A mathematical model for solving integer linear programming problems

A suggested algorithm to solve fully rough integer linear programming (FRILP) problems is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are rough intervals (RIs). In real-life situations, the parameters of linear programming problem model may not be defined precisely, because of globalization of the market, uncontrollable factors, etc., hence for that the FRILP problem solving methodology is presented using the slice-sum method with the branch and bound technique, where we will construct two integers linear programming (ILP) problems with interval coefficients and variables. One of these problems is an ILP problem, where all of its coefficients are upper approximations interval (UAI) of rough intervals and represents rather satisfactory solutions, the other is an ILP problem where all of its coefficients are lower approximations interval (LAI) of rough intervals and represents complete solutions. Thereafter, the two ILP problems are sliced into four crisp problems. Integer programming is used because many linear programming (LP) problems require that the decision variables should be integers. In addition, rough intervals are very important to tackle the uncertainty and imprecise data in decision making problems. Furthermore, the proposed algorithm enables us to search for the optimal solution in the largest range of possible solutions. A flowchart is also provided to illustrate the problem-solving steps. Finally, some examples are given to demonstrate the results.