1. Algorithms CISC 4080/CISC 5825
Linear Programming, Simplex Method and Duality Theory Dr Kamesam Graduate School Of Business Fordham University

2. Outline Elementary Row Operations Graphical Approach to solving LP
Solution of Maximization Problems with Simplex Method
Special Situations in the Simplex method.
Simplex method with artificial variables.
Solving LP with Excel Solver.
Duality
You may find the following free chapter helpful

3. SIMPLEX METHOD Simplex algorithm for standard maximization problems
Step-1
Write the standard maximization problem in standard form, introduce slack variables to form the initial system, and write the initial tableau.
Step-3
Select the pivot column
Step-5
Select the pivot element and perform the pivot operation
Step 4
Are there any positive elements in the pivot column above the dashed line?
Step 2
Are there any negative indicators in the bottom row?
STOP
The optimal solution has been found.
STOP
The linear programming problem has no optimal solution. UNBOUNDED
Simplex algorithm for standard maximization problems

4. To solve a linear programming problem in standard form, use the following steps.
1- Convert each inequality in the set of constraints to an equation by adding slack variables. 2- Create the initial simplex tableau. 3- Select the pivot column. ( The column with the “most negative value” element in the last row.) 4- Select the pivot row. (The row with the smallest non-negative result when the last element in the row is divided by the corresponding in the pivot column.) 5-Use elementary row operations to calculate new values for the pivot row so that the pivot is 1 (Divide every number in the row by the pivot number.) 6- Use elementary row operations to make all numbers in the pivot column equal to 0 except for the pivot number. If all entries in the bottom row are zero or positive, this is the Optimal Tableau. If not, go back to step If you obtain a Optimal tableau, then the linear programming problem has a maximum solution, which is given by the entry in the lower-right corner of the tableau.

5. Degeneracy and Cycling in Simplex Method
Sometimes, it may be possible that an extreme point has more than one algebraic representation. In n space, if more than n hyperplanes intersect at that point, then the vertex is called “degenerate, which could in theory lead to Simplex method Cycling infinitely.
In practice, Cycling does not seem to occur, so the interest in degenerate vertices is mostly for theoretical reasons.
There is a variation of the simplex method, known as Lexico- Simplex method, which ensures that Cycling can never occur.

6. Software for Solving LPs
There is a wide choice of software packages for solving LPs, including many open source packages.
High-end packages (ex: CPLEX from IBM ) can solve problems with tens of thousands of variables
Microsoft Excel has a LP solver built in, which is very easy to use and we will learn how to use it.

7. Example-2
A farmer owns a 100 acre farm and plans to plant at most three crops. The seed for crops A,B, and C costs $40, $20, and $30 per acre, respectively. A maximum of $3200 can be spent on seed. Crops A,B, and C require 1,2, and 1 workdays per acre, respectively, and there are maximum of 160 workdays available. If the farmer can make a profit of $100 per acre on crop A, $300 per acre on crop B, and $200 per acre on crop C, how many acres of each crop should be planted to maximize profit? Formulate the above problem as a LP. Solve it using Solver in Excel. Then interpret the solution. What should the former do?

8. Ex3: Media Selection for Advertising
Win Big gambling club promotes gambling junkets from a large Midwestern city to casinos in the Bahamas. The club has budgeted up to $8000 per week for local advertising. The money is to be allocated among 4 promotional media. TV spots, newspaper ads and two types of radio advertisements. Win- Big’s goal is to reach the largest possible high-potential audience through the various media. The table below shows the number of potential gamblers reached by one advertisement in each of the four media. The cost per advertisement is also shown in the table. Win Big requires that at least 5 radio spots be placed each week. To ensure a broad campaign, management also insists that no more than $1800 is spent on radio ads per week.

9. Problem Formulation

10. Duality in Linear Programming
Linear programming problems exist in pairs. That is in linear programming problem, every maximization problem is associated with a minimization problem. Conversely, associated with every minimization problem is a maximization problem. Once we have a problem with its objective function as maximization, we can write by using duality relationship of linear programming problems, its minimization version. The original linear programming problem is known as primal problem, and the derived problem is known as dual problem.

11. Primal dual problems
Linear programming problem specified with equalities
Linear programming in standard form
Dual problem
Primal problem y x
Primal problem
Dual problem y x

13. Primal and Dual – Example- 1 Standard Algebraic Form
Max Z = 3x1+ 5x2
s. to x1 ≤ 4
2x2 ≤ 12
3x1+ 2x2 ≤ 18
x1,x2 ≥ 0
Dual
Min W = 4y1+ 12y2+ 18y3
s. to y1+ 3y3 ≥ 3
2y2+ 2y3 ≥ 5
y1, y2, y3 ≥ 0

14. Primal and Dual – Example- 1 Matrix Form

15. Symmetry Property
For any primal problem and its dual problem, all relationships between them must be symmetric because…

16. Primal – Dual Example 2 Primal Minimize 3X1 + 2X2 -5X3
s.t X1 + X2 + X3 >= 12
- X1 + 3X2 – X3 >= 8
X1, X2, X3 >= 0
Dual: Maximize 12 Y1 + 8 Y2
s.t Y Y2 <= 3
Y Y2 <= 2
3Y Y2 <= -5
Y1, Y2 >= 0

17. Primal Dual Example 3 Primal Maximize 4 X1 – 5 X2 s.t. X1 + X2 = 8
Dual Minimize 8Y1+12Y2+9 Y3
s.t Y1 + 3Y2 – Y3 >= 4
1Y1 - 2Y2 +3Y3 >= -5
Y1 unrestricted
Y2,Y3 >= 0

18. Duality theorems
It is easy to show that we can move from one pair of primal-dual problems to the other.
It is also easy to show that the dual of the dual problem is the primal problem.
Thus we are showing the duality theorems using the pair where the primal primal is in the standard form:
primal
Dual

19. Duality theorems Weak duality theorem
If (i.e., x is feasible for the primal problem) and
if (i.e., y is feasible for the dual problem), then
Proof Indeed,

20. Duality theorems Corollary If and , and if
, then x* and y* are optimal solutions for the primal and dual problems, respectively..
Proof It follows from the weak duality theorem that for any feasible solution x of the primal problem
Consequently x* is an optimal solution of the primal problem.
We can show the optimality of y* for the dual problem using a similar
proof.

21. Duality theorems
Strong duality theorem If one of the two primal or dual problem has a finite value optimal solution, then the other problem has the same property, and the optimal values of the two problems are equal. If one of the two problems is unbounded, then the feasible domain of the other problem is empty.

22. Primal and Dual Primal Z=cx Dual W=yb superoptimal suboptimal
(optimal) Z*
W* (optimal)
suboptimal
superoptimal

23. Complementary slackness theory
We now introduce new necessary and sufficient conditions for a pair of feasible solutions of the primal and of the dual to be optimal for each of these problems.
Consider first the following pair of primal-dual problems.
primal
Dual x

24. Complementary slackness theory
Complementary slackness theorem 1
Let x and y be feasible solution for the primal and the dual, respectively. Then x and y are optimal solutions for these problems if and only if for all
j = 1,2,…,n

25. Complementary slackness theory
Now consider the other pair of primal-dual problems
Complementary slackness theorem 2
Let x and y be feasible solution for the primal and dual problems, recpectively. Then x and y are opyimal solutions of these problems
if and only if
for all j = 1,2,…,n for all i=1,2,…,m y x

26. Summary
Primal
Dual
(a) Maximize.
Minimize
(b) Objective Function.
Right hand side.
(c) Right hand side.
Objective function.
(d) i th row of input-output coefficients.
i th column of input output coefficients.
(e) j th column of input-output coefficients.
j the row of input-output coefficients.

27. Minimum Cost Flow Problem and Network Simplex Method
Next few charts are from MIT and James Orlin

28. Directed and Undirected Networks2 3 4 1 a b c d e
An Undirected Graph 2 3 4 1 a b c d e
A Directed Graph
Networks are used to transport commodities
physical goods (products, liquids)
communication
electricity, etc.
The field of Network Optimization concerns
optimization problems on networks
MIT and James Orlin © 2003

29. The shortest path problem1 2 3 4 5 6
The shortest path problem
x12
x13
x23
x25
x24
x35
x46
x54
x56 1 = 1 -1 = 1 -1 = 1 -1 = 1 -1 = -1 =
The constraint matrix is the node arc incidence matrix
MIT and James Orlin © 2003

30. A Transportation Problem
3 warehouses: nodes 1, 2, 3
3 retailers: nodes 4, 5, 6 1 2 3 W 4 5 6 R
Supplies of goods at the warehouses (6, 7 and 2) : Upper bound on flow 6 7 2 4 8 3 1 1
Demand for goods at the warehouses (4, 8 and 3) Lower bound on flow 1 2
Costs on shipping from each warehouse to each retailer 1 1
Objective: send at most the supply, send at least the demand, and minimize cost. 1
What is the linear program for this transportation problem?
MIT and James Orlin © 2003

31. The constraint matrix is a node-arc incidence matrix in disguise1 2 3 4 5 6 W R 7 8
x26
x34
x35
x36
x25
x14
x15
x16
x24 = 6 1 1 = 7 1 = 2 1 = 4 -1 = -4 -1 = -8 1 = 8 -1 = -3 1 = 3
The matrix is a node-arc incidence matrix
MIT and James Orlin © 2003