# Stochastic resource-constrained scheduling for repetitive construction projects with uncertain supply of resources and funding

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In this paper, we schedule construction projects that are repetitively performed by similar resources (labor and equipment) from

Housing communities, multistory buildings, high- period of time. This class of problem, so-called limited

the operation system of repetitive construction projects

and discuss an existing deterministic optimization model

with the objective of maximizing the system production

rate as well as the project completion time. The deter-ministic model is enhanced to a chance-constrained

* Corresponding author. Tel.: +886 2 226215656x3261; fax: +886 2

26209747.

E-mail addresses: ityang@mail.tku.edu.tw (I-Tung Yang), tavina.

chang@msa.hinet.net (C.-Y. Chang).

International Journal of Project Manag

INTERNATIONAL JOURNAL OF

PROJECT0263-7863/$30.00 2005 Elsevier Ltd and IPMA. All rights reserved.ways, and pipelines are good examples that exhibit

repetitive characteristics where crews perform similar

or identical tasks from one working location to another.

These projects are therefore called repetitive projects.

Since repetitive projects represent a large portion of the

construction industry, they need an eective scheduling

method to ensure the project can be completed in themost ecient manner.

Resources employed in repetitive projects, such as la-

bor and equipment, are often subject to limited avail-

ability. This leads to a practical concern of how to

resource allocation or resource-constrained schedul-

ing, has been an active subject for decades because of

its practical relevance [7].

The purpose of this paper is to present a chance-

constrained programming model to solve the limited re-

source allocation problem under uncertain supply

conditions of resources and funding. The paper is orga-nized as follows. In the next section, we review existing

approaches in scheduling scarce resources. The review

argues that it is needed for a resource-constrained model

to handle uncertain supply conditions. We then describeone location to another when the supply of resources and funding is not only limited but also subject to uncertainty. The supply

conditions are hence expressed as probabilistic distributions in lieu of single estimates. The problem is formulated as a chance-

constrained program, which is converted to a deterministic equivalent and solved by means of common linear programming

techniques. The proposed chance-constrained programming model is applied to a housing project with 100 units and a real-life

tunnel project to illustrate its practical usefulness in determining project acceleration strategies and in conducting risk management

analyses. Model verication is performed by Monte Carlo simulations to conrm that the probability of constraint satisfaction is

beyond the required condence level.

2005 Elsevier Ltd and IPMA. All rights reserved.

Keywords: Risk; Time; Cost; Chance-constrained programming

1. Introduction distribute the limited resources among various activities

so that the project can be completed within a minimalStochastic resource-constraconstruction projects with uncerta

I-Tung Yang a,

a Department of Civil Engineering, Tamkang Universityb Department of Construction Engineering

Received 15 September 2004; received in revis

Abstractdoi:10.1016/j.ijproman.2005.03.003d scheduling for repetitivesupply of resources and funding

hi-Yi Chang b

Yingchuan Road, Tamsui, Taipei County 251, Taiwan

oyang University of Technology, Taiwan

rm 7 December 2004; accepted 1 March 2005

www.elsevier.com/locate/ijproman

ement 23 (2005) 546553

MANAGEMENT

sources can be expressed as deterministic numbers. Yet

this assumption may be inappropriate when the supply

To address the concerns, it is necessary for the limited

resource allocation problem to incorporate uncertain

supply conditions of resources and funding so to help

project planners schedule their project and manage

inherent risks with better condence.

3. Deterministic optimization model

Within repetitive projects, scheduling is usually done

by considering crews owing through the whole pro-

ject, similar to manufacturing assembly lines [32]. In

repetitive construction projects, the product being built

tends to be stationary, whereas every crew proceedsfrom location to location and complete work that is pre-

requisite to starting work by the following crew, like a

parade of trades [29].

The ow model of repetitive projects leads to the use

urnal of Project Management 23 (2005) 546553 547of resources may in reality contain a great deal of uncer-

tainty when extending several months, or even years,

into the future. For instance, the situation of labour

shortage leads to a sharing of crews between multipleprojects. As a result, the actual manpower allocated to

the project may dier from what was originally planned.

Another practical concern is to schedule the project

under the limited supply of funding, which constitute

another limited resource allocation problem. Since the

budget of the project is usually estimated long before

the project commencement date, the supply condition

of funding is often subject to uncertainty and deviatesfrom the original plan due to nancing, political, and so-programming formulation. The chance-constrained pro-

gramming problem is converted to a deterministic equiv-

alent, which is then solved by common linear

programming techniques. A numerical example with

100 units of housing is used to illustrate the application

of the proposed model and to demonstrate how theproposed model can assist project managers in project

acceleration and risk management. We also briey

introduce the application of the proposed model in a

real-life tunnel project to demonstrate its practical

acceptance. The proposed model is veried through

Monte Carlo simulations. Conclusions are drawn in

the last section.

2. Review of existing approaches

Due to the assumption of unlimited resource avail-

ability, traditional scheduling techniques, such as the

critical path method (CPM) and barchart, have been

shown less than satisfactory in addressing the limited re-

source allocation problem [1]. Hence, three groups ofalternatives were proposed.

The rst group formulated the problem as a mathe-

matical optimization model. Operational research tech-

niques used to solve the mathematical model included

linear programming [28], mixed integer programming

[13,22], branch-and-bound [6,8], and dynamic program-

ming [10]. The second group specied a variety of heu-

ristics (a set of rules to prioritize activities in theassignment of resources) to provide a near-optimal solu-

tion in practical time. Examples of this group dated

back to the works of Kelley [18] and Wiest [31] while

some recent eorts can be found in [7,24]. The third

group capitalized on the development of evolutionary

computation techniques, such as genetic algorithms

[14,20], and local search techniques, such as simulated

annealing [17] and tabu search [16,30]. For a detailedsurvey of the literature, we refer to [2,15].

Previous methods typically assumed the limits of re-

I-Tung Yang, C.-Y. Chang / International Jocial risks [9].Duration

1 A BC

Duration of the First Unit Remaining Duration

Project Durationof production lines whose slopes represent individual

production rates. To accelerate the project, a well-

grounded treatment is to balance the production lines

(i.e., to make all the lines have the same slope) as de-creed in the line-of-balance procedure [21]. Fig. 1 illus-

trates a balanced situation where three consecutive

activities A, B, and C are repetitively performed from

unit 1 to unit u. Given the duration of the rst unit (cal-

culated by CPM), the rate of construction is then the re-

sult of dividing the remaining number of units (u 1) bythe dierence between project deadline and the duration

of the rst unit. Once the rate of construction has beencalculated, the start times of activities on dierent units

and the number of each required crew can be computed.

The traditional approach of average rate in the line-

of-balance procedure is underpinned by the assumption

that resource availability is unlimited. When the supply

of resources is constrained, a challenge emerges: how to

eciently allocate the working hours of available

Uni

t

uA CB

Rate ofConstructionFig. 1. Balanced condition.

548 I-Tung Yang, C.-Y. Chang / International Journal of Project Management 23 (2005) 546553resources to various activities when a resource may

work for dierent activities (e.g., carpenters construct

both column forms and exterior doors) and an activity

may employ multiple types of resources (e.g., earth mov-

ing needs both bulldozers and dump trucks). These is-

sues, unfortunately, cannot be resolved by thetraditional line-of-balance procedure. It is therefore

the goal of the proposed model to incorporate limited

supply conditions of resources and funding in the line-

of-balance procedure.

In what follows, we introduce a deterministic optimi-

zation model, which is a variant of Pereras model [27],for scheduling repetitive projects with the consideration

of limited resource and nancial availability. The fol-lowing model extends Pereras by introducing new coef-cients that are used to elucidate the logic behind the

model. In the next section, we take one step further to

consider the situation when the availability cannot be

expressed merely by a single estimate.

The general objective of the deterministic model is to

minimize the project duration, which can be computed

as follows:

D D1 M 1Q ; 1

where D is the project duration;M is the number of pro-

gress units; and D1 is the duration of the rst unit. Since

M is a constant, to minimize the project duration isequivalent to maximizing the system production rate,

Q (measured in unit/day), which can be expressed as

Q M 1D D1 . 2

The objective is thereby

Maximize Q. 3The constraints comprise six sets of equations. The

rst set describes that the system production rate is gov-

erned by the slowest activity. This is the underlying con-

cept of bottleneck in the Theory of Constraint [12]

Q 6 Qi 8i; 4where Qi denotes the production rate of activity i (mea-

sured in unit/day).The second set denes coecients of resource con-

sumption for individual activities (RCi)

RCi DiQi 8i. 5Since the unit duration of activity i (Di, measured in day/

unit) is a known constant, the coecient of resourceconsumption is directly proportional to the production

rate of activity i (Qi, measured in unit/day). The mecha-

nism is that an increase in the production rate of an

activity would require more resources working at dier-

ent units simultaneously. As a result of multiplication,the coecient of resource consumption is unitless.In the third set, we derive the daily working hours of

the resource j allocated to activity i (RDij, measured in

resource-hour/day) by dividing the working hours of

resource j required by activity i (Rij, measured in

resource-hour/unit) by the unit duration of activity i

(Di, measured in day/unit). The equation is

RDij RijDi 8i; j. 6

The fourth set of constraints ensures the feasibility of

resources. That is, the daily working hours of each re-

source (sum of the allocated amounts for dierent activ-

ities, i = 1,2, . . .,n) should be less or equal to theavailable amountXni1RDij RCi 6 T j 8j; 7

where Tj denotes the availability of resource j per day

(measured in resource-hour/day). Substitute the left-

hand side (LHS) of (7) by (5) and (6), the constraint

can be rewritten asXni1

RijQi 6 T j 8j. 8

Aggregating the costs of all the resources

(j = 1,2, . . .,m), the fth set considers the nancial feasi-bility by limiting the daily cost to be less or equal to the

available funding

Xmj1

Xni1

RijQi P j !

6 C; 9

where Pj is the unit price of resource j (measured in $/re-

source-hour) and C is the available funding.

The last set of constraints denes the upper and lower

bounds of the production rate for each activity. The for-

mer may be caused by technical constraints, such as lim-

its on equipment capacity or crew productivity whereas

the latter, being always positive, may be the result ofeconomical consideration

Qi6 Qi 6 Qi 8i; 10

where the upper bound Qi represents the fastest possiblepace at which activity i can be performed and the lower

bound Qidenotes the slowest possible pace.

To sum up, the deterministic model is

Maximize Q. 11Subject to

Q 6 Qi 8i; 12Xni1RijQi 6 T j 8j; 13

Xmj1

XnRijQi P j

!6 C; 14i1

~ujT jdT j aTj ; 18

urnalT j

where uj(Tj) is the distribution density function of Tj.Suppose the cumulative distribution function (CDF)

of Tj, denoted by Wj(Tj), is continuous and strictlymonotonic, ~T j can be computed directly based on the in-verse of the CDF

~T j W1j T j; 1 aTj . 19Qi6 Qi 6 Qi 8i. 15In the deterministic model, Qi are the decision vari-

ables. The required input includes Tj (available workinghours of resource j), C (available funding), Rij (necessary

working hours of resource j for activity i), Pj (unit price

of resource j), and Qi and Qi (upper and lower boundsof the production rate for activity i).

4. Stochastic optimization model

To deal with the uncertainty associated with the sup-

ply of resources, we modify (13) to a probabilistic

constraint

PrXni1

RijQi 6 T j

" #P aTj 8j; 16

where Pr[] is the probability of the event in []; Tj is nowa random value with a distribution of uncertainty; aTj isa pre-specied condence level. So a solution set is feas-

bile if and only if the probability measure of the set is at

least aTj . In other words, the constraint will be violatedat most 1 aTj of time.

Similarly, (14) is modied to address the uncertainty

inherent in the supply of funding

PrXmj1

Xni1

RijQi P j !

6 C" #

P aC; 17

where C is a random value; aC is a pre-specied con-dence level, which represents the lowest limit on theprobability of the constraint being satised. Again, the

constraint will be violated at most (1 aC) of time.The stochastic model can be solved by chance-

constrained programming (CCP), which was rst

developed by Charnes and Cooper [4] and has been since

applied to a wide variety of practical problems, such as

investment portfolio selection [23], railway reservation

[19], and water reservoir management [26].The basic technique of CCP is to convert the stochas-

tic constraints to their respective deterministic equiva-

lents. For (16), if we can estimate the distribution

density of Tj, the deterministic equivalent is to replace

the right-hand side (RHS) by ~T j such thatZ 1

I-Tung Yang, C.-Y. Chang / International JoThus, the deterministic equivalent of (16) isXni1

RijQi 6 W1j T j; 1 aTj 8j. 20

By the same token, we can convert the probabilistic con-

straint (17) to the following deterministic equivalent

Xmj1

Xni1

RijQi P j !

6 W1C; 1 aC. 21

Since (20) and (21) are linear combinations of the

decision variables, the deterministic equivalent of the

stochastic model is still a linear programming problem.

So it can be solved in polynomial time by the use of veryecient linear programming techniques, such as simplex

o...

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