LINEAR PROGRAMMING

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Silicon Valley Corporation (Silvco) manufactures transistors. An important aspect of the manufacture of transistors is the melting of the element germanium (a major component of a transistor) in a furnace. Unfortunately, the melting process yields germanium of highly variable quality. There are two methods that can be used to melt germanium; method 1 costs $50 per transistor, and method 2 costs $70 per transistor. The qualities of germanium obtained by methods 1 and 2 are shown in Table 1. Silvco can refire melted germanium in an attempt to improve its quality. It costs $25 to refire the melted germanium for one transistor. The results of the refiring process are shown in Table 2. Silvco has sufficient furnace capacity to melt or refire germanium for at most 20,000 transistors per month. Silvco's monthly demands are 1000 for grade 4 transistors, 2000 for grade 3 transistors, 3000 for grade 2 transistors, 3000 for grade 1 transistors. Use linear programming to minimize the cost of producing the needed transistors. NoTE: There is no need to refire grade 4 melted germanium as can be seen in Table 2. This is so because it is already of the highest possible quality. Refiring does not improve its quality any further. Defining the Decision Variables XMi= Amount of germanium melted by method Mi,i=1,2. YGj= Amount of melted germanium of grade j used to satisfy demand, j=1,2,3. ZGj= Amount of melted germanium of grade j to be refired in the, furnace, j=0 (for defective) ,1,2,3. \begin{tabular}{|l|r|c|c|} \hline 1 & \multicolumn{3}{|c|}{ Silicon Valley Corp. Germanium Melting Prob } \\ \hline 2 & \multicolumn{2}{|c|}{} \\ \hline 3 & & \multicolumn{2}{|c|}{ PERCENT YiELED BY MELTING } \\ \hline 4 & TABLE 1 & Method 1 & Method 2 \\ \hline 5 & GERMANIUM & & \\ \hline \hline 6 & Defective & 30 & 20 \\ \hline 7 & Grade 1 & 30 & 20 \\ \hline 8 & Grade 2 & 20 & 25 \\ \hline 9 & Grade 3 & 15 & 20 \\ \hline 10 & Grade 4 & 5 & 15 \\ \hline 11 & Cost per transistor & $50.00 & $70.00 \\ \hline 12 & Sum: & 100% & 100% \\ \hline \end{tabular}