Annals of Operations Research 98, 123–139, 2000 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Regulation of Overlaps in Technology DevelopmentActivities

JOHN LIU jjl@uwm.eduSchool of Business Administration, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA

Abstract. The innovation design and process engineering, two inevitable interrelated stages in the devel-opment of a new technology, are captured in this paper as a non-antagonistic leader-follower target-pursuitsystem subject to uncertainty in market and technology changes. The innovation design team (leader) isdriven by market needs, while the process engineering team (follower) then strives to attain suitable pro-duction means with a focus on technological availability and profitability. In face of uncertainty in marketchanges and technological advances, the proposed strategic regulation of overlaps (SRO) model addressesrelevant scheduling issues such as when to start, to overlap, and then to end the development activities. Weobtain the optimal timing of the overlaps in these development activities. An algorithm is developed tocalculate the optimal overlap schedule which is easy to be implemented for realistic applications.

Keywords: differential pursuit, stochastic scheduling, open-loop regulation

1. Introduction

Development of a new technology involves two inevitable interrelated stages of activ-ities, namely, innovation design (e.g., target, concept, product design, specifications,and prototype) and process engineering (e.g., process design, prototyping, integration,and transfer). The innovation (or product) design is driven by market needs and cus-tomer preferences, while the process engineering strives for suitable production meanswith a focus on technological availability and profitability [3,6]. Both researchers andpractitioners have long recognized that the success of an industrial innovation criticallydepends on regulation of the design and engineering activities. Figure 1 depicts a typicaldevelopment process of manufacturing technology. The total time spanTp − τd is re-ferred to astime-to-market. As generally observed, a delay in time-to-market will incurlosses in both market share and competitive potential. To shorten the time-to-market,the development process has evolved from a sequential (whenτp = Td in figure 1) to anoverlapped (including parallel) one.

In figure 1, the design team (leader) is to timely attain an innovation design so asto best reflect the customers’ needs, while the process engineering team (follower) thenintegrates accordingly the most productive means of diffusing the underlying technol-ogy. Each of the two teams is striving towards their respectivenon-antagonistictargets.A target is said to be antagonistic if it is independently controllable by an opponent (e.g.,an escaper). The targets in technology development are usually non-antagonistic, similarto the targets (e.g., runway on a carrier) in anairplane landingproblem of [13]. Such

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Figure 1. Overlapped phases in a concurrent development process (τd : design start;τp: process start;Td :design finish;Tp: process finish).

a non-antagonisticleader-followertarget-pursuit system is subject to inevitable uncer-tainty in the market changes and technology advances.

The regulation of the overlaps (e.g., when and how much to overlap) is quite in-tricate due to the uncertain nature of an innovation process. We address in this paperthe strategic regulation of overlaps (SRO) problem in the context of typical manufactur-ing applications as depicted in figure 1. First we characterize the SRO by a stochasticnon-antagonistic target-pursuit system as developed in [14]. With an objective of mini-mizing the expected total development cost which includes a linear innovation cost anda quadratic deviation loss, a strategic regulation of overlaps (SRO) model is developedto seek an optimal open-loop regulation policy. Notable results of the paper include:

(1) Provided that the law of diminishing return is prevalent, optimal regulation policyfollows astepwisestrategy, calledsurge-discharge(S-D) strategy. The S-D strategypromotes maximum flexibility in making design changes within a selected period oftime, and at the same time engages low-costs in the process engineering.

(2) The minimum time required for an innovation dependssolelyon technical inputs andconstraints of a firm. This implies that the optimal speed of the development can bedetermined by the internal structure of a firm, regardless of what market structure(e.g., monopoly versus oligopoly) the firm operates under.

(3) The optimal timing of the overlaps can then be determined upon a set of three criti-cal points along the optimal trajectory, namely, design finish, engineering start andfinish. An effective algorithm is developed for the computation and implementationof the SRO.

So far, the timing of an innovation has predominantly been studied througheconomical trade-offs, typically from the following perspectives: marketing/operat-

REGULATION OF OVERLAPS 125

ions [2,6,7], economics of innovation [9,11,12], and business strategy [4,5]. The market-ing (and lately operations) analysis of the timing has focused on the trade-off betweenmarket performance and R&D expenses such as sales and break-even-point. The tim-ing of innovation addressed in economics centers on the economic equilibrium on priceand/or quantity competition. Business strategists take the organizational learning andabsorptive capacity into account of the trade-off. From operations’ point of view, thefeasibility of the aforementioned trade-off analysis is often challenged by the availabil-ity of the necessary quantified data (e.g., measures of market performance and valuesof technical information, etc.), as they are either hardly available or difficult to evaluate.Stemming from realistic manufacturing applications (e.g., rapid prototyping at JohnsonControls, Inc.), the SRO model in this paper centers on the interrelation between marketchanges and technology advances, two essential factors in manufacturing that must beclosely observed and monitored.

The rest of paper is organized as following. Next in section 2, we present the SROmodel. The characteristics of optimal SRO policies are obtained in section 3, includingthe bang-bang principle. Section 4 presents the optimal S-D strategy for SRO, andsection 5 gives a computational algorithm of the optimal overlapping policy. In section 6,we extend the SRO model to a broader range of applications. We conclude the paper withconclusions and remarks in section 7.

2. The model

The development of a new technology entails a process of pursuing moving targets.Consider a targeted design attributemd(t) of a new product (e.g., traverse speed of a newmachine) that is established upon market analysis. Due to uncertain market changes andtechnology advances, the actual attainable speedis random. According to the technologyavailable at timet , the attainable speed is estimated asYd(t) (random) with a knowninitial estimate ofyd att = 0. An innovation design team is formed to attain a new designof the product that best reflects the targetmd(t) (assumed to be continuous int). Thedesign activities at timet are subject to regulations such as the allowable degree of designchanges, denoted byqd(t) ∈ [0, ud ] (and referred to asregulation later on) whereud isan upper limit of the regulation. The value ofqd(t) is dynamically adjusted according tothe design statusYd(t) in reference with the targetmd(t). At time t , qd(t) = ud gives amaximum allowable range of design changes, whileqd(t) = 0 indicates that no changesare applicable. The tasks in process engineering are then incurred. Similar formulationcan be derived for the engineering statusYp(t) (e.g., the assembly complexity or cost),its targetmp(t), and the associated regulationqp(t) ∈ [0, up] (e.g., allowable rangefor re-engineering in response to the design contents). In most industrial applications,the engineering targetmp(t) is determined upon the corresponding design targetmd(t)

and the degree of design changesqd(t). For example, it is reasonable to consider alinear engineering target in the form ofmp(t) = a0md(t) + a1qd(t), wherea0, a1 areappropriate coefficients. We will omit the time indext when convenient.

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With the applicable settings described above, next we develop an analyticalmodel for the strategic regulation of overlaps (SRO). We proceed in four steps:non-antagonisticleader-follower target-pursuit system, the moment-transformed state equa-tions, expected cost function, and the SRO model.

2.1. Non-antagonistic leader-follower target-pursuit system

The above development process can be characterized by a non-antagonisticleader-follower target-pursuit system as developed in [14] that consists of a pair ofqualitydiffusionequations (stochastic) of [15] and a pair of deterministic integral equations, assummarized below.

dYd = qd(md − Yd)dt + σdYd dZd, (2.1)

dYp = qp(mp − Yp)dt + σpYp dZp, (2.2)

Qd(t) = qd(t), (2.3)

Qp(t) = qp(t) (2.4)

with known initial states:

yd = Yd(0), yp = Yp(0), Qd(0) = 0, Qp(0) = 0,

whereYd(t), Yp(t): state of design and process engineering (valued inR), respectively;qd(t), qp(t): regulations of design and engineering activities, respectively;Qd(t),Qp(t): cumulative regulation efforts on design and engineering activities,

respectively;md(t),mp(t): design target and engineering target, respectively;σd, σp: disturbancefactors (constants);Zd,Zp: Weiner disturbances.The leader-followermechanism is captured in the interrelated targetsmd(t) and

mp(t). The design (leader) sets its targetmd(t) upon market needs (prescribed as afunction of time), while the engineering (follower) then determines its targetmp(t) ac-cording to the design target and design actions, for example

mp(md, qd) = a0md + a1qd (a0, a1 6= 0).

For generality, we refer the targets in general form asmd(t) andmp(md, qd), with∂mp/∂md = a0 = constant and∂mp/∂qd = a1 = constant. Apparently, the targets aremoving (i.e., time-variant) caused by the changes in the market, but are non-antagonistic(i.e., uncontrollable by an independent opponent).

2.2. Moment-transformed state equations

The processesYd andYp defined by (2.1) and (2.2) are of regulated target-reverting, ofwhich the mathematical moments can be obtained as following [15].

REGULATION OF OVERLAPS 127

Proposition 1. Let Sd = e−Qd , Sp = e−Qp , Kd = E{Yd}, Kp = E{Yp}, Vd = E{Y 2d }

andVp = E{Y 2p}. Then

Sd = −qdSd with Sd(0) = 1, (2.5)

Kd = mdqd − (Rd + yd)qdSd with Kd(0) = yd, (2.6)

Vd =(σ 2d − 2qd

)Vd + 2qdmdKd with Vd(0) = y2

d , (2.7)

Sp = −qpSp with Sp(0) = 1, (2.8)

Kp = mpqp − (Rp + yp)qpSp with Kp(0) = yp, (2.9)

Vp =(σ 2p − 2qp

)Vp + 2qpmpKp with Vp(0) = y2

p, (2.10)

where

Rd = qdmdS−1d with Rd(0) = 0, (2.11)

Rp = qpmpS−1p with Rp(0) = 0. (2.12)

Proof. It is easy to verify (2.5) by the definitions ofSd andQd . By taking integration(in Ito’s sense) of (2.1) in a way similar to [1, chapter 8], we can obtain

Kd = E{Yd} = e−Qd{y0+

∫ t

0qdmd eQd ds

}.

Taking derivative ofKd with respect tot we obtain (2.6). As shown by [1], functionVdmust satisfy the differential equation (2.7) withVd(0) = y2

d . With the same treatment,we obtain (2.8) and (2.12). �

By proposition 1 and with some mathematical transformations, we can develop thestate equations in vector form as elaborated below. LetWd = S−1

d andWp = S−1p . Then

we can writeWd = qdWd with Wd(0) = 1, andWp = qpWp with Wp(0) = 1. Letxd = (Sd,Kd, Vd, Rd,Wp)

t be the design state vector,xp = (Sp,Kp, Vp,Rp,Wp)t be

the engineering state vector, and(qd, qp) be the control vector subject to the respectiveadmissible domains ofqd ∈ �d = {q ∈ R: 0 6 q 6 ud , ∀t > 0} andqp ∈ �p = {q ∈R: 0 6 q 6 up, ∀t > 0}. Then the state equations (2.5) to (2.12) can be expressed in asuccinct vector form as

x =(xd

xp

)=(Fd(qd) 0

0 Fp(qd, qp)

)(xd

xp

)+(

Bd(xd, qd)

Bp(xp, qd, qp)

), (2.13)

with xd(0) = (1, yd, y2d ,0,1)

t , xp(0) = (1, yp, y2p,0,1)

t , where

Fd(qd) =

−qd 0 0 0 0−ydqd 0 0 0 0

0 2mdqd σ 2d − 2qd 0 0

0 0 0 0 mdqd0 0 0 0 qd

, (2.14)

Bd(xd, qd) = (0,mdqd − qdRdSd,0,0,0)t , (2.15)

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Fp(mp, qp) =

−qp 0 0 0 0−ypqp 0 0 0 0

0 2mpqp σ 2p − 2qp 0 0

0 0 0 0 mpqp0 0 0 0 qp

, (2.16)

Bp(xp,mp, qp) = (0,mpqp − qpRpSp,0,0,0, )t . (2.17)

2.3. Expected cost function

As the performance measure of the abovenon-antagonistictarget-pursuitsystem, weconsider the following expected cost function, which includes a regulatory costcdqd +cpqp (wherecd and cp are known unit control costs) and a quadratic deviation loss(deviated from the targets)Ld(Kd, Vd)+ Lp(Kp, Vp,mp):

Total Cost= θ(x, qd , qp, t) = cdqd + cpqp + Ld(Kd, Vd)+ Lp(Kp, Vp,md), (2.18)

wherex = (xdxp

), and

Ld(Kd, Vd) = E{Ad(m− Yd)2

} = Ad(Vd − 2mdKd +m2d

), (2.19)

Lp(Kp, Vp,mp) = E{Ap(mp − Yp)2

} = Ap(Vp − 2mpKp +m2p

), (2.20)

with constantsAd andAp given as the deviation losses in dollars with regard to designand engineering, respectively.

2.4. The SRO formulation

Within a finite time horizonT < ∞, let τd andτp be the starting times of innovationdesign and process engineering respectively, andTd andTp be the ending times of thedesign and the engineering, respectively (i.e., time-to-promotion and time-to-production,respectively). Now the strategic regulation of overlaps (SRO) problem can be stated as:find an regulation{qd, qp} and the timing{{τd, τp, Td, Tp}: 0 6 τd 6 τp 6 Td 6 Tp 6T }, so as to minimize the total expected development cost. Then in vector form we canwrite

SRO Model:

min J (x, qd , qp) =∫ T

0θ(x, qd, qp, t)dt

s.t. x =(xd

xp

)=(Fd(qd) 0

0 Fp(mp, qp)

)(xd

xp

)+(

Bd(xd , qd)

Bp(xp,mp, qp)

),

qd ∈ �d, qp ∈ �p, x(0) = (xd(0), xp(0))t (known).

Note that the minimization in SRO model is taken over the regulation{qd, qp} withthe timing {τd, τp, Td, Tp} as given parameters. The SRO is now expressed as anon-

REGULATION OF OVERLAPS 129

antagonisticleader-follower target-pursuit problem withleader’s strategyqd and fol-lower’s strategyqp.

3. The optimal regulation policy

As typical in optimal control, introducing adjoint vectorsvd = (λd, κd, νd, γd,$d),vp = (λp, κp, νp, γp,$p), andv = (vd, vp), the corresponding Hamiltonian can bewritten as

H = −θ + vd · xd + vp · xp, (3.1)

where

vd · xd = vd(Fd(qd)xd + Bd(xd , qd)

), (3.2)

vp · xp = vp(Fp(mp, qp) · xp + Bp(xp,mp, qp)

). (3.3)

Then introducing Lagrangian multipliersw1, w2, w3 andw4, we have the followingLagrangian:

H = H + w1(ud − qd)+ w2qd + w3(up − qp)+ w4qp. (3.4)

By Maximum Principle, the adjoint states must satisfy the following,

v = −∂H∂x= −∂H

∂xwith v(T ) = 0. (3.5)

From (2.13), we can write adjoint equations (3.5) as following

v = (vd, vp) = (vd , vp)(−Ed 0

0 −Ep), (3.6)

where

Ed = Fd + ∂

∂xdBd, Ep = Fp + ∂

∂xpBp,

andFd ,Bd , Fp,Bp are the matrices given by (2.14) to (2.17). For reference, we includethe detail expressions of (3.6) in the appendix.

Along the optimal trajectory(x∗, v∗), the optimality conditions of∂H/∂qd = 0and∂H/∂qp = 0 at optimal controlu∗ yield:

∂H∂qd= Hd(x∗,u∗, v∗, t)− w1+ w2 = 0,

w1(ud − qd) = 0 and w2qd = 0, (3.7)∂H∂qp= Hp(x∗,u∗, v∗, t)− w3+ w4 = 0,

w3(up − qp) = 0 and w4qp = 0, (3.8)

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where

Hd(x,u, v, t) = ∂H

∂qd= − ∂θ

∂qd+ ∂

∂qd

(vd · xtd

)+ ∂

∂qd

(vp · xtp

), (3.9)

Hp(x,u, v, t) = ∂H

∂qp= − ∂θ

∂qp+ ∂

∂qp

(vd · xtd

)+ ∂

∂qp

(vp · xtp

). (3.10)

Detailed expressions ofHd andHp are included in the appendix.To summarize it up, we have

Theorem 1 (optimal regulation conditions). Along the optimal trajectory(x∗, v∗), anoptimal regulationu∗ = (q∗d , q∗p) must satisfy the following simultaneously,

q∗d =

ud if Hd(x∗,u∗, v∗, t) > 0,

0 if Hd(x∗,u∗, v∗, t) < 0,

ud ∈ [0, ud ] if Hd(x∗,u∗, v∗, t) = 0,

(3.11)

q∗p =

up if Hp(x∗,u∗, v∗, t) > 0,

0 if Hp(x∗,u∗, v∗, t) < 0,

up ∈ [0, up] if Hp(x∗,u∗, v∗, t) = 0.

(3.12)

The functionsHd andHp represent the optimal rates of return oninnovation designandprocess engineering, respectively. The result of theorem 1 implies the optimalityof the so-calledbang-bang principle, where a switching in control is engaged wheneverHd and/orHp pass across zero. However, the optimal regulation is undetermined ifHdand/orHp are at zero which is referred to assingular conditions[10, chapter 5, p. 246]. Ifthe set ofsingular points, {t : Hd = 0,Hp = 0 | (x∗, v∗)}, is countable and no feedbackis considered, thenbang-bang principleapplies under thesingular conditionsas well.Intuitively, if along an optimal path(x∗, v∗) the return ratesHd andHp hit zero only bycountable times, then we can construct an optimalstepwiseopen-loop control by settingthe regulation either full or zero at these countable points in time. From both physicaland practical perspectives, the construction of astepwisestrategy seems to be the onlylogical choice. In fact, such strategy construction is rigorously justified by the so-calledstep-by-step strategy of[13]. It is important to note that the stepwise control in [13] ispositional (i.e., of feedback) as opposed to open-looped. Although a feedback controlusually performs better than an open-looped one, the implementation with feedback ismore difficult. That is why an open-loop control is still preferred when applicable. Forwhat is useful to the purpose of this paper, we confine the study herein to the followingopen-loopcontrol problem: find astepwiseopen-loop controlu(t) so as to attain adesired absolutely continuous trajectoryx(t) which is subject to theequations of motionx(t) = f (x,u, t). The existence and uniqueness of themotion controlare obtainedunder a Lipschitz continuousf . (See [13, chapter 1] for proofs and derivations.)

REGULATION OF OVERLAPS 131

4. Optimality of S-D regulation strategy

Hereafter, we confine the regulations to an open-looped (i.e., non-feedback) surge-discharge (S-D)stepwisestrategy for the SRO problem. See [13] and [16] for moreon positional (feedback) S-D regulation. An S-D regulation strategy consists of threephases: starting with asurgeto maximum design regulation, sustaining at the maximumfor a certain period of time, and then finishing with an irreversibledischarge. Figure 2depicts such an S-D strategy structure. For example, a full range of design changes areengaged from the design start timeτd = 0 until the design finish timeTd by which timethe design is finalized and ready for marketing. No design change (i.e.,qd = 0) is al-lowed afterTd . An S-D engineering strategy is to follow with a maximum regulation ofthe process engineering from the start timeτp > 0 until finish timeTp, and then withzero engineering regulation after the process finish timeTp. The surge is not recurrent asit is inapplicable to re-form the engineering team after it has been dismissed. By physi-cal terms, an S-D regulation requires 06 τp 6 Td 6 Tp. We denote an S-D regulationbyUSD(τp, Td, Tp) for convenience.

Since an S-D regulation herein actually is a piecewise constant control, the solutionof a SRO problem now depends on the analysis of the cases where the regulation is heldconstant, sayu(d, p) = {qd = d, qp = p; 06 d 6 ud , 06 p 6 up}. For convenience,we denoteHd(t) = Hd(x, u, v, t) andHp(t) = Hp(x, u, v, t), where(x, v) should beunderstood as the trajectory underu. By theorem 1, the start of an innovation can bejustified only ifHd(0) > 0. Also, the time horizonT must be sufficient for completingthe innovation project. That is, after the design is finalized (i.e.,qd = 0) there must beenough time for the engineering to finish up byT (i.e.,Hp(0) > 0). We formally define

Definition 1 (feasible start and horizon). The starting time (t = 0) is said to be feasibleif Hd(0) > 0. The time horizonT of an innovation project is said to be feasible ifHd(T ) 6 0 andHp(T ) 6 0.

To this end, we have the following results.

Figure 2. The S-D regulation strategy.

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Lemma 1 (lower-bound of feasible time horizon). Leta1 = ∂mp/∂qd = constant. Thetime horizonT is feasible only ifT > tm, where

tm = 1

upln

(2a1Apyp

cd

)and cd 6 2a1Apyp.

Proof. First, it is easy to verify from (A.II-2) of the the appendix thatHp(T ) = −cp <0 in any case. Under the extreme case where the design is finalized (qd = 0) from thebeginning (t = 0) and engineering is in full scale (qp = up) all the time, we can verifyfrom (A.II-1) of the appendix that the minimum time needed to haveHd(tm) = 0 iswhen

e−uptm = cd

2a1Apyp(6 1).

Thus, forT > tm we are assured that bothHd(T ) 6 0 andHp(T ) < 0. ThenT isfeasible by definition 1. �

Note that horizonT corresponds to thetime-to-marketas shown in figure 1.Lemma 1 gives an estimate of the minimum time required for an innovation. Inter-estingly, the determination of such lower bound is independent of exogenous factorssuch as market structure. Since the design target is the only item in the target-pursuitsystem that can contain the exogenous inputs, the SRO model can be extended to a hier-archicalnon-antagonistictarget-pursuit system. The detailed discussion on this subjectis beyond this paper. Furthermore, the innovation is not pursuable if either the start isnot justifiable or the horizon is not feasible. This gives a simple rule for the so-calledentrydecisions.

Theorem 2 (conditions for optimal S-D strategy). LetPd = {t : Hd > 0, t ∈ [0, T ]}andPp = {t : Hp > 0, t ∈ [0, T ]}. If both the start and horizonT are feasible bydefinition 1, and bothPd andPp are convex and nonempty, then there exists an S-DregulationUSD(τp, Td, Tp) with 0 6 τp 6 Td 6 Tp 6 T that satisfies the optimalregulation conditions of theorem 1.

Proof. Since the start is feasible (i.e.,Hd(0) > 0), we are assured with 06 τp 6 Tpin an optimal S-D control. SinceT is feasible as well, and bothPd andPp are convexand nonempty, hence: (1) there exists an optimalstepwiseregulation (by theorem 1);(2) Td 6 Tp 6 T in an optimalUSD(τp, Td, Tp) (by lemma 1). �

Obviously,Pd is convex if a feasible (by definition 1)Hd is quasi-concave or de-creasing convex int (similarly for Pp). Noting thatHd , Hp represent the rate of returnon the innovation, theorem 2 implies that an optimal S-D strategy can be devised as longas the rate of return on the innovation isdiminishingover time, which is indeed pervasiveby all means.

REGULATION OF OVERLAPS 133

Theorem 3 (SRO trajectory under constant control). Let

u(d, p) = {(qd, qp): qd = d, qp = p, 06 d 6 ud, 06 p 6 up, t ∈ [t0, tf ]}

be the control applied over a finite time interval[t0, tf ] ⊆ [0, T ], givenx(t0) andv(tf ).Then the original system of (2.13) can be expressed by

x =(xd

xp

)=(Fd(d, t) 0

0 Fp(d, p)

)(xd

xp

)+ B(d, p, t), (4.1)

whereFp is given by (2.16),

Fd(d, t) =

−d 0 0 0 0−ydd 0 0 −d e−dt 0

0 2mdd σ 2d − 2d 0 0

0 0 0 0 mdd

0 0 0 0 d

, (4.2)

B(d, p, t) = (0,mdd,0,0,0,0,2adp,0,0,0). (4.3)

Furthermore, the optimal trajectory(x, v) underu(d, p) can be obtained as

x(t) = 8(t, t0)x(t0)+∫ t

t0

8(t, τ )B(τ )dτ, 8(τ, τ) = I, (4.4)

v(t) = v(tf )9(tf , t)+∫ tf

t

Bt(τ )9(τ, t)dτ, 9(tf , tf ) = I, (4.5)

where8(t, t0) is the transition matrix of the state equations given by (2.13),9(tf , t) isthe inverse transition matrix of the adjoint equations given by (3.6).

Proof. It is known that the transition matrix solution to a linear dynamic system isunique, if the coefficient matrices of the system are piecewise continuous in time andadmissible controls are measurable (see [13]). Hence to prove the theorem, we needonly to show that both the state equations of (2.13) and the adjoint equations (3.6) arelinear dynamic systems underu(d, p). Underu(d, p), we haveSd = e−dt , Sp = e−pt ,

Rd =∫ t

0dmd(s)e

ds ds, Rp = ad ept , and RpSp = adp,

which are all specific functions solely dependent on timet . Substituting these functionsinto the coefficient matrices of (2.13), we can verify that the original system can berepresented by the linear system of (4.1). Similarly, we can verify the linearity of theadjoint equations (3.6). The theorem can then be concluded with the particular solutionsof these linear dynamic systems. �

If the design targetmd is time-invariant, we have

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Corollary 1 (solution with constant design target). Ifmd is time-invariant, then transi-tion matrices in theorem 3 can be written specifically as

8(t, t0) =(8d(t, t0) 0

0 eFp(t−t0)

). (4.6)

Proof. In fact with u(d, p) the coefficient matricesFp, B are constant matrices, andonly Fd in (2.1*) is a time-variant matrix. With8d denoting the transition matrix ofxd = Fd(d, t)xd , we have (4.6). �

We recall the followingknownproperties of a transition matrix:

(i) 8(t, τ ) is invertible fort, τ ∈ [0,∞), and8(t, τ ) = 8−1(τ, t);

(ii) 8(t, τ ) = 8(t, t )8(t, τ ).It is known that if coefficient matrix of a linear system is constant then its solution isdifferentiable (in this case,∞-th order differentiable). One can verify this from (4.6)where matrixFp is a constant matrix underu(d, p) and constantmd , and the resultingxp are differentiable with respect to timet (t ∈ (t0, tf ). The conclusion can be easilyextended to the cases of time-variant continuous coefficient matrices (e.g., using thetypical variation with Taylor expansions, as in [8]). That is, the resulting trajectory(x, v) of theorem 3 are differentiable fort ∈ (t0, tf ), and so areHd andHp by (A.II-1)and (A.II-2) of appendix. This conforms to the aforementioned theory ofmotion control,and justifies the construction of the S-D regulation strategy. To rest the argument, wesummarize it as the following theorem.

Theorem 4. If there exists an optimal S-D regulationUSD(τp, Td, Tp) of theorem 3, thenthe optimal S-D regulation can be represented by a piecewise function as

USD(τp, Td, Tp) =

u1 = u(ud,0) for t ∈ [0, τp),u2 = u(ud, up) for t ∈ [τp, Td),u3 = u(0, up) for t ∈ [Td, Tp),u4 = u(0,0) for t ∈ [Tp, T ].

(4.7)

According to the analysis above, the optimal regulation is astepwisefunction of fourphases as represented by (4.7). Therewith, the optimal timing of an innovation is char-acterized by three critical points in time, namely, engineering startτp, design finishTd ,and engineering finishTp. Obviously, an S-D regulationUSD(τp, Td, Tp) is in general anoverlapping, except eitherτp = 0 (parallel) or τp = Td(sequential).

5. Computation of optimal timing

In this section we consider the determination of optimal timing. By theorem 4, an op-timal regulationUSD(τp, Td, Tp) consists of four phases, within each of which the reg-ulation is fixed (see (4.7)). Let(xi , vi ) be the resulting trajectory of the phasei (i =

REGULATION OF OVERLAPS 135

1,2,3,4). Then the state trajectoryxi can be obtained from the forward solution equa-tion (4.4) of theorem 3, and the adjoint trajectoryvi can be obtained from the backwardsolution equation (4.5). Therewith, the following two algorithms are immediate.

Algorithm 1 (forward trajectoryxi). Given an initial statex0(t0) = x0 and withti (i =1,2,3,4; t1 = τp, t2 = Td , t3 = Tp, and t4 = T ) as independent variables, then theresulting trajectoryxi for i = 1,2,3,4 can be obtained from the forward solution (4.4)of theorem 3 as following

xi(t) = 8i(t, ti−1)xi−1(ti−1)+∫ t

ti−1

8i(t, τ )B i(τ )dτ, i = 1,2,3,4, (5.1)

xi+1(ti) = xi (ti), i = 1,2,3, (5.2)

where8i is fundamental transfer matrix of (4.4) underui of (4.7), B i correspondsto (4.3) underui , and (5.2) is needed to forward the initial state to the next phasexi+1(ti)

as a function ofti .

Algorithm 2 (backward trajectoryvi). Given an ending adjoint statev(t4) = 0, then theresulting trajectoryvi (i = 4,3,2,1) can be obtained from the backward solution (4.5)of theorem 3 as following

vi(t) = vi (ti)9i(ti , t), i = 4,3,2,1, (5.3)

vi−1(ti−1) = vi(ti−1), i = 4,3,2, (5.4)

where9i is the inverse fundamental transfer matrix of (4.5) underui of (4.7), and (5.4)is needed to backward transfer the ending adjoint state to the proceeding phasevi−1(ti−1)

as a function ofti−1.Note that algorithms 1 and 2 determine the trajectories(xi , vi ) as functions of

time ti (i = 1,2,3,4). By the optimality conditions (3.11) and (3.12), at each of thetime point ti we must have eitherHd = 0 and/orHp = 0 which are referred to assingular conditions. Thus using thesingular conditions, we can construct an iterativealgorithm for computing the optimal timing. We present below the main algorithm foroptimal timing and coordination in such a manner that it is self-explanatory so as to saveadditional explanations.

Main algorithm (optimal SRO solution). Suppose that(xi , vi ) is given by algorithms 1and 2,Hdi (t) = Hd(xi , vi , ui , t), andHpi (t) = Hp(xi , vi , ui , t) for i = 1,2,3,4. Thenoptimal SRO policyUSD(τp, Td, Tp) can be determined as following:

0. If Hd1(0) 6 0, then stop (infeasible start),else ifHd1(0) > 0 andHp1(0) > 0

thenτp = t1 = 0 (parallel development), go to step 2,else, continue to step 1.

136 LIU

1. Computetd1 = min{t : Hd1(t) = 0} and tp1 = max{t : Hp1(t) = 0}; Then, re-strict the td1 and tp1 to [0, T ] as following: td1 ⇐ min{max{td1,0}, T } andtp1⇐ min{max{tp1,0}, T }.If td1 < tp1,

then stop (unjustifiable process engineering);else iftd1 = tp1,

thent1 = t2 = td1, τp = Td = t1 (sequential development), go to step 3;else thenτp = t1 = td1.

2. Computetd2 = min{t : Hd2(t) = 0} and tp2 = max{t : Hp2(t) = 0}; Then, td2 ⇐min{max{td2, t1}, T } andtp2⇐ min{max{tp2, t1}, T }.If td2 > tp2,

then stop (infeasible design target, i.e.,Td > Tp);else iftd2 = tp2,

thent2 = t3 = td2, Td = Tp = td2 (parallel ending), go to step 4;else thenTd = t2 = td2.

3. Computetp3 = max{t : Hp3(t) = 0}; Then,tp3⇐ min{max{tp3, t2}, T },Tp = t3 = tp3.

4. Optimal timing and coordination found asUSD(τp, Td, Tp).

We can verify that the feasible S-D solutionUSD(τp, Td, Tp), if exists, generatedby the main algorithm indeed satisfies the necessary optimality conditions (3.11) and(3.12). Now walk through the main algorithm by an illustrative example. For the sakeof conciseness, we omit the tedious computation details such as matrix calculations.

An illustrative numerical example. Consider an innovation project of a new on-linetech-support system for CT scanners at a major medical diagnostic equipment manufac-turer in the U.S. The system is to be designed with a service level of 90% (i.e., 90%of the inquiries are covered automatically by the system). While current system coversabout 45% of the inquiries. That is,md = 0.90 andyd = Yd(0) = 0.45. The designcontrol qd ∈ {0,1} is given as percentage of design work to be completed per week(assuming 40 working hours a week). Each percentage point of design work will require20 hours (i.e., 0.5 week) of engineering work and each design control point will resultin 5 hours (i.e., 1/8 week) of additional engineering work, i.e.,mp = 1

2md + 18qd . The

engineering work completed at beginning is zero, i.e.,yp = Yp(0) = 0. The project isto be completed in 15 weeks (T = 15). The engineering controlqp ∈ {0,1} is specifiedas the percentage of engineering work to be completed each week. With a given set ofvalues for parametersσd , σp, cd , cp, Ad , andAp, the main algorithm can be carried outas following. First, for all the combination of controlUSD given in (4.7) the coefficientmatrices in (4.1) and (3.6) are evaluated. Secondly, solutions in (5.1) and (5.3) are ob-tained as a function ofti given in algorithm 1 (i = 1,2,3,4 with t4 = T ), using (4.6) toobtain8(t, t0). Note that the transition matrix8(t, t0) can be evaluated numerically ifits closed form is unattainable. Thirdly, with the solutions obtained, five equations can be

REGULATION OF OVERLAPS 137

formed according to the main algorithm, namely,Hd1(t) = 0,Hp1(t) = 0,Hd2(t) = 0,Hp2(t) = 0, andHp3(t) = 0. Lastly, solve these five equations and determine the opti-mal timing{τp, Td, Tp} according to the main algorithm. By theorem 2, it is guaranteedthat there exist feasible solutions to the five equations. The detailed numerical tests areavailable separately from this paper.

6. Generalization of SRO model

In this section, we shall give a brief description on possible generalization of SRO mod-eling approach in a less rigorous manner, and pose the rigorous treatment of the subjectfor future study. First of all, SRO model can be generalized to allow multiple levelsof development teams that are connected through some exogenous targets (e.g., pricequoted by suppliers, or acceptable price by buyers). With the exogenous targets as link-ages, the SRO model presents a multi-level non-antagonistic target-pursuit mechanismfor regulation of activities in a technology development chain.

It is clear from section 2 that the SRO modeling method is applicable as long asthe underlyingtarget-pursuitprocessesYd, Yp possess differentiable first and secondmoments (i.e.,Kd , Kp, Vd , Vp). Thus, SRO modeling can be easily extended to thecases of more general diffusion processes, such as

{dYd = qd(md − Yd)dt + σ (Yd)dZd,dYp = qp(mp(md, qd)− Yp)dt + σ (Yp)dZd,

where the targetsmd , mp(qd) are prescribed and continuous, and thedisturbancesareLipschitz continuous and state dependent. In our case, the time-variant design targetmdcan also contain explicit terms of exogenous inputs (e.g., market structure and competi-tors’ positions), and the engineering target is then a mapping (not necessarily linear) ofthe design actionqd .

7. Concluding remarks

Under a prescribed design target which reflects the expected market perception ofa concerned innovation, the SRO model developed in this paper suggests a practi-cal S-D policy for the regulation of overlaps between the design activities and en-gineering activities. The resulting S-D policy promotes flexibility in making designchanges while at the same time engages minimum costs in engineering of the necessarytechnical processes. In addition to solving an important problem of overlap regula-tion, SRO poses an effective mechanism for regulating multi-level development activi-ties.

138 LIU

Appendix

I. Expressions of adjoint equations (3.6)

From (2.14) to (2.17), we can derive the coefficient matrices of (3.6) as following:

−Ed =

qd 0 0 0 0

(Rd + yd)qd 0 0 qdSd 00 −2mdqd −(σ 2

d − 2qd) 0 00 0 0 0 −mdqd0 0 0 0 −qd

, (A.I-1)

−Ep =

qp 0 0 0 0

(Rp + yd)qp 0 0 qpSp 00 −2mpqp −(σ 2

d − 2qp) 0 00 0 0 0 −mpqp0 0 0 0 −qp

. (A.I-2)

II. Expressions of functionsHd andHp

Hd(x,u, v, t) = ∂H

∂qd= − ∂θ

∂qd+ ∂(vd · x

td )

∂qd+ ∂(vp · x

tp)

∂qd

= −(cd + 2a1Ap(mp −Kp))+ vd(∂Fd

∂qdxd + ∂Bd

∂qd

)+ vp

(∂Fp

∂qdxp + ∂Bd

∂qd

), (A.II-1)

Hp(x,u, v, t)= ∂H∂qp= − ∂θ

∂qp+ ∂(vd · x

td )

∂qp+ ∂(vp · x

tp)

∂qp

=−cp + vd(∂Fd

∂qpxd + ∂Bd

∂qp

)+ vp

(∂Fp

∂qpxp + ∂Bd

∂qp

)=−cp + vp

(∂Fp

∂qpxp + ∂Bd

∂qp

), (A.II-2)

whereFd ,Bd , Fp,Bd are given by (2.14) to (2.17).

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