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To recapitulate, the design algorithm requires the follow-ing steps. For an arbitrary pair (ko*,10*), (10) and (11) areused to find V(to,xo) (see (8)) when T is finite. When T isinfinite (14), (15), and (21) are used to find J' (see (23)). Theoptimal pair (k,*,l*) is then found by searching over theset of possible employment and inventory levels and com-paring the costs. As the example given in the precedingindicates, the monotonicity properties of the cost will aidin this search.The system specifications for the system are easier to

derive for the infinite planning horizon than they are forthe case of a finite horizon. This is basically due to the timeinvariant nature of the resulting control policy and the factthat the design equations are algebraic. Matrix equationsof a more complicated type than (10) were solved iterativelyin [7] with no problem related to either storage or com-putation time. Although no conclusive analytical resultsare now available, it is the authors' belief, based upon theirown experience, that systems with state vector dimension

less than ten should create little computational difficulty.Above that, care would have to be given to the solutionmethods used.

REFERENCES[1] R. S. Ratner and D. G. Luenberger, "Performance-adaptive

renewal policies for linear systems," IEEE Trans. Automat. Contr.,vol. AC-14, pp. 344-351, Aug. 1969.

[2] D. D. Sworder, "Uniform performance-adaptive renewal policiesfor linear systems," IEEE Trans. Auitomat. Contr. (Short Papers),vol. AC-15, pp. 581-583, Oct. 1970.

[3] W. M. Wonham, "Random differential equations in controltheory," in Probabilistic Methods in Applied Mathematics, vol. 2,A. T. Bharucha-Reid, Ed. New York: Academic Press, 1970.

[4] D. D. Sworder, "Feedback control of linear systems with jumpparameters," IEEE Trans. Automat. Conttr., vol. AC-14, pp. 9-14.Feb. 1969.

[5] K. L. Chung, Markov Chains w,ith Stationary Transition Proba-bilities. New York: Springer, 1967.

[6] D. R. Cox and H. D. Miller, The Theory of Stochastic Processes.New York: Wiley, 1965.

[7] B. D. Pierce and D. D. Sworder, "Bayes and minimax controllersfor linear systems with jump parameters," IEEE Trans. Automat.Contr., vol. AC-16, pp. 300-307, Aug. 1971.

Optimization and Simplification of SystemModels Characterizing an R&D Process

MOSHE TAMIR, MEMBER, IEEE

Abstract-Recent developments in modern control theory and high-speed computation techniques have enabled extensive treatment of com-plex processes. A method is presented for building a model of an R&Dprocess utilizing the state space approach. The specific model describedin this paper contains 21 state variables and a control vector of 6 com-ponents. Its formulation leads to the Mayer's problem with inequalityconstraints imposed on the control vector. An algorithm based on theadjoint system technique is used for simultaneous optimization andsimplification. The computation for the preceding model lasts 2.5 minand is completed within 5 iterations. It results in an improvement factorof three for the chosen index of performance. Simplification of the modelreduces the 21 state variables and 6 control components into 1 statevariable and 1 control component.

I. INTRODUCTION

A COMPLEX process contains many interrelated vari-tables. These interrelations may be algebraical, opera-

tional, or even nonanalytical, described only by means oftables. Recent developments in modern control theory andhigh-speed computation techniques have enabled extensive

Manuscript received July 16, 1970; revised February 1, 1972. Thispaper is part of a dissertation submitted to the Senate of the Technion-Israel Institute of Technology, Haifa, in partial fulfillment of therequirements for the D.Sc.Tech. degree.The author is with the Armaments Development Authority, Ministry

of Defense, Tel Aviv, Israel.

computational treatment of complex processes, an exampleof which is the R&D process. This process is the dynamicevolution of a research project under restricted amounts ofresources which limit its rate of progress. Like an ordinaryindustrial process, it contains flows of materials, money,man-power, and knowledge. Regulation of these flows de-termines the process quality. This is done according toadministrative and technical decisions fed back by informa-tion flows, the sources of which are various parametersdescribing the process.

In 1964 Roberts [1] introduced a mathematical model ofthe R&D process to be used in a digital computer simula-tion. This model was built according to the industrialdynamics technique, developed by Forrester [2] in 1961.The equations in this model are cumbersome and notamenable to analytical treatment. Hence the model istransformed to the state space presentation [3].An R&D model consisting of 21 state variables and a

control vector of 6 components is established. An index ofperformance is defined. The optimization problem appearsto be that of Mayer [4] with inequality constraints imposedon the control vector. The process duration and its initialstate vector are assumed to be fixed. No component of thefinal state vector is given (free endpoint [5]).

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The adjoint system optimization technique is used [4],[6], [7] with an algorithm which converts the originaloptimization problem into a single-variable maximizationsearch achieved by Fibonacci's method [8]. This search isestablished around a nominal trajectory and thus implieslocal optimization.Taking advantage of the fact that the sensitivity matrices

of the perturbed system equation must be computed forthe optimization algorithm, they are used to build a simpli-fication matrix which distinguishes those state variables andcontrol vector components which are dominant with respectto the chosen performance index. Thus the simplificationimplies reduction of the dimensionality of the state andcontrol vectors. The optimization and simplification of themodel are carried out simultaneously.

II. THE R&D MODEL

The R&D model is represented by the following discreteequations:

x(K) = f[x(K),u(K),K]

x(K + 1) = x(K) + i(K) AT

where

ATKx

u

x

basic time unit of computationdiscrete timestate vectorcontrol vectortime derivative of the state vector.

These discrete equations are equivalent to the continuousand more concise form which is given by

x = f[x(t),u(t),t] (1)

where t is the current time. It is a complicated model, theinterrelations of which are algebraical, operational, andnonanalytical; these are described by means of tables. Time-varying parameters and many feedbacks are included in it.The established R&D model has a state vector of 21

variables (Table I) and a control vector of 6 components(Table II). It consists of 9 submodels containing the variousvector variables, the interconnections of which are describedin Fig. 1. Owing to lack of space, the detailed descriptionof the model is omitted (for full description see [9]).

11. THE OPTIMIZATION PROBLEM

Among the numerous quantities which the R&D modelcontains are the estimate of the product future value ($)and investment cost ($). The ratio between these quantitiesat the final time t, of the R&D process expresses the worth-whileness of the R&D project and is defined as its index ofperformance 4(t,). The optimization problem is to deter-mine the time behavior of the control vector which willmaximize 0(t,).

Analytically, the problem is defined as follows. Theprocess set of equations is given (1):

x = f[X(t),U(t),t]

TABLE ISTATE VARIABLES OF THE R&D MODEL

Variable Description I. | so

xi yri eastmate of product future value * 20 x 10

Z2 Firm recognition of product present value $ 0 10

a3 Average of firm recopitio tate of product $ / mth 0value

4 1Firm knwledge tmennth 6

Z1 Average of firm knowledge mnamouth 6

Firm estimte of genral knowledge oeede6 msnsmonth 2500

for the project

Pirm estimte of present firm technicalX7 offectivenes percent 60

a8 Average of firm technical progrea rate perCeCnt 0onth

2 Ken_Iagemnt tat te of firm technical percent 10effectiveess

Average of excess egineers mployed in

Xl0 the project

Number of new engioners recruited to the 311 project

Number of nlgneers transferred In the fir _

x12 2to the project

X13 Numlr of enginoers leal4 the project 0

X14 Project salaries * 0

'is Project materials end equIpment expns $ 0

116 Customer estlmate of product future value O x 10

X17 Cuatomer estlmate of product present valu $ 0 x 10

x18 Average of customer value-recogSntion- s/ month 0rate

119 Customr estimte of general knowledge smnnth 140019 needed for the project

Cuatomr estimte of firm technlical120 effectivenses percent 25

AveraSe of custoer estimte rate of firm percentx21 technical progr"s th 0

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TAMIR: SYSTEM MODELS CHARACTERIZING R&D PROCESS

TABLE IICONTROL COMPONENTS OF THE R&D MODEL

Nominal a-Lower i-UpperComponent Description Dimension Value Bound Bound

Duration of firm knowledgeU1 acquisiton month 24 3 48

Allowable percentage of

U2 engineers transfer in percent 20 0 40

the firm

Recruitmemt delay of newu3 month 6 3 12

engineers

Engineers transferu4 month 3 1 6

delay

Unexploited percentageu5 percent 10 0 50

of annual budget

u6 1 Annual budget 6$0 106 0 60 X 106

Connections Entering

The initial state vector is defined (Table I) as

x(to) = xo. (2)There are constraints imposed on the control vector(Table II):

a < u < b. (3)

The index of performance is 4(tj), where t1 is the final timeof the process. The process duration is assumed to be fixed.The more general and real case involves unfixed duration.The optimization problem is

max O[x(t1),t,] (4)

where k is a function of the vector x and time t. 7 h.s isMayer's problem [4] with inequality constraints inqV{,(sedon the control vector.

IV. THE ADJOINT SYSTEM ALGORITHM

Among the various optimization methods, the adjointsystem technique [3] is preferred [4], [6], [7], owing to itsprogramming simplicity, reasonable computing time re-quirements, and nonexistence of convergence prerequisites.This technique is an iterative dynamic method for findingthe extremum of a functional.The system of equations defined by (1) is nonlinear, and

overall optimization is not feasible. Hence a nominal trajec-tory defined by unom(t) and xnom(t) is chosen. By a smallcontrol perturbation bu(t), an equation for the first variationis attained:

A B C D E F G E J2 2 1 2

2 3 3 2 7

I21 13 T2 1 T 1

- r rT21 T3 I1T 113 _ 2 1 _

3 1 2

2 2 1 2 3

1 2

1

d%x=Oaf 6x + af . udt Ox Xnom au Unom

= A(t) .5x + B(t) * au (5)

where A(t) and B(t) are known time-varying matricesevaluated along the nominal trajectory. Their correspondingdimensionality is n x n and n x m, where n is the numberof state vector variables, and m is the number of controlvector components.The costate vector A is defined by the adjoint system

equation [3]

A = -A'(t) * A (6)

Fig. 1. Interconnections of R&D model.

where

Ox -C. (7)

The dimensionality of A is n x 1, t, is the final time, andis the criterion functional of the process. A' is the trans-

pose matrice of A.The total increment of the performance index 4(tj) is the

sum of the increments c4(tl) occurring in each iteration. Inorder to attain a maximal total increment of the perfor-mance index, it is desired to maximize the increment 34(t,)in each iteration. The problem is then max [64(t,)]. It isproven ([3, p. 348]) that

tf

60(t1) = (B'(t) A)(t),6u(t)) dt (8)to

A

Connections B

Emerging C

D

E

H

J

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is given by

h*(K) = P hmax(K).q

(14)

K t

Fig. 2. Step factor calculation. uk(K)-component k of nominal u atdiscrete time K; ak-lower bound of Uk; bk-upper bound of Uk;Gk(K)--component k of G at discrete time K; (6uk)max-maximumperturbation of Uk constrained by the bounds ak, bk; and hmax-maximum step factor corresponding to (cUk)max.

where B' is the transpose of the matrice B. Defining thefollowing:

G(t)-=B '(t) * A(t) (9)

the increment 54(tl) is expressed by

34(t 1) = (G(t),bu(t)) dt. (10)

According to Schwarz's inequality, maximization of 64(t1)demands

The result p = q means that uk(K) attains its bound (bkin Fig. 2). h* is computed on at other discrete times in thesame iteration until a discrete time N is found at whichuk(N) does not touch the bound. The control programwhich is attained at the end of the iteration becomes thenominal one for the next iteration until the optimal controlprogram is attained. The algorithm block diagram for thevarious computational steps is described in Fig. 3.

V. SIMPLIFICATION OF THE R&D MODEL

The principle of the proposed simplification method isbased on the use of the perturbed system matrices A,B,Cdefined by (5) and (7) and the perturbations bu and bx.These quantities must in any case be computed, as men-

tioned in Fig. 3, for the adjoint system optimization.For the sake of clarity let us consider a state plane

instead of a multidimensional state space (Fig. 4). Assuming4[x(tl)] to be smooth in the vicinity of its local optimum4*(t,), its increment &/(t,) is independent of x2 under thefollowing conditions.

Condition 1):

bu= hG. (1 1)

The vector G determines the direction of the step in buwhile the factor h determines the magnitude of this step. Toconserve linearity (5), each step in bu should be small,leading to slow convergence toward the optimal solution.In order to increase the rate of convergence, it is suggestednot to limit h by linearity considerations but to search ineach iteration the magnitude of h which maximizes 60(t,).The direction of the search is given by G. Thus the op-

timization problem is converted into an ordinary max-

imization problem, max 3/(t,), which is much simpler as

the function to be maximized has only one variable h. Thecalculation of optimal h*, in each iteration, is achieved as

follows (Fig. 2):

(3Uk)max = [h * Gk(K)]max

= min [bk - Uk(K)], for Gk > 0

k,K

= min [uk(K) - ak], for Gk < 0. (12)k,K

Therefore,

(15)ax(t2) 6X2(tl) << ax(tl)*bx,(ti) -

This condition ensures that 60(tl) depends only on bx1(t,)(Fig. 4).To ensure that bx1(t,) depends only on x1(t), the follow-

ing condition should be kept.Condition 2): At all times, the variation of. I = fl(x1,x2)

in the direction of x2 is negligible compared to that in thedirection of xl:

le'*iIbX2i << 4dt*Xl*X I (16)

Under these conditions, movement in the direction of x2

causes negligible changes of 4(t1). The state variable x2 isthen reduced to its nominal time function:

X2 = X2 nom(t). (17)If

11X2 nom(t) - X2(t0)II X lX2(to)I (18)then

X2 = X2(t0) (19)

hmx = min bk Uk(K)max

k,K Gk(K)

Uk(K) ak,= min

k,K Gk(K)

for G. > O where to is the starting time of the process. If x2(tO) = 0,

then x2 = 0.

For the state space of n dimensions, all the componentsfor Gk < 0. (13) of x that fulfil conditions (15) and (16) have to be found.

For this end the following matrices are calculated:

The maximum step factor hmax is divided into q parts ac-

cording to the computational accuracy desired. By means

of Fibonacci's method [8], the pth division which ensures

max 60(t1) is found and accordingly the optimal step factor

Uk

I

350

[60(tA = Uc - bx(t0)] I x n

=00

- bx(tl) I j = 1,- - -,n (20)oxi II

TAMIR: SYSTEM MODELS CHARACTERIZING R&D PROCESS

Fig. 3. Adjoint system algorithm.

+ 4ti.)

*4,,(t) +d~4(

X(to)f3

Xi

Fig. 4. State plane description of dynamic process.

and

[6f(t)] = [(A * bx),(B * 1U)]nx(n+m)

= f[,(I ) ( h .U)]

i= 1, ,n, j= l, ,

k =n + 1, *, n + m. (21)

In each row of the matrices (20) and (21) the small terms(compared to the largest one) are neglected. If there appearsa column j or k which is negligible at all times and in all theiterations, the following simplification takes place:

Xi = Xi nom(t)' 1 .j<n

Uk = Uknom(t), n +l< k <n + m. (22)

The simplification implies dimensional reduction of thestate and control vectors. Any parameter of the system mayarbit. irily be chosen to be a control component in order todetermine its influence on the index of performance 4(t1).Thus sensitivity analysis may be accomplished.

Fig. 5. Influence tree of simplification example.

Simplification ExampleThe system in this example contains 5 state variables and

2 control components. Marking dominant terms by (+)and negligible terms by ( ), the following computed resultscorresponding to the matrices (20) and (21) are assumed:

-*j12 3 4 5

[54(t1)] = [+ * * * ' ]-*1

I 1 2 3 4 5 16 7ilI + + I+ -2 . . f .

[bf(t)] = 3 .+ . + . I . . .

4 * * I+ tS 5 + * + I * +

(23)

(24)

By definition (1), x; = Jfj dt. From (23) and (24) thetree of influence (Fig. 5) is established. f5 does not appearin the tree of influence, implying no effect on 4. Row 5 in(24) is thus erased. As a result, there exist two negligible

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columns: j = 5 and k = 7. The simplified system contains,therefore, only 4 state variables and 1 control componentrather than 5 and 2, respectively, assumed originally.

VI. COMPUTATIONAL RESULTS

The optimization and simplification algorithm has beenprogrammed in TAC and ALTAC languages on the PhilcoTRANSAC digital computer model 212. According to thecomputational results, the index of performance for theR&D model described in Section III has been improved bya factor of three within 5 iterations and during 2.5 min ofcomputation. The 21 state variables (Table I) and 6 controlcomponents (Table II) of the model are reduced to 1 statevariable (x15) and 1 control component (u6), owing to thespecial character of the model. The factor of improvementafter simplification is different by only one percent fromthat of the original model and is attained after 4 iterationsand 1.7 min.

VII. CONCLUSIONSA state space model of the R&D process is established,

based on Robert's model [1]. Thus analytical treatment ofthe model is enabled. The optimization problem appears tobe that of Mayer [4] with inequality constraints imposedon the control vector. The proposed algorithm, which isbased on the adjoint system method [3], converts theoriginal optimization problem into a single-variable max-

imization search achieved by Fibonacci's method [8]. Thissearch is established around a nominal trajectory, and thusthe optimization is local. Simplification of the R&D modelis carried out using the matrices of the perturbed systemequation computed for the optimization algorithm. Thussimplification and optimization of the R&D model areachieved simultaneously.

ACKNOWLEDGMENT

The author would like to thank Dr. Z. Bonen for hissupervision of the research reported in this paper.

REFERENCES[1] E. B. Roberts, The Dynamics of Research and Development. New

York: Harper and Row, 1964.[2] J. W. Forrester, Industrial Dynamics. Cambridge, Mass.: M.I.T.

Press, 1961.[3] L. A. Zadeh and A. D. Desoer, Linear System Theory. New York:

McGraw-Hill, 1963.[4] G. Leitmann, Optimization Techniques with Applications to Aero-

space Systems. New York: Academic Press, 1962.[5] M. Athans and P. L. Falb, Optimal Control. New York: McGraw-

Hill, 1966.[6] A. V. Balakrishnan and L. W. Neustadt, Computing Methods in

Optimization Problems. New York: Academic Press, 1964.[7] A. Lavi and T. P. Vogl, Recent Advances in Optimization Tech-

niques. New York: Wiley, 1965.[8] D. J. Wilde, "A review of optimization theory," Ind. Eng. Chem.,

vol. 57, pp. 19-31, Aug. 1965.[9] M. Tamir, "Optimization and simplification of complex systems

and their application to the R&D process," D.Sc.Tech. disserta-tion, Technion-Israel Inst. of Technol., Haifa, June 1967.

Modeling of an Operator's Performance in aShort-Term Visual Information

Processing TaskHOWARD A. SHOLL, MEMBER, IEEE

Abstract-A predictive model is presented that represents the humanas an information processor by mathematically simulating the results ofpast psychological research in choice reaction time, memory storage/retrieval, and perceptual information processing. Both discriminationand identification tasks are characterized by ideal decisions with additiveGaussian noise. In an information processing task both presented andstored information are processed using a sequential dimensional pro-cedure to identify a presented stimulus. This technique is shown toproduce a logarithmic variation in response time as a function of the

Manuscript received March 1, 1971; revised January 14, 1972.This work was supported by the Underwater Sound Laboratory underContract N00140-69-044.The author is with the Computer Science Program, Department of

Electrical Engineering, University of Connecticut, Storrs, Conn.06268.

number of stimuli being presented. The model's simulation of a three-dimensional visual identification task was compared with the experi-mental results of three subjects. The equipment used consisted of acomputer-controlled discrete dot display; the dimensions used werehorizontal and vertical extent and the percentage of dots present in agiven area. It was found that the simulation reflected the subjects'behavior in both decision accuracy and response time for different setsof a priori probabilities.

INTRODUCTION

IN AN INTERACTIVE man-machine signal processingsystem it is desirable to understand the information

processing capabilities of the human in order to use himeffectively. This paper presents an engineering orientedmodel of the human as a processor of sensory information

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