- Email: [email protected]
A procedure for optimizing development decisions
Automatiea, Vol. II, pp. 261 -269. Pergamon Press, 1975. Printed in Great Britain
A Procedure for Optimizing Development Decisions* Un Processus pour l'Optimisation de D~cisions de D6veloppement Eine Prozedur zur Optimierung von Entwicklungsentscheidungen STEPHEN KAHNEt
Qualitative and uncertain data play an important role in optimizing development decisions Summary--This paper addresses the problem of organizing data and formulating questions to be answered for the purpose of making planning and development decisions. The problem is separated into five distinct parts. Each part is discussed in the context of the planning process and each leads logically to the optimization of development decisions. The five parts are goal definition, establishment of criteria, criteria weighting, alternative rating and alternative ranking. The feature of making development decisions which distinguishes them from other optimization problems is what has been called 'fuzziness'. In any realistic problem formulation, the criteria are not precisely defined; they are fuzzy. The relative importance of each criterion is also fuzzy. Indeed, even when one attempts to rate a particular possible solution, he must deal with fuzzy information. The technique proposed in the paper accounts for this uncertainty in all aspects of the problem and yields probabilistic answers. Thus, when various alternative solutions are proposed for a development problem, the technique yields a probabilistic ranking of the alternatives. Sharper results are obtained if less uncertainty is present in certain parts of the data. However, even in the presence of great uncertainty, realistic problem solutions are obtained. Alternative solutions are rated independent of all others and only after the (fuzzy) ratings are complete are comparative rankings accomplished. Throughout the procedure the realistic uncertainties remain a prominent feature of the procedure.
More recently, decision theory and the more quantitative disciplines o f operations research and queuing theory have been brought to bear on such problems. But, inherent in all approaches to date has been a requirement to specify criteria, constraints, optimality goals and other parameters to an extent which one often finds to be unreasonable in actual decision making situations It is appropriate to expand on this particular point. In most real situations as contrasted with models of situations, the data which are at one's disposal are rather ill defined and imprecise. Intuitive feelings and inductive judgement play an important role in everyday life and decision making. The goal o f quantitative methods is to help focus on key issues and organize available data to assist the decision maker in his often unfortunate task of 'resolving' a problem and enunciating a decision. If these quantitative methods are properly applied they do indeed provide a structure for decision making. In the hands o f a competent executive, they can be an invaluable tool for problem solving. Probably one of the best examples of successful development of quantitative tools of this type is linear programming developed by a number of people at the end o f World War II. Chief among this group was George Dantzig whose work on the formulation of linear programming problems and a rather elegant solution, the simplex method, was reported in 1951 [1]. Numerous papers, books and monographs attest to the widespread utility and versatility o f linear programming. Applications ranging from transportation to machine utilization and covering a wide range o f other scientific and management-oriented problems have been found to be amenable to this quantitative tool. Given its widespread utility, attempts are made to couch problems in terms so that methods of linear programming can be brought to bear on them. This type of progression has been common in the development of other quantitative tools. First, a
INTRODUCTION THE PXOm-~ of organizing data and formulating appropriate questions to be answered for the purpose of making planning and development decisions is one which has received considerable attention in the management sciences. The disciplines of social psychology, group dynamics and other behavioral sciences have been discussed. * Received 10 January, 1974; revised 6 August 1974; revised 11 October 1974. The original version of this paper was presented at the IFAC/IFORS Symposium on Systems Approaches to Developing Countries which was held in Algiers, Algeria, during May 1973. The published Proceedings of this IFAC meeting may be ordered from ISA. It was recommended for publication in revised form by associate editor H. Kwakernaak. ? Associate Professor, Department of Electrical Engineering, Director, Hybrid Computer Laboratory, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
261
262
STEPHEN KAHNE
mathematical technique is invented or discovered. Often this occurs in the course of trying to solve a particular 'real world' problem. For example, in the case of linear programming, the problems involved resource allocation. Occasionally it is an independently studied procedure which is soon recognized to be of use in an interesting class of problems. The second phase of the progression is the careful refinement of the technique. Assumptions are examined. Rigorous-derivations are developed and extensions to the original method are discovered. A rather complete theory of the technique is established and many loose technical ends are tied up. Thii'dly, often in parallel with the second phase, the method -is used on a variety of problems. The class of problems for which the method is applicable is carefully searched. More and more interesting variations of the basic problem are discovered until finally a seemingly exhaustive collection of problems has been delineated and solved. As this step proceeds, it is not uncommon that continued advances are made in the theory of the technique and that subsequent refinements lead to subsequent variations in applicable problems. The fourth step is the final one. In the mature life of the technique, as many researchers become intimately familiar with it, attempts are made to change real world problems into problems which can be solved by 'the method'. Nonlinear problems are linearized. Stochastic problems are made deterministic. Distributed systems are considered to be lumped. Time-varying systems are considered stationary. The philosophy accompanying these asssumptions is understandable, if not valid. The technique exists, is well defined and well understood but, unfortunately, is not applicable to the problem at hand. So, if the problem were only approximated in such a way that the technique was useful, then a solution could be realized. Of course, the solution is a solution to the approximated problem, but one can hope that it will be an approximate solution to the real problem. This kind of reasoning, although pervasive, is often not persuasive. One of the biggest faults of modern automatic control education today is this lack of follow-through between real world problems and real world solutions [2]. The problem, of course, is that it is much harder to develop a solution technique ideally suited to a particular real world problem than it is to alter a real world problem so that it conforms to the assumptions of a well-developed technique. This interchanging of roles between problem and solution technique is at the heart of the issue. This confusion is not restricted to strictly scientific problems of automatic control. In this field, it has been noted that we have devoted much
time to 'problems of solution' and not enough to 'solutions of problems' [3]. However, this malady has certainly affected studies of problems of national development as well. Simulation and modelling of national economies [4] and various sectors of an urban society have resulted in glorious predictions of growth or decay. Indeed, the models described in these studies do behave as reported. The effect of running the model with certain given initial conditions does lead to the changes of model variables which are described in the literature. However, what that has to do with behavior of the urban or economic system under study is not at all clear, and how the results of the simulation studies should be applied to particular urban or national development programs is far from evident. The amount of idealism which is injected into the simulation most often completely obliterates the true nature of the process being studied, and the subtle dynamic processes at work which are the driving force of the system are often obscured beyond recognition. The author has alluded to certain of these difticulties in his review of a landmark book on urban dynamics [5]. The work described in this paper represents one approach to the class of problems described. It is designed specifically for these problems and has been used in several development applications. One could contrast it with, for example, Zadeh's 'fuzzy' approach which, at this time, must be characterized as being in the 'machinery' stage [11 ]. A great deal of mathematical machinery has been produced but few practical problems have yet been solved via 'fuzzy' system theory. The author happens to believe that these 'fuzzy' notions do, indeed, form the foundations for a basic and important approach to the solution of realistic development problems but that much work remains to bridge the 'fuzzy gap' between 'fuzzy theory' and practice. In much of this section we have at least suggested that a common underlying feature of decision processes is a model which may be either simulated or constructed~ the study of which leads to a recommended solution to a given problem. But is this a necessary assumption .9 Is a model always at the heart of a decision-making process? In the following section we suggest not; we observe that one may proceed along quite logical lines with no model at all; and, in fact, in many problems of development, such an approach is the preferred one. THE MODELLESS DECISION PROCESS Lot us examine the modelless decision process. This is characterized by an emphasis on the consequences of the decision, not the dynamics of the systems about which a decision is to be made.
A procedure for optimizing development decisions
263
Let us not overstate our case. The system dynamics will determine the consequences of a certain decision, but we will concentrate only on the decision and the consequences. This approach, called input--output analysis in the economic and social sciences and the transfer function approach in the study of automatic control systems, is certainly not new. The degree to which the system dynamics are ignored depends on the particular problem at hand. Practical measurements on physical systems are usually geared to I/O analyses either on a microscopic or macroscopic scale depending on the availability of certain data, instruments and techniques. Special theories have been established to show when I/O analyses are expected to work and when they are expected to fail. The notion of observability in systems theory [6] is specifically directed at this question of when I/O analysis is sufficiently rich to describe completely system behavior. The key point here is that there is no effort required to develop a detailed model and, in fact, no need to develop a model at all provided one is dealing with a real system. The emphasis is on the decision. But, how does one select the best decision? After all, many decisions are usually possible and each one has certain consequences which must be studied. Indeed, by studying in great detail a plethora of decisions and their consequences one might begin to construct a model of the process (or system) which is transforming these decisions into consequences. But this method building is clearly subordinate to the main task of decision making. This characteristic of emphasizing decision consequences rather than models seems to be a rather pragmatic and realistic one in the case of making decisions for regional or national development. With this explanation of a de-emphasis on modeling and with an understanding that development decisions must be made in existing systems of economy, politics and social order, we may now turn our attention toward a procedure for optimizing development decision.
while eliminating a host of unacceptable alternatives. A procedure used to select the best solution must compare only the feasible ones. (2) Well-defined goal--although the 'best' decision may be best with respect to a very complex goal, this goal must be defined well enough to allow comparisons of degrees of goal satisfaction for various feasible solutions. In the parlance of control theory, the goal may be a scalar or a vector performance index as long as this condition is satisfied. (3) Precision of data--unfortunately, when solving 'real world" problems such as those involved in development decisions, much important data are either missing or, at best, vague. It is because this imprecision exists that most methods developed have limited usefulness. There are several approaches usually taken in this case. If the dimension of the problem is very small, there are certain stochastic optimization approaches which may prove to be effective. By definition, a real world problem in the general area of development is not one of small dimension [7]. On the other hand, if the system to be optimized is large or if the data are too imprecise to fit the standard decisionmaking techniques, approximations to the system or refinement of the data are commonly tried. Unfortunately, the approximations and/or refinements are usually not warranted from the information at hand. The unwarranted refinements or approximations are made to fit the problem to the method, a rather unreasonable approach to take if it could be avoided. (4) Precision of criteria--we have seen that imprecise data are a characteristic of these development decisions. In the same way, the criteria for judging the success of one decision or another are, regrettably, imprecise. Any attempts which ignore this imprecision or approximate it are more or less unrealistic. Although one could continue the list of characteristics, by now the reader should see what is at the heart of the development decision problem. It is lack of precision of all aspects of the problem. Gresford says the same thing in another way
CHARACTERISTICS OF DEVELOPMENT DECISIONS For the purposes of this discussion, several characteristics of the development decision process are assumed. These characteristics have been found to hold over a large range of decision problems related to land development, social development, urban and regional development, fiscal and general economic development and planning. These characteristics are listed. (l) Finite number of possible decisionsmin any well-defined development problem preliminary analysis will indicate a variety of feasible solutions
[71: "To some people, the mention of systems analysis immediately suggests that one is referring to the application of particular techniques to an already well-defined problem or to tile application of various techniques related to computer applications. While computer applications and other techniques such as network analysis, critical path analysis, and PERT may play their part within an appropriate mode of analysis, initial emphasis is more clearly needed on reexamining the ways in which problems are defined for analysis and then in the methodology for analyzing systematically the implications of and
264
S~PH~:N KAHNE
the solutions to such problems. The vital importance of directing initial efforts to the accurate and useful definition of the problem requiring solution cannot be overemphasized. This pertains particularly to problems within the context of national development, as all types of distortions may e n t e r the analytic and subsequent implementation phases of programs suggested by the systems analysis efforts." One may conclude that a necessary condition for methodology for development decision problems is a quantitative technique for handling the type of imprecision which arises in all aspects of development. As well as the imprecision inherent in the formulation of the problem and its components, any realistic view of development must account for the nature of the decision-making process. This refers to the committee structure surrounding all decisionmaking activities,* the public scrutiny to which these decisions are subjected, the special interest groups whose 'contributions' must be included in the process and other such 'noise'. This may be called 'noise' not because it is not desirable, for, indeed, often it is. However, it does tend to camouflage the real issues and this has the effect of electrical noise which often camouflages electrical information signals. Any method devised which ignores the 'noise' is making unwarranted assumptions about the process of development. At this point, the reader may begin to sense that the problem is a hopeless one; that with such inherent imprecision in the data, criteria and other components there is no hope of obtaining the 'best' solution to a development problem. However, upon careful scrutiny, one should observe that it is only unreasonable to expect a precise selection of the 'best' solution. He should, however, be able to obtain an indication of which feasible decision is best with an imprecision no greater than that of the data. This is a rather qualitative discussion of an idea which can be quantified by studying one procedure for handling such problems. In this procedure, the mechanism exists for analyzing development decisions in which qualitative factors are as easily handled as quantitative ones. Again Gresford [7] comments on this situation. "One must stress that boundaries within which the problems are defined---especially complex and critical problems--are often initially unstructured, and quantification may not be readily possible. In dealing with qualitative problems or considerations, the means of taking into account qualitative values and putting them within a 'mix' of qualitative and quantitative variables requires much further effort * We will not discuss those characterized by an autocratic or dictatorial nature.
on the part of the management science communhy. Again, within a development context and in the light of the policy options that presumably art the end result of a systems analysis effort, adequate means must be found to take qualitative factors into the analytic process." Let us examine the procedure referred to above. THE PROCEDURE The procedure is comprised of five components: (1) Goal definition; (2) Criteria definition and measures; (3) Criteria weights; (4) Rating scheme; (5) Ranking scheme. Each component is described in this section. (I) Goal---one must know what he wishes to achieve by making a feasible decision in a development problem. It could be to maximize the standard of living, to maximize average income, to increase farm production, some combination of these, or any of thousands of other indications of level of development. In any case, the goal must be understood at the outset of the decision-making process for without this understanding, it is impossible to determine a rank ordering of consequences of various alternative decisions. (2) Criteria--a de.veiopment decision must be based on criteria, the satisfaction of which are important for successful achievement of the goal. The various feasible decisions will be judged according to the extent to which the various criteria are satisfied. As may be observed, the goal can be defined in terms of these criteria. These criteria may be qualitative or quantitative in nature and may be independent or mutually dependent. If a criterion is qualitative, it will be necessary to quantify it in a very special manner as described below. The quantification must not require unwarranted assumptions to be made regarding the precision of the data. Rather, it must mirror the extent of the imprecision inherent in the data. For each criterion, a scale of 'degree of goal satisfaction' must be developed to quantify what we mean by this 'degree of goal satisfaction.' If a particular alternative satisfies the criterion in question to a very great degree, with respect to the goal, of course, it should rate close to the top of the scale. If it is rather unsatisfactory, it should rate near the bottom of the scale. This general idea has been applied elsewhere and certainly does not need further explanation here. However, in development problems, as discussed above, there are further complications which do need further elaboration. Often, due to imprecision of data, even though one alternative is dearly very good, with respect to a particular criterion and goal, the precise 'degree of goodness' is not dear. The people who must rate a particular alternative solution on a particular criterion may have some difference of opinion.
A procedure for optimizing development decisions Indeed, if they do not, that would be an unusual situation. The range of differences conveys important information and this information must be used in the final decision-making process. To be m o r e specific, let us establish a scale from 0 to 10 so that a 10 indicates that the criterion is perfectly satisfied by one alternative and 0 denotes a completely unsatisfactory rating. Thus, various degrees of satisfaction are reflected by numbers between 0 and 10. If a 'grade' can be given with certainty (e.g. 6"5, 7.0, 9.0), it should be. However, development decisions surrounded by imprecision and uncertainty will often have 'grades' which are ranges of numbers between 0 and 10 (e.g. 2--4, 6-10, 8-9.5). With a lack of additional information one must assume that a-b represents a grade of somewhere between a and b but any number in that region is equally probable. In other words, a grade of a-b is a random number x, uniformly distributed so that xr[a,b]. There will be situations in which a normal or other distribution is more appropriate. However, for purposes of this paper we will always assume uniform distribution. Note that the grade ½(a+b) is a special case of this situation in which xr[~(a +b), ~(a + b)]. This latter comment is particularly significant when computer algorithms are developed to implement the procedure being described. Using this notion of grade, it is possible to rate each criterion. The certainty with which a rating can be assigned is related to the size of the range of rating. Thus, a rating of 0-10 denotes complete uncertainty about this particular grade while a rating of 0 or 2"5 or 8 indicates complete certainty. For group participation in the decision-making process, it has been found useful to predefine the ranges of possible grades. These predefined ranges are called 'measures' and are established, often by the group, prior to actual alternative rating. The task of the raters is then to select the appropriate measure which accurately describes the degree to which the criterion is satisfied. Of course, this must be done for each alternative decision. It would take us far from our objective in writing this paper if we were to totally describe a case study using this method. However, it may be interesting for the reader to see a set of criteria which were developed for a particular development-oriented project. Also, for a few of these criteria the measures are enumerated. The purpose of enumerating these criteria and measures is not so much to give a complete picture of the way the method was used on one project but to indicate the range of criteria and measures which may be developed and emphasize how they may be useful in aesthetic and qualitative-oriented studies as well as quantitative studies. There seems to be no reason to show all numerical ratings associated with the measures 20
265
since to justify them would require a much more thorough discussion of this particular project than is warranted here. The following set of criteria were developed for a project to determine the relative quality of over 150 outdoor recreation areas in the State of Minnesota. In this project there were several categories of recreation areas that were studied (e.g. State Park, State Wildlife Refuge, State Scientific and Historic Area, etc.) and each category of recreation area was evaluated using the same set of criteria and measures. However, the relative importance of the various criteria was determined individually for each category of recreation area. This is handled using numerical criteria weights discussed later in the paper. Here the 29 criteria used for the rating process are simply enumerated. CRITERIA (l) Distance of site from the population center of the Twin Cities Metropolitan area. (2) Location and relation to existing Minnesota resident population on a radius basis. (3) Location and relation to existing nonMinnesota resident population on a radius basis. (4) Conditions of existing access roads from nearest state or federal highway to the site boundary. (5) Relative cost and availability of site. C6) Amount of existing development within the site. (7) Compatibility of existing and anticipated future surrounding land uses within the site. (8) Impact of the site on surrounding land uses. (9) Degrees of pollution and its effect upon the site. (10) Man-made and natural constraints to expansion. (11) Availability of adequate drinking water. (12) Availability of sewage treatment and disposal. (13) Quality of views and visual character. (14) Diversity of topography (land and water, vertical and horizontal). (15) Unique charactersitics (e.g. water areas, land features, outeroppings, variety of environmental features, which enhance the overall quality of the site). (16) Tolerance of vegetation to site development. (17) Suitability of soils and topography for site development. (18) Suitability of land within site boundaries for buffer. (19) Appropriateness of water for water-based recreational activities. (20) Water quality.
STEPHEN K^HNE
266
(21) Significance of historical elements related to the site. (22) Significance of natural or scientific elements related to the site. (23) Historically distinctive characteristics of the site. (24) Integrity of the historical elements of the site. (25) Integrity of the natural or scientific elements of the site. (26) Endangered status of site. (27) Significance of site land form as an example of a landscape region. (28) Significance of site vegetation as an example of a landscape region. (29) Need by recreational planning region. Although some of these criteria are not selfexplanatory, they should be viewed as an indication of the breadth of considerations which can be incorporated in the development decision procedure described in this paper. To illustrate the development of measures for such criteria, several criteria are selected and their measures shown below. EXAMPLES OF M E A S U R E S
(I) Distance of site from the population center of the Twin Cities Metropolitan area. (a) Site located within a 50-mile radius of the population center of the Metropolitan area. (b) Site located within a lO0-mile radius of the population center of the Metropolitan area. (c) Site located within a 150-mile radius of the population center of the Metropolitan area. (d) Site located within a 200-mile radius of the population center of the Metropolitan area. (e) Site located within a 330-mile radius of the population center of the Metropolitan area. Note that the first measure in this list which most accurately describes the situation should be selected. (4) Coiaditions of existing access roads from nearest State or Federal Highway to the site boundary. (a) Site has an existing surfaced access road in good condition and of an adequate width to accommodate proposed traffic. (b) Site has an existing surfaced access road in need of resurfacing but of adequate width to accommodate proposed use. (c) Site has an existing gravel access road of adequate width but would require hard surfacing. (d) Site has an existing gravel or surfaced access road which would require major
renovation, including widening and hard surfacing. (e) Site has no existing access road. (8) Impact of the site upon surrounding land uses.
(a) The site and its intended use would greatly enhance the quality of the surrounding land uses. (b) The site and its intended use would tend to enhance the quality of the surrounding land uses. (c) The site and its intended use would tend not greatly to affect the actual quality of surrounding land uses. (d) The site and its intended use would tend to affect land use patterns in an adverse way. (e) The site and its intended use would tend greatly to affect land use patterns in an adverse way. (13) Quality of views and visual character. (a) The site offers a number of high quality on- and off-site views. It is apparent that these can remain as quality views in the future. (b) The site offers a number of high quality on- and off-site views. (c) The site offers a limited number of quality on- and off-site views. (d) Views from the site are of poor quality, but these views could be eliminated or screened with minimal expenditures. (e) Views from the site are of poor quality and could not be eliminated or screened without large expenditures. Ratings are assigned to these measures so that the satisfaction of a measure results in the selection of a range of numbers which represent the particular measure which is satisfied. In summary then, the rating scheme proposed allows one to indicate with certainty the amount of uncertainty which the data indicates. (3) Criteria weights--another 'degree of freedom' surrounding die development decision process is that various criteria have various relative importances. A decision-making procedure must take this into account. Moreover, often due to the fact that a group is involved in the decision-making process or that the relative criteria weights are not easy to agree on, uncertainty in these weights will exist. The situation is parallel to that for the criteria discussed in the section above. Ranges of numbers may be used to reflect uncertainty in precise criterion weights and the size of the ranges reflects the amount of uncertainty in the weights. For problems of development, it is not uncommon to have difficulty determining precisely how one criterion compares with another in importance.
A procedure for optimizing development decisions This often leads to assumptions about relative importance which are not supported by available data. Since our basic philosophy throughout this discussion is to reflect precisely the uncertainty in the decision process, the use of criterion weights is consistent with our earlier discussions. The question of criteria independence must be discussed here. Let us assume that the range of possible ratings extends over the same interval for each criterion. That is, the only rating which may be assigned to any goal with respect to any criterion is, for example, between zero and ten. Such ratings as 0-10, 0, 3, 4--6, 8.5-10, etc. are thus permitted while 8-11 is not. Then the maximum possible rating a particular alternative may receive with respect to one criterion is the same as with respect to another. But, let us assume that two criteria are dependent. Then a high rating on one may be coupled with a high rating on the other. It is not possible to state, in general, whether this is good or bad, but at least the effect of the dependency must be recognized. One way to avoid the problem is to reduce the weights on the dependent criteria so the overall effect of two (or more) dependent criteria is similar to that of one independent criterion. How this problem is handled depends on the particular situation in which it arises. It would not be fruitful to delineate such problems here. (4) Rating scheme--having defined the goal and indicated the relative importance and degrees of satisfaction of criteria, let us next look at the rating scheme. Each alternative is to be rated with respect to each criterion. The rating consists of selection of the appropriate measure for each alternative/criterion pair. Table I shows a simple example with two alternatives and two criteria. In the case of Table 1, a possible set of measures might be Criterion I:
Criterion II:
Superior
9-I0
Good Average
7-9 5-7
Poor
0-5
Indeed, in the presence of strong evidence justifying it, ranges between 0 and 10 may be assigned in particular situations. But, in any case, one must stick to the fundamental principal underlying the whole approach. That is, to be as precise as possible about the imprecision of data, criteria satisfaction or certainty of rating. In practice this rating step may be accomplished in a number of ways. One possibility is the 'committee of experts' approach. In this case, the rating is done by a group of people with considerable knowledge about the various alternatives. The average rating reflects the estimated degree of goal satisfaction for each alternative for each criterion. The size of the range of the ratings reflects the difference of opinion between the 'experts' or the uncertainty in the data or both. Another possibility is the 'analysis' approach. Here the rating is done by a team or an individual based upon careful research and analysis. In this case, the size of the range of ratings usually reflects the quality of the data or thoroughness of analysis rather than differences of opinion. Politics or 'extra-scientific' influences must be mentioned here. In development decisions, their influences are often prominent and always important. The approach under discussion is well suited to take such influences into account since they can be built into the rating ranges. Often these influences tend to make clear-cut choices or ratings somewhat less certain than they would be without these influences. That situation is precisely the reason for the introduction of the range concept. Using this method, it is possible to involve the political realities directly in the decision-making process rather than attempting to account for these inputs after the analysis is completed. (5) Ranking scheme--having assembled the data as described and prepared a table of these data as shown in Table 1, the remaining effort is largely mechanical. Let us establish a bit of terminology to study the ranking algorithm. Table 2 shows a rewriting of Table 1 for the general problem.
Same as Criterion I
TABLE 2
TABLE 1 Criterion Weight Criterion
Criterion weight
Alternative A
1 2
0-5 8-10
7-9 2--4
267
Alternative B 9-10 5-6
Note that the grade assigned for Alternative B, Criterion II, is 5-6. This indicates that the grade is in the low part of 'average' and illustrates how the actual rating assigned may not coincide with the given range of grades assigned to the given measure.
I 2 3
WL Wi W~
Alternative Alternative Alternative 1 2 n
RLz Rxl Rls
R11 Rn Rn
R~t RMj R.s
In Table 2, R~ is the rating of Alternative i with respect to Criterion j. It is a set of real numbers with elements p(j. Thus p~y~R~y. Generally R~y
STEPHEN KAHNE
268
will be a range of numbers between 0 and 10 (e.g. R u = [7, 9], then P~/might be 7 or 7.2 or 7"5 or 8"3 or ...). Similarly, We is a set of real numbers with elements ~o¢. R u and We may be considered to be a set of random variables distributed uniformly, for example, between specified limits. The form of the distribution is to be determined by the nature of the data made available. However, it should be selected to reflect the realistic state of knowledge about the data. The ranking of alternatives proceeds as follows. Using an appropriate random number generator to select the Pij and ~o~, from the sum of products
P~ = ~,~jp~j, ]-1
i = 1,2 .... ,m.
These are the sum of products for the ith alternative. When m of them have been calculated, one for each alternative, order the Pc with respect to magnitude and record the ordering. The process is then repeated many times, usually hundreds of times, and a cumulative record of the ordering of the P, is maintained. The P, which ranks first the largest percentage of runs indicates which alternative tends to be the 'best' solution to the problem. Table 3 shows a type of matrix which may result for the two alternative examples of Table 1. Clearly, since Alternative I must rank either first or second on each run and Alternative 1 or Alternative 2 must rank first on each run, we see that the row sums and column sums in Table 3 will always equal 100 per cent. This is, of course, true for the general case of M alternatives. TABLE 3
Alternative I Alternative 2
Rank first
Rank second
(%)
(%)
35 65
65 35
One's confidence in the result will depend upon how large a percentage of runs one alternative achieves a particular ranking. The interpretation of the results should now be obvious. No matter what value of P¢~ or w~ one chooses as his own, if he agrees that the sets R~j and We accurately represent the state of knowledge about the criterion and alternative under consideration, the degree of uncertainty of a given ranking of an alternative is indicated. In certain situations a rank of first 50 per cent of the runs might be considered a sufficiently positive result to establish confidvnea in the selection. In other situations, it may require a 90 per cent occurrence to establish this confidence.
A MATHEMATICAL CAVEAT
At the heart of the method is the requirement to weigh, with uncertain weighting, an uncertain rating. We have chosen to accomplish this by the multiplication of two uncertain factors and, mathematically, by the multiplication of two random variables. Each may be thought of as a sample from a distribution of possible variable values. In many types of problems the most reasonable distribution is uniform in both weights and ratings. By using the uniform distribution for a weight 'ten to eight', one is simply stating that "I am certain that the correct weighting factor lies somewhere between 10 and 8 but I have no idea what value in this range is most likely. Therefore, to ensure that I say precisely what I mean I will assume that all values between 10 and 8 are equally likely." In many development situations this is the precise state of affairs. Although one can say with certainty that the range of weights, or value of the measure, is correct, he cannot say more. So, we are confronted with the problem of multiplying two random variables, each from independent uniformly distributed sets. It is known that if the distributions for co~ and or Po are uniform, then the distribution of coep~j is not uniform and is, in fact, logarithmic. Therefore, the most likely value of the product is near the low end of possible products. In case the range of ~o~ or p~j includes zero, there is a high probability that the product will tend toward zero. If both ranges have minimum values of zero the mean value of the product distribution is about 20 per cent of the maximum value of this distribution. Therefore, one's lack of certainty about weight or rating will tend to reduce the value of the product. One way to reduce this bias is to shift the distribution of coi or p~j far from zero before the multiplication and then normalize the result to the correct interval. Since the logarithm changes rather slowly in narrow ranges far from zero, the effect of this is to make the distribution of the product more nearly uniform. In this case, the product distribution would more nearly conform to the desired shape from the point of view of the designer. That is, the distribution of the weighted ratings should reflect uniform uncertainty of the product within the appropriate range. CONCLUSIONS AND COMMENTS
A method has been proposed, and used in practice, which has the potential to resolve certain development decision problems. The data it requires and the type of results it produces are ideally suited to the development decision situation. The paper explains how the method fits into the decision-making process faced by organizations dealing with development problems.
A procedure for optimizing development decisions
The method is one step in the process called for by Doxiadis [9] to develop procedures for decision making in the presence of realistic constraints. It must be further refined and combined with other processes not considered here. One such process is effective data aggregation. Another is decision implementation. Many others are also needed. When comparing the approach described in this paper with other possible approaches, one is led to the conclusion thai few other proposals are effective in solving this class of problems. As mentioned earlier, the notion of fuzzy sets seems to reflect accurately the imprecisions inherent in development problems. Nevertheless, there have been no reported efforts to use this theory to solve such problems. One may surmise that it is a bit early to expect that a new theory will find application. This situation is to be contrasted with one in which the problem itself stimulates the development of a pragmatic, effective technique for solution. The latter more accurately reflects the contribution of this paper. Acknowledgements--The application of the procedure to outdoor recreational sites occurred in the author's position as Consulting Partner of InterDesign Inc., an environmental design firm in Minneapolis. He is indebted to his
269
colleagues at InterDesign for their contributions to the work of this project. Roger Martin, ASLA landscape architect, played a key role in the development of the criteria enumerated in the paper.
REFERENCES [I] T. S. KOOPMANS(ed.): Activity Analysis of Production and Allocation. Wiley, New York (1951). [2] S. KA~NE: Report from Dresden. Automatica 8, 513515 (1972). [3] O. RADEMAKERas quoted by S. KAHNE: Rel~ort from Dresden. Automatica 8, 513-515 (1972). [4] IFAC Symposium: Dynamic Modelling and Control of National Economics. University of Warwick, Coventry, England. June (1973). [5] S. K.~.MNE, Review of Urban Dynamics (by J. W. FORR~Tt=R): IEEE Trans. AC- 16, 214-215 (1971 ); also Automatiki i Telenlckhanika, 8b, 203-205 (1971). [6] E. B. LEE and L. MARKUS; Foundatwns of Optimal Control. Wiley, New York (1967). [7] G. B. GRESFORD:Systems approach for development. IEEE Trans. AMC-2, 311-318 July (1972). [8] PROJEC'r 80 Report--State Planning Agency, St. Paul, Minnesota (197 I). " [9] C. A. Doxt^ms: The two-headed eagle. Ekistics 33, No. 198 (1972). [10] R. C. FtSHaURN: Utility Theory for Decision Making. Wiley, New York (1970). [11] R. E. BELLMANand L. A. ZADEH: Decision making in a fuzzy environment. Management Sci. 17, pp. B-141B-164 (1970).
A Procedure for Optimizing Development Decisions* Un Processus pour l'Optimisation de D~cisions de D6veloppement Eine Prozedur zur Optimierung von Entwicklungsentscheidungen STEPHEN KAHNEt
Qualitative and uncertain data play an important role in optimizing development decisions Summary--This paper addresses the problem of organizing data and formulating questions to be answered for the purpose of making planning and development decisions. The problem is separated into five distinct parts. Each part is discussed in the context of the planning process and each leads logically to the optimization of development decisions. The five parts are goal definition, establishment of criteria, criteria weighting, alternative rating and alternative ranking. The feature of making development decisions which distinguishes them from other optimization problems is what has been called 'fuzziness'. In any realistic problem formulation, the criteria are not precisely defined; they are fuzzy. The relative importance of each criterion is also fuzzy. Indeed, even when one attempts to rate a particular possible solution, he must deal with fuzzy information. The technique proposed in the paper accounts for this uncertainty in all aspects of the problem and yields probabilistic answers. Thus, when various alternative solutions are proposed for a development problem, the technique yields a probabilistic ranking of the alternatives. Sharper results are obtained if less uncertainty is present in certain parts of the data. However, even in the presence of great uncertainty, realistic problem solutions are obtained. Alternative solutions are rated independent of all others and only after the (fuzzy) ratings are complete are comparative rankings accomplished. Throughout the procedure the realistic uncertainties remain a prominent feature of the procedure.
More recently, decision theory and the more quantitative disciplines o f operations research and queuing theory have been brought to bear on such problems. But, inherent in all approaches to date has been a requirement to specify criteria, constraints, optimality goals and other parameters to an extent which one often finds to be unreasonable in actual decision making situations It is appropriate to expand on this particular point. In most real situations as contrasted with models of situations, the data which are at one's disposal are rather ill defined and imprecise. Intuitive feelings and inductive judgement play an important role in everyday life and decision making. The goal o f quantitative methods is to help focus on key issues and organize available data to assist the decision maker in his often unfortunate task of 'resolving' a problem and enunciating a decision. If these quantitative methods are properly applied they do indeed provide a structure for decision making. In the hands o f a competent executive, they can be an invaluable tool for problem solving. Probably one of the best examples of successful development of quantitative tools of this type is linear programming developed by a number of people at the end o f World War II. Chief among this group was George Dantzig whose work on the formulation of linear programming problems and a rather elegant solution, the simplex method, was reported in 1951 [1]. Numerous papers, books and monographs attest to the widespread utility and versatility o f linear programming. Applications ranging from transportation to machine utilization and covering a wide range o f other scientific and management-oriented problems have been found to be amenable to this quantitative tool. Given its widespread utility, attempts are made to couch problems in terms so that methods of linear programming can be brought to bear on them. This type of progression has been common in the development of other quantitative tools. First, a
INTRODUCTION THE PXOm-~ of organizing data and formulating appropriate questions to be answered for the purpose of making planning and development decisions is one which has received considerable attention in the management sciences. The disciplines of social psychology, group dynamics and other behavioral sciences have been discussed. * Received 10 January, 1974; revised 6 August 1974; revised 11 October 1974. The original version of this paper was presented at the IFAC/IFORS Symposium on Systems Approaches to Developing Countries which was held in Algiers, Algeria, during May 1973. The published Proceedings of this IFAC meeting may be ordered from ISA. It was recommended for publication in revised form by associate editor H. Kwakernaak. ? Associate Professor, Department of Electrical Engineering, Director, Hybrid Computer Laboratory, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
261
262
STEPHEN KAHNE
mathematical technique is invented or discovered. Often this occurs in the course of trying to solve a particular 'real world' problem. For example, in the case of linear programming, the problems involved resource allocation. Occasionally it is an independently studied procedure which is soon recognized to be of use in an interesting class of problems. The second phase of the progression is the careful refinement of the technique. Assumptions are examined. Rigorous-derivations are developed and extensions to the original method are discovered. A rather complete theory of the technique is established and many loose technical ends are tied up. Thii'dly, often in parallel with the second phase, the method -is used on a variety of problems. The class of problems for which the method is applicable is carefully searched. More and more interesting variations of the basic problem are discovered until finally a seemingly exhaustive collection of problems has been delineated and solved. As this step proceeds, it is not uncommon that continued advances are made in the theory of the technique and that subsequent refinements lead to subsequent variations in applicable problems. The fourth step is the final one. In the mature life of the technique, as many researchers become intimately familiar with it, attempts are made to change real world problems into problems which can be solved by 'the method'. Nonlinear problems are linearized. Stochastic problems are made deterministic. Distributed systems are considered to be lumped. Time-varying systems are considered stationary. The philosophy accompanying these asssumptions is understandable, if not valid. The technique exists, is well defined and well understood but, unfortunately, is not applicable to the problem at hand. So, if the problem were only approximated in such a way that the technique was useful, then a solution could be realized. Of course, the solution is a solution to the approximated problem, but one can hope that it will be an approximate solution to the real problem. This kind of reasoning, although pervasive, is often not persuasive. One of the biggest faults of modern automatic control education today is this lack of follow-through between real world problems and real world solutions [2]. The problem, of course, is that it is much harder to develop a solution technique ideally suited to a particular real world problem than it is to alter a real world problem so that it conforms to the assumptions of a well-developed technique. This interchanging of roles between problem and solution technique is at the heart of the issue. This confusion is not restricted to strictly scientific problems of automatic control. In this field, it has been noted that we have devoted much
time to 'problems of solution' and not enough to 'solutions of problems' [3]. However, this malady has certainly affected studies of problems of national development as well. Simulation and modelling of national economies [4] and various sectors of an urban society have resulted in glorious predictions of growth or decay. Indeed, the models described in these studies do behave as reported. The effect of running the model with certain given initial conditions does lead to the changes of model variables which are described in the literature. However, what that has to do with behavior of the urban or economic system under study is not at all clear, and how the results of the simulation studies should be applied to particular urban or national development programs is far from evident. The amount of idealism which is injected into the simulation most often completely obliterates the true nature of the process being studied, and the subtle dynamic processes at work which are the driving force of the system are often obscured beyond recognition. The author has alluded to certain of these difticulties in his review of a landmark book on urban dynamics [5]. The work described in this paper represents one approach to the class of problems described. It is designed specifically for these problems and has been used in several development applications. One could contrast it with, for example, Zadeh's 'fuzzy' approach which, at this time, must be characterized as being in the 'machinery' stage [11 ]. A great deal of mathematical machinery has been produced but few practical problems have yet been solved via 'fuzzy' system theory. The author happens to believe that these 'fuzzy' notions do, indeed, form the foundations for a basic and important approach to the solution of realistic development problems but that much work remains to bridge the 'fuzzy gap' between 'fuzzy theory' and practice. In much of this section we have at least suggested that a common underlying feature of decision processes is a model which may be either simulated or constructed~ the study of which leads to a recommended solution to a given problem. But is this a necessary assumption .9 Is a model always at the heart of a decision-making process? In the following section we suggest not; we observe that one may proceed along quite logical lines with no model at all; and, in fact, in many problems of development, such an approach is the preferred one. THE MODELLESS DECISION PROCESS Lot us examine the modelless decision process. This is characterized by an emphasis on the consequences of the decision, not the dynamics of the systems about which a decision is to be made.
A procedure for optimizing development decisions
263
Let us not overstate our case. The system dynamics will determine the consequences of a certain decision, but we will concentrate only on the decision and the consequences. This approach, called input--output analysis in the economic and social sciences and the transfer function approach in the study of automatic control systems, is certainly not new. The degree to which the system dynamics are ignored depends on the particular problem at hand. Practical measurements on physical systems are usually geared to I/O analyses either on a microscopic or macroscopic scale depending on the availability of certain data, instruments and techniques. Special theories have been established to show when I/O analyses are expected to work and when they are expected to fail. The notion of observability in systems theory [6] is specifically directed at this question of when I/O analysis is sufficiently rich to describe completely system behavior. The key point here is that there is no effort required to develop a detailed model and, in fact, no need to develop a model at all provided one is dealing with a real system. The emphasis is on the decision. But, how does one select the best decision? After all, many decisions are usually possible and each one has certain consequences which must be studied. Indeed, by studying in great detail a plethora of decisions and their consequences one might begin to construct a model of the process (or system) which is transforming these decisions into consequences. But this method building is clearly subordinate to the main task of decision making. This characteristic of emphasizing decision consequences rather than models seems to be a rather pragmatic and realistic one in the case of making decisions for regional or national development. With this explanation of a de-emphasis on modeling and with an understanding that development decisions must be made in existing systems of economy, politics and social order, we may now turn our attention toward a procedure for optimizing development decision.
while eliminating a host of unacceptable alternatives. A procedure used to select the best solution must compare only the feasible ones. (2) Well-defined goal--although the 'best' decision may be best with respect to a very complex goal, this goal must be defined well enough to allow comparisons of degrees of goal satisfaction for various feasible solutions. In the parlance of control theory, the goal may be a scalar or a vector performance index as long as this condition is satisfied. (3) Precision of data--unfortunately, when solving 'real world" problems such as those involved in development decisions, much important data are either missing or, at best, vague. It is because this imprecision exists that most methods developed have limited usefulness. There are several approaches usually taken in this case. If the dimension of the problem is very small, there are certain stochastic optimization approaches which may prove to be effective. By definition, a real world problem in the general area of development is not one of small dimension [7]. On the other hand, if the system to be optimized is large or if the data are too imprecise to fit the standard decisionmaking techniques, approximations to the system or refinement of the data are commonly tried. Unfortunately, the approximations and/or refinements are usually not warranted from the information at hand. The unwarranted refinements or approximations are made to fit the problem to the method, a rather unreasonable approach to take if it could be avoided. (4) Precision of criteria--we have seen that imprecise data are a characteristic of these development decisions. In the same way, the criteria for judging the success of one decision or another are, regrettably, imprecise. Any attempts which ignore this imprecision or approximate it are more or less unrealistic. Although one could continue the list of characteristics, by now the reader should see what is at the heart of the development decision problem. It is lack of precision of all aspects of the problem. Gresford says the same thing in another way
CHARACTERISTICS OF DEVELOPMENT DECISIONS For the purposes of this discussion, several characteristics of the development decision process are assumed. These characteristics have been found to hold over a large range of decision problems related to land development, social development, urban and regional development, fiscal and general economic development and planning. These characteristics are listed. (l) Finite number of possible decisionsmin any well-defined development problem preliminary analysis will indicate a variety of feasible solutions
[71: "To some people, the mention of systems analysis immediately suggests that one is referring to the application of particular techniques to an already well-defined problem or to tile application of various techniques related to computer applications. While computer applications and other techniques such as network analysis, critical path analysis, and PERT may play their part within an appropriate mode of analysis, initial emphasis is more clearly needed on reexamining the ways in which problems are defined for analysis and then in the methodology for analyzing systematically the implications of and
264
S~PH~:N KAHNE
the solutions to such problems. The vital importance of directing initial efforts to the accurate and useful definition of the problem requiring solution cannot be overemphasized. This pertains particularly to problems within the context of national development, as all types of distortions may e n t e r the analytic and subsequent implementation phases of programs suggested by the systems analysis efforts." One may conclude that a necessary condition for methodology for development decision problems is a quantitative technique for handling the type of imprecision which arises in all aspects of development. As well as the imprecision inherent in the formulation of the problem and its components, any realistic view of development must account for the nature of the decision-making process. This refers to the committee structure surrounding all decisionmaking activities,* the public scrutiny to which these decisions are subjected, the special interest groups whose 'contributions' must be included in the process and other such 'noise'. This may be called 'noise' not because it is not desirable, for, indeed, often it is. However, it does tend to camouflage the real issues and this has the effect of electrical noise which often camouflages electrical information signals. Any method devised which ignores the 'noise' is making unwarranted assumptions about the process of development. At this point, the reader may begin to sense that the problem is a hopeless one; that with such inherent imprecision in the data, criteria and other components there is no hope of obtaining the 'best' solution to a development problem. However, upon careful scrutiny, one should observe that it is only unreasonable to expect a precise selection of the 'best' solution. He should, however, be able to obtain an indication of which feasible decision is best with an imprecision no greater than that of the data. This is a rather qualitative discussion of an idea which can be quantified by studying one procedure for handling such problems. In this procedure, the mechanism exists for analyzing development decisions in which qualitative factors are as easily handled as quantitative ones. Again Gresford [7] comments on this situation. "One must stress that boundaries within which the problems are defined---especially complex and critical problems--are often initially unstructured, and quantification may not be readily possible. In dealing with qualitative problems or considerations, the means of taking into account qualitative values and putting them within a 'mix' of qualitative and quantitative variables requires much further effort * We will not discuss those characterized by an autocratic or dictatorial nature.
on the part of the management science communhy. Again, within a development context and in the light of the policy options that presumably art the end result of a systems analysis effort, adequate means must be found to take qualitative factors into the analytic process." Let us examine the procedure referred to above. THE PROCEDURE The procedure is comprised of five components: (1) Goal definition; (2) Criteria definition and measures; (3) Criteria weights; (4) Rating scheme; (5) Ranking scheme. Each component is described in this section. (I) Goal---one must know what he wishes to achieve by making a feasible decision in a development problem. It could be to maximize the standard of living, to maximize average income, to increase farm production, some combination of these, or any of thousands of other indications of level of development. In any case, the goal must be understood at the outset of the decision-making process for without this understanding, it is impossible to determine a rank ordering of consequences of various alternative decisions. (2) Criteria--a de.veiopment decision must be based on criteria, the satisfaction of which are important for successful achievement of the goal. The various feasible decisions will be judged according to the extent to which the various criteria are satisfied. As may be observed, the goal can be defined in terms of these criteria. These criteria may be qualitative or quantitative in nature and may be independent or mutually dependent. If a criterion is qualitative, it will be necessary to quantify it in a very special manner as described below. The quantification must not require unwarranted assumptions to be made regarding the precision of the data. Rather, it must mirror the extent of the imprecision inherent in the data. For each criterion, a scale of 'degree of goal satisfaction' must be developed to quantify what we mean by this 'degree of goal satisfaction.' If a particular alternative satisfies the criterion in question to a very great degree, with respect to the goal, of course, it should rate close to the top of the scale. If it is rather unsatisfactory, it should rate near the bottom of the scale. This general idea has been applied elsewhere and certainly does not need further explanation here. However, in development problems, as discussed above, there are further complications which do need further elaboration. Often, due to imprecision of data, even though one alternative is dearly very good, with respect to a particular criterion and goal, the precise 'degree of goodness' is not dear. The people who must rate a particular alternative solution on a particular criterion may have some difference of opinion.
A procedure for optimizing development decisions Indeed, if they do not, that would be an unusual situation. The range of differences conveys important information and this information must be used in the final decision-making process. To be m o r e specific, let us establish a scale from 0 to 10 so that a 10 indicates that the criterion is perfectly satisfied by one alternative and 0 denotes a completely unsatisfactory rating. Thus, various degrees of satisfaction are reflected by numbers between 0 and 10. If a 'grade' can be given with certainty (e.g. 6"5, 7.0, 9.0), it should be. However, development decisions surrounded by imprecision and uncertainty will often have 'grades' which are ranges of numbers between 0 and 10 (e.g. 2--4, 6-10, 8-9.5). With a lack of additional information one must assume that a-b represents a grade of somewhere between a and b but any number in that region is equally probable. In other words, a grade of a-b is a random number x, uniformly distributed so that xr[a,b]. There will be situations in which a normal or other distribution is more appropriate. However, for purposes of this paper we will always assume uniform distribution. Note that the grade ½(a+b) is a special case of this situation in which xr[~(a +b), ~(a + b)]. This latter comment is particularly significant when computer algorithms are developed to implement the procedure being described. Using this notion of grade, it is possible to rate each criterion. The certainty with which a rating can be assigned is related to the size of the range of rating. Thus, a rating of 0-10 denotes complete uncertainty about this particular grade while a rating of 0 or 2"5 or 8 indicates complete certainty. For group participation in the decision-making process, it has been found useful to predefine the ranges of possible grades. These predefined ranges are called 'measures' and are established, often by the group, prior to actual alternative rating. The task of the raters is then to select the appropriate measure which accurately describes the degree to which the criterion is satisfied. Of course, this must be done for each alternative decision. It would take us far from our objective in writing this paper if we were to totally describe a case study using this method. However, it may be interesting for the reader to see a set of criteria which were developed for a particular development-oriented project. Also, for a few of these criteria the measures are enumerated. The purpose of enumerating these criteria and measures is not so much to give a complete picture of the way the method was used on one project but to indicate the range of criteria and measures which may be developed and emphasize how they may be useful in aesthetic and qualitative-oriented studies as well as quantitative studies. There seems to be no reason to show all numerical ratings associated with the measures 20
265
since to justify them would require a much more thorough discussion of this particular project than is warranted here. The following set of criteria were developed for a project to determine the relative quality of over 150 outdoor recreation areas in the State of Minnesota. In this project there were several categories of recreation areas that were studied (e.g. State Park, State Wildlife Refuge, State Scientific and Historic Area, etc.) and each category of recreation area was evaluated using the same set of criteria and measures. However, the relative importance of the various criteria was determined individually for each category of recreation area. This is handled using numerical criteria weights discussed later in the paper. Here the 29 criteria used for the rating process are simply enumerated. CRITERIA (l) Distance of site from the population center of the Twin Cities Metropolitan area. (2) Location and relation to existing Minnesota resident population on a radius basis. (3) Location and relation to existing nonMinnesota resident population on a radius basis. (4) Conditions of existing access roads from nearest state or federal highway to the site boundary. (5) Relative cost and availability of site. C6) Amount of existing development within the site. (7) Compatibility of existing and anticipated future surrounding land uses within the site. (8) Impact of the site on surrounding land uses. (9) Degrees of pollution and its effect upon the site. (10) Man-made and natural constraints to expansion. (11) Availability of adequate drinking water. (12) Availability of sewage treatment and disposal. (13) Quality of views and visual character. (14) Diversity of topography (land and water, vertical and horizontal). (15) Unique charactersitics (e.g. water areas, land features, outeroppings, variety of environmental features, which enhance the overall quality of the site). (16) Tolerance of vegetation to site development. (17) Suitability of soils and topography for site development. (18) Suitability of land within site boundaries for buffer. (19) Appropriateness of water for water-based recreational activities. (20) Water quality.
STEPHEN K^HNE
266
(21) Significance of historical elements related to the site. (22) Significance of natural or scientific elements related to the site. (23) Historically distinctive characteristics of the site. (24) Integrity of the historical elements of the site. (25) Integrity of the natural or scientific elements of the site. (26) Endangered status of site. (27) Significance of site land form as an example of a landscape region. (28) Significance of site vegetation as an example of a landscape region. (29) Need by recreational planning region. Although some of these criteria are not selfexplanatory, they should be viewed as an indication of the breadth of considerations which can be incorporated in the development decision procedure described in this paper. To illustrate the development of measures for such criteria, several criteria are selected and their measures shown below. EXAMPLES OF M E A S U R E S
(I) Distance of site from the population center of the Twin Cities Metropolitan area. (a) Site located within a 50-mile radius of the population center of the Metropolitan area. (b) Site located within a lO0-mile radius of the population center of the Metropolitan area. (c) Site located within a 150-mile radius of the population center of the Metropolitan area. (d) Site located within a 200-mile radius of the population center of the Metropolitan area. (e) Site located within a 330-mile radius of the population center of the Metropolitan area. Note that the first measure in this list which most accurately describes the situation should be selected. (4) Coiaditions of existing access roads from nearest State or Federal Highway to the site boundary. (a) Site has an existing surfaced access road in good condition and of an adequate width to accommodate proposed traffic. (b) Site has an existing surfaced access road in need of resurfacing but of adequate width to accommodate proposed use. (c) Site has an existing gravel access road of adequate width but would require hard surfacing. (d) Site has an existing gravel or surfaced access road which would require major
renovation, including widening and hard surfacing. (e) Site has no existing access road. (8) Impact of the site upon surrounding land uses.
(a) The site and its intended use would greatly enhance the quality of the surrounding land uses. (b) The site and its intended use would tend to enhance the quality of the surrounding land uses. (c) The site and its intended use would tend not greatly to affect the actual quality of surrounding land uses. (d) The site and its intended use would tend to affect land use patterns in an adverse way. (e) The site and its intended use would tend greatly to affect land use patterns in an adverse way. (13) Quality of views and visual character. (a) The site offers a number of high quality on- and off-site views. It is apparent that these can remain as quality views in the future. (b) The site offers a number of high quality on- and off-site views. (c) The site offers a limited number of quality on- and off-site views. (d) Views from the site are of poor quality, but these views could be eliminated or screened with minimal expenditures. (e) Views from the site are of poor quality and could not be eliminated or screened without large expenditures. Ratings are assigned to these measures so that the satisfaction of a measure results in the selection of a range of numbers which represent the particular measure which is satisfied. In summary then, the rating scheme proposed allows one to indicate with certainty the amount of uncertainty which the data indicates. (3) Criteria weights--another 'degree of freedom' surrounding die development decision process is that various criteria have various relative importances. A decision-making procedure must take this into account. Moreover, often due to the fact that a group is involved in the decision-making process or that the relative criteria weights are not easy to agree on, uncertainty in these weights will exist. The situation is parallel to that for the criteria discussed in the section above. Ranges of numbers may be used to reflect uncertainty in precise criterion weights and the size of the ranges reflects the amount of uncertainty in the weights. For problems of development, it is not uncommon to have difficulty determining precisely how one criterion compares with another in importance.
A procedure for optimizing development decisions This often leads to assumptions about relative importance which are not supported by available data. Since our basic philosophy throughout this discussion is to reflect precisely the uncertainty in the decision process, the use of criterion weights is consistent with our earlier discussions. The question of criteria independence must be discussed here. Let us assume that the range of possible ratings extends over the same interval for each criterion. That is, the only rating which may be assigned to any goal with respect to any criterion is, for example, between zero and ten. Such ratings as 0-10, 0, 3, 4--6, 8.5-10, etc. are thus permitted while 8-11 is not. Then the maximum possible rating a particular alternative may receive with respect to one criterion is the same as with respect to another. But, let us assume that two criteria are dependent. Then a high rating on one may be coupled with a high rating on the other. It is not possible to state, in general, whether this is good or bad, but at least the effect of the dependency must be recognized. One way to avoid the problem is to reduce the weights on the dependent criteria so the overall effect of two (or more) dependent criteria is similar to that of one independent criterion. How this problem is handled depends on the particular situation in which it arises. It would not be fruitful to delineate such problems here. (4) Rating scheme--having defined the goal and indicated the relative importance and degrees of satisfaction of criteria, let us next look at the rating scheme. Each alternative is to be rated with respect to each criterion. The rating consists of selection of the appropriate measure for each alternative/criterion pair. Table I shows a simple example with two alternatives and two criteria. In the case of Table 1, a possible set of measures might be Criterion I:
Criterion II:
Superior
9-I0
Good Average
7-9 5-7
Poor
0-5
Indeed, in the presence of strong evidence justifying it, ranges between 0 and 10 may be assigned in particular situations. But, in any case, one must stick to the fundamental principal underlying the whole approach. That is, to be as precise as possible about the imprecision of data, criteria satisfaction or certainty of rating. In practice this rating step may be accomplished in a number of ways. One possibility is the 'committee of experts' approach. In this case, the rating is done by a group of people with considerable knowledge about the various alternatives. The average rating reflects the estimated degree of goal satisfaction for each alternative for each criterion. The size of the range of the ratings reflects the difference of opinion between the 'experts' or the uncertainty in the data or both. Another possibility is the 'analysis' approach. Here the rating is done by a team or an individual based upon careful research and analysis. In this case, the size of the range of ratings usually reflects the quality of the data or thoroughness of analysis rather than differences of opinion. Politics or 'extra-scientific' influences must be mentioned here. In development decisions, their influences are often prominent and always important. The approach under discussion is well suited to take such influences into account since they can be built into the rating ranges. Often these influences tend to make clear-cut choices or ratings somewhat less certain than they would be without these influences. That situation is precisely the reason for the introduction of the range concept. Using this method, it is possible to involve the political realities directly in the decision-making process rather than attempting to account for these inputs after the analysis is completed. (5) Ranking scheme--having assembled the data as described and prepared a table of these data as shown in Table 1, the remaining effort is largely mechanical. Let us establish a bit of terminology to study the ranking algorithm. Table 2 shows a rewriting of Table 1 for the general problem.
Same as Criterion I
TABLE 2
TABLE 1 Criterion Weight Criterion
Criterion weight
Alternative A
1 2
0-5 8-10
7-9 2--4
267
Alternative B 9-10 5-6
Note that the grade assigned for Alternative B, Criterion II, is 5-6. This indicates that the grade is in the low part of 'average' and illustrates how the actual rating assigned may not coincide with the given range of grades assigned to the given measure.
I 2 3
WL Wi W~
Alternative Alternative Alternative 1 2 n
RLz Rxl Rls
R11 Rn Rn
R~t RMj R.s
In Table 2, R~ is the rating of Alternative i with respect to Criterion j. It is a set of real numbers with elements p(j. Thus p~y~R~y. Generally R~y
STEPHEN KAHNE
268
will be a range of numbers between 0 and 10 (e.g. R u = [7, 9], then P~/might be 7 or 7.2 or 7"5 or 8"3 or ...). Similarly, We is a set of real numbers with elements ~o¢. R u and We may be considered to be a set of random variables distributed uniformly, for example, between specified limits. The form of the distribution is to be determined by the nature of the data made available. However, it should be selected to reflect the realistic state of knowledge about the data. The ranking of alternatives proceeds as follows. Using an appropriate random number generator to select the Pij and ~o~, from the sum of products
P~ = ~,~jp~j, ]-1
i = 1,2 .... ,m.
These are the sum of products for the ith alternative. When m of them have been calculated, one for each alternative, order the Pc with respect to magnitude and record the ordering. The process is then repeated many times, usually hundreds of times, and a cumulative record of the ordering of the P, is maintained. The P, which ranks first the largest percentage of runs indicates which alternative tends to be the 'best' solution to the problem. Table 3 shows a type of matrix which may result for the two alternative examples of Table 1. Clearly, since Alternative I must rank either first or second on each run and Alternative 1 or Alternative 2 must rank first on each run, we see that the row sums and column sums in Table 3 will always equal 100 per cent. This is, of course, true for the general case of M alternatives. TABLE 3
Alternative I Alternative 2
Rank first
Rank second
(%)
(%)
35 65
65 35
One's confidence in the result will depend upon how large a percentage of runs one alternative achieves a particular ranking. The interpretation of the results should now be obvious. No matter what value of P¢~ or w~ one chooses as his own, if he agrees that the sets R~j and We accurately represent the state of knowledge about the criterion and alternative under consideration, the degree of uncertainty of a given ranking of an alternative is indicated. In certain situations a rank of first 50 per cent of the runs might be considered a sufficiently positive result to establish confidvnea in the selection. In other situations, it may require a 90 per cent occurrence to establish this confidence.
A MATHEMATICAL CAVEAT
At the heart of the method is the requirement to weigh, with uncertain weighting, an uncertain rating. We have chosen to accomplish this by the multiplication of two uncertain factors and, mathematically, by the multiplication of two random variables. Each may be thought of as a sample from a distribution of possible variable values. In many types of problems the most reasonable distribution is uniform in both weights and ratings. By using the uniform distribution for a weight 'ten to eight', one is simply stating that "I am certain that the correct weighting factor lies somewhere between 10 and 8 but I have no idea what value in this range is most likely. Therefore, to ensure that I say precisely what I mean I will assume that all values between 10 and 8 are equally likely." In many development situations this is the precise state of affairs. Although one can say with certainty that the range of weights, or value of the measure, is correct, he cannot say more. So, we are confronted with the problem of multiplying two random variables, each from independent uniformly distributed sets. It is known that if the distributions for co~ and or Po are uniform, then the distribution of coep~j is not uniform and is, in fact, logarithmic. Therefore, the most likely value of the product is near the low end of possible products. In case the range of ~o~ or p~j includes zero, there is a high probability that the product will tend toward zero. If both ranges have minimum values of zero the mean value of the product distribution is about 20 per cent of the maximum value of this distribution. Therefore, one's lack of certainty about weight or rating will tend to reduce the value of the product. One way to reduce this bias is to shift the distribution of coi or p~j far from zero before the multiplication and then normalize the result to the correct interval. Since the logarithm changes rather slowly in narrow ranges far from zero, the effect of this is to make the distribution of the product more nearly uniform. In this case, the product distribution would more nearly conform to the desired shape from the point of view of the designer. That is, the distribution of the weighted ratings should reflect uniform uncertainty of the product within the appropriate range. CONCLUSIONS AND COMMENTS
A method has been proposed, and used in practice, which has the potential to resolve certain development decision problems. The data it requires and the type of results it produces are ideally suited to the development decision situation. The paper explains how the method fits into the decision-making process faced by organizations dealing with development problems.
A procedure for optimizing development decisions
The method is one step in the process called for by Doxiadis [9] to develop procedures for decision making in the presence of realistic constraints. It must be further refined and combined with other processes not considered here. One such process is effective data aggregation. Another is decision implementation. Many others are also needed. When comparing the approach described in this paper with other possible approaches, one is led to the conclusion thai few other proposals are effective in solving this class of problems. As mentioned earlier, the notion of fuzzy sets seems to reflect accurately the imprecisions inherent in development problems. Nevertheless, there have been no reported efforts to use this theory to solve such problems. One may surmise that it is a bit early to expect that a new theory will find application. This situation is to be contrasted with one in which the problem itself stimulates the development of a pragmatic, effective technique for solution. The latter more accurately reflects the contribution of this paper. Acknowledgements--The application of the procedure to outdoor recreational sites occurred in the author's position as Consulting Partner of InterDesign Inc., an environmental design firm in Minneapolis. He is indebted to his
269
colleagues at InterDesign for their contributions to the work of this project. Roger Martin, ASLA landscape architect, played a key role in the development of the criteria enumerated in the paper.
REFERENCES [I] T. S. KOOPMANS(ed.): Activity Analysis of Production and Allocation. Wiley, New York (1951). [2] S. KA~NE: Report from Dresden. Automatica 8, 513515 (1972). [3] O. RADEMAKERas quoted by S. KAHNE: Rel~ort from Dresden. Automatica 8, 513-515 (1972). [4] IFAC Symposium: Dynamic Modelling and Control of National Economics. University of Warwick, Coventry, England. June (1973). [5] S. K.~.MNE, Review of Urban Dynamics (by J. W. FORR~Tt=R): IEEE Trans. AC- 16, 214-215 (1971 ); also Automatiki i Telenlckhanika, 8b, 203-205 (1971). [6] E. B. LEE and L. MARKUS; Foundatwns of Optimal Control. Wiley, New York (1967). [7] G. B. GRESFORD:Systems approach for development. IEEE Trans. AMC-2, 311-318 July (1972). [8] PROJEC'r 80 Report--State Planning Agency, St. Paul, Minnesota (197 I). " [9] C. A. Doxt^ms: The two-headed eagle. Ekistics 33, No. 198 (1972). [10] R. C. FtSHaURN: Utility Theory for Decision Making. Wiley, New York (1970). [11] R. E. BELLMANand L. A. ZADEH: Decision making in a fuzzy environment. Management Sci. 17, pp. B-141B-164 (1970).