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\title{Robust optimization }
\begin{abstract}
This paper develops robust optimization for the supplier selection problem and applies the fish swarm algorithm as solution procedure. This is because the return margin parameter is the main effective and in the real world has uncertainty. To show the application of this technique in the real world conditions, the selection of the supplier portfolio has been examined in a case study. Numerical results in the uncertain and deterministic conditions demonstrate the efficiency of the proposed solution procedures in finding the efficient limit of the supplier portfolio. Also, the return margin of investment in uncertain condition shows a higher risk than the deterministic condition, but these results are closer to the real world conditions of the problem. Comparison between artificial fish swarm algorithm gray wolf algorithm indicates that in both deterministic and uncertain conditions, gray wolf algorithm provides higher quality than fish swarm algorithm. In addition, in terms of speed of solution, the gray wolf algorithm has a slight advantage over artificial fish swarm optimization algorithm.
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\begin{keyword}
Supplier portfolio optimization\sep Gray wolf algorithm\sep Artificial fish swarm algorithm\sep Risk\sep Robust optimization.\\
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\section{Introduction}
\label{Intro}
Chain management includes communication and collaboration between chain members. These members can be suppliers, intermediaries, customers, etc. \cite{rushton2014handbook}
\section*{References}
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