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In vendor managed inventory (VMI) operations, it is necessary for both the suppliers and the customers to reach an agreement on the maximal and minimal inventory level. In practice, in VMI, inventory levels are decided manually by planning personnel, based on their experience and past operating records. From a system optimization point of view, the determination of the upper and lower inventory levels in storage areas is complicated, necessitating analysis of many factors. Consequently, it may not be possible to obtain the best planning results, to determine and maintain the optimum levels of inventory. In this study, we utilize network flow techniques to build a replenishment model to deal with the upper-and-lower inventory level control problem, with the objective of minimizing the total cost in short-term operations, subject to inventory level control and other operating constraints. The model is formulated as an integer network flow problem with side constraints and is characterized as NP-hard in terms of optimization. To efficiently and effectively solve the large-scale problems that occur in the real world, a solution algorithm is also developed. Finally, numerical tests are conducted using real data from a major retail company in northern Taiwan. The results demonstrate the usefulness of the proposed model and solution algorithm for replenishment planning in actual practice.

Stochastic disturbances occurring in real world operations could have a significant influence on the planned routing and scheduling results for inter-school buses. An optimal schedule might therefore lose its optimality when applied in actual operations where stochastic disturbances occur. In this study we employ network flow techniques to construct an inter-school bus routing and scheduling model, considering the characteristics of inter-school buses for class attendance, under stochastic travel times. The model is formulated as a special multiple commodity network flow problem which is characterized as NP-hard. To efficiently solve realistically large problems a problem decomposition and variable fixing method, coupled with the use of CPLEX mathematical programming solver, is employed to develop a solution algorithm. Finally, computational tests are performed using data from an allied university system in Taiwan. The test results show that the model and the solution algorithm are good and therefore could be useful for actual operations.