Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{ll} \text { Minimize } & C=200 x+150 y+120 z \\ \text { subject to } & 20 x+10 y+z \geq 10 \\ & x+y+2 z \geq 20 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

Step-by-Step Solution

Answer

The dual problem associated with the given primal problem is: \[ \begin{array}{ll} \text { Maximize } & \pi = 10u + 20v \\\ \text { subject to } & 20u + v \leq 200 \\\ & 10u + v \leq 150 \\\ & u+2v \leq 120 \\\ u & \geq 0, v \geq 0 \end{array} \] While we can use the graphical method to solve the primal problem, it might not be the most efficient with three decision variables. A better approach would be to use the Simplex method or other linear programming techniques to solve the problem.

Step 1: Constructing the dual problem

To construct the dual problem of the given primal problem, we will follow these steps: 1. Replace the objective function in the primal problem by a linear combination of the constraints: each constraint is multiplied by a new decision variable, which are called the dual variables. 2. Rewrite the coefficients of the primal decision variables, which are the x, y, and z variables in this case, as constraints in the dual problem. 3. Since we are minimizing in the primal problem, the objective function in the dual problem should maximize the linear combination of the constraints formed in step 1. 4. The inequality signs in the primal problem constraints are greater than or equal to (≥). This means that in the dual problem, we will use less than or equal (≤) constraints. Here is the dual problem: \[ \begin{array}{ll} \text { Maximize } & \pi = 10u + 20v \\\ \text { subject to } & 20u + v \leq 200 \\\ & 10u + v \leq 150 \\\ & u+2v \leq 120 \\\ u & \geq 0, v \geq 0 \end{array} \]

Step 2: Solving the Primal Problem using Graphical Method

To solve the primal problem, we can use the graphical method. 1. Plot the constraint inequalities in the form y = f(x) by treating z as a constant: For the first constraint, rewrite as \(y \geq -2x + 10 - 0.1z\): For the second constraint, rewrite as \(y \geq -x + 20 - 2z\). 2. Find the feasible region, which is the intersection of the constraint regions. 3. Depending on the feasible region and the objective function, choose corner points from the feasible region and substitute them in the objective function, C. 4. From these corner point values, select the minimum C as the solution. Since we are minimizing C, along with the solution, one should report the minimum obtained in this way. Note: The Graphical Method is done visually and is most suitable when solving problems with no more than two decision variables. In this exercise, the problem has three decision variables, which makes this method not the most efficient here. Therefore, I recommend using the Simplex method or other linear programming techniques to solve this problem.

Key Concepts

Primal Problem

In linear programming, the primal problem is the original problem we aim to solve. It involves either the minimization or maximization of a linear objective function. The primal problem is subject to a set of constraints typically expressed as inequalities. The primal problem in our exercise focuses on minimizing the cost represented by the expression \( C = 200x + 150y + 120z \).

We deal with constraints like:

• \( 20x + 10y + z \geq 10 \)
• \( x + y + 2z \geq 20 \)

These constraints also include non-negativity conditions: \( x \geq 0, y \geq 0, z \geq 0 \). Primal problems can be solved efficiently using different linear programming techniques, and being familiar with how the constraints interact is crucial for finding a solution that meets all conditions.

Linear Programming

Linear programming (LP) is a mathematical approach to solving problems that require an optimization of a linear objective function. This mathematical method finds the best outcome, such as minimum cost or maximum profit, given a set of linear constraints.

In essence, LP helps in decision-making where we fit all possible scenarios within feasible limits set by inequalities. It’s a powerful tool particularly used in fields like economics, business, engineering, and military applications.

Key elements of LP include:

• An objective function to be maximized or minimized.
• Constraints comprising linear inequalities.
• Decision variables that impact the outcome.

The exercise features this through minimizing the cost function with specific constraints, providing a classic example of a linear programming challenge.

Simplex Method

The Simplex Method is a well-known algorithm used in linear programming to find the optimal solution to problems. It's especially useful when graphical methods are impractical, which is often the case when there are more than two decision variables, as in our exercise.

The method works by moving along the edges of the feasible region defined by the constraints to reach an optimal corner point where the objective function has the best value. The process involves:

• Initialization, where a starting point is chosen from the set of constraints.
• Iterative improvement, where directions are chosen that improve the objective function until no further improvements are possible (optimal solution).

By employing the Simplex Method, complex problems that involve several variables and constraints can be solved efficiently and effectively. It’s a preferred method used in industries to optimize resource allocation and operational planning.