hierarchical modeling of regional total water resources systems

Automatica, Vol. 11, pp. 25-36. Pergamon Press, 1975. Printed in Great Britain

Hierarchical Modeling of Regional Total Water Resources Systems*

Modelage Hi6rarchique de Syst~mes R6gionaux des Ressources Totales en Eau

Hierarchische Modellierung der regionalen Wassergesamtreserven H e p a p x a q e c K o e Mo,aeyi r lpoBaHHe C y M M a p H o f i CttCTeMbI BO~IttbIX p e c y p c o a p e r n o H a

YACOV Y. H A I M E S t

The planning and management of regional water resources systems can be advantageously analysed by a hierarchy of coordinated models in recognition of the various associated noncommensurable and often conflicting goals and objectives.

Summary--The importance of modeling for the planning and management of regional total water resources systems is discussed. Due to the complexity and the high dimensionality of the resulting regional planning and management problem, hierarchical modeling is proposed where a decomposition and multilevel optimization approach is utilized. Four major water resource systems descriptions (decompositions) are discussed, namely: temporal, physical- hydrological, political-geographical, and goal or functional description. These four decompositions are demonstrated in two hierarchical structures. The first hierarchical structure is composed of a threedevel dynamic programming model aimed at the planning and management of agriculture systems. The second two-layer hierarchical structure is aimed at the management of water and related land resources for pollution control and ecolibrium. Higher level coordination procedures are presented and analyzed. In particular, the applications of the Surrogate Worth Trade-off (SWT) method, which is capable of analyzing multiple non-commensurable objective functions, as a higher level co-ordinator is discussed. This paper also presents a tutorial summary of water resource problems where hierarchical modeling can be utilized for the analysis and solution of complex water resources systems.

1. INTRODUCTION

THE increase of human population around the world has caused a critical increase in the water demand for domestic, industrial, agricultural and other uses. In addition, the acceleration in human activities of land use, resulting in the disruption of the natural hydrologic cycle and the self-purifica- tion process, have added a new dimension to the global problem of water supply and water quality. This new dimension has become known as the environmental crisis. It is a well-established thesis

* Received 6 September 1973; revised 27 February 1974; revised 2 July 1974. The original version of this paper was presented at the IFAC Symposium on Control of Water Resource Systems which was held in Haifa, Israel, during September 1973. It was recommended for publication in revised form by associate editor A. Sage.

t Associate Professor of Engineering, Systems Engineer- ing Department, Case Institute of Technology, Case Western Reserve University, Cleveland, Ohio, USA.

that, in order to restore the natural balance of our environmental system with respect to the water resources and related land use, one should carefully study and analyze the environmental impacts of any demand placed on the natural water and related land resources system. Consequently, a relatively new concept of relating water quantity, water quality and related land use has evolved during the last decade.

The fact that water resource projects are planned, designed, constructed, operated and expanded on a multi-purpose basis adds to the complexity of the problem. Consider the impoundment of a reservoir by one or more dams. The resulting reservoir may serve a variety of purposes, each of which alone may not fully justify the investment in the project (reservoir), but all together may make the project economically, and otherwise, feasible and tractable. For example, a reservoir with a high dam may generate hydroelectric power, supply water for municipal, industrial and agricultural use, provide ways and means for flood control, for proper flow of aquatic life, provide for water quality control, navigation, fishing and recreation to nearby streams, and provide a buffer for drought years and groundwater recharge, etc.

The planning horizon of water resource systems spans 30-50 yr. Most studies today address them- selves to the needs of the year 2020. Consequently, added to the complexity of the planning analysis are many time-dependent variables such as changes in water demands, supplies, levels of regional and national economic activities including cost of labor, material and resources, developments in science and technology, water quality standards and style of living, etc. The myth that all water demand has to be satisfied (supplied) at any cost at all times cannot be justified, cf. National Water Commission Report [56]. Thus concepts from dynamic cost-benefit

25

26 YACOV Y. HA1MES

analysis are useful in water resources planning, see Howe [32], Nainis [50], Haimes and Nainis [27], Nainis and Haimes [51].

The modeling of large-scale and comprehensive problems, which eventually yield efficient overall solution strategies, is obviously an intricate task. It is the availability of the hierarchical multi-level approach, however, which enables us to model and study small sub-systems, the local goals or objectives, at the lower level to yield ultimately an optimal solution to the system as a whole, that makes this task both feasible and tractable. Funda- mental to the hierarchical modeling of a total water resources system is the development, as well as the utilization and the co-ordination, of many of the existing models reported in the literature.

The objectives of this paper are to present: (I) a tutorial summary of water resource problems where hierarchical modeling can be ut i l izedfor the analysis and solution of complex water resources systems, and (2) a framework of hierarchical multilevel modeling structures by which regional water resources and related land systems can be studied and analyzed from a total point of view and with the considerations of a wide spectrum of non- commensurable objectives, goals and constraints.

2. REGIONAL APPROACH The analysis of water resources systems on a

regional basis has gained strong support in the last decade; in particular, a region is understood to include possibly several river basins. The following elements summarize some of the needs and the importance of a regional approach for the planning and management of water and related land resources:

(i) Jurisdictional power--jurisdictional powers of a small locality will be inadequate fully to implement comprehensive management policies.

(ii) Econom~ dependence---small localities cannot be considered economically independent from surrounding areas nor do they possess tile monetary resources to meet the capital expenditures required by water resource management policies. A local area typically does not have the tax support to finance its own projects and, therefore, must request outside support in the form of bond issues or goverfiment subsidies and grants.

(iii) Economic projections--a more meaningful aggregation of economic data may be possible at the regional level. Input-output models are more relevant at the regional level than at tlle local level.

(iv) Water quantity, quality and land--land and ~ater resources are best managed as a

(v)

(vi)

co-ordinated complex system, where the problems of water quantity, quality and land use are integrated. A regional approach may allow aggregation of required data. Water transfers--a local water demand is often met by water transfer from outside the locality. Therefore, it is impossible to solve the water supply problem at each locality in- dependently. A regional approach may be required to co-ordinate the actions of the various localities. Access to advanced technology--allowances for increased efficiency through use of tech- nological advances made feasible by pooling of funds for research and development expenditures can be made.

3. WHY TOTAL WATER RESOURCES SYSTEMS?

A basic concept being preached to students in the field of system analysis, (systems engineering, opera- tions research, etc.), is that for the analysis of a system to be meaningful, a total consideration of the whole system should be undertaken. This task is relatively easy for small industrial processes where the system boundaries can be well defined and the system can be isolated. On the other hand, when the water supply system to a community is under consideration, for example, the analysis of many elements related to the water supply system is essential. This may include the planning for capacity expansion of the existing water supply system and its network; the projection of future water needs based on expected population growth as well as industrial and agricultural economic growth; the planning for the capacity expansion of existing water and treatment plants and waste-water treatment plants with respect to size, the flow capacity, treatment efficiency, the removal of BOD, phos- phates, total dissolved solids, etc.; and future land use in the region since a considerable portion of pollution contributed to streams is attributed to non-point sources (also known as area sources or distributed sources), which are directly affected by land use.

If the water supplied is also coming from groundwater, a knowledge of the characteristics of the aquifer system is essential. Characteristics such as transmissivity and storage coefficients are critical for the prediction of the aquifer system response to demands placed upon it. In particular, water head projections in time and space can be obtained. The effect of the economic development due to water resources development on the ecosystem should not be overlooked. Also, the environmental impact, such as damaging side-effects, should be considered.

Hierarchical modeling of regional total water resources systems 27

Clearly, the consideration of all the above aspects and many others necessarily involve a large- scale and complex system.

4. MULTIOBJECTIVE FUNCTIONS In studying large-scale systems with a variety of

technical, societal, economic, environmental and other considerations, the usual problems of modeling and optimization become magnified and often overwhelm the analysis. This is due to the resulting models' high dimensionality, a very large number of variables, and complexity, including nonlinearity in the coupling of and interactions among the variables. Certainly most water resource projects are of this nature--it inevitably seems that a multitude of public and private agencies become involved or responsible, both directly and indirectly, for a program's assessment, justification, implementation, performance and ultimate success. In fact, associated with almost any water resource program are a myriad of conflicting, often competing, sometimes complementary and mostly non-commensurable goals and objectives.

It is evident that conflicting objectives and goals are associated with almost all water resources projects. Hydroelectric power generation may conflict with the planning for adequate water supply over a long period, which, in turn, may conflict with providing for flood control capability or with recreation, navigation, low flow augmentation needs, etc. Projects may also differ in the extent to which they satisfy different social welfare goals; for example, a project may provide a high level of regional income but cause extensive environmental degradation and essentially no increase in national income.

A fundamental and almost an axiomatic pre- requisite, then, for water resources models to be realistic, and thus be considered for an ultimate utilization by the decision-makers, is that they include multiple objective functions in their non- commensurable forms and units. Traditionally, of course, engineering planning included technical analysis of the relationship between expected performance, the value of output and monetary outlays. More recently, however, there is increasing recognition of the need to supplement these technical aspects of engineering planning and development with less customary, sometimes unfamiliar, but nonetheless important, 'other assessments'.

The classical notion of a simple scalar objective function, such as minimizing a cost function, cannot be acceptable any more: Optimal solution in the sense of vector optimization, pareto optimum or satisfaction is not only desirable but rather essential, of. Cohon [8], Cohon and Marks [9], Freedman [14], Haimes and Hall [23, 24], Haimes et al. [25], Major [46] and Miller and Byers [48].

The present absence of mathematical models with multiple objective functions can be attributed pri- marily to the past lack of operational methodologies capable of analyzing and optimizing multiple non- commensurable objective functions. The Surrogate Worth Trade-off (SWT) method (Haimes and Hall [23]) can be effectively utilized here. The major characteristics and advantages of the Surrogate Worth Trade-off Method are:

(i) Non-commensurable objective functions can be handled quantitatively.

(ii) The surrogate worth functions, which relate the decision-maker's preferences to the non- inferior solutions through the trade-off functions, are constructed in the functional space and only then are transformed into the decision space.

(iii) The decision-maker interacts with the mathematical model at a general and very moderate level. He expresses his subjective preference in the functional space, which is more familiar and meaningful to him, rather than in the decision space. This is particularly important since the dimensionality of the decision space, N, is generally much larger than the dimensionality of the functional space, n.

(iv) The availability of operational methodologies, such as the SWT method, encourages and enhances the systems modeling and pattern of thinking in multi-objective function terms. Thus, more realistic analyses may result in eliminating the needs for a single objective function formulation.

(v) The SWT method provides a significant contribution in the field of higher level co- ordination in hierarchical multi-level structures.

The SWT method is not currently applicable to dis- continuous functions. This method was further developed by Freedman [15] and Haimes et al. [25] to handle dynamic multiple objective functions.

5. HIERARCHICAL MODELING

Well-documented simulation and mathematical models which are aimed at investigating specific and selected aspects of regional water and related land resources are available in the literature. By the virtue of tackling one part of the problem while assuming knowledge of the others, these models, referred to as submodels hereafter, are often one- sided and usually do not represent the overall system behavior. There are many examples of such submodels in the literature, such as capacity expansion, planning for operation of multi-reservoir systems, identification of watershed and aquifer

28 YACOV Y. HAIMES

parameters, ground and surface water management, water quality management, etc. The reason for the above simplification in system modeling can be explained by the difficulties associated with solving large-scale and complex systems via the conventional systems engineering tools and methodologies, discussed by Hall and Dracup [30], Hufschmidt et al. [33], Isard [35], James and Lee [36], Kneese and Bower [38] and Maass et al. [44] and others. This is particularly true for nonlinear and dynamic water resources systems.

It is proposed in this paper that appropriate submodels be specified and existing submodels be modified and incorporated into the analysis of a complete system of water resources planning. This can be accomplished via a hierarchical multi-level structure which relates and co-ordinates the various submodels and objectives of the complete system. Cost-benefit analysis, cf. English [11], Howe [32] and James and Lee [36], can be utilized as a higher level co-ordination mechanism. Alternatively, multiobjective optimization methodologies, cf. Cohon [8], Haimes and Hall [23], etc., can be utilized for the same purpose. Furthermore, several hierarchical multi-level structures for the planning and management of a regional water resources system can be developed to account for the inter- and intra-system coupling and interaction. Four major descriptions are identified in water resources systems by Haimes and Macko [26]. These are:

(i) Temporal description. (ii) Physical-hydrological description.

(iii) Political-geographical description. (iv) Goal or functional description.

It is possible to incorporate all or several of the above four descriptions of the system through hierarchical modeling and overlapping co-ordination.

5.1. Temporal description A planning time horizon for water supply pro-

jects often spans 30-50 yr. This long-term planning is usually supplemented by an intermediate term of 10-15 yr, often referred to as planning for operation, and by a short term of 2-5 yr. Clearly the short, intermediate and long terms have to be com- patible with each other and, thus, co-ordinated since they relate to the same system. Planning horizons of water resources for crop and related land use can be of the order of 1-2 yr. However, when a crop has been selected and the water for its seasonal growth has been allocated, horizons of decisions with respect to the periodical irrigation within the season are of the order of weeks or days, see Hall and Butcher [29].

5.2. Physical-hydrological description A river basin is, by definition, a hydrologically

self-contained region, separated from adjacent basins by a ridge or other topographical divide. Often a water resources management system covers a region consisting of a complex of several river basins, e.g. the North Atlantic Region (NAR) [52]. Thus, a region can be divided into several subregions, six in the NAR study, the region can be further divided into several river basins, twenty-one river basins in the NAR study, and further into several sub-basins, fifty sub-basins in the NAR study, and even into a smaller sub-basin as was done in the NAR study.

5.3. Political-geographical description A regional water resources system often includes

different local governments such as metropolitan areas, counties and states. Modeling for water resource planning and management may consider the political-geographical description as a criterion for decomposing the regional area into subregions.

5.4. Goal or functional description Most water resources systems have been analyzed

with respect to their economic and functional goals. Various models following this pattern are available in the literature such as demand and supply models, models for hydroelectric power generation, irrigation, industrial and municipal use, recreation, etc.

6. THE HIERARCHICAL MULTI-LEVEL APPROACH

The concept of the multi-level approach is based on the decomposition of large-scale and complex systems and the subsequent modeling of the systems into 'indePendent' subsystems. This decentralized approach, by utilizing the concepts of strata, layers and echelons, enables the system analyst to analyze and comprehend the behavior of the subsystems at the lower level and to transmit the information obtained to fewer subsystems at a higher level, as described by Haimes [16-20], Mesarovic et al. [47] and Lefkowitz [42]. Whenever more decentraliza- tion is needed, the system is further decomposed. This decomposition is accomplished by introducing into the system new variables, which are called pseudo-variables. In general, the pseudo-variables uncouple the joint variables of two or more subsystems at a lower level in the hierarchy. They are then added to the overall system as equality constraints and satisfied at a higher level in the hierarchy via Lagrange multipliers or other co- ordination schemes as indicated by Haimes [18]. Each subsystem is then separately and indepen- dently optimized, with perhaps different optimization techniques being applied, based on the nature


of the subsystem models as well as on the objectives and constraints of the subsystems. This is termed a first level solution.

The subsystems are joined together by coupling variables manipulated at a second or higher level in order to arrive at the optimal solution of the whole system. This is termed the second or higher level solution. One way to achieve subsystem 'indepen- dence' is by first relaxing one or more of the necessary conditions for optimality, and then satisfying these conditions at the second level, cf. Haimes and Macko [26] and Lasdon [40].

Decomposition and multilevel optimization approaches have several significant advantages in the solution of large-scale and complex optimization problems over conventional optimization methods. For example, by decomposing the problem into several subproblems, or subsystems, a con- ceptual simplification of a complex system is achieved. This is especially important for highly coupled systems, where the outputs of one subsystem are the inputs to others. The decomposition yields a reduction in the dimensionality of the problem at hand at the expense of having to solve several subproblems of lower dimensions. This in turn reduces the computational effort involved, such as problem formulation time, programming effort, debugging effort and the number of cards to be punched, etc. With regard to establishing computer time requirements, there is no general rule of thumb as discussed by Haimes et al. [28]. A significant advantage of the multi-level approach is that. none of the system model functions need to be linear, and thus more flexible mathematical models can be constructed to represent the real system. Note that a major short- coming and deficiency of classical systems engineering practices is that they often result in an im- balance between system modeling and system optimization. This is reflected in the vast number of linearized models in the literature that take advantage of the simplex methods and its extensions. By applying the decomposition and multi-level optimization techniques, no such costly sacrifice of realism in modeling is needed, as more representa- tive and sophisticated nonlinear multi-variable dynamic mathematical models can be constructed. Furthermore, interactions among subsystems can be handled since at the lower levels the subsystems' 'independences' are achieved via pseudo-variables.

The above trade-off between system modeling and system optimization is minimized by the applicability of the approach to both static and dynamic systems. Thus the time domain, which plays an important role in water resources systems, need not be imbedded or ignored in the analyses as in the case in static models, e.g. linear programming. Therefore the water resource system can be modeled

by both static algebraic equations and dynamic differential equations. Both centralized and decentralized decision-making processes can be considered via the hierarchical multi-level approach. This is especially important for regional water resources management, including regional water quality control and pollution abatement, cf. Foley and Haimes [14] and Haimes et al. [22], and ground and surface water managements, see Yu and Haimes [58].

In order to illustrate the decomposition and multilevel approach, consider a company with N departments. Each department has a manager who reports to the president of the company. The sole objective of the ith manager is to maximize the performance of his department. The sole objective of the president is to maximize the performance of the whole company by imposing internal prices for transactions among the department products or services. The company's overall optimal policy is thus achieved through an iterative procedure. The separate optimization of each department does not necessarily imply the overall maximization of the company's performance, unless the performance of each department is co-ordinated at a higher level, i.e. by the president.

7. HIERARCHICAL STRUCTURES

Many hierarchical structures of co-ordinated models can be constructed for the purpose of regional planning and management of water and related land resources. These structures may con- sist of several major levels each of which can include several layers of detail. All four descriptions discussed in the previous section can be imbedded in the various levels of the hierarchy.

Two hierarchical structures will be discussed in this section as example structures.

7.1. Multilevel dynamic programming for planning and management of agriculture systems

A three-level hierarchy of models was proposed by Haimes [19] for the purpose of optimal timing and sequencing of project development for water supply, the optimal allocation and scheduling of land, water and funds for crop growth and the optimal timing for irrigation as illustrated in Fig. 1.

It is assumed that an initial estimate of projected water demand, D(t), over a planning time horizon T years is given. The function D(t) will be further modified based on the region's water demand and supply balance. The region R is divided into I subregions. One may identify a subregion by its soil type and consequently may view the region as composed of I different soil types. Let K be the number of seasons in the planning time horizon of

30

T years, J be the number of different candidate types of crops under consideration in region R during the planning time horizon T, Q~ be the number of units of water available for agricultural con- sumption at season k, S k be the amount of funds available for expenditure at season k, and M be the number of water supply projects available for construction in region R to meet future water demand for the planning time horizon T. Any constraints with respect to crop scheduling and production levels, financial funds, water availability, environmental quality levels, land use restrictions and other social, political and legal constraints should be included. Note, however, that some of the above constraints may be considered as decision variables.

L Balance cO~rdinohon mode~ I

i

OptimoJ timingof[ Optimal tithing of ilrlQotion model [ irrigation model tar suosystem • • • for subsystem

! I

YACOV Y. HAIMES

(ii) Optimal scheduling for the construction of water supply projects over the entire planning horizon T, for minimum present-value cost--this objective is considered at the second level of the hierarchy via the 'supply model' shown in Fig. 1. The Butcher- Haimes-Hall [4] model or the Nainis- Haimes [51] model can be utilized for this purpose. Both the above models utilize dynamic programming in searching for the optimal decision.

(iii) Maximize the overall return from crop yield for each season by determining the optimal allocation of water supply, funds and crop type for each subregion in R--this objective is considered at the second level of the hierarchy via the 'demand model', shown in

r.~rd,.~e, Fig. 1, and optimized via a dynamic pro- t gramming algorithm.

(iv) Optimal timing of irrigation for the maxi- s,canal mization of crop yield--this objective is con- level

1 sidered at the first level of the hierarchy. A dynamic programming model developed by

t Hall and Butcher [29] is utilized at this level. F~r,, The Hall-Butcher [29] and Windsor [57] level

I models recognize that plant growth as influenced by the irrigation of a specific crop in a given physical and biological environ- ment is a function of the energy required to extract water from the soil, the level of soil aeration throughout the growing season and the soil moisture and available nutrients. Once the amount of water allocated for crop growth in a season has been determined, the seasonal distribution in weeks of that quantity is very important. Hall and Butcher [29] found that "when soil moisture conditions are allowed to become less than optimum for plant growth, for any reason, a corresponding reduction in crop yield may be expected. Such reductions may exist locally in a field because of a lack of uni- formity of application of the irrigation water. Probably the most important reason for a reduction, from the point of view of water use planning, is a lack of adequate water supply for the entire season. Recent research suggests that the magnitude of losses may depend almost as much on when the soil- moisture deficiency occurs as it does on the total magnitude of the seasonal shortage .... " Since the model developed by Hall and Butcher [29] lends itself to a sequential decision process and Bellman's principle of optimality can be invoked as in Bellman and Dreyfus [3], dynamic programming was utilized for the solution of the problem. A season is divided into several periods and an

FIo. 1. Multilevel dynamic programming structure for planning and Management of agricultural systems,

The overall planning and management multiobjectives can be summarized as follows:

(i) Optimal scale of regional development--this objective is considered at the highest, the third level of the hierarchy along with the economic trade-offs derived from the supply and demand models. The SWT method proposed by Haimes and Hall [23] for the analysis of multi-objective functions in water resources can be readily utilized. At this level, the decision maker can analyze several non-commensurable objectives such as the level of regional development, environmental quality, cost and benefits via the SWT method. The successful applications of the SWT method can be achieved when all the objective functions are expressed as continuous analytical functions. Concepts from utility theory and decision theory can be utilized for the constructions of these functions, cf. Canada [5], Fishburn [13] and Raiffa [53]. As a supplementary and/or alternative approach for co-ordination at the third level, cost-benefit analysis can be utilized to co-ordinate the various decisions derived at lower levels of the hierarchy, as discussed by Haimes [20], Howe [32] and Nainis-Haimes [51 ].


optimal sequential decision is made as to the amount to irrigate a specific crop at each period.

The overall model's co-ordination of the three- level hierarchies can also be achieved via the balance co-ordination principle of Haimes-Macko [26] by utilizing the concepts of duality and shadow prices. Note that a temporal decomposition was imposed on the first level, geographical and physical-hydrological decompositions were imposed over the second level for the "demand model" and, finally, the second level itself was decomposed based on a goal or functional description.

7.2. Modeling and management of water and related land resources for pollution control and ecolibrium

Two major themes have dominated the direction and scope of this model's structure according to Haimes et al. [21], namely:

(i) Distributed pollution sources will play a major and critical role in the ultimate solution of the water pollution problem. Effluent discharges from waste-water treatment plants, industrial or municipal, are termed as point source pollution, otherwise they are termed as distributed source pollution, also known as area source, non-point source, unrecorded source or diffused source. The pollution from these sources is carried to the water body pri- marily by ~urface rurt-off.

(ii) The level of pollutio~ discharged to a water body attributed both to point and distributed sources depends on the economic activity of the region, recreational and esthetic goals, societal preference, as well as on the hydrologic characteristics of the region. A region is being defined to be a composite of several river basins and watersheds.

The above two major themes are modeled by a hierarchical structure consisted of two layers and two levels (see Fig. 2). The "Modified Leontief Input-Output Model" of Leontief [41 ] and Schaake [55] and the "Modified Stanford Streamflow Simulation Model" of Crawford and Linsley [10], Haimes et al. [21], Lombardo and Franz [43] and Ricca [54] constitute the lowest (first) layer. The "Modified Stanford Streamflow Simulation Model" provides for the prediction of the response of the watershed to an input of precipitation on it. The "Modified Leontief Input-Output Model" provides for the prediction of the pollutant levels of each sector of the economy in the region as a linear function of the production level of that sector. The quantity and quality of all effluent discharged, due

to both point and area sources, depend on both the level of economic activity and the hydrological characteristics of the region. Thus the predicted run-off hydrograph generated from one submodel will be superimposed on the predicted pollutant levels generated by the second submodel. This layer serves as an information base for the higher layer where no optimization procedure is required.

ir I Monogement and ceordlnatlo~ mo~d I recur optimization | coordJn4tor I

I i1 \ \ I I coordinator I \ \ I ,//---\\----y\

II ~¢st'Ymodeltm II II 'lcoit'modol II Jl "benefit" model I~cl

Second level

Second

! Fi t i t

FIG. 2. Hierarchical structure for the quality control of water and related land resources.

While Leontief models are linear, they do provide invaluable information on the complex interactions among the various sectors of the economy. The November 1973 issue of the IEEE Transactions on Systems Man and Cybernetics, which is devoted to the applications of Leontief input-output models to environmental and societal problems, clearly demonstrates the strong contribution that can be realized via these models.

The United States Clean Water Act of 1972, as amended, brings to focus the relations between point and distributed pollution sources. As the level of treatment in waste-water treatment plants improves with time, the absolute contribution of water pollution due to distributed sources will become domi- nant.* Furthermore, it can be rightfully argued that the investment of public funds in tertiary treatment is an unjustified waste extravagance unless a simul- taneous reduction of water pollution attributed to distributed sources is drastically reduced as well. This latter argument merits careful economic analyses (cost-benefit analyses) of the total invest- ments of public and private funds for the control and management of water pollution. In particular,

* By 1986 all point sources should have a tertiary treatment level, i.e. almost zero discharge. Also note that billions of dollars have been appropriated by the U.S. Congress and approved for expenditures by the Administra- tion for the construction, expansion and improvement of waste-water treatment plants.

32 YACOV Y. HAIMES

preliminary analyses of the diseconomies of scale of waste-water treatment plants associated with the last 10 per cent of pollutant removal, 90-100 per cent efficiency of pollutant removal, indicates that the marginal cost, S/additional unit of pollutant removal, is prohibitively high. Furthermore, this investment, in compliance with the U.S. Clean Water Act of 1972 as amended, cannot be technic- ally, economically, socially and politically justified if an equivalent effort for the abatement and control of pollution from distributed sources does not take place. This analysis is carried on at the second layer of the hierarchical structure shown in Fig. 2.

The second layer constitutes two levels. The first level is composed of three submodels, namely:

(i) Point source capacity expansion 'cost' model. (ii) Distributed source 'cost' model.

(iii) Ecosystem response 'benefit' model.

(i) Model 1. In this model the operational cost of the treatment plants is assumed to be negligible in comparison to the capital cost. Biological waste- water treatment plants can be included in this category.

The basic scheduling model for capacity expansion of waste-water treatment plants is presented by the general recursive dynamic programming relationship as follows

fnk"(q) = min [gi(qn) (1 + r)-¢ tq-q.)

+fn-lk"-'(q --qn)],

n

O<~qn <~q<~ Y~Q~, i = l

i = 1,2, . . . ,n but i~k~_l,

kn(q) = imin(q~ ® kn-l(q)

Submodels (i)-(ii) can be viewed as 'cost' models-- they provide the cost of point source and distributed source pollution control respectively. Submodel (iii) can be viewed as the 'benefit' model--it provides the benefit (improvement) of the ecosystem due to improvement in the control of all sources of pollution.

The second level in this hierarchy constitutes the management and co-ordination submodel, which respectively co-ordinates the 'cost' and 'benefit'. The second layer is hereby explained briefly.

where f,k,(q) is the minimum cost of treating waste water of quantity q at a specified level of pollutant removal with a sequence of n projects, kn; gi(qn) is the cost of treating waste water of quantity (qn) with project i; r is the interest rate; ~b(.) is the inverse time function of waste-water treatment demand; kn(q) is the sequence of n projects to meet the demand level q; Qi is the capacity of project i and @ means append to the sequence k,~_l(q). The above model is based on the Butcher-Haimes-Hall [4] model.

7.2.1. Point source capacity expansion "cost' model The planning for the scheduling for capacity expansion of waste-water treatment plants is, of course, important. Scheduling models for capacity expansion, which take into account the dynamics of the system as well as the capital cost and variable per unit cost as developed by Butcher- Haimes-Hall [4], Kolo-Haimes [39], Kaplan [37], Nainis [50], Nainis-Haimes [51], Muhich [49], Erlenkotter [12] and Cardenas [6] are readily available and can be utilized in this analysis. These scheduling models for capacity expansion provide essentially the optimal timing and sequencing of the construction of new plants and/or the expansion of existing plants to meet expected demand at minimum cost. Interbasin shipment of effluents can be taken into consideration in the above developed models.

Two models for the scheduling for capacity expansion of waste-water treatment plants are discussed here. Each of these models provides the higher level co-ordinator, second level in the srcond layer, with the needed information regarding the optimal scale of, and timing for the construction or expansion of, waste-water treatment plants at a minimum present value cost. These models are:

(ii) Model 2. In this model the operational cost, variable per unit cost, is considered to be an important element and thus is taken into account in the overall analysis.

The use of total treatment capacity q as the state variable, as in Model 1, can introduce a number of difficulties in actual practice when the per unit cost of operation is taken into account. In particular the state levels must be quantized so that a reason- able range of different values can be explored. If all treatment demand functions are to be accurately represented, the capacity grid size must be small. Additionally, when per-unit costs are included, the policy for each year must be determined and the cost calculated. When the state variable is capacity, it is difficult computationally to determine the time periods which occur between the two levels of capacity. For these two reasons the formulation presented in Model 1 is modified to utilize time in discrete years as the state variable. The use of time as the state variable also allows multiple types of demands to be considered without increasing the dimension of the dynamic programming state space. The following model is based on the work of Haimes and Nainis [24], Nainis [50] and Nainis and Haimes [51].


Let h~-.(t~, t o ffi Minimum present-vahe cost (PVC)

of per-unit costs satisfying the L demands from time t~, through year ts using a least-cost utilization of the projects in the sequence k,,.

t~ r.

hk'(tx, tO = min ~ ~ Z baZYa ~, (8) ~u |--lxt¢/¢.~ Z--1

subject to 2: (9)

~te k,.

t - - t 1 ....

1 = 1 , .... L, i f f i l . . . . ,N,

where b~ is the annual present-value per-unit cost of project i in year t for output l.

In general, define

ba z = b~(1 +r) -~. Let

5ff C a ffi C~(1 + r ) - t+ ~a~(1 +r ) -'r,

where b~: is the annual per-unit cost of project i in year t for output l; ya z is the utilization of project i in year t for output 1; qZ(t) is the demand at year t for output l; Q~ is the capacity of project i for output l; k~, is the set of n or fewer constructed projects; I is the output index; L is the total number of separate outputs and PVC is the present-value cost.

The capital construction costs must also be included in the PVC. At the first stage only one project is considered.

If a feasible solution to h~(tl, tO cannot be found, then

h~(tx, tO ~ oo.

The first stage requires

flk~(°(t) ffi m.in{C~:+h~(0,t)}, /~{I .... , N}, $

t x -- rain {~ I ql(~) > 0, any l},

where Ca~ is defined for t ffi tx. At the second stage,

ftk$(t)(t) = min {C¢,+flk~(~)(tO + hk~(to~i(t~, t)},

O <~ tz <~ t <~ T;

1 ¢ kx(t O, where Ca~ is defined for t = t2, or

i - -o ,

where O is the 'null' project symbol and @ is the 'append to sequence' operator.

The general recursive relationship is

f,,~(~)(t) ffi min {Ca. + f,,_~-~(~,)(t~

+ h~,-:(h)~(tn, t)},

O<~t~<~t<~T, i¢{1,2 ..... N}; $

i ¢ k,~, where Ca, is defined for t ffi t n, or

i - -O , Co, Q~ffiO.

7.2.2. Distributed source 'cost' model. This model is aimed at providing the cost functions for the various pollution abatement strategies. For a given level of agricultural, industrial or municipal activities, projected from the modified input-output model, and through the use of the modified Stanford Streamtiow Simulation Model, see Ricca [54], strategies for land use and reclamation policies can be studied and analyzed. The limitations of the Stanford Streamflow Simulation Model with respect to the aggregation of parameters, its need for adequate data, etc. must recognized. Most of

these limitations, however, are shared in other simulation models of this nature. Costs associated with these policies, which yield the reduction of the effluent discharged due to distributed sources, should be derived. Costs for abatement strategies of distributed source pollution can be derived from Becket and Mills [2], Haimes et al. [21], Miller et al. [48] and others. Consequently, the optimal selection and construction scheduling of pollution abatement projects for distributed sources can be achieved through many existing optimization methods, once the cost and performance functions are determined. A cost function can be associated with each abatement policy. Examples of such abatement policies are terraces, contour farming, diversion dikes and fertilization practices. It is more difficult, however, to assess the performance and utility of any specific abatement policy because of the dependency on the hydrological and physical characteristics of the region. Let the performance (utility) of abatement policy i at a level of investment C~ be f~(Cooq) where o~ denotes index associated with the local hydrological, geographical and environmental considerations. For example, the amount of soil erosion reduction per acre per year due to the ith abatement project may depend on the location of project construction, type of soil, slope, surface runoff, etc. Let bt(C~; be the annual operation, maintenance and replace- ment cost for the ith project with level of investment C~ and characteristic index oq. Let g~(C~, oq) ~< 0 be a vector of constraints associated with the ith project. Thus, for a given level of abatement requirements, an optimal selection of projects can be derived. Interaction among the ith project and others is assumed and must be handled through the optimization procedure.

Both cost models for point and distributed sources discussed in Sections 7.2.1. and 7.2.2 respectively will be coordinated at the second layer of the second level.

7.2.3. Ecosystem response "benefit" model. Environmental quality depends on many physical,

34 YACOV Y. HAIMES

chemical and biological variables. In particular water quality can be characterized by a vector of indices such as dissolved oxygen, pH, turbidity, coliform bacteria, algae, etc. A satisfactory lovel of environmental quality has not yet been defined and it may vary from region to region and with time. In the hierarchical structure discussed here, an ecosystem response model is envisioned. This model should relate the improvements of the indices characterizing the ecosystem due to pollution abatement policies exercised on both point and distributed sources. These improvements can be viewed as benefits resulting from the costs determined in the two 'cost' models at the first level of the hierarchy. In other words, the response of the ecosystem to different levels of discharges from both point and distributed sources should provide indices of performance for the proper levels of treatment and/or abatement of these' s o u r c e s .

The effect of effluent discharges, projected by other submodels previously discussed, on the ecosystem and the water quality of a water body can be estimated quantitatively by using models developed by AVCO [1 ], Macko and Teraguchi [45], Lombardo et al. [43], Chen and Orlob [7], Miller and Byers [48] and Hydroscience [34].

In order to circumvent the needs for developing ecosystem response functions in monetary terms which, in turn, are evaluated against the cost functions to yield an optimal investment policy for pollution control and abatement, the concept of multi-objective functions can be utilized. Each ecosystem or environmental quality index can be viewed as either an objective function to be minimized or maximized or as a constraint to he satisfied within acceptable limits. Subsequently, the Surrogate Worth Trade-off (SWT) method developed by Haimes and Hall [23] for the analysis of non-commensurable multi-objective functions in water resources can be utilized.

A major feature of the SWT method is its ability to generate and construct the trade-off functions 0~(x)/0~(x) =-)q.~ between the ith and the jth objective functions. This is done without the direct and explicit operation of a differentiation. The above trade-off function, ~j, is obtained directly from techniques based upon the duality concepts in the functional space, of the objective functions, and not in the decision space, x. Thus, if the units of the ith objective function are in parts per millions of dissolved oxygen and the units of the jth objective function are in dollars, then the trade- off function, )q.~, is given in units of dissolved oxygen per dollar.

It is not suggested here that environmental response functions need not be developed in the future. They are, of course, of primary importance,

and the more available and quantified these functions become, the better are the resulting analyses.

While these functions are missing or not fully developed at the present, the multi-objective approach with the SWT method suggested here represents a viable and promising alternative.

7.2.4. Management and co-ordination model. The second level within the second layer of the hierarchy consists of two major elements, namely, (i) overall minimum 'cost' co-ordinator and (ii) vector optimization co-ordinator.

(i) Overall minimum 'cost' co-ordinator: this co-ordinator determines an optimal combined strategy for an overall reduction of pollutants due to both point and distributed sources. A combined minimum cost policy is obtained based on a model developed by Hall-Haimes-Butcher [31]. This co- ordination method is briefly described below.

Assume there are n point sources in the watershed which, in turn, is decomposed into m subregions based on the soil type and utilization characteristics. Each subregion is assumed to constitute one distributed pollution source. It is assumed that n waste-water treatment plants have been optimally selected to the n point sources. This optimal scheduling solution is, of course, obtained from the capacity expansion 'cost' model at the first-levd optimization. It is also assumed that m abatement strategies, or m sets of abatement strategies, have been optimally selected for the m distributed sources. This solution is obtained from the distributed source 'cost' model at the first-level optimization. Consider the following two alternative policies:

Alternative 1: Only point source pollution is being treated to a high level of pollution removal.

Alternative 2: Both point and distributed pollution sources are being treated, where point sources are treated to satisfy a quality level determined at this second-level optimization.

It is assumed that the same quantity of effluent discharged from point sources is being treated in both alternative policies.

The difference of costs between alternative 2 and alternative 1 is maximized by the optimal choice of pollutant removal level for point sources and area sources, see Hall et al. [28].

(ii) Vector optimization" co-ordinator: the resul- tant overall minimal cost for the combined pollution control attributed to point and distributed sources is now viewed as one objective function and is optimized with other non-commensurate objective functions derived from the ecosystem response


benefit model. Thus the second element of this level is a vector optimization problem of non-commensurate objectives, namely, cost in dollars and environmental and societal quality level. The SWT method of Haimes and Hall [23] and Haimes et aL [25] can be effectively utilized for the overall co- ordination between the cost and benefit models of the first level. As was discussed in the previous section, the SWT method is based on two basic steps. The first step is the construction of the trade-off functions, ~j , for the ith and j t h objective functions that correspond to the non-inferior solutions. These trade-off functions, ~qi, which can be shown to be equal to - 0 f d ~ f ~ are derived on the basis of the duality in nonlinear programming. The second step is the construction of the surrogate worth functions via the interaction with the decision maker(s). All that is required of the decision maker is his ability to appraise the desirability of giving up one unit of objectivej to gain ~ j units of objective i, given the levels of attainment of all objectives corresponding to the numerical value of Ai~.

Since multiple non-commensurable and often conflicting objectives and goals are likely to charac- terize most water resource planning and management systems, vector optimization, as a higher level co-ordination, plays an important and almost an essential role in hierarchical modeling. Applica- tions of the Surrogate Worth Trade-off method to water resources problems can be found in Haimes et al. [23-25].

8. CONCLUSIONS

A great deal of professional manpower and other resources have been devoted to the problem of modeling for the planning, design, construction, development, operation and management of water resource systems. In addition, much experience f rom other fields applicable to these water resource problems is also available. Thus, it seems impera- tive that in dealing with today's and tomorrow's water problems, professionals in this field should not try to reinvent the wheel but rather make an extensive use of existing models pertinent to the problem at hand.

The thesis of these pages is to respond to this need by providing a f ramework for the immediate modi- fication and utilization of existing models. The two hierarchical structures, which are discussed in this paper, are just a sample of the potential integration and co-ordination schemes possible among several models.

Acknowledgements--This research has been supported in part by the National Science Foundation, Research Applied to National Needs Program, under the project: Multilevel Approach for Regional Water Resources Planning and Management.

The author wishes to thank W. Scott Nainis and S. O. Rosen for their review and comments.

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