Table 2 Mathematical symbols for PFSVRP.

\(n\)

Number of orders (here, the number of orders equals the number of customers)

\(i\),\(l\),\(h\)

The index of the order (here, the index of the customer is the same as the index of the order)

\(m\)

Number of machines

\(j\)

Index of the machine

\(K\)

Number of vehicles

\(k\)

Index of the vehicle

\(p_{ij}\)

The processing time of order \(i\) on machine \(j\)

\(c{\prime}\)

Unit time cost of order processing

\(G\)

A large positive number

\(c_{hj}\)

The completion time of the order \(h\) on the machine \(j\)

\(C_{i}\)

The completion time of the order \(i\)

\(C_{\max } = \mathop {\max }\limits_{1 \le i \le n} \{ C_{i} \}\)

Maximum completion time of the orders

\(d_{i}\)

The due date of the order \(i\)

\(dc\)

Unit time cost of order delay

\(b_{k}{\prime}\)

Unit distance cost of the vehicle \(k\)

\(Q\)

Maximum load capacity of factory vehicle \(k\)

\(d_{i}{\prime}\)

Distance between the depot and customer \(i\)

\(d_{il}^{^{\prime\prime}}\)

Distance from customer \(i\) to the customer \(l\)

\(S_{k}\)

Customer collection for the vehicle \(k\)

\(OR_{i}\)

The requirements of the customer \(i\)

\([a_{i} ,b_{i} ]\)

Time window of the customer \(i\); \(a_{i}\) is the allowed service start time and \(b_{i}\) is the allowed service end time

\(st_{i}\)

Actual service start time of the customer \(i\)

\(\lambda\)

A sequence of orders in a certain processing order

\(x_{ir}\)

If \(i\) is \(r\) order of the sequence \(\lambda\), \(x_{ir} = 1\); otherwise, \(x_{ir} = 0\). Here,\(r = 1, \ldots ,n\)

\(y_{ki}\)

If vehicle \(k\) has customer \(i\) as its first customer, \(y_{ki} = 1\); otherwise,\(y_{ki} = 0\)

\(y_{kil}{\prime}\)

If vehicle \(k\) serves customer \(i\) and transports directly to customer \(l\), \(y_{kil}{\prime} = 1\); otherwise,\(y_{kil}{\prime} = 0\)

\(y_{ki}^{^{\prime\prime}}\)

If the last customer of the vehicle \(k\) is customer \(i\), \(y_{ki}^{^{\prime\prime}} = 1\); otherwise,\(y_{ki}^{^{\prime\prime}} = 0\)