Understanding LP Solve Formula and Addressing Errors

LP Solve is a popular linear programming solver used to solve optimization problems. In this article, we will delve into the world of LP Solve and address errors in the provided formula.

Introduction to Linear Programming (LP) Solve

Linear Programming (LP) is a method used to optimize a linear objective function, subject to a set of linear constraints. The goal is to find the values of variables that maximize or minimize the objective function, while satisfying all the constraints.

LP Solve is a software package that can be used to solve LP problems. It uses an interior-point algorithm to find the optimal solution.

Formula and Objective Function

The provided formula uses the lp() function from the lpSolve package in R. The formula defines:

• An objective function obj.fun with three components, each representing a different aspect of the optimization problem.
• A constraint matrix constr with 11 rows and 9 columns, where each row represents one constraint and each column represents a variable.
• Constraint directions (constr.dir) indicating whether each constraint should be maximized or minimized.
• Right-hand side values (rhs) representing the target value for each variable.

The formula also sets compute.sens = TRUE, which computes sensitivity analysis for the optimal solution.

Analyzing Errors in the Formula

Upon examining the provided formula, we notice two errors:

1. Inconsistent constraint matrix: The second row of the constraint matrix has coefficients 2, which is inconsistent with the direction specified in the constr.dir vector.
2. Incorrect objective function: The objective function should have only three components, but it appears to have four.

Correcting Errors and Understanding LP Solve

To correct these errors, we need to understand how to manipulate constraints and objective functions to derive coefficients that satisfy all conditions.

Manipulating Constraints

LP Solve uses an interior-point algorithm to solve linear programming problems. To add new constraints, we need to modify the existing constraints to meet the required format.

In the provided example, the constraint matrix constr has 11 rows and 9 columns. The last two constraints have coefficients that are not consistent with the direction specified in constr.dir. We can rewrite these constraints to derive the correct coefficients:

# Last constraint (x1 + x2 - 3/450)
constr[-11, ] <- c(1, 1, 0, 1/750, 1/900, 0, 0, 0, 0)

# Second-to-last constraint (x1 + x2 - 4/450)
constr[-10, ] <- c(1, 1, 0, 2/750, 2/900, 0, 0, 0, 0)

Correcting Objective Function

The objective function should have only three components. We can modify the formula to achieve this:

# Corrected objective function
obj.fun <- c(420, 360, 300)

Applying LP Solve with Corrected Formula

Once we correct the errors and understand how to manipulate constraints and objective functions, we can apply LP Solve using the corrected formula.

library(lpSolve)

# Define objective function
obj.fun <- c(420, 360, 300)

# Define constraint matrix and directions
constr <- matrix(c(
    1, 1, 1, 1, 1, 1, 1, 1, 1,
    20, 15, 12, 20, 15, 12, 20, 15,
    12, 1, 1, 1, 1, 1, 1, 1, 1
), nrow = 11, byrow = TRUE)
constr.dir <- c("&lt;=", "&lt;=", "&lt;=", "&lt;=", "&lt;=", "&lt;=", "&lt;=",
                 "&lt;=", "&lt;=", "=")

# Define right-hand side values
rhs <- c(750, 900, 450, 13000, 12000, 5000, 900, 1200, 750, 0, 0)

# Solve linear programming problem using LP Solve
probsol <- lp("max", obj.fun, constr, constr.dir, rhs, compute.sens = TRUE)

# Print the solution
print(probsol$solution)

Conclusion

In this article, we explored the world of Linear Programming (LP) and addressed errors in a provided formula. We discussed how to manipulate constraints and objective functions using LP Solve and corrected the errors in the formula.

By understanding the basics of LP and how to apply LP Solve correctly, you can solve complex optimization problems efficiently.

References

• lpSolve Package in R

Last modified on 2024-01-08