# Geometric Programming Dealing With Neutrosophic Relational Equations Under the (Max-Min) Operation

Huda E. Khalid (Department of Scientific Affairs and Cultural Relations, Telafer University, Iraq)
DOI: 10.4018/978-1-7998-2555-5.ch004
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## Abstract

The neutrosophic relation equations are important elements of neutrosophic mathematics, and it can be widely applied in power systems, neutrosophic comprehensive evaluation. The aim of this chapter is to find the minimal solutions for the neutrosophic relation geometric programming having (V, Λ) operator. In this chapter, a max-min method has been built for finding an optimal solution in the neutrosophic relation equations, and a new characteristic matrix has been defined which is an important step to test the consistency of the system Aοx=b and for finding all effective paths that lead to the set of all quasi-minimum solutions. The gained results are reasonable and harmonized with those results in Khalid's work. The method overcomes the problems of poor convergence efficiency inherited from the stochastic hill-climbing method or genetic algorithms. The suggested algorithm has the intersection column method which was proposed to find the effective paths that neglect the fallacious paths, and two new theorems were presented to deal with the optimal solution for (NREGP).
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## 1. The Neutrality Concept Seeds At The Thought Of Ancient Arab Scholars

### 1.1. Neutrality in the Idiomatic Concept (Al-Baghdadi, 1928)

Neutrality is an expression that joins many fields: political, economic, social religious and scientific. Opinions and concepts varied according to each jurisdiction. Therefore, the Arab ancient scholars defined it as follow: If meant are individuals, then it means that the persons who adopt the average attitude between two conflicting parties and did not tend to one of the sides without the other. As an example the neutral countries are Not- Aligned to any of the Warring States, also the neutral school is against the sectarian school.

Neutrality in the conventional Concept, if the neutral has been fired to the one of the basic science, as an example in physics, it will be meaning that particle does not hold a negative or positive charge, while in the chemistry science the neutral matter means neither acid material nor basal.

### 1.2. Al-Muʿtazilah and Neutrality (Altai, 2016)

Muʿtazila (in Arabic: المعتزلة‎) is a rationalist school of Islamic theology that flourished in the cities of Basra and Baghdad, both in Iraq, appeared at the beginning of the second century AH when Wāṣil ibn ʿAṭā' (d. 131 AH/748 AD) left the teaching lessons of Hasan al-Basri after a theological dispute regarding the issue of a position between two positions.

The name Muʿtazili is derived from the reflexive of the root “separate, segregate”, “separate to withdraw from”. The name is derived from the founder's school of Al-Muʿtazilah when he withdrew from the studying circle of Hasan Al- Basri the wake of theological disagreement between them, Wāṣil ibn ʿAṭā' asked about the legal state of a person who has committed a serious sin, is he a believer or an unbeliever? Hasan answered the person remains a Muslim. Wasil dissented, suggesting that a sinner was neither a believer nor an unbeliever and withdrew from the study circle. Others followed Hasan to form a new circle, including ʿAmr ibn ʿUbayd. Hasan's remark, “Wāṣil has withdrawn from us”, is said to be the origin of the movement's name.

There are five basic tenets make up the Mu'tazilite creed:

• 1.

Justice and unity (monotheism).

• 2.

The inevitability of the threats and promises of God (or “the warning and the promise”).

• 3.

The intermediary position (i.e. Muslims who die without repentance after committing a grave sin are neither mu'mineen (believers), nor kuffar (non-believers), but in an intermediate position).

• 4.

The injunction of right.

• 5.

Promotion of Virtue and Prevention of Vice.

In this section, we will focus on the third tenet of their creed (Al-manzilah bayna al-manzilatayn) the intermediate position, that is, Muslims who commit grave sins and die without repentance are not considered as mu'mins (believers), nor considered as kafirs (non-believers), but in an intermediate position between the two. The reason behind this that a mu'min is a person who has faith and conviction in and about God, and who has his/her faith reflected in his/her deeds and moral choices. Any lacking on any of these two fronts makes him/ her not a mu'min. On the other hand, one does not become a kafir (i.e. rejecter; non-believer), for this entails, inter alia, denying the Creator's mercy full, something not necessarily done by a committer of a grave sin. The fate of those who commit grave sins and die without repentance is Hell. Hell is not considered a monolithic state of affairs but as encompassing many degrees to accommodate the wide spectrum of human works and choices, and the lack of comprehension associated to the Ultimate Judge (Al-Hakam is one of the other names of God in Islam). Consequently, those in the intermediate position, though in Hell, would have a lesser punishment because of their belief and other good deeds. Al-Muʿtazilah adopted this position as a middle ground between Kharijites and Murjites. In the words of ʿAbd al-Jabbar, the doctrine of the intermediate position is the knowledge that whoever murders, or commit adultery (Zina), or commits serious sins is a grave sinner (fasiq) and not a believer, nor in the same case of believers with respect to praise and attributing greatness, since he has to be cursed and disregarded. Nonetheless, he is not an unbeliever who cannot be buried in our Muslim cemetery or be prayed on him after died, or marry a Muslim. Rather, he has an intermediate position, in contrast to Kharijites sect who say that he is an unbeliever or the Delaying sect (Murjites) who say that he is a believer.

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