_Article_  
The Inventory Routing Problem with Priorities  
and Fixed Heterogeneous Fleet  
[PERSON] 1,*,+\({}^{\text{\textregistered}}\), [PERSON] 2,+\({}^{\text{\textregistered}}\) and [PERSON] 2,+\({}^{\text{\textregistered}}\)  
==WARNING: Truncated because of repetitions==
00% of clients with priorities is equivalent to 0% of clients with priorities).
+++  
The complete results for all benchmark problems are presented in Tables A1-A6 in Appendix A.  
The complete results show that, almost always, the total cost increases. For some instances with 5 and 10 customers, the cost remains the same. The cost remains the same when the customers with priorities are visited first (last) when no priorities are considered. For example, Figure 4 shows the obtained routes for an instance of 10 customers where the introduction of priorities did not increase the total cost \(TIC\). In both cases, vehicles 2 and 3 do not leave the depot, and also, in period 3, no distribution is made. The routes in period 2 \(P=2\) are equivalent. When priorities are introduced, the clients with priority are visited first instead of last, which is the case when no priority clients are considered. The complete results also show that the routing cost increment depends importantly on the location of customers with priority, their consumption rates, and the number of vehicles used. If the customers with priority have a high consumption rate and are located far away from the depot, the routing cost will increase importantly. Figure 5 shows how routes change when priority  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Subset** & \(n\) & _Priority_ & \(\Delta IC\) & \(\Delta RC\) & \(\Delta TC\) & \(\sigma\) \\ \hline \multirow{3}{*}{S4} & & 0.25 & \(-0.11\%\) & 10.35\% & 3.59\% & 1.44 \\  & 20 & 0.50 & 0.10\% & 12.91\% & 4.93\% & 1.22 \\  & & 0.75 & \(-0.06\%\) & 9.68\% & 3.41\% & 1.67 \\ \hline \multirow{3}{*}{S5} & & 0.25 & \(-0.09\%\) & 11.03\% & 3.28\% & 4.29 \\  & 25 & 0.50 & 0.01\% & 20.24\% & 6.33\% & 3.10 \\  & & 0.75 & 0.00\% & 18.91\% & 5.91\% & 3.41 \\ \hline \multirow{3}{*}{S6} & & 0.25 & \(-0.01\%\) & 19.99\% & 5.39\% & 1.52 \\  & 30 & 0.50 & 0.08\% & 21.94\% & 5.97\% & 1.27 \\ \cline{1-1}  & & 0.75 & 0.06\% & 23.59\% & 6.34\% & 1.07 \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._  
Figure 3: Average percentage increment for the total cost with 20, 25, and 30 customers (**left, middle**, and **right**, respectively). The whiskers show the minimum and maximum values.  
Figure 2: Average percentage increment for the total cost with 5, 10, and 15 customers (**left, middle**, and **right**, respectively). The whiskers show the minimum and maximum values.  
clients are introduced. In this case, the introduction of priorities modifies importantly routes \(P=1\) and \(P=2\) with the vehicle \(Ve=1\). For many of the test instances, the depot inventory costs \(DIC\) and customer inventory costs \(CIC\) increase or decrease when priority customers are introduced. However, it is important to notice that the total inventory cost \(TIC=DIC+CIC\) remains almost the same as mentioned before. In general, if \(DIC\) increases, \(CIC\) is reduced, and vice versa.