3.1. The PSI model

The PSI technique is relatively a new decisión-making approach (^{Maniya & Bhatt, 2010}). It does not require comparing the attribute importance. For each alternative, a PSI value is computed, in which the best alternative is that of the highest PSI value. The PSI method is explained in this section (Maniya & Bhatt, 2010).

The first step of the PSI method is identifying the objective, determining all possible criteria, its measures, and alternatives. A total of 20 attributes were selected as shown in Table 1 for constructing the PSI model of this study. The attribute, which is the same as the criterion, is referred to by the letter (C) as shown in Table 1. Selection of the attributes was based on the literature and on the results of interviewing a group of 65 industrial engineering students in Jordan seeking admission to a doctorate in industrial engineering in the U.S. Table 1 also classifies the attribute into a benefit or a cost.

The second step in the PSI approach in formulating the decision matrix (X_ij). An example of the decision matrix is shown in Table 2. Let A be a set of alternatives, where A= {A_i for i= 1, 2, 3, …, n}, C be a set of decision criteria where C= {C _j for j = 1, 2, 3, …, m} and X_ij is the performance of alternative A_i when it is evaluated by criteria C_j. A total of 37 alternatives (universities) were considered. The selection was based on the considered universities rankings organizations, the US News and World Report and Times Higher Education, and on the survey distributed to industrial engineering students in Jordan, who are seeking admission for a doctorate in this field. Table 1 information also was used in building the PSI decision matrix involving the 37 universities that all have a doctorate program in industrial engineering.

After constructing the decision matrix, data normalizing is performed. Data of the decision matrix are transformed into values in the 0-1 range. In the case of a positive expectancy (i.e. profit), the normalization formula is:

While in the case of negative expectancy (i.e. cost), the normalizing formula is:

Where X_ij is the attribute measures (i = 1, 2, 3, ..., n and j = 1, 2, 3, . . . , m) in the decision matrix.

Next, the preference variation value (PV_j) is calculated based on:

Where Rj- is the mean of the normalized values of attribute j and calculated as:

Afterward, for each attribute, the deviation (Φ_j) in preference variation value (PV_j ) is computed as:

Then, the overall preference value (Ψ_j) is computed for each attribute as:

Here, the overall summation of the preference value of all attributes must add to one.

Next, the preference selection index (I_i) is then calculated based on:

Finally, alternatives are ranked according to the I_i values; the one of the highest I_i value is of high preference.

3.2. The AHP model

To construct the AHP model, the following steps are required:

4.1. The PSI model

For the current study and the designed PSI decision matrix, Table 3 shows the computed values of the preference variation value (PV_j), the deviation (Φ _j) of the preference variation, and the overall preference value (Ψ _j) for each attribute being considered. These values were calculated using Equations 3, 5, and 6, respectively.

The preference selection index (I _i ) values are computed according to Equation 7 for the 37 universities, considering the 20 attributes involved in the PSI model, and are shown in Table 4. The alternatives now could be ranked based on their I _i value, such that the alternative of the highest I _i value is most preferable. This finding proves that the PSI approach is suitable and faster to deal with decision problems of a large number of attributes and alternatives.

4.2. The AHP model

Breaking down the decision problem into a hierarchy structure is an essential step. It helps to analyze the decision and increase the decisión maker understanding of the decision problem. The basic components of the hierarchy are shown in Figure 1.

Figure 1 The basic components of an AHP hierarchy (^{Taha, 2011}) 

The hierarchy upper level represents the decision goal, which was selecting an appropriate university in the U.S. for pursuing a doctorate in industrial engineering. The second level represents the criteria that will be used to decide the university selection. In the case of selecting a university, if the initial criteria were used directly to build the AHP model without starting with the PSI model, the resulted hierarchy will be as shown in Figure 2. Figure 2 shows that the hierarchy was complicated as it had several criteria, and some of these criteria have their sub-criteria. To reduce the number of criteria considered, the previous PSI model results were considered. This means instead of having all the 20 attributes and the 37 alternatives and studying them again in the AHP model, the most significant attributes and the highly preferred alternatives obtained in the PSI model will be considered in building the AHP model. Figure 3 shows the considered hierarchy for the AHP model after considering the PSI model outcomes.

Figure 2 The suggested AHP hierarchy without considering the PSI results. 

Figure 3 The AHP hierarchy considers the outcomes of the PSI model 

From the hierarchy shown in Figure 3, it is obvious that the main criteria level included only four criteria that were university reputation (CC1), university location (CC2), financial factors (CC3), and ease of admission (CC4). The university’s reputation was measured by two sub-criteria, the university rank (CC1.1) and the education quality(CC1.2). The education quality was divided into the facility resources (CC1.2a) and the research focus (CC1.2b). The university location (CC2) criterion included weather (CC2.1), location beauty (CC2.2), safety (CC2.3), and racism (CC2.4). The financial factors (CC3) criterion was divided into tuition cost (CC3.1), living and dining expenses (CC3.2), and student financial aid (CC3.3). Student financial aid was evaluated by scholarship opportunities. The fourth main criterion was the ease of admission (CC4) and it was measured by the admission rate. Table 7 summarizes the criteria sub-criteria that were used in this study. In addition, Table 7 provides similar research based on the literature considering the selected criteria or sub-criteria.

The next step in building the AHP model was deriving priorities. The relative weights (priorities) for the criteria were derived. These priorities were also called relative importance because the computed priorities of criteria were measured concerning each other. The priorities were derived using pairwise comparisons, which were made easily using the Saaty numeric scale of values between 1 and 9. To derive priorities for the main criteria, pairwise comparisons should be made between the levels of the main criteria level (university reputation, university location, financial factors, and ease of admission requirements).

In this study, the PSI was initially used to compare a total of 37 universities (alternatives) according to 20 different attributes. Based on the PSI results, a set of eight alternatives were selected from the PSI model as shown in Table 8. The alternatives had different PSI values (I_i) and varied performance on the AHP model sub-criteria. Data of the alternative’s performance on each sub-criteria were taken from the PSI model. In the AHP model, eight alternatives of the highest PSI values (I_i) were considered. In addition, the pairwise comparison that reflects the decisión-maker preferences of criteria and sub-criteria was made. Expert Choice © software was used to handle the AHP model. After obtaining the AHP model, a sensitivity analysis procedure was conducted to evaluate the AHP model.

For each level in the AHP hierarchy, pairwise comparisons for each node with its children nodes were made. To illustrate, Table 9 represents the pairwise comparisons of the main criteria considered in the AHP model concerning the decisive goal in this study. The inconsistency of this matrix was 0.03, which is an acceptable level as it is less than 10% (^{Taha, 2011}). The Expert Choice software compared the priority of the row concerning the column and put the values in the upper triangular of the matrix. For example, the reputation (the criterion in the row) is (7), which means that it is more important than the location (the criterion in the column). Similarly, comparisons were done for the other nodes.

As mentioned previously, the model rated each alternative concerning its performance on each criterion. From the PSI data, the performance of alternatives was evaluated regarding each criterion. Table 10 summarizes each alternative in detail, the first column shows the performance of alternatives regarding university rank (CC1.1). For example, alt1 (Georgia Institute of Technology) in the AHP model was ranked in the top 10 universities, the second column represents the perfor mance of alternatives regarding facility resources (CC1.2.a), and these numbers were taken from the facility resources attribute in the decision matrix of the PSI model, similarly for the research focus (CC1.2.b). Regarding the weather (CC2.1), the weather attribute in the PSI model was used such that the weather was categorized into excellent weather, average weather, and bad weather. For the location beauty (CC2.2), the data for the location beauty attribute in the PSI model was used. Regarding safety (CC2.3), the data for the safety attribute in the PSI model was also used. Similarly, for racism (CC2.4), the data for racism attribute in the PSI model was considered. For the financial factors sub-criteria; tuition fees (CC3.1), living and dining expenses (CC3.2), and student financial aid (CC3.3), the attributes data of the PSI model were used. Furthermore, the student’s scholarship opportunity attribute in the PSI model was used as a source of data for the student aid (CC3.3). Finally, for the last column ease of admission (CC4), data from the admission rate attribute were used. The relative weights of the AHP model’s main criteria are shown in Table 11.

The relative weights shown in Table 11 indicate that the university reputation is the most important criterion with a relative weight of 46.7%, followed by the ease of admission criterion with a relative weight of 34.9%, then by financial factors criterion with a relative weight of 13.5%, and the university location of a relative weight 4.9%. A higher weight means higher preferences of the decisión-maker.

Now, considering the university reputation sub criteria in the AHP model, the university rank occupied the highest priority of 83.3%, while the education quality occupied only 16.7%. For the education quality sub-subcriteria, the facility resources had a priority of 83.3%, followed by the research focus of 16.7% in weight. Regarding the financial factors sub criteria, a tie was found between the relative weights of the living and dining expenses and student financial of 45.45%, while that for the tuition of 9.1%. Regarding the university location sub-criteria, safety had the highest priority of 65.85%, followed by location beauty with a relative weight of 15.4%, weather with a relative weight of 13.15%, and finally the racism with a relative weight of 5.6%. The weight of each alternative regarding each main criterion was combined to obtain the overall weight as shown in Table 12.

Based on Table 12 and for the university reputation criterion, alt5 (Columbia University) had the highest priority of 18.9% followed by alt1 (Georgia Institute of Technology) with a priority of 18.5%, while alt8 (Montana State University) had the least priority of 2.1%. Considering the university location criterion, alt3 (University of Pittsburgh) occupied the highest priority of 14.9%, followed by a tie between alt5 (Columbia University) and alt7 (Binghamton University) with 14.5% in weight, while alt4 (Kansas State University) was of the least weight of 10.4%. Regarding the financial factors criterion, alt4 (Kansas State University) had the highest weight of 17.6% while alt5 (Columbia University) was of the least weight of 7.1%. Regarding the ease of admission criterion, three alternatives were tied regarding admission rate of 52.2%, these were alt6 (Auburn University), alt7 (Binghamton University), and alt8 (Montana State University) had the highest weight of 25.2%, while alt5 (Columbia University) had the least weight of 0.4%.

As mentioned previously, the AHP model combined all weights of the alternatives and provided an easy conclusion for the user. Table 13 shows the total weights and the rank of the eight alternatives in the AHP model.

According to Table 13, Auburn University (alt6) was the most preferred alternative as it had a maximum weight of 16%. Georgia Institute of Technology (alt1) was found to be the next students’ preference with a total weight of 14%, followed by the University of Pittsburgh (alt3) with a weight of 13.5%. Kansas State University (alt4) had the least weight at 9.5%.

4.2.1 The AHP sensitivity analysis

The total weights obtained previously in the AHP model could be changed if the priority of the criterion is changed. For this reason, the sensitivity of the decision to changes in the relative weights must be considered. The Expert Choice software was used to obtain a performance sensitivity graph for the model, as shown in Figure 4. This graph is a dynamic graph that is consisting of two axes. The horizontal axis that represents the as shown in Figure 4. This graph is a dynamic graph that is consisting of two axes. The horizontal axis that represents the criteria axes, which measu the total weight of each alternative as the priority given to criteria change; and the vertical axis is the objective axis, which represents the relative importance of each main criterion considered in the model. Figure 4 shows the AHP model results in that Alt6 (Auburn University) is the most preferred university.

Figure 4 Performance sernsitivity graph for the AHP model with alt6 (Auburn University) is the most favorable. 

To perform the sensitivity analysis, the priority of criteria was changed randomly and then the priority of alternatives was monitored. In Figure 5, more weight was given to the university’s reputation, the new priorities were 70%, 10%, 10%, and 10% for university reputation, university location, financial factors, and ease of admission, respectively. Under this scenario, alt1 (Georgia Institute of Technology) was the most recommended alternative. Note that alt6 (Auburn University) had a total relative weight of 16% in the previous model while alt1 (Georgia Institute of Technology) had a relative weight of 14%, which are close to each other.

Figure 5 Performance sensitivity graph for the AHP model with relative weights of 70%, 10%, 10%, and 10% for reputation, location, financial factors, and ease of admission, respectively. 

Now, considering a new scenario, of giving a higher priority for the ease of admission criterion. Figure 6 shows the result after modifying the new priorities into 10%, 10%, 10%, and 70% for university reputation, location, financial factors, and ease of admission, respectively. This change kept alt6 (Auburn University) as the preferred alternative.

Figure 6 Performance sensitivity graph for the AHP model with relative weights of 10%, 10%, 10%, and 70% for reputation, location, financial factors, and ease of admission, respectively. 

Another scenario was performed by giving the university location a higher priority, thus, the new priorities were 10%, 70%, 10%, and 10% for university reputation, university location, financial factors, and ease of admission, respectively. As shown in Figure 7, these priorities changed the decision by making alt3 (University of Pittsburgh) the most preferred. This implies that the model is sensitive to the change of criteria weights.

Figure 7 Performance sensitivity graph for the AHP model with relative weights of 10%, 70%, 10%, and 10% for reputation, location, financial factors, and ease of admission, respectively. 

Similarly assigning more weight to the financial factors, change the decision. Figure 8 shows that when the new priorities become 10%, 10%, 70%, and 10% for university reputation, university location, financial factors, and ease of admission, respectively, alt4 (Kansas State University) becomes the preferred alternative.

Figure 8 Performance sensitivity graph for the AHP model with relative weights of 10%, 10%, 70%, and 10% for reputation, location, financial factors, and ease of admission, respectively.