Linear Programming Tools

# Linear Optimization with Sensitivity Analysis Tools

This JavaScript Works Well in Netscape Navigator Version 4 (such as 4.7). If this is not feasible for you, you may download (free-of-charge) a software package that solves Linear Programs models by the Simplex Method and/or the Push-and-Pull Method:

Professor Hossein Arsham

This JavaScript E-labs learning object is intended for finding the optimal solution, and post-optimality analysis of small-size linear programs. It provides the optimal value and the optimal strategy for the decision variables. The necessary tools are produced to perform various sensitivity analyses on the coefficients of the objective function and on the right-hand-side values of the constraints.

Other JavaScript learning objects for decision making in this series are categorized under different areas of applications at the MENU section on this page.

In entering your data to move from cell to cell in the data-matrix use the Tab key not arrow or enter keys.

Instructions and Preliminaries:

1. This JavaScript is intended for experimentation in deepening understanding of LP concepts and techniques. Therefore it is designed for LP problems with at most 3-decision variables with at most 3-constraints. That is, 3-by-3 is the largest problem size.
2. Decision variable names must be single letters, e.g., X, Y, Z.
3. Convert the minimization problem into a maximization one (by multiplying the objective function by -1).
4. Every decision variable appear in any constraints must also appear in the objective function, possibly with zero coefficient if needed.
5. All decision variables must be non-negative.
To achieve this requirement, convert any unrestricted variable X to two non-negative variables by substituting T - X for X. This increases the dimensionality of the problem by only one (introduce one y variable) regardless of how many variables are unrestricted. For example, if Y is also unrestricted variable, the substitute - Y for Y.
If any variable, say X is restricted to be non-positive, substitute - X for every X. This reduces the complexity of the problem.
6. All decision variables must appear in the left side of the constraints, while the numerical values must appear on the right side of the constraints (that is why these numbers are called the RHS values).
7. All RHS values must be non-negative. Multiply both sides of the constraint by -1, if needed.
8. For non-integer coefficients for the decision variables, in the objective function, and the constraints, use fractional equivalent in bracket, e.g., for X/5 use (1/5)X.
9. All constraints must be in and/or form. You may enter in the non-negativity conditions, if you wish.
10. Any strictly equality constraint should be replaced by two simultaneous inequalities of form and with the same RHS value.
11. The output includes the optimal value and the optimal strategy for the decision variables. The optimal value of slack/surplus for each constraint can be found by evaluating each constraint using the available optimal strategy for the decision variables.
12. The output contains also the necessary tools to perform various sensitivity analyses on the coefficients of the objective function and on the right-hand-side values of the constraints. These tools are applicable to any LP problem having a unique optimal solution.

Solve the standard formatted problem, and then substitute these changes back to get the values for the original variables and optimal value.

An Example: Consider the following problem with an equality constraint:

Maximization 3x + 2y + z

subject to:
4x + 2y + 3z = 12
x + z 1
x, y, and z 0.

Converting the equality constraints to two inequality constraints, we have the following equivalent problem:

Maximization 3x + 2y + z

subject to:
4x + 2y + 3z 24
4x + 2y + 3z 24
x + z 1
x, y, and z 0.

Enter your standard LP problem in the following table, then click on the "Calculate" button.

In entering your data to move from cell to cell in the data-matrix use the Tab key not arrow or enter keys.
Enter your LP using Netscape Navigator. Click on the "Example" to see how:
Subject to

You May Enter the Non-Negativity Constraints Below:

The Optimal Value and The Optimal Strategic Decisions:

Sensitivity Analysis Tools for LP Having a Unique Solution:

For Technical Details on Linear Programming (LP), Back to:
Linear Optimization

For Technical Details on Construction of the Sensitivity Region, Back to:
Construction of the Sensitivity Region for LP Models

Kindly email your comments to:
Professor Hossein Arsham

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Kindly e-mail me your comments, suggestions, and concerns. Thank you.

Professor Hossein Arsham

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