Information Structure, Dynamic Team Decision, and an Economic Application

INFORMATION STRUCTURE, DYNAMIC TEAM DECISION, AND AN ECONOMIC APPLICATION H. Myoken Faculty of Economics, Nagoya City University, Mizuhoku, Nagoya 467, Japan

ABSTRACT The dynamic team decision problem specified by the decentralized control system with linear information structure and quadrati c co st function is considered in this paper. We derive a linear de centralized co ntrol scheme that follows person-by-person-satisfacto ry(P BPS) te a m decision for given information sharing pattern. An illustrativ e example is presented ba sed on the aggregate firm model of production planning i n manu f a c turin g i ndustr y .

I.

for o pt imal parameters and showed that the s € ?a r atio n theorem does not hold in such a case . Fo llowing to Chong and Athans(5), Yoshikawa[6j derived the same necessary conditi ons that are satisfied in the discretetime c a se, and, in addition, he obtained suff i c ient conditions for the problem to be weak separable by introducing an extended concept of the separation theorem. Myoken(13) modifie d the results given by Yoshikawa(6) in some way s to appl y for dynamic economic systems. The papers[8,9,lO,11,12) dealt mainly with the derivation of the optimal controller for t he LQC one-step-delay sharing informati on pattern problem. For example, Sandell and At hans( 8 ) performed a valuable service by showing that, in the LQG stochastic control probl em ca se, the team decision can be solved by me ans of mu l tistage decision approach.Also, Kurt a r a n( 12] c on s idered the co ntrol of disc ret e- time non l ine ar s tochastic systems under an n- step-d e l ay sharing pattern. However, it sho uld be no te d that the s tu d ie s along these lines emphas is o n the theoretica l a nd analytic al aspec ts of t he optimal contro l solution t o n onc l a ssica l LQG sto c hastic decision probl em. Co n seque ntl y , it be comes exceedingl y d i ff i cu lt t o a n aly tica lly solve the problem und e r co ns i de r a t io n, which is g i ven by numeri cal so l u tion or by su boptimal control s o lution. This paper i s concerned with the dyna mi c te am decis ion s pe c ified by th e decentr a li zed co ntr ol of a d isc rete-time linear stochasti c sy s tem with t wo control a gents. These two ag e n t s tr y t o min i mize t he same quadratic c os t func tion a nd ar e ass umed to be ab le to exchan ge their control vectors. The decentra l i zed co ntr ol s cheme obtained f ollows the pe r son- by-pers on-satis f a c tory(PBPS') te a m des io n unde r in fo rma ti on structure assumption. }~ n y inve st iga t i ons o f t he production planni ng pr oblems :\3Ve been und e rt aken by e conomic s

I NTRODUCTION

In r e cent years, a great de ~ l of att en tion has been paid to de centr a li ze d con tr o l sv stems that po sse s s limit ed a nd non-ide ntica l information s e ts on the s tates o f t he sy stem, inputs and its structure(see (1) ,f o r instan ce ). Different information sets und e r con s ideration play a central rol e in t e am (o r organization) decision problem, the di ffi c ult y of which also a ri se s from the s pe c i fic inf ormation pattern. Such s i tuations present th e mselves in various forms of r ea l i sti c e conomic systems. The co nstruct o f t e a m de c ision developed first by Mars c ha k and Radner( 2 ) pr ovided a useful framewor k f or t h e co ncep tu al ization and eva luation of in f orma tion within organi zations . Te a m decision pr oblems are the a r ea o f mutua l i nt e r e st s t o co n t r ol scien ti s t s and economi s t s . The so-call e d linea r-qu ad ra t i c- Cau ssia n ( LQC) pro bl em wi t h clas sica l informa ti on st ru c tur es r es ul t s in a li nea r op t imal decision r ule . Howe v er, t he famous Witsenhausen counte r- exampl e (3) showed t ha t fo r non- c l ass i cal i nf orma tio n struc tur e op timal decis i on ru les need no t be l i nea r. The r ecent works of Ho a nd Chu( 4 ) de mons tr a t ed t ha t li nea r op t imal dec is ion ru le s a l so r es ult if t he info rma t ion stru c tur e i s pa rt ia ll y nested : the dvnam i c te am dec i s i on pr ob lem is red uced to ~n eq ui vale n t s tati c pr oblem by Marsc~ak and Radn e r (2).

Further c ont r ib ut io ns have been mad e by man y a ut ho r s(see fer example [5 , 6 , 7 , 8 , 9, l O ,1~ 12 , 13 )). Chong and At 'lans(5j cons id e r ed a cons tr a ined op t imal con tr ol probl e ~ of continu o us -time linea r s t oc has ti c sy stem wit h two control age nt s t ha t do not exchan ge in f ormation and att emp te d t o determ i ne the op t imal values of coeffic ien t pa r amete r s , where t he stru c tur e s of a gen ts we r e assumed t o be l i nea r. T11en th ey de rived necessa r y co ndit i on s 357

H. Hyoken

358

and management scientists. In this paper a more general aggregate production problem consisting of inventory, sales and finance sectors is cast in a form amenable to solution by discrete-time optimal control theory. Then we present an analytical framework of a wide spectrum of production planning policies applicable to the method proposed in the present paper. 2.

THE MODEL AND PROBLEM

STATE~~NT

Any realistic models including economic systems are generally nonlinear. Sometimes we can treat nonlinear dynamic equations directly. However, very often we must resort to approximately linearizing the model in some small neighborhoods about the equilibrium point or about some specified reference time path, i.e., about tentative path as will be seen in Section 4. In this paper we assume the linearized approximation of nonlinear dynamic systems. Consider a linear time-invariant system dynamics and observation equations described by

criterion in economic system.s may be specified by N

(2.3)' J'

=

L [x(k)-~(k)]TW(k)[x(k)-~(k)] k=1

where i lk) is the desired value for the state vector x(k). Now let u(k) denote the desired control vector for u(k). When u(k) becomes a part of state variables, we can express the system involving ~(k) and u(k). In such case, the usual tracking problem can be formulated for the decentralized system by (2.1), (2.2) and (2.3)'. (For the discussion regarding this, see [15].) In this paper it is assumed that agents exchange their control values. Different informations under consideration play animportsnt role in the decentralized systems. Le t I i (k) be denoted as the informat ion set available to the control agent i at time k. In this paper we consider the information sets defined as

u

(2.1) x (k+l) = Ax(k) + B1ul(k) + B2 u 2 (k), k=O,1,2, Cix(k) + lli(k),

i = 1, ?

k = 1, 2, ... . where x(k)( Rn ~n the state vector, Uj (I,_. ) C RPi, the control vector applied by agent i, YiCk)

q.

the observation signal available to agent i. The noises ~i(k), i = 1,2 are taken to be mutually independent, their density distributions are ER \

P(~i(k»

= N(O, Ri)' i=1,2; k=l,2, ....

It is assumed that agent i has his O~1 prior distribution with mean ~ i and covariance Li, where Li, i=1,2 are symmetric nonegative definite matrices and known. Furthermore i t is assumed that a priori information on x(O) for each agent is different and is unknown to the other. Most of economic mod e ls c haracteristicall y become proper, since they ha ve much lar ge sampling intervals, and si n, e th e imp"ct multiplier often is not zero nu trix in policy implementation[14]. Thus th e direct part DuCk) is added t o the obs e rv a tion equ a tion. However, now define ~

(k-l) } ,

i = 1, 2.

A priori information of agent i consists of (2 . 4 . 2 ) I i (0) = { A. B 1 • B 2 ' Cl' C2' i' RI' R2 }

Various information sets may be considered: for examples, the case with information exchange on control and observation variables with one-time delay, or the case with information exchange on control laws used by the other, etc .. This paper derives the PBPS team decision rules under the information structure assumption (2.4). The results obtained are compared with these of informationally centralized control systems. Following to Marschak and Radner[2]. a team decision rule is defined to be PBPS if it cannot be improved by changing one of its components. A pair of decentralized control laws is PBPS if the expected cost cannot be reduced by changing any singlf' agent's control law. 3.

PBPS TEAH DECISION RULES

The optimal control vector at the last ti r.1t~ period under the centralized information assumpti on is given by T

C3.1) u~_ 1 = -(p+B \,'B) 1 I (N-l)

t uCk)

2

-I

T , I B h'AE \ x(k-1) ,

U 12 (~-l)

}

u(k+l) - u(k ) where

where ,~ u(k) is the input, y (k) is the output. Then the model be comes stri c tl y prop"r. The performan c e me a sure i s g ive n by

B

~

(2.3. l) J

=

p

:>: F (x (k) , u 1 (k -1 ) , u , (k -1) )

k=1

Let

where

wher e F 2, O is r ",:.; l valu2c.

Th e performan ce

~Ier e 'r i ( ~ )

stands for the cost ob s erve d by co n t r ol a gen t i. For agent 1. (3.2) can be

359

Information structure (ll (N-l)

written as 0.3) yl(N)

= IIXI(i,J) I! ~ + ll ul(N-l) i :~

where I

+llu 2 (N-l) ll p

+

:-1- 1 rCl! lC 2 J

ZI(:\)

2

where Eh(N) I 11 (N-l);

(3.4) XI (N)

A(ll (N-l) + BI U I (N-l) + B2 G2 (N· 0.5) (l1(N-l)

E{x(N-l) I II(N-l) ~

(3.6) ~2(N-l)

E{u 2 (N-l) I 11 (N-l) '

1)

and where (3.7) II (N) = E {II,x(N) -

+ 11 U (N-l) 2

A Xl

2 (N),11 w

u

(N-l) II, ?' p. ~II(:-;-l) )J

A

2

2

since II (N) is independent of u l (N-l). For E{FNII 2 (N-l) } , we obtain a similar expressior.. The unique PBPS decision rules at the last stage are obtained by the solution of 0.8)

llIin y. (N-l) , u (N-l) l. i

i = 1, 2

Then the problem 0.8) is equivalent to solving aYi(N)

(3.9)

---. 3xi(N)

d~i(N)

ay . (N)

aU i (N-l)

+

l.

0,

aU (N-l) i i

In his p3per[ 7 ], Aoki observed that the PSPB decisi un rules have a certain correction terms due t o the well-known optimal proportional feedback control signal obtainable under th~ information centralization assumption. A" indicated in the results (3.11) and (3.13), they do not rely on the information structur~, so that the PBPS decision rules under given information sharing pattern (2.4) are the same as that of the centralized systems. The estimates under consideration are the conditional means that are determined using Kalman filter, and then the equations (3.13) are satisfied. ~e can also deal with the PBPS decision rules for the case where the agents share the control law with each other. In this case, an infinite-dimensional filter would be required, when computing the conditional expectations 0.4), (3.5) and (3.6). Approximations to conditional expectations must be considered, from the practical point of view. For this purpose, a useful method may be found in the guaranteed cost synthesis approach proposed by Chang and Peng[16] and Jain[17, 18): The PBPS decision rules will be given so that the control law minimizes an upper bound on the performance cost function.

= 1, 2

4.

THE FIRM DECISION MODEL AND THE CONTROL SIMULATION HETHOD

where recall that FN is quadratic in all its arguments, and where (3.10.1) ~l (N)

4.1

A(l l (N-l) + Blu l (N-l)

=

+ B2 E{u 2 (N-l) I II(N-l) } (3.10. 2) ~2 (N)

Ap 2 (N-l) + B I E { u I (:-;-0

!

1 (N-l) } + B2 u 2 (N-l) 2

Therefore, the PBPS decision rules at the last stage are the solution of 0.11.1) Mllul(N-l)l+ H I2 G2 (N-l)

CI

0.11.2) N2l GI (N-l) + M22 u 2 (N-l)

C2;J 2 (N-l)

:-

I

(N-l)

where H ..

T R. + B l-IB i' 1. i

M21

B;\JB I ,

1.1.

C.

1.

T NI 2 = BI WB 2 _BTWA, 1.

i

=

,

1, 2

and are given by

The Basic Model

Nuch of the model formulations appeared in the literature on aggregate production planning has its origins in Holt et a1. [19). Very recently, considerable attention has paid to the application of discrete optimal control theor y to production planning policy. (see [20),[2 1] , for instances.) However, most of the previous models have in large part neglected the impact of policy decision under informatio nally decentralization. In this section, we present a d ynamic simultaneous mo del that considers the interaction between the sectors of a firm. The firm model developed here is a sound basis for the investigation of control scheme of informationally decentr a"li zed systems. Production sector on firm behavior interdependently is associated with other sectors consisting of at least inventory, sales and finance se c tors. (see Fig.1.) Fig.l:

(3.12.1) u I (N-l)

LI (N-lhl (N-l)

0.12.2) u 2 (N-l)

L2 (N-l) w2 (N-l)

if the following equations are satisfied:

A Simultaneous Firm Model

I

~ ~II-n-v-e-n-t-o-r":")~'I

'

,' production; Investment in ! plant & equipmentJ

!

~ j

~I ~~~~nce; I~

..-[S'::'a-l-e-s"'"

H. Myoken

360

Naturally, it is also possible to consider micro-aspects of firm behavior such as research development and labor management sectors. However, the model presented here devotes itself to the derivation of empirically testable hypotheses regarding macro-aspects of firms' activities on aggregative economic variables. The symbols in this model are as follol<'s: ~mbols

Q:

the desired production capacity(i.e. potential for production)

q:

change in production capacity

0:

production

~:

capacity utilization ratio

Cl: cost of investment in plant and equipment

a conservatio n of the number of unit items is assumed.

(Ui)

Sales Se c t or S(k) = S(a(k-l), P (k) , S(k-l) ,

(4.7)

i

(k-2) , S(k-2»

(4.8) C4 = C4(S (k» Sales durin g period k is assumed to be a function of the price during the same period, advertising cost and sales during time lagged period. This relation is a reasonable assumption. (iv)

Identities

= Cl(k) +

(4.9) C(k)

C2(k) + C3(k) + C4(k)

+ a(k)

C2: cost of production I:

inventory

S:

sales

C3: cost of inventory a:

advertising cost

p:

price total cost

T:

total sales amount

'I f:

profit

e:

the gap between supply and demand

d:

demand forecasts

( i)

P(k)S(k)

(4.11)

;,(k)

T(k)

C (k)

(4.12) e (k)

d(k)

S(k)

It should be noted that the demand forecast for the commodity is estimated in connected with macroeconometric model that is set up outside the model considered. d is generally expressed as

C4: marketing cost C:

(4.10) T(k)

d(t) =

Production and Investment in Plant and EguiEment Sector

(4.1) Q(k)

Q(k-l) + q(k-l)

(4.2) rI (k)

O(t)/Q(k)

(4.3) Cl (k)

CHq (k»

(4.4) C2 (k)

C2(Q(k),

o (k),

rI (k) )

d( ~ ,

D(k), ... )

where )J is the market share; D is the total demand forecast for the commodity estimated in the firm world. Thus d is assumed to be given as the data variable ,which is exogenous in the sense that it is determined by th e relation(i.e., macroeconometric model) outside the model. (On the other hand, the variables which are determined by the model are called endogenous variables.) The variables used in this model are partitioned as: Endogenous variabl e s

y

(Q,

~1 ,

I, S, e, T, Cl,

C2, C3, C4, C, "' )

In a manufacturing concern the management wants to determine how much the present capacity for production is to be changed in order to obtain the desired capacity for production. The capacity for production during period k indicates the relation with one time delay as shown in (4.1). Since Q(k) and 0 (k) are the fixed cost and the variable cost, respectively, rI (k) stands for the explainning factor regarding the adjustment cost. (ii)

Inventory Sector

(4.5) I(k) (4.6) C3(k)

= =

I(k-l) + O(k-l) - S(k-l) C30(k»

Equation (4.5) indicating the amount of inventory at beginning of period k can be expressed by the relation between one time delayed inventory, production, and sales, where

Control varaibl e s Data variables

z

u =

=

(q, 0, a, P)

d

State variables (= the number of predetermined variables - the number o f exogenous variables) x = (Q(k-l), I(k-l), S(k-l), q(k-l), O(k-l), a(k-l» where the predetermined variables are also classified as: .

(I lagged endo genous ! variables

predeterm~ned ~

variables I

(control variables*

! exogenous J

lvariables) . Idata (or ~nterac\ t ion) variables

Information structure

*

(current control variabl~s I

tlagged control variables

The basic model is represented Ioy the following vector form: (4.13) y(k)

a(y(k), x(k), u(k), z(k»

which is of the structural form. Economic systems generally are nonlinear in structure, but very often we must resort to some approximations. ~ow let us be y(k), ~(k), ~(k), ~(k) where superscript " ~ ,, indicates tentative paths. Then we can approximate the model (4.13) about tentative paths as follows: (4.14) y(k) = F(k)x(k) + G(k)u(k) + h(k) where (4 . 15)

(4.16) h(k) = y(k) Equation system (4.14) is called the reduced form or the output equation. Since the state variables are defined by the endogenous and control variables, we have

=

Ly(k) + Mu(k)

where elements of the matrices L and ~1 may be supposed to be zero or to be unity. Substitution of (4.14) into (4.17) yields (4.18) x(k+l)

= A(k)x(k)

+ B(k)u(k) + W(k)

wh~re

A(k)

LF(k)

l>(k)

~1

\.J(k)

Lh (k)

+ LB(k)

We call (4.18) the state equation system, which expresses the dynamics of the mode l. The dimensions of the vectors in the model under consideration are as follows: x:

~X

G

u:

;-,'V

4

y:

NY

12

z:

NZ

1

As indicated in the above examples, the dimensions of observable variables y are greater than these of state variables x. Very often this case appears in the usual economic systems. However, since the system dynamics can be experssed by the state variables;= tIle number of I t

1'. \' ! _

I~

prcdeter::lin"d variable,.; - the number of curr"nt exogenous \·,n iables), tlll' sta te variables are done .... ithout .l great number of variables. In addition, a!l they ca n be computed using (4.17). ~ounda rv

Condition,:

~ext

we suppose that

severJ.l. \'~!ri3bles used have some upper and lov..'er lir.1its: (-'+.1 'I)

q (k)

(4.20)

,(k)

q(k) < q(k) 1.2

(4.21) i S (k) -.: I(k) (4.22) S(k)

I (k)

(4.23) (k)

C(k)

(4.24 ) E (k) z P (k)

iIS(k)

P(k)

Performance Cost Function: Finally, wc suppose t-h a t the firm \"ishes to regulate production so to minimize the sum of the gap between supply and demand, total cost, and capacity util i za tion ratio deviated from a reasonable value over some fixed time interval [ 1, ,,:,]. h'e can then "Tite this performance criterion function as

[ F (k), G (k) ]

(4.17) x(k+l)

361

~~

(4.25) "

.. k ,(k)e 2 (k) + l...'z(k)C 2 (k) k=l .

+ wJ(k ) ('-,(k) - 1.0)2 ) where cC l , " '2' and "-' 3 are, respectively, the weights assessed against the gap, total cost, and the deviated capacity utilization ratio for the period. We first consider a control problem \,ith the cost function (4.25), where the constraints are subject to (4.1) through (4.12) and (4.19) through (4.24). Then the optimal paths are computed when obtaining the solutions so as to minimize the performandce criterion cost in (4.25). \,e call this compu t ation or the ope rati on th e control simul.Jtion. 4.2

Op ti mal Control Simulation

The model (4.1) through (4.12) is generally nonlinear, and the boundary conditions (4.1~ througil (':'.2:') involve the state and control variables. TI1US the optimal solution and the optimal path are obtained employing the iterative operation by the linearized approximation procedure. (For the discussions in details re g.Jrding this method, refer to [22].) Suppose now the optimal paths indicating superscript "*" are given using the method as x*(k), u*(k),

k = 1, 2, ... , ,,:;

The linearized approximations of the model in the neighborhood of optimal paths are as follows: (4.26) x(k+l) - x*(k+l) + B (k) (u (k) - u* (k»

= A(k)(x(k)

- x*(k»

+ W(k) (z (k) - z* (k»

The data variables for the period are assumed

362

H. Myoken

Cl 0)

cance "nd the empirical validity of the PilI'S team d"cisiOI1 rules for the aggregate prodlCtion plannill b • \lation. The plants and equipment allocated to the production for each commodity are transfered to these ior another commodity, according to the supply and demand for each commodity. It is assumed that the sales amount for each commodity is also influenced by the sales and advertiser.ent for another commodity. In this case, increase in the sales amount foranother commodity does not necessarily become positive. If it is a substitute, we have a negative effect. We now extend the basic model (4.1) through (4.12) used in the previous section to a multi-item formulation.

Cl 0)

(5.1)

to be given, i.e., z(k) = ~(k) = z(k), i

1, 2, ... ,

~T

The performance criteriol1 function is minimized by regarding the optimal paths as the target variables, i.e., NI<

(4.27) q, =

L {llx(k+l) - x*(k+l) I:~' (k+l) .

k=l

+ 11 u(k) - u* (k) 11 ~\(k); The weighted coefficients are defined as (4.28) r(k) = (4.29) A(k)

X*~k) 1

u*(k)

N

In doing so, each term in (4.27) is the squared error for each variable. Therefore, (4.27) is expressed as the squared sum of error. The optimal control solutions so as to minimize (4.27) subject to (4;26) are given as (4.30) u(k) = u*(k) -

~(k)(x(k)

- x*(k»

,\ .. (k-I)Q.(k-l) + 'l.(k-I)

1:

Q.(k) =

J=1

~

J

~J

1.

\,rhere ,0ij stands for the transfer from the pl.:lI1ts dnd equipmE'nt for the jth commodity to thes.:! for the ith commodity. (5.2)

' I . (k)

(5.3)

Cl.(k)

(5.4)

C2 . (k) = C2 . (Q. (k) ,

(5.5)

l.(k) = l.(k-l) + O.(k-l) - S.(k-l)

(5.6)

C3. (k) = C3. (I. (k»

(5.7)

S.(k)

1.

= 0.

~

(k)/Q. (k) ~

Cl. (q . (k) ,

1.

~

(I, .. ~J

1.

(k»

j=i; j=1,2, ... ,N where the state variables x(k) (= the number of predetermined variables - the number of current exogenous variables) are defined by (4.17) and perfectly observable. 5. MULTI-ITEH PRODUCTION PLAt'<:-lING A.'
~

1.

1.

1.

1.

11 • (k» ~

1.

~

1.

1.

1.

S.(a.(k-l), P.(k), S.(k-l),

11.1.

1.

1.

a. (k-2), S.(k-2» J J

A Hulti-Item Decision Hodel for Hanufacturing Concerns

Aconsiderable portion of the aggregate production planning problem can be traced to the traditional framework done by Holt-type model [19]. The model presented in Section 4 is in some part different from the aggregate Holt model: For examples, we first treat the production planning by inventory and sales as a typical prob lem 0 f s tochas t ic opt imal con t ro!, the impact of the resultant cos t or finance flows for each sector is considered, and the firm model is developed in combination of macroeconometri c model. Kriebel[23] investigated the quadratic team decision problems to the Holt-type model on optimal linear dedswn rules for aggregate planning. As Bergstrom and Smith[24] pointed out, however, because of the aggregate nature of this formulation, it is not possible to solve directly for the optimal production rates for individual products: Therefore in situations where no natural dimension for aggregate exists, the breakdown of an aggregate produ c tion plan into individual item plans may r es ult in a schedule which is far from optimum. In this case , specification of an aggregate production planning neglects the most interesting question. In this section, we extend the model presented in Section 4 to a multi-item formulation which solves directly for the optimal production, inventory, and sales for individual items in the future periods. Also, in doing so, it may be easier to observe the practical signifi-

o.1. (k) ,

j=1,2, ... ,N

(5.8)

C4 . ( k )

(5.9)

C. (k)

1.

1.

=

C4. (S . (k) ) ~

1.

Cl . (k) + C2 . (k) + C3 . (k) 1.

1.

1.

+ C4. (k) + a. (k) 1.

(5.10) T. (k) 1.

P. (k)S. (k) 1.

1.

(k)

T. (k) - c. (k)

(5.12) e. (k)

d. (k) - S. (k)

(5.1l)

~ . 1.

1.

1.

1.

1

1.

~

Similarly, the boundary const raints (4.19) thr ough (c. . 24) can be also re\rritten in connection \"it:\ multi-item formulation. Individual ~ptimization is implemen ted using submodels (5.1) thr ough (5.12), and after exchange of some values for interaction variables, the optimal design is made in such a ~av as the iterative ope ration is repeated until the optimal paths converge. Also, the submodels are set up as one firm model toge ther, and th en the optimal planning may be designed er:;ploying this node!. Assuming that the deternination of the op timal planning is made using the model involving many establishments, the decision-making for each establishmen t ir.~plemented according to the firm plan f rame. (See Fig.2.) The model for the ith establishment (that is c:~arge of th" ith commodity) is

Information structure

363

(x (k), k < k' -1 :

Suppose that the optimal paths are given by the firm plan frame as

(5.22) x(k !k')

the realized values

l

x (k t k), k = k' : the forecasting values

Yi(k), xi(k), ui(k), zi(k) k = 1, 2, ... , "N

It is apparent that

i

(Z(k), k
I I

(5.23) z(k i k')

z*(k), k,::,k'+1: the forecasting values The linearized approximation of (5.13) in the neighborhood of optimal paths "*" can be written in the state equation form as

(5.15) x (k+1) - xi(k+l) = A (k)[x . (k)-x~(k)] i ~ ~ + Bi(k)[ui(k) - ut(k)] + Ci(k)[zi(k) - zt(k)]

to carry out the rivision of (5.18). have

(5.24) u(k ik) =u*(k) -F(k) [A(k)[x(k lk) -x*(k)]

+ C(k)(z(k lk) - z*(k»] (2.25) x(k+l : k+l) =x*(k+1) +A(k)[x(k) -x*(k)]

Unless otherwise stated the subscript "i" is deleted. The performance measure assumed is given by kE/"x(k+1) - x*(k+l)llr(k+l) + lIu(k) - u*(k)il;,(k)} which indicates a reasonable specification form implementing the optimal contr0' . lsee (4.27), also.) 5.2

The Practical Control Simulation for Determining Optimal Planning

Assuming that the data-interaction variable vector

(5.17) z(k) = z*(k),

k = 1, 2, ... , NN

we consider the minimization of the performance cost in (5.16) subject to (5.15). The control solutions are then given as

(5.18) u(k) = u*(k) - F(k)A(k) [x(k) - x*(k)] where

(5.19) F (k) = [A(k) +B T (k) TT (k+ 1) B(k) ]-1 BT(k) n (HI), k

(5.20) TT(NN)

+ B(k)[u(k) - u*(k)] + C(k)[z(k) - z*(k)] where it is clear that

NN

(5.16)

Thenwe

= 1,2,

... , NN

= r(NN)

(5.21) TT(k)=A(k)+AT(k)TT(k+l) [I-B(k)F(k)]A(k), k = 1, 2, .•. , NN-l.

However, we here note that there are the two problems encountered. First, it is rarely the case that z(k) = z*(k), i=1,2, ... ,N:-.I. Second, it is very difficult to obtain the realized values for x(k) and z(k) at period k. Furthermore, it may ~e not possible to generally treat these problems. 111erefore, more realistic situations must be assumed as follows. I t is possible to obtain the realized values for x(k) and z(k) at one time period before, but their values at current period are obliged to forecast. Accordingly, we use

(5.26) u(k) = u(klk) = u(klk+l) and where in (5.24) through (5.26), x(klk), z(klk) and u(klk) are the forecasting values; the realized values are given by x(k), z(k) and u(k). Fig.3 describes the procedure for the implementation of the planning made in the interaction between the head office and individual establishment or branch in manufacturing concerns.

6.

CONCLUDING REMARKS

Host of studies on the decentralized control problems from a team decision approach have been prescriptive and very formal rather than empirical. The main point of this paper is to suggest a new practical and empirical method by illustrating the well-known aggregate production planning problem within the framework of the dynamic team decision under informationally decentralization. In particular, it is verified that the team decision of multiitem extension is one certain convenient framework for detailing the normative evaluation. In the realistic multi-item firm model, the information available at the various control agents on which decision are to be made is different. On the other hanJ, this multi-item model, indicates a multiple person situation in which no conflict of interest exists among members. The optimal team decision rules are generally bound to be nonlinear and comples, so some possible results are given in the linear solution case only. The computational device and its applicability are emphasized in order to hold an empirical validity of promise for integrating and extending our knowledge about information and decision. The approach proposed here may be a useful method adapted to analyze various aspects of decision problems in this direction.

H. Myoken

364

<03> I I

(Model Building; Specification

I

For e castin g Forecasts for Firm's Environment; i Simulation Decision on Firm Plan Frame ~----------------------~~ f o r Firm's - - - -, Environment*

--,

I I

~

I

_E__x_a_m_i_n_a_t_~_"o__n__b....y_E_S t ab 1 ishmen t ~_~_---,_ _____---,______.., . ~ '- or Branches Po l i c y Simulation ~ f o r De t e r mining Examination by the Head Office re71 ,' -_O_p_t_i_n_la_l_ P lann_i~~J 11

[~~~ '''J;::

[Completion of Firm's Plan Frame !

, I

~

* The course of the development of the f irm's environme nt including demand forecast for the commodity is estimated in th e combinati on o f mac r oe conometri c model. Fig. 2:

The Procedure for the Firm Plan Fr3me

k

IObtai;-th e i for z(k-l)

~~li zed Value s

The Head Off i ce: For ec as tin g of Firm's Environment. Co nt ac t with Observ e th e Re ali zed Values ~-----------------------------~)I I ndi vi dual for x (k-l ) Estab li shment or i Bran ch . !~F~o~r~e~c~a~s~t~~fo~r~x~(~k~"'~k~.~)!(---------------------------------~j D i ~~ : ss i o n and ____________________~~ nt I Fo r ec a s t fo r z (k "k)': ..--------- - - - -, I ----------------------~

I

Det e r mi na ti on or lI(kk)j

k < k+l

i

i

~0 ~:

Procedure for the Planning Impl ~ mentation for Individual Establishment or Branch

Information structure

365

REFERENCES [1]

Y.C.Ho and K.C.Chu: Information Structure in Dynamic Multi-Person Control Problems, Automatica 10(1974), 341-351.

[2]

J.Marschak and R.Radner: The Economic Theory of Teams, Yale Univ. Press, 1971.

[3]

H.Witsenhausen: A Counter-Example in Stochastic Control, SIAM J. Control 6 (1968), 131-147.

[4]

Y.C.Ho and K.C.Chu: Team Decision and Information Structure in Optimal Control Problems, IEEE Trans. Aut. Control AC-17 (1972), 15-22.

[~]

C.Y.Chong and M.Athans: On the Stochastic Control of Linear System with Different Information Sets, IEEE Trans. Aut. Control AC-16(197l) , 423-430.

[6]

T.Yoshikawa: Decentralized Control of Discrete-time Stochastic Systems inProc. 3rd IFAC Symp. Sensitivity, Adaptivity, and Optimality, 444-451, 1973.

[7]

M.Aoki: On Decentralized Linear Stochastic Control Problems with QuaJratic Cost, IEEE Trans. Aut. Control AC-17 (1973), 15-22.

[8]

N.R.Sandell and M.Athans: Solution of Some Nonclassical LQG Stochastic Decision Problems, IEEE Trans. Aut. Control AC-19(1974), 108-116.

[9]

B.Z.Kurtaran and R.Sivan: Linear Quadratic Gaussian Control with One StepDelay Sharing Pattern, IEEE Trans. Aut. Control, AC-19(1974) , 571-574.

[10] T.Yoshikawa: Dynamic Programming Approach to Decentralized Stochastic Control Problems, IEEE Trans. Aut. Control AC-~0(1975), 796-797. [11] B.Z.Kurtaran: A Concise Privation of the LQG One-Step-Delay Sharing Problem Solution, IEEE Trans. Aut. Control AC-20 (1975), 808-810. [12] B.Z.Kurtaran: Decentralized Stochastic Control with Delayed Sharing Information Pattern, IEEE Trans. Aut. Control AC-2l (1976), 576-58l. [13] H.Myoken: Optimal Control Problem of Decentralized Dynamic Economic Systems, Proc. the vm th International Congress on Cybernetics, Namur, 1975. [14] H.Myoken and Y.U c~ lida : A Minimal Canonical h Jrm Realization for a Multivariable Econometric Sys tem, Int. J. Sys tems Sci. 8(1977), 801-811. [15] H.Myoken: Controllability and Observability in Optimal Control of Linear ec onometric Models, Kybernetes 6(1977), 1-13. [16] S.S.L.Chang and T.K.C.Peng: Adaptive Guaranteed Cost Control of Systems with Uncertain Parameters, IEEE Trans. Aut. Control AC-17(197 2), 474-483. [17] B.N.Jain: Guaranteed Error Estimation in Uncertain Systems, IEEE Trans. Aut.

Control AC-20(197 5), 230-232. (18] B.N.Jain: Bounding Est:mators for Systems with Colour eJ Noise, IEEE Trans. Aut. Control AC-20(1975), 365-368. (19] C.C.Holt, F.Modigliani, F.Muth, and H.A.Simon: Planning Production, Inventories, and Work Force, Prentice Hall, 1960. [20] L.Taylor: The Existence of Optimal Distributed Lags, R. Econ. Studies, 37(1970) 95-11 l o. (21] P.R.Kleindorfer and K.Glover: Linear Convex Stochastic Optimal Control with Applications in Production Planning, IEEE Trans. Aut. Control AC-18(1973) , 56-59. (22] H.Myoken, H.Sadamichi and Y.Uchida: Decentralized Stabil1zation and Regulation in Large-Scale Macroeconomic-Environmental Models, and Conflicting Objectives: Proc. IFAC Workshop on Urban, Regional & National Planning: Environmental Aspects, Pergamon Press, 1977. [23] C.H.Kr1ebel: Quadratic Teams, Information Economics, and Aggregate Planning Decisions, Econometrica 36(1968), 530543. [24] G.L.BergstroDl and B.E.Smith: Multi-item Production Planning - an Extension of the HMMS Rule~, Management Science 16 (1970), 614-629.