exploring optimization through hierarchies and ratio scales

Socio-Econ. Plann. Sci. Vol. 20, No. 6, pp. 355-360, 1986 Printed in Great Britain

0038-0121/86 $3.00 + 0.00 Pergamon Journals Ltd

EXPLORING OPTIMIZATION THROUGH HIERARCHIES AND RATIO SCALES

THOMAS L. SAATY Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

Abstract-This paper explores the concept of optimization by solely using the AHP and compares outcomes with those obtained in traditional optimization theory without and with constraints. The difference is essentially in the absence of the traditional black box involving complex manipulations in algebra or the calculus on an assumed linear or nonlinear mathematical structure. This paper is an exploratory attempt to use an individual’s understanding together with a way to convert his judgments to ratios to deal with optimization. At first sight it may go contrary to the reader’s (learned) intuition, but in the end he must face the question of whether the magic of traditional manipulations gives rise to better answers than one’s actual and complete understanding in which one does not abdicate judgmental control of the solution, and why. This idea is ripe for deeper and more detailed exploration.

1. INTRODUCTION

Optimization is a goal directed activity. Goals depend on the perceptions and preferences of the people. It follows that in the final analysis optimization is an attempt to satisfy people’s goals. Ordinarily we use Cartesian geometry to solve an optimization problem by assuming independent variables represented or- thogonally. If some variables are more important, we take functions of them, e.g. we multiply them by appropriate constants and raise them to powers to reflect this importance in terms of larger or smaller numbers than represented by other variables. Out of habit and convenience the same basic model is used to represent a wide variety of problems interpreted to fit the model. This is particularly useful when the number of variables is very large, as it saves time, but could destroy relevance. But is this justified? How valid are our highly structured models and their solutions? How should the layman regard the black box manipulations of these models, particularly when they have no corresponding real life meaning?

In the book, Optimization in Integers and Related Extremal Problems, [l], there is contained a compre- hensive framework for optimization, organized to include formulations using equations and in- equalities. Classical optimization which has its ori- gins in physics and engineering is concerned with the maximization or minimization of a function (or a functional) subject to equality and/or inequality constraints expressed in algebraic or, more generally, in functional form, sometimes using derivatives and integrals. In programming theory the constraints also include the nonnegativity of the variables.

The purpose of optimization is to identify a “best” set of alternatives, but it need not be tied to a particular structure. For a large class of practical problems it is only with a great deal of simplification that a problem is formulated in a standard and familiar form so that manipulations can be carried out to derive a solution. For example, at times military or business problems are not concerned with the maximization (or minimization) of a well defined

and measurable quantity (such as dollar expenditure), but rather with the optimization of vague and intangible concepts such as military worth, loss of life, or trust in the business. How to deal with these is certainly not by starting with a simple exercise, and eventually coming to regard it as the real situation because of the complexity and hard work involved in formulating the problem.

It is useful to consider how rare it is that an individual attempts to maximize or minimize anything. Early in life we are trained for balance, “follow the happy medium”, to be satisfied. The concept of sufficiency can be found not simply in some modem works in economics and operations research but also in ethics and philosophy. Even risk avoidance is only minimized in theory. In practice people take varying degrees of risk to deal with the world.

2. COMMENTS ON THE ANALYTIC HIERARCHY PROCESS

The Analytic Hierarchy process (AHP) serves as a framework for people to structure their own problems and provide judgements based on knowledge, reason or feelings, to derive a set of priorities considered as an optimal solution to a decision problem, perhaps involving resource allocation. The AHP provides the opportunity for the following kinds of considerations:

(i) While in classical optimization one is restricted to measures of money, time, weight, temperature and other measurables, with the AHP one can use these tangibles along with relative measurements of intangibles derived through comparisons.

(ii) The structure of a problem is not set in advance, but is largely developed by those who experience it.

(iii) No assumptions of orthogonality and independence need to be made as the variables may be interdependent and hence the use of a Cartesian coordinate system may not be appropriate.

355

356 THOMAS L. SAATY

(iv) One is not restricted to the use of a euclidean metric.

(v) Different weights may be attached to different constraints or criteria.

(vi) As in real life the objective function is so intermeshed with the constraints that optimization is treated as a process of interaction between objectives and constraints without separating them.

(vii) One can optimize any number of objectives, and not simply one.

(viii) We need models that improve our understanding and ability to interpret the real world through interaction so that the line which separates what we think and what we experience disappears and reality becomes knowledge itself.

3. TWO ROLES FOR THE AHP IN OPTIMIZATION

There are two ways in which the AHP can be used in optimization. The first is natural for anyone concerned with including the measurement of intangibles in a model, for the AHP is simply a theory of generating and combining ratio scales. The resulting scale values or priorities are incorporated as coefficients of an objective function, of constraints or of a payoff matrix. This kind of application extends our reach in making applications.

The other is direct use of the AHP in optimization. It is useful when our existing framework of the optimization of an objective function subject to sep- arately written constraints is not easy to identify and construct because the two cannot be neatly separated in reality but are in the mind of the modeller. We said earlier that optimization is finding a best solution to a problem no matter how that problem is structured. The process avoids imposing a prior mathematical structure on the problem with specified parameters and variables and generates a numerical answer in a step by step fashion using judgment and concrete information.

Questions of independence and dependence among the entities being treated become academic in the AHP if we can treat the criteria and objectives as different but intimately related decompositions of the reality we experience. We must define them carefully to ensure that they capture what we intend them to mean and are related in desired ways. There is no harm in finding that there is overlap among them. We can still study both their independent and interdependent characteristics. Although a theory is available for this purpose (21, we need not go into such detail in this paper.

4. AN EXAMPLE OF A MAXIMIZATION PROBLEM

A calculus problem goes as follows: A farmer wishes to sell his crop of 120 bushels of potatoes. If he sells it now he would get $1.00 per bushel. However, if he waits his crop will increase, due to growth, by 20 bushels per week, but the price would decrease by $0.10 per week. When should he sell to maximize his profit?

The calculus answer to his problem is obtained by maximizing (120 + 20x) (1 - 0. lx), whose solution is x = 2 weeks.

Fig. 1. Hierarchy for maximizing profit.

Let us assume that the farmer has not studied calculus, nor does he know an expert consultant who could set up the problem for him. Even if he does, he has doubt that his crop will increase in such a systematic manner, or that the price predicted for the future would be as expressed in the statement of the problem. Of course, he can solve his problem using pure arithmetic if he buys into these clear cut assumptions, and has no need of calculus anyway. The complexity of his operations cannot be brought easily into a simple algebraic formula from which a precise answer is then obtained. The object is to assist him by unfolding the problem according to its uncertainties, enabling the farmer to get a good estimate for when he should sell his potatoes. It is hoped that for this simple exercise the answer would be close to the calculus answer by way of validating such an approach which may be useful for other complex problems he faces. Let us note that he may have several questions in mind and we can only assist him to deal with them one or at best a few at a time, but without simplifying the original version to obscurity.

To solve the problem we construct a hierarchy (not necessarily unique) illustrated below, whose goal is to maximize the farmer’s profit. Profit depends on yield and on the price per bushel which he can get when he harvests the crop. In the end profit is determined by the number of weeks that the farmer waits to sell his crop. The number of weeks he waits to harvest and sell, a period of no more than five weeks, represent his alternatives (0, 1, 2, 3, 4, 5) at the bottom or fourth level of the hierarchy. The other two levels below the Goal of profit are respectively, yield, whose elements are low, medium, high, and the price per bushel whose elements are small, medium, large. The alternatives at the bottom, indicating weeks will be compared with respect to the likelihood of obtaining a small, medium or large price per bushel.

Clearly, to obtain the maximum profit, crop size will have to be relatively large. However, eventually a very large harvest could be worthless, both because the price is low and because the crop is subject to the hazards of nature for a longer time.

According to this observation, the pairwise comparisons among the different degrees of production can now be given by answering the following question: Comparing crop sizes, which one produces more profit and how strongly? We use the following scale to make absolute comparisons as to how many times more one element dominates the other according to a common property. The judgment is first generated qualitatively and then the corresponding numerical

Exploring optimization 357

value is entered: Equal, 1; Moderate, 3; Strong, 5; Very Strong, 7; Extreme, 9: Intermediate Strengths, 2, 4, 6, 8; reciprocals of foregoing.

The answer, a composite vector of all the priorities obtained so far, is interpreted as the likelihood of selling in the given week for maximum profit:

Profit LOW Medium High Weights

Low 1 l/3 3 0.258 Medium 1 5 0.637 High 1;3 l/5 I 0.105

Consistency ratio 0.033

For brevity we will not give arguments to justify these judgements. The thoughtful reader should have no difficulty in understanding these judgements, although he may not agree with all of them. It has been shown in dealing with hierarchic structures that the outcome is relatively stable to anything but the most drastic change in the strategy of assigning judgements.

Now compare the price per bushel for each yield size. We answer the question: Given two category prices per bushel, which category is more likely to obtain under that level of yield? We have:

Low S M L Weiehts

s 1 l/3 115 0.109 M 3 1 112 0.309 L 5 2 1 0.582


Medium S M L Weights

S M L

High

I l/5 l/2 0.128 5 I 2 0.595 2 l/2 1 0.276


S M L Weights

S 1 4 5 0.691 M l/4 I 1 0.160 L l/5 1 1 0.149


It remains to compare the number of weeks in the fourth level with respect to the price per bushel in the third level. The question asked here is: of two sale weeks which provides more profit at the indicated level of price per bushel? We obtain the following three paired comparison matrices and their eigen- vector weights:

Small 0 I 2 3 4 5 Weights

0 1 I l/2 l/3 l/4 l/5 0.064 I 1 1 l/2 113 l/4 0.084 2 : 1 1 1 l/2 113 0.120 3 3 2 1 I I 112 0.173 4 4 3 2 I I 1 0.243 5 5 4 3 2 I I 0.315


Medium 0 1 2 3 4 5 Weights

0 1 113 115 l/5 l/3 1 0.053 I 3 1 l/3 l/3 I 2 0.122 2 5 3 I 1 3 5 0.321 3 5 3 1 1 3 0.321 4 3 2 1 1 1 2” 0.122 5 1 l/2 I 1 1 I 0.060


Large 0

0 I

I 2

3 4

3 4 5 Weights

5 6 I 0.426 I

1:2 1 3 4 6 0.254

2 I 1 3 : 5 0.150 3 l/3 l/2 I I 3 4 0.088 4 l/4 113 l/2 1 1 3 0.052 5 l/5 114 l/3 l/2 1 I 0.031


Week number 0 1 2 3 4 5

Likelihood of selling: 0.183 0.160 0.226 0.214 0.120 0.097

The farmer has two options. The first is to select the week with the likelihood of higher yield which is the second week. Week 2 agrees exactly with the answer from calculus. The second option, which is perhaps the better one because of the uncertainties, is to take the expected value:

(0 x 0.183 + 1 x 0.160 + 2 x 0.226 + 3 x 0.0214

+ 4 x 0.120 + 5 x 0.097) = 2.22

This answer recommends selling at week 2.22.

Had the question been concerned with price, the fourth level of the hierarchy would have considered different prices per bushel, as it also would, had the number of bushels (total yield) been the desired outcome.

5. COMPLEX CHOICE

Intuition and understanding of a complex situation should derive partly from numerical measurements associated with that situation. Exactly how feelings are shaped by numbers varies from individual to individual. Optimization becomes a relative matter. Here is a simple illustration of how a professor of mathematics, a friend of the writer, has explained how she allocates her money to shop for food. Several factors are involved, some of which are intangible such as taste, on which one cannot place a bound as in linear programming. Her hierarchy follows:

Fig. 2. Hierarchy for shopping for food.

She assigned preference weights of 0.4 to meat and 0.6 to vegetables in the second level. Her matrices for the elements in the third level with respect to those in the second and those in the fourth with respect to those in the third, follow. The meats and the vegetables listed in the fourth level are treated as two separate clusters with respect to each attribute in the third level.

Meat C T N CC Weinhts

C I l/3 l/3 l/3 0.099 T 3 I 2 2 0.413 N 3 l/2 I 1 0.244

cc 3 l/2 1 1 0.244

358 THOMAS L. SAATY

Veg c T N CC Weights

c I II5 II3 II3 0.079 T 5 I 2 3 0.477 N 3 l/2 I 2 0.270

cc 3 I13 l/2 I 0. I74

Note that there are no comparisons necessary for cost as actual store prices are available.

The weights in the two matrices below were obtained by taking the reciprocals of the “Per Serving Store Price” and normalizing it. We take the recip- rocal of the price to measure relative benefits because cheaper is better.

Per serving cost store price Weights

B $1.04 0.152 P $0.55 0.292 F $0.995 0.161

PY 60.404 0.395 A&B $0.99 0.088 C&P $0. I26 0.694

s $0.402 0.218

Taste Taste B P F PY Weights

B I 3 5 2 0.466 P l/3 I 3 I13 0.160 F l/5 l/3 I I13 0.08

Taste A&B C&P S Weights

A&B 1 4 3 0.623 C&P l/4 1 II2 0.137

S l/3 2 I 0.24

Nutrition N&t B P F PY Weights

B I 3 l/5 II3 0.125 P l/3 I l/7 II4 0.061 F 5 7 I 3 0.563

PY 3 4 l/3 I 0.251

Nutrit A&B C&P S Weights

A&B I 3 II2 0.334 C&P l/3 I l/3 0.142

S 2 3 I 0.525

Caloric content CC B P F PY Weights

B 3 I)3 I

II4 II3 0.141 P II5 II4 0.071 F 4 5 I 3 0.52

PY 3 4 l/3 I 0.268

cc A&B C&P S Weights

A&B I 4 2 0.557 C&P l/4 I I13 0.123

S I/2 3 I 0.32

On composing weights we find the following priorities: A&B, 0.292; S, 0.197; F, 0.123; PY, 0.114; B, 0.108; C&P, 0.108; P, 0.051. Although the mathe- matician considered this outcome as her “optimal” choice, the problem developed as to how to allocate dollars to these categories of food. In general in allocation problems, it may be that more dollars than indicated by the highest priority are needed or that the priority would lead to the allocation of too much money beyond a certain point. What is a basis for distributing a resource according to priority? One may sometimes need to identify minimum and maximum amounts of each category of food to be used over a period, and allocate the money available

according to the priorities (e.g. by maximizing the product of the cost per unit of each food and its priority together with the unknown amount to be purchased) subject to these constraints. There are other possibilities for allocation mentioned in Saaty and Kearns [3] investigated by the first author and J. P. Bennett. In the next section we illustrate this kind of allocation subject to constraints.

6. OPTIMIZATION SUBJECT TO CONSTRAINTS

Consider an individual who has available three types of food: meat (beef), bread, and vegetables (broccoli). He must determine the optimal mixture of these foods to eat, minimizing cost and satisfying minimum daily requirements for vitamins A, B, and the amount of calories intake.

Suppose that the cost of each food is as follows: meat, $2.50/lb (%0.0055/g); bread, $055/lb ($0.0012/g); vegetable, $0.62/lb (%0.0014/g). The individual knows, approximately, the amount of vitamin A, B,, and calories per gram of each food. They are:

Food Vit. A (I.U.) Vit. B, (mg) Calories (kcal)

Meat 0.3527/g 0.002 1 /g 2.86/g Bread 0.0000 0.0006 2.76 Veg 25 0.002 0.25

The minimum daily requirements are: Vitamin A: 7500 (I.U.), Vitamin B,: This amount varies from individual to individual, and it is measured in mg/kg. The minimum requirement for an individual who weighs 147 lb is 1.6338 mg.; Calories: 2050 kcal for the 147 lb individual.

We have the linear programming problem

Minimize z: (5.5 x, + 1.2 x2 + 1.4 x3) x 10m3 dollars Subject to

0.3527 x, + + 25.000 x3 > 7.5000

0.0021 x, + 0.0006 x2 + 0.02 x3 > 1.6338

2.86 x, +2.76 x,+ 0.25 x3 > 2.050

XI, x2, x3 2 0

Its solution is given by x, (meat) = 0, x2 (bread) = 687.44 g, x3 (vegetables) = 610.67, at a cost per day of z = $1.67.

Let us assume that an individual expresses his food preferences in terms of the amount he desires to eat per day as follows:

x, (meat > 140 g (5 ozs)

x2 (bread) < 56 g (2 slices)

x,(vegetables), no constraint

What is wrong with this example? There is no consideration of preference for the foods nor for the importance of vitamins and calories to one’s well being. They are all treated as equally important and cost is the overriding factor. However, if both nourishment and cost are considered to be important but not equally, how would one incorporate such information in the objective function and in the con-

Exploring optimization 359

straints, maintaining a contrast or a relationship The following composite vector of priorities is ob- between the two? tained by multiplying the above matrix by the

Let us now give an alternative approach to this weighted vector of nourishment and adding the re- problem with the AHP. We have the following sulting vector to the vector of costs weighted by 0.25 hierarchy: which is the priority of cost.

Foods Meat Bmad Vegetables Priorities 0.24 0.23 0.53

The total amount of food that an individual must eat to minimize cost and satisfy the daily requirements obtained earlier from the linear programming problem is x, + x, + x3 = 1298.11 g or 2.86 lb. If we take this amount, which in practice may be high, and distribute it as a total among the three foods according to the above priorities the number of grams of

Fig. 3. Hierarchy for choosing a diet. each food eaten per day is given by:

Foods Meat Bread Vegetables

For a person with an average income, to satisfy his Amount (g) 308.43 297.66 692.02

nourishment needs is more important than the cost of The cost of this diet is $3.0224, however if we check the food. Here we have for nourishment (N), and cost to see if the daily requirements are satisfied by 0: writing:

Daily Meat Bread Veg Requirements

Vit. A 0.3527 0.0

Vit. B2 0.0021 0.0006

Cals 2.86 2.76

Daily requirements N C Weights

N I 3 0.75 C l/3 1 0.25

Vitamin A and B, are considered to be equally important, but both are more important than calories. We have

Nourishment Vit. A Vit. B, Gals Weights

Vit. A I 1 2 0.4 Vit. B, Cals 112

1 2 0.4 112 1 0.2

The amount of Vitamin A, B2 and calories per gram of meat, bread and vegetables are obtained from the original data and then normalized in each column. We have:

Vit. A Vit. B, kcal Meat 0.0139 0.4468 0.4872 Bread 0.0000 0.1277 0.4702 Vegetables 0.9861 0.4255 0.0426

According to cost we have:

Cost

Meat 0.68 Bread 0.15 Vegetables 0.17

However, since we wish to minimize cost, we use the normalized reciprocals of the amounts given above. We have:

Cost-’

Meat 0.105 Bread 0.476 Vegetables 0.419

we find that the caloric intake would not be met and hence a larger diet would be necessary. However, if we change the nourishment factors so they are equally as important as cost, consumption would be 290.80 g of meat, 419.30 g of bread and 588.50 g of vegetables at a cost of $2.92 and the daily requirements would be met because we have:

Vit. A 14,815.08 Vit. Bz kcal 213:::

By way of sensitivity analysis we change the relative importance of cost and nourishment so that cost is considered to be extremely more important than nourishment. We have:

Nourishment Cost Weights

N 1 119 0.1 C 9 1 0.9

If we also assume that Vit. A, Vit. Bz and kcal are equally important, we obtain:

Foods Meat Bread Vegetables

Priorities 0.12 0.46 0.42 Grams of food eaten per day 157.07 598.04 543.13

with the following contribution to daily requirements:

Vit. A 13,633.65 Vit. B, 1.7749 kcal 2235.59

and again the requirements are met at a cost of $2.34. One conclusion that can be drawn from the com-

parison of the two approaches is that as we impose

360 THOMAS L. SAATY

a preference structure on the diet problem the minimum cost obtained from linear programming goes up from $1.67 to $3.80. The costs obtained by using the Analytic Hierarchy Process fell between these two values as is to be expected since this formulation includes additional variables. Another conclusion is that the AHP solution was obtained directly by expressing preferences and without the intervention of a complex algebraic structure that is forbidding to the layman. Still the AHP has not been tested for a large number of variables to determine its feasibility as a model in this case.

7. CONCLUSION

The Analytic Hierarchy Process is a multi-criterion decision method that is based on ratio scales and relative comparisons. The success of its use rests with the ability of the decision maker to express his preferences within an accessible hierarchic structure. The outcome of the process is a best or optimum choice, considering all the necessary objectives, criteria and alternatives in a single interrelated framework. Optimization problems can be represented in a

hierarchic model and the results make at least as much sense, and are justified on rigorous mathematical ground, as those obtained in traditional optimization. However, they have the advantage of greater realism, by relying on more data to derive their results and by avoiding elegant manipulations of quantities which give the impression that the solution is related to abstract operations that give the appearance of a magician’s wand in producing an answer. In sum, our ability to handle more complex problems with multiple objectives, and diverse and intangible criteria depends on our flexibility to consider adopting new methods without prejudice in favor of old ones merely because of our habituation and commitment.

REFERENCES

1. T. L. Saaty. Optimization in Integers and Related Ex- tremal Problems. McGraw-Hill, New York (1970).

2. T. L. Saaty. The Analytic Hierarchy Process. McGraw- Hill, New York (1980).

3. T. L. Saaty and K. P. Kearns. Analytical Planning. Pergamon Press, Oxford (1985).

exploring optimization through hierarchies and ratio scales

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