An Approximation Approach to Numerical Solution of Convex Cost Algorithm in Production Planning

The Convex Cost Algorithm is used to find the optimal solution to producion planning where the production cost, inventory cost and shortage cost functions are all assumed to be convex. When these convex costs are non-linear a large amount of computation is required to find the optimal solution.Whereasconvex piecewise linear production costs with linear inventory and shortage costs, the problem can be solved using the transportation tableau where the amount of computation is insignificant. In this paper, the convex non-linear production cost is approximated by two convex piecewise-linear functions, as two upper and lower bounds to the covex function.The estimated optimal cost for the inner piecewise-linear production cost represents a lower bound for the original optimal cost. Whereas, an upper bound for the optimal cost is computed using the outer piecewise linear production cost. In computing both of these bounds, an increment for the production level is used. Decreasing the amount of this increment will result in narrower bounds. Thus, the increment is decreased until the lower and upper bounds are tightened enough and the approximate optimal solution can attained.

Keywords:

Convex Cost Algorithm , Transportation Tableau , Linear Approximation , Prduction planning