Entropy-based scheduling of resource-constrained construction projects Symeon Christodoulou a, ⁎, Georgios Ellinas b , Pooyan Aslani c a Department of Civil and Environmental Engineering, University of Cyprus, Nicosia 1678, Cyprus b Department of Electrical and Computer Engineering, University of Cyprus, Nicosia 1678, Cyprus c Department of Civil Engineering, New York University's Polytechnic Institute, Brooklyn 11201, USA abstract article info Article history: Accepted 17 April 2009 Keywords: Resource-constrained construction scheduling Entropy A method is presented for resource allocation and scheduling of resource-constrained projects by use of a measure of the projects' entropy. The entropy, as a measure of the degree of disorder in a system, is an indicator of a project's tendency to progress out of order and into a chaotic condition and it can thus serve to forecast a project's development. The entropy metric used in the analysis is based on the resource assignments per activity (required vs. assigned resource units). © 2009 Elsevier B.V. All rights reserved. 1. Introduction Resource availability constraints are a typical real-life construction scheduling problem; a problem that limits a constructor's ability to execute and deliver a project as originally planned. It is, thus, imperative that the development of project schedules includes not only activity precedence relations and duration assignments, but also resource assignments for each activity in the network. Resource- loaded project schedules (and to that extent cost-loaded schedules) can furnish construction planners with the means for an effective incorporation and analysis of the possible effects of limited resource availability to a successful and timely project completion. Projects, the activities of which are assigned a resource (or a set of resources), with limited capacity or of limited availability, fall under a category of scheduling problems termed Resource-Constrained Scheduling Problems (RCSP). These projects are constrained in their capacity to meet the constructor's predefined objectives, usually to complete all necessary tasks in the minimum total finish time. Since the parameters defining RCSP could vary in nature and quantity, this is an NP-hard problem, the complexity of which increases substantially with the project's network size. NP-hard (nondeterministic polynomial- time hard) problems are a class of problems that are at least as hard as the hardest problems in NP (NP is the set of decision problems solvable in polynomial time by a non-deterministic Turing machine). For this class of problems, in order to arrive at the optimal solution a computationally intensive, exhaustive analysis is needed in which all possible outcomes are tested. 1.1. Outline Following the short introduction on resource-constrained sched- uling and on related research work, the paper presents an outline of the general concepts of entropy and the prevailing software-based heuristics utilized for solving RCSP (Sections 2 and 3). Section 4 provides the mathematical framework for the proposed entropy- based resource-constrained scheduling method, with an illustrative network example included in Section 5. The paper continues with two case studies of higher complexity (Sections 6 and 7). Concluding remarks derived from the application of the proposed method are included in Section 9. 2. Literature review Unlike resource-constrained scheduling problems encountered in typical day-to-day operations (such as employee shifts, manufac- turing, bus routing, etc.), resource-constrained construction schedules are also governed by network precedence relationships. This adds to the complexity of the problem since in searching for the minimum project duration under a set of resource constraints, activity precedence constraints should also be taken into account. In general, construction tasks require different amounts of resources of which only a certain amount is available at each time step [1]. A solution to RCSP can be sought through a number of possible methodologies. A very common approach is to utilize implicit enumeration and backtracking (the methods used by most commercially available exact solvers), such as branch and bound methods. Brucker et al. [2,3], Demeulemeester and Herroelen [4,5], Mingozzi et al. [6], Crawford [7] and Garey et al. [8] are some of the researchers who contributed to that genre. Branch and bound is a general algorithm for finding Automation in Construction 18 (2009) 919–928 ⁎ Corresponding author. E-mail address: schristo@ucy.ac.cy (S. Christodoulou). 0926-5805/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.autcon.2009.04.007 Contents lists available at ScienceDirect Automation in Construction journal homepage: www.elsevier.com/locate/autcon