Sheikh `Attiyah Saqr, former head of Al-Azhar Fatwa Committee, states that: “In fact, there is a great Divine wisdom behind the permissibility of marrying more than one wife. According to the dictate of man’s nature, his sexual desire may reach the stage that one wife may not be sufficient for him. As for woman, there are situations and natural phenomena that control and restrict her sexual desire; part of this is the nursing period, menstruation and in many other things that are pertaining to her nature.
In addition, if we are to allow such a degradation of allowing woman to marry more than a husband at one time, she will be no more than a mere object of sexual satisfaction. What makes the matter worse is the fact that conflict and dispute over the child lineage will emerge, and this will result in a total chaos and sexual deviation.
Viewed in another perspectives, polygyny is, in fact, a thing in favour of women themselves. If it happens that a wife becomes invalid or even disinterested in sexual intercourse, how does she expect her husband to behave? Also, if we suppose that a woman is infertile and her husband has a very great desire for having children, what should he do? Should he resort to adoption or engage in some illicit affairs? The best way to solve the problem is to seek a second wife that helps the first one and preserves the husband from resorting to deviation. Indeed, the society will be safe and secure, and both parties will experience a happy marital life.”
In addition, we would like to cite for you the following article on polygyny and polyandry; it reads:
“First, we’d address the issue of men having four wives before answering the question as to why women cannot have multiple husbands.
I think that among many other misconceptions that people have about Islam is their belief that Islam unconditionally allows a Muslim to have four wives. In this regard, some scholars have gone as far as to say that keeping four wives is a man’s essential physiological and psychological need. I am afraid that this point of view is in direct contradiction with the Qur’an. It is, in fact, a distortion of the stance of Islam.
According to the Qur’an, in normal circumstances, a family comes into being only through wedlock between man and woman. A subtle reference to this is made by the Qur’an in the verse that reads: “ O mankind! Be careful of your duty to your Lord Who created you from a single soul and from it created its mate and from them twain hath spread abroad a multitude of men and women.”(An-Nisa’: 1) where it alludes to the fact that when Almighty Allah created Adam, He created for him Eve as his wife.
Naturally, had Almighty Allah wished that a man should have more than one wife, He would have created more wives for Adam instead of just one. This shows us that as far as a man’s physiological and psychological needs are concerned, they are completely satisfied even if he has a single wife.
Let us now take a look at the verses, generally thought to be the basis of polygyny; But we’ll firstly make a brief word about their background.
A greater part of surat An-Nisa’, in which these verses occur, deals with the society and its reformation. In this regard, the foremost sphere in which directives were given was that of the welfare of the orphans since they are one of the weakest sections of society.
In Madinah, there came a time when many Muslims were martyred in various battles. As a result, many children were orphaned. In this situation, Almighty Allah called upon Muslims to look after these children and guard their wealth and property. In this regard, the already in-practice custom of polygyny in the Arabian society was resorted to. Muslims were told that if they were afraid that they would not be able to take proper care of these orphans, they should marry their widowed mothers.
We will now take a look at the following Qur’anic verse: “If you fear that you would not be able to deal justly with the orphans, marry two, three or four of the women who are lawful for you. But if you fear that you would not be able to deal justly [with them] then [restrict yourself to] one only.” (An-Nisa’: 3)

Three implications of this verse are very clear:
Firstly, polygyny is related to some social need.
Secondly, the number of wives should not exceed four in any circumstances.
Thirdly, if a person cannot maintain balance and do justice to his wives, he must restrict himself to one.
In other words, while in normal circumstances, a family comes into being through the union of a single man and woman, there may be certain exceptional circumstances in which the practice of polygyny may be benefited from with the restrictions mentioned in the above-mentioned verse.
For example, in our society, many young widows and divorcees with small kids experience a life full of misery with no one around to take them as wives. Such widows and children can lead normal lives if the dispensation of polygyny is benefited from. Similarly, many young women who, after embracing Islam, have been abandoned by their non-Muslim husbands need Muslims to marry them. So, a number of problems can be solved by using this permission, which would have been impossible to overcome had Islam totally forbidden polygyny.

With this background, we will now come to your question regarding polyandry (woman having many husbands at the same time). It is common sense that if a family is to come into being, not only should there be only one head but also one person should not be placed under the command of multiple heads, otherwise, great anarchy would result.
Since, in Muslim family, husbands takes the horn of leadership, if a wife has more than one husband, she would be placed under the authority of many husbands at the same time. This of course would only hasten to tear apart the fabric of a family unit. Furthermore, the lineage of the child borne by a woman having more than one husband cannot be ascertained. How would the father be ascertained?
Someone might say that a DNA test would be able to do so. However, even if this test is used, there still are great chances of dissent between all the husbands with one claiming to be the father and the other denying him. Owing to these reasons, Islam makes it forbidden for a woman to be married to more than one husband at a time.”