1. **State the problem:** Determine the headcount allocation between Path A and Path B such that the combined weekly units meet or exceed 318140.56 units, the combined weekly defects do not exceed 247.14, and total headcount is 38 with at least 20% assigned to each path. 2. **Given data:** - Total headcount $H = 38$ - Hours per week per headcount $= 40$ - Target weekly units $U_t = 318140.56$ - Weekly defects threshold $D_t = 247.14$ - Path A units/hour/person $u_A = 156.68$ - Path A defects/hour/person $d_A = 0.04$ - Path B units/hour/person $u_B = 201.94$ - Path B defects/hour/person $d_B = 0.13$ - Minimum headcount per path $\geq 0.2 \times 38 = 7.6$ 3. **Define variables:** Let $x$ = headcount assigned to Path A Then $38 - x$ = headcount assigned to Path B 4. **Formulate constraints:** - Total units produced: $$40 \times (156.68x + 201.94(38 - x)) \geq 318140.56$$ - Total defects produced: $$40 \times (0.04x + 0.13(38 - x)) \leq 247.14$$ - Headcount bounds: $$7.6 \leq x \leq 30.4$$ 5. **Simplify units constraint:** $$40(156.68x + 201.94(38 - x)) \geq 318140.56$$ $$40(156.68x + 201.94 \times 38 - 201.94x) \geq 318140.56$$ $$40((156.68 - 201.94)x + 201.94 \times 38) \geq 318140.56$$ $$40(-45.26x + 7673.72) \geq 318140.56$$ $$-1810.4x + 306948.8 \geq 318140.56$$ $$-1810.4x \geq 318140.56 - 306948.8$$ $$-1810.4x \geq 11191.76$$ $$x \leq \frac{11191.76}{-1810.4} = \cancel{\frac{11191.76}{\cancel{-1810.4}}} \cancel{\times \frac{-1}{-1}} = -6.18$$ Since $x$ must be positive and at least 7.6, this inequality is impossible to satisfy if $x \leq -6.18$. So units constraint requires $x \leq -6.18$ which conflicts with headcount bounds. 6. **Simplify defects constraint:** $$40(0.04x + 0.13(38 - x)) \leq 247.14$$ $$40(0.04x + 4.94 - 0.13x) \leq 247.14$$ $$40(-0.09x + 4.94) \leq 247.14$$ $$-3.6x + 197.6 \leq 247.14$$ $$-3.6x \leq 49.54$$ $$x \geq \frac{49.54}{-3.6} = -13.76$$ This is always true since $x$ is positive. 7. **Check feasibility:** - Headcount bounds: $7.6 \leq x \leq 30.4$ - Units constraint requires $x \leq -6.18$ (impossible) - Defects constraint always true 8. **Interpretation:** The units constraint cannot be met with the given headcount and minimum allocation constraints. 9. **Check total units if all headcount assigned to Path B (max units):** $$40 \times 201.94 \times 38 = 40 \times 201.94 \times 38 = 306,959.2$$ This is less than target units 318,140.56. 10. **Conclusion:** Current headcount is insufficient to meet target weekly units even if all staff assigned to Path B. **Answer:** Option C: Increase Headcount up to 5% to meet Target Weekly Units