\pdfpagewidth=8.5truein  \pdfpageheight=11truein
\hsize=5.5truein  \vsize=9truein
\pdfhorigin=1.5truein  \pdfvorigin=1truein

\headline={\hfil}  \footline={\hfil}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent Whiz-Bang Manufacturing produces four lines of products. Each unit
of each of the product lines requires one part called a hypergizmo, and
Whiz-Bang has 270~hypergizmos on hand. Each unit of each of the product lines
(except the first) also requires one or more parts called e-sprockets;
Whiz-Bang has 480~e-sprockets in stock. This week Whiz-Bang has 600~hours of
labor available for assembly and, separately, 566~hours available for
finishing. The per-unit profit and resource requirements for each of the
product lines are given in the table below:
$$\vbox{\offinterlineskip
\halign{\strut\quad\hfil#\hfil\quad&\vrule#&&\strut\quad\hfil#\hfil\quad\cr
Product line&&Profit&E-sprockets&Assembly hours&Finishing hours\cr
\omit&height2pt&\omit\cr
\noalign{\hrule}
\omit&height2pt&\omit\cr
1&&\$10&0&2&1\cr
2&&\$14&1&2&2\cr
3&&\$17&3&5&3\cr
4&&\$12&2&3&2\cr
}}$$
The marketing department has asked for at least 30~units of the first product
line to be produced this week, because that is the original Whiz-Bang product
and they are planning a ``throwback'' marketing campaign. How much of each
product line should Whiz-Bang produce this week in order to maximize total
profit?

\bigskip\hrule\bigskip

\noindent Here is the linear program:
$$\eqalignno{
	{\rm maximize}\quad10x_1+14x_2+17x_3+12x_4&&\hbox{[profit]}\cr
	{\rm subject\ to}\quad
	\phantom{01}x_1+\phantom{01}x_2+\phantom{01}x_3+\phantom{01}x_4&\le270&\hbox{[hypergizmos]}\cr
	x_2+\phantom03x_3+\phantom02x_4&\le480&\hbox{[e-sprockets]}\cr
	2x_1+\phantom02x_2+\phantom05x_3+\phantom03x_4&\le600&\hbox{[assembly hours]}\cr
	x_1+\phantom02x_2+\phantom03x_3+\phantom02x_4&\le566&\hbox{[finishing hours]}\cr
	x_1\phantom{{}+00x_2+00x_3+00x_4}&\ge\phantom030\qquad\qquad\qquad&\hbox{[production requirement]}\cr
\noalign{\smallskip}
	&\phantom{{}=000}\llap{$x_1\ge0,\quad x_2\ge0,\quad x_3\ge0,\quad x_4\ge0$}.&\cr
}$$

\noindent Here is the initial simplex tableau:
$$\vbox{\offinterlineskip
\halign{%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&\vrule#&%
	\strut\enspace\hfil$#$\enspace\cr
x_1&x_2&x_3&x_4&s_1&s_2&s_3&s_4&s_5&a_5&&\cr
1&1&1&1&1&0&0&0&0&0&&270\cr
0&1&3&2&0&1&0&0&0&0&&480\cr
2&2&5&3&0&0&1&0&0&0&&600\cr
1&2&3&2&0&0&0&1&0&0&&566\cr
1&0&0&0&0&0&0&0&-1&1&&30\cr
\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&height2pt&\omit\cr
\noalign{\hrule}
\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&height2pt&\omit\cr
-10&-14&-17&-12&0&0&0&0&0&0&&0\cr
-1&0&0&0&0&0&0&0&1&0&&-30\cr
}}$$

\noindent And here is the optimal simplex tableau:
$$\vbox{\offinterlineskip
\halign{%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&%
	\strut\enspace\hfil$#$\enspace&\strut\enspace\hfil$#$\enspace&\vrule#&%
	\strut\enspace\hfil$#$\enspace\cr
x_1&x_2&x_3&x_4&s_1&s_2&s_3&s_4&s_5&a_5&&\cr
0&1&0&2/3&5/3&0&-1/3&0&1&-1&&220\cr
0&0&0&1/3&1/3&1&-2/3&0&-1&1&&200\cr
0&0&1&1/3&-2/3&0&1/3&0&0&0&&20\cr
0&0&0&-1/3&-4/3&0&-1/3&1&-1&1&&36\cr
1&0&0&0&0&0&0&0&-1&1&&30\cr
\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&height2pt&\omit\cr
\noalign{\hrule}
\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&\omit&height2pt&\omit\cr
0&0&0&3&12&0&1&0&4&-4&&3720\cr
}}$$

\vfill\eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcount\qno \qno=0
\def\question{\global\advance\qno by1\par\item{\bf\the\qno.}\ignorespaces}

\question What is the optimal basic solution for the original linear program?

\vfill

\question A hypergizmo salesperson contacts Whiz-Bang and offers to sell them
some more hypergizmos. How much should Whiz-Bang be willing to pay per unit for
more hypergizmos? How many should they be willing to buy at that price?

\vfill

\question Suppose an employee were to open a dusty closet and discover a
long-forgotten box containing a dozen hypergizmos. How would this windfall
change Whiz-Bang's total profit and production schedule?

\vfill

\question A competitor, Kablooie Industries, has cut the selling price of one
of their product lines, and in response the Whiz-Bang sales department would
like to reduce the selling price (and hence the profit) of their second product
line by \$1 per unit, effective immediately. What effect would this change have
on Whiz-Bang's total profit and production schedule?

\vfill

\question Whiz-Bang receives an advertisement from E.~Spacely E-Sprockets,
Inc., announcing a special sale on e-sprockets this week only! How much should
Whiz-Bang be willing to pay per unit for more e-sprockets for use in this
week's production?

\vfill

\question The sales department is considering a change in the selling price of
the first product line. What range of changes to the selling price for this
product line will leave the current optimal basis unchanged?

\vfill

\question The sales department would like to increase the number of the fourth
product line sold. Assuming that Whiz-Bang can sell all of the products it
makes, what change to the selling price would be necessary to make it
profitable to do so?

\vfill

\question If Whiz-Bang has the opportunity to hire more labor, should it hire
more labor for assembly or for finishing? How much should it be willing to pay
for each hour of this labor? How many additional hours of labor should it be
willing to hire at this rate?

\vfill

\question What range of per-unit profit for the third product line will leave
the optimal basis unchanged?

\vfill

\question What is the effect of the marketing department's request?

\eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent{\bf Answers}

\bigskip

\item{\bf1.} $x_1=30$, $x_2=220$, $x_3=20$, $x_4=0$.

\smallskip

\item{\bf2.} Whiz-Bang should be willing to buy up to 27~more hypergizmos at
up to \$12 each.

\smallskip

\item{\bf3.} With twelve additional hypergizmos, Whiz-Bang should produce
30,~240, 12, and~0 units of the four product lines, respectively. Their total
profit will increase to \$3,864.

\smallskip

\item{\bf4.} If the per-unit profit on the second product line is reduced
by~\$1, the optimal production schedule remains the same (30,~220, 20, and~0
units of the four product lines, respectively), but total profit will be
reduced to~\$3,500.

\smallskip

\item{\bf5.} Whiz-Bang should not be willing to buy additional e-sprockets at
any (positive) price for use in this week's production.

\smallskip

\item{\bf6.} The selling price of the first product could be increased by~\$4
or decreased by any amount without changing the optimal basis.

\smallskip

\item{\bf7.} The selling price of the fourth product must be increased by at
least~\$3 before it becomes profitable to produce and sell.

\smallskip

\item{\bf8.} Whiz-Bang should not be willing to hire more labor for finishing.
It should be willing to hire up to 108~additional hours of assembly labor at up
to \$1~per hour (if it can possibly find labor that cheap---maybe
Oompa-Loompas).

\smallskip

\item{\bf9.} The per-unit profit for the third product line can range from
\$14 to~\$35 without changing the optimal basis.

\smallskip

\item{\bf10.} The total effect of the marketing department's request is to
decrease total profit by \$120; the optimal total profit could be \$3,840 if
that request were ignored.

\bye
