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Link concepts and partitioning in model formulation
Mathl. Comput. Modelling Vol. 17, No. 7, pp. 105-113, 1993 Printed in Great Britain. All rights reserved
LINK
CONCEPTS IN MODEL
Copyright@
08957177193 $6.00 + 0.00 1993 Pergamon Press Ltd
AND PARTITIONING FORMULATION
BOB
ANDER~SEN
CSIRO Division of Mathematics
and Statistics
GPO Box 1965, Canberra, ACT 2601, Australia RICHARD Department
of Economics,
MONYPENNY
James Cook University of North Queensland
Townsville QLD 4811, Australia (Received and accepted November
1992)
Abstract-For any endeavour (application), the goal is to ensure the chance is high that the conclusions, which flow from the underlying problem-solving, are valid and realistic. The extent and structure of the information available about the application and the degree to which this information is exploited in the modelling are the keys to making the chance high. At one level, this depends on the nature and amount of the available data. At another, it relates to the quality of the modelling which connects the available data to the indicators which characterize the relevant decision-making quantitatively. The interest of this paper is principally the latter aspect. The focus is link concepts which allow a model, which connects the available data to the indicators, to be partitioned into a number of interrelated sub-models. Such partitioning hss a dual role in problemsolving in that it is a strategy for the formulation process itself ss well as for the implementation of a model in terms of link concepts. The discussion is illustrated with a variety of examples including the modelling developed for a Decision Support System for the management of a beef cattle station in North Queensland.
1. INTRODUCTION Structures are the weapons of the mathematicians. -Bourbaki If you want to understand finctions,
skdy struckre. -F.H.C.
Crick
You decompose the whole into its parts, and you recombine the parts into a more or less different whole. -P6lya (1948) As the quotations imply, in any endeavour or application, improvements in the quality of the decision-making occur when the available information about the problem under consideration Such information includes the measured data, the nature of is fully and perceptively utilized. the indicators which characterize the relevant, decision-making quantitatively, known relationships between the data and the indicators, and problem-context-specific constraints such as nonnegativity and monotonicity. Some representative examples of the perceptive utilization of the available information are (a) the role of forensic science in solving crimes; (b) conditions which guarantee that a particular differential equation model has a unique solution; (c) exploiting the mathematical structure of a problem in order to derive better algorithms (e.g., sparse matrix methods; the linear functional strategy for improperly posed problems); Invited paper for the Department
“Modelling-A
Personal Viewpoint.”
105
106
B. ANDERSSEN,R. MONEYPENNY
(d) the use of electrical as well as magnetic data in geophysical exploration; and (e) the use of transmission as well as emission data in nuclear medical imaging. One approach to achieving such utilization is the application of experimental design techniques to determine optimal sampling strategies in terms of the spatial and temporal distribution and number of the data as well as data types. Another is to build models of the interrelationship between the data and the indicators which fully and perceptively exploit available information about the process under examination. The latter is the major concern of this paper. The focus is the identification and implementation of link concepts which allow the model of the interrelationship to be appropriately partitioned into a number of interrelated sub-models. The concept of partitioning in modelling and problem-solving is not new. Bransford and Stein’s [l, p. 1001 discussion of analysing the components represents an elementary exemplification of the basic idea. The concept of fractionation introduced by de Bono (2, p. 1311 represents a different characterization. Much of the discussion in [3] focuses on the need to identify the appropriate intermediate step or steps which link the data with the unknowns. However, the universality of the linking is not stressed. In one way or another, all these points of view support the need for breaking a problem into its relevant parts. However, the role of such strategies in modelling has not been given the emphasis it deserves. This paper therefore aims to make the role of link concepts and partitioning in modelling more explicit. A specific discussion of partitioning in problem-solving is contained in Section 4 along with some exemplification. As an illustrative example, the modelling used to build a decision support system to assist with the management of a beef cattle station is discussed in some detail in Section 5. In part, the current interest in modelling is motivated by the need for guidelines for the formulation of the modelling which underlies the construction of Decision Support Systems (DSS). The interrelationship between modelling and decision support is examined in Section 2. This yields a suitable framework for the examination in Section 3 of link concepts including their identification and implementation.
2. MODELLING
AND DECISION
SUPPORT
Because the success of a DSS is measured in terms of the accuracy, reliability and relevance of the information (outputs) it generates about various scenarios (inputs), and thereby of the quality and perceptiveness of the decisions which that information evokes, the accuracy, reliability and relevance of the outputs become the criteria for focussing the formulation of the quantitative model which describes the underlying process. Thus, when compared with the utilization of a DSS, the initial step of model formulation plays a hidden, but fundamental, role [4]. However, before turning to an examination of such model formulation, the relationship between decision support and the underlying modelling needs to be explored. The role of decision support is to supply the relevant information required for decision-making within a specific context such as the management of a beef cattle station [5]. The t)ssk of the modelling is to first identify the relevant factors, and to then quantify their interrelationship as inputs and outputs as well as the defining-relations which connect them. For example, for the management of a beef cattle station, the decision support becomes: given the current conditions on a beef cattle station and inductive/deductive models of how the relevant factors interrelate, the aim is to make a decision about the future management of the station on the basis of a careful assessment of the profitability of possible management scenarios (options). It therefore becomes clear that, though they depend on each other, the modelling and the decision support are separate activities [4]. One cannot have decision support without first constructing a suitable model. Once an appropriate model is available, then the decision support is an on-going activity. However, the modelling must be performed within the framework of the particular application for which decision support is being designed. This is reflected in the fact that the outputs of the model are the inputs to the decision support. Thus, for the management of a beef cattle station, profitability of the various management options in terms of the net cash flow (NCF) is both the output of the model and the input to the decision support.
Link concepts and partitioning
107
It follows that the success of a decision support activity depends crucially on the tacit assumptions that (a) the underlying model is realistic and relevant to the problem context; (b) the outputs depend continuously (and stably) on the input parameters defining the profitability. The first assumption reflects the need, already flagged above, to do the modelling within the framework of the particular application for which decision support is being designed. This requirement is driven by the fact that the more faithfully the model represents the situation under consideration, the more likely the outputs will reflect the outcome of that situation. The second assumption highlights the fact that the model, in order to be useful for decision support, must be such that small changes in the inputs produce only small changes in the outputs. The study of this aspect goes under a variety of names including trend surface analysis, stability analysis, etc. Usually, the validity of such assumptions is taken for granted. On the other hand, the aim of decision support is to allow the client to study and compare alternative solutions in terms of key indicators. This leads naturally to the use of comparative assessment [6] as the framework in which to construct a decision support system. However, this is a matter outside the scope of the present deliberations except that, in part, the choice of the indicators influences the modelling, and hence, any partitioning invoked.
3. LINK
CONCEPTS
In order to structure the subsequent discussion, it is first necessary to make some remarks about the nature of mathematical modelling. In one way or another, most texts on modelling draw a distinction between purely inductive (black box) models, where general inferences are derived from particular circumstances without knowledge of the underlying process, and fully deductive (white box) models, where particular conclusions are derived logically from general assumptions about the underlying process. Some even discuss modelling in terms of a spectrum of activities (gray boxes) ranging from the purely inductive (black, soft) to the completely deductive (white, hard). This is usually done in terms of the frameworks or contexts in which the models are being derived (e.g., black box models for social and ecological decision-making; gray box models for physiological and hydrological problems; and white box models for industrial and scientific processes). It reflects the extent to which deductive results can be utilized, and inductive steps are essential for generating structure. This aspect has been discussed at some length by Karplus [7]. However, in this paper, this is not the type of interpretation of modelling under consideration. The point which concerns us here is the actual construction of a model. For purely inductive modelling, it is simply a matter of identifying the measured input and outputs along with some suitable framework (e.g., parameterized least squares, convolution modelling, Kalman filtering, stationary stochastic processes) in which to model their interrelationship. Except possibly when choosing the framework, nothing concrete or deductive is exploited from the problem context. For deductive models, the process is much more complex and involved. Because the measured inputs and outputs are often determined by considerations which do not relate to the underlying process, any modelling must accommodate this dichotomy in some appropriate manner. Therefore, it is not simply a matter of deciding to relate the inputs and outputs deductively on the basis of laws and structure from the problem context. There is an initial need to analyse how the inputs and outputs relate to and can be modelled in terms of the underlying process, and then to seek appropriate laws and structure from the problem context for achieving this. Often, this is achieved by linking the inputs or the outputs or both context which can then be related deductively. For instance, the foliage discussed below illustrates just such a situation. This is the simplest complex forms of linking, the link concepts allow the modelling to be and inductive steps. A representative example is the DSS for a beef some detail below.
to concepts in the problem angle distribution problem form of linking. In more partitioned into deductive cattle station examined in
B. ANDEFLSSEN, R. MONEYPENNY
108
Thus, there are various ways in which models can be and are constructed using link concepts. They act as stepping stones between the inputs and the outputs. The use of such concepts has a number of advantages which can be summarized as follows: (i) Link concepts decouple the inputs from the outputs. In many problems, the form of the available inputs and outputs is determined by considerations dictated by the problem context rather than the process which relates them. The introduction of link concepts therefore circumvents this difficulty by allowing the inputs to be decoupled from the outputs. In addition, their introduction represents an opportunity to re-emphasize the underlying process connecting the inputs and the outputs. Thus, the introduction of link concepts can be used to both re-emphasize the underlying process and to suitably decouple (de-emphasize) the inputs from the outputs. (ii) Link concepts can enhance the relevance of the modelling. Through a judicious choice of link concepts, the underlying nature of the problem can often be exposed. For example, the link concept can be the phenomenon one would like to measure about the process but cannot. The foliage angle distribution problem considered below is a good illustration of this point. The output is then seen as the only available information about the phenomenon. In addition, the identification of such concepts often assists in focussing on how the inputs should or could be modelled deductively. (iii) The use of link concepts adds flexibility to the modelling. The model now becomes a multi-phase process, each component of which can be modelled either inductively or deductively. In fact, link concepts can be used to partition the deductive aspects of the model from the inductive. (iv) The link concept can be the key to solving the problem. In more complex mathematical problems, the identification of an appropriate link concept can be the key to obtaining its solution. Polya’s [3] discussion of “Decomposing and Recombining” is in part an illustration of this aspect, though he does not stress the role of the model which connects the data and the unknowns within the framework of the conditions. An intuitive and illustrative exemplification of (iv) is given in [3]. The problem examined is the determination of the length of the diagonal in a room, given the lengths of its sides. This is clearly a prototype problem for a variety of applications. The appropriate link concept is the length of the diagonal of any one of the walls, since it allows the problem to be solved in two separate steps using Pythagoras’ theorem as the model in each. Among other things, this problem illustrates that there can be a variety of choices for the link concept. From the point of view of a DSS for a beef cattle station, it is (ii) which is the crucial aspect. As explained in some detail in Section 5, the weight gain of the cattle has been used as the link concept for the construction of the DSS and thereby gives it its utility. In addition, it yields a natural partitioning of the deductive aspects from the inductive. 4. LINK CONCEPTS
AND PARTITIONING
IN PROBLEM-SOLVING
In situations where link concepts and partitioning are used on a routine basis, their presence and contribution are often taken for granted. It is for this reason that their role should be more explicitly articulated as an essential component of modelling and problem-solving. The spectrum of activities where link concepts and partitioning have an impact extends from cooking to theorem proving. In cooking, the linking (the recipe) is a dual process involving both the order and the way in which the ingredients are combined. In theorem proving, the link concepts are the key. Some representative examples are: - In analysis, the application of an appropriate mean value theorem can be the step which transforms the theorem proving to a framework which allows the required result to be established. An example is the proof that specified conditions on the derivative of a given function f are sufficient to guarantee, with respect to a given starting solution ~0, the convergence of the one-point iteration x,+1
=
f(s,),
n = 0, 1,. . .
Link concepts and partitioning
109
- The application of the Laplace transform in the solution of ordinary differential equations with constant coefficients. Its use transforms the given differential problem to an algebraic one. - The introduction of an intermediate solution in error analysis. This allows the comparison of the exact and approximate solutions to be replaced by the comparison of the exact with the intermediate and the approximate with the intermediate. Only through the judicious choice of appropriate intermediate solutions has it been possible to derive optimal error estimates for the numerical solution of partial differential equations. In additon, link concepts are the way in which partitioning in model formulation is implemented. The actual process of partitioning is one of a number of problem-solving strategies which include working backwards, appeal to simpler and specific situations [l, pp. 18-191. In this sense, the process of partitioning corresponds to breaking complex problems into a sequence of simpler ones. This is the key to systematic analysis discussed by Bransford and Stein. An instructive example is the model used to solve the foliage angle distribution problem [8,9]. The required information is the azimuthally averaged foliage angle density function g(o), where g(a) da defines the contribution to foliage density of leaves inclined to the horizontal at angles between (Y and (u + dcu. The data correspond to observations of the proportion of azimuthally averaged gap PO(Q) as a function of the elevation angle (Y, as obtained from hemispherical pho tographs of the canopy (taken from the ground). The link concept which allows a partitioning of the model into a useful form is the contact frequency f(o). Using it, the model becomes
PO(~) = exp
-(
f(P) =
f(o) since > ’
Jd’ K(au,P)d~) da,
OICY$'
where
W&P) =
cos cr sin p, coscrsinP{l
+ a(tan0
- e),
a 5 P, Q: 2 p,
with
e = cos-l -tari0
(principal value; f2 1 p). tan (Y In fact, the process by which one explains problem-solving is based on the use of partitioning. It is not simply a matter of saying that problem-solving is finding a solution for a problem. In order to explain the process, one uses appropriate link concepts of identification, formulation, solution and interpretation. However, there is no unique way of determining how the linking must be done. Consequently, Bransford and Stein [l] define problem-solving to be the sequential application of the link concepts of Identification, Definition, Examine, Analyse and Look (IDEAL). In heuristics (cf. [3]), link concepts and partitioning play a crucial role in defining strategies for problem solving, though they use other terminology and fail to stress the universal nature of the processes involved. 5. A DECISION
SUPPORT
SYSTEM
FOR
BEEF
CATTLE
STATIONS
The actual decision support system (DSS) constructed by Monypenny and colleagues [IO-U] is more complex than that described here. However, since the aim is to examine questions connected with modelling, it is only necessary (and in fact sensible and desirable) to consider a model which suitably encapsulates the essential quantitative features of the DSS under consideration. In any analysis of modelling, such action is crucial to minimizing the effort devoted to peripheral considerations. Topographically, a beef-cattle station (BS) consists of a large piece of land (2000 or more hectares) which has been partitioned into areas of (i) native pasture (in the hilly and/or poor soil regions); (ii) cleared native pasture (in undulating and/or moderate good soil regions); (iii) oversown legumes (in the flatter regions of good soil adjacent to (semi-) permanent This partitioning
defines the different
pasture
types p. More complete
water).
details can be found in [14].
B. ANDERSSEN, R. MONEYPENNY
110
As a commerical enterprise, the BS grazes different cattle types q on the various pasture types p, usually in a fairly well-defined stratified manner, in the sense that different cattle types are grazed on different pastures so that the model for each type decouples. For management purposes, the cattle types q are partitioned into (a) breeders and (b) non-breeders, with both groups subpartitioned on the basis of age. The role of the breeders is to maintain the stock levels so that the need for restocking (purchase of breeders or non-breeders) is minimized. The breeders therefore tend to play only a passive role in revenue generation. During the year, older non-breeders are sold to generate the revenue R, and hence, the net cash flow (NCF) for the BS. The aim then of the DSSBS is to determine possible stocking (management) strategies which will tend to ensure that NCF is maximized, within the framework of expected pasture growth, which is defined as the number of green weeks gt which summer rains in year t guarantee for that year. However, we are confronted by a situation where we have inadequate data for an inductive model, and the problem is too complex to derive a fully deductive model. Hence, the need arises for an alternative approach such as the use of a link concept. There are various ways in which these factors could be interrelated. A purely inductive model could be derived. But, the construction of such a model assumes the availability of appropriate data which highlights the underlying pattern which needs to be modelled. However, this is not the situation in the present circumstances. A fully deductive model is clearly out of consideration given the nature of the problem. Monypenny and colleagues constructed a model based on a link concept. For this concept, they chose the annual weight gain w = w(gt) of the cattle as a function of green weeks gt. The number gt of green weeks is determined at the end of summer on the basis of the nature and extent of that summer’s rainfall. It is an assessment of the growth potential the pasture has in the coming autumn and winter seasons. This turns out to be a highly perceptive choice for the following reasons: (a) It is realistic and appropriate for the matter under consideration, since the whole process is concerned with the weight gain of the animals. It gives a natural intuitive appeal to the model, since weight gain is a concept with which the station manager works regularly. It is also used as an indicator of other factors including the overall health of the animals. (b) It allows a natural decoupling of the inputs from the outputs in that (i) a deductive type model can be constructed relating weight gain to revenue; and (ii) an inductive/deductive
model can be contructed relating p, q and gt to w(gl).
The additional facts, definitions and constraints which fully characterise the problem quantitatively are: 1. The area of a pasture class p is denoted by A(p). 2. The weight gain ,w(gt) stratifies with respect to the pasture class p and the cattle type q. However, in the discussion below, this aspect is suppressed, since it adds nothing to the modelling considerations. 3. In year t, the amount of feed available is determined by the number of green weeks gt, which in turn is deterrnined by the extent of the summer rains at the beginning of that year. 4. In year t, the amount of feed available determines the carrying capacity of the BS, and hence, whether there are too few or too many stock for the coming season. A decision can then be made about stocking (i.e., whether cattle should be bought or sold) before the season commences. NOTE. For simplicity, we have invoked the Markovian assumption that the circumstances in year t only depend on those in t - 1. In fact, as we shall see below in equation (l), the connection is implicit because the cattle feed on pasture which has grown as the result of earlier rains. Clearly, the interrelationship is more complex in reality. One is now in a position to utilize the link concept, in the manner outlined above, in (b)(i) and (b)(ii) below.
Link concepts and partitioning
(b)(i) THE “INPUT-WEIGHT
111
GAIN” MODEL.
The model used to relate weight gain w(gt) to the various inputs has both a deductive and an inductive aspect. The deductive aspect defines, for a given cattle type q, the number Nt of the cattle alive at the end of year t given there were Nt-1 cattle alive at the end of year t - 1. In fact, we have Nt = Nt-111 +
b(w(d)lP - 44d)lD
- c(wbt))lP + T(ht))l,
(1)
where the terms modifying N,_1 correspond respectively to the effect of birth, death, culling of non-breeders for market, and the purchase of breeders and/or non-breeders for restocking. The inductive aspect defines, for a given cattle type q, the various functions b(.), d( .), c(.), and T(.), entering into the above deductive formula in terms of the weight gain w(gt). For example, on the basis of available data, the following models have been constructed for b(.) and d(.): non-breeders, i;(w(gt)
+ b),
breeders,
(2)
where G, & E consts, d(w(gt)) =
l/G + e(w(gt)+ f)),
non-breeders,
(3) l/(g + h(w(gt) + Ic)), breeders, where F, 6, f, g, h, k s consts, and where weight gain as a function of the number of green weeks gt is assessed inductively to be a, e, f = consts.
An equation for the culling proportion c(w(gt)) is based on whether or not the number of available breeders on the property exceed the maximum number required for the coming year. Culling only takes place when the number of available exceeds the number required. For simplicity, restocking is not pursued, which reduces to assuming that r(w(gt)) = 0. (b)(ii) THE “WEIGHT GAIN-OUTPUT”
MODEL.
This reflects one of the advantages of having made a judicious choice for the link concept. The model relating the weight gain to NCF takes a simple (pseudo-deductive) form (NCF)t = Rt - C,
(5) where C denotes the running “costs” of the cattle property, which can be assumed to be fixed ( i.e., C does not depend strongly on Nt); and Rt denotes the “revenue in year t” which is determined by Rt = HtWt$t, (6) where $t denotes the “price per kilo in year t” (which often has a dependence on the weight of the non-breeders sold); Wt denotes the “weight per non-breeder sold in year t,” which is determined by w, =
wt-1
+
w(gt);
and Ht denotes the “number of head sold in year t,” which is determined by n Ht = NtU - d(w(gt))) - Ht + Ht,
(7)
(8)
where I?, and fit denote respectively the number of non-breeders sold and bought throughout year t. The above modelling illustrates the clear advantages in the use of an appropriate link concept. One has in (1) a precise formula for predicting the number Nt of cattle in year t from that available in year t - 1 (assuming the individual terms b(.), d(.), etc., are known exactly). One has relegated to (2), (3) and (4) the inductive aspect of the modelling so that its role in the overall process is clearly differentiated. The perceptiveness of the choice of weight gain w(gt) as the link concept is reflected in (i) the fact that w(gt) is a concept regularly used on the BS for management purposes as well as the key factor in determining the market price; (ii) the economic and biological relevance of (5).-(8) m . modelling the revenue aspect of the process.
B. ANDERSSEN, Ft. MONEYPENNY
112
The Sensitivity Analysis of Outputs A major goal behind most modelling is to gain a more accurate understanding of the interdependence of the outputs (which characterize the performance of the process) on the inputs (which characterize how the process is driven). This is usually approached by some form of sensitivity analysis. Like the modelling itself, such an analysis can be accomplished in a variety of ways. One possibility is to simply apply the standard formula for the differentiation of a function of two or more variables to any component of the model as well as the full model. For example, consider equation (4), which defines weight gain as a function of green weeks. It is basically a function of two parameters g (= gt) and (Y (= N,/A) and can be written in the form
i; ( >
w(g,a)=ii-a
--E
.
9
It follows that
($+kY+(f$? +E)ba+($)6g,
(.wg, a) =
which allows one to conclude that 0 When the number of green weeks is held constant, the weight gain correlates negatively with cattle numbers. . When the number of cattle is held constant, the weight gain correlates positively with the number of green weeks. . Ignoring drought conditions (which is a pathological situation for the model), weight gain is more sensitive to changes in the number of green weeks than to changes in the number of cattle. An alternative way for studying sensitivity is to perform a more comprehensive stability analysis of the type discussed by Rheinboldt [15] for the analysis of parameterized non-linear equations. In fact, as indicated by Rheinboldt [15], “While the local stability is important in practice, it is equally important to understand those variations of the parameters which produce a change of behaviour.” This is achieved by simply reinterpreting in some appropriate way the model as parameterized non-linear equations. For example, one could examine the actual determination of the parameters in the weight gain equation (4). If the yj, j = 1,2,. . . , n, denote the observations of the weight gain (as a function of the green weeks j), which are used to determine the parameters in (4), then the least squares determination of these parameters reduces to solving
where F(a, e,
f; (yj}) =
e(w(a, e,f; d - yjj2 j=l
with Zu(a, e, f; j) denoting the right hand side of (4) with gt = j. REFERENCES 1. 2. 3. 4. 5. 6.
J.D. Bransford and B.S. Stein, The IDEAL Problem Solver, W.H. Freeman, New York, (1984). E. de Bono, Lateral Thinking, Harper and Row, New York, (1970). G. Pblya, How to Solve It, Princeton University Press, Princeton, NJ, (1948). R.S. Anderssen, Mathematics in action, SEARCH 22, 203-205 (1991). R.S. Anderssen and R. Monypenny, Model construction for decision support systems for beef stations, 1990 Mathematics-in-lnd~stry Study Group, CSIRO, Australia, (1990). R.S. Anderssen, Linking mathematics with applications: The comparative assessment process, Mathematics and Computers in Simulation 33, 469-476 (1992).
Link concepts and partitioning
113
7. W.J. Karplus, The spectrum of mathematical modelling and systems simulation, Math. Comp. in Simulation 19, 3-10 (1977). 8. J.B. Miller, An integral equation from phytology, J. Aust. Math. Sot. 4, 397-402 (1963). 9. R.S. Anderssen and D.R. Jsckett, Linear functionals of foliage angle density, J. Aust. Math. Sot. 25 (Ser. B), 431-442 (1984). 10. P. Gillard and R. Monypenny, A decision support approach for the beef cattle industry of tropical Australia, Agricultural System 26, 179-190 (1988). 11. P. Gillard and R. Monypenny, A Decision Support Model to Evaluate the Eflects of Drought and Stocking Rate on Beef Cattle Properties in Northern Austmlia, Agricultural System (to appear). 12. P. Gillard, J. William and R. Monypenny, Tree clearing from Australia’s semi-arid tropics, The Journal of the Australian Institute of Agricultural Science 2 (2), 34-39 (March 1989). 13. R. Monypenny and J. McIvor, Upsizing of pasture management experimental results, In Proceedings, Simulation Society of Austmlia 1989 Conference, Canberra, September, pp. 25-27, (1989). 14. R. Monypenny, Modelling and decision support on beef properties, Discussion Paper, Department of Economics, James Cook University, (January 1990). 15. W.C. Rheinboldt, Numerical Analysis of Pammeterixed Nonlinear Equations, A Wiley-Interscience Publication, John Wiley & Sons, (1986).
LINK
CONCEPTS IN MODEL
Copyright@
08957177193 $6.00 + 0.00 1993 Pergamon Press Ltd
AND PARTITIONING FORMULATION
BOB
ANDER~SEN
CSIRO Division of Mathematics
and Statistics
GPO Box 1965, Canberra, ACT 2601, Australia RICHARD Department
of Economics,
MONYPENNY
James Cook University of North Queensland
Townsville QLD 4811, Australia (Received and accepted November
1992)
Abstract-For any endeavour (application), the goal is to ensure the chance is high that the conclusions, which flow from the underlying problem-solving, are valid and realistic. The extent and structure of the information available about the application and the degree to which this information is exploited in the modelling are the keys to making the chance high. At one level, this depends on the nature and amount of the available data. At another, it relates to the quality of the modelling which connects the available data to the indicators which characterize the relevant decision-making quantitatively. The interest of this paper is principally the latter aspect. The focus is link concepts which allow a model, which connects the available data to the indicators, to be partitioned into a number of interrelated sub-models. Such partitioning hss a dual role in problemsolving in that it is a strategy for the formulation process itself ss well as for the implementation of a model in terms of link concepts. The discussion is illustrated with a variety of examples including the modelling developed for a Decision Support System for the management of a beef cattle station in North Queensland.
1. INTRODUCTION Structures are the weapons of the mathematicians. -Bourbaki If you want to understand finctions,
skdy struckre. -F.H.C.
Crick
You decompose the whole into its parts, and you recombine the parts into a more or less different whole. -P6lya (1948) As the quotations imply, in any endeavour or application, improvements in the quality of the decision-making occur when the available information about the problem under consideration Such information includes the measured data, the nature of is fully and perceptively utilized. the indicators which characterize the relevant, decision-making quantitatively, known relationships between the data and the indicators, and problem-context-specific constraints such as nonnegativity and monotonicity. Some representative examples of the perceptive utilization of the available information are (a) the role of forensic science in solving crimes; (b) conditions which guarantee that a particular differential equation model has a unique solution; (c) exploiting the mathematical structure of a problem in order to derive better algorithms (e.g., sparse matrix methods; the linear functional strategy for improperly posed problems); Invited paper for the Department
“Modelling-A
Personal Viewpoint.”
105
106
B. ANDERSSEN,R. MONEYPENNY
(d) the use of electrical as well as magnetic data in geophysical exploration; and (e) the use of transmission as well as emission data in nuclear medical imaging. One approach to achieving such utilization is the application of experimental design techniques to determine optimal sampling strategies in terms of the spatial and temporal distribution and number of the data as well as data types. Another is to build models of the interrelationship between the data and the indicators which fully and perceptively exploit available information about the process under examination. The latter is the major concern of this paper. The focus is the identification and implementation of link concepts which allow the model of the interrelationship to be appropriately partitioned into a number of interrelated sub-models. The concept of partitioning in modelling and problem-solving is not new. Bransford and Stein’s [l, p. 1001 discussion of analysing the components represents an elementary exemplification of the basic idea. The concept of fractionation introduced by de Bono (2, p. 1311 represents a different characterization. Much of the discussion in [3] focuses on the need to identify the appropriate intermediate step or steps which link the data with the unknowns. However, the universality of the linking is not stressed. In one way or another, all these points of view support the need for breaking a problem into its relevant parts. However, the role of such strategies in modelling has not been given the emphasis it deserves. This paper therefore aims to make the role of link concepts and partitioning in modelling more explicit. A specific discussion of partitioning in problem-solving is contained in Section 4 along with some exemplification. As an illustrative example, the modelling used to build a decision support system to assist with the management of a beef cattle station is discussed in some detail in Section 5. In part, the current interest in modelling is motivated by the need for guidelines for the formulation of the modelling which underlies the construction of Decision Support Systems (DSS). The interrelationship between modelling and decision support is examined in Section 2. This yields a suitable framework for the examination in Section 3 of link concepts including their identification and implementation.
2. MODELLING
AND DECISION
SUPPORT
Because the success of a DSS is measured in terms of the accuracy, reliability and relevance of the information (outputs) it generates about various scenarios (inputs), and thereby of the quality and perceptiveness of the decisions which that information evokes, the accuracy, reliability and relevance of the outputs become the criteria for focussing the formulation of the quantitative model which describes the underlying process. Thus, when compared with the utilization of a DSS, the initial step of model formulation plays a hidden, but fundamental, role [4]. However, before turning to an examination of such model formulation, the relationship between decision support and the underlying modelling needs to be explored. The role of decision support is to supply the relevant information required for decision-making within a specific context such as the management of a beef cattle station [5]. The t)ssk of the modelling is to first identify the relevant factors, and to then quantify their interrelationship as inputs and outputs as well as the defining-relations which connect them. For example, for the management of a beef cattle station, the decision support becomes: given the current conditions on a beef cattle station and inductive/deductive models of how the relevant factors interrelate, the aim is to make a decision about the future management of the station on the basis of a careful assessment of the profitability of possible management scenarios (options). It therefore becomes clear that, though they depend on each other, the modelling and the decision support are separate activities [4]. One cannot have decision support without first constructing a suitable model. Once an appropriate model is available, then the decision support is an on-going activity. However, the modelling must be performed within the framework of the particular application for which decision support is being designed. This is reflected in the fact that the outputs of the model are the inputs to the decision support. Thus, for the management of a beef cattle station, profitability of the various management options in terms of the net cash flow (NCF) is both the output of the model and the input to the decision support.
Link concepts and partitioning
107
It follows that the success of a decision support activity depends crucially on the tacit assumptions that (a) the underlying model is realistic and relevant to the problem context; (b) the outputs depend continuously (and stably) on the input parameters defining the profitability. The first assumption reflects the need, already flagged above, to do the modelling within the framework of the particular application for which decision support is being designed. This requirement is driven by the fact that the more faithfully the model represents the situation under consideration, the more likely the outputs will reflect the outcome of that situation. The second assumption highlights the fact that the model, in order to be useful for decision support, must be such that small changes in the inputs produce only small changes in the outputs. The study of this aspect goes under a variety of names including trend surface analysis, stability analysis, etc. Usually, the validity of such assumptions is taken for granted. On the other hand, the aim of decision support is to allow the client to study and compare alternative solutions in terms of key indicators. This leads naturally to the use of comparative assessment [6] as the framework in which to construct a decision support system. However, this is a matter outside the scope of the present deliberations except that, in part, the choice of the indicators influences the modelling, and hence, any partitioning invoked.
3. LINK
CONCEPTS
In order to structure the subsequent discussion, it is first necessary to make some remarks about the nature of mathematical modelling. In one way or another, most texts on modelling draw a distinction between purely inductive (black box) models, where general inferences are derived from particular circumstances without knowledge of the underlying process, and fully deductive (white box) models, where particular conclusions are derived logically from general assumptions about the underlying process. Some even discuss modelling in terms of a spectrum of activities (gray boxes) ranging from the purely inductive (black, soft) to the completely deductive (white, hard). This is usually done in terms of the frameworks or contexts in which the models are being derived (e.g., black box models for social and ecological decision-making; gray box models for physiological and hydrological problems; and white box models for industrial and scientific processes). It reflects the extent to which deductive results can be utilized, and inductive steps are essential for generating structure. This aspect has been discussed at some length by Karplus [7]. However, in this paper, this is not the type of interpretation of modelling under consideration. The point which concerns us here is the actual construction of a model. For purely inductive modelling, it is simply a matter of identifying the measured input and outputs along with some suitable framework (e.g., parameterized least squares, convolution modelling, Kalman filtering, stationary stochastic processes) in which to model their interrelationship. Except possibly when choosing the framework, nothing concrete or deductive is exploited from the problem context. For deductive models, the process is much more complex and involved. Because the measured inputs and outputs are often determined by considerations which do not relate to the underlying process, any modelling must accommodate this dichotomy in some appropriate manner. Therefore, it is not simply a matter of deciding to relate the inputs and outputs deductively on the basis of laws and structure from the problem context. There is an initial need to analyse how the inputs and outputs relate to and can be modelled in terms of the underlying process, and then to seek appropriate laws and structure from the problem context for achieving this. Often, this is achieved by linking the inputs or the outputs or both context which can then be related deductively. For instance, the foliage discussed below illustrates just such a situation. This is the simplest complex forms of linking, the link concepts allow the modelling to be and inductive steps. A representative example is the DSS for a beef some detail below.
to concepts in the problem angle distribution problem form of linking. In more partitioned into deductive cattle station examined in
B. ANDEFLSSEN, R. MONEYPENNY
108
Thus, there are various ways in which models can be and are constructed using link concepts. They act as stepping stones between the inputs and the outputs. The use of such concepts has a number of advantages which can be summarized as follows: (i) Link concepts decouple the inputs from the outputs. In many problems, the form of the available inputs and outputs is determined by considerations dictated by the problem context rather than the process which relates them. The introduction of link concepts therefore circumvents this difficulty by allowing the inputs to be decoupled from the outputs. In addition, their introduction represents an opportunity to re-emphasize the underlying process connecting the inputs and the outputs. Thus, the introduction of link concepts can be used to both re-emphasize the underlying process and to suitably decouple (de-emphasize) the inputs from the outputs. (ii) Link concepts can enhance the relevance of the modelling. Through a judicious choice of link concepts, the underlying nature of the problem can often be exposed. For example, the link concept can be the phenomenon one would like to measure about the process but cannot. The foliage angle distribution problem considered below is a good illustration of this point. The output is then seen as the only available information about the phenomenon. In addition, the identification of such concepts often assists in focussing on how the inputs should or could be modelled deductively. (iii) The use of link concepts adds flexibility to the modelling. The model now becomes a multi-phase process, each component of which can be modelled either inductively or deductively. In fact, link concepts can be used to partition the deductive aspects of the model from the inductive. (iv) The link concept can be the key to solving the problem. In more complex mathematical problems, the identification of an appropriate link concept can be the key to obtaining its solution. Polya’s [3] discussion of “Decomposing and Recombining” is in part an illustration of this aspect, though he does not stress the role of the model which connects the data and the unknowns within the framework of the conditions. An intuitive and illustrative exemplification of (iv) is given in [3]. The problem examined is the determination of the length of the diagonal in a room, given the lengths of its sides. This is clearly a prototype problem for a variety of applications. The appropriate link concept is the length of the diagonal of any one of the walls, since it allows the problem to be solved in two separate steps using Pythagoras’ theorem as the model in each. Among other things, this problem illustrates that there can be a variety of choices for the link concept. From the point of view of a DSS for a beef cattle station, it is (ii) which is the crucial aspect. As explained in some detail in Section 5, the weight gain of the cattle has been used as the link concept for the construction of the DSS and thereby gives it its utility. In addition, it yields a natural partitioning of the deductive aspects from the inductive. 4. LINK CONCEPTS
AND PARTITIONING
IN PROBLEM-SOLVING
In situations where link concepts and partitioning are used on a routine basis, their presence and contribution are often taken for granted. It is for this reason that their role should be more explicitly articulated as an essential component of modelling and problem-solving. The spectrum of activities where link concepts and partitioning have an impact extends from cooking to theorem proving. In cooking, the linking (the recipe) is a dual process involving both the order and the way in which the ingredients are combined. In theorem proving, the link concepts are the key. Some representative examples are: - In analysis, the application of an appropriate mean value theorem can be the step which transforms the theorem proving to a framework which allows the required result to be established. An example is the proof that specified conditions on the derivative of a given function f are sufficient to guarantee, with respect to a given starting solution ~0, the convergence of the one-point iteration x,+1
=
f(s,),
n = 0, 1,. . .
Link concepts and partitioning
109
- The application of the Laplace transform in the solution of ordinary differential equations with constant coefficients. Its use transforms the given differential problem to an algebraic one. - The introduction of an intermediate solution in error analysis. This allows the comparison of the exact and approximate solutions to be replaced by the comparison of the exact with the intermediate and the approximate with the intermediate. Only through the judicious choice of appropriate intermediate solutions has it been possible to derive optimal error estimates for the numerical solution of partial differential equations. In additon, link concepts are the way in which partitioning in model formulation is implemented. The actual process of partitioning is one of a number of problem-solving strategies which include working backwards, appeal to simpler and specific situations [l, pp. 18-191. In this sense, the process of partitioning corresponds to breaking complex problems into a sequence of simpler ones. This is the key to systematic analysis discussed by Bransford and Stein. An instructive example is the model used to solve the foliage angle distribution problem [8,9]. The required information is the azimuthally averaged foliage angle density function g(o), where g(a) da defines the contribution to foliage density of leaves inclined to the horizontal at angles between (Y and (u + dcu. The data correspond to observations of the proportion of azimuthally averaged gap PO(Q) as a function of the elevation angle (Y, as obtained from hemispherical pho tographs of the canopy (taken from the ground). The link concept which allows a partitioning of the model into a useful form is the contact frequency f(o). Using it, the model becomes
PO(~) = exp
-(
f(P) =
f(o) since > ’
Jd’ K(au,P)d~) da,
OICY$'
where
W&P) =
cos cr sin p, coscrsinP{l
+ a(tan0
- e),
a 5 P, Q: 2 p,
with
e = cos-l -tari0
(principal value; f2 1 p). tan (Y In fact, the process by which one explains problem-solving is based on the use of partitioning. It is not simply a matter of saying that problem-solving is finding a solution for a problem. In order to explain the process, one uses appropriate link concepts of identification, formulation, solution and interpretation. However, there is no unique way of determining how the linking must be done. Consequently, Bransford and Stein [l] define problem-solving to be the sequential application of the link concepts of Identification, Definition, Examine, Analyse and Look (IDEAL). In heuristics (cf. [3]), link concepts and partitioning play a crucial role in defining strategies for problem solving, though they use other terminology and fail to stress the universal nature of the processes involved. 5. A DECISION
SUPPORT
SYSTEM
FOR
BEEF
CATTLE
STATIONS
The actual decision support system (DSS) constructed by Monypenny and colleagues [IO-U] is more complex than that described here. However, since the aim is to examine questions connected with modelling, it is only necessary (and in fact sensible and desirable) to consider a model which suitably encapsulates the essential quantitative features of the DSS under consideration. In any analysis of modelling, such action is crucial to minimizing the effort devoted to peripheral considerations. Topographically, a beef-cattle station (BS) consists of a large piece of land (2000 or more hectares) which has been partitioned into areas of (i) native pasture (in the hilly and/or poor soil regions); (ii) cleared native pasture (in undulating and/or moderate good soil regions); (iii) oversown legumes (in the flatter regions of good soil adjacent to (semi-) permanent This partitioning
defines the different
pasture
types p. More complete
water).
details can be found in [14].
B. ANDERSSEN, R. MONEYPENNY
110
As a commerical enterprise, the BS grazes different cattle types q on the various pasture types p, usually in a fairly well-defined stratified manner, in the sense that different cattle types are grazed on different pastures so that the model for each type decouples. For management purposes, the cattle types q are partitioned into (a) breeders and (b) non-breeders, with both groups subpartitioned on the basis of age. The role of the breeders is to maintain the stock levels so that the need for restocking (purchase of breeders or non-breeders) is minimized. The breeders therefore tend to play only a passive role in revenue generation. During the year, older non-breeders are sold to generate the revenue R, and hence, the net cash flow (NCF) for the BS. The aim then of the DSSBS is to determine possible stocking (management) strategies which will tend to ensure that NCF is maximized, within the framework of expected pasture growth, which is defined as the number of green weeks gt which summer rains in year t guarantee for that year. However, we are confronted by a situation where we have inadequate data for an inductive model, and the problem is too complex to derive a fully deductive model. Hence, the need arises for an alternative approach such as the use of a link concept. There are various ways in which these factors could be interrelated. A purely inductive model could be derived. But, the construction of such a model assumes the availability of appropriate data which highlights the underlying pattern which needs to be modelled. However, this is not the situation in the present circumstances. A fully deductive model is clearly out of consideration given the nature of the problem. Monypenny and colleagues constructed a model based on a link concept. For this concept, they chose the annual weight gain w = w(gt) of the cattle as a function of green weeks gt. The number gt of green weeks is determined at the end of summer on the basis of the nature and extent of that summer’s rainfall. It is an assessment of the growth potential the pasture has in the coming autumn and winter seasons. This turns out to be a highly perceptive choice for the following reasons: (a) It is realistic and appropriate for the matter under consideration, since the whole process is concerned with the weight gain of the animals. It gives a natural intuitive appeal to the model, since weight gain is a concept with which the station manager works regularly. It is also used as an indicator of other factors including the overall health of the animals. (b) It allows a natural decoupling of the inputs from the outputs in that (i) a deductive type model can be constructed relating weight gain to revenue; and (ii) an inductive/deductive
model can be contructed relating p, q and gt to w(gl).
The additional facts, definitions and constraints which fully characterise the problem quantitatively are: 1. The area of a pasture class p is denoted by A(p). 2. The weight gain ,w(gt) stratifies with respect to the pasture class p and the cattle type q. However, in the discussion below, this aspect is suppressed, since it adds nothing to the modelling considerations. 3. In year t, the amount of feed available is determined by the number of green weeks gt, which in turn is deterrnined by the extent of the summer rains at the beginning of that year. 4. In year t, the amount of feed available determines the carrying capacity of the BS, and hence, whether there are too few or too many stock for the coming season. A decision can then be made about stocking (i.e., whether cattle should be bought or sold) before the season commences. NOTE. For simplicity, we have invoked the Markovian assumption that the circumstances in year t only depend on those in t - 1. In fact, as we shall see below in equation (l), the connection is implicit because the cattle feed on pasture which has grown as the result of earlier rains. Clearly, the interrelationship is more complex in reality. One is now in a position to utilize the link concept, in the manner outlined above, in (b)(i) and (b)(ii) below.
Link concepts and partitioning
(b)(i) THE “INPUT-WEIGHT
111
GAIN” MODEL.
The model used to relate weight gain w(gt) to the various inputs has both a deductive and an inductive aspect. The deductive aspect defines, for a given cattle type q, the number Nt of the cattle alive at the end of year t given there were Nt-1 cattle alive at the end of year t - 1. In fact, we have Nt = Nt-111 +
b(w(d)lP - 44d)lD
- c(wbt))lP + T(ht))l,
(1)
where the terms modifying N,_1 correspond respectively to the effect of birth, death, culling of non-breeders for market, and the purchase of breeders and/or non-breeders for restocking. The inductive aspect defines, for a given cattle type q, the various functions b(.), d( .), c(.), and T(.), entering into the above deductive formula in terms of the weight gain w(gt). For example, on the basis of available data, the following models have been constructed for b(.) and d(.): non-breeders, i;(w(gt)
+ b),
breeders,
(2)
where G, & E consts, d(w(gt)) =
l/G + e(w(gt)+ f)),
non-breeders,
(3) l/(g + h(w(gt) + Ic)), breeders, where F, 6, f, g, h, k s consts, and where weight gain as a function of the number of green weeks gt is assessed inductively to be a, e, f = consts.
An equation for the culling proportion c(w(gt)) is based on whether or not the number of available breeders on the property exceed the maximum number required for the coming year. Culling only takes place when the number of available exceeds the number required. For simplicity, restocking is not pursued, which reduces to assuming that r(w(gt)) = 0. (b)(ii) THE “WEIGHT GAIN-OUTPUT”
MODEL.
This reflects one of the advantages of having made a judicious choice for the link concept. The model relating the weight gain to NCF takes a simple (pseudo-deductive) form (NCF)t = Rt - C,
(5) where C denotes the running “costs” of the cattle property, which can be assumed to be fixed ( i.e., C does not depend strongly on Nt); and Rt denotes the “revenue in year t” which is determined by Rt = HtWt$t, (6) where $t denotes the “price per kilo in year t” (which often has a dependence on the weight of the non-breeders sold); Wt denotes the “weight per non-breeder sold in year t,” which is determined by w, =
wt-1
+
w(gt);
and Ht denotes the “number of head sold in year t,” which is determined by n Ht = NtU - d(w(gt))) - Ht + Ht,
(7)
(8)
where I?, and fit denote respectively the number of non-breeders sold and bought throughout year t. The above modelling illustrates the clear advantages in the use of an appropriate link concept. One has in (1) a precise formula for predicting the number Nt of cattle in year t from that available in year t - 1 (assuming the individual terms b(.), d(.), etc., are known exactly). One has relegated to (2), (3) and (4) the inductive aspect of the modelling so that its role in the overall process is clearly differentiated. The perceptiveness of the choice of weight gain w(gt) as the link concept is reflected in (i) the fact that w(gt) is a concept regularly used on the BS for management purposes as well as the key factor in determining the market price; (ii) the economic and biological relevance of (5).-(8) m . modelling the revenue aspect of the process.
B. ANDERSSEN, Ft. MONEYPENNY
112
The Sensitivity Analysis of Outputs A major goal behind most modelling is to gain a more accurate understanding of the interdependence of the outputs (which characterize the performance of the process) on the inputs (which characterize how the process is driven). This is usually approached by some form of sensitivity analysis. Like the modelling itself, such an analysis can be accomplished in a variety of ways. One possibility is to simply apply the standard formula for the differentiation of a function of two or more variables to any component of the model as well as the full model. For example, consider equation (4), which defines weight gain as a function of green weeks. It is basically a function of two parameters g (= gt) and (Y (= N,/A) and can be written in the form
i; ( >
w(g,a)=ii-a
--E
.
9
It follows that
($+kY+(f$? +E)ba+($)6g,
(.wg, a) =
which allows one to conclude that 0 When the number of green weeks is held constant, the weight gain correlates negatively with cattle numbers. . When the number of cattle is held constant, the weight gain correlates positively with the number of green weeks. . Ignoring drought conditions (which is a pathological situation for the model), weight gain is more sensitive to changes in the number of green weeks than to changes in the number of cattle. An alternative way for studying sensitivity is to perform a more comprehensive stability analysis of the type discussed by Rheinboldt [15] for the analysis of parameterized non-linear equations. In fact, as indicated by Rheinboldt [15], “While the local stability is important in practice, it is equally important to understand those variations of the parameters which produce a change of behaviour.” This is achieved by simply reinterpreting in some appropriate way the model as parameterized non-linear equations. For example, one could examine the actual determination of the parameters in the weight gain equation (4). If the yj, j = 1,2,. . . , n, denote the observations of the weight gain (as a function of the green weeks j), which are used to determine the parameters in (4), then the least squares determination of these parameters reduces to solving
where F(a, e,
f; (yj}) =
e(w(a, e,f; d - yjj2 j=l
with Zu(a, e, f; j) denoting the right hand side of (4) with gt = j. REFERENCES 1. 2. 3. 4. 5. 6.
J.D. Bransford and B.S. Stein, The IDEAL Problem Solver, W.H. Freeman, New York, (1984). E. de Bono, Lateral Thinking, Harper and Row, New York, (1970). G. Pblya, How to Solve It, Princeton University Press, Princeton, NJ, (1948). R.S. Anderssen, Mathematics in action, SEARCH 22, 203-205 (1991). R.S. Anderssen and R. Monypenny, Model construction for decision support systems for beef stations, 1990 Mathematics-in-lnd~stry Study Group, CSIRO, Australia, (1990). R.S. Anderssen, Linking mathematics with applications: The comparative assessment process, Mathematics and Computers in Simulation 33, 469-476 (1992).
Link concepts and partitioning
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7. W.J. Karplus, The spectrum of mathematical modelling and systems simulation, Math. Comp. in Simulation 19, 3-10 (1977). 8. J.B. Miller, An integral equation from phytology, J. Aust. Math. Sot. 4, 397-402 (1963). 9. R.S. Anderssen and D.R. Jsckett, Linear functionals of foliage angle density, J. Aust. Math. Sot. 25 (Ser. B), 431-442 (1984). 10. P. Gillard and R. Monypenny, A decision support approach for the beef cattle industry of tropical Australia, Agricultural System 26, 179-190 (1988). 11. P. Gillard and R. Monypenny, A Decision Support Model to Evaluate the Eflects of Drought and Stocking Rate on Beef Cattle Properties in Northern Austmlia, Agricultural System (to appear). 12. P. Gillard, J. William and R. Monypenny, Tree clearing from Australia’s semi-arid tropics, The Journal of the Australian Institute of Agricultural Science 2 (2), 34-39 (March 1989). 13. R. Monypenny and J. McIvor, Upsizing of pasture management experimental results, In Proceedings, Simulation Society of Austmlia 1989 Conference, Canberra, September, pp. 25-27, (1989). 14. R. Monypenny, Modelling and decision support on beef properties, Discussion Paper, Department of Economics, James Cook University, (January 1990). 15. W.C. Rheinboldt, Numerical Analysis of Pammeterixed Nonlinear Equations, A Wiley-Interscience Publication, John Wiley & Sons, (1986).