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Optimal control in homogeneous projects: analyticallysolvable deterministic casesKONSTANTIN KOGAN a b , TZVI RAZ c & RAMY ELITZUR da Department of Interdisciplinary Studies - Logistics , Bar-flan University , Ramat-Gan,52900, Israelb Department of Computer Sciences , Holon Center for Technological Education , Holon,Golomb 52, 58102, Israelc Faculty of Management, Leon Recanati Graduate School of Business Administration, Tel AvivUniversity , Ramat Aviv, Tel Aviv, 69978, Israeld The Rot man School of Management, University of Toronto , 105 St. George Street, Toronto,Ontario, M5S-3E6, CanadaPublished online: 17 Apr 2007.

To cite this article: KONSTANTIN KOGAN , TZVI RAZ & RAMY ELITZUR (2002) Optimal control in homogeneous projects:analytically solvable deterministic cases, IIE Transactions, 34:1, 63-75, DOI: 10.1080/07408170208928850

To link to this article: http://dx.doi.org/10.1080/07408170208928850

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IIE Transactions (2002) 34,63-75

Optimal control in homogeneous projects: analyticallysolvable deterministic cases

KONSTANTIN KOGAN', TZVI RAZ2 and RAMY ELlTZUR3

1Department oj Interdisciplinary Studies - Logistics. Bar-Ilan University. Ramat-Gan 52900. Israel and Department of ComputerSciences. Holon Center for Technological Education. Holan 58102. Golomb 52. IsraelE-mail: kogan@wine.cteh.ac.il2Faculty of Management. Leon Recanati Graduate School oj Business Administration, Tel Aviv University. Ramal AI'iv. Tel Aviv69978 IsraelE-mail: tzviraz@post.lau.ac.il)The Rotman School oj Management. University oj Toronto, 105 St. George Street. Toronto, Ontario. Canada M5S-3E6Ennail: eliIzu@rotman.utoronto.ca

Received November 1998 and accepted April 1001

We develop a model for determining the economically optimal amount of control effort required to manage a homogeneous project(one consisting of a large number of similar activities). The model is formulated in terms of the losses generated by deviations fromthe plan and of the costs associated with carrying out control activities. It accounts for changing levels of project activities andincludes parameters that represent control effectiveness and project management effectiveness. The model is studied by applyingoptimal control theory, which yields the optimal control effort described by a number of control functions and switching points atwhich they change over. We identify three analytically solvable cases for the most commonly used forms of control cost anddeviation loss functions. Our analysis of the model leads to several specific conclusions regarding tbe extent and timing ofmanagerial attention that should be devoted to keep projects on track. We also point out how the optimal off-line policy can beadapted for on-line control and real-time decision making throughout the project life cycle.

I. Introduction

A project is an undertaking with defined objectives andestablished starting and ending points that needs to becompleted within specified time, resource and budgetconstraints. Normally a detailed plan is prepared prior tostarting the project. The plan describes the dates at whichthe activities should start and end in order to reach theobjectives, the amounts of resources that should be ap­plied to implement the various activities, the budget al­located to support these resources, and the technical andoperational characteristics of the deliverables to be pro­d uced by each activity.

In most cases, however, actual execution tends to de­viate from the plan. This is due to a variety of factors,both internal (unforeseen difficulties, poor productivityand quality, unexpected dependencies, etc.) and external(supplier-induced delays, price and availability variations,customer requests, etc.). Project control is a mechanismdesigned to cope with the effects of these factors and tohelp complete the project as planned. Project controlconsists of measuring actual execution, comparing it tothe plan, analyzing the deviations, and initiating and

0740·817X © 2002 "liE"

implementing corrective action to bring the project backon course.

Two of the key issues involved in designing projectcontrol procedures are how much control should be ex­ercised, and how often. The project management litera­ture provides some general guidance. Bent (1988),suggests a hierarchy of control loops, including majorproject milestones, quarterly performance targets andweekly performance targets. He further distinguishes be­tween small projects « 100000 hours) requiring controlon a monthly basis, and large projects (> 1500 000 hours)req uiring control on a weekly basis, with intermediate sizeprojects presumably fitting in between. Meredith andMantel (1995) argue that control points should be linkedto the actual project plans and to the occurrence of eventsas reflected in the plan, and not only to the calendar.They provide the following list of attributes of goodcontrol systems: flexibility, cost effectiveness, usefulness,ethical integrity, timeliness, accuracy and precision, sim­plicity of operation, ease of maintenance, and full docu­mentation. However, they do not address the question ofhow to determine the extent and frequency of controlneeded. Turner (1993) mentions that the frequency of

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reporting in project control depends on the length of theproject, the stage of the project, the risks involved, andthe organizational level of the report recipient.

However, although the literature recognizes the need todetermine the timing and extent of project control in anefficient and effective manner, the analytical work in thisarea is in its infancy. Partovi and Burton (1993) carriedout a simulation study to compare the effectiveness of fivecontrol timing policies: equal intervals, front-loading,end-loading, random and no control. Their results sug­gest that although there are no significant differencesamong the policies in the terms of cost required to recoverfrom deviations from the plan, the end-loaded policyperforms best in preventing time overruns. De Falco andMaehiaroli (1998) proposed a model for the quantitativedetermination of the timing of control points. Their ap­proach is based on the definition of an effort function,which incorporates activity intensity and schedule slackaspects, and on the premise that control intensity is dis­tributed according to a bell shaped curve around thepoint of maximum effort.

In this paper we present an analytical framework fordetermining the timing and intensity of project controleffort throughout the life cycle of the project. The modelis designed for homogeneous projects, which we define asprojects that include a large number of relatively similaractivities with similar applicable control mechanisms. Thefollowing assumptions are required to obtain a tractableformulation that leads to important structural and quali­tative results on the optimal control effort:

• The project environment is deterministic, meaningthat we work with expected values rather than ran­dom variables.

• Deviations from the project plan have positive linearcorrelation with the planned intensity of project ac­tivities and negative linear correlation with the con­Irol effort.

• There arc enough internal and/or external resourceson hand to provide necessary control of project ac­tivities. Using these resources, however, might be­come extremely expensive.

The development is based on optimal control theory,which has been applied to a variety of production prob­lems - sec, for instance, Dixit (1990), Khmelnitsky et 01.(1995), Kogan and Khmelnitsky (1996)- but not as yet toproject control.

The paper is organized as follows. We start by intro­ducing and motivating the elements of the model: activityintensity, deviation from execution, deviation losses,control effort and control costs. This is followed by themathematical formulation as an optimal control problemand real life examples. We solve for the optimal controleffort over time for the following cases: linear deviationlosses and linear control costs; non-linear deviation lossesand linear control costs; and linear deviation losses and

Kogan et al.

non-linear control costs. We prove that the first two casesare analytically solvable for arbitrary forms of plannedproject intensities. These cases are characterized by twotypes of optimal control along the planning horizon, withone switching point where they change over. As for thethird case, we find that it is characterized bymultiple switching points. If the activity intensity func­tion is arbitrary, then this can only be treated numeri­cally. For this case we derive conditions that ensurethat the optimal control includes no more than oneswitching point, which makes the problem analyticallysolvable. We conclude with some management implica­tions of this work and with some directions for furtherresearch.

2. Model development

2.1. Activity intensity

We view projects as endeavors planned for execution overcontinuous-time. Continuous-time analysis in projectplanning has recently gained significant attention in theresearch literature. Weglarz (1981), Leachman et 01.(1990) and Kogan and Shtub (1998) considered optimalintensities of project activities. Leachman et 01. (1990)looked at scheduling projects with variable intensity un­der resource constraints. Kogan and Shtub (1998) sug­gested a number of optimal control-based models andmethods for numerical optimization of variable projectintensities for different types of precedence relationsamong the activities.

Consider a given project that has been planned to startat time zero and to end after T time units. The projectplan specifies which activities should be executed at thevarious points in time, along with the resources andbudget required. Similarly to Leachman et 01. (1990) andKogan and Shtub (1998), we will use the term activityintensity to denote the rate at which work should becarried out to attain the project objectives, according tothe project plan. Activity intensity can be measured interms of workload per time unit, person-hours per timeunit, dollars per time unit, or any other measure that isused to define the project plan. The activity intensity mayvary over time, depending on the nature of the tasks thatconstitute the project and on various planning consider­ations. We denote the activity intensity function at time 1

as 0(1). Since it is derived from the project plan, the ac­tivity intensity function is entirely known at the beginningof the project.

2.2. Deviation and deviation loss

It is quite likely that actual execution of the projectwill deviate from the plan. We now define a deviationfunction V(I), which represents the cumulative extent of

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Optimal control in homogeneous projects

deviations from the plan at time I. If the earned valueapproach is applied to measure progress, then the extentof deviation can be measured in monetary units. As analternative, the deviation function can represent thenumber of unresolved issues that the project manager hasto deal with, the number of variance reports or planchange requests, or any other measure of discrepancybetween the project plan and its actual execution.

In our modeling framework, deviation can have eitherfavorable or unfavorable implications with respect to thesuccessful completion of the project. We will use thefollowing convention to distinguish between favorableand unfavorable deviations: when the cumulative effect ofthe deviations from the project plan is unfavorable, thedeviation function will be positive; and when the cumu­lative effect of the deviations is favorable, the deviationfunction will be negative. However, in either case we in­cur a loss by carrying out the project in a manner thatdiffers from the original plan. This loss is the result ofinefficient utilization of resources, rescheduling down­stream activities, increased co-ordination and communi­cations, and so on.

The deviation function has the following properties:

I. At the beginning of the project (t = 0) there are nodeviations, i.e., V(O) = O.

2. The deviation at any point in time increases with thelevel of activity intensity, a(t), meaning that as morework is being done, the potential for tasks beingcarried out not exactly according to plan increases.

3. When the project is proceeding exactly according tothe plan, then V(I) = O.

We associate a loss function L(V(t)) with the deviationfunction. The loss function L(V(t)) represents the cost ofthe damage caused by the deviation from the project plan.It is, of course, a non-decreasing function of the devia­tion, i.e.,

65

invested in planning and implementing corrective actionsdesigned to bring the project back to plan. These controlactivities are not required in order to achieve the objec­tives of the project, but they are needed to ensure that theproject will not deviate from its plan. The control effortfunction is related to the deviation function in the fol­lowing manner: at any point in time the deviation func­tion decreases as the level of control effort, e(t), increases,meaning that as more effort is invested in measurement,analysis and correction, the number of deviations fromplan will decrease.

In our setting the project manager decides on the tra­jectory of optimal control intensity over time, e(t). Acontrol cost function is associated with the control effort.This control cost function, denoted as C(e(t)), has thefollowing properties:

I. Control cost is positive for positive control effort,i.e., C(e(t)) > 0 if e(t) > 0

2. Control cost is a non-decreasing function of controleffort: i.e.,

aC(e(t)) > O.8e(t) -

3. Additional control efforts may become increasinglymore expensive (Meredith and Mantel, 1995), or,in other words, we operate in the non-decreasingmarginal cost region, i.e.,

2.4. Mathematical formulation

The objective is to minimize the total cost, J, which is thesum of the cost resulting from plan deviations L(V(t)),and the cost of control, C(e(t)). The objective functionalis as follows:

and to the constraint that control effort cannot be nega­tive

subject to the equation of motion

V(t) = F(a(t),e(t)), V(O) = 0, (2)

8L(V(t)) > O.oV(I) -

Later on, when we will address the analytically solvablecases, we will distinguish between deviations with unfa­vorable consequences and those with favorable conse­quences by using two different loss coefficients: /"1- forpositive (unfavorable) deviations, and t: for negative(favorable) deviations, with typically t+ ~~ r .

J = I T

(L(V(t)) + C(e(t)))dt -> min,

e(t) ~ o.

(I)

(3)

2.3. Control effort

We define a function e(l) as the control effort at time t.The control effort represents the various activities en­tailed in measurement, analysis and correction of devia­tions from plan. Control effort reflects the level of detailand accuracy of the data collected, the investment oftime, expertise and other resources in the analysis of thedata, and the level of management attention and energy

As stated in the introduction, we assumed, for the sakeof simplicity, that there is a linear correlation betweennumber of deviations on one hand, and activity intensityand control effort on the other hand. This implies thatthe equation of motion is linear in a(t) and e(t),F(a(t),e(t)) = cw(t) - f3e(t). This provides for: (i) tract­ability of the results; and (ii) the estimation of the pa­rameters a. and f3 through regression analysis.

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The parameters ()( and Ii are interpreted in the followingmanner. The first parameter, «, represents the contribu­tion of the level of project activity at time I to the rate ofchange of the deviation. Larger values of ()( correspond toa project that is managed poorly, such that each unit ofactivity contributes a larger amount of deviation. Simi­larly, a smaller value of ()( would indicate a well-runproject, with work being carried out in close agreementwith the plan. The situation where ()( = 0 corresponds to aproject that is run precisely according to plan, withoutany additional deviation being generated. Thus, we mayrefer to ()( as a parameter that reflects the lack of effec­tivcncss in managing the project operations.

Parameter Ii represents the effectiveness of the controleffort. Larger values of Ii imply that the rate at whichcontrol efforts arc able to reduce the cumulative deviationis greater. A situation where Ii = 0 indicates that controlefforts are totally ineffective. Both parameters are as­sumed to be non-negative.

Finally, analytically solvable cases of this problem canbe found only if the form of the cost functions is explicitlygiven. Here we consider the two most commonly appliedtypes of convex cost functions: linear, as in Equation (4),and quadratic, as in Equation (5):

C(e(/)) = ce(/), L(V(/)) = { {+V(/), ~f V(/) > 0,-{- V(/), If otherwise,

(4)

C(e(/)) = ce2(/), L(V(/)) = { {+V,2(t), if V(t) >0,t: V-(t), If otherwise.

(5)

3. Examples

To illustrate the concepts developed in this paper andtheir practical importance, we will follow two real lifeexamples drawn from the professional involvement ofone of the authors. In order to safeguard company con­lidentiality, the actual numbers were slightly changed,while preserving the substance of the two cases. The firstexample deals with SE. a large software engineeringcompany situated in California. The company had severalcomplaints filed against them for delivery of unsatisfac­tory software products. In a meeting with the complain­ing clients and their lawyers, it was made clear if the suitswent to trial, the company would be forced into bank­ruptcy. The clients agreed to drop the complaints if thecompany develops a permanent in-house Quality Assur­ance department.

This case illustrates a typical software project envi­ronment. A project corresponds to the development of anew software product. Each product consists of a largenumber of functions, procedures and modules to develop.The development of each of these is considered a separate

Kogan et al.

project activity and is carried out following a standard­ized process. Specifically, all these activities are similar interms of the type of programming work and testingmechanisms (control).

Prior to the establishment of the QA department, aseries of statistical studies indicated that, for projectslasting 12 to 20 months, there was a linear relationshipbetween the amount of modifications or rework, and theproject intensity and the control effort, i.e., F(a(I),e(t)) = ()(a(t) - fJe(t). In particular, a value of ()( = 0.22was estimated from the regression between the plannedand actual project intensities, {a(t), V(t)}, measured forvarious projects at different points in time. Based on theinitial experience of the QA department, the value of thecoefficient fJ = 0.8 was estimated in a similar manner.

The activity intensity was measured in terms of devel­opment activities (functions, procedures, etc.) per month.Typically the activity intensity was either constant overthe project duration, or increased linearly or in a concavemanner, reflecting the fact that as progress is being made,more activities can be carried out in parallel.

Project control in this environment includes testing,debugging, and rewriting code as appropriate (correctiveaction). If the project is small, then the same program­mers who developed the code do the control, providingsufficient product quality without significant delays.

However, this is not the case in large projects, whichrequire special attention. For instance, in an earlier casethat went to trial, the client of SE claimed to have re­ceived software that was completely unusable. The law­suit was settled with SE agreeing to deliver conformingsoftware in 6 months and paying significant penalties forthe delay. This is a clear example where the losses L(V(t))associated with deviation from the plan V(I) (measured innumber of modules to rework) increased non-linearly.From historical data regarding delay costs, a quadraticloss coefficient (1+ = $126.3) was estimated using non­linear regression. The cost of control effort C(e(t)) =celt) was assumed to be linear, as follows. The QualityAssurance department had sufficient resources to test andrewrite modules. The control intensity e(l) was measuredin terms of modules per month. The cost of reworkingone module was calculated as c = $10000, based on50 hours at the rework rate of $200 per hour, which wassubstantially higher that the standard programming rate.

For our specific example we consider a project of du­ration T = 18months with estimated fixed activity in­tensity a(t) = 200 modules per month. The motionequation (2) takes the following form V(t) = 0.22 x200 - 0.8 x e(t). This implies that, if we apply a fixedcontrol intensity in order to achieve zero deviation overthe entire planning horizon, (i.e., V(t) = 0 modules torework for 0 ::; t ::; 18), then the resulting control level is

e(t) -_ -()( a(t) -_ 0.22 x 200 -_ 55 d hmo ules per mont .fJ 0.8

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Optimal control in homogeneous projects 67

4. Dual formulation

is non-negative, then it is optimal. Otherwise, no controlshould be undertaken at point I, i.e., e(l) = O.

According to the maximum principle, the Hamiltonianis the objective function of the dual problem and ismaximized at every point of time with respect to theadmissible controls e(I). This implies that if the controleffort function e(l) that satisfies the following equation:

DHi!l = _ BC(e(t)) _ "'(I)f3 = 0 (8)De(l) Be(l) 1

Problem (I )-(3) is stated in the canonical form of theoptimal control, where e(l) is a control variable and V(I)is a state variable the behavior of which is described bythe state (primal) differential Equation (2). In order tosolve this problem, a Hamiltonian, H, is formed andanalyzed based on the maximum principle:

H(I) = -L(V(I)) - C(e(I)) + "'(I)[a.a(l) - (Je(I)]. (6)

The co-state variable, "'(I) in (6) measures the sensi­tivity of the losses in each point in time I to changes in thedeviation V(I) and satisfies the following co-state (dual)differential equation with transversality condition:

(7)"'(T) = o.~( ) = BL(V(I))I DV(I)'

$1000 per week per unit of equipment. The two loss co­efficients, /+ and t: were identical, because the delayedinstallation (positive deviation) of equipment results inreduced customer fees for slow access as compared to thestandard fees during the transient period; and exceedingplanned installations (negative) deviation does not resultin increased fees levied since this was not anticipated.This implies the lost fees corresponding to the differencebetween the increased fees that could be requested forbetter service and standard transient period fees.

The planning horizon was set at 16 weeks, with theactivity intensity, measured in equipment units per week,planned to increase linearly, a(l) = 100 + 41. The ratioof discrepancies to activity intensity was estimated bythe project management consulting subcontractor asa. = 0.25. Therefore, the project had

l4 0.25( I00 + 41)dl = 108 units,

late installations after just I month of no special attentionto project control. At this point the project managementconsulting company started carrying out fixed, high in­tensity control over the remaining planning horizon. Themanagement of the telecommunications company wasanxious to optimize the trade-off between the high ex­penses of constant project control and the losses due to adecreasing level of customer service.

t 8

./0 10000 x 55dl = $9900000.

This is the approach taken by the Quality Assurancedepartment, which, at a cost of $10 000 per module,yields the objective function value (I),

Clearly, this approach is very expensive, and the questionsteadily raised by management is "what is the optimaltrade-off between providing good product quality and in­vesting in control intensity?" In other words, is there a needfor high intensity control over the entire planning horizon,or are there some points in time when the intensity could bedecreased to get better economic results for the companywithout sacrificing its clients or going to court.

The second example involves a project managementconsulting company working with a telecommunicationscompany. The project was concerned with the relocationof the customer's network operations center. Relocationof a computerized operations center is a typical project,involving co-ordination of the removal, transportationand installation of all trunks, circuits and equipment forthe center, while achieving minimal downtime. Consid­ering co-ordinated relocation of each piece of equipmentas an activity, we find that this is also a homogeneousproject consisting of a large number of similar activities.Even a small variance against the tight relocation plancould result in delayed installations and affect the cus­tomer's service. This makes the relocation process veryvulnerable, with delays being quite typical. In the par­ticular case we describe, though a detailed schedule formultiple transportation and installation teams was es­tablished, after just 4 weeks the project was more than100 installations behind the schedule. Therefore, a deci­sion was made to subcontract a project managementconsulting company in order to provide additional con­trol over the project implementation. As a result, thenetwork operations center was successfully relocated withminimum disruption to the customer's operations. Thecontrol process included an on-site project managementresource to assist in implementing the relocation plan(assessment of the project status, assigning critical tasksto individuals, reporting status to the management andtechnical expertise). The project management consultingcompany provided control of high efficiency (f3 = 0.9) butcharged at a marginally increasing rate for their services.

All system parameters were approximated with re­gression analysis similarly to that described in the previ­ous example. Specifically, the control effort was measuredin equipment units to relocate and install per week. Thecharge for on-site management set by the consultingcompany was approximated with a quadratic functionC(e(I)) = 120e2(1). The deviation was measured inequipment units not relocated with respect to the initialplan. Losses due to improper serving customers consti­tuted a piece-wise linear function (4), with /+ = t: =

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68 Kogan et al.

5. Project control with linear costs

We begin by studying the problem characterized by piece­wise linear losses and linear costs of contro\. This case istreated in Lemma I.

Lemma I. Given problem (1)-(3) with cost functions (4),the optimal control effort is determined as follows:

• if'

thenCI.

e(r) = pO(I), for 0 ::s: I ::s: 11,

and V(/) = O. However, the transversality condition in (7)requires i{t(T) = O. This implies that there must be aswitching point II before the end of the planning horizon,T, so that the co-state variable ends up at zero, e(/) = 0and V(/) > 0, II < I ::s: T. By integrating the correspon­ding condition from (11) we thus find the co-state variable:

ci{t(I) = -Ii' O::s: I::S: 11; i{t(I) = -I+(T -I), 11 ::s: I::S: T.

To find the unknown switching time point, we simplysolve the two equations for 11:

ci{t(/il = -Ii'

i{t(lil = -I+(T - lil·

• otherwise e(l) = 0for 0 ::s: I ::s: T.

Note that the optimal value of the control effort functionis found under the conditions

Proof. By considering only the control-dependent term ofthe Hamiltonian (6) and substituting the cost functions(4) we obtain:

We thus obtain:C

11 = T- 1+{3 '

The last expression is evidently feasible if

~::s: I+T.

[I' this is not the case then

i{t(0) > -~, ~(/) > 0, V(I) ~ 0,

and, as shown above, e(l) = 0 over the entire planninghorizon is the feasible solution for the stated primal anddual problems.

Finally, according to the maximum principle, a feasiblesolution of the system of primal and dual equations whichsatisfies the optimality condition (8) is an optimal solu­tion. Moreover, due to the fact that the objective functionof our problem is convex and the constraints are linear,the problem is unimodal, and thus the maximum princi­ple presents not only necessary, but also sufficient opti­mality conditions. •

The analysis of the optimal solution derived in Lemma Iprovides some insights into the allocation of the controleffort in typical projects. Specifically, unless the cost of thecontrol effort C is very high in comparison to the losses dueto the increasing deviation 1+ from the planned intensitya(I), the project execution must be controlled and cor­rected. This control (see Fig. I) is always exercised fromthe very beginning of the planning horizon and the lesscostly it is, the longer the portion of the project executionthat should be controlled. The value of the control effort isproportional to the planned intensity and to the ratio be­tween the contribution of the activity intensity CI. and con­trol effectiveness {3. It is interesting to note that there is acertain point in the life cycle of the project, II, beyondwhich it is not worthwhile to exercise any additional con­trol, as the cumulative effect of the additional deviationswill not be very significant. In other words, towards the endof the project, there is not much that is worthwhile doingto bring the project back to track.

(9)

(12)

(10)

(I I)

CI.e(l) = pO(/).

~(I) = O.

ci{t(I) = -P'

By differentiating the last equality we obtain

H(I) = -ce(l) - i{t(I){3e(I).

Expression (9) implies that if

CII = T-­

L+[3

and e(l) = 0 for 11 < I ::s: T;

ci{t(I»-p'

then e(r) = 0, otherwise Equation (8) must be satisfied,that is,

By substituting the deviation loss function from (4) intothe co-state equation (7) we find:

{

rl' if V(I) > 0,

~(I) = -1- if V(I) < 0,

1 E [-1-,1+], if V(/) = O.

From (II) it immediately follows that Equation (10) canbe satisfied only if V(I) = O. By differentiating the lastequality and substituting the state equation (2), we finallyobtain:

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Optima! control in homogeneous projects 69

then e(/) = 0, otherwise Equation (8) must be satisfied,implying that

c!{J(t)=-(j'

Similarly to Lemma I, by differentiating the last equalitywe again obtain Equation (10). The only change we findis a new expression for the co-state equation (7), whichnow takes the following form:

(13)if V(t) 2': 0,if V(t) < O.

a(l)

f3c+---------+--~=-'-

V(I)

If/(t)

e(l)

T

Fig. I. Optimal evolution of the state. co-state and controlvariables in the system with linear cost of control.

6. Project control with non-linear losses

Often, in practice small deviations from the plan are easierto tolerate, while larger deviations result in dispropor­tionally heavier losses, as illustrated by the SE example. Inthis section we consider non-linear deviation losses, ap­proximated by a quadratic function, in conjunction withlinear control costs. The following lemma proves that thisnon-linearity does not affect the optimal control value andthe maximal number of the switching points. In fact,Fig. I still illustrates typical optimal behavior for this casetoo. However, non-linearity does affect the timing for theoptimal control value, which now depends on almost allparameters of the problem: a, Ii, T, C, 1+ and art).

Lemma 2. Given problem (I )-(3) with control cost ac­cording to Equation (4) and deviation losses according toEquation (5), the optimal control effort is determined asfollows:

• if the equationT /

*= 2/+a11a(T)dTdt,II (\

has a non-negative root tl, thena

e(t) = (ja(t), for 0:::; t:::; t" e(t) = 0 for t1 < t:::; T;

• otherwise e(t) = 0 for 0 :::; t :::; T.

Despite this change, Equations (13) and (10) can be simul­taneously satisfied only if Vet) = O. Thus, again we have

ae(t) = part),

and Vet) = 0 fOT 0 :::; t :::; I, and e(t) = 0, Vet) > 0, fortl < t :::; T. Consequently, the single switching point t, canbe found by integrating the corresponding condition from(II ):

T

!{J(t) = -*,0:::; c-: II; !{J(t) = -2/+1V(T)dT,

/

«<r-:». (14)

By taking into account that/

V(t) = 1(aa(T) - {Je(T))dT,

II

we finally find an algebraic equation for the unknownswitching time point, t,:

T /

-!{J(tl)=*=2/+a11a(T)dTdt (15)(\ I]

Note that the right-hand-side of Equation (15) is a non­increasing function in t,. Thus, if there is no non-negativeroot of (l 5) with respect to tl then

c . .!{J(O) > -p' !{J(t) > 0, Vet) 2': 0,

and, as shown above, the feasible solution for the statedprimal and dual problems is e(r) = 0 over the entireplanning horizon. •

We can now apply Lemma 2 to find the switching pointfor the SE example, as follows:

Proof. First note that with the introduction of the qua­dratic loss function, the control-dependent term of theHamiltonian (9) does not change. Consequently, the op­timality condition proven in Lemma I remains un­changed. That is, if

T /

C 10000 1f{j = ----o.s == 21+a a(T)dTdt

II II

18 /

== 2 . 126.3 x 0.2211200dTdt =} I,

11 II

== 16 months.

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70 Kogan et al.

Thus, the optimal solution is

ae(t) = (ja(t) = 55 modules per month,

for 0::; t::; 16, and e(t) = 0 for 16 < t::; T.

7. Project control with non-linear control cost

In this section we consider the combination of the piece­wise linear deviation loss function and the non-linearcontrol cost function. This control function correspondsto the case when additional control efforts may becomeincreasingly more expensive (Meredith and Mantel,1995).

Note that the co-state equation (I I) from Lemma 2 isvalid for the present case. By considering only the con­trol-depcndcnt term of the Hamiltonian (6) and substi­tuting the cost functions (4), we obtain:

fI(t) = -ce2(t) - tjJ(t){Je(t). (16)

Expression (16) implies that if tjJ(t) 2: 0, then e(l) = O.Otherwise Equation (8) must be satisfied, that is

tensity up to that point is greater than the loss due tounfavorable deviations associated with the remainingpart of planning horizon, as stated in Lemma 3.

Lemma 3. Given problem (I )-(3) with control cost ac­cording to Equation (5) and deviation losses according toEquation (4), if

r

J {f/+a a(r)dr 2: -~«T _t)2 - T2) for 0::; t::; T,

o

then

Proof. Consider the case when there cannot be switchingpoints over an optimal solution, that is tjJ(t) < 0 foro::; t < T and V(t) 2: 0 for 0 ::; t ::; T. Then according to(II) and (17) we have:

2c +tjJ(t) = -pe(l) = -I (T -t),

This change complicates the optimal behavior of thesystem significantly (see Fig. 2). Unlike the previouscases, where optimal control was reswitched only onceand was characterized by either zero value (no control) ora function providing exactly zero deviation, this timethere can be difTerent non-zero control functions and alarge number of switching points. This implies that theproblem can be solvable only for very special cases. Thefirst solvable case is obtained when, for any point in timeI during the project duration, the cumulative aetivity in-

r

V(t) = V(O) + J(aa(r) - {J (J!+(~ - r») dro

To find the condition stated in this lemma, we determinewhen the control effort function as defined in (18) results ina feasible behavior of the state variable. We achieve this byintegrating its Equation (2) while requiring that V(I) 2: 0:

•

( 18)e(t) = ffc I+(T - I).

and thus(17)e(t) = - tjJ;l{J > O.

Fig. 2. Optimal evolution of the slate, co-state and control\'a;'iabks in the system with non-linear cost of control.

V(r)

1JI(t)

e(r)

t, t, t J T

Corollary 1. Given problem (I )-(3) with control costaccording to Equation (5) and deviation losses according toEquation (4), if a(l) is a concave function and

aa(O) 2: ffc 1+ T,

then there are no switching points on the optimal solutionalong the planning horizon.

Proof. The proof follows from the no switching pointcondition of Lemma 3, Equation (2) and control function(18). Indeed, given aa(O) 2: (Je(O) then with respect to (18)aa(T) 2: (ie(T) = 0 and hence due to the concavity of a(I),aa(l) 2: (Je(l) for 0 ::; I ::; T. The last inequality along withEquation (2) implies that V(t) 2: 0 along the entireplanning horizon, that is the first condition of Lemma 3 ismet. •

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Optimal control in homogeneous projects 71

The second solvable case is found when there is a singleswitching point such that the cumulative activity intensityup to this point is balanced by the losses for deviationsrelated to the remaining time interval, as stated in thefollowing equation.

e(l) = _!{J(1)/3 = .tl+(T -I) for II ::; I::; T. (23)2c 2c

Finally, to find the switching point we integrate (2) withcontrol (22) and take into account that at these points thedeviation is zero:

II II 2

V(lil = Jaa(l)cll + J~c W(II -I) - I+(T -lil)dl

o 0

Lemma 4. Given problem (I )-(3) with control COSf ac­cording fa Equation (5) and deviation losses according toEquation (4), if the smallest positive rOOI of the function

and

and

(24)

(25)r: /32 t: /32

--<a(I)<-.2ea - - 2ca

Proof. Let the optimal solution to the problem be suchthat:

V(I) = 0 for 0::; I::; II,

V(I) > 0 for II < I ::; T

1+/3e(l) 0= 2c (T - I) for II ::; I ::; T.

Thus, a non-negative root of the last equation withrespect to II is the switching point that we are look­ing for. Note that the found solution is correct onlyif after the switching point V(I)::::: 0 as stated in thelemma. •

The third and last solvable case is found when the in­tensity function .1(1) is not lumpy. We say that a functionis "lumpy" if its rate of change varies widely over smallintervals. More specifically, we say that the activity in­tensity function is not lumpy if its rate of change isbounded by the ratio of the negative/positive deviationlosses to the control cost according to the following re­lationship:

Lemma 5. Given problem (1)-(3) with control cost ac­cording 10 Equation (5) and deviation losses according 10

Equation (4), if/unction a(l) is continuous, almost every­where differentiable and there exists the smallest 1'001 If,

o::; II < T of the equation

a/+(T -I,) = 2c 2 o(II ),

/3

1+ /32 1- /32- 2ca ::; a(l) ::; 2ca '

for all I E [0, T], then the optimal solution is:. a

e(t) = pa(t) for 0 ::; I ::; II

is such Iha I

!{J(I) = !{J(O) - F't . 0::; I::; II, (19)

IjJ(I) = !{J(O) - rt l + 1+(1 - lil, II ::; I ::; T, (20)

!{J(I) = !{J(O) - rt l + I+(T -lil = O. (21)

I

JfJ'1+

aa(r)dr::::: ~(I-lil(2T - (I + II)),

I[

Proof. Let the optimal solution to the problem be char­acterized by a single switching point II:

V(I) ::; 0 for 0 ::; I ::; II, V(I)::::: 0 for II < I < T,

!{J(I) < 0 for 0 ::; I < T,

and thus (17) holds. Based on this behavior of V(I), weintegrate co-state equation (II) as follows:

I+To< II ::; 1_ + 1+ '

Note that according to the constructed solution, !{J(t) < 0and thus from (21) it follows that 1,/- ::; I+(T -lil musthold as stated in the lemma. Next, given !{J(I) along theentire planning horizon, optimal control is defined from(17) by substituting the corresponding equations for !{J(I)«19), (20)) and for !{J(O) (21):

e(l) = _!{J(1){3 = l.W(T -lil - '-(II -I))2c 2c

for 0 ::; I ::; II, (22)

and

for all I E [II, T]' then:

e(l) =: W(T -lil - 1-(11 -I)) for 0::; I::; II,_c

e(l) = :c I+(T - I) for I, ::; I ::; T.

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72

and1jJ(1) < 0 for 0 :s I < T.

Since V(I) is constant at the first interval of time,

exe(l) = pa(I),

must hold at this interval according to (12), and thusaccording to (17)

ex1jJ(1) = -2c za(I).fJ

Due to the co-state differential equation (II), the lastequation is feasible only if

r f32 t: fF-- < il(l) <-.2cex - - 2cex

To find thc co-state variable behavior at the second timeinterval, we integrate Equation (11) as follows:

1jJ(1) = 1jJ(IJl + /+(1 - IJl, II :s I :s T. (26)

Substituting IjJ(T) = 0 and

ex1jJ(IJl = -2c,a(IJl,

IJ"

in (26), we obtain the equation

ex/+(T - II) = 2c,a(IJl, (27)

fJ-lor the switching point II stated in the lemma. The controlvalue at the second interval of time immediately followsfrom (17) and (26). •

Considering the network operations center relocationproject described above, one can verify that conditions ofLemmas 3 and 4 are not met. On the other hand, theproject intensity is evidently not lumpy:

/+IF /-fF- -2- = -13.5 :s b(l) = 4 :s -2- = 13.5.

cex cex

This implies that the optimal solution can be deter­mined with the aid of Lemma 5. if the switching pointdefined by (27) is feasible, i.e., 0 :s II < 16:

ext+-(T-I I ) = 1000(16-IJl =2cfJ2a(IJl

0.25 .= 2.120 0.92 (100 + 41Jl =} II = 6.6 weeks.

Thus. the optimal control for the relocation project is

e(l) = ~a(l) = 27.8 + \.111 equipment units per week for

os I s 6.6,

and

() /+ fJ() ., ke I = 2c T - I = 60 - 3.751 equipment urnts per wee

lor 6.6:S I :s 16.

Kogan et a!.

Using this solution we find the optimal value J* for theobjective function (1):

T

J* = j(L(V(I)) + C(e(I)))dlo

= j /+ [}texa(r) - fJe(I)) dr] dr + j ce2(I)dl

o 0 0

16 I 16 I

= r « j j a(r)drdl -/+fJ j j (60 - 3.75r)drdl

6.6 6.6 6.6 6.6

~6 16

+ j c x (27.8 + 1.111)2dl + j c x (60 - 3.751)2dl

. 0 6.6

= $2042317.

This solution looks quite complex and not very easy toimplement. Therefore, we next compare it with the stan­dard simple, easy to implement approach of control effortconstant in time e(l) = e. To find the optimal controleffort value, we integrate the motion equation, taking intoaccount that /+ = t: for our example, and substitute theresult into the objective function (I):

T

J = .I(L(V(I)) + C(e(I)))dl

o

= 1/+ [I (exa(r) - f3e)d r] dt + ce2T

tl t 2

= -/+ex.f.f a(r)drdl + /+fJe~o 0

T I ,

+ r « .I .I a(r)drdl - /+IJe (T ~ IJl" + ce2T,

II 0

where II is determined by

II

V(tl) = .I(rta(r) - fJe)dr = O.

oHere we assumed that, for the optimal solution withconstant control, the following hold: V(I) < 0 before IIand V(I) > 0 after II. Then, this solution satisfies thefollowing necessary and sufficient condition of optimality:

8J = 2ceT _ /+fJ (T - IJl2 + /+fJ1 = 0 =}

8e 2 2

e = /+ fJ((T - IJl2 - If) = 0.12(256 _ 3211)'4cT

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Optimal control in homogeneous projects

In order to find t1, we substitute the value of the controleffort found above into

"J((1.a(r) - fJe)dr = 0,

owhich for our example results in

"J((25 + t) - (27.6 - 3.46t1))dl = 0 '* tl = 0.66 weeks.

o

This implies that t1 is feasible, e(t) = 28.2 and this is indeedoptimal when no change of control in time is allowed.

Using this solution we obtain the optimal value for theobjective function (I) with constant control effort:

T

J = J(L(V(t)) + C(e(t)))dt

o

= j1+ [}((1.a(T) - pe(t))dT] dt + } ce2(t)dt

o 0 0

0.66 t 0.66 t

= -1+(1. JJa(r)drdt + I+P JJ28.2drdto 0 0 0

16 I 16 f

+ 1+(1. J/ a(r)drdt - l+fJ.I .l28.2drdt0.66 0 0.66 0

16

+Jc x 28.22dt

o= $2164980.

Finally, one can observe, that even if the optimal valueof constant control effort is applied in the network centerrelocation example, there will be an amount of additionalIV = J - J* = $122663 spent on the project in compar­ison to the approach suggested in this paper.

Corollary 2. Given problem (I )-(3) with control costfunction according to Equation (5), deviation loss functionaccording to Equation (4), if condition (25) holds at K in­tervals of time ts, k = I, ... ,K, then the optimal solutioncontains at most K time intervals

{tlV(t) = 0, e~ = a(t) } s:;; Tk,

otherwise V(t) of 0 at any interval of time and optimalcontrol e(t) is a piece-wise linear function of time.

Proof. The proof immediately follows from Lemma 5.•

Remarks. From Lemmas 3 and 4 it follows that, unlessspecial relationships between system parameters hold orthe activity intensity function has a special form, deter­mined as, for example, in Corollary I, an optimal solu-

73

tion for problem (1)-(3) with control cost from (5),deviation losses from (4) and arbitrary activity intensitya(/), contains multiple switching points or even switchingintervals as determined in Corollary 2. To find such apoint, one needs to solve a system of equations corre­sponding to Equations (24) and (27). However, Equa­tions (24) and (27) are non-linear, and therefore in thegeneral case, will not be polynomially solvable. Thus, ifno feasible solution is found analytically for the zero orone switching point case as defined in Lemmas 3-5, thenthe problem can only be solved numerically. Conse­quently, this problem, unlike the other two considered inthe previous sections of the paper, is not always analyti­cally solvable. •

8. Management implications

The model presented in this paper serves to plan the al­location of resources in order to carry out the manage­ment control function during project execution. There is amain difference between the cases when control costs arelinear and when they are not. If control costs are linear,then we know that either there will be no need at all forcontrol, or that control will be done differently in twoperiods: in the first part of the project, up to a certainpoint in time, the extent of control effort will be pro­portional to the activity intensity, and afterwards noadditional control will be economically justified. Thelength of the period of time up to the point where controlstops increases with the duration of the project and de­pends on the cost coefficient of unfavorable deviationlosses 1+, the control costs c, and the control effectivenesscoefficient fJ. It is evident that the duration of the controlperiod does not depend on the cost coefficient of favor­able deviations 1-, because the optimal solution does notallow for favorable deviations over the entire project lifecycle.

Within the case of linear control costs, there is an im­portant distinction between linear and non-linear devia­tion losses. The optimal control effort over time does notdepend on the actual activity intensity function whendeviation losses are linear, while it does depend on itwhen the deviation losses behave in a non-linear manner.The implication is that when losses are linear, changingthe project plan does not affect the timing of the point atwhich control stops. Of course, the actual amount ofcontrol effort does depend on the activity intensity, asmentioned above. This allows the manager to carry outon-line adjustment of control effort if the activity inten­sity unexpectedly changes, while maintaining the point intime when control stops altogether.

When control costs are non-linear, we can only treat thesituation when losses due to deviation are linear. Theanalysis of this case is based on Lemmas 3,4 and 5, and isillustrated in Fig. 2. First of all, in contrast to the previous

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case, we distinguish between lumpy and not lumpy activityintensities as determined by condition (25). II'activity in­tensity changes significantly, i.e., it is lumpy (condition(25) is not met) as shown in Fig. 2, then the optimalamount or control effort is always a linear function oftime, and is exercised from the beginning or the project tillthe very end. Furthermore, the rate of change of controleffort is proportional to the ratio between the effectivenessor the control en'ort!1 and its cost c.

Another major difference between this case and thelinear control cost case is that both favorable and unfa­vorable deviations are possible, and thus it may not beeconomical to follow the plan precisely. The optimalsolution prescribes that control effort should be allocatedwhile taking into consideration significant future peaksand valleys in the lumpy activity intensity. During periodsor low activity intensity, management control effortsshould be intensified linearly to accumulate favorabledeviations in advance, in order to absorb unfavorabledeviations that will occur during upcoming activity in­tensity peaks. These unfavorable deviations will eventu­ally cancel out the favorable ones, and when the project isbrought back to the plan, the management policy isreversed, and control effort is diminished linearly inanticipation or upcoming activity valleys. Under non­linear deviation losses, this switching behavior thatalternates between favorable and unfavorable deviationsturns out 10 be more economic than a policy of expendingall the control effort necessary to follow the planprecisely.

On the other hand, if activity intensity is not lumpy,more exactly, if there are K intervals of time where con­dition (25) is met, there can exist up to K periods wherethe control should be exercised so as to provide no de­viations of planned intensities at all. Between such in­tervals, however, the optimal control will be similar tothat described for lumpy activity intensities.

Even if an analytical, off-line solution is not availablefor a particular activity intensity function, a feedbackpolicy could be effectively employed for on-line control ofproject execution and real-time decision making. Thispolicy should be based on the principles described above:increase control effort during periods of favorable devi­ations, switch over when deviation is zero, and decreasecontrol effort during periods or unfavorable deviations.Finally, project managers should be aware that projectplans that involve substantial fluctuations in activity in­tensity arc very likely to require significant changes incontrol procedures.

9. Concluding remarks

In this paper we extended our understanding of the in­terplay between the amount of work required to accom­plish the objectives of the project, and the control effort

Kogan et al.

needed to ensure that this work would be carried outaccording to plan.

We developed and solved a basic model for determin­ing the optimal amount of control effort that should beinvested throughout the life cycle of the project. Themodel accounts for changing levels of project activity andincludes two parameters that represent the effectivenessof the work management and control efforts. The objec­tive is to trace project execution as closely to the plannedintensity as possible while minimizing the cost of thecontrol effort and of the losses due to deviation from theplan. Lemmas I and 2 present two analytically solvablecases found when the cost of control is linear. Lemmas 3,4 and 5 lead to important managerial insights regardingthe optimal amount of control effort when the cost ofcontrol is non-linear. They also determine the conditionwhen the problem with quadratic cost of control is ana­lytically solvable and define a class of activity intensityfunctions satisfying this condition.

Future work in this area could proceed along twoparallel avenues: collection and analysis of empirical datain order to estimate the values of the various parametersof the model, and refinement of the model to reflect morecomplex project environments, such as those with feed­back from the cumulative deviations to the effectivenessparameters or to the activity intensity rate.

References

Bent, J.A. (1988) Project Management Ham/hook, 2nd edn., Cleland,D.l. and King, W.R. (eds). Van Nostrand-Reinhold, New York.p. 579.

De Falco, M. and Macchiaroli, R. (1998) Timing of control activities inproject planning. International Journal of Project Management, 16(1).51-58.

Dixit, A.K. (1990) Optimization in Economic Theory 2nd edn., OxfordUniversity Press, New York.

Khmelnitsky, E., Kogan, K. and Maimon, O. (1995) A maximum prin­ciple based combined method for scheduling in a flexible manu­facturing system. Discrete Event Dynamic Systems, 5, 343-355.

Kogan, K. and Khmelnitsky, E. (1996) An optimal control model forcontinuous time production and setup scheduling. InternationalJournal of Production Research, 34(3), 712-725.

Kogan, K. and Shtub, A. (1999) Scheduling projects with variableintensity activities: the case of dynamic carli ness and tardinesscosts. European Journal of Operational Research, 118(3), 65-80.

Leachman, R.C., Dincerler, A. and Kim, S. (1990) Resource-con­strained scheduling of projects with variable intensity activities.IlE Transactions, 22(1), 31-39.

Meredith, J.R. and Mantel, SJ. (1995) Project Management - AManagerial Approach, Wiley, New York.

Partovi, F.Y. and Burton, J. (1993) Timing of monitoring and controlofCPM projects. IEEE Transactions on Engineering Management,40( I). 68-75.

Turner, J. R. (1999) The Ham/hook of Project Based Management,McGraw-Hili, London.

Weglarz, J. (198 I) Project scheduling with continuously-divisible,doubly constrained resources. Management Science, 27(9), 1040­1053.

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Optima! control in homogeneous projects

Biographies

Konstantin Kogan holds a Ph.D. in Industrial Engineering from theCentral Institute of Mechanization and Power Engineering, Moscowwhere he worked from 1985 as a senior researcher. In 1990 he joinedthe Department of Industrial Engineering at Tel-Aviv University andfrom 1995, the Department of Computer Sciences, Halon Center forTeehnological Education affiliated with Tel-Aviv University. Since2000 he has been an Associate Professor at Bar-llan University, De­partment of Interdisciplinary Studies - Logistics. Konstantin Koganhas published over 40 research papers in refereed international jour­nals. His research interests are mainly in the area of production con­trol, scheduling and maintenance.

Tzvi Raz holds B.Sc, M.A.Se and Ph.D. degrees in Industrial andManagement Engineering. He is a faculty member with the Leon Rc­canati Graduate School of Business Administration at Tel Aviv Uni­versity. Previously, he managed a technology insertion program at an

75

IBM software development laboratory, and was on the IndustrialEngineering faculties of the University of Iowa and Ben Gurian Uni­versity. Professor Raz has published over 50 research papers in refereedinternational journals and is on the editorial boards of Computers andOperations Research and the Project Manugernent Journal.

Ramy Elitzur is an Associate Professor and the Executive Director,MBA Programs at the Rotman School of Management. The Universityof Toronto. Previously he taught at the Rccunati Graduate School ofBusiness Administration, Tel Aviv University and The Stern Sehool ofBusiness Administration, New York University. Ramy Elitzur earnedhis Ph.D. and M.Phil. degrees from The Stern School of BusinessAdministration, at New York University and an MBA from the Rc­canati School of Business, Tel Aviv University. Rarny's area of re­search is game theory applied to financial situations.

Contributed by the Manufacturing Systems Control Department

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