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Deriving utilities using the analytic hierarchy process
003%0121/86 $3.00 + 0.00 Pergamon Journals Ltd
Socio-&on. Plann. Sci. Vol. 20, NO. 6, pp. 393-395, 1986 Printed in Great Britain
DERIVING
UTILITIES USING THE ANALYTIC HIERARCHY PROCESS WARREN
Department
R. HUGHES
of Economics, University of Waikato, Hamilton, New Zealand
Alzstxact-The analytic hierarchy process is used to derive the utilities of outcomes in a decision problem. Such utilities may be verifiable or adjusted by the familiar standard gamble procedure. For multiattribute outcomes, the methodology may be easier to employ than scoring rule procedures.
never do. This latter category of people worry that this winter will not be a big snow year. A $1000 snow-blower is fairly conspicuous consumption. One is anxious that such a purchase be justified and quickly.”
1. INTRODUCTION Outcomes
in a decision
problem
involving
the inter-
action of discrete act/event spaces may require the derivation of utilities prior to determination of the optimal act using a criterion such as maximum expected utility. One technique of deriving utilities is to use the so-called “standard gamble” and this is a common procedure for single attribute money outcomes. If outcomes have multiattributeconsequences, the derivation is more complicated but is still possible using scoring rules for attributes and the standard gamble, Keeney and Raiffa [l]. In this paper, a direct approach for deriving utilities in both single and multiattribute situations is outlined utilizing analytic hierarchy process (AHP) methodology, Saaty [2]. Utilities derived using AHP can later be “verified” or fine-tuned by the standard gamble test.
2. ILLUSTRATIVE
Letting 0, denote the multiattribute outcome for act i=1,2,3andeventj=1,2inTablel,wefirstrank the outcomes in order of preference as follows:
EXAMPLE
011
To illustrate the approach, we will analyze a decision problem outlined in Bell [3]. There the customer
was faced with four choices when considering the purchase of a snow-blower. For our purposes, the problem can be adjusted slightly and reduced to a
o 32
three act problem as outlined in Table 1. The customer must decide on his optimal act out of the three 022
possibilities listed. We will assume that the discount deal is only available after the mid-point of the snow season. Choosing this act means a larger effort is required to shift snow in the early part of the season (if Big snow occurs) than if the usual deal had been chosen. As Bell [3] noted, the snow-blower manufacturer:
In ranking the outcomes this way, it has been assumed that non-justification of the purchase is less preferred to shifting a big snow unaided by the snow-blower. Saving $200 on the discount deal has a slight mitigating effect if the purchase is not justified. The rankings in Bell [3] reflect a one season perspective with 0,2 the best outcome. In the outcomes above, however, the decision maker also considers the durable nature of the snow-blower which will certainly be useful in future seasons. The “Discount deal” is the manufacturer’s attempt to exploit this sentiment. Having ranked the outcomes, the prospective purchaser must now deduce how much better or worse the outcomes are in relation to each other. The AHP
“knew that homeowners in the United States fall into three categories: those that own snowblowers, those that know they never will, and those who think about buying one every fall but Table I. Multiattribute
consequences to snow-blower customer Big snow
Don’t buy Usual deal Discount deal
$0, great effort SIOOO,low effort $800, medium effort
Purchase justified immediately; machine available in future; no more decision making necessary. Purchase made later in the season: would have been useful prior to purchase date; more effort expended than under 02, ; machine available in future; no more decision making necessary. Non-purchase decision justified in current year; future decision making necessary; machine will be needed at some future date. Purchase not made but would have been justified this year; great effort expended shifting snow; future decision making necessary; machine will be needed at some future date. Purchase at discount price but not justified this year; machine available in future; no more decision making necessary. Purchase at full price but not justified this year; machine available in future; no more decision making necessary.
Light snow $0, no effort $1000, no effort $800, no effort
393
394
WARREN R. HUGHES Table 2. AHP iudmental
assessments between
Number assigned
two outcomes
Assessment of outcomes
1 3 5 7 9
First outcome is
{ %?!k&?;re
} beneficial
Use 2.4.6. 8 for comaromises
Table 3. Hypothetical
AHP assessments for typical snow-blower customer
cost
0,
effort
02,
03,
01,
01,
03,
%
021
SlOOO
1
2
2
7
8
9
0.4031
1
$800 medium
l/2
I
1
5
5
6
0.2347
0.54
$0 none
l/2
I
I
4
4
5
0.2100
0.48
SO great
l/7
l/5
l/4
1
2
3
0.0694
0.10
$800 none
118
115
l/4
l/2
I
2
0.0491
0.04
$1000 none
l/9
l/6
115
113
l/2
I
0.0338
0
Utilities
IOW
022
Maximum eigenvalue = 6.14 Consistence ratio = 0.02
methodology developed in Saaty [2] can be applied to this situation. Using a nine point scale, the decision maker assesses in turn a pair of outcomes k, q and allocates a number l-9 in accordance with how much more beneficial one outcome is over another as outlined in Table 2. This number becomes the kqth element of the judgmental matrix and its reciprocal becomes the qkth element. Ones are placed on the diagonal. For WIacts and n events the dimension of the square judgmental matrix will be mn. This procedure results in quantitative weights from qualitative judgments. A hypothetical assessment is shown in Table 3 but each customer could differ with respect to the numbers assigned. The weights in the secondto-last column of Table 3 are the elements in a normalized (sum to unity) eigenvector associated with the maximum eigenvalue for the judgmental matrix. The consistency ratio of 0.02 being close to zero indicates the numbers assigned are consistent, Saaty [3]. The software package MATLAB was used on a VAX 1l/780 to derive the values in Table 3. Taking the weights in the second-to-last column, we use a positive linear transformation to derive utilities on a [0, l] scale. The transformation from weights (w) to utilities (a) is given by: u=a+bw where
I .OOOl
whatever the probability of a “Big snow” is. If this probability is assessed at less than 0.348 the optimal act is “Don’t buy”, otherwise the “Usual deal” is optimal. Bell [3] contains a fourth act labelled the “Sno-risk deal” which yielded utilities (1,0.48) respectively for the possible events. This dominates “Don’t buy” and the offer resulted in a significant sales increase for the manufacturer, Hesse [4]. 3.
CONCLUSIONS
If the outcomes are single attribute money outcomes, utility derivations using the AHP are verifiable by the standard gamble technique. That is, the utilities such as those in Table 4 should be probabilities that make the decision maker indifferent between the outcome represented by that utility and a gamble between the best and worst outcomes in the table where the utility is the probability of realizing the best outcome. Some revision of utilities may be necessary before indifference is achieved. It is possible that a decision maker may feel more comfortable initially making qualitative AHP judgments and then subsequently fine-tuning the probabilities necessary for indifference, than he would deriving the probabilities (or utilities) directly. Irrespective of whether or not fine-tuning is needed, it should be noted that
b = l/(best - worst) = l/(0.4031 - 0.0338) a = -b
(worst) = -0.0338/(0.4031
- 0.0338)
Using these utilities the decision problem for our hypothetical customer is shown in Table 4. From the utilities in Table 4 it is easy to show that the “Discount deal” is dominated by the other two acts
Table 4. Utility payoffs for snow-blower custnm,=r
Don’t buy Usual deal Discount deal
Bia snow
Light snow
0.10 I 0.54
0.48 0 0.04
Deriving
such indifference is required to justify a decision based on maximizing expected utility in accordance with the von Neumann-Morgenstern axioms. In the case of multiattribute outcomes, the above method has the advantage of requiring the decision maker’s judgment on all attributes collectively as between alternative outcomes. This may be more natural and less time consuming that trying to assess the relative merit of each individual attribute and then aggregating a score for each outcome. However, as the number of attributes grows beyond say three
395
utilities
or four, consistent judgments may only be possible with the aid of a scoring rule. REFERENCES 1. R. L. Keeney and H. Raiffa. Decisions With Multiple Objectives. Wiley, New York (1976). 2. T. L. Saaty. The Analytic Hierarchy Process. McGrawHill, New York (1980). 3. D. E. Bell. Putting a premium on regret. Mgmt Sci. 31(l), 117-120 (1985). 4. R. Hesse. Snow job. Decis. Line 16(l), 5 (1985).
Socio-&on. Plann. Sci. Vol. 20, NO. 6, pp. 393-395, 1986 Printed in Great Britain
DERIVING
UTILITIES USING THE ANALYTIC HIERARCHY PROCESS WARREN
Department
R. HUGHES
of Economics, University of Waikato, Hamilton, New Zealand
Alzstxact-The analytic hierarchy process is used to derive the utilities of outcomes in a decision problem. Such utilities may be verifiable or adjusted by the familiar standard gamble procedure. For multiattribute outcomes, the methodology may be easier to employ than scoring rule procedures.
never do. This latter category of people worry that this winter will not be a big snow year. A $1000 snow-blower is fairly conspicuous consumption. One is anxious that such a purchase be justified and quickly.”
1. INTRODUCTION Outcomes
in a decision
problem
involving
the inter-
action of discrete act/event spaces may require the derivation of utilities prior to determination of the optimal act using a criterion such as maximum expected utility. One technique of deriving utilities is to use the so-called “standard gamble” and this is a common procedure for single attribute money outcomes. If outcomes have multiattributeconsequences, the derivation is more complicated but is still possible using scoring rules for attributes and the standard gamble, Keeney and Raiffa [l]. In this paper, a direct approach for deriving utilities in both single and multiattribute situations is outlined utilizing analytic hierarchy process (AHP) methodology, Saaty [2]. Utilities derived using AHP can later be “verified” or fine-tuned by the standard gamble test.
2. ILLUSTRATIVE
Letting 0, denote the multiattribute outcome for act i=1,2,3andeventj=1,2inTablel,wefirstrank the outcomes in order of preference as follows:
EXAMPLE
011
To illustrate the approach, we will analyze a decision problem outlined in Bell [3]. There the customer
was faced with four choices when considering the purchase of a snow-blower. For our purposes, the problem can be adjusted slightly and reduced to a
o 32
three act problem as outlined in Table 1. The customer must decide on his optimal act out of the three 022
possibilities listed. We will assume that the discount deal is only available after the mid-point of the snow season. Choosing this act means a larger effort is required to shift snow in the early part of the season (if Big snow occurs) than if the usual deal had been chosen. As Bell [3] noted, the snow-blower manufacturer:
In ranking the outcomes this way, it has been assumed that non-justification of the purchase is less preferred to shifting a big snow unaided by the snow-blower. Saving $200 on the discount deal has a slight mitigating effect if the purchase is not justified. The rankings in Bell [3] reflect a one season perspective with 0,2 the best outcome. In the outcomes above, however, the decision maker also considers the durable nature of the snow-blower which will certainly be useful in future seasons. The “Discount deal” is the manufacturer’s attempt to exploit this sentiment. Having ranked the outcomes, the prospective purchaser must now deduce how much better or worse the outcomes are in relation to each other. The AHP
“knew that homeowners in the United States fall into three categories: those that own snowblowers, those that know they never will, and those who think about buying one every fall but Table I. Multiattribute
consequences to snow-blower customer Big snow
Don’t buy Usual deal Discount deal
$0, great effort SIOOO,low effort $800, medium effort
Purchase justified immediately; machine available in future; no more decision making necessary. Purchase made later in the season: would have been useful prior to purchase date; more effort expended than under 02, ; machine available in future; no more decision making necessary. Non-purchase decision justified in current year; future decision making necessary; machine will be needed at some future date. Purchase not made but would have been justified this year; great effort expended shifting snow; future decision making necessary; machine will be needed at some future date. Purchase at discount price but not justified this year; machine available in future; no more decision making necessary. Purchase at full price but not justified this year; machine available in future; no more decision making necessary.
Light snow $0, no effort $1000, no effort $800, no effort
393
394
WARREN R. HUGHES Table 2. AHP iudmental
assessments between
Number assigned
two outcomes
Assessment of outcomes
1 3 5 7 9
First outcome is
{ %?!k&?;re
} beneficial
Use 2.4.6. 8 for comaromises
Table 3. Hypothetical
AHP assessments for typical snow-blower customer
cost
0,
effort
02,
03,
01,
01,
03,
%
021
SlOOO
1
2
2
7
8
9
0.4031
1
$800 medium
l/2
I
1
5
5
6
0.2347
0.54
$0 none
l/2
I
I
4
4
5
0.2100
0.48
SO great
l/7
l/5
l/4
1
2
3
0.0694
0.10
$800 none
118
115
l/4
l/2
I
2
0.0491
0.04
$1000 none
l/9
l/6
115
113
l/2
I
0.0338
0
Utilities
IOW
022
Maximum eigenvalue = 6.14 Consistence ratio = 0.02
methodology developed in Saaty [2] can be applied to this situation. Using a nine point scale, the decision maker assesses in turn a pair of outcomes k, q and allocates a number l-9 in accordance with how much more beneficial one outcome is over another as outlined in Table 2. This number becomes the kqth element of the judgmental matrix and its reciprocal becomes the qkth element. Ones are placed on the diagonal. For WIacts and n events the dimension of the square judgmental matrix will be mn. This procedure results in quantitative weights from qualitative judgments. A hypothetical assessment is shown in Table 3 but each customer could differ with respect to the numbers assigned. The weights in the secondto-last column of Table 3 are the elements in a normalized (sum to unity) eigenvector associated with the maximum eigenvalue for the judgmental matrix. The consistency ratio of 0.02 being close to zero indicates the numbers assigned are consistent, Saaty [3]. The software package MATLAB was used on a VAX 1l/780 to derive the values in Table 3. Taking the weights in the second-to-last column, we use a positive linear transformation to derive utilities on a [0, l] scale. The transformation from weights (w) to utilities (a) is given by: u=a+bw where
I .OOOl
whatever the probability of a “Big snow” is. If this probability is assessed at less than 0.348 the optimal act is “Don’t buy”, otherwise the “Usual deal” is optimal. Bell [3] contains a fourth act labelled the “Sno-risk deal” which yielded utilities (1,0.48) respectively for the possible events. This dominates “Don’t buy” and the offer resulted in a significant sales increase for the manufacturer, Hesse [4]. 3.
CONCLUSIONS
If the outcomes are single attribute money outcomes, utility derivations using the AHP are verifiable by the standard gamble technique. That is, the utilities such as those in Table 4 should be probabilities that make the decision maker indifferent between the outcome represented by that utility and a gamble between the best and worst outcomes in the table where the utility is the probability of realizing the best outcome. Some revision of utilities may be necessary before indifference is achieved. It is possible that a decision maker may feel more comfortable initially making qualitative AHP judgments and then subsequently fine-tuning the probabilities necessary for indifference, than he would deriving the probabilities (or utilities) directly. Irrespective of whether or not fine-tuning is needed, it should be noted that
b = l/(best - worst) = l/(0.4031 - 0.0338) a = -b
(worst) = -0.0338/(0.4031
- 0.0338)
Using these utilities the decision problem for our hypothetical customer is shown in Table 4. From the utilities in Table 4 it is easy to show that the “Discount deal” is dominated by the other two acts
Table 4. Utility payoffs for snow-blower custnm,=r
Don’t buy Usual deal Discount deal
Bia snow
Light snow
0.10 I 0.54
0.48 0 0.04
Deriving
such indifference is required to justify a decision based on maximizing expected utility in accordance with the von Neumann-Morgenstern axioms. In the case of multiattribute outcomes, the above method has the advantage of requiring the decision maker’s judgment on all attributes collectively as between alternative outcomes. This may be more natural and less time consuming that trying to assess the relative merit of each individual attribute and then aggregating a score for each outcome. However, as the number of attributes grows beyond say three
395
utilities
or four, consistent judgments may only be possible with the aid of a scoring rule. REFERENCES 1. R. L. Keeney and H. Raiffa. Decisions With Multiple Objectives. Wiley, New York (1976). 2. T. L. Saaty. The Analytic Hierarchy Process. McGrawHill, New York (1980). 3. D. E. Bell. Putting a premium on regret. Mgmt Sci. 31(l), 117-120 (1985). 4. R. Hesse. Snow job. Decis. Line 16(l), 5 (1985).