A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
S e b a s tia n S ita r z *
STO C H A ST IC O R D E R S IN DISCRETE DY N A M IC PRO G RAM M ING
Abstract
T h is p a p er d eals w ith a pro b lem o f d ynam ic o p tim izatio n w ith values o f criteria fun ctio n in the set o f the ra n d o m variables. Precisely, there is a d y n am ic m odel w ith finite n u m b er o f stages, states and decision variables described. Such a d ynam ic process is evaluated regarding values o f the ra n d o m v ariables. T h e ra n d o m variables have to fulfil som e co n d itio n s, if they are to be applied to dy n am ie op tim izatio n . T hese c o n d itio n s a re described in presented paper. M o reo v er, th ere is given a review o f sto ch astic orders, w hich can be used in the m odel.
Key words: dy n am ic p ro g ram m in g , partially ordered set, sto ch astic orders.
I. IN T R O D U C T IO N
T he theory o f dynam ic program m ing was introduced by R. Bellman (1957). N ext Brown and S trauch (1965) generalized B ellm an’s principle to a class o f m ulti-criteria dynam ic program m ing with a lattical o rd er. T h en , the use of optim ality principle was the interest o f M itten (1974) who considered preferen ces relation, and Ilenig (1985) - infinite dynam ic process with values o f criteria function in a partially ordered set. O thers w ho to o k interest in the use of m ulticriterial m ethods in dynam ic program m ing have been: T rzaskalik (1998), Sobel (1975), Li and I laim es (1989). T h e continuo us dynam ic decision m odel w ith the m ethodology o f m ulti criteria-decision m aking is analyzed in G laser (2002). In the m eantim e theory o f com paring random variables have developed as well, R olski (1976), Shaked and S hanthikum ar (1993). It enabled us to use such stru ctu res in o u r dynam ic model.
O u r w ork considers the dynam ic discrete decision m ak in g m odel with returns in partially ordered set. B ellm an’s principle o f optim ality for such
* P h .D ., In stitu te o f M ath em atics, U niversity o f Silesia.
a problem will be show n. M oreover the m etho ds o f n arro w in g the set o f the solutions will be described. T hese m eth o d s are analogical to m ulti-criteria program m ing m ethods, but here, additionally, the dynam ic aspect o f the model m ust be taken to consideration. Such a general ap p ro ach , i.e. stru ctu re o f partially ordered set, let us to apply a lot m athem atical structures to describe practical problem s. As an interesting sample o f this, wc will show com bination o f dynam ic p rogram m ing and random variables with stochastic dom inance.
II. M O D E L
We consider a discrete dynam ic process, which consist o f T periods. Let fo r t = 1, T
- Y, is the set o f all feasible state variables at the beginning o f period I, - Уг + i is the set o f all states at the end o f th e process,
is the set o f all feasible decision variables for period I and state y t e Y r
We assum e th a t all above sets arc finite.
- D, = {dt = (у,, x t)\ y t e Y„ x , e X , ( y t)} is the set o f all period realizations in period t,
- Q t: D, —*• У(+1 are given transform ation s,
- D = { d *=(du . . . , d r ): V,6(1.... r} yt+1 = Clt(y„ x,) and x r e X r ( y T)} is the set o f all process realizations d,
D,{yt) = {{yv x,)\ x (e X t(yt)} is the set o f all realizations in period t which begin a t y t,
- d (yt) = ( y „ x t, y T, x T) is the p artial realization for given realization d, w hich begin a t y t,
D( yt) = {d(y,)' d e D } is th e set o f all p artia l re alizatio n s, which begin a t y t,
- D ( Y t) = {D (yt): y (e У,} is the set o f all p artial realizations, which begin a t period t,
P = {(Уг, У г+ i l ) , ■X’tO't), £it: t = l , . . . , T } deno tes discrete dynam ic process, w here sets У1(..., Y T+l , X ^ y j , X T( y T), fun ctions Ü U ...,£2T arc identified.
W e consider the follow ing structure, functions and o p erato rs to describe m ulti-period criteria function o f process realization.
- (W, < , ° ) - structure, where (W, < ) is partially ordered set (posct), and o p e ra to r o ^ x W —* W satisfies conditions
1) a° ( b° c ) — (flob)oc; a ,b ,c e W .
We den o te relatio n as follows
a < b * > a ^ b and а Ф b .
Remark 1. It is enough to assum e transitive co nd ition from the relation < in the set W. Jn this case, th ro u g h p roper o p eratio n s, we o b tain a poset; B irkhoff (1973).
F o r each finite subset / l c l f w e define
т а х ( Л ) = { a * e A : ~ 3 BEAa * < a } .
- f t : D, —*■ W, for t = 1 arc period criteria fun ctio ns with returns in partially ordered set W.
We assum e th a t for each period exist follow ing op erators: - o( : W x W —* W are m o n o to n e o p erato rs (i = 1 , . . . , T — 1).
, b , c c W U ^ b = > C o ,
F t : D ( Y t) —> W arc the functions defined in the follow ing way Ft ~ ( • • • ( /Y - i° r - t/r ) ) ) > •••> 1-- F = F t 1-- is called m ulti1--period criteria function.
In fu rth e r con sid eratio n we postulate to m axim ize fu nction F (in the sense o f relatio n ^ )
- (P , F) denotes discrete dynam ic decision process. It is given, if there are defined discrete dynam ic process P and m ulti-period criteria fu nction F.
- R ealization d * e D is said to be efficient, if F(cl*)e m ax F(D).
III. P R IN C IP L E O F O P T IM A L IT Y
In the discrete dynam ic decision process (P , F ) there holds following theorem: Theorem 1. Let (P , F) be decision dynam ic process,
F o r all t = T — 1, ..., 1 and all y t e Y t holds
a) m ax {F,(D (y,)} = т а х { / ^ > {т а х ( ^ +1д а ^ , ) ) ) ) : dt e D t(y t)}. b) m ax {.F(D)} = m ax ( m a x F j ^ y j ) ) : y ľ e Yj}.
IV. NARROWING THE SET OF SOLUTIONS
S olution o f such a dynam ic problem , in the sense o f m axim al values, m ay consist o f very num erous set. T o n arro w the set o f m axim al values, we m ay use m ethod presented below. Sources o f th e m eth od arc taken from the analogical m eth o d s o f m ulticriterial prog ram m in g, how ever, here, the p ro ced u re will proceed by stages. M oreover the m eth o d s are m uch m ore general considering the fact th at set o f values is m uch b ro a d e r th an vector o f real num bers.
W e will n arro w the set o f m axim al values with the help o f a new relation ^ ’ fulfilling the conditions:
1. ( W, ^ o ) is ordered structure. 2. (a <b=> a < ’b).
H aving in m ind the fact, th a t the relation ^ ‘ defines o th er m axim al elements, let’s m ark it m ax ’. We will m ark the relation si ’ fulfilling above conditions
1 and 2 (in reference to the relation < ) as follows ^ с T h e corollary o f the co n d itio n 2 is th a t for each A c: W
m a x ’i c m a x l .
Remark 2. L et’s notice th a t if certain relation < ’ fulfils w eaker condition then co ndition 2, nam ely
3. a ^ b=> a ^ "b,
then in troducing the relation < ’ o f the form a < Ъ о (a < "b) л ~ (b < a) we will get ^ с ^
Theorem 2. If the relation ^ ’ fulfills the con dition s 1 and 2, then applying the procedure describes in the p o int 3, we’ll get a subset o f m axim al values o f the process i.e.
m ax ’ F( R) cz m ax F(R).
Proof. C orrectness o f the procedure guarantees the co n d itio n 1, whereas the inclusion o f m axim al elem ents gives us the co nd itio n 2.
R em ark 3. W e use procedure o f calculating a narrow ed set o f m axim al values used to earlier obtained sets o f m axim al values in such a way th a t
wc solely consider m axim al values obtained w ith the help o f the procedure (we ignore o th er values o f stages criterion function).
V. STOCHASTIC ORDERS
N ow we present the n o tatio n o f stochastic do m in an ce and establish basic p roperities, which are uscfull in the dynam ic p rog ram m ing (O gryczak, 1997; R olski, 1976; Baccelli, 1991).
In th e stochastic dom inance approach, random variables are com pared by the pointw ise com parision o f their d istrib u atio n functions. F o r real ran dom variables t, function F{1) is the right-continous cum ulative distribution function
F ? \ x ) = f = P (£ x ), x e R . - 00
M oreover, the k -th function F f } is defined as follows:
F f \ x ) = J F f ~ l \ t ) dt , x e R .
— cjO
T h e relatio n o f k -th degree stochastic d om in ance is u n d ersto o d as follows:
^ « > / о ^ е 4 > ( х ) ^ ) ( х ) .
T he folowing properities o f relation ^ (k) are usefull in dynam ic program m ing. Let £, rj, у are rand om variables for which exist functions F f \ F ^ \ F f \ then
a w ľ = > £ < ( * ) ľ
^ ( « ^ ( í + ^ d n (т]+ у)
F o r m o re details on stochastic dom inance, p ro perties and p ro o fs see Rolski (1976).
A s we can see above, the properties o f stochasic do m in an ce let us use such a structure in dynam ic program m ing. We can ad o p t a stru ctu re (W, < , o )
preseneted in section II, which is the follow ing triple:
- FK is the set o f ran d o m varaibles č, for which exist F f \
- relatio n sg o f this stru ctu re is k-ih degree stochastic do m inance, - o p e ra to r о o f this stru ctu re is adding ran d o m variables (o r convolution o f d istru b u a tio n function).
VI. C O N C L U S IO N S
A m ulti-stage proccss with finite num ber o f periods, states and decision variables was considered. D ynam ic discrete decision m ak in g m odel with returns in partially ordered set and B ellm an’s principle fo r such a problem were show n. T h e m odel can be applied for m any m ulti-stage, m ulti-criteria decision m ak in g problem s. In the num erical exam ple, given in the paper, we considered o rdcrd structure, in which values o f criteria function were given by ran d o m variable. T h e ord er in the stru ctu re was defined by the stochastic dom inance.
R E F E R E N C E S
Bacclli F. (1991), S to c h astic o rd er o f ra n d o m proccss with an im bedded p o in t proccss, Journal o f A p p lied P robability, 28, 553-567.
Bellm an R . (1957), D ynam ic program m ing, P rin cen to n U niversity Press.
B irk h o ff G . (1973), L a ttice theory, A m erican M ath em atical Society. C o llo q u iu m Publications, vol. 25.
B row n T .A ., S trau ch R .E . (1965), D ynam ic p ro g ram m in g in m u ltiplicative lattices, Journal o f M ath em a tica l A nalysis and Applications, 12, 2, 364-370.
F u ch s L. (1963), P artially O rdered Algebraic S ystem s, In tern a tio n a l Scries o f M o n o g ra p h s on P ure and A pplied M ath em atics, vol. 28.
G laser B. (2002), E fficiency versus Sustainability in D ynam ic Decision M a kin g , L ecture N otes in E co n o m ics and M a th em atical System s, vol. 520, Springer V erlag, N ew Y ork.
H enig M .I. (1985), T h e p rinciple o f op tim ality in d ynam ic p ro g ram m in g with re tu rn s in p a rtially o rd ered sets, M athem atics o f O perations Research, 10, 3, 462-470.
Li D ., H aim cs Y .Y . (1989), M ultiobjective dynam ic pro g ram m in g : the state o f the a rt, Control T heory and Advanced Technology, 5, 4, 471-483.
M itte n L .G . (1974), Preference o rd er d ynam ic pro g ram m in g , M anagem ent Science, 21, 1, 43-46. O gryczak W. (1997), On Stochastic Doninance and M ean-Sem ideviation M odels, Interim R eport
IR -97-043, In te rn a tio n a l In stitu te for A pplied System s A nalysis, S alzburg, A ustria. R olski T . (1976), Order R elations in the S et o f Probability D istributions and their Applications
in the Queneining Theory, D issertatio n M a th em aticae, vol. 132, P A N , W arszaw a. S haked M ., S h a n th ik u m a r J.G . (1993), Stochatic Orders and Their Applications, A cadem ic
Press, H a rc o u rt B race & C o., B oston.
S obel M . J. (1975), O rd in a l dynam ic p ro g ram m in g , M anagem ent Science, 21, 9, 967-975. T rzaskalik T. (1998), M ultiobjective Analysis in Dynamic Environment, T h e A cadem y o f Econom ics,
K atow ice.
T rz a sk a lik T ., S itarz S. (2000), D y n am ic discrete p ro g ram m in g with p a rtially o rd ered criteria set, [in:] M ultiobjective and Goal Program ming, eds. T . T rzask alik , J. M ichnik, Springer V erlag, N ew Y o rk , 186-195.
Sebastian Sitarz PORZĄDKI STOCHASTYCZNE
W DYSKRETNYM PROGRAMOWANIU DYNAMICZNYM
Streszczenie
W p racy rozw ażane je s t zad an ie optym alizacji dynam icznej z w artościam i funkcji kryterium będącym i zm iennym i losow ym i. Ściślej opisany jest m odel d y n am iczn y ze sk o ń czo n ą liczbą etapów , stanów o raz decyzji. Proces taki oceniany jest ze względu n a osiągane wartości zmiennych losow ych. A by m o żn a było zastosow ać zm ienne losow e w optym alizacji d ynam icznej, m uszą one spełniać odp o w ied n ie w aru n k i, co op isan e je s t w pracy. P o d an y jest p rzy k ład m ożliwych d o w y k o rzy stan ia p o rz ąd k ó w stochastycznych, tzw. dom inacji stochastycznych.
