This paper presents a new mathematical model of the green closed-loop supply chain network (GCLSCN) during the COVID-19 pandemic. The suggested model can explain the trade-offs between environmental (minimizing CO 2 emissions) and economic (minimizing total costs) aspects during the COVID-19 outbreak. Considering the guidelines for hygiene during the outbreak helps us design a new sustainable hygiene supply chain (SC). This model is sensitive to the cost structure. The cost includes two parts: the normal cost without considering the coronavirus pandemic and the cost with considering coronavirus. The economic novelty aspect of this paper is the hygiene costs. It includes disinfection and sanitizer costs, personal protective equipment (PPE) costs, COVID-19 tests, education, medicines, vaccines, and vaccination costs. This paper presents a multi-objective mixed-integer programming (MOMIP) problem for designing a GCLSCN during the pandemic. The optimization procedure uses the scalarization approach, namely the weighted sum method (WSM). The computational optimization process is conducted through Lingo software. Due to the recency of the COVID-19 pandemic, there are still many research gaps. Our contributions to this research are as follows: (i) designed a model of the green supply chain (GSC) and showed the better trade-offs between economic and environmental aspects during the COVID-19 pandemic and lockdowns, (ii) designed the hygiene supply chain, (iii) proposed the new indicators of economic aspects during the COVID-19 outbreak, and (iv) have found the positive (reducing CO 2 emissions) and negative (increase in costs) impacts of COVID-19 and lockdowns. Therefore, this study designed a new hygiene model to fill this gap for the COVID-19 condition disaster. The findings of the proposed network illustrate the SC has become greener during the COVID-19 pandemic. The total cost of the network was increased during the COVID-19 pandemic, but the lockdowns had direct positive effects on emissions and air quality.

One of the most important aspects of supply chain management (SCM) is the recovery network (RN), which covers all activities associated with return products (such as collection, recovery, repair, recycling, and waste disposal). Our goal in this paper is to provide a new mathematical model of sustainable end-of-life management (SEOLM) during the COVID-19 pandemic for readers. The suggested recovery network model (RNM) can explain the trade-offs between economic (minimizing total costs), environmental (minimizing bad environmental impacts), and social (minimizing bad social impacts) aspects during the pandemic and the great lockdown. A new RN can be designed with a sustainable and hygienic design when taking environmental, economic, and social considerations into account. It proposes guidelines for managers and scholars on how to address recovery network design (RND) challenges during the pandemic through a mathematical article with a sustainable approach. The scalarization approach of a multi-objective mixed-integer programming (MOMIP) problem in this paper is the weighted sum method. The validation of the presented model and the related Pareto frontier has been illustrated by a case study and numerical example. To perform the optimization process, Lingo software is used.

A large number of engineering problems involve several conflicting objectives, which today are often solved through expensive simulation calculations. Methods based on meta-models are one of the approaches to solving this group of problems. In this paper, multiobjective optimization in the extraction system of a copper open-pit mine complex is presented by the modified-NBI optimization method and regression meta-model. For this purpose, two objective functions of maximizing the amount of total extraction, which is the sum of the extraction of sulfide, oxide, low-grade ores, and waste in this mine, and minimizing the transport time of haulage according to the limitation of its storage capacity, transport equipment, and budget are considered. The Central Composite Design (CCD) method is used to build the Design of Experiments (DOE) for the design variables. The considered design variables are the number of trucks of 120 tons, 240 tons, 35 tons, and 100 tons. The number of targets considered in each design combination is considered the response surface. The suitable meta-model to maximize the total extraction rate and minimize the transport time of the haulage, two modified functions of nonlinear regression have been determined. The accuracy of the models for selection has been done using PRESS and R 2 statistics. The most common PRESS error has also been used to validate the meta-models. Then the multiobjective optimization problem was solved using the modified-NBI method. Finally, Pareto and optimal solutions using the proposed approach were presented and discussed.

In this paper, we propose a new method to obtain eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix (Ã), which are the elements of the given fuzzy triangular matrix . To this purpose, we solve the 1 − cut of a fuzzy triangular matrix (Ã) to obtain the 1 − cut of eigenvalues and eigenvectors. Then, based on the results obtained in a 1 − cut mode, we use three new models to determine the left and right widths for those eigenvalues and eigenvectors. So, after some manipulation, in each of the models, the fully fuzzy linear systems (FFLSs) transformed to 2n crisp linear equations and some crisp linear non-equations (that, the first model includes 2(n + 1), the second model includes 2(n + 3) and the third model includes 6n + 2 crisp linear non-equation). Then, we suggest a nonlinear programming problem (NLP) to calculation simultaneous equations and non-equations. Furthermore, we define three other new eigenvalues (namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue) for a fuzzy triangular matrix (Ã) that does not have any suitable solution. the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set (TTFES), the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set (CTFES), and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set (ATFES). Finally, numerical examples are presented to illustrate the proposed method.

PurposeData envelopment analysis (DEA) is a significant method for measuring the relative efficiency of decision making units (DMUs) that use the least inputs, produce the most desirable outputs and emit the least undesirable outputs in order to maximize their profits. In DEA, detecting an optimal scale size (OSS) is also vital and could be more applicable in economic activities when there are integer and undesirable measures. The purpose of this research is to measure average-profit efficiency (APE) and OSSs with integer data and undesirable outputs.Design/methodology/approachThis study presents an alternative concept of APE using the concepts of most productive scale size (MPSS), profit efficiency and scales, containing desirable and undesirable outputs along with integer and non-integer measures. In fact, the OSS minimizes APE as the optimal scale, which is the ratio of the profit efficiency to the radial average output. Considering the prices of the inputs and desirable outputs, as well as the lack of any specific weight for the undesirable outputs, a two-step model for the numerical calculation of OSS is presented. In addition, the proposed approach is applied to a real data set of Iranian gas companies while there are integer measures and undesirable outputs.FindingsThe results show the introduced approach is beneficial to estimate OSSs from the aspect of maximizing profits of firms with undesirable outputs and integer values.Originality/valueEstimating OSSs is the significant issue for managers, but its investigation in the presence of integer measures and undesirable outputs is presently under-considered.