This paper introduces methods for formulating and solving a general class of nonpreemptive resource-constrained project scheduling problems in which the duration of each job is a function of the resources committed to it. The approach is broad enough to permit the evaluation of numerous time or resource-based objective functions, while simultaneously taking into account a variety of constraint types. Typical of the objective functions permitted are minimize project duration, minimize project cost given performance payments and penalties, and minimize the consumption of a critical resource. Resources which may be considered include those which are limited on a period-to-period basis such as skilled labor, as well as those such as money, which are consumed and constrained over the life of the project. At the planning stage the user of this approach is permitted to identify several alternative ways, or modes, of accomplishing each job in the project. Each mode may have a different duration, reflecting the magnitude and mix of the resources allocated to it. At the scheduling phase, the procedure derives a solution which specifies how each job should be performed, that is, which mode should be selected, and when each mode should be scheduled. In order to make the presentation concrete, this paper focuses on two problems: given multiple resource restrictions, minimize project completion time, and minimize project cost. The latter problem is also known as the resource-constrained time-cost tradeoff problem. Computational results indicate that the procedures provide cost-effective optimal solutions for small problems and good heuristic solutions for larger problems. The programmed solution algorithms are relatively simple and require only modest computing facilities, which permits them to be potentially useful scheduling tools for organizations having small computer systems.project management: resource constraints, programming: integer algorithms, enumerative, industries: construction

Green (vegetated) roofs have gained global acceptance as a technologythat has the potential to help mitigate the multifaceted, complex environmental problems of urban centers. While policies that encourage green roofs exist atthe local and regional level, installation costs remain at a premium and deter investment in this technology. The objective of this paper is to quantitatively integrate the range of stormwater, energy, and air pollution benefits of green roofs into an economic model that captures the building-specific scale. Currently, green roofs are primarily valued on increased roof longevity, reduced stormwater runoff, and decreased building energy consumption. Proper valuation of these benefits can reduce the present value of a green roof if investors look beyond the upfront capital costs. Net present value (NPV) analysis comparing a conventional roof system to an extensive green roof system demonstrates that at the end of the green roof lifetime the NPV for the green roof is between 20.3 and 25.2% less than the NPV for the conventional roof over 40 years. The additional upfront investment is recovered at the time when a conventional roof would be replaced. Increasing evidence suggests that green roofs may play a significant role in urban air quality improvement For example, uptake of N0x is estimated to range from $1683 to $6383 per metric ton of NOx reduction. These benefits were included in this study, and results translate to an annual benefit of $895-3392 for a 2000 square meter vegetated roof. Improved air quality leads to a mean NPV for the green roof that is 24.5-40.2% less than the mean conventional roof NPV. Through innovative policies, the inclusion of air pollution mitigation and the reduction of municipal stormwater infrastructure costs in economic valuation of environmental benefits of green roofs can reduce the cost gap that currently hinders U.S. investment in green roof technology.

In this paper, we report on a computational experiment designed to assess the efficacy of 26 heuristic decision rules which group work tasks into work stations along an assembly line such that the number of work stations required is minimized. The heuristic decision rules investigated vary from simple list processing procedures that consider a single attribute of each work task for assignment, to procedures which are optimal seeking, but which have had their search terminated through the imposition of a limit on the amount of computation time that can be devoted to each search. Also included are heuristic decision rules which backtrack in an attempt to locate an improved solution, and decision rules which probabilistically search for improved solutions. Our investigation differs from those reported previously, in that the objective in balancing each line is to determine the minimum number of work stations for a given limit on the time available for assembly at each work station (the cycle time). Previous approaches have investigated the problem of determining the minimum cycle time for a given line length. We also compare the results obtained with the optimal solution for a subset of the problems investigated. Both problems which have appeared in the open literature and problems which have been solved for the first time are included. Because a portion of our results differ from those reported previously, we suggest why these differences have occurred. Guidelines are also given to those balancing industrial assembly lines on the choice of the heuristic decision rule to use whether one is attempting to obtain a minimum station balance given a limit on the time available for assembly at each work station, or whether one is attempting to minimize the time devoted to assembly at a work station given a limit on the number of work stations available.production/scheduling: line balancing, networks/graphs: applications, facilities/equipment planning: design