Interpretation of Dual Variables in LP

Introduction

This document presents an analysis of a linear programming (LP) problem and its dual. The primary LP problem is focused on maximizing the profit of producing gentlemen’s and ladies’ hats, given constraints on cutting and knitting capacities. The dual problem, formulated based on standard procedures, introduces \(y\) variables representing the shadow prices or marginal values of the constraints in the primary LP problem.

LP Problem Formulation

The primary LP problem is formulated as follows:

• Objective Function: Maximize \(50x_1 + 75x_2\)
• Constraints:
  • Cutting Capacity: \(5x_1 + 10x_2 \leq 480\)
  • Knitting Capacity: \(9x_1 + 8x_2 \leq 480\)
  • Non-negativity: \(x_1, x_2 \geq 0\)

Where \(x_1\) and \(x_2\) are the volumes of gentlemen’s and ladies’ hats produced, respectively.

Dual Problem Formulation

The dual problem is formulated as:

• Objective Function: Minimize \(480y_1 + 480y_2\)
• Constraints:
  • \(5y_1 + 9y_2 \geq 50\)
  • \(10y_1 + 8y_2 \geq 75\)
  • Non-negativity: \(y_1, y_2 \geq 0\)

Interpretation of \(y\) Variables

The \(y\) variables in the dual problem represent the shadow prices of the constraints in the primary LP problem. Specifically:

• \(y_1\): Marginal value of one additional minute of cutting capacity.
• \(y_2\): Marginal value of one additional minute of knitting capacity.

These shadow prices indicate how the total profit would change with an additional unit of the respective resource, holding all else constant. Understanding these values is crucial for optimizing resource allocation and enhancing decision-making processes in production planning. These values indicate the increase in the objective function (profit) for each additional unit of the respective resource, assuming everything else remains constant. Essentially, \(y_1\) and \(y_2\) reflect how much the total profit would change if the company could use one more minute of cutting or knitting capacity, holding everything else constant.