Inventory control and its optimal solution is concerned with achieving a balance between two opposing objectives: minimizing the cost of holding inventory, and maximizing service to customers. The company can minimize Inventory costs by maintaining zero inventories. However, customer service may suffer, and customers may decide to take their needs elsewhere. This has a cost, which

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They employ Iterated Local Search (ILS) and Simulated Annealing (SA) to minimize the annual total cost. Torres et al. [15] presented a new supplier selection model to determine safety stocks based on the supplier delivery reliability using differential calculus. Guchhait et al. [16] established damageable items production inventory models with variable demands and inventory costs depends on reliability in an imperfect production process. using Euler–Lagrange function based on variational calculus and Newton–Raphson method to optimize the model and finding maximum total profit. Paul et al. [17] introduced a real time recovery plan from production disruptions either single or series of disruptions for a two-stage and single item batch production inventory system with reliability considerations. Using a pattern search and genetic algorithm to maximize the total profit function, which show that the pattern search is better. Lin and Srivastava [18] developed a new two-warehouse inventory model with quantity discounts and maintenance actions under an imperfect production process. The objective is to minimize the total expected cost per unit time. An efficient algorithm was developed to help the manager in accurately and quickly determining the order policy. Taleizadeh et al. [19] developed economic production quantity (EPQ) model with random defective

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[26] proposed an economic order quantity (EOQ) for items with imperfect quality similar to Salameh and Jaber [3] a screening process is adopted considering the errors in inspection and the defective items are sold at a discounted price. The objective is to maximize the total profit and the model is solved analytically. Roy et al. [27] present an EOQ model where imperfect quality items take place in each ordered lot and partial back ordering is done. The objective is to determine the optimum values of lot size and shortage period, which maximizes the expected average profit. Shah and Soni [28] proposed a multi-objective production inventory model with backorder for fuzzy random demand under flexibility and reliability of production process, the objective is to maximize the total expected profit incurred in each production cycle which is optimized using a multi-objective genetic algorithm (MOGA). Tripathy and Pattnaik [29] developed a model in a more general way to the work of Cheng [2], Tripathy et al. [30], Tripathy and Pattnaik [31], and assuming that demand exceeds supply, but the unit cost of production is inversely related to process reliability and directly related to the demand rate by a power function. Numerical example gives that this situation makes saving in the unit production cost than proposed by Tripathy et al. [30], and Sensitivity analysis, are performed to his model to find the effect of the constants of the power function on the unit cost of