Utility-based decision support system for schedule optimization
Introduction
Much research effort on schedule optimization has been devoted to the time–cost tradeoff analysis of projects. It is generally realized that accelerating the progress of a project would require more labor and better equipment, thus involving higher costs. Accordingly, a tradeoff exists between project duration and cost. The decision of choosing a combination of time and cost to suit specific needs is relevant and important. In the time–cost tradeoff analysis, there are three common objectives: to minimize the direct (or total) cost while meeting a specified deadline, to minimize the project duration within a prescribed budget, or to obtain the Pareto front by optimizing both time and cost simultaneously.
To solve the time–cost tradeoff problem, numerous attempts have been made to a wide variety of models that can be categorized in two main groups: heuristic approaches and mathematical programming techniques. A popular heuristic approach begins with a list of activities on the critical path ranked in accordance with the unit change in cost for a reduction in duration. The heuristic then proceeds by crashing activities in the order of their lowest impact on costs [11]. Another kind of heuristics is the local search method, which is inspired by the advancement of computational intelligence, such as genetic algorithms (GA), to perform optimization on time and cost [10], [16], [17]. Mathematical programming models, depends on the type of the time–cost relationships, can be further divided into several well-developed kinds: linear programming, mixed integer programming, dynamic programming, and chance-constrained programming [8], [18], [21], [27].
As a decision problem, the time–cost tradeoff analysis is embodied by subjective judgments in determining the optimality of a time–cost option. For example, whether it is preferred to have the project complete in 100 days, costing 10 million dollars or in 120 days, costing 9.5 million? Since both options cannot dominate each other, the question then becomes: “Can the decrease of 20 days offset the extra cost of 0.5 million?” The answer depends upon the perceived weights of 20 days and 0.5 million. The subjective judgments are often hard to measure and can vary from person to person and even time to time according to decision makers' individual preferences, implicitly influenced by environmental and personal factors, such as project immediacy, owner's financial condition, and consequences of late delivery.
The diversity in preferences can be addressed by individual's utility function, which measures satisfaction gained from the consequences of a project: total cost and project duration. In time–cost optimization, the utility functions are monotonically decreasing within the range of (0,1). Fig. 1 graphically depicts three common utility shapes: concave, convex, and combination. The shapes of the utility functions reflect the decision maker's nonlinear preference on time and cost, e.g., a 20 day reduction from 120 to 100 days would be preferred than that from 130 to 110 days in the case of a convex utility function. The absolute value of the curvature of the function indicates the strength of preferences. The higher the absolute value, the stronger the preference is. It is also possible that the decision maker switches his/her preference according to the magnitude of the variables. For example, the utility may be concave for a higher cost and convex when the cost is low.
Since preferences may vary and have a profound impact on schedule optimization, the diversity shall be accommodated to capture the true decision-making mechanisms. To address this need, the goal of the present study is three-fold: (1) to elicit decision makers' preference (utility) toward time and cost; (2) to quantify the impact of the utility function on time–cost optimization; and (3) to develop a decision support system (DSS) to help determine the optimal time–cost solution by maximizing the utility function.
The proposed DSS assesses decision makers' preference by measuring their utility functions for time and cost separately. The two one-dimensional utility functions are then combined into a multiattribute function, which is optimized (maximized) within the search space of feasible activity durations and costs. Note that the activity durations and costs handled in this paper are deterministic. The proposed DSS focuses on finding subjectively optimal solution based on nonlinear utility functions.
Because the search space is often of high complexity (nonlinear, discontinuous, and nonsmooth), the optimization model is solved by a new computational intelligence technique: particle swarm optimization (PSO). Given that genetic algorithms are readily available, the reasons to develop a new PSO algorithm are multi-fold. PSO has been shown competitive to GA [1], or even superior to GA in continuous optimization problems [25]. Furthermore, PSO is particularly attractive in its algorithmic simplicity since it demands no coding and decoding operation [15].
Finally, the optimization results are directly linked to a full-scale schedule to facilitate further planning. Fig. 2 outlines the procedural steps, whose theoretical foundation will be introduced in the remainder of this paper.
Section snippets
Assessment of utility: lottery equivalent
To measure one's utility function, the best and worst extremes of the range of possible outcomes are assigned utilities of 1 and 0, respectively. Intermediate points of the utility function can be determined by either of the following two methods: certainty equivalent and lottery equivalent. Due to limited space, details of these two methods can be found in [4].
The proposed DSS adopts the lottery equivalent method, not the certainty equivalent method. This is to prevent the problem of error
Multiattribute utility
The time–cost tradeoff analysis involves two criteria: project duration and total cost. To incorporate the preference of decision makers toward these two criteria simultaneously, the proposed DSS combines the two one-dimensional utility functions into a multiattribute utility function following the procedure suggested by Keeney and Raiffa [14].
The procedure, however, relies on two important assumptions: preferential independence and utility independence. Preferential independence is the
Optimization model
The next step of the proposed DSS is to maximize the utility function within the search space of feasible activity times and costs, which ultimately determine the optimal pair of project duration and cost. This section gives the formulation of the optimization model. The project duration iswhere D denotes the project duration, which is the maximum early finish time among all the activities i.
The early start time ESi is computed recursively by satisfying the precedence constraints
Particle swarm optimization algorithm
The complexity of the present optimization model is primarily determined by Eq. (7) as the function between activity time and cost may take several forms: piecewise linear [11], nonlinear [5], discontinuous [18], discrete [3], [10], and any mix of the above [28]. Furthermore, the MAX and MIN functions in Eqs. (9), (10) would make the problem nonsmooth and nonconvex, presumably causing the appearance of multiple local optimal solutions. This level of complexity makes traditional optimization
Illustrative application: concrete bridge
The proposed DSS was implemented and applied to a concrete bridge project described in [18]. Table 1 tabulates the descriptions of eight activities, their immediate predecessors, and discrete options of times and costs. The indirect cost of this project is $500/day. If the project is completed later than 100 days, the amount of liquidated damages is $400 per day up to 10% of the expense. If the project is completed before 80 days, the bonus is $200 per day up to 5% of the expense.
A group of
Data set with five cases
A data set is used to demonstrate the general performance of the proposed DSS against GA. The data set includes five cases from the literature [12], [17], [18], [20], [28]. We believe the data set is adequate because it includes all types of activity time–cost functions, as listed in Table 4: linear, nonlinear (quadratic, logarithmic, and exponential), discrete, and discontinuous.
The proposed DSS and GA are compared on the same ground: both with 40 candidate solutions evaluated for 10,000
Real-life application: highway maintenance
A larger case is used to investigate the practicability of the proposed DSS. This real-life project is to maintain an existing highway. The work section, originally built as a dual, 7.3 meter wide, 22.8 cm thick reinforced-concrete pavement, has developed severe step faulting at the transverse cracks in the outside lane. There are also many transverse cracks in the inside lane, but without faulting. The plan is to remove the old concrete for the first 6 km and build outside-lane concrete inlays
Conclusion
The present study develops a utility-based DSS to facilitate schedule optimization. The proposed DSS accounts for the diversity in preferences of decision makers and therefore can capture the true decision-making mechanisms. Specific steps of the proposed DSS and their theoretical grounds are presented in this paper. At the core of the proposed DSS is a new particle swarm optimization algorithm, which is used to obtain the optimal solution with respect to the maximal utility value. This study
Acknowledgment
The author is grateful to the financial support from National Science Council, Taiwan under Grant No. 95-2221-E-032-052-MY3. This study has benefited from the participation of the project managers and Sun, Ying-Shyuan, who develops the Evolver model for the highway maintenance project. Appreciation also goes to the anonymous reviewers for suggesting emphasis on the critical activities during optimization, which turns out to be advantageous.
I-Tung Yang is currently an Assistant Professor in the Department of Construction Engineering at National Taiwan University of Science and Technology. He earned his Ph.D. in Civil Engineering from the University of Michigan, Ann Arbor in 2002. He also holds two master’s degrees in Industrial and Operations Engineering and Construction Engineering and Management from the University of Michigan. He has research interests in stochastic optimization, computer simulation, and soft computing. He has
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I-Tung Yang is currently an Assistant Professor in the Department of Construction Engineering at National Taiwan University of Science and Technology. He earned his Ph.D. in Civil Engineering from the University of Michigan, Ann Arbor in 2002. He also holds two master’s degrees in Industrial and Operations Engineering and Construction Engineering and Management from the University of Michigan. He has research interests in stochastic optimization, computer simulation, and soft computing. He has served as a reviewer for 9 reputable journals and several international conferences, and on the boards of professional organizations. His publications have appeared in ASCE Construction Engineering and Management, IEEE Transactions on Engineering Management, Computer-aided Civil and Infrastructure Engineering, Construction Management and Economics, and International Journal of Project Management.