Invariance and robustness of the ordered inequality of aggregate sustainability indices by vector space theory

Abstract

Sustainability assessment of manufacturing sector has been described by a single aggregate index for each process. A set of production processes give rise to an inequality ordering of such indices. This ordered hierarchy must be representative of the physical system otherwise it may yield misleading results. A recent paper has treated processes as vectors whose components are the manufacturing factors in a Euclidean space generated by a set of orthonormal vectors. Aggregate indices are defined by the Euclidean magnitude of the process vectors. A subsequent paper considered an enlarged space where the Euclidean space is a particular case. In this increased space, the robustness of the Euclidean indices was analyzed. The present work in this broader space focuses on the robustness of the ordered inequality. The central aim consists in determining the conditions by which the Euclidean indices belong to an invariant hierarchy in this space. The formalism was confirmed by providing consistent results when applied to real production processes.

Appendices

Appendix I

Processes and factors for chlorine manufacturing

Source: Sikdar (2009)

P ₁ : Mercury cells

P ₂ : Diaphragm cells

P ₃ : Membrane cells

F ₁ : Energy intensity (MJ/kg Cl₂)

F ₂ : Material intensity (kg/kg Cl₂)

F ₃ : Potential chemical risk

F ₄ : Potential environment impact

Processes and factors for automobile fender production

Source: Sikdar et al. (2012)

P ₁ : Aluminum

P ₂ : Steel

P ₃ : PC/PBT

P ₄ : PP/EPDM

P ₅ : PPO/PA

F ₁ : Energy

F ₂ : Resources

F ₃ : Water

F ₄ : GWP

F ₅ : ODP

F ₆ : AP

F ₇ : EP

F ₈ : POCP

F ₉ : Htox air

F ₁₀ : Htox water

F ₁₁ : EcoTox

F ₁₂ : Waste

Processes and Factors for automotive shredder residue

Source: Sikdar et al. (2012)

P₁: Landfill

P₂: Recycle landfill (kg CO2-eq./tASR)

P₃: Energy recovery

P ₄ : Recycle energy (kg CO₂-eq./tASR)

F₁: Energy intensity (GJ/tASR)

F₂: Material intensity (kg Fe-eq./tASR)

F₃: Water consumption (m³/ASR)

F₄: Land use(m²a/tASR)

F₅: GW short term (kg CO₂-eq./tASR)

F₆: GW long term (kg CO₂-eq./tASR)

F₇: Human tox. Short term (kg C₆H₄C₁₂-eq./tASR)

F₈: Human tox. Long term (kgC₆H₄C₁₂-eq./tASR)

F₉: Treatment-costs (€/tAS)

Appendix II

By using a scientific calculator a simple evaluation of T: A[a _ij ] → M[x _ij ] and ||P _i || _p is sketched. Without the need of matrix multiplication it can be accomplished by taking each vector P _i one at a time. The vector coordinates must be expressed as a LIST, {a _i1 a _i2 …}, and not as a VECTOR, [a _i1 a _i2 …], because vectors do not divide. As an illustration we consider the first process P ₁ of the fender case. From the first row in Table 4, the list is

{P ₁ (a)} = {1290 15 36 104 1.0 28 4.4 6.7 3.9 0.66 2.9 3.7}

From the same Table the list of the normalization parameter s _j  = a ^> _j  + a ^< _j is

{s _j } = {2100 29 53 177 1.3 44 10.7 15.9 5.8 1.61 5.3 3.9}

The normalized vector is then {P ₁ (x)} = {P ₁ (a)}/{s _j }:

{P ₁ (x)} = {.614 .385 .679 .588 .769 .636 .411 .421 .672 .410 .547 .937}

that equals the first row in Table 5. Repeating this procedure to the other processes the matrix M is obtained. But it is more expedient to store as variables in the memory all these P _i vectors separately rather than the matrix M.

The norm ||P _i || _p is determined by first storing the exponent p as a variable, say, E, and then store as a variable the program that calculates Eq. (10) in the RPN (Reverse Polish Notation) mode:

||P _i || _p  = 〈〈E ˆ ∑LIST E INV ˆ 〉〉

In order to evaluate the hierarchies store the particular p value in E, recall the process {P _i (x)}, and press ||P _i || _p . For the ||P ₁ || _∞ norm one must simply pick out x ^> _j in {P _i }. In the P ₁ case, ||P ₁ || _∞  = 0.937.

Weighting coefficients c ₅  = c ₇  = 1.5 and c ₈  = c ₁₀  = 2.0 can be introduced by multiplying {P _i } by the list

{W ₁ } = {1 1 1 1 1.5 1 1.5 2.0 1 2.0 1 1}

The vector P ₁ becomes, with the weighted indicators highlighted,

{P ₁ (x)}_W = {.614 .385 .679 .588 1.153 .636 .617 .842 .672 .820 .547 .937}