**2. Research Gap and Contributions**

After a brief survey of the literature, it is conjectured that many works have been accomplished on production inventory (Table 1) for different types of products, such as food, electrical goods, garments, medicine, and so on, with various assumptions regarding the production rate, demand rate, deterioration rate, and so forth. On the other hand, the concepts of the mixing problem are essential in the production manufacturing of liquid products (like, medicine, juice, cosmetics, and so on). To the best of our knowledge, few works on the mixing process (viz. Essi [2], Ploypetchara et al. [3], Kim et al. [5], Jasikova et al. [9], and Fitschen et al. [12], among others) are available in the literature. However, very few researchers ([84,85]) considered the combination of the mixing process as well as manufacturing process in his/her work. Though Su et al. [84] accomplished their work on production inventory for mixed products, they did not consider the mathematical formulation of the mixing process.

To fill this gap, a production inventory model for mixed liquid was formulated by defining the mixing process mathematically. Here, the mixing process of liquids is considered as a part of the production process. The mixing process is presented mathematically by simultaneous differential equations. Then, in the manufacturing part of this modelling, the variable production rate (dependent on the stock level of mixed liquid) and preservation technology are considered. The mentionable contributions of this study are as follows:


All of the above represent the novelty of this work.
