## 1 Introduction  
As technological progress advances and generations change, the logistics industry has become an essential part of daily life [1]. In recent years, competition among businesses has intensified. To expand their customer base and improve profits, companies aim to cut costs, shorten delivery times, improve product quality, and offer personalized products. Therefore, the concept of the cold chain [2, 3] is currently trending in the logistics industry. In the past, the reliability of the cold chain network was considered multi-state based on the number of carriers, without taking other factors into account. However, a stochastic cold chain network with multi-state travel time (SCCNMT) takes into account two stochastic factors simultaneously. This research does not consider time as a fixed constant, but instead views it as a multi-state factor. A stochastic network is a real system, such as transportation systems, communication systems and supply chain systems [4, 5, 6]. In a cold chain network, we consider suppliers, logistics companies, and retail stores as nodes, and the path between each node as an arc. When the network demand is given, we can calculate the probability of successfully transmitting the flow from the source to the sink, which is the reliability of cold chain networks. To calculate the reliability of cold chain networks [7; 8; 9], we only consider the multi-state flow caused by the occupied orders. However, time factors such as travel time, are affected by personnel and environmental factors. In the previous study [10; 11], reliability was calculated without considering the multi-state travel time. Therefore, this research aims to develop a new reliability measure for a cold chain network that can simultaneously consider both the multi-state flow and travel time. The reliability of a cold chain network can be used as a performance indicator to assist cold chain businesses with order planning and decision-making. Previous research on stochastic supply chain networks has explored various aspects of network reliability. [PERSON] [8] studied the different states of available transportation on different roads and proposed a network performance evaluation algorithm for inventory issues. [PERSON] et al. [7] investigated the use of different transportation modes in the network and considered the characteristic of goods being damaged. [PERSON] et al. [12] further examined the cost concept by incorporating transportation and damage costs into the study and separately discussing the damage rates on different routes. [PERSON] et al. [9] specifically discussed perishable goods, calculated the flow rate that meets the damage rate and ensured that the product can meet demand under the possibility of damage. As stochastic networks have been studied by previous studies with only a single multi-state factor, this study attempts to extend the research by evaluating the network reliability under two multi-state factors. This study aims to evaluate the reliability of a cold chain throughout the entire production process. First, we model the entire cold chain as a network, with nodes representing cold chain suppliers, logistics companies, and retail stores, and each node connecting a pair of arcs. We consider the route for a product chain as a path and use the solution to the network problem to calculate the reliability of a cold chain network. In the past, this research addressed the issue of multi-state flow but did not account for the time factor as a multi-state variable [12; 13; 14]. Therefore, this study extends the approach to calculate the reliability of a cold chain network for two stochastic variables. The ultimate goal of this research is to determine the network reliability for actual products and account for both flow and time factors simultaneously. The paper is divided into five sections, organized as follows: In this section, the introduction is presented. In Section 2, a stochastic cold chain network with a multi-state travel time (SCCNMT) model is developed.