Attaining sustainability objectives has received wide attention in the supplier selection and order allocation (SSOA) literature. This paper aims to investigate an SSOA problem under multiple items, multiple suppliers, multiple price levels, and multiple period using a robust-fuzzy multi-objective programming in which: (a) transportation cost, delay penalty cost, and demand are uncertain; (b) four objectives are proposed to minimize total costs and the number of defective items and to maximize environmental and social impacts; and (c) all objectives of the problem have a fuzzy membership degree that is determined by the decision-makers. A robust optimization approach is elaborated as a solution procedure to address the uncertainty of the decision variables. The significance of each objective in practice is discussed based on seven distinct scenarios that produce a specific membership degree to help practitioners make efficient decisions in selecting the suppliers and allocating the orders. Two numerical examples with different sizes are conducted to validate the mathematical model. Thereafter, the sensitivity of each scenario on objectives and total satisfaction degree is analyzed. The results of the numerical solution compare the value of four objective functions under each developed scenario to provide a trade-off insight between different objectives for practitioners. Eventually, the credibility and efficiency of the proposed solution procedure are evaluated to validate the findings.@en