Linear Programming

Background

Linear Programming (LP) is a specialized field within mathematical optimization that focuses on maximizing or minimizing a linear objective function given a set of linear inequalities or equations known as constraints. Its relevance spans multiple disciplines, including economics, engineering, and operational research.

Historical Context

The development of linear programming is attributed to the advent of mathematical optimization in the mid-20th century, particularly through the work of George Dantzig, who introduced the Simplex method in 1947. This period marked the rapid growth of computational methods to address problems in logistics during and after World War II.

Definitions and Concepts

Objective Function: A linear function that represents the criterion to be optimized—usually maximized profits or minimized costs.
Constraints: A set of linear inequalities or equations that restrict the permissible solutions for the objective function.
Feasible Region: The solution space that satisfies all the constraints.
Optimal Solution: The feasible solution that maximizes or minimizes the objective function.

Major Analytical Frameworks

Classical Economics

Classical economics does not explicitly use modern linear programming but lays the foundations on resource allocation and optimization.

Neoclassical Economics

Neoclassical theories incorporate optimization, considering LP as a powerful tool for solving utility maximization and cost minimization problems under constraints.

Keynesian Economic

Keynesian models focus less on linear programming formally, but the concept of equilibrium and resource utilization are indirectly related.

Marxian Economics

Marxian economics focuses on the dynamic and distributive aspects of the economy. While Marxian models may deploy linear programming to address allocation efficiency under certain assumptions, it is not a core element.

Institutional Economics

Institutional economists may use LP to model the impacts of different institutional constraints on optimizing outcomes.

Behavioral Economics

Behavioral economics often questions the rationality assumptions underlying LP but can still use it to model bounded rationality scenarios.

Post-Keynesian Economics

This framework critiques mainstream assumptions about optimization but can utilize linear programming in empirical distributions of income and resources.

Austrian Economics

Austrian Economics emphasizes market processes over mathematical optimization but can recognize LP as useful in understanding individual cost-benefit analysis.

Development Economics

Linear programming plays a pivotal role in addressing economic development issues such as optimal resource allocation in different sectors.

Monetarism

Monetarism, with its focus on controlling money supply, might use linear programming to optimize policy implementations impacting multiple economic variables.

Comparative Analysis

Linear programming offers unique advantages like solving large-scale industrial problems and determining optimal allocation of limited resources, but it might lack consideration for non-linear realities and complex human behavior.

Case Studies

Production Planning: Companies use LP to determine optimal production levels to maximize profit margins given limitations of resources like labor and materials.
Diet Problems: LP finds the minimum cost diet that meets all nutritional requirements.
Transportation Problems: Optimization of shipping routes to minimize costs while meeting customer demands across various locations.

Suggested Books for Further Studies

“Linear Programming and Network Flows” by Mokhtar S. Bazaraa, John J. Jarvis, Hanif D. Sherali
“Introduction to Operations Research” by Frederick S. Hillier, Gerald J. Lieberman

Simplex Method: An algorithm for solving linear programming problems by moving along the edges of the feasible region to the optimal vertex.
Dual Problem: A linear programming problem derived from another LP problem, where the original problem’s constraints become the new problem’s objective function boundaries.
Shadow Price: In the context of LP, the value of the marginal utility of relaxing a constraint by one unit.