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Computerized regional planning for land disposal of wastewater
Urhun Swm~s.
Vol. 2. pp. l-14.
Pergamon
Press, 1977. Prmted m Great Britain
COMPUTERIZED REGIONAL PLANNING LAND DISPOSAL OF WASTEWATER? ROBERT E. MARKLAND,
L. DOUGLAS SMITH
School of Business Administration,
FOR
and JACK BECKER
University of Missouri, St. Louis, MO, U.S.A.
(Received 13 November 1975)
Abstract-This paper describes the development and utilization of a large scale mixed integer programming model for regional planning of land disposal of wastewater within the St. Louis Metropolitan Statistical area. This model is derived, and used to determine which land disposal sites should serve which treatment plants, when initial construction should be initiated and completed, and when capacity expansion should occur. Consideration is given to relevant construction and operating costs for land sites and transmission arteries, land acquisition costs, tangible benefits from land use, controls on pollution of aquifers, and various other engineering and technical constraints. The computational aspects of the model’s use are emphasized in the paper, and extensive test results are presented and discussed.
INTRODUCTION
and disposal of municipal wastewater is a serious environmental problem currently confronting the people of the United States [l, 21. One dimension of the problem involves the removal of pollutants from liquid effluents generated during wastewater treatment. Land application has recently been advocated as a potentially economical and ecologically beneficial method of reclaiming wastewater [3-73. This paper describes the development of a computerized mixed integer programming model for planning a land application system to dispose of sewage effluent from secondary treatment plants serving a large metropolitan area. Previously [8,9], the authors presented an historical perspective of land disposal methods, a brief description of features of alternative land disposal systems, a general background of the modeling problem for the St. Louis metropolitan area, and the derivation of a mixed integer programming model for selecting land sites to be developed for wastewater disposal. This paper summarizes our findings and emphasizes four aspects of computerized regional planning for land disposal of wastewater: THE TREATMENT
1. 2. 3. 4.
The data base required; Experience in solving the model; Presentation of a sample solution; Considerations in using the model.
The St. Louis Standard Metropolitan Statistical Area provided the setting within which the methodology was applied. Although many of the data sources cited are specific to this region, we do hope to convey a reasonable impression of the efforts required in making such a model operational in a typical urban environment. IDENTIFICATION WASTEWATER
OF POTENTIAL TREATMENT
LAND FACILITY
DISPOSAL SITES
The regional planning problem was approached in two phases. The first phase involved the identification of a set of suitable locations for land disposal wastewater treatment facilities utilizing a land-use forecasting simulation model. t Based on research supported by the US. Department Technology, under Title II Grant 14-31-0001-4221 (C5302). I I’.SS
2 1
A
of the Interior, Office of Water Research and
2
ROBERT E. MARKLAND, L. DOUGLAS SMITHand JACK BECKER
The land-use forecasting model Because of the potential political implications and the general lack of public awareness and understanding of land disposal wastewater treatment, it was considered necessary to avoid land which would be expected to experience concentrated industrial, residential, or commercial development in the future. Potential sites were to be selected from land which was projected to fall into agricultural, recreational (e.g. parks, golf courses), or vacant use categories. Projections of future land use were accomplished with the help of the U.S. Corps of Engineers’ (USCE) Land-Use Forecasting Model, developed for the St. Louis metropolitan area by Meyer et al. [lo, 111. This model forecasts industrial, residential, commercial, and public land use by census tract in ten year increments to the year 2030, for each county in the St. Louis Standard Metropolitan Statistical Area. Use of the model requires an exogenous forecast of county population and employment for each decade in the planning horizon, and specification of current-decade (1970) land use information. The four land uses are forecasted in the above order so that the same parcel of land cannot be allocated to more than a single use. The first allocation is made to industrial use, under the rationale that industry will generally be the first to acquire whatever new land it needs, since it has the economic resources to do so. Residential use is ranked second, followed by commercial use, primarily because the development of commercial areas follows the development of residential areas. Public land use is forecasted last, and represents a residual land use It consists of government facilities, streets, right-of-ways, and other similar uses. The forecasts by decade, by tract, for each of the four land uses are generated in four phases : 1. A desirability ranking on a county-wide basis; 2. Initial allocation to existing land (intensive land use); 3. Allocation to vacant and agricultural land (extensive land use); 4. A combination of intensive-extensive use on a countywide basis. The forecasts are made to distribute residents and employees among designated land units within each county based upon: 1. attractiveness for more intensive usage as measured by the number of residents and employees currently allocated to a land unit, considering also user-specified density limits; 2. attractiveness for more extensive development as measured by characteristics such as access to current and planned transportation arteries and physical features such as lack of severity of terrain. Output from the land use model consists of summary reports from each phase of the forecasting process, and a decade summary by land use for each county by census tract. Final selection-potential
land disposal wastewater treatment sites
Given the forecasts of population distribution and acreage usage for land units within each county, to the year 2000, a consolidated ranking of the land units was produced on the basis of total residential population; industrial, commercial, and public employment; and subjective estimates of the three sectors’ effective resistance to the establishment of land disposal sites in their vicinities. These rankings were subsequently combined with a Sparsity Index which reflected the percentage of ‘available land. The Sparsity Index was defined as the sum of all vacant, recreational, agricultural, and normal use land having five people or less per acre divided by the total acreage of the land unit. The Sparsity Index indicated the lack of development of a land unit, hence reflecting its potential as a land disposal wastewater treatment (LD/WT) site.
Computerized regional planning for land disposal of wastewater
Fig. 1. Secondary treatment facilities and potential land disposal wastewater treatment sites-St. Louis SMSA.
Finally, the more desirable sites according to the Sparsity Index were subjected to further scrutiny on the basis of their soil characteristics (e.g. water capacity, aerobic properties, soil color, and the possibility of groundwater contamination). Final site selection thus involved the consideration of four major factors: 1. The Sparsity Index ; 2. Soil suitability; 3. Special topographical features; 4. Provisions of a sufficient number of potential sites in each county. Areas outlined in black in Fig. 1 contain the primary potential sites, based upon joint consideration of these factors. A REGIONAL
PLANNING MODEL FOR OF WASTEWATER
LAND
DISPOSAL
Given a set of potential sites for land disposal wastewater treatment facilities, the modeling problem becomes that of developing a selection methodology’ for choosing between alternative sites within a regional cost-benefit framework. A large scale, mixedinteger programming model, based on the work of Kiihner and Harrington [12,13], was developed to assimilate the large quantities of relevant information and to enable the development of a regional plan to achieve specified objectives. This model accepts the set of potential land disposal sites and a set of plants planned for the secondary treatment of sewage. It determines which land disposal sites should be developed to serve which treatment plants, when initial construction should be completed, and when capacity expansion should occur. The plan developed by application of the model is one which minimizes the discounted present value of relevant construction and operating costs, net of tangible ongoing benefits and net of the anticipated change in land value at the developed sites. A specified quantity of wastewater from each of the secondary treatment plants in constrained to receive further treatment by land disposal methods.
4
ROBERT E. GARLAND,
L. DOUGLAS SMITH and JACK BECKER
The following factors were considered in the model: 1. Initial costs of constructing the land disposal facilities at the alternative sites; 2. Costs of increasing capacity at alternative sites; 3. Operating costs; 4. Capacity limitations at each site; 5. Acreage r~uirements; 6. External benefits derived from the land disposal facilities; 7. Costs of constructing and maintaining transmission arteries from secondary treatment facilities to land disposal sites; 8. Anticipated long term changes in values of acreage devoted to land treatment facilities; 9. Annual and cumulative budget constraints; 10. Controls on pollution of aquifers. Each of these factors is discussed in detail, in terms of the mathemati~l formulation of the model, which is presented in Appendix A. The mixed integer linear programming model provides an explicit statement of the information required in the development of a comprehensive regional plan for wastewater management. Furthermore, it offers a systematic procedure for selecting the best plan among many alternatives. Experiences in developing the data base and subsequently solving the mixed integer programming model using the MPSX-MIP software on an IBM 370/l@ are related in the following sections. Datu base for the regional planning model Projected locations of secondary treatment facilities and volumes of wastewater to be generated from each plant were obtained from the St. Louis District, U.S. Army Corps of Engineers. This information is presented in Table 1. Next it was decided which land sites could potentially serve which treatment plants, considering the costs of constructing connecting arteries for transmitting the e@uent, and also the anticipated land r~uirements for effluent from each treatment plant. In Fig. 1, iocations of the
Table 1. Forecasted locations and sizes of secondary wastewater treatment facilities Secondary treatment facility site number
Grid cell location JX-16 MX-30 SV-29 NZ45 NZ-51 02-64 NY-51
8 9 10 11
MZ-60 JZ-64 HW-34 MZ-39
12 13
LV-34 JY-11
Facility name
Population served (1974)
Actual usage-1974 (MGD)
Potential usage-2020 (MGD) 19.0 26.0 4.1 46.7 214.3 84.1 136.7
East St. Louis Grantie City Alton Bissel Point Treatment Plant (MSD) Coldwater Creek (MSD) St. Charles
61,200 40,400 40,000 587,000
9.6 3.0 161.0
173,600 19,000
21.0 2.0
Lemay Treatment Plant (MSD)
608.244
91.4
Suggested land disposal areas for the secondary treatment facilities: Jefferson County 1, 11, 12,13 St. Clair County 273 Madison and St. Clair Counties 4,5.6 S.W. Franklin County 7, 8, 10 9 N. St. Charles County. SKJRCE: St.Louis District. U.S. Army Corps of Engineers.
99.4 34.2 6.5 172.7 13.1 2.6
Computerized Table
Land disposal site
Secondary treatment facilities served
regional 2. Final
Population served
planning
for land disposal
land disposal
1974 requirement (MC=)
treatment
2020 requirement (MGD) 265.1
1
3,4, 5
101.200+
2 3 4
X6 2. 4
80,000 61,200+ 587.000
12.6 46.7+ 161.0
298.4 72.7 136.7
5
1, 8
760.600
182.0
236.1
6 I 8 9 10 11 12 13
7, 8 9 9 10 11,12 2, 11 2, 3, 11 10,12
760,600 19,000 19,000
182.0 2.0 2.0
236.1 34.2 34.2 6.5 185.8 198.7 202.8 19.6
14 15 16 17 18
1, 10, 13 1, 13 1, 10 10 11.12
608,244 +
19
7, 8
760,600
608,244 + 608,244+ 608,244 +
91.4+ 91.4+ 91.4+
91.4+ 182.0
28.1 21.6 25.5 6.5 185.5 236.1
of wastewater sites
Land units
Available land/unit (acres)
340 394 397 398 449 349 1 5 6 268 272 213 214 274 281 288 130 324 329 141 178 312 322 303 300 117 125 57
24,395 43,999 25,318 22,528 34,435 16,933 834 493 654 10,248 10,872 9523 58,844 58,844 38,840 64,629 6799 46,730 81,209 26,884 35,158 39,412 57,976 48,280 73,841 265 174 1459
Total land (acres) 116,240
34,435 16,933 1981
30,643
58,844 58,844 38,840 64,629 6799 46,730 81,209 62.042 62,042 39,412 57,976 48,280 73,841 439 1459
secondary treatment facilities are shown with respect to the potential land disposal sites. Possible linkages between the secondary treatment facilities and land sites are indicated by arrows. In Table 2, below, the final potential land disposal wastewater treatment sites, and their associated linkages are summarized. Note that 19 potential sites were considered, with most of these 19 sites involving a combination of the land units shown earlier in Fig. 1. Having developed the set of 19 land disposal wastewater treatment sites to be employed in the model, the data base for the model was constructed. This data base was composed of 38 data items, collected from various governmental, academic, and consulting firm sources. These sources are identified in the technical report by Markland et al. [3]. Matrix generation The contents of the data base were next employed as input to a matrix generator program. This matrix generator program written in FORTRAN IV, derived the coefficients for the objective function and the constraint set of the mixed integer programming model. Portions of the data summary from the matrix generator are provided in Appendix B. A brief discussion of the relation of the items in the data base (Appendix B) to the model formulation (Appendix A) follows. The number of possible land disposal sites, secondary treatment sites, and time periods determine the number of variables and constraints to be generated. The discount percentage is applied to the net value of all cash flows registered in the objective function. Indices of operating and construction costs are used to produce cost estimates for future cash flow constraints C(6) and (7) in Appendix A] and objective function coefficients. Secondary treatment facility data, including effluent volumes and percentage of effluent to be treated are incorporated into the treatment constraints [(4) and (5) in Appendix
6
ROBERT E. MARKLAND,L. D~UCLAS SMITHand JACK BECKER
A]. Budget data appear in the capital constraints [(6) and (7) in Appendix A]. Transmission artery cost data are utilized in the objective function, and in the capital constraints C(6) and (7) in Appendix A]; flow capacities are utilized in the arterial flow constraints [(12) in Appendix A]. Land disposal site data appear next. Consolidated cost coefficients for the objective function and capital constraints were derived for both initial development and expansion at a site, using the detailed components listed. The number of irrigating days per year and application rate per day were considered with the data for storage impoundments in deriving required lagoon acreage and then total acres required per million gallons per day at each site. These latter quantities are incorporated as coefficients in the land site capacity constraints. Estimates of the value of developed acreage at the end of the planning horizon are used to adjust the objective function coefficients for variables representing initial development (Xi) and expansion (Ei) at a land site. Aquifer absorption rates, the last entries among the land disposal site data, appear as coefficients in constraints upon pollution [(S) and (9) in Appendix A]. Benefits in period following development at a site are used to adjust objective function coefficients for land site development variables Xi and Eg. Projected land site value indices are reflected in both the cash flow constraints [(S) and (9) in Appendix A] and in the objective function for X; and Ef. Finally, the environmental protection data provide coefficients for the right hand sides of the pollution limit constraints [(S) and (9) in Appendix A]. One of the main difficulties encountered in compiling the data base was the separation of aggregated data into appropriate components. For the model, it is necessary to separate costs of facilities into fixed capital expenditures, variable capital expenditures, operating costs dependent upon the total acreage developed at a site, and operating costs dependent upon the volume of water processed. In contrast, published cost data was typically derived from examining total costs in connection with facilities which were designed to specific performance specifications. Naive ratios such as costs in dollars per million gallons per day or costs in dollars per acre had been calculated simply by taking total costs over the appropriate denominator. It is clear that improved methods in reporting the basis of cost estimates are required if planning for facilities is to proceed in an organized and efficient manner. One advantage of a formal decision model is that it encourages the development of standardized procedures for collecting and sharing relevant data. Solving the model The computerized version of the model exhibited approximately 1300 rows and 1300 structural variables, with matrix density 0.51%. The basic formulation contained 550 O-1 integer variables: 360 t&l variables for (36 possible 190 &l variables for (19 possible
construction of transmission arteries arteries x 10 time periods) initial site development sites x 10 time periods)
Steps in the solution process are represented schematically in Fig. 2 along with the files utilized during the solution exercise. By saving the optimal bases from previous runs, new continuous solutions for a revised version of the model (relaxing integer constraints) were obtainable in less than 2 min of CPU time, using the MPSX-MIP software on an IBM 370/168. Obtaining suitable integer feasible solutions, however, required considerably more computational effort. Solution effort is, of course, a function of the number of integer variables. Therefore, before undertaking the search for integer feasible solutions, the 550 O-l integer variables were examined to determine whether any could be treated as continous variables without rendering the solution unusable in a practical sense. Fractional values for Lfj, the construction of a transmission artery from plant i to land site j would be absolutely unacceptable in a final solution. Hence
Computerized regional planning for land disposal of wastewater
7
Basis fromwntinuous solution
Fig. 2 User files utilized in MPSX(LP) and MPSX(MIP) phases of solution process.
it was necessary to leave the constraints L:j = (0,l) intact. Xf = (O,l), however, could be relaxed to 0 I Xl I if the value of X: were redefined as the proportion of the maximum acreage available at site j which is to be developed during period t. X; relates to the proportion of maximum acreage available rather than the proportion of ‘initial development’ acreage because relaxation of the integer requirements on Xf enables site expansion E; to occur simultaneously whereas in the original formulation, E; would be forced to have a value of zero until initial site development is completed. This, in turn, implies that the version of the model with Xi = (OJ) relaxed to 0 I X: I 1 will allow the purely fixed costs in connection with initial site development to be delayed or even partially avoided for those land sites not completely developed. Fortunately, the fixed costs (unrelated to the amount of acreage to be developed) were overshadowed by land purchase cost and other variable costs. The resulting favorable cost variance was approximately 1% of the cost of initial acreage development. Thus, economic distortions due to the relaxations were not considered serious. The initial acreage to be developed was rather arbitrarily determined considering the anticipated application capacity which might be needed to justify construction at the site, and also the minimum size of land plot which might reasonably be involved in a real estate transaction. These arbitrary considerations could be waived provided the non-zero values of XJ in the final solution were sufficiently large to make the implied real estate transaction and extent of site development appear reasonable when reconsidered. The noninteger values for Xf were indeed of sufficient magnitude in the integer feasible solutions generated (i.e. solutions with integer values for the Lij variables) that further exploration to force Xf values to be integer was not deemed necessary. To obtain the first integer feasible solutions for the remaining 360 L~j variables required approximately 50 min of CPU time, working in 10 min intervals to facilitate the file management function. The maximum potential improvement in the objective function achievable from further exploration beyond the first integer feasible solution
8
ROBERT E. MARKLAND,L.
DOUGLAS SMITH and JACK BECKER
obtained was indicated to be about 30%. The bound was not judged to be very tight, however, and the solution developed showed no obvious potential for significant improvement. Further computation was therefore not undertaken beyond the first integer solutions for particular versions of the model. Table 3(a, b, c, d). A condensed sample solution to the mode1
(4
Time period 1975580
198&35
202tk25
Land disposal wastewater treatment sites*
Total acreage developed (cumulative)
Secondary treatment facilities* served
1 2 5 8 10 12 13 14 16 17
494 2124 9666 273 523 189 61 19 138 34
3,4 5,6 7, 8 9 11 2 12 13 1 10
1 2 5 8 10 12 13 14 16 17
986 5353 12,926 595 5219 3862 182 51 375 91
3,4 5,6 7, 8 9 11 11, 2 12 13
1 2 4 5 6 8 10 12 13 14 16 17 19
40,29 1 34,440 1980 29,968 18,464 7572 5219 52,946 2803 788 5758 1547 1459
3,4, 5 5, 6 7
Total effluent treated (MGD) (cumulative) 2.01 8.23 37.48 1.04 19.91 0.52 0.26 0.05 0.38 0.13 3.82 20.77 50.13 2.28 19.91 10.51 0.71 0.14 1.03 0.35
1
10
7, 8 I
9 11 11, 2 12 13 1 10 I
-
42.16 133.57 3.93 116.28 71.61 29.00 19.91 145.02 10.87 2.16 15.17 6.00 2.98
* Refer to Fig. 1. (b) Land site development summary Percentage of total available land developed (over time horizon) Land site No. 1 2 4 5 6 8 10 12 13 14 16 17 19
34.67 100.00 100.00 97.81 31.38 19.50 16.76 65.20 4.52 2.00 11.93 2.10 lOQ.00
Period capita1 expenditures ($ millions)
Cumulative capita1 expenditures (8 millions)
$656.8
$656.8
$210.2
$867.0
$800.0
$5064.6
9
Computerized regional planning for land disposal of wastewater (c) 2020-2025
1975-80
Artery
2
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Time period
From secondary treatment site
To land disposal site
1 1 1 2 2 2 3 3 4 4 5 5 6 7 7 I 7 8 8 8 9 9 10 10 10 10 10 11 11 11 11 12 12 12 13 13
14 15 16 3 11 12 1 12 1 3 1 2 2 4 5 6 19 5 6 19 7 8 9 13 14 16 17 10 11 12 18 10 13 18 14 15
-
0.00 0.00 0.38 0.00 0.00 0.52 0.08 0.00 1.83 0.00 0.00 6.01 2.22 0.00 31.72 0.00 0.00 5.76 0.00 0.00 0.00 1.04 0.00 0.00 0.00 0.00 0.13 19.91 0.00 0.00 0.00 0.00 0.26 0.00 0.05 0.00
Unused capacity (MGD)
Arterial flow
0.00 0.00 15.62 0.00 0.00 21.48 3.92 0.00 38.17 0.00 0.00 114.49 67.78 0.00 75.48 0.00 0.00 77.23 0.00 0.00 0.00 27.96 0.00 0.00 0.00 0.00 5.87 6.09 0.00 0.00 0.00 0.00 10.74 0.00 2.25 0.00
0.00 0.00 15.77 0.00 0.00 21.58 3.40 0.00 38.76 0.00 114.10 63.71 69.80 3.93 33.23 71.61 2.98 83.00 0.00 0.00 0.00 29.00 0.00 0.00 0.00 0.00 6.00 19.91 0.00 23.44 0.00 0.00 10.87 0.00 2.16 0.00
WGD)
...’
Unused capacity WGD) 0.00 0.00 0.23 0.00 0.00 0.42 0.60 0.00 1.24 0.00 65.90 56.73 0.20 2.87 73.97 42.39 2.62 0.00 0.00 0.00 0.00 0.00 OSKI 0.00 0.00 0.00
0.00 6.09 0.00 20.57 0.00 0.00 0.13 0.00 0.14 0.00
(MGD)
Regional % of effluent treated
Regional total of effluents absorbed by aquifers (MGD)
Regional % of effluents absorbed by aquifers
69.93 109.70 157.39 213.02 276.70 348.05 427.48 514.85 610.07 713.30
>20 >27 >34 241 >48 >55 262 >69 176 >83
3.50 5.48 7.87 10.65 13.83 17.40 21.37 25.74 30.50 35.66
5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
Regional total of effluent treated
1. 1975-80 2. lY8&85 3. 1985-90 4. 1990-95 5. 1995-2000 6.200@05 7. 2005-10 8. 2Ol(tl5 9. 2015-20 10. 202G25
Arterial flow WGD)
Illustration of a solution Information conveyed by a solution to the model is illustrated in Table 3. The first section indicates the proposed development of land disposal sites by period, cumulative acreage developed, the secondary treatment facilities which they serve, the amount of effluent treated at each, and the capital expenditures involved. The next section shows the extent to which each site is developed by the end of the 50-year planning horizon.
1.0
ROBERTE. MARKLAND, L. D~UCLAS SMITH and JACK BECKER
Then a summary of arterial flows and unused arterial capacities is provided for each period. Finally, regional totals of secondary effluent to receive tertiary treatment by land application, are provided for each period in the planning horizon. The net present value of the costs associated with the plan for the 50 year time horizon was estimated to be approximately 800 million dollars. A view toward implementation of the planning system A well-documented, user-oriented matrix generator facilitates reconstructing the model to reflect different assumptions about values of the model’s parameters. Only a few minutes are required for the user to change the data deck to make changes in expected costs of capital construction, operating costs, land values, EPA treatment requirements, maximum application rates, etc. For example, a computer run was performed to illustrate the costs of imposing uniform standards upon treatment of effluent from each secondary treatment facility. It was demonstrated that a given level of tertiary treatment for the region could be acquired much more economically by concentrating tertiary treatment at some facilities, and relaxing standards at others. Communications with personnel from the U.S. Army Corps of Engineers suggest that the model is a promising tool for regional planning. Some questions remain about whether all relevant technical information can be consolidated in such parameters as the proposed application rate at each site. Additional constraints may be required to consider specific aspects of the biological and chemical processes at the sites. Also, in spite of the fact that spray irrigation has been suggested as the most appropriate land application process for the Greater St. Louis area, the model may easily be expanded to include other land application processes such as overland runoff and infiltration-percolation. In addition, changes could be made to incorporate other types of processes such as advanced biological and chemical treatment. Among the benefits associated with the modeling approach described in this paper is that it provides a rational, rather than political, approach to decision making. Thus, it provides a means of structuring a massive amount of data into a very compact and workable form. All projects are considered on a relative basis, and an exhaustive series of separate analyses are not required. Furthermore, this approach is very comprehensive and facilitates an analysis which encompasses a long time horizon (50 years). The model offers the user extreme flexibility, in terms of making changes to the input data, or in terms of considering alternative sites. Finally, the methodology developed and tested in this paper could be used in other regions. A number of limitations to the model should also be noted. At present, the model is specific to the St. Louis metropolitan region, particularly with respect to secondary treatment facilities and pipeline configurations. The model obviously requires a large amount of derived data input. Some of this information is subjective; other information requires considerable analysis for its determination. Determination of the input information becomes more difficult as the time frame is lengthened. Solutions to this type of model are difficult, time consuming and hence, costly, to obtain. Indeed, pure optimality was not achieved, and efforts to develop efficient heuristic solution procedures would probably be worthwhile. Implementation of the results of this modeling effort would probably require further analysis on a site-by-site basis. As this occurred, the political ramifications of the work would have to be dealt with. The present solution to the model suggests that the majority of land disposal wastewater treatment sites would initially be located in areas that are sparsely populated. This would probably serve to soften public and community resistance towards this approach to wastewater treatment. CONCLUSION
This paper has discussed the use of a large scale mixed integer programming model to facilitate regional planning for land disposal of sewage effluent in an urban environment. Consideration was given to the data base required, identification of potential
Computerized
regional planning for land disposal of wastewater
I1
land disposal sites, experience in generating solutions to the model, and use of the model in developing alternative plans in response to different possible legislative constraints. Future work will investigate alternative approaches to the mixed integer (MIP) phase of the solution process. REFERENCES 1. Proceedings ofthe Joint Conference on Recycling Municipal Sludges and EfJluents on Land, Washington, DC., National Association of State Universities and Laud-Grant Colleges, p. vii (1973). 2. Metcalf and Eddy, Inc. ~~r~a~er ~~~neer~ng, McGraw-Hill, New York (1972). 3. Charles E. Pound and Ronald W. Crites, Wast~ater ~reat~nt and Reuse By Land Application, Environmental Protection Agency Report 6~/~-73-~b, p. 9. U.S. Environmental Protection Agency, Washington, D.C., August (1973). 4. Richard H. Sullivan, Morris M. Cohn and Samuel S. Baxter, Survey of Facilities Using Land Application of Wastewater, Environmental Protection Agency Report 430/9-73-G%, U.S. Environmental Protection Agency, Washington, D.C., July (1973). 5. A&essment of the Efictiveness-and
Effects
of Land
Disposal Methodologies
of
Wastewater Management,
Final Renort For Contract Nos. DACW 73-73-C-0@41-0@43,Office of the Chief of Engineers, U.S. Army Corps Engineers,Washington, D.C., January 14 (1972). 6. Wast~ater management By Disposal on Land, Cold Regions Research and Engineering Laboratory, U.S. Army Corps of Engineers, Hanover, NH, May (1972). 7. lmp~icaiions of Mu~t~bjective Plan Forn?ulation and EL~luation of Regional W~t~ater Ma~gement Systems, Contract No. DACW 73-72-C-0045, IRP 71-16, U.S. Army Corps of Engineers, Washington, D.C., February (1973). 8. Robert E. Markland, L. Douglas Smith, Jack D. Becker and Paul J. Haack, A Site Selection Modet ,for Land Disposal ofWastewater, working paper presented at the Chicago Joint Meeting of the Operations Research Society of America and The Institute of Management Sciences (1975). 9. Robert E. Markland, L. Douglas Smith and Jack D. Becker, Regional ‘Planning for Land Disposal of Wastewater Using Mixed integer Programming, working paper presented at the Las Vegas Joint National Meeting of the Operations Research Society of America and The Institute of M~agem~t Sciences (1975). 10. Carl F. Meyer, Andre B. Corbeau and Harold L. Mack, A water policy-land use computer simulation model, Wat. Resour. &II. 10, 952-968 (1974). 11. Carl F. Meyer, Andre B. Corbeau and Harold L. Mack, A computer-oriented land use forecasting model with mapping capability, Comput. & Urban Sot. 1, 3148 (1975). 12. Jochen Kiihner and Joseph J. Harrington, Mathematical models for developing regional solid waste management politics, J. Engng Optimization 2, (1974). 13. Jochen Kiihner and Joseph J. Harrington, Lurge Scale Mixed Integer Programming For Investigating
of
Multi-Party Public Investment Decisions: Application to a Regional Solid Waste Management
Problem,
working paper presented at the 45th National ORSA,TIMS Meeting, Boston, MA, April (19’74). 14. Robert E. Markland, L. Douglas Smith and Jack D. Becker, A Benefit-Cost Analysis of Alternative Land Disposal Waste Water ~et~luds in an Urban ~n~iro~ent, Final Technical Completion Report For Contract No. USDI 1~31-~1-4221 United States Department of the Interior, OlIice of Water Resources Research, December (1974). APPENDIX
A. MATHEMATICAL OF THE MODEL
FORMULATION
Variables used in the model. The variables used in the model are defined as follows: A; Eij Fij Lfj Xi
total acreage developed (thousands of acres) for land disposal at wastewater treatment site j by the end of period t; amount ofexpansion(thousands of acres) provided at the land disposal facility on site j at the beginning of period r; flow of wastewater (million gallons per day) from secondary treatment site i to land disposal site j during period f; bivalent (0,l) variable corresponding to completing construction of the transmission artery from treatment site ‘i to land disposal site j at the beginning of period t; bivalent (0,l) variable corresponding to completing construction of a land disposal facility on site j at the beginning of period t.
Parameters employed in the model. The following parameters
af
a,” b6
b”* I e;
were employed:
minimum acreage (lower bound) required (thousands of acres), including storage impoundment, for land disposal if site j is developed; (Note that ai is obtained by choosing the maximum of the minimum acreage specified for development at site j or the acreage requirement implied by the minimum flow capacity to be provided at the site if developed.) maximum available acreage (thousands of acres) at site j for development of a land disposal facility; aggregate benefits (thousand dollars/acre) in neriod k induced bv the initial canacitv _ _ of the land disposal facility located at site j at the beginning of period t; . benefits (thousand dollars/acre) in period k induced by site expansion which occurred at site j at the beginning of period t; cost (million dollars) of the initial capacity provided at land disposal site j at the beginning of period
ROBERT E.MARKLAND,L. DOUGLAS SMITH and JACK BECKER
cost (thousand dollars/acre) of capacity expansion at site j at the beginning of period t; operating costs (thousand dollars/acre) in period k induced by the initial capacity of a land disposal facility located at site j at the beginning of period t; operating costs (thousand dollars/acre) in period k induced by capacity expansion at site j at the beginning of period t; proportion of secondarily treated wastewater in period t which must be treated further by land disposal; salvage value (thousand dollars/acre) of development at land disposal site j; set of all possible transmission arteries, i-j; set of land disposal sites which can be connected to secondary treatment facility i; set of secondary treatment facilities which can be connected to land disposal site j: total wastewater produced by the region for secondary treatment in period t; discount factor applied to cash flows and valuations; capital available (million dollars) in period t for construction and expansion of land disposals sites; maximum cumulative capital investment (million dollars) in construction and expansion of land disposal sites by period t; upper bound upon acreage expansion (thousands of acres) beyond initial development at site j; proportion of wastewater treated by land disposal at site j in period t which will pollute aquifers; cost (million dollars) of constructing the transmission artery from i + j, during period t; cost (dollars per gallon per period) of wastewater flows from secondary treatment site i to land disposal site j during period t; m~imum amount of ~llution of aquifers (million gallons per day) to be tolerated by effluent from land disposal at site j in period t; total pollution of quifers (million gallons per day) to be tolerated from all land disposal sites in period t; lower bound on the amount (million gallons per day) of wastewater from secondary treatment site i which must be treated by land disposal in period I; m~imum flow (million gallons per day) which can be accommodated by tr~s~ssion artery, i -+ j: initial flow capacity (million gallons per day) of land treatment facility completed at site j period f; cost (dollars per gallon per period) of applying effluent at land disposal site j during period t; flow capacity (million gallons per day/thousand acres) of expansion at site j, period t. Objective function. The objective function for the model can now be stated, as follows: Minimize Net Present Value (NPV) = Initial Capital Investment site,j, period t Capacity Expansion Investment site j, period t + Transportationand Application Costs-Waste Water Movement From Secondary Treatment Facility i to Land Disposal Site j, period f Operating Costs, Initial Capita1 Investment, site j, period t Operating Costs, Capacity Expansion Investment, site j, period t Benefits, Initial Capita1 Investment, site j, period I
Benefits, Capacity Expansion Investment, site j, period t
t Pipeline Construction Costs, i-+j _
Salvage Value, terminus of the project, sitej, period T.
-(I+E)-*
This objective function thus considers all capital investment costs, operating costs, benefits, and the salvage value resulting from the construction of the various land disposal wastewater treatment sites. All costs and benefits are accumulated for both the initial site development and future capacity expansion. Constraints. The objective function presented above is minimized subject to the following sets of constraints. Initial development can occur at each land disposal site only once over the time horizon. Only one facility, per site, over the time horizon (1) j=l ,...,J. Average daily applications of effluent at any land disposal site cannot exceed its absorption capacity (which has been provided by initial development at the site and subsequent acreage expansions). Site capacity constraints for incoming flows
(2)
j= l,...,J
t=
l,...,T.
Computerized regional planning for land disposal of wastewater
13
Acreage expansions cannot occur at a site until initial development has taken place, and aggregate expansion cannot exceed the acreage available for expansion purposes after initial site development (i.e. ~7 = difference between total acreage at a site and the acreage utilized during initial development at the site).
i E; < yy j, *=* j=l
>...1
x; = {‘:
X;
J
t=
I,...,
Capacity upgrading constraints
(3)
T.
Each period, a designated minimum of the etIluent from each secondary treatment facility must be treated further by land application. Lower bound on effluent from each secondary treatment facility to be treated further by land application
c Ffj 2 (7;’ jtJ, i= i,...,I
t= l)...,
(4)
T.
Each period, similar constraints are stated for aggregate regional quantities of secondary effluent to be treated further by land disposal. Regional treatment constraint, each period
cc Fij 2 p’w’ fi,j&S t = l,...,T.
Capital expenditures for initial development or expansion of land disposal sites and construction pipelines in any period cannot exceed funds available.
(5)
of effluent
Period capital constraint
(6)
Cumulative capital expenditures cannot exceed designated limits. (7)
t = l,...,T. Pollution of groundwater at any site cannot exceed specified limits. c Ft 6; < rc; it,, j= l,...,J
Individual site pollution t=l,...,
(8)
T.
Pollution of regional aquifers must be restrained within prescribed limits. EC Ft 6; < n” (i&S
Regional pollution constraint
(9)
t = l,...,T. A ‘balance’ equation defines the amount of acreage developed at a site in terms of initial development and aggregate expansions to date, Acreage development at site j
j= I,...,J
(10)
t = i,...,T,
Acreage development at each site is constrained to available land (incorporated in the model as upper bounds upon A,! and not as explicit constraint rows). A: I a;
j=l
,,.-I J
Maximum acreage for development at site j t=
(11)
1,...,7-.
Daily flows through transmission arteries from secondary treatment facilities to land disposal sites cannot exceed the capacity of the pipelines. Arterial flow constraint
(ii) c S
t= l,...,
(12)
T.
Only one transmission artery (pipeline) can be constructed from each secondary treatment facility to a given land disposal site. 11 Lij I 1 (i&S t= l,...,T.
Lfj = {y
Arterial Construction--only
one artery between points
(13)
14
ROBERTE. MARKLAND. L. DOUGLAS SMITH and JACK BECKER
APPENDIX B. SELECTED PORTIONS OF FROM THE MATRIX GENERATOR
OUTPUT
Vol. 2. pp. l-14.
Pergamon
Press, 1977. Prmted m Great Britain
COMPUTERIZED REGIONAL PLANNING LAND DISPOSAL OF WASTEWATER? ROBERT E. MARKLAND,
L. DOUGLAS SMITH
School of Business Administration,
FOR
and JACK BECKER
University of Missouri, St. Louis, MO, U.S.A.
(Received 13 November 1975)
Abstract-This paper describes the development and utilization of a large scale mixed integer programming model for regional planning of land disposal of wastewater within the St. Louis Metropolitan Statistical area. This model is derived, and used to determine which land disposal sites should serve which treatment plants, when initial construction should be initiated and completed, and when capacity expansion should occur. Consideration is given to relevant construction and operating costs for land sites and transmission arteries, land acquisition costs, tangible benefits from land use, controls on pollution of aquifers, and various other engineering and technical constraints. The computational aspects of the model’s use are emphasized in the paper, and extensive test results are presented and discussed.
INTRODUCTION
and disposal of municipal wastewater is a serious environmental problem currently confronting the people of the United States [l, 21. One dimension of the problem involves the removal of pollutants from liquid effluents generated during wastewater treatment. Land application has recently been advocated as a potentially economical and ecologically beneficial method of reclaiming wastewater [3-73. This paper describes the development of a computerized mixed integer programming model for planning a land application system to dispose of sewage effluent from secondary treatment plants serving a large metropolitan area. Previously [8,9], the authors presented an historical perspective of land disposal methods, a brief description of features of alternative land disposal systems, a general background of the modeling problem for the St. Louis metropolitan area, and the derivation of a mixed integer programming model for selecting land sites to be developed for wastewater disposal. This paper summarizes our findings and emphasizes four aspects of computerized regional planning for land disposal of wastewater: THE TREATMENT
1. 2. 3. 4.
The data base required; Experience in solving the model; Presentation of a sample solution; Considerations in using the model.
The St. Louis Standard Metropolitan Statistical Area provided the setting within which the methodology was applied. Although many of the data sources cited are specific to this region, we do hope to convey a reasonable impression of the efforts required in making such a model operational in a typical urban environment. IDENTIFICATION WASTEWATER
OF POTENTIAL TREATMENT
LAND FACILITY
DISPOSAL SITES
The regional planning problem was approached in two phases. The first phase involved the identification of a set of suitable locations for land disposal wastewater treatment facilities utilizing a land-use forecasting simulation model. t Based on research supported by the US. Department Technology, under Title II Grant 14-31-0001-4221 (C5302). I I’.SS
2 1
A
of the Interior, Office of Water Research and
2
ROBERT E. MARKLAND, L. DOUGLAS SMITHand JACK BECKER
The land-use forecasting model Because of the potential political implications and the general lack of public awareness and understanding of land disposal wastewater treatment, it was considered necessary to avoid land which would be expected to experience concentrated industrial, residential, or commercial development in the future. Potential sites were to be selected from land which was projected to fall into agricultural, recreational (e.g. parks, golf courses), or vacant use categories. Projections of future land use were accomplished with the help of the U.S. Corps of Engineers’ (USCE) Land-Use Forecasting Model, developed for the St. Louis metropolitan area by Meyer et al. [lo, 111. This model forecasts industrial, residential, commercial, and public land use by census tract in ten year increments to the year 2030, for each county in the St. Louis Standard Metropolitan Statistical Area. Use of the model requires an exogenous forecast of county population and employment for each decade in the planning horizon, and specification of current-decade (1970) land use information. The four land uses are forecasted in the above order so that the same parcel of land cannot be allocated to more than a single use. The first allocation is made to industrial use, under the rationale that industry will generally be the first to acquire whatever new land it needs, since it has the economic resources to do so. Residential use is ranked second, followed by commercial use, primarily because the development of commercial areas follows the development of residential areas. Public land use is forecasted last, and represents a residual land use It consists of government facilities, streets, right-of-ways, and other similar uses. The forecasts by decade, by tract, for each of the four land uses are generated in four phases : 1. A desirability ranking on a county-wide basis; 2. Initial allocation to existing land (intensive land use); 3. Allocation to vacant and agricultural land (extensive land use); 4. A combination of intensive-extensive use on a countywide basis. The forecasts are made to distribute residents and employees among designated land units within each county based upon: 1. attractiveness for more intensive usage as measured by the number of residents and employees currently allocated to a land unit, considering also user-specified density limits; 2. attractiveness for more extensive development as measured by characteristics such as access to current and planned transportation arteries and physical features such as lack of severity of terrain. Output from the land use model consists of summary reports from each phase of the forecasting process, and a decade summary by land use for each county by census tract. Final selection-potential
land disposal wastewater treatment sites
Given the forecasts of population distribution and acreage usage for land units within each county, to the year 2000, a consolidated ranking of the land units was produced on the basis of total residential population; industrial, commercial, and public employment; and subjective estimates of the three sectors’ effective resistance to the establishment of land disposal sites in their vicinities. These rankings were subsequently combined with a Sparsity Index which reflected the percentage of ‘available land. The Sparsity Index was defined as the sum of all vacant, recreational, agricultural, and normal use land having five people or less per acre divided by the total acreage of the land unit. The Sparsity Index indicated the lack of development of a land unit, hence reflecting its potential as a land disposal wastewater treatment (LD/WT) site.
Computerized regional planning for land disposal of wastewater
Fig. 1. Secondary treatment facilities and potential land disposal wastewater treatment sites-St. Louis SMSA.
Finally, the more desirable sites according to the Sparsity Index were subjected to further scrutiny on the basis of their soil characteristics (e.g. water capacity, aerobic properties, soil color, and the possibility of groundwater contamination). Final site selection thus involved the consideration of four major factors: 1. The Sparsity Index ; 2. Soil suitability; 3. Special topographical features; 4. Provisions of a sufficient number of potential sites in each county. Areas outlined in black in Fig. 1 contain the primary potential sites, based upon joint consideration of these factors. A REGIONAL
PLANNING MODEL FOR OF WASTEWATER
LAND
DISPOSAL
Given a set of potential sites for land disposal wastewater treatment facilities, the modeling problem becomes that of developing a selection methodology’ for choosing between alternative sites within a regional cost-benefit framework. A large scale, mixedinteger programming model, based on the work of Kiihner and Harrington [12,13], was developed to assimilate the large quantities of relevant information and to enable the development of a regional plan to achieve specified objectives. This model accepts the set of potential land disposal sites and a set of plants planned for the secondary treatment of sewage. It determines which land disposal sites should be developed to serve which treatment plants, when initial construction should be completed, and when capacity expansion should occur. The plan developed by application of the model is one which minimizes the discounted present value of relevant construction and operating costs, net of tangible ongoing benefits and net of the anticipated change in land value at the developed sites. A specified quantity of wastewater from each of the secondary treatment plants in constrained to receive further treatment by land disposal methods.
4
ROBERT E. GARLAND,
L. DOUGLAS SMITH and JACK BECKER
The following factors were considered in the model: 1. Initial costs of constructing the land disposal facilities at the alternative sites; 2. Costs of increasing capacity at alternative sites; 3. Operating costs; 4. Capacity limitations at each site; 5. Acreage r~uirements; 6. External benefits derived from the land disposal facilities; 7. Costs of constructing and maintaining transmission arteries from secondary treatment facilities to land disposal sites; 8. Anticipated long term changes in values of acreage devoted to land treatment facilities; 9. Annual and cumulative budget constraints; 10. Controls on pollution of aquifers. Each of these factors is discussed in detail, in terms of the mathemati~l formulation of the model, which is presented in Appendix A. The mixed integer linear programming model provides an explicit statement of the information required in the development of a comprehensive regional plan for wastewater management. Furthermore, it offers a systematic procedure for selecting the best plan among many alternatives. Experiences in developing the data base and subsequently solving the mixed integer programming model using the MPSX-MIP software on an IBM 370/l@ are related in the following sections. Datu base for the regional planning model Projected locations of secondary treatment facilities and volumes of wastewater to be generated from each plant were obtained from the St. Louis District, U.S. Army Corps of Engineers. This information is presented in Table 1. Next it was decided which land sites could potentially serve which treatment plants, considering the costs of constructing connecting arteries for transmitting the e@uent, and also the anticipated land r~uirements for effluent from each treatment plant. In Fig. 1, iocations of the
Table 1. Forecasted locations and sizes of secondary wastewater treatment facilities Secondary treatment facility site number
Grid cell location JX-16 MX-30 SV-29 NZ45 NZ-51 02-64 NY-51
8 9 10 11
MZ-60 JZ-64 HW-34 MZ-39
12 13
LV-34 JY-11
Facility name
Population served (1974)
Actual usage-1974 (MGD)
Potential usage-2020 (MGD) 19.0 26.0 4.1 46.7 214.3 84.1 136.7
East St. Louis Grantie City Alton Bissel Point Treatment Plant (MSD) Coldwater Creek (MSD) St. Charles
61,200 40,400 40,000 587,000
9.6 3.0 161.0
173,600 19,000
21.0 2.0
Lemay Treatment Plant (MSD)
608.244
91.4
Suggested land disposal areas for the secondary treatment facilities: Jefferson County 1, 11, 12,13 St. Clair County 273 Madison and St. Clair Counties 4,5.6 S.W. Franklin County 7, 8, 10 9 N. St. Charles County. SKJRCE: St.Louis District. U.S. Army Corps of Engineers.
99.4 34.2 6.5 172.7 13.1 2.6
Computerized Table
Land disposal site
Secondary treatment facilities served
regional 2. Final
Population served
planning
for land disposal
land disposal
1974 requirement (MC=)
treatment
2020 requirement (MGD) 265.1
1
3,4, 5
101.200+
2 3 4
X6 2. 4
80,000 61,200+ 587.000
12.6 46.7+ 161.0
298.4 72.7 136.7
5
1, 8
760.600
182.0
236.1
6 I 8 9 10 11 12 13
7, 8 9 9 10 11,12 2, 11 2, 3, 11 10,12
760,600 19,000 19,000
182.0 2.0 2.0
236.1 34.2 34.2 6.5 185.8 198.7 202.8 19.6
14 15 16 17 18
1, 10, 13 1, 13 1, 10 10 11.12
608,244 +
19
7, 8
760,600
608,244 + 608,244+ 608,244 +
91.4+ 91.4+ 91.4+
91.4+ 182.0
28.1 21.6 25.5 6.5 185.5 236.1
of wastewater sites
Land units
Available land/unit (acres)
340 394 397 398 449 349 1 5 6 268 272 213 214 274 281 288 130 324 329 141 178 312 322 303 300 117 125 57
24,395 43,999 25,318 22,528 34,435 16,933 834 493 654 10,248 10,872 9523 58,844 58,844 38,840 64,629 6799 46,730 81,209 26,884 35,158 39,412 57,976 48,280 73,841 265 174 1459
Total land (acres) 116,240
34,435 16,933 1981
30,643
58,844 58,844 38,840 64,629 6799 46,730 81,209 62.042 62,042 39,412 57,976 48,280 73,841 439 1459
secondary treatment facilities are shown with respect to the potential land disposal sites. Possible linkages between the secondary treatment facilities and land sites are indicated by arrows. In Table 2, below, the final potential land disposal wastewater treatment sites, and their associated linkages are summarized. Note that 19 potential sites were considered, with most of these 19 sites involving a combination of the land units shown earlier in Fig. 1. Having developed the set of 19 land disposal wastewater treatment sites to be employed in the model, the data base for the model was constructed. This data base was composed of 38 data items, collected from various governmental, academic, and consulting firm sources. These sources are identified in the technical report by Markland et al. [3]. Matrix generation The contents of the data base were next employed as input to a matrix generator program. This matrix generator program written in FORTRAN IV, derived the coefficients for the objective function and the constraint set of the mixed integer programming model. Portions of the data summary from the matrix generator are provided in Appendix B. A brief discussion of the relation of the items in the data base (Appendix B) to the model formulation (Appendix A) follows. The number of possible land disposal sites, secondary treatment sites, and time periods determine the number of variables and constraints to be generated. The discount percentage is applied to the net value of all cash flows registered in the objective function. Indices of operating and construction costs are used to produce cost estimates for future cash flow constraints C(6) and (7) in Appendix A] and objective function coefficients. Secondary treatment facility data, including effluent volumes and percentage of effluent to be treated are incorporated into the treatment constraints [(4) and (5) in Appendix
6
ROBERT E. MARKLAND,L. D~UCLAS SMITHand JACK BECKER
A]. Budget data appear in the capital constraints [(6) and (7) in Appendix A]. Transmission artery cost data are utilized in the objective function, and in the capital constraints C(6) and (7) in Appendix A]; flow capacities are utilized in the arterial flow constraints [(12) in Appendix A]. Land disposal site data appear next. Consolidated cost coefficients for the objective function and capital constraints were derived for both initial development and expansion at a site, using the detailed components listed. The number of irrigating days per year and application rate per day were considered with the data for storage impoundments in deriving required lagoon acreage and then total acres required per million gallons per day at each site. These latter quantities are incorporated as coefficients in the land site capacity constraints. Estimates of the value of developed acreage at the end of the planning horizon are used to adjust the objective function coefficients for variables representing initial development (Xi) and expansion (Ei) at a land site. Aquifer absorption rates, the last entries among the land disposal site data, appear as coefficients in constraints upon pollution [(S) and (9) in Appendix A]. Benefits in period following development at a site are used to adjust objective function coefficients for land site development variables Xi and Eg. Projected land site value indices are reflected in both the cash flow constraints [(S) and (9) in Appendix A] and in the objective function for X; and Ef. Finally, the environmental protection data provide coefficients for the right hand sides of the pollution limit constraints [(S) and (9) in Appendix A]. One of the main difficulties encountered in compiling the data base was the separation of aggregated data into appropriate components. For the model, it is necessary to separate costs of facilities into fixed capital expenditures, variable capital expenditures, operating costs dependent upon the total acreage developed at a site, and operating costs dependent upon the volume of water processed. In contrast, published cost data was typically derived from examining total costs in connection with facilities which were designed to specific performance specifications. Naive ratios such as costs in dollars per million gallons per day or costs in dollars per acre had been calculated simply by taking total costs over the appropriate denominator. It is clear that improved methods in reporting the basis of cost estimates are required if planning for facilities is to proceed in an organized and efficient manner. One advantage of a formal decision model is that it encourages the development of standardized procedures for collecting and sharing relevant data. Solving the model The computerized version of the model exhibited approximately 1300 rows and 1300 structural variables, with matrix density 0.51%. The basic formulation contained 550 O-1 integer variables: 360 t&l variables for (36 possible 190 &l variables for (19 possible
construction of transmission arteries arteries x 10 time periods) initial site development sites x 10 time periods)
Steps in the solution process are represented schematically in Fig. 2 along with the files utilized during the solution exercise. By saving the optimal bases from previous runs, new continuous solutions for a revised version of the model (relaxing integer constraints) were obtainable in less than 2 min of CPU time, using the MPSX-MIP software on an IBM 370/168. Obtaining suitable integer feasible solutions, however, required considerably more computational effort. Solution effort is, of course, a function of the number of integer variables. Therefore, before undertaking the search for integer feasible solutions, the 550 O-l integer variables were examined to determine whether any could be treated as continous variables without rendering the solution unusable in a practical sense. Fractional values for Lfj, the construction of a transmission artery from plant i to land site j would be absolutely unacceptable in a final solution. Hence
Computerized regional planning for land disposal of wastewater
7
Basis fromwntinuous solution
Fig. 2 User files utilized in MPSX(LP) and MPSX(MIP) phases of solution process.
it was necessary to leave the constraints L:j = (0,l) intact. Xf = (O,l), however, could be relaxed to 0 I Xl I if the value of X: were redefined as the proportion of the maximum acreage available at site j which is to be developed during period t. X; relates to the proportion of maximum acreage available rather than the proportion of ‘initial development’ acreage because relaxation of the integer requirements on Xf enables site expansion E; to occur simultaneously whereas in the original formulation, E; would be forced to have a value of zero until initial site development is completed. This, in turn, implies that the version of the model with Xi = (OJ) relaxed to 0 I X: I 1 will allow the purely fixed costs in connection with initial site development to be delayed or even partially avoided for those land sites not completely developed. Fortunately, the fixed costs (unrelated to the amount of acreage to be developed) were overshadowed by land purchase cost and other variable costs. The resulting favorable cost variance was approximately 1% of the cost of initial acreage development. Thus, economic distortions due to the relaxations were not considered serious. The initial acreage to be developed was rather arbitrarily determined considering the anticipated application capacity which might be needed to justify construction at the site, and also the minimum size of land plot which might reasonably be involved in a real estate transaction. These arbitrary considerations could be waived provided the non-zero values of XJ in the final solution were sufficiently large to make the implied real estate transaction and extent of site development appear reasonable when reconsidered. The noninteger values for Xf were indeed of sufficient magnitude in the integer feasible solutions generated (i.e. solutions with integer values for the Lij variables) that further exploration to force Xf values to be integer was not deemed necessary. To obtain the first integer feasible solutions for the remaining 360 L~j variables required approximately 50 min of CPU time, working in 10 min intervals to facilitate the file management function. The maximum potential improvement in the objective function achievable from further exploration beyond the first integer feasible solution
8
ROBERT E. MARKLAND,L.
DOUGLAS SMITH and JACK BECKER
obtained was indicated to be about 30%. The bound was not judged to be very tight, however, and the solution developed showed no obvious potential for significant improvement. Further computation was therefore not undertaken beyond the first integer solutions for particular versions of the model. Table 3(a, b, c, d). A condensed sample solution to the mode1
(4
Time period 1975580
198&35
202tk25
Land disposal wastewater treatment sites*
Total acreage developed (cumulative)
Secondary treatment facilities* served
1 2 5 8 10 12 13 14 16 17
494 2124 9666 273 523 189 61 19 138 34
3,4 5,6 7, 8 9 11 2 12 13 1 10
1 2 5 8 10 12 13 14 16 17
986 5353 12,926 595 5219 3862 182 51 375 91
3,4 5,6 7, 8 9 11 11, 2 12 13
1 2 4 5 6 8 10 12 13 14 16 17 19
40,29 1 34,440 1980 29,968 18,464 7572 5219 52,946 2803 788 5758 1547 1459
3,4, 5 5, 6 7
Total effluent treated (MGD) (cumulative) 2.01 8.23 37.48 1.04 19.91 0.52 0.26 0.05 0.38 0.13 3.82 20.77 50.13 2.28 19.91 10.51 0.71 0.14 1.03 0.35
1
10
7, 8 I
9 11 11, 2 12 13 1 10 I
-
42.16 133.57 3.93 116.28 71.61 29.00 19.91 145.02 10.87 2.16 15.17 6.00 2.98
* Refer to Fig. 1. (b) Land site development summary Percentage of total available land developed (over time horizon) Land site No. 1 2 4 5 6 8 10 12 13 14 16 17 19
34.67 100.00 100.00 97.81 31.38 19.50 16.76 65.20 4.52 2.00 11.93 2.10 lOQ.00
Period capita1 expenditures ($ millions)
Cumulative capita1 expenditures (8 millions)
$656.8
$656.8
$210.2
$867.0
$800.0
$5064.6
9
Computerized regional planning for land disposal of wastewater (c) 2020-2025
1975-80
Artery
2
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Time period
From secondary treatment site
To land disposal site
1 1 1 2 2 2 3 3 4 4 5 5 6 7 7 I 7 8 8 8 9 9 10 10 10 10 10 11 11 11 11 12 12 12 13 13
14 15 16 3 11 12 1 12 1 3 1 2 2 4 5 6 19 5 6 19 7 8 9 13 14 16 17 10 11 12 18 10 13 18 14 15
-
0.00 0.00 0.38 0.00 0.00 0.52 0.08 0.00 1.83 0.00 0.00 6.01 2.22 0.00 31.72 0.00 0.00 5.76 0.00 0.00 0.00 1.04 0.00 0.00 0.00 0.00 0.13 19.91 0.00 0.00 0.00 0.00 0.26 0.00 0.05 0.00
Unused capacity (MGD)
Arterial flow
0.00 0.00 15.62 0.00 0.00 21.48 3.92 0.00 38.17 0.00 0.00 114.49 67.78 0.00 75.48 0.00 0.00 77.23 0.00 0.00 0.00 27.96 0.00 0.00 0.00 0.00 5.87 6.09 0.00 0.00 0.00 0.00 10.74 0.00 2.25 0.00
0.00 0.00 15.77 0.00 0.00 21.58 3.40 0.00 38.76 0.00 114.10 63.71 69.80 3.93 33.23 71.61 2.98 83.00 0.00 0.00 0.00 29.00 0.00 0.00 0.00 0.00 6.00 19.91 0.00 23.44 0.00 0.00 10.87 0.00 2.16 0.00
WGD)
...’
Unused capacity WGD) 0.00 0.00 0.23 0.00 0.00 0.42 0.60 0.00 1.24 0.00 65.90 56.73 0.20 2.87 73.97 42.39 2.62 0.00 0.00 0.00 0.00 0.00 OSKI 0.00 0.00 0.00
0.00 6.09 0.00 20.57 0.00 0.00 0.13 0.00 0.14 0.00
(MGD)
Regional % of effluent treated
Regional total of effluents absorbed by aquifers (MGD)
Regional % of effluents absorbed by aquifers
69.93 109.70 157.39 213.02 276.70 348.05 427.48 514.85 610.07 713.30
>20 >27 >34 241 >48 >55 262 >69 176 >83
3.50 5.48 7.87 10.65 13.83 17.40 21.37 25.74 30.50 35.66
5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00
Regional total of effluent treated
1. 1975-80 2. lY8&85 3. 1985-90 4. 1990-95 5. 1995-2000 6.200@05 7. 2005-10 8. 2Ol(tl5 9. 2015-20 10. 202G25
Arterial flow WGD)
Illustration of a solution Information conveyed by a solution to the model is illustrated in Table 3. The first section indicates the proposed development of land disposal sites by period, cumulative acreage developed, the secondary treatment facilities which they serve, the amount of effluent treated at each, and the capital expenditures involved. The next section shows the extent to which each site is developed by the end of the 50-year planning horizon.
1.0
ROBERTE. MARKLAND, L. D~UCLAS SMITH and JACK BECKER
Then a summary of arterial flows and unused arterial capacities is provided for each period. Finally, regional totals of secondary effluent to receive tertiary treatment by land application, are provided for each period in the planning horizon. The net present value of the costs associated with the plan for the 50 year time horizon was estimated to be approximately 800 million dollars. A view toward implementation of the planning system A well-documented, user-oriented matrix generator facilitates reconstructing the model to reflect different assumptions about values of the model’s parameters. Only a few minutes are required for the user to change the data deck to make changes in expected costs of capital construction, operating costs, land values, EPA treatment requirements, maximum application rates, etc. For example, a computer run was performed to illustrate the costs of imposing uniform standards upon treatment of effluent from each secondary treatment facility. It was demonstrated that a given level of tertiary treatment for the region could be acquired much more economically by concentrating tertiary treatment at some facilities, and relaxing standards at others. Communications with personnel from the U.S. Army Corps of Engineers suggest that the model is a promising tool for regional planning. Some questions remain about whether all relevant technical information can be consolidated in such parameters as the proposed application rate at each site. Additional constraints may be required to consider specific aspects of the biological and chemical processes at the sites. Also, in spite of the fact that spray irrigation has been suggested as the most appropriate land application process for the Greater St. Louis area, the model may easily be expanded to include other land application processes such as overland runoff and infiltration-percolation. In addition, changes could be made to incorporate other types of processes such as advanced biological and chemical treatment. Among the benefits associated with the modeling approach described in this paper is that it provides a rational, rather than political, approach to decision making. Thus, it provides a means of structuring a massive amount of data into a very compact and workable form. All projects are considered on a relative basis, and an exhaustive series of separate analyses are not required. Furthermore, this approach is very comprehensive and facilitates an analysis which encompasses a long time horizon (50 years). The model offers the user extreme flexibility, in terms of making changes to the input data, or in terms of considering alternative sites. Finally, the methodology developed and tested in this paper could be used in other regions. A number of limitations to the model should also be noted. At present, the model is specific to the St. Louis metropolitan region, particularly with respect to secondary treatment facilities and pipeline configurations. The model obviously requires a large amount of derived data input. Some of this information is subjective; other information requires considerable analysis for its determination. Determination of the input information becomes more difficult as the time frame is lengthened. Solutions to this type of model are difficult, time consuming and hence, costly, to obtain. Indeed, pure optimality was not achieved, and efforts to develop efficient heuristic solution procedures would probably be worthwhile. Implementation of the results of this modeling effort would probably require further analysis on a site-by-site basis. As this occurred, the political ramifications of the work would have to be dealt with. The present solution to the model suggests that the majority of land disposal wastewater treatment sites would initially be located in areas that are sparsely populated. This would probably serve to soften public and community resistance towards this approach to wastewater treatment. CONCLUSION
This paper has discussed the use of a large scale mixed integer programming model to facilitate regional planning for land disposal of sewage effluent in an urban environment. Consideration was given to the data base required, identification of potential
Computerized
regional planning for land disposal of wastewater
I1
land disposal sites, experience in generating solutions to the model, and use of the model in developing alternative plans in response to different possible legislative constraints. Future work will investigate alternative approaches to the mixed integer (MIP) phase of the solution process. REFERENCES 1. Proceedings ofthe Joint Conference on Recycling Municipal Sludges and EfJluents on Land, Washington, DC., National Association of State Universities and Laud-Grant Colleges, p. vii (1973). 2. Metcalf and Eddy, Inc. ~~r~a~er ~~~neer~ng, McGraw-Hill, New York (1972). 3. Charles E. Pound and Ronald W. Crites, Wast~ater ~reat~nt and Reuse By Land Application, Environmental Protection Agency Report 6~/~-73-~b, p. 9. U.S. Environmental Protection Agency, Washington, D.C., August (1973). 4. Richard H. Sullivan, Morris M. Cohn and Samuel S. Baxter, Survey of Facilities Using Land Application of Wastewater, Environmental Protection Agency Report 430/9-73-G%, U.S. Environmental Protection Agency, Washington, D.C., July (1973). 5. A&essment of the Efictiveness-and
Effects
of Land
Disposal Methodologies
of
Wastewater Management,
Final Renort For Contract Nos. DACW 73-73-C-0@41-0@43,Office of the Chief of Engineers, U.S. Army Corps Engineers,Washington, D.C., January 14 (1972). 6. Wast~ater management By Disposal on Land, Cold Regions Research and Engineering Laboratory, U.S. Army Corps of Engineers, Hanover, NH, May (1972). 7. lmp~icaiions of Mu~t~bjective Plan Forn?ulation and EL~luation of Regional W~t~ater Ma~gement Systems, Contract No. DACW 73-72-C-0045, IRP 71-16, U.S. Army Corps of Engineers, Washington, D.C., February (1973). 8. Robert E. Markland, L. Douglas Smith, Jack D. Becker and Paul J. Haack, A Site Selection Modet ,for Land Disposal ofWastewater, working paper presented at the Chicago Joint Meeting of the Operations Research Society of America and The Institute of Management Sciences (1975). 9. Robert E. Markland, L. Douglas Smith and Jack D. Becker, Regional ‘Planning for Land Disposal of Wastewater Using Mixed integer Programming, working paper presented at the Las Vegas Joint National Meeting of the Operations Research Society of America and The Institute of M~agem~t Sciences (1975). 10. Carl F. Meyer, Andre B. Corbeau and Harold L. Mack, A water policy-land use computer simulation model, Wat. Resour. &II. 10, 952-968 (1974). 11. Carl F. Meyer, Andre B. Corbeau and Harold L. Mack, A computer-oriented land use forecasting model with mapping capability, Comput. & Urban Sot. 1, 3148 (1975). 12. Jochen Kiihner and Joseph J. Harrington, Mathematical models for developing regional solid waste management politics, J. Engng Optimization 2, (1974). 13. Jochen Kiihner and Joseph J. Harrington, Lurge Scale Mixed Integer Programming For Investigating
of
Multi-Party Public Investment Decisions: Application to a Regional Solid Waste Management
Problem,
working paper presented at the 45th National ORSA,TIMS Meeting, Boston, MA, April (19’74). 14. Robert E. Markland, L. Douglas Smith and Jack D. Becker, A Benefit-Cost Analysis of Alternative Land Disposal Waste Water ~et~luds in an Urban ~n~iro~ent, Final Technical Completion Report For Contract No. USDI 1~31-~1-4221 United States Department of the Interior, OlIice of Water Resources Research, December (1974). APPENDIX
A. MATHEMATICAL OF THE MODEL
FORMULATION
Variables used in the model. The variables used in the model are defined as follows: A; Eij Fij Lfj Xi
total acreage developed (thousands of acres) for land disposal at wastewater treatment site j by the end of period t; amount ofexpansion(thousands of acres) provided at the land disposal facility on site j at the beginning of period r; flow of wastewater (million gallons per day) from secondary treatment site i to land disposal site j during period f; bivalent (0,l) variable corresponding to completing construction of the transmission artery from treatment site ‘i to land disposal site j at the beginning of period t; bivalent (0,l) variable corresponding to completing construction of a land disposal facility on site j at the beginning of period t.
Parameters employed in the model. The following parameters
af
a,” b6
b”* I e;
were employed:
minimum acreage (lower bound) required (thousands of acres), including storage impoundment, for land disposal if site j is developed; (Note that ai is obtained by choosing the maximum of the minimum acreage specified for development at site j or the acreage requirement implied by the minimum flow capacity to be provided at the site if developed.) maximum available acreage (thousands of acres) at site j for development of a land disposal facility; aggregate benefits (thousand dollars/acre) in neriod k induced bv the initial canacitv _ _ of the land disposal facility located at site j at the beginning of period t; . benefits (thousand dollars/acre) in period k induced by site expansion which occurred at site j at the beginning of period t; cost (million dollars) of the initial capacity provided at land disposal site j at the beginning of period
ROBERT E.MARKLAND,L. DOUGLAS SMITH and JACK BECKER
cost (thousand dollars/acre) of capacity expansion at site j at the beginning of period t; operating costs (thousand dollars/acre) in period k induced by the initial capacity of a land disposal facility located at site j at the beginning of period t; operating costs (thousand dollars/acre) in period k induced by capacity expansion at site j at the beginning of period t; proportion of secondarily treated wastewater in period t which must be treated further by land disposal; salvage value (thousand dollars/acre) of development at land disposal site j; set of all possible transmission arteries, i-j; set of land disposal sites which can be connected to secondary treatment facility i; set of secondary treatment facilities which can be connected to land disposal site j: total wastewater produced by the region for secondary treatment in period t; discount factor applied to cash flows and valuations; capital available (million dollars) in period t for construction and expansion of land disposals sites; maximum cumulative capital investment (million dollars) in construction and expansion of land disposal sites by period t; upper bound upon acreage expansion (thousands of acres) beyond initial development at site j; proportion of wastewater treated by land disposal at site j in period t which will pollute aquifers; cost (million dollars) of constructing the transmission artery from i + j, during period t; cost (dollars per gallon per period) of wastewater flows from secondary treatment site i to land disposal site j during period t; m~imum amount of ~llution of aquifers (million gallons per day) to be tolerated by effluent from land disposal at site j in period t; total pollution of quifers (million gallons per day) to be tolerated from all land disposal sites in period t; lower bound on the amount (million gallons per day) of wastewater from secondary treatment site i which must be treated by land disposal in period I; m~imum flow (million gallons per day) which can be accommodated by tr~s~ssion artery, i -+ j: initial flow capacity (million gallons per day) of land treatment facility completed at site j period f; cost (dollars per gallon per period) of applying effluent at land disposal site j during period t; flow capacity (million gallons per day/thousand acres) of expansion at site j, period t. Objective function. The objective function for the model can now be stated, as follows: Minimize Net Present Value (NPV) = Initial Capital Investment site,j, period t Capacity Expansion Investment site j, period t + Transportationand Application Costs-Waste Water Movement From Secondary Treatment Facility i to Land Disposal Site j, period f Operating Costs, Initial Capita1 Investment, site j, period t Operating Costs, Capacity Expansion Investment, site j, period t Benefits, Initial Capita1 Investment, site j, period I
Benefits, Capacity Expansion Investment, site j, period t
t Pipeline Construction Costs, i-+j _
Salvage Value, terminus of the project, sitej, period T.
-(I+E)-*
This objective function thus considers all capital investment costs, operating costs, benefits, and the salvage value resulting from the construction of the various land disposal wastewater treatment sites. All costs and benefits are accumulated for both the initial site development and future capacity expansion. Constraints. The objective function presented above is minimized subject to the following sets of constraints. Initial development can occur at each land disposal site only once over the time horizon. Only one facility, per site, over the time horizon (1) j=l ,...,J. Average daily applications of effluent at any land disposal site cannot exceed its absorption capacity (which has been provided by initial development at the site and subsequent acreage expansions). Site capacity constraints for incoming flows
(2)
j= l,...,J
t=
l,...,T.
Computerized regional planning for land disposal of wastewater
13
Acreage expansions cannot occur at a site until initial development has taken place, and aggregate expansion cannot exceed the acreage available for expansion purposes after initial site development (i.e. ~7 = difference between total acreage at a site and the acreage utilized during initial development at the site).
i E; < yy j, *=* j=l
>...1
x; = {‘:
X;
J
t=
I,...,
Capacity upgrading constraints
(3)
T.
Each period, a designated minimum of the etIluent from each secondary treatment facility must be treated further by land application. Lower bound on effluent from each secondary treatment facility to be treated further by land application
c Ffj 2 (7;’ jtJ, i= i,...,I
t= l)...,
(4)
T.
Each period, similar constraints are stated for aggregate regional quantities of secondary effluent to be treated further by land disposal. Regional treatment constraint, each period
cc Fij 2 p’w’ fi,j&S t = l,...,T.
Capital expenditures for initial development or expansion of land disposal sites and construction pipelines in any period cannot exceed funds available.
(5)
of effluent
Period capital constraint
(6)
Cumulative capital expenditures cannot exceed designated limits. (7)
t = l,...,T. Pollution of groundwater at any site cannot exceed specified limits. c Ft 6; < rc; it,, j= l,...,J
Individual site pollution t=l,...,
(8)
T.
Pollution of regional aquifers must be restrained within prescribed limits. EC Ft 6; < n” (i&S
Regional pollution constraint
(9)
t = l,...,T. A ‘balance’ equation defines the amount of acreage developed at a site in terms of initial development and aggregate expansions to date, Acreage development at site j
j= I,...,J
(10)
t = i,...,T,
Acreage development at each site is constrained to available land (incorporated in the model as upper bounds upon A,! and not as explicit constraint rows). A: I a;
j=l
,,.-I J
Maximum acreage for development at site j t=
(11)
1,...,7-.
Daily flows through transmission arteries from secondary treatment facilities to land disposal sites cannot exceed the capacity of the pipelines. Arterial flow constraint
(ii) c S
t= l,...,
(12)
T.
Only one transmission artery (pipeline) can be constructed from each secondary treatment facility to a given land disposal site. 11 Lij I 1 (i&S t= l,...,T.
Lfj = {y
Arterial Construction--only
one artery between points
(13)
14
ROBERTE. MARKLAND. L. DOUGLAS SMITH and JACK BECKER
APPENDIX B. SELECTED PORTIONS OF FROM THE MATRIX GENERATOR
OUTPUT