WATER RESOURCES RESEARCH, VOL. 17, NO. 5, PAGES 1339-1348, OCTOBER 1981 Multiperiod Design of Regional WastewaterSystems: Generating and Evaluating Alternative Plans MASAHISA NAKAMURA• Department of Chemical andEnvironmental Engineering, University of Louisville, Louisville, Kentucky 40292 E. DOWNEY BRILL, JR. Department of Civil Engineering and Institute for Environmental Studies, University of Illinois, Urbana,Illinois 61801 JON C. LIEBMAN Department of CivilEngineering, University of Illinois, Urbana,Illinois 61801 A simplified methodis proposed for analyzingmultiperiod design of regionalwastewater systems and othersystems possessing a similar networkflow structure. The primary objective of the methodis genera- tion of multiperiodlocations and facility expansion timings.The method is based on severalsimplifying assumptions. The solution procedure involvesa branch and bound algorithm specifically developedfor integrating a setof solutions obtained using the initial design flow with a setof solutions obtainedusing the terminaldesign flow. A methodfor comparing the generated multiperiodalternatives is also pro- posed. The usefulness of the methodis demonstrated, using asan example a hypothetical regional waste- water system. A Fortran program was written to implement the method. The program is not in a form that would permit its transfer,but interested readers may contactthe first author. INTRODUCTION The difficulty associated with developing a mathematical tool for analyzing multiperiod designof regional wastewater systems lies in the fact that the problemgenerallyexhibits an enormous multidimensionality with respect to temporal, spa- cial, and technical design options, not to mention the so- cioeconomic and environmental impacts of the resultingalter- native plans. Mathematical models dealing with the subject thereforehighlight the analysis of a specific aspect of design, generally cost, based on a setof rather restrictive planningas- sumptions. The currentwork is no exception. It is an attempt, however, to departfrom the conventional approach of seeking the mathematically optimal solution, however it may be de- fined. Severalmethods of analysis have been proposed in the past for the design of multiperiod regional wastewatertreatment systems. In one of the earliest of these works [Bl•alla and Rikker, 1971]a heuristic method of locatingfacilitiesand as- signing flowsdirectly to those facilitieswasproposed. The net- work flow structure,a feature characteristic to the designof wastewater systems, was disregarded for ease of computation. (Most multiperiod facility location methods proposed in the literature to date deal with problems of this type. See, as ex- amples, Wesolouski [1973] and $weeney and Tatham [1973]). The methods proposed more recently [e.g. Lauria, 19753 Chiang and Lauria, 1977; Rossman, 1978] take into account the network flow structure. All of the methods proposed thus far, however, emphasize the identification of the least cost solu- tion. As statedearlier, costis but one of many factorswhich af- fect the location of facilitiesin the planning of regional waste- • Now at WHOWestern Pacific Regional Centre forthe Promotion of Environmental Planning andApplied Studies, Kuala Lumpur, Ma- laysia. Copyright ¸ 1981 by the American Geophysical Union. water treatmentsystems. The utility of the conventional opti- mization methods which identify the least cost plan with precisefacility locationsand expansionschedules is limited unless provisions are made for the generationof alternatives with minimum additional effort. The method proposed herein takes a different approach. It generates alternative multi- period facility locationplansgiven a setof assumptions which restrictthe decisions both on the flow assignment over the de- sign period and on the choice of expansiontiming. While these assumptions limit the useof the method to special cases, the insightsgained through the use of this approach differs significantly from the insights gained through the use of con- ventional approaches. This paper is an extension of the pre- vious work by Brill and Nakamura [1978] and Nakamura and Brill [1979], which deal with, respectively, the generation of single-period alternative solutions and the cor0parison of al- ternative regional plans using an imputed value analysis method. BASIC ASSUMPTIONS FOR MULTIPERIOD ANALYSIS Cost functions for wastewater treatment Plantsand inter- ceptors are nonlinear with respect to capacityand the level of wastewater flow treated or transported. The assumptions listed below lead to simplified expressions of multiperiod costs, which involve combinations of fixed charges and linear cost terms. Thesesimplified expressions allow the application of a branch and bound method to obtain the least cost solu- tion (for a givensetof cost approximations), to generate alter- native solutions, and to calculate imputed values associated with regionalfacilities.Although slightmodifications of these assumptions may be possible without loss of generality of the method, these assumptions reflect the basic approach to the development of the mathematical theory. 1. Wastewater production at each sourceis a linear func- tion of time. 2. The interim design periods are predetermined, and Paper number IW0698. 0043-1397/81/001 W-0698501.00 1339