__Break-even Analysis__

Suppose Widgets & Gadgets is a start-up company. They are
confident of their costs and the market price, so that they are sure
that any widget they make and sell will return $100 above its variable
cost, and any gadget they make and sell will return $40 above its
variable cost. But they're not sure of the size of the market,

so their first concern is to break even; that is, to have their total
contribution
at least cover their fixed costs of $5500 per day.

Look at the spreadsheet **<click
here>, **

The tab "Contrib Goal" shows that one way they can break even is to
make
55 widgets and no gadgets. They way we find this is by setting a
goal
of $5,500 for total contribution in cell G3, making the "shortfall" in
cell
E3 a decision variable, and requiring that the sum of the actual
contribution
and the shortfall (cell F3) has to be greater than or equal to the
goal.
The objective function is to minimize the shortfall. The solution
sets
the shortfall to zero, and stops there. (Note that E3, like the
other
decision variables, is constrained to be non-negative.)

If you look at the Sensitivity Report 1, you can see that the decision
variable
"production gadgets" has a "final value" of zero and also a "reduced
cost"
of zero. This is a sign that there are other ways to break
even.
This is an opportunity to either raise the goal or add additional
coals.

If the goal is set high enough, the decision that minimizes the
shortfall will be the same as the decision that maximizes the
contribution.

__Employment Goal__

The "employment goal" tab ignores profitability, and sets as its goal
minimizing
the total number of hours below maximum capacity that the company
operates
at. It's a simple example of weighted sum goal programming in
which
there are two goals, assembly hours and painting hours, and the two
shortfalls
both have equal weight of 1.0. The result is to make 20 widgets
and
80 gadgets, even though this is not the most profitable choice when
gadgets
only return a contribution of $40.

__Sustained Employment Goal__

The "sustained employment goal" tab has a constraint that the firm must
break
even. We have already seen that there is more than one way to do
this,
so we use this flexibility to minimize the total hours of work
lost.
Note that an assembly hour and a painting hour have equal weight, but
all
the loss goes to the assembly department since giving up an hour in
painting
results in less, not more, work hours available in assembly when you
have
to break even. You can see this by adding a constraint that E7
>=
1 E8 >- 1.

__Weighted Compromise between Profit and
Employment__

The "weighted" tab shows a compromise between the goal of making a
profit
and the goal of providing jobs. If one extra hour of work
provided
is taken to be worth $10 in lost profits (cells H3 to H5), the solution
is
to make 20 widgets and 80 gadgets, which is the only choice that gives
full
employment.

Sensitivity Report 2,
cell H12, shows
If you run a sensitivity
analysis, the assembly line in the top "Objective Coefficients" section
will show that we
keep this decision as
long
as the dollar value of an hour of assembly work provided is at least
$1.67. If
you rerun the model with a tradeoff of $1.66 or less for an hour of
assembly work
provided, you'll see that the solution maximizes profits by making 60
widgets
and no gadgets.

Fuzzy Set Goal

This method tries to prevent any one goal from suffering a
substantially greater shortfall than the others. The membership
of each alternative solution in the fuzzy set of solutions that fall
very short on each individual goal goal is computed as a fuzzy
membership grade expressed as a percentage based on the importance and
the units of measure of each goal. The membership of the
alternative solution in the fuzzy set of alternatives that fall
significantly short on any goal is calculated as the greatest of its
memberships in the fuzzy sets corresponding to each individual
goal. The alternative solution whose membership in the fuzzy set
of alternatives that fall significantly short on any goal is less than
that of any other alternative solution is chosen as optimal.