Finding the Efficiency Scores

efficiencyScores(gI,gO,bO,bI;retToScale,prodStructure,dirGI,dirBI,dirGO,dirBO,startθ,startμ,startλ)

Compute efficiency indicators and convexity test results considering the joined production of good and bad outputs.

Given a set of measures of inputs, "good" ("desiderable") and "bad" ("undesiderable") outputs for different decision making units, compute their distance to the production frontier, i.e. their degree of efficiency.

Parameters:

• Positional
  • gI: "Good" inputs (3D matrix by DMUs, input items and periods)
  • gO: "Good" outputs (3D matrix by DMUs, input items and periods)
  • bO: "Bad" outputs (3D matrix by DMUs, input items and periods)
  • bI: "Bad" inputs (optional 3D matrix by DMUs, input items and periods) [default: empty array]
• Keyword
  • retToScale: Wheter the returns to scale should be assumed "constant" or "variable" (default). Non-convex distance and test is computed only under the "variable" assumption, except for the (0,0,1,-1) distance under multiplicative production function
  • prodStructure: Wheter the production structure should be assumed "additive" (default) or "multiplicative"
  • dirGI,dirBI,dirGO,dirBO: The directions toward where to measure the efficiency (see notes) [default: (0,0,1,0)]
  • startθ,startμ,startλ: Initial values in the convex optimisation problem [default: (0,0,1.1)]

Returns:

• A named tuple with the following items:
  • λs: The efficiency indicators coherent with the convexity test (2D matrix by DMUs and periods)
  • λs_convex: The efficiency indicators assuming a convex production frontier (2D matrix by DMUs and periods)
  • λs_nonconvex: The efficiency indicators assuming a non-convex production frontier (2D matrix by DMUs and periods)
  • nonConvTest: The result of the non-convexity test (2D matrix of boolean by DMUs and periods). Note that "true" here refers to non-convex.
  • nonConvTest_value: The value of the non-convexity test (2D matrix by DMUs and periods)

Description of the function

efficiencyScores defines environmental efficiency indicators for a set of Decision Making Units (ie., observations). The nature of the DMUs (eg. firms, countries etc.) depends on the aim of the environmental efficiency analysis.

All the DMUs are characterized by a set of inputs and outputs separated into undesirable (eg., polluting) and desirable (eg., no polluting) ones. Note that, inputs separation into desirable (eg., non emissions-causing) and undesirable (eg., emissions-causing) components is optional for the BDisposal package.

The BDisposal DEA modelling defines the smallest environmental production process that contains all DMUs informations (inputs/outputs) and satisfies the B-disposal axiomatic pattern.

The BDisposal algorithms for efficiency assessment compute the distance to the best environmental production procedures; ie., to the efficient production frontier. These efficiency indices inherit the structure of additive and multiplicative distance functions.

Notice that the BDisposal package computes general shape of additive and multiplicative distance functions. It follows that, the Bdisposal package introduces a flexible framework for environmental efficiency assessment, with the possibility to analyse different directions for the distance functions. In addition, the BDisposal package allows to test the (economists’) properties of convexity and disposability.

Interpretation of the results

Efficiency indicators

When the additive shape of the distance function is greater than 0 then, the production unit doesn’t operate efficiently. If the additive efficiency score is equal to 0 then, the DMU is efficient. It follows that, the DMU employs one of the best production technology to transform inputs into outputs; ie., the producer is a benchmark for the other observations. A similar reasoning holds for the multiplicative distance functions.

If the multiplicative efficiency index is greater than 1 then, the DMU doesn’t perform efficiently. When the multiplicative efficiency score is equal to 1 then, the production unit is a benchmark for the other observations.

Convexity test

When the multiplicative test of convexity is greater than 1 (respectively, equal to 1) then, the production process frontier is non convex (respectively, convex). If the multiplicative test of disposability is greater than 1 (respectively, equal to 1) then, the frontier of the production set displays B-disposability; ie., there exists costs for undesirable component decrease. A similar reasoning applies for the additive framework.

The informations provided by these tests define the shape of the production process boundaries.

Notes:

• Directions toward where to measure the efficiency distance can be tuned using the dirGI,dirBI,dirGO and dirBO parameters.
• The following directions are supported:
  • multiplicative pr. struct.: (-1,0,0,0), (0,-1,0,0), (0,0,1,0), (0,0,0,-1) or (0,0,1,-1)
  • addittive pr. struct.: (1,0,0,0), (0,1,0,0), (0,0,1,0) or (0,0,0,1)
  Other directions can be used to compute the direction under the convex frontier assumption, but the convexity test is not performed and, for the multiplicative case, no guarantee is given on solving the underliying (non-linear) problem (different initial value startθ,startμ,startλ can be attempted)