**4. Problem Description**

The problem of the proposed model has two parts: (i) the mixing problem and (ii) the production inventory problem. In the mixing problem, the process of mixing takes place on an instrument made by two containers (Figure 2). In this instrument, container-I is connected to container-II by a pipe line so that the liquid can pass from container-I to container-II, and vice versa. Initially, liquids of two different concentrations (*η* and *k*) are taken to make the initial mixture. During mixing, the liquid with a concentration *η* is passed through container-I at the rate *α* and then from container-I to container-II with the rate *β*. Again, the mixed liquid is returned back from container-II to container-I with the rate *γ*, and this process is continued to obtain the desired mixed liquid. Finally, the desired mixed liquid exits from container-II at the rate *δ*. The entire process of mixing is presented in Figure 2. Then, in the part of production process, the desired mixture is taken as a raw material and a single product is produced at the production rate *P*(*t*) *P*(*t*) = *<sup>δ</sup> <sup>B</sup> y*(*t*) . During the production period, owing to the customers' demand, the produced product is stored with the rate (*P* − *D*) per unit time and the level of inventory reaches its pick level at time *t* = *t*1. After that, the level of stock gradually decreases because of fulfilling the demand of the customer and the stock level reaches zero at time *t* = *T*. The variation in the level of inventory at any time *t* is shown in Figure 3.

**Figure 2.** Representation of the mixing procedure in the production process.

**Figure 3.** Changes in inventory level with respect to time.

#### **5. Mathematical Formulation**

Here, we have discussed the mathematical formulation of mixing and the production inventory system.
