Chapter 8
Hierarchies and Priorities
8.1. Introduction
IN every facet of life we are faced with making a choice among a set of alternatives. Some of these decisions are minor: what clothes to choose each morning, what we wish to eat for breakfast, etc. Other decisions are more important: which job to choose, whom we shall marry, which house we shall buy.
There are, however, decisions which affect many others beside ourselves. The managing director of a large company has to make many important decisions: where to locate a new plant, which products to concentrate on, which people to promote, how to apportion profits, etc. Public officials have to make major resource allocation decisions and other decisions about policies affecting thousands of people.
Even in situations where the decision about a desired final outcome has been made, decisions about intermediate policies to achieve this outcome are necessary.
All of this points up the necessity of a method for making decisions, particularly where a number of objectives have to be satisfied. The method of analytical hierarchies gives us a means of making such decisions in a rational manner.
Essentially there have been three ways of using logic to explain the phenomena of the real world. The first and time-honored method is that of cause and effect. It is known as the reductionist method upon which most scientific research so far has been based. The next is probabilistic and utilizes understanding of random process to explain occurrences by averages and deviations from them. The third and most recent is the systems approach which is concerned with the interactions of the parts within a system and the interactions of the system with its environment. This method of understanding is synthetic and looks at the overall purposes governing the design and functions of a system in order to explain its behavior. The systems approach is hierarchic in its nature, and goes from the particular to the general. Although synthesis cannot be separated from analysis and causality, it is different in its general outlook. Purpose and its fulfillment is its primary concern. Obviously then, priorities in the fulfillment of purpose become essential. Thus, the representation of a system according to its purposes, its environmental constraints, its actors, their objectives, the functions of the system, and the parts which perform these functions takes on a hierarchical form. One would then be concerned with the priority impact of any of these elements on the overriding purpose of the system.
8.2. Paired Comparisons
We first assume that n activities are being considered by a group of interested people and that the group's goals are:
(a) to provide judgments on the relative importance of these activities;
(b) to ensure that the judgments are quantified to an extent which also permits a quantitative interpretation of the judgments among all activities.
Clearly, goal (b) will require appropriate technical assistance.
Our goal is to describe a method for deriving, from the group's quantified judgments (i.e., from the relative values associated with pairs of activities), a set of weights to be associated with individual activities; in a sense defined below, these weights should reflect the group's quantified judgments. What this approach achieves is to put the information resulting from (a) and (b) into usable form without deleting information residing in the qualitative judgments.
Let C 1 , C 2 ,…, C n be the set of activities. The quantified judgments on pairs of activities C i , C j are represented by an n -by-n matrix
Suppose now we have n elements and we wish to compare them. We create the matrix of relative weights by asking the question: Of two elements i and j , which is more important with respect to some given property and how much more? We relate the importance of j by using the scale shown in Table 8.1 to give the values of a ij .
TABLE 8.1 The scale and its description
Intensity of importance
Definition
Explanation
1 *
Equal importance
Two activities contribute equally to the objective
3
Weak importance of one over another
Experience and judgment slightly favor one activity over another
5
Essential or strong importance
Experience and judgment strongly favor one activity over another
7
Demonstrated importance
An activity is strongly favored and its dominance demonstrated in practice
9
Absolute importance
The evidence favoring one activity over another is of the highest possible order of affirmation.
2, 4, 6, 8
Intermediate values between the two adjacent judgments
When compromise is needed
Reciprocals of above numbers
If activity i has one of the above numbers assigned to it when compared with activity j , then j has the reciprocal value when compared with i
Rationals
Ratios arising from the scale
If consistency were to be forced by obtaining n numerical values to span the matrix
* On occasion in 2-by-2 problems, we have used 1 + ε , 0 < ε ≤ to indicate very slight dominance between two nearly equal activities.
This means that we let a ij = 5 if element i (or activity i ) is essentially more important than activity j . Suppose we carry out this process to create the first row of the matrix A . If our judgments were completely consistent, the remaining rows of the matrix would then be completely determined (and the entries would not, in general, be integers). However, we do not assume consistency other than by setting a ji = 1/a ij . We repeat the process for each row of the matrix, making independent judgments over each pair.
It turns out in practice that this scale gives a workable representation of the way in which people think and compare similar elements [Saaty, J. Math. Psych , 1977].
Our concern now is to decide how the elements at each level should be weighted. We consider first a simpler problem. Assume that we are given n stones A 1 …A n , whose weights w 1 …w n , respectively, are known to us. Let us form the matrix of pairwise ratios whose rows give the ratios of the weights of each stone with respect to all others. Thus we have the matrix:
We have multiplied A on the right by the vector of weights w . The result of this multiplication is nw . Thus, to recover the scale from the matrix of ratios, we must solve the problem Aw = nw or (A – nI )w = 0. This is a system of homogeneous linear equations. It has a nontrivial solution if and only if the determinant of (A – nI ) vanishes, i.e., n is an eigenvalue of A . Now A has unit rank since every row is a constant multiple of the first row and thus all its eigenvalues except one are zero. The sum of the eigenvalues of a matrix is equal to its trace and in this case, the trace of A is equal to n . Thus n is an eigenvalue of A and we have a nontrivial solution. The solution consists of positive entries and is unique to within a multiplicative constant.
To make w unique we normalize its entries by dividing by their sum. Thus given the comparison matrix we can recover the scale. In this case the solution is any column of A normalized. Note that in A we have a ji = 1 / a ij the reciprocal property. Thus, also, a ii = 1. Also, A is consistent, i.e., its entries satisfy the condition a jk = a ik /a ij .
Thus the entire matrix can be constructed from a set of n elements which form a spanning tree across the matrix.
In the general case we cannot give the precise values of w i /w j but estimates of them. For the moment let us consider an estimate of these values by an expert who we assume makes small errors in judgment. From matrix theory we know that small perturbation of the coefficients implies small perturbation of the eigenvalues. Our problem now becomes A′w′ = λ max w ′ where λ max is the largest eigenvalue of A′ . To simplify the notation we shall continue to write Aw = λ max w where A is the matrix of pairwise comparisons. The problem now is how good is the estimate of w . Note that if we obtain w by solving this problem the matrix whose entries are w i /w j is a consistent matrix which is our consistent estimate of the matrix A . A itself need not be consistent. In fact, the entries of A need not even be ordinally consistent, i.e., A 1 may be preferred to A 2 , A 2 to A 3 , but A 3 is preferred to A 1 . What we would like is a measure of the error due to inconsistency. It turns out that A is consistent if and only if λ max = n and that we always have λ max ≥ n . This suggests using λ max – n as an index of departure from consistency. But
where λ i , i = 1,…, n , are the eigenvalues of A . We adopt the average value (λ max – n )/(n – 1) which is the (negative) average of λ i , i = 2,…, n (some of which may be complex conjugates).
Now we compare this value for what it would be if our numerical judgments were taken at random from the scale 1/9, 1/8, 1/7,…, 1/2,…, 1, 2,…, 9 (preserving the reciprocal relationship in order to improve consistency).
We have for different order random matrices and their average consistencies:
One rarely has to go beyond 7 × 7 matrices (in order to maintain connectedness between relations and keep the consistency relatively high). It is with these numbers that we divide the consistency index and recommend revisions if the ratio is considerably higher than 10%. This gives the consistency ratio CR.
We note that solution of the largest eigenvalue problem, when normalized, gives us a unique estimate of an underlying ratio scale.
8.3. The Calculation of the Weights and Priorities
We have noted that the normalized principal eigenvector of the paired comparison matrix A gives the weights of the elements being compared. An efficient program for finding this vector has been used in our applications. There is, however, a very simple method for finding an approximation to the vector of weights.
Suppose that the matrix A takes the following form:
where the ratios are not maintained exactly. Normalizing the first column by dividing each element by the sum of the elements in the column gives , normalizing the second column gives , while the third column normalizes to . Each of these columns can be regarded as a measure of the relative weights. A “reasonable” method of evaluating the weights would be to take the three columns
and to average across the rows to give
and to use this “average” column for the relative weights.
We can then multiply Aw to obtain an estimate of λ max w . We then divide each component of λ max w by the corresponding component of w to obtain an estimate of λ max ; we may average over these estimates to obtain an overall estimate of λ max . We then compute the consistency index (CI) = (λ max – n )/(n – 1).
This can be compared with its value derived from random entries and the vector w is accepted if the ratio CR is of the order of 10% or less.
Another quick way to obtain an estimate of the principal eigenvector is to take the geometric mean of each row and then normalize the elements of the resulting column. With perfect consistency one can also add the elements in each row and normalize the resulting column, but in general this is an unsatisfactory way.
Example . World Influence of Nations [Saaty and Khouja]
A number of people have studied the problem of measuring world influence of nations. We have briefly examined this concept within the framework of our model. We assumed that influence is a function of several factors. We considered five such factors: (1) human resources; (2) wealth; (3) trade; (4) technology; and (5) military power. Culture and ideology, as well as potential natural resources (such as oil), were not included.
Seven countries were selected for this analysis. They are the United States, the USSR, China, France, the United Kingdom, Japan, and West Germany. It was felt that these nations as a group comprised a dominant class of influential nations. It was desired to compare them among themselves as to their overall influence in international relations. We realize that what we have is a very rough estimate—mainly intended to serve as an interesting example of an application of our approach to priorities. We shall illustrate the method with respect to the single factor of wealth.
In Table 8.2 we give a matrix indicating the pairwise comparisons of the seven countries with respect to wealth. For example, the value 4 in the first row indicates that influence through wealth is between weak and strong importance in favor of the United States over the USSR. The reciprocal of 4 appears in the symmetric position, indicating the inverse relation of relative strength of the wealth influence of the USSR compared to the United States.
Note that the comparisons are not consistent. This basic inconsistency cannot be eradicated merely by changing scale. Moreover, there are also apparent numerical (or scale) inconsistencies. For example, the United States : China = 9 (not 28) despite the fact that the United States: USSR = 4 and USSR: China = 7.
TABLE 8.2. Wealth
Nevertheless, when the requisite computations are performed, we obtain relative weights of 43.3 and 21.7 for the United States and the USSR, and these weights are in striking agreement with the corresponding GNPs as percentages of the total GNP in Table 8.3 . Thus, despite the apparent arbitrariness of the scale, the irregularities disappear and the numbers occur in good accord with observed data.
Compare the normalized eigenvector column derived by using the matrix of judgments with the actual GNP fraction given in the last column. The two are very close in their values. Estimates of the actual GNP of China range from 74 to 128 billion.
TABLE 8.3. Normalized wealth eigenvector
*Billions of dollars.
8.4. Hierarchies in Decision Making
We shall not attempt to give a full theoretical development of the method of analytical hierarchies here, but will (in a later section) explain briefly why and how hierarchies work, and will then show how to use the method in practice.
The essential requirement for analysis by hierarchies is that one should be able to decompose a problem into levels, where each level consists of similar elements and has an impact on the levels above and below. We have found in practice that this is the way in which one instinctively structures a problem.
Consider a typical problem. Some decision will have to be made: this is placed at the apex of the hierarchy. Those people or groups who have an interest in the problem and in the final outcome will provide the next level of the hierarchy. Each of these groups will have a number of objectives which are of different degrees of importance to them. (Some of these objectives may be held in common.) The objectives will provide the next level. There are also the possible decisions or outcomes to be evaluated. Each of these will satisfy the objectives to a greater or lesser extent and, in consequence, they provide the final level of the hierarchy.
A typical hierarchy is shown in Fig. 8.1 .
FIG . 8.1.
8.5. Weighting through the Hierarchy
We may now apply the weighting process to the participants at the first full level of the hierarchy. By forming the matrix of paired comparisons of the power of each party to affect the outcome, we are able to derive the normalized eigenvector to obtain the power of each of the parties. The process may be repeated for the objectives of each party. A typical judgment which has to be made here is: For a given party, which of two of its objectives is it likely to pursue more, and how much more? This is a process of evaluating the objectives in pairs according to their contribution to each element, a party, in the level immediately above. A vector of objective weights is obtained for each party.
The final outcomes are now similarly weighted by a comparison matrix according to how well they would satisfy a given objective in the view of the party whose objective is under consideration. The process is repeated for each objective of each party.
In making the judgments about the power of a party to influence a decision, or about the relative importance of its objectives, etc., one either should seek advice from those who are actively involved or who are particularly knowledgeable about the situation, or should form one's own judgments based on a complete background study. The former is obviously preferable, but is not always possible.
The weights for each of the outcomes may now be obtained by composite weighting through the hierarchy. We follow a path from the outcome at the apex to each final outcome at the lowest level and multiply the weights along each segment of the path. The result is a normalized vector of final weights for the possible outcomes under consideration.
There are two possible (and related) ways to view this process. One is to consider a flow of power taking place downward along the paths of the hierarchy from the initial source; this flow splits according to the power of the parties. Each party then divides its available power among its objectives; this accounts for the first multiplication. The flow to each objective is further divided to allow for the way in which the outcomes satisfy each objective; this accounts for the second multiplication. Each outcome receives a contribution from each objective; these are added to give a final weight for each outcome. The initial unit flow has then moved down to the final outcomes to show how much power would be transferred to each. The second approach is to regard the weights as probabilities of independent events, so that the final weights indicate the probabilities that each final outcome will take place if each party used its power to satisfy its objectives.
The outcome to receive the highest weight then indicates an optimum which may be either a most likely or a most desirable result.
This is, of necessity, a very simplified account of the process, although it captures the essentials and indicates how the method works in practice. The interested readers will find a detailed exposition of the theoretical underpinning and of many ramifications and interesting results in the references.
Example . Job Decision
A student who had just received his PhD was interviewed for three jobs (A , B , and C ). His criteria (Fig. 8.2 ) for selecting the jobs and their pairwise comparison matrix are given along with the eigenvector (normalized) associated with the maximum eigenvalue (Table 8.4 ). In this matrix, the comparisons were made relative to which factor of a pair was more important (and by how much) when considering overall satisfaction with a job. Thus the value of 5 in row three column four indicates that benefits are strongly more important than colleague associations.
TABLE 8.4
Now each job is compared with each of the other jobs with respect to how they bear upon each of the six criteria (Table 8.5 ).
FIG . 8.2.
TABLE 8.5
The eigenvectors (one for each criterion) just obtained are each weighted by the eigenvector component of the associated criterion and the results summed and normalized. We obtain:
We have prioritized the jobs. The differences were sufficiently large for the candidate to accept the offer to job A .
The previous example was essentially a one-person conflict problem. Examples of conflicts and decisions involving more than one person may be found in the references.
There is another side to these problems. Suppose that a party to a conflict does not like the projected final outcome. Can he do anything to change this outcome? This question is examined next?
8.6. The Forward and Backward Processes in Planning and in Conflict Analysis
It is important to identify those outcomes which are likely to emerge and which, in large measure, satisfy the objectives of each party to a conflict or decision process. Such a process, which is descriptive, may be regarded as a one-point boundary problem fixed at the present state. It is the forward process. Given the present actors and their current objectives, capabilities, and policies, which outcome is the most likely to emerge? This outcome may be a composite of a number of pure outcomes which have been considered.
There is, however, an alternative approach to the solution of a conflict problem. Given a desired future outcome, what can or should be done to achieve such an outcome? Working backward, one assesses the problems and opportunities which affect this outcome and identifies those policies which would be most effective in producing the desired outcome. Such an approach, which is normative, may be regarded as a one-point boundary problem fixed at the future. It is the backward process.
A combination of the forward and backward processes is frequently used in planning. In order to apply the backward process, it is first necessary to find the desired outcome for each of the parties in the problem, and to evaluate their reactions to all of the outcomes. To do this, one may use the weights for the outcomes which were obtained from the first forward process and note how much of each weight came from each party.
The forward process indicates which outcome is likely to emerge, given the existing actors, objectives, and policies. In the backward process one tries to find ways in which a desired outcome may be made acceptable to those who do not give it a high weighting. This may enable a party which strongly desires a given outcome to see what it must do to induce other parties to move toward the desired outcome. The new policies which this process suggests are adjoined to the original policies and the forward process is repeated. The desired outcome then may be modified by some of the features of the newly emergent outcome and the process may be iterated.
If one of the parties realizes that its weighting is not very high and that, in consequence, it cannot influence the outcome as much as it would desire, that party may take action to increase its power by alliances or by the use of threat options, including a credible increase in determination matched by effective action.
This forward and backward process has been applied in a number of examples. [Alexander and Saaty, Emshoff and Saaty, McConney and Vrchota].
8.7. Design of a transport system for the Sudan: Priorities—Investment
Against a background of great potential agricultural riches, the Sudan, the largest country in Africa (967,491 square miles) but with only an 18.2 million population, is today a poor country with a GNP of about 2.8 billion dollars. Oil countries in the Middle East and international agencies, including the World Bank, recognize the capacity of the Sudan as a major provider of food for Africa and the Middle East, and have been investing in its development (see Saaty, 1977).
Incidentally, the oil-rich Arab countries’ populations do not exceed 20 million and, hence, their need does not even begin to make a difference in how many people the Sudan can feed. Even if its northern neighbor, Egypt, were to be included for one-half of its population (estimated at 50 million by 1985) to be fed by Sudan, there would still be land to feed perhaps 100 million more people. The entire economy of the Sudan and, in particular, the agricultural sector, suffer from lack of adequate transportation.
The Sudan is serviced by four major modes of transportation: rail, road, river, and air. These modes are combined together to provide a sparse and far-flung transportation infrastructure. The air network is centered at Khartoum and the rail and road systems are oriented for export through Port Sudan. The country is characterized by low transport connectivity. The object was to develop a transport plan for the Sudan by 1985.
The functions of a system can be. represented by a hierarchy with the most important “driving” purposes occupying the top level and the actual operations of the system at the lowest level. In the case of the Sudan, overall development occupied the top level, followed by a level of scenarios or feasible outcomes of the future. The third level consisted of the regions of the Sudan as it was desirable to know their impact on the scenarios and in turn the impact of the scenarios on the top level. The fourth level consisted of the projects for which it is desired to establish their priorities by studying their impacts on the regions. Thus, we had a four-level hierarchy in the study. By composing the impacts of the fourth level on the third, the third on the second, and the second on the first, we obtain the overall impact of each project in the fourth level on the overall development of the Sudan represented in the first.
Pairwise comparison of the four scenarios according to their feasibility and desirability by 1985 gave rise to the matrix presented in Table 8.6 .
TABLE 8.6. Priorities of the scenarios
The priorities of the scenarios in the order they are listed are: 0.05, 0.61, 0.25, 0.09. As can be seen, scenario II dominates, with scenario III next in importance. Since the future is likely to be neither one nor the other, but rather a composition of these scenarios—with emphasis indicated by the priorities—this information was used to construct a composite scenario of the Sudan of 1985. This scenario is intended as the anticipated actual state of the future, it being a proportionate mix of the forces which make up the four scenarios described above. The composite scenario takes the main thrust of scenario II, the future given by far the highest priority, and is enlarged and balanced with certain elements from scenarios III and IV. This composition indicates the likelihood of a synergistic amplification of individual features.
The Sudan has 12 regions whose individual economic and geographic identity more or less justifies political division into distinct entities. The regions were compared pairwise in separate matrices according to their impact on each of the scenarios. They comprise the third hierarchy level. The resulting eigenvectors are used as the columns of a matrix which, when multiplied by the eigenvector of weights or priorities of the scenarios, gave a weighted average for the impact of the regions (Table 8.7 ).
TABLE 8.7. Priority weights of regions (%)
Now the projects, of which there were 103 determined according to GNP growth rates which suggest supply–demand and flow of goods, comprised the fourth level of the hierarchy. They were compared pairwise in 12 matrices according to their impact on the regions to which they physically belonged. A project may belong to several regions and this had to be considered. The resulting matrix of eigenvectors was again weighted by the vector of regional weights to obtain a measure of the overall impact of each project on the future. This gave rise to the following kind of Table (8.8) of which there are nine in the final report:
We examined the 4.3%, the present growth rate, and found that most of the current facilities with the prevailing level of efficiency would be crammed to their limit. Obviously, a compromise with a rational justification for growth had to be made somewhere between these two extremes. When we examined the 6% GNP growth rate, found feasible by the econometric analysis, it provided excellent guidelines for those projects which were found to be needed at 4.3% and remained invariant with high priority at 7.3%. These were mostly the projects we recommended for implementation. The ratios of priorities to costs served as a measure of the effectiveness of investment. Six billion dollars have been earmarked for expenditure in the Sudan over the next few years closely following the recommendation of the study.
TABLE 8.8. The transportation development plan: phase I (1974 price level in £S000,000) (6% GNP growth rate)
* The priority rating of this project is based mostly on potential rather than present development. In view of its high cost relative to other road projects, it has been omitted. It is recommended that it be given urgent consideration in the following planning period.
Total investment requirements to achieve the composite scenario projected growth of real GNP at 4.3%, 6%, and 7.3% per year are given in Table 8.9 . For example, at 7.3% they are estimated to be approximately $5105 million at 1974 price levels, or $7647 million at current price levels (considering inflation between 1974 and 1985). The latter figure represents approximately 10% of the GNP each year over the planning period, 1972–1985. This will be divided among the major sectors as shown.
TABLE 8.9. Dollars (millions ), current prices
8.8. Rationing Energy to Industries: Optimization
Only a short time ago it was unthinkable and deemed as an academic exercise to speak of rationing energy. It was felt that there could not be a crippling energy crisis because our energy czars and planners would presumably take our needs into their projections. Today, things look very different. Witness the cases of the lack of natural gas in the cold winter of 1976–1977 which caused the shut down of some schools and industries and the coal strike of 1977–1978 (See Saaty and Mariano).
In any case we must face the needs of our homes, offices, industries, and massive transportation systems not simply by making additional supplies of energy available but also by replanning and redesign to improve efficiency and to diminish the 7% annual rise in energy consumption.
Besides improving efficiency, for the long range we need to consider several alternatives to prevent severe energy shortages. Among them are:
(1) Reduction of US consumption to the level of domestic oil production.
(2) Discovery or development of new forms of energy such as coal gasification, geothermal, nuclear fission, nuclear fusion, and solar. But these forms are now in short supply.
(3) Rationing. Although rationing is not an attractive alternative, we have seen that in cases of severe weather, energy had to be diverted from schools and industries in the Midwest to accommodate homeowner needs. Rationing can become a pressing alternative if supplies dry up or continuity in importing oil is seriously threatened.
In this application, we confined our analysis to manufacturing industries. Examples of the groups we considered are: (1) food and kindred products, (2) tobacco manufacturers, (3) textile mill products, (4) apparel and related products, (5) lumber and wood products, (6) furniture and fixtures, (7) paper and allied products, (8) printing and publishing, (9) chemicals and allied products, (10) petroleum and coal products, etc.
The optimal weights generated for these classes of industries are applicable on a yearly basis and, therefore, the actual scheduling of allocation on a day-to-day or a month-to-month basis is not made explicit by the model. Our approach can be extended to peak power demand considerations where shortage of power may occur in a short time duration. In this case, the optimal scheduling of power and its allocation will be determined as a function of time.
We used the following objectives which fall into two classes: class 1, characterized by two measurable indicators; contribution to economic growth (measured in dollars), and impact on the environment (measured in tons of pollutants); class 2, characterized by three qualitative indicators; contribution to national security, to health, and to education. The measures for these were derived using judgments and the eigenvalue procedure.
The results of the two classes were composed hierarchically to obtain an overall priority for each industrial group.
As the real-life problem is too long to work out here we have chosen an example which illustrates how one does a rationing problem. It combines priorities and optimization.
The problem in the energy demand allocation is concerned with finding allocation weights for several large users of energy according to their overall contribution to different goals of society. Let us assume the following conditions.
There are three large users of energy in the United States: C 1 , C 2 , and C 3 . The goals against which these energy users will be evaluated are: contribution to economic growth, contribution to environmental quality, and contribution to national security. Based on the overall objective of social and political advantage the matrix of paired comparisons of these three goals on the previously described scale from 1 to 9 is given by:
The normalized eigenvector corresponding to the dominant eigenvalue = 3 of this matrix is given by:
The decision-maker, after a thorough study, has made the following assessment of the relative importance of each user from the standpoint of the economy, environment, and national security. The matrices giving these judgments are respectively:
The corresponding normalized eigenvectors are respectively the three columns of the following matrix:
This matrix is multiplied by the vector P (0) yielding the following vector which is already normalized, giving the eigenvector priorities of the activities C 1 , C 2 , and C 3 :
We cannot allocate energy in proportion to the priorities of the industries as they may be interdependent. Material from a low priority industry may flow to a higher priority one. To express this relationship as a constraint we use the following input-output matrix:
When the coefficient in the (i , j ) position of the above matrix is weighted by α i and α j and summed over each row, we obtain the vector of dependence numbers:
Suppose that the energy requirements R i (in trillion BTU) of the three users are as follows:
Activity
(C i )
Energy requirements
(R i )
C 1
4616
C 2 7029
C 3
3297
Total
14942
Also assume that the total energy available has been cut back to a level of R = 12,000 BTU. We have the following linear programming problem:
Maximize
whose coefficients are the corresponding elements of the vector β , subject to:
in which the quantities on the right are respectively R i /R , i = 1, 2, 3 and to
The optimal allocation is given by
Thus only C 2 is not given its full requirement.
Note that here we have simplified the linear-programming problem to make it easier to grasp the procedure.
8.9. Oil Prices in 1985
8.9.1. THE PROBLEM
Today oil is the world's major energy resource. It accounts for about 54% of the world's total energy consumption. Because of conservation and the development of alternative sources in the industrialized countries, the share of oil in the world's total energy consumption is expected to decline. But the total volume of oil consumption will still rise and it will remain the largest single source of energy supply for the next two decades (See Saaty and Gholamnezhad).
Despite oil price hikes between 1974 and 1979, the real price of OPEC oil has not increased significantly when adjusted to inflation and depreciation of the dollar. Actually, the devaluation of the dollar against the Japanese yen and the West German mark caused the real price of oil to decline in these countries. However, because of depletion of the world's proven oil reserves, increasing demand, and possible political unrest in the major oil-producing countries, oil prices are expected to rise during the next decade.
There has been a number of projections of world oil prices by major oil companies and government agencies. Most are based on demand and supply. But in today's world, oil-market economics and politics are interwoven and political decisions increasingly influence the levels of oil production, consumption, and prices.
We now use the analytic hierarchy process to project the real price of oil in 1985. Figure 8.3 represents the hierarchical model used.
8.9.2. HOW TO COMPUTE PRICE INCREASES
1. Compute the relative weights of the factors (W 1 …W 5 ) according to their effectiveness in increasing the price of oil: W 1 , world oil consumption increase; W 2 , world excess production capacity; W 3 , oil discovery rate; W 4 , political factors; W 5 , development of energy sources alternative to oil.
2. For each W i , compute the relative likelihood of its corresponding subfactors. For i = 1, 2, 3 these are H i , M i , L i (H i , high; M i , medium; L i , low. See Fig. 8.3 for full details). For example, for oil-consumption increase W 1 we ask the question: Which one of the three levels of increase is more likely for the period under consideration: 4%, 2%, or 1% per year? For i = 4, the subfactors are: P 1 , instability of the Persian Gulf region; P 2 , continuation of Arab-Israeli conflict; P 3 , increasing Soviet influence in the Middle East and the relative importance of these is estimated. For i = 5, the subfactors are: V , vigorous; M , moderate; R , restrained; and the relative likelihood of those levels of development is estimated.
3. For the instability of the Persian Gulf region P 1 compute the relative importance of its three subfactors, namely: S 1 , social strains within countries; S 2 , tension between individual states; S 3 , continuing disorder in Iran.
4. Compute the composite weights for each subfactors and select subfactors with high relative weights.
5. Compute the relative likelihood for each level of price increase for each selected subfactor.
6. Compute the composite weights of the levels of price increases. The result will be a set of numbers representing the likelihood of each price increase.
7. Compute the expected value of price increase by multiplying each price increase level by its corresponding likelihood. Remark : Some of the judgments used here were initially provided by five experts from major oil companies and were later modified to enforce consistency and to cope with other factors not included in the first version of this model. It may be useful for the reader to examine these judgments closely.
FIG . 8.3.
W 1 World oil-consumption increase
W 2 World excess production capacity
W 3 Oil discovery rate
W 4 Political factors
W 5 Development of energy sources alternative to oil
S 1 Social strains within countries
S 2 Tension between individual states
S 3 Continuing disorder in Iran
V Vigorous
M Moderate
P 1 Instability of the Persian Gulf region
P 2 Continuation of Arab-Israeli conflict
P 3 Increasing Soviet influence in the Middle East
R Restrained
H i High i = 1, 2, 3
M i Medium
L i Low
bby Billion barrels per year
8.9.3. COMPUTATION OF OIL PRICE IN 1985
M 1 : Which factor would have a stronger effect on world oil prices by 1985?
M 2 – M 6 : Which development is more likely to take place by 1985?
Oil-consumption increase :
Excess capacity :
4 × 0.649 + 2 × 0.279 + 1 × 0.072 = 3.2%
10 × 0.105 + 5 × 0.63 + 1 × 0.258 = 4.5%
per year
above production level in 1985
Discovery rate:
20 × 0.072 + 10 × 0.649 + 5 × 0.279 = 9.3 billion barrels per year.
TABLE 8.10
M8–M16: Given the following developments, which price-increase level is more probable?
8.9.4. ANALYSIS OF THE RESULTS FOR 1985
We now analyze the factors influencing world oil prices according to their relative importance to show how the comparisons were made.
a. Political Factors (W 3 = 0.631)
Political factors play an extremely important role in the world oil market. The Arab oil embargo of 1973, the Iranian revolution, and consequent disruptions in the world oil supplies have demonstrated the significance of political factors in the supply, demand, and price of oil.
Political factors included in this analysis are the instability of the Persian Gulf region, continuation of the Arab-Israeli conflict and increasing Soviet influence in the Middle East. Although OPEC itself plays an important political role in the oil market, its stability is very dependent on developments in the Middle East.
P 1 : Instability in the Persian Gulf region (0.402). The region which will continue to be of extreme importance in the future supply and prices of oil is the Middle East, particularly the Persian Gulf states. The Persian Gulf is surrounded by a number of major oil-exporting countries such as Iran, Saudi Arabia, Iraq, Kuwait, Qutar, Bahrain, and the United Arab Emirates. These countries, excluding Bahrain, which is not a major oil exporter, are the members of OPEC and altogether account for over 80% of its proved oil reserves or nearly half of the world's total reserve. Over 30% of the world's oil supply comes from this region. Stability of the Persian Gulf itself depends on several other factors, particularly the social strains due to rapid economic developments, industrialization, unstable political systems, and religious movements in the region. Also, tensions between the individual states, as we are now witnessing between Iran and Iraq, could lead to a regional war. Another factor to be considered is the continuation of disorder in Iran which would not only keep its oil output as low as it is today, but may also increase instability in the region.
P 2 : Continuation of the Arab-Israeli conflict (0.163). The Arab oil embargo of 1973 demonstrated the impact of the Arab-Israeli conflict on the flow of oil to the industrialized world. Long delays in peace will discourage the major Arab oil producers from cooperating in meeting the demands of the industrialized world. This will put more pressure on the world oil market and consequently oil prices will rise drastically.
P 3 : Increasing Soviet influence in the Middle East (0.066). Although the Soviet Bloc is currently a net exporter of oil, because of decline in oil production in the Soviet Union, it is projected to become a net importer of oil in the near future. Therefore, the Soviet Union will be competing with Western countries for Middle Eastern oil. Some political observers believe that the purposes of Soviet intervention in Afghanistan and its assumed assistance to the rebels in Baluchistan include providing itself with secure oil and gas supply sources in the future by assured access to the Persian Gulf. Increasing Soviet influence in the Middle East will enhance its position in the oil market vis-à-vis the West. And, if it becomes to their advantage, the Soviets would not hesitate to use oil as an economic weapon against the West, particularly the United States. This action will lead to higher payments for oil for the Western countries.
b. World oil-consumption Increase (W 1 = 0.123)
In 1979, the United States, Japan, and Europe accounted for about 75% of the total world oil consumption. No substantial increase in demand is anticipated for these countries but, in the developing countries, particularly the oil-exporting countries, demand for oil is expected to increase significantly due to industrialization and development.
c. Oil Discovery Rate (W 3 = 0.099)
Before 1970 oil discovery rates were much higher than oil-production rates. Therefore, the volume of the world's discovered reserves was increasing. But, since the early 1970s, oil discovery declined steadily while production rates increased continuously. This downward trend for discovery rates is projected to continue slowly until 1985 and rather sharply thereafter.
d. Development of Energy Sources Alternative to Oil (W 5 = 0.051)
A substantial amount of oil could be replaced by synthetic fuels from large coal, oil shale, tar sands reserves, and biomass resources but, due to the long lead times (about 6 to 10 years), large capital requirements and environmental constraints, such fuels are not expected to make a significant contribution during the next decade. However, in the 1990s synfuels will play an important role in the world energy market.
e. World Excess Production Capacity (W 2 = 0.030)
Today, the world's excess production capacity is more than 10 million barrels per day (MBD), two-thirds of which is from the Middle East. At this level of excess capacity only large oil producers can affect oil prices by making their production levels fluctuate. However, when the excess capacity declines substantially, say to 2 to 3 MBD, even small producers can cause a sudden jump in oil prices by cutting back their production (or large producers by cutting back on a small portion of their production).
f. Projected Oil-Price Increase (1985)
Table 8.11 shows the probabilities of given oil-price increase for each level under consideration. According to these results, the price increase for 1985 would be:
TABLE 8.11. Probabilities of given levels of price increase by 1985
Level
%
Composite probability
E
80
0.080
H 40 0.281
S 20 0.389
M 10 0.190
L
5
0.059
given the present price of Arabian light crude (market crude) of $32 per barrel, a 27.6% increase by 1985 means that the real price of oil will be
Assuming an average inflation rate of 10% in the United States, Americans will pay 40.80 (1 + 0.10)5 = $65.70 (or more, depending on the quality of crude oil) for each barrel of imported oil by 1985.
Our results are higher than those projected by the Exxon Corporation (World Energy Outlook, 1980). Based on the price of Arab oil (light crude) in October 1979 of $18 per barrel, the Exxon study projected that the real price of oil would be $25 per barrel in 1985 and $28 per barrel in 1990. The Exxon results are below those assumed by the US Senate Finance Committee in its projections.
8.10. Architectural Design
In our final example, we combine the analytic hierarchy process with other considerations, such as aesthetic qualities (see Saaty and Beltran).
We want to design a house for a family of three (husband, wife, and child) who own a plot on which they want to build a custom-designed house. Their maximum budget is $105,000 to cover construction and landscaping but not legal and architectural fees, etc. The cost of construction is assured to be $45 per square foot and that of landscaping is $1.80 per square foot.
The members of the family are the decision-makers. We assume that they will seek to satisfy the same human needs, but may assign different weights to each objective. The objectives are (i) the need to eat (M ), (ii) the need to rest (R ), (iii) the need to entertain (including each other) (E ), (iv) the need to clean (people, clothes, etc.) (K ), (v) the need to store (S ), and (vi) the need to communicate within the house (by hallways, etc.) (C ). Certain rooms meet a number of these objectives: for example, the family room serves for resting, eating, and entertaining.
This gives us initially a hierarchy for allocation of space within the house according to the way in which each room satisfies the needs of family members. This hierarchy is shown in Fig. 8.4 and the abbreviations used in the judgmental matrices are taken from this diagram.
FIG . 8.4.
Our first calculation shows that the child will have little influence on the final outcome, even if he forms a coalition with his father, and so we consider only the needs of the husband and wife. The particular husband and wife in this situation agreed on the judgments as given; they would not necessarily be true for all families (Table 8.12 ).
TABLE 8.12. Power of parties—husband, wife, child
As before, we must now prioritize the importance of the objectives from the standpoint of both husband and wife, and we also include combined weights. This leads to Table 8.13 .
TABLE 8.13 (a ) Strength of objectives
We then allocate the space available for each activity from the composite outcomes. The family have decided, on the advice of an architect, to allocate about 85% or about $90,000 to construction and the remainder to landscaping. At $45 per square foot, this means that the house would have a maximum area of 2000 square feet. This leads to the following allocation:
Total area to allocate: 2000 sq. ft.
We then allocate the space in each category according to the relative importance given to each room by the parties. Since each views this differently, this must also be weighted according to their relative power to obtain a fair division of space. Note that some rooms will draw space for more than one category.
We give here in Table 8.14 the resulting allocation. (The reader is referred to Saaty and Beltran for details of judgmental matrices.)
TABLE 8.14. Allocation of areas to architectural spaces
We then have to decide on appropriate dimensions and this may be done by selecting varying sets of dimensions and prioritize: some adjustments may have to be made to maintain the totals for each objective. A final set of dimensions may then be obtained. In our example, the total area was 1982 sq. ft., which does not exceed the maximum permissible.
It remains to locate the different rooms. We focus on the three most important areas: entertainment, rest, and meals, and place these first:
Weights
Adjusted weights
E
0.4305
0.5113
R 0.2169 0.2576
M
0.1946
0.3211
0.8420
1.0000
FIG . 8.5.
The criteria used are front view FV, back view BV, left-side view LV, and right-side view RV (with a tree). The husband and wife again decide how they wish each area to be located by function. They then decide how to place the highest priority room, the living-room, in the highest priority location for entertainment needs, the front view. The house is divided into quadrants, as shown in Fig. 8.5 and blocks are prioritized. The quadrant C 6-10 H 6-10 received the highest weight and was selected. We position the remaining entertainment rooms and rooms in other areas similarly, in descending orders of priority. The remaining regions (storage, etc.) are then fitted in. The result is shown in Fig. 8.6 .
We have thus formalized a singularly complex decision process by using a number of simpler decisions and, by obtaining input at each stage, have combined both intuitive and more formal information in the same model.
FIG . 8.6.
Chapter 8—Problems
1 . Describe some decision which you have to make in the near future. Formulate an analytic hierarchy to help you make this decision and find the “best” outcome. Does this outcome surprise you? Why? Now enlarge the problem to include other people who may influence the final outcome. Does the answer change? Why?
2 . You have been asked to advise a company on the allocation of its resources between the development costs of two new products. Formulate an analytic hierarchy to solve this problem.
3 . A large insurance company has decided to diversify its portfolio. Formulate an analytic hierarchy to decide among a range of possible purchases, taking into account both the internal characteristics of the firm and the external factors of the market.
4 . You are entering the job market shortly. What factors are important in choosing a job? Formulate a decision model to help you in your choice.
5 . A family is deciding on a city in which to live. Develop a hierarchy to assist in the decision, considering physical, economic, and social factors.
6 . A manufacturing company has decided to open a plant overseas, but has not yet decided on the country. Formulate a hierarchy to assist in this decision. Remember to take into account such factors as managerial and operational control, tariffs, capital formation, labor conditions, social and cultural factors, and political risks.
7 . Using an appropriate hierarchy determine the expected cost of a compact automobile by 1990 allowing for inflation. It is the object of this exercise that one include the so called imponderables.
8 . The Saaty family considered buying a home computer, but the computer was not the only source of pressure on their resources at the time. At the time the kitchen was being remodelled causing considerable financial drain. Therefore even if the best computer were determined, it would have contended with saving the money in the bank, cementing the driveway or even donating the money to charity. Construct a hierarchy of benefits and a hierarchy of costs (not simply dollar costs but other sources of pain) to decide which of these computers, Radio Shack, Apple II, and Texas Instruments would have the highest benefit to cost ratio. Then take the best one along with keeping the money in the bank, and performing other urgent large expenditures to decide on the benefit to cost ratio. Note that the results may be suggesting an order of implementation over time.
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