Devising an efficient approach to determine the optimal sequence of from-to matrix - Laith, Watheq; Al-Salih, Rasheed (Eastern-European Journal of Enterprise Technologies, 2021)
Sequencing is the most impact factor in many production areas, such as assembly lines, batch production, Travelling Salesman Problem (TSP), product sequences, process sequences, etc. The flow and analysis from one item to another can be presented by the square matrix in which the number of rows or columns is equal to the number of manipulated items, this special matrix form is called “From-To matrix”. The matrix suffers from many drawbacks when it is applied to determine the optimal sequences, such as the number of variables must be as small as possible, there is no flexibility to determine the start or the end sequence to find the best sequencing with some conditions. Also, there is no possibility to add relations to point a variable as wanted or prevented from the sequence. In this paper, we solve the From-To matrix by binary linear programming (BLP). The proposed BLP approach has been applied in Ur company to solve the From-To matrix. This company has a production line that can manufacture four products: A, B, C, and D, the setup time matrix is considered as From-To matrix and the goal of this company is to get an optimum sequence of products with minimum time. The solution of state transition of the From-To matrix using BLP can be formulated in the following five model cases according to transition requirement condition and desired: the first case gives all possible sequence items, the second case lists the sequence items when the first sequence is known, the third case lists the sequence items when the last sequence is known, the fourth case gives all possible sequence items with a condition that prevents occurring of an undesired sequence, and the fifth case gives all possible sequence items with the condition of a wanted occurring of the desired sequence. Furthermore, we found the optimum sequences for states by determining the start or end sequences, and also add the wanted relations or prevented. The mathematical formulas for the number of all sequences under some conditions are derived and proved