Interval and fuzzy optimization. Applications to data envelopment analysis

Enhancing concern in the efficiency assessment of a set of peer entities termed Decision Making Units (DMUs) in many fields from industry to healthcare has led to the development of efficiency assessment models and tools. Data Envelopment Analysis (DEA) is one of the most important methodologies to measure efficiency assessment through the comparison of a group of DMUs. It permits the use of multiple inputs/outputs without any functional form. It is vastly applied to production theory in Economics and benchmarking in Operations Research. In conventional DEA models, the observed inputs and outputs possess precise and realvalued data. However, in the real world, some problems consider imprecise and integer data. For example, the number of defect-free lamps, the fleet size, the number of hospital beds or the number of staff can be represented in some cases as imprecise and integer data. This thesis considers several novel approaches for measuring the efficiency assessment of DMUs where the inputs and outputs are interval and fuzzy data. First, an axiomatic derivation of the fuzzy production possibility set is presented and a fuzzy enhanced Russell graph measure is formulated using a fuzzy arithmetic approach. The proposed approach uses polygonal fuzzy sets and LU-fuzzy partial orders and provides crisp efficiency measures (and associated efficiency ranking) as well as fuzzy efficient targets. The second approach is a new integer interval DEA, with the extension of the corresponding arithmetic and LU-partial orders to integer intervals. Also, a new fuzzy integer DEA approach for efficiency assessment is presented. The proposed approach considers a hybrid scenario involving trapezoidal fuzzy integer numbers and trapezoidal fuzzy numbers. Fuzzy integer arithmetic and partial orders are introduced. Then, using appropriate axioms, a fuzzy integer DEA technology can be derived. Finally, an inverse DEA based on the non-radial slacks-based model in the presence of uncertainty, employing both integer and continuous interval data is presented

Using slacks-based model to solve inverse DEA with integer intervals for input estimation

This paper deals with an inverse data envelopment analysis (DEA) based on the non-radial slacks-based model in the presence of uncertainty employing both integer and continuous interval data. To this matter, suitable technology and formulation for the DEA are proposed using arithmetic and partial orders for interval numbers. The inverse DEA is discussed from the following question: if the output of DMUo increases from Y-o to /beta(o), such the new DMU is given by (alpha(o)& lowast;, /3) belongs to the technology, and its inefficiency score is not less than t-percent, how much should the inputs of the DMU increase? A new model of inverse DEA is offered to respond to the previous question, whose interval Pareto solutions are characterized using the Pareto solution of a related multiple-objective nonlinear programming (MONLP). Necessary and sufficient conditions for input estimation are proposed when output is increased. A functional example is presented on data to illustrate the new model and methodology, with continuous and integer interval variables

Integer interval DEA: An axiomatic derivation of the technology and an additive, slacks-based model

The present paper studies the efficiency assessment of Decision-Making Units (DMUs) when their inputs and outputs are de- scribed under uncertainty of the type of integer interval data. An axiomatic derivation of the production possibility set (PPS) is presented. An additive, slacks-based data envelopment analysis (DEA) model is formulated, consisting of two phases. This has required the use of adequate arithmetic and LU-partial orders for integer intervals. This novel integer interval DEA approach is the first step towards DEA models under fuzzy integer intervals, with the extension of the corresponding arithmetic and LU-partial or- ders to fuzzy integer intervals. The proposed method is applied on a dataset, taken from the literature, that involves both continuous and integer interval variables.The first and second authors are partially supported by grant PID2019-105824GB-I00. The fourth author acknowledges the financial support of the Spanish Ministry of Science, Innovation and Universities, grant PGC2018-095786-B-I00

Efficiency assessment using fuzzy production possibility set and enhanced Russell Graph measure

This paper studies the efficiency assessment of Decision Making Units (DMUs) when their inputs and outputs are fuzzy sets. An axiomatic derivation of the fuzzy production possibility set is presented and a fuzzy enhanced Russell graph measure is formulated using a fuzzy arithmetic approach. The proposed approach uses polygonal fuzzy sets and LU-fuzzy partial orders, and provides crisp efficiency measures (and associated efficiency ranking) as well as fuzzy efficient targets. The proposed approach has been compared with other fuzzy DEA approaches on different datasets from the literature, and the results show that it has more discriminant power and more flexibility in modelling the input and output data.Ministerio de Economía y Competitividad MTM2017-89577-PMinisterio de Economía y Competitividad AYA2016-75931-C2-1-PJunta de Andalucía - Consejería de Educación y Ciencia TIC-101Ministerio de Ciencia, Innovación y Universidades PGC2018-095786-B-I0