## 3 Algorithm  
An algorithm to evaluate the network reliability for an SCCNMT model under time constraint \(I\) is proposed as follows (Algorithms 1-3):  
```
0:\(G\equiv(N,A,M)\), \(D=(d^{1},d^{2},\ldots,d^{p})\), \(v_{0}\), \(U,\)\(L,\)\(I\). 0: demand reliability \(R_{D}\)
0: Step 0. Find all feasible flow vectors \(F=\left(f_{1}^{1},f_{2}^{1},\ldots,f_{w}^{1},f_{1}^{2},f_{2}^{2},\ldots,f_{w}^ {2},f_{1}^{p},f_{2}^{p},\ldots f_{w}^{p}\right)\) satisfying \[\sum_{k=1}^{p}\sum_{o=1}^{w}f_{o}^{k}=d^{k}.\] (11)

Step 1. Check whether the flow vectors \(F\) exceeds the maximal capacity \(M_{t}\) of each arc or not. (1.1) Check whether the maximal capacity of each arc has been exceeded by the number of vehicles used via

\[\left[\sum_{o=1}^{w}\sum_{k=1}^{p}\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.\} \right]\leq M_{t},\text{ for }q=1,\ 2,\...,\ z. \tag{12}\]

(1.2) Check whether the truck departing from the logistics company can arrive within the given time via

\[\max\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.\}\leq\left[\frac{I-\Delta}{U} \right],\text{ for }o=1,\ 2,\...,\ w;k=1,\ 2,\...,\ p; \tag{13}\]

\[q=s+1,\ s+2,\...,\ z;b=1,\ 2,\...,\ s,\text{ where }\Delta=\max\{v_{0}\}+(U+L)+\max\{v_{q}\}\]

Step 2. Generate the capacity vectors \(X=(x_{1},\ x_{2},\...,\ x_{z})\) by following the procedure below. (2.1) Transform each \(F\) into capacity vector \(X\) before the logistics company via

\[x_{q}=\left[\sum_{o=1}^{w}\sum_{k=1}^{p}\{f_{o}^{k}\left|q_{o}\in P_{o}^{k} \right.\}\right],\text{ for }q=1,\ 2,\...,\ s. \tag{14}\]

(2.2) Transform each \(F\) into capacity vector \(X\) after the logistics company while taking into consideration time constraints via

\[x_{{}_{t}}=\left[\frac{U^{*}\max\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.}} {I-\Delta}\right],\text{ for }o=1,\ 2,\...,\ w;k=1,\ 2,\...,\ p; \tag{15}\]

\[q=s+1,\ s+2,\...,\ z;b=1,\ 2,\...,\ s,\text{ where }\Delta=\max\{v_{0}\}+(U+L)+\max\{v_{q}\}\]

(2.3) Obtain capacity vectors \(X\) in \(\Omega\) by steps 2.1 and 2.2, and eliminate other non-MCVs in \(\Omega\) using vector operations. Step 3. Assume there are \(z\) MCVs. The demand reliability \(R_{D}\) in an SCCNMT using the RSDP can be obtained via

\[R_{D}=\Pr\left\{\sum_{i=1}^{z}\left|\{X\}X\geq X_{i}\right.\right\} \tag{16}\]

**Input:**\(G\equiv(\textbf{N},\textbf{A},M)\), \(D=(d^{1},d^{2},\ldots,d^{p})\), \(v_{0}\), \(U\), \(L\), \(I\). **Output:** time reliability \(R_{T}\)

Step 1. Calculate the currently allocatable travel time by considering the average loading and unloading time from the time threshold \(I\) by Equation (17),

\[I^{*}=I-(U+2^{*}\,L) \tag{17}\]

Step 2. Find the travel time upper bound vectors. (2.1) Find the feasible travel time vectors \(V\) for each route using Equations (18) and (19). \[v_{q}^{\min}\leq v_{q}\leq v_{q}^{\max}\,,\,\text{for}\,q=1,\,2,\,\ldots,\,z,\]

\[\left[\sum_{o=1}^{w}\sum_{k=1}^{p}\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.\} \right]\leq M_{t},\text{ for }q=1,\ 2,\...,\ z. \tag{12}\]0

(2.2) Obtain travel time vectors \(V\) by step 2.1 and using vector operations to eliminate other non-travel time upper bound vectors. \[\left[\sum_{o=1}^{w}\sum_{k=1}^{p}\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.\} \right]\leq M_{t},\text{ for }q=1,\ 2,\...,\ z. \tag{12}\]1

\[\left[\sum_{o=1}^{w}\sum_{k=1}^{p}\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.\} \right]\leq M_{t},\text{ for }q=1,\ 2,\...,\ z. \tag{12}\]2

Step 3. Assume there are \(\rho\) travel time upper bound vectors. The time reliability \(R_{T}\) in an SCCNMT using the RSDP can be obtained via

\[\left[\sum_{o=1}^{w}\sum_{k=1}^{p}\{f_{o}^{k}\left|q_{o}\in P_{o}^{k}\right.\} \right]\leq M_{t},\text{ for }q=1,\ 2,\...,\ z. \tag{12}\]3

**Input:** demand reliability \(R_{D,\,\text{time}}\) reliability \(R_{T}\)

**Output:** network reliability \(R_{D,\,\text{T}}\)

\[R_{D,\,\text{T}}=R_{D}



## 4 A Numerical Example

We use a simple numerical example to demonstrate an SCCNMT model, including a simple cold chain network.