Pairwise comparison matrices and the error-free property of the decision maker

Pairwise comparison is a popular assessment method either for deriving criteria-weights or for evaluating alternatives according to a given criterion. In real-world applications consistency of the comparisons rarely happens: intransitivity can occur. The aim of the paper is to discuss the relationship between the consistency of the decision maker—described with the error-free property—and the consistency of the pairwise comparison matrix (PCM). The concept of error-free matrix is used to demonstrate that consistency of the PCM is not a sufficient condition of the error-free property of the decision maker. Informed and uninformed decision makers are defined. In the first stage of an assessment method a consistent or near-consistent matrix should be achieved: detecting, measuring and improving consistency are part of any procedure with both types of decision makers. In the second stage additional information are needed to reveal the decision maker’s real preferences. Interactive questioning procedures are recommended to reach that goal.

Pairwise comparison matrices: an empirical research

Our research focused on testing various characteristics of pairwise comparison (PC) matrices in controlled experiments. About 270 students have been involved in the test exercises and the final pool contained 450 matrices. Our team conducted experiments with matrices of different size obtained from different types of MADM problems. The matrix elements have been generated by different questioning orders, too. The cases have been divided into 18 subgroups according to the key factors to be analyzed. The testing environment made it possible to analyze the dynamics of inconsistency as the number of elements increased in a given case. Various types of inconsistency indices have been applied. The consequent behavior of the decision maker has also been analyzed in case of incomplete matrices using indicators to measure the deviation from the final ranking of alternatives and from the final score vector.

A simplified implementation of the least squares solution for pairwise comparisons matrices

This is a follow up to "Solution of the least squares method problem of pairwise comparisons matrix" by Bozóki published by this journal in 2008. Familiarity with this paper is essential and assumed. For lower inconsistency and decreased accuracy, our proposed solutions run in seconds instead of days. As such, they may be useful for researchers willing to use the least squares method (LSM) instead of the geometric means (GM) method.

On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper.

Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants.

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.

On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices

The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the case of 3 × 3 PRM, a differentiable one-to-one correspondence is given between Saaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4 × 4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n × n PRM is formulated, the elements aij (i < j) of which were randomly chosen from the ratio scale ... ... with equal probability 1/(2M − 1) and a ji is defined as 1/a ij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency.

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making forweighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.

Incomplete pairwise comparison matrices and weighting methods

A special class of preferences, given by a directed acyclic graph, is considered. They are represented by incomplete pairwise comparison matrices as only partial information is available: for some pairs no comparison is given in the graph. A weighting method satisfies the property linear order preservation if it always results in a ranking such that an alternative directly preferred to another does not have a lower rank. We study whether two procedures, the Eigenvector Method and the Logarithmic Least Squares Method meet this axiom. Both weighting methods break linear order preservation, moreover, the ranking according to the Eigenvector Method depends on the incomplete pairwise comparison representation chosen.

In vivo biocompatibility evaluation of composite polymeric materials for use in a novel artificial heart valve

A novel trileaflet polymer valve is a composite design of a biostable polymer poly(styrene-isobutylene-styrene) (SIBS) with a reinforcement polyethylene terephthalate (PET) fabric. Surface roughness and hydrophilicity vary with fabrication methods and influence leaflet biocompatibility. The purpose of this study was to investigate the biocompatibility of this composite material using both small animal (nonfunctional mode) and large animal (functional mode) models. Composite samples were manufactured using dip coating and solvent casting with different coating thickness (251μm and 50μm). Sample's surface was characterized through qualitative SEM observation and quantitative surface roughness analysis. A novel rat abdominal aorta model was developed to test the composite samples in a similar pulsatile flow condition as its intended use. The sample's tissue response was characterized by histological examination. Among the samples tested, the 25μm solvent-cast sample exhibited the smoothest surface and best biocompatibility in terms of tissue capsulation thickness, and was chosen as the method for fabrication of the SIBS valve. Phosphocholine was used to create a hydrophilic surface on selected composite samples, which resulted in improved blood compatibility. Four SIBS valves (two with phosphocholine modification) were implanted into sheep. Echocardiography, blood chemistry, and system pathology were conducted to evaluate the valve's performance and biocompatibility. No adverse response was identified following implantation. The average survival time was 76 days, and one sheep with the phosphocholine modified valve passed the FDA minimum requirement of 140 days with approximately 20 million cycles of valve activity. The explanted valves were observed under the aid of a dissection microscope, and evaluated via histology, SEM and X-ray. Surface cracks and calcified tissue deposition were found on the leaflets. In conclusion, we demonstrated the applicability of using a new rat abdominal aorta model for biocompatibility assessment of polymeric materials. A smooth and complete coating surface is essential for the biocompatibility of PET/SIBS composite, and surface modification using phosphocholine improves blood compatibility. Extrinsic calcification was identified on the leaflets and was associated with regions of surface cracks.

The use of hyphenated spectrometric techniques for the environmental forensic assessment of non-traditional pollutants and degradates in the greater Florida Everglades

A comprehensive investigation of sensitive ecosystems in South Florida with the main goal of determining the identity, spatial distribution, and sources of both organic biocides and trace elements in different environmental compartments is reported. This study presents the development and validation of a fractionation and isolation method of twelve polar acidic herbicides commonly applied in the vicinity of the study areas, including e.g. 2,4-D, MCPA, dichlorprop, mecroprop, picloram in surface water. Solid phase extraction (SPE) was used to isolate the analytes from abiotic matrices containing large amounts of dissolved organic material. Atmospheric-pressure ionization (API) with electrospray ionization in negative mode (ESP-) in a Quadrupole Ion Trap mass spectrometer was used to perform the characterization of the herbicides of interest. ^ The application of Laser Ablation-ICP-MS methodology in the analysis of soils and sediments is reported in this study. The analytical performance of the method was evaluated on certified standards and real soil and sediment samples. Residential soils were analyzed to evaluate feasibility of using the powerful technique as a routine and rapid method to monitor potential contaminated sites. Forty eight sediments were also collected from semi pristine areas in South Florida to conduct screening of baseline levels of bioavailable elements in support of risk evaluation. The LA-ICP-MS data were used to perform a statistical evaluation of the elemental composition as a tool for environmental forensics. ^ A LA-ICP-MS protocol was also developed and optimized for the elemental analysis of a wide range of elements in polymeric filters containing atmospheric dust. A quantitative strategy based on internal and external standards allowed for a rapid determination of airborne trace elements in filters containing both contemporary African dust and local dust emissions. These distributions were used to qualitative and quantitative assess differences of composition and to establish provenance and fluxes to protected regional ecosystems such as coral reefs and national parks. ^

The development of direct ultra-fast PCR for forensic genotyping using short channel microfluidic systems with enhanced sieving matrices

There are situations in which it is very important to quickly and positively identify an individual. Examples include suspects detained in the neighborhood of a bombing or terrorist incident, individuals detained attempting to enter or leave the country, and victims of mass disasters. Systems utilized for these purposes must be fast, portable, and easy to maintain. The goal of this project was to develop an ultra fast, direct PCR method for forensic genotyping of oral swabs. The procedure developed eliminates the need for cellular digestion and extraction of the sample by performing those steps in the PCR tube itself. Then, special high-speed polymerases are added which are capable of amplifying a newly developed 7 loci multiplex in under 16 minutes. Following the amplification, a postage stamp sized microfluidic device equipped with specially designed entangled polymer separation matrix, yields a complete genotype in 80 seconds. The entire process is rapid and reliable, reducing the time from sample to genotype from 1-2 days to under 20 minutes. Operation requires minimal equipment and can be easily performed with a small high-speed thermal-cycler, reagents, and a microfluidic device with a laptop. The system was optimized and validated using a number of test parameters and a small test population. The overall precision was better than 0.17 bp and provided a power of discrimination greater than 1 in 106. The small footprint, and ease of use will permit this system to be an effective tool to quickly screen and identify individuals detained at ports of entry, police stations and remote locations. The system is robust, portable and demonstrates to the forensic community a simple solution to the problem of rapid determination of genetic identity.

Elemental analysis of biological matrices by laser ablation high resolution inductively coupled plasma mass spectrometry (LA-HR-ICP-MS) and high resolution inductively coupled plasma mass spectrometry (HR -ICP -MS)

The need for elemental analysis of biological matrices such as bone, teeth, and plant matter for sourcing purposes has emerged within the forensic and geochemical laboratories. Trace elemental analyses for the comparison of materials such as glass by inductively coupled plasma mass spectrometry (ICP-MS) and laser ablation ICP-MS has been shown to offer a high degree of discrimination between different manufacturing sources. Unit resolution ICP-MS instruments may suffer from some polyatomic interferences including 40Ar16O+, 40Ar 16O1H+, and 40Ca 16O+ that affect iron measurement at trace levels. Iron is an important element in the analysis of glass and also of interest for the analysis of several biological matrices. A comparison of the analytical performance of two different ICP-MS systems for iron analysis in glass for determining the method detection limits (MDLs), accuracy, and precision of the measurement is presented. Acid digestion and laser ablation methods are also compared. Iron polyatomic interferences were reduced or resolved by using dynamic reaction cell and high resolution ICP-MS. MDLs as low as 0.03 μg g-1 and 0.14 μg g-1 for laser ablation and solution based analyses respectively were achieved. The use of helium as a carrier gas demonstrated improvement in the detection limits of both iron isotopes (56Fe and 57Fe) in medium resolution for the HR-ICP-MS and with a dynamic reaction cell (DRC) coupled to a quadrupole ICP-MS system. ^ The development and application of robust analytical methods for the quantification of trace elements in biological matrices has lead to a better understanding of the potential utility of these measurements in forensic chemical analyses. Standard reference materials (SRMs) were used in the development of an analytical method using HR-ICP-MS and LA-HR-ICP-MS that was subsequently applied on the analysis of real samples. Bone, teeth and ashed marijuana samples were analyzed with the developed method. ^ Elemental analysis of bone samples from 12 different individuals provided discrimination between individuals, when femur and humerus bones were considered separately. Discrimination of 14 teeth samples based on elemental composition was achieved with the exception of one case where samples from the same individual were not associated with each other. The discrimination of 49 different ashed plant (cannabis) samples was achieved using the developed method. ^

Environmental analysis of polar herbicides in complex organic-rich matrices by high performance liquid chromatography atmospheric pressure ionization mass spectrometry (HPLC-API-MS)

A comprehensive forensic investigation of sensitive ecosystems in the Everglades Area is presented. Assessing the background levels of contamination in these ecosystems represents a vital resource to build up forensic evidence required to enforce future environmental crimes within the studied areas. This investigation presents the development and validation of a fractionation and isolation method for two families of herbicides commonly applied in the vicinity of the study area, including phenoxy acids like 2,4-D, MCPA, and silvex; as well as the most common triazine-based herbicides like atrazine, prometyne, simazine and related metabolites like DIA and DEA. Accelerated solvent extraction (ASE) and solid phase extraction (SPE) were used to isolate the analytes from abiotic matrices containing large amounts of organic material. Atmospheric-pressure ionization (API) with electrospray ionization in negative mode (ESP-), and Chemical Ionization in the positive mode (APCI+) were used to perform the characterization of the herbicides of interest.