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On the dynamics of stochastic diffusion of manufacturing technology
European Journal of Operational Research 124 (2000) 601±614
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Theory and Methodology
On the dynamics of stochastic diusion of manufacturing technology 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 diusion (MTD), a close relative of demand diusion in marketing, refers to the transition 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: regularity (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 differential equations (SDEs), to overcome the limitations of invariance (e.g. a ®xed market size and the absence of disturbance) as suered 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 augmented 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 diusion; Optimal stopping; Stochastic control
1. Introduction Three distinct phases in the life cycle of a manufacturing technology have long been identi-
*
Tel.: 1-414-229-4235; fax: 1-414-229-6957. E-mail address: [email protected] (J. Liu).
®ed: research and development (R&D), transfer/ commercialization, and operation/regeneration (Cook and Mayes, 1996; Rogers, 1995). A ®rm's competitive strategic and operational position is critically dependent upon an accurate and objective understanding of the life cycle of underlying technology (Burgelman et al., 1995). A key characterization of the life cycle has been the transition
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
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of technology's economic value. For instance, the R&D of technology presents a phase of value build-up through technological innovations, while the transfer and operation of technology entail a value depletion process in accordance with inevitable technology obsolescence (Rosenberg, 1972). Manufacturing technology diusion (MTD) refers to the transition of economic value during the transfer and operation phases of a technology life cycle. There is a keen resemblance between technology and market diusions in that both cast a dynamic depletion process of their respective economic values. The depletion process of market diusion has been studied using the widely-known product life cycle (PLC) models (e.g. Bass, 1969; Kalish, 1983; Horsky and Simon, 1983; Teng and Thompson, 1985; Dockner and Jorgensen, 1988; Klepper, 1996; literature on another relevant topic, business life cycles by the economic logistics models, can be found in Aulin, 1996; Thore, 1991; Nagurney et al., 1996; Flam and Ben-Israel, 1990). Despite its obvious similarity to marketing methodology, the modeling of manufacturing technology diusion is fundamentally dierent in several ways. First, manufacturing technology is rooted in manufacturing activities (e.g. production control) as opposed to marketing alternatives (e.g. pricing and advertising). The valuation process associated with MTD is consistently in¯uenced by production decisions, such as whether or not to discontinue (or switch o) an extant production process, and what production level should be targeted, etc. As such, the transformative nature of manufacturing derives an immediate distinction between technology and market diusions. On the one hand, there hardly exists an MTD without exogenous regulations, while on the other hand it can be still reasonably argued (e.g. by Bass et al., 1994) as to why a PLC model will work without control. Second, the MTD will inevitably endure uncertain in¯uences and disturbances due to technological evolution and innovation. Therefore, it is no longer a justi®able modeling assumption to tolerate the limitations of invariance suered by a typical PLC model (e.g. the inelasticity of market size and the absence of disturbance). The PLC research bases its analysis on the
mean value (or the passive net present value) of market potential, and is known for its limitations of ®xed total market and deterministic variations of market potential. Such limitations and their alleviative solutions have long been discussed and addressed (e.g. Chatterjee and Eliashberg, 1990; Mesak and Berg, 1995; Liu and Chi, 1997; Feichtinger et al., 1994 for more details and extensive updates). In brief, MTD entails a stochastic value-depletion process where the uncertain status of the process (e.g. market potential) is continuously observed and regulated. Obviously, a rigorous modeling of MTD must give an accurate account for the aforementioned aspects: regularity and uncertainty. This paper attempts to achieve that. The methodology of stochastic dierential equations (SDEs) bodes well for the analysis of similar problems in the areas of physics, engineering, and lately ®nance. SDE modeling has also been used to study the R&D phase of technology's life cycle (Reinganum, 1981; Kulatilaka, 1988; Dixit and Pindyck, 1994; Kamrad, 1995; Chi et al., 1997). But as far as we can ascertain from current research literature, SDE modeling has not yet been applied to MTD. In this paper, we propose using an Ito's SDE formulation to characterize the in®nitesimal dynamics of the valuation process. Assuming full access to the past observable information up to current time t, the changes of the valuation in the next dt time consist of two components: a depletion rate prescribed from the past (drift), and a superimposed Wiener dierential (disturbance). Both drift and disturbance can be rather general and can include control variables such as production rate. The proposed SDE formulation overcomes the limitations of invariance by not only allowing random disturbances, but adopting a variable market as well. Furthermore, a controllable stopping time is introduced to allow a managerial decision to terminate the use of the technology before its PLC-de®ned natural ending time; that is, before total depletion of its underlying values. Life-cycleshortening has been pervasive worldwide (Wheelwright and Clark, 1995; Mans®eld, 1988), and it has become increasingly common to
J. Liu / European Journal of Operational Research 124 (2000) 601±614
discontinue a manufacturing process before its natural ending time. With the SDE characterization of the underlying depletion process, a general MTD model under production regulations is then formulated as a stochastic control problem with free terminal time (stopping time). The objective function of the MTD model is the expected total pro®t as derived from the total realization of the technology's value over the duration of the regulated MTD, including terminal payos, if applicable. The MTD seeks an admissible regulation policy joint with an exit (or stopping) decision that maximizes expected total pro®t. It is interesting to note that the MTD model becomes a basic optimal stopping problem when no other control variables are involved. With reference to a speci®c application of ¯exible manufacturing, it is shown in this paper that the optimal stopping policy suggests phasing out the underlying technology before its value vanishes, which coincides with the aforementioned life-cycle-shortening phenomena. Although the MTD model is mathematically suited for n-dimensional SDEs, we con®ne our analysis to the one-dimensional case for the sake of keeping within a realistic context. By variational calculus in combination with applications of Ito's formula, we derive augmented Hamilton±Jacobi variational equations for the MTD model, the solution of which presents a nonlinear free boundary problem of second-order ordinary differential equations (ODEs). Thereby, we show an analytical characterization of the MTD model to be the following: for a system with its state dynamics expressed by Ito's SDEs, and given a starting boundary and an ending boundary (both real functions on Rn ), we are to ®nd a twice differentiable functional pro®t that starts on the starting boundary and ends on the ending boundary, and in between satis®es the augmented HÿJ equations. Addressing such a problem is highly challenging and dicult. With the speci®cs of an application in ¯exible manufacturing, we derive a one-dimension Dirichlet solution method which is mathematically more tractable. Substantial theoretical and practical bene®ts can be realized by applying this well-known mathematical theory to the MTD problem.
603
Since drift (the expected regularity) is one of the model's two basic elements, it is critical to obtain a suitable drift function for the MTD. To achieve this, we ®rst derive a valuation forecast as the drift function and then we address in detail the analysis of the drift-only MTD (i.e. zero disturbance), so as to verify the applicability of the drift function. The rest of the paper is organized as follows. In Section 2 we introduce the SDE representation of the MTD process and construct the MTD model. Section 3 is devoted to the derivation of a drift function for the MTD model. A sample problem of the drift-only MTD is then solved, and an early phase-out policy is proved to be optimal. In Section 4 we analyze the regular MTD model, which includes both drift and disturbance. We obtain the Hamilton±Jacobi variational representation of the Maximum Principle for the MTD model. The solution of a sample stationary MTD problem (i.e. drift and disturbance do not depend on t) is then analyzed, and the optimality of the early phase-out policy is also obtained.
2. The MTD model Suppose that the process of adopting a new manufacturing technology starts at a known time t (0 6 t 6 T < 1). In a probability space X; F; P , assume that we have full access to all necessary observable information from the past up to current time s P t. Such sets of progressive information are expressed by an increasing family Fs of sub-r-algebras of F (see Bensoussan and Lions, 1982; Karatzas and Shreve, 1991 for mathematical details), based on which the underlying valuation process, denoted by fXs ; s P tg, is measured (i.e. fXs ; s P tg is Fs -measurable). Furthermore, the value depletion process fXs ; s P tg is de®ned as an R-valued Ito process with the following stochastic dierential (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 an admissible domain U Rm , real mapping f : Rm1 0; 1 ! R is termed drift, real mapping g:
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R 0; 1 ! R is termed disturbance, and dWs is the dierential of a one-dimension Wiener process. A non-random initial valuation of the technology is known at time t, denoted as x Xst > 0, which re¯ects that the underlying technology must have a positive value measured upon the best information available at the time, in order to start the transfer phase. Based on the information obtained by time t (such as required production capacity), the engineering and setup of the technology is planned and the necessary setup time is determined. Let r be a prescribed ready-time when setup is scheduled to be completed and the transfer (or operation) is ready to start (i.e. r P t and Xr P 0). By the nature of depletion, drift must have the characteristics of decreasing in time (i.e. of =ot 6 0). In (2.1), disturbance g is independent of the control us , which is representative of exogenous disturbances. Although (2.1) readily includes a control-dependent disturbance, its analysis is surprisingly intractable (see Bensoussan and Lions, 1982). The natural ending time, denoted by s, is a hitting time and is de®ned as the ®rst time when the technology value vanishes (see Karlin and Taylor, 1981 for more on hitting time). Formally we de®ne s inf fs P r: Xs 6 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 is non-random, it is assumed that fs > tg 2 Ft F. That is, the value of the technology which is strictly positive at t is stochastically decreasing in time, but cannot vanish in zero time right at the start. The stopping time, denoted by h, is the time to kill the MTD by discontinuing the use of technology before its value vanishes (i.e. r 6 h 6 s or equivalently the probability P fXh P 0g 1). Note that h is a control based on the observation of the latest state Xs and information Fs ( Ft ) such that the probability P fr 6 h 6 sjXs g 1. That is, after the MTD has started, it can be stopped (or killed) either automatically at the natural ending time s, or by control at the stopping time h before its natural ending. When appropriate we shall write h s as an implicit function of s and s Xs as an implicit function of Xs . For the objective function, we de®ne an expected total pro®t as follows:
p x; u; h; t 8 < Ex;u;h;t ÿ Z Xr ; reÿl rÿt V Xh ; heÿl hÿt : 9 Zh = u Xs ; us ; seÿl sÿt ds r 6 h 6 s ; ; t
2:3 where pro®t rate u, setup cost Z, and terminal payo V are all R-valued functions, and l is an applicable interest rate. The notation Ex;u;h;t represents the expected value given x, u, h, and t, and Efjh 6 sg gives the expectation conditioned on h 6 s. Then the MTD model can be de®ned as: ®nd a control u^s and a stopping time ^h (s 2 t; ^h), so that the expected total pro®t de®ned by (2.3) is maximized. That is, MTD Model: max p x; u; h; t u;h 8 < Ex;u;h;t ÿ Z Xr ; reÿl rÿt V Xh ; heÿl hÿt : 9 Zh = u Xs ; us ; seÿl sÿt ds r 6 h 6 s ; t
s:t: dXs f Xs ; us ; s ds g Xs ; s dWs ; s 2 t; T ; t P 0; x Xst > 0; us 2 U Rm : 2:4 As de®ned by (2.4), the MTD model is a stochastic control system joint with optimal stopping subject to an Ito's SDE as the state equation with a non-random positive initial state. Whether an MTD model is properly de®ned will depend critically on the existence of a solution to the SDE of (2.1). Omitting the proof and mathematical details, we present below the theorem of existence and uniqueness of solution to a standard Ito's SDE (i.e. the SDE as de®ned in (2.1), but without the control us ). Readers who are interested in further mathematical details are referred to the SDE textbooks (e.g. Chapter 6 of
J. Liu / European Journal of Operational Research 124 (2000) 601±614
Drift-only MTD Model:
Arnold, 1992; Appendix D of Fleming and Soner, 1993). Theorem 1 (existence and uniqueness). For an Ito's stochastic dierential 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, j f x; t ÿ f y; tj jg x; t ÿ g y; tj 6 K j x ÿ y j, and (b) (restrictive growth) 8s 2 t; T , x 2 R, 2 2 j f x; tj jg x; tj 6 K 2 1 j xj2 ; then Eq. (2.1) has on t; T a unique R-valued solution Xs , continuous with probability 1, that satis®es the initial condition Xst x.
605
max p x; u; h V Xh ; he
ÿlh
u;h
s:t: X_ t f Xt ; ut ; t;
Zh
u Xt ; ut ; teÿlt dt
0
t P 0;
x Xt0 > 0: 3:1
Corollary 1. Suppose that for an open and bounded domain D R 0; T , we have
In (3.1), the initial time t and the initial cost Z are both set to zero for convenience and the differential of the valuation process X_ t is now given by an ordinary dierential equation. The drift-only MTD model determines a pro®t-maximizing control u^t , together with an optimal stopping time ^h. That is, (3.1) is an optimal control problem with a free terminal condition. In the remainder of this section, the time index s associated with the state and control variables is replaced with t for convenience. Introducing a costate variable kt , a Hamiltonian function of the drift-only MTD can be constructed as
min g x; t > 0;
H Xt ; ut ; kt ; t u Xt ; ut ; teÿlt kt f Xt ; ut ; t:
D [ oD. Then Ex;t fsg < 1, 8 x; t 2 D. where D
Then, by the Maximum Principle, the solution f^ ut ; ^hg of the drift-only MTD model can be obtained from the following Hamiltonian conditions:
Another necessary component for an MTD model to be proper is a ®nite natural ending time s, which is indeed assured by the textbook results in SDEs, as included in the Corollary below.
x;t2D
Proof. (Karatzas and Shreve, 1991, p. 145).
In the next section, we will address the MTD model in more concrete context, so as to verify and validate the applicability of the modeling. First we examine the MTD with only the drift function f (i.e. the disturbance function g 0). Indeed, the drift derived therein satis®es the conditions (a) and (b) above (with g 0). Then in Section 4 we obtain an MTD model by amending a disturbance. One can verify that the SDE of the MTD model in Section 4 also satis®es the conditions of Theorem 1. 3. The drift-only MTD model and a solved example With zero disturbance, the value process Xs is deterministic, and from (2.4) we write:
oH X_ t f Xt ; ut ; t state; ok ii Xt0 x > 0 initial state; i
oH ou Xt ; ut ; t ÿlt ÿk_ e oX oXt of Xt ; ut ; t k costate; oXt oV Xt ; h terminal condition; iv kth oXt oV Xt ; h 0 v H Xt ; ut ; kt ; h oh free terminal condition; vi H Xt ; u^t ; k^t ; t max H Xt ; ut ; k^t ; t iii
u
optimality: 3:2
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The Hamiltonian conditions of (3.2) are not necessarily sucient. However if the functions u and f are both concave, then conditions of (3.2) are both necessary and sucient (see Mangasarian (1966) and Kamien and Schwartz (1978) for details; further generalized sucient conditions are obtained by Arrow and Kurz (1970)). In principle, hg of a drift-only MTD the optimal control f^ ut ; ^ can be obtained by solving the Hamiltonian equations given in (3.2), provided that proper function forms are speci®ed. Let us consider the valuation of a manufacturing technology in terms of its market potential, as a way to examine applicable forms of the drift function. At the start of the technology transfer (t 0), the total market potential of the underlying technology is estimated as x 0 < x < 1. Let Xt be a forecast at the current time t P 0 for the market potential of the technology over the rest of its life cycle. Let ut be a one-dimensional control variable, such as production rate. Corresponding to the rate of technology obsolescence, the market potential is decreasing at a rate of r Xt ; ut ; t (e.g. units of output per year). At the same time, value depletion is compensated by the learning eect in terms of knowledge, skill, competition, new developments, etc., denoted by n Xt ; ut ; t (a proportion). We shall elaborate more on r and n in the solved example as illustrated later in this section. Note that r and n are general R-valued functions. Thus during the next in®nitesimal time interval dt, the marginal depletion of the market potential, denoted by ÿdXt , can be expressed as ÿdXt r Xt ; ut ; t dt n Xt ; ut ; t dXt ; where the ®rst term on right-hand side is the timebased depletion, and the second term represents the volume-based learning eect. Rearranging the terms, we have r Xt ; ut ; t dt: dXt ÿ 1 n Xt ; ut ; t
3:3
Then the drift function under the circumstance is derived 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 not contain an explicit term of a ®xed market size. To verify the applicability of the derived drift function, we devote the rest of this section to solving a sample problem of the drift-only MTD. 3.1. A solved example of drift-only MTD problem The drift function in the example to be discussed in this section is derived from a project entitled ``Transfer and engineering of G&L FMC technology for industrial sewing machine production at Jiangwan (Shanghai)''. This is an application of ¯exible manufacturing technology to an automated batch production of fast-thread industrial sewing machines for China and Asia markets. The product mix consists of six types of applications from canvas-sewing to shoe-making, and the production level is continuously adjusted according to market changes. The FMC system is manufactured and engineered by Giddings & Lewis. Since the start of production in 1993 after a contracted 3-month of installation and setup, the production level has been regulated in proportion to the estimated market share. With Xt as potential market share and ut as proportional production level (controllable), an annual production rate can be determined as ut Xt . The selling price has been fairly stable at p ($/unit). The unit production cost has been a function of the annual production rate, and can be calculated as cut Xt , where c is a cost conversion coecient. The current data on timestandards and system utilization suggest a typical ®xed percentage of marginal learning. With constant b as a ®xed learning percentage, the volumebased learning eect can be expressed as bXt . Thus, by writing r Xt ; ut ; t ut Xt and n Xt ; ut ; t bXt , the drift function of (3.4) can be expressed as f Xt ; ut ; t ÿ
u t Xt : 1 bXt
3:5
By (3.5), drift herein describes the depletion of potential production output (e.g. in units of output per year). The terminal payo V Xt ; t here represents the salvage of the production capital at time t. In reference to the so-called exponential depreciation
J. Liu / European Journal of Operational Research 124 (2000) 601±614
607
(Dixit and Pindyck, 1994; Kamien and Schwartz, 1991), an example of salvage function is
With Lemma 1, the trajectory of costate k can be obtained from (iii) of (3.2) as shown below.
V Xt ; t V0 k ÿ cXt
Lemma 2 (optimal costate trajectory). Under the optimal production control u^, the costate trajectory ^k is q ^k 1 bX^ peÿlt kc ; 3:9
3:6
with c 2 1; 1 as the capital depreciation rate and k P 1 as a salvage factor. The ®nal salvage value drops to V0 k ÿ 1 at the natural ending time. Since the unit pro®t is p ÿ cut Xt and the production rate (units per year) is given by ut Xt = 1 bXt , we can write the unit pro®t u Xt ; ut ; t as u Xt ; ut ; t p ÿ cut Xt
u t Xt : 1 bXt
3:7
Next, with the speci®c function forms given above, we obtain the optimal control f^ u; ^ hg of the drift-only MTD by solving the Hamiltonian equations in (3.2). In addition to the production h, a complete optimal level u^t and stopping time ^ solution of (3.2) will include the costate ^ kt and the state trajectory (herein valuation curve) X^ t , all of which are obtained and presented below in a slate of four Lemmas. In what follows, we omit the index t when convenient. Lemma 1 (optimal production control). Let X^ ; u^; ^k be an optimal solution of drift-only MTD model. Then optimal production control u^ is determined as ( ) p ÿ ^kelt ;0 ; 3:8 u^ max 2cX^ Proof. By the Hamiltonian condition (vi) in (3.2), we have Hu0 ÿ cX eÿlt
uX 1 bX
p ÿ cuX e
ÿlt
X 0: ÿ k 1 bX
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 generates (3.8).
where kc ÿ
q ^ ^ 1 bX^ h ÿ peÿlh ÿ V0 ln c cX h :
3:10
Proof. By (iii) of (3.2), we have ÿk_ ÿ cueÿlt
uX 1 bX
p ÿ cuX eÿlt ÿ k
u 1 bX
2
:
Rearranging the above yields k_ ÿ
u 1 bX
2
k
! cu2 X cu2 X pu ÿ eÿlt : 1 bX 1 bX 2 1 bX 2 3:11
Using the state Eq. (3.3), the following can be derived ÿuX : k_ k0X X_ k0X 1 bX Substituting the above into (3.11), we have ÿ
uX u k0X k 2 1 bX 1 bX
cu2 X cu2 X pu ÿ 2 2 1 bX 1 bX 1 bX
! elt :
Factoring out common terms in the above equation yields 1 k ÿ X k0X ÿ 1 bX cuX p ÿ eÿlt : cuX 1 bX 1 bX
3:12
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J. Liu / European Journal of Operational Research 124 (2000) 601±614
By (3.8) of Lemma 1, we have cuX eÿlt 12 peÿlt ÿ k which will generate the following from (3.12) by substitution: 1 k ÿ X k0X ÿ 1 bX 1 1 p 1 eÿlt : peÿlt ÿ k ÿ 2 1 bX 1 bX Combining and then canceling the common terms in the above equation, we have b pb kÿ eÿlt : 3:13 1 bX 1 bX p It can be veri®ed that k 1 bX is a solution for the corresponding homogeneous equation of (3.13). Then solving the ordinary dierential (3.13) with terminal conditions in (3.2) shall conclude the Lemma.
2k0X ÿ
Lemma 3 can be concluded with the solution of (3.16), a simple ®rst-order ODE, and the details are therefore omitted. Lemma 4 (optimal stopping time). The optimal stopping time ^h satis®es the following equation: 1 ^ c^ u^h X^ ^h eÿlh V0 ln cX^h : c
3:17
Proof. First, from the salvage function of (3.6) we derived oV X ; h ÿ V0 ln ccXh X_ h : oh Thus, the terminal condition in (3.2), oV X ; h 0; oh generates the following:
H X ; u; k; h
Lemma 3 (optimal valuation curve). Under the optimal production control u^, the optimal valuation curve X^ is determined by the equation
1 ÿlh pe ÿ kh X_ th V0 lnccXh X_ th 0: 2 If X_ th 6 0 (i.e. Xth 6 0 by (3.5)), we have
1ÿ 2 1 lt 2kc lnz ÿ 4kc z z2 e X; b 2cl
peÿlh ÿ kh ÿ2V0 lnc cXh :
3:14
where
1ÿ 1 ; X 2k2c ln z0 ÿ 4kc z0 z20 ÿ b 2cl 1
pe From (3.8), it follows that 2cuX e with which (3.18) derives (3.17).
ÿlh
ÿ kh
The regeneration time ^h and the associated ending value X^h can be determined by solving (3.14) and (3.17) simultaneously. However, a closed form solution does not seem to be attainable yet. In principle, Lemmas 1, 2, 3 and 4 provide a complete set of optimal solutions for the drift-only MTD problem. Next, we discuss some useful properties.
1
z 1 bX^ 2 kc ;
z0 1 bX 2 kc ;
3:18 ÿlh
x Xt0 :
Proof. From (3.8) of Lemma 1, we have uX 2c1 peÿlt ÿ kelt with which the state equation (3.3) generating the following:
Proposition 1. ^k is positive and strictly increasing in t 2 0; ^h, and kc as in (3.10) is negative, i.e. kc < 0.
1 1 p ÿ kelt : X_ ÿ 2c 1 bX
Proof. By (3.11) in the proof of Lemma 2, we can write
3:15
Using (3.9) and substituting for k in (3.15), we have 1 1 kc 1 bX 2 3:16 elt : X_ 1 bX 2c
oH uk cu2 X eÿlt ÿ pueÿlt cu2 X ÿlt e k_ ÿ 2 oX 1 bX 1 bX
u k cuX eÿlt ÿ peÿlt 1 bX
2
cu2 X ÿlt e : 1 bX
J. Liu / European Journal of Operational Research 124 (2000) 601±614
Using 2cuX eÿlt peÿlt ÿ k shown in (3.8), we derive k_
ÿcuX 1 bX cbu2 X 2
2
ueÿlt
1 bX
2
eÿlt :
cuX 1 bX 1 bX
2
ueÿlt
Clearly, k_ > 0 8t 2 0; s. Since it is given by (ii) of (3.2) that x Xt0 > 0, thus k > 0 at t 0 by (3.9). Since kh V0 ln 1ccXh > 0, then we can verify from (3.9) and (3.18) that p kc kh ÿ 1 bXh ÿ peÿlh p 1 ÿ 1 bXh ÿ 2V0 ln cXh < 0: c This concludes the Proposition 1.
Proposition 2. For t 2 0; ^ h, the optimal production level is determined by u^ p ÿ ^ kelt = 2cX^ > 0; and u^ is strictly increasing in t, speci®cally, 3:19
where p ÿ ^k0 ; 2cx p ^k0 1 bx p kc ;
u^0
3:20 3:21
^
V0 ln 1c cX X^ : u^^h cX^ ^ eÿl^h
Using (3.9), it arrives that o ^ uX^ 1 ÿ ÿ lpeÿlt elt ÿ l^kelt ot 2c l p ÿ kelt > 0; 8t 2 0; ^h: 2c Thus, for any X^ , the production rate u^X^ is strictly increasing in t. Since X is decreasing in t (see (3.5)), it thus follows that u^ is strictly increasing 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 and 2 is an interesting optimal regularity as formally summarized in the following Proposition.
For the optimal production level u^, we obtain the following.
u^0 < u^ < u^^h ;
609
3:22
h
Proof. First, we show that u^ P 0 for all t 2 0; ^ h. From (3.8), it suces to show that p ÿ ^ kelt > 0 in t 2 0; ^h, which indeed holds since p ÿ ^ kelt is ÿlh strictly decreasing in t and pe ÿ kh > 0 by (3.18). Taking partial derivative of (3.8) with respect to t (2 0; ^h) yields ! o ^ uX^ 1 o^k lt ÿ e ÿ l^ kelt : ot 2c ot
Proposition 3. For the example problem of driftonly MTD, during the MTD period t 2 0; ^h the production should be kept at an increasing rate, and the underlying technology should be phased out at t ^h < 1 before its market potential vanishes (i.e. X^ ^h > 0). Proof. The optimality of an increasing production rate (^ uX^ ) is shown in Proposition 2. Note that the costate k measures the shadow price of the market potential (X). Thus, the Theorem follows immediately if the inequality ^kh > 0 holds, which is indeed the case as shown by Proposition 1. 4. Variational calculus formulation of the MTD problem Now, we consider the MTD model with both drift and disturbance as given by the SDE in (2.4), which can be characterized as an optimal stochastic control problem joint with optimal stopping. We recall the MTD model of (2.4) for convenience, MT Model: max p x; u; h; t u;h 8 < Ex;u;h;t ÿ Z Xr ; reÿl rÿt V Xh ; heÿl hÿt : 9 Zh = u Xs ; us ; seÿl sÿt ds r 6 h 6 s ; t
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Theorem 2. Let Xs be de®ned by (2.1) (in Ito's sense). Suppose that Theorem 1 holds. In addition, assume that
s:t: dXs f Xs ; us ; s ds g Xs ; s dWs ; s 2 t; T t P 0; x Xst > 0; us 2 U Rm : In fact, Z x; t represents the cost (or a fair price) of a new technology with a market potential of x, while V x; t is the payo if the technology is salvaged with a remaining market size of x. Clearly, we must have Z x; t > V x; t in order to have a meaningful case of technology diusion. hg, if exists, the pro®t Under an optimal policy f^ us ; ^ starts 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 is the pro®t generated from the technology in use and Z is the worth or price of a new one. In the meantime, we must have by optimality that p x; u^; ^h; t P V x; t, where V x; t represents a terminal payo if the MTD is terminated and the technology is salvaged. More precisely, on the one hand the diusion should continue if the expected pro®t is greater than the terminal payo (i.e. p x; u^; ^h; t > V x; t); on the other hand the diusion should be terminated immediately if the expected pro®t is no better than the terminal payo (i.e. p x; u^; ^h; t V x; t). With such, we de®ne a diusion set D as D f x; tj x > 0; t 2 0; T : V x; t < p x; u; h; t < Z x; tg:
4:1
Note that D is open and bounded. As de®ned, the MTD should continue if the system status at time t belongs to D. It has been identi®ed that a one-dimension stochastic control problem without optimal stopping can be represented by the so-called Hamilton±Jacobi equations of second-order ordinary dierential equations (ODEs). There have been many ways to derive such an H±J representation of the Maximum Principle for various problems including the optimal stopping (e.g. Bensoussan and Lions, 1982). For the MTD model which involves stochastic control joint with optimal stopping, we herein obtain augmented Hamilton± Jacobi equations of MTD model using the variational calculus in combination with applications of Ito's formula.
p 2 C 2 D;
u 2 L2 R 0; T U ;
4:2
where C 2 D is the space of all twice dierentiable functions on D of (4.1). Then an optimal policy f^ us ; ^hg of the MTD model satis®es the following augmented Hamilton±Jacobi equations: H±J Equations of MTD: ÿ
op _ x; t 0; ÿ B x; tp ÿ H p; p; ot
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 f x; t: p Zg; oD max f x; t: p V g;
4:6
1 o2 B x; t g2 x; t 2 ox 2 Brownian Differential Operator; 4:7
op _ x; t max f x; u; t ÿ lp u H p; p; u2U ox Hamiltonian Operator: 4:8 Proof. Theorem 1 and the Assumption (4.2) assure that an Ito's dierential is applicable (Karatzas and Shreve, 1991). Since t is known without regards to h and r is prescribed upon t, in combination with Ito's rule, we calculate the functional variation (as adopted in Gregory and Lin, 1992) of the total pro®t with control u ®xed, denoted by dpu h; x; t, as follows: dpu x; h; t dpu x; hjt dpu tj x; h; where the ®rst term on the right-hand side denotes the functional dierential with t ®xed, while the
J. Liu / European Journal of Operational Research 124 (2000) 601±614
second term is the dierential when x; h are ®xed. With each ®xed t 2 0; T , applying Ito's formula in a similar manner as by Arnold, 1992 we have dpu x; hjt op 1 2 o2 p op g x; h 2 f x; u; h dh oh 2 ox ox op 1 2 o2 p op g x; t 2 f x; u; t dt; ÿ ot 2 ox ox where a change of variable h ÿt is applied in the second equality above. Noting that p x; u; h; t by (2.4) involves a functional integral with t as a parameter, by variational calculus (e.g. Gregory and Lin, 1992) we have dpu tj x; h
d Ex;u;h;t ÿ Z Xr ; reÿl rÿt : dt V Xh ; heÿl hÿt Zh
Ex;u;h;t u Xs ; us ; se
ÿl sÿt
! ds
t
ÿ ÿ lEx;u;h;t Z Xr ; reÿl rÿt dt ÿ lEx;u;h;t V Xh ; heÿl hÿt dt 0h 1 Z @ lEx;u;h;t u Xs ; us ; seÿl sÿt dsA dt ÿ
t
ÿ Ex;u;h;t u Xt ; ut ; t eÿl tÿt dt lp x; u; h; t ÿ u x; u; t dt: Including all the terms, we obtain dpu h; x; t op 1 o2 p ÿ ÿ g2 x; t 2 ot 2 ox op dt ÿ f x; u; t ÿ lp u ox op _ u; h; x; t dt ÿ ÿ A x; tp ÿ L p; p; ot _ u; h; x; t f x; u; t op ÿ lp u. By where L p; p; ox the Maximum Principle and variational calculus, at the optimum we have dpu 0 and L is
611
maximized, with which we conclude the proof of the Theorem by denoting n o _ x; t max L p; p; _ u; h; x; t : H p; p; u;h
The augmented H±J equations of MTD in Theorem 2 involves a nonlinear dierential operator of the ®rst-order, the solution of which is quite involved and dicult. See Bensoussan and Lions, 1982 (Chapter 4) for a rigorous proof of the existence and uniqueness of a solution to the augmented H±J equations. In principle, the optimal control and optimal stopping time can be obtained by solving the H±J (4.3) and (4.5) which represent a free-boundary problem of a secondorder ODE. However, in most cases a closed form analytical solution of such a free-boundary problem is unattainable and thus numerical solution methods are usually called for. The details of the solution methods are beyond the scope of this paper and therefore are deferred to separate discussions. As an application of Theorem 2, let us consider an example of the MTD problem by amending a disturbance to the example of the drift-only MTD discussed in the previous section. For the disturbance of the example, we use g p gXs (g > 0 is scaling factor) as adopted in Pindyck (1993). With f as given by (3.5), the stochastic process de®ned by (2.1) is stationary since drift and disturbance are implicit of time. As discussed in Pindyck (1993) and Chi et al. (1997), a stationary disturbance should have the following characteristics: lim x!0 g x 0 and og=ox > 0. With f and g as selected, the Hamiltonian, as de®ned below, is dierentiable with respect to the control u 2 0; 1. That is, at optimum we have oH 0; ou where
op ÿ lp u: ox At the ready-to-start state xr , we adopt an engineering cost of the technology Z xr as an initial boundary condition. Based on the actual calculation used in the Jiangwan project, we use a quadratic cost function of initial market size xr , _ x; t f x; u; t H p; p;
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Z xr I axr bx2r , where I represents a ®xed setup cost, a and b are cost coecients. The salvage function is the same as given by (3.6). As such, the diusion set D becomes D f x: V x < p < Z xg: Now, the MTD problem can be stated as to ®nd a pro®t function p 2 C 2 D that satis®es V < p < Z and solves the diusion equation, with Z xr as the starting point and with V xh as the ending point. Since the example MTD model is stationary, we have op=ot 0, and we can set initial time to zero (i.e. t 0) for convenience. Thereby, the optimality conditions of the example MTD model can be warranted from Theorem 2 as below: H±J equations of example MTD model: 1 o2 p ux op ux gx 2 ÿ lp p ÿ cux 0 2 ox 1 bx ox 1 bx diffusion equation; oH o ux op ux ÿ lp p ÿ cux 0 ou ou 1 bx ox 1 bx Hamiltonian maximum; initial state; Xs0 x0 > 0 ^ 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±J equations, we construct two more terminal conditions using the so-called smooth-pasting method Dixit and Pindyck (1994) namely, op xr ; u^; h oZ xr oxr oxr
and
op xh ; u^; h oV xh : oxh oxh
Thereupon, the existence of a solution (in the strong sense) is assured by verifying with Bensoussan and Lions (1982, Theorem 1.1, p. 497). Thus, a much tractable solution algorithm of the example MTD problem can be obtained from Theorem 2. Dirichlet solution of example MTD problem: 1. Identify u^ from the Hamiltonian equation: ox ux op u^ u 2 U : ou 1 bx ox ux 0 : ÿ p ÿ cux 1 bx
2. With identi®ed control u^ as a parameter, determine p x; u, xr and xh by solving the following one-dimension Dirichlet problem: 1 o2 p ux op gx 2 ÿ lp 2 ox 1 bx ox ux 0; p ÿ cux 1 bx
x 2 D;
s:t: boundary conditions op xr ; u oZ xr ; oxr oxr op xh ; u oV xh ; p xh ; u V xh ; oxh oxh Z x I ax bx2 ; V x V0 kcx : p xr ; u Z xr ;
3. The optimal solution is obtained as fp; u^; xr ; xh g. Note that fxr ; xh g de®nes a start/stop policy that is practical and easy to implement: start to adopt the technology whenever its market value reaches xr or higher; while the technology should be discontinued as soon as its value falls to xh or below. Numerical methods are most likely to be needed for the steps 1 and 2 of the Dirichlet solution. Computational details are deferred to separate reports. We conclude this section with the proof of the optimality of an early phase-out policy for the example MTD model with the presence of disturbance. Proposition 4. For the example MTD model, if there exists a non-zero salvage value of the technology at the natural ending time (i.e. V 0 > 0), then it is optimal to phase out the technology before its market value vanishes (i.e. xh > 0). Proof. Let D f x P 0: p V g. Since V 2 C 1 R, for every x 2 D the salvage function V shall solve the diusion equation, that is, 1 ux 1 2 V0 ln cx ÿ lV0 k ÿ cx ÿ gxV0 ln c cx 2 1 bx c ux p ÿ cux 1 bx 0:
J. Liu / European Journal of Operational Research 124 (2000) 601±614
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 suggest to phase out the technology earlier than its natural ending time, either with or without uncertain disturbances. 5. Concluding remarks Herein, a manufacturing technology diusion (MTD) model is developed to characterize the depletion process of the valuation of an underlying technology over the phases of technology transfer and operation. The MTD model adopts an Ito's SDE as its state equation, so as to address two necessary aspects of a technology valuation process: the estimation of what is at stake, and the anticipation of future changes. The proposed MTD model represents a stochastic control problem joint with optimal stopping. The model is validated by an example MTD of ¯exible manufacturing. Under the speci®cs of the example, the optimal policy obtained from the MTD model suggests to phase out the underlying technology before its market value vanishes. The solution of an MTD model involves nonlinear free boundary problems of second order ODEs. Eective computation algorithms pose immediately as a future research topic. It is of our interest to extend the MTD model to the case of a Martingale as the initial state. Furthermore, a multidimensional MTD is another future topic of our particular interests. References Arnold, L., 1992. Stochastic Dierential 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 University 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.
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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 diusion 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 salvageability. 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 Uncertainty, Princeton University Press, Princeton, NJ. Dockner, E., Jorgensen, S., 1988. Optimal advertising policies for diusion models of new product innovation in monopolistic 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 Processes 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 diusion 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 manufacturing 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. Sucient conditions for the optimal control of nonlinear systems. SIAM Journal of Control 4, 139±152.
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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 replacement purchases in new product diusion 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 diusion of new technology: A game theoretic approach. Review of Economic Studies 48, 395±405.
Rogers, E.M., 1995. The Diusion of Innovations, 4th ed., The Free Press, New York. Rosenberg, J., 1972. Factors aecting the diusion of technology. 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 Development. Free Press, New York.
www.elsevier.com/locate/dsw
Theory and Methodology
On the dynamics of stochastic diusion of manufacturing technology 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 diusion (MTD), a close relative of demand diusion in marketing, refers to the transition 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: regularity (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 differential equations (SDEs), to overcome the limitations of invariance (e.g. a ®xed market size and the absence of disturbance) as suered 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 augmented 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 diusion; Optimal stopping; Stochastic control
1. Introduction Three distinct phases in the life cycle of a manufacturing technology have long been identi-
*
Tel.: 1-414-229-4235; fax: 1-414-229-6957. E-mail address: [email protected] (J. Liu).
®ed: research and development (R&D), transfer/ commercialization, and operation/regeneration (Cook and Mayes, 1996; Rogers, 1995). A ®rm's competitive strategic and operational position is critically dependent upon an accurate and objective understanding of the life cycle of underlying technology (Burgelman et al., 1995). A key characterization of the life cycle has been the transition
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
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of technology's economic value. For instance, the R&D of technology presents a phase of value build-up through technological innovations, while the transfer and operation of technology entail a value depletion process in accordance with inevitable technology obsolescence (Rosenberg, 1972). Manufacturing technology diusion (MTD) refers to the transition of economic value during the transfer and operation phases of a technology life cycle. There is a keen resemblance between technology and market diusions in that both cast a dynamic depletion process of their respective economic values. The depletion process of market diusion has been studied using the widely-known product life cycle (PLC) models (e.g. Bass, 1969; Kalish, 1983; Horsky and Simon, 1983; Teng and Thompson, 1985; Dockner and Jorgensen, 1988; Klepper, 1996; literature on another relevant topic, business life cycles by the economic logistics models, can be found in Aulin, 1996; Thore, 1991; Nagurney et al., 1996; Flam and Ben-Israel, 1990). Despite its obvious similarity to marketing methodology, the modeling of manufacturing technology diusion is fundamentally dierent in several ways. First, manufacturing technology is rooted in manufacturing activities (e.g. production control) as opposed to marketing alternatives (e.g. pricing and advertising). The valuation process associated with MTD is consistently in¯uenced by production decisions, such as whether or not to discontinue (or switch o) an extant production process, and what production level should be targeted, etc. As such, the transformative nature of manufacturing derives an immediate distinction between technology and market diusions. On the one hand, there hardly exists an MTD without exogenous regulations, while on the other hand it can be still reasonably argued (e.g. by Bass et al., 1994) as to why a PLC model will work without control. Second, the MTD will inevitably endure uncertain in¯uences and disturbances due to technological evolution and innovation. Therefore, it is no longer a justi®able modeling assumption to tolerate the limitations of invariance suered by a typical PLC model (e.g. the inelasticity of market size and the absence of disturbance). The PLC research bases its analysis on the
mean value (or the passive net present value) of market potential, and is known for its limitations of ®xed total market and deterministic variations of market potential. Such limitations and their alleviative solutions have long been discussed and addressed (e.g. Chatterjee and Eliashberg, 1990; Mesak and Berg, 1995; Liu and Chi, 1997; Feichtinger et al., 1994 for more details and extensive updates). In brief, MTD entails a stochastic value-depletion process where the uncertain status of the process (e.g. market potential) is continuously observed and regulated. Obviously, a rigorous modeling of MTD must give an accurate account for the aforementioned aspects: regularity and uncertainty. This paper attempts to achieve that. The methodology of stochastic dierential equations (SDEs) bodes well for the analysis of similar problems in the areas of physics, engineering, and lately ®nance. SDE modeling has also been used to study the R&D phase of technology's life cycle (Reinganum, 1981; Kulatilaka, 1988; Dixit and Pindyck, 1994; Kamrad, 1995; Chi et al., 1997). But as far as we can ascertain from current research literature, SDE modeling has not yet been applied to MTD. In this paper, we propose using an Ito's SDE formulation to characterize the in®nitesimal dynamics of the valuation process. Assuming full access to the past observable information up to current time t, the changes of the valuation in the next dt time consist of two components: a depletion rate prescribed from the past (drift), and a superimposed Wiener dierential (disturbance). Both drift and disturbance can be rather general and can include control variables such as production rate. The proposed SDE formulation overcomes the limitations of invariance by not only allowing random disturbances, but adopting a variable market as well. Furthermore, a controllable stopping time is introduced to allow a managerial decision to terminate the use of the technology before its PLC-de®ned natural ending time; that is, before total depletion of its underlying values. Life-cycleshortening has been pervasive worldwide (Wheelwright and Clark, 1995; Mans®eld, 1988), and it has become increasingly common to
J. Liu / European Journal of Operational Research 124 (2000) 601±614
discontinue a manufacturing process before its natural ending time. With the SDE characterization of the underlying depletion process, a general MTD model under production regulations is then formulated as a stochastic control problem with free terminal time (stopping time). The objective function of the MTD model is the expected total pro®t as derived from the total realization of the technology's value over the duration of the regulated MTD, including terminal payos, if applicable. The MTD seeks an admissible regulation policy joint with an exit (or stopping) decision that maximizes expected total pro®t. It is interesting to note that the MTD model becomes a basic optimal stopping problem when no other control variables are involved. With reference to a speci®c application of ¯exible manufacturing, it is shown in this paper that the optimal stopping policy suggests phasing out the underlying technology before its value vanishes, which coincides with the aforementioned life-cycle-shortening phenomena. Although the MTD model is mathematically suited for n-dimensional SDEs, we con®ne our analysis to the one-dimensional case for the sake of keeping within a realistic context. By variational calculus in combination with applications of Ito's formula, we derive augmented Hamilton±Jacobi variational equations for the MTD model, the solution of which presents a nonlinear free boundary problem of second-order ordinary differential equations (ODEs). Thereby, we show an analytical characterization of the MTD model to be the following: for a system with its state dynamics expressed by Ito's SDEs, and given a starting boundary and an ending boundary (both real functions on Rn ), we are to ®nd a twice differentiable functional pro®t that starts on the starting boundary and ends on the ending boundary, and in between satis®es the augmented HÿJ equations. Addressing such a problem is highly challenging and dicult. With the speci®cs of an application in ¯exible manufacturing, we derive a one-dimension Dirichlet solution method which is mathematically more tractable. Substantial theoretical and practical bene®ts can be realized by applying this well-known mathematical theory to the MTD problem.
603
Since drift (the expected regularity) is one of the model's two basic elements, it is critical to obtain a suitable drift function for the MTD. To achieve this, we ®rst derive a valuation forecast as the drift function and then we address in detail the analysis of the drift-only MTD (i.e. zero disturbance), so as to verify the applicability of the drift function. The rest of the paper is organized as follows. In Section 2 we introduce the SDE representation of the MTD process and construct the MTD model. Section 3 is devoted to the derivation of a drift function for the MTD model. A sample problem of the drift-only MTD is then solved, and an early phase-out policy is proved to be optimal. In Section 4 we analyze the regular MTD model, which includes both drift and disturbance. We obtain the Hamilton±Jacobi variational representation of the Maximum Principle for the MTD model. The solution of a sample stationary MTD problem (i.e. drift and disturbance do not depend on t) is then analyzed, and the optimality of the early phase-out policy is also obtained.
2. The MTD model Suppose that the process of adopting a new manufacturing technology starts at a known time t (0 6 t 6 T < 1). In a probability space X; F; P , assume that we have full access to all necessary observable information from the past up to current time s P t. Such sets of progressive information are expressed by an increasing family Fs of sub-r-algebras of F (see Bensoussan and Lions, 1982; Karatzas and Shreve, 1991 for mathematical details), based on which the underlying valuation process, denoted by fXs ; s P tg, is measured (i.e. fXs ; s P tg is Fs -measurable). Furthermore, the value depletion process fXs ; s P tg is de®ned as an R-valued Ito process with the following stochastic dierential (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 an admissible domain U Rm , real mapping f : Rm1 0; 1 ! R is termed drift, real mapping g:
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R 0; 1 ! R is termed disturbance, and dWs is the dierential of a one-dimension Wiener process. A non-random initial valuation of the technology is known at time t, denoted as x Xst > 0, which re¯ects that the underlying technology must have a positive value measured upon the best information available at the time, in order to start the transfer phase. Based on the information obtained by time t (such as required production capacity), the engineering and setup of the technology is planned and the necessary setup time is determined. Let r be a prescribed ready-time when setup is scheduled to be completed and the transfer (or operation) is ready to start (i.e. r P t and Xr P 0). By the nature of depletion, drift must have the characteristics of decreasing in time (i.e. of =ot 6 0). In (2.1), disturbance g is independent of the control us , which is representative of exogenous disturbances. Although (2.1) readily includes a control-dependent disturbance, its analysis is surprisingly intractable (see Bensoussan and Lions, 1982). The natural ending time, denoted by s, is a hitting time and is de®ned as the ®rst time when the technology value vanishes (see Karlin and Taylor, 1981 for more on hitting time). Formally we de®ne s inf fs P r: Xs 6 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 is non-random, it is assumed that fs > tg 2 Ft F. That is, the value of the technology which is strictly positive at t is stochastically decreasing in time, but cannot vanish in zero time right at the start. The stopping time, denoted by h, is the time to kill the MTD by discontinuing the use of technology before its value vanishes (i.e. r 6 h 6 s or equivalently the probability P fXh P 0g 1). Note that h is a control based on the observation of the latest state Xs and information Fs ( Ft ) such that the probability P fr 6 h 6 sjXs g 1. That is, after the MTD has started, it can be stopped (or killed) either automatically at the natural ending time s, or by control at the stopping time h before its natural ending. When appropriate we shall write h s as an implicit function of s and s Xs as an implicit function of Xs . For the objective function, we de®ne an expected total pro®t as follows:
p x; u; h; t 8 < Ex;u;h;t ÿ Z Xr ; reÿl rÿt V Xh ; heÿl hÿt : 9 Zh = u Xs ; us ; seÿl sÿt ds r 6 h 6 s ; ; t
2:3 where pro®t rate u, setup cost Z, and terminal payo V are all R-valued functions, and l is an applicable interest rate. The notation Ex;u;h;t represents the expected value given x, u, h, and t, and Efjh 6 sg gives the expectation conditioned on h 6 s. Then the MTD model can be de®ned as: ®nd a control u^s and a stopping time ^h (s 2 t; ^h), so that the expected total pro®t de®ned by (2.3) is maximized. That is, MTD Model: max p x; u; h; t u;h 8 < Ex;u;h;t ÿ Z Xr ; reÿl rÿt V Xh ; heÿl hÿt : 9 Zh = u Xs ; us ; seÿl sÿt ds r 6 h 6 s ; t
s:t: dXs f Xs ; us ; s ds g Xs ; s dWs ; s 2 t; T ; t P 0; x Xst > 0; us 2 U Rm : 2:4 As de®ned by (2.4), the MTD model is a stochastic control system joint with optimal stopping subject to an Ito's SDE as the state equation with a non-random positive initial state. Whether an MTD model is properly de®ned will depend critically on the existence of a solution to the SDE of (2.1). Omitting the proof and mathematical details, we present below the theorem of existence and uniqueness of solution to a standard Ito's SDE (i.e. the SDE as de®ned in (2.1), but without the control us ). Readers who are interested in further mathematical details are referred to the SDE textbooks (e.g. Chapter 6 of
J. Liu / European Journal of Operational Research 124 (2000) 601±614
Drift-only MTD Model:
Arnold, 1992; Appendix D of Fleming and Soner, 1993). Theorem 1 (existence and uniqueness). For an Ito's stochastic dierential 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, j f x; t ÿ f y; tj jg x; t ÿ g y; tj 6 K j x ÿ y j, and (b) (restrictive growth) 8s 2 t; T , x 2 R, 2 2 j f x; tj jg x; tj 6 K 2 1 j xj2 ; then Eq. (2.1) has on t; T a unique R-valued solution Xs , continuous with probability 1, that satis®es the initial condition Xst x.
605
max p x; u; h V Xh ; he
ÿlh
u;h
s:t: X_ t f Xt ; ut ; t;
Zh
u Xt ; ut ; teÿlt dt
0
t P 0;
x Xt0 > 0: 3:1
Corollary 1. Suppose that for an open and bounded domain D R 0; T , we have
In (3.1), the initial time t and the initial cost Z are both set to zero for convenience and the differential of the valuation process X_ t is now given by an ordinary dierential equation. The drift-only MTD model determines a pro®t-maximizing control u^t , together with an optimal stopping time ^h. That is, (3.1) is an optimal control problem with a free terminal condition. In the remainder of this section, the time index s associated with the state and control variables is replaced with t for convenience. Introducing a costate variable kt , a Hamiltonian function of the drift-only MTD can be constructed as
min g x; t > 0;
H Xt ; ut ; kt ; t u Xt ; ut ; teÿlt kt f Xt ; ut ; t:
D [ oD. Then Ex;t fsg < 1, 8 x; t 2 D. where D
Then, by the Maximum Principle, the solution f^ ut ; ^hg of the drift-only MTD model can be obtained from the following Hamiltonian conditions:
Another necessary component for an MTD model to be proper is a ®nite natural ending time s, which is indeed assured by the textbook results in SDEs, as included in the Corollary below.
x;t2D
Proof. (Karatzas and Shreve, 1991, p. 145).
In the next section, we will address the MTD model in more concrete context, so as to verify and validate the applicability of the modeling. First we examine the MTD with only the drift function f (i.e. the disturbance function g 0). Indeed, the drift derived therein satis®es the conditions (a) and (b) above (with g 0). Then in Section 4 we obtain an MTD model by amending a disturbance. One can verify that the SDE of the MTD model in Section 4 also satis®es the conditions of Theorem 1. 3. The drift-only MTD model and a solved example With zero disturbance, the value process Xs is deterministic, and from (2.4) we write:
oH X_ t f Xt ; ut ; t state; ok ii Xt0 x > 0 initial state; i
oH ou Xt ; ut ; t ÿlt ÿk_ e oX oXt of Xt ; ut ; t k costate; oXt oV Xt ; h terminal condition; iv kth oXt oV Xt ; h 0 v H Xt ; ut ; kt ; h oh free terminal condition; vi H Xt ; u^t ; k^t ; t max H Xt ; ut ; k^t ; t iii
u
optimality: 3:2
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The Hamiltonian conditions of (3.2) are not necessarily sucient. However if the functions u and f are both concave, then conditions of (3.2) are both necessary and sucient (see Mangasarian (1966) and Kamien and Schwartz (1978) for details; further generalized sucient conditions are obtained by Arrow and Kurz (1970)). In principle, hg of a drift-only MTD the optimal control f^ ut ; ^ can be obtained by solving the Hamiltonian equations given in (3.2), provided that proper function forms are speci®ed. Let us consider the valuation of a manufacturing technology in terms of its market potential, as a way to examine applicable forms of the drift function. At the start of the technology transfer (t 0), the total market potential of the underlying technology is estimated as x 0 < x < 1. Let Xt be a forecast at the current time t P 0 for the market potential of the technology over the rest of its life cycle. Let ut be a one-dimensional control variable, such as production rate. Corresponding to the rate of technology obsolescence, the market potential is decreasing at a rate of r Xt ; ut ; t (e.g. units of output per year). At the same time, value depletion is compensated by the learning eect in terms of knowledge, skill, competition, new developments, etc., denoted by n Xt ; ut ; t (a proportion). We shall elaborate more on r and n in the solved example as illustrated later in this section. Note that r and n are general R-valued functions. Thus during the next in®nitesimal time interval dt, the marginal depletion of the market potential, denoted by ÿdXt , can be expressed as ÿdXt r Xt ; ut ; t dt n Xt ; ut ; t dXt ; where the ®rst term on right-hand side is the timebased depletion, and the second term represents the volume-based learning eect. Rearranging the terms, we have r Xt ; ut ; t dt: dXt ÿ 1 n Xt ; ut ; t
3:3
Then the drift function under the circumstance is derived 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 not contain an explicit term of a ®xed market size. To verify the applicability of the derived drift function, we devote the rest of this section to solving a sample problem of the drift-only MTD. 3.1. A solved example of drift-only MTD problem The drift function in the example to be discussed in this section is derived from a project entitled ``Transfer and engineering of G&L FMC technology for industrial sewing machine production at Jiangwan (Shanghai)''. This is an application of ¯exible manufacturing technology to an automated batch production of fast-thread industrial sewing machines for China and Asia markets. The product mix consists of six types of applications from canvas-sewing to shoe-making, and the production level is continuously adjusted according to market changes. The FMC system is manufactured and engineered by Giddings & Lewis. Since the start of production in 1993 after a contracted 3-month of installation and setup, the production level has been regulated in proportion to the estimated market share. With Xt as potential market share and ut as proportional production level (controllable), an annual production rate can be determined as ut Xt . The selling price has been fairly stable at p ($/unit). The unit production cost has been a function of the annual production rate, and can be calculated as cut Xt , where c is a cost conversion coecient. The current data on timestandards and system utilization suggest a typical ®xed percentage of marginal learning. With constant b as a ®xed learning percentage, the volumebased learning eect can be expressed as bXt . Thus, by writing r Xt ; ut ; t ut Xt and n Xt ; ut ; t bXt , the drift function of (3.4) can be expressed as f Xt ; ut ; t ÿ
u t Xt : 1 bXt
3:5
By (3.5), drift herein describes the depletion of potential production output (e.g. in units of output per year). The terminal payo V Xt ; t here represents the salvage of the production capital at time t. In reference to the so-called exponential depreciation
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607
(Dixit and Pindyck, 1994; Kamien and Schwartz, 1991), an example of salvage function is
With Lemma 1, the trajectory of costate k can be obtained from (iii) of (3.2) as shown below.
V Xt ; t V0 k ÿ cXt
Lemma 2 (optimal costate trajectory). Under the optimal production control u^, the costate trajectory ^k is q ^k 1 bX^ peÿlt kc ; 3:9
3:6
with c 2 1; 1 as the capital depreciation rate and k P 1 as a salvage factor. The ®nal salvage value drops to V0 k ÿ 1 at the natural ending time. Since the unit pro®t is p ÿ cut Xt and the production rate (units per year) is given by ut Xt = 1 bXt , we can write the unit pro®t u Xt ; ut ; t as u Xt ; ut ; t p ÿ cut Xt
u t Xt : 1 bXt
3:7
Next, with the speci®c function forms given above, we obtain the optimal control f^ u; ^ hg of the drift-only MTD by solving the Hamiltonian equations in (3.2). In addition to the production h, a complete optimal level u^t and stopping time ^ solution of (3.2) will include the costate ^ kt and the state trajectory (herein valuation curve) X^ t , all of which are obtained and presented below in a slate of four Lemmas. In what follows, we omit the index t when convenient. Lemma 1 (optimal production control). Let X^ ; u^; ^k be an optimal solution of drift-only MTD model. Then optimal production control u^ is determined as ( ) p ÿ ^kelt ;0 ; 3:8 u^ max 2cX^ Proof. By the Hamiltonian condition (vi) in (3.2), we have Hu0 ÿ cX eÿlt
uX 1 bX
p ÿ cuX e
ÿlt
X 0: ÿ k 1 bX
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 generates (3.8).
where kc ÿ
q ^ ^ 1 bX^ h ÿ peÿlh ÿ V0 ln c cX h :
3:10
Proof. By (iii) of (3.2), we have ÿk_ ÿ cueÿlt
uX 1 bX
p ÿ cuX eÿlt ÿ k
u 1 bX
2
:
Rearranging the above yields k_ ÿ
u 1 bX
2
k
! cu2 X cu2 X pu ÿ eÿlt : 1 bX 1 bX 2 1 bX 2 3:11
Using the state Eq. (3.3), the following can be derived ÿuX : k_ k0X X_ k0X 1 bX Substituting the above into (3.11), we have ÿ
uX u k0X k 2 1 bX 1 bX
cu2 X cu2 X pu ÿ 2 2 1 bX 1 bX 1 bX
! elt :
Factoring out common terms in the above equation yields 1 k ÿ X k0X ÿ 1 bX cuX p ÿ eÿlt : cuX 1 bX 1 bX
3:12
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J. Liu / European Journal of Operational Research 124 (2000) 601±614
By (3.8) of Lemma 1, we have cuX eÿlt 12 peÿlt ÿ k which will generate the following from (3.12) by substitution: 1 k ÿ X k0X ÿ 1 bX 1 1 p 1 eÿlt : peÿlt ÿ k ÿ 2 1 bX 1 bX Combining and then canceling the common terms in the above equation, we have b pb kÿ eÿlt : 3:13 1 bX 1 bX p It can be veri®ed that k 1 bX is a solution for the corresponding homogeneous equation of (3.13). Then solving the ordinary dierential (3.13) with terminal conditions in (3.2) shall conclude the Lemma.
2k0X ÿ
Lemma 3 can be concluded with the solution of (3.16), a simple ®rst-order ODE, and the details are therefore omitted. Lemma 4 (optimal stopping time). The optimal stopping time ^h satis®es the following equation: 1 ^ c^ u^h X^ ^h eÿlh V0 ln cX^h : c
3:17
Proof. First, from the salvage function of (3.6) we derived oV X ; h ÿ V0 ln ccXh X_ h : oh Thus, the terminal condition in (3.2), oV X ; h 0; oh generates the following:
H X ; u; k; h
Lemma 3 (optimal valuation curve). Under the optimal production control u^, the optimal valuation curve X^ is determined by the equation
1 ÿlh pe ÿ kh X_ th V0 lnccXh X_ th 0: 2 If X_ th 6 0 (i.e. Xth 6 0 by (3.5)), we have
1ÿ 2 1 lt 2kc lnz ÿ 4kc z z2 e X; b 2cl
peÿlh ÿ kh ÿ2V0 lnc cXh :
3:14
where
1ÿ 1 ; X 2k2c ln z0 ÿ 4kc z0 z20 ÿ b 2cl 1
pe From (3.8), it follows that 2cuX e with which (3.18) derives (3.17).
ÿlh
ÿ kh
The regeneration time ^h and the associated ending value X^h can be determined by solving (3.14) and (3.17) simultaneously. However, a closed form solution does not seem to be attainable yet. In principle, Lemmas 1, 2, 3 and 4 provide a complete set of optimal solutions for the drift-only MTD problem. Next, we discuss some useful properties.
1
z 1 bX^ 2 kc ;
z0 1 bX 2 kc ;
3:18 ÿlh
x Xt0 :
Proof. From (3.8) of Lemma 1, we have uX 2c1 peÿlt ÿ kelt with which the state equation (3.3) generating the following:
Proposition 1. ^k is positive and strictly increasing in t 2 0; ^h, and kc as in (3.10) is negative, i.e. kc < 0.
1 1 p ÿ kelt : X_ ÿ 2c 1 bX
Proof. By (3.11) in the proof of Lemma 2, we can write
3:15
Using (3.9) and substituting for k in (3.15), we have 1 1 kc 1 bX 2 3:16 elt : X_ 1 bX 2c
oH uk cu2 X eÿlt ÿ pueÿlt cu2 X ÿlt e k_ ÿ 2 oX 1 bX 1 bX
u k cuX eÿlt ÿ peÿlt 1 bX
2
cu2 X ÿlt e : 1 bX
J. Liu / European Journal of Operational Research 124 (2000) 601±614
Using 2cuX eÿlt peÿlt ÿ k shown in (3.8), we derive k_
ÿcuX 1 bX cbu2 X 2
2
ueÿlt
1 bX
2
eÿlt :
cuX 1 bX 1 bX
2
ueÿlt
Clearly, k_ > 0 8t 2 0; s. Since it is given by (ii) of (3.2) that x Xt0 > 0, thus k > 0 at t 0 by (3.9). Since kh V0 ln 1ccXh > 0, then we can verify from (3.9) and (3.18) that p kc kh ÿ 1 bXh ÿ peÿlh p 1 ÿ 1 bXh ÿ 2V0 ln cXh < 0: c This concludes the Proposition 1.
Proposition 2. For t 2 0; ^ h, the optimal production level is determined by u^ p ÿ ^ kelt = 2cX^ > 0; and u^ is strictly increasing in t, speci®cally, 3:19
where p ÿ ^k0 ; 2cx p ^k0 1 bx p kc ;
u^0
3:20 3:21
^
V0 ln 1c cX X^ : u^^h cX^ ^ eÿl^h
Using (3.9), it arrives that o ^ uX^ 1 ÿ ÿ lpeÿlt elt ÿ l^kelt ot 2c l p ÿ kelt > 0; 8t 2 0; ^h: 2c Thus, for any X^ , the production rate u^X^ is strictly increasing in t. Since X is decreasing in t (see (3.5)), it thus follows that u^ is strictly increasing 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 and 2 is an interesting optimal regularity as formally summarized in the following Proposition.
For the optimal production level u^, we obtain the following.
u^0 < u^ < u^^h ;
609
3:22
h
Proof. First, we show that u^ P 0 for all t 2 0; ^ h. From (3.8), it suces to show that p ÿ ^ kelt > 0 in t 2 0; ^h, which indeed holds since p ÿ ^ kelt is ÿlh strictly decreasing in t and pe ÿ kh > 0 by (3.18). Taking partial derivative of (3.8) with respect to t (2 0; ^h) yields ! o ^ uX^ 1 o^k lt ÿ e ÿ l^ kelt : ot 2c ot
Proposition 3. For the example problem of driftonly MTD, during the MTD period t 2 0; ^h the production should be kept at an increasing rate, and the underlying technology should be phased out at t ^h < 1 before its market potential vanishes (i.e. X^ ^h > 0). Proof. The optimality of an increasing production rate (^ uX^ ) is shown in Proposition 2. Note that the costate k measures the shadow price of the market potential (X). Thus, the Theorem follows immediately if the inequality ^kh > 0 holds, which is indeed the case as shown by Proposition 1. 4. Variational calculus formulation of the MTD problem Now, we consider the MTD model with both drift and disturbance as given by the SDE in (2.4), which can be characterized as an optimal stochastic control problem joint with optimal stopping. We recall the MTD model of (2.4) for convenience, MT Model: max p x; u; h; t u;h 8 < Ex;u;h;t ÿ Z Xr ; reÿl rÿt V Xh ; heÿl hÿt : 9 Zh = u Xs ; us ; seÿl sÿt ds r 6 h 6 s ; t
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Theorem 2. Let Xs be de®ned by (2.1) (in Ito's sense). Suppose that Theorem 1 holds. In addition, assume that
s:t: dXs f Xs ; us ; s ds g Xs ; s dWs ; s 2 t; T t P 0; x Xst > 0; us 2 U Rm : In fact, Z x; t represents the cost (or a fair price) of a new technology with a market potential of x, while V x; t is the payo if the technology is salvaged with a remaining market size of x. Clearly, we must have Z x; t > V x; t in order to have a meaningful case of technology diusion. hg, if exists, the pro®t Under an optimal policy f^ us ; ^ starts 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 is the pro®t generated from the technology in use and Z is the worth or price of a new one. In the meantime, we must have by optimality that p x; u^; ^h; t P V x; t, where V x; t represents a terminal payo if the MTD is terminated and the technology is salvaged. More precisely, on the one hand the diusion should continue if the expected pro®t is greater than the terminal payo (i.e. p x; u^; ^h; t > V x; t); on the other hand the diusion should be terminated immediately if the expected pro®t is no better than the terminal payo (i.e. p x; u^; ^h; t V x; t). With such, we de®ne a diusion set D as D f x; tj x > 0; t 2 0; T : V x; t < p x; u; h; t < Z x; tg:
4:1
Note that D is open and bounded. As de®ned, the MTD should continue if the system status at time t belongs to D. It has been identi®ed that a one-dimension stochastic control problem without optimal stopping can be represented by the so-called Hamilton±Jacobi equations of second-order ordinary dierential equations (ODEs). There have been many ways to derive such an H±J representation of the Maximum Principle for various problems including the optimal stopping (e.g. Bensoussan and Lions, 1982). For the MTD model which involves stochastic control joint with optimal stopping, we herein obtain augmented Hamilton± Jacobi equations of MTD model using the variational calculus in combination with applications of Ito's formula.
p 2 C 2 D;
u 2 L2 R 0; T U ;
4:2
where C 2 D is the space of all twice dierentiable functions on D of (4.1). Then an optimal policy f^ us ; ^hg of the MTD model satis®es the following augmented Hamilton±Jacobi equations: H±J Equations of MTD: ÿ
op _ x; t 0; ÿ B x; tp ÿ H p; p; ot
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 f x; t: p Zg; oD max f x; t: p V g;
4:6
1 o2 B x; t g2 x; t 2 ox 2 Brownian Differential Operator; 4:7
op _ x; t max f x; u; t ÿ lp u H p; p; u2U ox Hamiltonian Operator: 4:8 Proof. Theorem 1 and the Assumption (4.2) assure that an Ito's dierential is applicable (Karatzas and Shreve, 1991). Since t is known without regards to h and r is prescribed upon t, in combination with Ito's rule, we calculate the functional variation (as adopted in Gregory and Lin, 1992) of the total pro®t with control u ®xed, denoted by dpu h; x; t, as follows: dpu x; h; t dpu x; hjt dpu tj x; h; where the ®rst term on the right-hand side denotes the functional dierential with t ®xed, while the
J. Liu / European Journal of Operational Research 124 (2000) 601±614
second term is the dierential when x; h are ®xed. With each ®xed t 2 0; T , applying Ito's formula in a similar manner as by Arnold, 1992 we have dpu x; hjt op 1 2 o2 p op g x; h 2 f x; u; h dh oh 2 ox ox op 1 2 o2 p op g x; t 2 f x; u; t dt; ÿ ot 2 ox ox where a change of variable h ÿt is applied in the second equality above. Noting that p x; u; h; t by (2.4) involves a functional integral with t as a parameter, by variational calculus (e.g. Gregory and Lin, 1992) we have dpu tj x; h
d Ex;u;h;t ÿ Z Xr ; reÿl rÿt : dt V Xh ; heÿl hÿt Zh
Ex;u;h;t u Xs ; us ; se
ÿl sÿt
! ds
t
ÿ ÿ lEx;u;h;t Z Xr ; reÿl rÿt dt ÿ lEx;u;h;t V Xh ; heÿl hÿt dt 0h 1 Z @ lEx;u;h;t u Xs ; us ; seÿl sÿt dsA dt ÿ
t
ÿ Ex;u;h;t u Xt ; ut ; t eÿl tÿt dt lp x; u; h; t ÿ u x; u; t dt: Including all the terms, we obtain dpu h; x; t op 1 o2 p ÿ ÿ g2 x; t 2 ot 2 ox op dt ÿ f x; u; t ÿ lp u ox op _ u; h; x; t dt ÿ ÿ A x; tp ÿ L p; p; ot _ u; h; x; t f x; u; t op ÿ lp u. By where L p; p; ox the Maximum Principle and variational calculus, at the optimum we have dpu 0 and L is
611
maximized, with which we conclude the proof of the Theorem by denoting n o _ x; t max L p; p; _ u; h; x; t : H p; p; u;h
The augmented H±J equations of MTD in Theorem 2 involves a nonlinear dierential operator of the ®rst-order, the solution of which is quite involved and dicult. See Bensoussan and Lions, 1982 (Chapter 4) for a rigorous proof of the existence and uniqueness of a solution to the augmented H±J equations. In principle, the optimal control and optimal stopping time can be obtained by solving the H±J (4.3) and (4.5) which represent a free-boundary problem of a secondorder ODE. However, in most cases a closed form analytical solution of such a free-boundary problem is unattainable and thus numerical solution methods are usually called for. The details of the solution methods are beyond the scope of this paper and therefore are deferred to separate discussions. As an application of Theorem 2, let us consider an example of the MTD problem by amending a disturbance to the example of the drift-only MTD discussed in the previous section. For the disturbance of the example, we use g p gXs (g > 0 is scaling factor) as adopted in Pindyck (1993). With f as given by (3.5), the stochastic process de®ned by (2.1) is stationary since drift and disturbance are implicit of time. As discussed in Pindyck (1993) and Chi et al. (1997), a stationary disturbance should have the following characteristics: lim x!0 g x 0 and og=ox > 0. With f and g as selected, the Hamiltonian, as de®ned below, is dierentiable with respect to the control u 2 0; 1. That is, at optimum we have oH 0; ou where
op ÿ lp u: ox At the ready-to-start state xr , we adopt an engineering cost of the technology Z xr as an initial boundary condition. Based on the actual calculation used in the Jiangwan project, we use a quadratic cost function of initial market size xr , _ x; t f x; u; t H p; p;
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J. Liu / European Journal of Operational Research 124 (2000) 601±614
Z xr I axr bx2r , where I represents a ®xed setup cost, a and b are cost coecients. The salvage function is the same as given by (3.6). As such, the diusion set D becomes D f x: V x < p < Z xg: Now, the MTD problem can be stated as to ®nd a pro®t function p 2 C 2 D that satis®es V < p < Z and solves the diusion equation, with Z xr as the starting point and with V xh as the ending point. Since the example MTD model is stationary, we have op=ot 0, and we can set initial time to zero (i.e. t 0) for convenience. Thereby, the optimality conditions of the example MTD model can be warranted from Theorem 2 as below: H±J equations of example MTD model: 1 o2 p ux op ux gx 2 ÿ lp p ÿ cux 0 2 ox 1 bx ox 1 bx diffusion equation; oH o ux op ux ÿ lp p ÿ cux 0 ou ou 1 bx ox 1 bx Hamiltonian maximum; initial state; Xs0 x0 > 0 ^ 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±J equations, we construct two more terminal conditions using the so-called smooth-pasting method Dixit and Pindyck (1994) namely, op xr ; u^; h oZ xr oxr oxr
and
op xh ; u^; h oV xh : oxh oxh
Thereupon, the existence of a solution (in the strong sense) is assured by verifying with Bensoussan and Lions (1982, Theorem 1.1, p. 497). Thus, a much tractable solution algorithm of the example MTD problem can be obtained from Theorem 2. Dirichlet solution of example MTD problem: 1. Identify u^ from the Hamiltonian equation: ox ux op u^ u 2 U : ou 1 bx ox ux 0 : ÿ p ÿ cux 1 bx
2. With identi®ed control u^ as a parameter, determine p x; u, xr and xh by solving the following one-dimension Dirichlet problem: 1 o2 p ux op gx 2 ÿ lp 2 ox 1 bx ox ux 0; p ÿ cux 1 bx
x 2 D;
s:t: boundary conditions op xr ; u oZ xr ; oxr oxr op xh ; u oV xh ; p xh ; u V xh ; oxh oxh Z x I ax bx2 ; V x V0 kcx : p xr ; u Z xr ;
3. The optimal solution is obtained as fp; u^; xr ; xh g. Note that fxr ; xh g de®nes a start/stop policy that is practical and easy to implement: start to adopt the technology whenever its market value reaches xr or higher; while the technology should be discontinued as soon as its value falls to xh or below. Numerical methods are most likely to be needed for the steps 1 and 2 of the Dirichlet solution. Computational details are deferred to separate reports. We conclude this section with the proof of the optimality of an early phase-out policy for the example MTD model with the presence of disturbance. Proposition 4. For the example MTD model, if there exists a non-zero salvage value of the technology at the natural ending time (i.e. V 0 > 0), then it is optimal to phase out the technology before its market value vanishes (i.e. xh > 0). Proof. Let D f x P 0: p V g. Since V 2 C 1 R, for every x 2 D the salvage function V shall solve the diusion equation, that is, 1 ux 1 2 V0 ln cx ÿ lV0 k ÿ cx ÿ gxV0 ln c cx 2 1 bx c ux p ÿ cux 1 bx 0:
J. Liu / European Journal of Operational Research 124 (2000) 601±614
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 suggest to phase out the technology earlier than its natural ending time, either with or without uncertain disturbances. 5. Concluding remarks Herein, a manufacturing technology diusion (MTD) model is developed to characterize the depletion process of the valuation of an underlying technology over the phases of technology transfer and operation. The MTD model adopts an Ito's SDE as its state equation, so as to address two necessary aspects of a technology valuation process: the estimation of what is at stake, and the anticipation of future changes. The proposed MTD model represents a stochastic control problem joint with optimal stopping. The model is validated by an example MTD of ¯exible manufacturing. Under the speci®cs of the example, the optimal policy obtained from the MTD model suggests to phase out the underlying technology before its market value vanishes. The solution of an MTD model involves nonlinear free boundary problems of second order ODEs. Eective computation algorithms pose immediately as a future research topic. It is of our interest to extend the MTD model to the case of a Martingale as the initial state. Furthermore, a multidimensional MTD is another future topic of our particular interests. References Arnold, L., 1992. Stochastic Dierential 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 University 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.
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