Production Planning for a Stochastic Demand Process

Production Planning for a Stochastic Demand ProcessAuthor(s): Nicholas J. Gonedes and Zvi LieberSource: Operations Research, Vol. 22, No. 4 (Jul. - Aug., 1974), pp. 771-787Published by: INFORMSStable URL: http://www.jstor.org/stable/169953 .

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Production Planning for a Stochastic Demand Process

Nicholas J. Gonedes

University of Chicago, Chicago, Illinois

and

Zvi Lieber

Tel Aviv University, Tel Aviv, Israel

(Received November 11, 1971)

This paper deals with planning production of a single good over a prescribed finite intervai of time [0, T]. The costs considered within the problexn are production and holding costs; holding costs are incurred for negative inventory (caused by backlogged sales) and positive inventory (caused by production in excess of demand). The demand for the product is treated as a stochastic process. Problem formulation and optimization use tools from continuous- time stochastic variational calculus and control theory. Among its results, the paper proves the uniqueness of the optimal solution and provides an algorithm for the problem.

T HIS PAPER deals with planning production of a single good over a prescribed finite interval of time [0, T]. The costs considered within the problem are

production and holding costs; holding costs are incurred for negative inventory (caused by backlogged sales) and positive inventory (caused by production in excess of demand). Several treatments of related deterministic production-planning problems (discrete-time and continuous-time) are available; these include, as examples, the treatments provided in: MODIGLIANI AND HOHN,(311 ARROW AND

KARLIN,121 BOWMAN,161 JOHNSON,1211 SPRZEUZKOUSKI,1351 EPPEN AND GOULD,1161 and LIEBER.126'271 In addition to invoking different assumptions regarding the applicable cost functional, this paper differs from the available work by allowing the demand for the product over the horizon [0, T] to be a stochastic process. One way of introducing stochastic features into a continuous-time problem is to invoke Markov-process asgumptions and related optimization tools, as has been done in some work on stochastic optimal control; see, e.g., ASTR6M,131 GONEDES,118,191

MARTIN-LoF, [291 and MILLER.101 An alternative approach (which may be used for a variety of stochastic processes) is to make direct use of tools from stochastic variational calculus and control theory, such as those discussed in KUSHNER,123-251 and BAUM;14'51 this work inspired the approach used in this paper.

The organization of the remainder of this paper is as follows: In Section I we introduce the notation and assumptions; problem formulation is treated in Section II and problem optimization in Section III; Section IV gives additional definitions, characterizations of production plans, and theorems; an algorithm for computing an optimal production plan is presented in Section V; finally, additional interpretive and analytical remarks are provided in Section VI.

771



772 Nicholes J. Gonedes and Zvi Lieber

I. NOTATION AND ASSUMPTIONS

Notation

The following notation will be used in this paper; all functions of time pertain to t4[O, T] and tildes denote (measurable) stochastic processes:

v (t) the rate of production at time t. F[v (t)] =the cost of production, per unit of time, when the rate of production is

v (t). 7(t) net inventory at time t. [Backlogging exists at t if I(t) <0.]

h[j (t)] =carrying cost per unit of net inventory per unit of time. If I(t) <0, h[I (t)] is the cost of backlogged sales; if I (t) >0, h[I (t)] is the inventory- holding (storage) cost.

'(t) =cumulative demand at t. y (t) =cumulative production at t, i.e., y (t) = f v (s) ds. + (t) = the expected value of cumulative demand at t. Q (t) = a vector consisting of the moments of + (t).

ft )= the distribution function of k (t); ft (k) = prf1 (t) < k], where pr() denotes probability.

In the sequel, primes will often be used to denote derivatives, e.g., F' (y) = &F ()/0y, F" (y) = 2F (- )/dy2, etc. A dot above a variable will often be used to denote a derivative with respect to time, e.g., v (t) av (* )/at, v (t) = &2v (. )/at2, etc. The notation O(w) will be used to denote a quantity that goes to zero faster than w, i.e., lim""o O (w ) /w = O.

Assumptions

A.1. The value of I(0) is known and equal to Io. A.2. The (measurable) stochastic process { J(t); 0< t< T3 is stochastically

continuous in the mean-squared sense; that is, for any tdfO, T], limh-OE[+ (t+h) -

4>(t)]2 = 0 for all t+hd[O, T], where E denotes the expectation operator. (See Note 1.)

A.3. The function h(.) is strictly convex [h"( )>0 and h( )>0] and there exists a finite N*> 2 such that hN(*) is continuous for all N N* and hM(* ) =0 for all M>N*.

A.4. Each rth moment of + (t) is continuous in t, for all t, for all r <5N*, where N* is as defined in (A.3).

A.5. The production-cost function is such that F (O) =0, F' ( ) >O, F" ( ) >O, and F"' (*) is continuous in v (t), for all v (t) > 0.

A.6. E{h[I (t)]2} < 00, for all tE[O, T].

II. PROBLEM FORMULATION

WE ASSUME THAT the objective function applicable to the production-planning process is minimization of expected costs. Thus, the problem may be expressed as



Production Planning 773

fT minfv(t); te[o,T]) E] fF[v(t)]+h[I(t)]B dt, (1)

for which we require v (t) _ 0, Vt. By definition, t

I(t)=Io+f v(s) ds-k(t), (2)

or, in symbolic form, di (t) = v (t) dt-dk (t), (3)

where (3) is the stochastic differential equation underlying (2). Note that (1 ) requires selection of the optimal production plan, { v* (t); tE[0, T] },

at time t=O. Accordingly, neither { v(t)} nor { F[v(t)]} are stochastic processes; rather they are known functions of time once the problem is solved. (See Note 2.)

Throughout this paper, we restrict attention to controls v (t); tE[O, T] } that are piecewise continuous. Any such control is an admissible control. (See Note 3.)

A variety of constraints may be added to (1) and (2). For example, one may add the constraint

T

EJ [v(t) dt-db(t)]+Io?0; (4)

i.e., the expected value of net inventory at time T should be nonnegative. As an alternative to (4) one may define an explicit penalty function for terminal net inventory (positive or negative) and append this penalty function to (1). In this paper we shall impose the chance constraint (see Note 4).

pr{I(T)_0} =pr{f(T)?< y(T)+Io} >a, (5)

where acE[O, 1]. Constraint (5) may be interpreted as follows: the probability that cumulative production plus initial inventory exceeds cumulative demand must be no less than a (i.e., the probability that terminal net inventory is nonnegative must be no less than a). For the moment, we simply state that the value of a must be determined by reference to managerial rules and policies. Operationally, a =1 would imply that management 'always' wants to satisfy cumulative demand. If a < 1, but 'close to' unity, the implication would be that management wants to have nonnegative terminal net inventory a 'high' proportion of the time, but not 'always.' Additional remarks on (5) will be provided in Section VI.

To slunm up, a complete statement of the problem under consideration is:

minjv(t);o< t_ T) EJ { F[v (t)]+h[I (t)] dt, (6.1)

subject to dy(t)=v(t) dt, (6.2)

dI (t) = dy (t) - d (t), I (O) = Io, (6.3)

v (t) >0 and piecewise continuous for tE[O, T], (6.4)

pr{4(T) <y(T)+Io} >a. (6.5)

Deterministic -Equivalent Formulation

As is usually done for stochastic programming problems, we attack (6.1)-(6.5) by deriving a deterministic-equivalent form. This is easily accomplished for (6.3)




by incorporating it into the objective function (6.1). Constraint (6.5) may be recast in a deterministic form as follows: Using the distribution function fT(&)

of +(T), (6.5) may be restated as

f[y (T) +Io] > a. (7)

Applying the inverse operator fWT (' ) to both sides of (7) yields the equivalent expression

y(T)+Io>f_'(a), (8)

where fTP (a) is the ath fractile of fT (*). Of course, this fractile need not be unique. We attain uniqueness by choosing the most favorable value of the ath fractile. That is, we let fTW (a) = min {gfT(g)>a}.

Next, note that EfT h[I (t)] dt= =f Eh[f (t)] dt because the (measurable) stochastic process {h[I(t)]; t4J0, T]f is such that fT Elh[I(t)II dt< oo, because of assumption A.6. A pertinent proof on interchanging expectation and integration is provided in BURRILL171 [Theorem 16-7-D, p. 443].

We now have the following deterministic equivalent for problem (6.1 )- (6.5):

rT

max,v(t);o0 t?t< TJ -F[v (t)]-Eh[Io+y (t) - (t)] dt, (9.1)

subject to dy (t) =v(t) dt, (9.2)

v (t) ?0 and piecewise continuous for t4o, T], (9.3)

y (T) >fT-' (a) -Io. (Terminal condition) (9.4)

Henceforth, we shall often refer to (9.1)- (9.4) as problem (9).

III. PROBLEM OPTIMIZATION

THE DETERMINISTIC-EQUIVALENT problem (9.1)-(9.4) can be attacked with the

usual tools of optimal-control theory. The appropriate Lagrangian function 2(t) is

? (t)-_aC(t)+ x(t)v (t) (10.1)

where X (t) is the Lagrange variable associated with constraint (9.3) and the Hamil- tonian 3C (t) is

3C (t) -='- {-F[v (t)] -Eh[] (t)] I +p (t)v (t), (10.2)

where p (t) is the adjoint variable associated with the differential equation y (t) =

v (t) underlying (9.2). In this problem the state variable and control variable at time t are y (t) and v (t), respectively.

Using assumptions A.S and A.5 we see that F (.) and Eh (.) are convex and differentiable in the variables [y (t), v (t)] for Vt4[O, T]. Moreover, the differential equation underlying (9.2) is linear in v (t), which can be shown to be bounded, and constraint (9.3) is a linear inequality. (See Note 5.) Finally, the terminal condition (9.4) is differentiable and convex in y (T). Consequently, the necessary and sufficient conditions stated in MANGASARIAN[281 may be applied to problem (9). That these conditions are necessary is proved in PONTRYAGIN ET AL.;[341 a proof of




sufficiency is provided by Mangasarian.128' Formally, using the notation of this paper, Mangasarian's results lead to (see Note 6): THEOREM 1 (Mangasarian). Let the admissible control v* (t), Vte[O, TI, be an optimal solution for problem (9) and let y* (t), Vtc4O, T], be the corresponding state variable. Then there exist piecewise continuous functions p (t) and X (t) such that, for all te[0, T]:

dS2 (t)/dv* (t) = - F[v* (t-)]+p (t) +) (t) =O (111

where ?(t) is as defined in (10.1);

x (t) a * (t)-O = \x(t) _t0; (11.2)

(t)-c (t)/Oy* (t)-=Eh[I (t)]/Oy* (t) =-Eh'[1 (t)], (11.3)

whenever p (t) is continuous, where S (t) is as defined in (10.1 ); and

p(T)[y(T)-fT1(a)+I(0)]=0 and p(T)>0. (11.4)

Note that in (11.3) the order of differentiation and expectation is reversed. Given the properties of the function h(- )-see A.3 and A.6-this interchange can be justified by applying Theorem 2 of DEGRooT1131 or the results in CRAME1R01 (p. 67).

Remark 1. Theorem 1 presupposes the existence of an optimal control. Since the objective function in (9.1) is strictly convex and since v(t), for all t40O, TI, is constrained to lie within a convex and compact region [see (9.3) and Note 5], the existence of an optimal control follows from Theorem 7.1 and Remark 7.6 in STRAuss.-163 Additional existence results are provided by CESARI. [81

The quantity Eh'[I(t)] plays an important role in the solution of problem (9) and-as will be seen in Section IV--in the characterization of solutions to (9). We proceed immediately to establish two of its properties that will be relied on in subsequent developments. LEMMA 1. Eh'[t (t)] is a continuous function of y (t) and Q (t). [Recall that Q (t) is defined to be a vector consisting of the moments of + (t).]

Proof. Expand h'[I (t)] about the mean I(t) of I(t):

h [f (t)]=h [I (t)]+h [I (t)][I (t)-I (t)I (12)

+ E8-3 I{h8[I (t)]/ (s-)! [ () I(t]-

Given assumption A.3, the upper index of summation in (12) may be changed from oo to N*, where N* is as in assumption A .3. Since I (t)-Io+y(t)-(t), 7(t)= Io+Y (t)-Ef (t), and thus [I (t)-I(t)] =- (t)++ (t)]. Using the latter expression in (12) and taking expectations throughout yield

Eh'[1 (t)]-=h'[i (t)]+ 8s*(l8t8It](-) Xfit+t]-} (13)

where EPi(t)- (t)]', i=2, 3, * is the ith central moment of +(t). Since all terms in (13) are continuous functions of these moments and since 7 (t) is a continuous function of y(t), Eh'[f (t)] is a continuous function of y1(t) and Q(t), as asserted in the lemma.

It should be stated that the continuity of EhJ[I (t)] with respect to t is of greatest importance for our purposes, as will be seen shortly. And since y (t) and all moments of $ (t) of order r < N* are continuous with respect to t (see A.4), the continuity of Eh'[J (t)] with respect to t follows automatically.




The conditions imposed on h(-) in order to attain the desired continuity of Eh'[ ] are, of course, not the only conditions that could have been used. As an alternative, we could have only required that h'[ ] be a uniformly bounded continuous function: Since 1 (t) = I (0) +y (t) - f (t) is stochastically continuous (in the mean-squared sense), any uniformly bounded continuous function of I (t) is also stochastically continuous. And the expected value of a stochastically continuous function must, by necessity, be continuous (see Note 7). Thus, the indicated alternative conditions on h() would also have led to the desired continuity of Eh'[I (t)].

In view of Lemma 1, we introduce the function G[y (t), Q (t)] _Eh'[f (t)], which will be employed extensively in Section IV. And we establish the following result for subsequent use. LEMMA 2. aG[y (t), Q (t)]/cy (t) >0.

Proof. By definition, G[y (t), Q (t)] = JA,h'[I (t)] dft, where ft is the distribution function of f (t) and At is the sample space for + (t). By assumption A.3, h[-] is at least twice continuously differentiable. Hence we may write

aG[, - d(t)= -h" [*[I(t)lay (t)] dft h" [ -Idft. (13.1) At At

By assumption A .3, h" [] > 0. Thus, *G[, ]/cy (t) > 0 as asserted.

IV. CHARACTERIZATION

THIS SECTION ESTABLISHES several propositions that serve to characterize problem (9). DEFINITION 1. A solution v(t), Vte[O, TI, to problem (9) will be called an extrapolation if the solution is a continuous function of time that satisfies conditions (11.1)- (11.3) of Theorem 1. Note that satisfaction of the transversality conditions, (11.4), is not required; thus, an extrapolation need not be an optimal solution.

As is indicated in (9.3), an optimal solution to problem (9) need only be piecewise continuous. The next proposition indicates that, in fact, an optimal solution to problem (9), must be continuous. PROPOSITION 1. If v(t), 1[0, T], is an optimal solution for problem (9), then v(t) is a continuous function of t, Vt.

Proof. This proposition will be established by showing that, for any (admissible) solution with at least one discontinuity, one can construct another (admissible) solution that yields lower total costs. As a result, no proposed (admissible) solution with at least one discontinuity can be optimal.

Suppose that the solution v (t), Vte[O, T], has discontinuity at t = t. First consider the case where lim,,o v (t- E) <lim,,o v (1+ e). Because of the discontinuity there exist At>0 and z> 0 such that

v(t^-r)+z<v(t+T)-z for TE(0, t). (14)

Note that z is a deterministic quantity. Now, consider another solution v* (t), where




fv(t), O<t_t-/vt,

v (t) _ Jv(t)+z, t-At<t<t;Y

v(t)-z, `t<t`+At, (v(t), t^+At<t_T.

Let y (t) and 1 (t) be the cumulative production and cumulative inventory, respectively, corresponding to v (t), and let y* (t) and 1* (t) correspond to v* (t). By definition, we have, for t > I+ At,

t,iAt ?+, &t?A

y (t)=1 v(s) ds+J [vl(s)+z] ds+Jg Iv(s)-z] ds+j v(s) ds, (15)

with obvious modifications for t<t?+ At, and

I(t) =Io+y(t)-4(t), I*(t) =Io+y*(t) -(t). (16)

Using expressions similar to (15) for various subintervals of [0, T], we have

Y (T)-y(T), for r <t-At

Y (T)=y(T)+z[T- (t- At)], for TE(1-At, 0, (17) y *(r)y(T)+Z(At+t)-ZT, for TE(', '+At), Y (T)-y (T), for r> t+ At.

Thus, using (16) and (17), we find that

(Tr) =I(T) +z[,- (1- At)],r Q'(- At, t), (18)

*(Tr) =I (T)+Z(At+I`)-ZTr, TIE(I, I+At),

I* (T) = I (T) , r _t + At.

Consider the values of the components of the objective function (9.1) associated with each of the two solutions v*(t) and v(t). First, consider the production-cost component; for v(t) we have

T t-id

t At

|F[v(s)] F[v(s)] ds+f F[v(t- T)I dT

At T (19)

+ F[v (1+r)I dr+f F[v(s)] ds,

and for v* (t) we have T A Ad At

|F[v*(s)] ds=t F[v(s)] ds+t F [v(1-Tr)+ z] dr

^ t T (20) + F[v (t^+T)-z] dr f F[v (s)] ds.

O ~~~~~~~~~~~~~~~t+A Upon subtracting (19) from (20), one finds that the difference in total production costs r is

t r= ({F[v(t-T)+z]+F[(vt+r) -z]l-{F[v(t- -r)]+F[v(t?r)]}) dT. (21)

From (14), v(t^-r)+z<v(t^+r)-z, z>O. Thus,




v(t--T)<v(t-r)+z<v(t1+Tr)-z<v(t'+Tr) for TE(O, At). (22)

Since F (.) is strictly convex,

F[v(t'-r)]+F[v(t+ r)]I>F[v(t'-r)+z]+F[v(t+ r)-z].

Consequently, P <0; i.e., the total production costs using v* (t) are less than those associated with v(t). And using the At established for (14), there exists a sufficiently small E> 0 such that P < - EAt. (Note that e is not a function of At.)

Now, consider the difference between the holding costs EfT h[I* (r)] dr and EfJo h[I (r)] dr. If these two integrals are written out as in (19) and (20), one finds that the difference in total holding costs A equals

4+At t+& t

A=Ef h[E*(s)] ds-EJ h[I (s)J ds. (23) t t t A

By using (18), the first term on the right-hand side of (23) may be restated as t+At At

E| h[I*(s)] ds=EJ h[I(s+t^-At)+zs] ds

at (24) +Ef h[f(s+0-z(s-At)] ds,

and the second term on the right-hand side of (23) may be re-expressed as t+,&t At At

EJ h[WI(s)] ds-=E h[f(s+t-At)] ds+E] h[I(s+t^)] ds. (25)

As indicated earlier, we may interchange the order of expectation and integration. After completing the interchange, one may expand each integral in a Taylor series about the lower limit of integration. For (24), the expansion is

Eh[f (t^-At)]At+Eh[I (t)+zAt]At+O? (At), (26)

and for (25) the expansion is

Eh[I (t^- At) ] At+Eh[I (t)] At+02 (At). (27)

In (26) and (27), O (At)/At->0 as At-)0, i=1, 2. Now A, from (23), may be rewritten as the difference between (26) and (27):

A= Eh[I (t)+zAt]At-Eh[I (t)]At+03 (At), (28)

where 03 (At) = 01 (At)-02 (At ). Expansion of (28) about f (1) yields

A = Eh'[I (t)] At (zAt) + 04 (At) = 0O (At). (29)

In (28) and (29), Oi(At)/At-+0 as At-*0, i=3, 4, 5. Consequently, the difference between the total costs of v* (t) and v (t) may be expressed as P+A< - eAt+O5 (At),

>0, At> 0. And, by making At sufficiently small in this inequality, we obtain r+A<0; i.e., the total change in costs, P+A, associated with using v*(t) rather than v (t) is negative. This completes the proof for lime.o v (t- E) < lim,,o v (t+ E).

The proof for lim,,o v (1- E) > lim,,o v (t+e), which is not presented here, is easily developed by using v** (t) as the alternative solution, where




[v(t), 0_!~t:g _t^At, v**(t)_)v(t)-,z, t-At<t<t,

v(t)+z, t1<t<t+At, (v(t), t^JrAt<5t<T,

and by imitating the proof given above for v* (t). In view of Theorem 1, Definition 1, and Proposition 1, the following corollary

is obviously true. COROLLARY 1. An optimal solution to problem (9) is an.extrapolation that satisfies the transversality conditions (11.4).

According to Theorem 1, p(t) and X (t), tE[0, T], need only be piecewise continuous functions. But, for problem (9), they are, in fact, continuous. COROLLARY 2. The functions p (t) and X (t), te[0, T], associated with the extrapolations to problem (9) are continuous functions of t.

Proof. From (11.1), (11.3), and the definition of G[, ], we have

p(t)=p(0)+f G[y(s), Q(s)] ds, (30.1)

X (t)=F'[v(t)]-p(t). (30.2)

Since the integral on the right-hand side of (30.1) is well defined, p (t) is continuous for all t. Since (a) extrapolations are, by Definition 1, continuous, and (b) F'(-) is continuous, X (t) is a continuous function of t, Vt. PROPOSITION 2. Let v (t), tE[O, T] be an extrapolation (see Definition 1). Then, for all t such that v (t) >0, we have

v (t) = G[y (t), Q (t)]/F '[v (t)]. (31 )

Proof. Since v (t) > 0, we have, from Theorem 1 [expressions (11.1 ) and (11.2)], p (t) =F'[v (t)]. From (11.3) and Corollary 2, it follows that p (t) exists for all t40, T]. Differentiating both sides with respect to t yields p(t)=F"[v(t)]i(t). Dividing this expression by F"[v (t)] and using (11.3) for p3(t) yield (31). LEMMA 3. Let to[O, T]. For every set of given values v (to) and y (to), there exists a unique solution v (t) and y (t), Vt[0, T], to the differential equation v (t) = G[y (t), Q (t)]/F"[v (t)] that passes through these values. [Recall that Q (t), Vt, is exogenous to problem (9).]

Proof. From the definition of y (t), it is a continuous function of time. By assumption A.3, Q (t) is also a continuous function of time. Thus, using Lemma 1, Eh'[-] is a continuous function of y (t) and t. By assumption A.5, F" ( *) is a continuous function of v (t). Since, in addition, h" ( * ) is continuous (by A.4), DG[ -, * ]/ 3y (t) is a continuous partial derivative [cf. expression (13.1 )]. Moreover, by A .5, F"' (. ) is continuous. As a consequence of these continuity properties:

(i) G[y(t), Q(t)]/F" [v(t)] tI4y(t), v(t), t]=v' is a continuous function of y(t), v(t), and t, Vt.

(ii) O4/ay (t) and l/cv (t) are continuous. The differential equation in (i) may be re-expressed as =(t)=1y (t), y (t), t],

where y(t)=v(t) and y(t)-v (t), because y(t)=fuv(s) ds. Given properties (i) and (ii) regarding 'I ( , *, ), it follows from Theorem 2 in Pontryagin'331 [pp. 18-22 and 25-30] that, for given v(to) and y(to) and for some to[0, T], a unique solution




v (t), Vte[O, T], exists for this differential equation. And thus a unique solution exists for i (t) = G[y (t), Q (t)]/F" [v (t)]. LEMMA 4. Let v (t) be an extrapolation and the interval [t', t2], 0 t1<t2 < T, be such that, for small At > 0, v (tl-e) > 0, v (t2+e ) > 0, Vee (0, At], and for all re4t , t2], V (r) = 0.

Then X(tl) =X(t2) =0. Proof. From Corollary 2, X (t) is a continuous function of t, VtE[O, T]. Since

v (tl-e) >0, Ves (0, At], we must have X (t'-e) = 0, VeE (0, At], by the complementary slackness condition (11.2) of Theorem 2. Since X (t) is continuous, X(tl)=0. [If, to the contrary, X(tl)>0, then 3aE(0, At) such that X(tl-b)>0, by the continuity of X(); and this would contradict the complementary slackness condition v (t'- 5)X(t1-5)=0, where for Vbe(0, At), v(t - )>O by assumption.] By applying similar arguments to X(t2) and X(t2+e), Vee(0, At], one also finds that X(t2) =0. PROPOSITION 3. Let v(t) be an extrapolation and let [ta, to] be an interval such that (i) v(Tr)=0, VT[ta, to], (ii) for sufficiently small At>O, v(ta-e)>O, Vee(O, At], if ta >O and v(t#+?)>O, V6E(O, At], if t <T. [In words, [ta, t] is amaximal interval over which v(t) =0.] Then, for all td[ta, to], we have

p(ta)+ G[y(ta), Q(s)] ds<F' (0). (32) ta

Moreover, if in fact t < T, then 3 Att> 0 such that, for all rE (to, t?+ At], we have v (r) >0 and

p (ta) + G[y(ta), a(s)] ds>F'(0). (33) ta

[Note that the first argument of G[, *] in (33) is y(ta), not y(T).] Proof. This proof is presented in two parts; first we establish (32), and then

(33). Part I. For all te[0, T], we have [from (9.2), (11.1), and (11.3)], y(t)=

fot v(s) ds, dp(t)=G[y(t), Q(t)] dt, and p(t)=F'[v(t)]-X(t). By definition of

[ta, t%], y(ta)=y(t')=y(r) and F'[v(r)]=F'(0)>0 for all rdEta, t']. Thus, since X(t)>0, Vt [see (11.2)], we have, for all redta, t],

p(T) =p(ta)?+ G[y(ta), Q(s)] ds<F'(0). ta

This establishes (32). Part II. Suppose, to the contrary of (33), that 3At>0, such that, for all

TrE (t,) t Jr At],

p )=p (ta)+ G[y (ta), (s)] ds<F' (0), (34) ta

which may be restated as

p (r) = p (t')+f G[y (t), Q (s)] ds<F' (0). (35)

[Recall from Part I that y (ta) = y (t).]

First suppose that the strict inequality in (35) holds, i.e.,

p(t)?+ G[y(t'), U(s)] ds<F'(0) (36)




for all rE(to, t+ At]. From Lemma 4, X (to) = 0. Thus, from (11.1 ) and A.5,

p (t') = F'[v (t)]I = F' (O ) >O0. (37)

Given (37), if (36) is to hold, then we must have, for all Te (to , t+? At],

J G[y (t'), Q (s)] ds < O. (38)

Expanding the left-hand side of (38) in a Taylor series about to yields, for small a>0,

pt#+6

G[y (t'), Q (s)]I ds = G [y (t'), Q (t')] 5+ 0 (5) < . (39) td+

And (39) implies that 35 >0 such that

G[y (t'), Q (t)] <0. (40)

Note that v(s)>0 for all sE(t, to+?]; thus,

p (s) = p (tl)f+ G[y (k), Q (k)] dk = F'[v (s)]. (41)

Since p (to) = F'(0) and F'[v (s)] > F'(0), we must have

f G[y (k), Q (k)] dk>0, (42)

for all se(to, to+?]. Expanding the left-hand side of (42) in a Taylor series about t' yields

JG[y (k), Q (k)] dk = G[Q (t'), y (t')]e+O0 (e) >O0. (43)

Thus, for small e > 0, (43) implies G[y (t'), Q (to)] >0, which contradicts (40). So the strict inequality in (35) cannot be true.

Finally, consider the strict equality in (35), i.e.,

p (to) Jr+ G[y (t'), Q (s)] ds = F'(0) (44)

for all ne (to, t?+ At]. From (I1.1 ) and (I1.2) we have p (t)-I-X (t) =F'[v (t)] and v(t)X(t)=O, VtE[O, T]. Since (by construction) v(r)>O, for all rE(to, t+?At], X(r)=O. So, (44) implies [see the argument of F'(.) in (44)] that v(r)=O is another solution for VTE (to, to+ At]. But this contradicts the uniqueness of v (t), VtE[O, T]; cf. Lemma 3. Thus, the strict equality part of (35) cannot be true. Consequently, expression (33) is true.

Remark 2. Using Propositions 1-3, the behavior of an extrapolation can be determined uniquely by selecting a value for p (0). Suppose that p (0) >F'(0); then, from (I1.1) and (I1.2), v(0)>0, v(0)=F'-'(0), and, from Proposition 2, the extrapolation proceeds in accordance with the differential equation

v (t ) = G [y (t ), Q (t )]/F" [v (t )] (45 )

so long as v(t)>0. If t=t* is such that v(T)>O, VT<t*, and v(t*)=0, then (using the notation of Proposition 3), t *=ta, i.e., ta is the initial instant of an interval over which v (t) =0. In order to find the terminal instant of this interval to, one may




use expression (33), which uses the 'dual variable' p (t): if to< T, then inequality (33) must hold for some interval (to, t+?At], Att>0. For all ee (t', t+?At], v (t+?e)> 0, and the extrapolation proceeds, once again, according to the differential equation (31), perhaps encountering a second ta.

If, on the other hand, p(O)<F'(0), then [from (11.1) and (11.2)], v(O)=O and ta=0. In this case we must search for a to at the outset [using (33)]. If t<T, then, for some interval (to, to+At], At>O and the extrapolation proceeds in accordance with (31), perhaps encountering a second ta.

The next proposition provides a characterization of the extrapolations corresponding to different values of p(0). PROPOSITION 4. (Nonintersection of extrapolations.) Let vi(t), v2(t) be two extrapolations for all tE[0, T], and let pi (t), P2 (t) be the dual solutions associated with vi(t) and v2(t), respectively. If pl(O)>p2(0), then vl(t)_v2(t), for VtE[0, T] and vl(t)>v2(t) for all t such that vi(t)>0. Moreover, yl(T)>Y2(T) and pi(T)>p2(T).

Proof. Recall, from (11.1), that pi(t)=F'[vi (t)]-Xi(t), i=1, 2, Vt, where Xi(t)>0 and vj(t)Xj(t)=0. If pi(O)>F'(0), then (see Remark 2), vj(0)>0 and pj(0)=F'[vi(0)], i=1, 2. Since (by assumption A.7), F" ( )>0, pl(O)>p2(0) and pi(0)>F'(0) imply v1(0)>v2(0). On the other hand, P1(O)>P2(0) and pi (0) <F' (0) imply v1 (0) =v2 (0) =0 (see Remark 2). So, in general, pl (0) > p2 (0) implies vl (O) _ V2 (0).

Let t1a be an instant such that v1 (t1a) = 0, and, if ta >0, v1 (tl - At) >0 for all AtE(0, 6] and small 3>0, and vl (T) > V2 (T) for VrE[O, tia]. (Note that tla may be equal to zero.) Since v2 (t) _ 0, Vt, these conditions imply v2 (tla) = 0. Also, from the definition of yi (t), i= 1, 2, we know that these conditions imply y, (r) >8 2 (r),

VTE[O, t1ai]. Moreover, we have [using (11.3)] ta

pi(tia)=pi(0)?+ G[yi(s), Q(s)] ds, i=1, 2, (46.1)

and (using Lemma 2)

G[yi(r), Q (r)]>G[Y2(T), Q (T)], VTE[0, t1a]. (46.2)

Since p1(O)>p2(0) by assumption, (46.1) and (46.2) imply pi(tia)>p2(t1a). Using expression (33) and the facts that Pl (tla) >P2 (tia) and Yl (tla) _ Y2 (tla), it is true that the first instant t2o after tia for which V2(t20+At)>0, for all AtE(O, 61],

1 >0, must be such that t2a > tip, where tio is the first instant after tia for which v1(ti0+At)>0, for all AtE(0, 62I, 52>0. In other words, after the instant t1a, if v2( () becomes positive, then the instant at which it begins to become positive must come after the instant at which v1 ) begins to become positive. Consequently, under the assumptions of the proposition, v1(t) >v2(t), VtE[O, TI and v1(t)>v2(t) whenever v1 (t) >0. Moreover, since yi (t)= f ot vi (s) ds, i= 1, 2, y1 (t) >-Y2 (t), Vt. If there exists an interval [t, t] over which v1 (* ) >0, then Yl (t) >Y2 (t), Vt >-t. And such an interval must exist if, in fact, the two extrapolations are not identical, for all tE[O, T]. Thus, y, (T) >Y2 (T) under the assumption that the extrapolations are different.

Finally, since

pi(T)3=pit(0) + G[yw (s), Q(s)] ds, i=Lm 21

Pl (?) > P2 (0) , and Yl (t) >_ Y2(t), Vt, we have (using Lemma 2) pi (T) > P2 (T).




Remark 3. As indicated in Remark 2, the behavior of an extrapolation is uniquely determined by selecting a value for p (0). The essential result of Proposi- tion 4 is that the different extrapolations generated during any process of assigning values to p (0) do not 'criss-cross,' or intersect, each other and these different extrapolations induce different values for p (T) and y (T). The fact that these different extrapolations induce different values for p (T) and y (T) leads to the following uniqueness result. PROPOSITION 5. The optimal solution to problem (9) is unique.

Proof. Recall that an optimal solution is an extrapolation that satisfies the transversality condition

p(T)[y(T)-fP1 (a)+Io]=0 and p(T)_O. (47)

Moreover, in order to be feasible, an extrapolation must satisfy the terminal condition

y (T) >fP1 (a)-Io. (48)

First note that one would never want a value of p (T) that is higher than necessary in order to satisfy (47), because such a value of p (T) would require production that is higher than necessary (cf. Proposition 4), and hence costs that are higher than necessary. And this, of course, would not be optimal for a cost-minimization problem. Observe also, using Proposition 4, that there is only one extrapolation satisfying y (T) =f71 (a) -0Io and there is only one extrapolation satisfying p (T) = 0. (These extrapolations may be one and the same.)

Suppose that the extrapolation yielding y (T) =f7(a) -Io gives p (T) >0. This extrapolation satisfies (47) and (48), and it is the only extrapolation with the lowest value of p (T) such that (47) and (48) are satisfied. Thus, this extrapolation is the optimal solution. To see this, note that, if an extrapolation satisfies (48) as a strict equality and (47) with p (T) _0, then, (1) all other extrapolations with lower values of p (T) such that p (T) _ 0 will induce lower values of y (T) and, hence, (48) would be violated, and (2) all other extrapolations with higher values of p (T) will induce higher values of y (T) and, hence, (47) will be violated. Thus, if the extrapolation yielding equality in (48) also yields p (T) _ 0, then it is the optimal solution.

Suppose that the extrapolation yielding y (T) =f1 (a) -Io results in p (T) <0. Since we must have p (T) > 0, this extrapolation cannot be optimal. But the unique extrapolation that yields p (T) = 0 will provide a higher value for y (T); thus it will satisfy (48) as a strict inequality, and it will satisfy (47); consequently, this extrapolation is the optimal solution.

V. ALGORITHM

THIS SECTION DESCRIBES the essence of an algorithm for determining the optimal production policy, at t = 0, for the horizon [0, T]. There are some obvious problems of numerical analysis involved here, such as computing Eh'[I(t)], but we shall not belabor these issues in this paper.

Stage I. Compute the extrapolation yielding y(T) =f-T1(a) -1o as follows:




1.1. Select a value for p(O); go to 1.2.

1.2. If t'he extrapolation corresponding to the selected p(O) yields y(T) >fT1(a) -1o, go to 1.3; otherwise go to 1.4.

1.3. Select a smaller value for p(O), go to 1.2.

1.4. If y(T) =fT1(a) -lo, go to 11.1. If y(T) <fTl(a), select a higher value for p(O); go to I.2.

Stage II.

II.1. Compute the value of p(T) associated with the extrapolation. If p(T) _O, the extrapolation is the optimal solution and the solution process is complete. If p(T) <0, go to 11.2.

11.2. Compute the extrapolation that yields p(T) =0 by searching for a value of p(O) that yields p(T) =0. Since the predecessor value of p(T) <0 yielded an extrapolation satisfying y(T) =f-1(a) -Io (see step 1.4), the current extrapolation must satisfyy(T) >fT1(a) -Io (see Proposition 5). Since the current extrapolation satisfies the terminal condition p(T) ?0 and the transversality condition on y(T), it is the optimal solution. So the solution process is complete.

Remark 4. A key aspect of the algorithm described here is the selection of higher or lower values of p (0). It should be observed that these selections involve searching, e.g., by a binary search procedure, over a decreasing interval of values for p (O).

VI. ADDITIONAL REMARKS

THIS SECTION PROVIDES some additional interpretive remarks on the chance constraint (5); they rely on the meaning of the dual solution p(t), VtE[O, T].

Recall that the problem at hand involves the chance constraint

pr{y (T)+Io?> (T) } _ aE[O, 1], (49)

which was shown to be equivalent to y (T) >fT (a) -Io. Earlier in this paper it was stated that the value of a may be determined by

reference to managerial rules and policies; (1- a) represents the maximum 'acceptable' probability of not satisfying cumulative demand over the horizon [0, T]. Alternatively, (1-a) is the maximum acceptable probability of having a negative terminal inventory. An alternative to the chance constraint (49) is specifying a penalty-cost function for nonsatisfaction of cumulative demand and incorporating it directly into the objective functional (1). Presumably, this approach would be preferable if the costs of not satisfying cumulative demand were well defined, and the chance-constraint approach might be preferable if such costs seemed so ill defined that all management can (or, is willing to) do is specify an acceptable 'risk level' (1-a). But, as will be indicated below, the chance-constrained approach need not be totally divorced from cost considerations, because there is an intimate relation between the chance constraint and the dual solution p (t), VtE[O, T].

Suppose that one first solves problem (9) with the restriction p (T) = 0 and suppose that the resultant value of y (T) is y (T) = K. From Proposition 4 (see Sec-




tion IV) we know that p* (T), the value of p (T) associated with the optimal solution to the problem, varies directly with the value of y (T), subject to the restriction p*(T)>O. Consequently, p*(T)=O for {aJf_1 (a)-Io<K}, p*(T)>O and monotone increasing as a function of a for { ajfT1 (a) - Io>K}. All of this simply says that the value of p* (T) is a function of a. But p* (T) is the change in the value of the objective functional associated with a change in y* (T), the value of y (T) associated with the optimal solution (see Note 8). Hence, p* (T) provides a measure of the marginal cost of, or the 'price' paid for, the restriction y (T) = y* (T). Since Io and fT ( ) are given, this imputed price may be directly associated with the specified value of a. That is, p* (T) represents the imputed marginal value fore- gone by management by not permitting cumulative production (plus initial inventory) to be less than cumulative demand with probability greater than (1--a). As such, p* (T) represents an imputed marginal penalty cost. And this imputed penalty cost may be used to assess the reasonableness of the original problem formulation, in particular, the selected value of a in the chance constraint. Management may wish to alter the value of a after being told the marginal penalty cost implied by this value.

NOTES

1. Stochastic continuity (in the mean-squared sense) is discussed in, e.g., CRAMER AND

LEADBETrER,[121 PAPOULIS1321 [pp. 312-3141, and JAZWINSKI(201 [Ch. 3, Sec. 4].

2. This requirement precludes allowance (at t =0) for stopping the production process and selecting a new control that is optimal, conditional on observed realizations of the stochastic processes.

3. That is, a control {v(t); 0 <t < TI, is admissible if it is continuous everywhere except for a finite number of instants, at which it may have discontinuities of the 'first kind,' i.e., if v(t) is discontinuous at, say, t =t*, where 0 <t* <T, then v(t) must have finite left- and right- hand limits at t*. At t =0 (t = T), v(t) must have a finite right- (left-) hand limit.

4. Chance constraints are discussed in, e.g., CHARNES AND COOPER,[?,1O] KIRBY,1221 and Gonedes.f11I

5. The boundedness of v(t), Vt, may be shown by imposing the constraint v(t) -AI, where M is some finite constant, and by observing that this constraint is nonbinding when Al is sufficiently large (and finite).

6. Our sign convention is not identical to Mangasarian's. Using his convention, constraint (9.3) would have to be stated as -v(t) ?0, for all t4[0, T].

7. See, e.g., Papoulis[32J [p. 313] or Jazwinski[20J [p. 61].

8. See, e.g., Arrow,R'] Mangasarian,[28I and DORFMAN.115]

ACKNOWLEDGMENTS

THE COMMENTS OF the participants in the Workshop on Information, Decision, and Control at the University of Chicago are gratefully acknowledged. A special




note of gratitude is due to two anonymous referees whose reviews impressed and aided the authors.

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