This paper explores inventory management issues for two mutually complementary products. The analysis begins by examining these products in a monopoly market and then extends to a perfectly competitive market. Using the concept of fuzzy sets, the paper investigates methods for determining the optimal ordering policy that minimizes total costs. Key findings are presented, along with numerical examples to clarify and illustrate these results. Additionally, the study compares its findings with previous research that did not use fuzzy logic to estimate total costs in a fuzzy context.

The main objective of the study is:

1. Study fuzzy set theory.
2. Mathematical modeling of the inventory problems for two mutually complementary merchandise.
3. To develop an inventory model with a fuzzy approach.
4. Develop fuzzy mathematical formulation.
5. Derivation of solution of the system under fuzzy approach.
6. Optimize the solution with a graded mean integration method
7. Minimize the total cost.

The researchers applied the fuzzy sets concept to deal with inventory problems. Zadeh, et al., (1965) developed the fuzzy set theory which is used in several research in different fields like mathematics, and statistics, used in the programming of vehicles and video games, robotics and driverless vehicles, etc. In this article, the fuzzy theory is used in inventory models. With the help of fuzzy numbers, we can control the vagueness of highly fluctuated demand factors. Petrovic and Sweeney [6] fuzzified the demand, lead time, and inventory level into triangular fuzzy numbers in an inventory control model, then decided the order quantity using fuzzy propositions. Chen and Wang [2] fuzzified the ordering cost, inventory cost, and backorder cost into trapezoidal fuzzy numbers and used the functional principle to obtain the estimate of total cost in the fuzzy sense. Vujosevic et al. [14] fuzzified the ordering cost into a trapezoidal fuzzy number in the total cost of an inventory without a backorder model and obtained the fuzzy total cost. Then they did the defuzzification by using centroid and gained the total cost in the fuzzy sense. Gen et al. [3] considered the fuzzy input data expressed by fuzzy numbers, where the interval mean value concept is used to help solve the problem. Roy and Maiti [7] then rewrote the problem of classic economic order quantity into a form of a nonlinear programming problem. They introduced fuzziness both in the objective function and storage area. It was solved by fuzzy both nonlinear and geometric programming techniques for linear membership functions. Ishii and Konno [4] fuzzified the shortage cost into  shape fuzzy number in a classical newsboy problem aimed to find an optimal ordering quantity in the sense of fuzzy ordering. Lee and Yao [5] and Yao and Lee [9] proposed the inventory without backorder models in the fuzzy sense, where the order quantity q is fuzzified as the triangular fuzzy number [5] and as the trapezoid fuzzy number [9]. Yao et al. [10] further presented the fuzzy inventory without backorder model in which both the order quantity q and the total demand R are fuzzified as triangular fuzzy numbers. We note that the above papers deal with inventory problems in the fuzzy sense, specifically, for a single merchandise. Rajput et al. (2019) proposed an inventory model with a different type of demand function and discussed the importance of fuzzy parameters in healthcare industries. They used triangular fuzzy number for the demand and defuzzified the model with the signed distance method and get the maximum profit for all three models. Rajput et al.(2021) presented an EOQ model for deteriorating items under inflation, this article derived a classical and fuzzy model for the effects of deterioration and inflation without any shortage. They used a pentagonal fuzzy number and then optimized the total inventory cost with the GMI defuzzification method.

On the other hand, Yao and Wu [11] discussed the best prices for two mutually complementary merchandise in a monopoly market and a perfect competitive market, such that the total revenue is maximum. However, in [11] the inventory problem of these two merchandise is not taken into account.

This paper addresses the issues of pricing and inventory management for two mutually complementary products within a fuzzy framework, building on the concepts employed in previous studies. Using the concept of fuzzy sets, we explore how to determine the optimal ordering policy for the given inventory problem to minimize total costs. The results are derived, and numerical examples are provided to illustrate the findings. The graded mean integration method is applied to estimate the total cost in the fuzzy sense.

This article is organized as follows. In Section 2, some definitions and properties of fuzzy sets, which will be needed later, are introduced. Section 3 explores the inventory problem for two mutually complementary merchandise. Specifically, in Section 3.1, we consider the crisp case and obtain the optimal order quantities for the merchandise in a monopoly market. In Section 3.2, for a perfectly competitive market, since the prices of two replaceable merchandise may vary a little, so we fuzzify the prices, P_i,i = 1,2, to be the fuzzy numbers and obtain the optimal ordering policies for two merchandise. Section 4 fuzzifies both the order quantities q_i,i = 1,2, and prices P_i,i = 1,2, to be fuzzy numbers, and and defuzzy by the graded mean integration method. Sections 4 and 5 are the result and discussion.

This paper addresses the pricing and inventory challenges for two mutually complementary products. For products in a monopoly market, we develop a clear total cost function and determine the optimal order quantity by minimizing the total cost. In a perfectly competitive market, we first fuzzify the prices of the two products as fuzzy numbers and then fuzzify the quantities received as fuzzy numbers. In these two fuzzy scenarios, we introduce the graded mean integration method to estimate the total cost in a fuzzy context.

Previous research on single-product inventory models, with or without backorders, has modified these models by fuzzifying the order quantity and total demand quantity. We demonstrate that the result obtained by using the graded mean integration method to denazify the fuzzy number is more accurate than the corresponding crisp value.

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