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

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 http://college.cengage.com/mathematics/larson/elementary_linear/4e/shared/downloads/c09s3.pdf

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

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 3. 7- 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.

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.

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.

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?

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.

Problem Formulation

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.

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

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

Primal and Dual – Example- 1 Matrix Form

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

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. Y1 - 1 Y2 <= 3 Y1 + 3 Y2 <= 2 3Y1 - 1 Y2 <= -5 Y1, Y2 >= 0

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

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

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,

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.

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.

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

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

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

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

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.

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

Directed and Undirected Networks 2 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

The shortest path problem 1 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

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

The constraint matrix is a node-arc incidence matrix in disguise 1 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