Feasible_region: Meaning, Criticisms & Real-World Uses

What Is Feasible Region?

The feasible region, also known as the feasible set or solution space, is the set of all possible points that satisfy all the constraints of a mathematical modeling problem. In quantitative finance and operations research, particularly in linear programming, this region represents all the valid combinations of decision variables that adhere to the problem's limitations. Any point within the feasible region is considered a "feasible solution," meaning it satisfies all the specified conditions, even if it may not be the optimal outcome for the problem's objective function.

History and Origin

The concept of a feasible region is intrinsically linked to the development of linear programming and optimization techniques. Its origins trace back to the mid-20th century, notably to the work of George Dantzig. In 1947, while working for the U.S. Army Air Forces, Dantzig developed the Simplex Method, a pivotal algorithm for solving linear programming problems. This breakthrough allowed for efficient resource allocation and planning in complex systems, effectively giving rise to the formal definition and systematic use of the feasible region. The ability to identify this set of permissible solutions was crucial for the widespread adoption of linear programming across various industries, from logistics to manufacturing and finance. Dantzig's work was foundational in establishing the field of linear programming.¹⁰, ¹¹

Key Takeaways

The feasible region encompasses all possible solutions that satisfy every constraint in an optimization problem.
It is a fundamental concept in linear programming and related quantitative methods.
Any point within the feasible region represents a valid, though not necessarily optimal, solution.
The shape and size of the feasible region are determined by the nature and number of constraints imposed on the problem.
Identifying the feasible region is the first step toward finding an optimal solution, which usually lies at one of its corners or vertices.

Formula and Calculation

The feasible region itself is not defined by a single formula but rather by a system of inequalities and equalities. For a linear programming problem, the feasible region is often described as the set of points ( (x_1, x_2, \ldots, x_n) ) that satisfy:

\begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n \le b_1 \\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n \le b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n \le b_m \\ x_j \ge 0 \quad \text{for } j=1, \dots, n \end{cases}

Where:

( x_j ) are the decision variables to be determined.
( a_{ij} ) are the coefficients representing the consumption or contribution of each variable to a given constraint.
( b_i ) are the right-hand side values, representing the limits or requirements for each constraint.
The inequalities (or equalities, depending on the problem) define the boundaries of the feasible region.
( x_j \ge 0 ) are non-negativity constraints, commonly applied in many real-world problems.

This system of linear constraints geometrically forms a convex polyhedron in n-dimensional space.⁸, ⁹

Interpreting the Feasible Region

Interpreting the feasible region involves understanding the landscape of possibilities within a given problem. Each point within this space represents a combination of decision variables that is permitted by all the established constraints. For instance, in a business context, if a company is trying to maximize profit subject to limited raw materials and labor hours, the feasible region would illustrate all possible production levels for different products that do not exceed the available resources. An unbounded feasible region, conversely, indicates that solutions can theoretically extend infinitely in one or more directions without violating constraints, though this is less common in practical applications where resources are finite. The optimal solution, whether maximizing or minimizing an objective function, will always lie on the boundary of the feasible region, specifically at one of its vertices or "corners."⁶, ⁷

Hypothetical Example

Consider a small investment firm aiming to allocate a client's funds between two types of assets: Asset A and Asset B.
The firm has the following constraints for this particular client:

The total investment must not exceed $100,000.
At least $20,000 must be invested in Asset A.
At least $30,000 must be invested in Asset B.
The investment in Asset A should be no more than twice the investment in Asset B.

Let ( x_A ) be the amount invested in Asset A and ( x_B ) be the amount invested in Asset B.
The inequalities defining the feasible region are:

( x_A + x_B \le 100,000 ) (Total investment constraint)
( x_A \ge 20,000 ) (Minimum Asset A investment)
( x_B \ge 30,000 ) (Minimum Asset B investment)
( x_A \le 2x_B ) (Asset A to Asset B ratio constraint)
( x_A \ge 0, x_B \ge 0 ) (Non-negativity constraints, though covered by minimums here)

The feasible region would be the area on a two-dimensional graph where all these conditions are simultaneously met. If the client also wants to maximize expected return, the firm would then define an objective function (e.g., ( \text{Maximize } 0.05x_A + 0.07x_B ) for a 5% and 7% return respectively) and use linear programming to find the point within this feasible region that yields the highest return.

Practical Applications

The concept of a feasible region is widely applied in various fields of quantitative finance and beyond, particularly wherever optimization is required.

Portfolio Optimization: Investors use the feasible region to identify all possible combinations of assets that satisfy their specific risk tolerance, return objectives, and other investment constraints, such as diversification requirements or regulatory limits. The optimal portfolio is then selected from this feasible set.
Production Planning: Businesses leverage the feasible region to determine optimal production schedules given limitations on raw materials, labor, machinery capacity, and market demand. This helps in efficient resource allocation and cost minimization.
Logistics and Supply Chain Management: Transportation companies and logistics providers define feasible regions based on vehicle capacity, delivery routes, time windows, and warehouse space to optimize delivery schedules and minimize shipping costs.
Financial Planning: Individuals and institutions might use the feasible region to explore viable savings or expenditure plans that meet budget constraints and financial goals.
Regulatory Compliance: Organizations often operate within a feasible region defined by regulatory constraints and policies. Understanding this region helps them ensure compliance while optimizing operations.
Digital Transformation Initiatives: In a broader business context, companies are increasingly focused on optimization to maximize the value of digital investments, which inherently involves identifying and operating within feasible sets defined by technological capabilities, talent, and strategic objectives.³, ⁴, ⁵

Limitations and Criticisms

While the feasible region is a powerful concept in optimization, it does have limitations. One primary criticism lies in the assumption that all constraints and the objective function are strictly linear. In real-world scenarios, relationships between variables are often non-linear, introducing complexities that linear programming, and thus the definition of a simple polyhedral feasible region, cannot fully capture. Non-linear constraints would result in a non-linear feasible region, requiring more advanced optimization techniques.

Another limitation is the challenge of accurately identifying and quantifying all relevant constraints. Omitting or misrepresenting a critical constraint can lead to a feasible region that includes unrealistic or unattainable solutions, rendering the subsequent optimization meaningless. Furthermore, the number of decision variables and constraints can grow exponentially in complex problems, making the computational determination of the feasible region and its vertices highly intensive, even for advanced algorithms like the Simplex Method. While the simplex method is effective, its computational complexity can be exponential in worst-case scenarios.² The practical application may also face challenges when dealing with uncertainty or dynamic environments, as the feasible region derived from static constraints might not hold true over time.

Feasible Region vs. Efficient Frontier

The feasible region and the efficient frontier are related but distinct concepts, primarily used within the domain of portfolio optimization under Modern Portfolio Theory (MPT).

The feasible region (or opportunity set) represents all possible portfolios that an investor can construct given a set of available assets and any defined constraints, such as minimum investment amounts, asset allocation limits, or non-negativity requirements for portfolio weights. It encompasses every combination of assets, regardless of whether they are optimal in terms of risk-return tradeoff. It is the entire set of "doable" portfolios.

The efficient frontier, on the other hand, is a subset of the feasible region. It comprises only those portfolios that offer the highest expected return for a given level of risk, or the lowest risk for a given expected return. Essentially, it's the "upper boundary" of the feasible region in a risk-return plot. Portfolios on the efficient frontier are considered "optimal" because no other portfolio within the feasible region can provide a better risk-return tradeoff. Investors typically aim to select a portfolio that lies somewhere along the efficient frontier, depending on their individual risk tolerance.¹

FAQs

What happens if there is no feasible region?

If there is no feasible region, it means that the set of constraints defined for the problem are contradictory or impossible to satisfy simultaneously. In such a case, there is no valid solution that meets all the specified conditions. This situation is referred to as an "infeasible" problem.

Can a feasible region be unbounded?

Yes, a feasible region can be unbounded. This occurs when the constraints do not completely enclose the solution space, meaning that one or more decision variables can increase indefinitely without violating any constraint. While mathematically possible, in practical resource allocation problems, feasible regions are usually bounded due to finite resources.

Is the optimal solution always found at a corner of the feasible region?

For linear programming problems, if an optimal solution exists, it will always occur at one of the vertices (or "corner points") of the feasible region. If multiple optimal solutions exist, they will form a line segment along an edge of the feasible region, connecting two optimal vertices. This property is a cornerstone of the Simplex Method for solving linear programs.

How is the feasible region used in portfolio management?

In portfolio management, the feasible region represents all possible combinations of assets that an investor can choose from while adhering to their investment constraints and goals. From this feasible region, investors then seek to identify the efficient frontier, which contains the optimal portfolios offering the best expected return for a given level of risk. This process is central to capital allocation and investment strategy.