An approximated solution to the constrained integrated manufacturer-buyer supply problem

In recent decades, the integrated manufacturer-buyer supply chain problem has been widely studied by many scholars, as it provides benefits to both parties in terms of planning flexibility, information sharing and joint costs. The manufacturer produces a production lot of size Q, with variable production rate P ≤ U, where U denotes the maximal production rate (e.g., due to technological restrictions or due to limited machinery and/or manpower capacities). The manufacturer delivers the lot to the buyer in n (integer) smaller shipments, each of size q. An upper bound on the production cycle length is assumed (e.g., to enable the scheduling of maintenance periods or idle windows of time in which workers are not required to work.) In order to solve the problem mathematically, we suggest a sub-optimal nested formulation of the problem that utilizes existing formulas for n*(r) and q*(r) (where r denotes the demand-to-production rate ratio, r = D/P) for the unconstrained problem. The optimal solution of the accurate formulation is obtained through numerical optimization, utilizing the IBM CPLEX solver, and is compared with the proposed sub-optimal method. Among the advantages of the suggested approach are that the solution is analytically derived, it is very simple to implement, and it yields a minimal joint cost that is close to the accurate formulation.