on the dynamics of stochastic diffusion of manufacturing technology

Theory and Methodology

On the dynamics of stochastic di�usion of manufacturingtechnology

John Liu *

School of Business Administration, University of Wisconsin±Milwaukee, Milwaukee, WI 53201, USA

Received 1 June 1998; accepted 1 January 1999

Abstract

Manufacturing technology di�usion (MTD), a close relative of demand di�usion in marketing, refers to the tran-

sition of technology's economic value during the transfer and operation phases of a technology life cycle. Since

manufacturing technology is rooted in manufacturing activities (e.g. production planning and control) as opposed to

marketing alternatives (e.g. pricing and advertising), the modeling of MTD must inevitably address two aspects: reg-

ularity (drift) and uncertainty (disturbance). The MTD model proposed herein addresses the problem of how to regulate

MTD in the face of uncertainty in order to maximize expected total pro®t. The MTD model adopts stochastic dif-

ferential equations (SDEs), to overcome the limitations of invariance (e.g. a ®xed market size and the absence of

disturbance) as su�ered by a typical product life cycle (PLC) model. First we derive a drift function in the context of

MTD and address the drift-only MTD model (i.e. with zero disturbance). With reference to a speci®c application of

¯exible manufacturing, we ®nd an optimal control for the regulation of MTD and in addition we prove the optimality

of early technology phase-out, which interestingly coincides with the pervasive phenomena of life-cycle-shortening in

manufacturing. Then, by variational calculus in combination with applications of Ito's formula, we obtain an aug-

mented Hamilton±Jacobi variational equation for the solution of the MTD model. An early phase-out policy is also

proved to be optimal for the ¯exible manufacturing case when disturbance is present. Ó 2000 Elsevier Science B.V. All

rights reserved.

Keywords: Technology di�usion; Optimal stopping; Stochastic control

1. Introduction

Three distinct phases in the life cycle of amanufacturing technology have long been identi-

®ed: research and development (R&D), transfer/commercialization, and operation/regeneration(Cook and Mayes, 1996; Rogers, 1995). A ®rm'scompetitive strategic and operational position iscritically dependent upon an accurate and objec-tive understanding of the life cycle of underlyingtechnology (Burgelman et al., 1995). A key char-acterization of the life cycle has been the transition

European Journal of Operational Research 124 (2000) 601±614www.elsevier.com/locate/dsw

* Tel.: 1-414-229-4235; fax: 1-414-229-6957.

E-mail address: [email protected] (J. Liu).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 7 6 - 9

of technology's economic value. For instance, theR&D of technology presents a phase of valuebuild-up through technological innovations, whilethe transfer and operation of technology entail avalue depletion process in accordance with inevi-table technology obsolescence (Rosenberg, 1972).Manufacturing technology di�usion (MTD) refersto the transition of economic value during thetransfer and operation phases of a technology lifecycle. There is a keen resemblance between tech-nology and market di�usions in that both cast adynamic depletion process of their respectiveeconomic values. The depletion process of marketdi�usion has been studied using the widely-knownproduct life cycle (PLC) models (e.g. Bass, 1969;Kalish, 1983; Horsky and Simon, 1983; Teng andThompson, 1985; Dockner and Jorgensen, 1988;Klepper, 1996; literature on another relevanttopic, business life cycles by the economic logis-tics models, can be found in Aulin, 1996; Thore,1991; Nagurney et al., 1996; Flam and Ben-Israel,1990).

Despite its obvious similarity to marketingmethodology, the modeling of manufacturingtechnology di�usion is fundamentally di�erent inseveral ways. First, manufacturing technology isrooted in manufacturing activities (e.g. productioncontrol) as opposed to marketing alternatives (e.g.pricing and advertising). The valuation processassociated with MTD is consistently in¯uenced byproduction decisions, such as whether or not todiscontinue (or switch o�) an extant productionprocess, and what production level should be tar-geted, etc. As such, the transformative nature ofmanufacturing derives an immediate distinctionbetween technology and market di�usions. On theone hand, there hardly exists an MTD withoutexogenous regulations, while on the other hand itcan be still reasonably argued (e.g. by Bass et al.,1994) as to why a PLC model will work withoutcontrol. Second, the MTD will inevitably endureuncertain in¯uences and disturbances due totechnological evolution and innovation. There-fore, it is no longer a justi®able modeling as-sumption to tolerate the limitations of invariancesu�ered by a typical PLC model (e.g. the inelas-ticity of market size and the absence of distur-bance). The PLC research bases its analysis on the

mean value (or the passive net present value) ofmarket potential, and is known for its limitationsof ®xed total market and deterministic variationsof market potential. Such limitations and theiralleviative solutions have long been discussed andaddressed (e.g. Chatterjee and Eliashberg, 1990;Mesak and Berg, 1995; Liu and Chi, 1997; Feich-tinger et al., 1994 for more details and extensiveupdates).

In brief, MTD entails a stochastic value-de-pletion process where the uncertain status of theprocess (e.g. market potential) is continuouslyobserved and regulated. Obviously, a rigorousmodeling of MTD must give an accurate ac-count for the aforementioned aspects: regularityand uncertainty. This paper attempts to achievethat.

The methodology of stochastic di�erentialequations (SDEs) bodes well for the analysis ofsimilar problems in the areas of physics, engi-neering, and lately ®nance. SDE modeling hasalso been used to study the R&D phase of tech-nology's life cycle (Reinganum, 1981; Kulatilaka,1988; Dixit and Pindyck, 1994; Kamrad, 1995;Chi et al., 1997). But as far as we can ascertainfrom current research literature, SDE modelinghas not yet been applied to MTD. In this paper,we propose using an Ito's SDE formulation tocharacterize the in®nitesimal dynamics of thevaluation process. Assuming full access to thepast observable information up to current time t,the changes of the valuation in the next dt timeconsist of two components: a depletion rate pre-scribed from the past (drift), and a superimposedWiener di�erential (disturbance). Both drift anddisturbance can be rather general and can includecontrol variables such as production rate. Theproposed SDE formulation overcomes the limi-tations of invariance by not only allowing ran-dom disturbances, but adopting a variable marketas well. Furthermore, a controllable stopping timeis introduced to allow a managerial decision toterminate the use of the technology before itsPLC-de®ned natural ending time; that is, beforetotal depletion of its underlying values. Life-cycle-shortening has been pervasive worldwide(Wheelwright and Clark, 1995; Mans®eld, 1988),and it has become increasingly common to

602 J. Liu / European Journal of Operational Research 124 (2000) 601±614

discontinue a manufacturing process before itsnatural ending time. With the SDE characteriza-tion of the underlying depletion process, a generalMTD model under production regulations is thenformulated as a stochastic control problem withfree terminal time (stopping time). The objectivefunction of the MTD model is the expected totalpro®t as derived from the total realization of thetechnology's value over the duration of the reg-ulated MTD, including terminal payo�s, if ap-plicable. The MTD seeks an admissible regulationpolicy joint with an exit (or stopping) decisionthat maximizes expected total pro®t. It is inter-esting to note that the MTD model becomes abasic optimal stopping problem when no othercontrol variables are involved. With reference to aspeci®c application of ¯exible manufacturing, it isshown in this paper that the optimal stoppingpolicy suggests phasing out the underlying tech-nology before its value vanishes, which coincideswith the aforementioned life-cycle-shorteningphenomena.

Although the MTD model is mathematicallysuited for n-dimensional SDEs, we con®ne ouranalysis to the one-dimensional case for the sakeof keeping within a realistic context. By variationalcalculus in combination with applications of Ito'sformula, we derive augmented Hamilton±Jacobivariational equations for the MTD model, thesolution of which presents a nonlinear freeboundary problem of second-order ordinary dif-ferential equations (ODEs). Thereby, we show ananalytical characterization of the MTD model tobe the following: for a system with its state dy-namics expressed by Ito's SDEs, and given astarting boundary and an ending boundary (bothreal functions on Rn), we are to ®nd a twice dif-ferentiable functional pro®t that starts on thestarting boundary and ends on the endingboundary, and in between satis®es the augmentedHÿJ equations. Addressing such a problem ishighly challenging and di�cult. With the speci®csof an application in ¯exible manufacturing, wederive a one-dimension Dirichlet solution methodwhich is mathematically more tractable. Substan-tial theoretical and practical bene®ts can be real-ized by applying this well-known mathematicaltheory to the MTD problem.

Since drift (the expected regularity) is one ofthe model's two basic elements, it is critical toobtain a suitable drift function for the MTD. Toachieve this, we ®rst derive a valuation forecast asthe drift function and then we address in detail theanalysis of the drift-only MTD (i.e. zero distur-bance), so as to verify the applicability of the driftfunction.

The rest of the paper is organized as follows. InSection 2 we introduce the SDE representation ofthe MTD process and construct the MTD model.Section 3 is devoted to the derivation of a driftfunction for the MTD model. A sample problemof the drift-only MTD is then solved, and an earlyphase-out policy is proved to be optimal. In Sec-tion 4 we analyze the regular MTD model, whichincludes both drift and disturbance. We obtain theHamilton±Jacobi variational representation of theMaximum Principle for the MTD model. The so-lution of a sample stationary MTD problem (i.e.drift and disturbance do not depend on t) is thenanalyzed, and the optimality of the early phase-outpolicy is also obtained.

2. The MTD model

Suppose that the process of adopting a newmanufacturing technology starts at a known time t(06 t6 T <1). In a probability space �X;F; P �,assume that we have full access to all necessaryobservable information from the past up to cur-rent time s P t. Such sets of progressive informa-tion are expressed by an increasing family Fs ofsub-r-algebras of F (see Bensoussan and Lions,1982; Karatzas and Shreve, 1991 for mathematicaldetails), based on which the underlying valuationprocess, denoted by Xs; s P tf g, is measured (i.e.

Xs; s P tf g is Fs-measurable). Furthermore, thevalue depletion process Xs; s P tf g is de®ned as anR-valued Ito process with the following stochasticdi�erential (in Ito's sense) for s P t:

dXs � f �Xs; us; s�ds� g�Xs; s�dWs; �2:1�where us represents an m-dimensional control in anadmissible domain U � Rm, real mapping f :Rm�1 � �0;1� ! R is termed drift, real mapping g:

J. Liu / European Journal of Operational Research 124 (2000) 601±614 603

R� �0;1� ! R is termed disturbance, and dWs isthe di�erential of a one-dimension Wiener process.A non-random initial valuation of the technologyis known at time t, denoted as x � Xs�t > 0, whichre¯ects that the underlying technology must have apositive value measured upon the best informationavailable at the time, in order to start the transferphase. Based on the information obtained by timet (such as required production capacity), the en-gineering and setup of the technology is plannedand the necessary setup time is determined. Let rbe a prescribed ready-time when setup is scheduledto be completed and the transfer (or operation) isready to start (i.e. r P t and Xr P 0). By the natureof depletion, drift must have the characteristics ofdecreasing in time (i.e. of =ot6 0). In (2.1), dis-turbance g is independent of the control us, whichis representative of exogenous disturbances. Al-though (2.1) readily includes a control-dependentdisturbance, its analysis is surprisingly intractable(see Bensoussan and Lions, 1982). The naturalending time, denoted by s, is a hitting time and isde®ned as the ®rst time when the technology valuevanishes (see Karlin and Taylor, 1981 for more onhitting time). Formally we de®ne

s � inf sf P r: Xs6 0g: �2:2�As de®ned by (2.2), it is obvious that Xs � 0. By

the context of MTD and the fact that Xt > 0 isnon-random, it is assumed that s > tf g 2Ft �F.That is, the value of the technology which isstrictly positive at t is stochastically decreasing intime, but cannot vanish in zero time right at thestart. The stopping time, denoted by h, is the timeto kill the MTD by discontinuing the use of tech-nology before its value vanishes (i.e. r6 h6 s orequivalently the probability P Xh P 0f g � 1). Notethat h is a control based on the observation of thelatest state Xs and information Fs (�Ft) suchthat the probability Pfr6 h6 sjXsg � 1. That is,after the MTD has started, it can be stopped (orkilled) either automatically at the natural endingtime s, or by control at the stopping time h beforeits natural ending. When appropriate we shallwrite h�s� as an implicit function of s and s�Xs� asan implicit function of Xs.

For the objective function, we de®ne an ex-pected total pro®t as follows:

p�x; u; h; t�

� E�x;u;h;t�

8<:ÿ Z�Xr; r�eÿl�rÿt� � V �Xh; h�eÿl�hÿt�

�Zh

t

u�Xs; us; s�eÿl�sÿt� ds

��r6 h6 s�9=;;�2:3�

where pro®t rate u, setup cost Z, and terminalpayo� V are all R-valued functions, and l is anapplicable interest rate. The notation E�x;u;h;t� rep-resents the expected value given x, u, h, and t, andE � �h6 s�jf g gives the expectation conditioned onh6 s. Then the MTD model can be de®ned as: ®nda control us and a stopping time h (s 2 �t; h�), sothat the expected total pro®t de®ned by (2.3) ismaximized. That is,

MTD Model:

maxu;h

p�x; u; h; t�

� E�x;u;h;t�


�Zh

t


��r6 h6 s�9=;

s:t:

dXs � f �Xs; us; s�ds� g�Xs; s�dWs; s 2 �t; T �;t P 0; x � Xs�t > 0; us 2 U � Rm:

�2:4�As de®ned by (2.4), the MTD model is a sto-

chastic control system joint with optimal stoppingsubject to an Ito's SDE as the state equation with anon-random positive initial state.

Whether an MTD model is properly de®nedwill depend critically on the existence of a solutionto the SDE of (2.1). Omitting the proof andmathematical details, we present below the theo-rem of existence and uniqueness of solution to astandard Ito's SDE (i.e. the SDE as de®ned in(2.1), but without the control us). Readers who areinterested in further mathematical details are re-ferred to the SDE textbooks (e.g. Chapter 6 of


Arnold, 1992; Appendix D of Fleming and Soner,1993).

Theorem 1 (existence and uniqueness). For an Ito'sstochastic di�erential equation

dXs � f �Xs; s�ds� g�Xs; s�dWs; �2:5�if there exists a constant K > 0 such that,

(a) (Lipschitz condition) 8s 2 �t; T �, x; y 2 R,f �x; t� ÿ f �y; t�j j � g�x; t� ÿ g�y; t�j j6K xÿ yj j,

and(b) (restrictive growth) 8s 2 �t; T �, x 2 R,

f �x; t�j j2 � g�x; t�j j26K2�1� xj j2�;then Eq. (2.1) has on �t; T � a unique R-valued so-lution Xs, continuous with probability 1, that satis-®es the initial condition Xs�t � x.

Another necessary component for an MTDmodel to be proper is a ®nite natural ending time s,which is indeed assured by the textbook results inSDEs, as included in the Corollary below.

Corollary 1. Suppose that for an open and boundeddomain D � R� �0; T �, we have

min�x;t�2 �D

g�x; t� > 0;

where �D � D [ oD. Then E�x;t� sf g <1, 8�x; t� 2 �D.

Proof. (Karatzas and Shreve, 1991, p. 145). �

In the next section, we will address the MTDmodel in more concrete context, so as to verify andvalidate the applicability of the modeling. First weexamine the MTD with only the drift function f(i.e. the disturbance function g � 0). Indeed, thedrift derived therein satis®es the conditions (a) and(b) above (with g � 0). Then in Section 4 we ob-tain an MTD model by amending a disturbance.One can verify that the SDE of the MTD modelin Section 4 also satis®es the conditions ofTheorem 1.

3. The drift-only MTD model and a solved example

With zero disturbance, the value process Xs isdeterministic, and from (2.4) we write:

Drift-only MTD Model:

maxu;h

p�x; u; h� � V �Xh; h�eÿlh �Zh

0

u�Xt; ut; t�eÿlt dt

s:t:

_X t � f �Xt; ut; t�; t P 0;

x � Xt�0 > 0:

�3:1�In (3.1), the initial time t and the initial cost Z

are both set to zero for convenience and the dif-ferential of the valuation process _X t is now givenby an ordinary di�erential equation. The drift-onlyMTD model determines a pro®t-maximizing con-trol ut, together with an optimal stopping time h.That is, (3.1) is an optimal control problem with afree terminal condition. In the remainder of thissection, the time index s associated with the stateand control variables is replaced with t for con-venience.

Introducing a costate variable kt, a Hamiltoni-an function of the drift-only MTD can be con-structed as

H�Xt; ut; kt; t� � u�Xt; ut; t�eÿlt � ktf �Xt; ut; t�:Then, by the Maximum Principle, the solution

fut; hg of the drift-only MTD model can be ob-tained from the following Hamiltonian conditions:

�i� oHok� _X t � f �Xt; ut; t� �state�;

�ii� Xt�0 � x > 0 �initial state�;

�iii� oHoX� ÿ _k � ou�Xt; ut; t�

oXteÿlt

� kof �Xt; ut; t�

oXt�costate�;

�iv� kt�h � oV �Xt; h�oXt

�terminal condition�;

�v� H�Xt; ut; kt; h� � oV �Xt; h�oh

� 0

�free terminal condition�;�vi� H�Xt; ut; kt; t� � max

uH�Xt; ut; kt; t�

�optimality�:�3:2�


The Hamiltonian conditions of (3.2) are notnecessarily su�cient. However if the functions uand f are both concave, then conditions of (3.2) areboth necessary and su�cient (see Mangasarian(1966) and Kamien and Schwartz (1978) for de-tails; further generalized su�cient conditions areobtained by Arrow and Kurz (1970)). In principle,the optimal control fut; hg of a drift-only MTDcan be obtained by solving the Hamiltonianequations given in (3.2), provided that properfunction forms are speci®ed.

Let us consider the valuation of a manufac-turing technology in terms of its market potential,as a way to examine applicable forms of the driftfunction. At the start of the technology transfer(t � 0), the total market potential of the underly-ing technology is estimated as x �0 < x <1�. LetXt be a forecast at the current time t P 0 for themarket potential of the technology over the rest ofits life cycle. Let ut be a one-dimensional controlvariable, such as production rate.

Corresponding to the rate of technology obso-lescence, the market potential is decreasing at arate of r�Xt; ut; t� (e.g. units of output per year). Atthe same time, value depletion is compensated bythe learning e�ect in terms of knowledge, skill,competition, new developments, etc., denoted byn�Xt; ut; t� (a proportion). We shall elaborate moreon r and n in the solved example as illustratedlater in this section. Note that r and n are generalR-valued functions. Thus during the next in®ni-tesimal time interval dt, the marginal depletion ofthe market potential, denoted by ÿdXt, can beexpressed as

ÿdXt � r�Xt; ut; t�dt � n�Xt; ut; t�dXt;

where the ®rst term on right-hand side is the time-based depletion, and the second term representsthe volume-based learning e�ect. Rearranging theterms, we have

dXt � ÿ r�Xt; ut; t�1� n�Xt; ut; t� dt: �3:3�

Then the drift function under the circumstance isderived as

f �Xt; ut; t� � ÿ r�Xt; ut; t�1� n�Xt; ut; t� : �3:4�

Note that the drift de®ned in (3.4) does notcontain an explicit term of a ®xed market size. Toverify the applicability of the derived drift func-tion, we devote the rest of this section to solving asample problem of the drift-only MTD.

3.1. A solved example of drift-only MTD problem

The drift function in the example to be dis-cussed in this section is derived from a projectentitled ``Transfer and engineering of G&L FMCtechnology for industrial sewing machine produc-tion at Jiangwan (Shanghai)''. This is an applica-tion of ¯exible manufacturing technology to anautomated batch production of fast-thread indus-trial sewing machines for China and Asia markets.The product mix consists of six types of applica-tions from canvas-sewing to shoe-making, and theproduction level is continuously adjusted accord-ing to market changes. The FMC system is man-ufactured and engineered by Giddings & Lewis.Since the start of production in 1993 after a con-tracted 3-month of installation and setup, theproduction level has been regulated in proportionto the estimated market share. With Xt as potentialmarket share and ut as proportional productionlevel (controllable), an annual production rate canbe determined as utXt. The selling price has beenfairly stable at p ($/unit). The unit production costhas been a function of the annual production rate,and can be calculated as cutXt, where c is a costconversion coe�cient. The current data on time-standards and system utilization suggest a typical®xed percentage of marginal learning. With con-stant b as a ®xed learning percentage, the volume-based learning e�ect can be expressed as bXt. Thus,by writing r�Xt; ut; t� � utXt and n�Xt; ut; t� � bXt,the drift function of (3.4) can be expressed as

f �Xt; ut; t� � ÿ utXt

1� bXt: �3:5�

By (3.5), drift herein describes the depletion ofpotential production output (e.g. in units of outputper year).

The terminal payo� V �Xt; t� here represents thesalvage of the production capital at time t. Inreference to the so-called exponential depreciation


(Dixit and Pindyck, 1994; Kamien and Schwartz,1991), an example of salvage function is

V �Xt; t� � V0�k ÿ cXt� �3:6�with c 2 �1;1� as the capital depreciation rate andk P 1 as a salvage factor. The ®nal salvage valuedrops to V0�k ÿ 1� at the natural ending time. Sincethe unit pro®t is �p ÿ cutXt� and the productionrate (units per year) is given by utXt=�1� bXt�, wecan write the unit pro®t u�Xt; ut; t� as

u�Xt; ut; t� � �p ÿ cutXt� utXt

1� bXt: �3:7�

Next, with the speci®c function forms givenabove, we obtain the optimal control fu; hg of thedrift-only MTD by solving the Hamiltonianequations in (3.2). In addition to the productionlevel ut and stopping time h, a complete optimalsolution of (3.2) will include the costate kt and thestate trajectory (herein valuation curve) X t, all ofwhich are obtained and presented below in a slateof four Lemmas. In what follows, we omit theindex t when convenient.

Lemma 1 (optimal production control). Let�X ; u; k� be an optimal solution of drift-only MTDmodel. Then optimal production control u is deter-mined as

u � maxp ÿ kelt

2cX; 0

( ); �3:8�

Proof. By the Hamiltonian condition (vi) in (3.2),we have

H 0u �ÿ cX eÿlt uX1� bX

� �p� ÿ cuX �eÿlt ÿ k� X�1� bX � � 0:

Factoring out the common term X=�1� bX �once, we have

ÿcXueÿlt � �p� ÿ cuX �eÿlt ÿ k� � 0

separating u subject to u P 0 immediately gener-ates (3.8). �

With Lemma 1, the trajectory of costate k canbe obtained from (iii) of (3.2) as shown below.

Lemma 2 (optimal costate trajectory). Under theoptimal production control u, the costate trajectoryk is

k ��1� bX �

q� peÿlt � kc; �3:9�

where

kc � ÿ��1� bX h�

qÿ peÿlh ÿ V0 lnc � cX h : �3:10�

Proof. By (iii) of (3.2), we have

ÿ _k �ÿ cueÿlt uX1� bX

� �p� ÿ cuX �eÿlt ÿ k� u

�1� bX �2 :

Rearranging the above yields

_kÿ u

�1� bX �2 k

� cu2X1� bX

� cu2X

�1� bX �2 ÿpu

�1� bX �2!

eÿlt:

�3:11�Using the state Eq. (3.3), the following can be

derived

_k � k0X � _X � k0XÿuX

1� bX:

Substituting the above into (3.11), we have

ÿ uX1� bX

k0Xu

�1� bX �2 k

� cu2X1� bX

� cu2X

�1� bX �2 ÿpu

�1� bX �2!

elt:

Factoring out common terms in the above equa-tion yields

ÿ Xk0X ÿ1

1� bXk

� cuX�

� cuX1� bX

ÿ p1� bX

�eÿlt: �3:12�


By (3.8) of Lemma 1, we have cuX eÿlt � 12�peÿlt ÿ

k� which will generate the following from (3.12) bysubstitution:

ÿ Xk0X ÿ1

1� bXk

� 1

21

�� 1

1� bX

��peÿlt ÿ k� ÿ p

1� bXeÿlt:

Combining and then canceling the commonterms in the above equation, we have

2k0X ÿb

1� bXk � ÿ pb

1� bXeÿlt: �3:13�

It can be veri®ed that k� � ��1� bX �pis a so-

lution for the corresponding homogeneous equa-tion of (3.13). Then solving the ordinarydi�erential (3.13) with terminal conditions in (3.2)shall conclude the Lemma. �

Lemma 3 (optimal valuation curve). Under theoptimal production control u, the optimal valuationcurve X is determined by the equation

1

b2k2

c lnzÿ ÿ 4kcz� z2

� � 1

2clelt � �X ; �3:14�

where

z � �1� bX �12 � kc;

�X � 1

b2k2

c lnz0

ÿ ÿ 4kcz0 � z20

�ÿ 1

2cl;

z0 � �1� bX �12 � kc; x � Xt�0:

Proof. From (3.8) of Lemma 1, we have uX � 12c ��peÿlt ÿ k�elt with which the state equation (3.3)

generating the following:

_X � ÿ 1

2c1

1� bX�p ÿ kelt�: �3:15�

Using (3.9) and substituting for k in (3.15), wehave

_X1

2c�1�� bX �12 � kc

1� bX

�elt: �3:16�

Lemma 3 can be concluded with the solution of(3.16), a simple ®rst-order ODE, and the detailsare therefore omitted. �

Lemma 4 (optimal stopping time). The optimalstopping time h satis®es the following equation:

cuhX heÿlh � V0 ln1

c� cXh : �3:17�

Proof. First, from the salvage function of (3.6) wederived

oV �X ; h�oh

� ÿ�V0 ln c�cXh _X h:

Thus, the terminal condition in (3.2),

H�X ; u; k; h� � oV �X ; h�oh

� 0;

generates the following:

1

2�peÿlh ÿ kh� _X t�h � �V0 lnc�cXh _X t�h � 0:

If _X t�h 6� 0 (i.e. Xt�h 6� 0 by (3.5)), we have

peÿlh ÿ kh � ÿ2V0 lnc � cXh : �3:18�From (3.8), it follows that 2cuX eÿlh � peÿlh ÿ kh

with which (3.18) derives (3.17). �

The regeneration time h and the associatedending value Xh can be determined by solving(3.14) and (3.17) simultaneously. However, aclosed form solution does not seem to be attain-able yet. In principle, Lemmas 1, 2, 3 and 4 pro-vide a complete set of optimal solutions for thedrift-only MTD problem. Next, we discuss someuseful properties.

Proposition 1. k is positive and strictly increasing int 2 �0; h�, and kc as in (3.10) is negative, i.e. kc < 0.

Proof. By (3.11) in the proof of Lemma 2, we canwrite

_k � ÿ oHoX� uk� cu2X eÿlt ÿ pueÿlt

�1� bX �2 � cu2X1� bX

eÿlt

� u�k� cuX eÿlt ÿ peÿlt��1� bX �2 � cu2X

1� bXeÿlt:


Using 2cuX eÿlt � peÿlt ÿ k shown in (3.8), wederive

_k � ÿcuX

�1� bX �2 ueÿlt � cuX �1� bX ��1� bX �2 ueÿlt

� cbu2X 2

�1� bX �2 eÿlt:

Clearly, _k > 0 8t 2 �0; s�. Since it is given by (ii)of (3.2) that x � Xt�0 > 0, thus k > 0 at t � 0 by(3.9). Since kh � �V0 ln 1

c�cXh > 0, then we can verifyfrom (3.9) and (3.18) that

kc � kh ÿ��1� bXh

pÿ peÿlh

� ÿ��1� bXh

pÿ 2V0 ln

1

c� cXh < 0:

This concludes the Proposition 1. �

For the optimal production level u, we obtainthe following.

Proposition 2. For t 2 �0; h�, the optimal productionlevel is determined by u � �p ÿ kelt�=�2cX � > 0;and u is strictly increasing in t, speci®cally,

u0 < u < uh; �3:19�where

u0 � p ÿ k0

2cx; �3:20�

k0 ��1� bx

p� p � kc; �3:21�

uh �V0 ln 1

c � cX X

cX heÿlh: �3:22�

Proof. First, we show that u P 0 for all t 2 �0; h�.From (3.8), it su�ces to show that p ÿ kelt > 0 int 2 �0; h�, which indeed holds since p ÿ kelt isstrictly decreasing in t and peÿlh ÿ kh > 0 by(3.18). Taking partial derivative of (3.8) withrespect to t (2 �0; h�) yields

o�uX �ot� 1

2c

ÿ ok

otelt ÿ lkelt

!:

Using (3.9), it arrives that

o�uX �ot

� 1

2c

�ÿ � ÿ lpeÿlt�elt ÿ lkelt

�� l

2cp� ÿ kelt� > 0; 8t 2 �0; h�:

Thus, for any X , the production rate uX isstrictly increasing in t. Since X is decreasing in t(see (3.5)), it thus follows that u is strictly in-creasing in t. With t� 0, (3.8) generates (3.20) and(3.9) in Lemma 2 gives (3.21). With t � h and using(3.18), (3.22) can then be derived from (3.8). �

Following immediately from Proposition 1 and2 is an interesting optimal regularity as formallysummarized in the following Proposition.

Proposition 3. For the example problem of drift-only MTD, during the MTD period t 2 �0; h� theproduction should be kept at an increasing rate, andthe underlying technology should be phased out att � h <1 before its market potential vanishes(i.e. X h > 0).

Proof. The optimality of an increasing productionrate (uX ) is shown in Proposition 2. Note that thecostate k measures the shadow price of the marketpotential (X). Thus, the Theorem follows immedi-ately if the inequality kh > 0 holds, which is indeedthe case as shown by Proposition 1. �

4. Variational calculus formulation of the MTD

problem

Now, we consider the MTD model with bothdrift and disturbance as given by the SDE in (2.4),which can be characterized as an optimal stochasticcontrol problem joint with optimal stopping. Werecall the MTD model of (2.4) for convenience,

MT Model:

maxu;h

p�x; u; h; t�

� E�x;u;h;t�


�Zh

t


��r6 h6 s�9=;


s:t:

dXs � f �Xs; us; s�ds� g�Xs; s�dWs;

s 2 �t; T �t P 0; x � Xs�t > 0; us 2 U � Rm :

In fact, Z�x; t� represents the cost (or a fairprice) of a new technology with a market potentialof x, while V �x; t� is the payo� if the technology issalvaged with a remaining market size of x.Clearly, we must have Z�x; t� > V �x; t� in order tohave a meaningful case of technology di�usion.Under an optimal policy fus; hg, if exists, the pro®tstarts with the maximum (i.e. p�x; u; h; r� � Z�x; r�)and is decreasing due to technology obsolescence(i.e. p�x; u; h; s� < Z�x; s�, 8s 2 �r; h�). Note that p isthe pro®t generated from the technology in useand Z is the worth or price of a new one. In themeantime, we must have by optimality thatp�x; u; h; t�P V �x; t�, where V �x; t� represents aterminal payo� if the MTD is terminated and thetechnology is salvaged. More precisely, on the onehand the di�usion should continue if the expectedpro®t is greater than the terminal payo� (i.e.p�x; u; h; t� > V �x; t�); on the other hand the di�u-sion should be terminated immediately if the ex-pected pro®t is no better than the terminal payo�(i.e. p�x; u; h; t� � V �x; t�). With such, we de®ne adi�usion set D as

D � �x; t� xjf > 0; t 2 �0; T �: V �x; t� < p�x; u; h; t�< Z�x; t�g: �4:1�

Note that D is open and bounded. As de®ned,the MTD should continue if the system status attime t belongs to D.

It has been identi®ed that a one-dimensionstochastic control problem without optimal stop-ping can be represented by the so-called Hamil-ton±Jacobi equations of second-order ordinarydi�erential equations (ODEs). There have beenmany ways to derive such an H±J representationof the Maximum Principle for various problemsincluding the optimal stopping (e.g. Bensoussanand Lions, 1982). For the MTD model which in-volves stochastic control joint with optimal stop-ping, we herein obtain augmented Hamilton±Jacobi equations of MTD model using the varia-tional calculus in combination with applications ofIto's formula.

Theorem 2. Let Xs be de®ned by (2.1) (in Ito'ssense). Suppose that Theorem 1 holds. In addition,assume that

u 2 L2�R� �0; T � � U �; p 2 C2�D�; �4:2�where C2�D� is the space of all twice di�erentiablefunctions on D of (4.1). Then an optimal policyfus; hg of the MTD model satis®es the followingaugmented Hamilton±Jacobi equations:

H±J Equations of MTD:

ÿ opotÿ B�x; t�pÿ H�p; _p; x; t� � 0; 8�x; t� 2 D;

�4:3�

p�x; u; h; t� � Z�x; t�; 8�x; t� 2 oDÿ; �4:4�

p�x; u; h; t� � V �x; t�; 8�x; t� 2 oD�; �4:5�where

oDÿ � min �x; t�: pf � Zg;oD� � max �x; t�: pf � V g; �4:6�

B�x; t� � 1

2g2�x; t� o2

ox2

�Brownian Differential Operator�;�4:7�

H�p; _p; x; t� � maxu2U

f �x; u; t� opox

�ÿ lp� u

��Hamiltonian Operator�:

�4:8�

Proof. Theorem 1 and the Assumption (4.2) assurethat an Ito's di�erential is applicable (Karatzasand Shreve, 1991). Since t is known withoutregards to h and r is prescribed upon t, incombination with Ito's rule, we calculate thefunctional variation (as adopted in Gregory andLin, 1992) of the total pro®t with control u ®xed,denoted by dpu�h; x; t�, as follows:

dpu�x; h; t� � dpu�x; h tj � � dpu�t x; hj �;

where the ®rst term on the right-hand side denotesthe functional di�erential with t ®xed, while the


second term is the di�erential when �x; h� are ®xed.With each ®xed t 2 �0; T �, applying Ito's formula ina similar manner as by Arnold, 1992 we have

dpu�x; h tj �

� opoh

�� 1

2g2�x; h� o

2pox2� f �x; u; h� op

ox

�dh

� ÿ opot

�� 1

2g2�x; t� o

2pox2� f �x; u; t� op

ox

�dt;

where a change of variable h � ÿt is applied in thesecond equality above. Noting that p�x; u; h; t� by(2.4) involves a functional integral with t as a pa-rameter, by variational calculus (e.g. Gregory andLin, 1992) we have

dpu�t x; h�j

� d

dt

E�x;u;h;t� ÿ Z�Xr; r�eÿl�rÿt��

:

� V �Xh; h�eÿl�hÿt��Zh

t

E�x;u;h;t� u�Xs; us; s�eÿl�sÿt�� ds

!� ÿ lE�x;u;h;t� Z�Xr; r�eÿl�rÿt�� ÿ �

dt

� lE�x;u;h;t� V �Xh; h�eÿl�hÿt�� ÿ �dt

�Zh

t

lE�x;u;h;t� u�Xs; us; s�eÿl�sÿt�� ds

0@ 1Adt

ÿ E�x;u;h;t� u�Xt; ut; t� eÿl�tÿt�� ÿ �dt

� lp�x; u; h; t�� ÿ u�x; u; t��dt:

Including all the terms, we obtain

dpu�h; x; t�

��ÿ op

otÿ 1

2g2�x; t� o

2pox2

ÿ f �x; u; t� opox

�ÿ lp� u

��dt

��ÿ op

otÿ A�x; t�pÿ L�p; _p; u; h; x; t�

�dt

where L�p; _p; u; h; x; t� � f �x; u; t� opox ÿ lp� u. By

the Maximum Principle and variational calculus,at the optimum we have dpu � 0 and L is

maximized, with which we conclude the proof ofthe Theorem by denoting

H�p; _p; x; t� � maxu;h

L�p; _p; u; h; x; t�n o

: �

The augmented H±J equations of MTD inTheorem 2 involves a nonlinear di�erential oper-ator of the ®rst-order, the solution of which isquite involved and di�cult. See Bensoussan andLions, 1982 (Chapter 4) for a rigorous proof of theexistence and uniqueness of a solution to theaugmented H±J equations. In principle, the opti-mal control and optimal stopping time can beobtained by solving the H±J (4.3) and (4.5) whichrepresent a free-boundary problem of a second-order ODE. However, in most cases a closed formanalytical solution of such a free-boundary prob-lem is unattainable and thus numerical solutionmethods are usually called for. The details of thesolution methods are beyond the scope of thispaper and therefore are deferred to separate dis-cussions.

As an application of Theorem 2, let us consideran example of the MTD problem by amending adisturbance to the example of the drift-only MTDdiscussed in the previous section.

For the disturbance of the example, we use g ��gXsp

(g > 0 is scaling factor) as adopted in Pin-dyck (1993). With f as given by (3.5), the stochasticprocess de®ned by (2.1) is stationary since driftand disturbance are implicit of time. As discussedin Pindyck (1993) and Chi et al. (1997), a sta-tionary disturbance should have the followingcharacteristics: lim x!0 g�x� � 0 and og=ox > 0.With f and g as selected, the Hamiltonian, as de-®ned below, is di�erentiable with respect to thecontrol u 2 �0;1�. That is, at optimum we have

oHou� 0;

where

H�p; _p; x; t� � f �x; u; t� opoxÿ lp� u:

At the ready-to-start state xr, we adopt an en-gineering cost of the technology Z�xr� as an initialboundary condition. Based on the actual calcula-tion used in the Jiangwan project, we use a qua-dratic cost function of initial market size xr,


Z�xr� � I � axr � bx2r , where I represents a ®xed

setup cost, a and b are cost coe�cients. The sal-vage function is the same as given by (3.6). Assuch, the di�usion set D becomes

D � x: V �x�f < p < Z�x�g:Now, the MTD problem can be stated as to

®nd a pro®t function p 2 C2�D� that satis®es V <p < Z and solves the di�usion equation, with Z�xr�as the starting point and with V �xh� as the endingpoint. Since the example MTD model is station-ary, we have op=ot � 0, and we can set initial timeto zero (i.e. t � 0) for convenience. Thereby, theoptimality conditions of the example MTD modelcan be warranted from Theorem 2 as below:

H±J equations of example MTD model:

1

2gx

o2pox2� ux

1� bxopoxÿ lp� �p ÿ cux� ux

1� bx� 0

�diffusion equation�;oHou� o

ouux

1� bxopox

�ÿ lp� �pÿ cux� ux

1� bx

�� 0

�Hamiltonian maximum�;Xs�0 � x0 > 0 �initial state�;p�xr; u; h� � Z�xr� �starting boundary�;p�xh; u; h� � V �xh� �ending boundary�:

In order to have a complete and solvable H±Jequations, we construct two more terminal con-ditions using the so-called smooth-pasting methodDixit and Pindyck (1994) namely,

op�xr; u; h�oxr

oZ�xr�oxr

andop�xh; u; h�

oxh� oV �xh�

oxh:

Thereupon, the existence of a solution (in thestrong sense) is assured by verifying with Ben-soussan and Lions (1982, Theorem 1.1, p. 497).Thus, a much tractable solution algorithm of theexample MTD problem can be obtained fromTheorem 2.

Dirichlet solution of example MTD problem:1. Identify u from the Hamiltonian equation:

u � u 2 U :oxou

ux1� bx

opox

��ÿ �p ÿ cux� ux

1� bx

�� 0

�:

2. With identi®ed control u as a parameter, deter-mine p�x; u�, xr and xh by solving the followingone-dimension Dirichlet problem:

1

2gx

o2pox2� ux

1� bxopoxÿ lp

� �p ÿ cux� ux1� bx

� 0; x 2 D;

s:t: �boundary conditions�

p�xr; u� � Z�xr�; op�xr; u�oxr

� oZ�xr�oxr

;

p�xh; u� � V �xh�; op�xh; u�oxh

� oV �xh�oxh

;

Z�x� � I � ax� bx2; V �x� V0�kcx�:

3. The optimal solution is obtained asp; u; xr; xhf g.

Note that fxr; xhg de®nes a start/stop policy thatis practical and easy to implement: start to adoptthe technology whenever its market value reachesxr or higher; while the technology should be dis-continued as soon as its value falls to xh or below.Numerical methods are most likely to be neededfor the steps 1 and 2 of the Dirichlet solution.Computational details are deferred to separatereports.

We conclude this section with the proof of theoptimality of an early phase-out policy for theexample MTD model with the presence of distur-bance.

Proposition 4. For the example MTD model, if thereexists a non-zero salvage value of the technology atthe natural ending time (i.e. V �0� > 0), then it isoptimal to phase out the technology before itsmarket value vanishes (i.e. xh > 0).

Proof. Let D� � x P 0: p � Vf g. Since V 2 C1�R�,for every x 2 D� the salvage function V shall solvethe di�usion equation, that is,

ÿ 1

2gxV0 lnc� �2cx � ux

1� bxV0 ln

1

ccx ÿ lV0 k� ÿ cx�

� �p ÿ cux� ux1� bx

� 0:


It can be veri®ed that fx � 0g 62 D� if k > 1 (i.e.V �0� > 0). Thus, we conclude that xh > 0. �

We note that both Propositions 3 and 4 suggestto phase out the technology earlier than its naturalending time, either with or without uncertain dis-turbances.

5. Concluding remarks

Herein, a manufacturing technology di�usion(MTD) model is developed to characterize thedepletion process of the valuation of an underlyingtechnology over the phases of technology transferand operation. The MTD model adopts an Ito'sSDE as its state equation, so as to address twonecessary aspects of a technology valuation pro-cess: the estimation of what is at stake, and theanticipation of future changes.

The proposed MTD model represents a sto-chastic control problem joint with optimal stop-ping. The model is validated by an example MTDof ¯exible manufacturing. Under the speci®cs ofthe example, the optimal policy obtained from theMTD model suggests to phase out the underlyingtechnology before its market value vanishes.

The solution of an MTD model involves non-linear free boundary problems of second orderODEs. E�ective computation algorithms poseimmediately as a future research topic. It is of ourinterest to extend the MTD model to the case of aMartingale as the initial state. Furthermore, amultidimensional MTD is another future topic ofour particular interests.

References

Arnold, L., 1992. Stochastic Di�erential Equations: Theory and

Applications. Krieger Publishing, Malabar, FL.

Arrow, K.J., Kurz, M., 1970. Public Investment, the Rate of

Return, and Optimal Fiscal Policy. Johns Hopkins Univer-

sity Press, Baltimore, MD.

Aulin, A., 1996. Causal and Stochastic Elements. In: Business

Cycles. Springer, New York.

Bass, F.M., 1969. A new product growth model for consumer

durables. Management Science 15, 215±227.

Bass, F.M., Krishnan, T., Jain, D., 1994. Why the bass model ®ts

without decision variables. Marketing Science 13, 203±223.

Bensoussan, A., Lions, J., 1982. Applications of Variational

Inequalities in Stochastic Control. Elsevier, Amsterdam.

Burgelman, R.A., Modesto, M.A., Wheelright, S.C., 1995.

Strategic Management of Technology and Innovation.

Irwin, Burr Ridge, IL.

Chatterjee, R., Eliashberg, J., 1990. The innovation di�usion

process in a heterogeneous population: A micromodeling

approach. Management Science 36, 1057±1079.

Chi, T., Liu, J.J., Chen, H., 1997. Optimal stopping rule for a

project with uncertain completion time and partial salvage-

ability. IEEE Transactions on Engineering Management 44,

54±66.

Cook, I., Mayes, P., 1996. Introduction to Innovation and

Technology Transfer. Artech House, Boston.

Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertain-

ty, Princeton University Press, Princeton, NJ.

Dockner, E., Jorgensen, S., 1988. Optimal advertising policies

for di�usion models of new product innovation in monop-

olistic situations. Management Science 34, 119±130.

Feichtinger, G., Hartl, R.F., Sethi, S.P., 1994. Dynamic optimal

control models in advertising. Management Science 40, 179±

226.

Flam, S.D., Ben-Israel, A., 1990. A continuous approach to

oligopolistic market equilibria. Operations Research 38,

1045±1051.

Fleming, W.H., Soner, H.M., 1993. Controlled Markov Pro-

cesses and Viscosity Solutions. Springer, New York.

Gregory, J., Lin, C., 1992. Constrained Optimization in the

Calculus of Variations and Optimal Control Theory. Van

Nostrand Reinhold, New York.

Horsky, D., Simon, L.S., 1983. Advertising and the di�usion of

new products. Marketing Science 2, 1±17.

Kalish, S., 1983. Monopolist pricing with dynamic demand and

production cost. Marketing Science 2, 135±159.

Kamien, M.I., Schwartz, N.L., 1991. Dynamic Optimization:

The Calculus of Variations and Optimal Control in

Economics and Management. North-Holland, Amsterdam.

Kamien, M.I., Schwartz, N.L., 1978. Self-®nancing of an R&D

project. American Economic Review 68, 252±261.

Kamrad, B., 1995. A lattice claims model for capital budgeting.

IEEE Transactions on Engineering Management 42, 140±

149.

Karlin, S., Taylor, H.M., 1981. A Second Course in Stochastic

Processes. Academic Press, San Diego, CA.

Karatzas, I., Shreve, S., 1991. Brownian Motion and Stochastic

Calculus. Springer, New York.

Klepper, S., 1996. Entry, exit, growth, and innovation over the

product life cycle. American Economic Review 86, 562±583.

Kulatilaka, N., 1988. Valuing the ¯exibility of ¯exible manu-

facturing systems. IEEE Transactions on Engineering

Management 35, 250±257.

Liu, J.J., Chi, T., 1997. Strategic Control of Product Life Cycle

Under Uncertainty, working paper. School of Business

Administration, University of Wisconsin-Milwaukee.

Mangasarian, O.L., 1966. Su�cient conditions for the optimal

control of nonlinear systems. SIAM Journal of Control 4,

139±152.


Mans®eld, E., 1988. The speed and cost of industrial innovation

in Japan and the United States: External vs internal

technology. Management Science 34, 1157±1168.

Mesak, H., Berg, W., 1995. Incorporating price and replace-

ment purchases in new product di�usion models for

consumer durables. Decision Sciences 26, 2425±2445.

Nagurney, A., Thore, S., Pan, J., 1996. Spatial market policy

modeling with goal targets. Operations Research 44, 393±

406.

Pindyck, R.S., 1993. Investments of uncertain costs. Journal of

Financial Economics 34, 53±76.

Reinganum, J., 1981. On the di�usion of new technology: A

game theoretic approach. Review of Economic Studies 48,

395±405.

Rogers, E.M., 1995. The Di�usion of Innovations, 4th ed., The

Free Press, New York.

Rosenberg, J., 1972. Factors a�ecting the di�usion of technol-

ogy. Explorations in Economic History 10, 3±33.

Teng, J.T., Thompson, G.L., 1985. Optimal strategies for

general price-advertising models. In: G. Feichtinger (Ed.),

Optimal Control Theory and Economic Analysis, vol. 2,

North-Holland, Amsterdam, 183-195.

Thore, S., 1991. Economic Logistics. Quorum Books, New

York.

Wheelwright, S.C., Clark, K.B., 1995. Leading Product Devel-

opment. Free Press, New York.


on the dynamics of stochastic diffusion of manufacturing technology

Documents