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


    E-mail addresses: (I-Tung Yang), tavina. (C.-Y. Chang).

    International Journal of Project Manag


    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

    ement 23 (2005) 546553


  • 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



    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



    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


    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



    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



    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



    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



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