Goal Programming at Widgets and Gadgets

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.

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.