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In this paper, two natural production inventory models based on fuzzy total production inventory cost with the preference of a decision maker are introduced, and combined by natural number parameters in which values are linguistic values in natural language, crisp real number variables, and fuzzy number variables. These are the one natural production inventory model for crisp production quantity, and the other natural production inventory model for fuzzy production quantity. The natural arithmetical operations of both natural numbers and fuzzy numbers by Function Principle are used to compute fuzzy total production inventory cost of each natural production inventory model. Graded k-preference integration representation method is discussed for defuzzifying fuzzy total production inventory cost by preference of decision maker. Furthermore, Extension of the Lagrangean method is used to solve inequality constrain problem in our natural production inventory model. We find out that our optimal solutions can be specified to the classical production inventory model when preference 0.5 of decision maker and natural numbers and fuzzy numbers in our models are crisp real numbers.

The work presented in this paper has been motivated primarily by Zadeh's idea of linguistic variables intended to provide rigorous mathematical modeling of natural language and CWW, Computing With Words. This paper reports some modeling of the linguistic inventory problems where CWW have been implemented: linguistic production inventory, linguistic inventory models under linguistic demand and linguistic lead time, linguistic production inventory models based on the preference of a decision maker, and linguistic inventory model for fuzzy reorder point and fuzzy safety stock. Only studies that focus on CWW, two linguistic inventory models and two linguistic backorder inventory models, which each model is combined by the heuristic fuzzy total inventory cost based on the preference of a decision maker, are proposed in this paper. The heuristic fuzzy total inventory cost of each model is modeled by linguistic values in natural language, fuzzy numbers, and crisp real numbers. It is also computed and defuzzified by using some fuzzy arithmetical operations by Function Principle and Graded k-preference integration representation method, respectively. In addition, Extension of the LaGrangean method is used for solving inequality constrain problem in the proposed linguistic inventory environments. Furthermore, we find that our heuristic optimal solutions of the new introduced modeling the linguistic inventory problems can also be specified to meet the classical inventory models, when all linguistic variables are crisp real numbers, such as the previous proposed linguistic inventory models.