# A Model for a Green, Lean Sustainable Improvement with Performance Measurement
## 2 Tropical Algebra  
In the proposed model that will be introduced in the next section, we will formulate optimization problems for two target functions, which will also be expressed in tropical algebra terminology. Tropical algebra is a term that unifies isomorphic algebraic mathematical systems (max-plus algebra, min-plus algebra, max-times algebra, and min-times algebra) and provides (together with its geometric counterpart--tropical geometry) an attractive way of describing a class of non-linear problems appearing, for instance, in the scheduling of manufacturing and transportation networks, discrete time event-dynamicsystems and their optimal control, combinatorial optimization, mathematical physics, information technology, system theory of computer networks, DNA analysis (the study of phylogenetic trees), machine learning and deep neural networks, medicine (the non-local diffusion of networks arising from medical data), differential equations with delay (the asymptotic study of slowly oscillating solutions of a class of differential equations with delay that arise in models from biology, social sciences, crystal growth, etc. ), and graph theory, etc. See, e.g., [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33], and the references cited therein. The usefulness of tropical algebra arises from the fact that these non-linear problems become linear when described in the language of tropical algebra. Its methods also turned out to be very useful for solving certain problems in linear algebra and graph theory--see, e.g., [25; 30; 31; 32; 33] and the references cited therein. Max-plus algebra is the (semialgebra) algebra over the ordered, idempotent semiring (in fact, semifield) \(\mathrm{R}_{\mathrm{max}}=\mathrm{R}\cup\{-\infty\}\) (where \(\mathrm{R}\) denotes a set of real numbers) equipped with the operations of the addition \(\mathrm{a}\oplus\mathrm{b}=\mathrm{max}\) {a,b} and the multiplication \(\mathrm{a}\odot\mathrm{b}=\mathrm{a}+\mathrm{b}\), with the unit elements \(-\infty\) (for max-plus addition \(\oplus\)) and 0 (for max-plus multiplication \(\odot\)). Here, we define \(-\infty\odot\mathrm{a}=\mathrm{a}\odot-\infty=-\infty\). The max-plus power is defined by a \(\odot\mathrm{k}\) = ka (the standard multiplication) for the real numbers a and \(\mathrm{k}\). This notation is used to write \(\oplus_{i=1}^{k}a_{i}=max\{a_{1},\ldots,a_{k}\}\) and \(\odot_{i=1}^{k}a_{i}=\sum_{i=1}^{k}a_{i}\) for \(a_{1},\ldots,a_{k}\in R_{max}\). As in standard arithmetic, the operations of addition and multiplication are associative and commutative, and multiplication is distributive over addition. Matrix and polynomial operations are defined similarly to their standard counterparts, with the max-plus operations replacing the standard operations. Max-plus algebra is isomorphic to min-plus algebra, which is the semifield \(R_{min}=R\cup\{\infty\}\), equipped with the operations of addition \(a\oplus^{\prime}\ b=min\{a,b\}\) and multiplication a \(\odot^{\prime}\ b=a+b\), with the unit elements \(\infty\) (for min plus addition \(\oplus^{\prime}\)) and 0 (for max-plus multiplication \(\odot\)). Here, we define \(\infty\odot^{\prime}a=a\ \odot^{\prime}\ \infty=\infty\). We also write \(\oplus^{\prime}\overset  
First, a company needs to identify the process to be analyzed. Starting from this decision, quality input data have to be collected, which will be used for green lean value stream mapping (GLVSM), which is the basis for all subsequent steps: the identification of opportunities for improvement and the elimination of waste and a reduction in environmental  
Figure 1: Proposed model for green, lean sustainable production. Figure 2: Process of green, lean sustainable improvement. burden. The green value stream mapping (GVSM) approach, introduced by the authors in [34], expands the standard value stream mapping by incorporating additional flows such as energy, material, transport, and emissions, alongside the standard time stream.