## Sensitivity analysis

Sensitivity analysis plays a fundamental role in decision making because it determines the effects of a change in a decision parameter on system performance. In this section, we attempt to provide an answer to the question of how to design policies for sustainable development. The SAFE model could aid decision makers to formulate sustainable policies by assessing sustainability for different scenarios of development. A scenario is defined by the available sustainability indicators, which largely reflect the results of policies and actions taken in a particular period. When these values are changed and the resulting changes on sustainability observed, we could identify the most important indicators contributing to sustainable development. This procedure is known as sensitivity analysis.

Sensitivity analysis entails the computation of the gradients of ECOS, HUMS, and OSUS with respect to each basic indicator. A gradient gives the increase of sustainability per unit increase of some basic indicator. To perform sensitivity analysis we follow the steps:

1.a. For a given country, normalize and smooth all indicator data using the methods described in previous sections. b. Fuzzify the basic inputs. c. Compute the membership grades of composite indicators to the fuzzy sets VB, B, A, G, and VG. Start from the inference engines that use only basic indicators as inputs and proceed successively to the ones that use composite indicators. Finally, compute the membership grades of OSUS to the nine fuzzy sets EL, VL, …, EH and compute a crisp value for OSUS by height defuzzification. 2.Calculation of OSUS:For some basic indicator, say,Introduction of perturbation:cincrease its normalized valuex∈ [0, 1] by some fixed amount_{c}δ, for example, 0.1 or 10%. If the result is greater than one, then truncate it to one to avoid overshooting permissible regions of indicators. 3.Assess the overall sustainability using the same set of data as in step 1 except for indicatorSensitivity analysis:cwhose value is nowx+_{c}δ. Denote the new assessment by OSUS(x+_{c}δ). The gradient of OSUS with respect toxis defined by the forward difference:_{c}Δ= OSUS(_{c}x+_{c}δ) - OSUS. Reset the basic indicatorcto its original valuex. 4._{c}Repeat steps 2 and 3 for all basic indicators. 5.Loop:Identify the gradients with the largest values, which correspond to the basic indicators that affect OSUS the most.Ranking:

An important feature of the SAFE model is monotonicity. Whenever a basic
indicator of sustainability is improved, the components of sustainability
that depend on this indicator as well as OSUS increase or at least do not
decrease, that is, if *δ* ≥ 0, then *Δ _{c}* ≥ 0. The use
of product-sum algebra in all inference engines ensures that the hierarchical
fuzzy system is monotonic (Kouikoglou and Phillis, 2009).

By changing several indicators simultaneously in step 3 we can compute
gradients of higher orders and formulate more comprehensive environmental
policies. For example, the second-order gradient of OSUS with respect to indicators *c* and *c'* is

*Δ _{c,c'}* = OSUS(

*x*+

_{c}*δ*,

*x*+

_{c'}*δ*) - OSUS.

Sensitivity analysis is biased towards indicators which belong to small
groups. For example, RE(AIR) depends only on renewable resources production.
Therefore, an increase in the latter directly affects the former. ST(AIR),
on the other hand, depends on four basic indicators, labeled 26–29. An improvement
of one of these indicators will result in a small improvement of ST(AIR).
To avoid this bias, a basic indicator *c* is ranked according to the
product

*D _{c}* = (1-

*x*)

_{c}*Δ*

_{c} where 1 - *x _{c}* is the distance of indicator

*c*from the sustainable value, and

*Δ*is the gradient of OSUS with respect to

_{c}*x*. Thus those indicators that affect OSUS the most and are farther in the unsustainable region are pinpointed and ranked accordingly.

_{c}