Linear Optimization Model

By Author: Pierce Brosnan
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A mathematical optimization model consists of an objective function and a set of constraints in the form of a system of equations or inequalities. Optimization models are used extensively in almost all areas of decision-making, such as engineering design and financial portfolio selection. This site presents a focused and structured process for optimization problem formulation, design of optimal strategy, and quality-control tools that include validation, verification, and post-solution activities.

How to Solve a Linear System of Equations by Lp Solvers?

In the Algebraic Method of solving LP problems, we have to solve some systems of equations. There is a link between LP solvers and the systems of equation solvers. Suppose we have a very large system of equations that we would like to solve and an LP solver package but we still have no solver computer package for a system of equations available. The question is "How to use an LP solver to find the solution to a system of equations?" The following steps outline the process of solving any linear system of equations using an available LP solver.

1- Because some ...
... LP solvers require that all variables be non-negative, substitute for each variable Xi = Yi - T everywhere.
2- Create a dummy objective, such as minimize T.
3- The constraints of the LP problem are the equations in the system after the substitutions outlined in step 1.

Numerical Example: Solve the following system of equations

2X1 + X2 = 3
X1 -X2 = 3

Since the WinQSB package accepts LP in various formats ( unlike Lindo), solving this problem by WinQSB is straightforward:

First, create an LP with a dummy objective function such as Max X1, subject to 2X1 + X2 = 3, X1 - X2 = 3, and both X1 and X2

unrestricted in sign. Then, enter this LP into the LP/ILP module to get the solution. The generated solution is X1= 2, X2= -1, which can easily be verified by substitution.

However, if you use any LP solver which requires by default (e.g., Lindo) that all variables be non-negative, you need to do some preparations to satisfy this requirement: First substitute for X1 = Y1 - T and X2 = Y2 - T in both equations. We also need an objective function. Let us have a dummy objective function such as minimize T. The result is the following LP:

Min T

Subject to:
2Y1 + Y2 - 3T = 3,
Y1 - Y2 = 3.

Using any LP solver, such as Lindo, we find the optimal solution to be Y1 = 3, Y2 = 0, T = 1. Now, substitute this LP solution into both transformations X1 = Y1 - T and X2 = Y2 - T. This gives the numerical values for our original variables. Therefore, the solution to the system of equations is X1 = 3 - 1 = 2, X2 = 0 - 1 = -1, which can easily be verified by substitution.

Dual Problem: Construction and Its Meaning

Associated with each (primal) LP problem is a companion problem called the dual. The following classification of the decision variable constraints is useful and easy to remember in construction of the dual.

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