progress payment timing

7/23/2019 Progress Payment Timing

http://slidepdf.com/reader/full/progress-payment-timing 1/15

Project Contracts and Payment Schedules:The Client’s Problem

Nalini Dayanand • Rema PadmanDialogos, Inc., 12 Farnsworth Street, Boston, Massachusetts 02210

The H. John Heinz III School of Public Policy and Management,

Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

[email protected] • [email protected]

Contractual agreements have assumed significant complexity in recent times because of

the emergence of strategies like outsourcing and partnering in the successful completion

of large software development, manufacturing, and construction projects. A client and con-

tractor enter into an agreement for a project either by bidding or negotiation. Effective andefficient bidding, negotiation, and subsequent monitoring are hindered by the lack of appro-

priate decision support tools for the management of project finances. Progress payments

to the contractor are an important issue in project management because of their potential

impact on project finances and activity schedules. In this paper, we consider the problem

of simultaneously determining the amount, location, and timing of progress payments in

projects from a client’s perspective. We develop three mixed-integer linear programming

models, based on some practical methods of determining payment schedules from different

types of project contracts. We discuss properties of the models and draw insights about the

characteristics of optimal payment schedules obtained with each model by an experimental

study on a sample of 10 small projects. Our analysis shows that, contrary to current practice,

the client obtains the greatest benefit by scheduling the project for early completion such thatthe payments are not made at regular intervals. It is also cost effective for the client to make

payments either in the early stages of the project or toward the end, even though this causes

considerable variation in the time gap between payments. We also evaluate the impact of

the client’s preferred payment schedules on the contractor’s finances and activity schedules,

and draw some conclusions on the interdependence of payment and project parameters on

the objectives of both parties entering the contractual agreement.

1. IntroductionWith outsourcing as a key business strategy in

manufacturing, software development, and construc-

tion operations, the formation and administration of

contracts has assumed considerable importance in

recent times (Martin and Webster 1988). According

to Oltman (1990), outsourcing contracts for informa-

tion technology alone were worth $100 billion in 1995.

Many organizations have turned to project manage-

ment theory and supporting software tools to control

costs and improve productivity in this process. There

is general consensus among project managers that the

client’s involvement in the management of a project

contributes significantly to its success (Cleland andKing 1983).

This paper develops decision tools to support a

client in the contract negotiation process. The tools

address the specific issue of scheduling payments in

projects. Payments are determined by a number of fac-

tors, and interdependence between these factors com-

plicates the determination of payments and schedules.

The relevant literature on project scheduling with NPV

Management Science © 2001 INFORMS

Vol. 47, No. 12, December 2001 pp. 1654–1667

0025-1909/01/4712/1654$5.00

1526-5501 electronic ISSN



DAYANAND AND PADMAN

Project Contracts and Payment Schedules

objective is reviewed in Dayanand (1996), as well as

Herroelen et al. (1997), and will not be repeated here.

Bey et al. (1981) were the first to observe that preferred

activity schedules for the contractor and client exhibitdifferences when NPV is used as the scheduling cri-

terion. Their observation resulted from an analysis of

the effect of bonuses and penalties on time schedules

for resource constrained projects with fixed cash flows.

Previous models of projects with cash flows have

focused primarily on obtaining a schedule of activi-

ties that maximizes financial performance. In contrast,

models of the payment scheduling problem discussed

here consider the simultaneous determination of both

the amount and timing of payments in projects. These

models are motivated mainly by the fact that mostproject environments allow some flexibility in setting

a schedule of progress payments. Properly structured

progress payments can reduce interest costs associ-

ated with the project, and allow the client to ensure

schedule compliance from the contractor by offering

these payments as incentives. This is especially valu-

able in the majority of project instances where uncer-

tainty associated with project parameters, such as

duration, expenses, and deadlines, can affect project

profitability significantly. Although effective schedul-

ing of fixed cash flows can enhance the financialperformance of a project, greater improvements can

be realized by allowing the decision maker control

over the amount of each payment and the schedule

of activities (Dayanand 1996; Dayanand and Padman

1993, 1998).

The paper is organized in the following major

sections. Section 2 highlights some modeling issues

for the client’s payment scheduling problem. In §3,

we develop a mathematical model of the problem

followed by two extensions based on commonly

observed practices of making payments in projects.

Each model is illustrated with a small example

project. Section 4 compares models and solutions by

describing model properties. Section 5 describes com-

mon features of payment schedules as observed from

a sample of 10 projects. In §6, we examine the impact

of the client’s payment schedule on the contractor’s

schedule of activities to highlight the key issues and

tradeoffs to be considered by the two parties during

the financial negotiations. The paper concludes with

a brief summary and possible extensions of this work

in §7.

2. Modeling the ProblemModels of the payment scheduling problem are based

on several common assumptions (Dayanand 1996).

These assumptions are as follows:

• Project activities, their durations and precedence

relationships are known.

• The project must be completed by a specified

deadline.

• Associated with each event/activity is a mone-

tary value that constitutes part of the criterion for

determining the amount of payment. As discussedin more detail later, this monetary value may repre-

sent expenses or value of work associated with the

event/activity.

• Payments in the project are also associated with

events or activities in the project.

• The client has the objective of maximizing the net

benefit from the project. The benefits from the project

can be estimated in monetary terms. All benefits are

associated with completion of the project. This can be

easily relaxed to allow benefits over the duration of

the project.

• The client and contractor negotiate either the

number of payments or the frequency of payment.

Three additional assumptions are particularly rele-

vant for models developed in this paper—the type of

contract, the method of distribution, and the retention

factor.

• The type of contract refers to the method of com-

pensating the contractor for undertaking the project.

We assume that the client pays the contractor a fixed

total amount for the project. While this suggests that

the models are applicable for lump sum, fixed price

contracts, they can be adapted to suit other types

of contracts (Dayanand and Padman 1998). It is also

worthwhile mentioning that fixed price contracts are

very popular for a number of reasons, and are often

regarded as the ultimate incentive contract (Martin

and Webster 1988).

• The method of distribution specifies the criterion

by which the amount of each payment is determined.

The amount of each payment is usually based on

Management Science/Vol. 47, No. 12, December 2001 1655



DAYANAND AND PADMAN


some measure of progress on the project. For the client

to ensure schedule and quality compliance, it is gener-

ally recommended that payments be based on actual

progress, which we measure by “earned value” at thetime of payment. Earned value is determined from

the value of work associated with each event/activity

in the project. Some proportion of the value of each

activity is earned when the activity commences, and

the rest when it is completed. Accounting practices

with respect to the allocation of activity values to the

beginning and end of activities vary from project to

project (Fleming and Koppelman 1994a, 1994b, 1995).

• Most project contracts include a provision for

retention of some portion of the payment that is

due to the contractor (Hendrickson and Au 1989,Gilbreath 1992). We assume that the client and con-

tractor negotiate the maximum percentage retention,

and this percentage remains constant for the duration

of the project. Varying retention percentages can be

easily accommodated.

Thus, models of the client’s payment scheduling prob-

lem utilize the earned value criterion and maximum

percentage retention to distribute a fixed total amount

in payments over the duration of the project. The con-

tractor and client also negotiate either the minimum

number of payments or the frequency of payment.The basic model described in this paper assumes that

the client negotiates a minimum number of payments

to be made to the contractor over the duration of the

project. We also propose an extension to this model

that is based on a specified frequency of payment.

3. Mathematical ModelsThe client’s payment scheduling problem is formu-

lated using event-based representations of project

networks.1 Activities are uniquely identified by the

starting and ending events, and monetary values areassociated with events in the project network.

Three models are described here. The three models

represent successive modifications that consider cur-

rent practices in distributing progress payments over

1 Some of the issues in data representation, mathematical formu-

lation of the payment scheduling problem for event and activity

oriented networks, and interpretation of corresponding solutions

are discussed in Dayanand and Padman (1997).

the duration of the project. All the models determine

an optimal payment and activity schedule from the

contract price, retention, and project network struc-

ture. The Basic Client (BC) model allows a fixednumber of payments and places no restrictions on

the permitted timing of the payments. The Equal

Time Intervals (EQ) model also allows a fixed num-

ber of payments, but forces these payments to occur

at approximately regular intervals, as is the common

practice. The Periodic Payment (PP) model schedules

payments at regular time intervals until the project is

completed so that the number of payments depends

on the frequency of payment and project completion

time. Periodic payments are useful when the payment

schedule has to accommodate the client’s paymentcycle.

We consider a project network with I events (1 rep-

resents the first event, and I the terminal event) and

J activities. The client pays a total of R dollars in K

progress payments over the duration of the project,

which must be completed by the deadline D. The

value of work associated with the occurrence of every

event is V i, and the maximum permitted retention is

0 < ≤ 1.

V i, the value associated with each event, is a criti-

cal parameter in the model and is determined by thevalue of work associated with each activity, prece-

dence relationships between activities, and account-

ing procedures that specify allocation of the value of

each activity to the beginning and end of the activity.

Definition 1. Let W ij represent the value of activ-

ity ijf ij the proportion of W ij allocated to event

i, and = gij the boolean adjacency matrix of the

project network.2 Then,

V i =I

j =1

gij f ij W ij +I

k=1

1 − f kigkiW ki (1)

Figure 1 is an example of the client’s payment

scheduling problem. In addition to activity dura-

tions and precedence relationships, the figure shows

the value of work associated with each activity and

earned value associated with each event. The value

2 For a network with I events and J activities, the boolean adjacency

matrix is a I × I matrix obtained by setting gvw equal to 1 for

every activity vw ∈ J , and 0 otherwise.

1656 Management Science/Vol. 47, No. 12, December 2001



DAYANAND AND PADMAN


Figure 1 The Client’s Payment Scheduling Problem—An Example

of every activity is earned only on completion of the

activity, i.e., f ij = 1 for every activity ij. The total

amount paid to the contractor is given by I

i=1 V i, and

equals $14,000 for this project. The project brings a

benefit of $50,000 to the client on completion.

All three models examined here are mixed-integer

programs consisting of four sets of variables. P t is

a continuous variable denoting the amount of pay-ment in time period t . P t takes nonzero values when

a progress payment is made in time period t . Binary

variables t indicate time periods in which payments

are made. xit is a binary variable that denotes the

occurrence of event i in time period t. zit is an addi-

tional binary variable that indicates the occurrence of

an event and a payment in time period t, i.e., zit is

1 when t = 1 and xit = 1. Thus, these models have

some nonlinear terms that can be transformed into

linear expressions with appropriate substitutions. The

rest of our notation is described below.

Notation.I = Number of events,

J = Number of activities,

K = Number of progress payments,

D= Project deadline,

R = Total amount paid by the client,

B = Monetary value of project benefits,

ij= Activity starting at event i and ending

at event j ,

dij = Duration of activity i j,

E i = Earliest occurrence time for event i,

Li = Latest occurrence time for event i,

= The rate of interest per time period,

ct = Discount factor for time period t ,

ct = 1/1 + t t = 0 D,

= Percentage retention, and

V i = Earned value associated with event i.

3.1. The Basic Client Model

Maximize

= BLI

t=E I

ctxIt −D

t=0

ctP t (2)

Subject to

Li

t=E ixit = 1 i = 1 I (3)

Lj

t=E j

txjt

−Li

t=E itx

it ≥ d

ij (4)

Dt=0 P t = R (5)

Dt=0 t = K (6)

T t=0 P t ≥ 1 −

I i=1

T t=0 V izit

T = 0 D (7)

iV i >0 xit ≥ t t = 0 D (8)

P t ≥ 0 t = 0 D (9)




DAYANAND AND PADMAN


xit ∈ 0 1 i = 1 I

t = 0 D (10)

t ∈ 0 1 t = 0 D (11)

f ij i = 1 I j = 1 I is not a parameter in our

model. However, it determines the distribution of

earned value over events in the project network. B is

the monetary value of project benefits, and is asso-

ciated with the final event in the project. B is much

larger in magnitude than R, typically several times

the order of R (Schnieders and Jameyson 1987). The

objective function represents the client’s net benefit

from undertaking the project, and is the present value

of all future benefits, less the present value of all

costs associated with the project. Constraints (3) force

assignment of an occurrence time for every event

in the project; (4) are precedence constraints. Con-

straint (5) restricts the total amount of payment to R,

and (6) controls the number of payments. Constraints

(7) are distribution constraints that ensure that the

total amount of each payment is at least as much as

the value earned at the time of payment less maxi-

mum allowed retention. Constraints (8) tie payments

to events in the project network by forcing at least one

event with nonzero earned value to occur at the time

of each payment. Additional constraints are requiredto link the values of xit t and zit . These constraints

are as follows:

zit ≤ xit i = 1 I t = 0 D (12)

zit ≤ t = 1 I t = 0 D (13)

xit + t − zit ≤ 1 i = 1 I t = 0 D (14)

zit ≥ 0 i = 1 I t = 0 D (15)

Constraints (12)–(15) are standard additions to models

in which one binary variable is determined by two

other binary variables (Glover and Woolsey 1974). The

complete model thus includes all Constraints (3)–(15).

Simple modifications to the model permit exten-

sions for some special cases. In practice, payments

are often linked to milestones that are defined by

completion of groups of activities based on the sig-

nificance of the activities to the entire project. Since

project progress is often monitored at these mile-

stones, the client’s model can be adapted to schedule

payments at milestone events. If represents the set

of milestone events, Constraints (8) can be modified

as follows:

i∈

xit ≥ t t = 0 D (16)

Some other extensions are discussed in Dayanand

(1996) as well as Dayanand and Padman (1998).

3.2. The Equal Time Intervals Model (EQ)

The EQ model is a modification of the BC model that

takes into account the common practice of making

payments at regular time intervals (Clough and Sears

1991, Gilbreath 1992). With this model, the client still

makes K payments for the project. The first K − 1 of these payments are scheduled at equal time intervals

over the duration of the project, and the final payment

is scheduled on project completion. The time inter-

val between payments is estimated from the num-

ber of payments and the project’s earliest completion

time (E I ).

Definition 2. If gK denotes the time gap between

payments, then

gK = E I /K + 05 (17)

Observation 1. BC can be modified to schedule K payments at equal time intervals by setting t equal

to 1 for the first K − 1 time periods that satisfy t

mod gK = 0.

Recall that t is the indicator variable for nonzero pay-

ments. Thus, forcing t to take a value of 1 guarantees

a payment in time period t .

3.3. The Periodic Payment Model (PP)

For various reasons, the contractor and client some-

times negotiate the frequency of payment instead of

the number of payments. Abernathy (1990), for exam-

ple, recommends that contractors submit requests for

payment to coincide with the client’s payment cycle

to avoid delays in realization of funds. Singh (1989)

also suggests that frequency of payment is one of the

parameters that determines the contractor’s cash flow

position over the duration of the project. Thus, the

third model we examine in this paper is formulated




DAYANAND AND PADMAN


to schedule payments at regular time intervals until

the project is completed. We refer to this model as the

Periodic Payment (PP) model. Unlike the EQ model,

in which a fixed number of payments occur at approx-imately equal time intervals, the PP model determines

the number of payments from the frequency of pay-

ment and the scheduled completion time.

Definition 3. Let gg ≤ D denote the time inter-

val between payments. Define two sets, S 1 and S 2,

where t ∈ S 1 i f (t mod g = 0 and t ∈ S 2 if t

mod g = 0 for all integer values of t < D.

Observation 2. The constraints

t −LI

t=E I

txIt + 05 ≥ −Dt ∀t ∈ S 1 (18)

t −LI

t=E I

txIt + 05 ≥ −D1 − t ∀t ∈ S 2 (19)

convert BC to P P .

The correction factor of 0.5 in Constraints (18) and

(19) is necessary to ensure that the values of t are

uniquely determined.

Observation 3. K is now given by D

t=0 t .

3.4. Solving the Problem

We may note that optimal solutions to BC always

exist, but cannot be guaranteed for EQ and PP. The

BC model allows payments to be scheduled at any

time. Thus, a feasible schedule can be found by using

the early time schedules for events in the project net-

work and assigning payments to the first K −1 events

and the terminal event. With EQ and PP, the network

structure may not permit the scheduling of events

at designated times of payment, or these schedules

may delay project completion beyond the deadline.

Both these conditions result in infeasibilities with EQ

and PP.

For projects of small size and short durations, the

problem can be solved using branch and bound pro-

cedures typically provided with commercially avail-

able software. We used GAMS (Brooke et al. 1988)

and OSL (Hung et al. 1994) to determine the client’s

optimal payment schedule for each of the three mod-

els described in this section. The payment and event

schedules are detailed in Table 1. The client retains a

maximum of 10% of the value of work completed at

the time of payment and the annual cost of capital is

20%. All activity durations are in weeks. The clientschedules 4 payments with BC and EQ. A payment is

scheduled every 8 weeks with the PP model.

4. Features of Optimal SchedulesAs seen from optimal schedules for the example prob-

lem, the three models result in significantly different

payment and activity schedules for a given project.

Some of the features of optimal payment schedules

follow from properties of the models. Others result

from project characteristics such as network structure,contract price, the distribution of earned value over

project events, the number of payments, and maxi-

mum percentage retention. Our analysis of optimal

payment schedules therefore comprises two parts. We

begin by examining some model properties. In the

second part of our analysis, we report features of

payment schedules as observed from a sample of 10

projects. We illustrate these features with solutions to

the example problem and summarize the characteris-

tics observed from the sample problems. The analy-

sis yields several useful insights into the features of the client’s optimal payment schedules, and permits a

comparison of schedules across models for variations

in the number of payments.

4.1. Model Properties

In the analysis that follows, ·· and · denote

the benefit, the expenses, and the net benefit associ-

ated with the project in present value terms, respec-

tively. Thus, · = · − ·. The (*) superscript

denotes optimal values.

Property 1. If T I BC = T I EQ∗EQ ≥ ∗BC.

With identical completion times under BC and EQ,

∗BC =

∗EQ. Thus, EQ ≥ BC.

Property 2. For fixed K and ∗BC ≥ ∗EQ.

Furthermore, when KBC = KPP∗BC ≥

∗PP K .

EQ schedules have K payments and satisfy distri-

bution and precedence constraints. Every EQ sched-

ule is therefore a feasible K -payment BC schedule.

Thus, ∗BC ≥ ∗EQ. Likewise, PP schedules with




DAYANAND AND PADMAN


Table 1 Optimal Solutions for the Example Project

Model/ Event

NPV (Expenses) → 1 2 3 4 5 6 7 8 9 10

BC Time 0 12 3 5 13 26 20 31 32 37

12,327.94 Amt. 604.80 403.20 1,260.00 12,732.00

EQ Time 0 18 3 9 17 27 24 35 36 41

12,488.74 Amt. 1,562.40 1,663.20 5,846.40 4,928.00

PP Time 0 17 6 8 16 24 23 32 35 40

12,613.79 Amt. 1,663.20 957.60 6,541.20 1,612.80 3,315.20

K payments are also feasible BC schedules, and

∗BC ≥

∗PP.

Property 3. Increases in contract price always

increase the client’s expenses.

Since R =

I i=1 V i, increases in contract price result

from increases in V i for at least one event i. This

increases the amount of the payment that covers the

event, thereby increasing the client’s expenses.

Property 4. Increased project duration decreases

benefits associated with the project.

Project duration is determined by the occurrence

time for the terminal event in the project. Benefits

from project completion are represented by a positive

cash flow B associated with the terminal event. The

net present value of these benefits, Be−T I , is decreas-

ing in T I . Since E I ≤ T I ≤ LI Be−E I ≤ ∗· ≤ Be−LI .Property 5. For a fixed time gap between pay-

ments, the number of payments in a PP schedule is

nondecreasing in project duration.

In PP schedules, K =D

t=0 t with t = 1 for all val-

ues of t < T I satisfying the condition t mod g = 0,

where g represents the time gap between payments.

Since the final payment always occurs on project com-

pletion, T I = 1. Suppose the project completion time

increases to T I , payments have to be scheduled at

g 2g K − 1g. For the remaining payments, we

consider three cases.Case 1. T I mod g = 0. Any increase in T I requires

at least one additional payment since T I = Kg.

Case 2. T I mod g = 0 and T I < Kg. Since T I <

Kg, the final payment occurs at T I so that K remains

unchanged.

Case 3. T I mod g = 0 and T I < Kg. At least one

additional payment occurs at Kg so that the number

of payments increases.

Property 6. For a fixed K and BCEQ dec-

rease as T I increases. Likewise, for a fixed gPP,

values of T I that do not result in increases in K

decrease PP.

With BC and EQ schedules, increasing projectduration delays the final payment, thereby reducing

expenses. The same is true for PP schedules with no

increase in the number of payments as can be seen

from Case 2 of Property 5.

Property 7. If B exp−D > RZ· is maximized

when T I = E I .

R > · for every payment schedule. Thus, B ×

exp−D>·. Since B is the only positive cashflow

in the project network, is maximized when T I = E I .

Property 8. BC schedules are unaffected by

increases in the project deadline when the project hasa positive NPV. The project is always scheduled for

completion at E I .

Property 9. While the project is in progress, the

client always retains the maximum amount permitted

under the terms of the contractual agreement with the

contractor.

In every payment schedule, the amount of each

payment (except the final payment) is (1 − ) times

the earned value since the previous payment. It is rel-

atively simple to show that any deviation from this

strategy results in suboptimal schedules.

Let T 1 T 2 T K be the optimal payment times

with T 1 < T 2 < · · · < T K , and P 1 P 2 P K the corre-

sponding optimal amounts of payment for a given

payment schedule. Let c1 c2 cK be the discount

factor associated with the timing of each payment.

Since T i < T i+1 ci > ci+1. Suppose that the client

reduces the retention on the j th payment so that a

payment of P j + , ( > 0) is now scheduled at time

T j . Since every payment P k k = 1 K − 1 k = j is




DAYANAND AND PADMAN


at its lower bound, and K

k=1 P k = R, the schedule can

remain feasible only if the final payment is reduced

by . The NPV of the new schedule is given by

= ∗· − cj − cK <

∗·

since cj − cK > 0, and > 0.

Property 10. If the project is scheduled for com-

pletion at the same time under an EQ and PP sched-

ule for g = gPP = gK EQ, the time gap to the last

payment is no more than g .

Let T 1 T 2 T K represent the timing of the pay-

ments under the EQ schedule. Since the PP sched-

ule is identical to the EQ schedule, these timings are

also the timing of payments under the PP schedule.Suppose that (T K − T K −1 > g. Under PP, a payment is

scheduled every g time periods. Thus, the time gap

T K − T K −1 requires additional payments and contra-

dicts the fact that the EQ and PP schedules are iden-

tical, since K PP > KEQ.

Property 11. Expenses associated with a BC, EQ,

or PP schedule decrease with increasing retention

over the duration of the project.

A payment schedule T 1· T 2· T K ·, and

P 1·, P 2· P K · that is optimal for = is

feasible for all > . Thus,

∗ ≤

∗ ∀ >

When changes in retention leave completion time

unchanged,

∗ ≥

∗ ∀ >

Property 12. Expenses associated with BC sched-

ules can be decreased by offering fewer payments

over the duration of the project.

From a schedule with K payments, we construct a

schedule with K − 1 payments and show that the

NPV of the reduced schedule is less than the NPV

of the K -payment schedule. Any BC schedule with

K payments can be reduced to a schedule with K −

1 payments by dropping one of the payments over

the duration of the project. Let T 1 T 2 T K be the

optimal payment times and P 1 P 2 P K the corre-

sponding payment amounts for a BC schedule with

K payments. t = 1 for t = T 1 T 2 · · · T K . The number of

payments can be reduced by setting t equal to zero

for some T j i = 1 · · · K − 1. The amount of payment at

time T j is zero, and the P j is added to the P j +1. The

resulting schedule satisfies precedence and distribu-tion constraints and has K − 1 payments. The change

in net present value of the expenses is given by

K −K − 1 = cj P j + cj +1P j +1 − cj +1P j + P j +1

= P j cj − cj +1

> 0 since P j > 0 and c j > c j +1

5. Observations fromSample Projects

While model properties allow us to make some infer-ences about general characteristics of optimal solu-

tions, the magnitude of the impact of changes in

model parameters can be better appreciated by exam-

ining optimal solutions to sample problems. Thus,

in the second part of our analysis, we study char-

acteristics of optimal solutions to 10 sample prob-

lems. The sample problems are randomly generated

projects with 10 activities each. Network structure and

distribution of earned value vary across projects. We

restricted our sample to projects of small size because

of the combinatorial nature of the models, and theconsequent difficulty in obtaining optimal payment

schedules for projects with a large number of activi-

ties, or long durations.

Projects in our test data set have activity durations between 1 to 9 time units, with a mean activity dura-

tion ranging from 3 to 4.7 for each project, and an

average duration of 3.73 across projects. The sample

projects have a positive or negative value associated

with each event. For our analysis, we aggregated all

positive values to obtain the contract price. We deter-

mined value of work associated with each activity

from the relation

W ij =RN ij

I i=1

I j =1 N ij

(20)

where N ij represents the expenses associated with

activity ij. Earned value was computed using

Equation (1). The contract price varies from $200–

$326, with an average of $264.70. Earned value asso-

ciated with each event ranges from a low of $2.23




DAYANAND AND PADMAN


Table 2 Optimal Objective Values

Mean NPVNumber of Number of Difference Percentage

Payments Projects BC EQ in NPV Delay Delay

2 10 255.27 256.59 1.32 2.56 14.06

4 9 263.44 264.81 1.37 2.59 13.80

6 6 266.17 267.09 0.92 1.79 12.96

to a high of $49.82. Distribution of earned value

over project events varies across projects; the standard

deviation of earned value over project events ranges

from $10.64–$24.22. We allowed a 10% retention for

all models and examined optimal solutions to the BC

and EQ models for 2, 4, and 6 payments. For the PP

model, we set gPP = g2EQg4EQ, and g6EQ.

We allowed the maximum possible time for comple-

tion of each project by setting the deadline equal to

the sum of the durations of all activities in the project.

Optimal solutions to the sample problems were

obtained using GAMS with OSL. As anticipated, we

were unable to obtain optimal solutions with EQ and

PP in some cases due mainly to the network struc-

ture. The number of infeasible cases increased as we

increased the number of payments for EQ, or equiv-

alently, as we decreased gK . For those cases in which

optimal EQ and PP schedules were obtained, a given

value of g produced identical EQ and PP schedules

in all but one project. Thus, our analysis focuses on a

comparison of BC and EQ schedules.

Our observations from these optimal solutions can

be summarized under two main categories. In the

first stage, we compared optimal solutions in terms

of the average NPV of the client’s expenses across all

projects and project completion times. In the second

stage, we examined payment patterns generated by

the two models.

5.1. Observations on NPV of the

Client’s Expenses

Table 2 summarizes our analysis in the first stage. The

main observations are as follows:

• Regardless of the number of payments, BC

always scheduled the project for early completion.

With EQ, 33%–40% of the projects were delayed

beyond early completion. The extent of delay ranged

from 5%–17% of the shortest possible completion

time.

• Columns 3 and 4 of Table 2 show average NPV of

the client’s expenses for 2, 4, and 6 payments. Becauseof the small values of V i and dij in the sample projects,

differences in average NPV between BC and EQ solu-

tions are small enough to be regarded as inconsequen-

tial.

• Small differences in NPV mask significant dif-

ferences in BC and EQ schedules. To evaluate the

impact of the schedule on the difference in NPV, we

express the observed difference in NPV as a differ-

ence in timing if the total payment for the project is

made as a single payment. With a single payment,

timing is the only factor that can cause a difference inNPV. We define a new measure, delay factor (T ), that

expresses the difference in timing that would result in

the observed difference in NPV between BC and EQ

schedules.

Definition 4. Delay Factor (T ): Let NPV be

the observed difference in NPV between two payment

schedules. Let T and T = T + T represent two dif-

ferent times at which a payment can be scheduled

for the project. Suppose that the total payment R is

scheduled at T and T , then T can be obtained from

R

cT − cT +T

= NPV (21)

so that

T = −NPV1 + T1 + T + T

R (22)

T is positive when NPV is positive and vice versa.

Table 2 also shows the mean value of T across

all projects for which we obtained both BC and EQ

schedules. We set T equal to E I /2 to obtain values of

T for each project. Column 7 of Table 2 shows the

average value of T as a percentage of the shortest

duration for the project. The difference in NPV trans-

lates to at least 13% of the project duration. Mean per-

centage delay shows a decreasing trend as the number

of payments increases.

5.2. Observations on Payment Patterns

Our observations, with respect to payment patterns,

are based on Tables 3 and 4. Table 3 shows, in chrono-

logical order, the mean and standard deviation of




DAYANAND AND PADMAN


Table 3 Summary of Optimal Payment Schedules

Payments(%) Time IntervalsNumber of

Payments Model 1 2 3 4 5 6 1 2 3 4 5 6

2 BC M 28 72 52 48

SD 31 31 36 36

EQ M 35 65 51 49

SD 15 15 1 1

4 BC M 23 29 13 35 36 34 12 18

SD 27 28 18 25 29 29 13 23

EQ M 13 28 30 29 26 26 26 22

SD 7 12 11 9 3 2 2 5

6 BC M 7 19 14 7 20 34 14 22 12 8 19 25

SD 6 19 20 2 23 21 10 25 11 4 22 26

EQ M 14 12 17 19 12 25 16 16 16 16 16 19

SD 10 13 12 11 11 5 1 1 1 1 1 6

the amount of each payment as a percentage of the

contract price, and the mean and standard devia-

tion of the time gap between successive payments as

a percentage of the project duration. Table 4 shows

average amount of payment when the payments are

considered in order of their amounts, starting with

the largest payment first.

• BC schedules show greater variation than EQ in

the amount and timing of payments across projects,

as well as over any given project.• Long time gaps between payments lead to large

payments since more activities can be completed.

• In most optimal BC schedules, payments tend

to be clustered either at the beginning or the end of

the project. The largest payment typically occurs at

the end of the project. Many of the remaining pay-

ments cover the earned value associated with one

event since this oftens results in a small payment.

Table 4 Maximum Payment

Payments Time Intervals

K Model 1 2 3 4 5 6 1 2 3 4 5 6

2 BC 84 (2) 16 (9) 84 16

EQ 68 (4) 32 (7) 51 49

4 BC 67 (1) 20 (1) 9 (2) 4 (7) 70 15 8 7

EQ 40 (2) 28 (2) 21 (3) 11 (4) 27 26 26 20

6 BC 53 (1) 20 (1) 12 (1) 8 (1) 5 (2) 2 (5) 47 20 14 7 6 6

EQ 32 (1) 26 (1) 20 (1) 12 (2) 6 (3) 4 (3) 16 16 16 16 16 19

• In general, the client reduces expenses by

scheduling slack events as late as possible. However,

when the earned value associated with a slack event

is small, optimal schedules indicate earlier scheduling

of these events so that events with higher value can

be delayed.

6. Impact on the Contractor’s

Activity ScheduleThe client’s payment schedule maximizes her net ben-

efit from the project, and the activity schedule min-

imizes expenses for the planned completion time.

Whether activities are completed according to the

client’s plan depends on the effect of the planned

schedule of payments on the contractor’s preferred

schedule of activities. The contractor views events

with associated progress payments as milestones. The




DAYANAND AND PADMAN


Figure 2 The Contractor’s Optimal Activity Schedule Under BC

timing of these milestone events, together with the

timing of the contractor’s expenses for the project,

determines profitability. The contractor’s objective

is to maximize NPV for the schedule of payments

offered by the client. Thus, the contractor determines

an optimal activity schedule from a model of the

project network with fixed cash flows. The net cash

flow at each event is the difference between the pay-

ment and expenses associated with that event. Ingeneral, the contractor increases NPV by scheduling

events with a net positive cash flow as early as possi-

ble, and events with a net negative cash flow as late

as possible. Just how early or late these cash flows

can be scheduled depends on the distribution of net

cash flows over project events, activity precedences,

and durations (Elmaghraby and Herroelen 1990).

Figure 2 shows the contractor’s optimal activity

schedule when the client determines the amount of

payment using the BC model. The value associated

with each event represents the expenses incurred by

the contractor on occurrence of that event. Net cash

flow at payment events is the difference between the

payment and the contractor’s expenses. The contrac-

tor can schedule activities to maximize NPV for this

set of net cash flows using one of several available

methods (Russell 1970, Grinold 1972, Elmaghraby and

Herroelen 1990). The contractor’s optimal schedule

is also shown in Figure 2. Table 5 shows the con-

tractor’s and client’s optimal activity schedules for

the example project with the client’s BC, EQ, and

PP payment schedules. Regardless of the model in

question, we observed differences in the contractor’s

and client’s optimal activity schedules for the exam-

ple project. The contractor fares poorly with BC. The NPV

improves dramatically with the client’s EQ and PP payment

schedules.

We compared the contractor’s and client’s optimal

activity schedules under the BC and EQ models for 2,4, and 6 payments on the sample projects.

6.1. Contractor NPV with the Client’s PP Schedule

Table 63 summarizes the differences in the contrac-

tor’s NPV under the client’s and the contractor’s own

activity schedule for BC and EQ schedules offered

by the client. Our analysis shows that as the number

of payments increases, the mean absolute difference

in the timing of payments increases. The contractor’s

activity schedule always leads to a higher NPV for the

contractor (column 8 of Table 6). A comparison of the

3 Columns 1–3 of the table are self explanatory. Column 4 of the

table reports the number of projects in which schedule differences

were observed. Column 5 is the mean absolute difference in the

timing of the payments. Columns 6 and 7 show the contractor’s

mean NPV with the client’s optimal activity schedule and the con-

tractor’s own optimal activity schedule. Column 8 converts the dif-

ference in NPV as the difference in timing described by Equation

(22), while column 9 reports this average difference in timing as a

percentage of the project’s shortest possible duration.




DAYANAND AND PADMAN


Table 5 The Contractor’s and Client’s Schedule of Activities

Event → 1 2 3 4 5 6 7 8 9 10

BC Amt. 604.80 403.20 1260.00 11732.000441.13 T Con

i 0 13 3 5 13 26 20 31 32 37

12,327.94 T Cl i

0 12 3 5 13 26 20 31 32 37

EQ Amt. 1562.40 1663.20 5846.40 4928.00

834.36 T Con i

0 15 3 5 13 21 20 31 32 37

12,488.74 T Cl i

0 18 3 9 17 27 24 35 36 41

PP Amt. 1663.20 957.60 6541.20 1612.80 3315.20

941.95 T Con i

0 13 3 5 13 21 20 26 32 37

12,613.79 T Cl i

0 16 3 8 16 24 23 32 35 40

Table 6 Comparison of Contractor-Client Activity Schedules

MAD NPV NPVK Model N # Diff Time (CL) (CT) Delay % Delay

2 BC 10 6 2.33 150.85 151.10 0.45 2.43

EQ 10 5 2.80 152.09 152.43 0.63 3.40

4 BC 10 10 4.80 151.68 152.22 0.92 5.26

EQ 9 9 5.22 159.71 160.20 0.88 4.76

6 BC 10 9 7.56 152.76 153.17 0.75 4.05

EQ 6 5 15.20 164.93 166.12 2.11 10.35

delay factor showed no trend with BC schedules, but

EQ schedules showed increasing average delay as K

increased.

6.2. Client Expenses with Contractor’s Preferred

Event Schedule

Table 7 shows the client’s expenses under the con-

tractor’s optimal schedule and client’s optimal pay-

ment schedule. For schedules with 2 and 6 payments,

on the average, the contractor’s optimal activity sched-

ule forces the client to incur expenses that are higher

than expenses incurred with her own optimal activity

schedule. However, with 4-payment schedules, the client’s

expenses are lower with the contractor’s activity sched-

ule. This occurs because of differences in the schedul-

ing of slack events. In many cases, the client sched-

ules slack events to prevent occurrence of multiple

events at the same time. However, the contractor’s

schedules sometimes indicate occurrence of multiple

events at the time of payment, even though the pay-

ment does not cover value associated with some of

the events. Many projects in the test data set have

slack events scheduled—as explained above—leading

to lower average expenses for the client. The differ-

ence in expenses translates to a “delay” ranging from

11% to 27% of project duration.

6.3. Comparison with Contractor’s Optimal

Payment and Activity Schedule

To complete the analysis, we also compared financial

performance of the project under the contractor’s best

schedule with the BC and EQ schedules offered by

the client. The contractor’s best schedule refers to the

amount and timing of payments that maximizes the

contractor’s NPV. The results are tabulated in Table

8. There is considerable disparity between the con-

tractor’s maximum profit and NPV obtained under

the optimal activity schedule with the BC payment

schedule as seen by the percentage delay in Column

5 of Table 8. Delay factors as a percentage of project

duration range from 17 to 24%. In comparison, EQ

payment schedules are closer to the maximum profit

that the contractor can make on a project. The varia-

tion with both BC and EQ schedules decreases with

increasing K .

Table 7 Comparison of Client’s Expenses

BC EQClient’s Exp.

K (Cont. Sch.) Avg. Exp. Delay % Delay Avg. Exp. Delay % Delay

2 257.60 255.27 2.71 13.95 256.59 2.11 11.19

4 258.83 263.44 5.02 26.74 264.81 2.71 14.78

6 259.07 257.09 3.43 18.25 267.09 3.02 14.76




DAYANAND AND PADMAN


Table 8 Comparison of Contractor’s Profit

BC EQ

K Max Avg. NPV Delay % Delay Max Avg. NPV Delay % Delay

2 153.25 151.12 4.04 21.92 153.25 152.43 1.53 8.10

4 154.20 152.21 3.42 17.87 160.70 160.20 0.92 4.92

6 154.21 153.17 1.94 24.95 166.20 166.12 0.18 1.86

7. ConclusionsThis paper discusses some of the issues in determin-

ing payment schedules from a client’s point of view.

Three models have been formulated for the client’s

payment scheduling problem. Model properties indi-

cate that project completion time is a critical factor

in comparing payment schedules across models. Theclient achieves the highest benefit and least expenses

from the BC model. Optimal BC schedules suggest

that the client benefits by paying small amounts

and/or paying them as late as possible. The client

can reduce the amount of payment by scheduling

only one event to occur at the time of any payment.

The client can also reduce her expenses by increas-

ing retention or by offering fewer payments over the

duration of the project. Although increased project

duration decreases the client’s expenses, extending

project deadlines does not offer any advantage to theclient in most projects, since the decrease in bene-

fit is generally much greater than the decrease in

expenses. BC schedules also show greatest variance

in the amount of payment or time gap between pay-

ments over a project. However, unlike the EQ and PP

models, feasible BC schedules can always be guaran-

teed, although they require the client to modify cur-

rent payment practices to meet the recommendations

of the model.

Several extensions of this work are possible. First,

the client’s model can be extended to include bonuses,

penalties, and constraints, such as on resource and

cash flows. Second, while we have examined some

properties of optimal schedules, no attempt has been

made to design efficient solution procedures for the

client’s payment scheduling problem. Given the dif-

ficult combinatorial nature of the problem, conver-

gence to optimality is a very slow process, while com-

mercially available software and heuristics may be

a more practical alternative. Finally, a joint view of

the payment scheduling problem that addresses the

diverse requirements of the multiple parties enter-

ing the project contract is a promising direction for

future work, especially given the cooperative natureof the partnering and outsourcing strategies. As indi-

cated by the analysis of the client’s payment sched-

ule on the contractor’s activity schedule, the two

parties can negotiate on many parameters of the

project, such as retention, number or frequency of

payments, distribution scheme, and deadline to arrive

at a satisfactory schedule of payments and activities.

These models and methods can be further embedded

in decision support environments to facilitate the bid-

ding and negotiation process, and to resolve payment

and financing conflicts in dynamic project situationsoften encountered when outsourcing large projects.

ReferencesAbernathy, E. S. 1990. Managing a contractor’s billings. J. Accoun-

tancy 197–110.

Bey, R. B., R. H. Doersch, J. H. Patterson. 1981. The net present value

criterion: Its impact on project scheduling. Project Management

Quart. 12(2) 35–45.

Brooke, A., D. Kendrick, A. Meeraus. 1988. GAMS: A Users Guide.

The Scientific Press, CA.

Cleland, D. I., W. R. King, eds. 1983. Project Management Handbook .

Van Nostrand Reinhold Company, NY.

Clough, R. A., G. A. Sears. 1991. Construction Project Management.

John Wiley and Sons, NY.

Dayanand, N. 1996. Scheduling Payments in Projects: An Optimiza-

tion Framework . PhD thesis, The Heinz School, Carnegie Mellon

University, Pittsburgh, PA 15213.

, R. Padman. 1993. Payments in projects: A contractor’s model.

Working Paper 93-71, The Heinz School, Carnegie Mellon Uni-

versity, Pittsburgh, PA 15213.

, . 1997. On modelling payments in project networks. J.

Oper. Res. Soc. 48(9) 906–918.

, . 1998. On payment schedules in contractor client nego-

tiations in projects: An overview of the problem and research

issues. J. Weglarz, ed., Recent Advances in Project Scheduling,Chapter 21. Kluwer Academic Publishers,

Elmaghraby, S. E., W. S. Herroelen. 1990. The scheduling of activi-

ties to maximize the net present value of projects. Eur. J. Oper.

Res. 49 35–49.

Fleming, Q. W., J. M. Koppelman. 1994a. The “earned value con-

cept:” Back to the basics. PMNetwork VIII(1).

, . 1994b. The “earned value concept:” Taking step two—

plan and schedule the project. PMNetwork VIII(9).

, . 1995. Taking step three with earned value: Estimate

and budget resources. PMNetwork IX(1).




DAYANAND AND PADMAN


Gilbreath, R. D. 1992. Managing Construction Contracts. John Wiley

and Sons, Inc.,

Glover, F., E. Woolsey. 1974. Converting a 0-1 polynomial pro-

gramming problem to a 0-1 linear program. Oper. Res. 22180–182.

Grinold, R. C. 1972. The payment scheduling problem. Naval Res.

Logist. Quart. 19(1) 123–136.

Hendrickson, C., T. Au. 1989. Project Management for Construction.

Prentice Hall, Englewood Cliffs, NJ.

Herroelen, W. S., P. V. Dommelan, E. L. Demeulemeester. 1997.

Project network models with discounted cash flows: A guided

tour through recent developments. Eur. J. Oper. Res. 100 97–121.

Hung, M. S., W. O. Rom. A. D. Warren. 1994. Optimization with IBM

OSL. Boyd and Fraser Publishing Company, CA.

Martin, M. D., F. M. Webster. 1988. Managing the future

through project contracting. Proceedings of the Annual Seminar/

Symposium of the Project Management Institute.

Oltman, J. R. April 16, 1990. 21st century outsourcing. Computer-world.

Russell, A. H. 1970. Cash flows in networks. Management Sci. 16(5)

357–373.

Schnieders, J. R., R. A. Jameyson. 1987. Financial analysis for

new formulation projects. Proceedings of the Annual Seminar/

Symposium of the Project Management Institute.

Singh, S. 1989. Sensitivity analysis of the effect of vari-

ables on cash flow for construction projects. Proceedings

of the Annual Seminar/Symposium of the Project Management

Institute.

Accepted by Awi Federgruen.


progress payment timing

Documents