## 1 Introduction
The grey system theory looks at each stochastic variable as a grey quantity that varies within a fixed region and within a certain time frame, and parallel to this, each stochastic process is a grey process. The engineering concept of a grey box is the basic concept of the grey system theory. Uncertain or not well-known relations among input and output parameters of the process refer to grey relations, or in other words, a relationship among the parameters of a grey system, where the proximity or closeness of grey data sequences represents different unbound parameters of the system. In this paradigm, greyness implies incompleteness [12]. In this work, TOC, COD, phenols and turbidity are taken as grey parameters that affect the amount of coagulant dosage to be used. In the engineering and technical literature, the most popular technique is [PERSON]'s grey relational analysis (GRA) model, permitting a multi-objective optimization and evaluation of process parameters. Some applications of the GRA model are as follows: (i) optimization of powder metallurgy processing parameters of the Al\({}_{2}\)O\({}_{3}\)/Cu composite using the GRA model coupled to the Taguchi method [13], in order to optimize electrical discharge machining process parameters; (ii) optimization of the process parameters for machining aluminum hybrid composites using electrical discharge machining (EDM) [14]; other researchers [15; 16] also led works on EDM using the GRA model coupled with the Taguchi method for the multi-optimization of the micromachining process parameters; (iii) principal component analysis combined with multi-objective optimization of process parameters for hard turning of AISI 52100 steel was carried out [11]; (iv) [PERSON] et al. [17] applied micro-milling to TI-6 Al-4V, whereas another researcher [18] used the analytic hierarchy process (AHP) integrated GRA model to analyze the integrated cascade utilization system of waste geothermal water; (v) the relationships between some parameters to study the degree of the relative importance of the influencing factors about fractal dimensions for super pulverized coal particles were studied [19]. The list is only a selection of case studies that involved the GRA model or variants to seek successful process optimization. Indeed, the common point of all processes is the high difficulty to select appropriate parameters to satisfy all the conditions. In such cases, the uncertainty of a system and the incompleteness of information could be related using grey relational analysis. Discrete data and multi-variable input can be processed for uncertainty and can generate discrete sequences for the correlation analysis of such sequences [20]. Therefore, it could be defined as a measurement method that discusses the consistency of an uncertain discrete sequence and its target. The present work reports multi-response optimization using GRA in the pretreatment of real olive mill wastewater through the coagulation-flocculation process by alum. The coagulation-flocculation method showed high removal rates of the targeted pollutants (TOC, turbidity, COD and total phenols). The experiments were performed by varying the coagulant dosages gradually from a low (100 mg/L) to a high amount (2000 mg/L), and the selection of the optimum amount of coagulant dosage was carried out using grey relational analysis. There is no literature available, so GRA is introduced for optimization of the coagulation and flocculation process to treat OMW. It was confirmed that GRA analysis is a potential and innovative statistical method to identify the optimal operating conditions without any errors.