Application of CPLEX Solver to Land Use Optimization in Gorgan Township

Although considering a varied mix of goals in environmental planning improves its results, it simultaneously increases the complexity of the process as well. This complexity forces researchers to use methods such as mathematical programming and optimization algorithms. In this context, linear programming as a sub-set of mathematical programming is one of the alternative methods for decision makers. As linear programming is not basically a spatial technique, its use in the context of spatial decision making problems such as land use planning is always considered in conjunction with GIS. This method has some limitations in terms of computational intensity and the time to achieve solution that is increased exponentially with increasing number of decision variables. This is a normal situation in land use planning. This study attempts to solve a land use planning problem with respect to optimization of four land uses: agriculture, forest, rangeland and development areas in Gorgan Township. In this regard, three objectives including minimizing allocating cost, land use conversion cost and maximizing compactness in the form of linear programming problem have been considered.
Introduction:
Inevitably, land use planning is the main clue in the process leading to sustainable development. Every unit of land cannot be allocated to more than one land use simultaneously. In this regard, land use planning defines the proportions and locations of special use for each spatial unit of land. With this definition place becomes an important issue when deciding about the most appropriate land use. It is the reason why we call land use planning a place-based decision making. Usually, in this process, the study area is modeled througha raster layer in which the cells act like land units waiting to be allocated their special uses. The important and crucial criterion in this process is suitability maps produced from overlaying several thematic maps. However, suitability maps are not sufficient for planning and optimization of land use for a region and without integration of other constraints, the result may become a fragmented land uses layer. In this study we attempt to solve a land use planning problem with respect to optimizing four categories including agriculture, forest, rangeland and development areas in GorganTownship. Here we incorporate three objectives including minimizing allocating cost, use conversion cost and maximizing compactness in the form of a linear programming problem.
Methods:
Linear programming is one of the well-known methods in decision making that in cases has been used integrated with GIS. Decision variables, constraints and objectives are three main and critical elements of linear programming. Decision variables are the questions of problems. In this domain, problem can be defined as minimizing or maximizing a problem based on suitability or unsuitability of the considered objectives. Land use planning or land use optimal allocation in the minimizing form and with respect to three objectives of this study including minimizing allocating cost, conversion cost and maximizing compactness can be considered as linear programming equations (1-8):
X_ijkis binary variable that equals 1 when land use k is allocated to cell (i, j) and equals 0 otherwise. C_ijk refers to the cost of cell (i, j) for kth land use, S_ijkis the cost of converting the current land use into a new one (land use k), Y_ijkis an integer variable introduced to the model to define compactness without violating the special linearity condition in linear programming problems.
It is clear that the problem is integer linear programming because decision variables (X_ijk)are binary. As linear programming is an exact method that enumerate all solutions to find optimal one, adding integer variables imposes a huge burden on computational processes and intensively increases the required processing time. This burden nearly makes it impossible to find optimal solutions. Therefore, we relaxed the problem from integrity and changed it to usual and classic linear programming. So, the values obtained for X_ijkwas ranged at [0 1]. The other difficulty was that one cell could not be allocated to more than one land use simultaneously. Therefore, we ranked the final responses that had been obtained from implementing linear model for every cell of the study area and then selected the maximum value in every cell and equaled other values with zero. After this selection, we had a solution map of the study area that showed land uses with the maximum value in every cell. We found that nearly none of the values were less than 0.5. So, we completely allocated every cell to land use with the maximum value. Although this result could not guarantee optimality but was very close to solution of the exact method of linear programming.
Results and Discussion:
In order to have an equal base for comparison, we implemented the problem with the same target area in MOLA algorithm in IDRISI. We selected MOLA because it works on the base of ranking the value of cells related to distance to ideal point which is the highest value possible after standardization of the suitability values. It is necessary to say that compactness and conversion cost objectives have not been considered in MOLA. Table (1) shows the number of cells (area) of every land use before and after implementing the above mentioned model (target area) and Table (2) shows the number of cells for every land uses before and after application of MOLA.
Conclusions
As figure (1) shows, fragmentation of land use in the suggested model is less than that for MOLA. Also, it is possible to change objectives or some other criterion in the linear programming model but MOLA in IDRISI is a crisp module that cannot accept other objectives like compactness for improving the results. If we accept the fact that land use planning is a main clue through achieving sustainable development, its importance becomes ever more clear. What improves land use planning and makes it more practical and powerful is paying attention to many aspects, stakeholders and sources that are involved and affected by the process which makes even more complex. Use of optimization algorithms is the way to address this complexity. The powerful feature of linear programming is achieving the exact solution which guarantees optimality. This feature besides its simplicity is sufficient to make this algorithm attractive and worthy of more in-depth studies. In this regard, this study attempts to introduce a way for application of linear programming in land use planning and optimization.