**1. Introduction**

The mixing problem has a great impact on different sectors of business management, viz. the medicine industry (Gautam et al. [1], Essi [2], Ploypetchara et al. [3]), cosmetics industry (Bernardo and Saraiva [4], Kim et al. [5], Zhang et al. [6]), chemical industry (Funt [7], Wu et al. [8], Jasikova et al. [9]), and so on, to produce essential commodities in our daily life. Thus, in the area of inventory control, investigation of the production inventory problem of a mixed product along with the mixing process is an intersecting research area. In this connection, Nienow et al. [10], Cheng et al. [11], Fitschen et al. [12], and many others have had a valuable influence in this area. As various inventory parameters like production rate, demand rate, deterioration rate, and preservation technology play a significant role to control a production inventory, researchers should take more care of those inventory parameters in the studying of the production inventory problem with the mixing process.

In production inventory, the production rate of the product is the key parameter that may be constant or dependent on customers' demand/stock level of the product, among others. On the other hand, owing to the failure of machines, sometimes imperfect production occurs during the production process. Thus, imperfect production is also an important factor for production firm/manufacturing firm. Several researchers developed

**Citation:** Rahman, M.S.; Das, S.; Manna, A.K.; Shaikh, A.A.; Bhunia, A.K.; Cárdenas-Barrón, L.E.; Treviño-Garza, G.; Céspedes-Mota, A. A Mathematical Model of the Production Inventory Problem for Mixing Liquid Considering Preservation Facility. *Mathematics* **2021**, *9*, 3166. https://doi.org/ 10.3390/math9243166

Academic Editor: Arsen Palestini

Received: 28 September 2021 Accepted: 8 November 2021 Published: 9 December 2021

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different production models by taking various production rates and imperfect production processes. De [13] analyzed a production problem with a variable rate of production. Su and Lin [14,15] investigated two production inventory models with a demand- as well as inventory-level-dependent production rate. After that, Giri et al. [16] analyzed an unreliable production system with variable production. Roy et al. [17] studied a production inventory model for defective products with rework policy. Considering an imperfect production process, Sana [18] formulated a production inventory model. A few years later, Sharmila and Uthayakumar [19] established the optimal policy of a production problem with three different production rates. Then, Patra and Maity [20] developed a production problem for defective items with a variable production rate. Dey [21] investigated an imperfect production model under an integrated system in an imprecise environment. Succeeding them, Mishra et al. [22] studied the sustainability of a production system under controllable carbon emission. Lu et al. [23] applied the Stackelberg gaming approach to determine the optimal policy of an imperfect production inventory model with collaborative investment policy for reducing emission. Recently, Öztürk et al. [24] studied an imperfect production process with random breakdowns, rework, and inspection costs and Khara et al. [25] formulated an imperfect production model considering advanced payment and trade credit facilities. Beside these, the works of Malik et al. [26], Lin et al. [27], and Rizky et al. [28] are valuable in this area.

Demand of customers is also an important factor in inventory control. It depends on several factors, such as selling price of the product, inventory level, frequency of the advertisements, time, and so on. In reality it is seen that, if the price of a commodity increases, the demand for that commodity must decrease, i.e., the selling-price-dependent demand rate is a decreasing function. On the other side, more customers are attracted because of the large number of items in stock, i.e., the stock-dependent demand rate is an increasing function of the stock level of the items. Sometimes, the customers' demand for a new product increases drastically owing to the advertisement of the product. Thus, advertisement frequency has a great impact on the demand rate. Resh et al. [29] first introduced the variable demand rate (selling-price-dependent) in the area of inventory control and modified Harris's EOQ model. Urban [30] analyzed an inventory model with stock-linked demand. Chang [31] studied a model for optimal lot sizing with a nonlinear stock-linked demand rate. Mukhopadhyay et al. [32] and You [33] studied different types of EOQ models with price-dependent demand. After a few years, Khanra et al. [34] constructed an inventory model with a time-dependent demand rate under trade credit policy. Further, Bhunia and Shaikh [35] studied a deterministic inventory model with pricedependent demand and a three-parameter Weibull distributed deterioration rate. Prasad and Mukherjee [36] proposed an inventory model where the demand rate is connected to stock and time, along with shortages. Manna et al. [37] investigated a production inventory model with imperfect production and advertisement-dependent demand. Jain et al. [38] investigated a fuzzy inventory model where the demand for an item is dependent on time. Recently, the contributions of Alfares and Ghaithan [39], Shaikh et al. [40], Rahman et al. [41], Cardenas-Barron et al. [42], Das et al. [43], Halim et al. [44], Rahman et al. [45], and others on this topic are worth mentioning.

Deterioration is also important in the control of inventory. Most of the commodities in our daily life deteriorate with the passing of time owing to the several factors. Thus, to study an inventory problem for deteriorating items, we cannot avoid the effect of deterioration. Naturally, the deterioration rate of an item cannot be predicted accurately. However, it was taken as constant or time-dependent or probabilistic by several researchers. In their work, for the first time, Ghare and Schrader [46] proposed the concept of deterioration (constant). Then, Emmons [47] proposed the concept of stochastic deterioration with twoparameter Weibull distribution. Since then, a number of research works have been reported in the existing literature. Among those, the works of Datta and Pal [48], Wee [49], Ouyang et al. [50], Min et al. [51], Dash et al. [52], Dutta and Kumar [53], Shah [54], Tiwari et al. [55], Shaw et al. [56], Mashud et al. [57], Khakzad and Gholamian et al. [58], Mishra et al. [59], Khanna and Jaggi [60], and Naik and Shah [61] are worth mentioning.

On the other side, the economy of an industry is badly affected by reckless deterioration. Thus, in the case of more deterioration, the control of deterioration is highly required. Usually, to prevent more deterioration, some policies/techniques are adopted, named preservation policies/technologies. For the first time, Hsu et al. [62] investigated the concept of preservation technology in the area of inventory control. After that, Dye [63] discussed the preservation investment effect on deterioration rate. Zhang et al. [64] solved an inventory problem for perishable goods by considering stock-dependent demand and investment in preservation technology. Yang et al. [65] proposed an inventory model under preservation technology and trade-credit policy. Tayal et al. [66] studied an inventory problem for a perishable product with a permissible delay in payment along with investment in preservation technology. Dhandapanin and Uthayakumar [67] analyzed the optimal policy of a multi-item inventory model under preservation technology. Recently, Shaikh et al. [68], Das et al. [69], Saha et al. [70], Mashud et al. [71], Sepehri et al. [72], and others contributed through their works on preservation technology.

The organization of the paper is according to Figure 1. In this work, a production problem for mixed liquid and price-dependent demand is formulated. In this formulation, at first, the mixing process is presented mathematically by the simultaneous differential equations under some restrictions. Then, the corresponding optimization problem related to this model is obtained as the profit maximization problem. Because of the high nonlinearity of the objective function (average profit), the mentioned maximization problem is solved by differential evolution and simulated annealing in Mathematica software. Then, to investigate the validation of the model, two numerical examples are solved. Finally, sensitivity analyses are performed graphically and this work is concluded with some future scopes. A summary of some of the literature is presented in Table 1.

**Figure 1.** Organization of the paper.


**Table 1.** Summary of the related literature.
