## 3 Problem Formulation
### Performance Analysis  
To evaluate the performance of the proposed approach, the \(d_{p}\) distance concept was used to develop an evaluation method; (19)-(21) were considered to determine the degree of closeness of the result of the extended TOPSIS approach to the ideal solution. The distance functions can be defined as follows.  
\[d\infty=\max\Bigl{\{}w_{j}\Bigl{|}z_{j}^{+}-zj(x)\Bigr{\}},j\in k\]  
and  
\[d_{p}=\Biggl{\{}\sum_{j=1}^{k}w_{j}^{p}\left[\frac{z_{j}^{+}-z_{j}(x)}{z_{j}^{ +}}\right]^{p}\Biggr{\}}^{1/p},p\geq 1,j\in k\]  
where \(d_{p}\) represents the degree of approximation of the preferred compromise solution vector X to the optimal solution vector with respect to the kth objective function. Here, \(w=(w_{1},w_{2},\cdots,w_{k})\) is the vector of objective demand levels. The power p represents a distance parameter \(1\leq p\leq\infty\).  
Finally, the solution of the illustrative example is considered using different methods. The target programming approach (a) yields the following results: \(Z_{1}\) = 4,750,000, \(Z_{2}\) = 2,616,120, and \(Z_{3}\) = 3785. The fuzzy goal programming approach (b) [16] gives the following results: \(Z_{1}\) = 3,350,000, \(Z_{2}\) = 2,679,250, and \(Z_{3}\) = 3785. The compromise programming (CP) approach (c) [41] yields \(Z_{1}\) = 7,530,000, \(Z_{2}\) = 4,706,250, and \(Z_{3}\) = 1019. On the other hand, the proposed approach (d) obtains the solution \(Z_{1}\) = 7,662,900, \(Z_{2}\) = 4,779,525, and \(Z_{3}\) = 10,458. Table 9 compares the results obtained by three existing methods and the proposed approach. In the given example, it is assumed that \(w_{1}\) = 0.4, \(w_{2}\) = 0.3, and \(w_{3}\) = 0.3.  
From Table 9 and Figure 1, it is clear that approaches (c) and (d) offer preferred solutions that are better than the solutions obtained by approaches (a) and (b) for all distance functions \(d_{1}\), \(d_{2}\), and \(d_{\infty}\). The profits and sales revenues resulting from approaches (c) and (d) are also higher than those obtained by the other two approaches. In addition, further comparison between approaches (c) and (d) reveals that the profit of the solution obtained by the proposed approach (d) is higher than that resulting from approach (c). The sales revenue (market share) in the solution is 132,900 units higher than that obtained  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{6}{c}{**Period (t)**} \\ \hline  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Product: 1 (Wireless LAN) & & & & & & \\ Regular time production & 734 & 753 & 128 & 633 & 485 & 305 \\ Overtime production & 10 & 102 & 194 & 207 & 26 & 127 \\ Subcontracting level & 2143 & 1662 & 1870 & 492 & 491 & 2978 \\ Inventory level & 0 & 0 & 0 & 0 & 0 & 0 \\ Backorder level & 0 & 0 & 0 & 0 & 0 & 0 \\ Product: 2 (Ethernet Switch) & & & & & & \\ Regular time production & 61 & 207 & 848 & 0 & 743 & 871 \\ Overtime production & 317 & 240 & 3 & 88 & 97 & 156 \\ Subcontracting level & 5122 & 5553 & 5661 & 3900 & 2721 & 2290 \\ Inventory level & 0 & 0 & 12 & 0 & 0 & 0 \\ Backorder level & 0 & 0 & 0 & 0 & 1439 & 1622 \\ Workforce level & 450 & 500 & 350 & 550 & 400 & 500 \\ Hiring & 0 & 50 & 0 & 200 & 0 & 100 \\ Laying off & 50 & 0 & 150 & 0 & 150 & 0 \\ \hline \end{tabular}
\end{table}
Table 8: Compromise Solution for the Production Plan.  
by approach (c). Moreover, the distances \(d_{1}\) and \(d_{2}\) are shorter in the solution, while \(d_{\infty}\) is the same for the two approaches. The shorter the distances \(d_{1}\) and \(d_{2}\) are, the better the performance. Thus, these outcomes show that the proposed approach has good performance. It can be concluded that the extended TOPSIS compromise solution is more suitable than the other approaches. The comparison between different approaches is illustrated in Figure 1.