The critical points of production rate under vendor–buyer coordination: explicit solutions

This paper revisits the deterministic joint vendor–buyer production-inventory problem and assumes that the production rate can be controlled. To guard against unwarranted extreme solutions, such as too abrupt production or, on the other hand, everlasting production aligned with demand, we place lower and upper bounds on the production rate. In addition, we incorporate an independent upper bound on the overall cycle of producing and remaining idle. Through identifying several critical points of the production rate, we solve the resulting problem for the optimal triplet (P, Q, n), where P is the constant production rate, a key decision variable, Q is the entire lot size produced in a cycle, and n is the number of equal successive shipments of this lot to the buyer. Our own treatment is purely analytical, which adds value from a theoretical perspective. Worst values of the production rate, when the other decision values are optimal for P, are found. We prove that only two production rates, the lowest and the highest, can yield minimal joint costs, and we identify which of the two is optimal under given relative positions of the defined critical points. Numerical illustration indicates that the joint cost sharply increases for small values of the bound on the overall cycle length. This study highlights the importance of solving variants of the suggested model and of developing managerial alternatives that relax this constraint as much as possible.