Maximal Closest Reference Set (MCRS)

• MCRS is a formal DEA construct that identifies all efficient peers forming the closest reference set for benchmarking inefficient DMUs.
• It employs a two-stage optimization process using mixed-integer and linear programming to minimize input/output adjustment distances.
• This approach enhances peer similarity and realistic target setting while overcoming limitations of classical DEA reference sets.

The Maximal Closest Reference Set (MCRS) is a formal construct in Data Envelopment Analysis (DEA) developed to address the challenge of identifying the most relevant benchmarking peers for an inefficient Decision Making Unit (DMU). Unlike classical DEA reference sets derived as a by-product of frontier projection, the MCRS aims to determine the largest group of efficient DMUs that are as similar as possible to the evaluated unit, based on minimal input/output adjustments. The MCRS formalism is motivated by practical benchmarking imperatives: attainable targets and maximally relevant comparators for realistic performance improvement. It is rigorously defined and computed through mixed-integer or linear programming models in the proximity-based DEA literature (Ruiz et al., 2020, Roshdi et al., 2014, Mehdiloozad et al., 2015).

1. Formal Definition and Motivation

The MCRS relates to the set of efficient DMUs (peers) used to benchmark a given inefficient unit, DMU₀, under closest-target DEA frameworks. For DMU₀ with input vector $x^0$ and output vector $y^0$, the Production Possibility Set (PPS) $T$ is constructed typically under variable returns to scale (VRS) as:

$$T = \{(x, y) \mid \exists \lambda \geq 0, \sum_j \lambda_j = 1, x \geq \sum_j \lambda_j x^j, y \leq \sum_j \lambda_j y^j \}$$

where $\{(x^j, y^j)\}_{j \in E}$ are extreme-efficient DMUs.

The classical DEA reference set for DMU₀ consists of units $j \in E$ with positive intensity $\lambda^*_j$ in an optimal solution projecting DMU₀ onto the frontier. However, degeneracy can lead to many possible such sets for a closest projection. The MCRS is formally defined as the union of all possible closest reference sets for a given closest projection point $P = (x^*, y^*)$: every efficient DMU with nonzero intensity in some convex combination representing $P$. In some formulations, an additional criterion is imposed: among candidates, select the peer set such that the maximal dissimilarity (distance in input/output space) between DMU₀ and any peer is minimized (Ruiz et al., 2020, Roshdi et al., 2014, Mehdiloozad et al., 2015).

2. Computation: Two-Stage and Single-Stage Frameworks

MCRS computation universally involves two principal stages:

Stage 1: Closest Target Projection.

A mathematical programming model (often a non-radial, non-oriented or lexicographic linear/mixed-integer program) identifies a point $P^C = (x^0 - s^{- *}, y^0 + s^{+ *})$ on the frontier that is closest to $y^0$0, minimizing a suitable norm such as (weighted) $y^0$1:

$y^0$2

subject to convex reproduction constraints and supporting hyperplane conditions (Roshdi et al., 2014, Ruiz et al., 2020, Mehdiloozad et al., 2015).

Stage 2: Maximal Closest Reference Set Extraction.

Given $y^0$3, the MCRS is determined by maximizing (over all convex representations) the number of efficient DMUs with strictly positive weight in the reconstruction of $y^0$4. Linear programs with auxiliary variables and strong complementary slackness conditions achieve this without enumerating all optimal bases:

• For each efficient DMU $y^0$5, introduce variables $y^0$6 and $y^0$7.
• Enforce $y^0$8 and maximize $y^0$9, guaranteeing that any $T$0 with $T$1 appears as an MCRS member (Roshdi et al., 2014).

Alternatively, a primal envelopment LP maximizes the sum of participation indicators (e.g., $T$2) subject to reproduction and normalization constraints, yielding $T$3 (Mehdiloozad et al., 2015).

A weighted single-stage MILP can trace the Pareto frontier of (target distance, maximal peer similarity) by optimizing $T$4, smoothly interpolating between purely closest targets and maximal peer similarity (Ruiz et al., 2020).

3. Distances, Similarity Metrics, and Lexicographic Programming

MCRS construction critically depends on rigorous similarity metrics:

• Projection Distance ($T$5): Typically a component-wise weighted $T$6-type distance measuring the minimal effort (in proportional input reduction and output augmentation) to move DMU₀ to the frontier (Ruiz et al., 2020, Roshdi et al., 2014).
• Peer Similarity Distance ($T$7): For each efficient peer $T$8, $T$9 assesses the (relative) proximity in input/output space (Ruiz et al., 2020). The second-stage objective may minimize the worst-case peer distance, $$T = \{(x, y) \mid \exists \lambda \geq 0, \sum_j \lambda_j = 1, x \geq \sum_j \lambda_j x^j, y \leq \sum_j \lambda_j y^j \}$$0, ensuring group homogeneity.

Lexicographic multi-objective formulations are employed to guarantee uniqueness in the closest projection, sequentially minimizing prioritized slack variables (Mehdiloozad et al., 2015).

4. Properties and Distinctions from Classical Reference Sets

• Maximality: The MCRS includes all efficient DMUs that can possibly serve as closest peers in any convex combination of the closest target. This yields the largest relevant peer group (maximal) for a given projection (Roshdi et al., 2014, Mehdiloozad et al., 2015).
• Hyperplane Support: All MCRS members lie on the supporting frontier hyperplane through the closest target; dual variables enforce this property (Roshdi et al., 2014).
• Homogeneity: The MCRS guarantees minimal worst-case dissimilarity from DMU₀, unlike classical methods that may select faraway extreme-efficient DMUs indifferent to proximity (Ruiz et al., 2020).
• Uniqueness: For a fixed closest projection, the MCRS is uniquely determined; for multiple equally optimal projections, multiple MCRSs may exist (Roshdi et al., 2014).
• No Enumeration Needed: The required models consolidate all possible reference sets for a single projection, obviating the need for combinatorial enumeration (Roshdi et al., 2014, Mehdiloozad et al., 2015).
• Scale: Models are tractable for hundreds of DMUs with state-of-the-art solvers; the dominant cost is the initial closest-projection MILP in larger instances (Roshdi et al., 2014).

5. Algorithms and Step-by-Step Procedures

A canonical approach to MCRS extraction is as follows (Roshdi et al., 2014, Mehdiloozad et al., 2015):

Step	Procedure	Purpose
1	Solve the closest-projection (MILP/lexicographic LP)	Obtain $$T = \{(x, y) \mid \exists \lambda \geq 0, \sum_j \lambda_j = 1, x \geq \sum_j \lambda_j x^j, y \leq \sum_j \lambda_j y^j \}$$1 on the efficient frontier
2	Identify efficient peers supporting $$T = \{(x, y) \mid \exists \lambda \geq 0, \sum_j \lambda_j = 1, x \geq \sum_j \lambda_j x^j, y \leq \sum_j \lambda_j y^j \}$$2 (via hyperplane conditions)	Determine candidate set for MCRS
3	Solve a consolidation LP (with participation or coupling variables)	Extract MCRS as all $$T = \{(x, y) \mid \exists \lambda \geq 0, \sum_j \lambda_j = 1, x \geq \sum_j \lambda_j x^j, y \leq \sum_j \lambda_j y^j \}$$3 with positive weight

This sequence ensures completeness and computational efficiency.

6. Illustrative Examples

Roshdi et al. (2015) (Roshdi et al., 2014) present a 9-DMU case with 2 inputs and 1 output, where for DMU₆, the MCRS = {1, 2} and for DMU₉, the MCRS = {4} (only the peer on the supporting face participates positively, as per dual-based selection). Similarly, Ruiz & Sirvent (Ruiz et al., 2020) demonstrate on university data that the MCRS for a unit may differ markedly from the classical reference set obtained by projection alone, with improved interpretability and peer relevance at the cost of a minor increase in modeling complexity.

7. Applications, Interpretive Significance, and Limitations

• Benchmarking: The MCRS framework provides actionable, attainable targets for performance improvement by focusing exclusively on nearest, most similar efficient DMUs (Ruiz et al., 2020).
• Sensitivity Analysis: MCRS structure delivers robust peer identification, facilitating stability assessments under data perturbation (Roshdi et al., 2014).
• Returns to Scale Measurement: MCRS methodology is compatible with precise returns-to-scale estimation at the closest projection (Mehdiloozad et al., 2015).
• Classification: The size and makeup of the MCRS can indicate natural peer groups for further segmentation or clustering analyses (Roshdi et al., 2014).
• Limitations: MCRS computation requires solving MILPs or sequences of LPs, which, although not prohibitive, can present computational challenges for very large-scale problems. Multiple equally closest projections may necessitate rerunning the second stage, and current frameworks are primarily formulated for constant or variable returns to scale (Roshdi et al., 2014, Mehdiloozad et al., 2015).

In summary, the MCRS paradigm advances benchmarking methodology by explicitly aligning the selection of peer groups with similarity and attainability criteria, distinguishing itself from traditional DEA approaches and enabling more informative, feasible, and interpretable managerial recommendations (Ruiz et al., 2020, Roshdi et al., 2014, Mehdiloozad et al., 2015).