Optimal Crashing and Buffering

of Stochastic Serial Projects

by

Dan TrietschISOM Department

University of AucklandOld Choral Hall

7 Symonds StreetAuckland, New Zealand

++649-373-7599 ext 87381fax: ++649-373-7430

d.trietsch@auckland.ac.nz

February, 2004

Optimal Crashing and Buffering

of Stochastic Serial Projects

by

Dan Trietsch

January, 2004

Abstract

Crashing stochastic activities implies changing their distributions to reduce the mean. This

can involve changing the variance too. We consider optimal crashing of serial projects

where the objective is to minimize total costs including crashing cost and expected delay

penalty. As part of the solution we determine optimal project buffers. We solve for

activities that are statistically dependent because they share an error element (e.g., when

all durations have been estimated by one person, when weather or general economic

conditions influence many activities, etc). The model requires minimal direct input from

process owners--a one point estimate per activity is sufficient.

Project Management: PERT/CPM stochastic Crashing, Project buffer;Production:

Stochastic lead times

1. Introduction

We study optimal or near-optimal crashing of stochastic activities in basic projects

with n (not necessarily independent) activities in series, and without intermittent idling.

Our purpose is to develop insights for more realistic project networks, where such chains

are often embedded. We assume continuous crashing with a linear cost per unit as in

classical CPM (e.g., see Fulkerson, 1961). A due date is also given, and delay is penalized

at a proportional rate, P. The objective is to minimize the total cost of crashing plus the

expected delay penalty. (We refer to this as thedefault objective.) If we define the

difference between the due date and the mean completion time as theproject buffer, then

optimal crashing yields optimal project buffers. For this reason we consider the two issues

together.

To review the literature related to stochastic crashing of project activities, unless

stated otherwise, the following statements are implied: (1) The model addresses general

PERT networks with statistically independent stochastic activities. (2) Crashing is

continuous. (3) The default objective is pursued (with the "cost of crashing" interpreted

according to the context). An alternative, to which we will refer explicitly as the

expectation objective, is to either minimize the expected duration given a crashing budget

or achieve a required expected duration most economically (the ability to solve one

suffices for the other). The expectation objective is a special case of the default objective,

obtained if we set a due date of zero and adjust the penalty rate. Finally, (4) no intentional

idling occurs--each activity starts as soon as it becomes feasible. In these terms, our own

problem is "serial networks with dependent activity distributions."

Arisawa and Elmaghraby (1972) considered GERT networks where only exclusive-

or nodes are allowed (thus excluding general PERT but including the serial case) with an

expectation objective, and presented a tractable fractional LP solution. Britney (1976)

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explicitly addressed both buffers and crashing with a sum of several default objectives as

the combined objective, and with possible idling between all activities. His buffers apply

to activities one-by-one and planned idling occurs unless the buffer is exceeded. Wollmer

(1985) pursued an expectation objective where crashing applies only to activity means. He

modeled stochastic variation by discrete random variables, and showed that this yields a

convex programming model. Gutjahr et al. (2000) considered discrete crashing. They

solved by a stochastic branch and bound approach that requires extensive simulation and

the use of a lower bound instead of the true objective value. This yields, therefore, a

computationally intensive approximate solution.

In general, it is not necessarily optimal to start all non-critical activities as soon as

possible. Usually, one can associate a holding cost with completed activities, and by

postponing them the total holding cost will be reduced. Therefore, postponement is

mathematically equivalent to negative crashing, and advancement yields positive crashing.

Thus, models that deal with the optimal start time of activities are directly relevant to our

subject. Ronen and Trietsch (1988) discussed the optimal times to place orders for n

project items, where the latest one determines the project’s completion time. Highly

overlapping results were developed independently by Kumar (1989) and by Chu et al.

(1993). Hopp and Spearman (1993) addressed the same problem with a step delay penalty

function (i.e., with a deadline). Yano (1987a) and Yano (1987b) analyzed a serial supply

chain with stochastic activities, with and without planned idling. Elmaghraby et al. (2000)

considered n activities that require the same resource. They allowed idling between

activities and solved for the best sequence by dynamic programming. Elmaghraby (2000)

considered renewable and non-renewable resource allocation, with deterministic or

stochastic resource requirements. He employed a dynamic programming approach that is

suitable for making dynamic decisions during a project. In the stochastic case (but not in

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the deterministic case) he specified an expectation objective, and proposed using the

solution with different parameters to study the implications for the default objective.

Finally, when projects are long, our default objective does not necessarily maximize net

present value. Elmaghraby and Herroelen (1990) discussed the state of the art in the

pursuit of maximal NPV, mostly for the deterministic case. Buss and Rosenblatt (1997)

maximized the expected NPV for a stochastic case with exponential activity time

distributions.

The idea of using planned buffers to achieve improved reliability and avoid

[implicit] project delay penalties had recently been popularized and many practitioners find

it useful. See, for example, Herroelen and Leus (2001) or Leach (2000). The size of the

protective buffer in this method is determined, quite arbitrarily, as a constant fraction of

the expected chain leading to the buffer. Leach (2000) suggested the maximum of a buffer

based on the traditional independence assumption and the constant fraction. Leach (2003)

listed 11 reasons why projects are typically longer than expected. He then suggested a

buffer based on the sum of the two elements mentioned before. Since for any positive

numbers the sum exceeds the maximum, Leach (2003) is more conservative than Leach

(2000). On the one hand, both are quite arbitrary in two senses: First, there is no

theoretical derivation. Second, there is no clear method how to tailor buffer sizes to

individual users. Thus, the theoretical question how to achieve the best buffer remains

open. On the other hand, arguably, both options are better than the prevailing alternatives.

Elmaghraby et al. (1999) studied the effect of changing the mean of an activity

(crashing) on the project variance, and highlighted some counterintuitive results that may

obtain. E.g., sometimes the project variance fluctuates up and down during progressive

crashing of some activity (due to a different mix of paths contending to be critical). They

also provided references for the effect of changing the variance of activities on the project

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mean and on the project variance. In all these cases counter-intuitive results may follow.

Only one result is completely intuitive: reducing the mean of an activity cannot increase

the mean of the project. (One may conclude that it is important to use an objective

function, such as ours, that can balance the costs and benefits associated with changing the

project mean and the project variance.)

Finally, Clark (1961) provided approximate calculations for the project duration

distribution. His model is based on the normal approximation, and its main strength is that

it can handle dependent random variables. Specifically, he modeled the effect of shared

activities on different paths by statistical dependence. Thus he obtained an approximation

that does not suffer from the bias associated with ignoring this effect. Clark did not

consider crashing, however (historically crashing was part of CPM, while Clark was on

the PERT team). It is interesting to note that most textbooks ignore Clark’s original

contribution and, unfairly, attribute to PERT an inferior calculation.

The results cited leave several questions open. Specifically, there is no explicit

treatment of cases where the variance changes during crashing. Even more importantly,

there is no explicit solution of the case where activities are dependent. This paper makes

three theoretical contributions to correct these shortcomings. We also discuss how to

collect data and estimate the parameters necessary to implement the main model, and

show that this places less burden on users than traditional PERT (single point estimates

are recommended instead of three-point estimates).

Sorted by importance, the first theoretical contribution is that we consider the

effects of systemic errors that apply to all activities, leading to a model where activities

are positively correlated. We model systemic error and show how to plan crashing under

the assumption that such error exists. This leads to a positive asymptotic lower bound on

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the ratio between the project buffer and the expected duration. In contrast, under the

independence assumption, this ratio approaches zero, which is highly counterintuitive.

Our second theoretical contribution is solving the case where the variance changes

along with the mean. Specifically we show that if the variance decreases with the mean,

then this calls for more crashing than would be suggested otherwise, and vice versa if

variance increases. We show by numerical examples that the difference can be substantial.

(Sequentially, this contribution comes first.)

Our third contribution concerns convexity. On the one hand, we show that if the

crashing cost function of each activity is convex and the standard deviation is a convex

function of the mean, then the objective function is convex (even though activities may be

correlated due to systemic error). On the other hand, we present a plausible model for

estimating the variance of project activities that can violate this convexity condition.

Finally, consider that in the analysis of general project networks, the first step is

often to reduce every embedded simple chain by convolution, as if independence applies.

Our results suggest that this may be ill advised. Thus our limited model can be used as

input for more general models, to correct this error. We leave for future research how to

apply the maximum operator to parallel correlated activities and how to address irreducible

structures, but note that the approximations suggested by Clark (1961) are directly

applicable. It is also likely that simulation would be necessary in those cases where

Clark’s approximation is not adequate. Simulation may also be preferred for convenience,

especially for practical applications. But it must take into account dependencies of the type

we discuss in this paper. Variance effects also need to be considered. Thus, even in its

present limited form, this paper provides important insights for the correct analysis or

simulation of general project networks.

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The remainder of the paper is organized as follows. Section 2 formulates the

models we address. Section 3 assumes statistical independence without variance effects, to

solve the basic model. In Section 4 we add variance effects. Section 5 presents the main

model (with correlation) and obtains the convexity result. Section 6 discusses how to

estimate the necessary model inputs. Section 7 discusses other loss functions, including the

case of a deadline. In Section 8 we discuss the implications of the model for the

determination of the due date. Section 9 provides numerical examples. Section 10 is the

conclusion.

2. Model Formulations

Let a project consist of n activities in series. We denote the activities by Yi, where

i=1,...n. Let Fi(a) be the CDF of the activity time. µi denotes the expected duration of Yi

(e.g., in weeks) and let µ=Σµi be the project’s expected duration (except in Section 6,

project parameters have no subscripts in our notation). Similarly,σi is the standard

deviation of Yi. Ci is the cost of crashing µi by one unit (e.g., if we can reduce µi by 0.2

weeks at a cost of $50, then Ci=$50/0.2wk=250$/wk). Let∆i denote the crashing amount

(by which the mean is decreased) such that∆i≤di (in our example di=0.2weeks), then the

cost is∆iCi. Let the project start at time zero with a due date, D, and a penalty of P per

week beyond D; i.e., the delay penalty is given by P(Y-D)+, where Y=ΣY i and

h+=max{0,h}. The objective is to minimize the total cost (TC),

We refer to the vector of planned crashing amounts, {∆i}, as thecrashing plan. Let

SL denote the service level, i.e., the probability the path will complete in time, then

SL=F(D). As a rule, we use stars to denote optimal values, e.g., SL* is the optimal service

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level that is associated with the optimal crashing plan. (While our objective is to find the

best crashing plan and project buffer, it is often convenient to do so by focusing on SL*.)

In the next three sections we solve three models.Model 1 assumes that activities

are statistically independent and crashing consists of shifting Fi(y) by a constant to the

left, thus decreasing µi by the crashing amount. We refer to this assimple crashing.

Another assumption, necessary for convexity, is that the cost of crashing is monotone non-

decreasing with∆i. Model 2 is based on Model 1 but allows for the possibility that the

distribution will change, and in particular that the standard deviation will change along

with the mean. We focus on the case where the standard deviation is a convex function of

the mean.Model 3 is based on Model 2 but without statistical independence. Instead, it

assumes that all activities are subject to a systemic error.

3. Model 1: Simple Crashing of Independent Activities

Under simple crashing the variance does not change and the project duration CDF

is shifted by the total crashing amount but retains its shape. In our analysis we also

assume that the cost per unit of crashing is constant. It is practically immediate to extend

this to a case with a non-decreasing crashing cost function, but we omit the details. When

Ci is viewed as a function of µi (instead of∆i), this is equivalent to a non-increasing cost

function (which is likely if activities are subprojects).

In this and the next section, by the independence assumption,σ2=Σσi2. Similarly,

since we have Fi(y), we can obtain F(y) by successive convolutions (often the normal

approximation may be adequate). For any given crashing plan, we can update F(y) and

thus update SL. For any such plan, let C=P(1-SL). C has a simple economic interpretation:

it is the marginal benefit associated with crashing the project further. To see this consider

that crashing by an infinitesimal amountδ reduces the delay penalty by Pδ with a

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probability of (1-SL), and makes no difference otherwise. So it reduces the expected delay

penalty by Pδ(1-SL). This leads to a necessary and sufficient optimality criterion: Suppose

we know C*=P(1-SL*), then any activity, say i, with Ci<C* should be crashed fully, any

activity, say j, with Cj>C* should not be crashed at all, and if there exists an index k such

that Ck=C*, then activity k should be crashed by some value between 0 and dk (inclusive),

until SL* is obtained. In Section 5 we prove a more general convexity result which

implies here that there always exists a crashing plan that satisfies the criterion. If all the Ci

values are distinct, then there is exactly one optimal crashing plan. If we sort all activities

by increasing Ci, we can identify an optimal crashing plan by a simple search (we omit

further details). We refer to this solution asProcedure 1.

The result is essentially identical to the newsboy model, using C as the "long" cost.

There are two main versions of the newsboy result, and perhaps the better known one uses

the form SL*=P/(P+C), while we are using the form SL*=(P-C)/P. But if "P+C" is

interpreted as "P" then "P" must be interpreted as "P-C" and the two results are identical.

4. Model 2: Crashing with Variance Effects

When crashing is not necessarily simple, the standard deviation of an activity may

change during crashing, so there exists a function,σi(∆i), whose derivative (when it exists)

yields the rate of change ofσi during crashing.Model 2 assumes thatσi(∆i) is convex.

The same applies toσi(µi) (since∆i+µi remains constant during crashing). For example, let

σi(µi)=σ0+aµi, whereσ0 is some non-negative value and a∈ R but such that if a<0σi≥0 for

any feasible µi. (If dσi/dµi>0, e.g., negative a in the example, then the variance decreases

during crashing.)

The following equation provides two ways to look at the expected penalty, EP (and

we will switch between them as convenient),

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The right hand side says that EP can be calculated by P times the area above the CDF

from D onwards. Let Z=(Y-µ)/σ be the standardized project duration and let G(z) and g(z)

be its CDF and density function (we may refer to these as G and g). Hence G(z)=F(zσ+µ)

and g(z)=σf(zσ+µ). A simplifying assumption is that crashing does not change the shape

of the final project distribution. In other words, only µ andσ will change, while G and g

remain the same for any z. This assumption is mild in practice because often we can

assume a normal chain distribution, and thus the present assumption would hold. (For the

normal variate we’ll substituteΦ andφ for G and g.) Rewriting Equation 1 in terms of G

and g we obtain,

where t=(D-µ)/σ is the standardized due date. For Model 1 we holdσ constant and

crashing reduces EP by changing t as a function of µ. But here, crashing influences EP in

two additional ways: 1)σ multiplies the integral and crashing changes it by changingσi,

2) t is a function ofσ, and EP is a function of t. The following basic results play a role,

Using these results and the Leibnitz rule, we obtain

and if we group together the elements involving∂σ/∂µi,

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Using Equation 2 we can show that

where the final equality follows by applying Bayes’ Rule to the conditional expectation

E(Z|Z>t). This is well-defined only for 1-SL>0, but any reasonable search procedure will

stop before violating this condition, and for the normal distribution, in particular, it is

always guaranteed. By combining Equations 3 and 4 and considering the cost of crashing

we obtain

where TC is the total cost. Equivalently, the expression yields the equilibrium condition,

which is achieved at the optimum where the gradient is zero. It should be noted that µi

decreases as we crash, so when Equation 5 is positive, crashing is beneficial.

An important special case is when we approximate the project completion time by

a normal variable with µ=Σµi, σ2=Σσi2. Portougal and Trietsch (2001) discuss a similar

problem in a different context (where the influence on the mean and variance is by

sequencing in a shop environment rather than by crashing, and the comparison is between

distinct distributions). They found that using the normal approximation even in cases

where the distribution was far from normal caused small errors of a few percentage points

in calculating the expected value of the objective function. But, more importantly, the

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difference in terms of the objective function itself (i.e., the cost increase associated with

using the approximation) was much smaller, typically less than one percent. In other

words, even in cases where the normality assumption caused some error in estimating the

expected penalty, it did not lead to bad decisions in terms of sizing buffers! The flatness

of the total cost function around the minimum is one of the reasons for this, and another

reason is that while distributions in this context are not truly normal they are often not far

from normal. Therefore, the recommendation to use this approximation can be made with

confidence. When this approximation is invoked, we can substitute the appropriate value

of E(Z|Z>t) for the normal distribution in Equation 6. A remarkable feature of the normal

distribution is that (1-SL)E(Z|Z>t)=φ(t). This leads to E(Z|Z>t)=φ(t)/(1-SL), but

1-SL=Φ(-t), and thus,

Equation 7 is explicitly circular (and thus requires iterations): we have 1-SL, orΦ(-t), on

both sides of the equation. This circularity appears because we solved E(Z|Z>t) explicitly.

Of course, the same circularity is implicit in Equation 6 for any other distribution as well.

We refer to the values

asmodified costs. For the normal distribution we can use

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If ∂σ/∂µi≥0, i.e., the variance decreases during crashing, then MCi≥Ci>0 and the

positiveness of the denominator is guaranteed, but if∂σ/∂µi<0 there is no such guarantee.

The economic interpretation of 1+(∂σ/∂µi)E(Z|Z>t)=0 is that the variance effect cancels

out the benefit of crashing (and thus we should not spend any resources for such

crashing). And if 1+(∂σ/∂µi)E(Z|Z>t)<0, crashing Yi is detrimental, and should not be

considered. Thus, if MCi is not defined then∆i*=0. Note now that simple crashing implies

MCi=Ci. Otherwise, either MCi<Ci (when∂σ/∂µi>0), or MCi>Ci (when∂σ/∂µi<0).

Proposition 1: ∂σ/∂µi>0 implies more crashing is beneficial than under Model 1 (higher

service level); symmetrically,∂σ/∂µi<0 calls for less crashing (lower service level).

In the next section we will show that Model 3 (and thus also Model 2) yields a

convex total cost function. So we can find an optimal solution by standard non-linear

search procedures. We refer to the use of any effective search for this purpose as

Procedure 2. Procedure 2 can be used as a heuristic even if the convexity conditions of

Model 2 do not hold, but in such a case we can only guarantee a local optimum.

5. Model 3: Positive Dependence Due to Systemic Error

Most academic sources treat activity durations as statistically independent. But if

we increase the number of activities along the chain, the independence assumption leads to

so-called "optimal" project buffers that, when expressed as a fraction of the mean,

approach zero asymptotically (Leach, 2003). In addition to leading to such a highly

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counterintuitive conclusion, the independence assumption should be challenged for another

reason: Even when activities are independent by nature, theirestimated durations are often

subject to the same estimation error. For example, when the same optimist estimates many

activities, they will all tend to be underestimated. Equivalently, pressure by management

may influence people to give too short estimates that cannot be met later. Similarly, bad

weather during the project or the loss of a key employee may impede several activities. A

booming economy may increase queueing time for more than one activity, etc. The same

or similar causes may also apply in the reverse direction; e.g., pessimists will tend to

overestimate durations (and severe punishment for missing due dates in the past will turn

optimists to pessimists). The result is a systemic bias across a project. But the magnitude

of this bias is a random variable and even its orientation is not known in advance. This

introduces a strong dependence as far asdeviations from plan are concerned. But,

operationally, the only variation that concerns us is relative to plan, so this is exactly what

counts! (Although one of the 11 causes of bias identified by Leach (2003)--namely,

omission of some ingredients of activities during estimation--contributes to systemic error,

we are effectively talking about a 12th cause here.)

In this section we introduce Model 3 for this type of dependence, and in the next

section we discuss how to collect and estimate data for its application. (Until further

notice, however, we assume that we already have estimates for all the necessary

parameters.) LetX={X i} be a vector of estimates of {Yi}, and note that we treat Xi as a

random variable. For convenience we will say that the nominal estimate is equal to

νi=E(Xi), and let V(Xi) denote the variance. Systemic error is modeled by the introduction

of an additional independent random variable, B, which multipliesX to obtain the true

activity times. We will useβ and Vb=σb2 to denote the mean and variance of B. If the

true activity times compose the random vectorY={Y i}, this implies Yi=bXi (where b is

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the realization of B that applies to a particular project). For consistency with the former

models, we reserve µi andσi2 for the mean and variance of Yi. In Models 1 and 2 we

assumed crashing applies without systemic error, e.g., Ci was the marginal cost of crashing

µi by one unit. Here, we must recognize that crashing information, by nature, applies to

the estimates,νi, and not directly to µi, the true mean. Thus the true crashing cost is bCi.

Because we do not know b, however, we useβCi instead. While knowledge of b would be

beneficial in terms of reducing variance, we can indeed reduce µi by one unit on average

at the marginal cost ofβCi, so the model remains correct. As in Model 1, we assume that

the crashing cost is monotone non-decreasing with∆i. As in Model 2 we assume that the

standard deviation of Xi is a convex function of∆i (where∆i is interpreted as a planned

amount, leading to B∆i in reality).

In broad terms, the randomness of Xi relates to chance events that are specific to

activity i; e.g., problems with raw materials or power supply.X also includes some (but

not all the) effects of randomness in estimation, since estimates are the result of processes

that are not deterministic. Even with the exact same data, different people at different

times may come up with different estimates. For example, they may forget or neglect

different aspects of the job. These two sources of variance--activity-specific chance events

and some estimation errors--are confounded with each other, so they must be represented

by the same random variable. In contrast, B models effects that are common to all

activities, such as pressure to produce attractive estimates quickly (leading to optimistic

bias and omissions), weather, economic conditions, etc. Note that B is important even if

β=1, i.e., if we take steps to remove bias on average, and thus achieve accuracy, B would

still capture relevant information about systemic imprecision. Specifically it will capture

those estimation errors that are not confounded with activity-specific chance events.

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Because B andX are independent, µi=βνi. But the multiplication by the same

realization, b, introduces [positive] dependence between the elements ofY. Specifically,

σi2=β2V(X i)+V(X i)Vb+Vbνi

2, and COV(Yi,Yj)=Vbνiνj; ∀ i≠j. We can separateσi2 to two

parts,β2V(X i)+V(X i)Vb and Vbνi2. The former equals E(B2)V(X i) and the latter is a

special case of Vbνiνj. Thus the VarCovar matrix ofY is the sum of a diagonal matrix

with elements V(Xi)E(B2) and a full matrix with elements Vbνiνj; ∀ i,j. The latter can be

expressed as the vector product {σbνi}{ σbνi}T. Finally, note that if B=1, we obtain Model

2. If B is deterministic butβ≠1, then after correcting for the bias we again obtain an

instance of Model 2. Therefore, Model 3 is a generalization of Model 2.

Theorem 1: σ2=E(B2)Σ∀ iV(X i)+(σbΣ∀ iνi)2.

Proof: Let V denote the Var-Covar matrix of Y, and let1 be a column vector in Rn with

elements of 1, then the result is obtained directly by the matrix product1TV1.

Corollary: Let q1(ν)=E(B2)Σ∀ iV(X i), and let q2(ν)=(σbΣ∀ iνi)2, whereν={νi}, then

Note that the central element in the inequality isσ. By the corollary, if we wish to

specify a buffer of kσ for some k>0, the two approaches provided by Leach (2000) and

Leach (2003) provide lower and upper bounds. (There is no theoretical reason to limit

ourselves to k>0, so we do not limit our analysis to this case. Nonetheless, most project

managers are uncomfortable with negative buffers and the low service levels they entail.)

The following lemma, although cast in more general terms, shows that if the square

root of each of several additive variance components is convex, then the standard

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deviation is convex. For this purpose interpret r() as the square root function and qk() as a

variance component.

Lemma 1: Let qk(x), k=1,2,...,K, be K functions from Rm to R and let r() be a monotone

increasing function from R to R such that r-1() exists. If

for any admissible x1, x2 andλ, then

Proof: Apply r-1() to the K conditions to obtain

Combine these K inequalities to one and apply r() to the sum. Since r() is monotone

increasing, the inequality is preserved.QED

Lemma 2: Let σ be a positive convex function of the vector {µi}, then the expected

penalty as a function of the vector {µi} is convex.

Proof: From Equation 5 (which applies for Model 3 since we made no independence

assumption to derive it) we obtain the gradient of EP,

Let Q={q i,j} be the hessian of EP, then qi,j is the partial derivative by µj of this

expression. We have to show thatQ is positive semi-definite (PSD). Note that

d[1-SL]/dt=-g(t), d[(1-SL)E(Z|Z>t]/dt=-tg(t) (see the last part of Equation 4), and

∂t/∂µj=-(1+t∂σ/∂µj)/σ. This leads to the following expression for all i,j,

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SeparateQ to a sum of two symmetric matrices,Q1 andQ2. Q2 includes the second

partial derivatives ofσ by {µi} (which are multiplied by P(1-SL)E(Z|Z>t)/σ). Q1 includes

the remainder.Q2 is PSD because P(1-SL)E(Z|Z>t)/σ is non-negative andσ is a convex

function of {µi}. Q1 is PSD if XTQ1X≥0 for any vectorX=(x1,x2,... xn)T. But Pg(t)/σ is

non-negative so this condition is demonstrated by,

Finally, Q is the sum of two PSD matrices, and thus it is also PSD.QED

Theorem 2: The total cost function is convex.

Proof: By showing that the crashing cost is convex and that the conditions of Lemma 2

are satisfied. This requires showing (1) that the crashing cost is a convex function of µi

and (2) thatσ is a positive convex function of the vector {µi}. However, the vectors {µi}

and {νi} are proportional to each other, so we can show the conditions in terms of {νi}.

(1) is true by the conditions we set for Ci (as in Model 1). For (2), using the notation of

the corollary, we first show that the square root of q1(ν) is convex. q1(ν) is the sum of n

elements E(B2)V(X i), and by invoking Lemma 1 we see that it suffices to show that the

square root of V(Xi) is convex. But this is true by assumption (as in Model 2). The square

root of q2(ν) is linear, and thus convex. Therefore, by Lemma 1,σ(ν)--the square root of

q1(ν)+q2(ν)--is also convex.QED

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When there is no systemic error, our assumptions in this section reduce to those of

Model 2, so the convexity result also holds there. As a special case of Model 2, the same

applies to Model 1.

6. Estimating Model Parameters

We now turn to the question of estimation, and we also discuss the implications for

data collection required to support it. A tempting approach is to produce Xi by the

classical 3-point method of PERT, thus yieldingνi and V(Xi). It would then only remain

to estimate B. However, it is notoriously difficult to get reliable 3-point activity estimates

from process owners. Anectdotal evidence to that effect was given by Woolsey (1992).

Woolsey’s account relates to work place politics. To this we must add systemic human

error as studied by Tversky and Kahneman (1974). Therefore we propose to limit the

information elicited from process owners to single point activity estimates,νi. So we must

estimate all the other necessary parameters from historical data. Operationally, if we

estimateβ, σb, and V(Xi), thenσi2 can be calculated explicitly. But because V(Xi) cannot

be estimated until we estimateβ, this requires a two-stage econometric model.

The historical data we need consists of pairs of activity estimates and realizations,

where we treat activity estimates as explaining variables and realizations as independent

variables. Assume we have J>1 projects in the history, and until further notice we amend

our former notation by the introduction of a double index, i,j where i=1,2,...,nj and

j=1,2,...,J. Here i denotes a specific activity and j a specific project with nj activities.

Optionally, we may elect to treat some subprojects as projects in their own right for this

purpose. This is reasonable sometimes, e.g., if the subprojects were managed relatively

independently or involved distinct sets of resources. For project j our data consists of nj

- 18 -

pairs (yij ,νij ), where yij is the realization andνij the original activity estimate. An

estimator of the mean systemic error of project j is given by

leading to

This completes the first stage.

Our decision to only employ single point activity estimates implies that V(Xij)

must be modeled as a function of the activity estimateνij itself. The parameters of this

function should be estimated from historical data, however. Suppose we start with a

quadratic approximation of the form V(Xij )=α0+α1νij+α2νij2. We first note that, because

V(X ij )≥0, αk≥0 is required. Furthermore, since the variance of a zero activity must be

zero,α0=0 is indicated. Accordingly, we propose the model V(Xij)=α1νij+α2νij2. Let CVij

denote the coefficient of variation of Xij , and recall that E(Xij)=νij , then CVij≥√α2 (∀ i,j),

with equality iff α1=0. So if, in reality, CV is approximately constant, our model will fit

with α2 dominating andα1 may be insignificant. But if long activities often include

several independent sub-activities,α1 will be significant. Letρij denote the residual of xij ,

then

and note that E(ρij)=0. We can now regress byνij andνij2 to obtain estimates for

- 19 -

α1 andα2. Since both must be non-negative, if either estimate is negative we repeat the

regression with the offender set to zero.

Another possibility is to use exponential smoothing to maintain the estimates ofβ,

σb, α1, andα2. Although we omit the details, arguably this is the best approach. On the

one hand, we need an adaptive model because the use of a model such as ours is likely to

cause shifts in the systematic error distribution over time. On the other hand, because of

the adjustment effect that is involved in this usage, the use of moving averages instead of

exponential smoothing can lead to severe harmful fluctuations (Trietsch, 1999, Ch. 4).

Once in place, the data collection burden this system places on project managers

and workers is very low. The estimates they are required to provide are limited to the bare

minimum--single point activity estimates. As for data to be collected during ongoing

projects, since the estimates are already in the data base, it is enough to record the actual

performance of each activity. One question remains: how to obtain initial values for such a

scheme. This should not be too difficult for most project organizations, since the data

required is very elementary. However, if such data is not available, we propose to use

generic data from the relevant industry. It’s important to distinguish between industries,

however, since it is well known that some tend to have more variance than others. For

example, construction of generic homes is much less volatile than R&D or software

development. The use of a weighted composite of several industries is also a possibility,

but we omit the details.

For convenience, since from here on out we are going to focus on the current

project exclusively, we now revert to our original notation, with a single index. Recall that

our convexity result required convexity of the square root of E(B2)V(X i) as a function of

νi. Since we interpret Ci as the cost of crashingνi, and we model V(Xi) as a function of

- 20 -

νi, then crashingνi has a variance effect. To quantify this effect we present the partial

derivative ofσ by µi.

Here, the central expression is true in general and the right expression assumes that V(Xi)

is modeled byα1νi+α2νi2. Implementation requires using our estimates forβ, E(B2), etc.

This makes possible the calculation of Equations 5 to 8 in the context of Model 3. So we

see that unless dV(Xi)/dνi is negative, Model 3 implies the existence of a positive variance

effect. Specifically, when V(Xi) is modeled as we suggest, the effect is positive. The

question is whether it is also convex. But ifα1 dominates, such convexity is not

guaranteed! There are several other elements in the hessian, e.g., the convexity of Ci

which is likely to be quite pronounced for smallνi, that make it highly unlikely that this

lack of convexity in one part will cause typical examples to have multiple local minima.

Nonetheless, it is possible to create theoretical instances where it does.

Finally, it is often reasonable to assume that B is lognormal (i.e., that it reflects the

multiplicative effects of many small independent causes) whileΣνi is often approximately

normal. The multiplication of a lognormal variate by a normal one is neither normal nor

lognormal, but depending on the variance its shape is in between the symmetric normal

and the skewed logmormal. Thus we may expect a skewed distribution, but not very

skewed ifσb is small. There is evidence that practical projects involve skewed

distributions (Leach, 2003). Our model provides an explanation for this phenomenon.

Furthermore, such skewness suggests thatσb is not likely to be small.

- 21 -

7. Convex Monotone and Other Delay Penalty Functions

Generally, the delay penalty can be expressed as a function of y, P(y). So far we

used a simple penalty function of 0 until D and (y-D)P afterwards. Using P(y), instead of

integrating by dt from D to∞ and multiplying by P we can integrate from 0 to∞ by

dP(y). If P(y) is a monotone non-decreasing convex function, then the convexity that we

showed still applies. To see this consider that it is always possible to approxiamte such

functions by piecewise linear ones, and any piecewise linear monotone increasing and

convex function can be presented as the sum of functions of the type for which we already

proved convexity.

An earliness bonus can also be modeled by P(y). If it is not higher than the delay

penalty, then P(y) is monotone and convex. Otherwise, P(y) is monotone but not convex

(which is only likely in a sellers market, since buyers typically prefer a convex structure).

In both cases the earliness bonus becomes a first "penalty" step covering the period from

zero to D. That is, P(y)=yPB (for y≤D) and P(y)=yPB+P(Y-D)+, where PB is the bonus

rate. By subtracting an initial credit of PBD from this penalty function the intended bonus

is obtained. (During a power crunch in the winter of 2003 in New Zealand, one utility

aggressively promoted a high-consumption penalty as a low-consumption bonus.)

P(y) is not always convex in practice, but it is usually non-decreasing. For

example, if the project has a deadline, the step loss function applies: missing a deadline by

a little is as expensive as missing it by a lot. In such cases, crashing is likely to reduce the

expected penalty (or increase profit in case P(y) is negative). But when we take the cost of

crashing into account there is no guarantee that a local minimum is globally optimal.

Thus, in such cases, the best we can do is to find a local optimum. Or we can use generic

wide-search methods to try to find the global optimum. It may also be worthwhile to

check whether zero crashing is a local minimum.

- 22 -

8. Using the Model to Negotiate the Due Date

Until now we assumed that the due date is given exogenously. In reality, the due

date is usually the outcome of negotiation. Ronen and Trietsch (1988) observed that such

negotiation can be supported by a model such as ours. The idea is to plan crashing

together with the determination of the due date and calculate TC for each such due date.

The role of the model here is to provide us with cost information that we can use to make

good decisions.

Conceptually, the customer has a time value, say Cc, although it may be implicit.

The penalty, P, is sometimes explicit and constitutes part of the contract, and sometimes

implicit. If we have these costs explicitly, it becomes possible to search for the due date

that minimizes the total cost as captured by the objective function of this paper plus the

"cost" of the due date to the customer, CcD. This due date, along with the crashing plan

associated with it, is optimal for the customer and the vendor together.

In such a situation, if Cc is very low and therefore the due date is very high, it

may not be optimal to start the project at time zero, to save on holding costs. Furthermore,

in such a case, holding costs may complicate the computation of Ci, as discussed in the

Appendix.

9. Numerical Examples

Example 1: A project involves 30 exponential activities in series, Y1 through Y10 with

µi=10 and Y11 through Y30 with µi=5 (we use the version where the parameter of the

exponential is the mean). Activities may be crashed by up to 50% and remain exponential

after crashing. Let the raw crashing costs of all Yi be 9.8, but let the holding cost per

period per activity be Hi=0.01. The appendix shows that the holding costs imply effective

- 23 -

(full) crashing costs given by the arithmetic sequence {Ci=9.8+(31-i)0.01}. Specifically,

C1=10.1, C11=10 and C30=9.81.

Solution: Here, Model 2 applies, but we start by solving the problem by Procedure 1 as a

heuristic. Observe that the project distribution is approximately normal by the central limit

theorem. If we select the cheapest activities to crash, namely Y11 through Y30 (in reversed

order), and crash them maximally, then the service level will be 0.5 (Y11, the most

expensive activity we crashed, implies SL=10/20 and this is achieved after crashing it

(10 10+2.5 20=150). The total cost associated with this solution is 762.9. This crashing

plan reduces the variance from 1500 to 1125. But if we crash Y1 through Y10 maximally

instead, and thus pay more for crashing but reduce the variance to 750, we obtain the

same service level and the objective function is reduced to 721.26. Therefore, selecting

activities to crash based on Ci alone is not optimal. Furthermore, the optimal value is

717.11 and it yields a higher service level. It involves crashing all activities to

progressively smaller µi values as per an arithmetic series with µ1=5.326 and µ30=4.345

(the maximal crashing constraints are inert). The optimal service level is 57.37%, instead

of 50%. The objective function value is 717.11. Thus it is very important to consider the

variance reduction effect and it is also useful to optimize the service level.

The example demonstrates our proposition that∂σ/∂µi>0 implies more crashing

than Procedure 1 would call for. Example 2 illustrates this point further.

Example 2: Let n=1, with an exp(5) activity time distribution, and assume crashing is by

switching to another exponential with lower mean. Finally, D=5, C=C1=10, and P=20.

Solution: Due to the memorylessness of the exponential random variable the expected

delaygiven a delay is µ, leading to an un-crashed expected conditional penalty of

5 20=100. Multiplying by 1-SL our total expected cost is 36.79. Note that the service

- 24 -

level, 0.6321, is already better than 0.5, as Procedure 1 would specify. So it may look like

there is no need to consider crashing in this case. Actually, if we could save money by

increasing µ we might be tempted to check this option instead (i.e., negative crashing).

However, the true optimal µ in this case is 2.979 (with 3 digits accuracy), leading to a

total cost of 31.33 and 1-SL=0.1867 (SL=0.8133). The reason such a high service level is

justified is that the gain by crashing is much higher than with simple crashing. The total

cost of crashing plus the expected penalty is given by

Taking the derivative by µ and setting it to zero we obtain

which is a special case of Equation 6. Equivalently, the modified cost is MC1=C1/(1+D/µ).

On the one hand, we need a search method to find the exact modification. On the other

hand, it is not important to find the exact optimum: The objective function is virtually

constant between 2.95 and 3.00 (yielding 31.3333 and 31.3325 respectively, as compared

with 31.3318 at 2.979). Furthermore, it changes by less than 0.2% in the range 2.795 to

3.17 (that is, we deviate from optimum by more than 6% but the objective function

changes by less than one fifth of a percent).

Similar flatness of the objective function near the optimum often applies for the

newsboy objective function when we deviate by a small fraction ofσ from optimum.

Example 3: This and the next example demonstrate the application of systemic error

correction to the estimation of project duration. Suppose that by analyzing past data we

obtain estimates of 1.25 and 0.15 forβ andσb. Let Σνi=28, andΣV(X i)=19.785. Find the

project buffer that provides SL=0.95.

- 25 -

Solution: The first action we take is to multiply 28 by 1.25, i.e., add 7 weeks to the

estimate to counteract the estimation bias. The variance contribution due toσb is

(0.15 28)2=4.22=17.64. Therefore, the standard deviation of the chain is now bounded

from below by 4.2 (even when we ignore all other variances). Now consider the

contribution ofΣV(X i)=19.785. Multiplying by E(B2)=1.585 we obtain an updated

contribution of 31.36. Finally, 31.36+4.22=49 (72). Since we require a project service level

of 95%, then we should set the buffer to achieve that. This requires perfect knowledge of

the distributions of B and ofΣXi. For the purpose of our examples (and as a practical

recommendation whenever it is difficult to obtain these exact distributions), we assume a

normal distribution instead. So we need a safety allowance of 1.645σ, or 11.52. Our due

date should therefore be set to 28 1.25+11.52=46.52 (of course, in practice such a result

would be rounded). This is a large increase but it is justified because our history suggests

optimism and sizable dependence, and also because 95% is a high service level.

Example 4: For the same B as in Example 3 Suppose a project comprises n activities,

each with a normal distribution withνi=4, V(Xi)=0.7942 (note that 1.585 0.7942=1). Let

P=4C. Determine the optimal project due date as a function of n. And determine the

optimal total buffer--including allowance for the expected expansion and safety time

(expressed as a fraction of the estimated project time) as a function of n.

Solution: P=4C leads to SL*=0.75. Using the normal approximation this implies a safety

time buffer of 0.6745σ. We first solve for three specific cases: n=1, n=5, and n=25. In all

cases, we need expansion allowances of n (0.25 4=1). For n=1, our variance is

1+42 0.152=1.36=1.1662 leading to a total buffer of 1.787 (compare to 0.6745 that might

be specified without accounting for the estimation bias error). This is 44.7% of the

estimated mean. The due date is 5.787 (of course, in practice such numbers should be

- 26 -

rounded). For n=5, the completion time variance is 5+202 0.152=14, leading to a due date

Figure 1: The optimal protective time cushion for n=1 to n=10 in Example 4.

of 27.524, and the safety time as a fraction of the estimated mean is 37.6%. n=25 leads to

a due date of 135.66, and the safety time is 35.7%. In the limit, we need an expansion

buffer of 25% and an additional safety buffer of 15 0.6745=10.1%, thus imposing a lower

limit of 35.1% on the size of the protection buffer as a fraction of the estimated mean. So

most of the protection for n=25 is due to expansion and dependence, while a sizable part

of the protection for n=1 is for processing time variation. Figure 1 depicts the optimal due

date and the size of the buffer for n=1 to n=10. We see that when expressed in percents

the buffer approaches the limit asymptotically. We also see that the optimal buffer as a

function of n follows an approximately straight line, but it has a positive intercept. Thus,

short chains require more protection.

- 27 -

10. Conclusion

The simplest non-trivial project structure is a serial chain of activities. Without

correct analysis of such chains it is impossible to analyze more realistic structures. In this

paper we discussed how to crash activities along such chains. We included a model that

assumes systemic error and showed that such error leads to statistical dependence. We also

showed that it is likely to lead to variance crashing effects, i.e., to changes in the variance

during crashing. Our model can take such variance effects into account, however. We

provided convexity conditions--essentially convexity of each activity--that guarantee

convexity for the model. (Previous convexity results in the literature do not cover variance

effects and systemic errors.) However, we also presented a plausible estimation model that

may lead to violation of these convexity conditions. The latter model assumes that process

owners only provide single-point estimates while variance and systemic error are estimated

from historical data, which is updated as new projects are completed. When necessary, this

historical data can initialized by using generic industry data.

An important by-product of optimal (or near-optimal) crashing is that the project

buffer is optimized too. But for the project buffer to be realistic we must take into account

positive dependence between activities due to systemic estimation errors and causes that

are shared by many activities. Applying the model we developed for this suggests larger

project buffers that tend to grow approximately linearly with the expected estimated

duration--i.e., they do not become negligible as the independence assumption erroneously

leads us to believe.

Selecting an optimal crashing plan is essentially a type of portfolio optimization.

We are selecting a portfolio of crashing investments. Indeed there are similarities between

our results and classical portfolio selection results. Specifically, the bias that prevents the

relative cancellation of variance when many activities are included is analogous to market

- 28 -

risk, which cannot be diversified away. Financial portfolio models exist that break the

market risk to components and assess each stock based on its sensitivity to each of these

components. A similar approach may be applied to our model, calling for more

sophisticated dependence estimation. This requires further research, however. Further

research is also required to extend the model to general networks, but our limited results

are already useful for simulations of general networks (where chains are embedded) and

can also be combined with the approximate analysis of Clark (1961). Clark’s

approximation--part of the original PERT effort--makes possible the explicit consideration

of statistical dependence.

Appendix A: Taking Holding Costs into Account

We begin our analysis for Model 1. Let Ck=min{Ci} and suppose that if we start

the project immediately we obtain SL0>(1-Ck)/P. Assume further that there is no economic

incentive to finish the project earlier than D. So we may have to postpone the start of the

project to save interest charges on funds that are invested in the project too early. We

refer to the these as holding costs, Hi. Assume that Hi is charged from the start of Yi until

the due date (other options, e.g., charging based on the activity’s progress, just add

complexity without making a substantial difference). Another reason to postpone the start

of a project is to save on possible fixed charges per week while the project is on the

books. We model this by adding astart activity, Y0 (with H0 modeling the fixed charges,

µ0=0, C0=∞, and d0=0). We also define apostponement activity, Y-1, which always starts

at time 0, with C-1=Σi=0,1,...,nHi (because postponement saves this holding cost per unit

just before the project ends), and H-1=0. Although we don’t need crashing to meet the due

date, it may reduce holding costs. But if we decide to crash Yi while Y-1>0, then we

should keep the service level constant while doing so (i.e., we increase Y-1 whenever we

- 29 -

crash a physical activity). Specifically, C-1 plays the role of C here and SL*=(P-C-1)/P.

Each time unit costs Ci for the crashing itself, but because Y-1 increases, activities Y0

through Yi start later and therefore our true crashing cost is Ci-Σj=0,...,iHj (we pay for

crashing but save on holding of previous activities). This value can be negative (especially

if H0 is large), which implies that we should crash Yi until Ci satisfies Ci≥Σj=0,...,iHj.

Henceforth, we assume that thispreliminary crashing has been done. Therefore, the

magnitudes Ci-Σj=0,...,iHj, to which we will refer aspreliminary full costs, may all be

assumed non-negative.

Suppose now that D is decreased gradually until the postponement is consumed. At

some stage (possibly but not necessarily immediately), crashing will become cost effective.

Once we do start crashing we again increase our holding costs by starting activities earlier.

For example, if we crash Yi by δ all the subsequent activities i+1 through n are advanced

by δ. Therefore, the effective crashing cost associated with Yi is Ci+Σi+1,...,nHi. We will

refer to these magnitudes asfull costs. (However, for convenience and where the context

permits to do so without confusion, we may denote the full costs by Ci.) Thus, regardless

of whether we have a free slack or not, the amount of crashing of an activity is not only a

function of project duration and the raw crashing cost, but also of the position of the

activity in the project sequence: in both cases, downstream activities are more likely to

justify crashing.

The difference between the full costs, Ci+Σi+1,...,nHi, and the preliminary full costs,

Ci-Σj=0,...,iHj, is Σk=0,...,nHk, or the C-1 value we used while the slack persisted. Since

Ci-Σj=0,...,iHj>0, we now see that Ci+Σi+1,...,nHi>Σj=0,...,nHj=C-1. Therefore, the optimal

service level following any physical crashing after the preliminary crashing must be lower

than (P-C-1)/P.

- 30 -

Now consider Model 2. As long as Y-1>0, SL*=(P-ΣHj)/P still holds, because for

any preliminary crashing plan we may select, we can adjust SL by advancing or

postponing all the activities as a block. Such an adjustment is associated with the costΣHj

and therefore must stop with SL* as asserted. But, unlike the case of Model 1, preliminary

crashing can still make a difference in EP even when SL is already optimal. This is

because EP is also a function of the project distribution from D onwards (see Equation 1),

and specifically of the variance.

We offer a solution under the assumption that the distribution retains its shape

under all possible preliminary crashes, e.g., if it is normal due to the central limit theorem.

Recall that t is the standardized value of the due date, and because SL* is determined in

advance by SL*=(1-ΣHj)/P, the optimal t, t*, is co-determined with SL*. By the middle

element of Equation 1, EP=PσE(Z|Z>t*)(1-SL*)-Pσt*(1-SL*), but by optimality

C=P(1-SL*), therefore the cost of the buffer, Cσt*, is equal to Pσt*(1-SL*). It follows that

the total cost of the buffer and the expected penalty is simply PσE(Z|Z>t*)(1-SL*).

Substituting C=ΣHj=P(1-SL*) in this expression we obtain

Therefore, every unit reduction inσ yields a gain ofΣHjE(Z|Z>t*) by reducing the buffer

and the expected delay penalty. This saves (∂σi/∂µi)ΣHjE(Z|Z>t*). Including the holding

cost implications, the effective costs of crashing Yi during the preliminary stage are given

by

where the last element is conducive to more crashing when dσi/dµi>0. If any of these

expressions is negative we should perform preliminary crashing until it turns positive.

- 31 -

Acknowledgment: The author is grateful to Candace Yano for substantial help during the

early stages of this highly stochastic and negatively-crashed project. The comments

of three anonymous referees on a previous version of the paper were equally

useful.

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