Sustainable Procurement

The proposed method is suitable for the assembled product where product variety is achieved by adding auxiliary parts/components to the main product or base product or core product as per demand of customer. To tackle uncertainty of demand a two-stage method is considered. The base product is assumed to be manufactured by deterministic demand and auxiliary parts/component is manufactured as per stochastic demand. Stage 1 is required to select suppliers for base product and stage 2 is required to select suppliers for auxiliary parts/components. In both stages analytic hierarchy process (AHP) is used with intuitionistic fuzzy set (IFS) to select and evaluate suppliers and multi-objective genetic algorithm (MOGA) is used to allocate order to selected suppliers for deterministic and/ stochastic demand. AHP is one of the most cited multi-criteria decision analysis (MCDA) tool and the fuzzy version of integrated AHP is mostly used for supplier selection to deal with uncertainties of the supplier selection process. Intuitionistic Fuzzy Set (IFS) is a generalized fuzzy set and is more suitable in selecting suppliers as it includes the degree of hesitation to measure uncertainty associated with each decision. Further, Shannon's Entropy is included in the proposed method to measure the discord or conflict in selecting suppliers.

Intuitionistic fuzzy set (IFS)

In 1986, Atanassov proposed a generalized concept of fuzzy set popularly known as the intuitionistic fuzzy set (IFS). If X be a universe of discourse, then IFS A can be defined as $A= {(x,µ_A(x),ν_A(x))|x∈X}$ where $µ_A(x), ν_A(x)$ denote membership and non-membership functions of A and satisfy $0 \leq µ_A (x) + ν_A \leq 1 \forall x \in X$. For every IFS A in X, the degree of hesitation can be defined as $πA(x)=1- µ_A (x) - ν_a (x)$ which express whether x belongs to A or not. If $A = {(x,µ_A(x),ν_A(x))|x∈X} and B= {(x,µ_B(x),ν_B(x))|x∈X}$ then the normalized hamming distance between A and B can be represented as

$l(A, B)=\frac{1}{2 n} \sum_{i=1}^{n}\left[\begin{array}{l}\left|\mathbf{\mu}_{A}(x)-\boldsymbol{\mu}_{B}(x)\right|+ \\\left|\nu_{A}(x)-\nu_{B}(x)\right|+ \\\left|\pi_{A}(x)-\pi_{B}(x)\right|\end{array}\right] \frac{1}{2}$

To rank three IFS, their normalized hamming distance from the ideal solution M (1,0,0) should be calculated. Lowest distance from M will give better solution.

Shannon's entropy

Shannon's entropy is a classical measure of discord in probability theory. Let $p=$ a probability defined on X. Then Shannon's entropy is defined as $S(p) = - \sum_ {x \in X} p_{x \; Log_2 \; p_x}$

In AHP, priority pi is the probability that i^th criterion is preferred by decision maker.

IF-AHP algorithm

1. 1 Prepare intuitionistic fuzzy pairwise comparison matrix for each criterion and alternative.
2. 2 Calculate the score (S_i) of all intuitionsitic fuzzy number with any of the given formula.
   

1. 3 Calculate the normalized score matrix with the given formula
   $\overline{s_{ij}}=\frac{\begin{array}{c}{s}_{ij}-\min \left\{s_{ij}\right\}\\ j\end{array}}{\begin{array}{c}\max \left\{s_{ij}\right\}-\min \left\{s_{ij}\right\}\\ j\end{array}}$
   (6)
   
   
2. 4 Normalize each row of $\overline S$ with the given formula
   $\dot{\bar{s}}=\frac{\overline{s_{i j}}}{\sum_{j=1}^{n} \overline{s_{i j}}} \forall i=1,2,3 \ldots . \text { mand } j=1,2,3 \ldots n$
   (7)
   
   
3. 5 Calculate entropy w.r.t ith attribute with the given formula
   $E_{i}=-\frac{1}{\ln n} \sum_{j=1}^{n} \overline{s_{i j}} \ln \dot{\overline{s_{i j}}}$
   (8)
4. 6 Calculate entropy weight wi with the given formula
   $w_{i}=\frac{1- E_{i}}{\sum_{i=1}^{m} (1-E_{i})}$
   (9)
   
   
5. 7 Calculate normalized entropy weight to rank criteria or alternative with the given formula
   $\overline{w_{i}}=\frac{W_{i}}{\sum_{i=1}^{m} W_{i}}$
   (10)