Confidence Neutrosophic Number Linear Programming Methods Based on Probability Distributions and Their Applications in Production Planning Problems

Abstract

A neutrosophic number linear programming (NN-LP) method reveals a useful tool for solving optimal production planning problems in an indeterminate environment. The optimal feasible solutions of the decision variables and the objective function in the NN-LP method are obtained only depending on the subjectively specified uncertain range of neutrosophic numbers without considering some distribution and confidence level of product sample data. Due to the lack of some probability distribution and confidence level/interval in indeterminate situations, existing indeterminate optimization methods are difficult to guarantee the reliability and credibility of the optimal interval feasible solutions. To solve these problems, this study proposes confidence neutrosophic number linear programming methods based on some probability distributions to objectively determine the confidence level/interval of a sample dataset/multivalued set in the NN-LP problems and strengthen the reliability and credibility of the optimal interval feasible solutions. Therefore, this article first introduces a transformation method from product sample datasets (multivalued sets) with normal and log-normal distributions to confidence neutrosophic numbers (CNNs) (confidence intervals) from a probabilistic point of view. Then, CNN linear programming (CNN-LP) models and solution methods are proposed according to the normal and log-normal distributions and confidence levels of product sample datasets to obtain the optimal interval feasible solutions. Furthermore, the proposed CNN-LP methods are applied to two real cases in a manufacturing company of Shaoxing City in China to perform the production planning problems under normal and log-normal distributions and 95% CNN/confidence interval of the product sample datasets. Compared with existing related linear programming methods, the proposed CNN-LP methods can make the optimal interval feasible solutions more reasonable and credible/reliable than the existing linear programming methods and overcome the defects of the existing linear programming methods in indeterminate situations.

MAGDM Model Using the Exponential Similarity Measure of Neutrosophic Confidence Cubic Sets in a Single-Valued Neutrosophic Multivalued Circumstance

“…Ten, s δ/2 in equations ( 2)-( 4) is a specifc value related to a confdence level of (1 − δ) × 100%, which is constructed as twosided CIs for the confdence level of (1 − δ) × 100% in the normal distribution situation of fuzzy data. In actual applications, the specifc values of s δ/2 are usually specifed as 1.645, 1.960, and 2.576 [27] subject to the confdence levels of 90%, 95%, and 99% under the normal distribution condition of fuzzy data.…”

“…Since the neutrosophic number (NN) (u � a + λI � [a + λI − , a + λI + ] for an indeterminacy I � [I − , I + ] and a, λ ∈ R) presented by Smarandache [1,25,26] shows the fexible representation merit of indeterminate information subject to diferent indeterminate ranges of I. Recently, from a probability perspective, the notion of a confdence neutrosophic number (CNN) or confdence interval (CI) [27] was presented in terms of the 95% confdence level and the normal and lognormal distributions of multivalued datasets to ensure the 95% confdence level of multivalued datasets falling within the CNN/CI, and then CNN linear programming methods were introduced subject to the confdence level and normal and lognormal distributions to carry out production planning problems in indeterminate scenarios. However, CNNs/CIs are not used for GDM issues in the neutrosophic multivalued setting.…”

“…Since there are the aforementioned insufciencies of the existing transformation techniques [23,24], by the motivation of the CI/CNN notion with some confdence level [27], this paper aims to propose a new transformation technique from SvNMVS to the neutrosophic confdence cubic set (NCCS) that consists of CIs and average values of true, false, and indeterminate fuzzy sequences in SvNMVS and a GDM model based on the exponential similarity measure (ESM) of NCCSs to carry out GDM issues subject to some confdence levels and the normal distribution (the most common distribution in the real world) in a SvNMVS circumstance.…”