Proportional Fair Division of Multi-layered Cakes

In our daily lives, there are many examples of time scheduling whereby we set our time in such a manner that we can do our daily essential work. Consider a group of students of a university who want to use multiple facilities, such as a seminar room or an indoor games room. The start and closed times of the two facilities are the same. Each student of the group has a different preferred time duration to get each facility room and everyone is also willing to take both facilities. The problem of fair division of a divisible heterogenous resource among different agents with their different preferences over divisible resource has been studied in the classical cake cutting model (Steinhaus [1949],Brams and Jones [2006],Procaccia [2013],Kurokawa et al. [2013]). We assume a cake as the unit interval [0, 1] that is divided among agents where each agents has different perference over the entire cake. The concept of fair division was given by the three mathematicians, Hugo Steinhaus, Bronis law Knaster and Stefan Banach to a meeting in the Scottish Cafe in Lvov (in Poland) to the end of the world war II.