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In multistage processes, two kinds of process capability indices (specific and total process capability indices) are defined in each stage. The total process capability index calculates the capability of each stage when it is affected by the previous stages and the specific process capability index calculates the capability of the stage when the effects of the previous stages are eliminated. Process capability indices in multistage processes are proposed under the assumption of no measurement errors. However, sometimes this assumption may be violated and leads to misleading interpretations. In this paper, the effects of measurement errors on the specific and total process capability indices in the second and third stages of the three-stage processes are statistically analysed. In addition, the effects of the measurement errors on the bias and the mean square error value of the total and specific process capability indices in the second and third stages are studied. Finally, the effects of the measurement errors on the specific and total process capability indices are shown through a numerical example. [ABSTRACT FROM AUTHOR]

• Proposing a stochastic model for two-layers time window assignment VRP. • Considering of time window violations penalty cost in the proposed model. • Using the Progressive Hedging algorithm to solve the TL-TWAVRP. • Sensitivity analysis to check the validity & efficiency of the model and algorithm. This paper presents an effective Progressive Hedging algorithm for vehicle routing problem with two-layers time window assignment and stochastic service times (2L-TWAVRPSST). Based on a predefined exogenous time window determined by the customers, an endogenous time window is assigned to each customer. Endogenous time windows have flexible width and composed of two-layers. The outer layer is wider than inner layer and is determined by violation variable. This approach aims to giving more flexibility to career companies for serving more customers using less vehicles. Customers could be visited even after the end of their assigned time windows by paying proportional penalty, while extra violation from the outer layer is not permitted. This problem is formulated as a two-stage stochastic model with the first-stage decisions of assigning inner and outer layers time window. Then in the second stage, routes are planned for each scenario combination of stochastic demand and service time. The validity and effectiveness of the proposed model was examined by various numerical examples. The problem was solved by a Progressive Hedging (PH) algorithm for large-scale instances. The results confirm efficiency of the considered solution approach in different instances. [ABSTRACT FROM AUTHOR]