## 3 Materials and Methods
### Mathematical Formulation
#### 3.2.4 The Model  
From the above analysis of the proposed intelligent logistics distribution service system, it is obvious that the optimization analysis of the distribution path of the swarm robots is the focus of our study. The purpose of the mission is to minimize the distribution cost while improving the service level. In this part, The MILP is proposed to support the  
Figure 3: Study on the influence of load weight. (**a**) Relationship between power and load; (**b**) Relationship between transport capacity factor and load. delivery of robots described by the system. Equation (4) is the objective function. The objective of the objective function is to minimize the weighted sum of two objectives, the total travel distance during the completion of the service to the customer, and the delivery of the service within the time window, where \(O_{gbf}\) is the penalty cost where the robot exceeds the customer service time in the \(fth\) transportation; the weighting factor a1 is used to provide a balanced solution between two goals; and a2 represents the penalty coefficient for exceeding the service time. \[ \begin{array}{cc}Min&a1\sum\limits_{g\in\Omega}&\sum\limits_{\begin{subarray}{ c}k\in\Omega_{K}\end{subarray}}\sum\limits_{f\in\Omega_{F}}d_{gh}\cdot X_{ ghkf}+\sum\limits_{g\in\Omega}\sum\limits_{k\in\Omega_{K}}\sum\limits_{f\in \Omega_{F}}O_{gbf}\\ &g\
eq h\end{array}  \tag{4}\]  
The relationship between decision variables \(X_{ghkf}\), \(Y_{gbf}\) is defined in Equation (5):  
\[\sum\limits_{h\in\Omega_{H}\cup\Omega_{SE}}X_{ghkf}=Y_{gbf},g\in\Omega_{H};k \in\Omega_{K};f\in\Omega_{F} \tag{5}\]  
The robot may end the work after completing the task as show in Equation (6). \[\sum\limits_{h\in\Omega}X_{ghkf}=\sum\limits_{g\in\Omega}X_{hgkf},g,h\in \Omega_{H};k\in\Omega_{K};f\in\Omega_{F} \tag{6}\]  
Equations (7) and (8) ensure that each robot starts the first transport from its initial service station, where \(S_{hk}\) represents the initial service station. In the same manner, Equation (9) indicates that each time the robot \(k\) enters a service station, it means that the \(fth\) transportation is completed. \[\sum\limits_{h\in\Omega_{H}\cup\Omega_{SE}}X_{s_{hk}hk1}=1,k\in\Omega_{K} \tag{7}\]  
\[\sum\limits_{s\in\Omega_{SE}}\sum\limits_{h\in\Omega_{H}\cup\Omega_{SE}}X_{ shkf}=1,k\in\Omega_{K},f\in\Omega_{F} \tag{8}\]  
\[\sum\limits_{s\in\Omega_{SE}}\sum\limits_{g\in\Omega_{H}\cup\Omega_{SB}}X_{ gskf}=1,k\in\Omega_{K};f\in\Omega_{F} \tag{9}\]  
All service tasks need to be completed. An idle robot must stay at a station to reduce unnecessary travel distance, and each task can be served by at most one robot, as follows:  
\[\sum\limits_{k\in\Omega_{K}}\sum\limits_{g\in\Omega}\sum\limits_{f\in\Omega_{ F}}X_{ghkf}=1\ \,h\in\Omega_{H} \tag{10}\]  
Equation (11) links previous and next shipments. With this constraint, if the robot \(k\) completes its \(fth\) transportation at the service station \(s\), its \((f+1)th\) transportation starts from the service station s-1:  
\[\sum\limits_{g\in\Omega_{H}\cup\Omega_{SB}}X_{gskf}=\sum\limits_{g\in\Omega_{ H}\cup\Omega_{SE}}X_{s-1,gkf+1},k\in\Omega_{K};f=1\ldots N_{f-1};s\in\Omega_{SE} \tag{11}\]  
In addition, because each station has two indexes as the starting station and the ending station, respectively, in order to avoid the robot from completing the transportation at the starting station, as show in Equation (12):  
\[\sum\limits_{g\in\Omega_{H}\cup\Omega_{SB}}X_{gskf}=0,k\in\Omega_{K},f\in \Omega_{F},s\in\Omega_{SB} \tag{12}\]  
A penalty cost function is used to represent the time window limit of each task, and each robot must perform the task within the allowable range of its power level, where M is a large positive number. The completion time of the \(fth\) of the robot is the start time of \((f+1)th\), and \(C_{skf}=C_{s-1,k,f+1}\):  
\[\left\{