Linear Programming

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The Simplex Method
of Linear Programming
Tutorial Outline
CONVERTING THE CONSTRAINTS TO
EQUATIONS

SOLVING MINIMIZATION PROBLEMS

SETTING UP THE FIRST SIMPLEX TABLEAU

KEY TERMS

SIMPLEX SOLUTION PROCEDURES

SOLVED PROBLEM

SUMMARY OF SIMPLEX STEPS FOR
MAXIMIZATION PROBLEMS

DISCUSSION QUESTIONS

ARTIFICIAL AND SURPLUS VARIABLES

SUMMARY

PROBLEMS

3

T 3-2 O N L I N E T U T O R I A L 3

THE SIMPLEX METHOD

OF

L I N E A R P RO G R A M M I N G

Most real-world linear programming problems have more than two variables and thus are too complex for graphical solution. A procedure called the simplex method may be used to find the optimal solution to multivariable problems. The simplex method is actually an algorithm (or a set of instructions) with which we examine corner points in a methodical fashion until we arrive at the best solution—highest profit or lowest cost. Computer programs and spreadsheets are available to handle the simplex calculations for you. But you need to know what is involved behind the scenes in order to best understand their valuable outputs.

CONVERTING THE CONSTRAINTS TO EQUATIONS
The first step of the simplex method requires that we convert each inequality constraint in an LP formulation into an equation. Less-than-or-equal-to constraints (≤) can be converted to equations by adding slack variables, which represent the amount of an unused resource. We formulate the Shader Electronics Company’s product mix problem as follows, using linear programming:

Maximize profit = $7X1 + $5X2
subject to LP constraints:
2 X1 + 1X2 ≤ 100
4 X1 + 3 X2 ≤ 240
where X1 equals the number of Walkmans produced and X2 equals the number of Watch-TVs produced. To convert these inequality constraints to equalities, we add slack variables S1 and S2 to the left side of the inequality. The first constraint becomes

2X1 + 1X2 + S1 = 100
and the second becomes
4X1 + 3X2 + S2 = 240
To include all variables in each equation (a requirement of the next simplex step), we add slack variables not appearing in each equation with a coefficient of zero. The equations then appear as 2 X1 + 1X2 + 1S1 + 0 S2 = 100

4 X1 + 3 X2 + 0 S1 + 1S2 = 240
Because slack variables represent unused resources (such as time on a machine or labor-hours available), they yield no profit, but we must add them to the objective function with zero profit coefficients. Thus, the objective function becomes Maximize profit = $7X1 + $5X2 + $0S1 + $0S2

SETTING UP THE FIRST SIMPLEX TABLEAU
To simplify handling the equations and objective function in an LP problem, we place all of the coefficients into a tabular form. We can express the preceding two constraint equations as SOLUTION MIX

X1

X2

S1

S2

QUANTITY (RHS)

S1
S2

2
4

1
3

1
0

0
1

100
240

The numbers (2, 1, 1, 0) and (4, 3, 0, 1) represent the coefficients of the first equation and second equation, respectively.

SETTING UP

THE

F I R S T S I M P L E X T A B L E AU

T3-3

As in the graphical approach, we begin the solution at the origin, where X 1 = 0, X 2 = 0, and profit = 0. The values of the two other variables, S1 and S2, then, must be nonzero. Because 2X1 + 1X2 + 1S1 = 100, we see that S1 = 100. Likewise, S2 = 240. These two slack variables comprise the initial solution mix—as a matter of fact, their values are found in the quantity column across from each variable. Because X1 and X2 are not in the solution mix, their initial values are automatically equal to zero.

This initial solution is called a basic feasible solution and can be described in vector, or column, form as
 X1   0 
X   0 
 2 = 

 S1  100 


 S2  240 
Variables in the solution mix, which is often called the basis in LP terminology, are referred to as basic variables. In this example, the basic variables are S1 and S2. Variables not in the solution mix—or basis—(X1 and X2, in this case) are...
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