Abstract
This paper develops an economic production quantity model for a multi-product multi-objective inventory control problem with fuzzy-stochastic demand and backorders. In this model, the annual demand is represented by trapezoidal fuzzy random numbers. The centroid defuzzification and the expected value methods are applied to defuzzify and make decisions in a random environment. In the case where the warehouse space is limited, the Lagrangian relaxation procedure is first employed to determine the optimal order and the maximum backorder quantities of the products such that the total inventory cost is minimized. The optimal solution obtained by the proposed approach is compared with that obtained by the traditional deterministic method. Moreover, a sensitivity analysis presents the rationality of the solution. Then, the model is extended to a multi-objective integer programming problem in which the optimal numbers of redundant production machines are determined to maximize the production system reliability. In the second proposed model, several constraints are considered to fit real-world situations. As the second model is developed for an NP-hard problem and hence cannot be solved using exact methods in a reasonable computational time, a multi-objective evolutionary algorithm called non-dominated sorting genetic algorithm-II (NSGA-II) is employed to provide Pareto front solutions. Due to non-availability of benchmark in the literature, another multi-objective evolutionary algorithm called non-dominated ranking genetic algorithm is implemented as well to validate the obtained results and evaluate the performance of NSGA-II. In addition, the Taguchi method is used to calibrate the parameters of both algorithms for better performance.
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Appendix
Appendix
Definition 1
A number referring to several values with varying weights between 0 and 1 are the fuzzy number, for which the weight is called degrees of membership [50].
Thus, the membership function for a trapezoidal fuzzy number defined by four parameters {a, b, c, d}, is
where \(0 \le \mu_{\,T} (x) \le 1\) is the membership degree of x in the fuzzy set \(\tilde{T}\) and x is fuzzy member. Figure 8 illustrates a trapezoidal fuzzy number.
Definition 2
Transferring a fuzzy number to a number is called defuzzification, for which the centroid method (or center of gravity) is a common defuzzification technique [18]. The following algebraic expression illustrates how the centroid method defuzzifies a fuzzy number.
where \(\smallint\) denotes an algebraic integration. Figure 9 shows a graphical representation.
Definition 3
A FRV (\(\hat{T}\)) is a random variable that takes fuzzy values [19].
Note 1 To make a FRV non-random, the expected value method can be applied so that the expectation of \(\hat{T}\) is obtained by
If \(\hat{T}\) is a discrete FRV, assuming \(P(\hat{T} = \tilde{t}) = p_{i} ,i = 1,2,3, \ldots ,z\), its fuzzy expectation is written as [51]
Definition 4
\(\hat{D}_{i}\) is a trapezoidal FRV satisfying the following equation [19].
where z is number of fuzzy sets for random variable with j = 1, 2, …, z and \(\sum\nolimits_{j = 1}^{z} {\Pr_{\text{i}}^{\text{j}} = 1}\).
Simple application Considering two TrFRVs \(\hat{D}_{1}\) and \(\hat{D}_{2}\) with parameters greater than zero [52]:
The basic arithmetic operations and their expected value are as follows.
The product and the division of two trapezoidal fuzzy numbers \(\hat{D}_{1}\) and \(\hat{D}_{2}\) are as follows.
As Eqs. (38, 39) do not exactly result in trapezoidal fuzzy numbers, some equivalence relations have been introduced in the literature to obtain trapezoidal fuzzy numbers [53]. This paper employs the equivalence relation in [54] for the division and the product of two trapezoidal fuzzy numbers as:
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Sadeghi, J., Niaki, S.T.A., Malekian, M.R. et al. A Lagrangian Relaxation for a Fuzzy Random EPQ Problem with Shortages and Redundancy Allocation: Two Tuned Meta-heuristics. Int. J. Fuzzy Syst. 20, 515–533 (2018). https://doi.org/10.1007/s40815-017-0377-z
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DOI: https://doi.org/10.1007/s40815-017-0377-z