


J. René Villalobos and Gary L. Hogg 

Arizona State University 

Paul M. Griffin 

Georgia Institute of Technology 





Almost any introductory Operations Research book has a chapter that covers the simplex algorithm. The students should read this chapter before coming to class. Two suggestions: 

Chapter 3 of Operations Research, by Hamdy A. Taha, Prentice Hall , 7^{th} Edition 

Chapter 4 of Introduction to Operations Research by Hillier and Liberman, 7^{th} Edition 

The student also should read the selfcontained module on the standard form of an LP problem. This module can be downloaded from: http://enpc2675.eas.asu.edu/lobos/iie476/stdform.doc 





At the end of the lecture each student should be able to: 

Express a Linear Programming problem in the standard form 

Set up the simplex tableau for a standard LP problem 

Find the optimal solution for an LP problem using the simplex tableau 




Introduction 3 minutes 

RAT 5 minutes 

Lecturing on standard problem and motivation for the Simplex Algorithm 8 minutes 

Individual exercise 5 minutes 

Lecturing on Simplex algorithm and tableau 20 minutes 

Team Exercise 5 minutes 

Wrap up 4 minutes 

Total lecture time 50 minutes 






Suppose that Team members A and C have decided to work together in their tasks. They have decided that their skills are complementary and by working together in both of the tasks assigned to them they can do a better job than if they work independently. A has very good writing speed, he can write 4 pages per hour but his rate of grammatical errors is very high (3 errors per page). On the other hand, C is slower (2.5 pages per hour) but his error rate is very low (1 error per page). They want to write a report that will earn them the highest grade. 

They have discovered that the grade given by the professor is very much influenced by the number of pages of the written assignments (the more pages the higher the grade assigned). However, they have also discovered that for every five errors in grammar, he deducts the equivalent of one written page, and also that the maximum number of mistakes the instructor will tolerate in an assignment is 80. Help this team of students to determine the best strategy (you need to define the meaning of this) given the total time constraint available to work in the project. The total time allocated to both of them is 8 hours. Also because of previous commitments C cannot work more than 6 hours in the project 





For the report strategy problem: 

You have found graphically the optimal solution. Based on that solution do the following exercise. 

You are given the alternative of increasing the number of errors allowed to 90 or to increase the number of total number of hours allowed to work in the project to 9 (for both team members). What option would you choose? Explain. 

At the end of the exercise each team should turn in the individual and team solutions in a single package for grading 

The instructor may select a member of the team to present the their solution to the rest of the class 






From last lecture the LP formulation of the problem is: 

Maximize z = 1.6X_{A} + 2.0X_{C} 

Subject to: 

12 X_{A} + 2.5X_{C} £ 80 

X_{A} + X_{C} £ 8 

X_{C} £ 6 

X_{A}, X_{C} ³ 0 










To answer the second question if we increase the number of errors to 90 the first constraint would become: 

12 X_{A} + 2.5X_{C} [ 90 

Then to find the point where the two constraints intersect we would have to solve the set of equations 

12 X_{A} + 2.5X_{C} =90 

X_{A} + X_{C} = 8 

Which would give us: X_{A}=7.368 X_{C} = 0.6315. The objective function evaluated at this point would be 13.05 (No increase of the optimal solution ànonbinding constraint) 

Doing the same with the other constraint we get 

12 X_{A} + 2.5X_{C} =80 

X_{A} + X_{C} = 9 

Which would give us : _{}X_{A}=6.05 X_{C} = 2.94. The objective function evaluated at this point would be 15.57àIncrease of the optimal solutionàBinding constraint 

We choose to increase the number of hours instead of the number of errors allowed 







Notice that by increasing (or decreasing) the availability of a scarce resource (hours or total number of errors allowed) the straight line (representing the constraint) moves in parallel to the original one (but note that the slope does not change) 

Also note that the totalerrors constraint is a nonbinding constraint. That is, it does not impact the current optimal solution. (at what point is this no longer true?) 





Unfortunately we can get the graphical solution of the LP problem only for problems with 2 and 3 decision variables 

Therefore we need to find an alternative method to get the optimal solution of LP problems 

Some observations: 

We know that the optimal solution must be in a corner point (why?) 

We also know that a corner point is defined by the intersection of two constraints (for the 2 decision variable case we solve a different subset of two linear equations for each corner point). What about when we have three or more decision variables? Can you identify all the corner points? 

So what we could do is solve the subsets of linear equations corresponding to all the corner points. Then we evaluate the objective function at those corner points and the one yielding the best objective value is our optimal solution. 

However, since there is a large number of corner points (how many?) this is impractical. We need a smart way to search for the optimal solution. This is what the the simplex algorithm does for us. 






Simplex Method . An algebraic, iterative method to solve linear programming problems. 

The simplex method identifies an initial basic solution (Corner point) and then systematically moves to an adjacent basic solution, which improves the value of the objective function. Eventually, this new basic solution will be the optimal solution. 

The simplex method requires that the problem is expressed as a standard LP problem (see module on Standard Form). This implies that all the constraints are expressed as equations by adding slack or surplus variables. 

The method uses Gaussian elimination (sweep out method) to solve the linear simultaneous equations generated in the process. 








6 X_{1} + 3X_{2} £ 10 (1) 

3 X_{1} + X_{2} = 7 (2) 

7 X_{1} + 4X_{2 } + X_{3} ³ 10 (3) 

Since (1) and (3) are inequalities we need to add a slack (1) and subtract a surplus variable (3) accordingly. Then the inequalities can be expressed as equations of the form: 

6 X_{1} + 3X_{2} + S_{1} = 10 (1) 

3 X_{1} + X_{2} = 7 (2) 

7 X_{1} + 4X_{2 } + X_{3} S_{3} =10 (3), 

Where S_{1} is a slack variable and S_{3} is a surplus variable. What is the physical meaning of these variables? 






Reddy Mikks Problem with slack variables 

Maximize z = 3X_{E} + 2X_{I} + 0S_{1} + 0S_{2} + 0S_{3}+ 0S_{4} 

Subject to: 

X_{E} + 2X_{I} + S_{1} = 6 (1) 

2X_{E} + X_{I} + S_{2} = 8 (2) 

X_{E} + X_{I} + S_{3} = 1 (3) 

X_{I} + S_{4}= 2 (4) 



X_{E} , X_{I,}, S_{1,}, S_{2,}, S_{3,}, S_{4} ³ 0 










Inspection Problem with slack and surplus variables 

Minimize Z = 40 X_{1} + 36X_{2}+ 0S_{1} + 0S_{2} + 0S_{3} 

Subject to: 

5X_{1} + 3X_{2}  S_{1} = 45 

X_{1} + S_{2 } = 8 

X_{2} + S_{3} = 10 



X_{1}, X_{2} S_{1,}, S_{2,}, S_{3,} ³ 0 






For the report strategy problem add the following constraint X_{A} + X_{C} ³ 2 

For the resulting LP Model introduce the appropriate slack and surplus variables in the constraints and objective function 

Find the maximum number of basic solutions and identify the subset of Basic Feasible Solutions on the graph. 






We define the standard form of a linear programming problem as: 

One whose objective function is maximization 

One whose constraints are expressed as equations and whose right hand side is nonnegative (after the introduction of slack and surplus variables) 

One in which all the decision variables are nonnegative 






A solution is any specification of values for the decision variables. 

A feasible Solution is a solution for which all the constraints are satisfied. 

The feasible region is the set of all feasible solutions. 

An Optimal Solution is a feasible solution that has the most favorable value of the objective function. 

A CornerPoint Solution corresponds to a solution of the subset of equations corresponding to the constraints meeting at that corner point (vertex) 

A Cornerpoint feasible (CPF) solution is a solution that lies at a corner of the feasible region . 

A Basic Solution is a cornerpoint solution 

A Basic Feasible Solution is a CPF solution for which all the variables are greater than or equal to zero. 




Step 0: Using the standard form determine a starting basic feasible solution by setting nm nonbasic variables to zero. 

Step 1: Select an entering variable from among the current nonbasic variables, which gives the largest perunit improvement in the value of the objective function.
If none exists stop; the current basic solution is optimal. Otherwise go to Step 2. 

Step 2: Select a leaving variable from among the current basic variables that must now be set to zero (become nonbasic) when the entering variable becomes basic. 

Step 3: Determine the new basic solution by making the entering variable, basic; and the leaving variable, nonbasic, and return to Step 1. 








The tableau format allows us to represent the problem compactly and more easily solve it. 

We merely record the coefficients of the problem only. 

In order to use the Tableau Method we need to do three things first: 

Represent the LP problem in standard form 

Represent the objective function as an equation 







where z is the value of the objective function. 

Determine the initial basic solution 





For Instance the ReddyMikks Problem: 

Maximize z = 3X_{E} _{}+ 2X_{I} 

Subject to: 

X_{E} _{}+ 2X_{I} £ 6 (1) 

2X_{E} _{}+ X_{I} £ 8 (2) 

X_{E} _{}+ X_{I} £ 1 (3) 

X_{I} £ 2 (4) 

X_{E} _{}, X_{I,} ³ 0 




The tabular form of the simplex method uses a simplex tableau to display the system of equations yielding the current basic feasible solution. 

There are different ways to design the tableau a common design is as follows 

Initial Tableau for the RM problem (Iteration 0) 







Step 0: Optimality test: the current basic feasible solution is optimal if every coefficient in row 0 of the tableau is nonnegative. If it is, stop; if not perform an additional iteration. 

Step 1: Determine the entering variable by selecting the variable having the most negative coefficient in row 0. The corresponding column is called the pivot column. 

Step 2: Determine the leaving variable by applying the minimum ratio test: 

Pick out each coefficient in the pivot column having a positive value. 

Divide the right hand of each row by each of these positive coefficients. 

Determine the smallest of these ratios 

The basic variable for the row corresponding to the smallest ratio is the leaving variable. Replace this variable with the entering variable in the next tableau. 

The row with the smallest ratio is called the pivot row. The number in the intersection of the pivot column and row is the pivot element. 

Step 3: Solve for the new BFS by using elementary row operations. 






To get the new BFS we perform the following steps 

Divide the pivot row by the pivot element and call it the new pivot row. Copy the result in the tableau for the next iteration 

For every row of the current tableau (excluding the pivot row) subtract the product of its pivotcolumn coefficient times the new pivot row. Copy the result in the corresponding row of the nextiteration tableau. Make sure to properly identify the new basic variable. 

MR Example: New pivot 1: 





Perform the operations needed to get the resulting equation 0 

The instructor might select a member of the team to present the teams solution to the rest of the class 







A minimization problem can be converted to a maximization problem just by multiplying the objective function by (1). 

Once this is done the problem is solved exactly the same as the maximization problem 

Example: 




Setup the tableau and perform one iteration of the Simplex Algorithm 

Answer the following question: How would you evaluate the efficiency of the simplex algorithm? 

The instructor may select a member of the team to present the teams solution to the rest of the class 








Efficiency of the simplex algorithm 

One way to assess the efficiency of the simplex algorithm is to count the number of iteration needed to arrive to the optimal solution and to compare this against the total number of corner point solutions given by the formula: 







(n) = number of variables, (m) = number of equations 






Setup the tableau and perform the first iteration for the report strategy problem 

Using the software provided with your textbook solve the ReddyMikks and the report strategy problem 




Previously we learned that a corner point solution corresponds to a solution of the subset of equations corresponding to the constraints meeting at that corner point (vertex) 

Thus if we have m constraints and n dimensions we use only n equations to find the corner point solution. 
