A mixed integer mathematical programming model solution using branch and bound techniques (abstract only)

Abstract

An optimal allocation of work release and labor assignments on a paced assembly line using the objective criterion of minimal standard hours has been formulated as a mixed integer mathematical programming model. The results of our research lead to the development of an algorithm which solves the mathematical model formulation and a computer implementation demonstrates its use in an actual operating environment. The algorithm developed for solving the problem uses a branch and bound approach as its basic solution technique.Basically, the ingredients of a branch and bound algorithm are a lower bounding strategy, a branching strategy (which partitions the set of feasible solutions into subsets), and a search strategy (which determines the order in which the subsets are investigated). The algorithm presented here uses the method of relaxing difficult constraints as a lower bounding strategy. The lower constraints are those which contain integer variables. This approach results in an ordinary linear programming problem model formulation which is used for computing lower bounds. The branching strategy is a binary partitioning in which a particular assembly mode is either in the solution subset, or not. The search strategy used is the backtracking procedure in which the branching variables are heuristically preordered.The mixed integer mathematical model follows: The objective function is V(X) = CX where C is (l x n) vector and X is a (n x 1) vector subject to AX = d where A is a (m x n) vector d is a (m x 1) vector HX ≤ h where H is a (1 x n) vector h is a (1 x 1) vector BX-DS ≤ O where B is a (r x n) matrix D is a (r x n) matrix S is a (r x 1) vector BX-PS ≥ O where R is a (s x n) matrix Os ≤ e where O is a (t x n) matrix e is a (t x 1) vectorAll variable are non-negative, the elements of the vector x arise from a continuous function, and the elements of the vector S are (0,1) integer values.The computer software used for solving the above model consists of two Fortran IV modules under the command of a language processor. Both software packages can be operated either interactively or under a batch processing mode. One of the computer software packages is labeled the “Initialization program” which is used to generate the coefficient vectors and matrices, i.e., C, A, d, H, h, B, D, R, O, and e presented in the above model. The other is labeled the “Action program” which is used to obtain an “optimal” solution for the problem. The computer software packages have been executed on a Xerox-560 computer in a limited fashion.In conclusion, there is an important by-product from our research so far; a way to find alternative optimal solutions. The branch and bound algorithm which we utilize is a partial enumeration technique in which subsets of feasible solutions are eliminated from consideration by comparing a lower bound on the objective function value (for a given subset) to the best solution obtained up to that point. Theoretically, alternative optimal solutions may be among those subsets of feasible solutions.