(a) Mean chart: CL $44.2$, UCL $47.56$, LCL $40.84$; four points out $\Rightarrow$ out of control. (b) VAM IBFS total cost $=202$.

(a) Mean ($\bar X$) chart. Given ten sample means and ranges ($n=5,\ A_2=0.58,\ D_3=0,\ D_4=2.115$).

$\bar{\bar X}=\frac{\sum \bar X}{10}=\frac{442}{10}=44.2,\qquad \bar R=\frac{\sum R}{10}=\frac{58}{10}=5.8.$

Control limits for the mean chart:

$\text{CL}=\bar{\bar X}=44.2,$

$\text{UCL}=\bar{\bar X}+A_2\bar R=44.2+0.58(5.8)=44.2+3.364=47.56,$

$\text{LCL}=\bar{\bar X}-A_2\bar R=44.2-3.364=40.84.$

Compare each sample mean $43,49,37,44,45,37,51,46,43,47$ with $[40.84,\,47.56]$:

• Sample 2 $(49)$ $>$ UCL, sample 7 $(51)$ $>$ UCL — above the upper limit.
• Sample 3 $(37)$ and sample 6 $(37)$ $<$ LCL — below the lower limit.

Comment: since four sample means fall outside the control limits, the process is not in statistical control (assignable causes of variation are present).

(b) VAM initial basic feasible solution. Balanced problem (supply $=$ demand $=80$). Applying Vogel's Approximation Method (allocate along the line of largest penalty to its least-cost cell each time):