# Inferior Education or Killing Grandma: The Dilemma Facing the Public School Systems in the United States

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Problem Setting

## 2. Description of the Alternatives and Criteria for This Study

#### 2.1. The Three Alternatives in This Study

#### 2.1.1. The School Alternative

#### 2.1.2. The Virtual Alternative

#### 2.1.3. The Hybrid Alternative

#### 2.2. The Three Criteria Used in This Study

- Safety: How safe is the school district’s system with respect to being infected by COVID-19? School districts wish to avoid students, faculty members, or other persons becoming infected and spreading the COVID-19 virus to others.
- Education: Are the students able to learn at an acceptable level to at least maintain their grade level?
- Economy: Can the economy grow when the school district removes impediments to business activities in (and around) the school district?

- Budget (or Cost) is a criterion that will play a significant role in analyzing a school district’s situation. Using pairwise comparisons, an analyst can compare the budgets of two alternatives as well as comparing the magnitude of a budget versus the other criteria that are part of the analysis.
- Number of dropouts (or percentage of students who drop out) is a criterion that concerns a school district. Dropouts can be a more severe situation in some school districts (for example, urban school districts) and less severe in other school districts.
- Hardware and software availability is a concern a school district faces, especially if the district is going to use the Virtual alternative. The issue with this criterion is how many students cannot attend the classes they want (or need) because they cannot obtain the appropriate hardware, software, and/or Internet access. As a result of these hardware and software shortages, these students are not able to attend their classes or do the required work for their classes at no fault of their own.

## 3. Urban School Districts

## 4. The Direct Comparison Model (DCM) of the Analytic Hierarchy Process (AHP)

- The Level 1 node is called the Goal Node (or Root Node) of the AHP tree.
- The Level 2 nodes for this AHP tree are the Safety, Education, and Economic criteria.
- The Level 3 nodes for this AHP tree are the Virtual, Hybrid, and School alternatives connected to each criterion listed in Level 2 of the AHP tree. In this example, the nodes in Level 3 are repeated for each criterion in the construction of the AHP tree.

#### 4.1. Verbal to Numeric Scale

**Question 1**: Are criterion A and criterion B of equal importance to the analyst? If the answer to this question is

**YES**, then the values of the pairwise comparison of A to B and B to A are equal to 1 and Question 2 is skipped. Otherwise, the analyst must answer Question 2 since the analyst believes either criterion A is more important than criterion B or criterion B is more important than criterion A.

**Question 2**: If the analyst believes criterion A is more important than criterion B, then the analyst must set the pairwise comparison value of A to B equal to a number between 1 and 9 and the pairwise comparison value of B to A is set equal to 1 divided by the pairwise comparison value of A to B. For example, if the analyst believes that criterion A is moderately preferred to strongly preferred to criterion B, the pairwise comparison value of A to B is equal to 4 and the pairwise comparison value of B to A is ¼. On the other hand, if the analyst believes that criterion B is moderately more important than criterion A, then the pairwise comparison value of B to A is equal to 3 and the pairwise comparison value of A to B is equal to 1/3.

#### 4.2. Properties of the Pairwise Comparative Matrix (PCM)

- The PCM has the same number of rows and columns.
- The PCM cannot be made up of a mixture of criteria and alternatives.
- The identification of the rows in the PCM must be in the same order as the identification of the columns in the PCM. Otherwise, entering the values of the different pairwise comparisons in the PCM can become messy.
- If j = k, then A(j, j) = 1 as long as the rows and columns are numbered in the same order. In other words, the value of the pairwise comparisons for each diagonal cell of the PCM is 1.
- The Reciprocal Rule, A(k,j) = 1/A(j,k), holds for every entry in the PCM.
- If the Reciprocal Rule holds for each entry in the PCM, then the PCM is called a Positive Reciprocal Matrix since all entries in the PCM are positive.
- Further details of a Reciprocal Matrix with Positive Entries can be found in [9].

#### 4.3. Finding the Value of the Weights for a PCM Using the Black Box

#### 4.4. Finding the AHP Weights for the DCM

#### 4.4.1. Determining the Weights for the Three Criteria Adjacent to the Goal Node

#### 4.4.2. Weights for the Three Alternatives under the Safety Criterion

#### 4.4.3. Weights for the Alternatives under the Education Criterion

#### 4.4.4. Weights for the Alternatives under the Economy Criterion

#### 4.5. Computing the Overall Score for the Alternatives Using the DCM

- The Virtual alternative score = 0.461 × 0.696 + 0.461 × 0.108 + 0.0769 × 0.134 = 0.3809.
- The Hybrid alternative score = 0.461 × 0.232 + 0.461 × 0.281 + 0.0769 × 0.290 = 0.2588.
- The School alternative score = 0.461 × 0.072 + 0.461 × 0.611 + 0.0769 × 0.576 = 0.3602.

#### 4.6. Matrix Representation of Determining the Weights of the Alternatives

#### 4.7. Conclusions and Implications

#### 4.8. Notes on Pairwise Comparisons and Using the AHP

## 5. The Ratings Model (The Search for Perfection)

#### 5.1. Key Aspects of the Ratings Model

- We assume in this paper that the same criteria can be used in analyzing a DCM and a Ratings Model. More general AHPs can have subcriteria and subalternatives.
- In the RM in this paper, there is a set of intensities ssociated with each criterion
- A criterion will have between 3 and 7 intensities. The set of intensities must cover the full range of possible outcomes of the criterion. Furthermore, there is a different collection of intensities for each criterion.
- The alternatives are not analyzed in the Ratings Model until late in the procedure.
- The intensities are processed one criterion at a time. A PCM is set up for the intensities associated with a criterion. The Black Box is then used to compute the AHP Scores for these intensities. This process is repeated for the intensities associated with each criterion.
- The Normalized Score for a criterion is equal to the value of the AHP score for this intensity divided by the value of the maximum AHP score for all intensities associated with the criterion. In this way, at least one Normalized Score is equal to 1 for each criterion and the Normalized Scores of all intensities are between 0 and 1.

#### 5.2. Pairwise Comparisons and Normalized Scores for the Safety Intensities

#### 5.3. Pairwise Comparisons and Normalized Score for the Education Criterion’s Intensities

#### 5.4. Pairwise Comparisons and Normalized for the Economy Criterion’s Intensities

#### 5.5. Computing the Ratings Score for Each Alternative

- Ratings Score for the Virtual Alternative = 0.461 × 1 + 0.461 × 0.22 + 0.0769 × 0.206 = 0.578.
- Ratings Score for the Hybrid Alternative = 0.461 × 0.234 + 0.461 × 0.49 + 0.0769 × 0.345 = 0.360.
- Ratings Score for the School Alternative = 0.461 × 0.75 + 0.461 × 0.49 + 0.0769 × 1 = 0.337.

#### 5.6. Analysis of the Results

## 6. The Team Approach

#### 6.1. Introduction to the AHP Team Approach

#### 6.2. Examples of the Geometric Average

- The Geometric Average (GA) of 4 and 9 is equal to square root of (4 × 9) = square root of 36 = 6. On the other hand, the Arithmetic Average of 4 and 9 = 13/2 = 6.5.
- The Geometric Average (GA) of 8, 27, and 64 = cube root of (8 × 27 × 64) = cube root of 13,824 = 24. On the other hand, the Arithmetic Average of 8, 27, and 64 is equal to (8 + 27 + 64)/3 = 33.

#### 6.3. Computing the Geometric Average in Excel

#### 6.4. Why Is the Geometric Average Used in the AHP?

#### 6.5. Interactive Features of the Geometric Average and the AHP

#### 6.6. An Example of the TEAM Approach

#### 6.6.1. Mo’s Pairwise Comparison Matrix for the Criteria

Safety | Education | Economy | |
---|---|---|---|

Safety | 1 | 1 | 6 |

Education | 1 | 1 | 6 |

Economy | 0.1667 | 0.1667 | 1 |

Weights | 0.461 | 0.461 | 0.078 |

#### 6.6.2. Shemp’s Pairwise Comparison Matrix

Safety | Education | Economy | |
---|---|---|---|

Safety | 1 | 3 | 8 |

Education | 0.3333 | 1 | 6 |

Economy | 0.125 | 0.1667 | 1 |

Weights | 0.6464 | 0.2895 | 0.0641 |

#### 6.6.3. Curly’s Pairwise Comparison Matrix

Safety | Education | Economy | |
---|---|---|---|

Safety | 1 | 0.3571 | 0.1333 |

Education | 2.8 | 1 | 0.2222 |

Economy | 7.5 | 4.5 | 1 |

Weights | 0.0826 | 0.1942 | 0.7232 |

#### 6.6.4. Geometric Average of the Pairwise Comparisons in Table 18, Table 19 and Table 20

#### 6.6.5. The Team2 Pairwise Comparison Matrix and Weights

#### 6.7. Summary

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Giannini, S. Time to Roll Out Education’s Recovery Package, UNESCO Report. 2021. Available online: https://en.unesco.org/news/time-roll-out-educations-recovery-package (accessed on 1 March 2021).
- Bender, R. Europe’s Schools Are Closing Again on Concerns They Spread COVID-19. The Wall Street Journal, 16 January 2021. [Google Scholar]
- Magome, M. Amid COVID-19 Surge, South Africa Delays Reopening Schools. Associated Press, 17 January 2021. [Google Scholar]
- Parth, M.N.; Joanna, S.; Niha, M. Schools in India Have been Closed Since March. The Costs to Children Are Mounting. The Washington Post, 30 December 2020; updated 12 February 2021. [Google Scholar]
- Saaty, T. The Analytic Hierarchy Process; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Saaty, T. Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World; RWS Publications: Pittsburgh, PA, USA, 2012. [Google Scholar]
- Saaty, T.; Vargas, L. Models, Methods, Concepts & Applications of the Analytic Hierarchy Process, 2nd ed.; Springer: New York, NY, USA, 2012. [Google Scholar]
- Reed, L.; Bowie, L. Baltimore-Area Schools Start the Year with Online Education. Thousands of Students Lack a Way to Engage. Baltimore Sun Newspaper, 8 September 2020. [Google Scholar]
- Gass, S.; Standard, S. Characteristics of Positive Reciprocal Matrices in the Analytic Hierarchy Process. J. Oper. Res. Soc.
**2002**, 53, 1385–1389. [Google Scholar] [CrossRef] - Wilford, A.; Bodin, L.; Gordon, L. Using the Analytic Hierarchy Process to Assess the Impact of Internal Control Weaknesses on Firm Performance. Int. J. Anal. Hierarchy Process
**2020**, 12, 203–224. [Google Scholar] [CrossRef] - Adams, B. The AHP Pairwise Process: How the Analytic Hierarchy Process Pairwise Comparison Process Works, DLProd Team. 2017. Available online: https://medium.com/dlprodteam/the-ahp-pairwise-process-c639eadcbd0e (accessed on 15 November 2017).
- Bodin, L.; Epstein, E. Who’s on First–with Probability 0.4. Comput. Oper. Res.
**2000**, 27, 205–215. [Google Scholar] [CrossRef] - Amenta, P.; Lucadamo, A.; Marcarelli, G. On the Choice of Weights for Aggregating Judgments in Non-Negotiable AHP Group Decision Making. Eur. J. Oper. Res.
**2021**, 288, 294–301. [Google Scholar] [CrossRef] - Bodin, L.; Gass, S. On Teaching the Analytic Hierarchy Process. Comput. Oper. Res.
**2003**, 30, 1487–1497. [Google Scholar] [CrossRef] - Bodin, L.; Gordon, L.; Loeb, M. Information Security and Risk Management. Commun. ACM
**2008**, 51, 64–68. [Google Scholar] [CrossRef] - Bodin, L.; Gordon, L.; Loeb, M. Evaluating Information Security Investments using the Analytic Hierarchy Process. Commun. ACM
**2005**, 48, 78–83. [Google Scholar] [CrossRef] - Forman, E.; Gass, S. The Analytic Hierarchy Process—An Exposition. Oper. Res.
**2001**, 49, 469–486. [Google Scholar] [CrossRef] - Wood, P.; Barker, J. With Maryland in ‘Danger Zone’, ‘Governor’ Hogan Tightens Coronavirus Restrictions on Restaurants, Strongly Discourages Large Gatherings. Baltimore Sun Newspaper, 10 November 2020. [Google Scholar]
- Smelkinson, M. Masks, Not Metrics, to Reopen Schools. Baltimore Sun Newspaper, 10 January 2021. [Google Scholar]

Numeric Scale | Verbal Scale |
---|---|

1 | Equal Importance |

3 | Moderate Importance |

5 | Strong Importance |

7 | Very Strong Importance |

9 | Extremely Strong Importance |

A | B | C | |
---|---|---|---|

A | 1 | 3 | 4 |

B | 0.333 | 1 | 2 |

C | 0.25 | 0.5 | 1 |

Safety | Education | Economy | |
---|---|---|---|

Safety | 1 | 1 | 6 |

Education | 1 | 1 | 6 |

Economy | 0.1667 | 0.1667 | 1 |

Virtual | Hybrid | School | |
---|---|---|---|

Virtual | 1 | 5 | 7 |

Hybrid | 0.2 | 1 | 5 |

School | 0.1433 | 0.2 | 1 |

Virtual | Hybrid | School | |
---|---|---|---|

Virtual | 1 | 0.3333 | 0.2 |

Hybrid | 3 | 1 | 0.4 |

School | 5 | 2.5 | 1 |

Virtual | Hybrid | School | |
---|---|---|---|

Virtual | 1 | 0.4 | 0.25 |

Hybrid | 2.5 | 1 | 0.5 |

School | 4 | 2 | 1 |

Safety | Education | Economy | |
---|---|---|---|

R = | 0.461 | 0.461 | 0.0769 |

Virtual | Hybrid | School | |
---|---|---|---|

S = | 0.696 | 0.108 | 0.134 |

0.232 | 0.281 | 0.29 | |

0.072 | 0.611 | 0.576 |

Virtual | Hybrid | School | |
---|---|---|---|

R × S = | 0.3809 | 0.2588 | 0.3602 |

Sa-Excellent | Sa-Good | Sa-Fair | Sa-Poor | |
---|---|---|---|---|

Sa-Excellent | 1 | 4 | 7 | 9 |

Sa-Good | 0.25 | 1 | 4 | 7 |

Sa-Fair | 0.143 | 0.25 | 1 | 3 |

Sa-Poor | 0.111 | 0.143 | 0.333 | 1 |

AHP Score | Normalized Score | |
---|---|---|

Sa-Excellent | 0.611 | 0.611/0.611 = 1 |

Sa-Good | 0.2 | 0.3273 = 0.2/0.611 |

Sa-Fair | 0.143 | 0.234 = 0.143/0.611 |

Sa-Poor | 0.046 | 0.0753 = 0.046/0.611 |

Ed-Excellent | Ed-Good | Ed-Fair | Ed-Poor | |
---|---|---|---|---|

Ed-Excellent | 1 | 2.5 | 5 | 7 |

Ed-Good | 0.4 | 1 | 2.5 | 5 |

Ed-Fair | 0.2 | 0.4 | 1 | 2.5 |

Ed-Poor | 0.143 | 0.2 | 0.4 | 1 |

AHP Score | Normalized Score | |
---|---|---|

Ed-Excellent | 0.549 | 1 |

Ed-Good | 0.269 | 0.49 |

Ed-Fair | 0.122 | 0.22 |

Ed-Poor | 0.06 | 0.109 |

Ec-Excellent | Ec-Good | Ec-Fair | Ec-Poor | |
---|---|---|---|---|

Ec-Excellent | 1 | 2 | 3 | 4 |

Ec-Good | 0.5 | 1 | 2 | 3 |

Ec-Fair | 0.333 | 0.5 | 1 | 2 |

Ec-Poor | 0.25 | 0.333 | 0.5 | 1 |

AHP Score | Normalized Score | |
---|---|---|

Ec-Excellent | 0.466 | 1 |

Ec-Good | 0.277 | 0.594 |

Ec-Fair | 0.161 | 0.345 |

Ec-Poor | 0.096 | 0.206 |

Safety | Norm. Score | Education | Norm. Score | Economy | Norm. Score |
---|---|---|---|---|---|

Sa-Excellent | 1 | Ed-Excellent | 1 | Ec-Excellent | 1 |

Sa-Good | 0.3273 | Ed-Good | 0.49 | Ec-Good | 0.594 |

Sa-Fair | 0.234 | Ed-Fair | 0.22 | Ec-Fair | 0.345 |

Sa-Poor | 0.0753 | Ed-Poor | 0.109 | Ec-Poor | 0.206 |

Safety | Education | Economy | |
---|---|---|---|

Virtual | 1 (Sa-excellent) | 0.22 (Ed-fair) | 0.206 (Ec-poor) |

Hybrid | 0.234 (Sa-fair) | 0.49 (Ed-good) | 0.345 (Ec-fair) |

School | 0.075 (Sa-poor) | 0.49 (Ed-good) | 1 (Ec-Excellent) |

Cell ID | Mo (A1) | Shemp (A2) | Curly (A3) | Geometric Average |
---|---|---|---|---|

(A1,A2) | 1 | 3 | 0.3571 | 1.0233 |

(A2,A1) | 1 | 0.3333 | 2.8 | 0.9769 |

(A1,A3) | 6 | 8 | 0.1333 | 1.8566 |

(A3,A1) | 0.1667 | 0.125 | 7.5 | 0.5386 |

(A2,A3) | 6 | 6 | 0.2222 | 0.5 |

(A3,A2) | 0.1667 | 0.1667 | 4.5 | 2 |

Safety | Education | Economy | |
---|---|---|---|

Safety | 1 | 1.02326 | 1.85662 |

Education | 0.97693 | 1 | 0.5 |

Economy | 0.53861 | 2 | 1 |

Weights | 0.4016 | 0.2620 | 0.3364 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bodin, L.; Frieman, B.
Inferior Education or Killing Grandma: The Dilemma Facing the Public School Systems in the United States. *Urban Sci.* **2021**, *5*, 29.
https://doi.org/10.3390/urbansci5010029

**AMA Style**

Bodin L, Frieman B.
Inferior Education or Killing Grandma: The Dilemma Facing the Public School Systems in the United States. *Urban Science*. 2021; 5(1):29.
https://doi.org/10.3390/urbansci5010029

**Chicago/Turabian Style**

Bodin, Lawrence, and Barry Frieman.
2021. "Inferior Education or Killing Grandma: The Dilemma Facing the Public School Systems in the United States" *Urban Science* 5, no. 1: 29.
https://doi.org/10.3390/urbansci5010029