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Stochastic resource-constrained scheduling for repetitive construction projects with uncertain supply of resources and funding
INTERNATIONAL JOURNAL OF
PROJECT MANAGEMENT International Journal of Project Management 23 (2005) 546–553 www.elsevier.com/locate/ijproman
Stochastic resource-constrained scheduling for repetitive construction projects with uncertain supply of resources and funding I-Tung Yang a
a,*
, Chi-Yi Chang
b
Department of Civil Engineering, Tamkang University, 151 Yingchuan Road, Tamsui, Taipei County 251, Taiwan b Department of Construction Engineering, Chaoyang University of Technology, Taiwan Received 15 September 2004; received in revised form 7 December 2004; accepted 1 March 2005
Abstract In this paper, we schedule construction projects that are repetitively performed by similar resources (labor and equipment) from one 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 chanceconstrained 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 verification is performed by Monte Carlo simulations to confirm that the probability of constraint satisfaction is beyond the required confidence level. Ó 2005 Elsevier Ltd and IPMA. All rights reserved. Keywords: Risk; Time; Cost; Chance-constrained programming
1. Introduction Housing communities, multistory buildings, highways, 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 effective scheduling method to ensure the project can be completed in the most efficient manner. Resources employed in repetitive projects, such as labor and equipment, are often subject to limited availability. This leads to a practical concern of how to * Corresponding author. Tel.: +886 2 226215656x3261; fax: +886 2 26209747. E-mail addresses: [email protected] (I-Tung Yang), tavina. [email protected] (C.-Y. Chang).
0263-7863/$30.00 Ó 2005 Elsevier Ltd and IPMA. All rights reserved. doi:10.1016/j.ijproman.2005.03.003
distribute the limited resources among various activities so that the project can be completed within a minimal period of time. This class of problem, so-called ‘‘limited resource allocation’’ or ‘‘resource-constrained scheduling’’, has been an active subject for decades because of its practical relevance [7]. The purpose of this paper is to present a chanceconstrained programming model to solve the limited resource allocation problem under uncertain supply conditions of resources and funding. The paper is organized 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 describe 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 deterministic model is enhanced to a chance-constrained
I-Tung Yang, C.-Y. Chang / International Journal of Project Management 23 (2005) 546–553
2. Review of existing approaches Due to the assumption of unlimited resource availability, traditional scheduling techniques, such as the critical path method (CPM) and barchart, have been shown less than satisfactory in addressing the limited resource allocation problem [1]. Hence, three groups of alternatives were proposed. The first group formulated the problem as a mathematical optimization model. Operational research techniques used to solve the mathematical model included linear programming [28], mixed integer programming [13,22], branch-and-bound [6,8], and dynamic programming [10]. The second group specified a variety of heuristics (a set of rules to prioritize activities in the assignment of resources) to provide a near-optimal solution in practical time. Examples of this group dated back to the works of Kelley [18] and Wiest [31] while some recent efforts 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 detailed survey of the literature, we refer to [2,15]. Previous methods typically assumed the limits of resources can be expressed as deterministic numbers. Yet this assumption may be inappropriate when the supply of resources may in reality contain a great deal of uncertainty when extending several months, or even years, into the future. For instance, the situation of labour shortage leads to a sharing of crews between multiple projects. As a result, the actual manpower allocated to the project may differ 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 deviates from the original plan due to financing, political, and social risks [9].
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 confidence.
3. Deterministic optimization model Within repetitive projects, scheduling is usually done by considering crews ‘‘flowing through’’ the whole project, similar to manufacturing assembly lines [32]. In repetitive construction projects, the product being built tends to be stationary, whereas every crew proceeds from location to location and complete work that is prerequisite to starting work by the following crew, like a ‘‘parade of trades’’ [29]. The flow model of repetitive projects leads to the use of production lines whose slopes represent individual production rates. To accelerate the project, a wellgrounded treatment is to ‘‘balance’’ the production lines (i.e., to make all the lines have the same slope) as decreed in the line-of-balance procedure [21]. Fig. 1 illustrates 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 first unit (calculated by CPM), the rate of construction is then the result of dividing the remaining number of units (u 1) by the difference between project deadline and the duration of the first unit. Once the rate of construction has been calculated, the start times of activities on different units and the number of each required crew can be computed. The traditional approach of average rate in the lineof-balance procedure is underpinned by the assumption that resource availability is unlimited. When the supply of resources is constrained, a challenge emerges: how to efficiently allocate the working hours of available
B
A
u
C
Unit
programming formulation. The chance-constrained programming problem is converted to a deterministic equivalent, 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 the proposed model can assist project managers in project acceleration and risk management. We also briefly introduce the application of the proposed model in a real-life tunnel project to demonstrate its practical acceptance. The proposed model is verified through Monte Carlo simulations. Conclusions are drawn in the last section.
547
Rate of Construction
Project Duration
1
A
B
C
Duration of the First Unit
Remaining Duration
Duration Fig. 1. Balanced condition.
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resources to various activities when a resource may work for different activities (e.g., carpenters construct both column forms and exterior doors) and an activity may employ multiple types of resources (e.g., earth moving needs both bulldozers and dump trucks). These issues, unfortunately, cannot be resolved by the traditional line-of-balance procedure. It is therefore the goal of the proposed model to incorporate limited supply conditions of resources and funding in the lineof-balance procedure. In what follows, we introduce a deterministic optimization model, which is a variant of Pereras model [27], for scheduling repetitive projects with the consideration of limited resource and financial availability. The following model extends Pereras by introducing new coefficients 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 ¼ D1 þ
ðM 1Þ ; Q
ð1Þ
where D is the project duration; M is the number of progress units; and D1 is the duration of the first unit. Since M is a constant, to minimize the project duration is equivalent to maximizing the system production rate, Q (measured in unit/day), which can be expressed as Q¼
M 1 . D D1
ð2Þ
The objective is thereby ð3Þ
Maximize Q.
The constraints comprise six sets of equations. The first set describes that the system production rate is governed by the slowest activity. This is the underlying concept of bottleneck in the Theory of Constraint [12] Q 6 Qi
8i;
ð4Þ
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 ¼
Rij Di
8i; j.
ð6Þ
The fourth set of constraints ensures the feasibility of resources. That is, the daily working hours of each resource (sum of the allocated amounts for different activities, i = 1, 2, . . ., n) should be less or equal to the available amount n X ðRDij RC i Þ 6 T j 8j; ð7Þ i¼1
where Tj denotes the availability of resource j per day (measured in resource-hour/day). Substitute the lefthand side (LHS) of (7) by (5) and (6), the constraint can be rewritten as n X Rij Qi 6 T j 8j. ð8Þ i¼1
Aggregating the costs of all the resources (j = 1, 2, . . ., m), the fifth set considers the financial feasibility by limiting the daily cost to be less or equal to the available funding ! m n X X Rij Qi P j 6 C; ð9Þ j¼1
i¼1
where Pj is the unit price of resource j (measured in $/resource-hour) and C is the available funding. The last set of constraints defines the upper and lower bounds of the production rate for each activity. The former may be caused by technical constraints, such as limits on equipment capacity or crew productivity whereas the latter, being always positive, may be the result of economical consideration Qi 6 Qi 6 Qi
8i;
ð10Þ
where Qi denotes the production rate of activity i (measured in unit/day). The second set defines coefficients of resource consumption for individual activities (RCi)
where the upper bound Qi represents the fastest possible pace at which activity i can be performed and the lower bound Qi denotes the slowest possible pace. To sum up, the deterministic model is
RC i ¼ Di Qi
Maximize Q.
8i.
ð5Þ
Since the unit duration of activity i (Di, measured in day/ unit) is a known constant, the coefficient of resource consumption is directly proportional to the production rate of activity i (Qi, measured in unit/day). The mechanism is that an increase in the production rate of an activity would require more resources working at different units simultaneously. As a result of multiplication, the coefficient of resource consumption is unitless.
ð11Þ
Subject to Q 6 Qi 8i; Xn Rij Qi 6 T j i¼1 Xm
n X
j¼1 i¼1
ð12Þ 8j;
ð13Þ !
Rij Qi P j
6 C;
ð14Þ
I-Tung Yang, C.-Y. Chang / International Journal of Project Management 23 (2005) 546–553
Qi 6 Qi 6 Qi
8i.
ð15Þ
In the deterministic model, Qi are the decision variables. The required input includes Tj (available working hours 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 bounds of the production rate for activity i).
4. Stochastic optimization model To deal with the uncertainty associated with the supply of resources, we modify (13) to a probabilistic constraint " # n X Pr Rij Qi 6 T j P aTj 8j; ð16Þ i¼1
where Pr[Æ] is the probability of the event in [Æ]; Tj is now a random value with a distribution of uncertainty; aTj is a pre-specified confidence level. So a solution set is feasbile if and only if the probability measure of the set is at least aTj . In other words, the constraint will be violated at most ð1 aTj Þ of time. Similarly, (14) is modified to address the uncertainty inherent in the supply of funding " ! # m n X X Pr Rij Qi P j 6 C P aC ; ð17Þ j¼1
i¼1
C
where C is a random value; a is a pre-specified confidence level, which represents the lowest limit on the probability of the constraint being satisfied. Again, the constraint will be violated at most (1 aC) of time. The stochastic model can be solved by chanceconstrained programming (CCP), which was first 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 stochastic constraints to their respective deterministic equivalents. 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 that Z þ1 uj ðT j Þ dT j ¼ aTj ; ð18Þ T~ 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 strictly monotonic, T~ j can be computed directly based on the inverse of the CDF T T~ j ¼ W1 j ðT j ; 1 aj Þ.
Thus, the deterministic equivalent of (16) is
ð19Þ
n X
T Rij Qi 6 W1 j ðT j ; 1 aj Þ
8j.
549
ð20Þ
i¼1
By the same token, we can convert the probabilistic constraint (17) to the following deterministic equivalent ! m n X X Rij Qi P j 6 W1 ðC; 1 aC Þ. ð21Þ j¼1
i¼1
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 very efficient linear programming techniques, such as simplex or interior point methods. In summary, the general form of the deterministic equivalent of the proposed chance-constrained programming model is ð22Þ
Maximize Q; Subject to Q 6 Qi 8i; Xn T Rij Qi 6 W1 8j; j ðT j ; 1 aj Þ i¼1 Xm
n X
j¼1
ð23Þ ð24Þ
! Rij Qi P j
6 W1 ðC; 1 aC Þ;
ð25Þ
i¼1
Qi 6 Qi 6 Qi
8i.
ð26Þ
Our formulation in (24) and (25) is applicable for commonly-used types of distributions, such as normal, lognormal, beta, gamma, uniform, or triangular. When Tj and C are normally distributed, the RHS of both constraints can be reduced to a combination of the estimated mean and standard deviation n X
Rij Qi 6 mTj þ kð1 aTj ÞrTj
8j;
ð27Þ
i¼1 m n X X j¼1
! Rij Qi P j
6 mC þ kð1 aC ÞrC ;
ð28Þ
i¼1
where m is the mean; k(Æ) is the normal distribution value corresponding to lower-tail probability of (Æ); r is the standard deviation. For other types of distributions, the computation of the RHS of (24) and (25) is straightforward when the inverse CDF can be expressed as a closed form (e.g., uniform and triangular distributions). Otherwise, numerical algorithms are readily available to approximate the inverse value, such as normal or lognormal [25] and beta distributions [5]. Despite the advantages, CCP is not without its own limitation. As shown by Elmaghraby et al. [11], CCP is not suitable for event realization in activity networks,
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i.e., project evaluation and review technique (PERT) analysis. This is primarily because traditional CCP relies on an implied assumption that all the constraints shall be independent. This assumption cannot be justified in PERT analysis because in an activity network (activity-on-arrow) the realization of an event is dependent on all the activities entering that particular node. In our model, however, the assumption can be valid if the availability of all the resources and funding is independent, e.g., when they are supplied by different sources.
Table 2 Unit prices of resources Resources
Unit price ($/resource-hour)
Carpenter Steelworker Laborer Mason Pump
11.67 13.33 8.33 15 25
Table 3 Supply conditions of resources and funding Resource type
5. Application example
Funding Carpenter Steelworker Laborer Mason Pump
The proposed chance-constrained programming model is applied to an example project documented in [27]. The project is to construct 100 units of housing, each of which consists of four consecutive activities: foundation, retaining wall, floor slab, and exterior wall. The necessary resources are carpenters, steelworkers, laborers, masons, and pumps. Table 1 lists the amount of resources required to complete each activity (Rij, measured in resource-hour/unit) and unit duration (Di, measured in day/unit). Table 2 lists the unit price for each resource (Pj, measured in $/resource-hour). Consider the situation when the supply of resources and funding is limited and normally distributed with the respective means and standard deviations shown in Table 3. Here, we use the coefficient of variation to estimate the level of uncertainty associated with the supply condition because it is a simple and direct measure of variability. If the required confidence level for every constraint is 90%, the chance-constrained programming model has the following deterministic equivalent:
Mean
Coefficient of variation (%)
SD
$3750 112 hours 57 hours 87 hours 25 hours 13 hours
10 5 20 20 5 5
$375 5.6 hours 11.4 hours 17.4 hours 1.25 hours 0.65 hours
Carpenter : 12 Q1 þ 150 Q2 þ 160 Q3 þ 130 Q4 6 112 þ 5:6 ð1:285Þ;
ð34Þ
Steelworker : 6 Q1 þ 36 Q2 þ 40 Q3 þ 24 Q4 6 57 þ 11:4 ð1:285Þ;
ð35Þ
Laborer : 16 Q1 þ 114 Q2 þ 156 Q3 þ 96 Q4 6 87 þ 17:4 ð1:285Þ; ð36Þ Mason : 6 Q1 þ 15 Q2 þ 36 Q3 þ 15 Q4 6 25 þ 1:25 ð1:285Þ;
ð37Þ
Pump : 0 Q1 þ 10 Q2 þ 24 Q3 þ 10 Q4 6 13 þ 0:65 ð1:285Þ;
ð38Þ
Funding : ð12 Q1 þ 150 Q2 þ 160 Q3 þ 130 Q4 Þ 11:67
ð29Þ
Maximize Q;
Supply condition
Subject to
þ ð6 Q1 þ 36 Q2 þ 40 Q3 þ 24 Q4 Þ
Q 6 Q1 ;
ð30Þ
Q 6 Q2 ;
ð31Þ
Q 6 Q3 ;
ð32Þ
þ ð6 Q1 þ 15 Q2 þ 36 Q3 þ 15 Q4 Þ 15 þ ð10 Q2 þ 24 Q3 þ 10 Q4 Þ 25
Q 6 Q4 .
ð33Þ
6 3750 þ 375 ð1:285Þ
13:3 þ ð16 Q1 þ 114 Q2 þ 156 Q3 þ 96 Q4 Þ 8:33
ð39Þ
Table 1 Resource amounts and unit durations of activities Activity
Foundation Retaining wall Floor slab Exterior wall
Resource (resource-hour/unit)
Unit duration (day)
Carpenter
Steelworker
Laborer
Mason
Pump
12 150 160 130
6 36 40 24
16 114 156 96
6 15 36 15
– 10 24 10
1.125 4 7.75 3.25
I-Tung Yang, C.-Y. Chang / International Journal of Project Management 23 (2005) 546–553
1000.0
ð41Þ
Q3 P 0;
ð42Þ
Q4 P 0.
ð43Þ
The optimal solution set is {Q1, Q2, Q3, Q4} = {0.169, 0.169, 0.169, 0.169}. This means all the activities should be performed at the same pace: 0.169 units/day. This should not be surprising because it is reasonable to keep all the activities at the same pace if the lowest one would eventually govern the production rate of the entire project. Given that the first unit takes 16.125 days (sum of the unit durations), the project can be completed within 601.9 days according to (1). It may be possible that activities cannot be performed at the same pace due to technical constraints. The proposed model can treat this practical concern directly by simply setting the bounds of the production rates for every activity as in (26). Suppose the fastest pace at which activities ‘‘foundation’’ and ‘‘retaining wall’’ can be performed is 0.18 units/day (i.e., Q1 and Q2 are no larger than 0.18) whereas the slowest pace at which activities ‘‘floor slab’’ and ‘‘exterior wall’’ can be performed is 0.2 units/day (i.e., Q3 and Q4 are no less than 0.2). The proposed model gives the optimal solution as {Q1, Q2, Q3, Q4} = {0.11, 0.11, 0.2, 0.2}, which brings down the optimal production rate of the project to 0.11 and the project duration will be prolonged to 916.1 days. The proposed model is helpful in determining possible strategies to expedite the project. The analysis starts with indicating the most influential resource by checking which resource constraint has zero slack. In this example, the binding resource is laborer. Therefore, the strategy is to hire more laborers (i.e., increase the mean) and/ or to decrease the uncertainty associated with the supply of laborers (i.e., reduce the standard deviation). If we double the mean and halve the standard deviation, the optimal production rate becomes 0.213 units/day, which leads to a much shortest project duration: 480.9 days. Taking the latter alternative alone (halve the standard deviation without increasing the mean) can bring down the project duration to 516.1 days. The significant difference of 85.8 days (601.9–516.1) stresses the importance of uncertainty management while serving as a convincing motivation for the contractor to take action in ensuring the stability of labor supply (e.g., sign a contract with labor unions in advance). The analysis above demonstrates the capability of the proposed model in quantifying the influence of uncertain resource availability on the project duration. The sensitivity analysis of the proposed model helps evaluate the impacts of different levels of uncertainty in the supply conditions of resources and funding. Sup-
908.0
0.25
900.0 800.0
0.2 671.8
700.0 600.0 500.0
0.15
534.4 442.8
444.7
400.0
0.1
300.0 200.0
0.05
100.0
System production rate (units/day)
Q2 P 0;
ð40Þ
Project duration (days)
Q1 P 0;
551
0
0.0 10%
20%
30%
40%
50%
Coefficient of variation (funding) Fig. 2. Sensitivity analysis on the level of funding uncertainty.
pose we have doubled the mean and halved the standard deviation for the supply of laborers. Fig. 2 compares the project durations caused by different levels of uncertainty of project financing, represented by the coefficients of variation of the funding. The comparison reflects the importance of maintaining a steady supply of funding. For instance, by reducing the coefficient of variation from 50% to 10%, the expected project duration can be significantly shortened from 908.0 to 442.8 days.
6. Model verification A Monte Carlo simulation is performed to verify whether the solution set generated by the proposed model, when exposed to uncertainty, can actually be satisfied beyond the required confidence level. For each constraint, a random number is generated in accordance with the underlying distribution of the RHS; for example, the RHS of (34) has a normal distribution with mean equal to 112, standard deviation equal to 5.6. On the other hand, the optimal solution set is inserted into the LHS of the constraint and the outcome is compared to the random number to check if the constraint can indeed be fulfilled. Among 1000 simulation runs, all the constraints can be fully fulfilled (1000 times) except the laborer constraint, which is fulfilled 903 times out of 1000 (probability of 90.3%). This confirms that the solution generated by the proposed model can be fulfilled beyond the 90% confidence level. To draw a comparison with the deterministic model, we compare the solution set generated by both models (stochastic and deterministic). In the deterministic model, the RHS of the constraints are only the mean (or a single estimate) for the supply of resources and funding. The optimal solution set is then {Q1, Q2, Q3, Q4} = {0.228, 0.228, 0.228, 0.228}, which is
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a lot more optimistic than the one considering uncertainty. By the same Monte Carlo simulation explained above, when this solution set is exposed to uncertainty, the probability for each constraint to be fulfilled is merely 0.941, 0.998, 0.5, 1.0, 1.0, 0.996. This means the entire system can be satisfied with only a probability of 46.8% (multiplication of all the probabilities if the supply conditions are mutually independent). In other words, it is more than half of the chance that the project will be in serious troubles due to neglecting the uncertainty in the supply conditions of resources and funding.
7. Real-life application The proposed model has also been applied to a reallife tunnel project, one among the 48 tunnels of Taiwan High Speed Rail (THSR). The tunnel has a crosssectional area of 120 m2 and is advanced using the Sequential Excavation and Support (SES) construction method. The section being considered has the length of 246 m, which is equivalent to 205 cycles. In each cycle, 11 activities are performed repetitively: (1) crosssection sealing; (2) drilling; (3) mucking; (4) shotcrete for this cycle; (5) steel mesh for this cycle; (6) steel mesh for previous two cycles; (7) steel support for this cycle; (8) shotcrete for this cycle again; (9) shotcrete for previous two cycles; (10) rock bolt for previous cycle; and (11) grouting. The machinery includes: (1) rock drill equipment, (2) shotcrete machine, (3) trencher, (4) loader, (5) breaker, (6) rock-bolting machine, and (7) grouting machine. The availability of these machines is subject to uncertainty due to occasional breakage and regular maintenance. The funding for this project is also anticipated to experience variability. Detailed formulations are to be found in [3]. The proposed model is used to find the minimum project duration with the stochastic constraints of equipment and funding. The optimal production rate is obtained to be 2.60 h/m and the project duration is estimated to be 26.65 days (working 24 h/day). These results are then compared to the actual dataset and the error is less than 8% (actual project duration is 28.91 days).
8. Closing remarks Repetitive construction projects utilize multifarious resources (labor and equipment) and require funding. While the amounts of resources and funding are subject to certain limits, their supply conditions may also be exposed to uncertainty in todays dynamic and complex business environment. Such uncertainty requires caution and should be incorporated into the decision-making
process. To achieve this goal, we present a chanceconstrained programming model, derive its deterministic equivalent, and solve the equivalent by classical linear programming techniques. The proposed model is applied to a project comprising 100 units of housing. For the example project, the proposed model has been shown useful in: (1) determining the acceleration strategy, (2) setting the technical bounds of individual activity production rates, and (3) evaluating the impact of different levels of uncertainty on the project duration. The solution set generated by the proposed model is verified by confirming that the constraints can be satisfied beyond the pre-specified confidence level. In contrast, the solution set generated by a model without considering uncertainty would fail to reach the required confidence level and therefore may put the project in great jeopardy. The proposed model has also been applied to a reallife tunnel project. The result shows that the proposed model can estimate the project duration under stochastic constraints of equipment and funding with a relatively small error. Further progress can be made to consider non-linear relationships between extra manpower and the decrease in activity production rate. More sophisticated optimization techniques may be in need to solve this class of problem.
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on a new mathematical formulation. Manage Sci 1998;44(5):714–29. Na¨slund B. Decision under risk. Publication No. 8162, Economic Research Institute, Stockholm School of Economics, Sweden; 1967. Neumann K, Schwindt C, Zimmermann J. Project scheduling with time windows and scarce resources: temporal and resourceconstrained project scheduling with regular and nonregular objective functions. 2nd ed.. Berlin: Springer; 2003. Odeh RE, Evans JO. The percentage points of the normal distribution. Appl Stat 1974;23:96–7. Ouarda TBMJ, Labadie JW. Chance-constrained optimal control for multireservoir system optimization and risk analysis. Stoch Environ Res Risk Assess 2001;15(3):185–204. Perera S. Resource sharing in linear construction. ASCE J Constr Eng Manage 1983;109(1):102–11. Reda RM. RPM: repetitive project modeling. ASCE J Constr Eng Manage 1990;116(2):316330. Tommelein ID, Riley D, Howell GA. Parade game: impact of work flow variability on trade performance. ASCE J Constr Eng Manage 1999;125(5):304–10. Tsai YW, Gemmill DD. Using tabu search to schedule activities of stochastic resource-constrained projects. Eur J Oper Res 1998;111:129–41. Wiest JD. A heuristic model for scheduling large projects with limited resources. Manage Sci 1967;13(6):B359–77. Yang IT, Ioannou PG. Scheduling system with focus on practical concerns in repetitive projects. Constr Manage Econ 2004;22(6):619–30.
PROJECT MANAGEMENT International Journal of Project Management 23 (2005) 546–553 www.elsevier.com/locate/ijproman
Stochastic resource-constrained scheduling for repetitive construction projects with uncertain supply of resources and funding I-Tung Yang a
a,*
, Chi-Yi Chang
b
Department of Civil Engineering, Tamkang University, 151 Yingchuan Road, Tamsui, Taipei County 251, Taiwan b Department of Construction Engineering, Chaoyang University of Technology, Taiwan Received 15 September 2004; received in revised form 7 December 2004; accepted 1 March 2005
Abstract In this paper, we schedule construction projects that are repetitively performed by similar resources (labor and equipment) from one 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 chanceconstrained 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 verification is performed by Monte Carlo simulations to confirm that the probability of constraint satisfaction is beyond the required confidence level. Ó 2005 Elsevier Ltd and IPMA. All rights reserved. Keywords: Risk; Time; Cost; Chance-constrained programming
1. Introduction Housing communities, multistory buildings, highways, 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 effective scheduling method to ensure the project can be completed in the most efficient manner. Resources employed in repetitive projects, such as labor and equipment, are often subject to limited availability. This leads to a practical concern of how to * Corresponding author. Tel.: +886 2 226215656x3261; fax: +886 2 26209747. E-mail addresses: [email protected] (I-Tung Yang), tavina. [email protected] (C.-Y. Chang).
0263-7863/$30.00 Ó 2005 Elsevier Ltd and IPMA. All rights reserved. doi:10.1016/j.ijproman.2005.03.003
distribute the limited resources among various activities so that the project can be completed within a minimal period of time. This class of problem, so-called ‘‘limited resource allocation’’ or ‘‘resource-constrained scheduling’’, has been an active subject for decades because of its practical relevance [7]. The purpose of this paper is to present a chanceconstrained programming model to solve the limited resource allocation problem under uncertain supply conditions of resources and funding. The paper is organized 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 describe 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 deterministic model is enhanced to a chance-constrained
I-Tung Yang, C.-Y. Chang / International Journal of Project Management 23 (2005) 546–553
2. Review of existing approaches Due to the assumption of unlimited resource availability, traditional scheduling techniques, such as the critical path method (CPM) and barchart, have been shown less than satisfactory in addressing the limited resource allocation problem [1]. Hence, three groups of alternatives were proposed. The first group formulated the problem as a mathematical optimization model. Operational research techniques used to solve the mathematical model included linear programming [28], mixed integer programming [13,22], branch-and-bound [6,8], and dynamic programming [10]. The second group specified a variety of heuristics (a set of rules to prioritize activities in the assignment of resources) to provide a near-optimal solution in practical time. Examples of this group dated back to the works of Kelley [18] and Wiest [31] while some recent efforts 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 detailed survey of the literature, we refer to [2,15]. Previous methods typically assumed the limits of resources can be expressed as deterministic numbers. Yet this assumption may be inappropriate when the supply of resources may in reality contain a great deal of uncertainty when extending several months, or even years, into the future. For instance, the situation of labour shortage leads to a sharing of crews between multiple projects. As a result, the actual manpower allocated to the project may differ 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 deviates from the original plan due to financing, political, and social risks [9].
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 confidence.
3. Deterministic optimization model Within repetitive projects, scheduling is usually done by considering crews ‘‘flowing through’’ the whole project, similar to manufacturing assembly lines [32]. In repetitive construction projects, the product being built tends to be stationary, whereas every crew proceeds from location to location and complete work that is prerequisite to starting work by the following crew, like a ‘‘parade of trades’’ [29]. The flow model of repetitive projects leads to the use of production lines whose slopes represent individual production rates. To accelerate the project, a wellgrounded treatment is to ‘‘balance’’ the production lines (i.e., to make all the lines have the same slope) as decreed in the line-of-balance procedure [21]. Fig. 1 illustrates 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 first unit (calculated by CPM), the rate of construction is then the result of dividing the remaining number of units (u 1) by the difference between project deadline and the duration of the first unit. Once the rate of construction has been calculated, the start times of activities on different units and the number of each required crew can be computed. The traditional approach of average rate in the lineof-balance procedure is underpinned by the assumption that resource availability is unlimited. When the supply of resources is constrained, a challenge emerges: how to efficiently allocate the working hours of available
B
A
u
C
Unit
programming formulation. The chance-constrained programming problem is converted to a deterministic equivalent, 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 the proposed model can assist project managers in project acceleration and risk management. We also briefly introduce the application of the proposed model in a real-life tunnel project to demonstrate its practical acceptance. The proposed model is verified through Monte Carlo simulations. Conclusions are drawn in the last section.
547
Rate of Construction
Project Duration
1
A
B
C
Duration of the First Unit
Remaining Duration
Duration Fig. 1. Balanced condition.
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resources to various activities when a resource may work for different activities (e.g., carpenters construct both column forms and exterior doors) and an activity may employ multiple types of resources (e.g., earth moving needs both bulldozers and dump trucks). These issues, unfortunately, cannot be resolved by the traditional line-of-balance procedure. It is therefore the goal of the proposed model to incorporate limited supply conditions of resources and funding in the lineof-balance procedure. In what follows, we introduce a deterministic optimization model, which is a variant of Pereras model [27], for scheduling repetitive projects with the consideration of limited resource and financial availability. The following model extends Pereras by introducing new coefficients 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 ¼ D1 þ
ðM 1Þ ; Q
ð1Þ
where D is the project duration; M is the number of progress units; and D1 is the duration of the first unit. Since M is a constant, to minimize the project duration is equivalent to maximizing the system production rate, Q (measured in unit/day), which can be expressed as Q¼
M 1 . D D1
ð2Þ
The objective is thereby ð3Þ
Maximize Q.
The constraints comprise six sets of equations. The first set describes that the system production rate is governed by the slowest activity. This is the underlying concept of bottleneck in the Theory of Constraint [12] Q 6 Qi
8i;
ð4Þ
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 ¼
Rij Di
8i; j.
ð6Þ
The fourth set of constraints ensures the feasibility of resources. That is, the daily working hours of each resource (sum of the allocated amounts for different activities, i = 1, 2, . . ., n) should be less or equal to the available amount n X ðRDij RC i Þ 6 T j 8j; ð7Þ i¼1
where Tj denotes the availability of resource j per day (measured in resource-hour/day). Substitute the lefthand side (LHS) of (7) by (5) and (6), the constraint can be rewritten as n X Rij Qi 6 T j 8j. ð8Þ i¼1
Aggregating the costs of all the resources (j = 1, 2, . . ., m), the fifth set considers the financial feasibility by limiting the daily cost to be less or equal to the available funding ! m n X X Rij Qi P j 6 C; ð9Þ j¼1
i¼1
where Pj is the unit price of resource j (measured in $/resource-hour) and C is the available funding. The last set of constraints defines the upper and lower bounds of the production rate for each activity. The former may be caused by technical constraints, such as limits on equipment capacity or crew productivity whereas the latter, being always positive, may be the result of economical consideration Qi 6 Qi 6 Qi
8i;
ð10Þ
where Qi denotes the production rate of activity i (measured in unit/day). The second set defines coefficients of resource consumption for individual activities (RCi)
where the upper bound Qi represents the fastest possible pace at which activity i can be performed and the lower bound Qi denotes the slowest possible pace. To sum up, the deterministic model is
RC i ¼ Di Qi
Maximize Q.
8i.
ð5Þ
Since the unit duration of activity i (Di, measured in day/ unit) is a known constant, the coefficient of resource consumption is directly proportional to the production rate of activity i (Qi, measured in unit/day). The mechanism is that an increase in the production rate of an activity would require more resources working at different units simultaneously. As a result of multiplication, the coefficient of resource consumption is unitless.
ð11Þ
Subject to Q 6 Qi 8i; Xn Rij Qi 6 T j i¼1 Xm
n X
j¼1 i¼1
ð12Þ 8j;
ð13Þ !
Rij Qi P j
6 C;
ð14Þ
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Qi 6 Qi 6 Qi
8i.
ð15Þ
In the deterministic model, Qi are the decision variables. The required input includes Tj (available working hours 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 bounds of the production rate for activity i).
4. Stochastic optimization model To deal with the uncertainty associated with the supply of resources, we modify (13) to a probabilistic constraint " # n X Pr Rij Qi 6 T j P aTj 8j; ð16Þ i¼1
where Pr[Æ] is the probability of the event in [Æ]; Tj is now a random value with a distribution of uncertainty; aTj is a pre-specified confidence level. So a solution set is feasbile if and only if the probability measure of the set is at least aTj . In other words, the constraint will be violated at most ð1 aTj Þ of time. Similarly, (14) is modified to address the uncertainty inherent in the supply of funding " ! # m n X X Pr Rij Qi P j 6 C P aC ; ð17Þ j¼1
i¼1
C
where C is a random value; a is a pre-specified confidence level, which represents the lowest limit on the probability of the constraint being satisfied. Again, the constraint will be violated at most (1 aC) of time. The stochastic model can be solved by chanceconstrained programming (CCP), which was first 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 stochastic constraints to their respective deterministic equivalents. 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 that Z þ1 uj ðT j Þ dT j ¼ aTj ; ð18Þ T~ 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 strictly monotonic, T~ j can be computed directly based on the inverse of the CDF T T~ j ¼ W1 j ðT j ; 1 aj Þ.
Thus, the deterministic equivalent of (16) is
ð19Þ
n X
T Rij Qi 6 W1 j ðT j ; 1 aj Þ
8j.
549
ð20Þ
i¼1
By the same token, we can convert the probabilistic constraint (17) to the following deterministic equivalent ! m n X X Rij Qi P j 6 W1 ðC; 1 aC Þ. ð21Þ j¼1
i¼1
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 very efficient linear programming techniques, such as simplex or interior point methods. In summary, the general form of the deterministic equivalent of the proposed chance-constrained programming model is ð22Þ
Maximize Q; Subject to Q 6 Qi 8i; Xn T Rij Qi 6 W1 8j; j ðT j ; 1 aj Þ i¼1 Xm
n X
j¼1
ð23Þ ð24Þ
! Rij Qi P j
6 W1 ðC; 1 aC Þ;
ð25Þ
i¼1
Qi 6 Qi 6 Qi
8i.
ð26Þ
Our formulation in (24) and (25) is applicable for commonly-used types of distributions, such as normal, lognormal, beta, gamma, uniform, or triangular. When Tj and C are normally distributed, the RHS of both constraints can be reduced to a combination of the estimated mean and standard deviation n X
Rij Qi 6 mTj þ kð1 aTj ÞrTj
8j;
ð27Þ
i¼1 m n X X j¼1
! Rij Qi P j
6 mC þ kð1 aC ÞrC ;
ð28Þ
i¼1
where m is the mean; k(Æ) is the normal distribution value corresponding to lower-tail probability of (Æ); r is the standard deviation. For other types of distributions, the computation of the RHS of (24) and (25) is straightforward when the inverse CDF can be expressed as a closed form (e.g., uniform and triangular distributions). Otherwise, numerical algorithms are readily available to approximate the inverse value, such as normal or lognormal [25] and beta distributions [5]. Despite the advantages, CCP is not without its own limitation. As shown by Elmaghraby et al. [11], CCP is not suitable for event realization in activity networks,
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i.e., project evaluation and review technique (PERT) analysis. This is primarily because traditional CCP relies on an implied assumption that all the constraints shall be independent. This assumption cannot be justified in PERT analysis because in an activity network (activity-on-arrow) the realization of an event is dependent on all the activities entering that particular node. In our model, however, the assumption can be valid if the availability of all the resources and funding is independent, e.g., when they are supplied by different sources.
Table 2 Unit prices of resources Resources
Unit price ($/resource-hour)
Carpenter Steelworker Laborer Mason Pump
11.67 13.33 8.33 15 25
Table 3 Supply conditions of resources and funding Resource type
5. Application example
Funding Carpenter Steelworker Laborer Mason Pump
The proposed chance-constrained programming model is applied to an example project documented in [27]. The project is to construct 100 units of housing, each of which consists of four consecutive activities: foundation, retaining wall, floor slab, and exterior wall. The necessary resources are carpenters, steelworkers, laborers, masons, and pumps. Table 1 lists the amount of resources required to complete each activity (Rij, measured in resource-hour/unit) and unit duration (Di, measured in day/unit). Table 2 lists the unit price for each resource (Pj, measured in $/resource-hour). Consider the situation when the supply of resources and funding is limited and normally distributed with the respective means and standard deviations shown in Table 3. Here, we use the coefficient of variation to estimate the level of uncertainty associated with the supply condition because it is a simple and direct measure of variability. If the required confidence level for every constraint is 90%, the chance-constrained programming model has the following deterministic equivalent:
Mean
Coefficient of variation (%)
SD
$3750 112 hours 57 hours 87 hours 25 hours 13 hours
10 5 20 20 5 5
$375 5.6 hours 11.4 hours 17.4 hours 1.25 hours 0.65 hours
Carpenter : 12 Q1 þ 150 Q2 þ 160 Q3 þ 130 Q4 6 112 þ 5:6 ð1:285Þ;
ð34Þ
Steelworker : 6 Q1 þ 36 Q2 þ 40 Q3 þ 24 Q4 6 57 þ 11:4 ð1:285Þ;
ð35Þ
Laborer : 16 Q1 þ 114 Q2 þ 156 Q3 þ 96 Q4 6 87 þ 17:4 ð1:285Þ; ð36Þ Mason : 6 Q1 þ 15 Q2 þ 36 Q3 þ 15 Q4 6 25 þ 1:25 ð1:285Þ;
ð37Þ
Pump : 0 Q1 þ 10 Q2 þ 24 Q3 þ 10 Q4 6 13 þ 0:65 ð1:285Þ;
ð38Þ
Funding : ð12 Q1 þ 150 Q2 þ 160 Q3 þ 130 Q4 Þ 11:67
ð29Þ
Maximize Q;
Supply condition
Subject to
þ ð6 Q1 þ 36 Q2 þ 40 Q3 þ 24 Q4 Þ
Q 6 Q1 ;
ð30Þ
Q 6 Q2 ;
ð31Þ
Q 6 Q3 ;
ð32Þ
þ ð6 Q1 þ 15 Q2 þ 36 Q3 þ 15 Q4 Þ 15 þ ð10 Q2 þ 24 Q3 þ 10 Q4 Þ 25
Q 6 Q4 .
ð33Þ
6 3750 þ 375 ð1:285Þ
13:3 þ ð16 Q1 þ 114 Q2 þ 156 Q3 þ 96 Q4 Þ 8:33
ð39Þ
Table 1 Resource amounts and unit durations of activities Activity
Foundation Retaining wall Floor slab Exterior wall
Resource (resource-hour/unit)
Unit duration (day)
Carpenter
Steelworker
Laborer
Mason
Pump
12 150 160 130
6 36 40 24
16 114 156 96
6 15 36 15
– 10 24 10
1.125 4 7.75 3.25
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1000.0
ð41Þ
Q3 P 0;
ð42Þ
Q4 P 0.
ð43Þ
The optimal solution set is {Q1, Q2, Q3, Q4} = {0.169, 0.169, 0.169, 0.169}. This means all the activities should be performed at the same pace: 0.169 units/day. This should not be surprising because it is reasonable to keep all the activities at the same pace if the lowest one would eventually govern the production rate of the entire project. Given that the first unit takes 16.125 days (sum of the unit durations), the project can be completed within 601.9 days according to (1). It may be possible that activities cannot be performed at the same pace due to technical constraints. The proposed model can treat this practical concern directly by simply setting the bounds of the production rates for every activity as in (26). Suppose the fastest pace at which activities ‘‘foundation’’ and ‘‘retaining wall’’ can be performed is 0.18 units/day (i.e., Q1 and Q2 are no larger than 0.18) whereas the slowest pace at which activities ‘‘floor slab’’ and ‘‘exterior wall’’ can be performed is 0.2 units/day (i.e., Q3 and Q4 are no less than 0.2). The proposed model gives the optimal solution as {Q1, Q2, Q3, Q4} = {0.11, 0.11, 0.2, 0.2}, which brings down the optimal production rate of the project to 0.11 and the project duration will be prolonged to 916.1 days. The proposed model is helpful in determining possible strategies to expedite the project. The analysis starts with indicating the most influential resource by checking which resource constraint has zero slack. In this example, the binding resource is laborer. Therefore, the strategy is to hire more laborers (i.e., increase the mean) and/ or to decrease the uncertainty associated with the supply of laborers (i.e., reduce the standard deviation). If we double the mean and halve the standard deviation, the optimal production rate becomes 0.213 units/day, which leads to a much shortest project duration: 480.9 days. Taking the latter alternative alone (halve the standard deviation without increasing the mean) can bring down the project duration to 516.1 days. The significant difference of 85.8 days (601.9–516.1) stresses the importance of uncertainty management while serving as a convincing motivation for the contractor to take action in ensuring the stability of labor supply (e.g., sign a contract with labor unions in advance). The analysis above demonstrates the capability of the proposed model in quantifying the influence of uncertain resource availability on the project duration. The sensitivity analysis of the proposed model helps evaluate the impacts of different levels of uncertainty in the supply conditions of resources and funding. Sup-
908.0
0.25
900.0 800.0
0.2 671.8
700.0 600.0 500.0
0.15
534.4 442.8
444.7
400.0
0.1
300.0 200.0
0.05
100.0
System production rate (units/day)
Q2 P 0;
ð40Þ
Project duration (days)
Q1 P 0;
551
0
0.0 10%
20%
30%
40%
50%
Coefficient of variation (funding) Fig. 2. Sensitivity analysis on the level of funding uncertainty.
pose we have doubled the mean and halved the standard deviation for the supply of laborers. Fig. 2 compares the project durations caused by different levels of uncertainty of project financing, represented by the coefficients of variation of the funding. The comparison reflects the importance of maintaining a steady supply of funding. For instance, by reducing the coefficient of variation from 50% to 10%, the expected project duration can be significantly shortened from 908.0 to 442.8 days.
6. Model verification A Monte Carlo simulation is performed to verify whether the solution set generated by the proposed model, when exposed to uncertainty, can actually be satisfied beyond the required confidence level. For each constraint, a random number is generated in accordance with the underlying distribution of the RHS; for example, the RHS of (34) has a normal distribution with mean equal to 112, standard deviation equal to 5.6. On the other hand, the optimal solution set is inserted into the LHS of the constraint and the outcome is compared to the random number to check if the constraint can indeed be fulfilled. Among 1000 simulation runs, all the constraints can be fully fulfilled (1000 times) except the laborer constraint, which is fulfilled 903 times out of 1000 (probability of 90.3%). This confirms that the solution generated by the proposed model can be fulfilled beyond the 90% confidence level. To draw a comparison with the deterministic model, we compare the solution set generated by both models (stochastic and deterministic). In the deterministic model, the RHS of the constraints are only the mean (or a single estimate) for the supply of resources and funding. The optimal solution set is then {Q1, Q2, Q3, Q4} = {0.228, 0.228, 0.228, 0.228}, which is
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a lot more optimistic than the one considering uncertainty. By the same Monte Carlo simulation explained above, when this solution set is exposed to uncertainty, the probability for each constraint to be fulfilled is merely 0.941, 0.998, 0.5, 1.0, 1.0, 0.996. This means the entire system can be satisfied with only a probability of 46.8% (multiplication of all the probabilities if the supply conditions are mutually independent). In other words, it is more than half of the chance that the project will be in serious troubles due to neglecting the uncertainty in the supply conditions of resources and funding.
7. Real-life application The proposed model has also been applied to a reallife tunnel project, one among the 48 tunnels of Taiwan High Speed Rail (THSR). The tunnel has a crosssectional area of 120 m2 and is advanced using the Sequential Excavation and Support (SES) construction method. The section being considered has the length of 246 m, which is equivalent to 205 cycles. In each cycle, 11 activities are performed repetitively: (1) crosssection sealing; (2) drilling; (3) mucking; (4) shotcrete for this cycle; (5) steel mesh for this cycle; (6) steel mesh for previous two cycles; (7) steel support for this cycle; (8) shotcrete for this cycle again; (9) shotcrete for previous two cycles; (10) rock bolt for previous cycle; and (11) grouting. The machinery includes: (1) rock drill equipment, (2) shotcrete machine, (3) trencher, (4) loader, (5) breaker, (6) rock-bolting machine, and (7) grouting machine. The availability of these machines is subject to uncertainty due to occasional breakage and regular maintenance. The funding for this project is also anticipated to experience variability. Detailed formulations are to be found in [3]. The proposed model is used to find the minimum project duration with the stochastic constraints of equipment and funding. The optimal production rate is obtained to be 2.60 h/m and the project duration is estimated to be 26.65 days (working 24 h/day). These results are then compared to the actual dataset and the error is less than 8% (actual project duration is 28.91 days).
8. Closing remarks Repetitive construction projects utilize multifarious resources (labor and equipment) and require funding. While the amounts of resources and funding are subject to certain limits, their supply conditions may also be exposed to uncertainty in todays dynamic and complex business environment. Such uncertainty requires caution and should be incorporated into the decision-making
process. To achieve this goal, we present a chanceconstrained programming model, derive its deterministic equivalent, and solve the equivalent by classical linear programming techniques. The proposed model is applied to a project comprising 100 units of housing. For the example project, the proposed model has been shown useful in: (1) determining the acceleration strategy, (2) setting the technical bounds of individual activity production rates, and (3) evaluating the impact of different levels of uncertainty on the project duration. The solution set generated by the proposed model is verified by confirming that the constraints can be satisfied beyond the pre-specified confidence level. In contrast, the solution set generated by a model without considering uncertainty would fail to reach the required confidence level and therefore may put the project in great jeopardy. The proposed model has also been applied to a reallife tunnel project. The result shows that the proposed model can estimate the project duration under stochastic constraints of equipment and funding with a relatively small error. Further progress can be made to consider non-linear relationships between extra manpower and the decrease in activity production rate. More sophisticated optimization techniques may be in need to solve this class of problem.
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