An alternative optimal solution technique for a single-vendor single-buyer integrated production inventory model

For the single-vendor single-buyer integrated production inventory problem, the lot is transferred by some researchers in shipments of equal and/or unequal sizes. For a given total number (n) of shipments, the minimal number (m) of unequal sized shipments is selected satisfying a constraint. Due to the complex dependence of m on n, there remained a conjecture about the convexity of the total cost (C) in n. Consequently, for a known n, lower and upper bounds on the minimal total cost are imposed and this cost is obtained through a search over n based on the convexity of C in the lot size (Q). Here it is demonstrated that m thus selected might not be the minimal and a method for obtaining the minimal m is developed. A formula for the minimal n based on the convexity of C in n (given m and Q) is also derived. Integrating the ways of finding the minimal m and n, along with the convexity of C in Q, an alternative minimal cost solution technique is presented. Finally, a comparative theoretical analysis of the methods along with numerical illustration is carried out to show the efficiency of our method.