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This study introduces an uncertain programming model for the integrated production routing problem (PRP) in an uncertain production-inventory-routing system. Based on uncertainty theory, an uncertain programming model is proposed firstly and then transformed into a deterministic and equivalent model. The study further probes into different types of replenishment policies under the condition of uncertain demands, mainly the uncertain maximum level (UML) policy and the uncertain order-up to level (UOU) policy. Some inequalities are put forward to define the UML policy and the UOU policy under the uncertain environments, and the influences brought by uncertain demands are highlighted. The overall costs with optimal solution of the uncertain decision model grow with the increase of the confidence levels. And they are simultaneously affected by the variances of uncertain variables but rely on the value of confidence levels. Results show that when the confidence levels are not less than 0.5, the cost difference between the two policies begins to narrow along with the increase of the confidence levels and the variances of uncertain variables, eventually being trending to zero. When there are higher confidence levels and relatively large uncertainty in realistic applications, in which the solution scale is escalated, being conducive to its efficiency advantage, the comprehensive advantages of the UOU policy is obvious.

The inventory replenishment policy in an uncertain production-inventory-routing system