mechanical engineering department 1 سورة النحل (78)

Mechanical Engineering Department

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النحل سورة(78)


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Chapter 2

Linear Programming (LP)

Lecture Objective

THE SIMPLEX METHOD


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SIMPLEX LINEAR SIMPLEX LINEAR PROGRAMMING PROGRAMMING SIMPLEX LINEAR SIMPLEX LINEAR PROGRAMMING PROGRAMMING


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1. INTRODUCTION1. INTRODUCTION

• The simplex method is a general-purpose linear programming algorithm that is widely used to solve large-scale problems.

• The simplex technique involves a series of iterations; successive improvements are made until an optimal solution is achieved.

• Most users of the technique rely on computers to handle the computations while they concentrate on the solutions.

• It is best to work with numbers in fractional form.


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Example Example

Consider the simplex solution to the following Consider the simplex solution to the following Problem:Problem:


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Graphical solutionGraphical solution

x1


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IntroductionIntroduction

• The simplex technique involves generating a series of solutions in tabular form, called tableaus.

• By inspecting the bottom row of each tableau, one can immediately tell if it represents the optimal solution.

• Each tableau corresponds to a corner point of the feasible solution space.

• The first tableau corresponds to the origin. • Subsequent tableaus are developed by shifting to an

adjacent corner point in the direction that yields the highest rate of profit.


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Simplex Method StepsSimplex Method Steps

1. Set up the initial tableau.2. Develop a revised

tableau using the information contained in the first tableau.

3. Inspect to see if it is optimum.

4. Repeat steps 2 and 3 until no further improvement is possible.


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Maximization Problem with only ≤ constraints

Maximization Problem with only ≤ constraints

1. Set up the initial tableau.A. Rewrite the constraints so that they become

equalities; add a slack variable to each constraint.

B. Rewrite the objective function to include the slack variables. Give slack variables coefficients of 0.

C. Put the objective coefficients and constraint coefficients into tableau form.

D. Compute values for the Z row.

E. Compute values for the C - Z row.


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Slack variables Slack variables

represent the amount of each resource that will not be used if the solution is implemented.


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Adding Slack variables to each Constraint

Adding Slack variables to each Constraint

In the initial solution, with each of the real variables equal to zero, the solution consists only of slack.

It is useful in setting up the table to represent each slack variable in every equation.


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Adding Slack variables to the Objective Function

Adding Slack variables to the Objective Function

The objective function can be written in similar form:

The slack variables are given coefficients of zero in the objective function because they do not produce any contributions to profits.


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The Problem After Adding Slack variables

The Problem After Adding Slack variables


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QuestionsQuestionsQuestionsQuestions


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The Initial Tableau The Initial Tableau


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Compute values for the Z row and C- Z row

Compute values for the Z row and C- Z row

• To compute the Z values, multiply the To compute the Z values, multiply the coefficients in each column by their respective coefficients in each column by their respective row profit per unit amounts, and sum within row profit per unit amounts, and sum within columns. To begin with, all values are zero.columns. To begin with, all values are zero.

• The last value in the Z row indicates the total The last value in the Z row indicates the total profit associated with a given solution (tableau). profit associated with a given solution (tableau). Since the initial solution has Since the initial solution has x1 x1 = 0 and = 0 and x2 x2 = 0, it = 0, it is not surprising that profit is 0.is not surprising that profit is 0.

• To compute in the C - Z row values, subtract the To compute in the C - Z row values, subtract the Z value in each column from the value of the Z value in each column from the value of the objective row for that column.objective row for that column.


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The Test for OptimalityThe Test for Optimality

If all the values in the C - Z row of any tableau If all the values in the C - Z row of any tableau are are zerozero or or negativenegative, the optimal solution has , the optimal solution has been obtained. been obtained.

In this case, the C - Z row contains two In this case, the C - Z row contains two positivepositive values, 4 and 5, indicating that improvement values, 4 and 5, indicating that improvement is possible.is possible.


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2. Set up subsequent tableaus.2. Set up subsequent tableaus.

A. Determine the entering variable (the largest positive value in the C - Z row).

B. Determine the leaving variable: Divide each constraint row's solution quantity by the row's pivot value; the smallest positive ratio indicates the leaving variable.


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The leaving and entering variablesThe leaving and entering variables


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2. Set up subsequent tableaus.2. Set up subsequent tableaus.

C. Form the new main row of the next tableau. Enter these values in the next tableau in the same row positions.

D. Compute new values for remaining constraint rows. Enter these in the new tableau in the same positions as the original row.

E. Compute values for Z and C - Z rows.F. Check to see if any values in the C - Z row are

positive; if they are, repeat 2A-2F Otherwise, the optimal solution has been obtained.


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Completed Second Tableau Completed Second Tableau


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Developing the Third TableauDeveloping the Third Tableau


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The Third Tableau (Optimum Solution) The Third Tableau

(Optimum Solution)


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Graphical analogies to Simplex Tableaus

Graphical analogies to Simplex Tableaus


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QuestionsQuestionsQuestionsQuestions


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A MINIMIZATION PROBLEM≥ and = Constraints


For = constraint, add an artificial variable.

For example, the equalities

Using artificial variables a1 and a2.

Slack variables would not be added.4


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The objective function, say, Z = 2x1 + 3x2 would be rewritten as:

whereM = A large number (e.g., 999)The artificial variables are not desired in the

final solution, selecting a large value of M (much larger than the other objective coefficients) will insure their deletion during the solution process.


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For a ≥ constraint, surplus variables must be subtracted instead of added to each

constraint. For example, the constraints

would be rewritten as equalities:


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As equalities, each constraint must then be adjusted by addition of an artificial

variable.

The final result looks like this:


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If the objective function happened to be

it would become


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Two Ways to Solve a Minimization ProblemTwo Ways to Solve a

Minimization ProblemThe first approach

We select the variable with the MOST NEGATIVE Cj - Zj; as the one to introduce next.

We would STOP when every value in the net evaluation row is ZERO or POSITIVE.


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Two Ways to Solve a Minimization ProblemTwo Ways to Solve a

Minimization ProblemThe second approach

Any minimization problem can be converted to an equivalent maximization problem by multiplying the objective function by -1.

Solving the resulting maximization problem wil1 provide the optimal solution to the minimization problem.


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ProblemProblem

Let us illustrate this second approach

The mathematical statement of the problem is


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SolutionSolution


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Solution (Continue)Solution (Continue)

The tableau form for this problem is


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The initial simplex tableau is:


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At the first iteration, x1 is brought into the basis and a1 is removed.

After dropping the a1 column from the tableau, The first iteration is:


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The final simplex tableau:

The minimum total cost of the optimal solution is $800


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Thank YouThank YouThank YouThank You

mechanical engineering department 1 سورة النحل (78)

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