## 4 Case Study
### Determination of the Optimal Solution Based on the Coalitional Game Model  
Step 1. The strategy sets of six conceptual solutions for warehouse logistics systems were obtained, which include (\(S_{P}\)(CS\({}_{1}\)), \(S_{E}\)(CS\({}_{1}\)))-(\(S_{P}\)(CS\({}_{6}\)), \(S_{E}\)(CS\({}_{6}\))). The interactions between CV and economic objectives are illustrated in Figure 3, and it is shown that the criteria weights of each objective will affect utility. Hence, the criteria weights were considered in utility calculations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**(a) Scheme Value Assessment of Criteria in the CV Objective**} \\ \cline{2-5}
**Evaluation Criteria** & \multicolumn{3}{c}{**Evaluation Value Assigned by Four DMs**} \\ \cline{2-5}
**DM1** & **DM2** & **DM3** & **DM4** \\ \hline P1 & 6 & 6 & 2 & 8 \\ P2 & 6 & 3 & 4 & 3 \\ P3 & 6 & 7 & 4 & 5 \\ P4 & 4 & 4 & 3 & 3 \\ P5 & 5 & 6 & 4 & 6 \\ P6 & 5 & 8 & 4 & 6 \\ P7 & 5 & 7 & 4 & 4 \\ P8 & 6 & 8 & 6 & 4 \\ P9 & 6 & 5 & 6 & 6 \\ P10 & 6 & 4 & 4 & 8 \\ \hline \multicolumn{5}{c}{**(b) Scheme Value Assessment of Criteria in the Economic Objective**} \\ \hline \hline \multicolumn{5}{c}{**Evaluation Value Assigned by Four DMs**} \\ \cline{2-5}
**Evaluation Criteria** & \multicolumn{3}{c}{**Evaluation Value Assigned by Four DMs**} \\ \cline{2-5}
**DM1** & **DM2** & **DM3** & **DM4** \\ \hline E1 (t) & 1.75 & 1.80 & 1.50 & 1.75 \\ E2 (km/h) & 13 & 13 & 16 & 13 \\ E3 (mm) & 3500 & 4000 & 4000 & 3000 \\ E4 (kw) & 4 & 6 & 6 & 4 \\ E5 (t) & 14,400 & 9600 & 12,000 & 12,500 \\ E6 & 6 & 2 & 5 & 2 \\ E7 & 6 & 7 & 7 & 6 \\ E8 (t) & 3.75 & 3.5 & 3.5 & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Initial evaluation matrix of the electric forklift PSS concept (taking the CS\({}_{1}\) as an example). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**CV Criteria** & \(cw_{j}\) & \(w_{j}\) & \(cw_{j}\) & **Economic Criteria** & \(cw_{j}\) & \(w_{j}\) & \(cw_{j}\) \\ \hline P\({}_{1}\) & 0.108 & 0.266 & 0.287 & E\({}_{1}\) & 0.154 & 0.059 & 0.097 \\ P\({}_{2}\) & 0.270 & 0.092 & 0.249 & E\({}_{2}\) & 0.086 & 0.188 & 0.172 \\ P\({}_{3}\) & 0.042 & 0.124 &Step 2. The normalized rough values of the warehouse logistics system solutions (Equations (23) and (24)) were converted into crisp values. Equations (14) and (15) were then adopted to calculate the utilities of the CV and economic objectives in each strategy set. The calculated utilities of the electric forklift solutions are presented in Table 10.  
Step 3. Based on Table 11, a min-max representation model (Equation (16)) was used to define the characteristic function for each possible coalition. This gave \(v([\text{P}])=0.901\) and \(v([\text{E}])=0.959\). The coalitional utility associated with the players cooperating with each other was calculated using a pessimistic criterion, i.e., \(v(\{\text{P},\text{ E}\})=\min\sum\limits_{c=1}^{2}U_{c}\). This gave a coalitional utility of \(v([\text{P},\text{E}])=1.9\). Concurrently, the Shapley value method (Equation (17)) was used to allocate the utility of each game player--that is, the optimal design expectation for CV and economy objective, thereby obtaining \(\varphi(v)=(\varphi_{\text{P}}(v),\,\varphi_{\text{E}}(v))=(0.923,0.982)\). Step 4. The deviation function (Equation (17)) was used to determine the conceptual solution closest to the Shapley value, and satisfied expectations relative to CV and economic utilities while maximizing overall utility. The deviations of the solutions from \(\varphi(v)\) were: \(D(\text{CS}_{1})=0.031\), \(D(\text{CS}_{2})=0.065\), \(D(\text{CS}_{3})=0.104\), \(D(\text{CS}_{4})=0.033\), \(D(\text{CS}_{5})=0.154\), and \(D(\text{CS}_{6})=0.186\). Therefore, the ranking of the electric forklift solutions was \(\text{CS}_{1}>\text{CS}_{4}>\text{CS}_{2}>\text{CS}_{3}>\text{CS}_{5}>\text{CS}_{6}\), and \(\text{CS}_{1}\) is the optimal solution. The CV and economic utilities of \(\text{CS}_{1}\) were (\(U_{P}\), \(U_{\text{E}})=(0.946,1.004)\), as presented in Figure 4.