In this paper, we discuss some of the concepts of Robustness for uncertain multi-objective optimization problems. An important factor involved with multi objective optimization problems is uncertainty. The uncertainty may arise fromthe estimation of parameters in the model, error of computation, the structure of a problem, and so on. Indeed, some parameters are often unknown at the beginning of solving a multi-objective optimization problem. One of the mostimportant and popular approaches for dealing with uncertainty is robust optimization. Markowitz's portfolio optimization problem is strongly sensitive to the perturbations of input parameters. We consider Markowitz's portfolio optimization problem with ellipsoid uncertainty set and apply set-based minmax and lower robust efficiency to this problem. The concepts of robust efficiency are used in the real stock market and compared to each other. Finally, the increase and decrease effects of uncertainty set parameters on these robust efficient solutions are verified.

According to dynamic core hypothesis, integration and dierentiation are main properties of consciousness. Hence, we expected that the consciousness neuronal correlate covers these properties in structural level. These properties could be captured in smallworldness properties, i.e. high clustering coe cient and low path-length. Thalamocortical (TC) loop and cortex are two main candidates for Neural Correlates of Consciousness (NCC). We studied small-worldness in these systems. For this purpose, we calculated clustering coe cients, characteristic path lengths and their Robustness against lesions. We simulated lesions in two ways: eliminating connections, and deleting nodes. We used anatomical connections of TC and cortex of macaque from the CoCoMac neuroinformatic database. Our results show that: 1) Lesions causes an increase in path length and decrease in clustering coe cient which cause the destruction of the integration and segregation capabilities of brain network, 2) Deleting the connections is more destructive than deleting the nodes, 3) During high levels of lesions, the thalamo-cortical connections are more important than cortico-cortical connections in the sense of clustering coe cient. In terms of path-length, during high levels of nodes' lesions, the thalamo-cortical connections are more important than cortico-cortical connections, while during edges' lesions cortico-cortical connections are more important than thalamo-cortical connections.

An Open-Pit Production Scheduling (OPPS) problem focuses on specifying block production scheduling to achieve the highest possible Net Present Value (NPV). This paper presents a new mathematical model for OPPS under uncertainty. To this end, a robust box and ellipsoidal counterpart approach was used. The proposed method was implemented in a hypothetical model. A Genetic Algorithm (GA) and an exact mathematical modeling approach were used to solve the model. It was shown that the scheduling of deterministic and robust models in various conditions is different. Considering the type of robust counterparts, different production plans under various conditions were scheduled. Furthermore, the price of Robustness was determined for various levels of conservation.

In this paper, we propose formulations and algorithms for robust portfolio optimization under both aleatory uncertainty (i. e., natural variability) and epistemic uncertainty (i. e., imprecise probabilistic information) arising from interval data. Epistemic uncertainty is represented using two approaches: (1) moment bounding approach and (2) likelihood-based approach. This paper first proposes a nested Robustness-based portfolio optimization formulation using the moment bounding approachbased representation of epistemic uncertainty. The nested robust portfolio formulation is simple to implement; however, the computational cost is often high due to the epistemic analysis performed inside the optimization loop. A decoupled approach is then proposed to un-nest the Robustness-based portfolio optimization from the analysis of epistemic variables to achieve computational efficiency. This paper also proposes a single-loop robust portfolio optimization formulation using the likelihood-based representation of epistemic uncertainty that completely separates the epistemic analysis from the portfolio optimization framework and thereby achieves further computational efficiency. The proposed robust portfolio optimization formulations are tested on real market data from five S&P 500 companies, and performance of the robust optimization models is discussed empirically based on portfolio return and risk.