have
D;
: : :)
x0 (D0 ; D1 ; : : : ; DT −1 ; D;
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 ):
By monotonicity of optimal plans:
x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
=x0∗ (D0 ; D1 ; : : : ; DT −1 )
139
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :):
In other words, for any forecast (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ;
DT′ +1 ; : : :) ∈ D such that Dt = Dt′ for 06t ¡ T :
x0 (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
¿x0∗ (D0′ ; D1′ ; : : : ; DT′ −1 )
=x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :):
=x0 (D0′ ; D1′ ; : : : ; DT′ −1 ; DT′ ; DT′ +1 ; : : :)
for every T ¿max{Ta ; Tb }.
5. Stopping rule
In view of Assumption 1, the rst period decision
map de ned above “inherits” optimality and motivates
the next straightforward result.
Proof. Let us consider forecasts (D0 ; D1 ; : : : ; DT −1 ; D;
: : :) and (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :):
D;
Then by Lemma 1, it follows that
The probable multiplicity of monotone optimal production plans leads to the existence of many forecast
horizons. The minimality of the forecast horizon identi ed in the above existence proof is thus of great
interest.
In his work, Topkis [9,10] and recently compiled in
[11] developed a general monotonicity theory of optimal solutions using lattice programming techniques
that not surprisingly encompasses many production
planning models. The application of this theory in
the context of deterministic production planning with
convex costs ensures the existence of a smallest and
a largest optimal solution that are monotone in demand. Recently, Hopenhayn and Prescott [3] have applied Topkis’s results to stochastic dynamic programming models such as problem (PT ). These results will
be the basis for our selection procedure that we now
introduce.
Assuming costs are uniformly bounded as follows:
D;
: : :)
x0 (D0 ; D1 ; : : : ; DT −1 ; D;
sup ct (·) 6 c(·);
Corollary. For every (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
∈ D; there exists a solution horizon.
4.1. Forecast horizon existence
By exploiting the monotonicity of the map x0 (:),
we prove the existence of a forecast horizon.
Theorem. Under assumptions 1 and 2; there exist
a forecast horizon for problem (P(D0 ; D1 ; : : : ; DT −1 ;
DT ; DT +1 ; : : :)).
¿x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; DT +1 ; : : :)
¿x0 (D0 ; D1 ; : : : ; DT −1 ; DT ; 0; 0; : : :)
t
sup ht (·) 6 h(·);
t
sup pt (·) 6 p(·);
t
140
A. Garcia, R.L. Smith / Operations Research Letters 27 (2000) 135–141
one can construct a pessimistic scenario, in which production and inventory holding costs are at their maximal levels, and random demand is exactly the one
associated with the supremum index, as de ned in
Section 2.2, namely
N
−1
X
lim sup
s:t
It+1 = f(It ; xt ; D);
N →∞ t=0
M ¿xt ¿0;
M ¿xt ¿0 integer;
t = 0; 1; 2; : : : ;
where
Z I +x
+ x − u) d F(u)
I; D)]
= c(x)
h(I
E[C(x;
+
0
Z ∞
p(−I
− x + u) d F(u):
+
I +x
The above problem is easy to solve by means of the
functional equation:
I; D)]
(DP) V (I ) = min {E[C(x;
M ¿x¿0
+ E[V (f(I; x; D))]}:
Let us now consider the next simpler nite-dimensional
problem:
#
"T −1
X
t Ct (xt ; It ; Dt ) + T +1 · V (IT )
min E
(P T )
t=0
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; : : : ; T − 1:
M ¿xt ¿0 integer;
By the application of Topkis [12] to problem (P T )
as in [3], there exist optimal plans to problem (P T )
such that their rst period production decisions are
monotone in monotone changes in stochastic demand.
Let us pick xT0 with the property that xT0 is the smallest
of such decisions.
Similarly, if we solve
"T −1
#
X
t
min E
Ct (xt ; It ; Dt )
t=0
(P T )
s:t
It+1 = f(It ; xt ; Dt );
t = 0; 1; : : : ; T − 1;
M ¿xt ¿0 integer;
xT0 +1 ¿xT0
and is also the largest of all such solutions. By the
Forecast Horizon Existence Theorem, we know that
these sequences must meet, in other words the algorithm we are to describe below must stop after a nite
number of steps:
I; D)]
E[C(x;
t
min
we know that there exists an optimal plan such that its
rst period action say, xT0 is monotonically increasing
in T , i.e.
Step 1: Solve functional equation (DP). T = 1.
Step 2: Solve (P T ) and (P T ) for xT0 and xT0 .
Step 3: If xT0 = xT0 then Stop.
Else T = T + 1; Go to Step 2.
Proposition 1. Let T ∗ be the forecast horizon detected by the above procedure. Then; T ∗ is also the
minimal forecast horizon.
Proof. By contradiction, let us assume there exists
T ¡ T ∗ such that T is the minimal forecast horizon.
By hypothesis,
xT0 ¿ xT0 :
But since xT0 is the rst period action of the smallest
optimal solution to problem (P T ) and xT0 is the rst period action of the largest optimal solution to problem
(P T ), this implies that the above inequality is valid
for any chosen pair of optimal solutions to problems
(P T ) and (P T ), but this contradicts T being a forecast
horizon.
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