In this paper

In this paper, the mathematical model is developed with the following assumptions
The planning horizon is finite.
Single item inventory control.
Demand and deterioration and break-ability rates are constant.
Production rate is constant.
Deterioration and break-ability occur as soon as the items are received into inventory.
There is no replacement or repair of breakable items during the period under consideration.
The shortage is not allowed.
The inflation rate is constant and the time value of money is considered.
The lead time is zero.
The inventory level at the end of the planning horizon will be zero.
The cost factors are deterministic.
The number of replenishment is restricted to one integer.
The total relevant cost consists of fixed ordering, purchasing, holding interest payable, interest earned from sales revenue during the permissible period.
The last order is only being placed to satisfy the balance in the stock of the last period.

2.3. Notation

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C_P = The present value of holding cost during the first replenishment cycle.
Q = The order quantity in the replenishment.
TC_A = The total fixed ordering cost (0, b).
TC_h = The total holding cost (0, b)
TC_P = The total purchasing cost (0, b).
TC_s = The total shortage cost (0, b).
TC = The total relevant cost (0, b).

2.4. Parameters
The mathematical model is representing the following parameters
A = The fixed ordering cost per replenishment, $order.
C = The unit purchasing price at time zero, $order.
C (t) = The unit purchasing price at time t, C(t)=Ce^(-RT).
D = The constant demand rate per unit time.
p = The production rate per cycle.
b = The length of the finite planning horizon.
i = The constant inflation rate.
I(t) = The inventory level at time t.
I_C = The interest charged per $per year by the supplier.
r = The discount rate representing the time value of money.
R=r-1, representing the net constant discount rate of inflation.
T = The length of each replenishment cycle.
T_j = The total time that elapsed up to, including interest charges.
? t?_j = The time at which the inventory level in the j^th
V = The unit selling price at time t.
C_S=The shortage cost pre-unit time ,$unitunit time
C_S (t)=The shortage cost pre-unit time ,$unitunit time at time t ,C_s (t)=C_s e^(-RT).
V(t) = The selling price per unit at time t, V(t)= Ve^(-RT)
? = The constant deterioration rate, units/unit time.
? = The constant break-ability rate, units/unit time.

3. Mathematical model
Let is the inventory level at any time t, Depletion due to demand and deterioration, break-ability will occur simultaneously. The first order differential equation that describes the instantaneous state of over the open interval (0, b) is given by.
(dI(t))/dt+?I(t)+?I(t)=-(P-D),0?t?t_1,0???1,0???1, P>D (1)

In this paper, we are going to thoroughly discuss about the alternative roles of a Monarch and a President. The reason for choosing this particular topic was that it takes into account the distinctive nature of the Head of States in various parts of the world. Some countries are ruled by the Monarch while the others are governed and administered by the President or Republic, whereby elected representatives hold the public offices and possess supreme power over all the others legislative policies, hold certain executive powers and might as well have control over certain judicial functions. An example of which could be United States of America that is ruled by the Presidential system. On the contrary, Saudi Arabia is an example of a country that is ruled by a Monarchy. In spite of the fact that individuals realize that they are different and distinctive, just a couple can tell precisely as to how and why they are different. For that reason, we are going to look into this matter by analyzing the differences through the spheres of power structures, tenure and legislation. We will try to cover every aspect this topic entails and analyze it to the best of our abilities.
Although, a Monarchy and a President are both regarded as forms of government as well as political systems, the two, they for the most part have a tendency to differ. The first difference one could talk about is the power that they possess in terms of the political base and political structure. A monarchy is a form of government whereby citizens are ruled by a king, who often claims to hold absolute power by the Divine Right or in other words, the Will of God whereas in a presidential system the government is elected by the general public through the process of voting and fair elections. In countries like, Thailand, for instance, the king is regarded as the Head of State, who is enthroned in a position of reverence, is respected by all and cannot be violated or defied. No one can expose the King to any sort of accusation or other royalties for that matter. On the other hand, Presidents, the chief executive and the ceremonial Head of State, can be chosen either directly by the people or indirectly by representatives of those elected by the people. When the president is elected indirectly, then we can call the regime a parliamentary republic. Indirectly elected presidents are selected by an electoral college. For example, in Hungary, the President is elected by members of the House of Parliament, and the National Assembly. However, the President of the United States or any other country for that matter would be likely to be impeached, followed by a trial, and, upon conviction of bribery, treason or committing other higher alleged crimes or misdemeanors, could be removed from the office legally; and would afterwards face the consequences and would then be liable to prosecution and punishment in the ordinary course of law. (Cole, J.P; Garvey, T.)
The second difference, which lies in these particular political systems, is the time period that they last to rule over their subjects. In a Monarchy, the crown is hereditary, the concept of Divine Rule that is the royal and political legitimacy is articulated here and the fact that it is passed down from one generation to another, whereas in a Presidential system, citizens of the country vote their representatives democratically through free and fair elections. In addition to that, a Monarchy allows for a life time rule through the passing of the crown down the family line contrary to a President, where government representatives are allowed to serve for a stipulated duration of time (term) for four to five years after which elections are held again. ( Rispa Akello)
Another difference that lies ahead is that of the legislation. A Monarch has the power to make, amend and repeal laws or bills. He might have advisors but it is still the King that interprets and implements a certain rule of law. However, it is not the same for the President. The President cannot enforce the laws or propose the bills, but he can veto it and prevent its adoption. The power lays with the Monarch for instance, in Saudi Arabia, the king drafts out the law and order based on Islamic Shariah, he decides on behalf of the citizen’s but with the President, the power to make laws lies with the Senate or the Parliament.
Over time, there has been a shift from monarchies to republics and, within republics, from parliamentary republics to semi-presidential and presidential regimes (Elgie, 2012).
One of the foremost exceptional systems found by the humans is the organization of the human society. This particular notion has its roots within the Darwinian theories of origin and evolution of the species that also embody human beings and therefore the evolution of the society as well (Claeys, Gregory 2000). The mere idea of man being a “social animal” tends to look at the humans as creatures that are highly dominated by the conception of animal instincts and desires, while also interested and inclined to create societies and live sociably. The development and the evolution of these societies, with all the intellectual activities too, converges at the ultimate purpose of satisfying carnal desires, albeit at anotherlevel.
In the primitive time period, humans were considered to be the hunter gathers or scavengers but as soon as they understood the environment around them they sought out in search for places to settle down. As time passed by, they realized the complexity of the world. That is how the governing system has come into existence, but in the set of governing being developed which one was more suitable for mankind to have? A Monarch or a President?
With this in mind, we realize that with the development of the society emerged clans often associated with the Monarchy. In this particular political system, a single ruler is the king. He manages all the affairs of the country from religion to education. The positive aspect of this is that the decisions are made swiftly since it does not go through different branches of the government and the power to propose laws lies with a single entity. However, due to growing gap between the subjects and the rulers and the issue of pride and hereditary rule, people revolted against the monarch and then emerged the concept of democratic form of government where the President ruled over the people. The positive aspect of this side is that it gives significance to equality. Equal opportunity for work, education and other social interactions amongst the people in the society is appreciated. However, due to establishment of different branches the decision making process is quite slow therefore the chances of corruption are high and conflicts between different opinions arises.
The Monarchy has lately been on the wane, in amidst of it emerged the constitutional monarchy whereby the king or queen sits back and a prime minister runs the affairs of the country. The Monarch has to stay neutral in other words they are apolitical. However, it is still a throwback to this era, which means it is seen as an elitist family in a position of privileged success that they did not work to earn. For this matter, we believe that a presidential form of government would help counter all the hassles that reside with the existence of monarchy and absolute power.
The President has been granted certain powers such as negotiating and signing treaties with foreign countries rather foreign diplomats with the consent of the Congress and has the power to veto laws as well. In addition to that, they also have the power to appoint ambassadors, the cabinet and federal judges. For instance, USA’s President George H.W. Bush nominated Clarence Thomas for the Supreme Court. Other constitutional powers include them to command the military, i.e. to commit troops as the Commander in Chief of all American military forces; the navy, the army, and the air force.
Alongside certain formal powers, the US president has multiple informal powers, powers which are not often enumerated in the Constitution, which by and large, in general may lead to his advantage when compared with constitutional monarchy. The President has the ability or rather the advantage of passing and carrying out official executive orders, makes executive agreements or in other words has an executive privilege, plans and devises agendas, sends out troops to protect the national interests without the declaration of war, as a crisis manager, and has an access to media or bully pulpit.
To conclude, which type of government is better than the other, in my opinion, the presidential system outweighs the monarchical form of government. Different countries aspire into different forms of governance based on their political values which are further based on certain democratic grounds in order to obtain national security, peace and prosperity within a nation. For instance, USA is a very stable Presidential democracy where power is shared at the federal level between the President (the executive body), the Congress (the legislative body) and the Supreme Court (the judicial body).

In this paper, the deterioration, breakability represented in the proposed model at the same time to reduce the total cost of eggs items and assumed the daily demand is constant, developed the mathematical model by the above assumptions and apply it in real life-based in the collected data. The output of the mathematical model is minimizing the total cost of eggs items and obtaining the economic order quantity for several breakable rate values which is including the calculated deterioration, breakable rates for eggs items. The proposed model is having two cases first case. The model is not fitted with less deterioration, breakability rates as percentage if and only if the percentage for deteriorating rate lies in the range 0%, 0.75 %, the percentage for breakability rate lies in the range 0%, 0.75% at same time the obtained economic order quantity is not satisfied the obtaining optimal total cost . The second case model is fitted when the risk of deterioration, breakability are high with a low net constant discount rate of inflation as 50 %, that makes it more significance in real life with high risk for the quality of inventory in the stock. The economic order quantity was decreasing when the deterioration, breakability rates increased so that the total cost of eggs is decreasing because the order economic quantity decreased. The difference between the total costs under assumptions of the mathematical model is increasing where deterioration, breakability rates were decreasing as the following figures. The output of the inventory model showed the optimal total cost, economic order quantity for the collected data, first cycle with 0.0000225 as deterioration rate, 0.0000448 as breakability rate obtained in maximum value with low risk level as percentages of deterioration, breakability rates lie between 0%, 0.75% respectively, which it is not satisfied the optimal total cost whereas it would be at minimum value when risk was increasing which is satisfied the optimal total cost, when the percentages of each deterioration, breakability are more than 0.75% respectively. The impression about the output of the proposed model for the tiles items is the committed policy of imported industry of eggs at the Republic of Yemen is useful with risk level as percentage of deterioration is more than 0.75%, percentage of breakability is more than 0.75% of the inventory of eggs in stock with 50% as representing the net discount rate of inflation. The optimal quantity was decreasing when optimal of first time run decreased


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