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Solving nonstationary innite horizon stochastic production
planning problems
Alfredo Garcia
∗, Robert L. Smith
1Department 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 dened as long enough planning horizons that ensure agreement of rst period optimal production decisions of nite and innite 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 monotone changes in stochastic demand. c2000 Elsevier Science B.V. All rights reserved.
Keywords:Production planning; Innite horizon optimal control; Decision and forecast horizons
1. Introduction
Production planners face substantial uncertainty in the demand for their products that comes from a va-riety of sources. For instance, demand could be sen-sitive 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.
1This work was supported in part by NSF grants DDM-9214894 and DMI-9713723.
Motivated by these issues, the concept of aforecast horizon(see for example, [1]) has long been studied in the literature. The idea is that problem parameters changes far enough o should not aect the optimal decisions of the rst few periods.
In this paper, we prove forecast horizon existence and provide computational procedures for produc-tion 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 dif-ferent 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
results. Rather, by assuming that monotonicity of optimal solutions is given, we show how the in-nite horizon non-stationary versions of these pro-duction 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 fore-cast horizons in a deterministic setting. Their ap-proach depends crucially on the monotonicity of the production decisions at all time periods, a fea-ture that may be too restrictive in many models (e.g. set-up costs), since higher inventory levels from earlier production decisions could substi-tute 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 fea-ture since longer forecasts are costlier and less accurate.
In this paper, we show that monotonicity of only therstproduction decision is sucient to prove ex-istence of forecast horizons. Moreover, our emphasis on extreme monotone optimal solutions allows us to eectively detect theminimalforecast horizon by means of a stopping rule with an embedded selection procedure.
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 dur-ingeach time interval. We will agree to call the “de-mand” realized during time period t, the total num-ber 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 periodt, there areIt units available of inventory and a production decision ofxtis made, the inventory–production system follows the equation
It+1=f(It; xt; Dt):
If Dt exceeds It + xt, the excess demand is ei-ther not satised (as in [4,6,12]) and f(It; xt; Dt) = max{It+xt−Dt;0}or backlogged (as in [5,8]) and
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});
wherect(·) is the production cost function, ht(·) in-ventory holding cost andpt(·) is either the opportu-nity cost for demand not satised or the backlogging cost function.
We assume that demand at time periodt,Dt is a non-negative random variable with probability distri-butionFt(·).
Hence, the expectedt-stage cost-to-go incurred is
E[Ct(x; I; Dt)] =ct(x) + Z I+x
0
ht(I+x−u) dFt(u)
+ Z ∞
I+x
pt(−I−x+u) dFt(u):
For a planning horizon ofT-periods, the production planning problem is then
(PT)
min E
"T−1 X
t=0
tCt(xt; It; Dt) #
s:t It+1=f(It; xt; Dt);
t= 0;1;2; : : : ; T−1;
M¿xt¿0 integer;
whereM is the maximal production capacity and∈ (0;1) is the discount factor.
2.1. Stochastic ordering
We say that a random variable X with probabil-ity distribution F(·) is stochastically larger than a random variableY with probability distributionG(·), writtenX ¡Y, if for everyx∈R:
F(x)6G(x):
LetAbe a poset. A parameterized family of random
variables{Xa}a∈Aisstochastically increasing ina, if
a¡a′ implies
Xa¡Xa′.
2.2. Notation for parametric analysis
Ais aniteposet. To this collection we associate the
family of distribution functions{Fa}a∈A. Moreover,
we will denote by aanda, the supremum and the in-mum of the setA, respectively, i.e.
a= sup a∈A
{a}; a= inf a∈A{a}:
We shall refer toaas the “zero” index to which we as-sociate the trivial distributionF(x)=1, forx∈[0;∞). In other words, to the “zero” index we associate zero demand. Similarly, we shall denote by Dand F(·), the random demand and distribution function associated to index a.
Let us denote byD
T the set of T-long sequences ofindependent random demandsfrom period 0 up to periodT−1. Formally,
DT=
DT we associate the production planning problem
(PT(D1; : : : ; DT−1)) dened as follows:
We endow the cartesian product setDT with the
prod-uct ordering “¡T”, i.e.
riod production decision in a production plan that solves problem (PT(D0; D1; : : : ; DT−1)).
2.3. Innite horizon production planning
We now introduce the innite horizon production planning problem as a suitable extension to problem (PT).
As above, in order to parametrize the innite hori-zon production planning problem we dene the in-nite productD=Q∞
In order to relate innite and nite horizon problems we dene the embedding,→T:DT 7→Das 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 horizonT.
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 innite horizon production planning problem.
This assumption is fairly standard (see for instance [2]) and holds for example when costs functions are uniformly bounded and decision spaces are compact.
Assumption 2(Monotonicity of optimal plans).
For everyT, there exists an optimal production plan to problem (PT(D0; D1; : : : ; DT−1)) whose rst period
straightforward to infer that rst period decisions are also monotone given that initial inventory is xed. This is not necessarily the case for production de-cisions at later stages where stochastically larger incoming inventories may neutralize the eect of greater thresholds.
3. Review of solution concepts
The gains in modeling accuracy aorded by an in-nite horizon are severely compromised by the tech-nical diculties in accurately forecasting problem pa-rameters. This consideration motivates the problem of nding a nite horizon such that the rst optimal deci-sion for such horizon coincide with the innite 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 hori-zon, but interestingly enough motivates a nite algo-rithm to solve an innite problem via a rolling horizon procedure.
Denition 1. Planning Horizon T∗
is called a Solu-theT -planning horizon production planning problem and the innite horizon production planning problem, respectively.
However, the solution horizon concept is of little practical interest, for its computation may potentially require an innite forecast of data. Thus, the concept of a forecast horizon (see for example, [1]), that is, a long enough planning horizon that entails the insensi-tivity 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 in-sensitive to changes in demand distribution at the tail.
Denition 2. Planning horizonT∗
is called a forecast
4. Forecast horizon existence and computation
We begin our analysis with the next straightforward observation:
x∗
0(D0; D1; : : : ; DT−1) =x0∗(D0; D1; : : : ; DT−1;0;0; : : :)
In words, solving the production planning problem with nite horizonT is equivalent to solving the in-nite horizon problem whereby we append “zero” de-mand afterT.
well-dened;continuous and monotone inD.
Proof. By the monotonicity of the optimal plan (As-sumption 2) we have that
x∗
0(D0; D1; : : : ; DT−1; DT)
¿x∗
0(D0; D1; : : : ; DT−1;0)
since this inequality is preserved by the embedding “,→T”, we have
x∗
0(D0; D1; : : : ; DT−1; DT;0;0; : : :)
¿x∗
0(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 mapx0(:) can also be seen as
the uniform limit of the functionsxT
0() dened as
xT
0(D0; D1; : : : ; DT−1; DT; DT+1; : : :) =x∗
0(D0; D1; : : : ; DT−1;0;0; : : :);
which are trivially continuous (by niteness ofDT)
As for monotonicity, let us pick (D0; D1; : : : ; DT−1;
By denition ofx0(:) there exists a planning horizon
Ta such that forT¿Tawe have
x0(D0; D1; : : : ; DT−1; DT; DT+1; : : :)
=x∗
0(D0; D1; : : : ; DT−1):
Similarly, there exists a nite planning horizonTbsuch that forT¿Tb we have By monotonicity of optimal plans:
x0(D0; D1; : : : ; DT−1; DT; DT+1; : : :)
In view of Assumption 1, the rst period decision map dened above “inherits” optimality and motivates the next straightforward result.
Corollary. For every(D0; D1; : : : ; DT−1; DT; DT+1; : : :) ∈D;there exists asolutionhorizon.
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; : : :)).
Proof. Let us consider forecasts (D0; D1; : : : ; DT−1;D;
D; : : :) and (D0; D1; : : : ; DT−1; DT;0;0; : : :): Then by Lemma 1, it follows that
x0(D0; D1; : : : ; DT−1;D; D; : : : )
Hence, there exists a nite T∗
such that for every
In other words, for any forecast (D′
0; D
The probable multiplicity of monotone optimal pro-duction plans leads to the existence of many forecast horizons. Theminimalityof the forecast horizon iden-tied 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 op-timal 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 asmallestand alargest optimal solution that are monotone in de-mand. Recently, Hopenhayn and Prescott [3] have ap-plied Topkis’s results to stochastic dynamic program-ming 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:
one can construct apessimisticscenario, in which pro-duction and inventory holding costs are at their max-imal levels, and random demand is exactly the one associated with the supremum index, as dened in Section 2.2, namely
min lim sup N→∞
N−1 X
t=0
tE[ C(x; I;D)]
s:t It+1=f(It; xt;D);
M¿xt¿0;
M¿xt¿0 integer; t= 0;1;2; : : : ;
where
E[ C(x; I;D)] = c(x) + Z I+x
0
h(I+x−u) d F(u)
+ Z ∞
I+x
p(−I−x+u) d F(u):
The above problem is easy to solve by means of the functional equation:
(DP)V(I) = min
M¿x¿0{E[ C(x; I;
D)]
+E[V(f(I; x;D))]}:
Let us now consider the next simpler nite-dimensional problem:
( PT)
min E
"T−1 X
t=0
tCt(xt; It; Dt) +T+1·V(IT) #
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 ( PT) as in [3], there exist optimal plans to problem ( PT) 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 thesmallest of such decisions.
Similarly, if we solve
(PT)
min E
"T−1 X
t=0
tCt(xt; It; Dt) #
s:t It+1=f(It; xt; Dt);
t= 0;1; : : : ; T−1;
M¿xt¿0 integer;
we know that there exists an optimal plan such that its rst period action say,xT
0 is monotonically increasing inT, i.e.
xT+10 ¿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 algo-rithm we are to describe below must stop after a nite number of steps:
Step1: Solve functional equation (DP).T= 1. Step2: Solve ( PT) and (PT) for xT0 andxT0. Step3: If xT0 =xT0 then Stop.
ElseT=T + 1; Go to Step 2.
Proposition 1. Let T∗
be the forecast horizon de-tected by the above procedure. Then; T∗
is also the minimal forecast horizon.
Proof. By contradiction, let us assume there exists
T ¡ T∗
such thatT is theminimal forecast horizon. By hypothesis,
xT0¿ xT0:
But since xT0 is the rst period action of thesmallest optimal solution to problem ( PT) andxT0 is the rst pe-riod action of thelargestoptimal solution to problem (PT), this implies that the above inequality is valid for any chosen pair of optimal solutions to problems ( PT) and (PT), but this contradictsT 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.
[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, Innite 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.
[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.
