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Modern Geomatics Technologies and Applications

multi-discrete. For example, the binary model is a discrete two-state model in which each cell can be defined as one of two states
at any given time and as an example for multi-continuous models. Also, it can have length (one dimension), length and width
(two dimensions) and length and width and depth (three dimensions) or in higher dimensions [14]. There are basic concepts in
automatic cell modelling, which include cells, networks, neighbourhoods, and rules.

4.3. Markov chain

It is a random process in which the transition from one state to another takes place in its random variables. Due to Markov's
feature, the next state of a variable depends only on the current state of that variable and does not depend on previous events.
The Markov chain is one of the memorable methods, and the length of the memory by which the possible values for the next
state are calculated is called the Markov order. In modelling where first-order Markov is used, the probability of a one-step state
change based on conditional probabilities is defined as follows[11][15][16] .

( → ) = [ = | −1 = ] (3)

The above equation expresses the possibility of changing the one-step state from time t-1 to time t. The probability of a
one-step state changing is very important in the Markov chain, and it is a very useful tool for presenting the probabilities of
changing the state of a Markov chain and it is equal to the probability of transfer from i mode to j mode [17]. The Markov chain
consists of three matrices, expressed in the following equations [18].

× = +1 (4)

where, the is the transfer matrix and and +1 are the pixels of each user at two times t and t+1.

 LC FF LC FB LC FP LC FO   F   F 
t
t 
1
 LC LC LC LC   B   B 

 BF BB BP BO   t    t  1  (5)
 LC LC LC LC   P   P 
 PF PB PP PO   t   t  1 
 LC OF LC OB LC OP LC OO   O t   O t  1

where F indicates a land use of agricultural lands and B indicates the use of construction and P indicates the use of trees and
parks and O indicates the use of wastelands and indicates the possibility of change from land use B in time t to F in time

t+1. Also, , , , and represents the number of pixels in F, B, P, and O at time t.

4.4. Multi Objective Land Allocation (MOLA)

To model multiple cell status changes, we need a way to identify the right areas for each cell to allocate. In this regard,
the MOLA method was used to assign a specific condition to the target cell. This is especially true when decisions need to be
made based on more than one situation, which may be conflicting. In other words, the MOLA method is a decision-making
support procedure whose goal is to create an optimal solution in allocating space to multiple and inconsistent situations, that is,
for one cell, more than one expected goal [19]. Given the problem in this study, which is expected in the situation for cells,
allocating several land targets will solve the task of solving the problem of assigning a specific situation to cells.
The MOLA method uses the concept of distance to the ideal point to solve the user problem in disputed areas. This method
considers an ideal point for each objective function, which indicates the most appropriate point for that goal and the most
inappropriate point for the other goal. The ideal point is generated by re-classify ordering. In this way, pixels are classified in
the order of competence, and pixels with the highest competence are considered as the ideal point. The decision line is then
drawn based on the logic of the shortest distance to the ideal point. The concept of distance here is not the Euclidean distance,
but the difference in the value, e.g. the grey level, of each pixel with the ideal point value. According to Fig. 5, this line divides
the decision space into two parts. the areas below the decision line in the disputed area belong to purpose 1 and the areas above
the decision line in the disputed area belong to purpose 2 [20][21].

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