*Article* **A Mathematical Model of the Production Inventory Problem for Mixing Liquid Considering Preservation Facility**

**Md Sadikur Rahman 1, Subhajit Das 1, Amalesh Kumar Manna 1, Ali Akbar Shaikh 1, Asoke Kumar Bhunia 1, Leopoldo Eduardo Cárdenas-Barrón 2,\*, Gerardo Treviño-Garza <sup>2</sup> and Armando Céspedes-Mota <sup>2</sup>**


**Abstract:** The mixing process of liquid products is a crucial activity in the industry of essential commodities like, medicine, pesticide, detergent, and so on. So, the mathematical study of the mixing problem is very much important to formulate a production inventory model of such type of items. In this work, the concept of the mixing problem is studied in the branch of production inventory. Here, a production model of mixed liquids with price-dependent demand and a stockdependent production rate is formulated under preservation technology. In the formulation, first of all, the mixing process is presented mathematically with the help of simultaneous differential equations. Then, the mixed liquid produced in the mixing process is taken as a raw material of a manufacturing system. Then, all the cost components and average profit of the system are calculated. Now, the objective is to maximize the corresponding profit maximization problem along with the highly nonlinear objective function. Because of this, the mentioned maximization problem is solved numerically using MATHEMATICA software. In order to justify the validity of the model, two numerical examples are worked out. Finally, to show the impact of inventory parameters on the optimal policy, sensitivity analyses are performed and the obtained results are presented graphically.

**Keywords:** mixing process; simultaneous differential equations; variable production rate; simulated annealing; differential evolution
