## 6 Conclusions  
In this paper, the TOPSIS compromise solution method was extended to solve programming problems with multiple objectives. The APP model, combined with a TOPSIS approach, provides an effective way to find a satisfactory solution to such problems. In generally, TOPSIS provides a broader principle of compromise for solving multiple-criteria decision making problems. It converts k-objectives, which are conflicting and incompatible with each other, into two-objectives (the shortest distance from PIS and the longest distance from NIS), which are compatible and conflicting with each other in most cases. The bi-objective problem can then be solved using membership functions to represent the satisfaction level for both criteria and obtain a TOPSIS compromise solution via a second-order compromise. The max-min operator is then considered a suitable approach to solve the conflict between the new criteria (the shortest distance from the PIS and the longest distance from the NIS) [39]. The proposed APP model provides useful information to the industry from the determined optimal overall production plan. Using this model, decision-makers can work out the optimal number of temporary workers to hire at the beginning of each period, the optimal number of temporary workers to be laid off at the end of each period, and the number of temporary workers that should be available in each period. This information is useful for managing the human resources of a manufacturing company. The output of this model also provides information on the optimal number of overtime man-hours of permanent and temporary workers during workdays and holidays, inventory levels, and  
Figure 1: Graphical representation of the solutions by different approaches. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **(a) GP** & **(b) Fuzzy GP** & **(c) CP** & **(d) Proposed Approach** & **(e) Optimal Solution** \\ \hline \(Z_{1}\) & 4,750,000 & 3,350,000 & 7,530,000 & 7,662,900 & 9,500,000 \\ \(Z_{2}\) & 2,616,120 & 2,679,250 & 4,706,250 & 4,779,525 & 1,642,785 \\ \(Z_{3}\) & 3785 & 3785 & 1019 & 10,458 & 0 \\ Profit & 2,130,095 & 666,965 & 2,822,731 & 2,872,917 & - \\ \(d_{1}\) & 0.8116 & 1.1504 & 0.5999 & 0.5928 & - \\ \(d_{2}\) & 0.5123 & 0.8017 & 0.3729 & 0.3714 & - \\ \(d_{\infty}\) & 0.4000 & 0.7343 & 0.3000 & 0.3000 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison of Solutions Obtained by Four Different Approaches. the number of regular working hours for some periods. The related cost elements and total costs are presented for financial consideration by top management. In this paper, four advantages of TOPSIS have been addressed: (i) sound logic that is the rationale in solving multi-objective decision problems; (ii) a scalar value that considers both the best and worst ideal solutions simultaneously; (iii) a simple calculation process that can be easily programmed into a spreadsheet; and (iv) the performance measures compared to other approaches can be visualized on a polyhedron. The proposed general APP model is applicable to a wide range of industries in which the output per period can be adjusted by changing the number of jobs and the number of workers, and by applying overtime. However, the proposed APP model is not suitable for multi-plant and multi-product problems because the model's design is suitable for single-product, single-plant problems. A new APP model can be constructed with different sets of constraints to handle problems with multiple plants and multi-product. We recommend that further studies be conducted to develop APP models that meet the specific requirements of multi-plant and multi-product problems. Appropriate methods for disaggregating the aggregate plan into the total production plan should also be devised based on the situations and requirements of the industries. Conceptualization, V.F.Y., H.-C.K. and F.-Y.C. ; methodology, H.-C.K. and F.-Y.C. ; software, H.-C.K. and F.-Y.C. ; validation, V.F.Y., H.-C.K., F.-Y.C. and S.-W.L. ; formal analysis, H.-C.K. and F.-Y.C. ; investigation, H.-C.K.