Building And Simplifying The Objective Function

From a Situation to an Objective

In an optimization problem, the objective is the quantity we want to make as small or as large as possible. It might represent cost, area, volume, distance, time, or profit. Writing down that quantity as a function of a decision variable is a mathematical model of the situation.

Constructing the formula is more than translating words into symbols. The formula must reflect how the real quantity is assembled, use variables consistently with the constraints, and be evaluated on the feasible domain. Once it is correct, algebra can often make it easier to work with. The key is to simplify without changing which feasible choices are best.

The feasible domain and the objective are related but distinct. Constraints determine which choices are allowed; the objective assigns a value to each allowed choice. A formula that is correct outside the feasible domain may be irrelevant there, while a formula that ignores a constraint may describe impossible choices.

Build the Objective in Pieces

A reliable construction starts by naming the decision variable and identifying what the objective counts. Then express each component in terms of that variable, combine the components according to the situation, and check that the resulting formula has the intended units. If an equality constraint has already been used to express one quantity in terms of another, use that relationship consistently rather than treating both quantities as independent.

For example, a total cost made from several charges is their sum. Profit is revenue minus cost. A total travel time from several route segments is the sum of the times for those segments. These relationships are part of the model, not optional algebraic choices.

What Simplification Can Preserve

For a fixed feasible domain, the important feature of an objective is its ranking of feasible choices. If a choice has a smaller value than another choice before simplification, a transformation is safe for minimization when it keeps that ordering. For maximization, it must keep the ordering in the direction of larger values.

Proof. Suppose \(x_*\) minimizes \(f\). Then \(f(x_*)\le f(x)\) for every \(x\in D\). Since \(g\) is increasing, applying \(g\) preserves this inequality: \(g(f(x_*))\le g(f(x))\). Therefore \(x_*\) minimizes \(g\circ f\). Conversely, if \(x_*\) minimizes \(g\circ f\), then \(g(f(x_*))\le g(f(x))\) for every \(x\in D\). Strict increase means that this inequality implies \(f(x_*)\le f(x)\). Thus \(x_*\) minimizes \(f\). The proof for maximizers uses the same argument with the inequalities reversed. This establishes equality of the optimizer sets, not just agreement at one particular choice.

Several useful simplifications follow. Adding a constant, \(g(u)=u+C\), is strictly increasing. Multiplying by a positive constant, \(g(u)=cu\) for \(c>0\), is also strictly increasing. Therefore, these operations preserve both minimizers and maximizers. Multiplication by a negative constant reverses the order: it turns minimizers into maximizers and maximizers into minimizers.

A strictly decreasing transformation reverses rankings as well. In particular, if \(f\) is positive, minimizing \(f\) is equivalent to maximizing \(1/f\), because \(u\mapsto 1/u\) is strictly decreasing for \(u>0\). This can be useful, but it changes the direction of the optimization. A transformation must be checked on the values actually taken by the objective, not assumed safe in every setting.

Equivalent Objectives for Nonnegative Quantities

Sometimes an objective includes a square root that makes the formula less convenient. If the quantity is known to be nonnegative, squaring it can simplify the expression without changing its minimizers: the function \(u\mapsto u^2\) is strictly increasing for \(u\ge 0\). The nonnegativity condition matters. On all real numbers, squaring is not strictly increasing, since, for example, \((-2)^2=2^2\). It can therefore merge distinct objective values and introduce additional minimizers.

Units, Constants, and Common Errors

Units are a practical check on an objective. A sum such as “distance plus time” is not a meaningful total unless the problem defines a conversion that makes the quantities comparable. Likewise, cost and revenue can be subtracted to define profit because both are monetary amounts over the same period. Expressions in an objective should have compatible units, and the variable should retain the meaning assigned to it when earlier constraints were used.

Adding a constant to an objective does not affect which feasible choice is best, but the constant must really be the same for every feasible choice. If a supposed “constant” depends on the decision variable, it is not constant and cannot simply be discarded. Similarly, multiplying an objective by a positive constant preserves its optimizer locations, while a negative constant changes a minimization into a maximization, or vice versa.

A frequent modeling error is forgetting that revenue is a rate per unit multiplied by the number of units. For the sales model above, a tempting but incorrect calculation is to write profit as \(80-3q-(200+20q)\). This subtracts weekly cost from a price per unit, so the units do not match; it also fails to calculate total revenue. The correction is to form revenue first as \(q(80-3q)\), then subtract the weekly cost:

Before relying on an objective, check that it counts the requested quantity, uses the constraints consistently, and has sensible units throughout. Then verify any simplified expression by reversing or checking the algebra, or by testing a feasible value in both forms. Testing a value does not prove two formulas identical, but it can expose an error.