Abstract:

In the context of global procurement， the fixed ordering costs involved in the procurement process， such as service charges， telecommunications costs， travel expenses of staff， intermediary costs for purchasing through agents， and shipping costs for international purchasing， have increased dramatically. Under such circumstances， the jointreplenishment （JR）policy is a highly effective cost-control measure， which not only decreases the fixed ordering cost but also procures a large order and achieves economies of scale. When a group of goods are supplied by the same supplier or are transported by the same freighter， JR policy can achieve the scale effect of purchasing and shipping by sharing the fixed transportation cost.Joint replenishment system， as a typical multi-item inventory system， also have some disadvantages. The heterogeneity of items is an in-negligible issue， in particular when there is a variety of items. Such heterogeneity may be caused by various reasons. For example， cold-chain transportation costs may be shared by items when fresh food and normal products are transported together； some chemical products may corrode other objects， and thus additional packaging costs can be charged in this case； joint transportation of bulky cargos and heavy cargos can lead to damage to bulky cargos.This heterogeneity often leads to additional logistics costs or unnecessary administrative difficulties. Although heterogeneous items are very important in joint replenishment system， researches about this consideration is very scarce. Moreover， the basic joint replenishment problem disregards the coordinated scheduling occurred in the delivery stage， which makes it very limited to be applied. In order to fill the above research gaps. The coordinated optimization of joint replenishment and delivery scheduling with heterogenous items （JRD-HI） is considered.The problem is formulated as a mixed integer nonlinear program. Since it has been proven strongly NP-hard， our problem at hand is very difficult to solve.Conventional methods for mathematical programming are basically inapplicable to JRD-HI.In addition， exact algorithms for this problem are very rare， and only problems of modest scale can be accurately solved. By analyzing mathematical properties of the model， a series of conclusions of optimal solutionsare obtained， including the Lipschitz continuity of the objective function， which gives rise to a new idea of solution algorithm. The new method adopts a dynamic-step-size policy to search on the continuous decision variable， while existing methods are all based on the search on integer decision variables. In an experiment of 400 randomized instances， our new algorithm shows competent performance： the optimality gap is below 0.5% averagely and the average running time is below 4 seconds. Compared to existing algorithms， the improvement on objective value can reach 11% to 26%. Furthermore， some managerial insights are found via a sensitivity analysis on the coordination effect of delivery coordination. It is found that the coordinated optimization of replenishment and delivery can achieve 8.83% cost savings in a practical instance.The reason for the cost savings mainly comes from two respects： （a） the replenishment and delivery scheduling are more coordinated； （b） the central warehouse have more cost advantage. One interesting finding we view as making important managerial implication is the boundary of coordinated delivery； It is found that coordinated delivery policy is necessary only if the inventory management level of the central warehouse is remarkably superior to that of retailers.Otherwise， coordinated optimization of replenishment and delivery cannot achieve effective cost savings. This result can be agood guide for a company's investment plan when applying the JRD model. Equipment and human resource of the central warehouse should be ensured preferentially. One should invest in warehouses first and then in retailers.

Key words: joint replenishment, coordinated delivery, heterogenous items, Lipschitz continuity, inventory management