What to Do? Let’s Think It Through! Using the Analytic Hierarchy Process to Make Decisions

Did you know you make an average of 35,000 decisions each day? Some of these choices are relatively unimportant, like deciding what to eat for breakfast. But others have a greater impact on your future, like choosing which university to attend. When faced with a difficult decision, how do you consider every option to pick the best one? The analytic hierarchy process is one method you can use. It helps you make hard decisions by comparing multiple alternatives to make the best choice.

a list of pros and cons. Another method that could help you process all the information and make your decision is called the analytic hierarchy process (AHP). A hierarchy is a system of ranking di erent

ANALYTIC HIERARCHY PROCESS
A decision making method that compares multiple alternatives, each with several criteria to help select the best option. options against each other based on importance.
The AHP is particularly helpful when you are making a decision with no clear best choice. Maybe there are more criteria involved in the decision-not just two as in the bike example. The AHP combines math and psychology to compare several options and select the best one [ ]. It does this by using a concept called pairwise comparisons. Instead

PAIRWISE COMPARISON
The process of comparing criteria two at a time.
of comparing several criteria at once, they are compared two at a time. That way, the choice is easier to make.
Linear algebra is a type of math that helps us calculate the importance

LINEAR ALGEBRA
A branch of mathematics that concerns linear functions, vector spaces, and matrices.
of criteria using matrices. The AHP uses linear algebra to assess the results of each pairwise comparison. Every criterion receives its own importance weight. The higher the weight, the more important the criterion is to the overall decision. This method of comparison can be applied to many decisions. For example, you can use it to select a product or service, like a car or airline flight. You can also use it to choose the most e cient process or route, such as evaluating whether to use private or public transit to get to school [ ].
The AHP is used daily in manufacturing, business, engineering, and other industries. For example, it can be used to prioritize the maintenance of publicly used structures, like bridges [ ]. There are currently , bridges in the United States in need of repair. These repairs would cost a total of $ billion. However, there is not enough money in the budget to fix every bridge. So, engineers used the AHP to develop a bridge rating system. This rating system decides which bridges to maintain, given a limited budget. The system considers each bridge's safety, use, riding comfort, and ability to withstand damage.
The engineers used pairwise comparisons to weigh and compare four criteria for , U.S. bridges. Based on the four criteria, each bridge received an overall rating, from "excellent" to "failing." The engineers found that none of the bridges they evaluated were rated above "good." In fact, % of all bridges in their study were just "satisfactory"-or worse! The ratings draw attention to problems with our country's infrastructure and could result in more money for maintenance and repair. The ratings also tell us which bridges need to be fixed first. This decision of which bridge to repair could even save lives! Figure   Figure Sample AHP matrices. (A) Matrix A is a × example matrix. It shows how pairwise comparisons are organized and referenced using subscripts: for example, x refers to the grid space in the first row, second column. (B) Matrix B is also a × matrix. It contains the three criteria in our university decision: cost, location, and rank. In this example, each grid space contains a score from the pairwise comparisons. These sample scores show that cost is the most important decision factor, followed by academic rank and lastly, location.

Decision, Options, and Criteria
Every decision begins with a question: which alternative is right for you? For example, you can use the AHP to select which university to attend. In this scenario, you may want to decide which of your top three options (Universities A, B, or C) is best for you. Once you understand the question and the options, you then decide which criteria to consider. Perhaps the three most important criteria for you are finding a school that is a ordable, located close to home, and well-ranked. Once you identify these criteria, you can begin the process of pairwise comparisons.

Pairwise Comparisons
The important criteria that you identified are then organized into matrices. A matrix is a grid with information arranged in rows and MATRIX A grid with information, such as numbers, letters, and symbols arranged in rows and columns.
columns. In AHP matrices, each criterion has its own row and column. The resulting square matrices allow pairwise comparison of all possible combinations of criteria ( Figure ).
The first step of pairwise comparisons is to assign a number to each grid space. This number is the relative importance of the two criteria. For example, a score of means both criteria are equally important. When a criterion is compared to itself, its relative importance is , because the criteria being compared are the same. Larger numbers show a criterion is increasingly more important, with being the highest score [ ]. In row of Figure B, cost is seven times as important as important (x = ) to the final decision as location and five times as important (x = ) as academic rank. This scoring process is repeated Sample utility functions for the university decision process. For a ordability, a school with free tuition has the maximum possible utility: utiles. If tuition exceeds your budget of $ , per year, it has the minimum possible utility: utiles. For location, if your goal is to be close to home, a school miles away earns utiles. A school more than miles away earns utiles. For academic rank, the top-ranked (# ) school earns utiles. A school that failed to make the top earns utiles. for all of the grid spaces to the right of the diagonal (the upper right half of the matrix).
Because cost is seven times as important as location, we can also say that location is / as important as cost (x ). Similarly, since cost is five times as important as academic rank, rank must be / as important as cost (x ). This process of using reciprocals, or inverses ( /x), to describe the opposite relationship between two criteria is repeated for all of the grid spaces to the left of the diagonal (the lower left half of the matrix). If criterion is x times as important as criterion , then criterion must be /x as important as criterion .

Importance Weights
The completed matrix is then used to calculate importance weights, which tell you how much each criterion will factor into your decision. The higher the importance weight of a given criterion, the more influence it will have on your final decision. This part of the AHP relies on linear algebra. The first step to determine the weight of a criterion is to find the geometric mean (V) of the row. The geometric mean

GEOMETRIC MEAN
The average value of a set of numbers found by using the product of their values, rather than the sum. is a type of average. You can find the geometric mean by multiplying all the relative importance scores from the row (x) and taking the nth root of this product (where n = total number of criteria). It is better to use the geometric mean than the arithmetic mean (sum/n) for this type of calculation [ ]. This sample equation shows you how to calculate the geometric mean of Criterion of the × generic matrix ( Figure A): Next, divide the criterion's geometric mean by the sum of the geometric means of all the criteria. The resulting decimal is the weight (W) of that criterion. This method is called normalization, because it ensures the sum of all weights equals , or %. The weights sum to because each criterion accounts for a portion of the entire decision. This sample equation shows you how to calculate the weight of Criterion of the × generic matrix ( Figure A):  Figure . These values were plugged into the utility function to calculate total utility. Each university's total utility is seen in the rightmost column. . .

Table
What happens if we do the same calculations for the criterion of cost, using the relative importance scores from Figure B?
This tells us cost accounts for % of the overall decision of which university to attend. Using this same method, we find that the calculated weight for location is % and rank is %.

Utility: Which Is the Best Option?
The final step of the AHP is to determine utility. Utility is a numerical value providing information on how useful something is to you, and it will help you to select the best option. The more beneficial or useful a criterion is, the higher its utility. Utility may be measured di erently for each criterion. In our example, a ordability is measured in dollars. Location is measured in miles from home. Academic ranking is given a score, where # represents the top-ranked (first-place) school and higher numbers correspond to less prestigious universities. When criteria have di erent units, we must first convert them into standardized units called utiles (a unitless measure of utility) using UTILE A unitless measure of utility used to standardize the units of di erent criteria. graphs called utility functions. In Figure , you can see how we use utility functions in the university decision. Total utility considers the utility of each individual criterion and its importance to the overall decision. Total utility equals the sum of the products of each criterion's weight and respective utility value: Once all three criteria are measured, the total utility of each option is found. A sample utility calculation for University A is: Cost Location Rank U total = . * + . * + . * = .
Using this same method, we find the total utility data for Universities B and C, shown in Table . Based on the results of the AHP, University B is the best choice. As seen in Table , University B has the highest total utility, . . This is because it is the most a ordable option, and cost accounts for % of the overall decision. Thanks to its great location, University A is a close second, with a total utility of . . Even though University C is kids.frontiersin.org June | Volume | Article | the highest-ranked school, it has a much lower total utility ( . ) than the other two due to its high cost, so it is the worst choice.

SUMMARY
The AHP is a powerful tool that can be used at home, at school, or at work. It removes bias from the decision-making process and ensures the decision you make reflects your values and priorities. The next time you are faced with a di cult decision, consider using the AHP to make your choice. The possibilities are endless-the AHP is limited only by your creativity in applying it.
. Bridge maintenance prioritization using analytic hierarchy process and fusion tables.
. Aggregation in the analytic hierarchy process: why weighted geometric mean should be used instead of weighted arithmetic mean. Expert Syst. Appl.

CONFLICT OF INTEREST:
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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