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A Procedure for Optimizing Development Decisions
A Procedure for Optimizing Development Decisions
Step hen Kahne Associate Professor. Electrical Engineering Director. Hybrid Computer Laboratory University of Minnesota Minneapolis. Minnesota 55455 ABSTRACT 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.
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 is at one's disposal is rather ill-defined and imprecise. Intuitive feelings and inductive judgement play an important role in everyday life and decision making. The goal of 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 of 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 of World War 11. Chief among thi s group was George Dantzi g whose \~ork 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 versitility of linear programming. Applications ranging from transportation to machine utilization and covering a wide range of 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.
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. INTRODUCTION
This type of progression has been common in the development of other quantitative tools. First. a mathematical technique is invented (discovered). Often this occurs in the course of trying to solve a particular "real world" problem. (In the case of linear programmin9. 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.
The problem 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. More recently, decision theory and the more quantitative disciplines of operations research and queuing theory have been brought to bear on such problems. But. inherent in all approaches to data has been a requirement to specify criteria. constraints. optimality goals and other parameters
*Superior numbers refer to similarly-numbered references at the end of the paper.
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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. Thirdly, 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 may researchers become intimately familiar with it, attempts are made to change real world problems into problems which can be solved by "the method". Non-linear 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 assumptions is understandable, if not valid. The technique exists, is welldefined 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.
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 difficulties in (5) his review of a landmark book on urban dynamics. In much 'of this section we have at least suggested that a connom 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? 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 Let us examine the model less 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. 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 micro-scopic or macro-scopic scale depending on the availability of certain data, instruments and techni ques. Speci a1 theori es have been established to show when I/O analyses are expected to work and when they are expected to fail. Th~ notion of observability in systems theory (6) is specifically directed at this question of when I/O analysis is sufficiently rich to completely describe system behavior.
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 betweeQ real world problems and real world solutions.t2) 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.
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 be9in 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.
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 solV~lon" and not enough to "solutions of problems".t) However, this malady has certainly affected studies of problems of national development as well, )Simulation and modelling of national economics\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
With this explanation of a deemphasis on modeling and with an understanding that development
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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. 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: 1. Finite number of possible decisions - In any well-defined development problem preliminary analysis will indicate a variety of feasible solutions 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 is 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 is too imprecise to fit the standard decision making techniques, approximations to the system or refinement of the data is 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 is 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.
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:(7) "To some people, the mention of systems analysis immediately suggests that one is referring to the application of particular techniques to an already welldefined problem or to the 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 ana1yzing systematically the implications of and 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 enter 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 struct~re surrounding all decision making 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, hO~/ ever, be able to obtain an indication of which feasible decision is best with an imprecision no *We will not discuss these characterized by an autocratic or dictatorial nature.
Although one could continue the list of characteristics, by now the reader should see what
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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 ea~ily handled as quantitative ones. Again Gresford t7) 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 varialbes requires much further effort on the part of the management science community. Again, within a development context and in the light of the policy options that presumably are the end result of a systems analysis effort, adequate means must be found to take qualitative factors into the analytic process".
For each criterion, a scale of "degree of goal satisfaction" must be developed to quantify what we mean by this Udegree of goal satisfaction". If a particular alternative satisfies the criterion in question to a very 9reat 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 doesn't need further explana~ion here. However, in development problems, as dlscussed above, there are further complications ~hich ~o.need further elaboration. Often, due to lmpreclslon of data, even though one alternative is clearly very good (with respect to a particular criterion and goal), the precise "degree of goodness" is not clear. The people who must rate a particular alternative solution on a particular criterion may have some difference of opinion. 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 more specific, let us establish a scale from 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 xE[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 a+b lS . a specla . 1 case of thlS .. gra de ~ sltuation in [a+b --2-a+b] . ThlS . latter comment is parwh·lC h XE~,
o to
Let us examine the procedure referred to above. THE PROCEDURE The procedure is comprised of five components: l.} Goal Definition 2.} Criteria Definition and Measures 3.} Criteria Weights 4.} Rating Scheme 5.} Ranking Scheme Each component is described in this section. 1. 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.
ticularly significant when computer algorithms are developed to implement the procedure being described. Using this notion of grade, it is possible to rate each feasible alternative decision with respect to 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 o - 10 denotes complete uncertainty about this particular grade while a rating of 0 or 2.5 or 8 indicates complete certainty.
2. Criteria - A development 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 crite;ia. These criteria may be qualitative or quantitative in nature and may be independent of 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 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 grOijp, 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.
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Let's look at an example which involved the se1ectiQn of a site for a state zoological garden.l 8) There is only one criterion shown: Problem: Select from among several sites the best site for a new state zoo.
of possible degrees of satisfaction of the goal by various alternatives with respect to this criterion. The ratings (i.e., 10-8, 8-6, etc.) are to be used in the absence of other detailed data but a range of numbers within these ratings may be used if more specific data is available. Thus, it is possible to replace 10-8 by 10-9.5 if very unusual enhancement takes place. Thus, even by prespecifying the measures it is not necessary to use only the ratings given. The key idea remains to keep the data and ratings comparable at each step in the process.
Criterion: Impact on natural, historic and scientific resources of adjacent land. Measures: Rating 10-8
8-6
6-4
4-2
2-0
Description
In summary then, the rating scheme proposed allows one to indicate with certainty the amount of uncertainty which the data indicates.
Land adjacent to the site is characterized by important natural historic or scientific resources which could be enhanced by an adjacent zoo function. The site is located in such a way that prevailing winds, traffic or noise as they relate to zoo demands would not be detrimental to the surrounding community. Land adjacent to the site is characterized by limited natural, historic or scientific resources which could be enhanced by adjacent zoo functions, and/or zoo function could provide a favorab1e background for fabric of surrounding community.
3. Cri teri a Wei ghts - Another "degree of freedom" surrounding the development decision process is that various criteria have various relative importance. 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. This often leads to assumptions about relative importance which is not supported by available data. Since our basic philosophy throughout this discussion is to precisely reflect the uncertainty in decision process, the use of ranges of criterion weights is consistent with our earlier discussions.
Land adjacent to the site has no natural, historic or scientific resources. The site is located in such a way that prevailing winds, traffic or noise as they relate to zoo demands would not be detrimental to the surrounding community. Land adjacent to the site has some natural, historic or scientific resources. The resources would not be destroyed by being adjacent to the zoo function, but there is a risk involved in that they may be damaged. Site is located in such a way that zoo function has strong possibilities of being detrimental to the surrounding community. Land adjacent to the site is characterized by important natural, historic or scientific resources which could be damaged or destroyed by an adjacent zoo function and/or the site is located in such a way that prevailing winds, traffic or noise as they relate to the zoo demands could be detrimental to the surrounding community.
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 ratings 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 can receive with respect to one criterion is the same as with respect to another. But, let's 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.
In this example, there are five measures for this criterion. The measures constitute a spectrum
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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 1 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: Superior 9-10 Good 7-9 Average 5-7 Poor 0-5 Criterion 11:
5. Ranking Scheme - Having assembled the data as described and prepared a table of this 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. In Table 2, Rij is the rating of Alternative i with respect to Criterion j. It is a set of real numbers with elements Pij' Thus PijER ij . Generally R' j will be a range of numbers between 0 and 10 (e.g~ Rij = [7,9], then Pij might be 7 or 7.2 or 7.5 or 8.3 ... ). Similarly, Wi is a set of real numbers with elements wi' Rij and Wi 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.
Same as Criterion I.
Note that the grade assigned for Alternative B, Criterion 11, is 5-6. This indicates that the grade is in the high part of "average" and illustrates how the actual rating assigned may not coincide with the given range of grades assigned to the given measure. 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.
The ranking of alternatives proceeds as follows. Using an appropriate random number generator to select the Pij and wi' form the sum of products n P =!: w.p .. i=1,2, ... ,m i j=l J lJ These are the sum of products for the ith alternative. When m of them have been calculated, one for each alternative, order the Pi 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 Pi is maintained. The Pi 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.
In practice this rating step may be accomplished in a number of ways. One possibility is the "corrrnittee 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.
Clearly, since Alternative 1 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%. This is, of course, true for the general case of Malternatives.
Another possibil ity 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.
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 Pij or wi one chooses as his own, if he agrees that the sets Rij and Wi accurately represent the state of knowledge about the criterion and alternative under consideration, the degree of certainty of a given ranking of an alternative is indicated. In certain situations a rank of first 50% of the runs might be considered a sufficiently positive result to establish confidence in the selection. In other situations, it may require a 90% occurence to establish this confidence.
Politics or "extra-scientific" influences must be mentioned here. In development decisions, their influences are often prominent and always important. The approach under discussion ;s 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.
39 0
A MATHEMATICAL CAVEAT
other processes not considered here. One such process is effective data aggregation. Another is decision implementation. Manyothers are also needed.
At the heart of the method is the requirement to weigh, with uncertain weighting, an uncertain rating. This involves the multiplication of two uncertain factors and mathematically, requires that two random variables be multiplied. Each may be thought of as a sample from a distribution of possible variable values. In many types of problems the most reasonable distributi on 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 say precisely what I mean I will assume that all values between 10 and 8 are equally likely." In ma~y 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 w., and a 1..J are uniform, then the distribution of w.a .. is not uniform and is, 1 lJ in fact, logarithmic. Therefore, the most likely value of the product is near the low end of possible products. In case the range of wi or aij include 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%of the maximum value of this distribution. Therefore, ones 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 wi or a lJ . . 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 .
Ratings Criterion
Crit. we i ht
A1t. A
Alt . B
Crit
0-5
7-9
9-10
Crit II
8-10
2-4
5-6
TABLE 1 Ratings Criterion
WeiQht
A1t.
Alt. 2
Alt. n
Wl
Rll
R21
Rml
2
W2
R12
R22
Rm2
3
W3
R13
R23
Rm3
n
Wn
R1n
R2n
Rmn
TABLE 2
Rank first
Rank second
Alt.
35%
65%
Alt. 2
65%
35%
TABLE 3
REFERENCES 1. T. S. Koopmans, Ed., Activity Analysis of Production and Allocation, N.V., Wiley, 1951.
CONCLUSIONS 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 deal i ng with development problems. The method i~ Qne 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
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2.
S. Kahne, "Report from Dresden", AUTOMATICA, Volume 8, Number 5, September 1972, Pergamon Press.
3.
O. Rademaker as quoted by S. Kahne, "Report from Dresden", AUTOMATICA, Volume 8, Number 5, September 1972, Pergamon Press
4.
IFAC Symposium, "Dynamic Modelling and Control of National Economies", June 1973, University of Warwick, Coventry, England.
5.
S. Kahne, Review of Urban Dynamics (by J. W. Forrester), IEEE Transactions, Volume AC-16, Number 2, April 1971. (Also Automatiki i Te1emekhanika, Number Bb, August 1971, pp. 203-205)
6.
E. B. Lee and L. Markus, Foundations of Optimal Control, Wiley, New York, 1967.
7.
G. B. Gresford, "Systems Approach for Development", IEEE Transactions, Volume AMC-2, Number 3, July 1972, pp. 311-318.
8.
Minnesota Zoological Garden - Mirror to the Environment, Inter Design Inc., Minneapolis, Mi nnesota, 1970.
9. C. A. Doxiadis, "The Two-Headed Eagle", Ekistics, Volume 33, Number 198, May 1972, Athens, Greece.
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Step hen Kahne Associate Professor. Electrical Engineering Director. Hybrid Computer Laboratory University of Minnesota Minneapolis. Minnesota 55455 ABSTRACT 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.
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 is at one's disposal is rather ill-defined and imprecise. Intuitive feelings and inductive judgement play an important role in everyday life and decision making. The goal of 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 of 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 of World War 11. Chief among thi s group was George Dantzi g whose \~ork 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 versitility of linear programming. Applications ranging from transportation to machine utilization and covering a wide range of 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.
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. INTRODUCTION
This type of progression has been common in the development of other quantitative tools. First. a mathematical technique is invented (discovered). Often this occurs in the course of trying to solve a particular "real world" problem. (In the case of linear programmin9. 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.
The problem 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. More recently, decision theory and the more quantitative disciplines of operations research and queuing theory have been brought to bear on such problems. But. inherent in all approaches to data has been a requirement to specify criteria. constraints. optimality goals and other parameters
*Superior numbers refer to similarly-numbered references at the end of the paper.
385
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. Thirdly, 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 may researchers become intimately familiar with it, attempts are made to change real world problems into problems which can be solved by "the method". Non-linear 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 assumptions is understandable, if not valid. The technique exists, is welldefined 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.
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 difficulties in (5) his review of a landmark book on urban dynamics. In much 'of this section we have at least suggested that a connom 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? 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 Let us examine the model less 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. 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 micro-scopic or macro-scopic scale depending on the availability of certain data, instruments and techni ques. Speci a1 theori es have been established to show when I/O analyses are expected to work and when they are expected to fail. Th~ notion of observability in systems theory (6) is specifically directed at this question of when I/O analysis is sufficiently rich to completely describe system behavior.
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 betweeQ real world problems and real world solutions.t2) 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.
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 be9in 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.
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 solV~lon" and not enough to "solutions of problems".t) However, this malady has certainly affected studies of problems of national development as well, )Simulation and modelling of national economics\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
With this explanation of a deemphasis on modeling and with an understanding that development
386
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. 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: 1. Finite number of possible decisions - In any well-defined development problem preliminary analysis will indicate a variety of feasible solutions 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 is 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 is too imprecise to fit the standard decision making techniques, approximations to the system or refinement of the data is 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 is 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.
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:(7) "To some people, the mention of systems analysis immediately suggests that one is referring to the application of particular techniques to an already welldefined problem or to the 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 ana1yzing systematically the implications of and 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 enter 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 struct~re surrounding all decision making 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, hO~/ ever, be able to obtain an indication of which feasible decision is best with an imprecision no *We will not discuss these characterized by an autocratic or dictatorial nature.
Although one could continue the list of characteristics, by now the reader should see what
387
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 ea~ily handled as quantitative ones. Again Gresford t7) 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 varialbes requires much further effort on the part of the management science community. Again, within a development context and in the light of the policy options that presumably are the end result of a systems analysis effort, adequate means must be found to take qualitative factors into the analytic process".
For each criterion, a scale of "degree of goal satisfaction" must be developed to quantify what we mean by this Udegree of goal satisfaction". If a particular alternative satisfies the criterion in question to a very 9reat 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 doesn't need further explana~ion here. However, in development problems, as dlscussed above, there are further complications ~hich ~o.need further elaboration. Often, due to lmpreclslon of data, even though one alternative is clearly very good (with respect to a particular criterion and goal), the precise "degree of goodness" is not clear. The people who must rate a particular alternative solution on a particular criterion may have some difference of opinion. 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 more specific, let us establish a scale from 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 xE[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 a+b lS . a specla . 1 case of thlS .. gra de ~ sltuation in [a+b --2-a+b] . ThlS . latter comment is parwh·lC h XE~,
o to
Let us examine the procedure referred to above. THE PROCEDURE The procedure is comprised of five components: l.} Goal Definition 2.} Criteria Definition and Measures 3.} Criteria Weights 4.} Rating Scheme 5.} Ranking Scheme Each component is described in this section. 1. 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.
ticularly significant when computer algorithms are developed to implement the procedure being described. Using this notion of grade, it is possible to rate each feasible alternative decision with respect to 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 o - 10 denotes complete uncertainty about this particular grade while a rating of 0 or 2.5 or 8 indicates complete certainty.
2. Criteria - A development 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 crite;ia. These criteria may be qualitative or quantitative in nature and may be independent of 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 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 grOijp, 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.
388
Let's look at an example which involved the se1ectiQn of a site for a state zoological garden.l 8) There is only one criterion shown: Problem: Select from among several sites the best site for a new state zoo.
of possible degrees of satisfaction of the goal by various alternatives with respect to this criterion. The ratings (i.e., 10-8, 8-6, etc.) are to be used in the absence of other detailed data but a range of numbers within these ratings may be used if more specific data is available. Thus, it is possible to replace 10-8 by 10-9.5 if very unusual enhancement takes place. Thus, even by prespecifying the measures it is not necessary to use only the ratings given. The key idea remains to keep the data and ratings comparable at each step in the process.
Criterion: Impact on natural, historic and scientific resources of adjacent land. Measures: Rating 10-8
8-6
6-4
4-2
2-0
Description
In summary then, the rating scheme proposed allows one to indicate with certainty the amount of uncertainty which the data indicates.
Land adjacent to the site is characterized by important natural historic or scientific resources which could be enhanced by an adjacent zoo function. The site is located in such a way that prevailing winds, traffic or noise as they relate to zoo demands would not be detrimental to the surrounding community. Land adjacent to the site is characterized by limited natural, historic or scientific resources which could be enhanced by adjacent zoo functions, and/or zoo function could provide a favorab1e background for fabric of surrounding community.
3. Cri teri a Wei ghts - Another "degree of freedom" surrounding the development decision process is that various criteria have various relative importance. 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. This often leads to assumptions about relative importance which is not supported by available data. Since our basic philosophy throughout this discussion is to precisely reflect the uncertainty in decision process, the use of ranges of criterion weights is consistent with our earlier discussions.
Land adjacent to the site has no natural, historic or scientific resources. The site is located in such a way that prevailing winds, traffic or noise as they relate to zoo demands would not be detrimental to the surrounding community. Land adjacent to the site has some natural, historic or scientific resources. The resources would not be destroyed by being adjacent to the zoo function, but there is a risk involved in that they may be damaged. Site is located in such a way that zoo function has strong possibilities of being detrimental to the surrounding community. Land adjacent to the site is characterized by important natural, historic or scientific resources which could be damaged or destroyed by an adjacent zoo function and/or the site is located in such a way that prevailing winds, traffic or noise as they relate to the zoo demands could be detrimental to the surrounding community.
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 ratings 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 can receive with respect to one criterion is the same as with respect to another. But, let's 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.
In this example, there are five measures for this criterion. The measures constitute a spectrum
389
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 1 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: Superior 9-10 Good 7-9 Average 5-7 Poor 0-5 Criterion 11:
5. Ranking Scheme - Having assembled the data as described and prepared a table of this 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. In Table 2, Rij is the rating of Alternative i with respect to Criterion j. It is a set of real numbers with elements Pij' Thus PijER ij . Generally R' j will be a range of numbers between 0 and 10 (e.g~ Rij = [7,9], then Pij might be 7 or 7.2 or 7.5 or 8.3 ... ). Similarly, Wi is a set of real numbers with elements wi' Rij and Wi 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.
Same as Criterion I.
Note that the grade assigned for Alternative B, Criterion 11, is 5-6. This indicates that the grade is in the high part of "average" and illustrates how the actual rating assigned may not coincide with the given range of grades assigned to the given measure. 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.
The ranking of alternatives proceeds as follows. Using an appropriate random number generator to select the Pij and wi' form the sum of products n P =!: w.p .. i=1,2, ... ,m i j=l J lJ These are the sum of products for the ith alternative. When m of them have been calculated, one for each alternative, order the Pi 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 Pi is maintained. The Pi 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.
In practice this rating step may be accomplished in a number of ways. One possibility is the "corrrnittee 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.
Clearly, since Alternative 1 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%. This is, of course, true for the general case of Malternatives.
Another possibil ity 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.
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 Pij or wi one chooses as his own, if he agrees that the sets Rij and Wi accurately represent the state of knowledge about the criterion and alternative under consideration, the degree of certainty of a given ranking of an alternative is indicated. In certain situations a rank of first 50% of the runs might be considered a sufficiently positive result to establish confidence in the selection. In other situations, it may require a 90% occurence to establish this confidence.
Politics or "extra-scientific" influences must be mentioned here. In development decisions, their influences are often prominent and always important. The approach under discussion ;s 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.
39 0
A MATHEMATICAL CAVEAT
other processes not considered here. One such process is effective data aggregation. Another is decision implementation. Manyothers are also needed.
At the heart of the method is the requirement to weigh, with uncertain weighting, an uncertain rating. This involves the multiplication of two uncertain factors and mathematically, requires that two random variables be multiplied. Each may be thought of as a sample from a distribution of possible variable values. In many types of problems the most reasonable distributi on 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 say precisely what I mean I will assume that all values between 10 and 8 are equally likely." In ma~y 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 w., and a 1..J are uniform, then the distribution of w.a .. is not uniform and is, 1 lJ in fact, logarithmic. Therefore, the most likely value of the product is near the low end of possible products. In case the range of wi or aij include 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%of the maximum value of this distribution. Therefore, ones 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 wi or a lJ . . 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 .
Ratings Criterion
Crit. we i ht
A1t. A
Alt . B
Crit
0-5
7-9
9-10
Crit II
8-10
2-4
5-6
TABLE 1 Ratings Criterion
WeiQht
A1t.
Alt. 2
Alt. n
Wl
Rll
R21
Rml
2
W2
R12
R22
Rm2
3
W3
R13
R23
Rm3
n
Wn
R1n
R2n
Rmn
TABLE 2
Rank first
Rank second
Alt.
35%
65%
Alt. 2
65%
35%
TABLE 3
REFERENCES 1. T. S. Koopmans, Ed., Activity Analysis of Production and Allocation, N.V., Wiley, 1951.
CONCLUSIONS 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 deal i ng with development problems. The method i~ Qne 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
391
2.
S. Kahne, "Report from Dresden", AUTOMATICA, Volume 8, Number 5, September 1972, Pergamon Press.
3.
O. Rademaker as quoted by S. Kahne, "Report from Dresden", AUTOMATICA, Volume 8, Number 5, September 1972, Pergamon Press
4.
IFAC Symposium, "Dynamic Modelling and Control of National Economies", June 1973, University of Warwick, Coventry, England.
5.
S. Kahne, Review of Urban Dynamics (by J. W. Forrester), IEEE Transactions, Volume AC-16, Number 2, April 1971. (Also Automatiki i Te1emekhanika, Number Bb, August 1971, pp. 203-205)
6.
E. B. Lee and L. Markus, Foundations of Optimal Control, Wiley, New York, 1967.
7.
G. B. Gresford, "Systems Approach for Development", IEEE Transactions, Volume AMC-2, Number 3, July 1972, pp. 311-318.
8.
Minnesota Zoological Garden - Mirror to the Environment, Inter Design Inc., Minneapolis, Mi nnesota, 1970.
9. C. A. Doxiadis, "The Two-Headed Eagle", Ekistics, Volume 33, Number 198, May 1972, Athens, Greece.
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