A new PDE-based approach for construction scheduling and resource allocation. Paul Gabet, Julien Nachef CE 291F Project Presentation Spring 2014
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1 A new PDE-based approach for construction scheduling and resource allocation Paul Gabet, Julien Nachef CE 291F Project Presentation Spring 2014
2 Problem Statement What is the schedule of a project? A chronological list of tasks (Task 1, Task 2, etc.) Constraints: Precedents and Following Tasks Nodes/Events Example of schedule CE 291F - Project Presentation 2
3 Cost of resources ($) Problem Statement Cost of a task i is a function of the desire duration C i = C i (d i ) Workers Time/Cost Tradeoff for an Activity Task completed quickly with massive resources involved Task completed slowly with minimum price Equipment Time (t) Payment depends on time of completion Cement mixer CE 291F - Project Presentation 3
4 Problem Statement Control: Resources allocated to all the future tasks Task duration d i = E i S i Time of events S i, E i and E node Time of project completion E end Objective: Subject to: C i = C i d i Maximize: P E end C i i for all tasks i d i = E i S i for all tasks i Precedence and Following tasks constraints CE 291F - Project Presentation 4
5 Industry State of the Art Widely-used of the Critical Path Method (CPM) Main limitations No systematic optimization of resource allocation Approximate iterative method Not designed for real-time adjustment When deviation occurs between actual and scheduled progress Addresses poorly criticality of tasks CE 291F - Project Presentation 5
![E 1 t = γ [C 1 E 1 S C 2 E E 1 ]](/docs-images/82/86239911/images/6-5.jpg "Tasks behave like springs of tension")
6 Our PDE approach Local resource allocation (sub-optimality) C E 1 = C 1 E 1 S + C 2 E E 1 Minimum for C E 1 = 0 C 1 E 1 S = C 2 E E 1 Dynamic D1 E 1 t = γ [C 1 E 1 S C 2 E E 1 ] Tasks behave like springs of tension c i (d i ) 6
7 General Case Local resource allocation (sub-optimality) Dynamic D1: E node t Dynamic D2: 2 E node 2 t = γ ( = γ i i c i E node S i c i E node S i j j c j (E j E node ), for all nodes c j (E j E node ) + f E t damping term Behaves as a real spring network dynamic, for all nodes 7
8 Literature Review Optimization of schedules without changing task duration Use of Genetic algorithms (Tarek Hegazy, 1999) Use of Back propagation models (Elazouni et al., 1997) Optimization of schedules and resource allocation simultaneously Use of Lyapunov function and a global dynamic (Adeli, 2001) What we do is different: We are using a local dynamic to optimize an entire schedule CE 291F - Project Presentation 8
9 Contribution Use PDE approach for resource allocation in project scheduling Proof of convergence of the model toward optimal schedule Proof of local exponential stability of the model Implementation of the model on Matlab (Oriented Object) CE 291F - Project Presentation 9
10 Contribution Demonstration of good behavior on An elementary project (9 tasks) A real construction project (~250 tasks) Provide a new approach to task criticality Simulation in real-time Benefits of continuous scheduling optimization along with task progress assimilation CE 291F - Project Presentation 10
11 Accuracy and exponential convergence Dynamic D1: For all node E node t Assumption: Cost functions (c i ) i are strictly convex. = γ ( i c i E node S i j c j (E j E node ) Proposition: Dynamic D1 has a unique equilibrium which is the optimal schedule. Theorem: Dynamic D1 converges locally exponentially towards optimum. CE 291F - Project Presentation 11
12 Accuracy and exponential convergence Dynamic D1: For all node Scheme of proof: Linearization of D1 near optimum (E ) node : d E dt E node t = γ ( i c i E node S i j c j (E j E node ) Theorem: Dynamic D1 converges locally exponentially towards optimum. = A E where ( E) node = (E) node (E ) node and matrix A = + positive diagonal matrix A is Hurwitz if cost functions (c i ) i are strictly convex near equilibrium. CE 291F - Project Presentation 12
13 Elementary Project Task 1: Obtain Building permit Task 2: Install temporary networks Task 3: Idle task Task 4: Order cable trays, work space outlets Task 5: Order router, cabling, etc. Task 6: Install cable trays Task 7: Install work space outlets Task 8: Install, test new file servers Task 9: Test communication systems CE 291F - Project Presentation 13
14 Elementary Project Time (days) Profit (k$) Initial Schedule Optimized Schedule % improvement 31% 108% CE 291F - Project Presentation 14
15 Task Criticality Low time pressure High time pressure How to characterize the criticality of task 1? Spring analogy: tension c 1 (d 1 ) c 1 (d 1 ): cost to finish task 1 one day earlier or how difficult it is to catch up a one day delay CE 291F - Project Presentation 15
16 Criticality on an elementary example Evolution of the criticality of each task while optimizing the schedule CE 291F - Project Presentation 16
17 Criticality on real construction project Change of the desired time completion of the project CE 291F - Project Presentation 17
18 Assimilation of progress data P=0% P estimated P=100% Progress P t CE 291F - Project Presentation 18
19 Assimilation of progress data P=0% P estimated P=100% P data Progress P t CE 291F - Project Presentation 19
20 Assimilation of progress data P=0% P estimated P=100% P data Progress P t CE 291F - Project Presentation 20
21 Assimilation of progress data P=100% P estimated = P data Progress P t End of task is loose: Future tasks can take more time and less resources. The project can be advanced. Apply dynamic D1 Only if continuous optimization of resource allocation. CE 291F - Project Presentation 21
22 Real-time simulation of project SIMULATION OF REALITY (TASK PROGRESS) Time t Random progress ΔP on current tasks Time t + Δt Random progress ΔP on current tasks Sparse progress data Time Assimilate progress data Update effort on current task Iterate dynamic D1 Adjust allocated resources SCHEDULING CE 291F - Project Presentation 22
23 Real-time simulation of project On-going work: quantify the gain of continuously optimizing the schedule CE 291F - Project Presentation 23
24 A new PDE-based approach for construction scheduling and resource allocation Paul Gabet, Julien Nachef CE 291F Project Presentation Spring 2014 Thank you
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