All the variables are nonnegative. 1

Real-Life Application—Optimization of Heart Valve Production

2. All the variables are nonnegative. 1

Converting inequalities into equations with nonnegative right-hand side. To convert a (

…

)-inequality to an equation, a nonnegative slack variable is added to the left-hand

99

1Commercial packages (and TORA) accept inequality constraints, nonnegative right-hand side, and unre- stricted variables. Preconditioning of the constraints and the variables to conform with the simplex method requirements is done internally in the software prior to solving the problem.

side of the constraint. For example, the M1-constraint of the Reddy Mikks model (Example 2.1-1) is converted into an equation as

6x

1 +

4x

2 + s₁ =

24, s

1 Ú

0

The nonnegative variable s

1

is the slack (or unused amount) of resource M1.

Conversion from 1

Ú2 to 1=2 is achieved by subtracting a nonnegative surplus

variable from the left-hand side of the inequality. For example, in the diet model (Example 2.2-2), the surplus variable S

11Ú

02 converts the 1

Ú2 feed mix constraint to

the equation

x₁ + x₂ - S₁ =

800, S

1 Ú

0

The amount of S

1

represents the excess tons of the mix over the required minimum

1= 800 tons2.

The only remaining requirement is for the right-hand side of the resulting equa- tion to be nonnegative. The requirement can be satisfied simply by multiplying both sides of the equation by

-

1, if necessary.

Dealing with unrestricted variables. The use of an unrestricted variable in an LP model is demonstrated in the multiperiod production smoothing model of Example  2.4-4, where the unrestricted variable S

i

represents the number of workers hired or fired in period i. In the same example, the unrestricted variable is replaced by two nonnegative variables by using the substitution

S_i = S_i^- - S_i⁺

, S

i- Ú

0, S

i+ Ú

0

In this case, S

i-

represents the number of workers hired and S

i+

the number of workers fired. As explained in Example 2.4-4, it is impossible (both intuitively and mathemati- cally) that S

i-

and S

⁺i

assume positive values simultaneously.

3.2 tRAnsitiOn FROm GRAPHicAL tO ALGebRAic sOLutiOn

The development of the algebraic simplex method is based on ideas conveyed by the graphical LP solution in Section 2.2. Figure 3.1 compares the two methods. In the graph- ical method, the solution space is the intersection of the half-spaces representing the constraints, and in the simplex method, the solution space is represented by m simultane- ous linear equations and n nonnegative variables. We can see that the graphical solution space has an infinite number of solution points, but how can we draw a similar conclusion from the algebraic representation of the solution space? The answer is that, in all non- trivial LPs, the number of equations m is always less than the number of variables n, thus yielding an infinite number of solutions (provided the equations are consistent).

²

For example, the equation x

+ y =

1 has m

=

1 and n

=

2 and yields an infinite number of solutions because any point on the straight line x

+ y =

1 is a solution.

2If the number of (independent) equations m equals the number of variables n (and the equations are con- sistent), the system has exactly one solution. If m is larger than n, then at least m -n equations must be redundant.

3.2 Transition from Graphical to Algebraic Solution 101

In the algebraic solution space (defined by m

* n equations, m 6 n), basic

solutions correspond to the corner points in the graphical solution space. They are determined by setting n

- m variables equal to zero and solving the m equations for

the remaining m variables, provided the resulting solution is unique. This means that the

maximum number of corner points is

C_mⁿ = n!

m!1n - m2!

As with corner points, the basic feasible solutions completely define the candidates for the optimum solution in the algebraic solutions space.

example 3.2-1

Consider the following LP with two variables:

Maximize z = 2x₁+ 3x₂ subject to

2x₁ + x₂… 4 x1 + 2x2… 5 x₁, x₂Ú 0

Figure 3.2 provides the graphical solution space for the problem.

Graphical Method Algebraic Method

Graph all constraints, including nonnegativity restrictions

Solution space consists of infinity of feasible points

Identify feasible corner points of the solution space

Candidates for the optimum solution are given by a finite number of corner points

Use the objective function to determine the optimum corner point from among all the candidates

Represent the solution space by m equations in n variables and restrict all variables to nonnegative values, m < n

The system has infinity of feasible solutions

Determine the feasible basic solutions of the equations

Candidates for the optimum solution are given by a finite number of basic feasible solutions

Use the objective function to determine the optimum basic feasible solution from among all the candidates

FiGure 3.1

Transition from graphical to algebraic solution

Algebraically, the solution space of the LP is represented by the following m= 2 equations and n= 4 variables:

2x₁ + x2 + s1 = 4 x1 + 2x2 + s2 = 5 x1, x2, s1, s2 Ú 0

The basic solutions are determined by setting n- m1= 4 - 2= 22 variables equal to zero and solving for the remaining m1= 22 variables. For example, if we set x1= 0 and x2= 0, the equa- tions provide the unique basic solution

s1 = 4, s₂ = 5

This solution corresponds to point A in Figure 3.2 (convince yourself that s₁ = 4 and s₂ = 5 at point A). Another point can be determined by setting s1= 0 and s2 = 0 and then solving the resulting two equations

2x₁ + x2 = 4 x1 + 2x2 = 5

The associated basic solution is 1x1 = 1, x2 = 22, or point C in Figure 3.2.

You probably are wondering which n- m variables should be set equal to zero to target a specific corner point. Without the benefit of the graphical solution space (which is available

0 1

1 2 3 4 5

2 3 4

A D

C

E B

F x₂

x₁ s₂ 5 0

s15 0

Optimum (x₁ 5 1, x₂ 5 2)

FiGure 3.2

LP Solution space of Example 3.2-1

3.3 The simplex Method 103 only for at most three variables), we cannot specify the 1n- m2 zero variables associated with a given corner point. But that does not prevent enumerating all the corner points of the solu- tion space. Simply consider all combinations in which n - m variables equal zero and solve the resulting equations. Once done, the optimum solution is the feasible basic solution (corner point) with the best objective value.

In the present example, the (maximum) number of corner points is C₂⁴ = 2!2!^4! = 6. Looking at Figure 3.2, we can spot the four corner points A, B, C, and D. So, where are the remaining two? In fact, points E and F also are corner points. But, they are infeasible, and, hence, are not candidates for the optimum.

To complete the transition from the graphical to the algebraic solution, the zero n - m variables are known as nonbasic variables. The remaining m variables are called basic variables, and their solution (obtained by solving the m equations) is referred to as basic solution. The following table provides all the basic and nonbasic solutions of the current example.

Nonbasic (zero)

variables Basic variables Basic solution

Associated

corner point Feasible?

Objective value, z

(x₁, x₂) (s₁, s₂) (4, 5) A Yes 0

(x₁, s₁) (x₂, s₂) (4, -3) F No —

(x₁, s₂) (x₂, s₁) (2.5, 1.5) B Yes 7.5

(x₂, s₁) (x₁, s₂) (2, 3) D Yes 4

(x₂, s₂) (x₁, s₁) (5, –6) E No —

(s₁, s₂) (x₁, x₂) (1, 2) C Yes 8

(optimum)

remarks. We can see from the preceding illustration that, as the size of the problem increases, enumerating all the corner points becomes a prohibitive task. For example, for m = 10 and n = 20, it is necessary to solve C₁₀²⁰1= 184,7562 sets of 10 * 10 equations, a staggering task, particularly when we realize that a 110 * 202-LP is a very small size (real-life LPs can include thousands of variables and constraints). The simplex method alleviates this computational burden dramatically by investigating only a subset of all possible basic feasible solutions (corner points). This is what the simplex algorithm does.

3.3 tHe simPLex metHOd

Rather than enumerating all the basic solutions (corner points) of the LP problem (as we did in Section 3.2), the simplex method investigates only a “select few” of these solutions. Section 3.3.1 describes the iterative nature of the method, and Section 3.3.2 provides the computational details of the simplex algorithm.

3.3.1 iterative nature of the simplex method

Figure 3.3 provides the solution space of the LP of Example 3.2-1. For the sake of

standardizing the algorithm, the simplex method always starts at the origin where all

the decision variables, x

j

, j

=

1, 2,

c

, n, are zero. In Figure 3.3, point A is the origin

1x₁ = x₂ =

0

2

and the associated objective value, z, is zero. The logical question now is

whether an increase in the values of nonbasic x

1

and x

2

above their current zero values

can improve (increase) the value of z. We can answer this question by investigating the objective function:

Maximine z

=

2x

1 +

3x

2

An increase in x

1

or x

2

(or both) above their current zero values will improve the value of z. The design of the simplex method does not allow simultaneous increases in vari- ables. Instead, it targets the variables one at a time. The variable slated for increase is the one with the largest rate of improvement in z. In the present example, the rate of improvement in the value of z is 2 for x

1

and 3 for x

2

. We thus elect to increase x

2

(the variable with the largest rate of improvement among all nonbasic variables). Figure 3.3 shows that the value of x

2

must be increased until corner point B is reached (recall from Figure 3.1 that stopping short of corner point B is not an option because a can- didate for the optimum must be a corner point). At point B, the simplex method, as will be explained later, will then increase the value of x

1

to reach the improved corner point C, which is the optimum.

The path of the simplex algorithm always connects corner points. In the present example the path to the optimum is A

SBSC. Each corner point along the path is

associated with an iteration. It is important to note that the simplex method always moves alongside the edges of the solution space, which means that the method does not cut across the solution space. For example, the simplex algorithm cannot go from

A to C directly.

0 1

1 2 3 4 5

2 3 4

A D

C

E B

F x₂

x₁ s₂ 5 0

s15 0

Optimum (x₁ 5 1, x₂ 5 2)

FiGure 3.3

Iterative process of the simplex method

3.3 The simplex Method 105

Aha! moment: the birth of Optimization, or How dantzig developed