strategic options in re-engineering of a manufacturing system with uncertain completion time

Theory and Methodology

Strategic options in re-engineering of a manufacturing system withuncertain completion time

John Liu *, Dongqing Yao

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

Received 1 May 1997; accepted 1 April 1998

Abstract

Re-engineering of an extant manufacturing process entails dynamic engineering decisions and continuous value

build-up with a high degree of uncertainty in completion time. Due to such uncertainty, the underlying project often

times cannot reach its design target in time. Therefore, upon updated assessment of remaining time to completion, the

manager of the re-engineering project may opt to either continue or stop the project. A pre-determined terminal payo�

can be realized if the project is completed as planned, while partial payo� can be salvaged if the project is stopped short.

In this paper, we address such strategic options in re-engineering of a manufacturing system with reference to a realistic

manufacturing application. We formulate the strategic engineering option problem (SEOP) by an optimal stopping

model with an objective of maximizing expected total pro®t. We show that the optimal policy for SEOP is either a bang-

bang control or a two-band control, depending on the nature of the underlying value build-up process. The managerial

implications of such optimal policy and their applications are reported. The solution for an optimal option policy

translates to a second-order free-boundary problem (SEOP-FB) which in general renders no closed-form analytical

solutions. Numerical tests with real data are conducted. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Options; Stochastic control; Re-engineering; Free-boundary problems

1. Introduction

The paper is based on our recent study of realworld applications of reengineering of an extantmanufacturing system, including such cases asHarley±Davidson Motors and Eaton Corporation.Reengineering of a manufacturing system entailsdynamic capital investment and continuous value

build-up aiming at improved productivity andpro®tability, often with a pre-determined targetand goal. The time to completion, i.e., the timeneeded to achieve the goal of the project as plan-ned, inevitably involves a high degree of uncer-tainty due to all the unexpected problems duringthe course of the reengineering project such asengineering di�culties, equipment delivery delaysand human errors. At any given point in timeduring the project, the time to completion (or theremaining time to completion) is a translation of

European Journal of Operational Research 115 (1999) 47±58

* Corresponding author. E-mail: [email protected]

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 6 8 - 4

all the past information combined with estimatesof the future potential. While as time proceeds,new information with disturbance unfolds in arandom fashion, and so does the updated time tocompletion. Due to such uncertainty, the under-lying project often times cannot reach its designtarget in time and is simply better o� to stop shortwith some penalty.

Our motivation for this project was a recentconsulting job performed for a major electricalapparatus company. A system, called ``ThermalExpansion Valve Assembly System'', was imple-mented by this company to replace an outdatedtechnology that could not meet modern safetyregulations. The re-engineering of the currentmanual assembly process into an automated sys-tem required signi®cant expenditures, 70% ofwhich were capital investments and engineeringcontracts. The technical and engineering details asspeci®ed in the ®nal project proposal called forcontracting of the master assembly system to anengineering ®rm through bidding. To protectagainst the uncertainty, the contract contains, as arequired item, a penalty clause for unable to passthe acceptance test on time. The acceptance testthat must be witnessed by both sides is set to assesswhether the delivered system meets the technicalobjectives as speci®ed in the contract. As soon asthe project starts, the time to completion is con-tinuously updated according to all informationunfolded by then, with which an updated projectstatus is then assessed. With the updates, themanager of the project will then assess the merit ofthe project potentials in comparison with theoriginal goal. Upon updated assessment, themanager may opt to either continue or stop theproject. To continue means to further invest in theproject with a given cost rate (e.g., labor costs). Apre-determined terminal payo� can be realized ifthe project is completed as planned, while partialpayo� can be salvaged if the project is stoppedshort.

As often the case, the company has nurtured agroup of engineers and technical personnel that iscompetent to take on all the design and technicalaspects of the project. However the key challengeduring the whole project, from proposal throughdesign to engineering, has been the life cycle

costing and the timing which has a lot to do withhuman and societal factors, such as delivery de-lays, resistance to change, and learning, etc. Astrategic engineering option problem (SEOP) of are-engineering project is referred to as how themanager should decide at any given time whetherto continue the project with some cost or terminatethe project to realize its accumulated value. In thispaper, we formulate the SEOP by an optimalstopping model with an objective of maximizingthe expected total pro®t of the project. The SEOPsuggests that there exist certain regularities (orscenarios) in a value build-up process upon whichthe strategic options policy shall be constructed.We identi®ed three typical value build-up scenar-ios termed as convex, concave and S-shaped, re-spectively, in one of which a re-engineering projectwill fall. The convex scenario represents a valueprocess with an increasing build-up rate, i.e., thevalue is built up faster towards the completion(e.g., a space shuttle or a construction project).While the concave one is the case where the valueis built up faster at the beginning and slower to-ward the end. One of the examples is a softwareengineering project where it may take a long timeto debug in the ®nal stage, while usually does notadd much value to the system. The S-shaped sce-nario characterizes a combined and more realisticcase where the build-up is convex at the beginningand becomes concave towards the end (e.g., engi-neering of a manufacturing system usually falls inthis category). The analysis of SEOP derives thefollowing optimal options: an entry-based bang-bang control for the convex case, an exit-basedbang-bang control for the concave case, and anentry/exit-based two-band control for the S-shaped case.

By Maximum Principle for stochastic control,we obtained a set of optimality conditions forSEOP, based on which we showed that the solu-tion of SEOP translates to a second-order free-boundary problem, denoted as SEOP-FB. Ingeneral, SEOP-FB renders no closed-form ana-lytical solutions. We proved the existence of ananalytical series solution to SEOP-FB, and deriveda recursive expression of a general solution toSEOP-FB. Then, the solution of SEOP-FB can beobtained by solving a system of non-linear as-

48 J. Liu, D. Yao / European Journal of Operational Research 115 (1999) 47±58

ymptotic equations, based on which a computa-tional solution algorithm is developed. For a spe-cial case where the investment discount rate iszero, a closed-form analytical solution is obtained.

Our paper is organized as follows. Section 2includes a literature review on the subject. ThenSection 3 gives the detailed formulation of theSEOP model with reference to the ExpansionValve project. Section 4 presents SEOP-FB byMaximum Principle, and obtains the existence ofan analytical series solution to SEOP. Section 5derives the general solution to SEOP-FB based onwhich a computational solution algorithm is de-veloped. In Section 6, we give an analytical solu-tion to the case where the discount rate is zero.Section 7 elaborates on managerial implications asapplied to the Expansion Valve project, includinghow the optimal control can be implemented invarious value build-up scenarios. In this section,we also present numerical examples and examinenumerically the e�ects of changes in the variousparameters on the optimal stopping rule for theSEOP-FB.

2. Literature review

The SEOP model developed in this paperadopts a similar method as by Chi et al. (1997) andextends their work on investment projects to theapplications of manufacturing re-engineering. Inaddition, new results on theoretical properties andsolution methods are developed in the paper. TheSEOP is also related to the work of Roberts andWeitzman (1981) and Pindyck (1993) on the opti-mal control of R&D projects. A key feature ofSEOP in this paper is that it allows the project'spotential value to be partially realizable if termi-nated before completion, whereas most other in-vestment models assume the project to have novalue in the case of termination prior to comple-tion. This less restrictive assumption of ours makesthe model more realistic in the context of manu-facturing system reengineering. In addition, ourmodel's focus on the remaining time to completionas the underlying stochastic variable also di�ersfrom theirs. The stochastic variable in Roberts andWeitzman (1981) is the project's potential value (p

in our model), and the stochastic variable in Pin-dyck (1993) is the overall cost of investment whoserandom evolution is partly due to the uncertaintyabout the input cost of the investment.

In recent years, the topic of business processreengineering (BPR) has drawn growing researchinterests in a wide range of contexts, particularlyrelating to information technology (Brynjolfsson,1994; Davenport and Short, 1990; Davenport andStoddard, 1994), organization management(Dowling and McGee, 1994; Scheer, 1994), andquality improvement (Gitlow, 1990). The researchon information technology (IT) has centered onthe modeling of the process that is strongly ap-plication-dependent. Grover and Kehinger (1995)included in his book reports on useful tools andexamples of IT process modeling, such as regres-sion analysis, multidimensional modeling, andmeasurement scales. The research on the organi-zational aspect of BPR is closely related to infor-mation technology in terms of its problem focus.Brynjolfsson examined the interactions between ITand organization structure based on contract the-ory. As indicated in the Scheer (1994) review, BPRis usually viewed as involving adjustments in thestructure of the organization and the businessprocesses of information and technology. Dowlingand McGee (1994) used regression models to an-alyze the performance of business organizations. Asummary of the current tools and models forquality improvement can be found in Gitlow, in-cluding regression analysis, control chart, Paretochart and time series.

Business process reengineering is a changephenomenon and a radical adjustment process(Davenport and Stoddard, 1994). As Davenportand Stoddard (1994) noted in their paper, there arevisible successes as well as many failures among®rms that initiated BPR. It is thus important toinvestigate not only how the process works ingeneral (such as by modeling the BPR process) butalso how to assess individual BPR projects in anuncertain and dynamically evolving industry en-vironment. Although there has been a growingbody of research on BPR, most of the extant lit-erature is devoted to process modeling and mea-surability, and little attention has been paid to thecontrol problem in the management speci®c BPR

J. Liu, D. Yao / European Journal of Operational Research 115 (1999) 47±58 49

projects. So far to our knowledge, there has notbeen a study on the evaluation and control of BPRprojects using optimal stopping analysis.

3. Strategic engineering option problem

3.1. Notation

3.2. The SEOP model

We choose completion time as the system statevariable which is random by nature, as elaborated inSection 1. Let the remaining time to completion attime t be denoted by a stochastic process Xt; t P 0f gwith the following in®nitesimal characteristics:

dXt � ÿdt � rq Xt� � dWt ; �1�where Wt is a Wiener process (i.e., a standardBrownian motion), r a constant representing themaximum standard deviation of the disturbance,and q(Xt) a scaling function for the standard de-viation of the disturbance. The underlying process

Xt; t P 0f g is stationary since Eq. (1) is implicit oftime t. Under this de®nition, the standard devia-tion of the disturbance diminishes as the remainingtime, Xt, approaches zero; and q(Xt) can be ageneral function of Xt, satisfying

06 q�Xt�6 1 and limXt!0

q�Xt� ! 0:

An example of such is

q�Xt� ��1ÿ eÿkXt

p; �2�

where k is a scaling coe�cient (Chi et al., 1997).The size of k determines how fast the functionapproaches its upper limit 1 as Xt grows large.

The stopping time h is the decision variable inour problem, denoting the time at which the pro-ject is terminated before completion. The com-pletion time s represents the time at which theproject is completed (in the case that it is notstopped before completion); as such, s is de®ned ass � infft P 0jXt6 0g: These de®nitions imply thats6 h if the project is completed, and that h < s ifthe project is terminated before completion. Ob-viously, we have Xs� 0 at the completion. Theterminal payo� u(Xt) represents the value built upin the project based upon the current Xt, and suchvalue can be realized as the payo� if the project isterminated right then. The total payo� from aproject that is completed as planned is known as p,i.e., p�u(0). In general, terminal payo� functionu(Xt) should bear the following characteristics:

du�Xt�dXt

6 0 and limXt!1

u�Xt� ! 0: �3�

An example of such is

u�Xt� � pcXt ; �4�where c 2 �0; 1�:

Suppose that x�Xt�0 is known and the oper-ational cost of the project is at a constant rate of k($/time unit). Let p(x) be the maximum expectedtotal pro®t of the project given initial state x. Thestrategic engineering option problem (SEOP) canbe expressed as follows:

SEOP:

p�x� � maxh

Jx�h�

� Ex peÿls1�s6 h�

8<: � u�Xh�eÿlh1�h<s�

ÿZh^s

0

keÿlt dt

9=;

Xt estimated time to com-plete the project from ton

qt(Xt) scaling function for thedeviation of Xt:06 qt�Xt�6 1, andlimXt!0 qt�Xt� ! 0

s � infft P 0 j Xt6 0g natural end of the pro-ject, i.e., time at whichthe project is completed

h time to stop the projectu(x) value buildup functionP terminal payo� at the

completion, i.e., u(0)� pl an applicable interest

ratek unit operation cost of the

project ($/time)


s.t.

dXt � ÿdt � rq�Xt� dWt ; x � Xt�0:

4. Optimality conditions and SEOP-FB problem

As shown by Bensoussan and Lion (1982), themaximum pro®t function p(x) of SEOP must sat-isfy

p�x� ÿ u�x�P 0; �5�1

2r2q�x�2 d2p�x�

dx2ÿ dp�x�

dxÿ lp�x� ÿ k6 0; �6�

p�x�� ÿ u�x�� 1


dx2

�ÿ dp�x�

dx

ÿ lp�x� ÿ k�� 0:

�7�

Intuitively by Eq. (5), the project ought to becontinued if the marginal pro®t by continuing theproject is greater than payo� with project termi-nation (i.e., p(x) ) u(x) > 0). With such observa-tion, a continuation set S can be de®ned as follows:

S � fx j p�x� > u�x�; x P 0g: �8�By Eqs. (6) and (7), it is immediate that left-handside of Eq. (6) equals to zero for all x in the con-tinuation set S, and that p�x� � u�x�; 8x 2 @S,where @S denotes the boundary of S. Let C00�n �1; 2; 3; . . .� denote the class of all the nth orderdi�erentiable real functions. Then a free-boundaryproblem SEOP-FB can be constructed to obtain asolution of SEOP.

SEOP-FB. Find a real function p�x� 2 C2 and aninterval �xa; xb� � S, such that 8x 2 �xa; xb�;1


dx2ÿ dp�x�

dxÿ lp�x� ÿ k � 0 �9�

s.t.

p�xa� � u�xa�; �10�p�xb� � u�xb�; �11�p0�xa� � u0�xa�; �12�p0�xb� � u0�xb�: �13�

Eqs. (10) and (11) are terminal conditions fordetermining two integral constants of the ordinary(9), while Eqs. (12) and (13) are the so-calledsmooth-pasting conditions needed for determiningthe interval �xa; xb�. From the solution of SEOP-FB, the optimal strategic engineering option canbe determined as a two-threshold band control,described as follows: pursue the project if the re-quired time to completion is not longer than xb;and continue the project until the remaining timeto completion falls to xa or below.

In order to have a meaningful project, the fol-lowing assumption is needed.

Assumption 1. There exist 06 xa6 xb, such thatu0�x�6 ÿ k; 8x 2 �xa; xb�:

The assumption assures that there is an intervalwithin which the payo� build-up rate is no lessthan the minimum cost rate )k so that it is justi-®able to continue the project.

Proposition 1. If u�x� 2 C2 is strict convex andmeets the characteristics of Eq. (3), and ifu0�0�P ÿ k, then the continuation set S �£, i.e.,xa� xb� 0.

Proof. By the de®nition of a strict convex function,if u0(0) P )k, then u�x� > ÿkx; 8x > 0. Therefore,assumption 1 can only be satis®ed at xa� xb� 0.Equivalently, we have S �£. This concludes theproof. h

By Proposition 1 the project is not worth pur-suing if there does not exist a non-zero intervalwhere it is pro®table to start the project.

Proposition 2. If u�x� 2 C2 is strict convex andmeets the characteristics of Eq. (3), and ifu0�0� < ÿk, then xa� 0 and xb > 0.

Proof. Similarly, by the de®nition of a strictconvex function, if u0�0� < ÿk, then there existsan xb > 0 such that u0�xb�6 ÿ k. Equivalently, wehave S 6�£: �

If the condition of Proposition 2 is met, thenxa� 0 and thus the condition (12) can be removed.


With xb solved from SEOP-FB, the optimal SEOPthen becomes a single threshold control, that is,the project shall be pursued when x6 xb, andthereafter the project is to be continued until ®nalcompletion.

Thus far, it has been shown that SEOPtranslates to solving the SEOP-FB. However, aclosed-form analytical solution is not attain-able in general. In Section 5, we derive a seriessolution for the SEOP-FB under speci®c formsof variance function q(x) and payo� functionu(x).

5. Series solutions of SEOP-FB

For convenience, we discuss the solutions ofSEOP-FB with speci®c function forms as assumedin the following.

Assumption 2. In the SEOP, it is assumed thatq�Xt� �

��1ÿ eÿkXtp

as given by Eq. (2) and u�Xt� �pcXt as given in Eq. (4).

The SEOP-FB centers on solving the non-ho-mogeneous second-order ordinary di�erentialequation (ODE) given by Eq. (9). Let h�x� �2=r2q2�x�, and denote

L�x� � d2p�x�dx2

ÿ h�x� dp�x�dxÿ lh�x�p�x�: �14�

Then, the original Eq. (9) is equivalent to L�x� �kh�x�, and its corresponding homogeneous equa-tion is L(x)� 0.

Lemma 1 (Existence of series solution). Thereexists, in x 2 �0; 2p�, a convergent series solution toL(x)� 0 as de®ned by Eq. (14).

Proof. By the theory of ODEs (ordinary di�eren-tial equations, e.g., Braun (1983)), it is su�cient toshow that h(x) has a convergent Taylor expansion.Since

h�x� � 2

r2q2�x� �2

r2�1ÿ eÿkx� ; �15�

we can write the Taylor expansion of h(x) as fol-lows:

h�x� � 2

r2�1ÿ ekx� �2

r2xx

1ÿ ekx

� 2

r2x1

�ÿ x

2� 1

12x2 ÿ 1

720x4 � � � �

� u� ÿ 1�n�1 Bn

�2n�! x2n � � � ��;

0 < x < 2p;

�16�

where Bn denotes a Bernoulli number. The Taylorexpansion in Eq. (16) converges over 0 < x < 2p,over which the existence of a convergent seriessolution to L(x)� 0 can thus be concluded. h

Lemma 2. There exists a series solution to L(x)� 0at x� 0.

Proof. It is su�cient to show that h(x) is analytic atx� 0 (see more in Braun (1983)). That is, x� 0 is aregular singular point of L(x)� 0, which is indeedthe case by checking that limx!0 xh�x� <1: �

Theorem 1. For x 2 �0; 2p�, there exists a seriessolution ~p�x� to SEOP-FB with the followingstructure:

~p�x� � c1 ~p1�x� � c2 ~p2�x� � ~p0�x�; �17�where c1 and c2 are two integral constants, while~p1�x� and ~p2�x� are two linearly independent generalsolutions to the homogeneous ODE L(x)� 0, and~p0�x� is a speci®c solution to the non-homogeneousODE L�x� � kh�x�. All ~p1�x�; ~p2�x� and ~p0�x� are inthe form of a Taylor series such asP1

n�0anxn�r �a0 6� 0�:

Proof. Note that Eq. (9) of a SEOP-FB (i.e.,L(x)� kh(x)) is a second-order ODE, to which thegeneral solutions have the structure of Eq. (17). ByLemmas 1 and 2, it is assured that there exists aseries solution (17) in the form ofP1

n�0anxn�r �a0 6� 0� which converges overx 2 �0; 2p�. The two integral constants c1 and c2,plus interval bounds xa and xb can be determined bythe terminal conditions (10)±(13) of SEOP-FB. h

Corollary 1. The coe�cient of ~p0�x� can be iterat-ively determined by setting to zero the coe�cients ofthe series H(x) where


H�x� � ÿk ÿ kk2

x

�X1n�0

r2k�n� r��n� r ÿ 1�2

�ÿ �n� r�

�anxn�rÿ1

ÿX1n�0

k�n� r�2

�� lan

�xn�r

�X1m�1

X1n�0

Bm�n� r��2m�!

� �anx2m�n�rÿ1

ÿX1m�1

X1n�0

lBm

�2m�!� �

anx2m�n�r ÿX1m�1

kBm

�2m�! x2m;

�18�

and the coe�cients of series solutions ~p1�x� and ~p2��x� can be iteratively determined by setting to zerothe coe�cients of the series H(x) of Eq. (18) withk� 0.

Proof. Let

p�x� �X1n�0

anxn�r; a0 6� 0;

and write

p0�x� �X1n�0

�n� r�anxn�rÿ1;

p00�x� �X1n�0

�n� r��n� r ÿ 1�anxn�rÿ2:

Plugging the above into L(x) ) kh(x)� 0 and re-arranging the terms, it derives as H(x)� 0, fromwhich a speci®c non-homogeneous solution ~p0�x�can be determined by setting coe�cients of H(x) tozero. Similarly, letting k� 0 in H(x)� 0, two lin-early independent general homogeneous solutions~p1�x� and ~p2�x� can be obtained. h

By Eq. (17), a general series solutions is deter-mined as long as the integral constants c1, c2 aredetermined. Thus, a solution to SEOP-FB can becompletely characterized by a set of four realnumbers, {c1, c2, xa, xb: xa6 xb}. A computationalalgorithm can be constructed under a pre-selectedorder of accuracy (i.e., the order of terms in a se-ries solution).

Algorithm 1 (Series SEOP).

Step 1. Select order of accuracy w (i.e., computeto the wth power in a series).

Step 2. Use Eq. (18) of Corollary 1 to determinethe coe�cients an�n � 0; 1; 2; . . . ;w� of ~p0�x�; ~p1�x�and ~p2�x�, respectively.

Step 3. Given terminal payo� function u(x). Let~p�xjc1; c2� � c1p1�x� � c2p2�x� � ~p�x�: Solve for{c1, c2, xa, xb: xa6 xb} from the following systemof equations:

�SEOP-series�~p�xajc1; c2� � u�xa�;~p�xbjc1; c2� � u�xb�;~p0�xajc1; c2� � u0�xa�;~p0�xbjc1; c2� � u0�xb�:

8>>><>>>:Step 4. The optimal pro®t function is found as

~p�xjc1; c2� � c1p1�x� � c2p2�x� � ~p�x�; and the op-timal continuation interval is determined as [xa,xb].

6. Analytical solution to SEOP with l� 0

When the interest rate l� 0 in the SEOPproblem with speci®c function forms of Assump-tion 2, a closed form solution of p(x) is attainable.In this section, we derive such solution. Althoughit does not bear much practical sense to have azero interest rate, an analytical solution underl� 0 presents useful characteristics even for thegeneral l¹0 cases, as concurred by numerical testsshown in Section 6.

Denote m�x� � p̂0�x�. Then, with l� 0, the ODEgiven in Eq. (9) can be expressed as

1

2r2q�x�2m0�x� � m�x� � k;

dm�x�m�x� � k

� 2

r2q�x�2 dx:

Taking integral on both sides, it becomes

ln�m�x� � k� �Z

2

r2q2�x� dx� c1:

As m�x� � p̂0�x� by de®nition, the above equationcan be rewritten as


p̂0�x� � ÿk � c1 exp2

r2

Z1

q2�x� dx� �

: �19�

Denoting G�x� � R �1=q2�x��dx and integratingEq. (19) on both sides, we obtain

p̂�x� � c2 ÿ kx� c1

Zexp

2

r2G�x�

� �dx:

Denoting F �x� � R exp 2r2 G�x��

dx; we can write

p̂�x� � c2 ÿ kx� c1F �x�:Note that u(x) in Assumption 2 is strict convexand therefore xa� 0 by Proposition 2. Since theboundary condition (10) implies p̂�0� � p, we havec2 � p ÿ c1F �0�. Substituting p ÿ c1F �0� for c2 inthe above expression, we get

p̂�x� � p ÿ kx� c1�F �x� ÿ F �0��which is equivalent to

p̂�x� � p ÿ kx� c1

Zx0

exp2

r2G�x�

� �dy:

Given q�Xt� ��1ÿ eÿkXtp

, we can obtain the spe-ci®c forms of G(x) and thus exp [2/r2)G(x)]:

G�x� �Z

1

1ÿ eÿkxdx � x� 1

kln�1ÿ eÿkx�;

exp2

r2G�x�

� �� exp

2

r2x�� 1

kln 1ÿ ÿ eÿkx

�� exp

2

r2x

� �1ÿ ÿ eÿkx

�2=r2k:

Then, by the boundary condition de®ned inEq. (13) and the result given in Eq. (19) above, wecan solve for c1 as a function of x̂:

c1�x̂� � k � pcx̂ ln c

exp��2=r2�x̂��1ÿ exp�ÿkx̂��2=r2k: �20�

Hence, the solution to SEOP with l� 0 can beobtained as

p̂�x� � p ÿ kx� c1�x̂�Zx

0

exp2

r2y

� �

1ÿ ÿ eÿky

�2=r2kdy:

�21�

Its ®rst and second derivatives are, respectively

p̂0�x� � ÿk � c1�x̂� exp2

r2x

� �1ÿ ÿ eÿkx

�2=r2k; �22�

p̂00�x� � 2

r2c1�x̂� exp

2

r2x

� �1ÿ ÿ eÿkx

��2=r2k�ÿ1: �23�

Given Eq. (21), the numerical value of x̂ for agiven set of parameter values can be obtained bysolving Eq. (11).

Corollary 2 (Rejection rule). With l� 0, thereexists a rejection rule such that if

ÿk6 p ln c �24�then reject the project immediately.

Proof. With u(x)� pcx, the continuation set S isempty by Proposition 1. Therefore, the Corollary 2follows immediately. h

7. Industrial applications and numerical examples

First, we report managerial implications of theSEOP as applied to the Expansion Valve project.At the initial stage of a re-engineering project, therejection rule of Corollary 2 provides a key crite-rion in evaluation of project proposals. Eachproposal contains either directly or indirectly the®gures such as estimated project horizon T, totalproject payo� p, cost rate k, and the early-stagevalue build-up rate c. For instance, based on thebest information available (e.g., the quotes fromseveral bidding vendors for the engineering con-tracts), the total time needed to complete theproject T, the total yield of the project p and anoperation cost rate k can be obtained throughroutine engineering calculations. The estimation ofthe rate of value build-up requires some extra ef-forts but is attainable. For example, historic datacan be gathered with the help of the engineering®rms who are bidding on the project, and the ex-pression for valuation process can be derived byidentifying the underlying ``scenario'' (e.g., con-vex, concave, or S-shaped), upon which best-®ttedcurves are determined. Then using Corollary 2, a


quick screening procedure can be applied to de-termine whether a proposal is worthwhile furtherpursuing. A proposal must be rejected if it couldnot pass the Corollary 2 (i.e., if (24) holds) evenwith an unrealistically conservative assumption ofzero discount rate.

According to the type of value build-up sce-nario, an optimal policy can be quickly deter-mined so as to carry out the project. Forinstance, if the payo� function u(x) is convex(i.e., an increasing build-up rate), then the opti-mal entry time xb is critical which can be cal-culated by SEOP. That is, the project shall bepursued if the estimated project time is on orbelow the entry time (i.e., T6 xb). While if u(x)indicates a decreasing build-up rate, the optimalexit time xa is critical, and the project should bestarted right away but must be stopped as soon

as the estimated remaining time falls below xa.Obviously, a two-band control based on xa andxb would be optimal if an S-shaped build-up isdetected where the value build-up is slow at bothbeginning and ending, while faster during themiddle section of the project.

Based on the actual data for the ExpansionValve project, a convex build-up scenario wasidenti®ed and an xb� 4.2 (in years) was deter-mined by SEOP. Thus, an entry-time basedcontrol should be used. That is, only considerthose proposals whose total time to completiondoes not exceed the maximum entry time xb (i.e.,4.2 years). The ®nal selected proposal for theExpansion Valve project required a total time ofthree years, which satis®es the optimal policy toenter (i.e., to start the project). In a similarmanner, the optimal control policy can be exer-

Table 1

Variation of xb in c, l and r (p� 5, k� 1 and k� 0.5)

c l

0.00 0.05 0.10 0.15 0.20 0.25

r� 3.0 0.8 5.1531 2.7031 0.0000 0.0000 0.0000 0.0000

0.7 7.7041 7.0204 6.3530 5.6630 4.8906 3.8688

0.6 8.3552 7.8655 7.4322 7.0372 6.6671 6.3118

0.5 8.5425 8.1088 7.7389 7.4147 7.1244 6.8598

0.4 8.5897 8.1738 7.8235 7.5212 7.2551 7.0171

0.3 8.5987 8.1872 7.8421 7.5458 7.2866 7.0564

0.2 8.5989 8.1889 7.8446 7.5495 7.2917 7.0631

0.1 8.5991 8.1890 7.8448 7.5497 7.2920 7.0636

0.0 8.5991 8.1890 7.8448 7.5497 7.2920 7.0636

r� 2.0 0.8 2.4810 0.0000 0.0000 0.0000 0.0000 0.0000

0.7 5.5275 4.6360 3.6752 2.3908 0.0000 0.0000

0.6 6.3578 5.7785 5.2552 4.7575 4.2585 3.7272

0.5 6.6396 6.1582 5.7503 5.3909 5.0636 4.7569

0.4 6.7318 6.2866 5.9203 5.6082 5.3346 5.0894

0.3 6.7569 6.3241 5.9727 5.6777 5.4235 5.2001

0.2 6.7615 6.3319 5.9846 5.6946 5.4465 5.2301

0.1 6.7619 6.3327 5.9860 5.6969 5.4498 5.2348

0.0 6.7619 6.3327 5.9860 5.6969 5.4499 5.2350

r� 1.0 0.8 1.2373 0.0000 0.0000 0.0000 0.0000 0.0000

0.7 4.0749 2.3976 0.0000 0.0000 0.0000 0.0000

0.6 4.9587 4.2892 3.6637 3.0097 2.1561 0.0000

0.5 5.2855 4.8010 4.3557 3.9553 3.5783 3.2067

0.4 5.4066 4.9876 4.6054 4.2792 3.9912 3.7294

0.3 5.4465 5.0519 4.6950 4.3982 4.1438 3.9204

0.2 5.4564 5.0694 4.7216 4.4358 4.1942 3.9204

0.1 5.4577 5.0721 4.7265 4.4434 4.2054 4.0013

0.0 5.4578 5.0722 4.7267 4.4438 4.2062 4.0025


cised during any time in a project as soon as anupdated remaining time to completion is ob-tained. Numerical examples for each of thesecases are presented next.

Now, we present numerical examples ofSEOP problems with speci®c function forms ofAssumption 2. The computations are conductedusing MathCad+ (version 6.0) installed on anIBM Pentium 120. First, we report numericalstudy on the regular case (i.e., l ¹ 0). Since theterminal payo� function u(x)� pcx is strict con-vex, we have xa� 0 by Proposition 2. In suchcase, the optimal option is simply characterizedby xb, that is, continue the project wheneverx 2 �0; xb�. Table 1 reports the variation of xb inparameters c, l and r.

As shown in Table 1, the interval [0, xb], inwhich the project should continue, becomes wideras the volatility increases (i.e., greater r). In otherwords, a longer project can be committed towhen a higher degree of uncertainty in comple-

tion time is encountered. Such rule coincides witha similar regularity as observed in an investmentproject under uncertainty (Dixit and Pindyck,1994).

Next, we study the numerical trajectory of anoptimal pro®t function p̂�x� and the positions ofthe associated continuation set. For simplicity, wecon®ne the numerical study to the special case ofl� 0. Let p̂r�x� be the pro®t function under therejection condition (24) of Corollary 2, while p̂c�x�the pro®t function otherwise (i.e., non-zero con-tinuation interval). Fig. 1 presents the numericaltrajectories of the pro®t function for both rejection(k� 2) and continuation (k� 1) with l� 0. For allx P 0, we have p̂r�x� underneath the payo� func-tion u(x)� pcx. Thus, the continuation set is emptyby Proposition 1 and the project must be rejected.On the other hand, there exists an interval overwhich the pro®t p̂c�x� is strictly above the payo�function. Since the payo� function u(x)� pcx isstrict convex and decreasing, the continuation in-

Fig. 1. Examples of empty and non-empty continuation intervals.


terval is completely determined by xb. The nu-merical result of xb is also indicated in Fig. 1.

Fig. 2 gives an example of non-zero xa when thepayo� function is not convex, speci®cally u�x� �p cos 0:5p 1ÿ exp�ÿaxg�� f g as depicted in Fig. 2.Clearly, in Fig. 2 we have xa > 0 and the contin-uation interval is presented by [xa, xb], over whichthe optimal pro®t p̂�x� is strictly above payo�function u(x).

Acknowledgements

We thank the editors for their rigorous reviewsand comments on the drafts of this paper. Ourthanks are also due to Professor Tailan Chi at theSchool of Business Administration, University ofWisconsin-Milwaukee, for his generous help oncomputation and numerical tests using MathCad+on an IBM PC.

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strategic options in re-engineering of a manufacturing system with uncertain completion time

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