Project scheduling under uncertainty and resource constraints

Proceedings in Applied Mathematics and Mechanics, 30 May 2017

Project scheduling under uncertainty and resource constraints

Veronika Hartmann1∗, Tom Lahmer1, and Kay Smarsly1

1 Bauhaus University Weimar, 99421 Weimar, Germany

This paper proposes a method to improve the reliability of construction schedules by optimizing schedule robustness ofconstruction projects. The schedule robustness is measured by a criterion evaluating the impact of deviations from the initialschedule on the makespan, which arise during the realization phase of a construction project.

Copyright line will be provided by the publisher

1 Construction scheduling

Construction project schedules provide information on the construction tasks to be executed, such as start and end dates aswell as precedence relations due to technical constraints. As many different stakeholders are involved in one project, a reliableschedule is of importance [2]. For a set of construction tasks with given discrete durations and precedence constraints,asshown in Fig. 1, the critical path method (CPM) generates one schedule, assuming unlimited resources [3], as shown in Fig. 2.In case of limited resources, the start dates scheduled according to CPM have probably to be pushed back in time if two ormore tasks are competing for the same resource, as indicated by Task 4 and 5 in Fig. 2.

0, 0

5, 2

3, 2 7, 31

2

3

4

5 6

7

3, 2 3, 1

0, 0n

d, r

n: Task number

d: Duration

r: Required amount of resource

Precedence constraint ( )

Fig. 1: Graph visualizing a set of tasks with precedence constraints, durations, and resource consumption.

3 5 6

0 10

2 4

5

3 5 6

0 10

2 4

5 15

3 5 6

0 10

2 4

5 15 20Duration [d] Duration [d] Duration [d]

Fig. 2: Schedule (left) with unlimited resources, (middle) for resource limited to 4 units, alternative 1, and(right) for resource limited to 4 units, alternative 2.

In practice, besides resource limitations, many delays occur during the realization phase constituting, in a mathematicalsense, uncertainty in the input data. In classical project scheduling, uncertainty in the input data is not taken into account.Uncertainty can be considered by defining the task durations by stochastic variables instead of discrete values. The makespanof a project as well as start and end dates of tasks are then described by distribution functions and can be determined in aMonte Carlo simulation.

2 Evaluating and improving schedule reliability

Using distribution functions for describing start and end dates is of little merit in construction practice. In this study, thedistribution of the makespan is used to quantify the ability of the schedule to absorb deviations during the realization phase.To this end, the makespan based on discrete process durations is compared to the distribution describing the makespan based onvariable process durations. The difference between the 95% quantile of the distribution and the deterministic makespan mref

∗ Corresponding author: [email protected]

Copyright line will be provided by the publisher

2 PAMM header will be provided by the publisher

0 20 40 60 80 100Duration [d]

0

0.01

0.02

0.03

0.04

0.05

PDF

(Dur

atio

n) [

-]

uncm

ref Q95

unc

Munc

Fig. 3: Distribution function of makespan with discrete value mref as reference.

is used as robustness measure R (Fig. 3):

R = Qunc95 −mref . (1)

Determining a resource and precedence feasible schedule that is optimal with respect to a certain criterion is a complex com-binatorial optimization problem known as Resource Constraint Project Scheduling Problem [1,4]. Moreover, the Multi-ModeResource Constraint Project Scheduling Problem considers different modes for each task, varying in duration and resourceconsumption. In construction projects, the variation of the order of tasks is restricted due to multiple technical boundaryconditions. Therefore, alternative schedules are generated by considering different modes for the tasks reflecting alternativeconstruction methods, and a schedule is defined by the combination of the modes of the tasks. A so-called combination vectoris introduced, defining the respective mode for each task. The robustness measure R introduced above serves as optimizationcriterion to improve the reliability of the schedule by varying the combination of modes. For the optimization procedure, agenetic algorithm is used, as shown in Fig. 4. Starting with generation i = 0 consisting of j = 1..n random individuals (i.e.combination vectors Cij), different combinations of modes are generated by crossing and mutating the combination vectors.After a certain number of generations, the procedure improves the schedule reliability with regard to the chosen robustnesscriterion.

Initial population with n

individuals

Pop0 = {C01, C02, ..., C0n}

Compute mref, Munc,

R = Qunc–mref for each

individual 95

[No]

[Yes]

Robust

solution

Selection of elite

children and parents

Mutation and

crossover of parents

[Stopping criteria

reached?]

New generation

Popi = {Ci1, Ci2, ..., Cin}

Fig. 4: Schematic representation of the optimization procedure.

References[1] P. Bruckner, A. Drexl, R. Möhring, K. Neumann and E. Pesch, Resource-constrained project scheduling: Notation, classification,

models, and methods. European Journal of Operational Research. 112 (1999), pp. 3-41.[2] V. Hartmann, T. Lahmer and K. Smarsly, Assessment and optimization of the robustness of construction schedules. In: Proceedings of

the 22nd EG-ICE Workshop. Eindhoven, The Netherlands, 07/13/2015.[3] J. Kelley and M. Walker, Critical-Path Planning and Scheduling. In: Proceedings of the Eastern Joint Computer Conference. Boston,

MA, USA, 12/01/1959.[4] A. Lova, P. Tormos, M. Cervantes and F. Barber, An efficient hybrid genetic algorithm for scheduling projects with resource constraints

and multiple execution modes. International Journal of Production Economics, 117 (2009), pp. 302-316.

Copyright line will be provided by the publisher