91Ó°ÊÓ

In linear programming, constraint inequalities are mathematical expressions that establish boundaries for the variables involved in an optimization problem.
These inequalities define conditions that the solutions must satisfy. In the given exercise, we have the following constraint inequalities:

• \(3s + 2t + u \geq 60\)
• \(2s + t + 3u \geq 60\)
• \(s + 3t + 2u \geq 60\)
• \(s \geq 0, t \geq 0, u \geq 0\)

These constraints essentially carve out a "space" in the solution domain that valid solutions must lie within.
In simpler terms, they tell us how the variables \(s\), \(t\), and \(u\) can interact with each other and the limits they are subject to. It is crucial to set the stage for feasible solutions which will be examined in the context of the Feasible Region.

The feasible region in linear programming is the set of all points that satisfy the constraint inequalities simultaneously.
This area is formed by the intersection of all the geographies created by each individual constraint. For our problem, with the constraints expressed in three-dimensional space, the feasible region becomes more of a shape, like a polyhedron.
Any point inside or on the surface of this polyhedron represents a potential solution where all constraints are satisfied. Understanding this visual representation aids in seeing which combinations of \(s\), \(t\), and \(u\) form legal solutions.Intersections of the planes described by each inequality define this space and assessment of solutions takes place here. Thus, identifying this "area" is crucial for finding answers to our problem, as only within this region can we find optimal solutions in terms of resource allocation.

In the context of linear programming, the objective function is the mathematical expression that we aim to optimize—either to maximize or minimize.
For this exercise, the given objective function is:\[c = s + t + u\]The goal is to minimize \(c\) under the constraints given. Objective function evaluation involves calculating this function for all significant points in our feasible region, mainly focusing on the vertices.
These calculations help determine how each combination of \(s\), \(t\), and \(u\) contributes to the function's value and subsequently guide us towards achieving the best possible outcome—here, the smallest total value of \(s + t + u\).
Let's move to see why vertices matter in this process next.

Vertices in optimization refer to the corner points of the feasible region. In linear programming problems, these points are critical.
They represent the combinations of \(s\), \(t\), and \(u\) where different planes (or constraint expressions) intersect.Since the objective function in a linear program changes linearly, it achieves its extremum values (maximum or minimum) at these vertices.
As such, we systematically identify them by solving simultaneous equations. Exploring these vertices tells us where exactly in the feasible region our sought-after minimum value of the objective function will occur.
It's often computational and visual efficiency in linear programs to move directly from one vertex to another to quickly find the optimal solution.